Planetary rings and other astrophysical disks
11Planetary rings and other astrophysical disks
H. N. LATTER, G. I. OGILVIE, H. REIN
Abstract
This chapter explores the physics shared by planetary ringsand the various disks that populate the Universe. It beginswith an observational overview, ranging from protoplanetarydisks to spiral galaxies, and then compares and contraststhese astrophysical disks with the rings of the Solar System.Emphasis is placed on fundamental physics and dynamics,and how research into the two classes of object connects.Topics covered include disk formation, accretion, collisionalprocesses, waves, instabilities, and satellite-disk interactions.
Disks are ubiquitous in astrophysics and participate in someof its most important processes. Most, but not all, feed a cen-tral mass: by facilitating the transfer of angular momentum,they permit the accretion of material that would otherwiseremain in orbit (Lynden-Bell and Pringle, 1974). As a con-sequence, disks are essential to star, planet, and satelliteformation (McKee and Ostriker, 2007; Williams and Cieza,2011; Papaloizou and Terquem, 2006; Peale, 1999). Theyalso regulate the growth of supermassive black holes andthus indirectly influence galactic structure and the intra-cluster medium (Volonteri, 2010; Fabian, 2012). Althoughastrophysical disks can vary by ten orders of magnitude insize and differ hugely in composition, all share the same ba-sic dynamics and many physical phenomena. This reviewexplores these areas of overlap.The prevalence of flattened astrophysical systems is a re-sult of dissipation and rotation (Goldreich and Tremaine,1982). A cloud of gas or debris in orbit around a centralmass conserves its total angular momentum but not its en-ergy, as there are numerous processes that may cool thecloud (inelastic physical collisions, Bremsstrahlung, molec-ular line emission, etc.). As a result, particles’ random ve-locities are steadily depleted — where ‘random velocity’ isunderstood to be the component surplus to the circular or-bit fixed by the angular momentum. The system contractsinto a flat circular disk, the lowest energy state accessible.The contraction ends, and an equilibrium balance achieved,once the cooling is met by heating (supplied by externalirradiation or an internal viscous stress).Let us define a cylindrical coordinate system with its ori-gin at the central mass and the vertical pointing in the direc- tion of the total angular momentum vector. We describe sys-tems as cold when the pressure gradient is weak and the finalequilibrium very thin: radially the dominant force balanceis between the centrifugal force and gravity, while verticallyit is between pressure and gravity. Systems described as hot have stronger pressure and settle into thick disks or tori. Atthe far extreme, when rotation is subdominant, spheroidalmorphologies ensue: examples include planets, stars, globu-lar clusters, elliptical galaxies, etc.What determines the importance of pressure, and whichstate is ultimately achieved, is the relative efficiency of heat-ing and cooling. In dense planetary rings, energetic lossesfrom strongly inelastic collisions predominate. Rings arehence exceptionally cold and thin (Colwell et al., 2009). Incomparison, radiative cooling in gaseous disks varies by or-ders of magnitude, depending on the temperature, dust lev-els, ionisation fraction, or other properties (e.g. Bell and Lin,1994; Abramowicz and Fragile, 2013). The heating rate alsovaries considerably, especially if turbulence or external radi-ation is present. Consequently, gaseous accretion disks canbe thin (though never as thin as dense rings) or so thickthey resemble doughnuts more than they do sheets.This fundamental paradigm accommodates a diversity ofdifferent astrophysical disk systems, ranging over an enor-mous variety of length scales, physical properties, and com-positions. In the next section we review the observationalliterature on these systems. We then make clear their keydistinguishing physics and physical scales, extending theschematic account above. The rest of the chapter visits anassortment of topics that provide enlightening comparisonsbetween planetary rings and other astrophysical disks. Inparticular, we dwell on instances of pollinisation betweenthe two fields of study. The topics covered include: forma-tion, accretion, collisional dynamics, waves, instabilities, andfinally satellite–disk interactions. We conclude by speculat-ing on further connections between planetary rings and otherdisks that future work might explore.
Since the Copernican revolution astronomers have recog-nised that the planets of the Solar System all orbit the Sun1 a r X i v : . [ a s t r o - ph . E P ] J a n Latter, Ogilvie, & Rein
Figure 1.1
ALMA image of the young star HL Tau and itsprotoplanetary disk in the mm continuum. Credit: ALMA/ESO in the same sense, and almost in the same plane. In the eigh-teenth century Swedenborg, Kant, and Laplace, recognisingthat this arrangement could not have arisen by chance, pro-posed that the planets condensed out of a flattened cloudof gas rotating around the Sun earlier in its life (Montmerleet al., 2006, and references therein). Their models of the so-lar nebula introduced the concept of the protoplanetary disk(hereafter ‘PP disk’) which, though abandoned briefly in theearly twentieth century (e.g. Jeans, 1917), lies at the heartof modern theories of star and planet formation.Originally inferred from the infrared excesses of youngstellar objects (e.g. Lada and Wilking, 1984), PP disks weredirectly imaged first in the sub-mm (Sargent and Beckwith,1987; Koerner et al., 1993), and then spectacularly in the op-tical, when the Hubble Space Telescope uncovered a num-ber of examples in the Orion Nebula (McCaughrean andO’Dell, 1996). PP disks consist of relatively cool gas, mostlyH , scattered with dust. Temperatures are ∼
100 K gen-erally, but can reach ∼ ∼ H/r ∼ .
05, where H is the disk’s vertical pressure scale-height and r is radius (Kenyon and Hartmann, 1995; Evanset al., 2009; Williams and Cieza, 2011).Observations of UV excess in the stellar spectrum allowastronomers to estimate the mass transfer rate through thedisk and onto the young star. Comparison of different sys-tems suggests that accretion is irregular, with some 50% ofthe disk mass falling upon its protostar during less than10% of the disk’s lifetime (Evans et al., 2009). Archetypalsystems that exhibit fast accretion events are the FU Orionisand EX Lupi variables (FUors and EXors), which undergoirregular outbursts of accretion on a range of long timescales,50–1000 years (Herbig, 1977, 1989; Hartmann and Kenyon,1996; Audard et al., 2014).More recent observations using infrared and radio wave-lengths (e.g. with Subaru, VLT and ALMA) reveal that PP Figure 1.2
The protoplanetary disk MWC 758 as mapped inpolarised scattered infrared with VLT/SPHERE. Credit:Benisty et al. (2015). disks are highly structured and exhibit gaps, asymmetries,and spirals (Andrews et al., 2011; Muto et al., 2012; P´erezet al., 2014; ALMA Partnership et al., 2015). Figure 1.1shows an early ALMA image of the disk around HL Tauri,a young Sun-like star, exhibiting a striking array of rings.Figure 1.2 presents a clear example of a spiral density wavein MWC 758. There is considerable research activity aim-ing to explain the features seen in these and similar images.Embedded planets are the focus of the most popular ideas(e.g. Tamayo et al., 2015), as theory predicts that they nat-urally carve gaps and excite spiral waves (Papaloizou andTerquem, 2006, see also Section 1.9). However, a panoplyof alternatives have been proposed that may bear on theobserved structures. These include vortices (Varni`ere andTagger, 2006; Lesur and Papaloizou, 2010a), gravitationalinstabilities (Durisen et al., 2007; Takahashi and Inutsuka,2014), snow lines (Zhang et al., 2015), stellar flybys (Clarkeand Pringle, 1993; Xiang-Gruess, 2016), and warps (Marinoet al., 2015a).
Most stars form in binary systems. The more massive (pri-mary) star evolves faster than its companion (the secondary)and thus ends up a white dwarf, a neutron star or a blackhole while its secondary is left behind on the main sequence.If the binary orbit is sufficiently close, the secondary over-flows its critical equipotential surface, or Roche lobe, andspills over towards the compact primary. Owing to its rota-tion in the binary orbit, the transferred gas has too muchangular momentum to fall directly on to the surface of theprimary. Instead, it forms an accretion disk around it. Underthe action of torques within the disk, the gas gradually losesangular momentum, spirals inwards, and is accreted. As thegas falls deeper into the potential well energy is liberated,and the system becomes luminous (Warner, 1995; Hellier,2001). lanetary rings and other astrophysical disks Because such disks possess radii similar to that of the Sun,they are impossible to resolve directly. Instead the abovepicture of disk formation and accretion was deduced fromdetailed analyses of spectra and the peculiarities of the sys-tems’ light curves, which on the orbital timescale ( ∼ Figure 1.3
Light curve of the dwarf nova IY UMa over roughlyfour binary orbits ( ∼ Systems with white dwarf primaries are known as cat-aclysmic variables because many of them exhibit dramaticoutbursts. These include the classical novae, in which a layeraccreted on the primary ignites in a thermonuclear runaway(Gallagher and Starrfield, 1978; Shara, 1989), and the dwarfnovae, in which outbursts occur cyclically and arise from astate change in the disk itself (Warner, 1995; Lasota, 2001).The latter outbursts feature an increase of brightness ofsome 2–5 magnitudes, last for a few days, and possess arecurrence interval of days to weeks. Figure 1.4 illustratesa clear sequence of dwarf nova outbursts in SS Cygni. Notethat some sources exhibit more complicated behaviour suchas ‘standstills’ (Z Camelopardalis variables) and ‘superout-bursts’ (SU Ursae Majoris variables) (Warner, 1995).Systems with neutron star or black hole primaries areknown as X-ray binaries because their emission is dominatedby high-energy photons. They were first discovered by X-ray satellites launched in the 1960s (Giacconi et al., 1962;Gursky et al., 1966; Sandage et al., 1966). In the following50 years, space-based X-ray observatories, such as
Einstein , Chandra , and
XMM-Newton , have uncovered the propertiesof many such systems, yet because of their weak emissionin the optical they are far less well constrained than dwarfnovae. Low-mass X-ray binaries, involving low-mass secon-daries, typically accrete by Roche-lobe overflow as describedabove. In contrast, the accretion disks in high-mass X-raybinaries typically capture their gas from the vigorous winds
Figure 1.4
Light curve of SS Cyg on long timescales, showingquasi-periodic outburst behaviour. Each panel represents oneyear, each small tic on the horizontal axis 10 days, and eachcross the daily mean in magnitude — thus small and fine-scalevariations (such as in Fig. 1.3) are removed. The first panelbegins at HJD 2,446,432, and the last panel ends at 2,450,082.Credit: Cannizzo and Mattei (1998). of the high-mass secondary stars (Charles and Coe, 2006;Tauris and van den Heuvel, 2006). In both cases the disktemperature is strongly affected by X-ray irradiation and,as a result, though they undergo outbursts, their dynamicscan differ markedly from dwarf novae (Lasota, 2001; King,2006).The disks associated with interacting binaries usually con-sist of hydrogen and helium in atomic or ionised form. Dwarfnovae disks are vertically thin (
H/r ∼ . ∼ ∼ ,
000 K at the midplane, respectively(Hellier, 2001). In the inner regions of X-ray binaries thedisk attains enormous temperatures, greater than 10 K atthe disk surface, and cycles through a number of poorly un-derstood spectral states, some of which are accompanied byjets of material launched perpendicular to the disk (Remil-lard and McClintock, 2006; Done et al., 2007).Finally, we touch on Be stars. These are rapidly spinningB stars in which material is periodically expelled from theequator and forms a centrifugally supported disk. Usuallythe disks are inferred observationally by their characteris-tic Balmer lines and their polarisation of the star’s contin-uum radiation (Rivinius et al., 2013), though in the caseof ζ Tauri, a map of the disk itself has been directly re-constructed using interferometry in the H α line (Quirren-bach et al., 1994), while peculiarities in the light curves ofthe eclipsing binaries (cid:15) Aurigae and EE Cephei are bestexplained by a tilted disk around a B star (Hoard et al.,
Latter, Ogilvie, & Rein
Debris disks consist of dust and larger solids usually orbit-ing a young or main-sequence star (Wyatt, 2008). They arethe end-points of protoplanetary disk evolution, the gas ei-ther accreted or swept away by a photoevaporative wind.Alternatively, they may be regarded as the ‘leftovers’ ofthe planet formation process (Alexander et al., 2006; Wyattet al., 2015).The first debris disk was discovered orbiting Vega by the
IRAS satellite, its presence betrayed by a large infrared ex-cess (Aumann et al., 1984). Thanks to recent surveys by
Spitzer and
Herschel , there are now several hundred debrisdisk candidates circling a wide variety of stars (Su et al.,2006; Eiroa et al., 2013). Debris disks have also been ob-served in sub-mm to optical wavelengths and may now beimaged directly, giving astronomers information about theirmorphology (Wyatt, 2008). These disks are believed to beextrasolar analogues of the asteroid belt, Kuiper belt, andexozodiacal dust in our own Solar System, and so the studyof debris disks straddles the two fields of planetary scienceand astrophysics.
Figure 1.5
This image shows the edge-on disk around BetaPictoris, taken by the Hubble Space Telescope. One can identifya primary disk and a secondary, slightly tilted, disk. Credit:ESA/Hubble.
The majority of observed debris disk dust lies in the 1–100 µ m size range, and hence their infrared emission is difficultto study from the ground. Typically, the total mass in thesegrains is significantly less than that of the Earth (Wyatt,2003, 2008), and is distributed between radii 10–100 AU.Rather than being primordial, the dust must be constantlyreplenished or else it would be eliminated by radiation pres-sure, collisional fragmentation, or stellar wind drag. Infre-quent impacts between larger bodies are thought to initiatea collisional cascade that supplies this material (Backmanand Paresce, 1993; Wyatt and Dent, 2002). Unfortunately,the population of large solids is difficult to observe, owingto their smaller total surface area. Planetary sized objects,however, have been directly imaged and also inferred fromperturbations in the dust (see below).In comparison, we know of several hundred bodies witha size of roughly 100 km in the Kuiper belt, and have con-strained some of their surface properties (Petit et al., 2008;Stansberry et al., 2008). Moreover, to explain the frequency of short-period comets, theoretical estimates show that thebelt must contain at least 10 bodies with sizes greater than1 km (Farinella and Davis, 1996; Jewitt and Luu, 2000).These estimates provide data on intermediate size bodies inone debris disk, at least.Direct imaging reveals that debris disks exhibit a rangeof intriguing morphologies: sharp edges, gaps, warps, rings,spirals, asymmetries, and clumps (Wyatt, 2008). Figure 1.5shows one of the nearest debris disks, around the star BetaPictoris. One can identify two disks slightly tilted with re-spect to each other. Planets can potentially sculpt and struc-ture debris disks (e.g. Mouillet et al., 1997; Wyatt, 2005a;Quillen, 2006; Su et al., 2009), and indeed the tilt evidentin Figure 1.5 is thought to be forced by a massive Jovianplanet (Lagrange et al., 2009). It is generally accepted that most galaxies contain a super-massive black hole, of up to a few billion solar masses, attheir centres (Ferrarese and Ford, 2005; Merritt, 2013). Asmall proportion of these are ‘active’, in that they produceimmense and persistent volumes of radiation, sometimes or-ders of magnitude greater than the total power output oftheir host galaxies. The origin of this spectacular luminos-ity is the accretion of matter, through which gravitationalenergy is converted into mechanical and electromagnetic en-ergy. Because of the intense gravity of supermassive blackholes, accretion of only one solar mass per year is required togenerate the observed luminosities (Salpeter, 1964; Lynden-Bell, 1969; Marconi et al., 2004).The spectrum of these ‘active galactic nuclei’ (AGN) isstrikingly broad, and can extend from the far infrared tohard X-rays; for example, the well studied case of NGC 4151emits roughly the same specific intensity over five orders ofmagnitude in frequency (Ulrich, 2000). The optical to ex-treme ultraviolet light emerges from the accretion disk sur-rounding the black hole, while the X-rays are generated bythe disk’s hot corona of dilute gas. Dust in the disk, or inthe systems’ enveloping torus, is responsible for the infrared(Ferrarese and Ford, 2005). The optical and UV emissiontells us that the disk surface is no hotter than about 10 K,though temperatures at the midplane near the black hole cangreatly exceed that. Variability on short timescales suggeststhat disk sizes are typically 1000 AU or less (e.g. Greensteinand Schmidt, 1964; Peterson, 2001). In addition, AGN ex-hibit strong and broad spectral lines: their large redshiftsreveal their distance from our galaxy, while their Doppler-broadened linewidths can constrain the mass of the centralblack hole, as in the case of M87 (Macchetto et al., 1997).Finally, a subclass of AGN is characterised by powerful ra-dio emission (e.g. Baade and Minkowski, 1954; Fanaroff andRiley, 1974; Miley and De Breuck, 2008), usually accompa-nied by non-thermal gamma rays (Hartman et al., 1999).This radiation issues from intense relativistic jets orientedperpendicular to the disk plane (e.g. Biretta et al., 1999).Figure 1.6 shows a radio image of such a jet emerging fromCygnus A. lanetary rings and other astrophysical disks Owing to this collection of varied and unusual properties,several decades passed before AGN were properly identifiedand understood. In fact, for some time they were classi-fied as separate and distinct sources: Seyfert galaxies, radiogalaxies, BL Lac objects, quasars, and blazars. By the early1980s it was becoming clearer that these classes were dif-ferent ‘faces’ of the same astrophysical object, an accretingsupermassive black hole, with the variation in their observedproperties attributed mainly to differing viewing angles, andthe presence, or not, of a jet (Rees, 1984; Antonucci, 1993).See Netzer (2015) for a recent discussion of AGN ‘unifica-tion’ theories.
Figure 1.6
A radio map of the galaxy Cygnus A (at awavelength of 6 cm). The two jets can be seen emerging fromthe nucleus of the galaxy and colliding with the intergalacticmedium in the two large radio lobes, each roughly 100 kpc fromthe AGN. Credit: NRAO/AUI.
AGN are of immense importance in galactic astronomyand cosmology. They impact on the structure and evolutionof their host galaxies, clearly demonstrated by the strongcorrelation between an AGN’s mass and the velocity dis-persion of its host galaxy’s stars, on one hand, and on thesize of the galactic central bulge, on the other (Kormendyand Richstone, 1995; McConnell et al., 2011). They control,to some extent, the dynamics of the intracluster medium ofgalaxies, via the deposition of mechanical energy throughjets (‘AGN feedback’), a process that bears directly on thecooling-flow problem in such systems (Fabian, 2012). AGNemission may also act as a probe of the intervening gas be-tween our galaxy and high redshifts (e.g. Fan et al., 2006). Atthe same time they pose a number of challenging problems:how do the black holes grow so large? How are relativisticjets launched, and why do only some AGN produce them?Why are they more numerous at large redshift, and doesthis mean that AGN represent a transient phase of galacticevolution?
Spectroscopic evidence indicates that some 20% of whitedwarf photospheres are polluted with metal elements (the‘DAZ phenomenon’, Zuckerman et al., 2003). Moreover,these stars must be continually accreting new pollutants be-cause of the short sinking time of some observed ions (e.g.Mg II) (Holberg et al., 1997; Koester et al., 1997). An addi-tional clue is that a fraction of the most contaminated exam-ples display evidence of a circumstellar dust ring from either infrared excess (e.g Farihi et al., 2009) or double-peaked op-tical emission lines (G¨ansicke et al., 2006). As a result, as-tronomers have deduced that DAZ white dwarfs are ringedby narrow accretion disks of debris and gas, probably theresult of the tidal disruption of an asteroid or minor planet(Graham et al., 1990; Debes and Sigurdsson, 2002; Jura,2003, 2008).One problem this scenario must overcome is how to supplythe white dwarf with a suitable body to disrupt. Asteroidbelts and/or planets on wide orbits can survive the giant-branch precursor to the white dwarf (Villaver and Livio,2007; Bonsor and Wyatt, 2010), but their orbits must besubsequently perturbed so that they plunge to sufficientlysmall radii. Scattering of asteroids by planets is the currentlyfavoured model, the planetary system wrought dynamicallyunstable in the post-stellar mass-loss stage (Bonsor et al.,2011; Debes et al., 2012). The intense interest driving thisfield centres on the make-up of the pollution itself, becauseit provides an opportunity to directly sample the composi-tions of rocky exoplanetary systems (e.g. Klein et al., 2010;G¨ansicke et al., 2012; Xu et al., 2014).Similarly, when stars stray too near supermassive blackholes they are tidally ripped apart. At early times the stel-lar material falls on the black hole at a tremendous rate,resulting in a flare whose luminosity approaches that of asupernova. After this initial period ( ∼
10 days), the remain-ing stellar mass accretes via a narrow disk over the course ofa few years, the emission at this stage peaking in the UV andsoft X-rays (Rees, 1988; Strubbe and Quataert, 2009). Firsthypothesised in the 1970s (Hills, 1975), such tidal disruptionevents were not observed until the
ROSAT
All-Sky survey20 years later (see Komossa and Bade, 1999; Donley et al.,2002). A dozen candidates have been revealed since, someaccompanied by short-lived relativistic jets (Bloom et al.,2011; Burrows et al., 2011).The astrophysical interest in these impressive events liesin their ability to identify and characterise quiescent super-massive black holes, which would otherwise lie undetectedat the centres of galaxies. They can help astronomers deter-mine black hole spin and study jet launching and accretiondisk physics; they may also provide an electromagnetic coun-terpart to the gravitational waves emitted during the initialaccretion of the star (e.g. Kobayashi et al., 2004; Kesden,2012).
It was Kant, again, who first proposed that the Milky Waywas a system of stars orbiting collectively in much the sameway as the Solar System. Moreover, he speculated that ob-served nebulae might be distant ‘island universes’, as vast asthe Milky Way. This hypothesis was partly verified by LordRosse in 1845, who resolved M51, and a number of othersources, into spiral patterns (Binney and Merrifield, 1998).But it was not till Hubble’s observations of Cepheid variablestars in M33 that it was firmly established that the nebulaewere in fact (far) outside the Milky Way and were indeedindependent ‘island’ galaxies (Hubble, 1925).
Latter, Ogilvie, & Rein
Figure 1.7
An HST image of the flocculent spiral galaxy NGC2841. Credit: ESA/HST.
Disk galaxies are flattened structures composed of stars,gas, and dust undergoing, for the most part, rotational mo-tion around the galactic nucleus. Originally classified byHubble (1936) as part of the Hubble sequence, they differfrom most astrophysical disks in that their orbital motion isnot entirely dominated by the central object (a supermas-sive black hole); their rotation curves hence depart signifi-cantly from Keplerian. Aside from the initial stage of galaxyformation, there is no accretion throughout the whole disk,per se, mainly on account of there being insufficient rotationperiods during a galactic lifetime. Most stars are clearly lu-minous in the optical and UV; the gas, on the other hand,falls into a variety of thermodynamic phases that emit incorrespondingly diverse frequency bands (Field, 1965; Mc-Kee and Ostriker, 1977; Draine, 2011).Galactic disks manifest large-scale features such as spiralarms, central bars, rings, and streams. They also generallypossess a central spheroidal bulge, containing older stars,and a large spherical dark matter halo which, according tolarge-scale cosmological simulations, plays an essential partin their formation (Springel et al., 2005). Spiral galaxies maybe grouped into various classes, the distinctions resting onhow tightly wound the spirals are (Binney and Merrifield,1998), and are sometimes labelled as ‘grand design’ or ‘floc-culent’ depending on the coherence of the spiral arms. Figure1.7 shows an example of a flocculent spiral galaxy. Lenticu-lar galaxies, on the other hand, are regarded as intermediatebetween elliptical and spiral galaxies. They do not exhibitspiral arms but can form bars and rings.
Disks encompass a vast range of scales and compositions butthey all share the same fundamental force balance: the cen-trifugal force matches the radial component of the central object’s gravity. The mutual cancellation of these powerfulforces releases into the dynamical arena a host of subdomi-nant processes that provide the inherent variety and interestof astrophysical disks (Gor’kavyj and Fridman, 1994). Someof these we cover later in this chapter. For now we providea list of key scales and dimensionless quantities that helpdistinguish different disks from each other and determinewhich physical models are appropriate.The physical quantities of most interest describe the ge-ometry of the disk and its microphysics. We have met H ,the vertical thickness or scaleheight of the disk, and r , theradius of the disk, earlier. In addition, the rotational angu-lar speed of the disk is denoted by Ω, the particles’ collisionfrequency by ω c , their velocity dispersion by c , and their sizeby a . Vertical hydrostatic balance implies that c ∼ H Ω. Ina frame moving with the bulk velocity of the fluid, the par-ticles’ mean free path is hence λ ∼ (Ω /ω c ) H . All of theseparameters vary within the disk, in particular with radiallocation.Sufficiently dense and cool gaseous disks feature colli-sion frequencies much greater than the rotation frequency, ω c (cid:29) Ω. It follows that the mean free path is significantlyless than the disk scale height. In other words, the (verti-cal) Knudsen number is small: Kn = λ/H (cid:28)
1. As a result,equilibrium microphysics is dominated by interparticle col-lisions; the phase space distribution of particles approachesa Maxwellian and the equations of fluid dynamics (or mag-netohydrodynamics) are appropriate (Shu, 1992).These relations are reversed in galactic disks, debris disks,and the tenuous plasma in the very hot regions aroundsome black holes (Binney and Tremaine, 2008; Rees et al.,1982; Narayan et al., 1998). In these settings, ω c (cid:28) Ω,and so we have Kn (cid:29)
1. Collisions between particles areinfrequent and the familiar continuum descriptions breakdown. In particular, the thermal relaxation timescale is oforder, or longer than, the dynamical timescale, leading to se-vere velocity anisotropies and unusual momentum and heattransport (e.g. Toomre, 1964; Braginskii, 1965; Lynden-Bell,1967). Especially extreme environments are the surpassinglyhot midplanes of AGN and X-ray binaries, where the ionsand electrons collisionally decouple and possess tempera-tures differing by 3 orders of magnitude (Rees et al., 1982).The collective behaviour in these systems is determined bylong-range gravitational interactions, in the cases of galacticand debris disks, and by electromagnetic fields, in the caseof a collisionless plasma, and researchers must resort to anappropriate kinetic theory or N -body simulations.Often planetary rings fall uncomfortably between theseframeworks. Ring particles undergo some tens of collisionsper orbit, and thus ω c ∼ Ω and Kn ∼ N -body simulations (see Chapters by Stewart and Salo),neither of which is straightforward to implement nor inter-pret. lanetary rings and other astrophysical disks Disk system r H/r T or c ω c / Ω n Protoplanetary disks >
100 AU 0 .
05 10 − − − cm − Dwarf novae 0.005 AU 0.01 10 − K 10 − − cm − AGN >
100 AU 0.001 10 − K 10 − − − cm − Debris disks: dust 10 − − cm/s 10 − − − − − − cm − Galactic disks: stars 10 −
100 kpc 0 . − . cm/s 10 − . −
100 pc − Galactic disks: gas 10 −
100 kpc 0 . − .
01 10 − K 10 − − cm − Saturn’s dense rings 0.001 AU 10 − − cm − Table 1.1.
Characteristic scales and parameters of selected disk categories. Estimates are lifted from references given inSection 1.2. Here T , c , ω c , and n refer to temperature, velocity dispersion (or sound speed), collision frequency, and numberdensity respectively. Ranges of protoplanetary disk properties are for radii between 0.1 AU and 100 AU. The AGN estimatesinclude the main disk but not the broad-line region at radii (cid:38) AU. The extremely hot temperatures (and consequent lowcollision frequencies) correspond to the radiatively inefficient inner disk around a solar mass black hole. The galacticestimates are for the Milky Way, and do not include the extremely hot diffuse phase of the ISM. Dense rings possess an additional peculiarity: not only isthe mean free path of order the vertical size of the system,so is the particle radius, a . Thus we have λ ∼ a ∼ H, or, formulated another way, R = a/H ∼
1, where R is theSavage–Jeffrey parameter of granular flow theory (Savageand Jeffrey, 1981). In short, the particles that constitutethe disk are macroscopic bodies, another impediment to acontinuum description. There is no astrophysical analoguefor this situation, as even the largest bodies in debris disks,and certainly stars in galaxies, possess R (cid:28) λ ∼ a ∼ H . Excluded-volume effects impose not only an unusualequation of state, but also precipitate the vertical ‘splashing’of particles out of the ring plane. In addition, the ring’s rhe-ology undergoes a dramatic alteration: the collisional trans-port of momentum (from one particle to another duringa collision) dominates over the standard free streaming or translational transport (by individual particles between col-lisions). The dependence of the latter on the system param-eters and state variables is markedly different and leads toalternative instabilities and dynamics (Araki and Tremaine,1986; Wisdom and Tremaine, 1988).Next we consider the range of scales upon which dynam-ical phenomena can manifest. Gaseous disks almost alwaysexhibit r (cid:29) H (cid:29) l, where l is the viscous scale, defined for most dynamical pur-poses by l = (cid:112) ν/ Ω, with ν the kinematic molecular viscos-ity. Using ν ∼ λc , we obtain the scalings l ∼ √ λH ∼ (Ω /ω c ) / H. Another way to put this is in terms of the Reynolds number,defined by Re = H Ω /ν ∼ H/λ (cid:29)
1. In most gaseous disks,the gulf separating the disk radius r from the disk thickness H is much smaller than that between H and the viscousdissipation length l . For instance, in a protoplanetary disk at1 AU, H/r ∼ .
05, and l/H ∼ − . The Reynolds numberis hence huge ∼ (Armitage, 2011). Contrast this with Saturn’s dense rings where r (cid:29) H ∼ l, and energy is dissipated on lengthscales of order the diskthickness H ∼ a . There simply are no shorter meaning-ful scales. Planetary rings are low Reynolds number flows,Re ∼
1, even if their viscosity is more than four orders ofmagnitude less than that of a protoplanetary disk: 10 cm /sversus 10 cm /s (Tiscareno et al., 2007; Armitage, 2011).How does this influence the dynamics? Consider the onsetof instabilities. In many cases, the input of energy is on alengthscale around H , and in gaseous disks, on scales con-sequently much larger than l . Because of this separation,instabilities typically saturate by initiating a turbulent cas-cade of energy to the distant dissipation scales, as this isthe most efficient way to rid the system of the excess en-ergy. Planetary rings make a striking contrast, because l isnot far from the input scale H . Though the ring viscosityis comparatively tiny, the system is ‘controlled’ by dissipa-tion, and instabilities saturate instead by generating struc-ture — and there is a huge range of scales between H and r available in which to do so. The result may be chaotic anddisordered, but it is categorically different from turbulence.Note that if there is a sufficient separation between H and r , gaseous disks can in principle exhibit structure on inter-mediate scales as well: for instance, modulations riding onsmall-scale turbulence (Johansen et al., 2009).Finally, an important distinguishing property of gaseousdisks, at least, is their ionisation fraction. Cold and poorlyionised disks undergo very different dynamics comparedwith partially and fully ionised disks (Blandford and Payne,1982; Balbus and Hawley, 1991; Blaes and Balbus, 1994;Wardle, 1999). This issue is less relevant in comparisonswith dense planetary rings, which are composed of mostlyuncharged macroscopic particles, and are thus closer to de-cidedly hydrodynamic systems, such as the cold neutral re-gions (‘dead zones’) of PP disks (Gammie, 1996). However,the low-collisional dust in faint rings when charged does suf-fer dramatic qualitative changes wrought by electromagneticeffects. The spoke phenomenon in the B-ring is perhaps themost famous example, but periodic structures are also forcedby Saturn’s magnetic field, and large-scale electromagnetic Latter, Ogilvie, & Rein instabilities have been postulated (Goertz and Morfill, 1988;Hor´anyi et al., 2004, 2009, Hedman chapter).
Most disk formation routes draw on either (a) the collapse ofa cloud of material, (b) the tidal disruption of a body, due toits close approach to another massive body, or (c) a physicalcollision. Similarly, ring formation scenarios fall into one ofthese three camps, and historically have strongly influenced,and been influenced by, the question of disk formation gen-erally. In this section we briefly review the three ideas. Seethe Charnoz chapter for a more in-depth discussion.The first scenario is particularly relevant for proto-planetary disks (Section 1.2.1), which form from collapsingdense cores within gravitationally unstable molecular clouds(McKee and Ostriker, 2007). Because the collapsing mate-rial usually carries non-zero angular momentum, it flattensout and ultimately forms a disk orbiting the protostar. Tur-bulence in the gas then processes the angular momentumand permits the remaining mass to fall upon the young star.Note that magnetic fields and non-ideal MHD probably con-trol the early collapse phase, and possibly the ensuing tur-bulent state (e.g., Joos et al., 2012). A similar scheme wasoriginally postulated for disk galaxies (Eggen et al., 1962),though now it is generally accepted that they form via asequence of mergers followed by gas contraction into a disk(Searle and Zinn, 1978).The early stages of Saturn’s formation involved a collaps-ing spinning blob of gas in the protosolar nebula, some ofwhich must have found its way into a circumplanetary disk.One theory posits that Saturn’s rings are the icy leftoversof this disk, the gas long accreted or blown away (Pollacket al., 1976). In this scenario the rings are primordial, almostas old as the Solar System, and share a similar dynamicalorigin (on a much smaller scale) to their host protoplan-etary disk. As explained in the Charnoz chapter, however,the theory has problems explaining their icy composition, inaddition to the relative absence of darker material borne byimpacting meteroids (an indication of relative youth) (e.g.Estrada and Cuzzi, 1996). Perhaps most problematic is thequestion of angular momentum, which is being drained fromthe system by ballistic transport (amongst other processes),and which would ensure a lifetime shorter than that of theSolar System (Durisen, 1984).The second class of disk formation scenarios appeals tothe full or partial tidal disruption of a secondary object. Asdescribed by Roche in the 19th century, a body that orbitstoo close to a central mass will be pulled apart by the com-bined action of gravitational and centrifugal forces (Chan-drasekhar, 1969). Examples of astrophysical disks that formfrom the disrupted material include those around supermas-sive black holes after a stellar disruption, and the debris sur-rounding polluted white dwarfs (described in Section 1.2.5).Related, less dramatic, examples are those associated withclose binaries, where the secondary star overflows its Roche lobe, and the resulting stream feeds an accretion disk aroundthe primary (Section 1.2.2).Roche’s theory, in fact, was originally applied to the for-mation of Saturn’s rings; he proposed that a moon hadveered too close to Saturn and had been tidally disrupted.Recent work augmenting this theory includes that of Harris(1984) and Canup (2010). The latter, in particular, positsthe tidal disruption of the icy shell of a differentiated Titan-sized body and the subsequent accretion of its rocky core;the theory then naturally accommodates the striking homo-geneity of the rings. An explanation similar in some detailshas been proposed by Leinhardt et al. (2012) for the denseUranian ringlets. The inward migration of the moons maybe driven by interactions with the circumplanetary disk ortidal interactions with the planet. A recent passing cometmay also have been disrupted in a similar fashion (Dones,1991), but estimates of the cometary flux at Saturn suggestthat this would be an exceptionally rare occurrence in thelast billion years (Lissauer et al., 1988).The third formation scenario involves collisions. The as-trophysical objects most closely associated with this scenarioare dusty disks around stars (Section 1.2.3). Because of col-lisional destruction and radiation pressure, this dust mustbe constantly replenished by continual but infrequent colli-sions between the larger bodies in the belt. Collision-basedformation theories for Saturn’s dense rings have appealedto a single cataclysmic impact between two moons or be-tween a moon and a comet, the latter event occurring duringthe late heavy bombardment (Pollack, 1975; Charnoz et al.,2009). Both scenarios inherit the problems of the homoge-neous composition and apparent youthfulness of the rings.It is plausible, however, that the F-ring is the result of a re-cent impact between Prometheus and Pandora (Hyodo andOhtsuki, 2015), and a similar event may be responsible forthe Neptunian arcs (Smith et al., 1989).A more direct analogy with debris disks is provided by thedusty rings in the Solar System. These include Saturn’s F-ring, whose dust is continually supplied by the collisions be-tween embedded larger bodies (Cuzzi and Burns, 1988; Bar-bara and Esposito, 2002). They also include Saturn’s G-ring,Methone, Pallene, and Anthe rings, in addition to the Joviandust rings. For the most part, the dust in the latter disksissues from collisions between extrinsic impactors (microme-teroids) and embedded large bodies, such as atmosphere-freemoons (Hedman et al., 2009, 2010; Burns et al., 1999).Finally, we mention Saturn’s E-ring and Io’s plasma torus,both vulcanic in origin, for which there are no perfectextrasolar analogues. Very recent observations, however,show short-lived arcs of material around close-in exoplan-ets, ejected from their atmospheres or surfaces (van Lieshoutet al., 2014; Sanchis-Ojeda et al., 2015).
Once an astrophysical disk has formed, its subsequent evo-lution and ultimate dissipation are, to a large extent, dom-inated by accretion. Disks have a finite lifetime and ulti- lanetary rings and other astrophysical disks mately fall onto their central object via a process of angularmomentum redistribution. The exceptions are galactic anddebris disks. In this section we focus on accretion drivenby local angular momentum transport, such as turbulenceor interparticle collisions. Note that other mechanisms existthat are important in certain circumstances, such as mag-netocentrifugal winds, or large-scale waves (Pudritz et al.,2007; Balbus and Papaloizou, 1999). The classical theory of accretion disks (e.g. Lynden-Bell andPringle, 1974; Pringle, 1981) describes the evolution of a(more or less) continuous disk or ring of matter in circularorbital motion around a massive central body. Any dissipa-tive process that acts similarly to a frictional or viscous forcecauses the inner parts of the disk, which rotate more rapidly,to transfer angular momentum to the outer parts. The diskspreads as its inner and outer parts move to smaller andlarger orbital radii consistent with their specific angular mo-mentum. Alongside this outward transport of angular mo-mentum there is a net inward transport of mass. Meanwhilethe orbital energy of the disk is lowered and heat is gener-ated which, once radiated and intercepted by astronomicalinstruments, yields some of the observations described inSection 1.2. Ultimately the disk will fall upon the centralobject in a time of roughly ∼ r /ν , where r is the disk’souter radius. According to this estimate Saturn’s rings willbe gone in 10 years, longer than the age of the Solar Sys-tem. In contrast, if the only momentum transport processin PP disks was molecular viscosity, they would survive forsome 10 years, longer than the age of the Universe! Toexplain the observations in PP disks at least, an effective,or ‘anomalous’, viscosity must be present at a much greatermagnitude, some 10 cm /s.Within the astrophysical disks mentioned in Section 1.2,several different mechanisms of angular-momentum trans-port may be operating. Some gaseous disks are sufficientlyhot that the main constituents are substantially ionised;these include the disks around black holes and compact starsin interacting binary systems (at least during their activelyaccreting phases), and the protostellar disks of FU Orionissystems. In these systems the magnetorotational instability(MRI; Balbus and Hawley, 1991, 1998) is expected to sus-tain a dynamically significant magnetic field and to providemeasurable angular-momentum transport through magne-tohydrodynamic turbulence. In cooler disks, such as typicalprotoplanetary systems, where the degree of ionisation ismuch lower, the MRI may be restricted only to special re-gions of the disk (see for example Gammie, 1996; Armitage,2011). Other, purely hydrodynamic mechanisms have beenproposed to permit sustained activity in magnetically deadregions. These include turbulence instigated by gravitationalinstability (‘gravitoturbulence’; see Section 1.8.1), whichmay attack the more massive early stages of PP disks (Pa-paloizou and Savonije, 1991; Durisen et al., 2007), subcriticalbaroclinic instability (Lesur and Papaloizou, 2010b), verti-cal shear instability (Nelson et al., 2013; Barker and Latter, 2015), and vertical convection (though it may be difficult toself-sustain; Lesur and Ogilvie, 2010).In situations where a plausible mechanism of angular-momentum transport has been identified, the difficulty re-mains of quantifying its efficiency. Most transport processesare stochastic in nature, but what is needed for the globalevolution of the disk is the mean value of the shear stressand its dependence on relevant quantities such as the den-sity, pressure, shear rate, etc. Shakura and Sunyaev (1973)introduced a useful parametrisation in which the shear stressis written as the pressure multiplied by a dimensionless pa-rameter α . In the case of hydrodynamic or magnetohydro-dynamic turbulence, α is expected to lie between 0 and 1 ifthe perturbations of the fluid velocity (and Alfv´en velocity,in the MHD case) are related to the sound speed (but typ-ically less than it). This ‘alpha-disk’ prescription has domi-nated accretion-disk theory for some 40 years, as it permitsresearchers to construct models of disk evolution and struc-ture, and hence generate synthetic emission spectra that canbe compared with observations. Figure 1.8
A snapshot from a simulation of MRI turbulenceconducted in the shearing box model of a gaseous accretion disk.The field plotted is the magnetic field strength | B | . Credit:Tobias Heinemann. The turbulent state can be investigated through numer-ical simulations of a relevant system of hydrodynamic ormagnetohydrodynamic equations. If the disk is thin and thecorrelation length of the turbulence is small compared to theorbital radius, then a local simulation based on the shearingbox (e.g. Hawley et al., 1995) is usually sufficient; the shearstresses can be measured directly from the simulation. Fig-ure 1.8 presents a snapshot from a local simulation of theMRI. Attempts have also been made to describe the turbu-lent state analytically by means of a set of moment equa-tions derived from the basic hydrodynamic or magnetohy-drodynamic equations using a closure model, such as a sim-ple modelling of the triple correlations of velocity and mag-netic fluctuations (Kato and Yoshizawa, 1993, 1995; Ogilvie,2003). Latter, Ogilvie, & Rein
In the case of planetary rings, angular momentum is trans-ported in part by a viscous stress associated with ananisotropic velocity distribution of the particles, and there-fore with an anisotropic pressure tensor (free-streamingtransport). Indeed, the α viscosity parameter of a diluteplanetary ring (and also a dilute plasma) can be linked to thedegree of anisotropy in the pressure tensor. In dense rings,however, there is an additional contribution from the trans-port of angular momentum during (rather than between)collisions.There is a close analogy between the behaviour of the pres-sure tensor in a dilute planetary ring and the Reynolds stresstensor that describes the correlations of velocity fluctuationsin a turbulent gaseous disk (Quataert and Chiang, 2000).These tensors obey evolutionary equations with some identi-cal terms, describing the interaction of the fluctuations withthe Keplerian orbital motion (which tend to make the tensoranisotropic), as well as some differing terms, describing thecollisional and nonlinear dynamics. In the case of a diluteplanetary ring, collisions make the pressure tensor isotropicand damp the fluctuations. Goldreich and Tremaine (1978b)showed that a circular dilute equilibrium can be sustained ifthe collisions are sufficiently elastic. Similarly, in purely hy-drodynamic turbulence, the nonlinear effects tend to bothisotropise and damp the velocity fluctuations. Sustained hy-drodynamic turbulence is possible only if the isotropisa-tion is relatively strong compared to the damping, a con-dition that is not believed to be satisfied in a Keplerian disk(Ogilvie, 2003). Note that additional thermal gradients maypermit sustained activity.An additional complication is that planetary rings, likegaseous disks, can suffer instabilities that lead to disorderedflows on large scales ( > H ), e.g. gravitational instability(Salo, 1992, 1995). The correlated motions of the associatedturbulence transport angular momentum up to an order ofmagnitude greater than both the free-streaming and col-lisional stresses (Daisaka et al., 2001). This transport canalso be parametrised by an alpha prescription, though sucha model must omit the complicated interplay between theturbulent wakes and the collisional dynamics of the ringparticles, the length and timescales of which are not wellseparated. Plasma systems that undergo analogous mixedbehaviour involving ‘micro-turbulence’ are the solar wind,the intracluster medium, and the inner regions of black-holeaccretion disks (e.g. Kunz et al., 2014).The alpha model, and more sophisticated approaches, aremean-field theories in which the details of the small-scalephysics (turbulent motions, collisions) are deemed quasi-steady and averaged away. However, for certain astrophys-ical processes this microphysics cannot be treated so indif-ferently. Planet formation and dust production, in proto-planetary and debris disks respectively, rely on the detailsof collisional disruption and agglomeration, which are alsorelevant for both dense and dusty planetary rings. In thenext section we review these processes. The collisional dynamics of Saturn’s and Uranus’s macro-scopic ring particles are unusual in astrophysics on accountof their low impact speeds ∼ mm/s. In fact, these are of or-der the escape speeds from the larger particles, and only en-hanced tidal forces (due to their orbit) prevent gravitationalcollapse. Solid bodies in debris or protoplanetary disks col-lide more violently on the whole, at speeds ranging betweenmm/s and km/s. Nevertheless, there is significant overlap-ping physics that is instructive to review. We also discuss inthis section the connections between the dynamics of dustin debris disks and the tenuous rings of the Solar System. The theory of planet formation tracks the agglomeration ofsolid particles in disks from micron-sized dust to the 10 km cores of giant planets. Vaulting this tremendous gulf,spanning some 12 orders of magnitude, requires multiplegrowth mechanisms and physical processes.Microgravity experiments show that impacts between sub-centimetre dust grains are controlled by their coupling withthe ambient gas and generally result in sticking collisions,whether their motions are Brownian or induced by turbu-lence (Blum and Wurm, 2008). Typical impact speeds areof order 1 m/s (Brauer et al., 2008) and attractive surfaceforces are relatively strong in this regime. Larger particlesare only weakly coupled to the surrounding gas flow andthus achieve greater impact speeds, meaning surface forcesbecome less dominant. As a consequence, collisions are morelikely to result in bouncing or fragmentation, not sticking,and agglomeration to sizes larger than centimetres is diffi-cult. Note that these details are complicated by particles’structure (how ‘fluffy’, how compacted, etc.) and their com-position (silicate versus icy, for instance) (Testi et al., 2014).Especially interesting are collisional outcomes between solidsof disparate sizes, which sometimes result in a net masstransfer to the larger particle. Hence there may exist a nar-row route by which a small number of ‘lucky’ aggregates cangrow to km sizes, leaving behind a swarm of small ‘unlucky’grains (Wurm et al., 2005; Testi et al., 2014).In conjunction with laboratory experiments, numericalsimulations, using N -body molecular dynamics and SPH,have determined collisional outcomes between aggregatesof up to decimetre sizes (Wada et al., 2008; Geretshauseret al., 2010; Ringl et al., 2012; Seizinger and Kley, 2013).Figure 1.9 shows the outcome of such a simulation. Addi-tionally, statistical information has been gleaned from in-tegration of suitable coagulation equations (e.g. Safronov,1969; Dohnanyi, 1969; Tanaka et al., 1996) that evolve for-ward in time the distribution function of a swarm of collidingdust grains (e.g. Dullemond and Dominik, 2005; Windmarket al., 2012; Garaud et al., 2013). In such calculations, colli-sions are parametrised in a mathematically convenient butalso physically motivated way.The growth of larger bodies, from centimetre to kilometresizes, is an especially active area of research. Most theoriesrely at some point on the gravitational collapse of many lanetary rings and other astrophysical disks Figure 1.10
Snapshots before, during, and after a collisionbetween two planetesimals using an N -body simulation. Credit:Leinhardt and Richardson (2002). Figure 1.9
Snapshots before and after a simulated collisionbetween dust aggregates. The red shading in the right panelindicates the amount of energy dissipated during the impact.The number of spherical components in each cluster is 2000 and128,000 respectively. The impact speed is 52 m/s. Credit: Wadaet al. (2013). particles, usually in collaboration with aerodynamical ef-fects: streaming instability; accumulation in ‘dust traps’,such as zonal flows and disk vortices; ‘pebble accretion’, etc.(Youdin and Goodman, 2005; Barge and Sommeria, 1995;Lambrechts and Johansen, 2012). The robustness and effi-ciency of these various mechanisms are still unclear.Solid bodies above roughly a kilometre (called planetes-imals) are decoupled from the gas and further coagulationis achieved by direct collisions, the frequency of which isaided significantly by gravitational focusing (Papaloizou andTerquem, 2006). Numerical SPH and N -body simulationshave probed the outcomes of their collisions and show thatthey fall into a variety of regimes: cratering, merging, frag-mentation, ‘hit-and-run’, and annihilation (Leinhardt andStewart, 2012). These regimes partly depend on whetherthe bodies are held together by tensile strength or self-gravity. Figure 1.10 shows snapshots of a simulation of ahigh-velocity collision between two planetesimals. The collective dynamics of a swarm of planetesimals maybe modelled with a suitable coagulation equation or N -bodysimulation (see e.g. Wetherill and Stewart, 1993; Kokuboand Ida, 1996; Weidenschilling et al., 1997; Richardson et al.,2000). These typically indicate runaway growth of a few ag-gregates which halts upon reaching 1000 km sizes (Green-berg et al., 1978; Wetherill and Stewart, 1989). The resultingplanetary ‘embryos’ or ‘protoplanets’ continue to acrete, al-beit at a much reduced rate, via what is termed oligarchicgrowth (Kokubo and Ida, 1998). If researchers in planet formation focus almost exclusivelyon how large objects are assembled from dust grains, onecould say researchers of debris disks take the diametricallyopposed viewpoint: how do large bodies produce the ob-served tiny grains? In fact, the process of runaway accre-tion in planetesimal belts produces not only larger objects,but also significant quantities of dust (Kenyon and Brom-ley, 2004a,b). Dust production hence continues well afterthe disk gas dissipates and throughout the intermittent col-lisional evolution of the ‘leftover’ planetesimals and proto-planets (Wyatt, 2008; Matthews et al., 2014).In contrast to the coagulation equations employed inplanet formation, researchers compute the statistical distri-bution of debris disk solids with collisional cascade models.The largest bodies ( ∼ −
100 km) are input as ‘fuel’, andmass is lost at the smallest sizes due to radiation effects andcollisional destruction. The resulting wide dynamical range,some 40 orders of magnitude in mass, makes these calcu-lations especially difficult. The simplest models assume asteady-state size distribution but with decreasing total mass(Dominik and Decin, 2003). For detailed comparison withobservations, however, more advanced variants are neededthat include, for instance, the particles’ orbital elements, anduse kinetic theory (Krivov et al., 2006; L¨ohne et al., 2012)or ‘particle in a box’ methods (e.g. Th´ebault and Augereau,2007). N -body codes that track explicitly each member of a smallsubset of the solids have been useful in simulating the effectsof large perturbations on disk structure, such as those aris-ing from an embedded planet. They struggle, however, to Latter, Ogilvie, & Rein explain the overall distributions. Hybrid codes have emergedrecently that comprise N -body simulations coupled to dustevolution, thus describing both dynamics and collisions ac-curately and concurrently (Kral et al., 2013; Nesvold et al.,2013). The codes evolve forward in time the properties ofa cloud of similarly sized particles on the same orbit (‘su-perparticles’). Collisions between these groups generate newsuperparticles that represent the post-collisional fragments(Matthews et al., 2014). As in the fields of planet formation and debris disks, thecollisional dynamics of planetary ring particles has been ex-plored with laboratory experiments and statistical methods.The overwhelming majority of work, however, has been un-dertaken with N -body simulations. Owing to numerical lim-itations, these have focused on shorter time-scale dynamicalphenomena such as the onset of instabilities and satellitewakes (Salo, 1991; Salo et al., 2001; Lewis and Stewart, 2000,2009), rather than on the slower processes that shape theparticles’ size distribution. Most N -body studies of denserings involve hard indestructible spheres, either identical orwith a fixed distribution of sizes. Collisions are controlledby a normal coefficient of restitution, and a tangential co-efficient of restitution when including the particle spin. SeeChapters by Stewart and Salo for further details. Needlessto say, the regime of frequent and gentle collisions, whichcharacterises dense rings, is very remote from the contextsdescribed previously in this section.Because of the low impact speeds, proximity to the planet,and the properties of the particles’ regolith, a myriad of pro-cesses control the evolution of the size distribution in denserings. In addition to inelastic ‘bouncing’ (modelled in N -body simulations), collisions may lead to: adhesion (due tothe meshing of micron-sized surface structures or more dras-tic structural reconfigurations); surface compaction of loosefrost (‘polishing’); mass transfer between impacting parti-cles; as well as the more familiar collisional fracture. Lab-oratory experiments have been essential in uncovering andcharacterising these various effects (e.g. Bridges et al., 1984;Hatzes et al., 1988, 1991; Supulver et al., 1995, 1997, Col-well Chapter). A raft of non-collisional processes also con-tribute. These include gravitational recapture of collisionallydislodged fragments, tidal fragmentation, rotational frag-mentation, and erosion by micrometeoroid impacts (Weiden-schilling et al., 1984). This miscellany of physics includeseffects present in protoplanetary dust dynamics (bounc-ing, adhesion, polishing, fragmentation) and planetesimaldynamics (rotational fracture, gravitational recapture), aswell as new effects (tidal fracture, micrometeoroid bombard-ment).In parallel to laboratory experiments, theoretical descrip-tions of individual collisions have been developed that em-ploy viscoelastic theories (e.g. Spahn et al., 1995; Albers andSpahn, 2006a). On the other hand, computing the energet-ics of variously packed aggregates can characterise collisionaloutcomes as a function of impact speed (Guimar˜aes et al.,2012). Future work, involving N -body or molecular dynam- ics simulations (as with planetesimals; Leinhardt and Stew-art, 2012), may categorise collisional events more securely.There exist a small number of statistical studies explor-ing the cumulative effect of collisional coagulation and frag-mentation, such as Weidenschilling et al. (1984), Longaretti(1989), and more recently Bodrova et al. (2012) and Bril-liantov et al. (2015). Finally, ‘sticky’ collisions have beenmodelled in a restricted set of N -body simulations that nu-merically produce self-consistent size distributions in rel-atively good agreement with observations (Perrine et al.,2011; Perrine and Richardson, 2012). In comparison withthe field of planet formation, however, this area of research,though very promising, is underdeveloped. For example,no well-defined ‘barriers’ have been computed above whichgrowth of large aggregates halts, and below which small par-ticles are efficiently swept up by larger ones. Nor have cal-culations been attempted that could decide if it is statisti-cally likely that a few ‘lucky’ aggregates could grow to verylarge sizes ∼
100 m (as in planet formation theories). If suchgrowth was possible, it may provide an explanation for theobserved propellers in Saturn’s A-ring (see Spahn chapter).The collisional dynamics of Saturn’s F-ring differs fromthat of the inner dense rings. The F-ring consists of a belt oflarge ∼ ∼ km sized moonlets and per-turbing the overall structure of the ring in much the sameway that embedded or nearby planets shape the dust in de-bris disks (Murray et al., 2005; Beurle et al., 2010; Attreeet al., 2014).One of the essential features of the F-ring is its weakertidal environment, as compared to the inner dense rings.Gravitational aggregates at smaller radii have difficultygrowing to large sizes before they are tidally disrupted (aninteresting contrast to debris disks and planetesimal belts).This has stimulated alternative theories for F-ring dynam-ics that posit that the population distribution is quasi-static,the number of large bodies regulated by fragmentation andgravitational accretion (Barbara and Esposito, 2002). In-deed it is true that N -body simulations show F-ring par-ticles readily clump into gravitationally bound aggregates,akin to ‘rubble piles’ (Karjalainen and Salo, 2004; Latteret al., 2012b), whose further growth and collisional destruc-tion characterise the general dynamics (Karjalainen, 2007;Hyodo and Ohtsuki, 2014). The collective outcome of aggre-gation and disruption has been theoretically explored usingstatistical methods similar to those employed in other astro-physical disks (Barbara and Esposito, 2002; Esposito et al.,2012). There remains, however, plenty of scope to further ap- lanetary rings and other astrophysical disks ply the well developed techniques of debris disks and planetformation to this problem. In debris disks, radiation forces (Burns et al., 1979) playan important role in the dynamics of small particles. Of-ten a distinction is made between radiation pressure andPoynting–Robertson (PR) drag, although the origin of thetwo effects is the same (photons transferring energy and mo-mentum to dust grains). Radiation pressure removes micron-sized dust grains from the system on a dynamical timescale,thus enforcing a strict lower cut-off in the size distribu-tion of solids (Wyatt, 2008). PR drag, on the other hand,causes small particles to spiral into the central star, but on atimescale longer than collisional erosion. It is hence far lessimportant (Wyatt, 2005b).In contrast, the PR drag on planetary ring particlescan lead to rapid inward migration of dust (Goldreich andTremaine, 1982), especially an issue in the F-ring and Sat-urn’s diffuse rings (Sfair et al., 2009; Verbiscer et al., 2009).This is less important in the inner dense rings where thedust is partially shielded from the Sun and its dynamicscontrolled by collisions with larger particles. The scarcityof sub-mm dust in dense rings can be attributed to theirefficient absorption by larger particles (Becker et al. 2016,French and Nicholson, 2000).Ring dust is also subject to drag forces issuing from boththe host planet’s exosphere and the dilute plasma coorbit-ing with its magnetosphere. As a result, dust can feel eithera headwind or tailwind that causes radial migration andeventual loss from the system, an effect remarkably simi-lar to the aerodynamical migration experienced by centime-tre to metre-sized particles in protoplanetary disks (Wei-denschilling, 1977). Uranus’s outer atmosphere is especiallyextended (Broadfoot et al., 1986), explaining the paucityof dust in its ring system, and which may also have somedynamical consequences on ringlet confinement (Goldreichand Porco, 1987a). Charged dust elevated above Saturn’smain rings, on the other hand, interacts with the Saturnianmagnetosphere leading to radial drifts, angular momentumexchange, potential instability, and the striking spoke phe-nomenon (Goertz and Morfill, 1988; Hor´anyi et al., 2004,2009).
The study of waves (and instabilities) in astrophysical disksbegan with attempts to explain the appearance of spiralstructure in galactic disks (Toomre, 1964; Lin and Shu,1964; Goldreich and Lynden-Bell, 1965). The observed spi-rals may be interpreted as ‘density waves’, a collectivephenomenon combining inertial (epicyclic) forces and self-gravity, but strongly influenced by the orbital shear. Densitywaves have also been studied in gaseous disks, where theycan be thought of as inertial–acoustic waves because pres-sure usually dominates self-gravity, leading to qualitative
Figure 1.11
The dispersion relations of the first fewaxisymmetric p, f, g, and r modes in a local model of a disk.The frequency of the modes ω , scaled by Ω, is plotted versusradial wavenumber k . The disk is a stably stratified polytrope.Credit: Ogilvie (1998). differences in their propagation (the group velocity differsin sign, for instance).If the viscosity parameter α (cid:28)
1, then gaseous disks man-ifest a variety of additional wave modes with wavelengthscomparable to or less than H . Such waves are only accu-rately studied in three-dimensional models that resolve thedisk’s vertical structure (Loska, 1986; Okazaki et al., 1987;Lubow and Pringle, 1993; Korycansky and Pringle, 1995;Ogilvie, 1998). The disk may then be understood as a waveguide through which the various modes propagate. In somemodels the hydrodynamic waves can be classified into f (fun-damental), p (pressure) and g (gravity) modes by analogywith stellar oscillations (Ogilvie, 1998). Rotation also intro-duces low frequency r modes (also called ‘inertial waves’).Example dispersion relations of these modes are plotted inFigure 1.11. Global oscillations can be formed when thesewaves reflect from radial boundaries of the disk or internalturning points. The study of this rich assortment of modes issometimes called ‘diskoseismology’. One of its aims has beento explain the curious quasiperiodic oscillations (QPOs) ex-hibited by certain X-ray sources (Remillard and McClintock,2006), then use these to probe the relativistic gravitationalfields associated with black holes and neutron stars (e.g.Wagoner, 1999).The symmetric f modes exhibit the least vertical struc-ture and correspond to the spiral density waves observed ingalaxies and PP disks, in addition to the large-scale eccen-tric modes inferred in both close binaries and certain PPdisks. In addition, antisymmetric f modes can manifest asbending waves, which transmit a warp (or vertical deforma-tion) through the disk (Papaloizou and Lin, 1995; Ogilvie,1999, 2006). Vertical tilts and warps, in fact, have been ob-served in X-ray binaries, AGN, and protoplanetary disks, Latter, Ogilvie, & Rein and are usually driven by a misaligned companion (Katz,1973; Kotze and Charles, 2012; Miyoshi et al., 1995; Marinoet al., 2015b).In planetary rings the smallest meaningful lengthscale isthe particle size ∼ H and hence wave modes express little tono vertical structure: the disk is effectively two-dimensional.As a consequence, planetary rings cannot support nearly thesame variety of waves as gaseous disks, and in fact only thef modes are present. The Voyager and
Cassini spacecraftimaged many examples of spiral density and bending wavesexcited in Saturn’s A-ring and B-ring by the planet’s moons(Colwell et al., 2007). Their physics is very similar to thecase of close binaries and embedded planets in PP disks(see Section 1.9).In contrast, some density waves located in the C-ring mayhave been generated by low-order normal-mode oscillationswithin Saturn itself (Marley, 1991; Hedman and Nicholson,2013; Fuller, 2014). The study of such waves (‘kronoseis-mology’) may provide clues about the internal structure ofthe central planet. This effort provides a nice parallel todiskoseismology’s attempts to characterise black holes andneutron stars.Large-scale ‘corrugation waves’ with wavelengths >
30 kmhave also been observed in the C and D-rings generated bya cometary impact dating from the 1980s (Hedman et al.,2007, 2011). Similar waves were excited in Jupiter’s mainring by Shoemaker–Levy 9 in 1994 (Showalter et al., 2011).In fact, an analogue of this process occurs in young proto-stellar disks: infalling material from the star’s natal envelopecan cause spiral shocks in the disk, which may transport anon-trivial amount of angular momentum, while thermallyprocessing chondrule precursors (Boss and Graham, 1993;Lesur et al., 2015). An important distinction, however, isthat the collective forces are far more significant in gaseousdisks, so that the disturbances are bonafide waves, unlike inthe ring context, where the structures are essentially kine-matic.The observed spiral waves in planetary rings possess ra-dial wavelengths that are much greater than the thicknessof the rings >
10 km, and this means that self-gravity dom-inates pressure (or velocity dispersion) in their propagation,while the viscosity of the rings usually damps the waves. Butdensity waves have been observed on much shorter scalesas well. Both axisymmetric and non-axisymmetric densitystructures appear with roughly 100 m wavelengths in RSS,UVIS, and VIMS observations (Thomson et al., 2007; Col-well et al., 2007; Hedman et al., 2014). The small-scalenon-axisymmetric wakes, in particular, give rise to a strik-ing large-scale effect, the azimuthal brightness asymmetry(Camichel, 1958; Colombo et al., 1976; Thompson et al.,1981; Salo, 1992).In order to reach dynamically important (and observ-able) amplitudes, waves must grow to nonlinear strengths.As mentioned, periodic forcing due to an orbiting compan-ion naturally excites waves and other, non-propagating, dis-turbances. This is reviewed in Section 1.9. Conversely, nu-merous instabilities can drive wave (and other) activity toobservable levels. Relevant instabilities are discussed in thefollowing section.
Because of their physical complexity, gaseous accretion disksaccommodate a large number of instabilities, usually draw-ing their energy from the background orbital shear but some-times also from vertical shear, thermodynamic gradients, ordirectly from the self-gravitational potential of the disk. Itmight be said there are more instabilities than observationsthey could feasibly explain! This is in contrast to Saturn’sdense rings where there is an abundance of observed struc-ture, much of it presumably generated by instabilities notyet identified or understood. In the following subsection wereview only those processes that are shared by, or have some-thing in common with, those appearing in planetary rings.
As explained in the Stewart chapter, self-gravity decreasesthe squared frequency of density waves, leading to ax-isymmetric instability on intermediate wavelengths whenToomre’s parameter Q = κc/πG Σ < κ is the disk’s epicyclic frequency and Σ is its surface density.Note that the criterion given here is for a two-dimensionalgaseous disk, and differs slightly in other models. The abovecondition can plausibly be met in spiral galaxies, more mas-sive PP disks, accretion disks in active galactic nuclei, and,of course, dense planetary rings.Gravitational instability (GI) is thought to power the den-sity waves observed in flocculent spiral galaxies (and possi-bly grand design spirals), and thus controls a crucial aspectof their structure. Observed spirals in protostellar disks mayshare the same origin, though embedded planets could alsodrive these features. GI also features in the ‘disk instability’theory of planet formation, by which gas giants are formedby direct collapse of the disk (Kuiper, 1951; Cameron, 1978;Boss, 1998). In planetary rings, GI is responsible for ‘wake’activity on much smaller relative scales, on account of the ex-treme thinness of the rings. Because unstable waves emergeon scales (cid:38) H , they are usually global features in gaseousdisks and local features in planetary rings.Unstructured disks are linearly stable for Q >
1, as theycannot support non-axisymmetric GI modes. However, forsomewhat larger Q ≈
2, finite-amplitude perturbations in-stigate sustained spiral density waves and the system settlesinto a ‘gravitoturbulent’ state. Figure 1.12 shows a snapshotof GI-induced turbulence in a local model of a gaseous disk;here the mean Q is 2.5. At least locally, this is a ‘subcritical’transition: the disk can support both a laminar and a turbu-lent state for 1 < Q (cid:46) N -body simulations is always sufficient to disrupt the laminarstate.The nonlinear outcome of gravitational instability is sensi-tive to heating and cooling because Q is proportional to thevelocity dispersion, or sound speed, of the disk. Typicallythe instability leads to enhanced dissipation that tends toincrease Q , so a thermostatic regulation can be achieved lanetary rings and other astrophysical disks − − − − − − − − y . . . . . . . . . . S u rf ace d e n s it y Σ Figure 1.12
Gravitoturbulence in a 2D shearing box model ofa gaseous disk. The fractional surface density perturbation isplotted, while the unit of length is the initial unperturbed scaleheight c/ Ω, which here is five times the Jeans length G Σ / Ω ,where Σ is the mean surface density. The (linear) cooling timeis 9 / Ω. Credit: Antoine Riols-Fonclare. in which Q ∼ a is ∼ H , and clumpsare thus easier to be ripped apart by tidal forces and/or col-lisions with other particles (or wakes) than the much smallergaseous cores. Another way to think about this is in termsof the equation of state, or a change of phase. Gravitationalcollapse, in any system, should end once material becomes sodense that pressure resists infall and/or its cooling radicallydiminishes (via an opacity jump, for example). In gaseous disks the lengthscale upon which collapse halts is excep-tionally small (cid:28) H , whereas in planetary rings it is ∼ H .Once ring material clumps it almost immediately becomes‘incompressible’: it changes state from a ‘granular gas’ to a‘granular liquid’. In addition, cooling is minimised becausewithin a crowded aggregate ‘collisions’ are very gentle (if notabsent) and hence more elastic. Nonetheless, clump forma-tion is an important feature in the outer A-ring and in theF-ring, where there is indeed a population of larger objects,‘propellers’ and ‘kittens’ respectively (see Section 1.6.3 andchapters by Spahn and Murray). A viscous stress need not just damp density waves (or fmodes), it can also, somewhat counterintuitively, cause suchwaves to grow. A density wave produces stress perturbationsthat couple the background orbital shear to the wave ve-locities. Energy is extracted and the wave amplified if thevelocity and stress oscillations are sufficiently in phase. Theinstability typically saturates in a quasi-steady state, withthe viscous driving of the waves balanced by their viscous de-struction (see Stewart chapter). The process was first discov-ered in the accretion-disk context (Kato, 1978; Blumenthalet al., 1984), where it was hoped it might explain observedluminosity fluctuations around compact objects.As with density waves excited by GI, viscously overstablemodes, and their saturation, are essentially global in theaccretion disk context. Thus the inner and outer bound-aries, Lindblad resonances, and the disk’s vertical structureall feature in the evolution of the instability (Papaloizouand Stanley, 1986; Kley et al., 1993; Miranda et al., 2015).In contrast, viscous overstability in planetary rings can bea very local phenomenon — certainly when axisymmetric.The fastest growing modes favour scales of some 100 m, atiny fraction of r (Schmit and Tscharnuter, 1995, 1999; Saloet al., 2001; Schmidt et al., 2001), and as a consequence, thesaturation mechanism is controlled by nonlinear travellingwaves (Schmidt and Salo, 2003; Latter and Ogilvie, 2009,2010; Rein and Latter, 2013). It is generally accepted thatthe fine-scale axisymmetric density waves in Saturn’s A andB-rings are generated and sustained by viscous overstability(Thomson et al., 2007; Colwell et al., 2007; Hedman et al.,2014).Viscous overstability can generate large-scale features,such as eccentric modes (Borderies et al., 1985; Papaloizouand Lin, 1988; Longaretti and Rappaport, 1995). It is likelythat the structure of eccentric ringlets and the outer B-ringedge are partly sculpted by this process (Spitale and Porco,2010; Nicholson et al., 2014). Similarly, certain gaseous ac-cretion disks may have obtained their eccentricity in this way(Lyubarskij et al., 1994; Ogilvie, 2001; Latter and Ogilvie,2006b), though the issue is far less clear cut than in theplanetary ring context.One reason why the onset of viscous overstability is prob-lematic in gaseous accretion disks, as opposed to planetaryrings, concerns the nature of the viscous stresses. In gaseousdisks these are presumed to be supplied by hydrodynamic(or magnetohydrodynamic) turbulence — but it is unclear if Latter, Ogilvie, & Rein the turbulence responds in the desired way when a densitywave propagates through the disk. The stress may not beenhanced in high density-wave crests, and even if it is en-hanced the stress may be out of phase with the dynamicaloscillation (Ogilvie, 2003). In either case, viscous overstabil-ity may fail to occur. Note that the collisional stress in denseplanetary rings does not suffer from this shortcoming (Arakiand Tremaine, 1986; Salo et al., 2001; Latter and Ogilvie,2008).
The viscous instability (also called the ‘inflow instability’)occurs when the stresses in a disk decrease with increasingsurface density. The equation for small perturbations in thesurface density becomes, as a result, a diffusion equationwith a negative diffusivity. Physically, a localised bump ofdensity accretes less vigorously than its surroundings andmass piles up at its outer edge, enhancing the overdensity.In the early 1970s the viscous instability was shown toafflict certain disk models of X-ray binaries, in particularwhen the disk is assumed optically thick and its stressesproportional to the radiation pressure (Lightman and Eard-ley, 1974; Shakura and Sunyaev, 1976). But there has alwaysbeen a question mark regarding the applicability of viscousinstability to real accretion disks because of the last assump-tion. It is by no means assured that the turbulent stressesin radiatively dominated accretion flows behave in the wayrequired (but see Hirose et al., 2009).Inspired by the
Voyager images, early theories of plan-etary ring structure appealed to the viscous instability(Ward, 1981; Lukkari, 1981; Lin and Bodenheimer, 1981),but again doubts were raised about the properties of theviscous stress, and interest dwindled. Kinetic theory indi-cates that dilute rings possess a stress that decreases withsurface density (Goldreich and Tremaine, 1978b; Shu andStewart, 1985) but also that dense rings emphatically donot (Araki and Tremaine, 1986). If viscous instability is tooccur in dense rings, it must attack only the smallest parti-cles selectively (Salo and Schmidt, 2010).
Gaseous accretion disks may fall into one of many ther-mal/ionisation equilibria for a given set of parameters. Notall of these are thermally stable and so it is possible thata disk cycles between different states over time, generatingquasi-periodic variability in accretion luminosity and non-thermal emission.The classic and best understood examples are dwarf no-vae which straddle the temperature threshold for hydrogenionisation (about 5000 K). Because the opacity increasesdramatically in the partially ionised phase, the gas’s coolingrate is a complicated function of the temperature and per-mits the disk to support three possible thermal equilibria fora given surface density (Meyer and Meyer-Hofmeister, 1981;Faulkner et al., 1983). In the phase space of surface densityand temperature, disk equilibria sketch out a characteris-tic ‘S-curve’, an example of which we plot in Figure 1.13.
Figure 1.13
A characteristic dwarf nova S-curve, describingthermal equilibria in the phase space of surface density andtemperature. The arrows indicate a limit cycle that the diskmay follow. Credit: Latter and Papaloizou (2012).
The disk may then enter a limit cycle, oscillating betweenthe hot, well-ionised, and luminous high state (an outburst)and the cool, poorly ionised, and dim low state. The tran-sition between the two states takes place via thermal frontsthat rapidly sweep through the disk. The story is compli-cated by a raft of ancillary physics, but in general the con-tact with observations is relatively good. Similar physics isshared with low-mass X-ray binaries, but these systems arenot so well understood and the model enjoys less success(Lasota, 2001).Disks dominated by radiation pressure can support a va-riety of interesting equilibria partly because of the relativeinefficiency of cooling in very hot plasma. In addition to thestandard thin disk solution of Shakura and Sunyaev (1973),there exist ‘slim’ and ‘thick’ disks in which radiative cool-ing is supplanted by radial advection of energy by the gas(Abramowicz et al., 1988). The most extreme case is theadvection-dominated accretion flow (ADAF) where little ofthe dissipated energy is radiated locally and the rotationprofile deviates significantly from Keplerian (Narayan andYi, 1994). If the turbulent stresses are assumed proportionalto the total pressure, these solutions are thermally unsta-ble because the heating depends on temperature to a muchgreater power than cooling (Shakura and Sunyaev, 1976;Piran, 1978). Even though recent MRI simulations indicatethat thermal instability can arise in such disks (Jiang et al.,2013), X-ray observations of strongly accreting black holesystems fail to exhibit unstable or cyclic dynamics on theexpected timescales (Gierli´nski and Done, 2004; Done et al.,2007). Only the exceptionally luminous black hole binary,GRS 1915+105, shows anything like an expected limit cycledriven by thermal instability (Done et al., 2004). In contrastthere are abundant observations of other kinds of variability,especially in the spectroscopy, and a panoply of well-definedstates exist in the phase space of intensity, hardness, andrms fluctuation (Remillard and McClintock, 2006; Belloni,2010). To date there is no convincing explanation for thecycling between these phases. lanetary rings and other astrophysical disks Finally, protoplanetary disks are thought to jump aperi-odically between high and low accreting states on a timescale of 100+ years. This outbursting behaviour is exempli-fied by the archetype FU Orionis, though other classes mayexist, notably the shorter timescale EXors (Audard et al.,2014). The current theoretical paradigm posits that the lowstate corresponds to a disk whose central region (the ‘deadzone’) is cold, laminar and inefficiently accreting and thatthe high state corresponds to one in which this region isengulfed in MRI turbulence. The build-up of mass at theouter edge of the dead zone and a consequent gravitationalinstability are the trigger for an outburst (Gammie, 1996;Armitage et al., 2001).Saturn’s rings also exhibit multiple phases. In the innerB-ring there exist adjacent flat and undulatory states, whilethe middle and outer B-ring breaks up into disjunct bands ofintermediate and high optical depth; both phenomena occuron scales of order 100 km (Colwell et al., 2009). In accretiondisks the different phases are distributed over time, but inplanetary rings, with their achingly slow timescales, adjoin-ing states are distributed spatially. For example, during adwarf nova outburst the high state overwhelms the disk in ∼ days (Latter et al.,2012a).The origin of the observed states in planetary rings is notwell understood. It is perhaps unlikely that they issue from abistability in the thermal equilibrium itself (as in dwarf no-vae). For a constant coefficient of restitution (cid:15) , there is onlyever one equilibrium for a given set of parameters (Arakiand Tremaine, 1986), while experimentally derived laws de-scribing (cid:15) ’s dependence on impact speed yield the same re-sult (Wisdom and Tremaine, 1988; Salo, 1991; Latter andOgilvie, 2008). Note, however, that if (cid:15) is a non-monotonicfunction of impact speed then bistability could be possi-ble. Such a coefficient of restitution may correspond to colli-sion models that incorporate particle sticking at low impactspeeds (Albers and Spahn, 2006b).The undulatory and flat states in the inner B-ring couldbe competing outcomes of the ballistic transport instability(Durisen, 1995), whose nonlinear dynamics supports bista-bility (Latter et al., 2014a,b). The 100 km structures deeperin the B-ring are more mysterious. Tremaine (2003) pro-posed that these correspond to shearing and shear-free re-gions, the latter held together by strong inter-particle ad-hesion. While it may be possible to hold together 100 kmshear-free bands if the disk were restricted to the orbitalplane, in three-dimensions the rings’ tensile strength will betoo weak because the rings can vertically ‘relax’. Astrophysical bodies that are surrounded by continuousdisks often possess discrete satellites as well. Examples in-clude the moons of giant planets, which then interact with their planetary rings, protoplanets interacting with a cir-cumstellar disk, and binary stars and black holes interact-ing with accretion disks. Various geometrical configurationsare possible, as a satellite can orbit fully interior or exteriorto the disk, be confined within an annular gap, or be fullyembedded in the disk.The gravitational interaction between an orbital compan-ion and a disk is a problem of general interest in astro-physics. By generating waves and other disturbances in thedisk, the satellite both undergoes orbital evolution and influ-ences disk properties. In addition, observations of the struc-tures induced in the disk can be used to constrain physicalproperties of both the disk and the perturber.One important application, discussed in greater detail be-low, is to planets formed in a gaseous disk around a youngstar. The planet–disk interaction causes the planet to mi-grate radially through the disk, a process that needs to beunderstood and quantified in order to interpret the observedproperties and architectures of exoplanetary systems (as wellas the Solar System). Ever since the first planets were discov-ered on orbits very close to solar-type stars, it was suggestedthat planetary migration brought them to their current lo-cations (Mayor and Queloz, 1995). As more and more hotJupiters have been found, it is generally accepted that theseplanets formed at locations beyond 1 AU and have sincemigrated inwards.Orbital migration can also be important in AGN. Whentwo galaxies merge, the central black hole of the smallergalaxy interacts with the accretion disk surrounding thelarger black hole, in a way analogous to a planet interact-ing with a circumstellar disk. Inward orbital migration leadseventually to a compact binary black hole that merges as aresult of gravitational radiation, as spectacularly confirmedby recent observations (Abbott et al., 2016).Finally, many of the observed structures in astrophysi-cal disks and planetary rings can be attributed to satellite–disk interactions. Examples include the Cassini division, theEncke and Keeler gaps, the spiral waves in Saturn’s rings, aswell as the spiral arms in interacting galaxies and the tidaltruncation of accretion disks in binary stars. Planet–disk in-teraction can also create annular gaps and spiral structuresin PP disks, some examples of which may have been recentlyobserved (see section 1.2.1).
The simplest situation involves a satellite on a circular orbitthat is coplanar with the disk. The satellite exerts a peri-odic gravitational force on the disk and excites its epicyclicmotion. The forced motion is resonant at a series of radii,located both interior and exterior to the satellite’s orbit, andthe disk responds by launching a spiral density wave at eachof these resonances. To the extent that the epicyclic fre-quency corresponds to the orbital frequency, these Lindbladresonances can be identified with mean-motion resonancesinvolving commensurabilities of the form m : m ± Latter, Ogilvie, & Rein from Keplerian motion due to pressure gradients, disk self-gravity, oblateness of the central body, etc.The act of launching a wave involves a transfer of en-ergy and angular momentum from the satellite’s orbit. Asthe waves propagate radially away from the Lindblad res-onances, their radial wavelength decreases and they aredamped by viscosity or other dissipative processes. Thedamping may be enhanced if the waves attain nonlinear am-plitudes. The angular-momentum flux carried by the waveis transferred to the disk as the wave is damped and thusa resonant torque is exerted between the satellite and thedisk.Numerous examples of these density waves in Saturn’s A-ring were observed by the
Voyager and
Cassini spacecraft,and have been identified with specific Lindblad resonanceswith various moons that orbit outside the A-ring. A differ-ent type of density wave is seen near the edges of the Enckeand Keeler gaps in the outer A-ring. These are excited bythe satellites that orbit within these gaps and are thereforecloser to the ring material than the larger external moons.In these cases the wake cannot be identified with a singleLindblad resonance, although it can be thought of as a su-perposition of waves generated at many such resonances ofhigh order.In the case of planets interacting with PP disks, relatedphenomena have manifested chiefly in theoretical work (al-though there is now some observational evidence of spiralwaves in PP disks, which can be explained best by embed-ded planets; Dong et al., 2015, 2016), and in recent yearshydrodynamic simulations have led the way in determiningthe nonlinear dynamics of planet–disk interactions. Embed-ded satellites that are not massive enough to open a gapin the disk’s density profile undergo what is called type-I migration (Goldreich and Tremaine, 1978a, 1980; Ward,1997). The density waves launched at different Lindbladresonances constructively interfere to produce a coherentone-armed spiral wake (Ogilvie and Lubow, 2002), a nar-row overdense region. The wake is not symmetric about thesatellite’s location, because of the circular geometry and pos-sible radial gradients in the properties of the unperturbeddisk. The satellite therefore experiences a net gravitationaltorque, which under most circumstances is negative, lead-ing to inward migration of the satellite. A snapshot of thesurface density in a hydrodynamic disk simulation with anembedded planet of 10 Earth masses is shown in Figure 1.14This description is incomplete because a different type ofinteraction happens closer to the satellite’s orbit, where diskmaterial approximately corotates with the satellite. In thisregion the relative motion of the disk and satellite is tooslow for density waves to be excited, but the satellite caninstead generate non-wavelike disturbances in the potentialvorticity and entropy of the disk, each of which involves anasymmetric rearrangement of the surface density and there-fore a torque. In distinction to the Lindblad torques associ-ated with the launching of density waves, the torques arisingfrom this region are known as coorbital, corotation or horse-shoe torques. The last name comes from the property that,in the frame rotating with the satellite’s orbit, disk materialin the coorbital region has streamlines that librate rather
Figure 1.14
Hydrodynamic simulation of a planet of 10 Earthmasses undergoing type-I migration in a protoplanetary disk.The surface density is plotted in grey-scale. than circulate, and involve horseshoe-shaped turns near thesatellite’s longitude.The entropy-related corotation torque can lead to stallingof planets exceeding roughly 3-5 earth masses at specificdisk locations that vary with time (Baruteau et al., 2014).Ultimately, however, these torques are sustained by dissi-pative effects, such as viscosity, thermal diffusivity, and ra-diative cooling which are all uncertain (Paardekooper andPapaloizou, 2008, 2009). In addition, the physics of the coor-bital region is strongly nonlinear, and thus difficult to de-scribe accurately. The net (Lindblad plus coorbital) torqueis subject to similar uncertainties.Nevertheless, there is general agreement that planets ofEarth mass typically migrate towards the star on a timescaleof a few hundred thousand years. This poses a problem forplanet formation because the viscous timescale, i.e. the life-time of the accretion disk, is thought to be significantlylonger than that. The inconvenient conclusion is that everyEarth-mass planet in a protoplanetary disk should have mi-grated into the star. Various ways to prevent this from hap-pening have been proposed. These ideas include the stochas-tic torques arising from turbulence in the disk (see later),positive torques associated with asymmetric heating in theplanet’s neighbourhood (Ben´ıtez-Llambay et al., 2015), andso-called planet traps due to dead zones, snow lines or otherfeatures in the disk (Masset et al., 2006). Similar physicsmay also control the migration of propellers in Saturn’s rings(Tiscareno, 2013). We still, however, lack a general theorycapable of predicting the speed and direction of type-I mi-gration, which is vitally needed to link together the earlyand late stages of planet formation. lanetary rings and other astrophysical disks In the type-I regime, while the fractional surface densityperturbation in the wake can become of order unity at somedistances from the satellite (involving a shock, in the case ofa gas disk), the azimuthally averaged surface density of thedisk is not significantly perturbed. More massive satellites,however, are able to deplete the disk locally by opening anannular gap. This happens because the Lindblad torquesexerted by the satellite on the interior and exterior parts ofthe disk are negative and positive, respectively, causing bothto recede from the satellite’s orbit.The scaling laws for gap opening have been formulated byLin and Papaloizou (1986, 1993). Note that it is difficult toquantify the process exactly because gaps can be of differ-ing degrees of cleanliness. In order to open a gap, the massratio M s /M of the satellite and the central object should besufficiently large. A first criterion is that M s /M (cid:38) ( H/r ) ,or, equivalently, that the Hill radius R H (cid:38) H . This is thecondition for the disturbance generated in the disk to benonlinear close to the satellite, which allows it to be dis-sipated locally. A second criterion, which is usually morestringent, is that M s /M (cid:38) (81 π/ ν/r Ω). This is the con-dition that the tidal torque exceed the viscous torque in theundisturbed disk (if indeed it can be adequately describedby a simple kinematic viscosity). When applied to a proto-planetary disk, the latter criterion implies that a planet ofabout Saturn’s mass or greater is able to open a gap. Whenapplied to the outer part of Saturn’s A-ring, the criteria im-ply that a moon of about Daphnis’s mass or greater is ableto open a gap.The characteristics of satellite migration change signifi-cantly when a gap opens — a situation referred to as type-IImigration (Figure 1.15). The gap effectively divides the diskinto interior and exterior disks, making it difficult (or impos-sible) for material to pass through. The satellite is effectivelylocked to the viscous evolution of the disk, and may even re-tard that evolution if it is more massive than the interiordisk. The timescales of type-II migration are therefore ingeneral longer than those of type-I migration. HL Tau (seeFigure 1.1) might be a system where planets were able toopen gaps, although confirmation of this interpretation isstill pending.
Classical theories of type-I and type-II migration assume asmooth, laminar disk as a background state. Since disks aretypically active with a number of instabilities, this may be apoor approximation. Instabilities can lead to turbulence andcause density perturbations on many scales. This can, espe-cially for small embedded satellites, lead to migration thatresembles a random walk in the orbital parameters (Nel-son and Papaloizou, 2004; Rein and Papaloizou, 2009). This stochastic migration is relevant for low-mass planets in PPdisks in the presence of the MRI or GI, and for moonlets inplanetary rings (discussed in detail in the Spahn chapter). Ifthe moonlets are small enough, their migration will be dom-inated by the stochastic component (Rein and Papaloizou,
Figure 1.15
Hydrodynamic simulation of a planet of 5 Jupitermasses opening a gap and undergoing type-II migration in aturbulent protoplanetary disk. Again the surface density isplotted. Credit: Nelson & Papaloizou (2003).
Satellites that are significantly more massive than those thatare able to open a gap can truncate a disk at great distance.In Saturn’s rings the outer edge of the B-ring is associatedwith the 2:1 Lindblad resonance with Mimas, while the outeredge of the A-ring coincides with the 7:6 resonance withthe coorbital moons Janus and Epimetheus. Related phe-nomena occur in accretion disks in binary stars; for typicalmass ratios, however, the 2:1 resonance is so strong thatthe disk is truncated well inside it (Papaloizou and Pringle,1977; Paczynski, 1977). On the other hand, the (cid:15) -ring ofUranus appears to be ‘shepherded’, i.e. radially confined, bythe satellites Cordelia and Ophelia on either side, via dis-crete Lindblad resonances (Goldreich and Porco, 1987b).In order to explain the extreme sharpness of the edgesof planetary rings, in particular those mentioned above, ithas been found necessary to appeal to a modification of theviscous torque that occurs when the ring carries a nonlin-ear density wave (Borderies et al., 1982, 1989, Chapters byLongaretti and Nicholson). The presence of the wave altersthe velocity gradient associated with circular orbits in sucha way that the angular-momentum flux is reduced to zerowhen the wave has a critical amplitude, and is reversed forwaves of higher amplitude. Without this effect, the edgesof planetary rings would be smoothed out by viscosity and Latter, Ogilvie, & Rein it would be difficult or impossible to account for the shep-herding process. It is not known whether the modificationor reversal of the angular-momentum flux plays a role ingaseous disks.
If the satellite’s orbit is eccentric or inclined with respectto the disk, then the gravitational force on the disk con-tains additional components that launch density or bendingwaves at Lindblad or vertical resonances. The eccentricity orinclination of the satellite evolves as a result of these inter-actions. For satellites embedded in the disk, a small ( (cid:46)
H/r )eccentricity e or inclination i is damped on a timescale thatis shorter than that for type-I migration. This interaction isdominated by disk material close to the planet. For larger e or i the damping is less efficient and the direction and rateof migration may also be affected. When the planet’s speedrelative to the local gas exceeds the sound speed or velocitydispersion (as happens for example when the inclination ofthe satellite brings it above the disk, i.e. i (cid:38) H/r ) the in-teraction is dominated by gas drag similar to that of a starpassing through the interstellar medium (Rein, 2012).More massive satellites that are well separated from thedisk material interact through more distant orbital reso-nances and the net effect can, under some circumstances, bea growth in e and/or i (Goldreich and Tremaine, 1981; Bor-deries et al., 1983, 1984). In fact, the disk can also becomeelliptical and/or warped through these interactions, and therelative magnitude of the e or i of the disk and the satel-lite depends on their coupled dynamics (Lubow and Ogilvie,2001; Teyssandier and Ogilvie, 2016).A good example of satellite–disk interaction producing agrowth of eccentricity occurs in SU Ursae Majoris binarystars, where the accretion disk around a white dwarf is un-derstood to become elliptical as a result of an interactionwith the companion star at the 3:1 orbital resonance, anexample of an eccentric Lindblad resonance (Lubow, 1991).This is the ‘superhump’ phenomenon, so called because ofthe associated modulation of the light curve during a super-outburst.A similar process may be responsible for maintaining theeccentricity of Uranus’s (cid:15) -ring; the 47:49 eccentric Lindbladresonance with its inner ‘shepherd’ satellite, Cordelia, lieswithin the ring (Goldreich and Porco, 1987b). However, itshould be noted that other narrow rings around Saturn andUranus are also found to be eccentric even though they haveno observed shepherds (see Nicholson chapter). In this chapter we make explicit connections between thestudy of planetary rings and of other astrophysical disks,putting an emphasis on dynamics. Disk systems exhibit anenormous physical and dynamical diversity, but one an-chored upon the fundamental balance between radial grav-ity and the centrifugal force. A relic of formation, their or- bital angular momentum is inherited from the collapse of acloud, the disruption of body by a massive companion, orthe collision of two bodies around a more massive object.This ‘excess’ angular momentum thwarts the simple accre-tion of disk material upon the central body, irrespective ofthe different formation routes, and leads to planetary ringsand astrophysical disks.Through the diversity of composition and physics one candiscern recurrent themes. To begin, most gaseous disks sup-port hydrodynamical (or magnetohydrodynamical) activitythat permits disk material to slowly shed or redistribute itsangular momentum and thus ultimately accrete. By liberat-ing orbital energy this activity also causes the disk to radi-ate — the key to understanding certain high energy sources.The outward transport of angular momentum (whether me-diated by turbulent motions in gas or collisions between par-ticles) controls the evolution and lifetime of the disk or ring.Though the ‘viscous’ lifespans of planetary rings and gaseousdisks differ by orders of magnitude, accretion represents oneof their key connections.Collisional dynamics presents a link between rings andother particulate disks, such as debris disks and the beltsof dust and planetesimals orbiting young stars. In the ringcontext, collisions not only help transport angular momen-tum but also control the composition, structure, and sizeevolution of the constituent particles themselves. The sameprocesses govern PP and debris disks, leading to the forma-tion of planetesimals and planets in the former, and dust inthe latter.Both rings and disks support the passage of waves andthe growth of instabilities, which contribute to the activityrequired to sustain accretion and angular-momentum trans-port. Regarding self-excited instabilities, only in the caseof gravitational instability is (nearly) the same process re-liably occurring in both rings and gaseous disks, althoughviscous overstability may be a second example. Satellite–disk interactions, on the other hand, provide the richest setof dynamics shared by the two classes of systems. Spiraldensity waves, gap formation, and satellite migration, allnow directly observed in detail around Saturn, have impor-tant analogues around young stars that are beginning toyield observational manifestations (with instruments suchas ALMA). In that respect, observations of satellite–disk in-teractions in planetary rings are ahead of those in PP disksby a few decades. But just as the Voyager data proved soexciting and fruitful during the 1980s, so should ALMA ob-servations in the following decade, as its full capabilities arebrought to bear on the problem of exoplanets and their hostdisks.Given these overlaps, it is no surprise the fields of disksand of rings have connected in mutually beneficial ways.The clearest instance is in the study of satellite–disk inter-actions, where work on binary stars and Saturn’s rings con-verged fruitfully in the theoretical understanding of planet–disk coupling in the 1980s. Another example is the researchin disk instabilities, first introduced in gaseous disks in the1970s but matured in the planetary ring context over thenext few decades. Despite this historically close connection,there yet remain a number of correspondences that have yet lanetary rings and other astrophysical disks to be fully capitalised upon and which form the basis forseveral appealing research directions.The size-distribution dynamics of dense rings is an un-derdeveloped area compared with that for debris disks andplanet formation. Though arguably a more difficult prob-lem, the techniques and tools of the latter could be prof-itably adapted to help explain the distributions in Saturn’srings, which are observationally well constrained in compari-son to those more distant particulate systems. Note that thestatistical approaches used in debris disks may be appliedwithout too much complication to the F-ring, as it is proba-bly the closest ring analogue. On the other hand, this is alsoan opportunity to determine how well these techniques andtools perform on an object so much better constrained thana debris disk. An especially compelling sub-question is theprovenance of the propellers in the A-ring. Is it possible thatthese large objects are related to the ‘lucky’ planetesimalsthat grow to large sizes in PP disks, their smaller brethrenlanguishing in the cm to m size classes?Another area of fruitful overlap is in the detailed micro-physics of collisions. Only recently have studies of denseplanetary rings moved away from the bouncing hard-spheremodel of ring particles. In contrast, the numerical treatmentof collisions in planet formation is much better developed, al-lowing for the full gamut of physical processes (compaction,mass transfer, fragmentation, reaccretion, etc.). Is it possibleto establish clear barriers to growth in dense rings, as in PPdisks? Can we construct a typology of collisional outcomesin dense rings as clearly as in planetesimal belts?Gravitational instability (and gravitoturbulence) is sharedby planetary rings, PP disks, and galactic disks. However,the details of its onset and saturation are still unclear ineach context. A more unified approach would nail down itsmanifestation in all three classes, and indeed uncover moreprofound connections with other subcritical transitions toturbulence in shearing and rotating systems. Another areaworth exploring is the production of tidal streams aroundwhite dwarfs, which clearly shares the same physics of cer-tain ring formation scenarios, especially of narrow rings. Fi-nally, the field of satellite–disk interactions, though mature,could but undoubtedly support further connections betweenrings and disks. As observations of PP disks become moreand more detailed due to ALMA, the intricate and variedmorphologies supported by Saturn’s rings will provide valu-able analogues with which to understand them. Acknowledgments
The authors thank the anonymous reviewer and the editorsMatthew Tiscareno and Carl Murray for a set of helpfulcomments. They are particularly indebted to the generosityof colleagues and friends who read through earlier stages ofthe manuscript, in particular Julia Forman, Cleo Loi, Pierre-Yves Longaretti, John Papaloizou, and Mark Wyatt. Thechapter was much improved by their insightful and help-ful remarks. We also thank Tobias Heinemann and AntoineRiols-Fonclare for providing figures.
E F E R E N C E S
Abbott, B. P., Abbott, R., Abbott, T. D., Abernathy, M. R., Ac-ernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P.,Adhikari, R. X., and et al. 2016. Observation of Gravita-tional Waves from a Binary Black Hole Merger.
PhysicalReview Letters , (6), 061102.Abramowicz, M. A., and Fragile, P. C. 2013. Foundations of BlackHole Accretion Disk Theory. Living Reviews in Relativity , .Abramowicz, M. A., Czerny, B., Lasota, J. P., and Szuszkiewicz,E. 1988. Slim accretion disks. ApJ , , 646–658.Albers, N., and Spahn, F. 2006a. The influence of particle ad-hesion on the stability of agglomerates in Saturn’s rings. Icarus , , 292–301.Albers, N., and Spahn, F. 2006b. The influence of particle ad-hesion on the stability of agglomerates in Saturn’s rings. Icarus , , 292–301.Alexander, R. D., Clarke, C. J., and Pringle, J. E. 2006. Photoe-vaporation of protoplanetary discs - II. Evolutionary modelsand observable properties. MNRAS , , 229–239.ALMA Partnership, Brogan, C. L., P´erez, L. M., Hunter, T. R.,Dent, W. R. F., Hales, A. S., Hills, R. E., Corder, S., Fo-malont, E. B., Vlahakis, C., Asaki, Y., Barkats, D., Hirota,A., Hodge, J. A., Impellizzeri, C. M. V., Kneissl, R., Li-uzzo, E., Lucas, R., Marcelino, N., Matsushita, S., Nakan-ishi, K., Phillips, N., Richards, A. M. S., Toledo, I., Al-adro, R., Broguiere, D., Cortes, J. R., Cortes, P. C., Espada,D., Galarza, F., Garcia-Appadoo, D., Guzman-Ramirez, L.,Humphreys, E. M., Jung, T., Kameno, S., Laing, R. A.,Leon, S., Marconi, G., Mignano, A., Nikolic, B., Nyman,L.-A., Radiszcz, M., Remijan, A., Rod´on, J. A., Sawada, T.,Takahashi, S., Tilanus, R. P. J., Vila Vilaro, B., Watson,L. C., Wiklind, T., Akiyama, E., Chapillon, E., de Gregorio-Monsalvo, I., Di Francesco, J., Gueth, F., Kawamura, A.,Lee, C.-F., Nguyen Luong, Q., Mangum, J., Pietu, V., San-hueza, P., Saigo, K., Takakuwa, S., Ubach, C., van Kempen,T., Wootten, A., Castro-Carrizo, A., Francke, H., Gallardo,J., Garcia, J., Gonzalez, S., Hill, T., Kaminski, T., Kurono,Y., Liu, H.-Y., Lopez, C., Morales, F., Plarre, K., Schieven,G., Testi, L., Videla, L., Villard, E., Andreani, P., Hibbard,J. E., and Tatematsu, K. 2015. The 2014 ALMA Long Base-line Campaign: First Results from High Angular ResolutionObservations toward the HL Tau Region. ApJL , , L3.Andrews, S. M., Wilner, D. J., Espaillat, C., Hughes, A. M., Dulle-mond, C. P., McClure, M. K., Qi, C., and Brown, J. M. 2011.Resolved Images of Large Cavities in Protoplanetary Tran-sition Disks. ApJ , , 42.Antonucci, R. 1993. Unified models for active galactic nuclei andquasars. ARAA , , 473–521.Araki, S., and Tremaine, S. 1986. The dynamics of dense particledisks. Icarus , , 83–109. Armitage, P. J. 2011. Dynamics of Protoplanetary Disks. ARAA , , 195–236.Armitage, P. J., Livio, M., and Pringle, J. E. 2001. Episodic accre-tion in magnetically layered protoplanetary discs. MNRAS , , 705–711.Attree, N. O., Murray, C. D., Cooper, N. J., and Williams, G. A.2012. Detection of Low-velocity Collisions in Saturn’s FRing. ApJL , , L27.Attree, N. O., Murray, C. D., Williams, G. A., and Cooper, N. J.2014. A survey of low-velocity collisional features in Saturn’sF ring. Icarus , , 56–66.Audard, M., ´Abrah´am, P., Dunham, M. M., Green, J. D., Grosso,N., Hamaguchi, K., Kastner, J. H., K´osp´al, ´A., Lodato, G.,Romanova, M. M., Skinner, S. L., Vorobyov, E. I., and Zhu,Z. 2014. Episodic Accretion in Young Stars. Protostars andPlanets VI , 387–410.Aumann, H. H., Beichman, C. A., Gillett, F. C., de Jong, T.,Houck, J. R., Low, F. J., Neugebauer, G., Walker, R. G.,and Wesselius, P. R. 1984. Discovery of a shell around AlphaLyrae.
ApJL , , L23–L27.Baade, W., and Minkowski, R. 1954. Identification of the RadioSources in Cassiopeia, Cygnus A, and Puppis A. ApJ , ,206.Backman, D. E., and Paresce, F. 1993. Main-sequence starswith circumstellar solid material - The VEGA phenomenon.Pages 1253–1304 of: Levy, E. H., and Lunine, J. I. (eds), Protostars and Planets III .Balbus, S. A., and Hawley, J. F. 1991. A powerful local shearinstability in weakly magnetized disks. I - Linear analysis. II- Nonlinear evolution.
ApJ , , 214–233.Balbus, S. A., and Hawley, J. F. 1998. Instability, turbulence, andenhanced transport in accretion disks. Reviews of ModernPhysics , , 1–53.Balbus, S. A., and Papaloizou, J. C. B. 1999. On the DynamicalFoundations of α Disks.
ApJ , , 650–658.Barbara, J. M., and Esposito, L. W. 2002. Moonlet Collisionsand the Effects of Tidally Modified Accretion in Saturn’s FRing. Icarus , , 161–171.Barge, P., and Sommeria, J. 1995. Did planet formation begininside persistent gaseous vortices? A&A , , L1–L4.Barker, A. J., and Latter, H. N. 2015. On the vertical-shearinstability in astrophysical discs. MNRAS , (June), 21–37.Baruteau, C., Crida, A., Paardekooper, S.-J., Masset, F., Guilet,J., Bitsch, B., Nelson, R., Kley, W., and Papaloizou, J. 2014.Planet-Disk Interactions and Early Evolution of PlanetarySystems. Protostars and Planets VI , 667–689.Bell, K. R., and Lin, D. N. C. 1994. Using FU Orionis outburststo constrain self-regulated protostellar disk models.
ApJ , , 987–1004. eferences Belloni, T. M. 2010. States and Transitions in Black Hole Bina-ries. Page 53 of: Belloni, T. (ed),
Lecture Notes in Physics,Berlin Springer Verlag . Lecture Notes in Physics, BerlinSpringer Verlag, vol. 794.Benisty, M., Juhasz, A., Boccaletti, A., Avenhaus, H., Milli, J.,Thalmann, C., Dominik, C., Pinilla, P., Buenzli, E., Pohl, A.,Beuzit, J.-L., Birnstiel, T., de Boer, J., Bonnefoy, M., Chau-vin, G., Christiaens, V., Garufi, A., Grady, C., Henning, T.,Huelamo, N., Isella, A., Langlois, M., M´enard, F., Mouillet,D., Olofsson, J., Pantin, E., Pinte, C., and Pueyo, L. 2015.Asymmetric features in the protoplanetary disk MWC 758.
A&A , , L6.Ben´ıtez-Llambay, P., Masset, F., Koenigsberger, G., andSzul´agyi, J. 2015. Planet heating prevents inward migrationof planetary cores. Nature , (Apr.), 63–65.Beurle, K., Murray, C. D., Williams, G. A., Evans, M. W.,Cooper, N. J., and Agnor, C. B. 2010. Direct Evidence forGravitational Instability and Moonlet Formation in Saturn’sRings. ApJL , , L176–L180.Binney, J., and Merrifield, M. 1998. Galactic Astronomy .Binney, J., and Tremaine, S. 2008.
Galactic Dynamics: SecondEdition . Princeton University Press.Biretta, J. A., Sparks, W. B., and Macchetto, F. 1999. HubbleSpace Telescope Observations of Superluminal Motion in theM87 Jet.
ApJ , , 621–626.Blaes, O. M., and Balbus, S. A. 1994. Local shear instabilitiesin weakly ionized, weakly magnetized disks. ApJ , , 163–177.Blandford, R. D., and Payne, D. G. 1982. Hydromagnetic flowsfrom accretion discs and the production of radio jets. MN-RAS , , 883–903.Bloom, J. S., Giannios, D., Metzger, B. D., Cenko, S. B., Perley,D. A., Butler, N. R., Tanvir, N. R., Levan, A. J., O’Brien,P. T., Strubbe, L. E., De Colle, F., Ramirez-Ruiz, E., Lee,W. H., Nayakshin, S., Quataert, E., King, A. R., Cucchiara,A., Guillochon, J., Bower, G. C., Fruchter, A. S., Morgan,A. N., and van der Horst, A. J. 2011. A Possible RelativisticJetted Outburst from a Massive Black Hole Fed by a TidallyDisrupted Star. Science , , 203–.Blum, J., and Wurm, G. 2008. The Growth Mechanisms of Macro-scopic Bodies in Protoplanetary Disks. ARAA , , 21–56.Blumenthal, G. R., Lin, D. N. C., and Yang, L. T. 1984. On theoverstability of axisymmetric oscillations in thin accretiondisks. ApJ , , 774–784.Bodrova, A., Schmidt, J., Spahn, F., and Brilliantov, N. 2012. Ad-hesion and collisional release of particles in dense planetaryrings. Icarus , , 60–68.Bonsor, A., and Wyatt, M. 2010. Post-main-sequence evolutionof A star debris discs. MNRAS , , 1631–1646.Bonsor, A., Mustill, A. J., and Wyatt, M. C. 2011. Dynamicaleffects of stellar mass-loss on a Kuiper-like belt. MNRAS , , 930–939.Borderies, N., Goldreich, P., and Tremaine, S. 1982. Sharp edgesof planetary rings. Nature , (Sept.), 209–211.Borderies, N., Goldreich, P., and Tremaine, S. 1983. The dynam-ics of elliptical rings. AJ , (Oct.), 1560–1568.Borderies, N., Goldreich, P., and Tremaine, S. 1984. Excitationof inclinations in ring-satellite systems. ApJ , (Sept.),429–434.Borderies, N., Goldreich, P., and Tremaine, S. 1985. A granularflow model for dense planetary rings. Icarus , , 406–420.Borderies, N., Goldreich, P., and Tremaine, S. 1989. The forma-tion of sharp edges in planetary rings by nearby satellites. Icarus , (Aug.), 344–360. Boss, A. P. 1998. Evolution of the Solar Nebula. IV. GiantGaseous Protoplanet Formation. ApJ , , 923–937.Boss, A. P., and Graham, J. A. 1993. Clumpy disk accretion andchondrule formation. Icarus , , 168.Braginskii, S. I. 1965. Transport Processes in a Plasma. Reviewsof Plasma Physics , , 205.Brauer, F., Dullemond, C. P., and Henning, T. 2008. Coagula-tion, fragmentation and radial motion of solid particles inprotoplanetary disks. A&A , , 859–877.Bridges, F. G., Hatzes, A., and Lin, D. N. C. 1984. Structure,stability and evolution of Saturn’s rings. Nature , , 333–335.Brilliantov, N., Krapivsky, P. L., Bodrova, A., Spahn, F.,Hayakawa, H., Stadnichuk, V., and Schmidt, J. 2015. Sizedistribution of particles in Saturn’s rings from aggregationand fragmentation. Proceedings of the National Academy ofScience , , 9536–9541.Broadfoot, A. L., Herbert, F., Holberg, J. B., Hunten, D. M.,Kumar, S., Sandel, B. R., Shemansky, D. E., Smith, G. R.,Yelle, R. V., Strobel, D. F., Moos, H. W., Donahue, T. M.,Atreya, S. K., Bertaux, J. L., Blamont, J. E., Mcconnell,J. C., Dessler, A. J., Linick, S., and Springer, R. 1986. Ul-traviolet spectrometer observations of Uranus. Science , ,74–79.Burns, J. A., Lamy, P. L., and Soter, S. 1979. Radiation forceson small particles in the solar system. Icarus , , 1–48.Burns, J. A., Showalter, M. R., Hamilton, D. P., Nicholson, P. D.,de Pater, I., Ockert-Bell, M. E., and Thomas, P. C. 1999. TheFormation of Jupiter’s Faint Rings. Science , , 1146.Burrows, D. N., Kennea, J. A., Ghisellini, G., Mangano, V.,Zhang, B., Page, K. L., Eracleous, M., Romano, P.,Sakamoto, T., Falcone, A. D., Osborne, J. P., Campana, S.,Beardmore, A. P., Breeveld, A. A., Chester, M. M., Corbet,R., Covino, S., Cummings, J. R., D’Avanzo, P., D’Elia, V.,Esposito, P., Evans, P. A., Fugazza, D., Gelbord, J. M., Hi-roi, K., Holland, S. T., Huang, K. Y., Im, M., Israel, G., Jeon,Y., Jeon, Y.-B., Jun, H. D., Kawai, N., Kim, J. H., Krimm,H. A., Marshall, F. E., P. M´esz´aros, Negoro, H., Omodei,N., Park, W.-K., Perkins, J. S., Sugizaki, M., Sung, H.-I.,Tagliaferri, G., Troja, E., Ueda, Y., Urata, Y., Usui, R., An-tonelli, L. A., Barthelmy, S. D., Cusumano, G., Giommi,P., Melandri, A., Perri, M., Racusin, J. L., Sbarufatti, B.,Siegel, M. H., and Gehrels, N. 2011. Relativistic jet activityfrom the tidal disruption of a star by a massive black hole. Nature , , 421–424.Cameron, A. G. W. 1978. Physics of the primitive solar accretiondisk. Moon and Planets , , 5–40.Camichel, H. 1958. Mesures photom´etriques de Saturne et de sonanneau. Annales d’Astrophysique , , 231.Cannizzo, J. K., and Mattei, J. A. 1998. A Study of the Outburstsin SS Cygni. ApJ , (Sept.), 344–351.Canup, R. M. 2010. Origin of Saturn’s rings and inner moons bymass removal from a lost Titan-sized satellite. Nature , ,943–946.Chandrasekhar, S. 1969. Ellipsoidal figures of equilibrium .Charles, P. A., and Coe, M. J. 2006.
Optical, ultraviolet andinfrared observations of X-ray binaries . Pages 215–265.Charnoz, S., Morbidelli, A., Dones, L., and Salmon, J. 2009. DidSaturn’s rings form during the Late Heavy Bombardment?
Icarus , , 413–428.Clarke, C. J., and Pringle, J. E. 1993. Accretion disc response toa stellar fly-by. MNRAS , (Mar.), 190–202. References
Colombo, G., Goldreich, P., and Harris, A. W. 1976. Spiral struc-ture as an explanation for the asymmetric brightness of Sat-urn’s A ring.
Nature , (Nov.), 344.Colwell, J. E., Esposito, L. W., Sremˇcevi´c, M., Stewart, G. R.,and McClintock, W. E. 2007. Self-gravity wakes and radialstructure of Saturn’s B ring. Icarus , , 127–144.Colwell, J. E., Nicholson, P. D., Tiscareno, M. S., Murray, C. D.,French, R. G., and Marouf, E. A. 2009. The Structure ofSaturn’s Rings . Page 375.Cuzzi, J. N., and Burns, J. A. 1988. Charged particle depletionsurrounding Saturn’s F ring - Evidence for a moonlet belt?
Icarus , , 284–324.Daisaka, H., Tanaka, H., and Ida, S. 2001. Viscosity in a DensePlanetary Ring with Self-Gravitating Particles. Icarus , ,296–312.Debes, J. H., and Sigurdsson, S. 2002. Are There Unstable Plan-etary Systems around White Dwarfs? ApJ , , 556–565.Debes, J. H., Walsh, K. J., and Stark, C. 2012. The Link be-tween Planetary Systems, Dusty White Dwarfs, and Metal-polluted White Dwarfs. ApJ , , 148.Dohnanyi, J. S. 1969. Collisional Model of Asteroids and TheirDebris. JGR , (May), 2531–2554.Dominik, C., and Decin, G. 2003. Age Dependence of the VegaPhenomenon: Theory. ApJ , , 626–635.Done, C., Wardzi´nski, G., and Gierli´nski, M. 2004. GRS1915+105: the brightest Galactic black hole. MNRAS , ,393–403.Done, C., Gierli´nski, M., and Kubota, A. 2007. Modelling thebehaviour of accretion flows in X-ray binaries. Everythingyou always wanted to know about accretion but were afraidto ask. AARv , (Dec.), 1–66.Dones, L. 1991. A recent cometary origin for Saturn’s rings? Icarus , , 194–203.Dong, R., Zhu, Z., Rafikov, R. R., and Stone, J. M. 2015. Ob-servational Signatures of Planets in Protoplanetary Disks:Spiral Arms Observed in Scattered Light Imaging Can BeInduced by Planets. ApJL , (Aug.), L5.Dong, R., Zhu, Z., Fung, J., Rafikov, R., Chiang, E., and Wag-ner, K. 2016. An M Dwarf Companion and Its Induced Spi-ral Arms in the HD 100453 Protoplanetary Disk. ApJL , (Jan.), L12.Donley, J. L., Brandt, W. N., Eracleous, M., and Boller, T. 2002.Large-Amplitude X-Ray Outbursts from Galactic Nuclei: ASystematic Survey using ROSAT Archival Data. AJ , ,1308–1321.Draine, B. T. 2011. Physics of the Interstellar and IntergalacticMedium .Dullemond, C. P., and Dominik, C. 2005. Dust coagulation in pro-toplanetary disks: A rapid depletion of small grains.
A&A , , 971–986.Durisen, R. H. 1984. Transport effects due to particle erosionmechanisms. Pages 416–446 of: Greenberg, R., and Brahic,A. (eds), IAU Colloq. 75: Planetary Rings .Durisen, R. H. 1995. An instability in planetary rings due toballistic transport.
Icarus , , 66–85.Durisen, R. H., Boss, A. P., Mayer, L., Nelson, A. F., Quinn,T., and Rice, W. K. M. 2007. Gravitational Instabilitiesin Gaseous Protoplanetary Disks and Implications for GiantPlanet Formation. Protostars and Planets V , 607–622.Eggen, O. J., Lynden-Bell, D., and Sandage, A. R. 1962. Evidencefrom the motions of old stars that the Galaxy collapsed.
ApJ , , 748.Eiroa, C., Marshall, J. P., Mora, A., Montesinos, B., Absil, O.,Augereau, J. C., Bayo, A., Bryden, G., Danchi, W., del Burgo, C., Ertel, S., Fridlund, M., Heras, A. M., Krivov,A. V., Launhardt, R., Liseau, R., L¨ohne, T., Maldonado, J.,Pilbratt, G. L., Roberge, A., Rodmann, J., Sanz-Forcada, J.,Solano, E., Stapelfeldt, K., Th´ebault, P., Wolf, S., Ardila,D., Ar´evalo, M., Beichmann, C., Faramaz, V., Gonz´alez-Garc´ıa, B. M., Guti´errez, R., Lebreton, J., Mart´ınez-Arn´aiz,R., Meeus, G., Montes, D., Olofsson, G., Su, K. Y. L., White,G. J., Barrado, D., Fukagawa, M., Gr¨un, E., Kamp, I.,Lorente, R., Morbidelli, A., M¨uller, S., Mutschke, H., Nak-agawa, T., Ribas, I., and Walker, H. 2013. DUst aroundNEarby Stars. The survey observational results. A&A , ,A11.Esposito, L. W., Albers, N., Meinke, B. K., Sremˇcevi´c, M., Mad-husudhanan, P., Colwell, J. E., and Jerousek, R. G. 2012.A predator-prey model for moon-triggered clumping in Sat-urn’s rings. Icarus , , 103–114.Estrada, P. R., and Cuzzi, J. N. 1996. Voyager Observations ofthe Color of Saturn’s Rings. Icarus , , 251–272.Evans, II, N. J., Dunham, M. M., Jørgensen, J. K., Enoch, M. L.,Mer´ın, B., van Dishoeck, E. F., Alcal´a, J. M., Myers, P. C.,Stapelfeldt, K. R., Huard, T. L., Allen, L. E., Harvey, P. M.,van Kempen, T., Blake, G. A., Koerner, D. W., Mundy,L. G., Padgett, D. L., and Sargent, A. I. 2009. The Spitzerc2d Legacy Results: Star-Formation Rates and Efficiencies;Evolution and Lifetimes. ApJS , , 321–350.Fabian, A. C. 2012. Observational Evidence of Active GalacticNuclei Feedback. ARAA , , 455–489.Fan, X., Carilli, C. L., and Keating, B. 2006. Observational Con-straints on Cosmic Reionization. ARAA , , 415–462.Fanaroff, B. L., and Riley, J. M. 1974. The morphology of extra-galactic radio sources of high and low luminosity. MNRAS , , 31P–36P.Farihi, J., Jura, M., and Zuckerman, B. 2009. Infrared Signaturesof Disrupted Minor Planets at White Dwarfs. ApJ , ,805–819.Farinella, P., and Davis, D. R. 1996. Short-Period Comets: Pri-mordial Bodies or Collisional Fragments? Science , , 938–941.Faulkner, J., Lin, D. N. C., and Papaloizou, J. 1983. On the evo-lution of accretion disc flow in cataclysmic variables. I - Theprospect of a limit cycle in dwarf nova systems. MNRAS , , 359–375.Ferrarese, L., and Ford, H. 2005. Supermassive Black Holes inGalactic Nuclei: Past, Present and Future Research. SSRv , , 523–624.Field, G. B. 1965. Thermal Instability. ApJ , , 531.French, R. G., and Nicholson, P. D. 2000. Saturn’s Rings II.Particle Sizes Inferred from Stellar Occultation Data. Icarus , , 502–523.Fuller, J. 2014. Saturn ring seismology: Evidence for stable strati-fication in the deep interior of Saturn. Icarus , , 283–296.Ga(cid:32)lan, C., Miko(cid:32)lajewski, M., Tomov, T., Graczyk, D., Apos-tolovska, G., Barzova, I., Bellas-Velidis, I., Bilkina, B., Blake,R. M., Bolton, C. T., Bondar, A., Br´at, L., Bro˙zek, T.,Budzisz, B., Cika(cid:32)la, M., Cs´ak, B., Dapergolas, A., Dimitrov,D., Dobierski, P., Drahus, M., Dr´o˙zd˙z, M., Dvorak, S., Elder,L., Fr¸ackowiak, S., Galazutdinov, G., Gazeas, K., Georgiev,L., Gere, B., Go´zdziewski, K., Grinin, V. P., Gromadzki, M.,Hajduk, M., Heras, T. A., Hopkins, J., Iliev, I., Janowski,J., Koci´an, R., Ko(cid:32)laczkowski, Z., Kolev, D., Kopacki, G.,Krzesi´nski, J., Kuˇc´akov´a, H., Kuligowska, E., Kundera, T.,Kurpi´nska-Winiarska, M., Ku´zmicz, A., Liakos, A., Lister,T. A., Maciejewski, G., Majcher, A., Majewska, A., Marrese,P. M., Michalska, G., Migaszewski, C., Miller, I., Munari, eferences U., Musaev, F., Myers, G., Narwid, A., N´emeth, P., Niar-chos, P., Niemczura, E., Og(cid:32)loza, W., ¨Oˇgmen, Y., Oksanen,A., Osiwa(cid:32)la, J., Peneva, S., Pigulski, A., Popov, V., Pych,W., Pye, J., Ragan, E., Roukema, B. F., R´o˙za´nski, P. T.,Semkov, E., Siwak, M., Staels, B., Stateva, I., Stempels,H. C., St¸e´slicki, M., ´Swierczy´nski, E., Szyma´nski, T., Tomov,N., Waniak, W., Wi¸ecek, M., Winiarski, M., Wychudzki, P.,Zajczyk, A., Zo(cid:32)la, S., and Zwitter, T. 2012. International ob-servational campaigns of the last two eclipses in EE Cephei:2003 and 2008/9.
A&A , (Aug.), A53.Gallagher, J. S., and Starrfield, S. 1978. Theory and observationsof classical novae. ARAA , , 171–214.Gammie, C. F. 1996. Layered Accretion in T Tauri Disks. ApJ , , 355.Gammie, C. F. 2001. Nonlinear Outcome of Gravitational Insta-bility in Cooling, Gaseous Disks. ApJ , , 174–183.G¨ansicke, B. T., Marsh, T. R., Southworth, J., and Rebassa-Mansergas, A. 2006. A Gaseous Metal Disk Around a WhiteDwarf. Science , , 1908–.G¨ansicke, B. T., Koester, D., Farihi, J., Girven, J., Parsons, S. G.,and Breedt, E. 2012. The chemical diversity of exo-terrestrialplanetary debris around white dwarfs. MNRAS , , 333–347.Garaud, P., Meru, F., Galvagni, M., and Olczak, C. 2013. FromDust to Planetesimals: An Improved Model for CollisionalGrowth in Protoplanetary Disks. ApJ , , 146.Geretshauser, R. J., Speith, R., G¨uttler, C., Krause, M., andBlum, J. 2010. Numerical simulations of highly porous dustaggregates in the low-velocity collision regime. Implemen-tation and calibration of a smooth particle hydrodynamicscode. A&A , , A58.Giacconi, R., Gursky, H., Paolini, F. R., and Rossi, B. B. 1962.Evidence for x Rays From Sources Outside the Solar System. Physical Review Letters , , 439–443.Gierli´nski, M., and Done, C. 2004. Black hole accretion discs:reality confronts theory. MNRAS , (Jan.), 885–894.Goertz, C. K., and Morfill, G. 1988. A new instability of Saturn’sring. Icarus , , 325–330.Goldreich, P., and Lynden-Bell, D. 1965. II. Spiral arms assheared gravitational instabilities. MNRAS , , 125.Goldreich, P., and Porco, C. C. 1987a. Shepherding of the UranianRings. II. Dynamics. AJ , , 730.Goldreich, P., and Porco, C. C. 1987b. Shepherding of the UranianRings. II. Dynamics. AJ , (Mar.), 730.Goldreich, P., and Tremaine, S. 1978a. The excitation and evo-lution of density waves. ApJ , , 850–858.Goldreich, P., and Tremaine, S. 1980. Disk-satellite interactions. ApJ , (Oct.), 425–441.Goldreich, P., and Tremaine, S. 1981. The origin of the eccentric-ities of the rings of Uranus. ApJ , (Feb.), 1062–1075.Goldreich, P., and Tremaine, S. 1982. The dynamics of planetaryrings. ARAA , , 249–283.Goldreich, P., and Tremaine, S. D. 1978b. The velocity dispersionin Saturn’s rings. Icarus , , 227–239.Gor’kavyj, N. N., and Fridman, A. M. 1994. Physics of planetaryrings. Celestial mechanics of continuous medium.
Graham, J. R., Matthews, K., Neugebauer, G., and Soifer, B. T.1990. The infrared excess of G29-38 - A brown dwarf ordust?
ApJ , , 216–223.Greenberg, R., Hartmann, W. K., Chapman, C. R., and Wacker,J. F. 1978. Planetesimals to planets - Numerical simulationof collisional evolution. Icarus , , 1–26.Greenstein, J. L., and Schmidt, M. 1964. The Quasi-Stellar RadioSources 3c 48 and 3c 273. ApJ , , 1. Guimar˜aes, A. H. F., Albers, N., Spahn, F., Seiß, M., Vieira-Neto,E., and Brilliantov, N. V. 2012. Aggregates in the strengthand gravity regime: Particles sizes in Saturn’s rings. Icarus , , 660–678.Gursky, H., Giacconi, R., Gorenstein, P., Waters, J. R., Oda,M., Bradt, H., Garmire, G., and Sreekantan, B. V. 1966. AMeasurement of the Location of the X-Ray Source SCO X-1. ApJ , , 310–316.Harris, A. W. 1984. The origin and evolution of planetary rings.Pages 641–659 of: Greenberg, R., and Brahic, A. (eds), IAUColloq. 75: Planetary Rings .Hartman, R. C., Bertsch, D. L., Bloom, S. D., Chen, A. W.,Deines-Jones, P., Esposito, J. A., Fichtel, C. E., Friedlan-der, D. P., Hunter, S. D., McDonald, L. M., Sreekumar,P., Thompson, D. J., Jones, B. B., Lin, Y. C., Michelson,P. F., Nolan, P. L., Tompkins, W. F., Kanbach, G., Mayer-Hasselwander, H. A., M¨ucke, A., Pohl, M., Reimer, O., Knif-fen, D. A., Schneid, E. J., von Montigny, C., Mukherjee, R.,and Dingus, B. L. 1999. The Third EGRET Catalog of High-Energy Gamma-Ray Sources.
ApJS , , 79–202.Hartmann, L., and Kenyon, S. J. 1996. The FU Orionis Phe-nomenon. ARAA , , 207–240.Hatzes, A. P., Bridges, F. G., and Lin, D. N. C. 1988. Collisionalproperties of ice spheres at low impact velocities. MNRAS , , 1091–1115.Hatzes, A. P., Bridges, F., Lin, D. N. C., and Sachtjen, S. 1991.Coagulation of particles in Saturn’s rings - Measurements ofthe cohesive force of water frost. Icarus , , 113–121.Hawley, J. F., Gammie, C. F., and Balbus, S. A. 1995. Lo-cal Three-dimensional Magnetohydrodynamic Simulationsof Accretion Disks. ApJ , , 742.Hedman, M. M., and Nicholson, P. D. 2013. Kronoseismology: Us-ing Density Waves in Saturn’s C Ring to Probe the Planet’sInterior. AJ , , 12.Hedman, M. M., Burns, J. A., Showalter, M. R., Porco, C. C.,Nicholson, P. D., Bosh, A. S., Tiscareno, M. S., Brown, R. H.,Buratti, B. J., Baines, K. H., and Clark, R. 2007. Saturn’sdynamic D ring. Icarus , , 89–107.Hedman, M. M., Murray, C. D., Cooper, N. J., Tiscareno, M. S.,Beurle, K., Evans, M. W., and Burns, J. A. 2009. Threetenuous rings/arcs for three tiny moons. Icarus , , 378–386.Hedman, M. M., Cooper, N. J., Murray, C. D., Beurle, K., Evans,M. W., Tiscareno, M. S., and Burns, J. A. 2010. Aegaeon(Saturn LIII), a G-ring object. Icarus , , 433–447.Hedman, M. M., Burns, J. A., Evans, M. W., Tiscareno, M. S.,and Porco, C. C. 2011. Saturn’s Curiously Corrugated CRing. Science , , 708.Hedman, M. M., Nicholson, P. D., and Salo, H. 2014. ExploringOverstabilities in Saturn’s A Ring Using Two Stellar Occul-tations. AJ , , 15.Hellier, C. 2001. Cataclysmic Variable Stars .Herbig, G. H. 1977. Eruptive phenomena in early stellar evolu-tion.
ApJ , (Nov.), 693–715.Herbig, G. H. 1989 (Sept.). FU Orionis eruptions. Pages 233–246 of: Reipurth, B. (ed), European Southern ObservatoryConference and Workshop Proceedings . European SouthernObservatory Conference and Workshop Proceedings, vol. 33.Hills, J. G. 1975. Possible power source of Seyfert galaxies andQSOs.
Nature , , 295–298.Hirose, S., Blaes, O., and Krolik, J. H. 2009. Turbulent Stresses inLocal Simulations of Radiation-dominated Accretion Disks,and the Possibility of the Lightman-Eardley Instability. ApJ , , 781–788. References
Hoard, D. W., Howell, S. B., and Stencel, R. E. 2010. Taming theInvisible Monster: System Parameter Constraints for epsilonAurigae from the Far-ultraviolet to the Mid-infrared.
ApJ , (May), 549–560.Holberg, J. B., Barstow, M. A., and Green, E. M. 1997. The Dis-covery of Mg II λ ApJL , , L127–L130.Hor´anyi, M., Hartquist, T. W., Havnes, O., Mendis, D. A., andMorfill, G. E. 2004. Dusty plasma effects in Saturn’s mag-netosphere. Reviews of Geophysics , , 4002.Hor´anyi, M., Burns, J. A., Hedman, M. M., Jones, G. H., andKempf, S. 2009. Diffuse Rings . Page 511.Hubble, E. P. 1925. Cepheids in Spiral Nebulae.
Popular Astron-omy , , 252–255.Hubble, E. P. 1936. Realm of the Nebulae .Hyodo, R., and Ohtsuki, K. 2014. Collisional Disruption of Grav-itational Aggregates in the Tidal Environment.
ApJ , ,56.Hyodo, Ryuki, and Ohtsuki, Keiji. 2015. Saturn/’s F ring andshepherd satellites a natural outcome of satellite system for-mation. Nature Geoscience , (9), 686–689.Jeans, J. H. 1917. Some problems of astronomy (XXIV. Theevolution of rotating masses). The Observatory , , 196–203.Jewitt, D. C., and Luu, J. X. 2000. Physical Nature of the KuiperBelt. Protostars and Planets IV , 1201.Jiang, Y.-F., Stone, J. M., and Davis, S. W. 2013. On the ThermalStability of Radiation-dominated Accretion Disks.
ApJ , ,65.Johansen, A., Youdin, A., and Klahr, H. 2009. Zonal Flows andLong-lived Axisymmetric Pressure Bumps in Magnetorota-tional Turbulence. ApJ , , 1269–1289.Johnson, B. M., and Gammie, C. F. 2003. Nonlinear Outcomeof Gravitational Instability in Disks with Realistic Cooling. ApJ , , 131–141.Joos, M., Hennebelle, P., and Ciardi, A. 2012. Protostellar diskformation and transport of angular momentum during mag-netized core collapse. A&A , , A128.Jura, M. 2003. A Tidally Disrupted Asteroid around the WhiteDwarf G29-38. ApJL , , L91–L94.Jura, M. 2008. Pollution of Single White Dwarfs by Accretion ofMany Small Asteroids. AJ , , 1785–1792.Karjalainen, R. 2007. Aggregate impacts in Saturn’s rings. Icarus , , 523–537.Karjalainen, R., and Salo, H. 2004. Gravitational accretion ofparticles in Saturn’s rings. Icarus , , 328–348.Kato, S. 1978. Pulsational instability of accretion disks to axiallysymmetric oscillations. MNRAS , , 629–642.Kato, S., and Yoshizawa, A. 1993. A model of hydromagneticturbulence in accretion disks. PASJ , , 103–112.Kato, S., and Yoshizawa, A. 1995. A Model of HydromagneticTurbulence in Accretion Disks. II. PASJ , , 629–637.Katz, J. I. 1973. Thirty-five-day Periodicity in Her X-1. NaturePhysical Science , , 87–89.Kenyon, S. J., and Bromley, B. C. 2004a. Collisional Cascades inPlanetesimal Disks. II. Embedded Planets. AJ , , 513–530.Kenyon, S. J., and Bromley, B. C. 2004b. Detecting the DustyDebris of Terrestrial Planet Formation. ApJL , , L133–L136.Kenyon, S. J., and Hartmann, L. 1995. Pre-Main-Sequence Evo-lution in the Taurus-Auriga Molecular Cloud. ApJS , ,117. Kesden, M. 2012. Tidal-disruption rate of stars by spinning su-permassive black holes. PhRvD , (2), 024037.King, A. R. 2006. Accretion in compact binaries . Pages 507–546.Klein, B., Jura, M., Koester, D., Zuckerman, B., and Melis, C.2010. Chemical Abundances in the Externally PollutedWhite Dwarf GD 40: Evidence of a Rocky Extrasolar Mi-nor Planet.
ApJ , , 950–962.Kley, W., Papaloizou, J. C. B., and Lin, D. N. C. 1993. Two-dimensional viscous accretion disk models. II - On viscousoverstability. ApJ , , 739–747.Kobayashi, S., Laguna, P., Phinney, E. S., and M´esz´aros, P. 2004.Gravitational Waves and X-Ray Signals from Stellar Disrup-tion by a Massive Black Hole. ApJ , , 855–865.Koerner, D. W., Sargent, A. I., and Beckwith, S. V. W. 1993. Arotating gaseous disk around the T Tauri star GM Aurigae. Icarus , (Nov.), 2.Koester, D., Provencal, J., and Shipman, H. L. 1997. Metals inthe variable DA G29-38. A&A , , L57–L59.Kokubo, E., and Ida, S. 1996. On Runaway Growth of Planetes-imals. Icarus , , 180–191.Kokubo, E., and Ida, S. 1998. Oligarchic Growth of Protoplanets. Icarus , , 171–178.Komossa, S., and Bade, N. 1999. The giant X-ray outbursts inNGC 5905 and IC 3599:() hfill Follow-up observations andoutburst scenarios. A&A , , 775–787.Kormendy, J., and Richstone, D. 1995. Inward Bound—TheSearch For Supermassive Black Holes In Galactic Nuclei. ARAA , , 581.Korycansky, D. G., and Pringle, J. E. 1995. Axisymmetric wavesin polytropic accretion discs. MNRAS , , 618–624.Kotze, M. M., and Charles, P. A. 2012. Characterizing X-raybinary long-term variability. MNRAS , , 1575–1589.Kraft, R. P. 1962. Binary Stars among Cataclysmic Variables. I.U Geminorum Stars (dwarf Novae). ApJ , , 408.Kraft, R. P. 1964. Binary Stars among Cataclysmic Variables.III. Ten Old Novae. ApJ , , 457.Kral, Q., Th´ebault, P., and Charnoz, S. 2013. LIDT-DD: A newself-consistent debris disc model that includes radiation pres-sure and couples dynamical and collisional evolution. A&A , , A121.Krivov, A. V., L¨ohne, T., and Sremˇcevi´c, M. 2006. Dust distri-butions in debris disks: effects of gravity, radiation pressureand collisions. A&A , , 509–519.Kuiper, G. P. 1951. On the Origin of the Solar System. Proceed-ings of the National Academy of Science , , 1–14.Kunz, M. W., Schekochihin, A. A., and Stone, J. M. 2014. Fire-hose and Mirror Instabilities in a Collisionless ShearingPlasma. Physical Review Letters , (20), 205003.Lada, C. J., and Wilking, B. A. 1984. The nature of the embeddedpopulation in the Rho Ophiuchi dark cloud - Mid-infraredobservations. ApJ , , 610–621.Lagrange, A.-M., Gratadour, D., Chauvin, G., Fusco, T., Ehren-reich, D., Mouillet, D., Rousset, G., Rouan, D., Allard, F.,Gendron, ´E., Charton, J., Mugnier, L., Rabou, P., Montri,J., and Lacombe, F. 2009. A probable giant planet imagedin the β Pictoris disk. VLT/NaCo deep L’-band imaging.
A&A , , L21–L25.Lambrechts, M., and Johansen, A. 2012. Rapid growth of gas-giant cores by pebble accretion. A&A , , A32.Lasota, J.-P. 2001. The disc instability model of dwarf novae andlow-mass X-ray binary transients. NewAR , , 449–508.Latter, H. N., and Ogilvie, G. I. 2006a. The linear stability ofdilute particulate rings. Icarus , , 498–516. eferences Latter, H. N., and Ogilvie, G. I. 2006b. Viscous overstabil-ity and eccentricity evolution in three-dimensional gaseousdiscs.
MNRAS , , 1829–1839.Latter, H. N., and Ogilvie, G. I. 2008. Dense planetary rings andthe viscous overstability. Icarus , , 725–751.Latter, H. N., and Ogilvie, G. I. 2009. The viscous overstability,nonlinear wavetrains, and finescale structure in dense plan-etary rings. Icarus , (Aug.), 565–583.Latter, H. N., and Ogilvie, G. I. 2010. Hydrodynamical simula-tions of viscous overstability in Saturn’s rings. Icarus , ,318–329.Latter, H. N., and Papaloizou, J. C. B. 2012. Hysteresis andthermal limit cycles in MRI simulations of accretion discs. MNRAS , , 1107–1120.Latter, H. N., Ogilvie, G. I., and Chupeau, M. 2012a. The ballistictransport instability in Saturn’s rings - I. Formalism andlinear theory. MNRAS , , 2336–2348.Latter, H. N., Rein, H., and Ogilvie, G. I. 2012b. The gravita-tional instability of a stream of co-orbital particles. MNRAS , (June), 1267–1276.Latter, H. N., Ogilvie, G. I., and Chupeau, M. 2014a. The ballistictransport instability in Saturn’s rings - II. Non-linear wavedynamics. MNRAS , , 2760–2772.Latter, H. N., Ogilvie, G. I., and Chupeau, M. 2014b. The bal-listic transport instability in Saturn’s rings - III. Numericalsimulations. MNRAS , , 2773–2781.Leinhardt, Z. M., and Richardson, D. C. 2002. N-Body Simula-tions of Planetesimal Evolution: Effect of Varying ImpactorMass Ratio. Icarus , , 306–313.Leinhardt, Z. M., and Stewart, S. T. 2012. Collisions betweenGravity-dominated Bodies. I. Outcome Regimes and ScalingLaws. ApJ , , 79.Leinhardt, Z. M., Ogilvie, G. I., Latter, H. N., and Kokubo, E.2012. Tidal disruption of satellites and formation of narrowrings. MNRAS , , 1419–1431.Lesur, G., and Ogilvie, G. I. 2010. On the angular momentumtransport due to vertical convection in accretion discs. MN-RAS , , L64–L68.Lesur, G., and Papaloizou, J. C. B. 2010a. The subcritical baro-clinic instability in local accretion disc models. A&A , ,A60.Lesur, G., and Papaloizou, J. C. B. 2010b. The subcritical baro-clinic instability in local accretion disc models. A&A , ,A60.Lesur, G., Hennebelle, P., and Fromang, S. 2015. Spiral-drivenaccretion in protoplanetary discs. I. 2D models. A&A , ,L9.Lewis, M. C., and Stewart, G. R. 2000. Collisional Dynamics ofPerturbed Planetary Rings. I. AJ , , 3295–3310.Lewis, M. C., and Stewart, G. R. 2009. Features around embed-ded moonlets in Saturn’s rings: The role of self-gravity andparticle size distributions. Icarus , , 387–412.Lightman, A. P., and Eardley, D. M. 1974. Black Holes in BinarySystems: Instability of Disk Accretion. ApJL , , L1.Lin, C. C., and Shu, F. H. 1964. On the Spiral Structure of DiskGalaxies. ApJ , , 646.Lin, D. N. C., and Bodenheimer, P. 1981. On the stability ofSaturn’s rings. ApJL , , L83–L86.Lin, D. N. C., and Papaloizou, J. 1986. On the tidal interac-tion between protoplanets and the protoplanetary disk. III -Orbital migration of protoplanets. ApJ , (Oct.), 846–857.Lin, D. N. C., and Papaloizou, J. C. B. 1993. On the tidal in-teraction between protostellar disks and companions. Pages 749–835 of: Levy, E. H., and Lunine, J. I. (eds), Protostarsand Planets III .Lissauer, J. J., Squyres, S. W., and Hartmann, W. K. 1988. Bom-bardment history of the Saturn system.
JGR , , 13776–13804.L¨ohne, T., Augereau, J.-C., Ertel, S., Marshall, J. P., Eiroa, C.,Mora, A., Absil, O., Stapelfeldt, K., Th´ebault, P., Bayo,A., Del Burgo, C., Danchi, W., Krivov, A. V., Lebreton,J., Letawe, G., Magain, P., Maldonado, J., Montesinos, B.,Pilbratt, G. L., White, G. J., and Wolf, S. 2012. Modellingthe huge, Herschel-resolved debris ring around HD 207129. A&A , , A110.Longaretti, P.-Y. 1989. Saturn’s main ring particle size distribu-tion - an analytic approach. Icarus , , 51–73.Longaretti, P.-Y., and Rappaport, N. 1995. Viscous overstabilitiesin dense narrow planetary rings. Icarus , , 376–396.Loska, Z. 1986. Three-dimensional waves in disks. AcA , , 43–61.Lubow, S. H. 1991. A model for tidally driven eccentric instabil-ities in fluid disks. ApJ , (Nov.), 259–267.Lubow, S. H., and Ogilvie, G. I. 2001. Secular Interactions be-tween Inclined Planets and a Gaseous Disk. ApJ , (Oct.),997–1009.Lubow, S. H., and Pringle, J. E. 1993. Wave propagation inaccretion disks - Axisymmetric case. ApJ , , 360–371.Lukkari, J. 1981. Collisional amplification of density fluctuationsin Saturn’s rings. Nature , , 433–435.Lynden-Bell, D. 1967. Statistical mechanics of violent relaxationin stellar systems. MNRAS , , 101.Lynden-Bell, D. 1969. Galactic Nuclei as Collapsed Old Quasars. Nature , (Aug.), 690–694.Lynden-Bell, D., and Pringle, J. E. 1974. The evolution of viscousdiscs and the origin of the nebular variables. MNRAS , ,603–637.Lyubarskij, Y. E., Postnov, K. A., and Prokhorov, M. E. 1994.Eccentric Accretion Discs. MNRAS , , 583.Macchetto, F., Marconi, A., Axon, D. J., Capetti, A., Sparks, W.,and Crane, P. 1997. The Supermassive Black Hole of M87and the Kinematics of Its Associated Gaseous Disk. ApJ , , 579–600.Marconi, A., Risaliti, G., Gilli, R., Hunt, L. K., Maiolino, R., andSalvati, M. 2004. Local supermassive black holes, relics ofactive galactic nuclei and the X-ray background. MNRAS , , 169–185.Marino, S., Perez, S., and Casassus, S. 2015a. Shadows Castby a Warp in the HD 142527 Protoplanetary Disk. ApJL , (Jan.), L44.Marino, S., Perez, S., and Casassus, S. 2015b. Shadows Cast bya Warp in the HD 142527 Protoplanetary Disk. ApJL , ,L44.Marley, M. S. 1991. Nonradial oscillations of Saturn. Icarus , ,420–435.Masset, F. S., Morbidelli, A., Crida, A., and Ferreira, J. 2006.Disk Surface Density Transitions as Protoplanet Traps. ApJ , (May), 478–487.Matthews, B. C., Krivov, A. V., Wyatt, M. C., Bryden, G., andEiroa, C. 2014. Observations, Modeling, and Theory of De-bris Disks. Protostars and Planets VI , 521–544.Mayor, M., and Queloz, D. 1995. A Jupiter-mass companion toa solar-type star.
Nature , , 355–359.McCaughrean, M. J., and O’Dell, C. R. 1996. Direct Imaging ofCircumstellar Disks in the Orion Nebula. AJ , , 1977.McConnell, N. J., Ma, C.-P., Gebhardt, K., Wright, S. A., Mur-phy, J. D., Lauer, T. R., Graham, J. R., and Richstone, D. O. References
Nature , , 215–218.McKee, C. F., and Ostriker, E. C. 2007. Theory of Star Forma-tion. ARAA , , 565–687.McKee, C. F., and Ostriker, J. P. 1977. A theory of the inter-stellar medium - Three components regulated by supernovaexplosions in an inhomogeneous substrate. ApJ , , 148–169.Meinke, B. K., Esposito, L. W., Albers, N., and Sremˇcevi´c, M.2012. Classification of F ring features observed in CassiniUVIS occultations. Icarus , , 545–554.Merritt, D. 2013. Dynamics and Evolution of Galactic Nuclei .Meyer, F., and Meyer-Hofmeister, E. 1981. On the Elusive Causeof Cataclysmic Variable Outbursts.
A&A , , L10.Miley, G., and De Breuck, C. 2008. Distant radio galaxies andtheir environments. AARv , , 67–144.Miranda, R., Hor´ak, J., and Lai, D. 2015. Viscous driving of globaloscillations in accretion discs around black holes. MNRAS , , 240–253.Miyoshi, M., Moran, J., Herrnstein, J., Greenhill, L., Nakai, N.,Diamond, P., and Inoue, M. 1995. Evidence for a blackhole from high rotation velocities in a sub-parsec region ofNGC4258. Nature , , 127–129.Montmerle, T., Augereau, J.-C., Chaussidon, M., Gounelle, M.,Marty, B., and Morbidelli, A. 2006. From Suns to Life: AChronological Approach to the History of Life on Earth 3.Solar System Formation and Early Evolution: the First 100Million Years. Earth Moon and Planets , , 39–95.Mouillet, D., Larwood, J. D., Papaloizou, J. C. B., and Lagrange,A. M. 1997. A planet on an inclined orbit as an explanationof the warp in the Beta Pictoris disc. MNRAS , , 896.Murray, C. D., Chavez, C., Beurle, K., Cooper, N., Evans, M. W.,Burns, J. A., and Porco, C. C. 2005. How Prometheus createsstructure in Saturn’s F ring. Nature , , 1326–1329.Muto, T., Grady, C. A., Hashimoto, J., Fukagawa, M., Hornbeck,J. B., Sitko, M., Russell, R., Werren, C., Cur´e, M., Currie, T.,Ohashi, N., Okamoto, Y., Momose, M., Honda, M., Inutsuka,S., Takeuchi, T., Dong, R., Abe, L., Brandner, W., Brandt,T., Carson, J., Egner, S., Feldt, M., Fukue, T., Goto, M.,Guyon, O., Hayano, Y., Hayashi, M., Hayashi, S., Henning,T., Hodapp, K. W., Ishii, M., Iye, M., Janson, M., Kan-dori, R., Knapp, G. R., Kudo, T., Kusakabe, N., Kuzuhara,M., Matsuo, T., Mayama, S., McElwain, M. W., Miyama,S., Morino, J.-I., Moro-Martin, A., Nishimura, T., Pyo, T.-S., Serabyn, E., Suto, H., Suzuki, R., Takami, M., Takato,N., Terada, H., Thalmann, C., Tomono, D., Turner, E. L.,Watanabe, M., Wisniewski, J. P., Yamada, T., Takami, H.,Usuda, T., and Tamura, M. 2012. Discovery of Small-scaleSpiral Structures in the Disk of SAO 206462 (HD 135344B):Implications for the Physical State of the Disk from SpiralDensity Wave Theory. ApJL , , L22.Narayan, R., and Yi, I. 1994. Advection-dominated accretion: Aself-similar solution. ApJL , , L13–L16.Narayan, R., Mahadevan, R., and Quataert, E. 1998. Advection-dominated accretion around black holes. Pages 148–182 of:Abramowicz, M. A., Bj¨ornsson, G., and Pringle, J. E. (eds), Theory of Black Hole Accretion Disks .Nelson, R. P., and Papaloizou, J. C. B. 2004. The interaction of gi-ant planets with a disc with MHD turbulence - IV. Migrationrates of embedded protoplanets.
MNRAS , , 849–864.Nelson, R. P., Gressel, O., and Umurhan, O. M. 2013. Linearand non-linear evolution of the vertical shear instability inaccretion discs. MNRAS , , 2610–2632. Nesvold, E. R., Kuchner, M. J., Rein, H., and Pan, M. 2013.SMACK: A New Algorithm for Modeling Collisions and Dy-namics of Planetesimals in Debris Disks. ApJ , , 144.Netzer, H. 2015. Revisiting the Unified Model of Active GalacticNuclei. ARAA , , 365–408.Nicholson, P. D., French, R. G., Hedman, M. M., Marouf, E. A.,and Colwell, J. E. 2014. Noncircular features in Saturn’srings I: The edge of the B ring. Icarus , , 152–175.Ogilvie, G. I. 1998. Waves and instabilities in a differentiallyrotating disc containing a poloidal magnetic field. MNRAS , , 291–314.Ogilvie, G. I. 1999. The non-linear fluid dynamics of a warpedaccretion disc. MNRAS , , 557–578.Ogilvie, G. I. 2001. Non-linear fluid dynamics of eccentric discs. MNRAS , , 231–248.Ogilvie, G. I. 2003. On the dynamics of magnetorotational tur-bulent stresses. MNRAS , , 969–982.Ogilvie, G. I. 2006. Non-linear bending waves in Keplerian accre-tion discs. MNRAS , , 977–990.Ogilvie, G. I., and Lubow, S. H. 2002. On the wake generated bya planet in a disc. MNRAS , (Mar.), 950–954.Okazaki, A. T. 1991. Long-term V/R variations of Be stars dueto global one-armed oscillations of equatorial disks. PASJ , , 75–94.Okazaki, A. T., Kato, S., and Fukue, J. 1987. Global trappedoscillations of relativistic accretion disks. PASJ , , 457–473.Paardekooper, S.-J. 2012. Numerical convergence in self-gravitating shearing sheet simulations and the stochastic na-ture of disc fragmentation. MNRAS , , 3286–3299.Paardekooper, S.-J., and Papaloizou, J. C. B. 2008. On disc pro-toplanet interactions in a non-barotropic disc with thermaldiffusion. A&A , (July), 877–895.Paardekooper, S.-J., and Papaloizou, J. C. B. 2009. On coro-tation torques, horseshoe drag and the possibility of sus-tained stalled or outward protoplanetary migration. MN-RAS , (Apr.), 2283–2296.Paczynski, B. 1977. A model of accretion disks in close binaries. ApJ , (Sept.), 822–826.Papaloizou, J., and Pringle, J. E. 1977. Tidal torques on accretiondiscs in close binary systems. MNRAS , (Nov.), 441–454.Papaloizou, J. C., and Savonije, G. J. 1991. Instabilities in self-gravitating gaseous discs. MNRAS , (Feb.), 353–369.Papaloizou, J. C. B., and Lin, D. N. C. 1988. On the pulsationaloverstability in narrowly confined viscous rings. ApJ , ,838–860.Papaloizou, J. C. B., and Lin, D. N. C. 1995. On the dynamicsof warped accretion disks. ApJ , , 841–851.Papaloizou, J. C. B., and Stanley, G. Q. G. 1986. The structureand stability of the accretion disc boundary layer. MNRAS , , 593–610.Papaloizou, J. C. B., and Terquem, C. 2006. Planet formation andmigration. Reports on Progress in Physics , , 119–180.Patterson, J., Kemp, J., Jensen, L., Vanmunster, T., Skillman,D. R., Martin, B., Fried, R., and Thorstensen, J. R. 2000.Superhumps in Cataclysmic Binaries. XVIII. IY Ursae Ma-joris. PASP , (Dec.), 1567–1583.Peale, S. J. 1999. Origin and Evolution of the Natural Satellites. ARAA , , 533–602.P´erez, L. M., Isella, A., Carpenter, J. M., and Chandler, C. J.2014. Large-scale Asymmetries in the Transitional Disks ofSAO 206462 and SR 21. ApJL , , L13. eferences Perrine, R. P., and Richardson, D. C. 2012. N-body simulationsof cohesion in dense planetary rings: A study of cohesionparameters.
Icarus , , 515–533.Perrine, R. P., Richardson, D. C., and Scheeres, D. J. 2011. Anumerical model of cohesion in planetary rings. Icarus , ,719–735.Peterson, B. M. 2001. Variability of Active Galactic Nuclei. Page3 of: Aretxaga, I., Kunth, D., and M´ujica, R. (eds), AdvancedLectures on the Starburst-AGN .Petit, J.-M., Kavelaars, J. J., Gladman, B., and Loredo, T. 2008.
Size Distribution of Multikilometer Transneptunian Objects .Pages 71–87.Piran, T. 1978. The role of viscosity and cooling mechanisms inthe stability of accretion disks.
ApJ , , 652–660.Pollack, J. B. 1975. The rings of Saturn. SSRv , , 3–93.Pollack, J. B., Grossman, A. S., Moore, R., and Graboske, Jr.,H. C. 1976. The formation of Saturn’s satellites and rings,as influenced by Saturn’s contraction history. Icarus , ,35–48.Pringle, J. E. 1981. Accretion discs in astrophysics. ARAA , ,137–162.Pudritz, R. E., Ouyed, R., Fendt, C., and Brandenburg, A. 2007.Disk Winds, Jets, and Outflows: Theoretical and Computa-tional Foundations. Protostars and Planets V , 277–294.Quataert, E., and Chiang, E. I. 2000. Angular Momentum Trans-port in Particle and Fluid Disks.
ApJ , , 432–437.Quillen, A. C. 2006. Predictions for a planet just inside Fomal-haut’s eccentric ring. MNRAS , , L14–L18.Quirrenbach, A., Buscher, D. F., Mozurkewich, D., Hummel,C. A., and Armstrong, J. T. 1994. Maximum-entropy mapsof the Be shell star zeta Tauri from optical long-baselineinterferometry. A&A , , L13–L16.Rees, M. J. 1984. Black Hole Models for Active Galactic Nuclei. ARAA , , 471–506.Rees, M. J. 1988. Tidal disruption of stars by black holes of 10to the 6th-10 to the 8th solar masses in nearby galaxies. Nature , , 523–528.Rees, M. J., Begelman, M. C., Blandford, R. D., and Phinney,E. S. 1982. Ion-supported tori and the origin of radio jets. Nature , , 17–21.Rein, H. 2012. Planet-disc interaction in highly inclined systems. MNRAS , , 3611–3616.Rein, H., and Latter, H. N. 2013. Large-scale N-body simulationsof the viscous overstability in Saturn’s rings. MNRAS , ,145–158.Rein, H., and Papaloizou, J. C. B. 2009. On the evolution of meanmotion resonances through stochastic forcing: fast and slowlibration modes and the origin of HD 128311. A&A , ,595–609.Rein, H., and Papaloizou, J. C. B. 2010. Stochastic orbital mi-gration of small bodies in Saturn’s rings. A&A , , A22+.Remillard, R. A., and McClintock, J. E. 2006. X-Ray Propertiesof Black-Hole Binaries. ARAA , , 49–92.Rice, W. K. M., Lodato, G., and Armitage, P. J. 2005. Investi-gating fragmentation conditions in self-gravitating accretiondiscs. MNRAS , , L56–L60.Rice, W. K. M., Paardekooper, S.-J., Forgan, D. H., and Ar-mitage, P. J. 2014. Convergence of simulations of self-gravitating accretion discs - II. Sensitivity to the implemen-tation of radiative cooling and artificial viscosity. MNRAS , , 1593–1602.Richardson, D. C., Quinn, T., Stadel, J., and Lake, G. 2000. Di-rect Large-Scale N-Body Simulations of Planetesimal Dy-namics. Icarus , , 45–59. Ringl, C., Bringa, E. M., Bertoldi, D. S., and Urbassek, H. M.2012. Collisions of Porous Clusters: A Granular-mechanicsStudy of Compaction and Fragmentation. ApJ , , 151.Rivinius, T., Carciofi, A. C., and Martayan, C. 2013. ClassicalBe stars. Rapidly rotating B stars with viscous Kepleriandecretion disks. AARv , , 69.Safronov, V. S. 1969. Evolution of the Protoplanetary Cloud andFormation of the Earth and the Planets. Nauka, Moscow(NASA technical Translation TTF-677) .Salo, H. 1991. Numerical simulations of dense collisional systems.
Icarus , , 254–270.Salo, H. 1992. Gravitational wakes in Saturn’s rings. Nature , , 619–621.Salo, H. 1995. Simulations of dense planetary rings. III. Self-gravitating identical particles. Icarus , , 287–312.Salo, H., and Schmidt, J. 2010. N-body simulations of viscousinstability of planetary rings. Icarus , , 390–409.Salo, H., Schmidt, J., and Spahn, F. 2001. Viscous Overstabilityin Saturn’s B Ring. I. Direct Simulations and Measurementof Transport Coefficients. Icarus , , 295–315.Salpeter, E. E. 1964. Accretion of Interstellar Matter by MassiveObjects. ApJ , (Aug.), 796–800.Sanchis-Ojeda, R., Rappaport, S., Pall`e, E., Delrez, L., DeVore,J., Gandolfi, D., Fukui, A., Ribas, I., Stassun, K. G., Al-brecht, S., Dai, F., Gaidos, E., Gillon, M., Hirano, T., Hol-man, M., Howard, A. W., Isaacson, H., Jehin, E., Kuzuhara,M., Mann, A. W., Marcy, G. W., Miles-P´aez, P. A.,Monta˜n´es-Rodr´ıguez, P., Murgas, F., Narita, N., Nowak, G.,Onitsuka, M., Paegert, M., Van Eylen, V., Winn, J. N., andYu, L. 2015. The K2-ESPRINT Project I: Discovery of theDisintegrating Rocky Planet K2-22b with a Cometary Headand Leading Tail. ApJ , , 112.Sandage, A., Osmer, P., Giacconi, R., Gorenstein, P., Gursky,H., Waters, J., Bradt, H., Garmire, G., Sreekantan, B. V.,Oda, M., Osawa, K., and Jugaku, J. 1966. On the opticalidentification of SCO X-1. ApJ , , 316.Sargent, A. I., and Beckwith, S. 1987. Kinematics of the circum-stellar gas of HL Tauri and R Monocerotis. ApJ , (Dec.),294–305.Savage, S. B., and Jeffrey, D. J. 1981. The stress tensor in a gran-ular flow at high shear rates. Journal of Fluid Mechanics , , 255–272.Schmidt, J., and Salo, H. 2003. Weakly Nonlinear Model forOscillatory Instability in Saturn’s Dense Rings. PhysicalReview Letters , (6), 061102.Schmidt, J., Salo, H., Spahn, F., and Petzschmann, O. 2001.Viscous Overstability in Saturn’s B-Ring. II. HydrodynamicTheory and Comparison to Simulations. Icarus , , 316–331.Schmit, U., and Tscharnuter, W. M. 1995. A fluid dynamicaltreatment of the common action of self-gravitation, colli-sions, and rotation in Saturn’s B-ring. Icarus , , 304–319.Schmit, U., and Tscharnuter, W. M. 1999. On the Formation ofthe Fine-Scale Structure in Saturn’s B Ring. Icarus , ,173–187.Searle, L., and Zinn, R. 1978. Compositions of halo clusters andthe formation of the galactic halo. ApJ , , 357–379.Seizinger, A., and Kley, W. 2013. Bouncing behavior of micro-scopic dust aggregates. A&A , , A65.Sfair, R., Winter, S. M. G., Mour˜ao, D. C., and Winter, O. C.2009. Dynamical evolution of Saturn’s F ring dust particles. MNRAS , , 2157–2161.Shakura, N. I., and Sunyaev, R. A. 1973. Black holes in binarysystems. Observational appearance. A&A , , 337–355. References
Shakura, N. I., and Sunyaev, R. A. 1976. A theory of the instabil-ity of disk accretion on to black holes and the variability ofbinary X-ray sources, galactic nuclei and quasars.
MNRAS , , 613–632.Shara, M. M. 1989. Recent progress in understanding the erup-tions of classical novae. PASP , , 5–31.Showalter, M. R., Hedman, M. M., and Burns, J. A. 2011. TheImpact of Comet Shoemaker-Levy 9 Sends Ripples Throughthe Rings of Jupiter. Science , , 711.Shu, F. H. 1992. The physics of astrophysics. Volume II: Gasdynamics.
Shu, F. H., and Stewart, G. R. 1985. The collisional dynamics ofparticulate disks.
Icarus , , 360–383.Smak, J. 1971. Eruptive Binaries. II. U Geminorum. AcA , ,15.Smith, B. A., Soderblom, L. A., Banfield, D., Barnet, C.,Basilevksy, A. T., Beebe, R. F., Bollinger, K., Boyce, J. M.,Brahic, A., Briggs, G. A., Brown, R. H., Chyba, C., Collins,S. A., Colvin, T., Cook, A. F., Crisp, D., Croft, S. K., Cruik-shank, D., Cuzzi, J. N., Danielson, G. E., Davies, M. E., deJong, E., Dones, L., Godfrey, D., Goguen, J., Grenier, I.,Haemmerle, V. R., Hammel, H., Hansen, C. J., Helfenstein,C. P., Howell, C., Hunt, G. E., Ingersoll, A. P., Johnson,T. V., Kargel, J., Kirk, R., Kuehn, D. I., Limaye, S., Ma-sursky, H., McEwen, A., Morrison, D., Owen, T., Owen, W.,Pollack, J. B., Porco, C. C., Rages, K., Rogers, P., Rudy,D., Sagan, C., Schwartz, J., Shoemaker, E. M., Showalter,M., Sicardy, B., Simonelli, D., Spencer, J., Sromovsky, L. A.,Stoker, C., Strom, R. G., Suomi, V. E., Synott, S. P., Ter-rile, R. J., Thomas, P., Thompson, W. R., Verbiscer, A., andVeverka, J. 1989. Voyager 2 at Neptune: Imaging ScienceResults. Science , , 1422–1449.Spahn, F., Hertzsch, J.-M., and Brilliantov, N. V. 1995. Therole of particle collisions for the dynamics in planetary rings. Chaos Solitons and Fractals , , 1945–1964.Spitale, J. N., and Porco, C. C. 2010. Detection of Free UnstableModes and Massive Bodies in Saturn’s Outer B Ring. AJ , , 1747–1757.Springel, V., White, S. D. M., Jenkins, A., Frenk, C. S., Yoshida,N., Gao, L., Navarro, J., Thacker, R., Croton, D., Helly, J.,Peacock, J. A., Cole, S., Thomas, P., Couchman, H., Evrard,A., Colberg, J., and Pearce, F. 2005. Simulations of theformation, evolution and clustering of galaxies and quasars. Nature , , 629–636.Stansberry, J., Grundy, W., Brown, M., Cruikshank, D., Spencer,J., Trilling, D., and Margot, J.-L. 2008. Physical Propertiesof Kuiper Belt and Centaur Objects: Constraints from theSpitzer Space Telescope . Pages 161–179.Strubbe, L. E., and Quataert, E. 2009. Optical flares from thetidal disruption of stars by massive black holes.
MNRAS , , 2070–2084.Su, K. Y. L., Rieke, G. H., Stansberry, J. A., Bryden, G.,Stapelfeldt, K. R., Trilling, D. E., Muzerolle, J., Beichman,C. A., Moro-Martin, A., Hines, D. C., and Werner, M. W.2006. Debris Disk Evolution around A Stars. ApJ , ,675–689.Su, K. Y. L., Rieke, G. H., Stapelfeldt, K. R., Malhotra, R., Bry-den, G., Smith, P. S., Misselt, K. A., Moro-Martin, A., andWilliams, J. P. 2009. The Debris Disk Around HR 8799. ApJ , (Nov.), 314–327.Supulver, K. D., Bridges, F. G., and Lin, D. N. C. 1995. Thecoefficient of restitution of ice particles in glancing collisions:Experimental results for unfrosted surfaces. Icarus , ,188–199. Supulver, K. D., Bridges, F. G., Tiscareno, S., Lievore, J., andLin, D. N. C. 1997. The Sticking Properties of Water FrostProduced under Various Ambient Conditions. Icarus , ,539–554.Takahashi, S. Z., and Inutsuka, S.-i. 2014. Two-component Sec-ular Gravitational Instability in a Protoplanetary Disk: APossible Mechanism for Creating Ring-like Structures. ApJ , , 55.Tamayo, D., Triaud, A. H. M. J., Menou, K., and Rein, H. 2015.Dynamical Stability of Imaged Planetary Systems in Forma-tion: Application to HL Tau. ApJ , , 100.Tanaka, H., Inaba, S., and Nakazawa, K. 1996. Steady-State SizeDistribution for the Self-Similar Collision Cascade. Icarus , (Oct.), 450–455.Tauris, T. M., and van den Heuvel, E. P. J. 2006. Formation andevolution of compact stellar X-ray sources . Pages 623–665.Testi, L., Birnstiel, T., Ricci, L., Andrews, S., Blum, J., Carpen-ter, J., Dominik, C., Isella, A., Natta, A., Williams, J. P.,and Wilner, D. J. 2014. Dust Evolution in ProtoplanetaryDisks.
Protostars and Planets VI , 339–361.Teyssandier, J., and Ogilvie, G. I. 2016. Growth of eccen-tric modes in disc-planet interactions.
MNRAS , (May),3221–3247.Th´ebault, P., and Augereau, J.-C. 2007. Collisional processes andsize distribution in spatially extended debris discs. A&A , , 169–185.Thompson, W. T., Lumme, K., Irvine, W. M., Baum, W. A., andEsposito, L. W. 1981. Saturn’s rings - Azimuthal variations,phase curves, and radial profiles in four colors. Icarus , ,187–200.Thomson, F. S., Marouf, E. A., Tyler, G. L., French, R. G., andRappoport, N. J. 2007. Periodic microstructure in Saturn’srings A and B. GeoRL , , L24203.Tiscareno, M. S. 2013. A modified Type I migration model forpropeller moons in Saturn’s rings. PSS , (Mar.), 136–142.Tiscareno, M. S., Burns, J. A., Nicholson, P. D., Hedman, M. M.,and Porco, C. C. 2007. Cassini imaging of Saturn’s rings. II.A wavelet technique for analysis of density waves and otherradial structure in the rings. Icarus , , 14–34.Toomre, A. 1964. On the gravitational stability of a disk of stars. ApJ , (May), 1217–1238.Tremaine, S. 2003. On the Origin of Irregular Structure in Sat-urn’s Rings. AJ , , 894–901.Ulrich, M.-H. 2000. The active galaxy NGC 4151: Archetype orexception? AARv , , 135–178.van Lieshout, R., Min, M., and Dominik, C. 2014. Dusty tails ofevaporating exoplanets. I. Constraints on the dust composi-tion. A&A , , A76.Varni`ere, P., and Tagger, M. 2006. Reviving Dead Zones in ac-cretion disks by Rossby vortices at their boundaries. A&A , , L13–L16.Verbiscer, A. J., Skrutskie, M. F., and Hamilton, D. P. 2009.Saturn’s largest ring. Nature , , 1098–1100.Villaver, E., and Livio, M. 2007. Can Planets Survive StellarEvolution? ApJ , , 1192–1201.Volonteri, M. 2010. Formation of supermassive black holes. AARv , , 279–315.Wada, K., Tanaka, H., Suyama, T., Kimura, H., and Yamamoto,T. 2008. Numerical Simulation of Dust Aggregate Colli-sions. II. Compression and Disruption of Three-DimensionalAggregates in Head-on Collisions. ApJ , , 1296–1308.Wada, K., Tanaka, H., Okuzumi, S., Kobayashi, H., Suyama, T.,Kimura, H., and Yamamoto, T. 2013. Growth efficiency eferences of dust aggregates through collisions with high mass ratios. A&A , , A62.Wagoner, R. V. 1999. Relativistic diskoseismology. PhR , ,259–269.Ward, W. R. 1981. On the radial structure of Saturn’s rings. GeoRL , , 641–643.Ward, W. R. 1997. Protoplanet Migration by Nebula Tides. Icarus , (Apr.), 261–281.Wardle, M. 1999. The Balbus-Hawley instability in weakly ionizeddiscs. MNRAS , , 849–856.Warner, B. 1995. Cataclysmic variable stars. Cambridge Astro-physics Series , .Warner, B., and Nather, R. E. 1971. Observations of rapid bluevariables - II. U Geminorum. MNRAS , , 219–229.Weidenschilling, S. J. 1977. Aerodynamics of solid bodies in thesolar nebula. MNRAS , , 57–70.Weidenschilling, S. J., Chapman, C. R., Davis, D. R., and Green-berg, R. 1984. Ring particles - Collisional interactions andphysical nature. Pages 367–415 of: Greenberg, R., andBrahic, A. (eds), IAU Colloq. 75: Planetary Rings .Weidenschilling, S. J., Spaute, D., Davis, D. R., Marzari, F., andOhtsuki, K. 1997. Accretional Evolution of a PlanetesimalSwarm.
Icarus , , 429–455.Wetherill, G. W., and Stewart, G. R. 1989. Accumulation of aswarm of small planetesimals. Icarus , , 330–357.Wetherill, G. W., and Stewart, G. R. 1993. Formation of plane-tary embryos - Effects of fragmentation, low relative veloc-ity, and independent variation of eccentricity and inclination. Icarus , , 190.Williams, J. P., and Cieza, L. A. 2011. Protoplanetary Disks andTheir Evolution. ARAA , , 67–117.Windmark, F., Birnstiel, T., G¨uttler, C., Blum, J., Dullemond,C. P., and Henning, T. 2012. Planetesimal formation bysweep-up: how the bouncing barrier can be beneficial togrowth. A&A , , A73.Wisdom, J., and Tremaine, S. 1988. Local simulations of plane-tary rings. AJ , , 925–940.Wurm, G., Paraskov, G., and Krauss, O. 2005. Growth of plan-etesimals by impacts at 25 m/s. Icarus , , 253–263.Wyatt, M. C. 2003. Resonant Trapping of Planetesimals byPlanet Migration: Debris Disk Clumps and Vega’s Similarityto the Solar System. ApJ , , 1321–1340.Wyatt, M. C. 2005a. Spiral structure when setting up pericentreglow: possible giant planets at hundreds of AU in the HD141569 disk. A&A , , 937–948.Wyatt, M. C. 2005b. The insignificance of P-R drag in detectableextrasolar planetesimal belts. A&A , , 1007–1012.Wyatt, M. C. 2008. Evolution of Debris Disks. ARAA , , 339–383.Wyatt, M. C., and Dent, W. R. F. 2002. Collisional processes inextrasolar planetesimal discs - dust clumps in Fomalhaut’sdebris disc. MNRAS , , 589–607.Wyatt, M. C., Pani´c, O., Kennedy, G. M., and Matr`a, L. 2015.Five steps in the evolution from protoplanetary to debrisdisk. ApSS , , 103.Xiang-Gruess, M. 2016. Generation of highly inclined protoplane-tary discs through single stellar flybys. MNRAS , (Jan.),3086–3100.Xu, S., Jura, M., Koester, D., Klein, B., and Zuckerman, B. 2014.Elemental Compositions of Two Extrasolar Rocky Planetes-imals. ApJ , , 79.Youdin, A. N., and Goodman, J. 2005. Streaming Instabilities inProtoplanetary Disks. ApJ , , 459–469. Zhang, K., Blake, G. A., and Bergin, E. A. 2015. Evidence ofFast Pebble Growth Near Condensation Fronts in the HLTau Protoplanetary Disk. ApJL , , L7.Zuckerman, B., Koester, D., Reid, I. N., and H¨unsch, M. 2003.Metal Lines in DA White Dwarfs. ApJ ,596