Surface Layer Accretion in Transitional and Conventional Disks: From Polycyclic Aromatic Hydrocarbons to Planets
DDraft version September 25, 2018
Preprint typeset using L A TEX style emulateapj v. 8/13/10
SURFACE LAYER ACCRETION IN TRANSITIONAL AND CONVENTIONAL DISKS:FROM POLYCYCLIC AROMATIC HYDROCARBONS TO PLANETS
Daniel Perez-Becker
Department of Physics, University of California, Berkeley, CA 94720, USA andEugene Chiang
Departments of Astronomy and Earth and Planetary Science, University of California, Berkeley, CA 94720, USA
Draft version September 25, 2018
ABSTRACT“Transitional” T Tauri disks have optically thin holes with radii (cid:38)
10 AU, yet accrete up to themedian T Tauri rate. Multiple planets inside the hole can torque the gas to high radial speeds overlarge distances, reducing the local surface density while maintaining accretion. Thus multi-planetsystems, together with reductions in disk opacity due to grain growth, can explain how holes can besimultaneously transparent and accreting. There remains the problem of how outer disk gas diffusesinto the hole. Here it has been proposed that the magnetorotational instability (MRI) erodes disksurface layers ionized by stellar X-rays. In contrast to previous work, we find that the extent to whichsurface layers are MRI-active is limited not by ohmic dissipation but by ambipolar diffusion, the lattermeasured by Am : the number of times a neutral hydrogen molecule collides with ions in a dynamicaltime. Simulations by Hawley & Stone showed that Am ∼
100 is necessary for ions to drive MRIturbulence in neutral gas. We calculate that in X-ray-irradiated surface layers, Am typically variesfrom ∼ − to 1, depending on the abundance of charge-adsorbing polycyclic aromatic hydrocarbons,whose properties we infer from Spitzer observations. We conclude that ionization of H by X-rays andcosmic rays can sustain, at most, only weak MRI turbulence in surface layers 1–10 g/cm thick, andthat accretion rates in such layers are too small compared to observed accretion rates for the majorityof disks. Subject headings: accretion, accretion disks — instabilities — ISM: molecules — MHD — planetarysystems: protoplanetary disks — stars: pre-main sequence INTRODUCTION
On the road from molecular clouds to planetary sys-tems, transitional disks are among the brightest sign-posts. Encircling T Tauri and Herbig Ae/Be stars hav-ing ages of 1–10 Myr, these disks have large inner holesnearly devoid of dust. Identified by spectral energy dis-tributions (SEDs; e.g., Strom et al. 1993; Calvet et al.2005; Kim et al. 2009) and imaged directly (e.g., Ratzkaet al. 2007; Hughes et al. 2007; Brown et al. 2009), tran-sitional disk cavities have radii on the order of 3–100 AU.Transitional disks are so named (Strom et al. 1990) be-cause they might represent an evolutionary link betweenoptically thick disks without holes (e.g., Watson et al.2007) and debris disks containing only rings of opticallythin dust (e.g., Wyatt 2008). They are of special inter-est not least because their central clearings may harbornascent planets, potentially detectable against relativelyweak backgrounds.
The Need for Companions
The idea that transitional disk holes are swept cleanby companions, possibly of planetary mass, is natural.We adhere to this interpretation, although not all ourarguments as given below are the ones usually discussed.
Stellar-mass Companions
Roughly half of all transitional disks—6 out of 13 inthe sample of Kim et al. (2009)—are already known to
Electronic address: [email protected] contain stellar-mass companions. A prototypical exam-ple is CoKu Tau/4: a transitional system whose hole ispractically empty, both of dust (D’Alessio et al. 2005)and gas (G. Blake, private communication, 2007). Insideits hole of radius ∼
10 AU resides a nearly equal mass K-star binary having a projected separation of 8 AU (Ire-land & Kraus 2008). Gravitational torques exerted bythe binary can easily counteract viscous torques in thedisk (Goldreich & Tremaine 1980), staving off accretiononto either star (Artymowicz & Lubow 1994; Ochi et al.2005). Indeed stellar accretion rates for CoKu Tau/4are unmeasurably small, (cid:46) − M (cid:12) yr − (Najita et al.2007).Not every hole, however, is as empty as that of CoKuTau/4. In many cases there is a sprinkling of dust: opti-cal depths at 10 µ m wavelength for many sources rangefrom 0.01–0.1 (Calvet et al. 2002, 2005). Observationsof rovibrational emission from warm, optically thick COimply that gas fills many disk holes (Salyk et al. 2007),albeit with surface densities that may be far below thoseof conventional disks (we will argue below that this is infact the case). Most germane to our work, the host starsof many transitional systems actively accrete, at ratesthat on average are somewhat lower than those of con-ventional disks (Najita et al. 2007), but which in severalinstances approach 10 − M (cid:12) yr − , the median T Taurirate. The holes of these systems must contain accretinggas.How can we reconcile the fact that many holes contain a r X i v : . [ a s t r o - ph . E P ] N ov Perez-Becker & Chianggas, accreting at rates approaching those of conventionaldisks, with the fact that the holes contain only traceamounts of dust? To explain the paradox of simultaneousaccretion and hole transparency, appeals are sometimesmade to grain growth, or the filtering of dust out of gasby hydrodynamic mechanisms (Paardekooper & Mellema2006; Rice et al. 2006) or radiation pressure (Chiang &Murray-Clay 2007).These proposals, which invoke changes in disk opacity,may be part of the solution. But they cannot alone ex-plain the observations. Ward (2009) has criticized thehydrodynamic filter. The force of radiation pressure de-pends on uncertain optical constants and grain porosi-ties, and is likely to expel only grains having a narrowrange of sizes (Burns et al. 1979). Even if grains have theright properties to be blown out by radiation pressure invacuum, the inward flow of accreting gas may be strongenough to carry as much as half of the grains that leakfrom the rim into the hole (Chiang & Murray-Clay 2007).Grain growth does not explain why transitional disk ac-cretion rates ˙ M tend to be several times smaller thanfor conventional T Tauri systems (Najita et al. 2007).Finally, none of these proposals predicts gapped (“pre-transitional”) disks, which are optically thick at stel-locentric distances a (cid:46) .
15 AU (e.g., LkCa 15; Espaillatet al. 2007a).An alternative explanation for why holes can be si-multaneously transparent and still contain accreting gasinvolves the special way in which disk gas accretes in thepresence of companions, particularly those on eccentricorbits. If the hole rim leaks gas—and for some combina-tions of disk viscosity, disk pressure, and binary param-eters the rim can be quite leaky (Artymowicz & Lubow1996; Ochi et al. 2005)—the gas can suddenly plunge in-ward at rates approaching freefall velocities. The catas-trophic loss of angular momentum is enabled by the non-axisymmetric and time-dependent potential of the eccen-tric binary, which directs gas streamlines onto radial or-bits that may intersect and shock. Artymowicz & Lubow(1996) explained the paradox of simultaneous accretionand transparency:“ . . . [for some circumbinary disk parame-ters] gravitational resonant torques are ableto open a fairly wide gap [hole], while con-currently the accretion flow proceeds throughthat gap in the form of time-dependent, well-developed or efficient gas stream(s) carryingvirtually all the unimpeded mass flux. Theradial velocity of the stream is of order [theKepler velocity], i.e., ∼ Re [Reynolds number]times faster than in the disk. By mass conser-vation, the axially averaged surface densitymust differ by a factor of Re > betweenthe gap and the disk edge region. . . . Thespectroscopic ramification of this is a deficitof the observed radiation flux emitted at tem-peratures appropriate for the gap location.”(italics theirs)Thus reductions in the dust-to-gas ratio by graingrowth or dust filtration are not the only processes thatcan render accreting gas transparent in transitional diskholes. Companions can accelerate disk gas to such high radial speeds that, by mass continuity, the surface den-sity in both gas and dust is reduced by orders of magni-tude. According to this explanation, the reduction in to-tal surface density is not necessarily due to consumptionof gas by companions, but is rather due to gravitationalforcing.
Multiple Planetary Mass Companions
Though a stellar-mass companion can exert torquesstrong enough to maintain holes of large size, a singleplanet-mass companion on a circular orbit cannot do thesame job. For observationally reasonable values of thedisk viscosity, a single companion having of order ∼ (cid:46)
10 (Lubow & D’Angelo 2006). Thismodest reduction in ˙ M , combined with the narrownessof the gap seen in simulations (∆ r/r ∼ . v (but not accretion rates˙ M ) can be performed, not by a single planet, but bya system of multiple planets. We imagine a series ofplanets, with the outermost lying just interior to andshepherding the hole rim. Gas that leaks from the rimis torqued from planet to planet, all the way down tothe central star, its optical depth decreasing inversely asits radial speed. The more massive the planets and themore eccentric their orbits, the fewer of them should berequired.Such a picture is supported by numerical simulationsof Jupiter and Saturn embedded within a viscous disk(Masset & Snellgrove 2001; Morbidelli & Crida 2007). Inthese simulations the two planets were close enough thattheir gaps overlapped. Gas outside Saturn’s orbit exe-cuted half a horseshoe turn relative to Saturn, and thenanother half-horseshoe turn relative to Jupiter, therebycrossing from the outer disk through the Jupiter-Saturncommon gap into the inner disk. Morbidelli & Crida(2007) found that the surface density in the gap regionwas reduced by 1–2 orders of magnitude, at least nearJupiter.As our paper was being reviewed, we became awareof planet-disk simulations by Zhu et al. (2010, submit-ted) which included as many as 4 Jupiter-mass planetsand whose results supported those of Morbidelli & Crida(2007). Depending on the assumed efficiency with whichplanets consumed disk gas, a set of four planets wasfound to reduce surface densities in their vicinity by upto 2 orders of magnitude, while disk accretion rates werereduced by factors (cid:46)
10 (see their run P4A10). However,such surface density suppressions are not by themselveslarge enough to explain the observed low optical depthsof disk holes. Zhu et al. (2010) concluded that reduc-tions in gas opacity by some means of dust depletionAHs, MRI, and Planets 3(e.g., grain growth) are still required.Companions can also accommodate gapped or “pre-transitional” disks in which optically thin holes containoptically thick annuli. As inferred from spatially unre-solved spectra, these annuli are narrow and abut theirhost stars, extending mere fractions of an AU in radius(Espaillat et al. 2007a; but see also Eisner et al. 2009who showed using spatially resolved observations thatthe gapped disk interpretation of SR 21 is incorrect). Inregions far removed from secondary companions—in par-ticular, in those regions closest to the primary star wherethe potential is practically that of a point mass—the in-fall speeds of accreting gas must slow back down to thenormal rate set by disk viscosity. By continuity, the sur-face density must rise back up, and optical thickness isthus restored.
Companions are Not Enough: The Case for theMagnetorotational Instability for the Origin of DiskViscosity
Our case for companions presumes a source of diskviscosity. While a stellar-mass companion or a systemof multiple planets can transport gas quickly, effectivelygenerating an enormous viscosity in their vicinity (i.e.,inside the hole), they cannot cause the outer disk to dif-fuse in the first place. An inviscid outer disk will notleak. Another source of viscosity has to act in the outerdisk, causing it bleed inward and supply the observedaccretion rates ˙ M . We now turn to the main subjectof this paper, the possibility that the magnetorotationalinstability is the source of viscosity in the outer disk.The magnetorotational instability (MRI) amplifiesmagnetic fields in outwardly shearing disks and drivesturbulence whose Maxwell stresses transport angular mo-mentum outward and mass inward (for a review, see Bal-bus 2009). Gas must be sufficiently well ionized for theMRI to operate. For the most part, T Tauri and Her-big Ae disks are too cold at their midplanes for ther-mal ionization to play a role there. The hope instead isthat X-rays emitted by host stars can provide the requi-site ionization in irradiated disk surface layers (Glassgoldet al. 1997). The basic picture was conceived by Gam-mie (1996), who proposed that disk surface layers ionizedby some non-thermal means may accrete, leaving behindmagnetically “dead” midplane gas. Like other workers(e.g., Bai & Goodman 2009, hereafter BG; and Turneret al. 2010, hereafter TCS), we focus in this study onionization of H by X-rays. Ionization of trace species byultraviolet (UV) radiation is also potentially important—we discuss this topic briefly at the close of our paper.The exposed rim of a transitional disk constitutes akind of surface layer. X-rays may penetrate the rim wall,activate the MRI there, and dislodge a certain radial col-umn of gas every diffusion time (Chiang & Murray-Clay2007, hereafter CMC). Within the MRI-active column,both the magnetic Reynolds number Re ≡ c s hD ≈ (cid:16) x e − (cid:17) (cid:18) T
100 K (cid:19) / (cid:16) a AU (cid:17) / (1)and the ion-neutral collision rate (normalized to the or- bital frequency) Am ≡ x i n H β in Ω ≈ (cid:16) x i − (cid:17) (cid:16) n H cm − (cid:17) (cid:16) a AU (cid:17) / (2)must be sufficiently large for magnetic fields to cou-ple well to the overwhelmingly neutral disk gas. Here T is the gas temperature, c s is the gas sound speed, h = c s / Ω is the gas scale height, Ω is the Kepler or-bital frequency, D = 234 ( T / K) / x − cm s − is themagnetic diffusivity, x e(i) is the fractional abundance ofelectrons (ions) by number, n H is the number densityof hydrogen molecules, β in ≈ . × − cm s − is thecollisional rate coefficient for ions to share their momen-tum with neutrals (Draine et al. 1983), and a is the diskradius.Dimensionless number (1) governs how well magneticfields couple to plasma, while (2) assesses how wellplasma couples to neutral gas. Both these numbers mustbe large for good coupling between magnetic fields andneutral gas. Numerical simulations have suggested crit-ical values Re ∗ of ∼ –10 (Fleming et al. 2000), de-pending on the initial field geometry, and Am ∗ of ∼ (Hawley & Stone 1998, hereafter HS). Some studies (e.g.,TCS) assumed Am ∗ ∼ Am ∗ may be 2 orders of magnitude higher (HS).The value of Am ∗ is critical to our work.For typical T Tauri parameters, CMC found active ra-dial column densities N ∗ ∼ × cm − or equivalentlymass columns of Σ ∗ ∼ − —essentially the stoppingcolumn for 3 keV X-rays. When they combined their de-rived value for N ∗ with an assumed value for the dimen-sionless disk viscosity α ∼ − , the accretion rates ofmany transitional systems were successfully reproduced.According to this model, the maximum accretion rate˙ M inside the hole is set by conditions at the rim wall,i.e., by how large a radial column N ∗ the MRI can drawfrom the rim. Stellar or planetary companions, knownor suspected to be present (Section 1.1), regulate howquickly this leaked material spirals onto the host star—these companions modulate the radial inflow speed v ( a )and thus the surface density Σ( a ) = ˙ M / (2 πva ). But thecompanions inside the hole do not initiate disk accretion.They may reduce ˙ M by exerting repulsive torques to keepmaterial in the rim wall from leaking in, or by accretingmaterial that flows past (e.g., Lubow & D’Angelo 2006;Najita et al. 2007). But they do not generate a non-zero˙ M in the first place. That fundamental task is left to theMRI operating at the rim—or whatever source of anoma-lous viscosity must be present in the outer disk to makeit bleed. The magnetic Reynolds number as we define it is notas accurate a predictor of MRI turbulence as the Elsasser(a.k.a. Lundquist) number, which is given by (1) with c s replacedby the vertical Alfv´en speed v A z . Self-consistent resistive MHDsimulations by Turner et al. (2007) found that MRI-active regionscoincide with Elsasser numbers greater than unity (see also Sano& Inutsuka 2001; Sano & Stone 2002). Our criterion Re (cid:38) –10 offsets some of the inaccuracy because v A z (cid:46) − c s in simulationsof MRI turbulence. In any case we will find that the limiting factorfor active surface layers is not Re but rather Am . Perez-Becker & Chiang
The Threat Posed by Polycyclic AromaticHydrocarbons to the MRI
One concern raised by CMC but left quantita-tively unaddressed is the degree to which ultra-smallcondensates—macromolecules whose sizes are measuredin angstroms—may thwart the MRI. In planetary atmo-spheres, aerosols can strongly damp electrical conduc-tivities (e.g., Schunk & Nagy 2004; Borucki & Whitten2008). Most studies of active layers neglect aerosolsand fixate on roughly micron-sized grains, despite thefact that in many particle size distributions, the small-est particles collectively present the greatest geometricsurface area and therefore the greatest cross section forelectron adsorption and ion recombination. Exceptionsinclude Sano et al. (2000), who in one model considereda grain size distribution extending down to 0.005 µ m =50 ˚ A , and BG, who considered grain sizes as small as0.01 µ m = 100 ˚ A . Both studies found that in principlesmall grains can be deadly to the MRI.Notwithstanding their possibly decisive role, smallgrains are sometimes wishfully dismissed as being de-pleted in number by grain growth, i.e., assimilated intolarger grains. Undeniably grains grow (Blum & Wurm2008; Chiang & Youdin 2010), so much so that their col-lective mass may be concentrated in particles millimetersin size. But the question relevant for ionization chemistryis not where the mass is weighted in the size spectrum ofparticles, but rather where the collective surface area forcharge neutralization is weighted. Determining the grainsize distribution in disks seems a problem that cannot beforward modeled with confidence. Sano et al. (2000) andBG instead parameterized the population of small grainsand studied the effects of varying their numbers, leav-ing undecided the question of whether their parameterchoices were favored by observation or theory.Like Sano et al. (2000) and BG, this paper consid-ers the effects of small condensates on the MRI. Whatis new about our contribution is that we consider thesmallest imaginable condensates that are still accessi-ble to observation: polycyclic aromatic hydrocarbons(PAHs). These molecules, typically containing severaldozens of carbon atoms, are excited electronically byultraviolet radiation and fluoresce vibrationally at 3.3,6.2, 7.7, 8.6, 11.3, and 12.7 µ m, the signature bands oftheir constituent C-C and C-H bonds (e.g., Li & Draine2001; Pendleton & Allamandola 2002). Spitzer satellitespectra and ground-based adaptive optics imaging revealPAHs to be fluorescing strongly in Herbig Ae/Be and TTauri disk surface layers directly exposed to stellar ul-traviolet radiation (Geers et al. 2006, 2007; Goto et al.2009). Thus PAHs help to constrain the aerosol abun-dance where magnetically driven accretion is thought tooccur: in disk surface layers.In this work we incorporate PAHs into a simple chemi-cal network to assess the proposal that X-ray driven MRIoperates in disk surface layers, either on the top and bot-tom faces of conventional hole-less disks, or at the rimsof transitional disks. We make as realistic an estimate as Some fire alarms work on this principle. A radioactive sourceinside the alarm drives ionization currents in the air which normallycomplete an electrical circuit. When smoke particles from a firereduce the density of free ions and electrons in air, the circuit isbroken and the alarm is triggered. we can of the PAH abundance based on observations, togauge how deep the X-ray-irradiated, MRI-active layermight actually be.To summarize this introduction: companions—eitherstars or a system of multiple planets, but not a singleJupiter-mass planet—can help clear the extensive holesof transitional disks. The outermost companion serves toestablish the location of the rim where viscous torques inthe disk and gravitational torques from the companionseek balance. If gas leaks inward from the outer disk, itis driven onto the host star so quickly by gravitationaltorques from companions that its optical depth may bereduced by orders of magnitude. Companions, togetherwith reductions in disk opacity by grain growth, thusmaintain the transparency of the hole while still permit-ting stars to accrete gas. But companions do not, inand of themselves, cause gas in the outer disk to dif-fuse inward. That responsibility may be reserved for theMRI—whose ability to operate despite the presence ofcharge-neutralizing PAHs is the subject of this paper.Our paper is organized as follows. The ingredients ofour numerical model for X-ray-driven ionization chem-istry in disk surface layers are laid out in Section 2.There we gauge what PAH abundances in disks may be.Results—principally, how Am and Re vary with the col-umn density penetrated by X-rays, and the extent towhich PAHs reduce these numbers—are presented in Sec-tion 3. Analytic interpretations of our numerical results,and direct comparison with previous calculations (BG,TCS), are given there as well. We discuss our main re-sults for X-ray driven MRI in Section 4, and close bydiscussing the possibility of UV-driven MRI. MODEL FOR DISK IONIZATION
In this paper we are interested in the degree to whichstellar X-rays and Galactic cosmic-rays can ionize H gasin T Tauri disks. In this respect our study is similarto many others, and we make direct comparisons of ourwork to BG and TCS in Section 3.4. For simplicity ourmodel neglects ionization of trace species like C and Sby stellar UV radiation. Omitting UV-driven chemistryrenders our model inconsistent because our model alsoincludes PAHs, whose abundances we constrain in Sec-tion 2.4.3 by using observed PAH emission lines excitedby stellar UV radiation. We will discuss the critical issueof UV ionization in Section 4.1. S i d e w a y s N, hN, X − r a y s h ≈ N/n H Σ= Nµ c o s m i c r a y s Fig. 1.—
X-ray ionized surface layers, located either on the topand bottom faces of a flared disk, or at its inner rim. Our calcu-lations apply to both situations, although they are more accuratefor the former. We assume in this work that the lengthscale (cid:96) overwhich gas is distributed radially at the hole rim is equal to h , thevertical gas scale height. Other sources of ionization are interstel-lar cosmic-rays and ultraviolet radiation from the star. At largestellocentric distances ( a (cid:38)
30 AU), cosmic rays may penetrate thedisk from the side. For most of our paper, we neglect ionizationof trace species by far ultraviolet radiation, but in Section 4.1 webriefly discuss this important topic.
AHs, MRI, and Planets 5
X-ray and Cosmic-ray Ionization Rates and GasDensitiesChandra spectra of pre-main-sequence stars in theOrion Nebula can be fitted by a pair of thermal plas-mas with characteristic temperatures kT X ∼ k is Boltzmann’s constant, and compara-ble luminosities L X ∼ –10 erg s − (Wolk et al.2005; Preibisch et al. 2005). The softer component isbelieved to be emitted by shock-heated accreting gas(Stelzer & Schmitt 2004), and the harder component by astrongly magnetized and active stellar corona (Wolk et al.2005). X-ray luminosities tend to increase with increas-ing stellar mass and decreasing accretion rate (see Fig-ure 1 of Telleschi et al. 2007, and Figure 17 of Preibischet al. 2005). These correlations are relevant for transi-tional disks because some transitional disks are hostedby higher mass Herbig Ae stars, and accretion rates fortransitional disks tend to be lower than for conventionaldisks (Najita et al. 2007). For our standard model wewill adopt L X = 10 erg s − , but we will also experi-ment with L X = 10 erg s − . We fix the temperatureof the X-ray emitting plasma at kT X = 3 keV, an as-sumption that ignores how X-ray spectra harden withincreasing L X (Preibisch et al. 2005). Although none ofour numerical models explicitly considers kT X > kT X = 8 keV) X-ray spectrum; we will see therethat the effects are not large. See Table 1 for a list ofall model parameters.We derive X-ray ionization rates ζ X as a function ofpenetration column N from Igea & Glassgold (1999,hereafter IG), who constructed a Monte Carlo radiativetransfer model that accounts for Compton scattering andphotoionization. Compton scattering enables X-ray pho-tons to penetrate to deeper columns than would other-wise be possible. For our standard model we use IG’sFigure 3 for a thermal plasma of L X = 10 erg s − and kT X = 3 keV. Their ionization rates were computed forstellocentric distances of 5 and 10 AU; we scale theserates to the stellocentric distances of our model, a = 3and 30 AU, using the geometric dilution factor a − . Wetest the accuracy of this approach by scaling their resultsinternally using this dilution factor, finding (by neces-sity) excellent agreement at low columns where materialis optically thin to X-rays, and agreement better than afactor of two at the highest columns calculated. For thecase L X = 10 erg s − , we increase all ionization ratesfrom our standard values by a factor of 100.Interstellar cosmic-rays can also ionize disk gas, but areattenuated by magnetized stellar winds blowing acrossdisk surface layers. Even the contemporary solar wind,characterized by a mass loss rate of ∼ − M (cid:12) yr − ,modulates the cosmic-ray flux at Earth by as much as ∼
10% with solar cycle (Marsh & Svensmark 2003). TTauri winds, having mass loss rates up to 5 orders of A minority of sources surveyed by
Chandra exhibited superhotX-ray flares with peak L X ∼ erg s − and kT X ∼
15 keV(Getman et al. 2008a; Getman et al. 2008b). The degree to whichsuperhot flares enhance ionization rates depends on the uncertainflare duty cycle. Because only ∼
10% of the
Chandra sources flaredonce or twice over a 15-day observing period, and because eachflare lasted less than ∼ magnitude higher than that of the solar wind today, seemlikely to shield disk surfaces from cosmic-rays directednormal to the disk plane (cf. Turner & Drake 2009).Nevertheless, cosmic-rays may reach disk gas from the“side,” striking the disk edge-on from the outside. At a = 3 AU we estimate that these “sideways cosmic-rays”are too strongly attenuated by intervening disk gas to besignificant. The same is not true on the outskirts of thedisk at a = 30 AU, where column densities measured ra-dially outward may be smaller than the cosmic-ray stop-ping column of 96 g cm − (Umebayashi & Nakano 1981).Thus we consider another model at a = 30 AU where inaddition to our standard X-ray source we include side-ways cosmic-rays with a constant, column-independentionization rate of ζ CR ∼ (1 / × − s − (Caselli et al.1998, hereafter C98). The factor of 1 / ζ = ζ X + ζ CR .In all our simulations we neglect ionization by energeticprotons emitted by the stars. As discussed by Turner &Drake (2009), estimates of the stellar proton flux rely onextrapolated scaling relations, and the ability of parti-cles to reach the disk surface in the face of strong stel-lar magnetic fields is uncertain. Moreover, protons areemitted in flares which may occur too infrequently tosustain disk ionization. In one of their models, Turner &Drake (2009) used a time-steady stellar particle luminos-ity whose ionization rate exceeded, by a factor of 40 at amass column of Σ = 8 g cm − , that of an X-ray sourcehaving L X = 2 × erg s − and kT X = 5 keV. Thismodel probably yields a hard upper limit on the stellarproton ionization rate, derived under a set of generousassumptions. Our L X = 10 erg s − case produces ion-ization rates ζ that approach those of the aforementionedmodel to within an order of magnitude. In any case wewill see in Section 3.3.1 how our results can be scaled toany ζ .Figure 1 depicts schematically how X-rays irradiatedisk surface layers, usually pictured in the vertical di-rection as ensheathing the disk on its top and bottomfaces. But in a transitional disk, a surface layer may alsobe present in the radial direction, at the rim of the centralhole. We consider each of these environments in turn, es-timating local number densities n H [H cm − ] from thecolumn density N [H cm − ] penetrated by X-rays. Surface Layers I: Top and Bottom Faces of aConventional Flared Disk
When considering the surface layers of a conven-tional non-transitional disk, we describe our results asa function of the vertical column density N of hydro-gen molecules, measured perpendicular to and towardthe disk midplane. Thus our N coincides with N ⊥ ofIG, save for a factor of 2 because IG count hydrogen nu-clei whereas we count hydrogen molecules. An equivalentmeasure of vertical column density N is the mass surfacedensity Σ ≡ N µ , where µ ≈ × − g is the meanmolecular weight of gas. In this paper, ionization rates ζ , column densities N , and frac-tional densities x are referred to hydrogen molecules, not hydrogennuclei. Perez-Becker & Chiang
TABLE 1Model parameters.
Parameter Variable Value ReferenceDisk radius a
3, 30 AU . . .X-ray source luminosity a L X (10 ) erg s − Section 2.1X-ray source temperature kT X ζ CR
0, (0 . × − ) s − Caselli et al. (1998)Initial CO abundance b x CO − Aikawa et al. (1996)Total metal abundance a,b x M − (0 , − ) Section 2.4.1Total grain abundance b x grain × − (cid:15) grain Section 2.4.2Grain settling (depletion) factor (cid:15) grain − ≤ (cid:15) grain ≤ − Section 2.4.2Total PAH abundance b x PAH − (cid:15) PAH
Section 2.4.3PAH depletion factor (cid:15)
PAH − ≤ (cid:15) PAH ≤ − Section 2.4.3Central stellar mass M ∗ M (cid:12) . . .Gas temperature T
80, 30 K Section 2.2 a Values in parentheses correspond to test cases different from our standard model. b All abundances are relative to H by number. To good approximation, the local number density n H ≈ N/h (3)where the vertical scale height h = c s / Ω = ( kT /µ ) / / Ω.For gas temperature T ≈ a = 3(30) AU (seeSection 2.2 for how we derive these temperatures), wefind h = 0 .
09 (1 .
8) AU.Equation (3) underpins all our calculations of chemicalequilibrium. For a typical N ∼ H cm − , we have n ∼ × (4 × ) H cm − at a = 3 (30) AU. Surface Layers II: Gap Rim of Transitional Disk
We assume that the gap rim is not shadowed fromthe star by gas interior to the rim. We cannot provethat the rim is not shadowed, but disk models based onthe infrared SED suggest it is not (e.g., Calvet et al.2005). Possibly gas at the rim wall “puffs up” becauseit is heated by X-rays and can maintain a larger verticalheight than gas inside the hole (e.g., Dullemond et al.2001).For the case of the rim of a transitional disk, we reinter-pret N (equivalently Σ) as the radial column of hydrogenmolecules traversed by X-rays (Figure 1). To estimatethe local number density n H , we need to know the ra-dial lengthscale (cid:96) over which material at the rim wall isdistributed. Plausibly h (cid:46) (cid:96) (cid:46) a . Chiang & Murray-Clay (2007) take (cid:96) ∼ a , but models based on SEDs andimages suggest the rim is much sharper. Here we assumethat (cid:96) ∼ h so that Equation (3) applies equally well totransitional disks as to conventional disks—keeping inmind that N should be measured radially for the formerand vertically for the latter.To calculate ionization rates in the rim, we still usethe results of IG, reinterpreting their N ⊥ in their Figure3 as our radial column N . Clearly the scattering geom-etry differs between the case of a transitional disk rimand the case of the top and bottom faces of IG’s con-ventional disk. Where material is optically thin to stel-lar X-rays, the two cases match in ionization rate (permolecule), but where it is optically thick, we underes-timate the ionization rate in transitional disk rims byusing IG because more X-rays escape by scattering ver-tically out of conventional disk surface layers than fromthe rim. Another reason we underestimate the ionizationrate at high column density is because the total X-ray flux per unit surface area of the disk is lower for conven-tional surface layers—which are illuminated at grazingincidence—than for the rim, which is illuminated at nor-mal incidence. Nevertheless we estimate that these er-rors are of the order of unity, insofar as the columns thatmight possibly be MRI-active are not too optically thickto X-rays (Section 3.3), because the Thomson scatteringphase function is fairly isotropic, and the cross-sectionfor scattering is only comparable to that for photoion-ization at the relevant photon energies. In any case wewill explore the effects of higher ionization rates by run-ning a model with higher L X = 10 erg s − (Section3.3.1). Gas Temperature
The surface layers of protoplanetary disk atmospheresvary widely in temperature, from ∼ (cid:46) a = 1 AU at column densities of interest ( N (cid:38) cm − ), thermal balance is controlled primarily by repro-cessing of starlight by dust, and gas and dust tempera-tures are nearly equal at ∼
130 K (Glassgold et al. 2004,their Figure 2). We adjust this result for the disk radii ofour standard model using the dust temperature scalinglaw for the midplane of a passive flared disk, T ∝ a − / (e.g., Chiang & Goldreich 1997). Thus at a = 3 AU wehave T = 80 K, and at a = 30 AU we have T = 30 K.Note that these temperatures are lower—and arguablymore realistic—than those assumed by BG and TCS, whoinvoked temperatures of the traditional Hayashi nebulawithout justification. Chemical Network
Following Ilgner & Nelson (2006, hereafter IN), CMC,and BG, we apply a simple network of chemical reactionsbased on that designed for molecular clouds by Oppen-heimer & Dalgarno (1974, hereafter OD). Ilgner & Nel-son (2006) and BG compared the results of OD-basedschemes to those of more complex networks extractedfrom the UMIST (University of Manchester Institute ofScience and Technology; Woodall et al. 2007, hereafterAHs, MRI, and Planets 7W07; Vasyunin et al. 2008) database. Fractional elec-tron abundances derived by IN using the simple networkwere greater than those derived using the complex net-work, whereas BG, who used a more recent version of theUMIST database, found that the sign of the differencevaried from case to case. The magnitude of the differenceranged up to a factor of 10, but was often (cid:46)
3. Using thesimple network seems the most practical approach, if weare content with order-of-magnitude answers. In Section3.4, we test the results of our code against those of BGand TCS.All reactions in our OD-based network are listed inTable 2 and shown schematically in Figure 2. Rate co-efficients and their temperature dependences are takenfrom the UMIST database. The chain of events basi-cally proceeds as follows. X-rays ionize H to H +2 , whichrapidly reacts with H to produce H +3 . The H +3 ion com-bines with CO to form HCO + . Most HCO + ions disso-ciatively recombine with free electrons, but some transfertheir charge to gas-phase metal atoms such as Mg. Freemetals tend to be abundant positive charge carriers, asthey recombine with free electrons only by a slow radia-tive channel. Charged particles in the network (e − , H +3 ,HCO + , metal + ) can neutralize by collisionally transfer-ring their charge to PAHs and grains. The collisionalcharging process is described in Section 2.5.The reaction loop is closed by the formation of H ongrain surfaces. To compute the rate of this reaction, wetake neutral H atoms to collide with grains using the ge-ometrical cross section for grains, and adopt from BGthe uniform probability η = 10 − for a pair of adsorbedhydrogen atoms to form a hydrogen molecule (see theirEquation 27). The precise rate of this reaction is not im-portant for us, as it only sets the equilibrium abundanceof H, which is irrelevant for the ionization fraction, aslong as n H (cid:28) n H .In their original study OD included ionization of Heand reactions involving atomic and molecular oxygen.We neglect these for simplicity. Most reactions involvingoxygen initiate with the formation of the hydroxyl ion(H +3 + O → OH + ), which proceeds at a rate only com-parable to the formation of HCO + , which we do accountfor. Thus our neglect of oxygen within the OD frame-work is not expected to alter our results for the fractionalionization by more than a factor of 2. In any case, in Sec-tion 3.4 we will compare our results with those of morecomplex networks considered by BG and TCS. Properties and Abundances of Trace Species
The trace ingredients of our model include gas-phasemetals (Section 2.4.1), a monodispersion of micron-sizedgrains (Section 2.4.2), and PAHs (Section 2.4.3).
Gas-phase Metals (Magnesium)
For gas-phase metals which serve importantly as elec-tron donors (OD; Fromang et al. 2002), we are guidedby Mg, whose solar abundance is 3 . × − atoms perhydrogen nucleus (Lodders 2003). The fraction of Mgthat is in the gas phase—neither incorporated into graininteriors nor adsorbed onto grain surfaces—might be atmost 3–30% by number, its value in the diffuse interstel-lar medium (Jenkins 2009). In the dense environmentsof protoplanetary disks, the gas-phase fraction should be much smaller because magnesium is used toward buildinggrains.Nominally, our model temperatures of 30–80 K are solow that almost all of the Mg not incorporated into graininteriors should be adsorbed onto grain surfaces, leavingbehind only a tiny fraction in the gas phase (Turner et al.2007, their Section 2.2; see also Equation 26 of BG). Justhow tiny is uncertain, given how sensitive the adsorptionfraction is to gas temperature, and how steep temper-ature gradients can be in disk surface layers (Glassgoldet al. 2004). Turbulent mixing of hot, high altitude, nor-mally metal-rich layers with cold, low altitude, normallymetal-poor layers can also complicate matters (Turneret al. 2007; TCS).We adopt a standard metal abundance of x M = 10 − metal atoms per H , which corresponds to a gas-phasefraction of ∼ − by number relative to solar. Our choiceis similar to those of IN and BG. We also experimentwith a metal-free case in which all metals have been ad-sorbed onto grain surfaces ( x M = 0), and a metal-richcase for which x M = 10 − . Although the metal-rich caseis not especially realistic and is not justified by our modelparameters—in particular our low gas temperatures—weconsider it anyway because we would like to understandthe effects of metals in principle, and to connect withother studies that consider similarly large metal abun-dances (CMC; Turner et al. 2007; TCS). Grains
The number of grains per H molecule is x grain = µ πs ρ s ρ dust ρ gas , (4)where µ ≈ × − g is the mean molecular weightof gas and ρ dust /ρ gas is the dust-to-gas mass ratio. Forsimplicity we consider grains of a single radius s = 1 µ mand internal density ρ s = 2 g cm − . There is ample evi-dence that micron-sized grains abound in surface layers,both from mid-infrared spectra of silicate emission lines(e.g., Natta et al. 2007) and from scattered light imagesat similar wavelengths (e.g., McCabe et al. 2003).The dust-to-gas ratio in surface layers may differ con-siderably from its value in the well-mixed diffuse inter-stellar medium (ISM): ρ dust ρ gas ≡ (cid:15) grain ρ dust ρ gas (cid:12)(cid:12)(cid:12)(cid:12) ISM , (5)where for the ISM of solar abundance ρ dust /ρ gas | ISM =0 .
015 (Lodders 2003). Based on model fits to observedfar-infrared SEDs (Chiang et al. 2001; D’Alessio et al.2006; Dullemond & Dominik 2004), there is consensusthat surface layer grains directly illuminated by opti-cal light from their host stars have settled toward themidplane into regions of denser gas. Thus (cid:15) grain < (cid:15) grain by a factor of 10.Table 3 lists fitted values of (cid:15) grain for some transitionaldisks, drawn from the literature. At best they are ac- Perez-Becker & Chiang TABLE 2Chemical Reactions Including Collisional Charging of PAHs and Grains.
Number Reaction Rate Coefficient a Value Reference1 b H + hν → H +2 + e − ζ X Taken from radiative transfer model IG2 H + Cosmic-ray → H +2 + e − ζ CR (1 / × − s − C983 H +2 + H → H +3 + H α H +2 , H × − W074 H +3 + CO → HCO + + H α H +3 , CO × − W075 c M + H +3 → M + +H + H α M , X + × − W076 M + HCO + → M + + H + CO α M , X + × − W077 H +3 + e − → H + H α H +3 , e × − ( T/ − . W078 HCO + + e − → H + CO α HCO + , e × − ( T/ − . W079 M + + e − → M + hν α M + , e × − ( T/ − . W0710 PAH(Z) + e − → PAH(Z − α PAH , e Section 2.5 DS11 d PAH(Z) + X + → PAH(Z+1) α PAH , X + Section 2.5 DS12 grain(Z) + e − → grain(Z − α grain , e Section 2.5 DS13 grain(Z) + X + → grain(Z+1) α grain , X + Section 2.5 DS14 PAH(Z= −
1) + PAH(Z=1) → × PAH(Z=0) α PAH , PAH
Section 2.5 DS15 H + H + grain → H + grain α physisorption Section 2.3 BG a ζ has units of s − α has units of cm3s − b hν denotes a photon. c M represents a gas-phase atomic metal, e.g., Mg. d X+ can be either H+3 , HCO+, or M+. curate to order of magnitude. For our calculations weconsider 10 − ≤ (cid:15) grain ≤ − (Table 1). TABLE 3Dust Settling Parameter (cid:15) grain for Some Transitional Disks
Source (cid:15) grain
ReferenceLkCa 15 10 − Espaillat et al. (2007a); Chiang et al. (2001)UX Tau A 10 − Espaillat et al. (2007a)CS Cha 10 − Espaillat et al. (2007b)GM Aur 10 − Calvet et al. (2005)DM Tau 10 − Calvet et al. (2005)
PAHs
For simplicity we model PAHs as spheres, each havinga radius s = 6˚ A and internal density ρ s = 2 g cm − .Although in reality carbon atoms in PAHs are arrangedin sheets and not spheres (e.g., Allamandola et al. 1999),the difference in cross section arising from geometry isonly on the order of unity. Each of our model PAHs hasabout as much mass as a real PAH containing N C = 100carbon atoms. A PAH of this size is estimated to be justlarge enough to survive photo-destruction around HerbigAe stars (Visser et al. 2007).The central wavelengths of PAH emission lines fromHerbig Ae/Be (HAe/Be) disks are observed to trendwith the effective temperatures of their host stars (Sloanet al. 2005; Keller et al. 2008). This correlation indi-cates that PAHs in disks are not merely PAHs from thediffuse ISM transported unadulterated into circumstellarenvironments. Rather, PAHs in disks have been photo-processed, their chemical bonds altered by radiation fromhost stars. Possibly PAHs are continuously created anddestroyed by local processes, e.g., sublimation of grainmantles and photodestruction (Keller et al. 2008). In thispaper we do not account explicitly for such processes, i.e.,we do not attempt to calculate the abundance of PAHsfrom first principles. Rather we fix the abundance of PAHs using observations, as detailed in the remainder ofthis subsection.Emission from PAHs is detected in an order-unity frac-tion of HAe/Be stars, but is rarely seen in T Tauristars (e.g., Geers et al. 2006, hereafter G06). In prin-ciple this could mean that PAHs are less abundant in TTauri disks, but the more likely explanation is that thisis an observational selection effect: Herbig Ae/Be starsare more luminous in the ultraviolet (UV) and thereforecause their associated PAHs to fluoresce more strongly(see, e.g., Figure 9 of G06, which shows how the PAH in-tensity drops below the Spitzer detection threshold withdecreasing stellar effective temperature). Another cluethat PAHs are just as abundant in T Tauri disks as inHAe/Be disks is that those few T Tauri stars with pos-itive PAH detections tend to have unusually low mid-infrared continua, allowing PAH emission lines to standout more clearly (G06). In other words, those T Taurisystems where PAHs have been detected are transitionalsystems, and their PAH abundances seem no differentthan in their HAe/Be counterparts.Geers et al. (2006) used radiative transfer models tofit the intensities of the 11.2 µ m PAH fluorescence linein eight Herbig Ae and T Tauri disks, concluding thatthe PAH abundance is 10 − –10 − per H (see their Fig-ure 9). This result is highly model dependent. Perhapsthe chief source of uncertainty lies in the grain opacity.Inferred PAH abundances relative to gas are sensitiveto assumptions about the local grain size distributionand dust-to-gas ratio because the soft ultraviolet radia-tion ( ∼ A wavelength) which causes PAHs tofluoresce is also absorbed by ambient grains. Thus theintensity of PAH emission depends on how many grainsare competing with PAHs for the same illuminating pho-tons. The grain opacity, in turn, decreases by orders ofmagnitude as dust settles (Section 2.4.2). Because G06did not account for dust sedimentation and instead as-sumed the dust-to-gas ratio in disks was similar to thatof the well-mixed ISM, the PAH abundances relative toAHs, MRI, and Planets 9 M ++ H e − e − e − e − + HCO grains e − H M PAHs +2 H H z grains z CO grain + H + H → grain + H H +3 + CO → HCO + + H α = 2 × − cm s − HCO + + e − → H + COH +3 + e − → H + H α = 2 × − cm s − cm s − α = 6 × − cm s − M + X + → M + + X α = 9 × − α ∼ − cm s − α = 5 × − cm s − H +2 + H → H +3 + HM + + e − → M H + h ν → H +2 + e − Fig. 2.—
Our chemical reaction network, derived from Oppenheimer & Dalgarno (1974). Rate coefficients as shown in this Figure areevaluated at T = 80 K. See Table 2 for a comprehensive list of all modeled reactions and precise rate coefficients. − –10 − per H ,the ratio of PAH line intensity to dust continuum wouldbe larger than observed (Dullemond et al. 2007).In our model the number of PAHs per H is x PAH ≡ (cid:15) PAH × − , (6)where (cid:15) PAH < (cid:15) grain ∼ . − (cid:46) (cid:15) PAH (cid:46) − (Table 1). In our calculations we select the pa-rameter combinations ( (cid:15) grain , (cid:15) PAH ) = (10 − , − ) and( (cid:15) grain , (cid:15) PAH ) = (10 − , − ) which bracket the range ofpossibilities.A final point to consider is whether the observed PAHsare present at the same column depths that are rele-vant for X-ray driven MRI. The X-ray stopping columnshould be compared with the column that presents op-tical depth unity to the soft UV radiation driving PAHemission. In the model of G06 in which dust has not set-tled, photons at wavelengths of 1000–3000 ˚ A are stoppedby submicron-sized silicate/carbonaceous grains withina hydrogen column of ∼ − (V. Geers, privatecommunication, 2010; see also Habart et al. 2004 whoused similar dust opacities). After we account for grainsettling ( (cid:15) grain ), the UV absorption column increases to ∼ − . Although model-dependent, our esti-mate of the UV absorption column corresponds well toX-ray stopping columns, and thus to columns that mightpossibly be MRI-active. Collisional Charging of PAHs and Grains
Grains and PAHs are modeled as conducting spheresfor simplicity. Electrons and ions collide with and stickto grains and PAHs, charging them. When the total elec-tron capture rate by grains and PAHs matches the totalion capture rate, the distribution of charges carried byPAHs and grains reaches dynamical equilibrium. Theaverage charge state on a PAH/grain (cid:104) Z (cid:105) < (cid:104) Z (cid:105) in various limits were given by Draine & Sutin (1987,hereafter DS). We will find that (cid:104) Z (cid:105) ranges between − A , while for our micron-sized grains (cid:104) Z (cid:105) ≈ −
22. The remainder of this subsec-tion details how we compute the electron and ion capturerates.The rates at which ions or electrons collide with PAHsor grains are enhanced by Coulomb focusing betweenstatic charges, as well as by the induced dipole force(Natanson 1960; Robertson & Sternovsky 2008). Thecross sections can be derived from kinetic theory by con-sidering the potential between a conducting sphere ofradius s and charge Ze , located at a distance r from acharge q : φ ( Z, r ) = qZer − q s r ( r − s ) (7) (e.g., Jackson 1975). The first term is the usual monopoleinteraction, while the second arises from the induceddipole (image charges). For a neutral sphere, thevelocity-dependent cross section derives from applyingconservation of energy and momentum to the secondterm of (7). Multiplying this cross section by either theelectron or ion velocity, and averaging over a Maxwellianspeed distribution at temperature T , yields the rate co-efficient (units of cm s − ) α = πs Sc (cid:32) (cid:114) πq skT (cid:33) for Ze/q = 0 (8)where k is the Boltzmann constant, c = (cid:112) kT /πm isthe mean speed for either electrons of mass m = m e orions of mass m = m X + , and S is the probability that theelectron/ion sticks to the PAH/grain. We will discussthe sticking coefficient S shortly.For a charged sphere, the cross section is enhanced byboth terms in (7). There is no analytical solution forthis case, but DS provided the following approximateformulae: α = πs Sc [1 − Ze/ ( qτ )](1+ (cid:112) / ( τ − Ze/q )) for
Ze/q < α = πs Sc [1+(4 τ +3 Ze/q ) − / ] exp( − β/τ ) for Ze/q > τ ≡ skT /q , β ≡ Zeqg − g ( g − , (11)and g is the solution to the transcendental equation2 g − g ( g − = Zeq . (12)Upon colliding with a PAH or grain, the electron or ionsticks with probability S . For ions, we set S = S X + = 1(DS; IN; BG). For electrons colliding with PAHs, S = S e depends on the detailed molecular structure of the PAH.Allamandola et al. (1989) calculated how the electronsticking coefficient increases with both the number ofcarbon atoms and the electron affinity. The dependenceon electron affinity is especially strong. A PAH hav-ing N C = 32 and an electron affinity of 0.7 eV has S e ≈ × − (see their Figure 25), while the same-sizedPAH with an electron affinity of 1 eV has S e ≈ − (seepage 769 of their paper). Estimated electron affinitiesof real N C = 32 PAHs (e.g., ovalene and hexabenzo-coronene) exceed 1 eV. Allamandola et al. (1989) statedthat “only for pericondensed PAHs [which are more sta-ble than catacondensed PAHs] containing considerablymore than 20 C atoms will the electron sticking coeffi-cient approach unity.” Based on these considerations,we take S e = 0 . N C = 100 PAHs. For the muchlarger grains we set S e = 1.In our code, the range of charges a grain can possessextends from Z = −
200 to +200. We have verified thatthis range is large enough to accommodate the entireequilibrium charge distribution, which for our µ m-sizedgrains peaks at −
22 (see Figure 4). Accounting only for afew charges—up to | Z | = 3 as did Sano et al. (2000), IN,BG, and TCS—is not necessarily adequate for micron-sized grains which have fairly large capacitances. ForAHs, MRI, and Planets 11PAHs we consider charges Z between −
16 and +16. MostPAHs will turn out to have either Z = 0 or −
1. Becauseof their smaller size, a single PAH will be less chargedthan a single grain; electrons collide less frequently witha negatively charged sphere as the radius of the spheredecreases and the Coulomb potential steepens.We neglect adsorption of neutral gas-phase species ontograin surfaces, and any mass increase of grains and PAHsfrom collisions with ions. Grain-grain and PAH-grain col-lisions are negligible and ignored. We do account for thepossibility that a PAH with a single negative charge canneutralize by colliding with a PAH with a single positivecharge (reaction 14 in Table 2), though in practice thisreaction is not significant.
Numerical Method of Solution
The time-dependent rate equations for the abundancesof species are readily constructed from the reactionslisted in Table 2. For example, the number density ofelectrons n e obeys dn e dt = n H ζ − n e (cid:88) X + α X + , e n X + − n e 16 (cid:88) Z = − α PAH , e n PAH − n e 200 (cid:88) Z = − α grain , e n grain (13)where the index X + runs over reactions 7, 8, and 9 inTable 2. The first sum over Z occurs over the chargestates of PAHs, while the second sum occurs over thecharge states of grains, with rate coefficients α given inSection 2.5.The charge distributions of PAHs and grains aregoverned by recurrence equations (Parthasarathy 1976;Whitten et al. 2007), e.g., for PAHs: dn PAH ,Z dt = ( n PAH α PAH , e ) Z +1 n e + (cid:88) X + (cid:0) n PAH α PAH , X + (cid:1) Z − n X + − ( n PAH α PAH , e ) Z n e − (cid:88) X + (cid:0) n PAH α PAH , X + (cid:1) Z n X + . (14)The right-hand side of Equation (14) accounts for allthe ways in which PAHs of charge Z can be created ordestroyed by collisions with electrons and ions (reactions10 and 11 in Table 2). When Z = ±
1, Equation (14)is supplemented by an extra loss term accounting forreaction 14.All rate equations are discretized to first order and ad-vanced simultaneously using a forward Euler algorithmwith a fixed timestep ∆ t ≤ × − s. At t = 0, all PAHsand grains have Z = 0 and all hydrogen is in the formof H . In principle we could simply advance the networkforward until the system equilibrates, i.e., until the timerates of change of the abundances fall below some speci-fied tolerance. However the reaction rates in our network span almost 5 orders of magnitude. Thus, our equationsare stiff and a brute-force integration would require aninordinate number of timesteps. Metals are typically theslowest constituent to reach equilibrium because they re-act with electrons only slowly by radiative recombination(reaction 9).To circumvent the bottleneck posed by metals, we pro-ceed as follows. We run R versions of the code having R evenly spaced initial abundances for charged metals n M + .For each run, we initially set n e = n M + to ensure chargeneutrality. We run each code until the abundances of allspecies drift only because of slow changes in n M + . Weevaluate dn M + /dt at the end of each run. The equilib-rium value of n M + is bracketed by the two runs havingopposing signs for dn M + /dt . We then start a new itera-tion with R runs having initial metal abundances evenlyspaced between the two bounding runs of the previous it-eration. In this way we refine our initial guesses for n M + until we arrive at two sets of initial conditions that differby less than 30%. The equilibrium value of n M + we re-port lies at the intersection of the two curves for n M + ( t ),linearly extrapolated forward in time. Other variables( n e , n HCO + , and n H +3 ) are also extrapolated. The num-ber of runs R at each iteration varies from 2 to 5.We use the time t eq at which the two extrapolatedcurves for n M + intersect as an estimator of the equili-bration time of the chemical network. For t eq to be arobust estimator, it should be independent of initial con-ditions. We found that the value of t eq remained constantto within a factor of 3 when initial conditions varied over2 orders of magnitude. Our values for t eq will be com-pared to dynamical timescales Ω − in Section 3.3.2.Errors are estimated by monitoring conservation ofcharge and conservation of the total number density ofPAHs + grains. Over 10 timesteps, the charge re-mains constant (at zero) to better than one part in 10 ,with similar results for the number density of PAHs +grains. As a test of our code, we reproduced the normal-ized charge distribution on PAHs computed by Jensen &Thomas (1991, see their Figure 2a). RESULTS
In Section 3.1, we describe how charges distributethemselves on PAHs and grains in dynamical equilib-rium. In Section 3.2, we explore how the free electronand ion abundances vary with increasing PAH abun-dance. In Section 3.3, we show what all this implies forthe degree of magnetic coupling in disk surface layers, in-terpreting our numerical results whenever possible withsimple analytic estimates. In that section we also com-pute timescales for the chemical network to equilibrate,and compare to the dynamical timescales over which theMRI may act. In Section 3.4, we test the validity of oursimple network/code by seeing how closely we can repro-duce the results of more complex networks/codes by BGand TCS.
Charge Distributions on PAHs and Grains
Figures 3 and 4 show the charge distributions onPAHs and grains, respectively, for the case a = 3 AU,Σ = 0 . − , x M = 10 − (standard metal abundance), (cid:15) PAH = 10 − (low PAH abundance), and (cid:15) grain = 10 − (low grain abundance). Most of the PAHs either have2 Perez-Becker & Chiang Z PAH = 0 or Z PAH = −
1. For grains, the averagecharge state (the peak of the distribution) is (cid:104) Z grain (cid:105) ≈−
22. The shape of the charge distribution for grains ap-proaches the Gaussian given by Equation (4.15) of DS.We may understand (cid:104) Z (cid:105) simply. Consider the PAHs;identical considerations apply to grains. We take thelimit that the dominant ions are metals and the limitthat the total charge carried by PAHs is much less thanthe free charge. Together these limits imply that x e ≈ x M + . Then detailed balance between forward and reversereaction rates dictates that (cf. Equation 14):( n PAH α PAH , e ) Z +1 = (cid:0) n PAH α PAH , M + (cid:1) Z . (15)From this equation it is evident that if ever the rate co-efficients ( α PAH , e ) Z +1 and ( α PAH , M + ) Z were to be equal,the densities ( n PAH ) Z +1 and ( n PAH ) Z would be equal,i.e., the charge distribution would be at an extremum.Thus we may estimate the average charge (cid:104) Z (cid:105) by merelyplotting the rate coefficients α PAH , M + and α PAH , e against Z and seeing where the curves intersect. This exerciseis performed in Figures 3 and 4. Indeed what the fullnumerical model gives for (cid:104) Z (cid:105) is close to the Z for whichthe curves for the rate coefficients intersect. (Of course,perfect agreement cannot be obtained because it is neverstrictly true that ( α PAH , e ) Z +1 = ( α PAH , M + ) Z .)There is another, even simpler limit where (cid:104) Z (cid:105) may beestimated. In the extreme case that the gas is so satu-rated with grains or PAHs that practically no free chargesare left, we must have (cid:104) Z (cid:105) →
0. Figure 5 shows the re-sults of an experiment using our full code in which we in-crease (cid:15)
PAH until this regime is reached. For this Figure,the grain abundance is set to zero to isolate the effects ofPAHs. Figure 6 is analogous; (cid:15) grain is increased while thePAH abundance is held fixed at zero. Both figures followthe transition from (cid:104) Z (cid:105) (cid:54) = 0 to (cid:104) Z (cid:105) →
0. Observationallyinferred values for (cid:15) grain (see the shaded region of Figure6) are never so high as to cross into the (cid:104) Z grain (cid:105) → x (cid:63) PAH dividing the (cid:104) Z PAH (cid:105) (cid:54) = 0 limit from the (cid:104) Z PAH (cid:105) →
Ionization Fraction vs. PAH Abundance
Figure 7 plots the fractional electron and ion densi-ties, x e and x i , against the PAH abundance x PAH , for a = 3 AU, Σ = 0 . − , x M = 10 − (standard metalabundance), and (cid:15) grain = 10 − –10 − (see the figure cap-tion for how (cid:15) grain is assigned to each (cid:15) PAH ). Figure 8is identical except that it considers the metal-rich case x M = 10 − . The primary ions in both cases are atomicmetals and HCO + molecules. At low PAH abundances,charged metal ions are the most abundant. As the num-ber of PAHs is increased, HCO + becomes the dominantion. See Table 4 for a precise breakdown of componention densities for our standard metal abundance case.According to Figures 7 and 8, the electron and ion den-sities are nearly equal and constant with x PAH as longas x PAH is not too large. In going from the standardmetal abundance of x M = 10 − to the metal-rich caseof x M = 10 − , the free charge abundance increases by − − − − − − α P A H , e o r α P A H , M + [ c m s − ] − − − − − N o r m a li z e d c h a r g e d i s t r i bu t i o n − − − Z PAHs C h a r g e d i s t r i bu t i o n α P A H , e α PAH , M + Fig. 3.—
Equilibrium charge distribution on PAHs (solid cir-cles, left axis) for a = 3 AU, Σ = 0 . − , x M = 10 − (stan-dard metal abundance), (cid:15) PAH = 10 − (low PAH abundance), and (cid:15) grain = 10 − (low grain abundance). The distribution peaks at Z = 0, approximately where the attachment coefficients (dashedlines, right axis) for electrons with PAHs and metal ions with PAHscross. − − − α g r a i n , e o r α g r a i n , M + [ c m s − ] − − − − − N o r m a li z e d c h a r g e d i s t r i bu t i o n − − − −
10 0 Z grains Chargedistribution α g r a i n , e α g r a i n , M + Fig. 4.—
Same as Figure 3 but for grains. an order of magnitude. Once x PAH exceeds some criticalabundance x (cid:63) PAH , the electron and ion densities diverge—the ion density is higher, and the balance of negativecharges is carried by PAHs. In the limit x PAH (cid:29) x (cid:63) PAH ,both the electron and ion densities decrease with increas-ing PAH abundance in an approximately inverse linearway. All of this behavior can be understood analyticallyas follows.
Analytical Model for Ionization Fraction vs. PAHAbundance
Our code’s results for x e ( x PAH ), x i ( x PAH ), and x (cid:63) PAH may be understood using the following simple model.The model consists only of X-rays, molecular hydrogen,electrons, PAHs, and one ion species—either HCO + forAHs, MRI, and Planets 13 TABLE 4Densities of Charged Species for our Standard Model ( x M = 10 − , L X = 10 erg s − ) a = 3 AU , (cid:15) PAH = 10 − , (cid:15) grain = 10 − Σ n H n e n M + n HCO + n H +3 x e x i x PAH x grain (cid:104) Z PAH (cid:105) (cid:104) Z grain (cid:105) × − × × × × × − × − × − × − × − − × − − × − × × × × × − × − × − × − × − − × − − × − × × × × × − × − × − × − × − − × − − × × × × × − × − × − × − × − × − − × − − × × × − × × − × − × − × − × − × − − × − − × × × − × − × − × − × − × − × − × − − × − − a = 3 AU , (cid:15) PAH = 10 − , (cid:15) grain = 10 − Σ n H n e n M + n HCO + n H +3 x e x i x PAH x grain (cid:104) Z PAH (cid:105) (cid:104) Z grain (cid:105) × − × × × − × × − × − × − × − × − − × − − × − × × × × × − × − × − × − × − − × − − × − × × − × − × × − × − × − × − × − − × − − × × × − × − × − × − × − × − × − × − − × − − × × × − × − × − × − × − × − × − × − − × − − × × × − × − × − × − × − × − × − × − − × − − × − a = 30 AU , (cid:15) PAH = 10 − , (cid:15) grain = 10 − Σ n H n e n M + n HCO + n H +3 x e x i x PAH x grain (cid:104) Z PAH (cid:105) (cid:104) Z grain (cid:105) × − × × − × − × − × − × − × − × − × − − × − − × − × × − × − × − × − × − × − × − × − − × − − × − × × − × − × − × − × − × − × − × − − × − − × × × − × − × − × − × − × − × − × − − × − − × × × − × − × − × − × − × − × − × − − × − − × × × − × − × − × − × − × − × − × − − × − − a = 30 AU , (cid:15) PAH = 10 − , (cid:15) grain = 10 − Σ n H n e n M + n HCO + n H +3 x e x i x PAH x grain (cid:104) Z PAH (cid:105) (cid:104) Z grain (cid:105) × − × × − × − × − × − × − × − × − × − − × − − × − × × − × − × − × − × − × − × − × − − × − − × − × × − × − × − × − × − × − × − × − − × − − × × × − × − × − × − × − × − × − × − − × − − × × × − × − × − × − × − × − × − × − − × − − × × × − × − × − × − × − × − × − × − − × − − × − Note . — The surface density Σ has units of g cm −
2; number densities n have units of cm −
3; and fractional densities x are measured per H2. The ion density x i = x M+ + x HCO+. our standard metal abundance case, or ionized metalsM + for the metal-rich case. In the simplified model, X-ray ionization of a hydrogen molecule produces a freeelectron and—skipping the entire reaction chain—oneion. The system reduces to the rate equations dn e dt = ζn H − n e n i α i , e − n e x PAH n H ( α PAH , e ) (cid:104) Z (cid:105) (16) dn i dt = ζn H − n i n e α i , e − n i x PAH n H ( α PAH , i ) (cid:104) Z (cid:105) (17)for the electron and ion densities, n e and n i . The sub-script i denotes either HCO + or M + .In the simplified model, all PAHs with abundance x PAH n H are assumed to be identically charged. We setthis common charge equal to the average charge state (cid:104) Z (cid:105) , results for which were given in Section 3.1.We exclude our large, micron-sized grains from the an-alytic model. Although these grains were useful for infer- ring PAH abundances from observations (Section 2.4.3),their collective surface area is too low to significantly in-fluence the electron chemistry in any of our model runs.Of course, because PAHs and grains are both modeledthe same way, i.e., as spherical conductors, all of theequations below would still be valid were we to replacePAHs with grains. Standard metal abundance. — As stated above, we assumefor this case that all ions are HCO + molecules and neglectM + . We may solve for x e and x i = x HCO + in the limits oflow and high PAH abundance. In the limit of low x PAH ,the rightmost terms in Equations (16) and (17) can be4 Perez-Becker & Chiang − − − − − x e o r x i − h Z P A H i − − − − PossiblePAHabundances h Z PAH i x e x i − − (cid:15) PAH − x PAH
Fig. 5.—
Average charge state of PAHs as a function of PAHabundance (solid diamonds, left axis). Dashed lines show simu-lation results for fractional electron abundance x e (solid circles,right axis) and fractional ion abundance x i (open squares, rightaxis). The shaded region marks observationally inferred PAHabundances, measured by number either relative to H ( x PAH ,bottom axis) or relative to the PAH abundance in the diffuse ISM(depletion factor (cid:15)
PAH ≡ x PAH / − , top axis). Parameters forthis run are a = 3 AU, x M = 10 − , Σ = 0 . − , and (cid:15) grain = 0(kept at zero to isolate the effect of PAHs). The shift to (cid:104) Z PAH (cid:105) = 0occurs when there are so many PAHs that they begin to adsorbmost of the free charge. At this point x i and x e diverge; see alsoFigure 7. − − − − − − x e o r x i − − − h Z g r a i n i − − − − P o ss i b l e g r a i n a bund a n c e s h Z grain i x e x i x grain − (cid:15) grain Fig. 6.—
Same as Figure 5 but for grains. Parameters for this runare a = 3 AU, x M = 10 − , Σ = 0 . − , and (cid:15) PAH = 0 (keptat zero to isolate the effect of grains). Our grains all have radiiof 1 µ m, which is so large that their corresponding abundance asinferred from observation (shaded region) is too low to significantlyaffect the amount of free charge. ignored, yielding in steady state: x e = x HCO + = (cid:113) ζ/n H α HCO + , e (18) ∼ − (cid:18) L X erg s − (cid:19) / × (cid:18) Σ0 . − (cid:19) − / (cid:16) a (cid:17) − . (19)for standard metals and low PAHs . In going from Equations (18) to (19) we account for thedistance dependence of temperature but assume materialis optically thin to X-rays. The square-root law of Equa-tion (18) is often used by other workers (e.g., Gammie1996; Glassgold et al. 1997). It is plotted as a horizontaldashed line in Figure 7, and should be compared withthe curves for x e and x i from our code, plotted as solidlines. In the limit of low x PAH , the electron and ionabundances computed from the code are nearly constantwith x PAH , as predicted by the analytic model. However,the results from the code sit above the line for Equation(18) by about an order of magnitude. The factor of 10offset arises because Equation (18) ignores ionized met-als, which recombine with electrons much more slowlythan does HCO + and which remain abundant comparedto HCO + in our standard model. The offset also impliesthat reducing the total metal abundance below that ofour standard model ( x M = 10 − ) can only decrease x e and x i by at most a factor of ∼
10. In this sense ouruncertainty in the metal abundance (Section 2.4.1) hasonly a limited impact on the ionization fraction, assum-ing x M < − . See also Section 3.3.1 where we considerthe case x M = 0.In the limit of high PAH abundance, electron recombi-nation on PAHs dominates electron recombination withHCO + . Low equilibrium abundances of free electrons im-ply the average charge on PAHs (cid:104) Z (cid:105) → x e = ζx PAH n H ( α PAH , e ) (cid:104) Z (cid:105) =0 (20a) x HCO + = ζx PAH n H ( α PAH , HCO + ) (cid:104) Z (cid:105) =0 (20b)for high PAHs . Note that in this limit of high PAH abundance, x e
Possible PAHabundances li m x P A H (cid:29) x ? P A H x H C O + li m x P A H (cid:29) x ? P A H x e lim x PAH → x e , x HCO + x M = 10 − (cid:15) PAH x PAH − − − − x M + + x HCO + x e Fig. 7.—
Ionization fraction as a function of PAH abundance for x M = 10 − (standard metal abundance), a = 3 AU, and Σ = 0 . − . Dashed lines: asymptotic values for x e and x HCO + of thesimplified model of Section 3.2.1. Solid lines: simulation results forfractional electron abundance x e (solid circles) and fractional ionabundance x M + + x HCO + (open squares). The dotted vertical linemarks x (cid:63) PAH (Equation 21), which roughly divides the regime of“low PAH abundance” where electron and ion densities are equaland insensitive to PAH abundance, from the regime of “high PAHabundance” where the ion density exceeds that of electrons andboth decrease approximately as 1 /x PAH . The behavior at highPAH abundance is independent of the metal abundance; comparewith Figure 8. The shaded region marks observationally inferredPAH abundances and happens to span the transition from low tohigh PAH regimes. Simulation data use (cid:15) grain = 10 − for (cid:15) PAH ≤ − ; (cid:15) grain = 10 − for (cid:15) PAH = 10 − . ; and (cid:15) grain = 10 − for (cid:15) PAH ≥ − . Metal-rich case. — Analogous results are obtained for themetal-rich case as shown in Figure 8, with the only dif-ference that M + replaces HCO + as the dominant ion. Inthe limit of low PAH abundance, the abundance of free charges is x e = x M + = (cid:113) ζ/n H α M + , e (22) ∼ × − (cid:18) L X erg s − (cid:19) / × (cid:18) Σ0 . − (cid:19) − / (cid:16) a (cid:17) − . (23)for high metals and low PAHs . In going from Equations (22) to (23) we account for thedistance dependence of temperature but assume materialis optically thin to stellar X-rays. Just as assuming allions took the form of HCO + in the standard model gavea lower limit (Equation 18) for the ionization fraction,assuming that all ions take the form of metals gives anupper limit (Equation 22) because fast recombination ofelectrons with HCO + is neglected.The analogous asymptotic solutions in the high PAHlimit are practically unchanged from Equations (20a) and(20b) because the charging rates of PAHs by HCO + andM + are similar; the mass of the HCO + molecule and thatof a metal ion like Mg + are similar. The critical PAHabundance at which PAHs begin to reduce the numberof free charges is x (cid:63) PAH = (cid:115) ζα M + , e n H ( α PAH , e ) (cid:104) Z (cid:105) =0 (24) ∼ × − (cid:18) L X erg s − (cid:19) / × (cid:18) Σ0 . − (cid:19) − / (cid:16) a (cid:17) − . for high metalsand is confirmed by the code. − − − − − − x e o r ( x M + + x H C O + ) − − − − − x ? PAH
Possible PAHabundances li m x P A H (cid:29) x ? P A H x M + li m x P A H (cid:29) x ? P A H x e lim x PAH → x e , x M + x M = 10 − − (cid:15) PAH x PAH − − − x M + + x HCO + x e Fig. 8.—
Same as Figure 7 but for the metal-rich case ( x M =10 − ). The curves for x e and x i at high PAH abundance ( x PAH >x (cid:63) PAH , where x (cid:63) PAH is now given by Equation 24) are essentiallythe same as in Figure 7: when PAHs dominate charge balance, themetal abundance ceases to matter.
Degree of Magnetic Coupling: Re and Am We measure the extent of the MRI-active column bymeans of the magnetic Reynolds number Re (Equation 1)and the ion-neutral collisional frequency Am (Equation2). Figures 9 and 10 show both dimensionless numbersas a function of the surface density Σ = N µ penetratedby X-rays at a = 3 and 30 AU, respectively, over therange of observationally inferred PAH abundances. Weoverplot for comparison the solution obtained when weomit PAHs completely.In both Figures 9 and 10, the middle panels displayresults for our standard model parameters: x M = 10 − per H , L X = 10 erg s − , and ζ CR = 0. In each of thepanels on the left and on the right, we vary one of theseparameters. We describe here results for our standardmodel and compare with other test cases in Section 3.3.1.The middle top panels of Figures 9 and 10 show that if Re were the only discriminant, MRI-active surface layerscould well exist, even with PAHs present. At surface den-sities Σ ∼ . − (column densities N ∼ cm − ), Re lies comfortably above the critical values of 10 –10 (Section 1.2) required for plasma to couple to the mag-netic field, for a wide range of possible PAH abundances.If the critical Re ∼ (as assumed by BG), and if PAHsare at their lowest possible abundance as inferred fromobservation ( (cid:15) PAH = 10 − relative to the ISM; invertedtriangles), then the MRI-active layer could extend as faras Σ ∼
20 g cm − —if ohmic dissipation were the onlylimiting factor for the MRI.But ohmic dissipation is not the only factor. The samemargin of safety enjoyed by Re does not at all apply tothe ambipolar diffusion number Am , for any surface den-sity. Even in the unrealistic case that there are no PAHs, Am stays <
10 in the middle bottom panels of Figures 9and 10. By comparison, values of Am exceeding 10 arereported by Hawley & Stone (1998) as necessary for theMRI to excite turbulence in predominantly neutral gas.When PAHs are present, Am barely exceeds 1, and thenonly for the low end of possible PAH abundances. Com-pared with ohmic dissipation, ambipolar diffusion seemsthe much greater concern for the viability of the MRI indisk surface layers.Values of Am (Σ) and Re (Σ) vary only slightly as thestellocentric distance increases from a = 3 AU (Figure9, middle) to 30 AU (Figure 10, middle); the former de-creases while the latter increases, each typically by fac-tors of a few. This behavior is readily understood. Firstrecognize that x i ∝ a − . approximately; this is an aver-age scaling between the low PAH limit, which implies x i ∝ a − . according to Equation (18), and the highPAH limit, which implies x i ∝ a − / ≈ a − . accordingto Equation (20b). Combining this result with n H ∝ h − ∝ Ω /T / ∝ a − / , we find that Am = x i n H / Ω ∝ a − . . Similarly, Re = c s h/D ∝ x e T / / Ω ∝ a . .In computing Am , we have omitted the contribu-tion from collisions between neutral H and negativelycharged PAHs. The latter are as well coupled to mag-netic fields as molecular ions are—see, e.g., the ion andgrain Hall parameters calculated in Section 2.2 of BG.Thus, collisions between H and charged PAHs shouldincrease Am . However, we find that in practice the gainis negligible. We estimate that the collisional rate co- efficient β in that enters into Am is about the same forcharged PAHs as for ions; in both cases a collision withan H molecule is mediated by the induced dipole in H ,and the relative velocity is dominated by the thermalspeed of H . At Σ (cid:38)
10 g cm − , charged PAHs areabout as abundant as ions and thus raise Am by a factorof 2—but at these Σ’s, Am is already too low for theMRI to be viable. At Σ (cid:46)
10 g cm − —i.e., at thosecolumns where Am peaks—charged PAHs are much lessabundant than ions and thus hardly affect Am . Higher L X and T X , Higher and Lower x M , andCosmic-ray Ionization In each of the leftmost and rightmost panels of Figures9 and 10, we vary one model parameter away from itsstandard value. We begin with the case of higher L X .Increasing L X certainly raises Re and Am , but as the leftpanels of Figure 9 show, even a fairly high L X = 10 ergs − only causes Am to just exceed 10 at the lowest PAHabundance. In the limit of low PAH abundance, x e = x i ∝ L / , as predicted by Equation (18). In the limit ofhigh PAH abundance, x e and x i scale linearly with L X ,according to Equations (20a) and (20b). Thus the spaceof possible values of Re and Am narrows with increasing L X , as the lower envelope increases as L X while the upperenvelope increases as L / .These same scaling relations, with L X replaced by theionization rate ζ at fixed distance, enable us to estimatethe effects of a higher T X , a case we did not explicitlycompute using our numerical model. According to IG,raising kT X from 3 keV (our standard value) to 8 keVincreases the ionization rate ζ by factors of 2–4 at Σ = 1–30 g cm − . Thus in the extreme case that L X = 10 ergs − and kT X = 8 keV—parameters appropriate only fora small minority of young stars (Telleschi et al. 2007;Preibisch et al. 2005)—we apply the low-PAH scalingrelation x i ∝ ζ / to the bottom left panel of Figure9 to find that the largest possible value of Am is ∼ x M = 10 − per H . As discussedin Section 2.4.1, the higher metal abundance is not espe-cially realistic and is considered primarily as an exercise.Comparing the middle and right panels of Figure 9, wesee that increasing the metal abundance by a factor of100 raises Re and Am at low PAH abundance by a fac-tor of ∼
10. At high PAH abundance, Re and Am alsoincrease with increasing metal abundance, but the gainis less. This same behavior is reflected in the solid curvesof Figures 7 and 8: at low PAH abundance, increasing x M by a factor of 100 leads to a factor of ∼
10 increase in x e = x i , but at high PAH abundance, the now divergentcurves for x e and x i are essentially independent of metalabundance. Equations (20a) and (20b) from our analyticanalysis reflect this insensitivity to metal abundance athigh PAH abundance.At Σ (cid:38) − , gas temperatures may be so lowthat all of the metals condense onto grains. The case x M = 0 is shown in the leftmost panels of Figure 10.Here Am (cid:46) . (cid:38) − , and it seems safe toconclude that X-ray driven MRI is unviable under theseconditions.Additional ionization by “sideways cosmic-rays” at a =AHs, MRI, and Planets 1730 AU is considered in the rightmost panels of Figure10. These cosmic-rays, which we have imagined enterthe disk edge-on from the outside, dominate stellar X-rays at large Σ. At Σ ∼
10 g cm − —comparable to thefull surface density of the disk at a = 30 AU—sidewayscosmic-rays raise the maximum value of Am to ∼
2. Wehave verified that the gains in Am afforded by cosmic-rays are consistent with our scalings of x e and x i with ζ as derived above. Chemical Equilibration Timescales vs. DynamicalTimescales
In assessing whether disk surface layers are MRI-active,we have relied on the critical value Am ∗ ∼ reportedby HS. As a simplifying assumption, HS held fixed theglobal (box-integrated) ion abundance in each of theirsimulations. A fixed ion abundance would apply if thechemical equilibration timescale t eq exceeds the dynam-ical timescale t dyn = Ω − over which HS’s simulationsran. A fixed ion abundance would also apply if ion re-combination occurs predominantly on condensates, re-gardless of t eq /t dyn (e.g., Mac Low et al. 1995). This laststatement follows from our Equation (20b), which shows x i n H does not depend on n H in the high condensatelimit.The high condensate limit applies for PAH abundancesnear the high end of those inferred from observation (Fig-ure 7). For this high PAH case we expect the assumptionof constant ion abundance, and by extension the resultsof HS, to hold. For high PAH abundance and our stan-dard X-ray luminosity, Am < a (Figure9), and our conclusion that X-ray-driven MRI shuts downeverywhere seems safe.For PAH abundances at the low end of those inferredfrom observation, Equation (20b) for the high conden-sate limit does not apply. Moreover, as shown in Figure11, t eq /t dyn < (cid:46)
10 g cm − . Because t eq /t dyn (cid:38) . Am ∗ would be even higherthan the reported value of ∼ . Comparison with Previous Work: IonizationFractions
The ionization chemistry in disks remains inherentlyuncertain, with rate coefficients for many reactions inthe UMIST database determined to no better than fac-tors of ∼ x e . We should reproduce their resultsby at least this margin, as a validation of our code. Inthe following we directly compare our results to thoseof BG and TCS, adjusting the input parameters of ourcode to match theirs. Once we match these input param-eters, any difference in our codes’ outputs should resultprimarily from our different chemical networks (ours isthe simplest of the three), and not from differences inradiative transfer, as all our codes rely on the ionizationrates calculated by Igea & Glassgold (1999).We start with BG by computing x e as a functionof density n H at a fixed ionization rate ζ = 10 − s − , following their Figure 3. We reset T = 280 K, x M = 2 . × − per H , and the electron-grain stickingcoefficient S e = 0 .
03 to match their standard parame-ters. To compare to their “grain-free” case, we run ourcode without any PAHs or grains. To compare to theirstandard monodispersion of grains, we run our code witha single population of grains having s = 0 . µ m, internaldensity ρ s = 3 g cm − , and a mass fraction of 1% rela-tive to gas. Figure 12 shows the comparison. Our resultsfor the condensate-free case track those of BG, but arehigher by factors of 3–10 depending on whether the com-parison is made with their simple or complex network.For the case with grains, the agreement with the simplemodel is excellent and that with the complex model isgood to a factor of 2.In Figure 13, we make a similar comparison with TCS,computing x e as a function of N at a distance of a = 5AU from an X-ray source of L X = 2 × erg s − and kT X = 5 keV, for T = 125 K and a metal abundanceof x M = 6 . × − per H . We consider the two casesof their Figure 1, one without any grains or PAHs, andanother with a single population of grains having s =1 µ m, ρ s = 5 g cm − , and a mass fraction of 1%. Forboth cases our computed electron abundances are higher,but only by factors of 2 or less.These comparisons with BG and TCS give us confi-dence that we have computed ionization fractions aboutas well as they did. Where our ionization fractions differ,ours are often higher. Our higher values will only bolsterthe conclusion we make in Section 4 that thicknesses ofX-ray-ionized MRI-active surface layers have been over-estimated by them and others. SUMMARY AND DISCUSSION
In Section 1, we presented the evidence that holes andgaps of transitional disks are cleared by companions totheir host stars. Residing within the hole, these compan-ions could either be stars—already observed in about halfof all transitional systems—or multi-planet systems. Asingle Jupiter-mass planet on a circular orbit carves outtoo narrow a gap to explain the large cavities inferredfrom observations. But multiple planets can shuttle gasquickly from one planet to the next, all the way down tothe central star. Surface densities fall in inverse propor-tion to radial infall speeds, and radial infall speeds canapproach freefall speeds for sufficiently many and mas-sive planets. In this way, multi-planet systems mighthelp to clear holes and simultaneously sustain stellar ac-cretion rates that approach those in disks without holes.The more eccentric the planets’ orbits, the fewer of them8 Perez-Becker & Chiang . . A m = ( x M + + x H C O + ) β i n n H / Ω .
01 0 . Σ [g cm − ] Am ∗ n H [cm − ] R e = c s h / D Re ∗ L X = 10 erg s − .
01 0 . Σ [g cm − ] Am ∗ N o P A H L o w P A H H i g h P A H P o ss i b l e P A H a bund a n c e s n H [cm − ] Re ∗ Std . Model ( a = 3 AU) .
01 0 . Σ [g cm − ] Am ∗ n H [cm − ] Re ∗ x M = 10 − Fig. 9.—
Magnetic Reynolds number Re and ambipolar diffusion number Am as a function of surface density Σ at a = 3 AU. Themiddle panels show results for our standard model ( x M = 10 − , L X = 10 erg s − , ζ CR = 0). The side panels have the same parametersas our standard model, except for a 100 × more luminous X-ray source (left panels), and a 100 × greater metal abundance (right panels).Oppositely pointing triangles bracket values for Am and Re corresponding to possible PAH abundances. These abundances were inferredin Section 2.4.3 from observations. The dashed curve refers to the case with no PAHs and is shown for comparison only. Polycyclicaromatic hydrocarbons reduce ionization fractions and thus the degree of magnetic coupling by an order of magnitude or more. The dottedlines mark the critical values Am ∗ and Re ∗ above which coupling between magnetic fields and neutral gas is sufficient to drive the MRI(Section 1.2). The curves for Am first rise as Σ increases—a consequence of the increasing number density—and then fall as the ion fractiondecreases, never reaching Am ∗ . Ambipolar diffusion threatens the MRI more than ohmic dissipation does. If the critical Am required forgood collisional coupling between ions and neutrals is Am ∗ = 10 , as evidenced in simulations by Hawley & Stone (1998), then even a diskwithout PAHs cannot sustain X-ray-driven MRI at any Σ. AHs, MRI, and Planets 19may be required to explain a given hole size. Accretionin the presence of multi-planet systems has not receivedmuch attention and seems an interesting area for futuresimulation (e.g., Zhu et al. 2010, submitted).Stellar or planetary companions regulate accretion ve-locities v but do not give rise to mass accretion rates ˙ M in the first place. A planet orbiting just inside the cir-cumference of a disk hole exerts torques to repel gas inthe hole’s rim away from the star. Thus, the shepherd-ing planet may reduce ˙ M —and indeed accretion ratesin transitional systems tend to be smaller than those inconventional disks (Najita et al. 2007)—but the planetdoes not initiate disk accretion. A separate mechanismmust act to pull or diffuse gas inward from the hole rim tosupply the stellar accretion rates that are observed. Thatmechanism may be turbulence driven by the magnetoro-tational instability (MRI), activated by stellar radiationionizing rim gas. Whether the MRI can operate dependson how well ionized the gas is. The greater the free elec-tron fraction, the greater the magnetic Reynolds number Re , and the less ohmic dissipation dampens the MRI.The greater the atomic and molecular ion densities, thegreater the collisional rate Am between neutral particlesand ions, and the less ambipolar diffusion weakens theMRI.A principal threat to the MRI is posed by dust grains,which adsorb electrons and ions. The smallest grainsmay present the biggest danger, because in many parti-cle size distributions the smallest grains have the great-est surface area for attachment. The smallest grainsthat can also be detected observationally are polycyclicaromatic hydrocarbons (PAHs), each several angstromsacross and containing of order a hundred carbon atoms.These macromolecules may reside in the very disk sur-face layers that promise to be MRI-active. Excited bysoft ultraviolet radiation, PAHs fluoresce in a distinctiveset of infrared emission lines detectable from Spitzer andfrom the ground. The hydrocarbon molecules are proba-bly generated locally, photo-sputtered off larger particlesexposed to hard UV and X-ray radiation from host stars.To assess the impact of PAHs on the MRI, we needto know PAH abundances relative to gas. These canbe inferred from observed PAH emission lines. Unfortu-nately such inferences are model dependent; they dependon knowing the local grain opacity, because the soft UVradiation that excites PAHs is also absorbed by grains.In other words, observed PAH line intensities depend onPAH-to-dust ratios. It follows that the quantity of inter-est to us—the PAH-to-gas ratio—depends on knowingthe dust-to-gas ratio. The latter can vary widely withthe degree to which grains settle toward disk midplanes.The more grains have settled, the lower are local dust-to-gas ratios, and the lower the PAH-to-gas abundance thatis needed to explain a given set of PAH emission spectra.By compiling a few lines of model-dependent evidencefrom the literature, we estimated that disk PAHs haveabundances anywhere from 10 − –10 − per H , withlower values corresponding to a greater degree of dustsettling.Such PAH abundances, although depleted relative tothe ISM by 10 − –10 − , are still large enough to signif-icantly weaken the MRI in disk surface layers. In fact,they might even shut off X-ray-driven MRI altogether, everywhere. For stellar X-ray luminosities of L X = 10 –10 erg s − and X-ray stopping columns of Σ ∼ − , PAHs reduce electron and ion densities—which arenot equal when PAHs are present—by factors of ∼
10 ormore. At these surface densities, the collisional couplingfrequency Am ≈ − –10, depending on PAH abundanceand X-ray luminosity. These values fall short, by 1–5orders of magnitude, of the critical value Am ∗ ∼ re-quired for good coupling between ions and neutrals, asmeasured in simulations by Hawley & Stone (1998, HS).The potentially catastrophic effect that small grains canhave on the MRI was highlighted by Bai & Goodman(2009, BG). Our study grounds their concern in real-lifeobservations.Other studies reported X-ray-driven MRI-active sur-face layers to be alive and well (e.g., Chiang & Murray-Clay 2007, CMC; Turner, Carballido, & Sano 2010, TCS;and BG, in many of whose models the active layer ex-tended to ∼ − , even with grains present). Weshould understand why our conclusions differ from theirs.In part, the difference arises because previous studies ne-glected PAHs. A further difference with TCS is that theyassumed a metal abundance of x M = 6 . × − per H ,nearly 2 orders of magnitude higher than our standardmodel value, and one that we find difficult to justify. Stillanother difference, as significant as any of the ones justmentioned, is the criterion used for whether ambipolardiffusion defeats the MRI. Turner et al. (2010) assumed Am ∗ ∼ Am . Using their data, we com-puted the Am values characterizing their claimed activelayers. At a (cid:38) Am values of BG’s grain-freeactive layer are at most on the order of unity. For BG’sstandard models containing grains, Am ≈ . a = 50 AU anda population of grains having two sizes, and the highestvalue corresponding to a = 1 AU and a single-sized grainpopulation (we computed both limits using results fromtheir complex chemical network). Hawley & Stone (1998)showed that when Am (cid:46) .
01, ions and neutrals were ef-fectively decoupled. Even when Am ∼
1, HS showedthat the MRI saturation amplitude scaled with the ionand not the neutral density, with the neutrals acting todamp out MRI turbulence in the ions. If the MRI drivesturbulence only in the ions of protoplanetary disks, itmight as well not operate at all, given how overwhelm-ingly neutral such disks are.
Future Directions
We have shown in this paper that the MRI cannotdrive surface layer accretion under typical circumstancesin protoplanetary disks, either transitional or conven-tional, if the critical Am ∗ ∼ and if stellar X-rays andGalactic cosmic-rays are the dominant source of ioniza-tion. These two “if”s are subject to further investigation.We discuss each in turn.We are not aware of more modern estimates of Am ∗ apart from that given by HS. As these authors cautioned,numerical resolution is a greater concern for two-fluidsimulations than for single-fluid ones, and HS did notdemonstrate convergence of their results with resolution.In addition, the value of Am ∗ was not as precisely deter-mined by HS for toroidal field geometries as for verticalones—although Am ∗ ∼ did seem to apply equally0 Perez-Becker & Chiang . . A m = ( x M + + x H C O + ) β i n n H / Ω .
01 0 . Σ [g cm − ] Am ∗ n H [cm − ] R e = c s h / D Re ∗ x M = 0 .
01 0 . Σ [g cm − ] Am ∗ N o P A H L o w P A H H i g h P A H P o ss i b l e P A H a bund a n c e s n H [cm − ] Re ∗ Std . Model ( a = 30 AU) .
01 0 . Σ [g cm − ] Am ∗ n H [cm − ] Re ∗ ζ CR = 1 / × − s − Fig. 10.—
Same as Figure 9, but at a stellocentric distance of a = 30 AU. The middle panels show results for our standard model( x M = 10 − , L X = 10 erg s − , ζ CR = 0). The side panels have the same parameters as our standard model, except that gas-phasemetals are omitted in the left panels ( x M = 0), and sideways cosmic-rays are added in the right panels ( ζ CR = 1 / × − s − ). AHs, MRI, and Planets 21 n H [cm − ]10 − − − − t e q / t d y n .
01 0 . − ] a = 3 AU x M = 10 − N o P A H L o w P A H H i g h P A H P o ss i b l e P A H a bund a n c e s Fig. 11.—
Ratio of the chemical equilibration timescale t eq , com-puted according to the procedure described in Section 2.6, to thedynamical time t dyn = Ω − , for a = 3 AU and x M = 10 − . Valuesof t eq /t dyn at a = 30 AU are typically lower than those shownhere by factors of 3 or less. Simulations by HS assumed a con-stant (volume-integrated) abundance of ions, a condition satisfiedif t eq /t dyn >
1. The ion abundance is also constant if ion re-combination occurs primarily on condensates (see Equation 20b),a situation that obtains for PAH abundances near the high endof those inferred from observations. For the lowest possible PAHabundances, 0 . < t eq /t dyn < . − . In thislow PAH case, the results of HS might still be expected to applyto order unity. Even if they do not, we argue in the main text thatwhen t eq /t dyn < − − − − − − − x e = n e / n H n H [cm − ] m o n o d i s p e r s i o n o f g r a i n s c o nd e n s a t e – f r ee Test comparison with BG(fixed ζ = 10 − s − H − )Our codeBG simpleBG complex Fig. 12.—
Test comparison with BG: electron abundance as afunction of gas density at a fixed ionization rate of ζ = 10 − s − H − . The upper set of lines are for a condensate-free system,and the lower set are for a monodispersion of grains. Results fromBG were drawn from their Figure 3. To generate our results, theparameters of our code were reset to those of BG: temperature T =280 K, metal abundance x M = 2 . × − per H , electron-grainsticking coefficient S e = 0 .
03, grain radius s = 0 . µ m, internalgrain density ρ s = 3 g cm − , and a mass fraction in grains relativeto gas of 1%. well to the cases of uniform vertical field and zero net − − − − − − x e = n e / n H N [cm − ] c o nd e n s a t e – f r ee m o n o d i s p e r s i o n o f g r a i n s Test comparison with TCS( x M = 6 . × − per H )Our codeTCS Fig. 13.—
Test comparison with TCS: electron abundance asa function of column depth penetrated by X-rays. The upperpair of lines are for a condensate-free system, and the lower pairare for a monodispersion of grains. Ionization rates are fromIG. Results from TCS are taken from their Figure 1. To gen-erate our results, we reset the parameters of our code to matchthose of TCS: stellocentric distance a = 5 AU, X-ray luminosity L X = 2 × erg s − , temperature T = 125 K, metal (magne-sium) abundance x M = 6 . × − per H , electron-grain stickingcoefficient S e = 0 .
03, grain radius s = 1 µ m, internal grain density ρ s = 5 g cm − , and a mass fraction in grains relative to gas of 1%. vertical field. Perhaps higher resolution simulations willreveal that Am ∗ < —although accounting for ion re-combination in these simulations should only increase Am ∗ (Section 3.3.2).The second possibility is that our model has neglecteda significant source of ionization. Stellar radiation justlongward of the Lyman limit—so-called far ultraviolet(FUV) radiation at photon energies between ∼ x i would be a few × − , or 5 orders ofmagnitude higher than the largest values of x i reportedin this paper! At disk midplanes which are shielded fromphotodissociating radiation, an order-unity fraction ofthe full solar abundance of C is expected to take theform of CO ( x CO = 10 − ; Aikawa et al. 1996). As com-puted in chemical models by Gorti & Hollenbach (2004;see also Tielens & Hollenbach 1985 and Kaufman et al.1999), CO near disk surfaces photodissociates nearly en-tirely by FUV radiation into a layer of neutral C. At thehighest altitudes, nearly all of this carbon is photoionizedby FUV radiation. The column density of C + depends onhow many small grains having sizes (cid:46) . µ m are present,as grains compete to absorb the same FUV photons thatphotodissociate CO and photoionize C.We may estimate maximum FUV-ionized column den-sities by neglecting such dust extinction, and by neglect-ing shielding of FUV radiation by molecular hydrogen.Consider a trace species T whose total number densityregardless of ionization state is f T n H . Take all of T tobe singly ionized within a Str¨omgren slab at the disk sur-2 Perez-Becker & Chiangface: n T + = n e = f T n H . Per unit surface area of slab,the rate of photoionizations balances the rate of radiativerecombinations: L FUV E FUV πa ∼ n T + n e α T + , e h ∼ f n α T + , e h , (25)where the FUV luminosity capable of ionizing T is L FUV ∼ erg s − (Gorti et al. 2009), the photon en-ergy E FUV ∼
10 eV, the rate coefficient α T + , e ∼ × − cm s − at an FUV-heated gas temperature of 300 K,and the slab thickness h ∼ . a . Solve for the hydrogencolumn N FUV = n H h (26) ∼ × (cid:18) L FUV erg s − (cid:19) / × (cid:18) a (cid:19) / (cid:18) − f T (cid:19) cm − , or equivalentlyΣ FUV = N FUV µ (27) ∼ . (cid:18) L FUV erg s − (cid:19) / × (cid:18) a (cid:19) / (cid:18) − f T (cid:19) g cm − . In Equation (27), we have normalized f T to its highestplausible value, appropriate for C. An MRI-active surfacedensity Σ FUV ∼ .
07 g cm − is modest, and would drivemass accretion rates only barely observable. Lowering f T would increase Σ FUV . But accounting for extinctionof FUV radiation by dust and molecular hydrogen woulddecrease Σ
FUV . We are currently undertaking a morecareful study of FUV ionization to quantify these effects.Neal Turner provided invaluable feedback during for-mative stages of this work. We thank Xue-Ning Bai,Kees Dullemond, Josh Eisner, Vincent Geers, Al Glass-gold, Uma Gorti, Lee Hartmann, David Hollenbach, An-ders Johansen, Yoram Lithwick, Dimitri Semenov, GregSloan, Jim Stone, Marten van Kerkwijk, Yanqin Wu,and Andrew Youdin for discussions. Xue-Ning Bai andJim Stone provided encouraging feedback that led to ad-ditional analyses such as that in section 3.3.2. Zhao-huan Zhu and Lee Hartmann generously shared theirpreprint which impressed upon us the need for graingrowth to explain the low optical depths of transitionaldisk holes. An anonymous referee provided a thoughtfuland thorough report that alerted us to the possibility of“sideways cosmic-rays,” and that motivated us to con-sider the effects of UV ionization. We also thank oureditor, Eric Feigelson, for additional comments on ourmanuscript. E.C. acknowledges the hospitality of theKavli Institute for Astronomy and Astrophysics in Bei-jing, China, where a portion of this work was carried out.This work was funded by the National Science Founda-tion, in part through a Graduate Research Fellowshipawarded to D.P.-B.
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Note added in proof. — In a private communication, Xue-Ning Bai reports that in new, unstratified MRI simulations,the Shakura-Sunyaev transport parameter α is at most ∼ − when Am ∼
1. Combining his result with the results ofour paper, we estimate that the mass accretion rate in the surface layer of a conventional disk at 3 AU is ∼ − M (cid:12) yr − . At the rim of a transitional disk, we would predict an accretion rate that is lower by a factor of ∼ h/a , or ∼ − M (cid:12) yr − at 3 AU. These theoretical accretion rates are too low to explain the observed accretion rates of mostdisks. The situation is similar at 30 AU, where the surface layer accretion rate in conventional (transitional) disks canonly be as high as ∼ − (10 − ) M (cid:12) yr −1