Fomalhaut b as a Dust Cloud: Frequent Collisions within the Fomalhaut Disk
aa r X i v : . [ a s t r o - ph . E P ] F e b Draft version July 10, 2018
Preprint typeset using L A TEX style emulateapj v. 04/17/13
FOMALHAUT B AS A DUST CLOUD: FREQUENT COLLISIONS WITHIN THE FOMALHAUT DISK
S. M. Lawler , S. Greenstreet , and B. Gladman Draft version July 10, 2018
ABSTRACTThe planet candidate Fomalhaut b is bright in optical light but undetected in longer wavelengths,requiring a large, reflective dust cloud. The most recent observations find an extremely eccentricorbit ( e ∼ . d ≃
100 km bodies in the high-eccentricityscattering component is several per decade. This model paints a picture of the Fomalhaut systemas having recently (within ∼ INTRODUCTION
Fomalhaut is a nearby (7.70 pc), widely-spacedtriple star system (Mamajek et al. 2013) with an ageof 440 Myr (Mamajek 2012). Fomalhaut A (Fo-malhaut hereafter), is an A4 star that dominatesthe system and possesses a beautiful, eccentric de-bris ring that is resolved at optical (Kalas et al. 2005),infrared (Fajardo-Acosta et al. 1998; Stapelfeldt et al.2004; Su et al. 2013; Acke et al. 2012), submillimeter(Holland et al. 1998, 2003; Marsh et al. 2005), and mil-limeter wavelengths (Boley et al. 2012).Fomalhaut b was directly imaged using the
HubbleSpace Telescope (Kalas et al. 2008) and, based on twoobservation epochs, plausibly consistent with orbital pre-dictions for a massive planet dynamically constrain-ing the eccentric dust ring (Quillen 2006; Chiang et al.2009). However, follow-up observations in infrared wave-lengths failed to detect Fom b (Marengo et al. 2009;Janson et al. 2012), indicating a significantly lower massthan Jupiter.The nature of Fom b is uncertain. The most recentobservations of Fom b suggest that it is on a highly ec-centric, possibly ring-crossing orbit (Kalas et al. 2013;Beust et al. 2014). Fom b is also brighter at optical wave-lengths than predicted by planetary atmosphere models,requiring a cloud of optically reflective small dust grains.Kennedy and Wyatt (2011) suggest a swarm of collision-ally grinding irregular satellites, gravitationally bound toa super-Earth planet. However, dynamical analyses sug-gest that a planet of this mass on such an eccentric orbitwill destroy the apsidally aligned debris ring within a Department of Physics and Astronomy, University of Victo-ria, PO Box 1700, STN CSC Victoria, BC V8W 2Y2, Canada NRC-Herzberg, 5071 West Saanich Road., Victoria, BC V9E2E7, Canada Department of Physics and Astronomy, University of BritishColumbia, 6224 Agricultural Road Vancouver, BC V6T 1Z1,Canada very small fraction ( < COMPARISON WITH THE KUIPER BELT
As Kalas et al. (2013) point out, Fomalhaut’s debrisring and our Solar System’s main Kuiper belt are simi-lar dynamically, despite their different ages. A perhapsimportant difference is that while the classical Kuiperbelt has very low eccentricity, the Fomalhaut debris ringhas a significant forced eccentricity of e ≃ .
1. Look-ing at the large (diameter d &
100 km) objects in theKuiper belt, the only apparent structure is the dense,relatively narrow ring of the main classical belt (Lawler2014). Due to the high optical depth of the dust grainsin the Fomalhaut ring ( ∼ − ; Marsh et al. 2005) com-pared to the Kuiper Belt ( ∼ − ), the maximal dust dis-tribution should co-located with the ring (see simulationsby Kuchner and Stark 2010).Fom b’s orbit is placed in context in Figure 1, where theKuiper belt, as measured by the Canada-France EclipticPlane Survey (CFEPS; Petit et al. 2011; Gladman et al.2012) had its semimajor axis multiplied by three tomatch the Fomalhaut system. Though many KBO dy-namical sub-classes exist, here (Figure 1) we only dis- Lawler et al. e cc e n t r i c i t y resonant scatteringclassical detachedFomalhaut b100 150 200 250 300 350 400 semimajor axis [AU] i n c li n a t i o n [ ◦ ] Fig. 1.—
Orbital element distributions for the debiased KuiperBelt as measured by CFEPS, where the semimajor axis has beenscaled so that the classical KBOs match Fomalhaut’s main debrisring. The different populations are shown by different colors, la-belled in the legend. Also shown on the plot is the best-fit orbitalelements for Fom b (Kalas et al. 2013, large purple square). Fo-malhaut b most clearly belongs in the scattering population. cuss four broad categories: resonant objects (red) whichare in mean-motion resonances with Neptune, scatteringobjects (green) whose orbits significantly change due toclose encounters with the giant planets within a few Myr,detached objects (blue) which have eccentricities corre-lated with their semimajor axes, implying past planetaryinteractions, and lastly classical objects (black) which in-clude the rest of the KBOs (see Gladman et al. 2008).The resonant semimajor axes in the Kuiper Belt dependon the location of Neptune, thus we do not expect to findobjects in these exact locations in the Fomalhaut disk.With a pericenter well inside the ring, Fom b is clearlya member of the scattering population. Our hypothesisis that Fom b is a newly disrupted member of Fomal-haut’s scattering planetesimal population, in a recentlyperturbed Fomalhaut system, similar to the Nice Modelinstability that occurred early in our Solar System’s his-tory. MODELLING THE FOMALHAUT DISK
Our model of the Fomalhaut disk is based on theknown structure and populations of the Kuiper Belt.The Kuiper Belt’s scattering disk in particular is heav-ily eroded from its state just after the migration of thegiant planets. Simulations show that today’s scatteringpopulation is ∼
1% of the original (Duncan and Levison1997), which has been eroded away mainly by planetaryclose encounters causing ejection from the Solar System.Because Fomalhaut is significantly younger than the Sun,in order to transform our Kuiper Belt into an approxima-tion of the Fomalhaut disk, we first must correct for thiserosion. This brings the number of scattering objects toabout the same as the classical population, though in amuch more widely dispersed orbital distribution. Addi- tionally, very massive planets ( & M Jup ) with projectedseparations >
15 AU have been ruled out (Currie et al.2013), so the timescale for depletion of the scatteringpopulation shouldn’t be much faster than in our SolarSystem. Thus, the number of scattering objects is scaledup by 100 times relative to other Kuiper Belt popula-tions.The orbital distribution of objects in the young scatter-ing disk was different from today’s distribution, as someorbits are more likely to be ejected than others. We pro-duced a simple “young scattering” distribution for ourSolar System by starting with a flat distribution of ob-jects with a = 4 −
35 AU, e = 0 .
1, and i = 15 ◦ , andrunning a numerical integration with the four giant plan-ets for 100 Myr. The surviving objects, many of whichshow more deeply crossing perihelia than for the scatter-ing population shown in Figure 1, had their semimajoraxes tripled to match the Fomalhaut system.Next, one must also scale up the system mass untilthe main classical belt matches the Fomalhaut ring’smuch larger observed mass compared to our Kuiper Belt.CFEPS measured the absolute number of d &
100 kmKuiper Belt objects, so matching the mass of anotherdisk is accomplished by scaling the number of d >
100 kmobjects in the model. Coincidentally, d ≃
100 km is theminimum size required to produce the observed Fom bdust cloud (Galicher et al. 2013; Kenyon et al. 2014).We settled on a scaling factor of 1000. This is very ap-proximate, as mass estimates for the Kuiper Belt vary byan order of magnitude ( ∼ M ⊕ ; e.g., Fraser et al.2014; Vitense et al. 2010), and mass estimates for theFomalhaut ring vary by two orders of magnitude ( ∼ M ⊕ ; e.g., Boley et al. 2012; Acke et al. 2012).Orbital elements for simulated planetesimals in the Fo-malhaut model are based on the CFEPS L7 syntheticmodel of the Kuiper Belt (available at ).The heart of the classical Kuiper Belt sits at ∼
45 AU,while the peak of the Fomalhaut disk as measured byALMA (Boley et al. 2012) is at ∼
135 AU, so we mul-tiplied the semimajor axes of each synthetic object bythree. To match the main ring shape and higher eccen-tricity, the eccentricities of the main classical belt wereincreased by 0.05, so that the median eccentricity be-came e ≃ .
1. To match the apsidal aligment of theplanetesimals relative to each other, the simulated clas-sical objects were set with Ω = − ω ± ◦ .This orbital distribution of synthetic objects is thestarting point for our collisional probability simulations. Collisional Probability Simulations of theFomalhaut System
We use an Opik collision probability code based onDones et al. (1999) to compute the collision probabilityfor a Fom b progenitor (radius r = 50 km, a = 177 AU, e = 0 . i = 17 ◦ ) with various small body populations,ignoring any other planets in the system. The code calcu-lates the collision probability for a population of projec-tiles by numerically integrating over the precession cycleof the nodal longitude Ω and argument of pericenter ω for both the impactor and target bodies.The code was modified to bin the collision probabilityinto individual impact velocity bins (as opposed to bin-ning the total collision probability into an average impactvelocity bin computed from all possible impact orienta-omalhaut b as a Dust Cloud 3tions over a full precession cycle of the orbital nodes), aswell as individual impact distance bins. This producesdetailed impact velocity distributions for each projectilepopulation. Catastrophic Collisional Probability
Using the probabilities for a range of impact velocitiesin each distance bin, we calculate the total catastrophicdisruption probability. Closer to the star, relative speedsare generally higher allowing smaller objects to producethe same collision energy as a slower, larger object fartherfrom the star.Equation 2 from Leinhardt and Stewart (2012) relatestarget mass, projectile mass, collisional velocity, and thecatastrophic disruption energy for an icy body (whichwe take from Figure 11 in Leinhardt and Stewart 2009,using r = 50 km). This gives the minimum projectilesize required to catastrophically disrupt a d = 100 kmtarget body at a given velocity. To properly weight thecollisional probabilities in each velocity bin, we multiplyby the number of bodies of this minimum size, assuming aprojectile size distribution of dn/dr ∝ r − q , with q = 3 . d = 100 km progenitor ona Fom b-like orbit.The full disk collisional model (black line in Figure 2)shows the likelihood of collisions between an object ona Fom b-like orbit and anything in the Fomalhaut sys-tem model. The total integrated catastrophic collisionalprobability is 14/decade, and the most likely place forcollisions is just sunward of the main ring. The portionof the catastrophic probability for just the non-scatteringobjects is also shown separately (red line); as expectedthe main ring dominates at its distance, but most scat-tering object disruptions are mutual and occur between50-110 AU. We also confirmed that our model producesa catastrophic disruption rate for collisions between ob-jects in the main ring that approximately reproduces theobserved dust production rate (Acke et al. 2012).Though the probabilities for disruption of a Fom b-likeorbit are high within the main belt, collisions here wouldnot generate a dust cloud because the high dust densitywithin the ring would effectively absorb and/or collision-ally destroy the dust cloud almost instantly. Also, due toFom b’s orbital direction, we know Fom b has not beeninside the main belt for at least ∼
250 years. Therefore,this is not our favoured projectile population.The most likely population to cause a catastrophic col-lision is the scattering population (blue line in Figure 2).This curve peaks at about 95 AU from Fomalhaut, sim-ilar to Fom b’s distance ∼ d = 100 km ob-jects of 11/decade. The most likely collision is a ∼
10 kmprojectile destroying a 100 km target. PLAUSIBILITY OF FOM B AS A DUST CLOUD
Galicher et al. (2013) estimate that within the mainbelt, mutual collisions between d = 100 km bodies
40 60 80 100 120 140 160 180 distance from Fomalhaut [AU] -3 -2 -1 c a t a s t r o p h i c c o ll . p r o b . [ y r − b i n − ] main ring scattering full disk M a i n r i n g Fom b distance
Fig. 2.—
The catastrophic collision probability per year per2 AU-wide bin for different projectile populations onto a targetwith Fom b’s current orbit. Black shows the full disk model, redshows just the non-scattering component (the main ring), and blueshows just the scattering component, with total integrated catas-trophic disruption probabilities stated for each population. Yellowband shows the pericenter-apocenter range of main ring particles,the mutual main ring collision rate (not shown) is a factor of sev-eral larger. Measured stellocentric distances of Fom b in 2000 and2012 are shown. happen once per century (they do not calculate catas-trophic disruption rate), while Currie et al. (2012) andKalas et al. (2013) argue that the dust cloud scenario isunlikely because of the long collisional timescale and rel-atively quick orbital shearing. Our analysis shows thatcloser to the star than the main ring, higher mutual colli-sion speeds result in smaller objects being able to catas-trophically disrupt the same size target. Because of thesize distribution, there are abundantly more smaller ob-jects, which shifts the peak destruction probability sun-ward of the main ring.Kenyon et al. (2014) use size distribution models toargue that a dust cloud resulting from a collision isonly possible for a limited range of parameters. A50 km radius progenitor can reproduce the observedproperties for a dust power-law size index ( q ) of 4.7or less, with the largest remaining fragment less than1% the size of the initial body (otherwise there isnot enough mass in dust to reproduce observations).While similar q values are often reproduced in simu-lations of catastrophic disruptions, near-complete de-struction of the target is only rarely produced in sim-ulations (Leinhardt and Stewart 2012). Additionally,Kenyon et al. (2014) point out that the expansion of thedust cloud is constrained by the eight year observationalbaseline. If the dust cloud is unresolved (Currie et al.2012; Kalas et al. 2013; Kenyon et al. 2014), this limitsthe expansion to .
300 m/s, while if it is marginally re-solved (Galicher et al. 2013), the expansion is limited to . ∼
100 km) depends noton the collision velocity, but on the escape velocity of thetotal mass (projectile+target; Benz and Asphaug 1999). Lawler et al.For a d = 100 km icy body, the escape velocity is ap-proximately 30 m/s, suggesting that there is close to acentury before the dust cloud would expand enough tobe resolved by Hubble , greatly increasing the window oftime the dust cloud remains unresolved.This theory implies that there are many possibly vis-ible dust clouds in the Fomalhaut system at any giventime. However, we note the difficulty of these observa-tions: only a limited range of distances and longitudeshave deep enough magnitude limits to detect such a cloud(Kalas et al. 2013; Currie et al. 2012). Also, only a frac-tion of the catastrophic collisions will result in a visibledust cloud (Kenyon et al. 2014). Lastly, 11/decade isprobably an overestimate since our scattering populationestimate is appropriate for just after the system’s pertur-bation; if instead the perturbation occurred ∼
30 Myr ago(perhaps a more reasonable ∼
10% of the system’s age)our integrations show that the catastrophic collision ratefor the scattering population would drop to ∼ ∼ µ m) grains expanded the fastest ( ∼
250 m/s),creating a large shell that was the most easily visible,and a second cloud composed of larger ( ∼ µ m) grainsexpanding more slowly ( ∼ ∼ µ m foricy grains) is much larger than in our Solar System. Anygrains smaller than this would immediately be placed onhyperbolic orbits. The relative fraction of large versussmall dust grains produced by the collision would dic-tate which portion would be more easily visible: eitherthe larger grains that continue on roughly the same orbitas the parent body, or the small grains which find them-selves on either an extremely elliptical or even hyperbolicorbit.The latter offers another possible explanation for theextreme orbit of Fom b, as small grains with high β val-ues will instantaneously have their orbits shift to high e .While Beust et al. (2014) rule out e > .
98 in their or-bital fits for Fom b, Kalas et al. (2013) do not specificallyrule out unbound orbits. If Fom b’s orbit is found to beunbound in future observations, a dust cloud made ofsmall grains is a more plausible explanation than an ex-tremely fortuitous observation of a planet being ejectedfrom the system. DISCUSSION: EVOLUTION OF THE FOMALHAUTSYSTEM
The theory that Fom b is a collisional dust cloud fitsinto the following possible narrative of the Fomalhautsystem. Fomalhaut hosts a young planetary system. Ashas been proposed for the early few hundred Myr of ourown Solar System (Gomes et al. 2005), the Fomalhautsystem experienced a relatively recent ( ∼
100 Myr) dy-namical instability that resulted in a dramatic reshuffling of the orbits of its planets and smaller bodies. Simula-tions of our own Solar System show that a dynamicallycold population (like the debris ring) can survive such aviolent event (Batygin et al. 2011). Fomalhaut c (as yetunobserved) was scattered or migrated onto its currentmoderately eccentric orbit ( e ≃ .
1) and is currently sec-ularly driving the eccentricity of the main debris ring,as discussed in previous works. Any planetesimals thatwere on orbits interior to the main ring ( a ≃ β values (e.g., Wyatt 2008). Thus, Fomalhaut couldbe hiding a significant population of KBO-analogues onhighly eccentric orbits similar to that of Fomalhaut b.More detailed modelling is required to determine the or-bital population and spectral energy distribution of dustresulting from these collisions within the scattering pop-ulation. A likely outcome would be a population of warmdust grains, as found by recent IR observations (Su et al.2013; Acke et al. 2012).There are two predictions of our Fom b-as-a-dust-cloudhypothesis. The first is that the ∼ SUMMARY AND CONCLUSION
We have shown that the catastrophic disruption rateswithin the Fomalhaut debris disk can be much higherthan previously assumed. This is primarily due to inclu-sion of a high eccentricity scattering component, equiv-alent to the scattering population in our Solar System’sKuiper Belt. Fomalhaut’s young age is also an importantfactor permitting a more massive scattering component.This calculation shows that the possibility that Fom b isjust a cloud of dust should not be dismissed.The relatively high collision rate that we calculate herewould mean that another Fom b-like object should ap-pear within the next decade, and Fom b itself will fadeover the coming years, possibly becoming resolved. In or-der to test these two predictions, continued follow-up ob-servations capable of detecting objects as faint as Fom bare vital. For now, the only telescope capable of de-tecting Fom b is
Hubble , but the upcoming
James WebbSpace Telescope will be able to resolve the dust cloud, andprovide some additional constraints on the dust compo-sition with near-IR measurements.While perhaps disappointing to think of Fom b asmerely a cloud of dust and not an actual planet, thisscenario tells us about the structure of the Fomalhautdebris disk, and strongly implies the presence of multi-ple planets, just as the scattering component in our Solaromalhaut b as a Dust Cloud 5System is constrained and limited by the presence of gi-ant planets interior to the Kuiper Belt.S.M.L. wishes to thank Wes Fraser, Kat Volk, AaronBoley, Yanqin Wu, Dan Tamayo, Christa Van Laerhoven, and Henry Ngo for enlightening discussions and advicethat made this project possible. S.M.L. acknowledges anNSERC Discovery Accelerator Supplement which fundedthis work.
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