Transience of hot dust around sun-like stars

8/14/2019 Transience of hot dust around sun-like stars

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arXiv:as

tro-ph/0610102v2

2Nov2006

Submitted to ApJ 16 July 2006, Accepted 3 October 2006, Revised 2 November 2006Preprint typeset using LATEX style emulateapj v. 04/21/05

TRANSIENCE OF HOT DUST AROUND SUN-LIKE STARS

M. C. WyattInstitute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK

R. SmithInstitute for Astronomy, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK

J. S. GreavesScottish Universities Physics Alliance, University of St. Andrews, Physics & Astronomy, North Haugh, St Andrews KY16 9SS, UK

C. A. Beichman1 and G. BrydenJet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

C. M. LissePlanetary Exploration Group, Space Department, Johns Hopkins University Applied Physics Laboratory, 11100 Johns Hopkins Rd,

Laurel, MD 20723, USA

Submitted to ApJ 16 July 2006, Accepted 3 October 2006, Revised 2 November 2006

ABSTRACT

There is currently debate over whether the dust content of planetary systems is stochasticallyregenerated or originates in planetesimal belts evolving in quasi-steady state. In this paper a simplemodel for the steady state evolution of debris disks due to collisions is developed and confronted withthe properties of the emerging population of 7 sun-like stars that have hot dust at < 10 AU. The modelshows that there is a maximum possible disk mass at a given age, since more massive primordial disksprocess their mass faster. The corresponding maximum dust luminosity is fmax = 0.16103r7/3t1age,where r is disk radius in AU and tage is system age in Myr. The majority (4/7) of the hot disksexceed this limit by a factor 1000 and so cannot be the products of massive asteroid belts, ratherthe following systems must be undergoing transient events characterized by an unusually high dustcontent near the star: Corvi, HD69830, HD72905 and BD+20307. It is also shown that the hotdust cannot originate in a recent collision in an asteroid belt, since there is also a maximum rate atwhich collisions of sufficient magnitude to reproduce a given dust luminosity can occur in a disk ofa given age. For the 4 transient disks, there is at best a 1:105 chance of witnessing such an eventcompared with 2% of stars showing this phenomenon. Further it is shown that the planetesimal beltfeeding the dust in these systems must be located further from the star than the dust, typically at 2 AU. Other notable properties of the 4 hot dust systems are: two also have a planetesimal beltat > 10 AU ( Corvi and HD72905); one has 3 Neptune mass planets at < 1 AU (HD69830); allexhibit strong silicate features in the mid-IR. We consider the most likely origin for the dust in thesesystems to be a dynamical instability which scattered planetesimals inwards from a more distantplanetesimal belt in an event akin to the Late Heavy Bombardment in our own system, the dust beingreleased from such planetesimals in collisions and possibly also sublimation. Further detailed study ofthe planet, planetesimal and dust populations in these rare objects has the potential to uncover thechaotic evolutionary history of these systems and to shed light on the history of the solar system.

Subject headings: circumstellar matter planetary systems: formation

1. INTRODUCTION

Planetesimal belts appear to be a common feature ofplanetary systems. There are two main belts in the solarsystem: the asteroid belt and the Kuiper belt. Thesebelts inhabit the regions of the solar system where plan-etesimal orbits can remain stable over the 4.5 Gyr ageof our system (Lecar et al. 2001). The larger planetes-imals in the belts are continually grinding down feedingthe smaller bodies in a process known as a collisional

Electronic address: [email protected] Michelson Science Center, California Institute of Technology,

M/S 100-22, Pasadena, CA 91125, USA

cascade which is slowly eroding the belts (Bottke et al.2005). The smallest dust in the asteroid belt is acted onby radiation forces; P-R drag makes the dust spiral in to-ward the Sun making a disk known as the zodiacal cloudthat the Earth sits in the middle of (Leinert & Grun1990). A dust cloud is also predicted to arise from colli-sions amongst Kuiper belt objects (Liou & Zook 1999),although our information on this population is sparse(Landgraf et al. 2002) because its emission is maskedby the zodiacal emission (Backman, Dasgupta & Stencel1995) and few dust grains make it into the inner solarsystem (Moro-Martn & Malhotra 2003).

Many extrasolar systems also have such planetesimal
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2 Wyatt et al.

belts, known as debris disks. These have been detectedfrom their dust content (Aumann et al. 1984) from whichit has been inferred that larger planetesimals must existto replenish the dust disks because of the short lifetimeof this dust (Backman & Paresce 1993). The collisionalcascade scenario is supported by modeling of the emis-sion spectrum of the dust which shows a size distributionsimilar to that expected for dust coming from a colli-

sional cascade (Wyatt & Dent 2002, hereafter WD02).However, the issue of how these disks evolve has recentlycome under close scrutiny.

From a theoretical point view, Dominik & Decin (2003;hereafter DD03) showed that if P-R drag is not importantthen a planetesimal belt evolving in quasi-steady statewould lose mass due to collisional grinding down giving adisk mass (and dust luminosity) that falls off t1. Thisis in broad agreement with the observed properties ofdebris disks: the mean dust luminosity at a given age fallsoff t1.8 (Spangler et al. 2001); the mass inferred fromdetection statistics falls off t0.5 (Greaves & Wyatt2003), while the mass of the detected disks falls off t1(Najita & Williams 2005); the upper limit in luminosity

of the detected disks also falls off t

1 (Rieke et al.2005). While these trends can be viewed as a success ofthe steady-state model, it has yet to be proved that asteady state evolution model fits the data in more than

just general terms (Meyer et al. 2006). Several puzzlingobservations also remain to be explained.

Decin et al. (2003) noted that the maximum frac-tional luminosity of debris disks remains constant atf = Lir/L 103 up to the oldest stars, where Lirand L are the disk and stellar luminosities respectively(see also Table A1 for definitions of the parameters usedin the text), and this was explained by DD03 as a con-sequence of delayed stirring. A delay in the ignition ofa collisional cascade is expected if it is the formation of

Pluto-sized objects which trigger the cascade, since suchmassive bodies take longer, up to several Gyr, to formfurther from the star (Kenyon & Bromley 2002). How-ever, that interpretation predicts that the radius of thebelts should increase with stellar age, and this is notobserved (Najita & Williams 2005). There is also re-cent evidence that the dust content of some systems istransient. The discovery of a population of dust grainsaround Vega in the process of removal by radiation pres-sure indicates that this system cannot have remained insteady state for the full 350 Myr age of the star (Su etal. 2005). Rieke et al. (2005) used their statistics on Astars, which showed a wide variety of properties amongthe debris disks, to suggest that much of the dust wesee is produced episodically in collisions between largeplanetesimals. There is also an emerging population ofdebris disks detected around sun-like stars with dust ata few AU (Gaidos 1999; Beichman et al. 2005; Song etal. 2005; Smith, Wyatt & Dent in prep.). There is de-bate over whether these are atypically massive asteroidbelts or the consequence of a rare transient event (e.g.,Beichman et al. 2005).

A stochastic element to the evolution of debris diskswould fit with our understanding of the evolution of thedust content of the inner solar system. This is believedto have been significantly enhanced for timescales of afew Myr following collisions between objects 100 kmin size in the asteroid belt (Nesvorny et al. 2003; Far-

ley et al. 2006). However, it is not known whether theaftermath of individual collisions would be detectable ina debris disk, or indeed whether such events would hap-pen frequently enough to explain the statistics (WD02;Telesco et al. 2005). Such events have a dramatic effecton the amount of dust in the solar system because thereis relatively little around during the quiescent periods.Planetesimal belts of equivalent mass to those in the so-

lar system would not have been detected in the currentdebris disk surveys. However, there is evidence to sug-gest that both belts were 200 times more massive inthe past (e.g., Stern 1996; Bottke et al. 2005). Periodsanalogous to the heavy bombardment experienced in thesolar system up to 700 Myr after its formation havealso been invoked to explain the fact that debris disks aremost often detected around stars < 400 Myr old (Habinget al. 1999).

In the light of this controversy we revisit a simple an-alytical model for the steady state collisional evolutionof planetesimal belts which was originally explored inDD03. The model we derive for that evolution is givenin

2, and differs in a subtle but important way from that

of DD03, since it affects the dust production as a func-tion of collision velocity. This model shows that thereis a maximum possible disk mass (and dust luminosity)at any given age. In 3 confrontation with the few hotplanetesimal belts discovered recently shows that the ma-

jority of these cannot be explained as massive asteroidbelts, rather these must be systems undergoing a tran-sient event. The possibility that these are caused bya recent collision within a planetesimal belt is also dis-cussed, as is the possibility that the dust originates ina planetesimal belt in the terrestrial planet region. Theimplications of these results are discussed in 4. Appli-cation of the model to the statistics of detected debrisdisks will be considered in a later paper (Wyatt et al., in

prep.).2. ANALYTICAL COLLISIONAL EVOLUTION MODEL

In this section a simple analytical model is developedfor the evolution of a planetesimal belt due to collisionsamongst its members. The parameters used in this modelare summarized in the table A1 which also gives the unitsassumed for these parameters throughout the paper.

2.1. The planetesimal belt size distribution

The planetesimal belt is assumed to be in collisionalequilibrium with a size distribution defined by:

n(D) = KD 23q, (1)

where q = 11/6 in an infinite collisional cascade

(Dohnanyi 1969) and the scaling parameter K is called faby DD03. That distribution is assumed to hold from thelargest planetesimal in the disk, of diameter Dc, down tothe size below which particles are blown out by radiationpressure as soon as they are created, Dbl. If we assumethat q is in the range 5/3 to 2 then most of the mass is inthe largest planetesimals while the cross-sectional area isin the smallest particles such that:

tot = 3.5 1017K(3q 5)1(109Dbl)53q (2)Mtot = 8.8 1017K(6 3q)1D63qc , (3)

= 2.5 109

3q 56 3q

totDbl

109Dc

Dbl

63q,(4)


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Transience of hot dust around sun-like stars 3

where spherical particles of density have been assumedand Mtot is in M if the units of table A1 are used forthe other parameters.

The planetesimal belt is assumed to be at a radius r,and to have a width dr (in AU). One of the observableproperties of a planetesimal belt is its fractional lumi-nosity, f = Lir/L, i.e., the infrared luminosity from thedisk divided by the stellar luminosity. Assuming that

the grains act like black bodies and so absorb all theradiation they intercept we can write:

f = tot/(4r2). (5)

In other words, in this model tot, Mtot and f are allproportional to each other and just one is needed to de-fine the scaling factor K in equation (1). Assuming theparticles act like black bodies also allows us to derive thefollowing relation:

Dbl = 0.8(L/M)(2700/), (6)

where Dbl is in m, L and M are in solar units, and is in kg m3.

Relaxing the black body assumption is easily achieved

(e.g., WD02). However, this would result in relativelysmall changes in the way f scales with Mtot, and so forits heuristic simplicity we keep this assumption through-out this paper. Probably the most important simplifi-cation within this model is that of the continuous sizedistribution. For example, we know that the cut-off inthe size distribution at Dbl would cause a wave in thesize distribution at sizes just larger than this (Thebault,Augereau & Beust 2003), that large quantities of blow-out grains can also affect the distribution of small sizeparticles (Krivov, Mann, & Krivova 2000), and that thedependence of planetesimal strength on size can result inq = 11/6 as well as a wave in the distribution at largesizes (Durda et al. 1998; OBrien & Greenberg 2003).

Also, since the largest planetesimals would not be in col-lisional equilibrium at the start of the evolution, theirinitial distribution may not be the same as that of a col-lisional cascade, although distributions with q 11/6have been reported from planet formation models (e.g.,Stern & Colwell 1997; Davis & Farinella 1997; Kenyon& Luu 1999) meaning this is a reasonable starting as-sumption. Despite these simplifications, we believe thismodel is adequate to explore to first order the evolutionof planetesimal belts which can later be studied in moredepth.

2.2. Collisional evolution

In a collisional cascade material in a bin with a givensize range D to D + dD is replaced by fragments fromthe destruction of larger objects at the same rate thatit is destroyed in collisions with other members of thecascade. The long-timescale evolution is thus determinedby the removal of mass from the top end of the cascade.In this model the scaling factor K (and so the total massand fractional luminosity etc) decreases as the number ofplanetesimals of size Dc decreases. The loss rate of suchplanetesimals is determined by their collisional lifetime,which in the terminology of WD02 is given by:

tc =

r3/M(rdr/tot)[2I/f(e, I)]/fcc, (7)

where maintaining the units used previously gives tc inyears, I is the mean inclination of the particles orbits

(which determines the torus height), f(e, I) is the ratioof the relative velocity of collisions to the Keplerian ve-locity (= vrel/vk, also called by DD03), and fcc is thefraction of the total cross-sectional area in the belt whichis seen by planetesimals of size Dc as potentially causinga catastrophic collision.

From hereon we will use the assumption that f(e, I) =1.25e2 + I2, where e is the mean eccentricity of the par-

ticles, which is valid for Rayleigh distributions of e andI (Lissauer & Stewart 1993; Wetherill & Stewart 1993).An expression for fcc was given in WD02, however, herewe will ignore the gravitational focussing effect, which isimportant in the accumulation phase but not during thedestruction phase of a planetesimal belt (see 3.2), andso derive an expression that is the same as that given inWyatt et al. (1999):

fcc = (109Dbl/Dc)

3q5G(q, Xc), (8)

where Xc = Dcc/Dc, Dcc is the smallest planetesimalthat has enough energy to catastrophically destroy aplanetesimal of size Dc (which is called in DD03), and:

G(q, Xc) =[(X53qc 1) + (6q 10)(3q 4)

1

(X43qc 1)

+(3q 5)(3q 3)1(X33qc 1)]. (9)The factor Xc can be worked out from the dispersal

threshold, QD, defined as the specific incident energy re-quired to catastrophically destroy a particle such that(WD02):

Xc = (2QD/v

2rel)

1/3, (10)

= 1.3 103[QDrM1 f(e, I)2]1/3, (11)where QD is in J kg

1 (called S in DD032)Combining the above equations gives for the collisional

lifetime of the planetesimals of size Dc:

tc =

r2.5drM0.5 tot

2[1 + 1.25(e/I)2]0.5

G(q, Xc)

109Dbl

Dc

53q(12

=

3.8r2.5drDc

M0.5 Mtot

(12q 20)[1 + 1.25(e/I)2]0.5

(18 9q)G(q, Xc)

.(13

Assuming that collisions are the only cause of mass lossin the belt, the evolution of the disk mass Mtot(t) (orequivalently ofK, tot, or f) can be worked out by solv-ing dMtot/dt = Mtot/tc to give:

Mtot(t) = Mtot(0)/[1 + t/tc(0)], (14)

where Mtot(0) is the initial disk mass and tc(0) is thecollisional lifetime at that initial epoch; this solution isvalid as long as mass is the only parameter of the plan-

etesimal belt that changes with time. This results in adisk mass which is constant at Mtot(0) for t tc(0),but which falls off 1/t for t tc(0) (as noted, e.g., inDD03).

However, another interesting property of this evolutionis that, since the expression for tc(0) includes a depen-dence on Mtot(0), the disk mass at late times is indepen-dent of initial disk mass. This is because more massive

2 Equation 25 in DD03 differs from our equation (10) becausewe define QD to be the specific incident kinetic energy so that

0.5M2v2rel = M1Q

Dwhereas DD03 define S to be the specific bind-

ing energy of the two objects (giving their equation 24). In the limitof S v2rel/8 the two equations are the same, since Xc 1.


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Fig. 1. The dependence of (left) G(11/6, Xc) and (right) Xc on planetesimal eccentricity (e) for planetesimals of different strengths(QD) and at different distances from the star (r).

disks process their mass faster. This means that for anygiven age, tage, there is a maximum disk mass Mmax(and also infrared luminosity, fmax) that can remain dueto collisional processing:

Mmax =

3.8 106r3.5(dr/r)Dc

M0.5 tage

(12q 20)[1 + 1.25(e/I)2]0.5(18 9q)G(q, Xc)

, (15)

fmax =

106r1.5(dr/r)

4M0.5 tage

109Dbl

Dc

53q

2[1 + 1.25(e/I)2]0.5

G(q, Xc)

. (16)

In this model, the present day disk mass (or luminosity)is expected to be equal to this maximum disk mass (or

luminosity) for disks in which the largest planetesimalsare in collisional equilibrium. This corresponds to disksaround stars that are older than the collisional lifetimeof those planetesimals given in equation (13).

For example, with the further assumptions that q =11/6, e I, and = 2700 kg m3, we find:

Mmax = 0.009r3.5(dr/r)DcM

0.5 t

1age/G(11/6, Xc),(17)

fmax = 0.004r1.5(dr/r)D0.5c L

0.5 t

1age/G(11/6, Xc),(18)

where Mmax is in M, r in AU, Dc in km, tage in Myr,and G(11/6, Xc) = X0.5c + 0.67X

1.5c + 0.2X

2.5c 1.87,

with Xc = 103(rQD/e2)1/3 (QD is in J kg

1).Plots of G(11/6, Xc) and Xc for typical planetes-

imal belts are shown in Fig. 1. However, formany disks the approximation that Xc 1 isvalid, and so G(11/6, Xc) 0.2X2.5c = 6.3 106r5/6QD

5/6e5/3M5/6 , giving:

Mmax = 1.4 109r13/3(dr/r)DcQD5/6 e5/3M

4/3 t

1age, (19)

fmax = 0.58 109r7/3(dr/r)D0.5c QD5/6 e5/3M

5/6 L

0.5 t

1age. (20)

2.3. Comparison with DD03

Since DD03 produced a very similar analytical model,our results were compared with those of DD03. Theresults of disk evolution for a planetesimal belt closeto their nominal model were computed using the pa-

rameters: r = 43 AU, dr = 15 AU, Dc = 2 km , = 2700 kg m3, f(e, I) = 0.1, e/I = 1, QD = 200 Jkg1, Mtot(0) = 10M, A0 star (for which L = 54L,M = 2.9M, Dbl = 15 m). Each of the parametersMtot(0), r, f(e, I), Dc and spectral type were also variedto make the plots shown in Fig. 2 which are equivalentto Figs 1b-1f of DD03.

The results are very similar in most regards: moremassive disks start out with higher f, but the turnoverfrom constant to 1/t evolution is later for lower massdisks meaning that at late times all disks converge to thesame maximum value (Fig. 2a); putting the same massat larger distances reduces the initial dust luminosity f,but the resulting lower surface density and longer orbital

timescales there combine to make the turnover happenlater which means that at late times more distant beltsare more massive (Fig. 2b); putting the same mass intolarger planetesimals reduces the cross-sectional area ofdust (equation 4) and so the initial dust luminosity f,but increases the collisional lifetime of those planetesi-mals (equation 13) which means that at late times beltswith larger planetesimals retain their mass for longer(Fig. 2d); later spectral types have higher starting dustluminosities because the cascade extends down to smallersizes (equation 6), and the longer orbital times mean thatthey keep their mass for longer (Fig. 2e).

Where the models differ is in the exact way Mtot isused to get f and tc, and in the way the evolution is af-

fected by changing vrel/vk (Fig. 2c). This is because themodels make different assumptions. Here we assume thatthe size distribution is continuous between Dc and Dbl,whereas in DD03 the large planetesimals feeding the cas-cade are seen as separate from the cascade. This meansthat for us Mtot gives a direct estimate of K (equation3) and so the amount of dust f, while for DD03 theyequate the mass flow through the cascade with the massinput from the break-up of planetesimals meaning thatwhile their scaling parameter is proportional to Mtot (asis ours), it also includes a dependence on the parameterwe call Xc which affects the mass flow rate in the cas-cade. This explains all of the differences: the details of


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Fig. 2. The collisional evolution of a planetesimal belt with parameters similar to the nominal model of DD03 [ r = 43 AU, dr = 15AU, Dc = 2 km, = 2700 kg m3, f(e, I) = 0.1, e/I = 1, QD = 200 J kg

1, Mtot(0) = 10M, A0 star] showing the effect of changing:(top left) starting disk mass Mtot(0), (top right) disk radius r, (middle left) collision velocity vrel/vk, (middle right) maximumplanetesimal size Dc, and (bottom left) stellar spectral type. These plots can b e directly compared to figs. 1b-f of DD03.

the scaling explain the slightly different initial f valuesin all the figures, and the fact that for us planetesimals ofsize Dc are destroyed by planetesimals down to size XcDcmeans that our collisional lifetimes are always shorterthan those in DD03, since they assume that planetes-imals only collide with same size planetesimals. For uschanging vrel/vk does not affect the initial f parameter asdescribed above, but it does affect the collisional lifetimeof the largest planetesimals which can survive longer ifvrel/vk is reduced (since this means that fewer planetes-imals in the cascade cause destruction on impact). Theopposite is the case for the DD03 model: changing vrel/vkdoes not affect the collisional lifetime of the largest plan-

etesimals, since they only collide with each other, but alower collision velocity does increase the initial dust lu-minosity because the cascade must have more mass in itto result in a mass flow rate sufficient to remove mass in-troduced by the large planetesimals. While the differenceis subtle, it is important, since vrel/vk may be importantin determining the presence of dust at late times (DD03;section 3).

On the face of it, it seems that our model provides amore accurate description of the disk. The reason is thatin a collisional cascade the mass flow does not need to betaken into account, since it results in the q = 11/6 sizedistribution (Tanaka et al. 1996). In other words the


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6 Wyatt et al.

dependence of the scaling of the cascade with Xc foundby DD03 should have been removed if the largest plan-etesimals had been allowed to collide with smaller plan-etesimals (since increasing Xc would have both restrictedmass flow within the cascade and slowed down the massinput from the destruction of large planetesimals). How-ever, it is also true that the q = 11/6 distribution onlyapplies in an infinite cascade, and since both models have

truncated the size distribution at Dc, this would affectthe evolution. Also, the effect of the variation of QD withD on the size distribution and its evolution are not yetclear, and neither is the evolution of the size distributionwhile the collisional cascade is being set up. These is-sues will be discussed only briefly in this paper, in whichthe simple evolution model described above is applied tosome of the latest observational results on debris disks.

3. APPLICATION TO RARE SYSTEMS WITH HOT DUST

Very few main sequence stars exhibit hot dust within 10 AU, i.e., in the region where we expect planets mayhave formed. Four surveys have searched for hot dustaround sun-like stars (main sequence F, G or K stars) by

looking for a 25 m flux in excess of photospheric levelsusing IRAS (Gaidos 1999), ISO (Laureijs et al. 2002) andSpitzer (Hines et al. 2006; Bryden et al. 2006). All con-cluded that only 22% of these stars have hot dust withinfrared luminosities f = Lir/L > 10

4, finding a totalof 3 candidates. Other hot dust candidates exist in theliterature, however some IRAS excess fluxes have turnedout to arise from chance alignments with background ob-

jects (e.g., Lisse et al. 2002), including the candidateHD128400 from the hot dust survey of Gaidos (1999)(Zuckerman, priv. comm.). Thus confirmation of thepresence of dust centred on the star using ground- andspace-based mid-IR imaging is vitally important (Smith,Wyatt & Dent, in prep.). The tally of confirmed hot dust

sources now stands at seven, and these are summarizedin Table 1 which also gives the estimated radial locationof the dust based on fitting of the spectral energy dis-tribution of the excess emission; for all stars the dust ispredicted to lie at < 10 AU.

While the frequency of the presence of such emissionis low, there is as yet no adequate explanation for its ori-gin and why it occurs in so few systems. Analogy withthe solar system suggests that these are systems in whichwe are witnessing the collisional grinding down of atyp-ically massive asteroid belts. However, other scenarioshave also been proposed in which the dust is transient,having been produced in some stochastic process. Sucha process could be a recent collision between two massiveprotoplanets in an asteroid belt (Song et al. 2005), thesublimation of one supercomet (Beichman et al. 2005),or the sublimation of a swarm of comets, possibly scat-tered in from several tens of AU in an episode analogousto the period of Late Heavy Bombardment in the solarsystem (Gomes et al. 2005).

3.1. Are these massive asteroid belts?

Here we consider the possibility that these are atypi-cally massive asteroid belts, and show that for the major-ity of the known systems this is unlikely to be the case.The reason is that given in 2.2, which is that more mas-sive asteroid belts are not necessarily more dusty at latetimes, and there is a maximum dust luminosity we can

expect for a belt of a given age, given its radial loca-tion (equations 15-20). To arrive at a rough estimate ofthe maximum possible fmax we assume the following pa-rameters: the largest possible planetesimal is Dc = 2000km, since this is above the largest members of the aster-oid and Kuiper belts, and fits with the expectation thatplanetesimal growth is halted once the largest planetes-imals reach this size due to the resulting gravitational

perturbations (Kenyon & Bromley 2002); belt width isdr = 0.5r; planetesimal strength is QD = 200 J kg

1, thecanonical value used in DD03, although gravity strength-ening can give rise to higher values for planetesimalslarger than 1 km (see 3.2); planetesimal eccentricityis e = 0.05, typical for planetesimal belts like the asteroidbelt that are undergoing a collisional cascade, and closeto that expected from stirring by 2000 km planetesimalswithin such belt. 3 Substituting in these nominal val-ues into equation (20) and approximating M = L = 1gives:

fmax = 0.16 103r7/3t1age. (21)

Plots analogous to those in Fig. 2 are presented in Fig. 3which shows the evolution for a planetesimal belt withthe nominal parameters described above (and with anominal starting mass ofMtot(0) = 1M) along with theconsequence for the evolution of changing any of thoseparameters. Note that it is most appropriate to refer toFig. 3, rather than Fig. 2, when considering the evolutionof planetesimal belts close to sun-like stars.

The value of fmax is quoted in Table 1 under the as-sumption that the planetesimal belt has the same ageas the star. The quoted value for each star is that fromequation (18) for its spectral type, but is within a fac-tor of three of that given in equation (21), indicatingthat this equation may be readily applied to observedbelts in the future. The four oldest systems (BD+20307,HD72905, Corvi and HD69830) have fobs 103fmax.We show in 3.2 that even with a change in parametersit is not possible to devise asteroid belts in these systemsthat could survive to the age of the stars giving rise tothe observed dust luminosities. Thus we conclude thatthis period of high dust luminosity started relatively re-cently. The timescale over which a belt can last abovea given luminosity, fobs, is tagefmax/fobs, since collisionswould grind a belt down to this level on such a timescale.This implies that belts this luminous only last between afew thousand years (BD+20307 and HD72905) and a fewMyr ( Corvi and HD69830). However, the true durationof this level of dust luminosity depends on the details ofthe process causing it, and moreover there is still up to

two orders of magnitude uncertainty in fmax (see 3.2).Thus this calculation should not yet be used to infer fromthe 2% of systems with hot dust that, e.g., every sun-like star must undergo 10-1000 such events in its lifetime(or fewer systems must undergo even more events). Fornow the conclusion is that these systems cannot be plan-etesimal belts that have been evolving in a collisionalcascade for the full age of the star.

This leaves open the possibility that the collisional cas-cade in these systems was initiated much more recently,

3 Equating the velocity dispersion in the belt with the es-cape velocity of a planetesimal of size Dc gives e 2.6 1070.5r0.5M0.5

Dc.


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TABLE 1Main sequence sun-like (F, G and K) stars in the literature with evidence for hot dust at < 10 AU.

Star name Sp. Typ e Age, Myr Radius, AU fobs = Lir/L fmax Transient? Reference

HD98800c K4/5V 10 2.2 220 103 270 106 Not req Low et al. (2005)HD113766ac F3V 16 3 2.1 103 45 106 Not req Chen et al. (2005)

HD12039 G3/5V 30 4-6 0.1 103 200 106 Not req Hines et al. (2006)

BD+20307a

G0V 300 1 40 103

0.36 106

Yes Song et al. (2005)HD72905a G1.5V 400 0.23b 0.1 103 0.011 106 Yes Beichman et al. (2006a) Corvia F2V 1000 1-2b 0.5 103 0.15 106 Yes Wyatt et al. (2005)

HD69830a K0V 2000 1 0.2 103 0.13 106 Yes Beichman et al. (2005)

ainfrared silicate featurebalso has cool dust component at > 10 AUcbinary star

perhaps because a long timescale was required to formthe 2000-3000 km sized planetesimals necessary to stirthe planetesimal belt and cause the switch from accre-tion to collisional cascade (Kenyon & Bromley 2004).However, we consider this to be unlikely, because thetimescale for the formation of objects of this size at 1

AU from a solar mass star was given in Kenyon & Brom-ley (2004) to be 0.6dr/Mtot Myr, where Mtot is themass of material in an annulus of width dr, just as in therest of the paper. This means that the cascade can onlybe delayed for 100-1000 Myr at 1 AU for planetesimalbelts of very low mass, which would also be expected tohave low dust luminosities when the cascade was even-tually ignited. For example, a delay of > 500 Myr wouldrequire < 0.6 103M in the annulus at 1 AU of 0.5AU width, a mass which corresponds to a fractional lu-minosity of< 5106 (equations 4 and 5 with = 2700kg m3 and q = 11/6), much lower than that observed inall systems. One can also consider the same argument inthe following way: the observed luminosity fobs implies

a planetesimal belt mass which current planet formationtheories indicate would result in the growth of 2000 kmplanetesimals which would ignite a collisional cascade ona timescale of 3103(dr/r)/fobs Myr if this was placedat 1 AU from a solar mass star. The conclusion at theend of the last paragraph also considers the collisionalcascade to evolve in quasi-steady state, and it is possi-ble that collisions between large members of the cascademay have recently introduced large quantitites of smalldust; that possibility is discussed in 3.3.

For the three youngest systems the conclusions are lessclear. The dust luminosities of HD12039 and HD113766are, respectively, close to and fifty times higher than themaximum allowed value for collisionally evolved plan-etesimal belts. However, given the uncertainties in theparameters in the model (described in 3.2), we concludethat it is not possible to say that these could not bemassive asteroid belts. The main reason that firm con-clusions cannot be drawn is the large radial location ofthe dust at > 2 AU. The strong dependence of fmaxon r means that it is easiest to constrain the nature ofbelts within a few AU which evolve very rapidly. Forthe youngest system (HD98800), while its dust luminos-ity lies a factor of 800 above the maximum for the ageof the star, we do not infer that this must be transient,since the high dust luminosity and low age imply thatthis system is in a transitional phase and the collisionalcascade in this debris disk is likely to have only recently

been ignited. Rather we note that this model impliesthat due to collisional processing this debris disk can-not maintain this level of dust emission beyond the next 10, 000 years (albeit with an additional two orders ofmagnitude uncertainty, 3.2).

3.2. Possible caveatsGiven the large number of assumptions that went into

the estimate for fmax, it is worth pointing out that thismodel is in excellent agreement with the properties of theasteroid belt in the solar system, since for a 4500 Myrbelt at 3 AU the model predicts Mmax = 0.4 103M,which is close to the inferred mass of the asteroid belt of0.6 103M (Krasinsky et al. 2002). The model alsopredicts fmax = 5 107, which is consistent with theestimate for the zodiacal cloud of Lir/L = 0.8 107(Backman & Paresce 1993). 4 It is also necessary to ex-plore if there is any way in which the parameters of themodel could be relaxed to increase fmax and so changethe conclusions about the transience of the hot dust sys-

tems. Equation (20) indicates one way in which fmaxcould be increased, which is by either reducing the ec-centricities of the planetesimals, e, or increasing theirstrength, QD, both of which could increase Xc and sodecrease the rate at which mass is lost from the cascade(e.g., fig. 3). The other way is to change the size distri-bution so that a given disk mass results in a significantlylarger dust luminosity, e.g., by increasing q.

In fact Benz & Asphaug (1999) found a value of QDthat is higher than 2 105 J kg1 for planetesimalsas large as 2000 km for both ice and basalt composi-tions. This would result in an increase in fmax by afactor of 170 (e.g., Fig. 3). However, such a high valueof QD is possible only due to gravity strengthening of

large planetesimals, and the dependence in this regimeofQD D1.3 (Benz & Asphaug 1999) would result in anequilibrium size distribution with qg 1.68, since whenQD Ds then q = (11 + s)/(6 + s) (OBrien & Green-berg 2003). If such a distribution was to hold down tothe smallest dust grains the net result would be a de-crease in fmax by 200. This is not the case, how-

4 In planetesimal belts as tenuous as the asteroid belt, the effectof P-R drag is important (Wyatt 2005) meaning that the cross-sectional area of dust in the zodiacal cloud is dominated by 100 m sized grains rather than grains of size Dbl as assumed inthe simple model of 2.1. Taking this into account would reducethe fractional luminosity predicted by the model by an order ofmagnitude.


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8 Wyatt et al.

Fig. 3. The collisional evolution of a planetesimal belt at r = 1 AU around a sun-like star (L = M = 1) of initial mass Mtot(0) = 1Massuming that belt can be described by the parameters used in 3.1 (i.e., dr/r = 0.5, Dc = 2000 km, = 2700 kg m3, e = 0.05, e/I = 1,QD = 200 J kg

1). All panels show dust luminosity f = Lir/L as a function of time, and the evolution with the above nominal parametersis shown with a solid line. The different panels show the effect of changing the following parameters: (top left) starting disk mass Mtot(0),

(top right) disk radius r, (middle left) planetesimal eccentricity e, (middle right) maximum planetesimal size Dc, (bottom left)planetesimal strength Q

D, and (bottom right) stellar spectral type.

ever, since objects in the size range D < Dt = 0.15 kmare in the strength scaled regime where QD D0.4leading to a size distribution with qs = 1.89 in thisrange. According to OBrien & Greenberg (2003) thesize distribution of a collisional cascade with a realisticQD prescription should have two components (charac-terized by qg and qs), but there is a discontinuity atthe transition size Dt with the strength scaled compo-nent shifted down by an appropriate amount xt (see theirFig. 3b). This means that fmax should be higher thanthat derived using equation (16) with q = qg by a factor

xt(3qg5)(3qs5)1(Dbl/Dt)3(qgqs). Since xt < 1, thensubstituting the values from Benz & Asphaug (1999)given above implies that Table 1 underestimates fmax byat most a factor of 50-100 (possibly much less). In otherwords, we anticipate that by including a more realisticprescription for QD and the resulting size distribution,this would change the inferred fmax but not upwards byan amount more than two orders of magnitude. For thisreason, transience is only inferred for those systems forwhich fobs/fmax 100.

A lower eccentricity is, however, one potential av-


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enue for increasing the amount of dust remaining at latetimes. Equation (20) shows that since G(11/6, Xc) e5/3 (Fig. 1a), this means that reducing e from 0.05to 0.01 or 0.001 gives a decrease in G(11/6, Xc) of 15or 680, and so an increase in fmax by a correspondingamount (e.g., Fig. 3). In fact the increase can be muchmore than this, since when e is reduced to levels below

4.7105

QDrM

1 [1.25 + (I/e)

2

]1

then Xc > 1 (e.g.,Fig. 1b). In such a regime mutual collisions do not resultin the destruction of planetesimals, rather in their mergerand growth. At this point G(11/6, Xc) < 0, i.e., fmax isinfinite since, in this simple model, whatever the start-ing conditions there is no evolution (although in practicethe size distribution would evolve due to planetesimalgrowth). At 1 AU, this means e must be larger than0.0005 (for QD = 200 J kg

1, appropriate for Dc = 0.15km) or 0.014 (for QD = 2 105 J kg1, appropriate forDc = 2000 km) to initiate a collisional cascade, valueswhich are consistent with those quoted by more detailedplanet formation models (e.g., Kenyon & Bromley 2002).Such eccentricities would be expected through stirring

either by > 1000 km planetesimals which formed withinthe belt, or by more massive perturbers which formedoutside the belt, both of which can be expected to occurwithin 10-100 Myr (Kenyon & Bromley 2006). This wasconsidered in 3.1 where it was shown that the cascadewould be initiated following the growth of 2000 kmplanetesimals on timescales that are much shorter thanthe age of the system for the disk masses required toproduce a dust luminosity at the observed level.

The only route which could plausibly maintain the hotdust systems in Table 1 in collisional equilibrium overthe age of the stars might be to invoke some mecha-nism which maintains the eccentricity at a level whichthe cascade is only just being eroded. However, Fig. 1a

shows that G(11/6, Xc) is a strong function of e whenG(11/6, Xc) < 1, since the range G(11/6, Xc) = 0 1is covered by a factor of less than two in eccentricity.Thus we consider it reasonable to assume that the bestpossible combination of QD and e in this regard wouldresult in G(11/6, Xc) 1 (corresponding to Xc = 0.69);lower values ofG(11/6, Xc) are possible, but only withina very narrow range of eccentricity. Since in the aboveexample with a realistic QD prescription extending up to2000 km we assumed e = 0.05 which already resulted inG(qg, Xc) < 1, we consider that it is not reasonable tofine tune the eccentricity further to increase fmax; e.g.,decreasing to e = 0.03 results in some disks not evolvingand the rest with fmax higher than that quoted in Table

1 by a factor 150. Thus we conclude that the estimategiven in Table 1 (and e.g., equation 16) underestimatesfmax by at most a factor of 100, unless the eccentricityhappens to lie within 10% of a critical value.

It is also worth noting that low levels of eccentricitywould result in large gravitational focussing factors forlarge planetesimals which would enhance fcc and so de-crease the time for these planetesimals to be catastroph-ically destroyed, something which is compounded by thehigher collision velocity in gravitationally focussed colli-sions which reduces Xc because collisions with smallerplanetesimals can cause catastrophic disruption (e.g.,equation 11). However, we do not need to account forthis here, since gravitational focussing becomes impor-

tant when vrel < vesc

(2/3)G(103D) and so

when e < 4 107

rM1 [1.25 + (I/e)2]1D (where

D is in km); i.e., when e < 2 106 for D = 0.15 kmand e < 0.027 for D = 2000 km at 1 AU from a 1 M star,both of which occur close to or below the level at whichcollisions result in accumulation rather than destruction.

3.3. Are these the products of single collisions?One possible origin for the hot dust which is quoted in

the literature is that it is the product of a single colli-sion (Song et al. 2005). Our model can be used to makefurther predictions for the likelihood of massive collisionsoccurring within an asteroid belt. The maximum num-ber of parent bodies (i.e., planetesimals) larger than Dpbremaining at late times occurs when Mtot = Mmax andso is given by:

n(D > Dpb) =

5.6 1010r3.5(dr/r)

M0.5 D2c tage

Dc

Dpb

3q3 1

(3q

5)[1 + 1.25(e/I)2]0.5

(3 3q)G(q, Xc) . (22)The collision timescale for planetesimals of size Dpb is

tc(Dpb) = tc(Dc)fcc(Dc)/fcc(Dpb)

= 106tage(Dpb/Dc)3q5, (23)

noting that the collisional lifetime of the largest plan-etesimals, tc(Dc), is the age of the star for a planetes-imal belt at maximum luminosity for this age. Thesecan be combined to give the destructive collision rate forplanetesimals larger than Dpb:

dNc(D > Dpb)/dt =1000r13/3(dr/r)t2ageDcD

3pb

M

4/3

Q

D

5/6

e

5/3

, (24)in Myr1, where the assumptions that q = 11/6, e = Iand Xc 1 have been used in deriving this equation.

We now assume that we are considering collisions capa-ble of reproducing the observed dust level, fobs, so thatthe lifetime of the resulting collision products can be esti-mated from the collisional lifetime of that dust, assumedto be of size Dbl (WD02):

tc(Dbl) = 0.04r1.5M0.5 (dr/r)f

1obs, (25)

in years, noting that collisions would remove the dust ona faster timescale than P-R drag (Wyatt 2005; Beichmanet al. 2005). Combining equations (24) and (25) givesthe fraction of time that collisions are expected to result

in dust above a given level of fobs:

P(f > fobs) = 4 105r35/6(dr/r)2t2ageDcD3pb M11/6 f

1obsQ

D

5/6e5/3. (26)

To estimate the minimum size of the parent body, Dpb,responsible for this dust, we consider how large a plan-etesimal must be to reproduce fobs if a destructive colli-sion resulted in one fragment with half the mass of theoriginal planetesimal (i.e., the definition of a destructivecollision), with the remaining mass in particles of sizeDbl:

Dpb = 890[Dblr2fobs]

1/3. (27)


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10 Wyatt et al.

TABLE 2Parameters in the model for the hot dust systems of Table 1 used to determine whether the observed dust can be the

outcome of a single collision in a massive asteroid belt which is itself not normally bright enough to be detected.

Star name Dpb, km N(D > Dpb) dNc(D > Dpb)/dt, Myr1 tc(Dbl), yr P(f > fobs) Single coll ision?

HD98800 530 200 41 0.36 15 106 NoHD113766 280 890 150 41 6100 106 Not imposs

HD12039 110 77,000 12,000 2300 27*

Not impossBD+20307 320 0.47* 0.0039 0.49 0.0019 106 NoHD72905 15 1.4 0.036 22 0.79 106 No Corvi 110 7.8 0.033 59 2.0 106 No

HD69830 39 19 0.068 110 7.7 106 No

*For disks with P(f > fobs) > 1, this value indicates the number of collisions at that level we can expect to see in the disk at any one time.Likewise, for disks with N(D > Dpb) < 1, this value indicates the probability that there is an object of this size remaining in the disk.

Table 2 lists the parameters for the hot dust systemsassuming the canonical parameters of QD = 200 J kg

1,Dc = 2000 km and e = 0.05. To determine whethera system could have been reproduced by a single colli-sion, the final value of P(f > fobs) was compared withthe statistic that 2% of systems exhibit hot dust (which

therefore considers the optimistic case where all starshave planetesimal belts at a few AU). For the systemswhich were inferred in Table 1 to be transient, all areextremely unlikely (< 0.001%) to have been caused by asingle collision amongst planetesimals in a planetesimalbelt which has undergone a collisional cascade since thestar was born.

While this statistic is subject to the uncertainties inthe model parameters described in 3.2, and so couldbe in error by around two orders of magnitude, it mustalso be remembered that the most optimistic assump-tions were used to arrive at this figure. For example, itis unlikely that the destruction of planetesimals of sizeDpb would release half of the mass of the planetesimal

into dust Dbl in size.

5

On the other hand, one mightconsider that the lifetime of the observed dust, tc(Dbl),is an underestimate of the duration of dust at the levelof f > fobs, since the dust could be replenished fromthe destruction of larger particles. Indeed Farley et al.(2006) model the destruction of a 150 km planetesimalin the asteroid belt and inferred a dust peak that lasted 1 Myr, precisely because large fragments produced inthe collision replenished the dust population. However,it should be cautioned that the dust peak inferred byFarley et al. (2006) would not have been detectable asan infrared excess since it only caused a factor 10 en-hancement in the luminosity of the zodiacal cloud (i.e.,to f 0.8106), and that in the context of our model,invoking a population of larger grains that result fromthe collision would lead to a larger parent body (i.e., alarger Dpb) required to reproduce the observed luminos-ity fobs and so less frequent collisions; i.e., it may bepossible (even desirable) to increase tc(Dbl), but onlyat the expense of decreasing dNc(D > Dpb)/dt leadingto little change in P(f > fobs). We note that tc(Dbl)

5 Such an optimistic assumption should not be dismissed out ofhand, however, since the large amount of collisional processing thatmust have taken place means that planetesimals more than a fewkm would be rubble piles. These would have undergone shatteringand reaccumulation numerous times meaning that they could havedeep dusty regolith layers which could be preferentially ejected ina collision.

given in Table (2) is sufficiently short that a measure-ment of the variability of the infrared excess on realistic(few year) timescales could lead to constraints on the sizeof the grains feeding the observed phenomenon, since ifa population of larger grains existed then the luminositywould fade on much longer timescales.

A further argument against the transient disks beingcaused by single collisions is the fact that the probabilityof seeing the outcome of a collision, P(f > fobs), falls off t2age, which means that we would expect to see moretransient disks around younger stars than around olderstars (because young stars have more massive disks withmore large planetesimals and so more frequent collisions).There is some evidence from Table 1 that transience ismore common around young systems, since none of thetransient systems is older than 2 Gyr, whereas sun-likestars in the solar neighborhood would be expected tohave a mean age of 5 Gyr. However, while the statisticsare poor, a t2age dependence does seem to be ruled out;e.g., we would have expected to have detected 10 times

more transient disks caused by single collisions in theage range 50-500 Myr6 than in the age range 0.5-5 Gyr,whereas 2 transient disks are known in the younger agebin, and 2 in the older age bin which is more consistentwith a t1age dependence.

In fact, within the context of this model, all of the diskswhich we infer to be transient would also be inferred tonot be the product of single collisions. This is evidentby substituting Dpb from equation (27) and fmax fromequation (18) into equation (26) to get:

P(f > fobs) = 0.2 106(fmax/fobs)2(Me2r1QD1)5/6,(28)

which reduces to P(f > fobs) =16(fmax/fobs)2(Mr1)5/6 for the canonical param-

eters used before. Since transient disks are definedby fobs/fmax 1000, this means they cannot alsohave a high probability of having their origin in singlecollisions. It would only be inferred that disks withfobs/fmax 1000 could have their origin in singlecollisions, but since it is also possible that these disksare the result of steady state collisional evolution thereis no need to invoke a single collision to explain theirpresence, which is why Table 2 simply concluded thatit is not impossible that the disks of HD113766 and

6 It is not reasonable to extend the age range to younger systems,since, as noted in 3.1 it is hard to discern whether or not dustdetected in such systems is transient.


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TABLE 3Parameters in the model for the transient hot dust systems of Table 1 used to determine whether the observed dust

could originate in the destruction of a planetesimal belt coincident with the dust: dMloss/dt is the observed mass lossrate, Mmax is the maximum mass of a planetesimal belt that is coincident with the dust given the age of the star,

t(f > fobs) is the length of time such a planetesimal belt could sustain the observed dust luminosity, and rout(100Myr) isthe radius of a planetesimal belt which would still have enough mass to sustain the observed dust luminosity for 100

Myr.

Star name dMloss/dt, 106

MMyr1

Mmax, 106

M t(f > fobs), Myr rout(100Myr), AUBD+20307 8.0 106 53 6.7 106 45HD72905 19 0.072 3.7 103 2.4 Corvi 2500 57 0.023 9.6

HD69830 64 12 0.18 4.5

HD12039 are the product of single collisions. Whatequation (28) does indicate, however, is that it ispossible for single collisions to cause disks to spend somefraction of their time at a luminosity enhanced abovethe nominal maximum value fmax, and that this occursmore readily for disks at smaller radii and around highermass stars. However, whether single collisions really doachieve an observable increase in luminosity depends

on the size distribution of the collisional fragments, forwhich it must be remembered that equation (28) usedan unrealistically optimistic estimate.

3.4. Are parent planetesimals coincident with dust?

For similar reasons to those in 3.3 it is also possibleto show that the parent planetesimals of the dust areextremely unlikely to originate in a planetesimal belt thatis coincident with the dust. The reason is that the massremaining in such a belt would be insufficient to replenishthe dust for a length of time commensurate with thestatistic that 2% of stars show this phenomenon. Theobserved dust luminosity, assuming this is comprised ofdust of size Dbl which has a lifetime oftc(Dbl) (equation

25), implies a mass loss rate due to mutual collisionsbetween the dust grains of:

dMloss/dt = 1700f2obsr

0.5LM0.5 (r/dr), (29)

in M/Myr, and this is independent of the collisionalevolution model of 2. However, due to the collisionalevolution of a planetesimal belts largest members, thereis a maximum mass that can remain in a belt at thesame radius as the dust at this age, and this is given inequation (15). This means that if the observed dust orig-inates in an event which, for whatever reason, is causingplanetesimals in a belt at the same radius as the dustto be converted into dust, then this can last a maximumtime of t(f > fobs) = Mmax/dMloss/dt before the plan-etesimal belt is completely exhausted. These figures aregiven in Table 3 which shows that the longest the typeof transient event observed could be sustained in thesesystems is under 1 Myr, under the assumptions aboutthe planetesimal belts employed in the rest of the paper.

A maximum duration of 1 Myr is not sufficient to ex-plain the statistic that 2% of sun-like stars exhibit thisphenomenon, since the median age of such stars is 5Gyr, indicating a typical duration (even if this occursin multiple, shorter, events) of around 100 Myr. Clearlya reservoir of mass is required in excess of that which itis possible to retain so close to the star.

3.5. Constraints on parent planetesimal belt

If we assume that the observed mass of hot dust orig-inates in planetesimals that were initially in a belt at aradius rout which has properties like those assumed inthe rest of the paper, and a fractional luminosity of fout,then there are two main constraints on that belt. First,assuming that this belt has been collisionally evolving forthe age of the star, then this belt cannot have more mass(or luminosity) than the maximum that could possibly

remain due to collisional processing, i.e., fout < fmax(equation 16). Second, it must have sufficient mass re-maining to feed the observed mass loss rate for longenough to reproduce the statistic that 2% of stars ex-hibit this phenomenon which implies a total duration of> 100 Myr. For a belt to have enough mass to feed theobserved hot dust luminosity of fobs at a radius r fora total time of thot in Myr requires the belt to have aluminosity of:

fout > 710thotf2obsr

2outr

0.5D0.5c L0.5 (dr/r)

1, (30)

or rather, this is the luminosity it must have had beforeit was depleted.

Comparing this with the maximum mass possible at

this age indicates that the parent belt must have a min-imum radius of:

rout(thot) > 615t3/13hot t

3/13age f

6/13obs r

3/26(dr/r)6/13 D3/13c Q

D5/26e5/13L

3/13 M

5/26 . (31)

Table 3 gives an estimate of the minimum radial locationof such a planetesimal belt, under the assumption thatthe event (or multiple events) of high hot dust luminositylast thot = 100 Myr. These values indicate that the plan-etesimal belts must be at least a few AU from the star.It must be cautioned that this conclusion is relativelyweak in the case of HD69830, since the uncertainty inthe properties of the planetesimals still leaves two orders

of magnitude uncertainty in the maximum luminosity,fmax, and so also in the maximum mass Mmax (see 3.2).This means that, with suitable planetesimal belt prop-erties, a belt in this system that is coincident with thedust at 1 AU may be able to replenish the observed phe-nomenon for 20 Myr. However, we still consider this tobe an unlikely scenario, since it would require that themass of the planetesimal belt is depleted at a constantrate for the full 100 Myr, whereas most conceivable sce-narios would result in a mass loss rate which decreaseswith time as the planetesimal population is depleted thusrequiring an even larger starting mass.

These two constraints are summarized for the 4 sys-tems with transient hot dust in Fig. 4, which shows the


12/16

12 Wyatt et al.

Fig. 4. Constraints on the fractional luminosity and radius of the planetesimal belt feeding the observed transient hot dust (shadedregion) for the systems: (top left) Corvi, (top right) HD72905, (bottom left) HD69830 and (bottom right) BD+20307. The solidlines are the constraints imposed by the far-IR detection limits (assuming black body emission), the maximum luminosity possible in a beltat this radius due to erosion by collisional processing, and the luminosity from a belt of sufficient mass to feed the observed mass loss ratefor 100 Myr. The properties of the hot dust in these systems is shown with a diamond, and those of the cold dust, where known (the toptwo figures), shown with a triangle.

shaded region of parameter space in fout and rout wherethe parent planetesimal belt can lie. This figure alsoshows the location of the hot dust at fobs and r, illus-trating the conclusion of 3.1 that this lies significantlyabove fmax, the maximum fractional luminosity expectedfor a planetesimal belt at the age of the parent star. Notethat the value rout(100Myr) given in Table 3 denotes theintersection of the limits from fmax and from equation(30).

A third constraint for the parent planetesimal beltcomes from far-IR observations of these systems. For2/4 of the transient dust systems a colder dust compo-nent has already been detected: Corvi has a planetes-imal belt with a resolved radius of

100 AU (Wyatt et

al. 2005), and HD72905 has one inferred to be at 14AU (Beichman et al. 2006a). In both cases these outerplanetesimal belts have been inferred to be at a differ-ent spatial location from the hot dust either because ofimaging constraints (Wyatt et al. 2005) or from anal-ysis of the SED (Beichman et al. 2006a). The proper-ties inferred for these planetesimal belts are indicated onFig. 4 and lie within the shaded region, implying thatthese planetesimal belts do not have to be transientlyregenerated, and also provide a plausible source popula-tion for the hot dust found closer in. However, no suchexcess emission has been seen toward HD69830 at either70 m (Beichman et al. 2005) or 850 m (Sheret, Dent &

Wyatt 2004) indicating a planetesimal belt with a massat most 5-50 times greater than our own Kuiper belt.Likewise, BD+20307 does not have a detectable excessin IRAS 60 m observations (Song et al. 2005).

A low mass reservoir of planetesimals does not neces-sarily rule out the presence of an outer planetesimal beltwhich is feeding the hot dust for two reasons. First, theshaded region of Fig. 4 actually constrains the proper-ties of the planetesimal belt at the time at which de-pletion started; i.e., this population may have alreadybeen severely depleted by the same event which is pro-ducing the dust and we are now nearing the end of thehot dust episode. Second, the constraints imposed by anon-detection in the far-IR do not eliminate the wholeof the parameter space in which an outer planetesimalbelt can lie. Fig. 4 includes the constraints on the outerplanetesimal belt imposed by the non-detection of excessin the far-IR, assuming that the dust emits like a blackbody. The resulting detection limit is then given by:

fdet = 3.4 109Fdet()d2r2out/B(, Tbb), (32)where Fdet is the detection limit in Jy, d is the distance tothe star in pc, and B(, Tbb) is in Jy/sr. For BD+20307the non-detection is limited by the sensitivity of IRAS,and so lower limits should be achievable with Spitzer. Forthe two systems with non-detections, the shaded regionalready takes the far-IR constraint into account.


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The simplification that the emission comes from blackbody-type grains means that equation (32) underesti-mates the upper limit from the far-IR fluxes. This isbecause the majority of the luminosity comes from smallgrains which emit inefficiently at long wavelengths. In-deed, the black body assumption would require the hotdust of HD69830 and BD+20307 to have been detectedin the far-IR, whereas this is not the case. We modeled

the emission from non-porous silicate-organic refractorygrains in a collisional cascade size distribution at 1 AUfrom these stars to find that the black body assumptionused in equation (32) underestimates the limit by a fac-tor of 3 5 meaning that non-detection of the hot dustin these systems in the far-IR is to be expected. Thisalso means that slightly more luminous outer planetesi-mal belts than those indicated by the shaded region onFig. 4 may still have escaped detection in the far-IR.

Until now we have not proposed a mechanism whichconverts the planetesimals into dust. Whereas Beichmanet al. (2005) invoke sublimation of comets as the originof the hot dust, and use this to estimate the mass of theparent planetesimal belt, we consider a scenario in which

a significant fraction of material of all sizes in the parentplanetesimal belt is placed on orbits either entirely coin-cident with the hot dust, or with pericenters at that loca-tion. In this scenario the dust is reproduced in collisionsand the material maintains a collisional cascade size dis-tribution. Simply moving material from rout to r wouldresult in an increase in fractional luminosity from fout tofout(rout/r)2. This indicates that the parent planetesi-mal belt responsible for the hot dust could have originallybeen on the line on Fig. 4 traced by fout = fobs(r/rout)2.Since this is parallel to the mass loss limit line (equa-tion 30), and for all but HD69830 the observed hot dustcomponent lies below this line, this indicates that par-ent planetesimal belts in the shaded region could be re-

sponsible for the hot dust observed, as long as a largefraction of their mass is scattered in to the inner regions.However, it is to be expected that only a fraction of theouter planetesimal belt ends up in the hot dust region,and so it is more likely that the parent planetesimal beltstarted on a line which falls off less steeply than r2out,and this is consistent with the ratio of the hot and coldcomponents of Corvi and HD72905 which indicate adependence of fout = fobs(r/rout)

0.50.2; it is also inter-esting to note that both have rout/r = 6070. We deferfurther consideration of the expected properties of theparent planetesimal belt to a more detailed model of thedynamics of the types of events which could cause such aperturbation, but simply note here that the existence ofan outer planetesimal belt is not ruled out by the currentobservational constraints in any of the systems.

4. DISCUSSION

A simple model for the steady state evolution of dustluminosity for planetesimal belts evolving due to colli-sions was described in 2. This showed how at late timesthe remaining planetesimal belt mass and so dust lumi-nosity is independent of the initial mass of the belt. Thishas important implications for the interpretation of theproperties of detected disks. This paper discussed theimplications for the population of sun-like stars with hotdust at < 10 AU; the implications for the statistics willbe discussed in a forthcoming paper (Wyatt et al., in

prep.).It was shown in 3.1 that for 4/7 of the systems with

hot dust their radius and age are incompatible with aplanetesimal belt which has been evolving in quasi-steadystate over the full age of the star, and in 3.2 it wasshown that this is the case even when uncertainties inthe model are taken into account. This implies eitherthat the cascade was started recently (within the last

Myr or so), or that the dust arises from some other tran-sient event. Recent ignition of the collisional cascadeseems unlikely, since the mass required to feed the ob-served luminosity would result in the growth of 2000 kmplanetesimals which would stir the belt and ignite thecascade on timescales much shorter than the age of thestars. Possible origins for the transient event that havebeen proposed in the literature are: recent collision be-tween massive planetesimals in a planetesimal belt whichintroduces dust with a size distribution q 11/6 and socan be detected above a collisional cascade which is toofaint to detect; one supercomet 2000 km in diameterthat was captured into a circular orbit in the inner sys-tem replenishing the dust through sublimation (Beich-

man et al. 2005); a swarm of comets scattered in fromthe outer reaches of the system (Beichman et al. 2005).In 3.3 the collisional model was used to show that thetransient disks are very unlikely (< 0.001% for the mostoptimistic estimate for any of the stars compared witha detection probability of 2% for transient hot dust) tohave their origin in a recent collision; such collisions oc-cur too infrequently. In 3.4 it was also shown that theparent planetesimals of the observed dust must originatein a planetesimal belt much further from the star thanthe observed dust, typically at 2 AU. This is becausecollisional processing means that the mass that can re-main so close to the star at late times is insufficient tofeed the observed phenomenon.

The most likely scenario is thus a recent event whichprovoked one or more planetesimals to be scattered infrom further out in the disk (Beichman et al. 2005).The observed dust could have been produced from sucha scattered planetesimal population through their grind-ing down in mutual collisions (3.5), although sublima-tion close to the pericenters of the planetesimals orbitsis a further possible source of dust. More detailed studyof the scattering and consequent dust production pro-cesses is required to assess these possibilities. However,this scenario is supported by the presence of far-IR emis-sion originating from a colder outer planetesimal beltcomponent in 2/4 of the transient dust systems. Theconstraints on the outer planetesimal belt which is feed-ing the phenomenon are discussed in

3.5, showing that

the outer planetesimal belts already found in Corviand HD72905 provide a plausible source population forthe hot dust found closer in, and that the current non-detection of cold dust around the remaining two systemsdoes not rule out the presence of an outer planetesimalbelt capable of feeding the observed hot dust luminosity.

One clue to the origin of the parent planetesimals ofthe dust may be the composition of that dust. Silicatefeatures have been detected in the mid-IR spectrum ofall of the transient hot dust stars (Song et al. 2005; Be-ichman et al. 2005; Beichman et al. 2006a; Chen et al.2006). Detailed modeling of the spectrum of HD69830indicates that the mineralogical composition of its dust is


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TABLE A1Symbols employed in this paper and their units.

Symbol Units Meaning

B(, T) Jy/sr black body emission spectrumd pc distance to star

dMloss/dt M /Myr rate of mass loss assuming observed dust has size Dbldr AU planetesimal belt width

Dbl m diameter of smallest dust in cascadeDc km diameter of largest planetesimal in cascade

Dcc km smallest planetesimal capable of destroying planetesimals of diameter DcDpb km minimum diameter of parent body required to produce observed dustDt km planetesimal diameter at transition between strength and gravity regimes

e mean orbital eccentricity of planetesimalsf fractional luminosity (= Lir/L) in model

fdet fractional luminosity for emission from belt to be detectedfmax maximum fractional luminosity of cascade after time tagefobs fractional luminosity observedfout fractional luminosity of putative outer planetesimal belt feeding the dust

f(e, I) ratio of collision velocity to Keplerian velocityfcc see equation (8)

Fdet() Jy detection limit at wavelength G(q, Xc) see equation (9)

I rad mean orbital inclination of planetesimalsK scaling factor in size distributionL L stellar luminosityLir L infrared luminosity of material in the cascadeM M stellar mass

Mmax M maximum mass remaining in cascade after time tageMtot M total mass of material in cascaden(D) size distribution of material in the cascade

n(D > Dpb) number of objects in cascade larger than DpbdNc(D > Dpb)/dt Myr

1 destructive collision rate for planetesimals larger than DpbP(f > fobs) fraction of time collisions result in f > fobs

q slope of size distributionqg slope of size distribution expected in the gravity regimeqs slope of size distribution expected in the strength regime

QD J kg1 specific incident energy required to catastrophically destroy a planetesimal

r AU planetesimal b elt radius, assumed to b e coincident with dustrout(thot) AU outer planetesimal belt radius required to maintain fobs for thot

rout AU radius of putative outer planetesimal b elt feeding the dusts exponent in relation QD D

s

t(f > fobs) Myr time a planetesimal belt at r can sustain f > fobs

tage Myr time since cascade initiated (assumed to b e stellar age)tc yr collisional lifetime of planetesimals of size Dctc(D) yr collisional lifetime of material of size D

thot Myr total duration of hot episodes throughout stellar l ifetimevrel m s

1 relative velocity of collisionsvesc m s1 escape velocityvk m s

1 Keplerian velocityxt jump in size distribution expected at DtXc = Dcc/Dc kg m3 planetesimal density

tot AU2 total cross-sectional area of material in cascade

APPENDIX

SUMMARY OF SYMBOLS

The symbols which are employed in this paper are summarized in Table A1 along with the units assumed throughoutthe paper.

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Transience of hot dust around sun-like stars