Stochastic accretion of planetesimals onto white dwarfs: constraints on the mass distribution of accreted material from atmospheric pollution

0.85 MB·

by  M. C. Wyatt

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed27January2014 (MNLATEXstylefilev2.2)
Stochastic accretion of planetesimals onto white dwarfs:
constraints on the mass distribution of accreted material
from atmospheric pollution
M. C. Wyatt1⋆, J. Farihi1†, J. E. Pringle1, A. Bonsor2,3,
4 1 Institute of Astronomy, Universityof Cambridge, Madingley Road, Cambridge CB3 0HA, UK
1 2 Institut de Plan´etologie et d’Astrophysique de Grenoble, Universit´e Joseph Fourier, CNRS, BP 53, 38041 Grenoble, France
0 3 H.H. WillsPhysics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK
2

n
27January2014
a
J

3
ABSTRACT
2
This paper explores how the stochastic accretion of planetesimals onto white dwarfs

] would be manifested in observations of their atmospheric pollution. Archival obser-
P vations of pollution levels for unbiased samples of DA and non-DA white dwarfs are
E usedtoderivethedistributionofinferredaccretionrates,confirmingthatratesbecome
. systematically lower as sinking time (assumed here to be dominated by gravitational
h
settling) is decreased, with no discernable dependence on cooling age. The accre-
p
- tion rates expected from planetesimals that are all the same mass (i.e., a mono-mass
o distribution) are explored both analytically and using a Monte Carlo model, quanti-
r fying how measured accretionrates inevitably depend on sinking time, since different
t
s sinking times probe different times since the last accretion event. However, that de-
a
pendence is so dramatic that a mono-mass distribution can be excluded within the
[
context of this model. Consideration of accretion from a broad distribution of plan-

1 etesimal masses uncovers an important conceptual difference: accretion is continuous
v (rather than stochastic) for planetesimals below a certain mass, and the accretion of
3 such planetesimals determines the rate typically inferred from observations; smaller
7
planetesimals dominate the rates for shorter sinking times. A reasonable fit to the
1
observationally inferred accretion rate distributions is found with model parameters
6
consistentwith a collisionallyevolvedmassdistribution upto Pluto-mass,andanun-
.
1 derlyingaccretionratedistributionconsistentwiththatexpectedfromdescendantsof
0 debrisdiscsofmainsequenceAstars.Withtheseparameters,whilebothDAandnon-
4
DA white dwarfs accrete from the same broad planetesimal distribution, this model
1
predicts that the pollution seen in DAs is dominated by the continuous accretion of
:
v < 35 km objects, and that in non-DAs by > 35 km objects (though the dominant
i size varies between stars by around an order of magnitude from this reference value).
X
Further observations that characterise the dependence of inferred accretion rates on
r sinking time and cooling age (including a consideration of the effect of thermohaline
a
convection on models used to derive those rates), and the decadal variability of DA
accretion signatures, will improve constraints on the mass distribution of accreted
material and the lifetime of the disc through which it is accreted.
Key words: circumstellar matter – stars: planetary systems: formation.
1 INTRODUCTION orbiting0.05 5AUfromtheirstars,butanewpopulation
−
oflowmassplanets(2 20timesthemassofEarth)orbiting
Our understanding of the planetary systems around main −
within 1 AU has been found in transit and radial velocity
sequence Sun-like stars has grown enormously in the past
surveys,aswellamoredistant8 200AUpopulationofgi-
few years. Not only do we know about planets like Jupiter −
antplanetsfoundinimagingsurveys(Udry&Santos2007).
Ourunderstandingofthedebrisdiscs,i.e.beltsofplanetes-
imalsanddust,orbitingmainsequencestarshasalsogrown
⋆ Email:[email protected] rapidly; surveys show that > 50% of early-type stars host
† STFCErnestRutherfordFellow
(cid:13)c 0000RAS
2 M. C. Wyatt et al.
debris (Wyatt 2008). Most of this debris lies 10 AU in accretion process. While others have recently shown that
≫
regions analogous to the Solar System’s Kuiper belt, but a the previously unmodelled stellar process of thermohaline
few%ofstarsexhibitdustat 1AUthatmayoriginatein convection can lead to substantial revision in the accretion
∼
an asteroid belt analogue. ratesinferredtowardsomewhitedwarfs,potentiallyremov-
Much less is known about the planetary systems and ingthedifferenceintheinferredaccretionratedistributions
debris of post-main sequence stars, though these should be between the two populations (Deal et al. 2013), we show
directdescendantsofthemainsequencepopulation.Several herethat such a differenceis not unrealistic, rather it is al-
post-mainsequenceplanetarysystemsarenowknown(e.g., mostunavoidablewithinthecontextofthemodelpresented
Johnson et al. 2011), but the debris discs of post-main se- here.
quencestarshaveremainedelusive(thoughthereareexam- In 2 we compile observations from the literature and
§
plesaround subgiants,e.g., Bonsor et al. 2013). Theclosest use these to derive the distribution of inferred accretion
to a counterpart of the Kuiper belt-like discs found around rates1 toward white dwarfs of different atmospheric prop-
main sequence stars may be the 30 150 AU disc at the erties (notably with different sinkingtimes for metals tobe
−
centre of the Helix nebula (Su et al. 2007) and a few oth- removedfrom theatmosphere) and ages. Asimple model is
ers like it (Chu et al. 2011; Bilikova et al. 2012). However, then presented in 3 that quantifies what we would expect
§
a more ubiquitous phenomenon is that a large fraction of toobserveiftheplanetesimalsbeingaccretedontothewhite
cool (< 25,000K) white dwarfs show metals in their atmo- dwarfs all have the same mass; 4 demonstrates that such
§
spheres. This is surprising because their high surface gravi- a model is a poor fit to the observationally inferred accre-
tiesandsmall(ornon-existent)convectionzonesmeanthat tion rate distributions, even if different stars are allowed to
suchmetals sinkon short (daytoMyr) timescales implying havedifferent accretion rates and if themodel is allowed to
that material is continuously accreted onto the stars with include a disc lifetime that moderates the way accretion is
polluted atmospheres. It has been shown that this material recorded on stars with short sinking times. In 5 the model
§
does not originate from the interstellar medium (Farihi et is updated to allow stars to accrete material with a range
al. 2009; 2010), and its composition has been derived from of masses, showing that this provides a much better fit to
atmospheric abundancepatterns tobe similar to terrestrial theobservationallyinferredaccretionratedistributions.The
material in the Solar System (Zuckerman et al. 2007; Klein results are discussed in 6 and conclusions given in 7.
§ §
etal.2010;G¨ansickeetal.2012).Theprevailinginterpreta-
tionisthatasteroidalorcometarymaterialisbeingaccreted
fromacircumstellarreservoir,i.e.,fromtheremnantsofthe
2 DISTRIBUTION OF ACCRETION RATES
star’s debrisdisc and/or planetary system.
INFERRED FROM OBSERVATIONS
Meanwhile a complementary set of observations pro-
vides clues to the accretion process, since around 30 white The accretion rate onto a white dwarf can be inferred
dwarfsalsoshownear-IRemissionfromdust(Zuckerman& from observations of its atmosphere, since its thin (or non-
Becklin 1987; Graham et al. 1990; Reach et al. 2005) and existent) convection zone means that a metal (of index i)
sometimes optical emission lines of metallic gas (G¨ansicke sinkson arelatively short timescale t .Theexact sink-
sink(i)
et al. 2006; Farihi et al. 2012a; Melis et al. 2012) that is ingtimescaledependsonthemetalinquestionandtheprop-
located within 1R⊙ from the stars. Given its close prox- ertiesofthestar,butcanbereadilycalculated(e.g.,Paque-
∼
imity to the tidal disruption radius, and the fact that all tteetal.1986).Inthispaperthesinkingprocessisassumed
whitedwarfswithevidenceforhotdustorgasalsoshowev- to be gravitational settling, and so the sinking timescale is
idence for accretion in their atmospheric composition, it is the gravitational settling timescale. However, to allow for
thought that both the dust, gas and atmospheric pollution the possibility that other processes act to remove metals
all arise from tidally disrupted planetesimals (Jura 2003). fromtheconvectivezone(suchasthermohalineconvection),
However, the exact nature of the disc formation process, or indeed to replenish it (e.g., radiative levitation), we re-
and of the accretion mechanism are debated, which could fer to sinking timescales rather than gravitational settling
forexamplebethroughviscousprocessesorradiationforces timescales throughout.
(e.g.,Rafikov2011;Metzger,Rafikov&Bochkarev2012).It Thus observations of photospheric absorption lines,
isalso debatedwhetherthepollution is causedbyacontin- which can be used to infer the abundance of an element
uous rain of small rocks (Jura 2008), or by the stochastic atthestellarsurfaceandbyinferencethetotalmassofthat
accretion of much larger objects (Farihi et al. 2012b). elementintheconvectionzoneM ,canbeconvertedinto
cv(i)
Inthispaperwepresentasimplemodeloftheaccretion an inferred mass accretion rate (assuming steady state ac-
ofplanetesimals inmultipleaccretion eventstoexplorehow cretion, Dupuiset al. 1992, 1993a, 1993b) of
such events are manifested in observations of the star’s at-
M˙ =M /t . (1)
mospheric metal abundance.The aim isto understandhow obs(i) cv(i) sink(i)
such observations can be used to derive information about Note that M˙ is expected to differ significantly from
obs(i)
the mass (or mass distribution) of accreted objects, and the actual accretion rate, depending on the time variabil-
about whethermetal-polluted atmospheres are theproduct ity of the accretion, as outlined in this paper; thus we use
ofsteadystateaccretionofmultipleobjectsortheaccretion M˙ primarily as a more convenient way of expressing
obs(i)
of single objects. A central motivation for this study is the
recentclaimthatthedistributionofinferredaccretionrates
isdifferenttowardstarswithdifferentprincipalatmospheric 1 Note that the rates we use here do not include the effect of
compositions (Girven et al. 2012; Farihi et al. 2012b), and thermohalineconvection,theeffectsofwhichhaveyettobefully
we show how this is an important clue to determining the characterisedinthiscontext.
(cid:13)c 0000RAS,MNRAS000,000–000
Stochastic accretion of planetesimals onto white dwarfs 3
Mcv(i)/tsink(i). Measurements of different elements provide slightlymoremassive(0.65M⊙ versus0.60M⊙;Kleinmanet
information on the composition of the accreted material, al. 2013). The low ratio of DB to DA white dwarfs in glob-
which generally looks Earth-like (Zuckerman et al. 2007; ular clusters (Davis et al. 2009) also suggests that the two
Klein et al. 2010; G¨ansicke et al. 2012), and extrapolation populations could have different distributions of formation
toanyundetectedmetalscan beusedtoinferatotalaccre- environments; our assumption requires that this difference
tionrateM˙ .Itisworth emphasisingthattheseaccretion doesnotsignificantlyaffecttheplanetarysystemproperties
obs
rates are not direct observables, rather they need to be de- (Zuckermanetal.2010).Practically,thisassumptionmeans
rivedfromstellarmodels(togetbothM andt ).As that we expect the observationally inferred distribution of
cv(i) sink(i)
such, changes in stellar models can potentially lead to sig- accretion rates, f(> M˙ ), to depend both on stellar age
obs
nificant changes in inferred accretion rates (e.g., Deal et al. (because of evolution of the circumstellar material) and on
2013). The modelswe usein 2.2 are thosemost commonly sinking time (because that affects how the accretion rate is
§
employed in the white dwarf literature, though these have sampled),butnotonthedetailsofwhetherthestarisaDA
yet to incorporate theeffects of thermohaline convection. or a non-DA.
Althoughtheliteratureincludesmanystudiesthatmea-
sureaccretionratestowardswhitedwarfs(e.g.,Fig.8ofGir-
venetal.2012),forourpurposeswewillrequirethedistribu-
2.2 Uniform analysis
tionofaccretionrates,i.e.,thefractionofwhitedwarfsthat
exhibitaccretion rateslargerthanagivenvaluef(>M˙ ), The uniform analysis consists of using reported measure-
obs
for which information about non-detectionsis as important ments of atmospheric Ca/H (for DAs) or Ca/He (for non-
as that about detections. Thus here we perform a uniform DAs) for stars for which their effective temperature Teff is
analysis of data available in the literature for samples cho- also known. These abundance measurements had been de-
sen to be unbiased with respect to the processes that may rived from modelling of stellar spectra and were multiplied
becausing atmospheric pollution. by the total convection zone mass (or that in the envelope
From the outset it is important to note that this pa- above an optical depth τR = 5; Koester 2009) to get the
perwilldistinguishbetweentwodifferentatmospherictypes: massofCainthatregion.Theeffectivetemperatureisused
DA white dwarfs that have H-dominated atmospheres, and todeterminethesinkingtimescaleofCaduetogravitational
non-DAwhitedwarfs(comprisedofbasicsub-typesDBand settling, tsink(Ca), for the appropriate atmospheric type us-
DC)thathaveHe-dominatedatmospheres.Thisdistinction ing the models of Koester (2009), and then the convection
isnecessary,becausemetalshaveverydifferentsinkingtimes zonemassisconvertedintoamassaccretionrateofCa.This
in the two different atmospheres, and observations toward rateisscaled upbyassuming thattheCa represents1/62.5
co-evalDAandnon-DAwhitedwarfshavedifferentsensitiv- of the total mass of metals accreted, like the bulk Earth,
ities to convection zone mass. This distinction is discussed whichappearsbroadlysupportedbydataforstarswithCa,
furtherin 2.1,then 2.2describestheuniformanalysisem- Fe, Mg, Si, O and other metals detected (Zuckerman et al.
§ §
ployed, 2.3describestheunbiasedDAandnon-DAsamples, 2010).
§
andthedistributionsofaccretionratesinferredfromtheob- The other parameter of interest is the star’s cooling
servations are described in 2.4, while 2.5 discusses uncer- age tcool.Althoughcooling age is actually afunction ofTeff
§ §
taintiesin theinferred accretion ratedistributionsfrom the and logg, in practise the surface gravity is poorly known
choice of model used to derive those rates. due to insufficient observational data and a lack of good
parallaxmeasurements.Thusthroughoutthispaperwehave
assumed all stars to be of typical white dwarf mass2 with
2.1 DA vs non-DA stars
logg=8.0,sothatT mapsuniquelyontoacorresponding
eff
t , which also then maps onto a corresponding t .
Animplicit assumption adopted hereis that populations of cool sink(i)
bothDAandnon-DAwhitedwarfsundergothesamehistory Using this assumption, Fig. 1 reproduces the sinking times
of mass input rate into the convection zone; i.e., two white duetogravitationalsettlingofafewmetalsasafunctionof
coolingagefromKoester(2009)forbothDAsandnon-DAs.
dwarfs that are thesame age can have different mass input
rates, but the distribution of mass input rates experienced Fig.1showsthatsinkingtimesvaryonlybyafactorof
by white dwarfs of the same age is independent of their at- afewfordifferentmetalsinthesamestar,butthatthereis
mospheric type. There are several channels by which both alargedifferenceinsinkingtimescaleofagivenmetalwhen
DA and non-DA white dwarfs might form. However, most put in the atmosphere of the same star at different ages,
whitedwarfswithHe-dominatedatmospheres(i.e.,thenon- and for stars of the same age but of different atmospheric
DAs) are thought to form from very efficient H-shell burn- type.FortheDAwhitedwarfstsink canbeasshortasafew
ing in the latter stages of post-main sequence evolution, or days(e.g.,Koester&Wilken2006),whereasforthenon-DA
late thermal pulses that dilute the residual H-rich envelope whitedwarfstsinkismoretypically0.01 1Myr(e.g.Koester
−
with metal-rich material from the interior (e.g., Althaus et 2009). The dependence of sinking time on cooling age is
al. 2010). So, as long as these processes are not biased in similar for both atmospheric types in that it is shorter at
termsofstellar mass,orintermsofplanetary systemprop- younger ages (i.e., at high effective temperatures), followed
erties, then it is reasonable to expect that the parent stars
(and circumstellar environments) of DA and non-DA white
dwarfpopulationsshouldbesimilar.Indeed,observationally 2 Giventhenarrowdistributioninwhitedwarfmassesestimated
the mean mass of DB white dwarfs is very close to that of fromgravitational redshifts(Falcon et al.2010), theuncertainty
theirDAcounterparts(e.g.,Bergeronetal.2011),thougha in cooling age from this assumption would be expected to be
smalldifferencehasrecentlybeendiscernedwithDBsbeing <6%.
(cid:13)c 0000RAS,MNRAS000,000–000
4 M. C. Wyatt et al.
against sinking time (Fig. 2c). The sense of the detection
bias is evident from the lower envelope of the detections in
Fig. 2a; e.g., there are far fewer detections in the younger
agebinsduetothehighertemperatureof thesestarswhich
makes Ca lines harder to detect for a given sensitivity in
equivalent width (see Fig. 1 of Koester & Wilken 2006).
The right panels use the information in the left pan-
els to determine the distribution of inferred accretion rates
f(>M˙ ) for different sub-samples as outlined in the cap-
obs
tions.Forexample,Fig. 2b keepsthesplit between DAand
non-DAand furthersub-dividesthese samples according to
stellarage,usingagebinsof100-500Myr(here-ontheyoung
bin) and 500-5000 Myr (here-on the old bin). Fig. 2d com-
bines the DA and non-DA samples, but then makes sub-
samplesaccordingtosinkingtimebinsof0.01-100 yr(here-
ontheshortbin),100yr-0.1Myr(here-onthemediumbin)
and 0.1-1 Myr (here-on the long bin), though overlap be-
Figure1.Sinkingtimescalesduetogravitationalsettlingatthe
base of the convection zone (or at an optical depth of τR =5 if tween the DA and non-DA samples is confined to a small
thisisdeeper)ofdifferentmetals(shownwithdifferentline-styles fraction (4.4%) of non-DAsin the medium bin.
as indicated in the legend) as a function of the star’s cooling Identifyingthemostaccuratewaytodeterminetheun-
age (from tables 4-6of Koester 2009) both for DA white dwarfs derlying distribution of f(> M˙ ) for the different sub-
obs
(i.e.,thosewithH-dominatedatmospheres,showninred)andfor samples (i.e., that which would be measured with infinite
non-DAwhitedwarfs(i.e.,thosewithHe-dominatedatmospheres, sensitivity and sample size) is complicated by the fact that
showninblue).Weadopttheparametersformoreefficientmixing
the observations only result in upper limits for many stars,
inDAscoolerthan13,000K.
and the sample size is finite, a problem encountered many
times in astrophysics though without a definitive solution
(e.g., Feigelson & Nelson 1985; Mohanty et al. 2013). Two
by a transition to longer sinking times once temperatures
bounds on the underlying distribution can be obtained by
arecoolenoughforasignificantconvectionzonetodevelop.
considering that the most pessimistic assumption for the
stars that have upper limits is that they are not accreting
2.3 DA and non-DA samples (i.e.,thatwithinfinitelydeepobservationsM˙obs =0),while
the most optimistic assumption is that those stars are ac-
Weconsider twosamples, oneof DAsandtheotherofnon- cretingatalevelthatisattheupperlimitinferredfromthe
DAs.TheDAsampleiscomprisedof534DAwhitedwarfsof observations.TheseboundsareplottedonFigs2band2dfor
which38havedetectionsofCa,whiletheremaining496have the different sub-samples with dotted lines, and one might
upperlimitsonthepresenceofCa.Thesedatacomprisetwo expecttheunderlyingdistributionstofallbetweenthesetwo
surveys:a Keck survey that specifically searched about 100 bounds.However,whileinstructive,theseboundsencounter
cool DA white dwarfs for Ca absorption (Zuckerman et al. two problems. First, the optimistic limit requires the im-
2003), and the SPY survey which took VLT UVES spectra probableoccurrenceofmanydetectionsatthe3σlimit.This
of > 500 nearby white dwarfs to search for radial velocity problem is particularly acute when a significant fraction of
variations from double white dwarfs (SN Ia progenitors); the sample only has upper limits, such as the short sinking
these data are also sensitive to atmospheric Ca (Koester et time sub-sample on Fig. 2d, because not only is it statis-
al.2005).Themoreaccuratedatawerechoseninthecaseof tically unlikely that the observer recorded an upper limit
duplication.Thesestarsarerandomlychosenbasedonbeing for each star when the true accretion level was as high as
nearby and bright, and not biased in terms of the presence assumed in the optimistic case, but also the small number
or absence of metals. ofactualdetectionsalready suggests thatonlyasmall frac-
Thenon-DAsampleisasmall, butuniformly-sampled, tion of stars should have detections at such a high level.
set of DB stars searched for metal lines with Keck HIRES In other words, the optimistic limit is unrealistically opti-
(seeTable1ofZuckermanetal. 2010). Starsinthissample mistic.Thesecondproblemisthatthisdoesnotaccountfor
are predominantly young, with 50 500 Myr cooling ages, small number statistics, which affects in particular the dis-
−
butareotherwiseunbiasedwithrespecttothelikelihood to tribution at high accretion rates, where the optimistic and
detect metal lines. Although additional accretion rate mea- pessimistic lines converge, but where the rates have been
surements exist in the literature for DB stars, these would estimated from veryfew detections.
onlybesuitableforinclusioninthisstudyifthesamplewas Here we adopt an alternative method for estimating
unbiased with regard to the presence of a disc, and if non- f(> M˙ ) that circumvents these two problems. The idea
obs
detectionswerereported with upperlimitson theaccretion is that if we want to know the fraction of stars in a sub-
rates. sample of size N that have accretion levels above say
s
M˙ = 107 gs−1, then we should only consider the sub-
obs
set of N stars within that sub-sample for which accretion
2.4 Distribution of inferred accretion rates ss
couldhavebeendetectedatthatlevel.Thefractionofstars
TheleftpanelsofFig.2showtheinferredaccretionratedata withaccretion abovethatlevelisthenthenumberofdetec-
for the two samples, plotted both against age (Fig. 2a) and tions in that subset N (noting that this may be lower
ssdet
(cid:13)c 0000RAS,MNRAS000,000–000
Stochastic accretion of planetesimals onto white dwarfs 5
(a) (b)
(c) (d)
Figure 2. Inferred accretion rates for unbiased samples of DA white dwarfs (shown in red) and for non-DA white dwarfs (shown in
blue).Theleftpanels(aandc)showaccretionratesinferredfromCameasurementsassumingaterrestrialcomposition.Detections are
shownwithasterisksandupper limitswithasmallplus.In(a)thex-axisisthecoolingageofthewhitedwarfinferredfromthestar’s
effectivetemperature(assuminglogg=8.0),whereasin(c)thex-axisisthesinkingtimeofCainferredfromtheeffectivetemperature.
The right panels (b and d) show the fraction of white dwarfs indifferent sub-samples that have inferredaccretion rates above a given
level.Thesesub-samplesaresplitbycoolingagein(b)intoyoungandoldagebins,andbysinkingtimein(d)intoshort,mediumand
longsinkingtimebins;thebinboundaries arenoted inthelegends andnodistinctionismadeforthesub-samplesin(d)between DAs
and non-DAs. The dotted lines give the range of distributions inferred for each sub-sample for optimistic and pessimistic assumptions
about the stars with upper limits (see text for details). The solidlines give the best estimate of the distributions for each sub-sample,
andthedashedlinesandhatchedregionsshowthe1σ uncertaintyduetosmallnumberstatistics (seetextfordetails).
than the number of stars in the whole sub-sample with ac- long as stars are included in the subset in a way that does
cretionabovethatlevel)dividedbyN .Theuncertaintyon not introduce biases with respect to the level of accretion.
ss
that fraction can then be determined from N and N In this case Fig. 2a shows that as we try to measure the
ssdet ss
using binomial statistics (see Gehrels 1986), and it is evi- distribution down to lower levels of accretion, theonly bias
dentthatsmallnumberstatisticswillbeimportantbothfor is that the subset becomes increasingly biased toward the
large accretion rates where there are few detections (small older stars in the sub-sample. So, the distribution we infer
N ), and for small accretion rates where few of the sub- inthiswayisonlyagoodrepresentationofthatofthewhole
ssdet
sample can be detected at such low levels (small N ). In sub-sample as long as the inferred accretion rate distribu-
ss
Figs2band2dweshowthefraction determinedinthisway tion is not strongly dependent on cooling age, a topic we
with a solid line, and the hatched region and dashed lines address below.
indicate the 1σ uncertainty. 3 This method only works as While Figs 2b and 2d provide the best estimate of the
underlying inferred accretion rate distributions in the sub-
samples,wewillalsouseFisher’sexacttesttoassignaprob-
abilitytothenullhypothesisthattwosub-sampleshavethe
3 Note that these errors apply only to the measurement of f(>
M˙obs) at a specific accretion rate and so the points on this line same inferred accretion rate distribution. To do so we just
arenotindependentofeachother.Thisisrelevantwhenassigning need four numbers, Nssdet and Nss for the two sub-samples
aprobabilitythatagivenmodelprovidesagoodfittothedata, measured at an appropriate accretion level, and the proba-
aswillbediscussedlater. bility quoted will be that for the observations of these sub-
(cid:13)c 0000RAS,MNRAS000,000–000
6 M. C. Wyatt et al.
samples resulting in rates that are as extreme, or more ex- the4.3% (6/140) rate in the short bin if thetwo are drawn
treme, if the nullhypothesis were true. from the same distribution. This probability becomes 0.4%
The first thing to note from Fig. 2b is that the distri- when comparing the rate in the long bin with the 13.4%
butions of inferred accretion rates in the young age bin are (13/97)rateinthemediumsinkingtimebin,and1.1%when
significantly different between the DA and non-DA popula- comparing the rates in the short and medium sinking time
tions. For example, for the subsets corresponding to accre- bins (this latter probability is further reduced to 0.6% if
tion above 107 gs−1, there is only a 0.002% probability of larger accretion rates up to 108 gs−1 are considere∼d). That
obtaining rates as extreme as, or more extreme than, the is, as expected from above, there is a significant difference
4.6% (6/131) of young DAs and the 39% (9/23) of young between the sinking time bins, though the confidence level
non-DAs if the two are drawn from the same distribution. that all three sinking time bins have distributions that are
Ifasassumedin 2.1theonlydifferencebetweentheunder- different from each other, and hence that there is a mono-
§
lying distribution of inferred accretion rates toward these tonic change in inferred accretion rates across a wide range
stars is the sinking timescale on which the accretion rate is of sinking times, is slightly below 3σ.
measured, then this indicates that the longer sinking times While the above analysis is not sufficient to make a
ofthenon-DApopulation(withamedianlevelof0.37Myr) strong statement about the difference between (say) the
have lead to a distribution with higher inferred accretion short and medium sinking time bins, we take the near 3σ
rates than the DA population (with a median sinking time significancetoindicatethatfutureobservationswillsoonbe
of 5 days). abletofindsuchadifference,ifitexists.Thuswetailor the
Concentrating now on the inferred accretion rate dis- models in the following sections to reproduce as good a fit
tributions for the DA sub-samples in Fig. 2b we conclude to the solid lines in Fig. 2d as possible. This approach al-
that there is no strong evidence that these vary with age. lowsusdemonstratethequalitativebehaviourofthemodels,
For example, taking again subsets corresponding to accre- andhowthedifferentparametersaffecttheirpredictionsfor
tionabove107 gs−1,thereisa2.6%probabilityofobtaining the dependence of the inferred accretion rate distribution
ratesasextremeas,ormoreextremethan,the4.6%(6/131) onsinkingtime.However,indoingsowerecognisethatthis
ofyoungDAsandthe12.4% (13/105) ofoldDAsifthetwo approach may appear to constrain the model in ways that
are drawn from the same distribution. While the small dif- willnotbeformallysignificantgiventhelimitationsofsmall
ferenceinratesbetweenthepopulationscouldbeindicative numberstatistics, and note in future sections where that is
of an age dependence in the inferred accretion rate (higher thecase.
rates around older stars), this is of low statistical signifi- Note that while we have assumed that there is no de-
cance. Moreover, since age is correlated with sinking time pendence of accretion rate on age, the lack of evolution is
intheDAsub-samples(Fig.1),andthepreviousparagraph not well constrained, and the different sinking time bins
concluded that longer sinking times lead to higher inferred have different age distributions; the median ages are 140,
accretion rates, it is possible that the (marginally) higher 840 and 220 Myr for the short, medium and long bins, re-
accretion rates around the older DA sub-sample are due to spectively. If there was a dependence of accretion rate on
their longer sinking times relative to the younger DA sub- age, the most significant effect would likely be on the po-
sample, and have nothing to do with the evolution of the sition of the medium sinking time bin with respect to the
underlying accretion rate distribution. However, it is not other bins. For example, a decrease in accretion rates with
possible to conclude that age is not an important factor in agewouldmeanthedistributionf(>M˙obs)forthemedium
determiningtheinferredaccretionratedistribution,asthere bin would be higher if plotted at a comparable age to that
could even be a strong decrease in accretion rate with age of thelong and short bins.
that has been counteracted in the sub-samples of Fig. 2b
by the sinking time dependence. To assess the effect of age
2.5 Caveats
properly would require comparison of sub-samples of DAs
andnon-DAswiththesamesinkingtimesbutdifferentages, Themethoddescribedabovetoderiveaccretionratesmakes
but this is not available to us for now (see Fig. 2c). Never- some simplifications about the evolution of accreted met-
theless, since we do not see any evidence for a dependence als. Specifically the assumption is that metals are removed
on age (see also Koester 2011), our analysis in this paper from the observable outer atmosphere over a sinking time,
willassumetheunderlyingdistributionofaccretionratesto wherethesinking timeis that duetogravitational settling.
beindependentofage(notingthatanagedependenceinthe Thisisthestandardapproachintheliterature(e.g.,Koester
distributionofinferredaccretionratesmayarisethroughthe 2009).Howeveroneimportantprocessthatisomittedhereis
sinking time). thermohaline (or fingering) convection. Thermohaline con-
Given that sinking time is likely the dominant factor, vection is triggered by a gradient in metallicity in the stel-
themostimportantplotisFig.2d.Thepicturethatemerges lar atmosphere that decreases toward the centre, such as
reinforcesthepreviousconclusionontheimportanceofsink- would beexpectedif highmetallicity material had beenac-
ingtimeintheinferredaccretionratedistributions,andfur- creted at the surface. In such a situation, the metals can
thermorepointstoamonotonicchangeinthedistributionof be rapidly mixed into the interior through metallic fingers,
inferred accretion rates, with longer sinking times resulting analogous to salt fingers studied in the context of Earth’s
in higher inferred accretion rates. To quantify the signifi- oceans(e.g.,Kunze2003).Applicationofthisprocesstogen-
cance of thedifference between thesub-samples,takeagain eralastrophysicalsituations,suchasmixinginstellaratmo-
subsetscorresponding toaccretion above 107 gs−1; thereis spheres, has been characterised using 3D numerical simula-
a 0.0006% probability of obtaining levels as extreme as, or tions(Traxleretal.2011;Brownetal.2013).Thermohaline
moreextremethan,the43%(9/21)rateinthelongbinand convectionhasbeenshown tohaveimportant consequences
(cid:13)c 0000RAS,MNRAS000,000–000
Stochastic accretion of planetesimals onto white dwarfs 7
formixingof planetary material accreted bymain sequence theobservableatmosphericsignaturesatanyonetime.Thus
stars (Vauclair 2004; Garaud 2011), for stars that accreted by m we really mean the mass of planetesimal i that
atm,i
material from an AGB companion (Stancliffe & Glebeek remains in the convective zone, which can be determined
2008), and possibly for low mass RGB stars (Denissenkov through observations of abundances in the white dwarf’s
2010). atmosphereusingastellarmodeltodeterminethetotalmass
Arecentstudyalso foundthatthisprocess maybeim- oftheconvectivezoneoverwhichthatabundanceisassumed
portantforaccretionontowhitedwarfatmospheres(Dealet to apply.
al. 2013), in that accretion rates inferred from observations The total mass of pollutants that are present in the
of DAwhitedwarfs may actually behigherthan previously convective zone, and hence potentially visible in the white
considered. The rates for non-DA white dwarfs would be dwarf’s atmosphere at any one time, which we call the at-
unaffected by this process leading to the interesting possi- mosphericmass, isthesumof allpreviousaccretion events,
bilitythatthedistributionofratesforbothpopulationsare depleted appropriately by thedecay,i.e.,
the same. However, for now the model has been only been
appliedto6stars,andtheimplicationshaveyettobechar- Matm = matm,i. (3)
acterised across the range of stellar and pollution param- Xi
eters required in this study. As such it is premature (and Wealsodefinetheaccretionratethatwouldbeinferredfrom
not possible with published information) to use rates that such an atmospheric mass as
account for thermohaline convection in this paper. Never-
M˙ =M /t . (4)
theless,sincethisprocesshasthepotentialtoaffectinferred atm atm sink
accretion rates, and may also do so in a way that depends
Notethesimilaritywitheq.(1),whichisbecausewewillbe
on sinking time, a caveat is required when interpreting the comparingM˙ withM˙ ,andunderscorestheimportance
atm obs
conclusions in 2.4 about how accretion rate distributions
ofusingthesamevalueoft inthemodellingasthatused
§ sink
depend on sinking time. If the rates need to be modified as
to obtain accretion rates from theobservations.
a result of this process, the analysis in this paper could be
Since the mass can only arrive in units of m , this is a
p
repeated, and we note below the potential implications if Poisson process, and M˙ is not necessarily equal to M˙ .
atm in
therate was toturn out to be independentof sinking time.
Rathertheinferred accretion rate hasa probability density
function P(M˙ ), and an associated cumulative distribu-
atm
tion function that we characterise by
3 SIMPLE MODEL: STOCHASTIC ∞
ACCRETION OF MONO-MASS f(>M˙ )= P(x)dx, (5)
atm
PLANETESIMALS ZM˙atm
whichisthefractionofthetimewewouldexpecttomeasure
The dependence of inferred accretion rates on sinking time
an accretion rate larger than a given value.
has previously been noted by Girven et al. (2012) and dis-
The set-up of this problem is exactly the same as that
cussed further in Farihi et al. (2012b) from a difference be-
forshotnoise,thenatureofwhichdependsontheparameter
tween the accretion rates inferred toward DA and non-DA
n, the mean number of shots per unit time (see Appendix
populations. It is interpreted as evidence of the stochastic
A).For our problem,
natureof theaccretion process, with theshort sinkingtime
DAsprovidingameasureoftheinstantaneous levelofaccre- n=M˙ t /m (6)
in sink p
tion being experienced by the star, and the longer sinking
time of non-DAsproviding evidencefor historical accretion is the mean number of accretion events per sinking time,
events, such as the accretion of a large comet which can and theshots have theform
leave mass in the atmospheres of non-DAs for long periods F(τ)=H(τ)e−τ, (7)
after the event. In this section we use a pedagogical model
toillustrate thenatureof stochasticprocesses andtoquan- where τ = t/tsink is time measured in units of the sinking
tify how different mass accretion rates (of objects of finite timescale, H(τ)istheHeavisidestepfunction,andtheshot
mass) would beexpectedtobeinferred toward whitedwarf amplitude discussed in the appendix and references therein
populations with different sinking times. should be scaled by mp/tsink to get this in terms of the
inferred accretion rate.
Herewederivethecumulativedistributionfunctionus-
3.1 Pedagogical model
ingaMonteCarlo model( 3.2), and applyresults from the
§
literature for shot noise to explain the shape of the distri-
Consider a white dwarf at which planetesimals are being
thrownatameanrateM˙ .Hereitisassumedthatallplan- bution function analytically ( 3.3).
in §
etesimalshavethesamemassm ,andthatonceaccretedat
p
time t , the mass from planetesimal i that remains poten-
i 3.2 Monte Carlo model
tiallyvisibleinobservationsofthewhitedwarf’satmosphere
decaysexponentiallyonthesinkingtimetsink,i.e.,fort>ti For a white dwarf with a given tsink, and accretion defined
m =m e(ti−t)/tsink. (2) by M˙in and mp, we first define a timestep dt=tsink/Nsink,
atm,i p whereN isthenumberoftimestepspersinkingtime(this
sink
Note that after being accreted the planetesimal is mixed shouldbelargeenoughtorecovertheshapeoftheexponen-
nearly instantaneously within the white dwarf’s convective tial decay of atmospheric mass, and is set to 10 here). We
zone, and only a small fraction of that mass contributes to then set a total number of timesteps, N (set to 200,000
tot
(cid:13)c 0000RAS,MNRAS000,000–000
8 M. C. Wyatt et al.
here),andusePoissonstatisticstoassignrandomlythenum-
ber of planetesimals accreted in each timestep (using the
poidev routine,Pressetal.1989,andameanofM˙ dt/m ).
in p
TheN timestepsareconsideredasa(looped)timeseries,
tot
andsothemassaccretedineachtimestepiscarriedforward
to subsequent timesteps with the appropriate decay (eq. 2)
to determine the mass in the atmosphere and inferred ac-
cretion rate as a function of time.
Fig. 3shows theresult ofthisprocessfor canonical pa-
rametersofM˙ =1010gs−1andm =3.2 1019 g.Thisac-
in p
×
cretion ratecorrespondstothemass ofthecurrentasteroid
belt(Krasinskyetal.2002) beingaccreted every 10Myr.
∼
This planetesimal mass corresponds to a 27 km diameter
planetesimalforadensityof3gcm−3,andhasbeenchosen
so that a sinking time of 100 years corresponds to a mean
rateofoneplanetesimalbeingaccretedpersinkingtime(i.e., (a)
n = 1). This process has been repeated for seven different
sinkingtimesthatcorrespond ton=0.001, 0.01, 0.1, 1,10,
100and1000planetesimalsbeingaccretedpersinkingtime.
Fig. 3a shows how longer sinking times (larger n) re-
sult in larger quantities of mass accreted in one sinking
time. However,decreasing thesinking timerunsintoa bar-
rier since the accreted mass cannot be less than the mass
of a single planetesimal. Thus as n is decreased to 1 and
below, the mass accreted in any one sinking time becomes
morenoticeablyprobabilistic.Thesameeffectisalsoseenin
Fig. 3b, except that the mass remaining as potentially vis-
ible in the atmosphere can be less than m . Indeed for the
p
shortest sinking times of 0.1 and 1 years, the atmospheric
mass spends most of its time at insignificantly small lev-
els,increasingtothelevelofm onlyimmediatelyfollowing
p
an accretion event, with exponential decay thereafter. In (b)
constrast, for the longest sinking times (n 1), the atmo-
spheric mass is approximately constant at a≫level M˙ t .
in sink
The distributionof atmospheric masses is quantifiedin
Fig. 3c, which shows the fraction of time the accretion rate
wouldbeinferredtobeaboveagivenlevel.Forlongsinking
timescales (n 1), this is close to a step function, transi-
tioning from 1≫to 0 close to M˙ ; i.e., the inferred accretion
in
rate is always very close to the mean level. For short sink-
ing timescales (n 1) however, the inferred accretion rate
≪
coversabroadrange,fromaroundm /t justafteranac-
p sink
cretion event,which is significantly higher than M˙ in this
in
regime, down to levels far below M˙ . As noted in 3.3, the
in
§
distribution at levels just below m /t in this regime is
p sink
to a reasonable approximation dictated by the exponential
decayfunction,sinceintermediateaccretionratesaresimply
thevestiges of earlier accretion events. (c)
Figure 3. Monte carlo simulations of accretion of 3.2×1019 g
planetesimals at a mean rate 1010 gs−1 onto white dwarfs with
sevendifferentsinkingtimestsinklogarithmicallyspacedbetween
0.1 yr and 0.1 Myr shown with different colours. (a) The total
massaccretedinonesinkingtime,asafunctionoftime,withonly
3.3 Analytical
the first500sinkingtimes shown forclarity. (b)The total mass
The distribution of shot noise characterised in the manner remaining as potentially visible in the atmosphere as a function
of equations (6) and (7) is given in section 6.1 of Gilbert oftime.(c)Thefractionofalltimestepsforwhichtheaccretion
& Pollack (1960) (see Appendix A). There they derive rateismeasuredtobeabovetherategivenonthex-axis;i.e.,the
the exact form of the probability density distribution for cumulative distribution function f(> M˙atm). The top axis gen-
M˙ <m /t (or equivalentlyfor M˙ /M˙ <n−1) as eralises this plot to dimensionless accretion rate (M˙/M˙in) when
atm p sink atm in usedinconjunctionwiththenumberofaccretioneventspersink-
ingtime(n)giveninthelegend.
t n e−nγ
P(M˙ )= sink M˙n−1, (8)
atm (cid:18) mp (cid:19) Γ(n) atm
(cid:13)c 0000RAS,MNRAS000,000–000
Stochastic accretion of planetesimals onto white dwarfs 9
m
f(>M˙ )=nln p (11)
atm (cid:20)M˙ t (cid:21)
atm sink
in the range 1 to e−1/n times m /t .
p sink
Thereisnoexactsolutionforthedistributionathigher
accretion rates (M˙ >m /t ), however Gilbert & Pol-
atm p sink
lack provide a differential difference equation that can be
solvedtodetermineP(M˙ )(seeeq.A5).Weshowthere-
atm
sulting solution in green on Fig. 4, but only over a limited
region of parameter space as validation of the technique,
and of the Monte Carlo model, since these are essentially
different numerical methods of obtaining thesame answer.
However, there is an asymptotic solution in the large
n regime (i.e., large t ). Campbell’s theorem (Campbell
sink
1909) can be applied to show that the probability density
function in this limit becomes a Gaussian with a mean of
M˙ (see eqs. A8and A9)
Figure 4.Simulationsofaccretionof3.2×1019 gplanetesimals in
tastinak.mTeahnerliantees1s0h1o0wgtsh−e1doinsttoribauwtihointeodfwinafrefrrweidthacacsrientkioinngratitmese; P(M˙atm)=(cid:18)tmsinpk(cid:19)√21πσ2e−2σ12(M˙atm−M˙in)2, (12)
e.g.,thetoplinecorrespondstothelevelthatwouldbeexceeded
in 1% of measurements, while the f(> M˙atm) = 0.5 line is the where the variance σ2 = n(mp/tsink)2/2. This means that
median of the distribution. The blue solidlineshows the results thecumulativedistribution function is
of an expanded set of Monte Carlo simulations similar to those
shown in Fig. 3. The dashed lines show various analytical esti- f(>M˙ )=[1 erf(x)]/2, (13)
atm
mates discussed in the text: the M˙atm < mp/tsink solution in −
purple, the solution to the Gilbert & Pollack (1960) differential where erf(x) is the error function of x = M˙atm−M˙in .
difference equation ingreen, and Campbell’s theorem inorange. √M˙inmp/tsink
Equation (13) can be solved to get the appropriate points
The top and right axes generalise this plot to dimensionless ac-
cretionrate(M˙/M˙in)asafunctionofnumberofaccretionevents in the distribution shown in orange on Fig. 4 for n > 1.
persinkingtime(n). This shows that Campbell’s theorem provides an adequate
approximation for large n, but that discrepancies become
noticeable as n approaches 1.
where γ 0.577215665 is Euler’s constant and Γ(n) is the
≈
gamma function (see eq. A6).This means that thecumula- 4 CAN A MONO-MASS PLANETESIMAL
tivedensity distribution is DISTRIBUTION FIT THE OBSERVATIONS?
e−nγ M˙ t n It is clear from 3 that even with a very simple model, in
f(>M˙atm)=1− nΓ(n)(cid:18) atmmpsink(cid:19) . (9) which planetesim§als have the same mass around all stars,
andinwhichallstarsareaccretingmatteratthesamemean
Rather than compare this prediction directly with the rate, it is expected that a broad distribution of accretion
distributionderivedfromtheMonteCarlomodelinFig.3c, rates could beinferred observationally, and that this distri-
weinsteadusethosedistributionstofindthe1%,10%,50% butioncouldbedifferenttowardwhitedwarfswithdifferent
and90%pointsinthedistribution,repeatforalargernum- sinking times. However, in 4.1 we explain why such a sim-
§
ber of sinking times, and plot these as a function of t in ple model cannot explain the observationally inferred rates
sink
Fig.4.Abbreviatingf(>M˙ )tof fornow,theprediction of 2. Then in 4.2 we explore the possibility that all stars
atm § §
is that haveplanetesimals that are thesame mass, but that differ-
ent stars have different mean accretion rates, again ruling
m
M˙atm(f)=(cid:16)tsinpk(cid:17)[(1−f)nΓ(n)enγ]1/n, (10) athffiescoteudt.iIfnp§la4n.3e,tewseimcaolnssiadreerphroowcetshseedsetchornoculguhsioandsimscaoynbae
whichwillbevalidaslongasthequantityinsquarebrackets timescalethatcanexceedthesinkingtimescalebeforebeing
is less than 1 (e.g., for n = 1, i.e. t = 100 yr, this is accreted.
sink
valid for f > 1 e−γ 0.44). This is plotted in purple on Throughout the paper we quantify the goodness-of-fit
− ≈
Fig. 4 showing excellent agreement with the Monte Carlo for a model in a given sinkingtime bin s as
model,notingthatdeviationsfromtheanalyticalprediction
f(>M˙ ) f(>M˙ ) 2
are expected dueto small numberstatistics. χ2 = obs(j,s) − atm(j,s) , (14)
Forheuristicpurposes,itisalsoworthpointingoutthat s (cid:18) σ[f(>M˙ )] (cid:19)
the distributions in the limit of n 1 for f(> M˙ ) 1 Xj obs(j,s)
atm
are asymptotically the same as wo≪uld be expected had≪we where f(> M˙ ) is the best estimate from the ob-
obs(j,s)
imagined planetesimals to arrive at regularly spaced inter- servations of the fraction of stars in bin s with accretion
vals of t /n in time. In that case, thefraction of time we abovealeveldenoted bytheindexj,wherethesumis per-
sink
wouldexpecttoinferaccretionratesofdifferentlevelswould formed for j corresponding to 107, 108, 109 and 1010 gs−1,
bedetermined bythe exponentialdecay, and so σ[f(>M˙ )]isthelarger ofthepositiveornegative1σ
obs(j,s)
(cid:13)c 0000RAS,MNRAS000,000–000
10 M. C. Wyatt et al.
uncertainties plotted on Fig. 2d, and f(> M˙ ) is the
atm(j,s)
corresponding model distribution. Since the observables in
a cumulative distribution (i.e., f(> M˙ )) are not in-
obs(j,s)
dependent at the different indices, the absolute value of χ2
s
should not be used to determine the formal significance of
themodelfittothedata.Ratherwewillbeusingithereas
a relative measure of thegoodness-of-fit of different models
for a given bin.
4.1 Mono-mass, mono-rate accretion
Thedistributionofinferredaccretionratesforamono-mass
mono-rate model will always have a dependence on sink-
ing time that is similar in form to that shown in Fig. 3c.
Varyingthemeanaccretionrateparameter,M˙ ,wouldsim-
in
plychangethex-axisscaling suchthatthedistributionsfor
thelongestsinkingtimesallhaveaccretion ratesinferredat
the M˙ level (see top axis). Varying the planetesimal mass
in Figure 5.Simulationsofaccretionof2.2×1022 gplanetesimals
would change thesinkingtimes corresponding tothediffer-
atamean rate1.7×108 gs−1 onto populations of whitedwarfs
ent lines on the figure, but these lines would always corre-
withdistributionsof sinkingtimes that match that ofthe corre-
spondtothesamengiveninthelegend(e.g.,thegreenline
sponding observed populations in each of the sinking time bins.
corresponds to n = 1), and so eq. (6) can be used to work
The dashed lines show the distribution inferred from the obser-
out the corresponding sinking time which just scales with vations,whilethehatchedregionsanddottedlinesshowthe±1σ
planetesimal mass (e.g., the pale green line corresponds to rangeof possibledistributions givensmall number statistics (re-
t =m /M˙ ). produced from Fig. 2d). The model predictions are shown with
sink p in
Fig.2dprovidesseveralcluesastowhatcombinationof solid lines in the corresponding colour. The model for the short
M˙ and m would be required to reproduce any given dis- sinkingtimebinisindistinguishablefrom0onthisplot.
in p
tribution. For example, the fact that the M˙ distribution
obs
is broad for all of the sinking time bins means that n 1
≪
forallofthebins.Thebreadthofthedistributionisindica-
having n 1 in the long bin means that such timescales
tive of the n required to fit any of the relevant lines, and ≪
are already sampling the vestiges of past events (i.e., such
theappropriatevalueforthelongsinkingtimescale bincan
events happen much less frequently than once per Myr).
beinferred readilyfromFig.4usingthetopandrightaxes.
Thismeansthat,whileitispossibleformeasurementswith
That is, for thereto bea range of around 1000 in accretion
shortersinking times toinfer high accretion rates just after
ratesbetweenthe10%and50%pointsinthedistributionre-
the event, such measurements would be extremely rare. By
quiresn 0.09.Theinputaccretionratecanthenbefound
by scalin≈g the 50% point to be close to 106 gs−1 (Fig. 2d) consequencewewouldexpecttoseeessentially noaccretion
signatures in thesamples with t <0.1 Myr(see Fig. 5).
givingan M˙ of around1.7 108 gs−1,and soaplanetesi- sink
in
×
malmassm ofaround2.2 1022 g(forthemediansinking
p
×
time of 0.37 Myr in thisbin).
4.2 Mono-mass, multi-rate accretion
Fig. 5 reproduces the inferred accretion rate distribu-
tionsofFig.2dandalso makespredictionsformodelpopu- One conclusion from 4.1 is that, for a mono-mass distri-
§
lationsinwhichstarshavethesamedistributionsofsinking bution of planetesimal masses, a model that fits all sinking
timesas thatoftheobservedpopulation in thecorrespond- time bins simultaneously requires n 1 for (the majority
≫
ing bin, under the assumption that all stars are accreting of) thelong sinkingtimebin.Thebroad distribution of ob-
2.2 1022 gplanetesimals(i.e.,roughly240kmdiameteras- servationallyinferredaccretionratesinthisbin,f(>M˙ ),
obs
tero×ids)atameanrate1.7 108 gs−1 (equivalenttoaround thus implies that different stars accrete at different rates,
×
1 asteroid belt every 680 Myr). This model population was and that the observationally inferred distribution is repre-
implemented by taking each star in the corresponding ob- sentativeofthatofthemeanrateatwhichmaterialisbeing
served population and running the Monte Carlo model of accreted, f(>M˙ ). At least this must be the case for high
in
3.2withthesinkingtimeforthatstar,thencombiningthe accretion rates, but it is possible that the lowest accretion
§results for all stars into one single population. The number rates, say below M˙ = 107 gs−1, are in the n < 1 regime.
in
oftimestepsusedforeachstar,N ,waschosensothatthe This also sets a constraint on the planetesimal mass, since
tot
total number of accretion rates used for the model popu- requiring n > 1 in the 0.37 Myr sinking time bin for
lation (i.e., N times the number of stars in the observed 107 gs−1 means that m∼ <1020 g.
tot p
∼
population) was close to 105. Herewemodifythepopulationmodelof 4.1byassum-
§
Asexpectedfromtheargumentstwoparagraphsago,a ingthatdifferentwhitedwarfsaccreteatdifferentrates,i.e.
model with these parameters gives a decent fit to the long that there is a distribution f(> M˙ ), but from the same
in
sinkingtimebin(forreferenceχ2 =1.0asdefinedineq.14). mono-massdistributionofplanetesimalmasses, m .Practi-
s p
However, the same model provides a very poor fit to the cally this is implemented in the model population by each
shorter sinking time bins (χ2 =12 and 21 in the short and of the observed stars in the appropriate sample having its
s
mediumsinkingtimebins,respectively).Theproblemisthat accretion rate chosen randomly from the given distribution
(cid:13)c 0000RAS,MNRAS000,000–000