A Search for Asteroids, Moons, and Rings Orbiting White Dwarfs
aa r X i v : . [ a s t r o - ph . E P ] D ec A Search for Asteroids, Moons, and Rings Orbiting White Dwarfs
Rosanne Di Stefano , Steve B. Howell , Steven D. Kawaler Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 National Optical Astronomy Observatory, 950 N. Cherry Ave, Tucson, AZ 85719 Department of Physics and Astronomy, Iowa State University, Ames, IA 50011
ABSTRACT
Do white dwarfs host asteroid systems? Although several lines of argument sug-gest that white dwarfs may be orbited by large populations of asteroids, transits wouldprovide the most direct evidence. We demonstrate that the
Kepler mission has the ca-pability to detect transits of white dwarfs by asteroids. Because white-dwarf asteroidsystems, if they exist, are likely to contain many asteroids orbiting in a spatially ex-tended distribution, discoveries of asteroid transits can be made by monitoring onlya small number of white dwarfs, compatible with
Kepler’s primary mission, which isto monitor stars with potentially habitable planets. Possible future missions that sur-vey ten times as many stars with similar sensitivity and minute-cadence monitoringcan establish the characteristics of asteroid systems around white dwarfs, such as thedistribution of asteroid sizes and semimajor axes. Transits by planets would be moredramatic, but the probability that they will occur is lower. Ensembles of planetarymoons and/or the presence of rings around planets can also produce transits detectableby
Kepler . The presence of moons and rings can significantly increase the probabilitythat
Kepler will discover planets orbiting white dwarfs, even while monitoring only asmall number of them.
1. Introduction
Stars are orbited by dust, asteroids, planets and their moons. As each star evolves, its planetarysystem evolves as well, through a combination of stellar expansion, mass loss, and dynamicalinteractions. In spite of the fact that the first planet to be discovered orbits a pulsar (Wolszczan &Frail 1992), we still know little about the planetary systems around stellar remnants. Fortunately,a variety of methods are poised to change this. Pulsar timing studies in combination with HSTimages have found a planet in a circumbinary orbit around a binary consisting of a millisecondpulsar and a white dwarf (Sigurdsson et al. 2003). Timing measurements of pulsating compact 2 –stars can also identify candidate planetary systems around highly evolved stars (i.e. Silvotti et al.2007), with complementary
Spitzer observations able to detect or place limits on possible planetarycompanions of white dwarfs (Mullally et al. 2009). In this paper we point out that the
Kepler mission has the sensitivity and cadence needed to detect transits of white dwarfs by asteroids.
Calculations show that asteroid or cometary systems can survive stellar evolution (Alcock etal. 1986). While asymmetries in the mass loss can influence survivability, some white dwarfscould experience asteroid impacts at a rate of 10 - yr - (Parriott and Alcock 1998). Several in-dependent lines of evidence suggest that some white dwarf stars do host circumstellar materialranging from dust to asteroids and planets (Howell et al. 2008; Jura 2008 and references therein;Farihi et al. 2009; Jura, et al. 2009; Jura, Farihi, & Zuckerman 2009). Asteroid-sized objects,with diameters ranging from several tens of kilometers up to and including dwarf planets, are theprimary focus of this paper. Their instantaneous orbital distance from the white dwarf ranges fromthe tidal limit out to the equivalent of the Sun’s Oort Cloud. We will also consider the possibleeffects of rings and moons orbiting planets in white-dwarf systems.White dwarfs with infrared excesses have been studied by a number of groups. A commonconclusion is that circumstellar dust is present. These white dwarfs tend to exhibit unusuallystrong metal lines in their photospheres (von Hippel et al. 2007), perhaps showing the signs ofenrichment by recent accretion of material from an asteroid. Indirect evidence of planetary materialaround warmer white dwarfs includes several stars that show metal lines in their photosphericspectrum (Zuckerman et al. 2007, and references therein). In white dwarfs, metals sink below thephotospheres on extremely short time scales (of order days), so the presence of elements such ascalcium argues that the stars have recently accreted metal-rich material.As an example of possible asteroidal material surrounding a white dwarf, consider the recentwork by Gansicke et al. (2007). They report observations of CaII and FeII double-peaked emissionlines, interpreted to be from a circumstellar, metal-rich gaseous disk. The white dwarf itself is hotenough ( T eff ∼ K ) to burn off dust from the disk, accounting for the lack of an infraredexcess. The disk itself is the likely remnant of a tidally disrupted rocky body of asteroid-sizedmass (Gansicke et al. 2007). The case for a metal-rich disk is strengthened by observations ofMgII absorption lines in the stellar spectrum. Dynamical modeling of the system by Gansicke etal. (2007) constrain the outer edge of the disk to be at about 1 . R ⊙ , and places the inner edge atapproximately 0 . R ⊙ . Zuckerman et al. (2007) provide an analysis of the white dwarf GD 362, including an estimate 3 –of the abundance of 17 elements accreted by that star. They conclude that an asteroid-mass object(either a remnant asteroid or the residual of a disrupted terrestrial planet) is needed to explain theabundance pattern. A very rough estimate of the size of an object needed to account for the metalsseen in that star (10 g) is of order 80 km; such an object would produce a transit with a depth of0 . For every asteroid that is tidally disrupted, there must be many more with perihelia locatedmuch farther from the white dwarf. Direct detection of these asteroids is challenging. They havevery little gravitational influence on their star and cannot presently be detected through eitherDoppler or ground-based photometric transit methods. Transits of white dwarfs by large asteroidscan, however, be detected by
Kepler , a space mission designed to detect the transits of Sun-likestars by Earth-like planets. Consider a white dwarf with a radius of 8000 km. An asteroid witha radius of 100 km (1000) km will produce a fractional decrease in the amount of light receivedof 150 ppm (15,000 ppm), within Kepler’s detection limit. Many such asteroids of this size areknown in our own solar system [an estimated 80 ,
000 in the Kuiper Belt alone (Trujillo, Jewett, &Luu 2001)], so it is reasonable to expect that they exist elsewhere as well.In §2 we show that the
Kepler observatory can discover asteroids orbiting white dwarfs byidentifying short-lived downward deviations from the baseline flux in white dwarfs associated withtransits by asteroids. In § 3 we discuss what we can learn through
Kepler monitoring of a small setof white dwarfs. Asteroid transits or significant limits on white-dwarf asteroid systems are a certainscience return. In addition, depending on the structure of white-dwarf planetary systems, transitsby rings and/or moons may also be detected by monitoring a modest number of white dwarfs.
2. Kepler Detections of Asteroid Transits2.1. Detection of Transits
The depth of a transit and its time duration determine its level of detectability. If A ast is theprojected area of the asteroid, and A wd = π R the cross-sectional area of the white dwarf, the depthof the transit is ( A ast / A wd ) . The calculations for asteroids transiting white dwarfs mirror the resultsfor an Earth-like planet transiting a Sun-like star, because the relative size scales are similar. Foran asteroid of a given size, the depth is greatest for more massive white dwarfs, which are smaller.Asteroid transits against a white dwarf will be of short duration and may have distinctive 4 –profiles. The time required for the asteroid’s center of mass to cross the diameter of the whitedwarf is τ cross = 2 R wd / v , where v is approximately equal to the orbital velocity. . τ cross = 9 . (cid:16) R wd . × cm (cid:17) (cid:16) a AU (cid:17) (cid:16) . M ⊙ M wd (cid:17) (1)Because the orbital speed, v , decreases with increasing a , an asteroid of fixed size produces a longerevent when it is farther from the white dwarf. The crossing time is larger for less massive whitedwarfs because the orbital speed is smaller (for a given a ) and because the radius of a low-masswhite dwarf is larger. The ingress and egress may be distinctive because many asteroids will notbe massive enough to have been pulled into a spherical shape by self gravity.We have carried out a set of calculations to determine how large an asteroid must be in orderfor its transit against a white dwarf of given brightness to be detectable by Kepler.
We use the in-formation on the
Kepler web pages to estimate the number of detected photoelectrons per minute: N = 1 . × (10 - . M Kepler - ) per minute. For representative examples, we used the Kepler mag-nitudes of 3 relatively bright white dwarf candidates in the
Kepler field: M Kepler = 12 . , . , . . For each candidate white dwarf we carried out two sets of calculations. In the first set, we assumedthat the mass of the white dwarf was M wd = 0 . M ⊙ , and used the corresponding radius. In thesecond set we used M wd = 1 . M ⊙ , and decreased the value of R wd accordingly. Then, for each of60 values of the orbital separation a , we computed the diameter of an asteroid for which we wouldhave a 1 σ , σ , and 3 σ detection of the transit, by integrating over the crossing time. The resultsare shown in Figure 1. For lower-mass white dwarfs, asteroids of smaller diameter can producedetectable transits. For the brightest white dwarf, transits of asteroids in the 100-km class can bedetected even when they are relatively close to the white dwarf. The dimmer the white dwarf, thefarther from it must a 100-km asteroid be in order for its transit to be detected by Kepler.
In thesolar system, the Kuiper Belt extends from roughly 30 AU to 50 AU. Large interlopers from theOort Cloud, such as 2006 SQ372, are also found in this region, while the bulk of the Oort cloudlies beyond 1000 AU. Although the white dwarf systems we target for
Kepler study may be verydifferent, the example of the solar system indicates that it is good to be sensitive to asteroids atlarge values of a . Equation 1 neglects the size of the asteroid, the proper motion of the white dwarf, and assumes that the transitoccurs along a diameter of the disk. http://kepler.nasa.gov/ -2 0 2 40200400600800 Log[a (AU)] -2 0 2 4Log[a (AU)] -2 0 2 4Log[a (AU)] Fig. 1.—
The asteroid diameter (in km) necessary in order to detect a transit vs the orbital separation (in AU) atthe time of transit. The significance of the detection was estimated by integrating over the crossing time. The mass ofthe white dwarf was taken to be 0 . M ⊙ in the green (lighter) curves and 1 . M ⊙ in the blue (darker) curves. For eachwhite dwarf, the (lowest, middle, top) curve corresponds, respectively, to a (1 σ, σ, σ ) detection. Each panel refersto the Kepler magnitude of a specific candidate white dwarf known to be in the
Kepler field.
The probability of detecting a transit by an individual asteroid is small, because it is propor-tional to the maximum angle of orbital inclination for which a transit can be observed: ( R wd + R ast ) / a , where R wd is the radius of the white dwarf, R ast is the radius of the asteroid, and a is theorbital separation. If a typical value of ( R wd + R ast ) is 8 × cm, and the average value of a atthe time of transit is 1 AU, then taking inclination alone into account, we would have to observealmost 30 ,
000 white dwarfs to have a good chance of detection. If each white dwarf is orbited bymany asteroids, the probability of detection increases. Should all of the asteroids orbit in a com-mon plane, however, we must still monitor a large number of white dwarfs to have a good chanceof detection.Fortunately, our own solar system offers hope, and star and planet formation theory alsosuggest that planetary systems each host large numbers of asteroids and that the orbits are notaligned. The Oort Cloud ( a > In this discussion of orientation effects we neglect duty cycle issues, but note that if the orbital period is largecompared to the total time during which observations occur, there is a further diminuation of the detection probability. -
50 AU), while others could make even closer approaches. If each whitedwarf we observe has a distributions of asteroids with the geometry of the Oort Cloud, we have agood chance to detect transits with
Kepler monitoring of even a single white dwarf. The KuiperBelt itself has a scattered component in which the average orbital inclination angle, i , is 12 o , while individual orbits can be even more inclined. (See Sheppard 2006 and references therein.)Therefore, if the asteroid systems of white dwarfs have a geometry similar to that of the scatteredKuiper Belt, and if we could monitor 10 -
15 white dwarfs, we would have a good chance that theequivalent of the scattered “Kuiper” Belt of one or more of them was inclined toward our line ofsight.Assuming a spherical distribution, we can quantify the probability of detecting a transit as afunction of a as follows. We compute the number of asteroids that would have to have periastronsat a particular value of a in order to have a probability near unity of detecting the transit. Given( R wd + R ast ) = 8 × cm, then for a = (0.1, 1, 10, 100) AU, the number of such asteroids is (1 . × , 1 . × , 1 . × , 1 . × ). These numbers assume that the interval T during whichcontinuous observations occur spans the orbital periods of the asteroids. In fact, T will be longerthan P orb for small orbital apastrons, and the scaling above holds for those separations. Evenbetter, for T > n P orb , n transits will be detected; confidence that the photometric dips were causedby transits can therefore be high, just as multiple planetary transits enhance confidence in thediscovery of planets. For wider separations, however, the probability is reduced by a factor T / P orb . For circular orbits, the number of asteroids needed to have a transit detection probability near unityis N = 1 . × (cid:18) yr T (cid:19) (cid:18) M ⊙ M (cid:19)(cid:18) R wd + R ast × cm (cid:19)(cid:18) a AU (cid:19) . (2)If T = P orb for a = 1 AU , then the numbers of asteroids needed to ensure detection for a = (0.1, 1,10, 100) AU are (1 . × , 1 . × , 6 . × , 1 . × ).These numbers are modest enough to suggest that, unless the white dwarf systems are depletedin 100-km class asteroids relative to the solar system, Kepler can discover asteroid transits bymonitoring a handful of white dwarfs. Furthermore, these numbers above are small enough that anull result would represent a meaningful limit. 7 –
The characteristics of transit light curves are related to the properties of the asteroid. If theingress and egress can be resolved in time, there will be an interval δ t in during which the fluxdeclines by ∆ F , a second interval ∆ T , during which the flux remains at its steady minimum value F - ∆ F , and a third interval δ t out during which the transit ends and the flux returns to the level, F it would have had without the transit.It is important to note that, particularly for the bright white dwarfs likely to be monitored by Kepler, the white dwarf’s radius and mass can be estimated to high accuracy using good qualityoptical spectra and model fits to T eff and log ( g ). The depth of the transit (the maximum downwarddeviation, ∆ F ), therefore measures the projected area of the asteroid. The value of ∆ T , combinedwith the estimated white dwarf’s radius, provides an estimate of the asteroid’s speed, hence itsdistance from the white dwarf. If δ t in and δ t out cannot be measured, then we can use the timeresolution of the observations to place upper limits on the linear dimensions of the asteroid. If theycan be measured, then we can (1) determine the projected linear size of the asteroid at the time ofingress and (2) also at the time of egress. If they are different, then the asteroid may have beenspinning; we can (3) check for consistency to determine if a realistic spin period is consistent withthe observed change. If they are the same, we can (4) determine if the shape of the light curveis consistent with a disk-like structure. Using the area (from the depth of the transit) to estimatethe possible mass, we can (5) check if we expect the asteroid to be spheroidal. Finally, we can(6) check if the linear dimensions as estimated during ingress and egress are consistent with theprojected area, as estimated from the depth of transit.Note that if an event fails consistency checks, we can rule it out as a transit candidate. Passingthe checks does not however confirm the transit interpretation. When transit candidates are iden-tified, an exhaustive analysis is required of any effects that could have produced a false positivesignal. In an individual case, if the asteroid orbit happens to not be highly eccentric and if a is notmuch larger than an AU, we may see a repeat during the lifetime of Kepler; this will confirm thetransit model. Overall, it is likely that an asteroid population large enough to produce a transit byone asteroid will produce transits by several independent asteroids, and that a set of self-consistentresults will increase confidence in the transit interpretation. Note that the white dwarf may be intrinsically variable. If the variability is periodic, it will not interfere with ourability to detect a transit. Nevertheless, when analyzing the light curves, care must be taken to consider the influence ofany intrinsic variability because if the variability is complex, detectability may require a deeper, longer-lasting transit.To simplify the in the text, we discuss the flux F as if it is constant when a transit is not occurring. The nature of the results that can be obtained by
Kepler depends on the characteristics ofplanetary systems. Because the progenitor of a white dwarf was a giant, it seems likely that theregion within roughly an AU was cleared of planets and asteroids. Yet, the evidence sited in theintroduction of this paper indicates that asteroids can and do approach close to white dwarfs.If the monitored white dwarfs have close-in asteroids with small semi-major axes, we willdiscover “repeats”in a sense: multiple transits with similar characteristics. If the monitored whitedwarfs have large asteroids, the photometric dips during transits will be highly significant. Ifthe monitored white dwarfs have ∼ asteroids in a Kuiper-like belt, several events caused bydifferent asteroids will be observed. Even if individual events are detected with low confidence(e.g., 1 - σ photometric dips), we may be able to derive significant results when several such aredetected. This is because the probability of detecting multiple dips due to random processes (whichwe can assess through observations of other stars) is expected to be low. Thus, multiple transitscaused by one or by several asteroids, or deep transits, or long-lasting transits would provideinformation about some characteristics of the asteroid system.It is certainly possible, however, that for one or more white dwarfs, no highly significantevents are discovered. In this case, given the estimated efficiency, which will be well knownbased on Kepler’s observations of hundreds of thousands of other stars, we can place quantitativelimits on the presence of close-in, and/or large, and/or numerous asteroids around any given whitedwarf. We therefore expect monitoring of each white dwarf to produce significant results of eithera positive or negative nature.
3. Prospects3.1. Asteroids
The
Kepler team is about to announce the first year’s results. The mission is scheduled to takedata for 3 . Kepler can trans-mit data on approximately 170 ,
000 targets, most main-sequence stars which are being monitoredin hope of detecting transits by Earth-like planets. A limited number of data slots are availableto monitor other targets suggested by the community. Two modes of monitoring are available:30-minute cadence and 1-minute cadence. Equation 1 shows that the 1-minute mode is neededto detect transits by close-in asteroids. Although asteroids making close approaches to the whitedwarf must be larger if their transits are to be detectable, it is important to be sensitive to closeapproaches for two reasons. First, the probability that the orientation is favorable scales as 1 / a . Kepler field are known. Fortunately,the large numbers of asteroids expected per white dwarf will almost certainly make it possible todiscover asteroids by monitoring almost any white dwarf that has them.Because planets are likely to form in a bottom-up approach, stellar formation seems likely toalways produce small masses that will be gravitationally bound to the star, regardless of whetherlarge planets form. Even though stellar evolution is associated with mass loss from the system,a large number of asteroids should remain bound. In addition, the dynamical evolution of plane-tary orbits during stellar evolution seems likely to yield collisions and additional space debris inthe form of asteroids. This line of argument is consistent with the data summarized in the intro-duction, which argues independently that asteroids orbit white dwarfs. Nevertheless, some whitedwarfs may be less likely to host asteroid systems, at least the ones associated with planet forma-tion. Consider, for example, a white dwarf that emerged from a common envelope episode. Thissuggests that the white dwarfs most suitable for the first monitoring program are those with massesnear or above 0 . M ⊙ ; in addition, they should not have close stellar companions. If white-dwarf asteroid systems occupy a region similar to the Solar System’s scattered disk,then by monitoring a set of white dwarfs, we can sample a random distribution of possible orien-tations. If therefore,
Kepler can monitor (for approximately one year each) roughly a dozen brightwhite dwarfs, it should discover asteroids and begin to quantify the fraction of white dwarfs withasteroids in the 100-km class.Future projects that can take this study further are under consideration. Since missions with
Kepler ’s sensitivity can detect asteroids around white dwarfs, a more comprehensive all-sky surveymonitoring ∼ . × stars (such as that proposed for the Transiting Exoplanet Survey Satellite TESS ) will be able to establish the statistics of asteroid systems around white dwarfs: the fre-quency as a function of white dwarf properties, and the distributions of asteroid sizes and orbitalseparations. White dwarfs in close binaries and those that have been involved in prior mass transfer may also be interesting,but the science to be explored in those cases is different. We therefore suggest that a limited program focus on isolatedwhite dwarfs.
10 –
White dwarfs may well be orbited by planets, but the probability P that the orientation of aplanetary orbit is favorable for the detection of a transit is small. P = ( R wd + R pl ) a = 2 . × - (cid:16) R wd + R pl × cm (cid:17) (cid:16) AU a (cid:17) . (3)This implies that thousands of white dwarfs would have to be monitored in order to discover transitsby planets, which is not compatible with the primary goal of the Kepler mission. Nevertheless,other signatures of planets are more likely to be detected by
Kepler .Although planetary rings are generally composed of bits of debris that are individually toosmall to produce detectable transits, the combined effect can be to absorb and scatter enough lightthat the transit of the ring will be detectable (e.g., Ohta, Taruya, & Suto 2009 and referencestherein). In this case, the ingress and egress profiles are likely to be symmetric and distinguishablefrom the patterns produced by an isolated mass. Strategies for seacrhing for evidence of rings andmoons in transit light curves have been developed (Barnes & Fortney 2004; Barnes 2004). HSTobservations of the planetary transit of HD 189733 were able to dervie a convincing null result,ruling out the presence of rings or moons around around HD 189733b through a detailed lightcurve analysis (Pont et al. 2007).
Kepler could do the same, or else discover moons and ringsaround white dwarf planets, should they exist.To compute the probability of detecting a transit by a ring, the term R pl in Equation 2 must bereplaced by R ring sin ( θ ) In this expression R ring is the outer diameter of the ring system, and can besignificantly larger than R pl . Thus, even though the planet itself may out of our line of sight, the ringcould transit. In the case of Saturn, for example, the outer radius of its ring system (4 . × cmfor the E ring) is almost ten times larger than the radius of the planet. The angle θ in Equation 2is the angle between the plane of the ring and the orbital plane of the planet. If θ = 0 , then transitsby the ring will only be detected in some cases in which the planet transits as well. In those casesthe effect of the ring will be striking, because it will produce a diminuation in light from the whitedwarf that lasts significantly longer than the transit by the planet. If, however, the plane of the ringis oriented at a non-zero angle θ relative to the orbital plane, then the probability of a transit bythe rings can be greater than the probability of a transit by the planet. ( R ring sin ( θ )) > ( R wd + R pl )If a Saturn-like planet were to orbit a white dwarf at a = 0 . θ = 90 , the value of P would be 0 . . If such systems are common, then a ring transit could be discovered by monitoringone to two dozen white dwarfs.Planets are also orbited by moons. Our own solar system contains more than 150 moons,many large enough to produce transits of a white dwarf that would be detectable by
Kepler.
Thecases in which a moon is likely to produce a transit, even if the planet does not transit, are those in 11 –which the planet orbits the star many times during the course of the monitoring observations and inwhich the moon also orbits the planet many times during the same interval. Let a m be the distancebetween the planet and its moon, and let θ be the angle between the orbital planes of the moonand planet. To compute P , the term R pl in Equation 2 must be replaced by a m sin ( θ ) . Consider aSaturn-like planet orbited by a moon at 10 cm (an 11 day orbit). If the distance of the Saturn-likeplanet from an 0 . M ⊙ white dwarf is equal to 0 . . θ = 90 , then P = 0 .
04 As with the case of rings, the probability of that a detectable transit will occur during ayear of monitoring one to two dozen white dwarfs is significant.Altough we do not have the a priori knowledge needed to assess the likelihood of transitsby rings or moons, the discovery that exoplanets commonly have properties that were unexpectedleads us to consider a range of possibilities for planets orbiting white dwarfs. The considerationsabove show that, if ‘ white dwarfs tend to be orbited by close-in planets, and i if ‘ these planetshave rings and/or moons, there is a chance that Kepler will discover them by monitoring a modestnumber of white dwarfs. An all–sky survey with the sensitivity of Kepler would either discoversuch systems or definitively rule them out.Consider the possibility that white dwarfs host both asteroid systems and close-in planetswith rings and/or moons. Dynamical stability arguments place constraints on the number of close-in planets and on the linear dimensions of the system of moons orbiting each. Unless, therefore,the asteroid systems are deficient in large asteroids relative to what we might expect based on thesolar system, transits by asteroids should provide the dominant signal.