FINDING ROCKY ASTEROIDS AROUND WHITE DWARFS BY THEIR PERIODIC THERMAL EMISSION

8/11/2019 FINDING ROCKY ASTEROIDS AROUND WHITE DWARFS BY THEIR PERIODIC THERMAL EMISSION

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Draft version September 1, 2014Preprint typeset using LATEX style emulateapj v. 12/16/11

FINDING ROCKY ASTEROIDS AROUND WHITE DWARFS BY THEIR PERIODIC THERMAL EMISSION

Henry W. Lin1, Abraham Loeb2

Draft version September 1, 2014

ABSTRACTSince white dwarfs are small, the contrast between the thermal emission of an orbiting object and a

white dwarf is dramatically enhanced compared to a main sequence host. Furthermore, rocky objectsmuch smaller than the moon have no atmospheres and are tidally locked to the white dwarf. We showthat this leads to temperature contrasts between their day and night side of order unity that shouldlead to temporal variations in infrared flux over an orbital period of 0.2 to 2 days. Ground basedtelescopes could detect objects with a mass as small as 1% of the lunar massMLaround Sirius B witha few hours of exposure. The James Webb Space Telescope (JWST) may be able to detect objects assmall as 103ML around most nearby white dwarfs. The tightest constraints will typically be placedon 12,000 K white dwarfs, whose Roche zone coincides with the dust sublimation zone. Constrainingthe abundance of minor planets around white dwarfs as a function of their surface temperatures (andtherefore age) provides a novel probe for the physics of planetary formation.

Subject headings: white dwarfs

1. INTRODUCTION

The search for extrasolar planets around main se-quence stars has already harvested thousands of can-didates over the past decade (see e.g. Batalha et al.2013). Although the number of confirmed exoplanetsis now well over a thousand, no exomoons have yet beenconfirmed, though the smallest exoplanet detected hastwice the mass of the moon(Konacki & Wolszczan 2003).To date, no detections of extrasolar rocky objects withmasses an order of magnitude lower than a lunar massML= 7.35 1025 g have been reported.

In this Letter, we propose a method to detect objectswith masses several orders of magnitude smaller than

ML in close orbits around white dwarfs (WDs). WDsare ideal for two reasons. First, the small size of a WDcompared to that of a main sequence star makes the de-tection of faint companion objects and their propertiesmuch easier (Burleigh et al. 2002;Loeb & Maoz 2013; Linet al. 2014). Second, evidence for circumstellar rockymaterial already exists. The presence of metal enrich-ment on the surface of many WDs suggests accretion ofrocky debris, since the sedimentation timescale of WDsis often orders of magnitude shorter than their age. Theinferred metal accretion rates cannot be accounted forby the interstellar medium (Farihi et al. 2010a). Fur-thermore, at least 30 WDs show evidence for dusty disksthat are within the Roche zone of the host star (Farihiet al. 2010b; Koester et al. 2014; Bergfors et al. 2014).

The standard model (see reviews byFarihi(2011) andDebes (2011)) for these disks is that they originate froma close-in minor planet that was perturbed by an outerplanet into the Roche zone, where it disintegrated intoa disk (Jura 2003). Alternatively, a small fraction of thedisks might be accounted for by scrambled comets (Par-riott & Alcock 1998; Debes & Sigurdsson 2002; Bonsoret al. 2011; Stone et al. 2014), though the metal abun-

[email protected] Harvard College, Cambridge, MA 02138, USA2 Harvard Astronomy Department, Harvard University, Cam-

bridge, MA 02138, USA

dance of most of the dusty disk WDs favors tidal disrup-tion of rocky objects.

Consistency with the dusty disk model places lowerlimits on the population of rocky objects around WDs.Wyatt et al.(2014) developed a theoretical model wherean underlying mass distribution of asteroids graduallyaccrete onto WDs, and concluded that to explain pho-tospheric absorption lines, asteroids of masses 5102MLare required for typical WDs. The minimum to-tal mass required to explain the metal pollution alone inone DBZ WD is 0.01ML (Dufour et al. 2010), roughlythe mass of Ceres, the largest asteroid in the asteroidbelt. With recent work estimating that 30% WDs withan age of 20200 Myr accrete planetary debris (Koesteret al. 2014), a method to directly detect such objectsshould either lead to many detections or a revision of thecurrent paradigm.

This paper is organized as follows. In2, we studythe properties of objects with masses much less than MLarounds WDs. In3, we calculate the minimum size ofobjects that could be detected with existing telescopesand the future James Webb Space Telescope (JWST)3.Finally, we discuss possible issues as well as the signifi-cance of detection (or upper limits) in 4, and summarizeour conclusions in5.

2. PROPERTIES OF WD COMPANIONS WITHM

ML

The tidal locking timescale for a gravitationally boundrocky object with M < M at an orbital radii < 0.02AU around a WD is 1000 yr (Agol 2011), i.e., muchshorter than the planet formation timescale of 1 10Myr for the objects under consideration. We thus expectthat virtually all asteroids should be tidally locked to thehost WD.

Over the age of the solar system, only objects witha mass 0.1ML can retain volatiles at 50 K (Schaller& Brown 2007). At higher temperatures, a much larger

3 http://www.stsci.edu/jwst/
mailto:[email protected]://www.stsci.edu/jwst/http://www.stsci.edu/jwst/mailto:[email protected]


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mass is needed, since the timescale for the atmosphereto leak away is dominated by an exponential in inversetemperature, tleak exp(GM/kTR), where is theatomic weight andR is the radius of the object4. There-fore, the objects under consideration here are unlikely tohave atmospheres.

In the absence of an atmosphere, the temperature dif-ferences between the night and day sides of an object

must be of order unity, because thermal conduction isinefficient. The situation is more dramatic than in thecase of our moon, because one side of the object willbe tidally locked with the WD illumination. The powertransmitted by thermal conduction is

Pc QR2 T R2 TR, (1)whereas the power radiated due to blackbody radiationis of order

Pr T4R2. (2)For an object at 600 K with mass 103MLand a thermalconductivity 4 103 ergs1 cm1K1, radiation is 7orders of magnitude more efficient than conduction.

Another way to decrease the contrast between the dayand night side is by tidal heating of the objects. ThepowerPtides generated by the oscillating tidal forces asthe object cycles from apoapsis to periapsis is (Barneset al. 2009)

Ptides= 63

16

(GM)3/2MR5

Qpa15/2

e2, (3)

where Qp 102 is the minor planets tidal dissipationfunction (Goldreich & Soter 1966). Fore


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pared to transiting systems. Therefore, we can targetWDs which are closer by a factor 1001/3 than in tran-sit surveys, with a flux higher by 1002/3, improving theresultingS/N (Loeb & Maoz 2013).

3.1. Targeting nearby WDs with ground basedtelescopes

The most abundant WDs are a few Gyr old and havetemperatures in the 5000 6000 K range. However,emission from asteroids would be difficult to observe fromthe ground for cold WDs, since the sky is bright at IRwavelengths > 3 m, whereas the peak emission from arocky asteroid is at max 12 m (Tp/300 K)1. Forinstruments sensitive to < 3 m, only a few systemsare hot enough and close enough to be observed fromthe ground.

The brightest WD in the sky is Sirius B, with V mag-nitude 8.3 and an effective temperature of 26, 000 K.The estimated age of Sirius B, 120 Myr (Liebert et al.2005), is sufficient to allow for planetesimal formation, aprocess which occurred in our solar system on a 1 10 Myr time scale (see e.g. Yin et al. 2002). AlthoughSirius B orbits the main sequence star Sirius A, theyare separated by 3 11.5 arcsec (8.2 31.5 AU), mak-ing it possible to resolve Sirius B alone. With a 5%duty cycle over 50 hrs (for a few hours of total in-tegration time), it will be possible to detect the vari-able emission from an object with a mass 102MLwith the Folded-port InfraRed Echellette (FIRE) on oneof the twin 6.5 meter Magellan telescopes. of these ob-

jects requires a photometric stability of 100 ppm on afew hours timescale. RMS fluctuations of 1000 ppmhave already been achieved on one minute timescales (seeFigure 6 ofBean et al. (2013)); fluctuations on 2 hourlong timescales should be 1000/120 assuming whitetemporal noise, giving us the desired sensitivity. How-

ever, atmospheric effects such as a time-dependent air-mass could lead to systematic errors on hour-long timescales, though calibration against nearby stars may al-leviate such issues. Although we simulate observationswith FIRE, an imager such as the MMT and Magellan In-frared Spectrograph Detection (MMIRS) would achievevirtually identical limits and allow for real-time photo-metric calibration against other stars.

Alternatively, the odds of finding an object may be in-creased around dusty, metal-rich WDs. The most wellstudied dusty WD is G29-38 (see e.g. Gansicke et al.(2006)) which has a temperature Twd = 11700 K andis 14 pc away. G29-38s Roche zone coincides with itssublimation zone, implying that periodic flux variations

can occur over exceedingly short timescales 4 hr. InFigure 1, we show the minimum mass Mmin that a 1hr observation with FIRE on one of the Magellan tele-scopes could detect on both Sirius B and G29-38. As thetemperature of the objects drops, the minimum mass in-creases dramatically since FIRE is only sensitive out to awavelength of 2.5 m. For G29-38, the required photo-metric stability is only 1000 ppm, as the white dwarfis cooler than Sirius B and Poisson-limited size of theobjects is much larger.

In calculating detectability thresholds, we assumedthat the signal to noise for detecting an asteroid is lim-ited by the Poisson noise of photon counts from the host

Fig. 2. Minimum detectable asteroid mass for nearby WDs ata given temperatureTwd for a 1 hr exposure with JWST MIRI.The solid red curve represents the minimum detectable mass foran object as close to the WD as possible. Dashed pink curvesrepresent constraints on objects with longer orbital periods P. AtTwd = 1200 K there is a transition from a Roche limited radius,rR, to a sublimation limited radius, rs. The tightest constraintswill typically be placed on 12, 000 K WDs, where rs rR.

WD:

(S/N)2 = A2p

Awd

df(, T

p)2f(, Twd)

(7)

where f(, T) tint is the number of photons per unitwavelength per unit emitted surface area received by thetelescope from a blackbody at temperature T over anintegration time tint. Ap and Awd are the effective sur-face areas of the planet and white dwarfs, respectively.The normalization of f is fixed by the brightness of aWD. RequiringS/N= 5, we solve for the minimum sizeof a asteroid that can be detected as a function of WDparameters. For real observations, the required exposuretime may be greater by a factor of order unity dependingon the spacing of the observations in the time domain.Once evidence is found for a asteroid, follow up observa-tions with a similar integration time but with a cadenceoptimized for the period of the asteroid would give thebest statistics.

Submillimeter observations with ground-based experi-ments like the Atacama Large Millimeter/submillimeterArray (ALMA)5 could constrain the population of coolerasteroids, though the signals would be periodic on alonger timescale as the orbital period scales P T3p .Hotter asteroids could also be targeted, but ALMAwould then probe the Rayleigh-Jeans part of the spec-trum where it is not possible to constrain the tempera-ture of the asteroid and therefore difficult to distinguishit from atmospheric features on the surface of a WD (see

4).

3.2. Targeting cool WDs with JWST

A space-based mid-infrared telescope will dramaticallyimprove our constraints on rocky bodies, allowing detec-tion of asteroids around older, cooler WDs. In particular,the Mid Infrared Instrument (MIRI) onboard JWST issensitive in the 5 28.3 m spectral range, which willdramatically decrease the minimum observable mass, es-pecially at lower temperatures where the peak emissionis at >2.5 m.

5 http://almascience.eso.org
http://almascience.eso.org/http://almascience.eso.org/


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In Figure 1, we show the minimum mass that a 1hr observation with JWST could detect on both SiriusB and G29-38. Note that the logarithmic slope is muchshallower for JWST than Magellan. At Tp = 800 K,JWST will improve constraints on asteroid masses bythree orders of magnitude if a photometric stability of 50 ppm per one-minute exposure is achieved. Forcooler, more distant white dwarfs, the required photo-

metric stability is less stringent.

4. DISCUSSION

A potential source of error in future observations mightbe distinguishing asteroids from atmospheric features onthe WD such as spots, although there is currently no ev-idence (Vornanen & Berdyugin 2012). One simple wayto break the degeneracy is to constrain the effective tem-perature of the object. In particular, the period of theflux oscillation allows one to predict (up to a factor oforder unity due to uncertainty in albedo) the tempera-ture of the asteroid. For a fixed integration time, thefractional uncertainty in the temperature of the minorplanet is Tp/Tp T4p in the limit where the spectro-scopic band is much wider than the wavelength spread inthe emitted spectrum of the asteroid. Unless experimen-tal constraints force us to observe only in the Rayleigh-Jeans regime of the spectrum, only a factor of order unitymore integration time will be necessary to constrain theasteroids temperature to a value much cooler than anyplausible spot on the WD surface.

If multiple asteroids are orbiting close in, Fourier anal-ysis in the time domain should distinguish the variousobjects, as asteroids on different orbits will have differentperiods. Note that the discovery of objects at multipleperiods would require be increasingly harder to explainwith spots, as a naive model would have all spots rotate

with the same period. Furthermore, detection of multipleblackbody components with effective temperatures givenby the inferred orbital radii would be a way to confirmthe multiple asteroid interpretation. On the other hand,if there are multiple asteroids with very similar periodsbut different orbital phases, S/Nwill be reduced as thediffering phases wash out the amplitude of the periodicsignal. In the limit of a thin dusty disk, the observableperiodicity is lost. Consequently, our observational strat-egy is optimal if asteroids do not spend too long near theRoche zone (such that many asteroids could be in similarorbits) nor too little time near the Roche zone (such thatfew asteroids will ever be detected).

The detection of asteroids around WDs would be a

powerful probe of planetary physics. A survey of nearbyWDs by JWST could constrain the distribution functionnp(Tp, Twd, Mp), where Tp and Mp are the temperatureand mass of asteroids, respectively. np(Twd) can then beused to infer the statistical evolution of asteroids withtime, since Twd is linked to the age of the WD by theWD cooling function. For example, the tidal disruptionmodel implies that np must be intimately tied to the

abundance of dusty disks nd(Twd, Md). A generic pre-diction of this model is that any changes in the massof debris disks should be accompanied by metal accre-tion onto the surface of the WD and/or a non-trivialnp(Tp) dependence. Since constraints have already beenplaced on nd, observations of np could falsify or refinethis model.

In addition to detecting minor planets, this techniquecould also harvest exoplanet detections, though wewould expect Tp/Tp 1 for planets with atmospheres.To distinguish a large planet with a small Tp/Tpfrom a small object with a larger Tp/Tp, the black-body spectrum of the object could be obtained andused to infer the effective surface area. Furthermore,

the mass of close-in exoplanets could in principle beconstrained by observations, as tidal forces should pullthe planet into an ellipsoid. As a planet orbits theWD, the effective area of the planet will change over aperiod T /2, as opposed to flux oscillations with a pe-riodT from the planets day/night temperature gradient.

5. CONCLUSION

We have shown that the periodic flux variability fromthe night/day oscillations of close-in asteroids allows ex-isting and future telescopes to detect objects with a smallfraction of the moons mass. A possible strategy wouldbe to use an existing-ground based telescope to search forasteroids around nearby WDs withT 12,000 K. JWSTcan then be used to follow up ground-based observationsin order to improve the associated constraints by a feworders of magnitude on already-observed targets and tosample the asteroid environments around the entire pop-ulation of local WDs. The unique environment of WDsthus affords an unprecedented window into the physicsof planet formation.

We thank Sean Andrews, Jonathan Grindlay, ScottKenyon, Dani Maoz, and Brian McLeod for useful dis-cussions. This work was supported in part by NSF grantAST- 1312034 and the Harvard College Program for Re-search in Science and Engineering (PRISE).

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FINDING ROCKY ASTEROIDS AROUND WHITE DWARFS BY THEIR PERIODIC THERMAL EMISSION