WATER FRACTIONS IN EXTRASOLAR PLANETESIMALS

In order to assess the relative amounts of accreted hydrogen and heavy elements, we assemble a volume-limited sample of DBs that is as bias-free as possible. We cannot use the available survey of white dwarfs within 20 pc of the Sun that is nearly complete because it contains only one DB (Holberg et al. 2008). As a compromise between having enough stars to perform an analysis yet not being too incomplete, we adopt the outer boundary of our sample at 80 pc. Because it is difficult to distinguish helium in atmospheres cooler than 13,000 K and such cool DBs are difficult to characterize, we only consider stars warmer than this temperature.

Using the available literature, we list in Table 1 57 DBs with luminosities greater than 3 × 10⁻³ L_☉ or M_Bol = 11 mag, along with their distances, temperatures, galactic latitudes, masses, and accretion rates. According to Bergeron et al. (2011), the space density of such DBs is 5.15 × 10⁻⁵ stars pc⁻³, and we might expect to identify 110 appropriate objects within 80 pc. We now assess the incompleteness of our sample in more detail.

Table 1. DBs within 80 pc of the Sun

Star	Name	d	b	Teff	Mwd	log  $\overline{{\dot{M}(H)}}$	log  $\overline{{\dot{M}(Z_{{\rm tot}})}}$	Notes
(WD)		(pc)	(°)	(K)	(M☉)	(g s−1)	(g s−1)
0002+729	GD 408	32	+11	14,410	0.75	⩽4.39	7.83	(1), (2)
0017+136	Feige 4	75	−48	18,130	0.65	⩽5.24		(1)
0100−068	G270-124	43	−69	19,800	0.64	⩽3.63	7.70	(1), (3)
0125−236	G274-39	61	−81	16,610	0.75	⩽4.70	7.50	(1), (2)
0138−559	BPM 16571	51	−60	15,800				(4)
0300−013	GD 40	64	−49	15,300	0.67	⩽4.51	9.44	(1), (5)
0308−565	BPM 17088	50	−52	23,000	0.63	<0.50	<	(1), (6)
0414−0434	HE	61	−36	13,470	0.67	⩽5.14	<	(1), (7)
0418−539	BPM 17731	76	−44	19,050	0.66	<4.70	<	(1), (7)
0435+410	GD 61	52	−04	16,810	0.70	⩽5.91	8.81	(1), (8)
0437+138	LP475-242	45	−21	15,120	0.74	⩽5.63		(1)
0503+147	KUV	31	−15	15,610	0.64	⩽5.20		(1)
0517+771	GD 435	69	+22	13,150	0.67	⩽4.80	<5.92	(1), (2)
0615−591	NLTT 16355	37	−27	15,750	0.61	<4.32	<	(1), (7)
0716+404	GD 85	59	+22	17,150	0.64	<4.16	<6.8	(1), (9)
0840+262	Ton 10	49	+35	17,770	0.78	⩽5.61		(1)
0845−188	NLTT 20260	75	+15	17,470	0.68	< 3.97	<7.53	(1), (2)
0948+013	PG	77	+40	16,810	0.65	⩽4.93	<	(1), (7)
1009+416	KUV	69	+55	16,480	1.00	<4.00		(1)
1011+570	GD 303	48	+49	17,350	0.67	<5.05	8.84	(1), (2)
1046−017	GD 124	70	+48	14,620	0.68	<4.14	<6.20	(1), (2)
1107+265	GD 128	78	+67	15,060	0.65	⩽5.31	7.17	(1), (2)
1129+373	PG	74	+70	13,030	0.68	⩽4.67	<6.07	(1), (2)
1333+487	GD 325	35	+67	15,320	0.61	<5.35		(1)
1336+123	NLTT 34784	51	+71	15,950	0.60	<4.41	<	(1), (7)
1352+004	PG	69	+59	13,980	0.62	⩽5.60	7.37	(1), (2)
1403−010	G64-43	80	+56	15,420	0.65	⩽4.48	<6.44	(1), (2)
1411+218	PG	38	+71	14,910	0.62	<5.34	<6.26	(1), (2)
1425+540	G200-40	56	+58	14,490	0.56	⩽6.86	7.73	(1), (2)
1444−096	PG	52	+44	17,040	0.75	⩽4.11	<	(1), (7)
1459+821	G256-18	49	+34	15,850	0.65	<5.20	<6.36	(1), (2)
1542+182	GD 190	67	+49	22,630	0.63	<0.58	<	(1), (6)
1557+192	KUV	78	+46	19,570	0.68	⩽4.48	<	(1), (7)
1610+239	PG	50	+45	13,360	0.69	<5.13	<6.11	(1), (2)
1644+198	PG	56	+36	15,190	0.66	<5.17	6.76	(1), (2)
1645+325	GD 358	46	+39	24,940	0.57	<1.14		(1), (10)
1703+319	PG	67	+35	14,430	0.88	⩽4.37		(1)
1708−871	BPM 921	58	−26	23,980	0.64	<1.31		(1)
1709+230	GD 205	65	+32	19,610	0.65	⩽4.87	8.73	(1), (2)
1726-578	BPM 24886	51	−13	14,320	0.71	⩽5.05		(1)
1822+410	GD 378	45	+22	16,230	0.60	⩽6.20	8.58	(1), (2)
1919−362	SCR J1920-3611	42	−21	27,800	0.59			(11)
1940+374	NLTT 48137	47	+07	16,630	0.64	<5.22	<6.73	(1), (2)
2034−532	BPM 26944	35	−37	17,160	0.90	<3.63		(1)
2129+000	G26-10	49	−35	14,380	0.75	<3.88	<6.01	(1), (2)
2130−047	GD 233	50	−38	18,110	0.66	⩽4.07	<7.40	(1), (2)
2144−079	GJ837.1	49	−42	16,340	0.70	<4.00	8.08	(1), (2)
2154−437	BPM 44275	61	−52	16,700	0.60	⩽5.68	<	(7)
2222+683	G241-6	65	+10	15,230	0.71	<4.97	9.30	(1), (2)
2224−344	LTT 9031	72	−58	19,000				(12)
2229+139	PG	76	−37	14,940	0.70	⩽5.75		(1)
2236+541	KPD	79	−04	15,470	0.78	<4.76		(1)
2253−062	GD 243	63	−55	17,190	0.64	⩽5.90	<	(1), (7)
2253+8023	HS	71	+19	14,400	0.84	⩽4.43	9.95	(13)
2310+175	PG	64	−39	15,170	0.82	<4.55		(1)
2328+510	GD 410	53	−10	14,460	0.63	<5.30		(1)
2334−4127	HE	80	−69	18,250	0.61	⩽4.55	<	(7)

Notes. The entries in the column for $\overline{{\dot{M}(H)}}$ may be blank (no measurement), < (no hydrogen detected), or ⩽ (hydrogen detected), but the rate of accretion is an upper limit because there could be some primordial hydrogen. We enter "<" in the log  $\overline{{\dot{M}(Z_{{\rm tot}})}}$ column for those stars examined in the SPY survey that do not display any calcium absorption but no quantitative upper limit is provided (Voss et al. 2007). References. (1) Bergeron et al. 2011; (2) Zuckerman et al. 2010; (3) Desharnais et al. 2008; (4) Sion et al. 1988; (5) Klein et al. 2010; (6) Petitclerc et al. 2005; (7) Voss et al. 2007; (8) Farihi et al. 2011a; (9) Dupuis et al. 1993; (10) Provencal et al. 2000; (11) Subasavage et al. 2008; (12) Castanheira et al. 2006; (13) Klein et al. 2011.

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If the stars are distributed isotropically in the sky, we would expect 50% to lie at |b| ⩽ 30°. However, we only identify 17 stars in this zone. The incompleteness of Table 1 is at least partly because some stars near the Galactic plane are missed. As demonstrated by Lepine et al. (2011) and as illustrated by the recent discovery of a DAZ at a distance of 55 pc with b = −8° and T_eff = 20,900 K (Vennes et al. 2010), there are nearby warm white dwarfs yet to be found in this region.

In actuality, we expect fewer than half the stars lie at |b| > 30° because the space density of stars decreases away from the Galactic plane with a scale height, h, which depends upon the mass of the main-sequence progenitor. Assume that

$$\begin{equation} n\;=\;n_{0}\;e^{-|r\,\sin b|/h}, \end{equation} \tag{ 1 }$$

where n₀ is the local density in the midplane. If so, then the number of stars in a spherical volume of radius D centered on the Sun assumed to lie in the midplane of the Galactic plane, N(D), can be found from a Taylor series expansion as

$$\begin{equation} N(D)\;{\approx }\;\frac{4{\pi }\,n_{0}}{3}\,D^{3}\left(1\,-\frac{3}{8}\,\frac{D}{h}\,+\,\frac{1}{10}\frac{D^{2}}{h^{2}}\,-\,\cdots \right). \end{equation} \tag{ 2 }$$

With D = 80 pc and h no larger than 150 pc (Gilmore & Zeilik 2000), we expect to find no more than 91 DBs. Because we identify 57 stars in Table 1, our sample is at least 60% complete.

Most of the stars in Table 1 are listed in Bergeron et al. (2011) and Voss et al. (2007). When a star is considered in both samples, to be as consistent as possible, we adopt the stellar parameters in Bergeron et al. (2011). For a few stars, only very limited data are available.

One of the most important DBs for our purposes is HS 2253+8023 because it has a large amount of external pollution. The star's gravity is not well constrained in the detailed study of Klein et al. (2011); we adopt their most probable value. The distance to this star is derived from its radius, effective temperature, and J-band 2MASS magnitude. We adopt a similar approach to determine distances to the other stars whose distances are not provided by Bergeron et al. (2011).

The average mass of the DBs in Table 1 is 0.67 M_☉. Using the initial–final mass relation of Williams et al. (2009), this implies a typical main-sequence progenitor mass of 2.5 M_☉.

We now consider a DB's hydrogen budget. For the ith star, we compute the upper bound to the hydrogen accretion rate averaged over the entire cooling age of the star, $\overline{{\dot{M}_{i}(H)}}$, simply as the total mass of accumulated hydrogen, M_i(H), divided by the white dwarf's cooling age, t_{cool, i}:

$$\begin{equation} {\overline{\dot{M}_{i}(H)}}\;=\;\frac{M_{i}(H)}{t_{{\rm cool},i}}. \end{equation} \tag{ 3 }$$

To determine M_i(H), we have used reported values of [H]/[He] and models for the mass of the star's convective zone as a function of stellar gravity and effective temperature. Because Koester (2009) only presented results for one value of white dwarf gravity and hydrogen composition, D. Koester kindly provided a grid of models for DB stars with a sufficient range of temperature, mass, and [H]/[He] to enable us to interpolate for the mass of the convective zone for each star individually in Table 1. The estimates of M_i(H) are strongly sensitive to the star's total mass. For example, for DBs with an effective temperature of 13,000 K, the mass of the convective zone is nearly a factor of 200 smaller in a star of 1.1 M_☉ compared to a star of 0.6 M_☉ (Dufour et al. 2010). In most cases, we take [H]/[He] from the optical study of Bergeron et al. (2011), but for a few stars hotter than 20,000 K, we use upper bounds from published ultraviolet observations which are appreciably more sensitive to the hydrogen abundance. All our estimates for $\overline{{\dot{M}_{i}(H)}}$ are upper bounds because the white dwarf could possess primordial hydrogen (Bergeron et al. 2011).

As shown in Figure 1, for the seven stars in Table 1 for which both Bergeron et al. (2011) and Voss et al. (2007) derived masses spectroscopically, instead of just adopting an assumed mean, the values of $\overline{{\dot{M}_{i}(H)}}$ are systematically lower from our use of the Bergeron et al. (2011) study compared to those in Voss et al. (2007) by an average factor of four. This substantial discrepancy is the result of four factors all operating in the same direction. First, Bergeron et al. (2011) typically found larger stellar masses. The typical 15% increase in derived mass means that the mass of the convective zone is smaller by approximately a factor of two. Second, stars with a larger mass take a longer time to cool to the observed effective temperature; this effect might contribute a factor of 1.2 to the estimate of $\overline{{\dot{M}_{i}(H)}}$. Third, because the star's gravity is estimated to be higher in the analysis of Bergeron et al. (2011), the ratio of the hydrogen to helium may be found to be as much as a factor of two lower than derived by Voss et al. (2007). Finally, there is a different theoretical treatment of convection such that the values of the mass of the convective zone are typically a factor of 1.5 lower in the models of Koester (2009) compared to those used by Voss et al. (2007). The spectroscopically derived masses derived by Bergeron et al. (2011) usually agree very well with those derived from trigonometric parallaxes, and therefore it seems likely that the values of $\overline{{\dot{M}_{i}(H)}}$ derived from their analysis are realistic. It must be recognized, however, that there are uncertainties.

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A few stars in Table 1 have remarkably low upper limits for $\overline{{\dot{M}_{i}(H)}}$. These stars can be used to place a bound on the space density of interstellar comets (Jura 2011).

We wish to compare the hydrogen and heavy atom accretion rates. For the ith star and the jth element, the time-averaged mass accretion rate from the disk onto the star, $\overline{{\dot{M}_{i}(Z_{j})}}$, we take

$$\begin{equation} \overline{\dot{M}_{i}(Z_{j})}\;=\;\;\frac{M_{i}(Z_{j})}{t_{i}(Z_{j})}, \end{equation} \tag{ 4 }$$

where M_i(Z_j) is the inferred mass of element Z_j in the star's mixing zone where the gravitational settling time is t_i(Z_j). Using Z_tot to denote the mass of all the heavy elements, we define for each star the time-averaged total heavy element accretion rate, $\overline{{\dot{M}_{i}(Z_{{\rm tot}})}}$, as

$$\begin{equation} \overline{{\dot{M}_{i}(Z_{{\rm tot}})}}\;=\;{\sum _j} {\overline{\dot{M}_{i}(Z_{j})}}. \end{equation} \tag{ 5 }$$

In many cases, only calcium is measured because this element is the easiest to detect in optical spectra; the total heavy element accretion rate is extrapolated as described by Zuckerman et al. (2010). The values of $\overline{{\dot{M}_{i}(Z_{{\rm tot}})}}$ are relatively insensitive to the white dwarf mass because, as illustrated in Klein et al. (2010), the settling time scales approximately with the mass of the convective zone.

Three sources of hydrogen in DBs have been proposed: interstellar (Voss et al. 2007), circumstellar (Jura & Xu 2010), and primordial (Bergeron et al. 2011). Because the heavy elements settle while the hydrogen does not, the absence of a correlation in Figure 2 is not a decisive test of any particular model for the source of the DBs' hydrogen.

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Although there are nearby DBs that are missed, we assume that in identifying the stars, there is no bias related to their external pollution. If so, then we can treat the set of stars as a well-defined ensemble and therefore compare the average hydrogen and heavy atom accretion with each other in a meaningful manner. We define ${\dot{M}_{{\rm Sum}}}(H)$ as

$$\begin{equation} {\dot{M}_{{\rm Sum}}\,(H)}\;=\;{\sum _i}{\overline{\dot{M}_{i}(H)}}. \end{equation} \tag{ 6 }$$

For comparison, the summed heavy atom accretion rate, ${\dot{M}_{{\rm Sum}}(Z_{{\rm tot}})}$, is

$$\begin{equation} {\dot{M}_{{\rm Sum}}\,(Z_{{\rm tot}})}\;=\;{\sum _i}\;\;{\overline{\dot{M}_{i}(Z_{{\rm tot}})}}. \end{equation} \tag{ 7 }$$

With estimates in Table 1 of ${\overline{\dot{M}_{i}(Z_{{\rm tot}})}}$ for 2/3 of the stars, and ignoring any contribution from the remaining 1/3 of the stars, ${\dot{M}_{{\rm Sum}}\,(Z_{{\rm tot}})}$ is 1.6 × 10¹⁰ g s⁻¹. Nearly all stars in Table 1 have been examined for hydrogen, and ${\dot{M}_{{\rm Sum}}\,(H)}$ is ⩽1.4 × 10⁷ g s⁻¹, approximately a factor of 1000 smaller than the rate for heavy atoms. Because some currently unknown fraction of this hydrogen is derived from the interstellar medium or is primordial, we conclude that in the entire ensemble less than 1% of the accreted mass is carried in water. This bound is only strengthened if there is appreciable accretion of heavy atoms in those stars which have not been examined with high spectral sensitivity. Because we examined 57 stars, the upper bound to the average accretion rate for an individual star is 2.5 × 10⁵ g s⁻¹.

For both heavy elements and hydrogen, the assumed accretion is dominated by a few stars. If we remove the top three accretors in each category, then ${\dot{M}_{{\rm Sum}}\,(Z_{{\rm tot}})}$ and ${\dot{M}_{{\rm Sum}}\,(H)}$ are ⩽2.6 × 10⁹ g s⁻¹ and 4.4 × 10⁶ g s⁻¹, respectively. In this analysis, the ratio of the two accretion rates is approximately a factor of 600. Considered as an aggregate and interpreting the data with the best available parameters, we find that extrasolar asteroids accreted onto the warm DB stars are appreciably drier than CI chondrites.

For the four stars in Table 1 for which comprehensive abundance studies have been performed, we show in Figure 3 a plot of the fraction of the total mass of each of the major individual constituents—O, Mg, Si, Ca, and Fe—and the upper bound to the fraction of the total mass that could have been water. In this figure, we assume the steady state approximation given by Equation (4). It can be seen that there is a relatively small scatter in most of the mass fractions—especially oxygen, consistent with our treatment of the extrasolar asteroids orbiting separate stars as one population. We also see that the upper limit derived from the analysis of the aggregate population is significantly lower than the limit for any individual star. The individual bounds plotted in Figure 3 are derived by using the lower of the two values for the water fraction in the parent body. The first value, which is usually the stronger constraint, assumes that all the oxygen in excess of that which could be bound into MgO, SiO₂, and Fe₂O₃ was contained in water. The second value assumes that all the atmospheric hydrogen was bound into water. As explained in Section 3, the evidence that GD 61 has accreted ice-rich material is ambiguous, and we therefore plot the water fraction as an upper limit rather than as a measured quantity.

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We see in Figure 3 that the extrasolar asteroids are in aggregate drier even than CV chondrites—the most refractory rich of common meteorites (Wasson & Kallemeyn 1988). We also see that commonly the extrasolar asteroids also are more rich in the refractory element Ca than the CV chondrites. We can understand the dryness of extrasolar asteroids as being a result of their having formed interior to the snow line. We do not yet have a good model to explain the relative enhancement of Ca that is measured in these systems.

In the outer mixing zone of a white dwarf, the balance between accretion and settling is governed by the expression (Koester 2009)

$$\begin{equation} \frac{{\it dM}_{*}(Z_{j})}{dt}\;=\;-\frac{M_{*}(Z_{j})}{t_{j}}\,+\,{\dot{M}_{{\rm PB}}}(Z_{j}), \end{equation} \tag{ 8 }$$

where ${\dot{M}_{{\rm PB}}}(Z_{j})$ denotes the accretion rate from the circumstellar disk and is a measure of the composition of the tidally disrupted parent body.⁴ Performing an integration on both sides and assuming that M_*(Z_j) = 0 at t = 0, the solution to Equation (8) is

$$\begin{equation} M_{*}(Z_{j})\;=\;e^{-t/t_{j}}\,{\int }_{\!\!\!\! 0}^{t}\,e^{t^{\prime }/t_{j}}\,{\dot{M}_{{\rm PB}}(Z_{j})}\,dt^{\prime }. \end{equation} \tag{ 9 }$$

Inverting the integral in Equation (9) is required to determine the parent body composition. One approach is to Fourier analyze the accretion rate. Because of the large unknowns, for simplicity, we consider just the illustrative special case of a single Fourier term:

$$\begin{equation} {\dot{M}_{{{{\rm PB}}}}(Z_{j})}\;=\;{\overline{\dot{M}_{{{\rm PB}}}(Z_{j})}}\,(1\,+\,r\,\sin {\omega }\,t^{\prime }). \end{equation} \tag{ 10 }$$

Here, r and ω are free parameters that characterize the amplitude and frequency of the fluctuations in the accretion rate of the jth element whose time-average value is ${\overline{\dot{M}_{{{{\rm PB}}}}(Z_{j})}}$. In addition to accretion rate variations intrinsic to the disk, there may be multiple disruptions of parent bodies that also may lead to time variations of the accretion rate (Jura 2008). In all cases, we require r < 1.

For convenience, define the dimensionless term:

$$\begin{equation} B_{j}\;=\;t_{j}\,{\omega }. \end{equation} \tag{ 11 }$$

Then,

$$\begin{eqnarray} M_{*}(Z_{j})\;&=&\;{\overline{\dot{M}_{{{{\rm PB}}}}(Z_{j})}}\;\;t_{j}\;\Bigg([1\,-\,e^{-t/t_{j}}]\, \nonumber\\ &+&\,\left[\frac{r}{1\,+\,B_{j}^{2}}\right][\sin {\omega }\,t\;-\;B_{j}\cos {\omega }\,t\;+\;B_{j}\,e^{-t/t_{j}}]\Bigg).\nonumber\\ \end{eqnarray} \tag{ 12 }$$

It is instructive to consider some limiting case solutions for Equation (12). When t ≪ t_j, then

$$\begin{equation} M_{*}(Z_{j})\;{\approx }\;{\overline{\dot{M}_{{{{\rm PB}}}}(Z_{j})}}. \end{equation} \tag{ 13 }$$

In this circumstance, the element's mass in the stellar atmosphere directly corresponds to the accretion rate, the initial buildup phase discussed by Koester (2009). When t ≫ t_j, then the solution to Equation (12) is

$$\begin{eqnarray} M_{*}(Z_{j})\;&{\approx }&\;{\overline{\dot{M}_{{{{\rm PB}}}}(Z_{j})}}\;\; t_{j}\;\Bigg(1\,+\,\left[\frac{r}{1\,+\,B_{j}^{2}}\right]\nonumber\\ &&\times[\sin {\omega }\,t\;-\;B_{j}\cos {\omega }\,t]\Bigg). \end{eqnarray} \tag{ 14 }$$

In the exact steady state where the accretion rate is constant and r = 0, then the abundance of an element in the parent body is just given by the abundances in the photosphere divided by the settling time and from Equation (14) we recover Equation (4). However, if r > 0, the relationship between the photospheric concentration of an element and its relative abundance in the parent body is not the simple proportionality of Equation (4).

By examination of Equation (14), we can see that the quasi-steady state approximation usually agrees with the exact steady state model by better than a factor of two. That is, if B_j ≫ 1, then clearly M_*(Z_j) is essentially constant and the limit of Equation (4) is reached. In this case, variations in the accretion time are short compared to the settling time and the pollution mass in the mixing zone is constant. If B_j ≪ 1, then M_*(Z_j) might vary substantially, but the variations in the accretion rate are so slow that the system's behavior is very similar to the exact steady state. The only situation when there is a significant difference between the quasi-steady state and the exact steady state is when B_j ≈ 1. In this case, the settling and accretion times are comparable and the interplay between the two factors can be complex, each element behaving differently according to its specific value of B_j.

We now use the model of quasi-steady accretion to address the question of whether the parent body accreted onto GD 61 was ice-rich as proposed by Farihi et al. (2011a). We see in Figure 3 that both G241-6 and GD 61 have relatively high fractions of oxygen in their contaminants, and therefore both systems are candidates for the accreted parent body having had a substantial amount of water. However, G241-6 has little photospheric hydrogen, and therefore less than 10% of the mass fraction of the pollution is water (Klein et al. 2011). Because G241-6 does not have a circumstellar dust disk (Xu & Jura 2011), plausibly, it is in a late phase of an accretion event and the oxygen is especially abundant because it lingers longer in the outer settling zone. In contrast, GD 61 has a substantial amount of hydrogen in its photosphere and a circumstellar dust disk (Farihi et al. 2011a). It is likely that there is ongoing accretion from the disk onto the star, and Farihi et al. (2011a) assume that the GD 61 system is in an exact steady state. If so, then by mass there is about 30% more oxygen than can be locked into oxide minerals and this implies that the parent body contained a substantial amount of ice. This oxygen excess is sufficiently small that the analysis is sensitive to whether the system is in an exact steady state or only a quasi-steady state.

One difficulty with using the exact steady state model to interpret the data for GD 61 is that iron displays a relatively low abundance. In an exact steady state model, as shown in Figure 3, it is inferred to have an unusually low fractional abundance compared to other polluted white dwarfs. One possible way to explain simultaneously both the low iron and the high oxygen fractional abundances is that the system simply is in a late phase where only settling occurs (Jura & Xu 2010; Klein et al. 2011). However, this hypothesis is inconsistent with the presence of a dust disk and therefore the likelihood that accretion is ongoing. Farihi et al. (2011a) suggested that the accretion onto GD 61 is iron-poor because the circumstellar disk is composed of the outer portion of a planetesimal where most of the iron was concentrated into a central core.

GD 61 is deficient in carbon relative to the Sun by about a factor of 1000 (Desharnais et al. 2008; Farihi et al. 2011a). According to Lee et al. (2010), in planet-forming disks, carbon should be treated as a volatile which therefore explains why this element is commonly observed to have a low abundance in extrasolar asteroids (Jura 2006). If, in fact, the parent body accreted onto GD 61 was ice-rich, it would be a puzzle to understand why it was simultaneously carbon-poor.

Here, we suggest that a quasi-steady state model may explain the data without any requirement that the accreted parent body contained ice. Therefore, our input parameters are chosen to find the largest effect of the quasi-static model; they are not necessarily physically realistic. Consequently, using Equation (14), we compute a model with ω = 0.8 × 10⁻⁵ yr⁻¹ and r = 0.95. Because the gravitational settling times typically are ∼10⁵ yr (Farihi et al. 2011a), this choice of ω means that the values of B_j are nearly 1. The relative mass fractions of O, Mg, Si, and Fe are taken to equal 0.40, 0.18, 0.20, and 0.22; these relative abundances are close to the values for bulk Earth (Allegre et al. 2001) except for iron which is about 50% too low. However, because the error in the photospheric abundance of Fe in GD 61 is 0.2 dex (Farihi et al. 2011a), the true fractional abundance of this element in the parent body is not required to be anomalously low compared to bulk Earth.

We display in Figure 4 the results of our model calculation for the fractional abundances in the photosphere at different times. We see that there is a phase when the model matches the data. It therefore seems that the argument that there must be water in the parent body accreted onto GD 61 is provisional.

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