CHEMICAL ABUNDANCES IN THE EXTERNALLY POLLUTED WHITE DWARF GD 40: EVIDENCE OF A ROCKY EXTRASOLAR MINOR PLANET

Data reduction was carried out with standard IRAF⁴ tasks for echelle data. A PyRAF code meta-structure controls the data processing flow, maintains file organization, and adds versatility. The data have undergone typical echelle reduction procedures: bias removal, flat-fielding, variance-weighted ("optimal") extraction, dispersion calibration with linearization, and heliocentric Doppler correction. Much of the echelle blaze function was removed by dividing each spectrum by a low-order polynomial fit to the extracted spectrum of the quartz lamp flat exposure. It was desirable to increase the S/N by combining orders of overlapping wavelength coverage, and to reduce the known HIRES artifacts (waves and bends in the continuum of a wavelength scale greater than ∼10 Å) most likely due to variable vignetting (Suzuki et al. 2003). Since we do not require absolute flux calibrated information, and we are mainly interested in heavy-element absorption lines (equivalent widths, EW < 3 Å), the echelle orders were continuum normalized to unity with a low-order polynomial (typically fifth-order cubic spline) to reduce artifacts and align the echelle orders. An error associated with this action was estimated by varying the order of the normalization polynomial, which resulted in a ≲1% variation in EW measurements of typical heavy element absorption lines. Adjacent orders of overlapping wavelength coverage were combined in an average, weighted by the blaze function to suppress the contribution from the noisier edges of the CCD chip, and a final order-merged, normalized spectrum was obtained.

We employed different instrumental setups to achieve our desired wavelength coverage and sensitivity. For the 2008 red data, an additional re-normalizing processing step had to be applied to calibrate and remove second (diffraction) order flux contamination in the region λ8200–9000 Å. The 2006 red data had a different instrumental setup, providing an un-contaminated comparison to check our re-normalization procedure. The final all-combined GD 40 HIRES spectrum has S/N in the range 35–50 for the spectral interval 3300–4500 Å, and between 60 and 90 for 4500–7000 Å. Shorter than 3300 Å, the S/N is ≃20; near 7000 Å, the S/N is ≃50; redward of 8000 Å, the S/N is ⩽ 35.

GD 40's optical spectrum is dominated by large (EW ∼ 1–10 Å) pressure broadened helium lines, a signature of the spectral class "DB." The high-resolution and high-S/N Keck/HIRES spectrum displays absorption lines of many more species. In addition to He we identify the presence of nine elements, H, O, Mg, Si, Ca, Ti, Cr, Mn, and Fe (Figures 1–8), whose detections are securely established through multiple (⩾4), unambiguous lines for all elements heavier than hydrogen. Hydrogen is detected via a weak but clearly visible Hα feature, also observed by Voss et al. (2007). Except for the Ca ii H&K lines and two Ti ii lines (3242.0 Å and 3383.8 Å), GD 40's absorption lines arise from excitation levels above the ground state, and are therefore not of interstellar origin. The very large and broad profiles of the Ca ii H&K lines indicate their atmospheric origin. The carbon detection from UV data (Shipman et al. 1995; F99W02) brings the number of known elements aside from He in GD 40's atmosphere to 10.

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Each absorption line was visually inspected. Line identifications made use of the Kurucz database⁵ and the Vienna Atomic Line Database. We used IRAF (splot) to fit the line profiles in the continuum-normalized spectrum with the line center, core depth, FWHM, and the local continuum value as free fit parameters. The EW and a fitting error were calculated in each profile fit. All of the lines analyzed were best fit with a Lorentzian profile and the fit-derived EWs were checked for agreement with direct numerical integration. Three different measurements were performed for each line, varying the interval of nearby continuum data points from which the continuum level was determined. The reported EW is the average of the three measured EW values, and the total EW uncertainty includes the rms of the three fit values added in quadrature with the average fitting error and the 1% error from continuum normalization (Section 2.1). Measurement uncertainties range from ∼3% to 50%, with the peak of the distribution at ∼10%, depending on the strength of the line and the ability to establish a well-defined continuum. Our line identification and measurement results are listed in Table 2. From 99 high-Z absorption lines, the average-measured (heliocentric) radial velocity of GD 40 is 10 km s⁻¹, with a standard deviation of 2 km s⁻¹ (Figure 9). The average high-Z line velocity agrees with the average helium line velocity (9 ± 4 km s⁻¹) implying a coincident physical position with helium in the gravitational well, an additional confirmation of atmospheric location for the absorbers.

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Shipman et al. (1995) tentatively identified lines of neutral oxygen in the 1 Å resolution UV HST (FOS) spectrum, noting a possible blend with silicon. However, Friedrich et al. (1999) re-analyzed those spectra to derive abundances, and attributed the uncertain features entirely to Si i and Si ii. Our ground-based optical/near-IR detection of oxygen in GD 40 clears up the previous ambiguity regarding this element. Oxygen is an important element for study since it is the most abundant constituent of rocky planetary material and is a component of water.

Nickel is not unambiguously detected in the HIRES data, but two of the stronger expected transition lines are possibly associated with noisy features (e.g., λ3465.6 in Figure 5). Aluminum was also tentatively identified previously in the FOS spectrum (Shipman et al. 1995), and fit for abundance by Friedrich et al. (1999), although the latter authors regard its detection as unconvincing.

First, we note that GD 40 has accreted at least the mass of a large asteroid. With a helium-dominated convective zone (CVZ) that contains a fraction q = log(M_cvz/M) = −5.86 (Koester 2009) of the WD's total mass of 0.59 M_☉, there is 1.7 × 10²⁷ g of helium in the CVZ. From this and Table 3 abundances, we find a total mass of pollutant material presently detected in the CVZ of 3.6 × 10²² g (Table 5). For comparison, the mass of asteroid Juno is 3.0 × 10²² g with a radius of 117 km (Konopliv et al. 2006), although Konopliv et al. (2006) note that Juno has a near metallic density, 4.43 ± 0.43 g cm⁻³. A 3.6 × 10²² g asteroid of density 3.0 g cm⁻³ has a radius of 142 km.

Table 5. Gravitational Settling Times, Contaminant Masses and Steady State Mass Accretion Rates for GD 40's Convective Zone

Atomic No.	Z	log (tset/yr)	Mcvz/(1021 g)	$\dot{M}_{\rm acc}/(10^{8}$ g s−1)
1	H	floats	0.29 ± 0.11	...
6	C	5.794	0.32 ± 0.19	0.16 ± 0.10
7	N	5.772	<29.7	<15.9
8	O	5.759	16.6 ± 3.2	9.2 ± 1.8
11	Na	5.668	<1.0	<0.7
12	Mg	5.662	5.9 ± 1.9	4.1 ± 1.3
13	Al	5.662	<0.91	<0.63
14	Si	5.652	2.06 ± 0.35	1.46 ± 0.25
15	P	5.599	<0.083	<0.066
20	Ca	5.469	2.2 ± 0.9	2.4 ± 0.8
21	Sc	5.439	<0.003	<0.004
22	Ti	5.437	0.050 ± 0.018	0.058 ± 0.021
23	V	5.440	<0.022	<0.03
24	Cr	5.432	0.11 ± 0.03	0.13 ± 0.04
25	Mn	5.411	0.056 ± 0.019	0.069 ± 0.023
26	Fe	5.401	7.8 ± 1.9	9.8 ± 2.3
28	Ni	5.372	⩽ 0.50	⩽ 0.67
38	Sr	5.209	<0.003	<0.006
Total detected			35.5	27.4

Notes. Settling times, t_set (≡ diffusion times, τ_diff) are defined in Koester (2009). Contaminant masses are calculated from Table 3 and Table 4 abundances and a helium-dominated convection zone mass of M_cvz(He) = 1.7 × 10²⁷ g. In the steady state limit, $\dot{M}_{\rm acc}$(Z) = M_cvz(Z)/t_set(Z). Some upper limits, nitrogen in particular, are not very restrictive but are included nonetheless for completeness.

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Second, we identify the material to be in a category of terrestrial–planetary composition. Figure 14 depicts the bulk composition of GD 40's pollutants and a variety of solar system objects for comparison. It is immediately apparent how GD 40's volatile-poor abundances are very different from primitive solar system abundances, and we find the closest agreement with an Earth-like composition. Ninety-four percent of Earth's mass comes from just four elements: O, Mg, Si, and Fe (Allegre et al. 1995). Ca, Ni, and Al contribute another ∼5%, and all other elements are minor constituents. Similarly, by bulk mass, the material polluting GD 40's atmosphere is 88% O, Mg, Si, and Fe. Ca contributes another 6%. Upper limits for the CVZ mass percentages of Al (2.5%) and Ni (1.3%) have been included in the totals.

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Jura (2006) and Farihi et al. (2009) have pointed out that, for a handful of polluted WDs with carbon measurements, including GD 40, the polluting abundance of carbon relative to iron is down by a factor of 6 or more from the solar ratio. The deficiencies of carbon and hydrogen are consistent with rocky matter. By similar arguments as apply for GD 362, which have been described in detail by Zuckerman et al. (2007), based on the quantity and constitution of pollutants, an interstellar or cometary origin for the high-Z elements is "exceedingly unlikely."

The origin of the hydrogen is less certain because unlike the high-Z elements, it does not settle below the outer CVZ, which results in an accumulation of all the primordial and accreted hydrogen over the WD lifetime. Both Voss et al. (2007) and Dufour et al. (2007) find a trend of increasing hydrogen mass with increasing age in helium-dominated WDs which argues strongly in favor of an accreted, rather than primordial, origin. GD 40's hydrogen abundance is consistent with these populations of helium-dominated WDs (Figure 1 of Jura et al. 2009b). Since we do not have a constraint on its arrival epoch, it is possible that some, or maybe all, of the presently observed hydrogen is a result of previous accretion of other planetary matter (whose high-Z components have settled), or due to accretion from the interstellar medium, and may not even be associated with the most recently accreted material. Alternatively, some, or possibly all, of GD 40's hydrogen could be tied to the currently observed heavy element pollution. At best we can place some upper limits. One path by which hydrogen may survive processes that result in a refractory-rich environment is through H₂O. Jura & Xu (2009) propose that water from accreted minor planets may have a role in the hydrogen "pollutions" of helium-dominated WDs. Our measured hydrogen abundance (at its upper error limit) imposes an upper limit to the amount of water or ice which could have been accreted (in total for GD 40's entire WD cooling time) at 3.6 × 10²¹ g. If all of this water was accreted along with the present high-Z material, then an upper limit to the amount of water in the bulk pollutant composition is 10%. We show below, in Section 4.3, that due to settling of the heavier elements, the upper limit is likely to be much less than this. In either case, GD 40 has not accreted a very large amount of water, or hydrogen, along with its high-Z material.

GD 40's atmospheric pollution is consistent with having accreted rocky material. However, the atmospheric abundances, n(Z)/n(He), need not be the same as the accreted abundances, and going from the measured abundances to the accreted ones requires some insight and modeling.

The relationship between observed and accreted abundances is governed by the interplay of accretion and diffusion. Key inputs are the characteristics of the encounter with the reservoir, the rate that matter is falling into the star and the rates by which heavy elements are settling (diffusing) downward. Fundamentals of accretion—diffusion theory have been described in detail by Dupuis et al. (1993). More recently, Koester & Wilken (2006) and Koester (2009) have quantified mixing zone mass fractions and settling times of commonly accreted elements for a variety of WD atmospheres in the temperature range of interest. Table 5 lists settling times, t_set(Z) (≡ diffusion time, τ(el) in Koester 2009), calculated specifically for GD 40's CVZ.

Following the models of Jura et al. (2009b) and Koester (2009), accretion is occurring from the observed disk of material and may be represented by an exponentially decaying mass function with a characteristic lifetime, t_disk (≡τ_acc in Koester 2009). We are observing the system at a time t_obs, which we would like to constrain, with respect to t_set and t_disk. While general expressions for the behavior of the abundance ratios can be found from Equation (8) of Koester (2009) and Equation (8) of Jura et al. (2009b), here we summarize the characteristics of three limiting cases. (1) In the early phase of accretion, t_obs ≪ t_set, the accreted abundance ratios are directly the observed atmospheric ones. (2) For times on the order of the diffusion timescales a steady state is approached, and if the reservoir is long-lived, t_set ≲  t_obs ≪  t_disk, the steady state values of the accreted abundances are modified from the measured abundances by the ratios of the settling times (Equation (7) of Koester 2009, equivalently Equation (16) of Jura et al. 2009b):

$$\begin{equation} \left({n(Z_1) \over n(Z_2)}\right)_{\rm accreted} = {t_{\rm set}(Z_2) \over t_{\rm set}(Z_1)} \left({n(Z_1) \over n(Z_2)}\right)_{\rm CVZ}. \end{equation} \tag{ 1 }$$

The ratios of settling times are at most a factor of 2.6 between detected elements studied in this work. (3) After the accretion reservoir is spent, t_disk < t_obs, the atmospheric element ratios deviate exponentially with time from the accreted ratios, as gravitational settling causes the heavier elements to sink more rapidly than the lighter ones. This case leads to a significant departure from the accreted abundances.

Referring to GD 40 CVZ abundances shown in Figure 15, the high-Z ratios do not differ from the Earth or CI chondrite ratios by more than a factor of 4, (except for carbon), mostly the factor is 1–2 as is the case for O, Mg, Ti, Cr, Mn, and Fe. This is not at all the exponential deviations between observed and accreted abundances that are expected from gravitational settling if accretion has ended, paused, or significantly decreased. Also, there is no trend of increasing depletion of element abundances (relative to Earth or chondrite abundances) with decreasing element settling times. Thus, the relative atmospheric abundances are consistent with accretion being presently ongoing, which is further supported by the observational existence of a disk. This implies that we are either observing GD 40 in a steady state, at an early phase of accretion, or somewhere in between. Figures 14 and 15 depict these two limiting extremes of the accreted abundances.

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The Earth-like composition of GD 40's pollutants is almost certainly not coincidental, but rather, it is probably linked to the stoichiometric balance of rock-forming oxides. In the solar system, rocky planetary bodies are primarily composed of mineral oxides (MgO, Al₂O₃, SiO₂, CaO, TiO₂, Cr₂O₃, MnO, FeO, Fe₂O₃, NiO, and minor constituents) and possibly metals, e.g., Fe, Ni, Cr, etc., particularly, if a core has formed.

It is a remarkable result that the quantities of heavy elements in GD 40's atmosphere are consistent with a rocky composition wherein heavy elements are largely contained within oxides. In other words, the observed oxygen is of sufficient quantity to form minerals with the other observed elements, as is evident from computing the number of oxygen atoms needed to form oxides, Z_p(Z)O_q(Z), (measured relative to He):

$$\begin{equation} {n(\mbox{O})_{\rm oxides} \over n(\mbox{He})} = \sum \limits _Z {q(Z) \over p(Z)} {n(Z) \over n(\mbox{He})}. \end{equation} \tag{ 2 }$$

With Table 3 observed element (Z) abundances, which directly represent early-phase accretion as defined in Section 4.2, Equation (2) adds up roughly to 1.5 × 10⁻⁶, significantly below the measured oxygen abundance, n(O)/n(He) = 2.5 × 10⁻⁶. Additionally, as discussed in Section 4.1, some of the oxygen may be associated with water/ice, since a portion, or possibly all, of the atmospheric hydrogen could be coeval with the current accretion episode. Constrained by the relatively low hydrogen abundance, at most 0.5 × 10⁻⁶ of the observed n(O)/n(He) can originate from water.

Now we have the question of accounting for all the excess oxygen, since the refractory-rich, volatile-poor element abundances imply that unbound, volatile oxygen is not a likely vehicle. What happens in the steady state? In order to calculate the steady state abundances from Equation (1), pollutant-to-pollutant ratios are needed. We list early phase and steady state abundance ratios, relative to oxygen, in Table 6. To evaluate the oxygen budget, let us look at the oxide balance in greater detail. As is suitable for a rocky parent body, we assume oxygen is mainly carried in oxides or water. Removing the helium dependence of Equation (2) and working with the oxygen ratios, a balanced oxygen budget requires

$$\begin{equation} {1 \over 2}{n(\mbox{H}_{\rm par}) \over n(\mbox{O})} + \sum \limits _Z {q(Z) \over p(Z)} {n(Z) \over n(\mbox{O})} = 1. \end{equation} \tag{ 3 }$$

We use H_par to represent the undetermined portion of the observed hydrogen that is associated with water in the parent body causing the current accretion. Another ambiguity comes from the fact that there are three principle states of iron in the Earth: metallic Fe (as in Earth's core), FeO, and Fe₂O₃, but in the WD's atmosphere we are measuring the sum of Fe atoms from all states. Since we do not have a clear determination of the partitioning of Fe in the GD 40 parent body, in Table 6 we show sums for three different distributions of Fe in order to better understand the range of effects on the oxide balance. Iron is potentially one of the main carriers of oxygen, and in order to account for the observed O abundance, we find that the Fe cannot be all metallic. From the steady state oxide balance, we need at least 45% of the iron atoms in FeO (at least 30% if the more oxygen-rich species, Fe₂O₃, is the principle state) in order to carry the oxygen. This constraint means that by total mass the accreted material was likely less than 24% metallic iron. For comparison, the bulk Earth is 25%–28% metallic iron (McDonough & Sun 1995), with the uncertainty due to the uncertain light-element composition of Earth's core.

Table 6. Limits of Parent Body Abundances

Z	n(Z)/n(O)	n(Z)/n(O)
	Early Phase	Steady State
H	0.28 ± 0.11	< 0.2
C	0.026 ± 0.016	0.024 ± 0.015
Mg	0.24 ± 0.06	0.29 ± 0.08
Al	< 0.032	< 0.04
Si	0.071 ± 0.016	0.09 ± 0.02
Ca	0.053 ± 0.015	0.10 ± 0.03
Ti	0.0010 ± 0.0002	0.0021 ± 0.0004
Cr	0.0020 ± 0.0004	0.0042 ± 0.0008
Mn	0.0010 ± 0.0003	0.0022 ± 0.0006
Fe	0.13 ± 0.02	0.30 ± 0.05
Ni	< 0.008	< 0.02
Oxygen budget
H2O	< 0.2	< 0.1
Oxides (three cases)
All Fe in Fe2O3	0.69 ± 0.09	1.12 ± 0.12
All Fe in FeO	0.62 ± 0.09	0.97 ± 0.10
${1 \over 2}$ FeO, ${1 \over 2}$ Fe metal	0.56 ± 0.09	0.82 ± 0.09

Notes. Upper portion: two limiting extremes for the accreted abundance ratios relative to oxygen. Early phase and steady state are defined in Section 4.2. The pollutant-to-pollutant ratios are less sensitive to variations in T_eff than are the pollutant-to-helium ratios, which results in smaller errors (30%–60% smaller) listed here than what one obtains in propagated errors from Table 3. With the exception of hydrogen, the steady state values have been calculated using the settling times from Table 5 according to Equation (1). For hydrogen, the steady state fraction relative to O depends on the number of settling times that have passed. An upper limit is estimated assuming t_obs ⩾ 2 t_set, i.e., n(H)/n(O) < (0.28+0.11)/2 (= early phase upper limit/two settling times). Lower portion: the oxygen budget, which includes Al and Ni at their upper limits, is the fraction of the observed oxygen that may be accounted for by the total sum of oxides (MgO, Al₂O₃, SiO₂, CaO, TiO₂, Cr₂O₃, MnO, Fe in Fe₂O₃, FeO or Fe metal, NiO) and water. Three different compositions involving iron are shown. Uncertainties from the element ratios are propagated in quadrature. The oxide, Na₂O can also contribute, but we do not have an abundance measurement, or meaningful upper limit, for Na. In the silicate Earth, Na₂O ≈ Cr₂O₃ ≈ 0.4% by mass, within the errors of our oxygen budget. The expectation is that the contribution from oxides and water will add up to unity (Equation (3)). The steady state abundances account for all of the observed oxygen, in agreement with a mineral stoichiometry.

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Referring to the oxygen budget of Table 6, if one includes all of the hydrogen as water and assumes that all of the iron is in the more oxygenated form, Fe₂O₃ (no metallic Fe), then, even at the upper limit of the uncertainty, the fraction of oxygen accounted for by early accretion phase abundances gets close to, but does not quite add up to 1. In other words, the parent body abundances that would be associated with early phase accretion have too much oxygen compared to what can be carried by high-Z elements and water. A more likely scenario is that accretion has been going on long enough for the system to be in a steady state with heavier elements settling more rapidly than the oxygen. Indeed, we see in Table 6, the steady state abundances are in balance with respect to oxides (i.e., the oxygen budget adds up to unity), and in this configuration, the problem of too much oxygen in the parent body disappears.

We learn from this exercise that the oxide balance favors the steady state accretion situation (represented by the steady state bulk composition in Figure 14 and the steady state abundances in Figure 15, with element-to-element ratios derivable from the steady state column of Table 6). In order for the system to achieve a steady state, the disk lifetime must be longer than the settling timescales t_disk ≫ t_set ∼ 6 × 10⁵ yr. By these arguments, a lower bound on GD 40's disk lifetime is t_disk > 10⁶ yr. With GD 40 in, or near, an accretion—diffusion steady state, then after just a few settling times, the total accreted mass would be ∼ a few × 10²³g, or about the mass of Vesta, the second most massive asteroid in our solar system (Konopliv et al. 2006). We have constrained t_set < t_obs < t_disk, but we do not know how many settling times have passed. Also, since the disk is opaque (Jura et al. 2007a) we cannot estimate its mass, so we do not know how much more mass may be waiting to accrete from there.

Returning to the question of water content in the parent body, since we have argued in favor of accretion taking place for at least a couple of settling times, then in the steady state, an upper limit to the percent of water by mass is 3%, decreasing nearly linearly with t_obs ≪ t_disk (Equation (6) of Jura et al. 2009b). The water content of a planetary body can be a distinctive measure. By mass, the Earth's mantle,⁶ Ceres and Callisto are 0.03%–0.1%, 25%, and 50% water, respectively (van Thienen et al. 2007; McCord & Sotin 2005; Canup & Ward 2002, respectively). Meteorites originating from the outer asteroid belt (2.5–4 AU) are up to 10% water by mass, while those originating from the inner asteroid belt (∼2 AU) range from 0.05 to 0.1% water by mass (Morbidelli et al. 2000).

GD 40 has a disk within its tidal radius, circumstellar silicate dust emission, the mass of a large asteroid in its CVZ, and pollutant abundances consistent with rocky material. The most plausible origin of the atmospheric heavy elements is a tidally disrupted planetary parent body, such as an asteroid/minor planet. Presumably, this object was part of an ancient planetary system, had its orbit perturbed (Debes & Sigurdsson 2002), such that it journeyed within the star's tidal radius and was disrupted (Jura 2003), forming a disk of material that is accreting onto the star.

In Figure 15, the element abundances are shown normalized to the primitive material CI chondrites to assist our interpretation of how the GD 40 parent body may have evolved from a primitive constitution. We see an overall trend of increasing abundances with increasing condensation temperature indicative of thermal processing. The close agreement with bulk Earth values for O, Mg, Cr, Mn, and Fe anchors the Earth-like abundance nature of the polluting body, permitting us to interpret the deviations, most notably the depletion of silicon and the build up of refractory elements, in terms of evolutionary processes.

A distinctive result we find is n(Si)/n(Mg) = 0.30 ± 0.11 in significant contrast to the ratio near unity for the bulk Earth, the Sun, chondrites, and nearby stars. This is not part of the pattern of the thermal trend, nor can it be caused by differential gravitational settling, since Si and Mg have nearly the same diffusion times (see also Figure 15). Our result is consistent with the F99W02 value of n(Si)/n(Mg) = 0.3^+0.4_−0.2. In this paper, we have reduced the uncertainties over previous work enough to recognize that the primordial elemental mix in this evolved planetary system has been altered within the parent body.

Figure 16 depicts the distinction from cosmic abundances. The GD 40 star and proto-planetary system was probably not born with a peculiar Si/Mg ratio. Presently, no solar system object has an observed Si/Mg abundance with such a low ratio. Perhaps the closest comparison to a measured material in our own solar system comes from the Pallasite meteorites, which are thought to derive from the mantle–core boundary of a differentiated body (i.e., physically stratified into mantle, core, and/or crust), and measure n(Si)/n(Mg) = 0.6 (Buseck 1977). One possibility is that the distinctive Si/Mg ratio may be associated with mantle/crust differentiation, in which Si was concentrated in a crust, and Mg in a mantle, and the crust was later lost. Evidence for eroding collisions exists in our solar system in the form of the Moon and the diversity of the population of asteroids and meteorites such as the Pallasites. The so-called hit-and-run process (Asphaug et al. 2006), thought to play a role in terrestrial planet formation, results in increasingly exotic characteristics for the unaccreted remnants of planetesimal collisions. If differentiation takes place in a planetary body, Si-rich minerals tend to rise into the crust, whereas Mg-rich minerals prefer the mantle. In Earth's crust n(Si)/n(Mg) = 11, but in a large rocky planetary body such as Earth, the 35 km crust is a relatively thin layer and the mantle/crust mass ratio is 169:1. Thus, even though Si is highly concentrated in the crust, a relatively small amount by mass has been depleted from the Earth's mantle. In an asteroid-sized body, such as Vesta (radius = 270 km), the crust is thought to be ∼27 km, and a mantle/crust mass ratio has been estimated at 3:1 (Ghosh & McSween 1998; Jones 1984), making a substantial Mg and Si differentiation in an asteroid plausible.

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A difficulty of this explanation is with the observed enhancements of Ca, Ti, and possibly Al in GD 40. Similar to silicon, the elements Ca, Ti, and Al also tend to concentrate in the crust of a differentiated body. For instance, Si, Ca, Ti, and Al are respectively, 1.3, 1.8, 3.4, and 3.5 times more abundant in Earth's crust than in Earth's mantle (Palme & O'Neill 2007). If GD 40's parent body lost its crust, then should not Ca, Ti, and Al also be depleted along with Si? Something else must be going on. Since Ca, Ti, and Al are refractory, most likely their enhancements are attributable to a significant thermal process. Indeed, Earth's mantle, while depleted in Si relative to primitive CI chondrites, is modestly enhanced in Ca, Ti, and Al (see Table 7), notably due to the refractory lithophile nature of these elements. Given the strong enhancements of Ca and Ti in GD 40's polluting planetary body, it likely experienced a more extreme thermal history than Earth, as may have come about for example from intense heating during the red giant phase or through the energy of collisions. Another consideration is that exposure to high temperatures during planetary formation and/or evolution could be important. With an estimated WD mass of 0.59 M_☉ for GD 40, its main sequence progenitor was likely greater than 1.5 M_☉ (Weidemann 2000). As previously suggested by Jura et al. (2009b), the circumstellar environments around stars more massive than our Sun may support conditions of more intense heating of the protoplanetary material, planetesimals and planets than in the solar nebula, which could explain the dramatically high abundances of calcium polluting the WDs GD 362 and GD 40. One possibility for GD 40 is that both heating and crust loss have occurred, and that Ca and Ti are enhanced because the heating effect on their abundances dominates over the crust loss. In that case, if the GD 40 parent body had not lost its crust, the Ca, Ti, and Al abundances would be even more enhanced than what are measured now.

Fegley & Cameron (1987) describe another mechanism—a silicate vaporization process—which removes substantial quantities of material (e.g., ∼80% of Mercury's silicate) from the outer layers of a differentiated body due to prolonged heating. Their calculations were designed to explain the unusually high density of planet Mercury and are successful at doing so by modeling the protoplanet's exposure to a high-temperature (2500–3500 K) phase of the primitive solar nebula. More generally we obtain a sense of the constitutional evolution of a planetary body with an initially chondritic composition after experiencing a period of intense heating. The models predict a unique resulting abundance pattern such that, relative to chondritic abundances, the mantle+crust becomes depleted in SiO₂, while MgO, CaO, TiO₂, and Al₂O₃ are enhanced. In particular, model 4 (from Table 4 of Fegley & Cameron 1987) yields Si/Mg, Ca/Mg, and Ti/Mg ratios in agreement with GD 40's measured values, as shown in Table 7. We do not refer to the iron abundances from the mantle+crust results of Fegley & Cameron (1987) since in their model most of the iron resides in a core. This silicate vaporization model can simultaneously reproduce the Si depletion, and the Ca and Ti enhancements in GD 40's polluting parent body; however, the calculated high density planet has an iron-to-silicate mass ratio twice that of Earth, which is appropriate for planet Mercury, but is not what we observe in GD 40. GD 40's parent body probably had a metallic iron-to-silicate mass ratio that was less than (possibly near) that of Earth (Section 4.3). Although this model involves differentiation of the parent body and essentially an erosion of the outer layers—mechanisms which we think are important—it is highly idealized and its application to real systems is uncertain.

Finally, we note that a crust/mantle differentiated body is likely to have also formed a core composed primarily of Fe metal as in Earth's core. As discussed in Section 4.3, within the uncertainties of GD 40's steady state pollutant abundances, the oxide balance allows up to 70% of the iron to originate from a metallic form (<55% if the iron oxide is predominantly FeO), although a rocky composition with no metallic iron is also allowed by the oxygen budget (Table 6). This translates to an upper limit on the size of a metallic core in the accreted parent body at ∼30% of its total mass (<27% assuming a Vesta-like core composition of 0.89 Fe by mass, as modeled by Ghosh & McSween 1998). For comparison, Earth's core is 32% of its mass (Allegre et al. 1995), and estimates of core sizes in Mars, Vesta, and the Moon are 30%, 10%, and 5% of their total masses, respectively (Righter & Drake 1996).