MagAO IMAGING OF LONG-PERIOD OBJECTS (MILO). II. A PUZZLING WHITE DWARF AROUND THE SUN-LIKE STAR HD 11112

We observed HD 11112 using the Magellan Clay telescope at Las Campanas Observatory in Chile on the nights of UT 2014 November 8–10, (epoch 1), and approximately one year later on the night of UT 2015 November 30, (epoch 2). Observations were made using both the visible camera VisAO (Kopon et al. 2010) and the infrared camera Clio-2 (Sivanandam et al. 2006). VisAO has a plate scale of 00079 and a field of view (FOV) of ∼8'' (Males et al. 2014). For Clio-2, we used the narrow camera, which has a plate scale of 001585 and an FOV of ∼9 × 15'' (Morzinski et al. 2015). The instrument rotator was turned off for both cameras to enable angular differential imaging (ADI; Marois et al. 2006). We observed in the r' (0.63 μm), i' (0.77 μm), z' (0.91 μm), Ys (0.99 μm), J (1.1 μm), H (1.65 μm), Ks (2.15 μm), and L' (3.76 μm) filters, detecting a faint point-source ∼22  away from the host star at a position angle of ∼226° in all images. Unsaturated photometric images were acquired after each imaging sequence; for VisAO at r' and i', due to the brightness of the star, a neutral density (ND) filter was required to obtain unsaturated calibration images. For epoch 1, the observing conditions were fair, with variable seeing, strong wind, and intermittent cirrus clouds throughout both nights. For epoch 2, the conditions were much better, with clear skies and steady seeing under 1'' for most of the night. See Table 1 for a summary of all MagAO observations.

The VisAO images were dark-subtracted, registered, divided by the integration times to give units of counts/s, cropped, and had their 2D radial profiles removed (since ADI was not used to reveal the companion). The images were then rotated to north-up, east-left, and median-combined into final images (shown in Figure 1). The Clio-2 images were sky-subtracted, divided by the coadds and integration times to give units of counts/s, registered, cropped, and corrected for nonlinearity (Morzinski et al. 2015). Due to the sky brightness at 1–4 μm, we used ADI and a Principal Component Analysis (PCA, Soummer et al. 2012) to reveal HD 11112B. The number of PCA modes used at a given wavelength was determined by maximizing the SNR of the faint point source. For epoch 1, the number of modes at J, H, Ks, and L' was 6, 18, 10, and 4, respectively; for epoch 2, the number of modes at J, H, and Ks was 4, 7, and 4, respectively. After PCA PSF subtraction, the images were rotated to north-up, east-left, and then median-combined into final images (shown in Figure 1). The unsaturated calibration images of the star were reduced in analogous ways for both VisAO and Clio-2.

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Observations of HD 11112 were obtained using the UCLES echelle spectrograph (Diego et al. 1990) at the AAT by the Anglo-Australian Planet Search (AAPS) team. The observations for this star began in 1998 January. The AAPS uses a 1'' slit to obtain spectra with a resolution of ∼45,000 in the iodine region (5000–6300 Å). A temperature-controlled iodine absorption cell (Marcy & Butler 1992; Butler et al. 1996) is mounted in front of the instrument's entrance slit, imprinting the reference iodine spectrum directly on the incident starlight and providing a wavelength scale and measurement of the effective PSF for every observation (Butler et al. 1996). The median internal uncertainty achieved for HD 11112 using the iodine-cell technique with UCLES was 1.92 m s⁻¹. The RVs are reported in Table 2 and shown in Figure 2.

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The age of HD 11112A has been estimated by several studies and ranges from ∼4 to 8 Gyr (Feltzing et al. 2001; Valenti & Fischer 2005; Holmberg et al. 2009; Ghezzi et al. 2010a; Ramírez et al. 2012; Bensby et al. 2014). The star's space velocity (UVW from Holmberg et al. 2009) points to an old age, as it is likely in the Hercules stream (Ramya et al. 2016), and the star is chromospherically older than the Sun (log R'HK ∼ −5.0; Jenkins et al. 2006; Pace 2013). HD 11112A is likely slightly evolved, since its log g (∼4.2; Valenti & Fischer 2005; Ghezzi et al. 2010b; Ramírez et al. 2012; Bensby et al. 2014) is smaller than what is typical for main-sequence dwarfs (log g ∼ 4.3–4.5). Using the new MIST tracks with a metallicity of [Fe/H] = 0.25 (close to the metallicity of HD 11112A, [Fe/H] = 0.20, Bensby et al. 2014), we find that HD 11112A is consistent with being a ∼1.12 M_⊙ star with an age of ∼7 Gyr (see Figure 3). Thus multiple lines of evidence point to HD 11112A being old (likely older than the Sun) and to it evolving off the main sequence right now. We adopt as the age of HD 11112A the average of the previous measurements, excluding the three isochronal ages that are younger than the Sun (Valenti & Fischer 2005; Bensby et al. 2014; Ghezzi et al. 2010a); this leads to an age of ${7.2}_{-1.2}^{+0.78}\,{\rm{Gyr}}$, consistent with our own new estimate using the MIST tracks (∼7 Gry).

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The SED of HD 11112A, computed by comparing the star's apparent magnitudes at B, V (converted from BT and VT from the Tycho catalog, Høg et al. 2000), R, I (from Cousins 1980), J, H, Ks (from 2MASS, Skrutskie et al. 2006), and W1, W2, W3, and W4 (from ALLWISE) is shown in Figure 4. We fit the photometry to a grid of Kurucz/Castelli stellar models¹⁴ using different $\mathrm{log}g$ values (which made little difference). We also tested different log metallicity values relative to the Sun and found that these also made little difference. The final best-fit model yielded an effective temperature of 6000 K on the 250 K temperature grid, which is consistent with previous measurements (∼5900 from, e.g., Valenti & Fischer 2005; Ghezzi et al. 2010a; Ramírez et al. 2012; Bensby et al. 2014). As is evident in Figure 4, this fit underpredicts the NIR and mid-infrared photometry. One explanation for this would be that the star has an infrared excess due to hot, close-in dust. The best-fitting blackbody model for the excess with emissivity ∝ν yields a dust temperature of 1465 K and an infrared fractional luminosity (L_ir/L_*) of 0.01, both of which are quite high for such an old Sun-like star. However, Ertel et al. (2014) found that ∼10% of Sun-like stars have bright exozodiacal dust, and the frequency of excesses may actually increase with stellar age (for Sun-like stars); HD 11112A may fall into this category. Another possibility is that the dust resides around the imaged companion, which we reveal later to be a white dwarf.

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Astrometry of the faint point-source was computed by calculating the photocenter inside circular apertures with radius = 01 (for VisAO images) or 008 (for Clio-2 images). Uncertainties were assumed to be 5 mas in the north and east directions, based on previous results with MagAO from Rodigas et al. (2015). Figure 5 shows that, based on the primary star's high proper motion (0415 east, 0152 north, van Leeuwen 2007) over the course of the year between the two imaging epochs, the faint companion is inconsistent with being a zero proper motion background object at more than 60σ confidence. Therefore, we consider the faint point-source to be a gravitationally bound object and henceforth refer to it as HD 11112B. At a projected separation of ∼22 (99.7 au), the companion does not show any statistically significant orbital motion over the course of one year. The final astrometry is reported in Table 3.

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Table 3.  HD 11112B Photometry and Astrometry

	Epoch 1	Epoch 2
Julian Date	2456970.50000	2457356.50000
Δr' (0.63 μm)	${10.23}_{-0.23}^{+0.19}$	${9.94}_{-0.19}^{+0.16}$
Δi' (0.77 μm)	${10.24}_{-0.16}^{+0.14}$	${10.09}_{-0.10}^{+0.09}$
Δz' (0.91 μm)	${10.04}_{-0.23}^{+0.19}$	${10.19}_{-0.10}^{+0.09}$
ΔYs (0.99 μm)a	${9.68}_{-0.32}^{+0.24}$	⋯
ΔJMKO (1.1 μm)	${11.02}_{-0.03}^{+0.03}$	${10.87}_{-0.06}^{+0.06}$
ΔHMKO (1.65 μm)	${10.85}_{-0.07}^{+0.07}$	${10.89}_{-0.13}^{+0.12}$
ΔKsBarr (2.15 μm)	${10.81}_{-0.31}^{+0.24}$	${10.91}_{-0.09}^{+0.08}$
ΔL'MKO (3.76 μm)	${10.43}_{-0.19}^{+0.16}$	⋯
Mr'	13.99 ± 0.29	13.70 ± 0.25
Mi'	13.83 ± 0.22	13.69 ± 0.15
Mz'	13.59 ± 0.29	13.74 ± 0.16
MJ	13.73 ± 0.10	13.57 ± 0.12
MH	13.34 ± 0.13	13.38 ± 0.19
MK	13.17 ± 0.37	13.27 ± 0.15
${M}_{{L}^{{\prime} }}$	12.75 ± 0.24	⋯
ΔR.A. ('')	−1.59 ±  0.01	−1.58 ±  0.01
ΔDecl. ('')	−1.52 ±  0.01	−1.53 ±  0.01
ρ ('')	2.20 ±  0.01	2.20 ±  0.01
P.A. (°)	226.4 ±  0.2	225.9 ±  0.2

Note.

^aYs data are not used in the SED modeling.

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Calculating accurate photometry for HD 11112B was not a straightforward task considering that some images used NDs for calibration, some used unsaturated short exposures, and others were reduced using ADI+PCA, which introduces self-subtraction, over-subtraction, and other biases. In the following, we describe in detail the methods used to obtain photometry for each image in each epoch.

At r', the image quality is very sensitive to the observing conditions since the Strehl ratio is low (∼10%–30%). Because of the unfavorable observing conditions during epoch 1, the first r' image was not of high quality (characterized by a "blobby," non-spherical PSF). Further complicating matters was the use of the ND, which at the time of the observations only had a few calibration measurements. To mitigate these concerns, when computing the photometry we used a very large circular aperture, with radius = 7×  the full-width half-maximum (FWHM) at r' (radius = 014) to ensure that most of the flux from HD 11112B was included in the aperture, while still being as small as possible to maximize SNR. The flux inside the aperture was averaged, as was the flux inside the same aperture centered on the star in the unsaturated (ND) image. The stellar flux was then scaled up by the ND's diminution factor of 7176 ± 332. The Δ magnitude at r' was computed by dividing the companion's flux by the scaled stellar flux. The uncertainty on the companion flux was computed by placing the same circular aperture at 12 equally spaced azimuthal angles around the star at the same radius as the companion, averaging the fluxes inside these apertures, then computing the standard deviation of all the fluxes. The final uncertainty was the sum in quadrature of this uncertainty with the ND calibration uncertainty. All of the above steps were repeated for the epoch 2 r' images.

For i', the procedure was identical to the above, except an aperture with radius = 3 × the FWHM (radius = 007) was used and the ND scaling was 1317.99 ± 52.85. For z', a 3 × FWHM (radius = 009) aperture was used and the ND scaling was not required. For Ys, a 3 × FWHM (radius = 010) aperture was used and the ND scaling was not required; in addition, to mitigate the effects of the bright ghost near the companion, we subtracted the average flux inside an annulus with inner radius = 3 × FWHM and outer radius = 6 × FWHM.

For the Clio-2 images, the procedure required that we account for the biases introduced by the ADI+PCA reductions. At each wavelength, a circular aperture with radius = 2 × FWHM (0.5 × FWHM for L' due to the lower SNR detection) was used to calculate the average flux of the companion. This flux was then scaled up by a correction factor, which was determined by inserting and recovering scaled-down replicas of the unsaturated stellar PSF at 12 equally spaced azimuthal angles around the star, calculating the average fluxes inside the same apertures centered on the recovered point sources, then comparing these to the expected average flux inside the same aperture centered on the pre-inserted scaled-down point source. Uncertainties were calculated as the sum in quadrature of the standard deviation of the average fluxes inside the same apertures placed at the 12 position angles (of the final image without any artificial sources inserted), and the standard deviation of the correction factors.

To convert the Δ magnitudes into absolute magnitudes, we converted the catalog 2MASS photometry for HD 11112A into the MKO system using the color transformation relations in Carpenter (2001) and used the derived catalog SDSS photometry for HD 11112A from Ofek (2008), since the VisAO filters are very similar. We then added the Δ magnitudes to the primary's absolute magnitudes. We converted the absolute magnitudes into F_λ (e.g., see Faherty et al. 2013) using the Hipparcos parallax of 22.07 ± 0.57 mas (van Leeuwen 2007). All photometry is reported in Table 3, and the SED is shown in Figure 6.

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While the NIR colors of HD 11112B point to it possibly being a cool brown dwarf (Leggett et al. 2010), the fact that it is bright at optical wavelengths suggests instead that it is a white dwarf (e.g., Crepp et al. 2013a). Therefore, we set out to fit the companion's photometry to cool white dwarf models. While the quality of the epoch 1 data was poor, we included these data in the analysis for completeness (and the final fitting results are not markedly different compared to epoch 2). For cool white dwarfs, the atmospheric parameters (T_eff and log g) and chemical compositions can be measured accurately using the photometric technique developed by Bergeron et al. (1997). To fit the SED of HD 11112B, we first converted the optical and infrared photometric measurements into observed fluxes using the procedure outlined in Holberg & Bergeron (2006), including the zero points for the various photometric systems. The transmission functions in the optical for the ugriz photometry is described in Holberg & Bergeron (2006) and references therein, while those for the Mauna Kea Observatories (MKO) photometric systems are taken from Tokunaga et al. (2002).

We then related the average observed fluxes f_λ and the average model fluxes H_λ—which depend on T_eff, $\mathrm{log}g$, and chemical composition—by the equation

$$\begin{eqnarray}&&{f}_{\lambda }=4\pi {(R/d)}^{2}{H}_{\lambda },\end{eqnarray} \tag{ 1 }$$

where R/d defines the ratio of the radius of the white dwarf to its distance from Earth. Next, we minimized the χ² value defined as the difference between observed and model fluxes over all bandpasses with weights determined by the photometric uncertainties. We used the nonlinear least-squares method of Levenberg–Marquardt (Press et al. 1986), which is based on a steepest descent method. Only T_eff and the solid angle π (R/d)² were considered free parameters, while the uncertainties of both parameters were obtained directly from the covariance matrix of the fit. We started with $\mathrm{log}g=8$ and determined the corresponding effective temperature and solid angle, which combined with the distance, d (obtained from the known parallax) gives us the radius of the white dwarf, R. We then converted the radius into mass using evolutionary models similar to those described in Fontaine et al. (2001) but with C/O cores $q(\mathrm{He})\equiv \mathrm{log}{M}_{\mathrm{He}}/{M}_{\star }={10}^{-2}$ and q(H) = 10⁻⁴, which correspond to hydrogen-atmosphere white dwarfs, and q(He) = 10⁻² and q(H) = 10⁻¹⁰, which correspond to helium-atmosphere white dwarfs. Since the atmospheric composition of HD 11112B is unknown, we assumed pure hydrogen and pure helium model atmospheres and used the synthetic spectra described in Bergeron et al. (1995), Tremblay & Bergeron (2009), Bergeron et al. (2011), and references therein. In practice, the $\mathrm{log}g$ value obtained from the inferred mass and radius (g = GM/R²) is different from our initial guess of $\mathrm{log}g=8$. Therefore, the fitting procedure was repeated until an internal consistency in $\mathrm{log}g$ was reached.

The best-fitting models for the epoch 1 and epoch 2 photometry are shown in Figure 6 as well as in Table 4. The reduced χ² (${\chi }_{\nu }^{2}$) ranges from ∼1.45 to 2.5, indicating overall good fits. Interestingly, the L' flux is 1.6–2σ larger than the models predict. Pure He atmospheres provide marginally better fits than pure H. The estimated white dwarf masses and effective temperatures range from ∼0.9 to 1.1 M_⊙ and 7300–9750 K, respectively. These correspond to progenitor masses ranging from 4.3 to 6.5 M_⊙ (Kalirai et al. 2008; Williams et al. 2009), which have main-sequence lifetimes ≲160 Myr (Bertelli et al. 2009). This is insignificant compared to the expected cooling age for the white dwarf, which we modeled explicitly.

Table 4.  HD 11112B SED Fitting Results

		Epoch 1			Epoch 2
		Pure H	Pure He		Pure H	Pure He
${\chi }_{\nu }^{2}$		2.47	2.37		1.62	1.47
Mass (M⊙)		${1.06}_{-0.02}^{+0.02}$	${0.90}_{-0.02}^{+0.021}$		1.08${}_{-0.02}^{+0.02}$	${0.96}_{-0.02}^{+0.02}$
Teff (K)		${8400}_{-2000}^{+2000}$	${7300}_{-1900}^{+1900}$		${9800}_{-1700}^{+1700}$	${8700}_{-1800}^{+1800}$
$\mathrm{log}g$ (cm s−2)		${8.73}_{-0.03}^{+0.03}$	${8.50}_{-0.03}^{+0.03}$		${8.77}_{-0.03}^{+0.03}$	${8.59}_{-0.03}^{+0.03}$
$\mathrm{log}L/{L}_{\odot }$		−3.61${}_{-0.02}^{+0.02}$	−3.69${}_{-0.02}^{+0.02}$		−3.39${}_{-0.02}^{+0.03}$	−3.46${}_{-0.02}^{+0.02}$
Age${}_{\mathrm{cool},50 \% C/O}$ (Gyr)a		${3.17}_{-1.27}^{+1.90}$	${3.53}_{-1.41}^{+0.92}$		${2.43}_{-0.70}^{+1.03}$	${2.88}_{-1.22}^{+1.02}$
Age${}_{\mathrm{cool},100 \% C}$ (Gyr)b		${3.58}_{-1.63}^{+2.38}$	${3.92}_{-1.61}^{+1.45}$		${2.65}_{-1.04}^{+1.43}$	${3.15}_{-1.54}^{+1.28}$

Notes.

^aThe cooling age of the white dwarf, assuming a 50% C/O core and taking into account the uncertainties in mass and temperature. ^bThe same, but for a 100% C core.

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Determining the cooling age of a white dwarf by fitting its observed mass, atmospheric composition, and temperature using evolutionary sequences requires making an assumption about its core composition; the effect of this assumption is most pronounced in the case of older white dwarfs (Fontaine et al. 2001). Because the thermonuclear burning rate of He is uncertain, the exact core compositions of white dwarfs are generally unknown. Attempts to obtain such measurements have shown that white dwarf cores must at least be partly composed of oxygen and are perhaps even dominated by it (e.g., see Salaris et al. 1997; Althaus et al. 2010). More recently, Fields et al. (2016) performed a Monte Carlo simulation to estimate the core composition of a 0.64 M_⊙ white dwarf using the STARLIB reaction rate library and the MESA evolutionary code. It is the first time that such an analysis accounted for uncertainties in the ¹²C(α, γ)¹⁶O, the triple-α, and the ${}^{14}N{(\iota p,\gamma )}^{15}O$ nuclear reaction rates. They found that uncertainties in these reaction rates dominated variations in the central C/O ratio of the white dwarf. However, they were able to show that the core compositions of their simulated white dwarfs were of at least 25% oxygen at 95% confidence. Giammichele et al. (2016) have produced the only reliable direct measurement of the core composition of a white dwarf to date, using asteroseismology to deduce that the 0.65 M_⊙ white dwarf Ross 548 has a fractional oxygen core composition of X(O) = 0.70 ± 0.06.

Based on these previous results, for HD 11112B, we assumed a core composition of 50% C and 50% O and used the evolutionary models described in Bergeron et al. (2001). The resulting cooling ages ranged from ∼2.4 to 3.5 Gyr. If we assume an unrealistic 100% C core, then the cooling ages are slightly larger, ranging from ∼2.6 to 4 Gyr.

The RVs for HD 11112 show no statistically significant curvature; therefore, it is difficult to estimate the period of the companion HD 11112B. However, its mass can be estimated following the procedure outlined in Torres (1999) and Rodigas et al. (2016). Briefly, we used a Bayesian Markov chain Monte Carlo (MCMC) approach to produce posterior distributions of the allowed parameter values (Ford 2005). The likelihood function L, which contains the Keplerian model and nuisance parameters, is simpler in this case because data were obtained with only one instrument and there are no obvious planetary signals in the data (just a long-period linear trend). The likelihood function is given by

$$\begin{eqnarray}&&L=\displaystyle \frac{1}{\sqrt{2\pi }}\displaystyle \frac{1}{\sqrt{{\epsilon }_{i}^{2}+{s}^{2}}}\exp \left[-\displaystyle \frac{1}{2}\displaystyle \frac{{({v}_{i}-v(t))}^{2}}{{\epsilon }_{i}^{2}+{s}^{2}}\right]\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{v}_{i}=\gamma +\displaystyle \sum _{p}\,m({\hat{\kappa }}_{p};t)+{\dot{v}}_{r}(t-{t}_{0}),\end{eqnarray} \tag{ 3 }$$

where i indexes the individual observations, _i is the nominal uncertainty of each RV measurement, γ and s are the zero-point and extra noise parameters (also called jitter), and the Doppler signal from a companion on the star is encoded in the model $m({\hat{\kappa }}_{p};t)$, which is a function of time t and the Keplerian parameters $\hat{{\kappa }_{p}}$. The Keplerian parameters of the p^th companion in the system are the orbital period P_p (in days), the semi-amplitude K_p (in m/s), the mean anomaly μ_0,p at the reference epoch t₀ (in degrees), the eccentricity e_p, and the argument of periastron ω_p. The second term in Equation (3) accounts for the possible presence of a long-period candidate whose orbit is only detected as a trend (acceleration, ${\dot{v}}_{r}$). A third term, (jerk, ${\ddot{v}}_{r}$) was initially included, but fits to the data showed that there was no statistically significant curvature, so this term was later dropped. Additional details on our MCMC fitting can be found in Section 3.3 of Rodigas et al. (2016).

For HD 11112, initial periodogram results for a single planet showed that there was a peak at 1.46 days with a false alarm probability of 5%, making the signal dubious. Therefore, the planetary term (the second term in Equation (3)) was dropped, leaving only the long-period term (the slope). Fits to the data were then computed. The significance of the slope (median = −2.82 m s⁻¹) being nonzero was 8σ (see Figure 7). We used the fitted slope terms to compute the posterior mass distribution of HD 11112B using

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{B}}{{M}_{\odot }}=5.341\times {10}^{-6}\,{\dot{v}}_{r}{\left(\displaystyle \frac{\rho }{{\rm{\Pi }}}\right)}^{2}\displaystyle \frac{1}{{\rm{\Psi }}},\end{eqnarray} \tag{ 4 }$$

where M_B is the mass of HD 11112B in solar masses, ${\dot{v}}_{r}$ is the slope term generated by the MCMC procedure, ρ is the current projected separation of HD 11112B, Π is the parallax, and Ψ contains the angle and time terms:

$$\begin{eqnarray}&&{\rm{\Psi }}={[(1-e)(1+\cos E)]}^{-1}(1-e\cos E)\sin i\,\times \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&(1-{\sin }^{2}(\nu +\omega ){\sin }^{2}i)(1+\cos \nu )\sin (\nu +\omega )\end{eqnarray} \tag{ 6 }$$

where E is the eccentric anomaly, i is the inclination, ν is the true anomaly, and ω is the argument of periastron.

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We generated the posterior distribution for the mass of HD 11112B using Equation (4) while randomly drawing from Gaussian distributions centered on the measured values and including the corresponding uncertainties for Π and ρ, and also randomly drawing from uniform distributions for the other variables ($\cos i$ and e between 0 and 1, ω between 0 and 2π). The resulting distribution, shown in Figure 8, is sharply peaked at ∼0.42 M_⊙. Given the high likelihood that HD 11112B is a white dwarf (see Section 3.3), we imposed a mass cutoff of 1.4 M_⊙  (the Chandrasekhar limit; Mazzali et al. 2007). Based on this constraint, at 99% confidence, the minimum mass of HD 11112B is 0.28 M_⊙ and the median mass is 0.49 M_⊙. We can also examine the resulting posterior distributions for the inclination (top right panel of Figure 8), finding at 99% confidence a minimum inclination of 98. The other orbital parameters were fully unconstrained; however, some inferences can be gleaned by examining the relationship between the mass, inclination, eccentricity, and argument of the periastron (Figure 9 bottom panels). For the best-fitting SED mass range (0.9–1.1 M_⊙), the white dwarf companion should have a near edge-on, high-eccentricity orbit.

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We can also make some inferences on its likely semimajor axis (a) using the information at hand. Since the object is most likely near apoastron, r ∼ r_apo = a (1 + e), where r_apo is the apoastron distance. We also know that the object's current projected separation is the closest it could be to the primary, or r > r_proj. Therefore, assuming e ∼ 0.9 from Figure 9, a ≳ 50 au (period longer than ∼300 years). This means that we should not expect much orbital motion over the next few years.