THE CHARA ARRAY ANGULAR DIAMETER OF HR 8799 FAVORS PLANETARY MASSES FOR ITS IMAGED COMPANIONS

The observed quantity of an interferometer is fringe contrast or "visibility," more formally defined as the correlation of two wave fronts whose amplitude is the visibility squared (V²), which is fit to a model of a uniformly illuminated disk (UD) that represents the face of the star. Diameter fits to V² were based upon the UD approximation given by V² = ([2J₁(x)]/x)², where J₁ is the first-order Bessel function and x = πBθ_UDλ⁻¹, where B is the projected baseline at the star's position, θ_UD is the apparent UD angular diameter of the star, and λ is the effective wavelength of the observation (Shao & Colavita 1992). A more realistic model of a star's disk involves limb-darkening (LD), and the relationship incorporating the linear LD coefficient μ_λ (Hanbury Brown et al. 1974) is

$$\begin{eqnarray} V^2 &=& \left({1-\mu _\lambda \over 2} + {\mu _\lambda \over 3} \right)^{-2} \nonumber\\ && \times \left[(1-\mu _\lambda) {J_1(x) \over x} + \mu _\lambda {\left(\frac{\pi }{2} \right)^{1/2} \frac{J_{3/2}(x)}{x^{3/2}}} \right]^{2}. \end{eqnarray} \tag{ 1 }$$

Table 3 lists the date of observation, the calibrator used, λ/B, the calibrated visibilities (V²), and errors in V² (σV²).

The LD coefficient was obtained from Claret & Bloemen (2011) after adopting the effective temperature (T_eff) and surface gravity (log g) values from Allende Prieto & Lambert (1999), which were 7586 K and 4.35, respectively. The resulting UD and LD angular diameters are listed in Table 6. Figures 1–8 show the LD diameter fits for HR 8799 by night and by calibrator and Figure 9 shows all the data combined.

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Table 4 lists the resulting angular diameter fits for each night using each calibrator, and Figure 10 shows a graphical version of the results. Though there is some scatter in the diameter fit from each night/calibrator combination, the scatter from the 2011 data is less pronounced than from 2010 data, which is seen in the diameter fits shown in Figures 4–7. In particular, for observations obtained in 2010, the data exhibit sinusoidal-like variations about the best angular diameter fit; the variations are not seen in 2011 data. This is almost certainly due to coating asymmetries between CHARA Array telescopes (particularly overcoated silver versus aluminum) that were present in 2010 but removed in 2011. Attempts were made to search for polarization effects that would cause these residuals, but no conclusive evidence was found. Visibilities were, however, never measured in full Stokes parameters in a configuration that showed these residuals.

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Because the 2011 data show none of the sinusoidal residuals, our final θ_LD incorporates the data from 2011 only. This yielded an angular diameter of 0.341 ± 0.008 mas. When the data from all nights and using all calibrators are fit together, θ_LD is 0.347 ± 0.007 mas, which is within the uncertainty of the adopted value.

The uncertainty for θ_LD was calculated as described in the supplemental material from Derekas et al. (2011): for each one of 10,000 Monte Carlo simulated Gaussian distributions, we calibrated the instrumental visibilities and fit an angular diameter to the calibrated data using a least-squares minimization. We accounted for random errors by adding random numbers generated from the empirical covariance matrix scaled by the reduced χ² of the original fit, and then repeated the procedure. The final uncertainty was the resulting standard deviation of the total distribution.

We observed HR 8799 with multiple calibrator stars every night except one to check on the behavior of the calibrators themselves. For example, when three calibrator stars were used, we used calibrator 1 to determine the angular diameters of calibrator 2 and calibrator 3 to make sure the stars were reliable. The calibrator HD 214698 shows the largest scatter, which is expected because it is the smallest calibrator of the three. There will naturally be more uncertainty when measuring its calibrated visibilities using the other two larger stars. The scatter seen in the night-to-night measurements is expected because the stars are small and very nearly unresolved.

At a distance of 39.4 ± 1.1 pc (van Leeuwen 2007), HR 8799's θ_LD of 0.342 ± 0.008 mas corresponds to a stellar radius⁹ of 1.44  ±  0.06 R_☉, corresponding to a precision of 4%. We note that this radius is 8% larger than the radius inferred from photometric and temperature considerations in Gray & Kaye (1999) and subsequently adopted in recent asteroseismic analyses Moya et al. (e.g., 2010a, 2010b); only 1% of this discrepancy can be attributed to the pre-revised HIPPARCOS distance used in the calculation of Gray & Kaye (1999).

In order to determine the luminosity (L) and T_eff of HR 8799, we first constructed its photometric energy distribution (PED) from the averages (when multiple measurements were available) of Johnson UBV magnitudes, Strömgren uvby magnitudes, and Two Micron All Sky Survey (2MASS) JHK magnitudes, as published in Crawford et al. (1966), Breger (1968), Eggen (1968), Mermilliod (1986), Schuster & Nissen (1986), Mendoza et al. (1990), Hauck & Mermilliod (1998), Gezari et al. (1999), Høg et al. (2000), Cutri et al. (2003), and Olsen (1983, 1993, 1994). Table 5 summarizes the adopted magnitudes; the assigned uncertainties for the 2MASS infrared measurements are as reported, and for optical measurements are standard deviations of multiple measurements. The assigned uncertainties of the optical measurements therefore account for the low amplitude optical variability (±0.02 mag) observed for this variable star (Zerbi et al. 1999).

The bolometric flux (F_BOL) of HR 8799 was determined by finding the best-fit (via χ² minimization) stellar spectral template from the flux-calibrated stellar spectral atlas of Pickles (1998). This best PED fit allows for extinction, using the wavelength-dependent reddening relations of Cardelli et al. (1989). The best fit was found using an F0 V template with an assigned temperature of 7211 ± 90 K, an extinction of A_V = 0.00 ± 0.01 mag, and an F_BOL of 1.043 ± 0.012 × 10⁻⁷ erg s⁻¹ cm⁻² (a 1.1% precision). Figure 11 shows the best fit and the results are listed in Table 6. To check for possible systematic biases in our adopted prescription, we also fit the PED using synthetic Kurucz spectral models,¹⁰ assuming no extinction. The resulting F_BOL estimate is consistent to within the 1.1% error reported above, further validating our technique and measured F_BOL.

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The bolometric flux was then combined with the distance to HR 8799 to estimate its luminosity (L = 4π d²F_BOL), which yielded a value of 5.05 ± 0.29 L_☉. The uncertainty in the luminosity (6%) is predominantly set by the uncertainty in the distance. The F_BOL was also combined with the star's θ_LD to determine its effective temperature by inverting the relation,

$$\begin{equation} F_{\rm BOL} = {1 \over 4} \theta _{\rm LD}^2 \sigma T_{\rm eff}^4, \end{equation} \tag{ 2 }$$

where σ is the Stefan–Boltzmann constant. This produces an effective temperature of 7203 ± 87 K, determined to a precision of 1%. Because μ_LD is chosen using a given T_eff, we used our new T_eff value to select μ_LD and iterated. μ_LD increased by only 0.01 to 0.49 ± 0.02, θ_LD increased by 0.001 mas to 0.342 ± 0.008 mas, and T_eff decreased by 10 K to 7193 K. The very slight change in θ_LD did not affect the radius calculation at all.

We note that this T_eff is nearly identical to the assigned temperature of the best-fit stellar templates used to calculate F_BOL. However, our interferometric T_eff is significantly less model dependent, and it is important to verify the accuracy of the T_eff predicted by the PED. Moreover, our measured T_eff is also consistent with the detailed spectral-type classification by Gray & Kaye (1999) of kA5 hF0 mA5 v λ Boo, following the spectral-type temperature scale for dwarf stars assembled in Kraus & Hillenbrand (2007). The hydrogen lines are a better tracer of the stellar T_eff (hF0; 7200 K); the metal lines suggest a higher T_eff (mA5; 8200 K) only because its atmosphere is slightly metal depleted.

A potential bias in the size measurement of any A-type star is oblateness caused by rapid rotation. Royer et al. (2007) measured a vsin i of 49 km s^-1 and assuming the planets orbit in the same plane as stellar rotation, the actual rotation velocity will increase from 49 to 104 km s^-1 when the 28° inclination is taken into account (Soummer et al. 2011). For an A5 V star with an approximate mass of 2.1 M_☉ (Cox 2000), our measured radius of 1.44 R_☉, and the relation between M, oblateness, and vsin i described in van Belle et al. (2006), the predicted oblateness of HR 8799 is only ∼2% and thus within the 1σ errors in θ_LD. Using the F0 V spectral type, which is a closer fit to our measured T_eff, the estimated mass is 1.6 M_☉ and the predicted oblateness is 2.6%, still a small effect.

HR 8799 is currently the only directly imaged multiple planet system and there may be smaller, more Earth-like planets orbiting the star or even moons orbiting the imaged planets that have not yet been detected. This would present the possibility of life if the planets were in the habitable zone (HZ) of the star, so we used our new precise measurements to improve the estimate of the system's HZ. We used the following equations from Jones et al. (2006):

$$\begin{eqnarray} &&S_{b,i}(T_{\rm eff}) = \big(4.190 \times 10^{-8} \; T_{\rm eff}^2 \big) - (2.139 \times 10^{-4} \; T_{\rm eff}) + 1.296 \nonumber\\ \end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray} S_{b,o}(T_{\rm eff}) &=& \big(6.190 \times 10^{-9} \; T_{\rm eff}^2 \big) - (1.319 \times 10^{-5} \; T_{\rm eff}) \nonumber\\ &&+ 0.2341, \end{eqnarray} \tag{ 4 }$$

where S_{b, i}(T_eff) and S_{b, o}(T_eff) are the critical fluxes at the inner and outer boundaries in units of the solar constant. The inner and outer physical boundaries r_{i, o} in AU were then calculated using

$$\begin{equation} r_i = \sqrt{ \frac{L/L_\odot }{S_{b,i}(T_{\rm eff})} } \; \; \; \; \; {\rm and} \; \; \; \; \; r_o = \sqrt{ \frac{L/L_\odot }{S_{b,o}(T_{\rm eff})} }. \end{equation} \tag{ 5 }$$

These equations assume that the HZ is dependent on distance only and do not take other effects, such as tidal heating, into account. The inner boundary is the limit where a runaway greenhouse effect would evaporate any surface water while the outer boundary is the limit where a cloudless atmosphere could maintain a surface temperature of 273 K. Jones et al. (2006) note that these equations produce a conservatively small HZ and the actual HZ may be wider.

We obtained HZ boundaries of 1.62 AU and 3.32 AU. HR 8799's planets have semimajor axes of 14.5–68 AU (Marois et al. 2008, 2010). There is no chance the planets orbit anywhere near the HZ unless they are highly eccentric, which is a configuration more likely to be unstable.

The accurately determined stellar properties of HR 8799 also allow for detailed comparisons with stellar evolutionary models from which a mass and age can be inferred. However, correctly interpreting masses and especially ages depends critically upon knowing the internal abundances of this peculiar abundance star. As noted in Section 1, the atmospheric abundances of Fe-peak elements are distinctly subsolar ([Fe/H] ∼ − 0.4 dex) while the abundances of C, N, O, and S are essentially solar. Because these surface abundances may not directly trace the internal abundances, it is unclear which, if any, of the available uniformly scaled abundance models to adopt for these comparisons; we subsequently refer to the abundance of the uniformly scaled model that predicts properties most consistent with observational constraints as the effective abundance.

As an example of the effect on the inferred ages, comparisons of HR 8799 with models that assume subsolar abundances throughout yield ages of 1.05 ± 0.26 Gyr for [Fe/H]= −0.27 or 1.71 ± 0.18 Gyr for [Fe/H] = −0.43, while those that assume near-solar abundances ([Fe/H] = +0.05) yield an age of <0.1 Gyr; the specific physical assumptions used in these models are described in Section 4.3. Obviously, the polyabundic atmospheres of λ Boo stars and their uncertain internal abundance compromises the validity of these comparisons and the inferred values. Nevertheless, we present the available observational evidence for constraining the metallicity of λ Boo stars, and HR 8799 in particular, and suggest the effective abundance of HR 8799, and perhaps all λ Boo stars, is near solar.

The observationally favored theory to explain the λ Boo phenomenon, originally proposed by Venn & Lambert (1990), is that the atmospheres have been polluted by the accretion of Fe-peak depleted gas. Depletion is believed to occur because of grain formation; a similar depletion of Fe-like elements has been observed in interstellar clouds (Morton 1974). The grains that themselves accrete Fe-elements are consequently inhibited from accreting on to the star because of the stronger radiation pressure they experience. It is not clear why this accretion onto the grains occurs, or where the accreting material comes from. It may be interstellar, but in many cases appears to be circumstellar. As noted by Gray & Corbally (2002), all four λ Boo stars within 40 pc (including HR 8799) exhibit an infrared excess that is interpreted as the presence of circumstellar dust, while only ∼18% of non-λ Boo A-type stars exhibit such excesses. The apparent depletion of these elements requires only relatively low accretion rates (10⁻¹³ M_☉ yr⁻¹), because of the shallow convective zones in A-type stars (Charbonneau 1991). Accretion at these rates will quickly establish abundance anomalies within a few Myr, but these anomalies will likewise disappear in as little as 1 Myr once the accretion has terminated (Turcotte & Charbonneau 1993); the interesting implication is that all λ Boo stars are either currently accreting, or have only recently terminated their accretion. If this favored theory is correct, it suggests that the depleted Fe-peak abundances do not represent the internal abundances of these stars. Given this and the typical limiting main-sequence lifetime of ∼2 Gyr for these intermediate mass Population I stars,¹¹ one would expect their internal abundance to be on average close to solar.

We note that an alternative mechanism to explain the λ Boo phenomenon proposed by Michaud & Charland (1986) suggests that the depletion of Fe-peak elements is a result of diffusion and mass loss. In this case, λ Boo stars are much closer to the end of their main-sequence lifetimes. To be effective, the star would need to be losing mass at a rate of the order of 10⁻¹³ M_☉ yr⁻¹ for a few times 10⁸ yr and thus implies that the λ Boo phenomenon be restricted to the end of the main-sequence evolutionary phase for A stars. This predicted timescale is difficult to reconcile with the discovery of very young λ Boo stars in the Orion OB1 associations (age <10 Myr), and the lack of any λ Boo stars in intermediate age open clusters (Gray & Corbally 2002). Evidence such as the higher fraction of these stars with circumstellar debris disks, noted above, instead suggest that the λ Boo phenomenon is more common among young A stars. If true, this would strengthen the case that the internal abundances of these Population I stars, at least on average, should be close to solar. In particular, we note that the existence of λ Boo stars in the Orion star-forming region implies that their primordial abundances are solar, consistent with other stars in this cluster (Nieva & Przybilla 2012). If the depletion of Fe-like elements is restricted to only the outer atmospheres, then one can conclude that in these cases the internal abundances should likewise be solar.

As described in Marois et al. (2008), the space motions of HR 8799 are consistent with those of young stars in the solar neighborhood. We investigated this further by comparing the UVW space motions of HR 8799 to those of nearby moving groups assembled in Torres et al. (2008); kinematic association with a moving group would not only help constrain the age of the system but also the primordial abundance of the star. We calculated UVW space motions using HR 8799's HIPPARCOS distance and parallax (van Leeuwen 2007), and a radial velocity of −12.6 ± 1.3 km s⁻¹ from Barbier-Brossat & Figon (2000). This yielded velocities of U = −12.24 ± 0.37 km s⁻¹, V = −21.22 ± 1.10 km s⁻¹, and W = −7.15 ± 0.86 km s⁻¹; these are all within ∼1 km s⁻¹ of the values reported in Marois et al. (2008). Figure 12 illustrates the space motion of HR 8799 relative to that of several kinematically similar moving groups after adopting moving group velocities and uncertainties from Torres et al. (2008). Although the space motion of HR 8799 is not consistent to within 1σ of any of these groups, it is consistent to within 3σ of two of these groups: Columba (1.2σ), and Cha (2.2σ). We thus confirm the report by Marois et al. (2008) of the potentially youthful kinematics of HR 8799, which favors but cannot confirm an age less than ≲ 1 Gyr, and likely a solar abundance consistent with many of these stars.

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As described in Section 1, there are two other aspects of HR 8799 that argue for a young age for the star and system. The strong infrared excess of HR 8799 statistically favors an age less than ∼500 Myr (Gáspár et al. 2009), despite the challenge raised by Moya et al. (2010a). An even stronger youthful age restriction is imposed by the dynamical considerations of Moro-Martín et al. (2010). Although their conclusions require use of the predictions of poorly constrained planetary evolutionary models, even considerable uncertainties in the mass estimates require the age of the system to be less than a few 100 Myr, otherwise the planets would be too massive to be orbitally stable. The ages much less than 1 Gyr for HR 8799 can only be reconciled if the adopted effective abundances are near solar.

Recently, Moya et al. (2010b) used the γ Dor-type pulsations of HR 8799 to asteroseismically constrain the star's internal metallicity. They find a best-fit internal metallicity of −0.32 ± 0.1, assuming that the rotation axis of the star, which affects the rotation speed of the star, is somewhat highly inclined relative to our line of sight (50°). This high inclination is inconsistent with the more face-on orientation of HR 8799's disk (Su et al. 2009) and the apparent orbital plane of its companions (Lafrenière et al. 2009), although these need not be coplanar. The asteroseismology analysis is compromised by the use of an assumed radius that is too small by 8% and a temperature that is too hot by 3% relative to the precisely determined values in this study. The measured R and T_eff presented here will be valuable to constrain the model parameter space in future asteroseismic modeling efforts, which will possibly constrain the internal metallicity.

Moya et al. (2010b) do find possible—though less likely—solutions with internal abundances close to solar ([Fe/H] =−0.12) for closer to pole-on orientations; this is the most metal-rich metallicity reported in their study. Overall, we find the results of this metallicity analysis to be inconclusive primarily because of the large uncertainty in the inclination of the star's rotation axis, which Moya et al. acknowledge is the limiting factor in their analysis. If anything, the result that they find some acceptable solutions with near-solar metallicities and young ages corroborated previous asteroseismology studies of λ Boo stars. Using densities inferred from stellar pulsations, Paunzen et al. (1998) found the location of most λ Boo stars showing δ Scuti pulsations on a plot of average density versus period to be consistent with the positions of δ Scuti stars with solar abundances. They interpret this as evidence that the low Fe-peak abundances are restricted to the surface of λ Boo stars and do not represent the state of the interior composition. Although the variations in HR 8799 are compatible with γ Dor pulsations, the large number of discovered hybrid δ Scu–γ Dor pulsators (e.g., Rowe et al. 2006; Grigahcène et al. 2010) suggests that this argument could be applicable for all λ Boo stars.

In summary, based on the observational evidence supporting the accretion of clean gas theory explaining the abundance pattern of λ Boo stars, the existence of at least some λ Boo stars in an OB Association, the evidence of youth for many λ Boo stars (and especially HR 8799), the restriction of ages less than ∼2 Gyr for Population I stars, and the available asteroseismic constraints on internal metallicity, we conclude that the most appropriate effective abundances for HR 8799, and quite possibly all λ Boo stars, is near solar. Until more sophisticated evolutionary models that account for the polyabundic atmospheres and possibly interiors of these stars are developed, we are hopeful that λ Boo stars in clusters or with lower mass companions will be discovered, which will enable improvements in both assigning effective abundances and tests of the validity of using uniformly scaled abundances.

Following the above arguments, we estimated the mass and age of HR 8799 using the stellar evolutionary models of the Yonsei-Yale group (Y²; Yi et al. 2001), updated by Demarque et al. (2004) to account for convective core overshoot. The models handle convection using mixing length prescription, adopting a mixing length of 1.7432 times the pressure scale height that is set by comparisons with a solar model. The model uses the solar abundances of Grevesse et al. (1996) and OPAL radiative opacities (Rogers & Iglesias 1995; Iglesias & Rogers 1996) for the interior and the Alexander & Ferguson (1994) opacities for the cooler, outer regions of the star. Additional information regarding the input physics assumed in these models can be found in Yi et al. (2001) and Demarque et al. (2004). However, since a set solar metallicity models is not provided, we followed the recommendation of the modelers themselves and generated a set of solar metallicity models by linearly interpolating isochrones and mass tracks between nearest the metallicity models of [Fe/H] = −0.27 dex and [Fe/H] = +0.05 dex (models x74z01 and x71z04 models, respectively). The solar metallicity models are illustrated in Figure 13.

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At HR 8799's temperature, stars are predicted to have fully contracted to their smallest size, or the zero-age main sequence (ZAMS) at 40 Myr, and then begin expanding due to stellar evolutionary effects at a much slower rate afterward. As such, HR 8799 is either contracting onto the ZAMS or expanding from it, and we use the 40 Myr isochrone as the upper limit for the age in the pre-main-sequence scenario and as the lower limit for the age in the post-main-sequence scenario. More specifically, we find that if HR 8799 is contracting onto the ZAMS, it has an age of 33⁺⁷_−13.2 Myr and a mass of 1.516^+0.038_−0.024 M_☉. If HR 8799 is expanding from the ZAMS, we find it to have an age of 90⁺³⁸¹₋₅₀ Myr and a mass of 1.513^+0.023_−0.024 M_☉. These masses and ages are found by interpolating between the solar metallicity models described above. The masses are consistent with the mass used in Marois et al. (2008), which was 1.5 ± 0.3 M_☉. We remind the reader that these quoted errors on the star's mass and age are statistical and therefore do not take into account uncertainties in the metallicity and/or models themselves.

With an age of ≲0.1 Gyr, the companions that HR 8799 harbors are more likely to have planetary masses. As explained in Marois et al. (2008), planetary mass objects could only be as bright as these observed companions are if they are young. That inferred age, and thus companion masses, depends critically upon adopted evolutionary model abundance. While near solar seems likely, even slightly subsolar abundances can give uncertainties in age that extend to above ∼500 Myr, increasing the inferred companion masses by at least a factor of two.