DIRECTLY DETERMINED LINEAR RADII AND EFFECTIVE TEMPERATURES OF EXOPLANET HOST STARS

The calibration of the target star visibility (V²) data is performed by estimating the interferometer system visibility (V²_SYS) using the calibration sources with model angular diameters and then normalizing the raw target star visibility by V²_SYS to estimate the V² measured by an ideal interferometer at that epoch (Mozurkewich et al. 1991; Boden et al. 1998; van Belle & van Belle 2005). Uncertainties in the system visibility and the calibrated target visibility are inferred from internal scatter among the data in an observation using standard error-propagation calculations (Boden et al. 1999). Calibrating our pointlike calibration objects against each other produced no evidence of systematics, with all objects delivering reduced V² = 1.

Visibility and uniform disk (UD) angular size (θ_UD) are related using the first Bessel function (J₁): V² = [2J₁(x)/x]², where the spatial frequency x = πBθ_UDλ⁻¹. We may establish UD angular sizes for the target stars observed by the interferometer since the accompanying parameters (projected telescope-to-telescope separation, or baseline, B and wavelength of observation λ) are well characterized during the observation. The UD angular size can (and should) be connected to a more physical limb darkened (LD) angular size (θ_LD); however, this is a minor effect since θ_LD/θ_UD is small in the near-infrared (<1.5%; see, for example, Scholz & Takeda 1987; Tuthill 1994; Dyck et al. 1996, 1998; Davis et al. 2000).

Strictly speaking, LD angular size is utilized here as a reasonable proxy for the Rosseland angular diameter, which corresponds to the surface where the Rosseland mean optical depth equals unity, as advocated by Scholz & Takeda (1987) as the most appropriate surface for computing an effective temperature. The dense, compact atmospheres of the stars considered in this investigation are well characterized by an UD fit, and the small correction factors tabulated in Davis et al. (2000) will be used to convert our θ_UD sizes into the appropriate LD θ_LD numbers. The number of visibility points N(V²), derived θ_UD sizes, associated goodness-of-fit χ²_ν and residuals (δV²), Davis et al. (2000) correction factors θ_LD/θ_UD, and the resultant θ_LD sizes are found in the first column of Table 2.

PTI is an 85–110 m H- and K band 1.6 μm and 2.2 μm interferometer located at Palomar Observatory in San Diego County, California, and is described in detail in Colavita (1999). It has three 40 cm apertures used in pairwise combination for detection of stellar fringe visibility on sources that range in angular sizes up to 5.0 milliarcseconds (mas), being able to resolve individual sources with angular diameter (θ) greater than 0.60 mas in size. PTI has been in nightly operation since 1997, with minimum downtime throughout the intervening years. The data from PTI considered herein cover the range from the beginning of 1998 (when the standardized data collection and pipeline reduction went into place) until the beginning of 2008 (when the analysis of this manuscript was begun). In addition to the target stars discussed herein, appropriate calibration sources were observed as well and can be found en masse in van Belle et al. (2008). Additional calibration sources of minimal angular size, as discussed in Section 4.3, were also selected and are listed in Table 1.

As discussed by Boden et al. (1998, 1999), PTI has an empirically established fundamental limiting visibility measurement error of $\sigma _{V^2_{{\rm SYS}}}\approx 1.5\%$. The source of this limiting night-to-night measurement error is most likely to be a combination of effects: uncharacterized atmospheric seeing (in particular, scintillation), detector noise, and other instrumental effects.

This night-to-night repeatability limit restricts the ultimate resolution of the instrument. This is at odds with the desire to measure stellar diameters which, for a given brightness, are quite small in an angular sense relative to PTI's resolution. Main-sequence stars are squarely in this regime for PTI, with only a few examples—those considered in this investigation—that creep out of the nether regions of pointlike obscurity into the realm of resolvability. Attempting to resolve stars at the edge of PTI's performance envelope requires careful consideration of the demonstrated limits of the instrument, using the techniques described in van Belle & van Belle (2005, henceforth Paper VB2).

For PTI, operating at the K band with its 109 m N-S baseline, a target of 0.60 mas in size should have a normalized visibility of V² = 94.89% (as introduced in Section 4.1). As discussed in VB2, there is a strong motivation toward using calibration sources that are as pointlike as possible—generally speaking, one wishes to have calibration sources that are significantly smaller than the targets being observed. For this investigation, to reach the regime of 0.60 mas targets, we restricted our use of calibrators to those that are, on average, 0.35 mas or less in size. These two size limits are selected to have sufficient numbers of sufficiently bright targets and calibrators, respectively.

For such calibrators, observed by PTI, the visibility calibration limit is $\sigma _{V^2}=0.186\%$ (from VB2, Equation (7)), which contributes an angular size error due to calibration of roughly 0.012 mas. The night-to-night limiting V² measurement error of $\sigma _{V^2_{{\rm SYS}}}\approx 1.5\%$, however, contributes an angular size error of 0.086 mas. This is significant in that the measurement error dominates any possible calibration bias, which is particularly important when considering smaller targets. If we were instead to have selected calibrators closer to ∼0.70 mas in size—more typical of PTI investigations that observe larger targets that are >1 mas in size—then the calibration angular size error be ∼0.045 mas, and would start to compete with the measurement error in dominating the error budget. This would put our results at a substantial risk of directly reporting any measurement bias inherent in the process we used to estimate the angular sizes of our calibration sources. Since our goal is direct measurement of the target angular sizes, we have taken great care to ensure that this is not the case.

A second aspect of this consideration of PTI-limiting performance is the reported angular sizes of our target stars. For stars that, after calibration, report formal errors that are sufficiently small to be in violation of PTI's known night-to-night repeatability, we increased their reported angular size errors to the level consistent with that repeatability. As a function of target angular size, we show the limits of angular size accuracy possible with PTI's repeatability limit in Table 4. The first column shows various target angular sizes, followed by the corresponding visibilities. A calibrator of 0.35 mas, as noted above, contributes a limit on knowledge of visibility of $\sigma _{V^2}=0.186\%$; the associated limit in angular size knowledge is then listed in column 3. The next two columns list the night-to-night repeatability limit of V², and the associated angular size error. The final column combines the calibration limit and the night-to-night limit in quadrature.

Table 4. Calibration Floor by Target Angular Size as Discussed in Section 4.3.

Target θ (mas)	Target V2	Calibration $\sigma _{V^2}$	Calibration σθ (mas)	Night-to-Night $\sigma _{V^2}$	Night-to-Night σθ (mas)	σθ Floor (mas)
0.600	0.94893	0.00186	0.012	0.01500	0.085	0.086
0.650	0.94028	0.00186	0.010	0.01500	0.079	0.080
0.700	0.93103	0.00186	0.010	0.01500	0.075	0.076
0.750	0.92116	0.00186	0.010	0.01500	0.071	0.072
0.800	0.91072	0.00186	0.009	0.01500	0.068	0.069
0.850	0.89971	0.00186	0.008	0.01500	0.064	0.065
0.900	0.88815	0.00186	0.008	0.01500	0.062	0.063
0.950	0.87607	0.00186	0.008	0.01500	0.060	0.060
1.000	0.86348	0.00186	0.008	0.01500	0.058	0.058

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Additional angular diameters of EHSs were obtained with the Georgia State University CHARA Array (Baines et al. 2008a) with an intent of detecting possible face-on binarity masquerading as planetary companionship (Baines et al. 2008b). The CHARA Array is a optical/near-infrared interferometer similar to PTI (ten Brummelaar et al. 2005), but with longer baselines (up to 330 m), allowing for resolution of smaller objects. For inclusion of the appropriate CHARA data into our data set, we will apply observation criteria similar to the PTI data: first, the calibration sources must be sufficiently unresolved, which we set for CHARA to be 0.50 mas or less. Second, the ratio of angular sizes of science targets and their calibrators must be greater than 1.5. In applying these two criteria, we are confident that the resulting measured angular sizes are sufficiently independent of the calibrator angular sizes predicted by SED fitting.

The resulting data set for inclusion in this analysis consists of seven EHS angular sizes from the CHARA investigation, of which four stars are common to both the PTI and CHARA samples (as noted in Section 2). The ratios of the CHARA to PTI UD angular sizes for those four stars (HD3651, HD75732, HD143761, HD217014) are 1.15 ± 0.14, 1.05 ± 0.10, 0.99 ± 0.13, 1.08 ± 0.13, respectively, with an overall weighted average ratio of 1.06 ± 0.06, indicating possibly a slight tendency for the PTI sizes to be too small (or the CHARA sizes to be too large), but this is a weak 1σ result.

As a further check on the consistency of the CHARA results and our techniques, we modeled the predicted SED sizes of the calibrators found in Baines et al. (2008a). These results are seen in Table 5; on average, our calibrator predictions are within 0.5σ, and no individual results are more than 1.9σ away from Baines et al. (2008a). Overall, we find that the CHARA and PTI results are excellent agreement with each other, despite independently developed methodologies.

Stellar effective temperature, T_EFF, is defined in terms of the star's luminosity and radius by L = 4πσR²T⁴_EFF. As noted in Section 1, rewriting this equation in terms of angular diameter (θ_LD) and bolometric flux (F_BOL), T_EFF can be expressed as T_EFF = 2341 × (F_BOL/θ²_LD)^1/4, where F_BOL is in 10⁻⁸ erg cm⁻² s⁻¹ and θ_LD is in mas (van Belle et al. 1999). The derived temperature values for the resolved stars of this study are found in Table 6, along with the (V − K)₀ color. These temperatures are plotted versus (V − K)₀ in Figure 1, and to explore any potential difference between the EHSA stars and the control sample, a fit of the T_EFF versus (V − K)₀ trend is performed.

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For the control sample, the initial fit reveals HD87901 as a significant outlier. This is most likely due to two factors: (1) HD87901 is the bluest and hottest star, at (V − K)₀ = −0.352 and T_EFF = 14231 ± 314 K, and (2) HD87901 is a rapid rotator with vsin i = 300 km s⁻¹ (Abt et al. 2002), and will show departures from sphericity that induce gravity darkening which render individual T_EFF determinations meaningless (Aufdenberg et al. 2006). Omitting HD87901 from the fit, the best fit for the control sample stars is

$$\begin{eqnarray} T_{\rm EFF}&=&(2832 \pm 239) + (6511 \pm 225)\nonumber\\ &&\times 10^{(-0.2204 \pm 0.0255) \times (V-K)_0} \end{eqnarray} \tag{ 1 }$$

with χ²_ν = 1.72, with the fitting and error ellipses following the techniques described in Press et al. (1992). (Inclusion of HD87901 in this fit returns χ²_ν = 4.98.)

If we include the EHSA stars in the fit, we find the CHARA data point for 55 Cnc (HD75732) a significant outlier as well, which we will discuss further in Section 5.4.1. Omitting 55 Cnc from the unified fit, we find a single fit gives

$$\begin{eqnarray} T_{\rm EFF}&=&(2974 \pm 199) + (6368 \pm 208)\nonumber\\ &&\times 10^{(-0.2362 \pm 0.0227) \times (V-K)_0} \end{eqnarray} \tag{ 2 }$$

with χ²_ν = 1.82. This fit line is plotted in Figure 1. These fits indicate there is no statistically significant difference between the two populations (noting that the EHSA fit is poorly constrained with a small number of data points over a small range of (V − K)₀, preventing a fit to those data alone). We revisit the question of population similarity in further detail in Section 5.3.

For the fit in Equation (2), the median value of the differences between the T_EFF values predicted by this fit and the measured T_EFF values is $\overline{\Delta T}_{\it {(V-K)}_0} = 138$ K. Since the median value of the errors in the individual T_EFF measurements is $\overline{\sigma _T}= 164$ K, we believe the limit of precision in the line fit is not due to any intrinsic astrophysical scatter in the T_EFF versus (V − K)₀ relationship, but rather the limits of the current measurements.

Alternatively, a fit may be made for a cubic relationship between T_EFF and (V − K)₀, (see, for example, the corresponding equation in Levesque et al. 2005) but this produces no significant improvement:

$$\begin{eqnarray} T_{\rm EFF} &=& (9455 \pm 313) + (-3590 \pm 483) \times (V-K)_0\nonumber\\ &&+ (891 \pm 222) \times (V-K)_0^2 + (-89 \pm 33) \times (V-K)_0^3\nonumber\\ \end{eqnarray} \tag{ 3 }$$

with only χ²_ν = 1.68, in spite of the extra degree of freedom.

For those spectral types for which we have more than one stellar angular size measurement, we can compare the resultant weighted mean T_EFF values to the "canonical" values cited in Cox (2000), which can be traced back to the investigation by de Jager & Nieuwenhuijzen (1987). This comparison is seen in Table 7. It is interesting to note that our values of T_EFF all track increasingly lower between types F8V to G2V in comparison with the de Jager & Nieuwenhuijzen (1987) values, before returning to agreement with those values at G5V and cooler.

Finally, given the large number of individual samples of our data set between types F6V and G5V, we present an empirical calibration of T_EFF versus spectral type for this full range, also in Table 7. Spectral types that have no measurements (e.g., F7V) have T_EFF values interpolated from the adjoining spectral types. The average error by spectral type is $\overline{\Delta T}_{\rm SpType}= 105$ K. This table and Equation (2) represent a direct calibration of the T_EFF scale for solarlike main-sequence stars for the spectral-type range F6V–G5V and color range (V − K)₀ = 0.0–4.0. No attempt was made for T_EFF calibration for the later types due to the sparseness of the data, although our data at K1V, K7V, and M2V represent T_EFF calibration for those specific spectral types.

From the parallax values found in Table 2 from Hipparcos (Perryman et al. 1997), linear radii are derived for the resolved stars of this investigation and are found in Table 6. A cubic relationship fit to the combined EHSA and control samples is

$$\begin{eqnarray} R &=& (2.263 \pm 0.026) + (-1.261 \pm 0.016) \times (V-K)_0\nonumber\\ &&+ (0.347 \pm 0.011) \times (V-K)_0^2 + (-0.036 \pm 0.010)\nonumber\\ &&\times (V-K)_0^3 \end{eqnarray} \tag{ 4 }$$

with a χ²_ν = 15.1. Clearly this metric indicates a poor fit, which is consistent with some of the stars beginning to evolve well off of the zero-age-main-sequence (ZAMS) line. This effect is seen in a plot of the data in Figure 2, with the presumably older stars being situated to the right of the line fit. As such, Equation (4) should be regarded as only a rough indication of stellar radius, and not applicable in any general sense to determining linear radii of random field stars.

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As detailed in Press et al. (1992), the Kolmogorov–Smirnov (KS) test can be executed to compare two arrays of data values, and examine the probability that the two arrays are drawn from the same distribution. The KS test returns two values: the KS statistic D, which specifies the maximum deviation between the cumulative distribution of the two samples of data, and probability p, giving the significance of the KS statistic. Small values of p(<0.20) show that the two distributions differ significantly.

Examining the T_EFF versus (V − K)₀ data of the EHSA stars versus the control sample stars, we find that D = 0.25 with p = 0.54—strong indication that two data sets are indeed statistically indistinguishable. The astrophysical implication is that, within the limits of our measurements, the effective temperature scale of stars with known planets does not differ from those without known planets.

The corresponding R versus (V − K)₀ KS test, however, reports D = 0.50 and p = 0.01, which seems to indicate the two samples are inconsistent with each other. However, the significance of this result is simple: our control sample is specifically selected to be main-sequence stars, whereas the EHSA sample includes a number of evolved sources, as clearly seen in Figure 2. One corollary implication of these two KS tests is that stars on main sequence and those evolving off of it do not differ significantly in their T_EFF versus (V − K)₀ relationships.

There is a variety of data available for the known EHSs in the literature, derived from different methods by different authors. Thus, discrepancies, though sometimes small, exist. In order to be as consistent as possible, we chose the following two catalogs as data sources for astrophysical parameters: (1) mass, age, T_EFF, and [Fe/H] from Valenti & Fischer (2005). and (2) linear radius from Takeda et al. (2007).

A comparison of the T_EFF values measured in this investigation can be directly contrasted against those found in Valenti & Fischer (2005). Combining our EHSA and control star samples, we find

$$\begin{equation} T_{\rm EHS} = (-123 \pm 693) + (1.023 \pm 0.122) \times T_{\rm FV05} \end{equation} \tag{ 5 }$$

with χ²_ν = 1.66. As illustrated in Figure 3, there is no significant difference between the T_EFF values obtained with interferometry and spectroscopy.

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A marginal offset is found between our R and the radii of Takeda et al. (2007):

$$\begin{equation} R_{\rm EHS} = (0.071 \pm 0.047) + (0.930 \pm 0.059) \times R_{\rm T+07} \end{equation} \tag{ 6 }$$

with χ²_ν = 1.87—roughly a 2σ offset between the line slope and intercept values for R from theory versus those determined interferometrically. The general trend is for the larger (R > 1.2 R_☉) stars to have a larger theoretical, rather than interferometric, linear size. These values and the general trend can be seen in Figure 4.

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5.4.1. Discussion of 55 Cnc (HD 75732)