Predicting stellar angular diameters from V, I_C, H and K photometry

Abstract

1 INTRODUCTION

Precise stellar radius measurements are important for many subfields of astronomy, especially for exoplanet characterization. While precise radii are most readily applicable to transiting exoplanet characterization, they also correspond directly to stellar angular diameters. One notable application for such angular diameters is in constraining the physical properties of microlensing events, for example in distinguishing cases of self-lensing from those of Massive Compact Halo Objects lensing (Calchi Novati et al. 2010; Fukui et al. 2015).

Microlensing systems are often far too distant for direct measurements of the stellar angular size, prompting empirical means to determine stellar sizes from photometry alone.

Wesselink (1969) demonstrates a strong empirical correlation between surface brightness and (B − V) colour; a more general correlation between surface brightnesses and colour indices has been shown in Barnes & Evans (1976). Therefore, we expect to be able to construct relations between stellar angular size, colour and an apparent magnitude from the given colour. Barnes & Evans (1976) further demonstrate that surface brightness is independent of stellar luminosity class, which implies that such an angular size–colour–magnitude relation should hold regardless of whether stars have evolved off the main sequence. Di Benedetto (2005) proposed, through a comparison of the empirical relations for both dwarf and giant stars, that there was enough overlap in the then available data to motivate a combined fit across evolutionary stages.

The use of interferometry to measure the angular diameters of stars has played a major role in empirically constraining the radii of nearby stars. Our new relations benefit from recent precise angular diameter measurements of both main-sequence and evolved stars using optical interferometry. They extend the results of Boyajian, van Belle & von Braun (2014) with new data and more precise relations for both main-sequence and evolved stars, constructed for a more limited range of spectral types.

Section 2 describes the criteria for data selection and sources of angular diameters and photometry, and the methodology for fitting the data is presented in Section 3. We analyse the results in Section 4, including a comparison with previous works (Section 4.1).

2 DATA

We compile a list of stars with both V, I_C, H and/or K magnitudes and precise angular diameters in Tables 1 and 2. A total of 57 distinct main-sequence stars are selected among all relations, with effective temperatures of 3927–9553 K (spectral types A1–M0), a mean angular diameter uncertainty of 0.013 mas and apparent V magnitudes of 0.03–7.70. The evolved sample contains 50 stars with effective temperatures of 3972–10 330 K (spectral types A1–M0), a mean angular diameter uncertainty of 0.043 mas and apparent V magnitudes of 1.16–6.18. The following subsections outline both the source information as well as selection and classification criteria.

2.1 Stellar classification

We restrict included stars to an effective temperature range of 3900 < T < 10 500 K, which approximately captures spectral types A–K.

Evolved stars are selected not based on their listed spectral classes, but by a stellar radius cut of 6 < R_⋆/R_⊙ < 100. This is done in an attempt to disambiguate the luminosity classes of stars which might have inconsistent classifications in the boundary between subgiants and giants.

Some stars in our sample are known to be in multiple star systems. The presence of additional stars can introduce an offset in flux and visibility of the target star. We adopt the selection precedent from Boyajian et al. (2008); we exclude any binary systems where a secondary star is both separated from the target by at most 5 arcsec and is within 3 mag of the target in any bands used in the analysis.

2.2 Angular diameters

All stars are required to have limb-darkened angular diameters with mean random errors ≤2 per cent, and must have been observed on at least two separate occasions. The measurements come from a variety of sources, which are detailed in Boyajian et al. (2012b) and Boyajian et al. (2013) and listed for reference in Tables 1 and 2. Stars with inconsistent diameters (here defined as any two sources differing by at least three times the maximum uncertainty of any measurement) were excluded. We take the uncertainty-weighted means of the remaining measurements for our quoted angular diameters. Angular diameter source instruments include the Palomar Testbed Interferometer, the Very Large Telescope Interferometer, the Sydney University Stellar Interferometer, the Narrabri Stellar Intensity Interferometer, the Mark III interferometer, the Navy Prototype Optical Interferometer and especially the CHARA Array.

2.3 Magnitudes

2MASS photometry (Cutri et al. 2003) is saturated for most of the stars in our sample due to brightness. Therefore, we rely on earlier photometric catalogues for reliable magnitudes. All magnitudes used are listed in Tables 1 and 2. For the I magnitudes, we use Cousins I_C photometry converted from Johnson I_J sources (Mallama 2014), as well as magnitudes from Koen et al. (2010) for our reddest stars. We use Gezari et al. (1999) for H magnitudes, querying the catalogue for all magnitude measurements centred at 1.65 μm. Here, errors of 0.05 mag are assumed as a conservative estimate. The K magnitudes are taken from a combination of Neugebauer & Leighton (1969), Kidger & Martín-Luis (2003) and Kimeswenger et al. (2004). Since the filter profiles for the magnitudes in these catalogues differ appreciably, we choose to convert all into the 2MASS system. Neugebauer & Leighton (1969) magnitudes are originally in the California Institute of Technology (CIT) system (Iyengar et al. 1982), and the Kimeswenger et al. (2004) are listed as Deep Near-Infrared Survey (of the Southern sky) (DENIS) K_S magnitudes. Carpenter (2001) provides transformations from both the CIT and DENIS systems into the 2MASS system (with updated transformations available on the 2MASS website). For the Kidger & Martín-Luis (2003) magnitudes, we first transform into an intermediate system, the Koornneef system (Koornneef 1983). Kidger & Martín-Luis (2003) compare their photometry to the system described in (Koornneef 1983) and find a constant offset in magnitude. From this, we convert to 2MASS via the relation in Carpenter (2001). While significant colour dependence exists for the DENIS and Koornneef transformations, the CIT transformation exhibits only a very weak colour dependence. In light of this, and the lack of accompanying J magnitudes for the Neugebauer & Leighton (1969) K magnitudes, we choose to neglect the colour term, incorporating the error from this omission into our final uncertainty propagation.

In order to calculate colour indices, all stars must have at least one available magnitude in any of the I_C, H and K bands. We exclude stars with inconsistent magnitudes, i.e. magnitudes from different sources whose values disagree by at least triple the largest uncertainty of any one value. The listed uncertainties in the resulting colours are propagated from both the uncertainties from conversion as well as assumed 0.02 mag errors in the original Johnson V magnitudes (Mallama 2014).

3 FITTING PROCEDURE

We choose to construct relations for V − I_C, V − H, V − K, I_C − H and I_C − K (Fig. 1). We start with a constant-only fit of log θ_Q=0 and add polynomial terms in colour, following the form of equation (3). The fitting procedure uses a Levenberg–Markquardt least-squares algorithm provided by the MINPACK-1 Least Squares Fitting routine (Markwardt 2009). For each solution, we perform an F-test (Press et al. 1992) to determine whether the improvement to the relation by adding a polynomial term is statistically significant. Once the functional form is obtained, we run a Monte Carlo simulation by generating 10⁴ simulated data sets, randomly choosing colours and diameters drawn from Gaussian probability distributions of each star's true colour and diameter. The means and standard deviations are given by the initial fit coefficients and their associated uncertainties, respectively. This allows us to incorporate all measurement uncertainties into the relation.

Figure 1.

Top panels: Angular diameter–colour relations for both dwarf/subgiants (blue line) and giants (red line), as well as a combined fit (black line). The functional form of the fits is described in equation (3) with coefficients listed in Table 3. The data are introduced in Section 2 and catalogued in Tables 1 and 2. The fitting methodology is described in Section 4. All panels show previous relations from Boyajian et al. (2014). For V − I_C, we include the result for dwarfs from Kervella & Fouqué (2008), and for V − K we include the result for dwarfs from Kervella et al. (2004b). Bottom panels: The residuals in dex are shown with respect to the combined relation.

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4 ANALYSIS

For each colour, we construct independent fits of unevolved and evolved stars (Fig. 1). Table 3 shows the number of stars, range of colours and fit coefficients for each fit. These relations are only valid for the colour ranges for which we have data. We then construct fits for each colour using all stars, unevolved and evolved. The combined fits test that variations in surface gravity with stellar evolution will not affect the relations (as noted in Section 1). The derived relations with the combined sample have similar root mean square (RMS) errors to the separated fits, which is consistent with the result of Barnes & Evans (1976) that surface brightness is independent of luminosity class. The RMS in the residuals ranges from 0.017 to 0.03 dex, corresponding to the minimum expected uncertainties in log θ_LD before uncertainties in magnitudes are considered. The I_C − K relations have the smallest spread for all fits.

To estimate the uncertainties in limb-darkened diameters, we propagate uncertainties for assumed 0.03 errors in both magnitudes of a given colour. The most precise relations for all stars are those for V − H and V − K, which have estimated uncertainties of 1.8–2.9 per cent. (The range is due to the dependence of the uncertainty on the colour, which we have varied within the range of the sample.) The least precise results are in V − I_C, where the corresponding uncertainties could be as high as 6.5 per cent.

Initially, M dwarfs were included in our sample to see if the derived relation would change drastically with their inclusion. The addition of dwarfs at T < 3900 K adds a statistically significant quadratic coefficient to our fits in the V ��� I_C relation. Our temperature cut therefore provides a more precise relation for FGK stars in particular. In contrast, the V − I_C relation in giants has a marginally significant quadratic term, even excluding stars below 3900 K. On the other end of our temperature range, inclusion of the A dwarfs (and one A giant) did not significantly change the fits, and so we are less hesitant to include them here.

We also test whether the angular diameter relations have a statistically significant dependence on stellar metallicity. Metallicity has shown to be a factor in relations of stellar radii to colour indices (Boyajian et al. 2012b), since changes in metallicity tend to affect bluer parts of the stellar spectra due to line blanketing (McNamara & Colton 1969). Such effects would propagate to stellar angular diameter relations, but in all colours considered in this paper the relations were insensitive to metallicity. This is consistent with the findings of Boyajian et al. (2014), who found that metallicity in their angular diameter relations was strongest for the colours with the shortest wavelength bands (B − V and g − r), where the bluer colours would be affected more strongly by line blanketing.

4.1 Comparison with previous works

We directly compare our relations to those of Boyajian et al. (2014), and for V − I_C to the relation in table 3 of Kervella & Fouqué (2008), and the V − K relation to equation (23) of Kervella et al. (2004b), as seen in Fig. 1. All the mentioned relations are valid for dwarfs and subgiants, and it should be noted that all extend through the M spectral type (not shown here). In V − I_C, the largest offset in angular diameter between the data and the Kervella & Fouqué (2008) prediction is 0.08 dex, and for V − K the largest offset with respect to Kervella et al. (2004b) is 0.06 dex. We expect at least a reasonable agreement with the results of Boyajian et al. (2014) and Kervella & Fouqué (2008) (see Fig. 1) by construct, since we include the subset of angular diameters used in these works which meet our uncertainty constraint (≤2 per cent). Nevertheless, our sources differ from these works for I_C (Koen et al. 2010; Mallama 2014), H (Gezari et al. 1999) and K (Neugebauer & Leighton 1969; Kidger & Martín-Luis 2003; Kimeswenger et al. 2004) band photometry, as well as a larger sample within the FGK colour range. Hence, differences in the predicted angular diameters exist, particularly on the blue end of the V − I_C relation, where both Kervella & Fouqué (2008) and Boyajian et al. (2014) use higher-order polynomial fits which underestimate the diameters of the bluest dwarfs, while fitting well for dwarfs well beyond the red end of our colour range.

5 CONCLUSIONS

This work describes new relations linking stellar angular diameter to photometric colours. We use a data set with roughly twice the precision in angular diameter measurements compared to previous papers. We use empirical evidence that predictions of angular diameters from colour and magnitude are insensitive to luminosity class to construct, for the first time, a prediction of angular diameters at fixed magnitude for A–K stars across the stages of stellar evolution. We find that there is no dependence on stellar metallicity for the colours tested.

Further improvement to the relations will require additional angular diameter measurements to fill in parameter space for the earlier-type giants. Additionally, the lack of demonstrably consistent I_C photometry for M stars of any luminosity class limits us from extending the red end of our relations. Transformations among systems are susceptible to differences in zero-points, susceptibility of filters to red leaks and correlated errors in the filter profiles (Mann & von Braun 2015). Nevertheless, for FGK stars the continuity in the relation between colour and angular size is well constrained by the regions of overlap of the spectral classes.

Acknowledgements

This work was supported by a graduate fellowship from the Gruber Foundation. Additional gratitude to Anthony Mallama for providing converted Cousins I magnitudes for the dwarfs in our sample.

This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.

REFERENCES