STELLAR DIAMETERS AND TEMPERATURES. II. MAIN-SEQUENCE K- AND M-STARS

All of the observed stars are bright enough to have precise trigonometric parallaxes measured by Hipparcos (van Leeuwen 2007). These values are converted to distances d, which range from 1.82 to 9.75 pc, and are known with an average precision of 0.5%. The physical stellar radii R are trivially calculated via

$$\begin{equation} \theta _{\rm LD} =\frac{2 R}{d}. \end{equation} \tag{ 1 }$$

We produce stellar energy distribution (SED) fits for all targets using flux-calibrated photometry published in the literature (see Table 5). We fit the photometry to spectral templates in the Pickles (1998) library, providing us with a robust and empirical approach to measuring the bolometric flux F_BOL (average errors of ∼1.3%). General details of this method are given in van Belle et al. (2008), although contrary to the approach in van Belle et al. (2008) where interstellar reddening is a free variable in the SED fitting, this work sets the value of the interstellar extinction identical to zero since all our targets are nearby. This ensures consistency across all targets in our survey. However, we note that this approach does not attempt to take into account any potential circumstellar reddening that may be present in individual systems. Thus, it is possible that previously published stellar parameters for stars may be slightly different (within 1σ), even though they are based largely on the same data (e.g., GJ 581; von Braun et al. 2011). We were not able to measure a bolometric flux for visual binary GJ 702A and GJ 702B because the photometry for these two objects is blended due to their close separation. Thus for these two stars, we use the luminosity values computed from Eggenberger et al. (2008) using bolometric corrections.

With knowledge of θ_LD and F_BOL, we calculate stellar effective temperature T_EFF via the rewritten Stefan–Boltzmann Law

$$\begin{equation} T_{\rm EFF} ({\rm K}) = 2341 \big(F_{\rm BOL}/\theta _{\rm LD}^2\big)^{{1}/{4}}, \end{equation} \tag{ 2 }$$

where F_BOL is in units of 10⁻⁸ erg cm⁻² s⁻¹ and θ_LD is in units of mas. The total luminosity given by L = 4πd²F_BOL is also derived for the stars in the sample. We present these results and associated errors in Table 6.

One of the main contributions of this paper is to provide empirical relations linking observables, namely, broadband photometry and metallicity, to the measured physical properties of late-type stars. Due to the fact that these objects are very bright in the infrared, we are unable to get reliable photometry from the 2MASS Catalog (Cutri et al. 2003a). Alternatively, we list in Table 7 the collection of broadband Johnson BVRI JHK photometry for all the program stars, along with the reference for each measurement.

Metallicities for M-dwarfs are difficult to calibrate accurately due to their complex spectral features and molecular bands contaminating the continuum. [Fe/H] can be used for diagnostic purposes to understand the effects of a star's metallicity on relations between other stellar parameters. This is a crucial point of interest when comparing the stars to models, in which opacities in the atmospheres of M-dwarfs may not have properly been accounted for. The metallicity values for the stars in this paper were preferentially collected from a uniform source—so that we could minimize any systematic offsets with respect to different methods/references in the metallicity scales. Unfortunately, a single catalog of stellar metallicities does not contain all of the stars in this survey.

For this study, we cite the metallicities of M-dwarfs from the calibration presented in Rojas-Ayala et al. (2012), wherever available. Alternatively, we quote the metallicity values from Neves et al. (2011), e.g., for GJ 15A and GJ 880. We find that all of the K-dwarfs have entries in Anderson & Francis (2011), who provide average metallicity values of several available literature references. While the Anderson & Francis (2011) metallicities are an average—and thus not of uniform origin—we cross check these averaged metallicities with those in the SPOCS catalog (Valenti & Fischer 2005), where entries are available for the majority of our stars, and we find that the values are within 0.04–0.05 dex each other, revealing no systematic offset. The [Fe/H] values used in this study and references can be found in Table 6.

In order to compute stellar masses for the stars in the sample, we employ an absolute K-band mass–luminosity relation (MLR). While the relation from Delfosse et al. (2000) is commonly used for M-type stars, it does not extend to be valid for K-type stars. Due to this, we chose to use the relation from Henry & McCarthy (1993) because it is good throughout the whole mass range of the sample presented in this paper. We note that Figure 1 in Delfosse et al. (2000) illustrates consistency in masses derived via the piece-wise solution derived in Henry & McCarthy (1993) and the polynomial form of their solution in Delfosse et al. (2000). We apply the conversion from Leggett (1992) to transform the Johnson K magnitudes in Table 7 into the CIT system (Elias et al. 1982) so the relations in Henry & McCarthy (1993) could be applied. These corrections are on the order of 0.02 mag. The resulting estimates of the stellar mass are included in Table 6 and assume a 10% uncertainty.

Lastly, we use Equation (1) in López-Morales (2007) to convert hardness ratios from the ROSAT All-Sky Survey Bright Source Catalogue (Voges et al. 1999) to X-ray flux for each object and compute the X-ray to bolometric luminosity ratio L_X/L_BOL, a useful diagnostic to the activity levels in late-type stars. These values are presented in the last column of Table 6.

At present, there are 10 published papers that present interferometrically determined radii of K- and M-dwarf stars with precision to better than 5%: Lane et al. (2001), Ségransan et al. (2003), Berger et al. (2006), di Folco et al. (2007), Kervella & Fouqué (2008), Boyajian et al. (2008), van Belle & von Braun (2009), Demory et al. (2009), and von Braun et al. (2011, 2012). These results within these papers amount to 22 measurements of 17 unique sources and are compiled in Table 6, which also contains the values and references for the stellar metallicities (Section 3.2).

We perform an SED fit for each of these stars to determine the bolometric flux, luminosity, and effective temperature in the same manner as we did for our targets in Section 3.1. The stellar mass is again derived from the K-band MLR in Henry & McCarthy (1993). Lastly, we also list the derived values of the derived bolometric to X-ray luminosity ratio L_X/L_BOL.

Table 6 lists each individual measurement in our survey, as well as in the literature in the top two partitions. We also mark in Table 6 stars that have multiple measurements with a † or a ^††. The † marked stars include the stars discussed in Section 2 that we used for consistency checks across different instruments and interferometers: GJ 15A, GJ 166A, GJ 380, GJ 411, and GJ 699. Note that both GJ 380 and GJ 411 have two measurements in the literature. The ^†† marked stars indicate stars that have multiple literature measurements, but are not included in our survey: GJ 820A, GJ 820B, and GJ 887. We determine the weighted mean of these stars' radii and temperatures, and list them in the third portion of Table 6. The other properties of these stars, i.e., the spectral type, metallicity, bolometric flux, luminosity, mass, and L_X/L_BOL, are not affected by the combination of multiple measurements, and we do not reprint them in this section of Table 6 for clarity.

In summary, there are 17 late-type dwarfs in the literature with high-precision (better than 5%), direct measurements their radii, of which 10 are K-dwarfs and 7 are M-dwarfs. This publication adds equivalent (or higher precision) measurements of 16 additional late type stars (7 K-dwarfs and 9 M-dwarfs), plus 5 additional measurements of the 17 dwarfs in the literature for the purpose of consistency checks across facilities and techniques. The total number of directly measured radii of late-type dwarfs with precision better than 5% thus stands at 33: 17 K-dwarfs and 16 M-dwarfs.

In this section, we derive empirical color–metallicity–temperature relations with the combined data set (Table 6), using the indices of the Johnson photometric system: (B − V), (V − R), (V − I), (V − J), (V − H), and (V − K) (Table 7). In order to properly model the relations at the blue end of our data range, we add an additional seven G-type stars from our Paper I (Boyajian et al. 2012), whose properties were determined in the same manner as this work. We provide a table of the G-stars from Paper I in Table 8. While other G-type star angular diameters are available in Paper I, we find that only the seven listed in Table 8 have a complete photometric collection of all BVRI JHK magnitudes. So in order to ensure consistency of the improvement that the inclusion of additional measurements of G-stars will have on the color–metallicity–temperature relations we derive, we selected only these seven G-stars that will contribute equally in the relations for all color indices applied in this work.

The solution for temperature as a function of color and metallicity is in the form

$$\begin{eqnarray} T_{\rm EFF} ({\rm K}) &= & a_0 + a_1 X + a_2 X^2 + a_3 X Y + a_4 Y + a_5 Y^2, \end{eqnarray} \tag{ 3 }$$

where variable X is the color index and Y is the stellar metallicity [Fe/H]. The coefficients a₀–a₅ and a statistical overview containing the number of data points used in the fit (n), the approximate color range over which the fit is valid, and the median absolute deviation about the best fit for each of the relations are presented in Table 9.¹⁵ The data and solutions are plotted in Figure 6. We color code the data plotted in these figures to reflect the metallicity of the star: blue indicating metal poor and red metal rich (an [Fe/H] range of −0.68 to +0.35). We also show the solution to the polynomial fit as colored solid lines illustrating a range of solutions for fixed metallicity values.

Zoom In Zoom Out Reset image size

All solutions give a median absolute deviation in temperature between 43 K and 70 K. The tightest of these relations are the V band to infrared color indices ((V − H), and (V − K)), perhaps because they provide the most sensitivity as to where the peak of the SED is located, and hence a better determination of the effective temperature. However, we find that the (V − J) color gives the largest scatter. We are uncertain about the source of the deficiency in this bandpass index, but speculate that the systematics seen for 1.5 < (V − J) < 2.0, where all the points fall beneath the curve, and 1.2 < (V − J) < 1.5, where all the points fall above the curve, is indicative of the fact that a higher order function might be needed to properly model the rapidly changing slope of the data.

In Figure 6, we show our solution as iso-metallicity lines for fixed values of metallicity in 0.25 dex increments. It can be seen that each of these iso-metallicity curves converge for the earlier-type stars. We find that adding metallicity as an extra parameter in the color–metallicity–temperature relations only improves the median absolute deviation of the data to the fit by ∼30%, with the most prominent effect between 3500 and 4000 K, where the solution for [Fe/H] = −0.5 is about 100 K cooler than a solar abundance of [Fe/H] = 0. This is true even for the color indices with long baselines (i.e., (V − J), (V − H), and (V − K)), which we find to be just as sensitive to metallicity as the shorter baseline color indices examined here. This could be caused by the redder apex of the SED in these types of stars and the formation of molecular features in the atmosphere that contribute to larger flux variations in the IR band fluxes. Although this is a weak detection to the influence of the metallicity to color and temperature, it can perhaps be refined with a better sampled range of metallicities, especially for high and low values of metallicity at the cool endpoint of the relations.

We include the (B − V) color–metallicity–temperature relation but caution against using this relation for stars with peculiar abundances since (B − V) color is known to be strongly affected by stellar metallicity. In fact, Figure 6 shows one of the most extreme outliers toward the bluest of (B − V) colors and several sigma under the curve is the metal-poor star GJ 53A (μ Cas A). We note that GJ 53A does not fall on the (B − V)–metallicity–temperature relation, but it is not an outlier in any of the other color–metallicity–temperature relations (see Figure 6). We chose to include it in our color–metallicity–temperature analysis to be complete, and we note that it does not change our results when removing it and re-fitting the data. In fact, all solutions except for the (B − V) color–metallicity–temperature relation are shown to converge at temperatures hotter than ∼4000 K.

In Table 9, we show that the median absolute deviation of the fits using the extended GKM sample is within 5 K of the KM sample. This is shown graphically in Figure 6, where we plot the G-dwarfs used to constrain the fit on the hotter side, along with the K- and the M-dwarf sample, showing that there is a smooth transition of temperature extending to the bluer colors. Thus, the extrapolation of the solutions on the hotter side is valid through spectral type G0.

On the other hand, extrapolation to stars redder than the ranges specified in Table 9 is not recommended. Figure 6 shows that past this specified range, there is a large gap in color index before the reddest endpoint, the star GJ 551, is encountered. So, although we use the data for GJ 551 in the fitting procedure, the paucity of data in this range makes the results unreliable for such red colors, as for the relations presented here, the position of GJ 551 deviates several hundred Kelvin from the modeled curve.

In Figure 6, we show the (V − K)–temperature solution for dwarfs (expressed as a power function) derived from interferometric measurements in van Belle & von Braun (2009), displaying excellent agreement with our fit. Also for comparison, we show the model-based temperature scale for the solar metallicity (B − V), (V − J), (V − H), and (V − K) color–temperature solution from Lejeune et al. (1998) in Figure 6. While the temperatures from Lejeune et al. (1998) for the earlier type stars show excellent agreement with our own, the (V − J), (V − H), and (V − K) solution diverges at temperatures <4500 K (approximately the K5 spectral type) to produce temperatures ∼200 K cooler than the ones that we measure with interferometry. For the (B − V) colors, the Lejeune et al. (1998) scale diverges to predict higher temperatures from 5000 < T_EFF < 4000 K. Below this temperature, the data are modeled more satisfactorily with the Lejeune et al. (1998) scale, although with a slope much different than our own. The difference in slope is no surprise, for the low-order polynomial we use in this work is not capable of modeling any rapidly changing slope and/or kinks in the data. Lastly, we show the BT-Settl PHOENIX model atmosphere color–temperature curves (Allard et al. 2012) for solar metallicity in the (B − V), (V − J), and (V − K) plots, expressing by far the best agreement with the temperatures we derive here for all ranges of temperatures. It is only in the (B − V)–temperature solution that for stellar effective temperatures less than 4000 K, the BT-Settl model predicts a steep drop in temperature at colors too red by about (B − V) ∼ 0.2 mag. This is due to troubles in determining the B magnitudes from the synthetic PHOENIX model grid (see the discussion in Dotter et al. 2008).

Specific references citing temperatures of these stars in Table 6 are plentiful. For instance, for any given target in the sample, there are, on average, tens of references to the effective temperature in the literature, estimated using a number of different techniques. As such, a quantitative comparison of our temperature values to all temperature measurements in the literature is beyond the scope of this paper. Instead, for each star we prepare an overall qualitative assessment of how our temperatures compare to previously obtained ones by examining the full range of temperature estimates listed in Vizier.¹⁶ Figure 7 shows the results of this exercise, where the x-errors indicate our measurement precision and the y-errors indicate the range in temperatures found for each object. Note that a search in Vizier may not gather all references to the temperature in the literature, but rather a collection of available online material. In fact, a search in Vizier resulted in no references for temperature estimates of GJ 702B, and only one for GJ 551. We thus performed a more thorough literature search to gather a range of temperature estimates for these stars to make this comparison (the additional references include Eggenberger et al. 2008 and Luck & Heiter 2005 for GJ 702B, and Morales et al. 2008, Demory et al. 2009, and Wright et al. 2011 for GJ 551).

Zoom In Zoom Out Reset image size

Similar to Section 4.1, we use the data in Table 6 to derive empirical color–metallicity–radius relations in the form of Equation (4), where X is the color index, and Y is the stellar metallicity [Fe/H]:

$$\begin{eqnarray} R ({\rm R_{\odot }}) &= &a_0 + a_1 X + a_2 X^2 + a_3 X Y + a_4 Y + a_5 Y^2. \end{eqnarray} \tag{ 4 }$$

However, as opposed to the temperature relations, we refrain from including the G-type type stars in Paper I to constrain the bluer end of the relations, as their radius may be influenced by stellar evolution (we assume that a star with radius less than 0.8 R_☉ has not had time to evolve off the main sequence). The inclusion of GJ 551 is also omitted from this analysis, for the adopted form of the equation we use does not allow for extrapolation of the curve to such red colors.

The coefficients and statistical overview of the solutions are in Table 10. We find that, when adding metallicity as an extra parameter adds significant improvement to the significance of our fits, cutting the median absolute deviation of the observed versus calculated radius in half. Figure 8 displays the data and our fit using the same color scheme as identified in the temperature relations to illustrate the metallicity. The data show a large spread in radii for a given color index toward the coolest stars (e.g., for (B − V) ∼1.5 and (V − K) ∼4, the radii range from 0.4 to 0.6 R_☉). We discuss this effect in more detail in Section 5.4.

Zoom In Zoom Out Reset image size

The ranges of color indices where the relations are valid are listed in Table 10. The ranges listed are hard limits, and do not allow for extrapolation. There are several reasons for these strict boundaries. On the blue end, we are not able to observationally constrain the form of the upper part of this curve due to stellar evolution. This causes the unphysical behavior of the curve to (1) fold over as a parabola and (2) display cross-overs among fiducials of different metallicities (see Figure 8). In actuality, a zero-age main-sequence radius curve would have an inflection point near this upper boundary we impose on the ranges in Table 10. Past this point, the stellar radius will continue to rise with decreasing color index (but unfortunately also changes with increasing age). The use of the color–metallicity–radius relation (Equation (4)) on the blue end, despite the cross-over, will still lead to results within the stated errors, so long as the object in question is non-evolved. The red ends of the relations are strictly hard limits as well, where, by visual inspection of Figure 8, extrapolation would result in null radius estimates approaching the late M-type star regime.

We show 5 Gyr Dartmouth isochrones (Dotter et al. 2008) based on the UBV(RI)_CJHK_S synthetic color transformations (Bessell 1990; Cutri et al. 2003a) within the panels displaying the (B − V), (V − J), (V − H), and (V − K) data and relations in Figure 8.¹⁷ These isochrones are for two metallicities, [Fe/H] = 0 and −0.5, roughly encompassing the metallicity range of the data included in this paper. The (B − V)–radius plot also shows the Dartmouth isochrones under the same conditions, but with the semi-empirical BV(RI)_C color-transformations (VandenBerg & Clem 2003, blue lines). Dartmouth models for Johnson R and I colors are not provided, so we are unable to compare these color indices to our observations.

There is acceptable agreement of the models and data with the earlier type stars, and the isochrones also show the sharp change in slope to increasing radii when extending beyond our data set to bluer colors (G-type stars). Extending to the later-type stars, however, we find that for stars with radii around 0.6 R_☉ and below, the models predict a drop in radius at colors too blue by several tenths of a magnitude compared to what we observe in the (V − J), (V − H), and (V − K) colors. The (B − V) panel shows that at this point, the solar metallicity model unexpectedly drops and crosses over the model −0.5 dex curve. This is a well-documented failure of the synthetic color grid's ability to reproduce accurate colors for short wavelengths (e.g., see discussion in Dotter et al. 2008 for the accuracies in the color conversions). For instance, the use of the semi-empirical color transformations in the (B − V) panel (blue lines) do not show this flaw, demonstrating the contrast of using different color-table conversions.

Overall, the models show a pattern of divergence with the observations in the color–radius plane. The reason behind this disagreement is not easily identified, for it could be a consequence of underpredicted radii for a given color index or an offset in the color tables used to calculate the colors of the later-type stars (see Section 5).

In this section, we use the data in Table 6 to derive relations between a star's color and metallicity to its luminosity, which can be inverted to solve for the distance, if unknown, so long as one can assume that the star is on the main sequence. In the same way as the radius and temperature relations, we express luminosity as a function of color and metallicity using a polynomial in the form of

$$\begin{eqnarray} \log L ({L_{\odot }}) &= &a_0 + a_1 X + a_2 X^2 + a_3 X Y + a_4 Y + a_5 Y^2, \end{eqnarray} \tag{ 5 }$$

where the variable X is the color index and Y is the stellar metallicity [Fe/H]. Table 11 lists the coefficients and overview of each relation, and Figure 9 shows the data and solutions for our fits. Similar to the radius relations in Section 4.2, we find that the addition of metallicity as an extra parameter reduces the median absolute deviation by a factor of two. The plots in Figure 9 show that the solutions converge for the earlier type stars regardless of the metallicity. However, for the later-type stars there is a spread in luminosity for a given color index, confirming that the metallicity is a necessary factor in estimating a star's luminosity using color relations.

Zoom In Zoom Out Reset image size

We impose the same hard limits for the range of color indices over which the color–metallicity–luminosity relations are valid, just as we do in the radius relations. That is, no extrapolation of the curve is warranted for stars bluer or redder than the indicated color ranges listed in Table 11. Once again there is a tendency for some of the iso-metallicity solutions displayed in Figure 9 to cross over each other. This effect is non-physical and is a product of the metallicity-dependent solutions converging at the blue end of the relations. Despite the appearance of the cross over, any application of the relations with real data will still lead to reliable results with the quoted errors—so long as the boundary conditions are carefully regarded (Table 11).

The luminosity versus color plots can be interpreted as empirical isochrones. Thus, the panels displaying the (B − V), (V − J), (V − H), and (V − K) relations in Figure 9 also show Dartmouth isochrones (Dotter et al. 2008) for an age of 5 Gyr and two metallicities 0.0 and −0.5, roughly bracketing the metallicity range of the data included in this paper. Within the (B − V) panel, the Dartmouth model solutions using both the synthetic (black) and semi-empirical (blue) color conversions are shown, as described above in Section 4.2. Two items are immediately clear based on the comparison of the models to our data. First, the models reproduce the properties of the bluer stars (i.e., stars with log L/L_☉ > −1). Below this limit, the models predict an observed steep drop in luminosity, however, this drop is predicted at colors bluer than what we observe. This is very similar to our color–radius results (Section 4.2; Table 10; Figure 8) and provides substantial motivation for modelers to identify whether or not the source of this disagreement is within the computed astrophysical properties or in the tabulated colors of the later-type stars. We discuss this more in Section 5.

For each spectral type observed, we use the data in Table 6 in order to build a look-up table referring a spectral type to the star's temperature and radius. These results are in Table 12. The quantity n indicates the number of stars that fall into that spectral type bin. The value of the parameter given is the average value of all stars within the spectral type bin, and the σ is the standard deviation of the parameter uncertainties for each spectral type bin. The spectral types with only one measurement (n = 1) simply lists the individual value and the measured error of that measurement.

We use the 33 stars in Table 6 to derive a relation between the luminosity as a function of temperature expressed as

$$\begin{eqnarray} \log L ({L_{\odot }}) &= &-5960.5710 (\pm 207.7541) \nonumber \\ &&+4831.6912 (\pm 172.6408) X \nonumber\\ &&- 1306.9966 (\pm 47.8104) X^2\nonumber\\ &&+ 117.9716 (\pm 4.4125) X^3, \end{eqnarray} \tag{ 6 }$$

where the variable X stands for the logarithmic effective temperature log (T_EFF) and is valid for non-evolved objects within a temperature range of 3200–5500 K. A metallicity-dependent solution did not show any improvement to the fit, therefore this relation is metallicity independent. The median absolute deviation of this fit is 0.0057 L_☉, similar to the abundance-dependent color–metallicity–luminosity relations (Equation (5), Table 11). The top plot in Figure 11 shows the data and the solution to this fit. The middle plot in Figure 11 shows the fractional deviation of observed luminosities compared to the empirical relation expressed in Equation (6).

Zoom In Zoom Out Reset image size

The fractional residuals in the middle panel of Figure 11 (calculated as (L_Obs − L_Fit)/L_Fit, where L_Fit is the luminosity of the empirical relation at the star's observed temperature) show that for stars with temperatures above ∼3500 K, the observed luminosities lie within ±10% of our empirical solution. However, we note that for stars with temperatures below ∼3500 K, the scatter about the fit is much higher. The plotted residuals also show no pattern with respect to the stellar metallicity. This reinforces the fact that we were not able to solve for a metallicity-dependent solution for Equation (6) using our data.

In the top plot, we overlay Dartmouth 5 Gyr isochrones for [Fe/H] = 0, −0.5 (dash-dotted and dotted lines, respectively). We find that the observed agreement of our data and the Dartmouth models in Figure 11 is fairly decent, although much more favorable in the earlier type stars. To evaluate this agreement of our data and the Dartmouth models more quantitatively, we generated Dartmouth model isochrones for each of our stars at its respective metallicity. Each isochrone was chosen at an age of 5 Gyr (appropriate for non-evolved low-mass field stars) with the default helium mass fraction (Y = 0.245 + 1.5*Z), and alpha-enhanced elements scaled to solar ([α/Fe] = 0).¹⁹ We then interpolate through the model grid to find the model temperature, T_Mod, at the observed luminosity of each star. The results are plotted in the bottom panel of Figure 11. The results show that for a given luminosity, models overestimate the stellar temperature by an average of 3% (∼120 K) for stars with temperatures under 5000 K. For the seven stars with temperatures greater than 5000 K, we detect a trend that the temperatures of the more metal-rich stars (orange points; [Fe/H] ∼0.0) are underestimated by the models by a few percent. On the other hand, the more metal-poor stars (green points; [Fe/H] ∼ − 0.25) with temperatures greater than 5000 K show that their temperatures are overestimated by the models by a few percent. This effect is also seen for stars cooler than 5000 K, where the less abundant the star is in iron, the more offset the star's predicted temperature will be with the observed temperature. This pattern of metallicity within the residuals in the bottom panel of Figure 11 suggests that the models could be overpredicting the influence of metallicity on the global stellar parameters.

We use the 33 stars in Table 6 to derive a solve for a solution on the radius–luminosity plane expressed as

$$\begin{eqnarray} \log L ({L_{\odot }}) &= &{-}3.5822 (\pm 0.0219) + 6.8639 (\pm 0.1562) X \nonumber \\ &&- 7.1850 (\pm 0.3398) X^2 + 4.5169 (\pm 0.2265) X^3,\nonumber\\ \end{eqnarray} \tag{ 7 }$$

where X is the stellar radius in solar units. The median absolute deviation about the fit is 0.008 L_☉, and the data and solution are plotted in the top panel of Figure 12. The middle panel in Figure 12 shows the fractional deviation in luminosity about the fit. Stars with radii between ∼0.4 R_☉ and ∼0.67 R_☉ show the tightest correlation, deviating from the fit by less than ±10%. Outside this range in radii, the scatter is worse, where on the lower end the scatter increases to double this level and on the upper end the scatter increases to triple. We observe no pattern with metallicity in the fractional residuals compared to our fit, enforcing the fact that a metallicity-dependent solution is not appropriate on the luminosity–radius plane.

Zoom In Zoom Out Reset image size

We plot Dartmouth 5 Gyr isochrones for [Fe/H] = 0, −0.5 on the top portion of Figure 12. By inspection of these curves, the solar metallicity model agrees well with our empirical fit (Equation (7)), while the metal-poor isochrone appears slightly offset from the data, similar to the luminosity–temperature plots in Figure 11. To make a fair assessment of the comparison to each star's unique composition, we use the custom-tailored Dartmouth isochrones generated for the 33 stars (discussed above in Section 5.1) and interpolate through the model grid to determine the model radius R_Mod at the observed luminosity for each star. The fractional deviations of the observed radii from the model radii are presented in the bottom panel of Figure 12. This shows a clear offset on the order of 5% for models to underpredict the stellar radii at a given observed luminosity.

The deviations in observed radii to those predicted by models become prevalent for stars with radii <0.7 R_☉ (bottom panel of Figure 12). For the full range of stars in our sample, however, we also detect the same pattern in the residuals shown in the bottom panel of Figure 11, where the stars with lower metallicities have radii that are more deviant than model predictions than the stars with more solar type metallicity values. While the effect is less pronounced on the radius plane than on the temperature plane, the conclusions remain the same: models appear to be overpredicting the impact stellar metallicity has on the physical stellar properties.

Our results are consistent with the findings in the literature (e.g., Popper 1997; Ribas et al. 2007; López-Morales 2007; Morales et al. 2008), where models are shown to predict temperatures too high and radii too small while still being able to correctly reproduce the luminosity. Such references, however, are typically referring to binary star properties, and we note that this result is quite novel to detect such a distinct pattern for single stars. Additionally, a unique contribution to our project's analysis is that we are able to reveal the pattern of the offset in connection with the stellar metallicity, an observable often-times unreachable for binary stars due to the complexity of the blended spectrum from each component.

Casagrande et al. (2008) use the infrared-flux method in order to determine properties of M-dwarfs. Their conclusions show that there is a radius offset present when comparing their observations of single stars to models. However, while similar to our own conclusions, they estimate that this offset is on the order of 15%–20%, unlike the 5% we claim from our analysis. This difference is possibly caused by the errors introduced in their semi-empirical determinations of the stellar properties. They also identify a discontinuity at the transition from K- to M-dwarfs, most apparent on the luminosity–temperature plane that is not reproduced by models. We do not observe this feature, however, this region from 4200 to 4300 K is not well populated within our interferometrically observed sample, as we only have one data point that lies in this range of temperatures. Perhaps what is more interesting is a region not explicitly identified in Casagrande et al. (2008); the point where the deviations of their data compared to models begin. The data in their Figure 12 show that this is at ∼5000 K (M_BOL ∼ 6.5). Their Figure 13(a) and (c) show this point at ∼0.7 R_☉ and ∼0.7 M_☉, the exact positions we identify in the beginning of the trend in our data.

In order to link effective temperature to stellar radius, we use the 33 stars in Table 6 to derive a single parameter solution in the form of a third-order polynomial, valid for non-evolved objects with temperatures between 3200 and 5500 K:

$$\begin{eqnarray} R ({R_{\odot }})\! &=&{-} 10.8828 (\pm 0.1355) + 7.18727 (\pm 0.09468)\times 10^{-3} X \nonumber \\ &&- 1.50957 (\pm 0.02155) \times 10^{-6} X^2 \nonumber\\ &&+ 1.07572 (\pm 0.01599) \times 10^{-10} X^3, \end{eqnarray} \tag{ 8 }$$

where the variable X is the effective temperature. Like the radius–color plane (Section 4.2), there is a lot of scatter in the radius of a star for a given temperature (see Figure 13). However, the data on the temperature–radius plane does not show the same spread in the measurements due to differences in metallicity as in the color–metallicity–radius relations, as its solution is independent of metallicity. We find that the median absolute deviation of the observed versus calculated radius for Equation (8) is 0.031 R_☉. This value is similar to ones derived via the abundance-dependent color–metallicity–radius relations of Equation (4) shown in Table 10.

Zoom In Zoom Out Reset image size

Stars more massive than the sample of K- and M-dwarfs (i.e., masses ≳ 0.8 M_☉) are sensitive to evolutionary effects. For example, with increasing age, a more massive star will first begin to evolve off its zero-age main-sequence position to larger radii and hotter temperatures. We illustrate this in Figure 13, showing the expected model predictions of a 0.8 M_☉ star (blue *), a 0.9 M_☉ star (red +), and a 1.0 M_☉ star (black ×) at the age of 1 Gyr and 4.5 Gyr from solar-metallicity Dartmouth model isochrones. Arrows connecting these positions point to the top left of the plot, i.e., the direction of evolution to larger radii and hotter temperatures. We also show in Figure 13 the Sun's present position. To model the data beyond our sample of K- and M-dwarfs to hotter temperatures, we re-fit the data adding the Sun as a point of reference to more massive stars, effectively making a solar-age-calibrated isochrone. This relation is expressed as

$$\begin{eqnarray} R ({R_{\odot }}) &= &{-}8.133 (\pm 0.226) + 5.09342 (\pm 0.16745) \times 10^{-3} X \nonumber \\ &&- 9.86602 (\pm 0.40672) \times 10^{-7} X^2 \nonumber\\ &&+ 6.47963 (\pm 0.32429) \times 10^{-11} X^3, \end{eqnarray} \tag{ 9 }$$

where X is the effective temperature. We show this solution in Figure 13 as a blue line, as well as the Sun's position (farthest left point in Figure 13, assuming T_☉ = 5778 K, R = 1 R_☉). The extrapolation of this curve past 5500 K is shown as a dotted line that intersects the Sun's present position. A comparison of the temperature–radius relation excluding the Sun (Equation (8)) and including the Sun (Equation (9)) shows that while the hottest of the K-stars in our sample are modeled better by Equation (8), there is an inflection point to larger radii beyond our data sample range of spectral type K0 to meet the Sun's position at its age today.

What is more intriguing, is whether or not the models are able to reproduce the trend to the temperatures and radii of single stars, with the omission of the luminosity as a constant like structured in the last two Sections of 5.1 and 5.2. The bottom left panel in Figure 10 shows that the Dartmouth models predict the sharp drop in stellar radii to happen at hotter temperatures than what we observe. This is also seen in the temperature–luminosity (Section 5.1) and radius–luminosity (Section 5.2) relations displayed in Figures 11 and 12, where compared to our observations, the Dartmouth models predict temperatures too high and radii too low for a given luminosity.

Similar to the method described in Sections 5.1 and 5.2, we use the custom-tailored Dartmouth isochrones for each star to evaluate expected parameters from the model compared to our observations. First, we assume that the temperature is a constant within the observations and model grid. For each star, we then interpolate within the model grid for a model radius at the observed temperature. The second approach is the reverse, and we assume that the radius is constant within the observations and model grid. A model temperature is then solved for by interpolation through the model grid at the observed radius. We show the results in Figure 14.

Zoom In Zoom Out Reset image size

The top plot in Figure 14 shows the fractional deviation of temperature with the observations to the Dartmouth models as a function of temperature. The bottom plot shows the fractional deviation of the observed to Dartmouth model radii as a function of radius. The fractional temperature offsets for stars with temperatures less than 5000 K show that models predict temperatures higher than we observe by an average of 6%. The fractional radius offsets for stars with radii less than 0.7 R_☉ show that models predict radii smaller than we observe by 10%, where this offset increases with decreasing radii to ∼50% for stars with radii ∼0.4 R_☉. The three stars with temperatures below 3300 K (GJ 725B, GJ 551, and GJ 699) produce null results when interpolating at the observed temperatures to derive model radii, and thus their results are not plotted in the bottom plot of Figure 14. The computed offsets in temperature and radius given here reflect doubly deteriorating conditions compared to the results in Sections 5.1 and 5.2, where the stellar luminosities were held as a constant. These compounded errors should be regarded with caution when using models to relate stellar temperatures and radii for stars <5000 K and <0.7 R_☉.

Data from double-lined spectroscopic EB systems are standards for the mass–radius relations available today. For the following discussion, we collect the binary star parameters presented in Table 2 of Torres et al. (2010) together with the low-mass binaries and secondaries in binaries compiled in López-Morales (2007). We impose the same criteria as for the single stars in our analysis, limiting the sample to only allow stars with mass and radius <0.9 M_☉ and R_☉, and where the radius is measured to better than a 5% error, leaving a total of 24 individual stars. These properties of the single stars are discussed in Section 3, where we have a total of 33 stars in this range of masses and radii. Note that although we limited the literature measurements of single stars to only include those with angular diameters measured to better than 5%, the sample remains unchanged if we filter with respect to the error in the linear radii to be better than 5%, due to the close proximity of the objects (within ∼10 pc) and thus well-known distances.

To determine a mass–radius relation, we use a second-order polynomial to fit the data for single stars (Equation (10)). We experimented with higher order polynomials and found that they did not improve the fit. We solve for a solution for the EB's in the same manner (Equation (11)). Data and fits are displayed in Figure 15. The resulting relations are

$$\begin{eqnarray} {\rm R_{\rm SS}} (R_{\odot }) &= & 0.0906 (\pm 0.0027) + 0.6063 (\pm 0.0153) M_{\ast } \nonumber \\ && + 0.3200 (\pm 0.0165) M_{\ast }^2 \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} {\rm R_{\rm EB}} (R_{\odot }) &= & 0.0135 (\pm 0.0070) + 1.0718 (\pm 0.0373) M_{\ast } \nonumber \\ &&- 0.1297 (\pm 0.0367) M_{\ast }^2, \end{eqnarray} \tag{ 11 }$$

where M_* is the stellar mass in solar units.

Zoom In Zoom Out Reset image size

In Figure 15, we show the data and solutions for single (circles, solid line) and binary stars (squares, dotted line). These relations are both consistent with a 1:1 relation to the mass and radius of a star (dashed line). One can see from the figure that the radii of single and binary stars are indistinguishable for a given mass. This shows that tidal influences on binary component radii may not be a concern when viewed on this scale. In other words, the scatter in the measurements in EB systems may wash out the effect the binary period might have in the radii of the binary star components (for instance, see discussion in Kraus et al. 2011).

Historically, large discrepancies have been observed when comparing observed radii with radii predicted by models for low-mass stars (for example, see discussion in Section 1 and López-Morales 2007) in the sense that the models tend to underestimate the stellar radii at a given mass. We compare our measured radii to the model radii from the 5 Gyr model isochrones of Padova, Dartmouth, BCAH98, and Yonsei-Yale (Girardi et al. 2000; Dotter et al. 2008; Baraffe et al. 1998; Demarque et al. 2004) in Figure 10. Points for the masses and radii from EBs are also plotted.

Due to the density of information from showing several model predictions along with the data and errors, the visibility of the claimed radius discrepancy with models in Figure 10 is difficult to see clearly. While it is abundantly documented in the literature that such a discrepancy in the predicted and observed radii exists for binary stars, the intentions of this work is to show its equivalent—if present—for single stars. Already in Sections 5.1 and 5.2 we have shown proof of this discrepancy in the luminosity–temperature and luminosity–radius plane, where models overpredict temperatures by ∼3%, and underpredict radii by ∼5% compared to our observations. These results presented on the luminosity–temperature and luminosity–radius plane are more robust for single stars because temperatures and radii are properties of single stars we can directly measure to high precision with interferometry. However, the ability to comprehensively show the radius discrepancy from this data set is quite difficult on the mass–radius plane, since the mass errors for single stars are characteristically large (as shown on the bottom axis of the plot). To first order, however, we find it useful to present Figure 16, a series of plots showing the fractional deviation of the observed to theoretical radii for single stars.

Zoom In Zoom Out Reset image size

The theoretical radii, R_Mod, are computed by using the mass (calculated from the MLR, Table 6) and interpolating through the custom-tailored mass–radius grid of values computed for each star's metallicity in a 5 Gyr Dartmouth model isochrone. The result is a value for a model radius at the given mass and metallicity. The difference is then between the observed radius (from interferometry, Table 6) and the model radius (from interpolation of the mass from the M–L relation in the model mass–radius grid). Since the single star mass errors are quite large, the errors in the theoretical radii dominate the y-errors shown in Figure 16. Again, we caution that the absence of any signal in an (R_observed–R_model) plot could be misleading when using single stars as clues into this discrepancy on the mass–radius plane due to the large mass errors.

On the top panel of Figure 16, we show the difference in observed versus model radii as a function of mass. The two middle and two bottom panels show the fractional deviation of radius as a function of metallicity [Fe/H] and stellar activity level L_X/L_BOL (Table 6), since previously proposed diagnostics to explain any disagreement with models have been linked to abundances and stellar activity levels for stars in this mass regime. The bottom four panels are segregated to show stars of all masses <0.9 M_☉ and only admit stars with mass <0.6 M_☉ (left panel and right panel, respectively).

The top panel on the plot shows that even with the stated large errors, stars with masses below 0.42 M_☉ (eight total) have bigger radii than those of the Dartmouth models, where six of the eight deviate by several sigma. We identify these eight stars with masses below 0.42 M_☉ as blue points within the right-hand side plots of Figure 16. Regarding the fractional deviation in radii for the blue points with respect to metallicity, we find that for metallicities [Fe/H] < − 0.35, four of the five blue points deviate by more than 1σ. Any pattern with respect to L_X/L_BOL is less clear within the data. The fact that we do not see any trend with activity is not a surprise, due to the low levels of these single field stars compared to their active binary star counterparts (typically a decrease in X-ray flux by two to three orders of magnitude; see López-Morales 2007).

Our conclusion supports the known discrepancy in observed radii compared to models for stars less than 0.42 M_☉, showing a slight dependence on metallicity. This conclusion is different from previous works that focused on interferometric radii of single stars. For instance, although Berger et al. (2006) show the presence of the radius discrepancy with models, they observe an opposite trend compared to our own: that stars with the most deviant stellar radii have larger metallicities. Contradictory to the Berger et al. (2006) conclusion as well as the one presented here is the analysis from Demory et al. (2009), where they find that the radii of single stars are consistent with predictions of stellar models.

The differences in the previously published results and the one presented here are multi-fold. A primary difference is the sample used in the analysis of each work. Berger et al. (2006) applied no filter to the diameter precision allowed in their analysis, and at the time of publication, this amounted to 10 interferometric radii of M-dwarfs available for their analysis. Minding our filter of using only data with sizes determined to better than 5%, this constitutes to only 5 of the 10 stars used in the Berger et al. (2006). In addition to limiting the precision of measurements allowed in our analysis, we also tripled the sample size of M-dwarfs (14), making for better statistics and thus giving more weight to our results. Our work also uses recently defined metallicity calibrations for M-dwarfs (see references in Table 6). These new references use techniques to measure metallicity that are more refined than the ones used to derive the metallicities for the stars cited in Berger et al. (2006), and thus could explain the different conclusions presented here. However, by comparing the metallicities used in Berger et al. (2006) to the ones we use in this work, we find an agreement within 0.02–0.14 dex. This is less than the 1σ uncertainty in metallicity (∼0.2 dex) and thus should not be the cause of new trend we observe.

The sample of M-dwarfs at the time of the Demory et al. (2009) publication was identical to the time of Berger et al. (2006).²⁰ In their analysis, they include both K- and M-dwarf measurements, choosing to disregard all but one measurement of the Berger et al. (2006) data, and limit published measurements to have a precision better than 10%.

The comparison of the radius offsets examined in Demory et al. (2009) comes from an analysis of an expanded mass range encompassing both K- and M-dwarfs, similar to the one presented here on the left-hand side panels of our Figure 16. In the full mass range, we come to the same conclusion as Demory et al. (2009) in their Figure 9 that we see no deviation in single star radii with respect to metallicity.²¹ However, we find that by adding the stellar mass as an additional constraint, the higher-mass stars that do not show offsets from models are more likely to hide signals in the low-mass stars that do contain offsets (for example, see the right-hand side panels of Figure 16). It is thus a misleading claim to pronounce the conclusions in Berger et al. (2006) void, since they are analyzing the wrong mass range (the full span of K- and M-dwarfs, as opposed to just M-dwarfs in Berger et al. 2006), in addition to not bringing any new measurements for interpretation. Lastly, as opposed to the methods in both Berger et al. (2006) and Demory et al. (2009) where a solar metallicity model is assumed, we compare the observations to models that are specifically tailored to each star's composition.

We find that the empirical relations we derive relating the global properties of luminosity–temperature, luminosity–radius, and temperature–radius show no dependence on the metallicity of a star. This conclusion, while unexpected, challenges the Vogt–Russell theorem, which states that the properties of a star should exhibit a unique solution for a star of given mass and chemical composition (see Weiss et al. 2004, and references therein). Generally speaking, when regarding relations to global properties of stars, models predict that the resulting scatter we observe is caused from varying the stellar metallicity. However, the majority of metallicities used in this work are from a uniform source and should only have systematic offsets with respect to each other. So although we do not see this trend for the scatter in our data to be caused by metallicity (e.g., see Figure 13), we propose that it could be a residual effect from underestimated errors in metallicity measurements for these types of stars (average metallicity error is ∼0.2 dex). It is important to note, however, that we are only reviewing this lack of dependence with the composition on the stellar physical parameters with respect to iron abundances. Alternate elemental abundances such as helium that are not accounted for in the models can have an effect on a star's physical properties. However, our data not only show no correlations with differences in iron abundances, but also suggest that models overpredict the dependence on the stellar global parameters with metallicity as illustrated in the pattern of the plotted residuals in the observations versus models of Figures 11 and 12.

With our data, we are able to develop an empirical method of quantifying the effect a star's metallicity has on the observed color index. For instance, our analysis shows that metallicity does not effect the stellar radius when relating the global properties of stars (luminosity–radius, temperature–radius), while on the other hand, the analysis of the color–metallicity–radius relations in Section 4.2 (Figure 8) identifies a substantial spread in radii for a given color index, noting a dependence on metallicity. This leads us to deduce that the metallicity only directly affects the observed color index of a star. The effect can be visualized in the color–metallicity–radius relations (Figure 8) as a shift in the horizontal direction for our solutions with different metallicity, as opposed to the vertical direction, where at a fixed color index, the radii change with respect to metallicity. We can quantify this shift by defining a zero metallicity color index: we find that for a differential metallicity of 0.25 dex, the (B − V), (V − R), (V − I), (V − J), (V − H), and (V − K) color indices change by ∼0.05, 0.15, 0.2, 0.3, 0.3, and 0.3 mag, respectively. This argument, while described with respect to our observed radii, also holds using the stellar luminosities as a proxy.

In this work, the masses for single stars are derived indirectly via the MLR in Henry & McCarthy (1993) (Section 3.2). This relation, as well as others in the literature (e.g., Delfosse et al. 2000), do not incorporate metallicity in their formulation and analysis. Therefore, the application of an MLR may produce spurious results for an object with a different composition than the objects used to develop the MLRs. To alleviate this concern, we used the K-band form of the MLR, which is thought to be insensitive to metallicity: a fortunate by-product to a proper balancing act of equalizing luminosities and flux redistributions in the infrared with changing metallicity (see Delfosse et al. 2000 for their discussion and empirical verification). However, unlike luminosity–temperature, luminosity–radius, and temperature–radius planes, we are not able to confirm or deny any metallicity dependence on the mass–radius plane with our data because the mass errors are too large for the single stars in our sample (Section 5.5, Figure 15).

Within Sections 4.1–4.3, we use a multi-variable function to relate the observed colors and metallicities to the global properties of temperature, radius, and luminosity. We found that inclusion of metallicity as a variable was essential to properly relate the astrophysical quantities (T_EFF, R, and L) to the observed colors of our stars. Contrary to the color solutions, we find that linking the global properties (temperature, radius, luminosity, mass) of a star (Sections 5.1–5.4) do not appear to be sensitive to the stellar metallicity within the observational errors used in this work.²² In other words, our data show that the metallicity only affects the observed color index of a star, and thus using the metallicity-dependent transformations in order to convert colors into the stellar properties of temperatures, radii, and luminosities is essential. However, the analysis clearly shows that for the range of stars that we observe, the metallicity does not impact the physical properties of a star in a way that we are able to measure: throwing a bucketful of metals in a star does not make it expand in size or cool its surface temperature, it simply morphs the observed color index.