El Estado no ateo ni agnóstico, y no indiferente a los sentimientos religiosos de los

It is fortuitous that ground-truth measurements of radius are possible by “simply” looking at stars in the sky and combining their angular sizes and measured distances. Thus, we begin by comparing our results to radius measurements from LBI, which currently are the gold standards for direct measurements of stellar radii.1 The left panel of Figure 5.1 and Table 5.1 illustrate our ability to replicate radius results obtained via LBI to a median absolute difference of 6% from the most recent results for 26 stars.

Table 5.1 maps our consistency star by star with results from 12 interferometry efforts using CHARA and the Very Large Telescope Interferometer (VLTI), namely Lane et al. (2001); Ségransan et al. (2003); Berger et al. (2006); van Belle & von Braun (2009); Demory

1_{Whereas many radii for red dwarfs in eclipsing binaries have been determined, they generally do not}

et al. (2009); von Braun et al. (2011, 2012); Boyajian et al. (2012b); von Braun et al. (2014); Kane et al. (2017); Schaefer et al. (2018); Rabus et al. (2019). Column 1 lists the star name. The same star may be repeated throughout the table if studied by another group. Columns 2–5 contain radius and angular results from the publication and their errors. Columns 6–7 contain the radius and corresponding error calculated using the angular diameter listed in the publication scaled by the parallax used in our study. This practice accounts for the fact that differences in radii may stem from difference in parallaxes, especially in the cases of older papers that may contain outdated results. Our radius and angular diameter results and errors are listed in Columns 8–11. In the final two columns we compare our radius results to those in Column 6 derived using our parallaxes and published angular diameters. The absolute value of the difference between the two radii is listed in Column 12 and the percent difference between our result and that from interferometry is listed in Column 13, where the “correct” value in the denominator is declared to be the value from LBI. The resultant values in Column 13 range from 0.393% to 22.4% with the values generally decreasing, down to an average 5.6% difference from results by Rabus et al. (2019).

Because interferometric observations are generally limited to the biggest, brightest stars, predictably 25 of these 26 stars pass through our “warm door” (discussed in §4.2). We are therefore confident that our procedure produces realistic results for our warmer subsample of stars (V −K < 7). The single star that passes through our “cool door”, as discussed in §4.2, is GJ 406 with V − K = 7.53. Even in this case, our result is offset from the interferometric result of Rabus et al. (2019) by only 2.6%, illustrating the capability of our

cooler star procedure to produce results matching those from direct observations.

In the right panel of Figure 5.1 we consider the distances to the stars and compare our derived angular diameters to interferometric measurements. We attribute the larger errors for stars with larger angular sizes to the increased levels of saturation and therefore error in the photometry. Although the points shift from their locations in the left panel, the result is the same — the median difference between our angular diameters and those measured interferometrically is again 6%. We notice a small systematic offset in favor of our having larger radii, though we do not know its cause.

Figure 5.1: For validation of our method to determine radius, we compare our results to the most model-independent method for resolved single stars, long-baseline interferometry. (Left) We compare our radii to LBI radii calculated by scaling published angular diameters with our parallaxes. Note that these radius values and therefore the associated calculations for Boyajian et al. (2012b) (B12) and Rabus et al. (2019) (R19) are different from their original values presented later in Figure 6.3. (Right) We make a similar comparison using angular diameter results. In both panels, our results match to within median absolute differences of 5.9%, with closer matches to more recent publications. This is especially reassuring because the more recent results are expected to use more developed and refined instruments, data reduction pipelines, and/or telescope systems compared to their earlier- published counterparts. Note that we include only the most recent results for each star, but comparison to each star by publication is listed in Table 5.1.

Table 5.1: Comparisons of Angular Diameter and Radius to Interferometric Values

Star Publication Publication with ourπ∗ This Work Comparison∗∗

θ σθ R σR R σR θ σθ R σR |∆Rπ| %Dif fRπ (′′) (′′) (R⊙) (R⊙) (R⊙) (R⊙) (′′) (′′) (R⊙) (R⊙) (R⊙) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) Lane et al. (2001) BARNARDS 1.026 0.04 0.2009 0.0002 0.202 0.008 0.956 0.053 0.188 0.010 0.014 6.848 GJ0015 1.014 0.05 0.3890 0.0004 0.389 0.019 0.934 0.112 0.359 0.043 0.031 7.917 Ségransan et al. (2003) GJ0191 0.692 0.06 0.291 0.025 0.291 0.025 0.617 0.035 0.260 0.015 0.031 10.792 GJ0205 1.149 0.11 0.702 0.063 0.702 0.067 1.005 0.097 0.614 0.059 0.088 12.534 PROXIMA 1.044 0.08 0.145 0.011 0.146 0.011 1.237 0.043 0.173 0.006 0.027 18.491 Berger et al. (2006) GJ0015 0.988 0.016 0.379 0.006 0.379 0.006 0.934 0.112 0.359 0.043 0.021 5.494 GJ0514 0.753 0.052 0.611 0.043 0.615 0.042 0.615 0.032 0.502 0.026 0.113 18.324 GJ0526 0.845 0.057 0.493 0.033 0.492 0.033 0.889 0.054 0.517 0.031 0.025 5.157 GJ0687 1.009 0.077 0.492 0.038 0.492 0.038 0.904 0.046 0.441 0.022 0.051 10.37 GJ0752 0.836 0.051 0.526 0.032 0.525 0.032 0.802 0.044 0.504 0.028 0.021 4.069 GJ0880 0.934 0.059 0.689 0.044 0.686 0.043 0.808 0.035 0.594 0.025 0.092 13.457

van Belle & von Braun (2009)

GJ0526 0.823 0.069 0.481 0.040 0.479 0.040 0.889 0.054 0.517 0.031 0.038 7.968

Demory et al. (2009)

PROXIMA 1.011 0.052 0.141 0.007 0.141 0.007 1.237 0.043 0.173 0.006 0.032 22.358

von Braun et al. (2011)

GJ0581 0.446 0.014 0.299 0.010 0.301 0.009 0.478 0.010 0.323 0.006 0.022 7.225

von Braun et al. (2012)

GJ0436 0.417 0.013 0.455 0.018 0.453 0.014 0.422 0.012 0.459 0.008 0.006 1.292

Boyajian et al. (2012b)

BARNARDS 0.952 0.005 0.1867 0.0012 0.188 0.001 0.956 0.053 0.188 0.010 0.001 0.393

GJ0015 1.005 0.005 0.3874 0.0023 0.386 0.002 0.934 0.112 0.359 0.043 0.027 7.092

Comparisons of Angular Diameter and Radius to Inter- ferometric Values

Star Publication Publication with ourπ1 _{This Work Comparison}2

θ σθ R σR R σR θ σθ R σR |∆Rπ| %Dif fRπ (′′) (′′) (R_⊙) (R_⊙) (R_⊙) (R_⊙) (′′) (′′) (R_⊙) (R_⊙) (R_⊙) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) GJ0412 0.764 0.017 0.3982 0.0091 0.399 0.009 0.725 0.038 0.379 0.020 0.020 5.047 GJ0526 0.835 0.014 0.4840 0.0084 0.486 0.008 0.889 0.054 0.517 0.031 0.031 6.416 GJ0687 0.859 0.014 0.4183 0.0070 0.419 0.007 0.904 0.046 0.441 0.022 0.022 5.281 GJ0809 0.722 0.008 0.5472 0.0067 0.548 0.006 0.761 0.041 0.578 0.031 0.030 5.396 GJ0880 0.744 0.004 0.5477 0.0048 0.547 0.003 0.808 0.035 0.594 0.025 0.047 8.644

von Braun et al. (2014)

GJ0176 0.448 0.021 0.4525 0.0221 0.438 0.021 0.506 0.013 0.495 0.009 0.057 12.933 GJ0649 0.484 0.012 0.5387 0.0157 0.535 0.013 0.473 0.011 0.522 0.010 0.013 2.355 GJ0876 0.746 0.009 0.3761 0.0059 0.374 0.005 0.760 0.026 0.381 0.013 0.007 1.898 Kane et al. (2017) GJ0628 0.695 0.018 0.3207 0.0088 0.319 0.008 0.758 0.026 0.348 0.012 0.029 9.072 Schaefer et al. (2018) GJ0393 0.564 0.021 0.428 0.017 0.432 0.016 0.576 0.019 0.441 0.013 0.009 2.046 Rabus et al. (2019) GJ0001 0.812 0.005 0.379 0.002 0.379 0.002 0.824 0.055 0.385 0.025 0.006 1.481 GJ0273 0.783 0.010 0.320 0.005 0.316 0.004 0.854 0.033 0.345 0.013 0.029 9.120 GJ0406 0.582 0.020 0.159 0.006 0.150 0.005 0.597 0.018 0.154 0.004 0.004 2.550 GJ0447 0.540 0.029 0.196 0.010 0.195 0.010 0.596 0.017 0.215 0.006 0.020 10.428 GJ0581 0.476 0.007 0.322 0.005 0.321 0.005 0.478 0.010 0.323 0.006 0.002 0.467 GJ0628 0.661 0.014 0.306 0.007 0.303 0.006 0.758 0.026 0.348 0.012 0.045 14.682 GJ0674 0.737 0.037 0.360 0.018 0.360 0.018 0.760 0.031 0.371 0.015 0.011 3.167 GJ0729 0.642 0.020 0.205 0.006 0.205 0.006 0.652 0.018 0.208 0.006 0.003 1.605 GJ0832 0.814 0.010 0.435 0.005 0.433 0.005 0.844 0.056 0.449 0.030 0.016 3.717 GJ0876 0.705 0.009 0.354 0.005 0.353 0.005 0.760 0.026 0.381 0.013 0.028 7.824 PROXIMA 1.103 0.007 0.154 0.001 0.154 0.001 1.237 0.043 0.173 0.006 0.019 12.153

* These radii and errors are calculated by scaling the published angular diameter values by the parallaxes used in this work.

5.3 Trends in the Radius-Effective Temperature Relationship of Red Dwarfs —