• NASA ADS Record
• Simbad Objects
• About Related Links

Table 1. Long-baseline Optical Interferometry Diameter Surveys of Five or More Stars

Note. See Table 3.1 in von Braun & Boyajian (2017) for a detailed list of stars with interferometrically determined radii. See Section2 for details.

Download table as: ASCIITypeset image

Given the highly efficient queue-scheduled nature of observing with PTI, a large program of observing as many evolved stars as possible was undertaken at the facility. These were targets of opportunity, available for observing when the facility was not tasked with other observing. The scope of this manuscript will focus on the field giant stars. This leaves objects such as S-type stars and Miras for forthcoming papers, and objects such as supergiants (van Belle et al.2009) and carbon stars (van Belle et al.2013) have already been published. This broad sweep meant target selection was largely limited by the “sweet spot” of PTI sensitivity for both target brightness and angular size.

Angular size range. Angular sizes at PTI for robust size determination, noting the night-to-night precision cited above, were typically intended to be in the range of 1.5–4.0 mas. This range is consistent withK-band operation of a 109 m baseline, resulting in aV² contrast below roughly 90% (to ensure target resolution) and above roughly 10% (to avoid degenerate diameter solutions). A priori estimators, such as theV −K technique found in van Belle (1999), allowed reasonable expectations of the gross size of a given target prior to observing. The number of resulting targets over the range of angular sizes is found in Table2 and plotted in Figure1. Some angular sizes in excess of 4.0 mas are seen from large objects observed with PTI’s shorter 85 m baseline. Based on the minimum expected night-to-nightV² repeatability discussed above, the limiting fractional error expected for various angular sizes is presented in Table3.

Zoom InZoom OutReset image size

Brightness range. The PTI’s limiting magnitude ofK < 5 primarily limited the available pointlike calibrator observations interleaved with resolved target star observations. Generally speaking, targets that satisfied the angular size range constraints for resolvability above were quite bright, withK < 3. The bigger impacts on operations were those targets that were extremely red (corresponding to the targets with the lowest effective temperature); a lack of visible-light photons made it difficult for the facility’s tip-tilt tracker to follow atmospheric turbulence. However, this was more of a concern for studies that targeted even redder targets than this particular investigation, such as carbon stars (van Belle et al.2013).

Spectral types. Spectral types were initially taken from various literature references (e.g., see Skiff2014, and references therein); care was taken to select those that had been previously typed as luminosity class III or were otherwise indicated to be off the main sequence (e.g., based on parallax values). As will be presented below in Section4, these types were taken as starting points for our own solutions for fitting spectral types. These initial spectral types are presented in Table4, as well as our best-fit values.

Completeness. A cursory examination of the Two Micron All Sky Survey (2MASS) and Skiff catalogs (Cutri et al.2003a; Skiff2014) can give an indication of the completeness of our giant star sample. First, the 2MASS catalog was sampled for all objects brighter thanK < 3.5, resulting in approximately 13,000 objects. For stars in this brightness range, 2MASS photometry is saturated and increasingly unreliable, although for this assessment, it is more than sufficient. Second, from theB andK magnitudes in 2MASS, all such objects were selected for angular sizes in the range ofθ = {1.0, 4.75} mas using theB −K size estimator in van Belle (1999). Finally, those ∼11,600 objects were compared against the Skiff catalog for objects previously typed to be luminosity class III objects, resulting in ∼4500 targets, of which 1723 are northern hemisphere objects. As such, our sample herein constitutes a sample of ∼10% of all possible giant stars that PTI could have observed.

This is an ex post facto assessment and could not have guided target selection at the time. The Skiff catalog was not available at the time of the PTI target selection, and 2MASS was available only for the latter portion of PTI operations. During PTI operations, our guideposts were the Bright Star Catalog (Hoffleit & Jaschek1982), the recently released (for that era) Hipparcos catalog (Perryman et al.1997), and the Catalog of Infrared Observations (CIO; Gezari et al.1993). During that era, luminosity classification was somewhat less clearly defined during PTI operations, even if, in the course of this investigation, our targets will be diligently sorted for luminosity class III objects.

Table 5. Sources of Photometry for Our Program Stars withN ≥ 20 Data Points

Note. In the system column, “Johnson-visible” is used to denote any passbands fromUBVRI, and “Johnson-IR” denotes passbands fromJHKLM. Additional sources are listed on a star-by-star basis in Table6. For more details, see Section4.2.

Download table as: ASCIITypeset image

A significant question that we frequently encountered during this study was whether to fit our photometric data with empirical spectral templates such as the INGS library or work with model spectral templates based on models such as the PHOENIX. The INGS (IUE/NGSL/SpeX) library ¹¹ is a refinement of the empirical templates originally found in Pickles (1998) by the same author; it has the virtue of being a model-independent approach to providing a stellar spectral template. Unfortunately, the drawback of the INGS templates is that they are, relative to model grids such as the PHOENIX models, less densely available in a grid of stellar spectral type versus luminosity class.

In contrast to the INGS empirical templates, a dense grid of PHOENIX model templates is available (Husser et al.2013) ¹² that offers a significantly denser sampling inT_eff (i.e., spectral type) versus surface gravity (i.e., luminosity class) and also sidesteps any potential concerns about uncorrected interstellar extinction in the INGS empirical templates.

Our solution to get the best of these complementary approaches to stellar spectral templates was to utilize a set of PHOENIX models that had been calibrated against the INGS spectral data. Using the INGS data as input spectrophotometry into oursedFit code (described in Section4.3, with the reddening option forsedFit disabled), the best-fit PHOENIX model spectra from a grid of PHOENIX models were matched to each INGS empirical template. The results in log(g)–T_eff space of fitting of INGS spectra against PHOENIX models can be seen as a Kiel diagram (see Fuhrmann1998) in Figure2. Comparing the PHOENIXT_eff model values against the PicklesT_eff values, we find an averageT_eff difference of −12 K (−0.2%) with no particular evident trend against spectral type of those differences. Overall, this approach allows us the density of the PHOENIX grid while still being based on the empirical INGS grid. Table7 outlines the INGS-to-PHOENIX mapping and indicates where spectral types not present in the INGS are mapped onto PHOENIX.

Zoom InZoom OutReset image size

Execution of thesedFit code, and its ability to provide meaningful results on our program stars, was wholly dependent upon providing the code with meaningful photometry. A large number of archival sources were researched for narrow- to broadband photometry. Meta-references such as SIMBAD, ¹³ the General Catalogue of Photometric Data ¹⁴ (GCPD; Mermilliod et al.1997), and the CIO (Gezari et al.1993) were useful in locating primary data sources; the sources that provided the largest number of points are noted in Table5, with the individual data points called out explicitly in Table6. The meta-references are uniquely empowering in collecting a large number of archival data points; thus, for the sake of traceability and proper attribution credit, we were scrupulous in Table6 in providing the original source reference.

Table 7. PHOENIX Model Parameters Fit to INGS Spectral Templates

Spectral Type	Teff
	12,000	3.0
	11,800	3.0
	11,600	3.0
B9III	11,400	3.0	0.34
	11,200	3.0
	11,000	3.0
	10,800	3.0
	10,600	3.0
	10,400	3.0
	10,200	3.0
	10,000	3.0
	9800	3.0
A0III	9600	3.0	0.48
	9400	3.0
	9200	3.0
{A1III}	9000	3.0
	8800	3.0
	8600	3.0
{A2III}	8400	3.0
	8200	3.0
	8000	3.0
A3III	7800	3.0	0.46
A5III	7600	3.0	0.16
A7III	7400	3.0	0.56
{A8III}	7200	3.0
{A9III}	7000	3.0
	6900	3.0
F0III	6800	3.0	0.51
	6700	3.0
{F1III}	6600	3.0
	6500	3.0
F2III	6400	3.0	0.55
	6300	3.0
{F3III}	6200	3.0
{F4III}	6100	3.0
{F5III}	6000	3.0
{F6III}	5900	3.0
{F7III}	5800	3.0
{F8III}	5700	2.5
{F9III}	5600	2.5
{G0III}	5500	2.5
{G1III}	5400	2.5
{G2III}	5300	2.5
{G3III}	5200	2.5
{G4III}	5100	2.5
G5III	5000	2.5	0.96
{G6III}	5000	2.5
{G7III}	4900	2.5
G8III	4900	2.5	0.85
{G9III}	4800	2.5
K0III	4800	2.5	1.21
{K0.5III}	4700	2.5
K1III	4700	2.5	0.71
	4600	2.0
{K1.5III}	4500	2.0
K2III	4400	2.0	0.93
{K2.5III}	4400	2.0
K3III	4300	2.0	1.16
	4200	2.0
{K3.5III}	4100	2.0
K4III	4000	2.0	3.13
{K4.5III}	3900	1.5
K5III	3900	1.5	2.21
{K6III}	3900	1.5
{K7III}	3900	1.5
M0III	3800	1.5	2.49
{M0.5III}	3800	1.5
M1III	3800	1.5	3.19
{M1.5III}	3800	1.5
M2III	3700	1.5	2.43
{M2.5III}	3700	1.5
M3III	3600	1.5	2.77
{M3.5III}	3500	1.0
M4III	3500	1.0	5.54
{M4.5III}	3400	1.0
M5III	3400	1.0	4.06
{M5.5III}	3300	1.0
M6III	3200	1.0	8.44
{M6.5III}	3200	1.0
M7III	3100	0.5	11.28
{M7.5III}	3000	0.5
	2900	0.5
M8III	2800	0.5	16.57
{M8.5III}	2700	0.5
M9III	2600	0.0	26.35
{M9.5III}	2500	0.0
M10III	2400	0.0	229.16
	2300	0.0

Note. Spectral types in brackets do not have INGS spectral templates and are linearly inferred interpolations associated with the corresponding PHOENIX models. For more details, see Section4.1.

Download table as: ASCIITypeset images:12

In addition to the archival sources, we engaged in an extensive program of photometric observing at the Lowell 31 inch telescope located at Anderson Mesa. These visible data included broadband JohnsonUBVRI, as well as narrowband photometry taken in the Hale–Bopp (HB) system of Farnham et al. (2000). The telescope and CCD characteristics for this observing are described in Schleicher & Bair (2011) and Knight & Schleicher (2015), with HB system filters that isolate emission bands in OH, NH, CN, C₃, and C₂, as well as narrowband continuum points in the UV, blue, and green.

For best results with thesedFit code, we found that three elements were needed in the photometric input data. First, visible data were necessary to broadly characterize the Wien short-wavelength side of the SED. Second, near-infrared data were essential for characterizing the Rayleigh–Jeans long-wavelength side of the SED. Since most of our targets’ spectral peaks are located between 3000 and 6000 K, data bracketing the peak atλ ∼ 1μm were necessary. Third, for the resolution of the degeneracies between interstellar extinction, spectral types, and luminosity classes, narrowband (Δλ ∼ 60–100 Å) photometric data were particularly helpful and motivated our observing program at the 31 inch.

A note on V- and K-band magnitudes. Given the extensive use ofV andK magnitudes (and their dereddened counterparts,V₀ andK₀) in the analyses of Sections7 and9, it is worthwhile to note in detail the nature of these parameters. Both symbols are intended to denote the standard Johnson bandpasses and zero-points; for the visibleVband (Johnson & Morgan1951; Arp1958; Bessell1990), this is to be expected, but for the near-infraredKband (Mendoza v1963; Bessell & Brett1988), a likely alternative would beK_s found in 2MASS (Cutri et al.2003a). Unfortunately, for the majority of our stars—which, particularly in the near-infrared, were rather bright—we found that theK_s data was saturated, and that use of theK_s data in oursedFit process led to poor fits. Since most of our program stars had JohnsonK-band data present in the CIO, which did not make for problematicsedFit fits, we elected to use this variant of theK band in our study.

We have previously used thesedFit code in similar investigations (van Belle et al.2007,2009,2013,2016; van Belle & von Braun2009) for determinations ofF_bol andA_V. The process is rather straightforward: input photometry is matched against an input spectral template via amplitude scaling of that template, optimized through a Marquardt–Levenberg (M-L) least-squares technique. As an option, the overall spectral template can also be optimized for reddeningA_V. Reddening corrections are based upon the empirical determination by Cardelli et al. (1989), which is only marginally different from van de Hulst’s theoretical reddening curve No. 15 (Johnson1968; Dyck et al.1996a).

If the input spectral template has an assigned effective temperature estimate, then an angular size prediction can also be produced. ¹⁵ A multithreaded wrapper script enables many (∼dozens) of input spectral templates to be tested against the same input photometry for establishing which template is most appropriate for that photometry. An important recent improvement to thesedFit code has been the ability to incorporate detailed spectral profiles and optimized zero-points for individual filters of the photometric systems; a number of specific filters have significantly skewed and/or side-weighted profiles (e.g., JohnsonR band) that benefit significantly from this improvement. As a result, the detailed empirical characterization of those filters based upon HST NGSL data computed by Mann & von Braun (2015) were included as part of the SED fitting.

ThesedFit code works as follows. Input photometry (consisting of values and their uncertainties) is read in, matched to its photometric system, and converted, if necessary, from astronomical magnitudes to flux values using standard values for its photometric system. These points are then scaled in amplitude for comparison against a reference spectral template (e.g., an INGS-linked PHOENIX model). That spectral template can be reddened as noted above, and the whole process converges to an optimum solution using an M-L least-squares technique. Once optimized, the code explores the relevant reducedχ² space to establish uncertainties on template amplitude and reddening. Finally, the code computes the indicated bolometric flux via a sum across the wavelength of the spectral template. This process can be automatically repeated to explore a grid of templates for the best-fit template against a set of photometry, as indicated by the reducedχ² values for each template.

A significant element of thesedFit process is establishing the correct input spectral template for SED fitting. Our technique for arriving at the optimal template began with a literature search for previously published spectral types for a given star; ideally, this was reported from at least two sources. These spectral types and original references are given in Table4.

Following the mapping of that spectral type to PHOENIX models from our analysis in Section4.1, a grid of spectral templates inT_eff and about that location was examined for the optimum spectral template. The best fit in, and the corresponding spectral type, is also given in Table4, along with the resultant estimates forF_bol andA_V. While the range ofA_V estimates is small—typically a few tenths of a magnitude—corrections at this level are important in attempting to achieveF_bol measurements and derivedT_eff values at our optimum levels of accuracy and precision. Estimation of the appropriate reddeningA_V for a given object also allowed us to establish dereddened photometric values forV andK, as well as the dereddened colorV₀ −K₀.

The resulting values forF_bol are evenly distributed as a function of source angular size, as seen in Figures3 and4; we have used Equation (1) to show “standard”F_bol versusθ curves for given values ofT_eff.

Zoom InZoom OutReset image size

Zoom InZoom OutReset image size

TheF_bol estimation process has the following sources of error. First, the observables—the photometry—have inherent measurement errors, as well as errors in the corrections applied for Earth’s atmospheric extinction. Next, we interpret those observables radiometrically, using zero-point values, which themselves have uncertainty. Finally, we model those radiometric points with a model SED—the PHOENIX models—while applying interstellar extinction corrections. Once thesedFit process has determined the optimum template for a given star, the median 1σ uncertainty reported by the code after exploration of the reducedχ² space for that optimum template in ourF_bol values is 2.1%. This level of uncertainty is consistent with (a) a limiting photometric (observable) accuracy of 1%–2%, as seen in Table6; (b) zero-point errors, which, as seen in Mann & von Braun (2015), have a median error of 0.5%; and (c) the SED modeling with its interstellar extinction corrections. On that latter point, the overall flux levels are set by the agreement between the SED models, with the notable caveat that application of interstellar extinction could potentially lead to spurious results when a mismatched spectral template is overly reddened. To break this degeneracy, we found that narrowband photometry was particularly useful in ensuring that the correct template was utilized for a given star. Specifically, when using narrowband photometry, we found that the reducedχ² values for spectral templates offset from the optimum template increased, on average, by a value of for wideband photometry alone, this value was less than half (i.e., not statistically significant).

Taking these elements into account, the SED fitting process established the estimates of uncertainty forF_bol andA_V simultaneously through the appropriate exploration ofχ² space; the median uncertainty inA_V of 2% is consistent with our medianF_bol uncertainty. Finally, one minor potential source of systematic error could result from our spectral type mapping exercise of Section4.1, with disagreement between the Pickles templates and PHOENIX models. However, the matched Pickles–PHOENIX pairs showed disagreement in overall bolometric flux at only the ∼0.1% level, well below the general level of uncertainty in theF_bol estimates.

To convert the wavelength-specific UD angular size measurements from PTI into more general LD Rosseland mean angular diameters, we applied the standard procedure of estimating a multiplicative factor for converting theK-band UD diameters to LD diameters. One advantage of theK-band PTI data is that this correction is small (on the order of 2%–6%) relative to corrections for observations at shorter wavelengths; systematic uncertainties, if any, that exist in these corrections will therefore have a smaller impact on the results. For example, the visible-light study of Mozurkewich et al. (2003) had UD-to-LD factors that averaged ∼9% and went as high as ∼20%.

Previous estimates of such conversion factors by Davis et al. (2000, hereafterDTB00) are now superseded by those in Neilson & Lester (2013, hereafterNL13), the latter of which we used in our investigation. The principal improvement between these two investigations is a transition from plane-parallel to spherically symmetric model atmospheres. Also,NL13 reported results from a plane-parallel evaluation, which is similar toDTB00, with a slight decrease in the magnitude of the correction. The spherically symmetric results for the stars hotter than >4000 K remain similar to the two plane-parallel results; but for stars cooler than 4000 K, the UD-to-LD correction trends upward from ∼1.03 to ∼1.06. This is consistent with the expectation that, for these cooler extended atmospheres, the spherically symmetric model atmospheres start to deviate from plane-parallel models and improve predictions of the wavelength dependence of the structure (Scholz & Takeda1987; Hofmann & Scholz1998). Values taken fromNL13 were for 2.5M_⊙ stars (out of a choice of 1.0, 2.5, and 5.0M_⊙ stars) and interpolated betweenNL13 temperature data point spacing of 100 K for use in our study; see the last column of Table8. As demonstrated in Section8.2, this mass range is a reasonable choice for our program stars.

To quantify the numerical impact of these corrections if we were to not utilize them, we impose a uniform UD-to-LD conversion value of 1.026 (the average of theDTB00 values in Table8 for our typical program starT_eff range of 3000–5200 K), resulting in a shift downward inT_eff for the hottest stars in the range by −4 K and the cooler stars by −60 K relative to our final adopted answer using theNL13-fit values. To quantify the impact of uncertainties in this correction factor, we assume that the correction factors presented in Table8 have an uncertainty of ±0.01 and the finalT_eff values presented in Table9 would be impacted by ±20 K, well below the errors in those values from other sources.

Incorporating the limb-darkening corrections of the previous subsection, the final values for effective temperature can be computed, as seen in Table9. The UD angular sizes as measured by PTI are given with their formal and systematic errors (as noted in Table3). An initial estimate ofT_eff is then computed using angular sizes and the computed bolometric fluxes (Table4) in the standard relationship,

where the units ofF_bol are 10⁻⁸ erg s⁻¹ cm⁻², andθ is in milliarcseconds. Equation (1) is used withθ_UD for an initial estimate ofT_eff to select the UD-to-LD correction, which is consequently applied toθ_UD to obtain the LD Rosseland mean diameterθ_R. Additional iterations on theT_eff-to-correction step are not necessary, since this would have the effect of revisingT_eff further by 10⁻⁴, which is far below our measurement error. Withθ_R andF_bol in Equation (1), a final value forT_eff is then computed and presented in the last column of Table9.

The median uncertainty of ourT_eff values is 49 K (1.25%). Of this error, the principal contribution is error inθ_UD; if there were no uncertainty inF_bol, the total random error inT_eff values is still 42 K (1.00%). Conversely, ifθ_UD uncertainty were eliminated, the total random error inT_eff values would be 22 K (0.55%).

Table 10. LuminositiesL and RadiiR for the Program Stars

Note. We discuss dereddened colors in Section4.3, spectral typing in Sections4.1 and4.3, distances and radii in Section6, and luminosities in Section8.1.

Only a portion of this table is shown here to demonstrate its form and content. Amachine-readable version of the full table is available.

Download table as: DataTypeset image

OurT_eff results as indexed byV₀ −K₀ color are seen in Figure5, with a zoom on the central results in the range ofV₀ −K₀ = {2, 6} in Figure6. Unsurprisingly,T_eff is tightly correlated with this color. The general progression ofT_eff withV₀ −K₀ is summarized in Table11, wherein we step through our sample in steps of ΔV₀ −K₀ = 0.1, increasing the bin sizes on the red end to account for the increasing sparsity of data. For this table, we estimate the unbiased sample variance, taking into account the varying weights for the parameters {V₀ −K₀,T_eff},

whereμ* is the weighted mean for each of these parameters,w_i = 1/σ²,, and. Given the occasional large spread in measurement error ofT_eff for a given star, as seen in Table9, an approach such as this was necessary to give proper consideration to the varying weights of each pair of data points in determining the bin weight (Kendall et al.1987). For the data in Table11, the medianT_eff uncertainty is 66 K bin^–1.

Zoom InZoom OutReset image size

Zoom InZoom OutReset image size

7.1.1. Gap Analysis:T_eff versusV₀ −K₀

A striking characteristic of theT_eff versusV₀ −K₀ plots (Figures5 and6) that merited further investigation was the apparent gaps in the continuum of data points in the region of densest sampling ofV₀ −K₀ color space,V₀ −K₀ = {1.9, 6.5}. Two such gaps are apparent to visual inspection, located at {T_eff,V₀ −K₀} = {4350 K, 3.10 mag} and {3600 K, 4.55 mag}. However, to test our intuition in this regard, we developed a Monte Carlo test to examine the likelihood that these gaps in our data were simply the product of random fluctuation.

The initial step in this analysis was to collect the observational data in the region of interest,V₀ −K₀ = {1.9, 6.5}, which encompasses a subsample ofN = 183 of our stars. A polynomial fit ofT_eff versusV₀ −K₀ gave the following result:

For our subsample in this range of interest, the median individual measurement errors of our observational data are K and mag; the median absolute difference between the individualT_eff measurements and the fit in Equation (4) is 50 K, which agrees well with our measurement error.

Given these data, we were able to construct synthetic populations of observational data points in {T_eff,V₀ −K₀} space. Each population consisted of the same number of observations (N = 183), and randomly generated values forV₀ −K₀ were generated in the range of {1.9, 6.5} from a uniform distribution, with a correspondingT_eff value determined from Equation (4). Each synthetic data point was then scattered by a typical measurement error by random generation of a from a normal distribution of width equal to the median error of the true observational data (49 K), which was then added to the synthetic data point. An identical step added a measurement error to the synthetic data point’sV₀ −K₀ value. Each of the synthetic data points was then assigned values for their measurement error equal to the median errors in our true observational data.

Once constructed, these synthetic populations could be “gap tested” by scanning a test particle along the Equation (4) fit line in steps of ΔV₀ −K₀ = 0.01 mag in the range of interest, {1.9, 6.5}. At each step, a region around the test particle would be examined for the presence of data points from the test population; if at any point along the scan region, a test particle’s examination region was found to be free of data points, that population would be considered to have a gap. For sufficiently small scan regions, nearly all populations would be found to have gaps; for sufficiently large regions, all populations would fail that test. (It is worthwhile to note that this test broadly seeks gaps anywhere along the population’sV₀ −K₀ sequence, not just at specific locations.) The statistics of gap likelihood for various region sizes can then be built by running a large number of synthetic populations; for our investigation, we foundN_pop = 1000 was a reasonable size for each test run.

Gap 1 testing. Examination of the first gap that we noted in our data at {T_eff,V₀ −K₀} = {4350 K, 3.10 mag} shows a gap of size ΔT_eff = ±88 K and ΔV₀ −K₀ = ±0.15 mag. Taking that as our scan region envelope, we find that a run ofN_pop = 1000 synthetic populations turns up only 53 as having gaps of this size anywhere along the range ofV₀ −K₀ values.

Gap 2 testing. Examination of the second gap that we noted in our data at {T_eff,V₀ −K₀} = {3500 K, 4.55 mag} shows a gap of size ΔT_eff = ±400 K and ΔV₀ −K₀ = ±0.10 mag. For this sample, we find that anN_pop = 1000 synthetic population run indicates only 70 as having gaps of this size.

“Control” run. Finally, as a check on the overall likelihood of being able to conceal a data point of “typical” size in the sample, we tested our process against a gap matched in size to the median errors in bothT_eff andV₀ −K₀, namely, ΔT_eff = ±49 K and ΔV₀ −K₀ = ±0.07 mag. Interestingly, in this only slightly smaller scan region, only one of the 1000 synthetic populations has no gaps.

Overall, we took these results from our first and second gap analyses as motivating evidence that one or more astrophysical phenomena are quite likely causing these overall discontinuities in what would otherwise be a continuum of points in {T_eff,V₀ −K₀} space.

Gaps being noted in photometric color sequences are not a new phenomenon. Böhm-Vitense (1970a,1970b,1981,1982) noted that the onset of surface convection in hotter stars (T_eff ∼ 7250 K) would result in a discontinuity inM_V versusB −V data, given how the temperature differences between convective and radiative layers were expected to affectB-band filters. Evidence for detection of these gaps was initially sparse (Böhm-Vitense & Canterna1974; Rachford & Canterna2000), and their existence was disputed (Mazzei & Pigatto1988; Newberg & Yanny1998) until gaps in HipparcosM_V versusB −V data were noted by de Bruijne et al. (2000,2001). Böhm-Vitense (1995a,1995b) further predicted a second gap atT_eff ∼ 6400 K, associated with “a sudden increase in convection zone depths,” for which evidence was found in the de Bruijne et al. (2001) investigation.

Recently, the detection of a possible gap in Gaia data forM_G versusG_BP −G_RP for cooler stars was noted by Jao et al. (2018), who attributed it to the luminosity–temperature regime where M dwarf stars transition from partially to fully convective (roughly M3.0V). This detection is particularly relevant to our finding here, in that the second of our gaps—at {T_eff,V₀ −K₀} = {3500 K, 4.55 mag}—corresponds well with the gap detected by Jao et al.

For examining the relationship betweenT_eff and spectral type, we took two approaches to arranging the data. First, we sorted by straight spectral type, and second, we sorted byV₀ −K₀ color, with an analysis of the resulting relationship seen betweenT_eff and spectral type in each approach.

7.2.1. Sorted into Spectral Type Bins

Although stellar spectral typing may at times be considered to be subjective (e.g., different practitioners can arrive at different spectral types of individual stars, as discussed in Section4.3 and seen in the range of results for individual stars in Skiff2014), it remains a useful shorthand for quickly referencing stellar properties. We have collected the generalT_eff properties using the spectral type values assigned during thesedFit process in Section4.3. The rawT_eff versus spectral type values, and general trends, can be seen in Figure7.

Zoom InZoom OutReset image size

For each spectral subtype, the weighted averageT_eff and weighted varianceσ_T (as described in Section7.1) were computed; these values are seen in Table12 and Figure8. A simple linear fit (T_eff =a ×ST +b) was applied to the data seen in Figure7 and can be seen as well in Figure8; the fit values are presented in Table13. Minimization of theχ² metric was used not only to optimize the line fits but also to correctly select when to bracket spectral type ranges. A continuity requirement was enforced at spectral type values that joined two linear fit regions.

Zoom InZoom OutReset image size

Table 12. Spectral Type as a Function of Effective Temperature for Our Program Stars

Sp. Type	Sp. Type Num.	N	Teff	σT	Tfit	χ2/dof
G0III	50	1	4963	0	0	0.00
G1III	51	1	5120	0	5166	⋯
G3III	53	2	5062	24	5061	⋯
G4III	54	6	5023	46	5008	0.11
G5III	55	13	4963	71	4955	0.01
G8III	58	11	4781	122	4797	0.02
K0III	60	8	4715	43	4692	0.30
K1III	61	4	4634	107	4639	⋯
K1.25III	61.25	5	4528	47	4537	0.03
K1.5III	61.5	9	4424	80	4487	0.62
K2III	62	4	4387	32	4387	⋯
K3III	63	1	4452	0	4188	⋯
K3.25III	63.25	10	4183	59	4138	0.61
K3.5III	63.5	8	4063	98	4088	0.06
K4III	64	8	3951	63	3988	0.35
K5III	65	18	3911	54	3902	0.03
M0III	66				3816
M1III	67	13	3777	102	3730	0.21
M2III	68	10	3617	92	3644	0.09
M3III	69	8	3559	120	3558	⋯
M4III	70	27	3466	114	3472	⋯
M5III	71	14	3381	69	3386	0.01
M5.5III	71.5	4	3301	147	3343	0.08
M6III	72	2	3112	29	3134	0.60
M7III	73	1	3114	0	3134	⋯
M7.5III	73.5	1	3121	0	3134	⋯
M7.75III	73.75	2	3145	22	3134	0.27

Note. Spectral types of our program stars as determined by our SED fitting approach as a function of effective temperatures for our program stars, with italicized values for where interpolation provided aT_eff value. For more details, see Section7.2.

Download table as: ASCIITypeset image

Table 13. Linear Fits ofT_eff vs. Spectral Type for Specific Spectral Type Ranges

Spectral Type	Slope	Intercept	χ2	Median
Range	a	b		ΔT (K)
51–61	−52.74	7856	0.44	52
61–64	−199.41	16751	1.68	53
64–71.5	−85.98	9491	0.77	52
72–74	0.00	3134	0.87	28

Note. These fits are plotted in Figure8. Spectral type indices range from G0 = 50, K0 = 60 and M0 = 66. Median ΔT is the median average difference between the linear fit and the individual data points in the spectral type range. For more details, see Section7.2.

Download table as: ASCIITypeset image

7.2.2. Sorted byV₀ −K₀ Color into Bins of Five

As a secondary check against the possible subjectivity of spectral typing in exploring theT_eff versus spectral type trends seen in Figures7 and8, we explored arranging our spectral type data in the following alternative way. First, we sorted the stars byV₀ −K₀ color and then binned the data into bins of five. Once done, we then determined the average spectral type and weighted averageT_eff values for each bin of five stars. These data can be seen in Table14 and Figure9 and qualitatively duplicate the results seen in Figures7 and8. The principal difference between the latter figure and the first two is a measure of spectral type “smearing” in each of theV₀ −K₀ bins (e.g., the bin widths in spectral type are one to two subtypes), illustrating that spectral type,T_eff, andV₀ −K₀ color do not uniquely map in 3D space for these stars.

Zoom InZoom OutReset image size

Table 14. T_eff vs. Spectral Type

V0 −K0	Sp. Type Num.	Teff
1.974 ± 0.031	53.20 ± 0.54	5060 ± 62
2.081 ± 0.032	54.60 ± 0.79	5011 ± 43
2.111 ± 0.029	54.00 ± 0.58	5016 ± 60
2.167 ± 0.028	55.40 ± 0.62	4901 ± 126
2.223 ± 0.033	57.40 ± 0.65	4870 ± 124
2.258 ± 0.030	57.00 ± 0.45	4867 ± 134
2.369 ± 0.029	58.40 ± 0.46	4773 ± 111
2.454 ± 0.025	60.00 ± 0.71	4721 ± 75
2.524 ± 0.033	60.05 ± 0.68	4668 ± 100
2.622 ± 0.031	61.65 ± 0.79	4513 ± 52
2.745 ± 0.028	61.05 ± 0.57	4488 ± 41
2.874 ± 0.028	61.45 ± 2.24	4410 ± 67
2.983 ± 0.031	61.80 ± 1.29	4396 ± 90
3.166 ± 0.030	63.10 ± 0.95	4178 ± 95
3.269 ± 0.035	63.30 ± 2.24	4179 ± 93
3.360 ± 0.039	63.40 ± 1.83	4110 ± 115
3.456 ± 0.032	63.95 ± 0.93	4033 ± 66
3.520 ± 0.040	64.40 ± 0.91	3947 ± 76
3.627 ± 0.035	64.80 ± 1.12	3929 ± 67
3.759 ± 0.032	65.00 ± 0.25	3916 ± 46
3.849 ± 0.029	65.00 ± 0.71	3921 ± 114
3.981 ± 0.032	65.80 ± 0.65	3842 ± 47
4.181 ± 0.035	66.60 ± 0.79	3744 ± 85
4.285 ± 0.028	67.20 ± 1.12	3723 ± 77
4.390 ± 0.029	67.80 ± 0.79	3725 ± 91
4.510 ± 0.033	67.50 ± 0.50	3603 ± 117
4.772 ± 0.039	68.80 ± 0.79	3558 ± 132
4.867 ± 0.033	69.20 ± 0.79	3549 ± 93
4.952 ± 0.028	69.80 ± 1.12	3530 ± 94
5.045 ± 0.036	69.60 ± 0.79	3484 ± 40
5.159 ± 0.036	70.20 ± 1.12	3453 ± 32
5.257 ± 0.030	70.00 ± 0.25	3426 ± 87
5.397 ± 0.037	69.60 ± 0.79	3442 ± 68
5.523 ± 0.024	70.60 ± 0.91	3440 ± 54
5.786 ± 0.033	70.80 ± 1.12	3340 ± 144
6.113 ± 0.028	71.10 ± 1.58	3356 ± 54
6.542 ± 0.032	71.40 ± 1.12	3242 ± 117
8.109 ± 0.029	73.20 ± 0.85	3116 ± 33

Note. TheT_eff results are sorted byV₀ −K₀ and then collected into bins ofN = 5. Spectral type indices range from G0 = 50, K0 = 60, M0 = 66. For more details, see Section7.2 and Figure9.

Download table as: ASCIITypeset image

7.2.3. Calibration of Spectral Type versusV₀ −K₀

One useful tangential result from the analyses performed in support of this investigation is that a calibration ofV₀ −K₀ versus spectral type can be explored. “Intrinsic” values for giant stars of this color were presented in Bessell & Brett (1988, hereafter BB88), who provided a comparison basis for our own values, presented in Table15 and Figure10. Interestingly, our calibration of this relationship shows a consistently redward trend; the difference between ourV₀ −K₀ colors for a given spectral subclass and those ofBB88 is that our values are redder by a median value of ΔV₀ −K₀ = +0.11.

Zoom InZoom OutReset image size

Table 15. V₀ −K₀ by Spectral Type from Our SED Fitting

Spectral	Sp. Type	N	V0 −K0	V0 −K0
Type	Num.		(BB88)	(This Work)
G0III	50	1	1.75	2.024 ± 0.073
G1III	51	1	1.83	2.121 ± 0.072
G2III	52		1.90	2.078 ± 0.088
G3III	53	2	1.98	2.035 ± 0.051
G4III	54	6	2.05	2.156 ± 0.024
G5III	55	13	2.10	2.126 ± 0.019
G6III	56		2.15	2.207 ± 0.028
G7III	57		2.16	2.288 ± 0.028
G8III	58	11	2.16	2.369 ± 0.021
G9III	59		2.24	2.393 ± 0.032
K0III	60	8	2.31	2.416 ± 0.025
K1III	61	4	2.50	2.519 ± 0.027
K1.25III	61.25	5	2.55	2.694 ± 0.030
K1.5III	61.5	9	2.60	2.865 ± 0.022
K2III	62	4	2.70	2.990 ± 0.034
K3III	63	1	3.00	2.753 ± 0.101
K3.25III	63.25	10	3.07	3.194 ± 0.023
K3.5III	63.5	8	3.13	3.373 ± 0.028
K4III	64	8	3.26	3.612 ± 0.026
K5III	65	18	3.60	3.785 ± 0.019
M0III	66		3.85	3.974 ± 0.026
M1III	67	13	4.05	4.163 ± 0.018
M2III	68	10	4.30	4.603 ± 0.024
M3III	69	8	4.64	4.718 ± 0.024
M4III	70	27	5.10	5.222 ± 0.013
M5III	71	14	5.96	5.842 ± 0.019
M5.5III	71.5	4	6.40	6.527 ± 0.034
M6III	72	2	6.84	7.435 ± 0.068
M7III	73	1	7.80	7.632 ± 0.073
M7.5III	73.5	1	...	7.674 ± 0.064
M7.75III	73.75	2	...	8.473 ± 0.040

Note. For comparison, the values fromBB88 are also presented. Our values are in the last column, with italicized values for where interpolation providedV₀ −K₀ values. These data are plotted in Figure10. For more details, see Section7.2.3.

Download table as: ASCIITypeset image

In contrast to the tight correlations seen inT_eff versusV₀ −K₀ in Section7.1, we find that values for stellar radiusR show considerably more scatter. As discussed in Section6, there are uncertainties in our targets’ distance determinations and, consequently, significant intrinsic scatter forR as a function ofV₀ −K₀; the raw data are seen in Figure11.

Zoom InZoom OutReset image size

In order to assess any possible general trend in the relation between stellar radius and dereddened color, we binned the data into sections of width ΔV₀ −K₀ = 0.5. The first and final bins (ΔV₀ −K₀ = {1.5, 2.5} and ΔV₀ −K₀ = {6.0, 9.0}) are wider to capture the color outliers. For each one of the bins, a weighted average radius for all of the stars in that given bin was computed. To assess the impact of significant outliers, a second step was taken; the data for a given bin were assessed with a linear fit (R =a × (V₀ −K₀) +b), and the 10σ outliers were excluded from a second weighted average computation of meanR for that bin. In most bins, this did not result in a significant change in the expectedR value for that bin. These results are presented in Table16.

The general trend shows a relatively flat size distribution for the bluest ΔV₀ −K₀ = {1.5, 2.5} stars at ∼11R_⊙ and then a steady increase in linear size as theV₀ −K₀ color becomes redder. This becomes more gradual as the stellar linear sizes exceed 100R_⊙ but still trends upward with increasingV₀ −K₀ as the expectation for the stellar linear size hits ∼150R_⊙ for the reddest stars. Overall,V₀ −K₀ is indicative of stellar linear size at only the ∼30% level.

Similar to Section7.2, we examinedR versus spectral type by arranging the data by spectral type and then byV₀ −K₀ color.

7.4.1. Sorted into Spectral Type Bins

7.4.2. Sorted byV₀ −K₀ Color into Bins of Five

Similar to the second part of Section7.2, to check against the possible subjectivity of spectral typing in exploring theR versus spectral type trends seen in Figure12, we again arranged our spectral type data into bins of five. We then determined the average spectral type and weighted averageR values; these data are presented in Table18 and Figure13.

Zoom InZoom OutReset image size

Zoom InZoom OutReset image size

Table 18. R vs. Spectral Type Sorted byV₀ −K₀ and Collected into Bins ofN = 5

Sp. Type Num.	V0 −K0	R	Rfit
53.2 ± 0.5	1.97 ± 0.03	9.2 ± 1.8	9.8
54.6 ± 0.8	2.08 ± 0.03	11.1 ± 2.6	10.1
54.0 ± 0.6	2.11 ± 0.03	10.0 ± 2.0	10.0
55.4 ± 0.6	2.17 ± 0.03	11.4 ± 5.1	10.2
57.4 ± 0.6	2.22 ± 0.03	12.0 ± 3.0	10.5
57.0 ± 0.4	2.26 ± 0.03	11.1 ± 3.4	10.4
58.4 ± 0.5	2.37 ± 0.03	9.8 ± 1.9	10.7
60.0 ± 0.7	2.45 ± 0.03	11.5 ± 2.1	10.7
60.1 ± 0.7	2.52 ± 0.03	11.5 ± 3.0	11.0
61.7 ± 0.8	2.62 ± 0.03	14.2 ± 4.1	22.9
61.1 ± 0.6	2.74 ± 0.03	16.5 ± 5.5	18.4
61.5 ± 2.2	2.87 ± 0.03	22.2 ± 4.8	21.4
61.8 ± 1.3	2.98 ± 0.03	24.4 ± 7.7	24.0
63.1 ± 1.0	3.17 ± 0.03	30.2 ± 12.5	33.6
63.3 ± 2.2	3.27 ± 0.04	34.6 ± 7.0	35.1
63.4 ± 1.8	3.36 ± 0.04	26.0 ± 8.5	35.8
64.0 ± 0.9	3.46 ± 0.03	43.3 ± 10.7	39.9
64.4 ± 0.9	3.52 ± 0.04	37.6 ± 8.4	43.2
64.8 ± 1.1	3.63 ± 0.04	37.2 ± 12.6	46.2
65.0 ± 0.3	3.76 ± 0.03	43.2 ± 10.9	47.6
65.0 ± 0.7	3.85 ± 0.03	52.3 ± 23.0	47.6
65.8 ± 0.6	3.98 ± 0.03	65.5 ± 6.2	53.6
66.6 ± 0.8	4.18 ± 0.03	61.2 ± 12.0	61.6
67.2 ± 1.1	4.28 ± 0.03	62.5 ± 13.6	68.3
67.8 ± 0.8	4.39 ± 0.03	82.3 ± 8.6	75.0
67.5 ± 0.5	4.51 ± 0.03	76.9 ± 22.8	71.6
68.8 ± 0.8	4.77 ± 0.04	75.4 ± 33.5	86.1
69.2 ± 0.8	4.87 ± 0.03	67.4 ± 12.1	90.5
69.8 ± 1.1	4.95 ± 0.03	103.2 ± 13.5	97.2
69.6 ± 0.8	5.04 ± 0.04	102.0 ± 12.2	95.0
70.2 ± 1.1	5.16 ± 0.04	75.1 ± 19.7	101.6
70.0 ± 0.3	5.26 ± 0.03	90.1 ± 11.8	99.4
69.6 ± 0.8	5.40 ± 0.04	90.8 ± 18.7	95.0
70.6 ± 0.9	5.52 ± 0.02	104.2 ± 13.9	106.1
70.8 ± 1.1	5.79 ± 0.03	100.6 ± 33.3	108.3
71.1 ± 1.6	6.11 ± 0.03	124.8 ± 4.6	111.6
71.4 ± 1.1	6.54 ± 0.03	138.8 ± 40.7	115.0
73.2 ± 0.8	8.11 ± 0.03	172.0 ± 37.4	135.0

Note. The fit values in the last column are from the linear fits noted in Table19. Spectral type indices range from G0 = 50, K0 = 60 and M0 = 66. Data are shown in Figure13. For more details, see Section7.4.

Download table as: ASCIITypeset image

It is apparent in the figure that arranging our data in this fashion results in qualitatively “cleaner” data, exhibiting clear trends inR versus the binned spectral type values. For each of the GKM spectral types in this data set, we performed a linear fit to assess theR trend as a function of binned spectral type; the linear fit parameters are presented in Table19. A continuity constraint was enforced for the fitting procedure so that there were no “jumps” inR at the boundaries of the linear fit regimes. The results in this table are consistent with the previous sections: the G-type giants show an almost flat linear radius across all subtypes at ∼11R_⊙; the K-types present a trend of increasing size with subtype, hitting a maximum of ∼50R_⊙ at K5III; and the M-types have an increased slope in their size–subtype relationship, hitting a size of ∼150R_⊙ for the reddest in this group. The average deviation of the data with respect to these linear fits across the GKM range is 11%.

Table 19. Linear Fits forR vs. the Spectral Type Numbers in Table18 and shown Figure13

Sp. Type Range	a	b	χ2
G	0.16	1.41	0.85
K	7.39	−432.90	5.41
M	11.11	−678.55	7.05

Note. For more details, see Section7.4.

Download table as: ASCIITypeset image

Given the values of stellarF_bol and distance, stellar luminosityL can be calculated. Note that, as discussed in detail in theAppendix,L is not dependent upon our determinations ofR andT_eff but ratherF_bol andπ; see Equation (A3). Our target luminosity values are seen in Table10 and Figure14.

Zoom InZoom OutReset image size

For further illustration, we took stellar evolutionary tracks inL versusT_eff from the BaSTI models ¹⁶ in Pietrinferni et al. (2006), using the solar abundance tracks ([M/H] = 0.058,Z = 0.0198,Y = 0.273) for two representative mass tracks,M = 1.2 and 2.4M_⊙. Our choice of solar abundance was rather perfunctory and motivated only by there being no significant impetus to choose any other track. The mass values were chosen to represent values expected to be characteristic of the lowest-mass giants atM = 1.2M_⊙, with twice that to explore the impact of mass on the track path. Additionally, we cross-referenced our target list against those stars called out in Gontcharov (2008) as red giant clump stars. Clump giants are post–helium flash giants undergoing core helium burning, with an inert hydrogen envelope surrounding the core (Gontcharov2017). These tracks, along with our program stars, are shown in Figure14. Along the evolutionary tracks, time intervals of Δt = 10 Myr are indicated, and there is a pileup of those interval ticks in the lower left corner of the plot, characteristic of red giant clump stars.

Interestingly, when identified as clump giants, it is qualitatively apparent that there is an overdensity of our program stars in that region of our HR diagram, which is consistent with the increased loiter time of the evolutionary tracks in that area of the plot. Overall, the BaSTI evolutionary tracks bracket our program stars well, with most outliers coming to the upper left of theM = 2.4M_⊙ stars, indicative of higher-mass objects.

A precision measurement of stellar linear radius enables the calculation of stellar mass, provided a value for a given star’s surface gravity is available. For that purpose, we utilized the most recent version of the PASTEL catalog of Soubiran et al. (2016), using the 2020 January 30 version, which features an inventory of such values from the literature. Ninety-one of our program stars can be found in this catalog; the values we used from PASTEL are seen in Table21, along with their respective primary references.

Table 21. Reported Values for the Program Stars Found in the PASTEL Catalog of Soubiran et al. (2016) along with Resulting Mass DeterminationsM

Star ID		References	R	M
			(R⊙)	(M⊙)
HD 3546	2.83 ± 0.19	da Silva et al. (2015)	9.37 ± 0.26	1.20 ± 0.53
HD 3574	1.44	McWilliam (1990)	105.52 ± 6.59	6.21 ±3.66
HD 3627	2.56	Zhao et al. (2001)	14.24 ± 0.44	1.49 ±0.86
HD 6186	2.99	McWilliam (1990)	11.59 ± 0.30	2.66 ±1.54
HD 7087	2.77	McWilliam (1990)	20.54 ± 0.76	5.03 ±2.92
HD 8126	1.93	McWilliam (1990)	27.82 ± 0.77	1.33 ±0.77
HD 9927	2.17 ± 0.14	Maldonado & Villaver (2016)	20.19 ± 0.45	1.22 ± 0.40

Note. For references that did not report a formal error in, a value of 0.25 dex was assumed (and the resulting error inM is italicized). For more details, see Section8.2.

References. Kyrolainen et al. (1986), Smith & Lambert (1986), Smith & Lambert (1990), McWilliam (1990), Mallik (1998), Zhao et al. (2001), Hekker & Meléndez (2007), Thygesen et al. (2012), Mortier et al. (2013), Matrozis et al. (2013), Ramírez et al. (2013), Adamczak & Lambert (2014), Luck (2015), Jofré et al. (2015), da Silva et al. (2015), Heiter et al. (2015), Maldonado & Villaver (2016), Deka-Szymankiewicz et al. (2018), Lomaeva et al. (2019).

Only a portion of this table is shown here to demonstrate its form and content. Amachine-readable version of the full table is available.

Download table as: DataTypeset image

For stars that haveR and, (g =GM/R²) can be trivially rearranged to solve forM, minding the usual conversion factors and error propagation, so long as errors are provided for. For the PASTEL catalog measurements that had no value for the uncertainty in, a value of 0.25 was assigned as a placeholder.

Overall, this approach is quite crude, and limited as well. On the former point, our average mass error was typically 50%; this improved only slightly, by about 10%, if the measures were restricted to those that reported formal errors. For this approach’s limitations, it is worth noting that few of our program stars have measures if they are belowT_eff < 4000 K, which is indicative of the difficulty currently found in determining for these lower-temperature stars.

Nevertheless, this approach is at least qualitatively illustrative of mass effects on our results. If we replot our Figure14 with the corresponding mass information, in Figure15, we can see that the points that lie above and to the left of the BaSTI evolutionary tracks are, as expected, the stars that indicate higher masses. Quantitatively, the values in Table21 are consistent with expectations. The median value of our program stars for which we can calculate a value for mass isM = 1.9M_⊙; these program stars are bracketed well by theM = {1.2, 2.4}M_⊙ evolutionary tracks in Figure14.

Zoom InZoom OutReset image size

If stellar measures can be (a) improved to better than 0.05 dex and (b) extended to the lower-temperature ranges, the community will have a useful tool for ∼10% or better mass determinations of stars in this region of the HR diagram. In so doing, the measures will approach the current noise floor on the uncertainty inM that is contributed by the linear radius measures, which is approximately 8.5%.

All data produced for this investigation will be made available at the NASA Exoplanet Science Institute (NExScI) PTI archive. ¹⁷ This includes all tables in this paper andsedFit figures for each of our program stars.