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On the Use of Empirical Bolometric Corrections for Stars

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On the Use of Empirical Bolometric Corrections for Stars Draft version August 25, 2010 A Preprint typeset using LTEX style emulateapj v. 11/10/09 ON THE USE OF EMPIRICAL BOLOMETRIC CORRECTIONS FOR STARS Guillermo Torres Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA; e-mail: [email protected] Draft version August 25, 2010 ABSTRACT When making use of tabulations of empirical bolometric corrections for stars (BCV ), a commonly overlooked fact is that while the zero point is arbitrary, the bolometric magnitude of the Sun (Mbol,⊙) that is used in combination with such tables cannot be chosen arbitrarily. It must be consistent with the zero point of BCV so that the apparent brightness of the Sun is reproduced. The latter is a measured quantity, for which we adopt the value V⊙ = −26.76 ± 0.03. Inconsistent values of Mbol,⊙ are listed in many of the most popular sources of BCV . We quantify errors that are introduced by not paying attention to this detail. We also take the opportunity to reprint the BCV coefficients of the often used polynomial fits by Flower (1996), which were misprinted in the original publication. Subject headings: stars: fundamental parameters — stars: general — Sun: fundamental parameters — Sun: general 1. INTRODUCTION practical use of BCV tables. Bolometric corrections are widely used in Astronomy 2. COEFFICIENTS FOR THE FLOWER (1996) BCV AND to infer either luminosities or absolute magnitudes of log TEFF FORMULAE stars. Empirical corrections in the visual band, BCV , Flower (1996) expressed the bolometric corrections as are perhaps the most frequently needed, and numerous a function of the effective temperature of the star, and tabulations exist in the literature that differ sometimes presented polynomial fits for BC of the form significantly. This has been a persistent source of confu- V 2 sion among users. The most common way in which these BCV = a + b (log Teff )+ c (log Teff ) + ··· , (1) tables are employed is in combination with the bolomet- for three different temperature ranges. The coefficients ric magnitude of the Sun, M ,⊙, which is not a directly bol a, b, c, ··· were given in his Table 6 for each interval, but measured quantity. Many different values of Mbol,⊙ can be found in the literature as well, adding to the con- are missing powers of ten. We rectify this situation here fusion. For a given table of bolometric corrections the and present them with a larger number of significant dig- its (see Table 1). While this formulation is certainly choice of Mbol,⊙ is not arbitrary, however, since there is an observational constraint that must be satisfied, given very handy for many applications, it must be kept in by the measurement of the visual brightness of the Sun. mind that bolometric corrections become less reliable for This fact is often ignored, and as a result it is common cooler stars due to the limited number of observational constraints, and they break down completely for the M to see inconsistent uses of BCV that can lead to errors dwarfs. Two of the very few such constraints available in the luminosity of 10% or larger, or errors in MV of are for the low-mass eclipsing binaries CU Cnc (compo- 0.1 mag or more. The primary motivation for this pa- ≈ ≈ per is to call attention to this fact, to illustrate common nents with M 0.42 M⊙) and CM Dra (M 0.24 M⊙), which have known parallaxes. For these stars one may misuses of some of the most popular BCV tables, and to offer some perspective on the problem. compute the BCV values for the components directly One frequently used source of bolometric corrections from the estimated MV values and the bolometric mag- is the tabulation by Flower (1996), which is an update nitudes based on the absolute radius and temperature de- arXiv:1008.3913v1 [astro-ph.SR] 23 Aug 2010 on his earlier work (Flower 1977) based on a compila- terminations (Ribas 2003; Morales et al. 2009). Compar- ing the measured values with the predicted ones shows tion of effective temperatures (Teff ) and BCV determi- that the Flower relations give a BCV that is too negative nations for a large number of stars. Many authors find ∼ ∼ this source convenient because, in addition to the table, by 1.2 mag for CU Cnc, and 1.7 mag for CM Dra. Similar discrepancies are found in other tabulations of it presents simple polynomial fits for BCV valid over a wide range of temperatures. Unfortunately the original empirical bolometric corrections. publication had misprints in the coefficients of those for- A useful color/temperature calibration was also pre- mulae that have prevented their use, and an erratum was sented by Flower (1996) in his Table 5, separately for never published. The present author has received numer- supergiants and for main-sequence, subgiant, and giant ous inquiries about this problem over the years. Thus, stars, but unfortunately the formula given there was printed incorrectly, and should have expressed the tem- a second motivation for this paper is to make the cor- − rect coefficients available to the community, as well as perature (log Teff ) as a function of the B V color index, to amend the form of the equation and one of the coef- rather than the reverse. The polynomial fits are similar ficients for a color/temperature calibration in the same to those used for BCV : 2 paper that were also misprinted. We begin by present- log Teff = a + b (B−V )+ c (B−V ) + ··· . (2) ing these corrections, and follow with a discussion of the One of original coefficients was also misprinted, and a 2 Torres TABLE 1 Bolometric corrections by Flower (1996) as a function of temperature: 2 BCV = a + b(log Teff )+ c(log Teff ) + ··· Coefficient log Teff < 3.70 3.70 < log Teff < 3.90 log Teff > 3.90 a −0.190537291496456E+05 −0.370510203809015E+05 −0.118115450538963E+06 b 0.155144866764412E+05 0.385672629965804E+05 0.137145973583929E+06 c −0.421278819301717E+04 −0.150651486316025E+05 −0.636233812100225E+05 d 0.381476328422343E+03 0.261724637119416E+04 0.147412923562646E+05 e ··· −0.170623810323864E+03 −0.170587278406872E+04 f ··· ··· 0.788731721804990E+02 reference. By noting that TABLE 2 Effective temperature as a function of color − − 2 BCV,⊙ = mbol,⊙ V⊙ = Mbol,⊙ MV,⊙ , (6) (Flower 1996): log Teff = a + b(B−V )+ c(B−V ) + ··· it is immediately clear that setting a value for the bolo- Main-Sequence Stars, metric correction of the Sun is equivalent to specifying Coefficient Supergiants Subgiants, Giants the zero point of the bolometric magnitude scale, since a 4.012559732366214 3.979145106714099 V⊙ is a known quantity. A common practice when using b −1.055043117465989 −0.654992268598245 one of the many available tables of empirical bolometric c 2.133394538571825 1.740690042385095 corrections is to adopt a value for Mbol,⊙, but all too d −2.459769794654992 −4.608815154057166 e 1.349423943497744 6.792599779944473 often this is done without regard for whether the cho- f −0.283942579112032 −5.396909891322525 sen value is consistent with BCV,⊙ from the same table. g ··· 2.192970376522490 From eq.(6) we have h ··· −0.359495739295671 Mbol,⊙ = MV,⊙ + BCV,⊙ = V⊙ +31.572+ BCV,⊙ , (7) correction was issued by Prˇsa & Zwitter (2005). We where the numerical constant (with the opposite sign) present all coefficients again in Table 2 to higher pre- corresponds to the distance modulus of the Sun at 1 AU. cision. This formulation shows that once a particular tabula- tion of BCV values is adopted, the absolute bolometric magnitude of the Sun is no longer arbitrary. This fact has been emphasized by Bessell et al. (1998), and other authors, but is still largely overlooked. 3. THE USE OF EMPIRICAL BOLOMETRIC CORRECTIONS Direct measurements of V⊙ are difficult to make be- cause of the extreme difference in brightness between the As is well known, the apparent bolometric magnitude Sun and the stars that are used as the reference in the V of a star is defined as system, and also because the Sun is spatially resolved. ∞ Nevertheless, many careful determinations have been − mbol = 2.5 log fλdλ + C1 , (3) carried out over the years (although not very recently, Z 0 as far as we are aware), and the most reliable of the pho- where fλ is the monochromatic flux from the object per toelectric measurements have been reviewed by Hayes unit wavelength interval received outside the Earth’s at- (1985).1 Among them, one that carries particularly high mosphere, and C1 is a constant. The bolometric correc- weight is that of Stebbins & Kron (1957), which was ad- tion is usually defined as the quantity to be added to justed slightly (by −0.02 mag) by Hayes (1985) for an the apparent magnitude in a specific passband (in the error in the treatment of horizontal extinction, and fur- absence of interstellar extinction) in order to account for ther updated by Bessell et al. (1998) using modern V the flux outside that band: magnitudes for the reference stars. The result is V⊙ = − ± 2 − − 26.76 0.03. Two additional determinations discussed BCV = mbol V = Mbol MV . (4) by Hayes (1985) are those of Nikonova (1949), trans- We focus here on the visual band because that is the formed to the standard Johnson system by Martynov − ± context in which bolometric corrections were historically (1960), and Gallou¨et (1964): V⊙ = 26.81 0.05 and − ± defined, although of course the definition can be general- V⊙ = 26.70 0.01, respectively. Because systematic ized to any passband. Note that this definition is usually 1 interpreted to imply that the bolometric corrections must A useful compilation and discussion of solar data may be found in “Basic Astronomical Data for the Sun (BADS?)”, always be negative, although many of the currently used maintained by Eric Mamajek, Univ.
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