ON THE USE OF EMPIRICAL BOLOMETRIC CORRECTIONS FOR STARS

Bolometric corrections are widely used in astronomy to infer either luminosities or absolute magnitudes of stars. Empirical corrections in the visual band, BC_V, are perhaps the most frequently needed, and numerous tabulations exist in the literature that sometimes differ significantly. This has been a persistent source of confusion among users. The most common way in which these tables are employed is in combination with the bolometric magnitude of the Sun, M_bol,☉, which is not a directly measured quantity. Many different values of M_bol,☉ can be found in the literature as well, adding to the confusion. For a given table of bolometric corrections the choice of M_bol,☉ is not arbitrary, however, since there is an observational constraint that must be satisfied, given by the measurement of the visual brightness of the Sun. This fact is often ignored, and as a result it is common to see inconsistent uses of BC_V that can lead to errors in the luminosity of 10% or larger, or errors in M_V of 0.1 mag or more. The primary motivation for this paper is to call attention to this fact, to illustrate common misuses of some of the most popular BC_V tables, and to offer some perspective on the problem.

One frequently used source of bolometric corrections is the tabulation by Flower (1996), which is an update on his earlier work (Flower 1977) based on a compilation of effective temperatures (T_eff) and BC_V determinations for a large number of stars. Many authors find this source convenient because, in addition to the table, it presents simple polynomial fits for BC_V valid over a wide range of temperatures. Unfortunately, the original publication had misprints in the coefficients of those formulae that have prevented their use, and an erratum was never published. The present author has received numerous inquiries about this problem over the years. Thus, a second motivation for this paper is to make the correct coefficients available to the community, as well as to amend the form of the equation and one of the coefficients for a color/temperature calibration in the same paper that were also misprinted. We begin by presenting these corrections, and follow with a discussion of the practical use of BC_V tables.

Flower (1996) expressed the bolometric corrections as a function of the effective temperature of the star, and presented polynomial fits for BC_V of the form

$$\begin{equation} \mbox{BC}_V = a + b\thinspace (\log T_{\rm eff}) + c\thinspace (\log T_{\rm eff})^2 + \cdots, \end{equation} \tag{ 1 }$$

for three different temperature ranges. The coefficients a, b, c, ⋅ ⋅⋅ were given in his Table 6 for each interval, but are missing powers of ten. We rectify this situation here and present them with a larger number of significant digits (see Table 1). While this formulation is certainly very handy for many applications, it must be kept in mind that the bolometric corrections become less reliable for cooler stars due to the limited number of observational constraints, and they break down completely for M dwarfs. Two of the very few such constraints available are for the low-mass eclipsing binaries CU Cnc (components with M ≈ 0.42 M_☉) and CM Dra (M ≈ 0.24 M_☉), which have known parallaxes. For these stars, one may compute the BC_V values for the components directly from the estimated M_V values and the bolometric magnitudes based on the absolute radius and temperature determinations (Ribas 2003; Morales et al. 2009). Comparing the measured values with the predicted ones shows that the Flower relations give a BC_V that is too negative by ∼1.2 mag for CU Cnc and ∼1.7 mag for CM Dra. Similar discrepancies are seen in other BC_V sources.

A useful color/temperature calibration was also presented by Flower (1996) in his Table 5, separately for supergiants and for main-sequence, subgiant, and giant stars, but unfortunately the formula given there was printed incorrectly, and should have expressed the temperature (log T_eff) as a function of the B − V color index, rather than the reverse. The polynomial fits are similar to those used for BC_V:

$$\begin{equation} \log T_{\rm eff} = a + b\thinspace (B-V) + c\thinspace (B-V)^2 + \cdots. \end{equation} \tag{ 2 }$$

One of original coefficients was also misprinted, and a correction was issued by Prša & Zwitter (2005). We present all coefficients again in Table 2 to higher precision.

As is well known, the apparent bolometric magnitude of a star is defined as

$$\begin{equation} m_{\rm bol} = -2.5 \log \left(\int _{0}^{\infty }\! f_{\lambda } d\lambda \right) + C_1, \end{equation} \tag{ 3 }$$

where f_λ is the monochromatic flux from the object per unit wavelength interval received outside Earth's atmosphere, and C₁ is a constant. The bolometric correction is usually defined as the quantity to be added to the apparent magnitude in a specific passband (in the absence of interstellar extinction) in order to account for the flux outside that band:

$$\begin{equation} \mbox{BC}_V = m_{\rm bol} - V = M_{\rm bol} - M_V. \end{equation} \tag{ 4 }$$

Here we focus on the visual band because that is the context in which bolometric corrections were historically defined, although of course the definition can be generalized to any passband. Note that this definition is usually interpreted to imply that the bolometric corrections must always be negative, although many of the currently used tables of empirical BC_V values violate this condition. We return to this below. Equation (4) may also be written as

$$\begin{equation} \mbox{BC}_V = 2.5 \log \left(\int _{0}^{\infty }S_{\lambda }(V) f_{\lambda } d\lambda \bigg/ \int _{0}^{\infty } f_{\lambda } d\lambda \right) + C_2, \end{equation} \tag{ 5 }$$

where S_λ(V) is the sensitivity function of the V magnitude system. The constant C₂ contains an arbitrary zero point that has been a common source of confusion. This zero point has traditionally been set using the Sun as the reference. By noting that

$$\begin{equation} \mbox{BC}_{V,\odot } = m_{\rm bol,\odot } - V_{\odot } = M_{\rm bol,\odot } - M_{V,\odot }, \end{equation} \tag{ 6 }$$

it is immediately clear that setting a value for the bolometric correction of the Sun is equivalent to specifying the zero point of the bolometric magnitude scale, since V_☉ is a known quantity. A common practice when using one of the many available tables of empirical bolometric corrections is to adopt a value for M_bol,☉, but all too often this is done without regard for whether the chosen value is consistent with BC_V,☉ from the same table. From Equation (6) we have

$$\begin{equation} M_{\rm bol,\odot } = M_{V,\odot } + \mbox{BC}_{V,\odot } = V_{\odot } + 31.572 + \mbox{BC}_{V,\odot }, \end{equation} \tag{ 7 }$$

where the numerical constant corresponds (with the opposite sign) to the distance modulus of the Sun at 1 AU. This formulation shows that once a particular tabulation of BC_V 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.

Direct measurements of V_☉ are difficult to make because of the extreme difference in brightness between the Sun and the stars that are used as the reference in the V system, and also because the Sun is spatially resolved. Nevertheless, many careful determinations have been carried out over the years (although not very recently, as far as we are aware), and the most reliable of the photoelectric measurements have been reviewed by Hayes (1985).¹ Among them, one that carries particularly high weight is that of Stebbins & Kron (1957), which was adjusted slightly (by −0.02 mag) by Hayes (1985) for an error in the treatment of horizontal extinction, and further updated by Bessell et al. (1998) using modern V magnitudes for the reference stars. The result is V_☉ = −26.76 ±  0.03.² Two additional determinations discussed by Hayes (1985) are those of Nikonova (1949), transformed to the standard Johnson system by Martynov (1960) and Gallouët (1964): V_☉ = −26.81 ±  0.05 and V_☉ = −26.70 ±  0.01, respectively. Because systematic effects likely dominate the differences, we follow Hayes (1985) and adopt as a consensus value a simple average of the three estimates, giving V_☉ = −26.76 ±  0.03, with an error that is probably realistic. The absolute visual magnitude of the Sun then becomes M_V,☉ = 4.81 ±  0.03.

Measurements of V_☉ based on absolute flux-calibrated spectra of the Sun (Colina et al. 1996; Thuillier et al. 2004, and others) or synthetic spectra based on model atmospheres such as ATLAS9 and MARCS have also been made by many authors, and generally range from −26.74 to −26.77, with minor differences depending on the author even when using the same spectrum (see, e.g., Bessell et al. 1998; Casagrande et al. 2006). These estimates agree well with the direct measurements.

While the zero point of BC_V is completely arbitrary, and no particular scale has been officially endorsed by the International Astronomical Union (IAU) (but see Section 5), it is common for some authors of these tabulations to define the scale by adopting a certain value for BC_V,☉, sometimes for historical reasons. For example, the widely used reference by Cox (2000), the successor to Allen's Astrophysical Quantities (Allen 1976, and earlier editions) indicates that it adopts BC_V,☉ = −0.08 (p. 341, but see also the next section), a value inherited from earlier compilations. Similar scales have been chosen in many other empirical tables. On the other hand, the theoretical BC_V values that are incorporated in the stellar evolution models of Pietrinferni et al. (2004) are based on BC_V,☉ = −0.203, while the Yi et al. (2001) isochrones use either BC_V,☉ = −0.109 or BC_V,☉ = −0.08, for the two different color transformation tables offered with their models (Lejeune et al. 1998; Green et al. 1987). The more negative values above tend to come from the practice by many stellar modelers (starting with Buser & Kurucz 1978, if not earlier) of computing theoretical bolometric corrections from a grid of model atmospheres for a large range of metallicities, temperatures, and surface gravities, and then shifting all values by arbitrarily setting the smallest BC_V to zero. This usually corresponds to late A type stars of low surface gravity.³

The scale adopted by Flower (1996) is such that BC_V,☉ = −0.080. As a result, his bolometric corrections for stars between T_eff ≈ 6400 K and T_eff ≈ 8500 K (spectral type approximately F5 to A5) are positive, seemingly conflicting with the idea that bolometric magnitudes ought to be brighter than V magnitudes, according to Equation (4). Other tables with a similar zero point as Flower's share the same problem. In reality, however, the contradiction is of no consequence because of the arbitrary nature of the zero point. Luminosities inferred for stars are never affected if a consistent value of M_bol,☉ is used, because the bolometric magnitudes are always compared to the Sun. To see this more clearly one may make use of

$$\begin{equation} M_{\rm bol} - M_{\rm bol,\odot } = -2.5 \log (L/L_{\odot }) \end{equation} \tag{ 8 }$$

along with Equations (4) and (7) to express the luminosity of a star in terms of its absolute visual magnitude and bolometric correction as

$$\begin{equation} \log (L/L_{\odot }) = -0.4[ M_V - V_{\odot } - 31.572 + (\mbox{BC}_V - \mbox{BC}_{V,\odot })]. \end{equation} \tag{ 9 }$$

Alternately, if seeking to determine the absolute magnitude from knowledge of the luminosity (e.g., in eclipsing binaries if the temperature and absolute radius are known), one has

$$\begin{equation} M_V = -2.5 \log (L/L_{\odot }) + V_{\odot } + 31.572 - (\mbox{BC}_V - \mbox{BC}_{V,\odot }). \end{equation} \tag{ 10 }$$

The last two equations involve only the difference between two bolometric corrections, so that the zero point cancels out. The apparent contradiction is thus irrelevant since any table of BC_V may be shifted arbitrarily with no impact on the results from these expressions.

As an illustration of the confusing state of affairs brought about by the proliferation of zero points, and to make readers aware of the sometimes serious inconsistencies lurking in these BC_V sources, here we examine several of the widely used tables of empirical bolometric corrections and spell out their assumptions. It is not uncommon for tabular information of this kind to be copied over from earlier sources, but assumptions are sometimes changed along the way, so that each table has its own problems.

• 1.  
  In the chapter on the Sun, the latest edition of the popular Allen's Astrophysical Quantities (Cox 2000, p. 341) adopts an internally consistent set of solar parameters given by BC_V,☉ = −0.08, M_bol,☉ = 4.74, M_V,☉ = 4.82, and V_☉ = −26.75. However, inspection of the table of bolometric corrections for dwarfs in the chapter on normal stars (p. 388), which is the one employed in practice, reveals that the BC_V,☉ there is −0.20 rather than the value advocated earlier. This implies V_☉ = −26.63. Therefore, the use of this table together with M_bol,☉ = 4.74 will introduce a systematic error of 0.13 mag, which is the same as produced by a difference of 500 K in the input temperature. In order to be consistent with the measured value of V_☉ = −26.76 discussed in the previous section, the bolometric magnitude that should be used for the Sun is M_bol,☉ = 4.61. An earlier edition of Allen's Astrophysical Quantities has inconsistencies of its own and adopts a rather different BC_V scale which, interestingly, does not have as serious a problem. For example, the solar parameters in the third edition by Allen (1976) are also internally consistent, and again use BC_V,☉ = −0.08, but with M_bol,☉ = 4.75 (p. 162 of that work). The BC_V table on p. 206, however, lists a bolometric correction for a normal star with the solar temperature as BC_V,☉ = −0.05. Nevertheless, if used in conjunction with the solar bolometric magnitude advocated, this table implies V_☉ = −26.77, which is much more accurate than in the later edition.
• 2.  
  The solar values adopted by Schmidt-Kaler (1982) (p. 451) are BC_V,☉ = −0.19, M_bol,☉ = 4.64, M_V,☉ = 4.83, and V_☉ = −26.74, which are internally consistent. Schmidt-Kaler's BC_V table for main-sequence stars on p. 453 gives a slightly different value of BC_V,☉ = −0.21 for a star of solar temperature. This implies V_☉ = −26.72 rather than their adopted value. To be consistent with V_☉ from the previous section, the bolometric magnitude to be used for the Sun is M_bol,☉ = 4.60.
• 3.  
  The extensive compilation by Lang (1992) adopts the following values for the Sun (p. 103): V_☉ = −26.78, M_V,☉ = 4.82, and M_bol,☉ = 4.75. The first two are slightly inconsistent with the distance modulus of the Sun. The last two imply BC_V,☉ = −0.07, yet the listing of bolometric corrections on p. 138 gives BC_V,☉ = −0.20 for a star of solar temperature. According to Equation(7), the proper value of M_bol,☉ to use with this table is M_bol,☉ = 4.61. Consequently, the systematic error incurred by using the Lang (1992) table in combination with their M_bol,☉ is 0.14 mag.
• 4.  
  A BC_V table still often used mainly in the binary star field is that of Popper (1980). The bolometric correction and absolute visual magnitude adopted there for the Sun are BC_V,☉ = −0.14 and M_V,☉ = 4.83, from which V_☉ = −26.74. These imply M_bol,☉ = 4.69. To be in exact agreement with our apparent magnitude for the Sun, the solar bolometric magnitude should be adjusted slightly to M_bol,☉ = 4.67.
• 5.  
  The BC_V table in the textbook by Gray (2005) gives BC_V,☉ = −0.09 for a star of solar temperature, and adopts V_☉ = −26.75 (and the corresponding value of M_V,☉ = 4.82). Together these imply M_bol,☉ = 4.73. For exact consistency with V_☉ from the previous section, we recommend using M_bol,☉ = 4.72.
• 6.  
  The work of Straižys & Kuriliene (1980) contains many useful tables of average stellar properties, and adopts a zero point for the BC_V scale that is adjusted to give BC_V,☉ = −0.07. These authors also adopt M_bol,☉ = 4.72, which leads to M_V,☉ = 4.79 and V_☉ = −26.78. Perfect agreement with our V_☉ requires M_bol,☉ = 4.74.
• 7.  
  Kenyon & Hartmann (1995) presented a table of bolometric corrections that is often used in the field of pre-main-sequence stars, and has been incorporated into some model isochrones for young stars such as those by Siess et al. (2000). It is compiled from a variety of sources, and the zero point is such that a star of solar temperature has BC_V,☉ = −0.21. No value of M_bol,☉ is specified.
• 8.  
  Finally, as mentioned earlier, the BC_V table by Flower (1996) gives BC_V,☉ = −0.08 for a star of solar temperature, but there is no associated value of M_bol,☉ given in the text. For consistency with V_☉ from the previous section, the number to be used is M_bol,☉ = 4.73.

Table 3 summarizes the various empirical BC_V scales, along with the M_bol,☉ values listed by each source, as well as the M_bol,☉ recommended here to maintain consistency with the adopted V_☉. The systematic error introduced when using the former instead of the latter is presented in the last column.

From a practical point of view one may approach the use of published tables of empirical BC_V values in several ways, but it is essential to always maintain consistency with the measured brightness of the sun, V_☉. Errors introduced when overlooking this requirement are seen rather often in the literature, and are quantified in Table 3. Bessell et al. (1998) chose to define M_bol,☉ = 4.74, from which one derives BC_V,☉ = −0.07. If one follows this path then it is necessary to adjust the BC_V table one is using, in order to match this zero point. In general, each table will require a different offset. However, most users find it more convenient to adopt a particular BC_V table "as is," in which case care must be taken to use the proper M_bol,☉ as listed in Table 3, and not an arbitrary value from another source (or even from the same source if it is inconsistent). Yet another approach is to use Equation (9) or Equation (10) directly, as advocated by Gray (2005), which dispenses with having to find a formal M_bol,☉ value, and requires only to read off BC_V for the Sun from the same table used for the star of interest. These three approaches are of course completely equivalent.

Somewhat surprisingly, the IAU has not issued a formal resolution on the matter of BC_V zero points, although two of its Commissions did agree at the Kyoto meeting of 1997 (Andersen 1999, pp. 141 and 181) on a preferred scale that is equivalent to adopting a value for M_bol,☉.⁴ The scale was set by defining a star with M_bol = 0.00 to have an absolute radiative luminosity of L = 3.055 × 10²⁸ W (see also Cayrel 2002). The rationale was that this value together with the nominal bolometric luminosity of the Sun adopted by the international Global Oscillation Network Group project (L_☉ = 3.846 × 10²⁶ W, according to the IAU Commission reports cited above) leads exactly to M_bol,☉ = 4.75, which is the bolometric magnitude for the Sun listed in the 1976 edition of Astrophysical Quantities (Allen 1976). This was a widely used source at the time (and still is, by some), so it was thought to be a logical choice. Effectively, therefore, the scale is set by this value of M_bol,☉. Combined with our adopted solar brightness of V_☉ = −26.76, it implies BC_V,☉ = −0.06. As it turns out, however, the most recent edition of Astrophysical Quantities (Cox 2000) did not follow that recommendation and adopted a slightly different zero point. Neither have some other recent BC_V compilations (see Table 3). In practice, the adoption of a value of BC_V,☉ or a value of M_bol,☉ may not be the most convenient way to solve the immediate problem faced by users. The first alternative would not be of much help to those wishing to make use of an existing BC_V table, and the second would force them to adjust the table to match V_☉. To conclude, one may argue that it would perhaps be more useful instead to agree on the best value for the apparent visual magnitude of the Sun, which is directly measured.

I am indebted to Phillip Flower for providing me with the correct coefficients for his BC_V and color/temperature relations, which were misprinted in his original work (Flower 1996). They are presented here with his permission. I also thank Gene Milone for stimulating discussions and motivation for this paper, Todd Henry for alerting me to the significant errors in BC_V for cool stars, and the anonymous referee for helpful comments. Correspondence with Martin Asplund, Dainis Dravins, Arlo Landolt, Pedro Martínez, Gene Milone, and Chris Sterken regarding IAU deliberations on the matter of BC_V is also acknowledged. This work was partially supported by NSF grant AST-0708229. The research has made use of NASA's Astrophysics Data System Abstract Service.