On the Use of Empirical Bolometric Corrections for Stars

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02138, MA Cambridge, ., V BC ⊙ ucino h ffcietmeaueo h tr and star, the of for temperature fits effective polynomial the presented of function a r o h o-aselpigbnre UCc(compo- Cnc available CU constraints binaries such eclipsing with few nents low-mass M very the the the for for are observational of completely of Two down number break in dwarfs. limited they kept the and certainly to be constraints, for due must is reliable stars less it formulation become cooler corrections applications, this bolometric many that While mind for handy 1). dig- very significant Table here of number situation (see larger this its a rectify with We them present ten. and of powers missing are hc aekonprlae.Frteesasoemay one estimated stars the from these For the parallaxes. compute known have which o he ieettmeauerne.Tecoefficients The ranges. c, b, temperature a, different three for ahrta h ees.Teplnma t r similar are fits for polynomial used The those reverse. to the than rather n h esrdvle ihtepeitdoe shows ones predicted a give the relations Flower with the values that Compar- measured 2009). the al. et ing de- Morales temperature 2003; and (Ribas radius terminations absolute the on based nitudes n foiia offiinswsas ipitd n a and misprinted, also was coefficients original of One rcia s of use practical by tr,btufruaeytefruagvnteewas tem- there the given expressed (log have formula perature should for the and incorrectly, separately giant unfortunately printed and 5, but subgiant, Table main-sequence, his stars, for in and (1996) supergiants Flower by sented of tabulations other in corrections. bolometric found empirical are discrepancies Similar = V 2. lwr(96 xrse h ooerccretosas corrections bolometric the expressed (1996) Flower sflclrtmeaueclbainwsas pre- also was calibration color/temperature useful A − eqatf rosta r nrdcdby introduced are that errors quantify We . ∼ OFIINSFRTEFOE (1996) FLOWER THE FOR COEFFICIENTS 26 . a o UCc and Cnc, CU for mag 1.2 · · · BC log . 76 eegvni i al o ahitra,but interval, each for 6 Table his in given were V T ± M eff = 0 T BC . i antd fteSn( Sun the of magnitude ric ≈ = a 3 nossetvle of values Inconsistent 03. eff BC + di h rgnlpublication. original the in ed 0 a iy tms ecnitn with consistent be must It rily. safnto fthe of function a as ) V BC pitthe eprint . 42 b + rsas( stars or V (log M log ausfrtecmoet directly components the for values b V M tables. V ( : T B ⊙ EFF T ausadteblmti mag- bolometric the and values − n MDa( Dra CM and ) eff V + ) FORMULAE BC BC + ) etlparameters mental BC c BC V V c ∼ (log V ,acommonly a ), ( . a o MDra. CM for mag 1.7 offiinsof coefficients B V fteform the of − T hti o negative too is that eff B V M − ) e sa is ter ) M M 2 2 V bol + + bol ≈ oo index, color , · · · BC · · · ⊙ , 0 ⊙ ) . edu 24 V , . AND M ⊙ (1) (2) ), 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. of Rochester (NY), at tables of empirical BCV values violate this condition. We http://www.pas.rochester.edu/∼emamajek/sun.txt. return to this below. Eq.(4) may also be written as 2 The uncertainty we assign is slightly larger than the ±0.02 mag given by Bessell et al. (1998) because it includes contributions from ∞ ∞ systematic effects described by Stebbins & Kron (1957). We also BCV =2.5 log Sλ(V )fλdλ / fλdλ + C2 , (5) 0 0 re-examined the corrections made in the later work to reduce R R
the measurements to the distance of 1 AU, since variations in where Sλ(V ) is the sensitivity function of the V magni- the Earth-Sun distance throughout the year lead to non-negligible tude system. The constant C2 contains an arbitrary zero brightness changes of ±0.036 mag. This leads to only a very mi- point that has been a common source of confusion. This nor difference in the third decimal place compared to the value zero point has traditionally been set using the Sun as the reported by Bessell et al. (1998), which we have ignored here. Bolometric corrections 3 effects likely dominate the differences, we follow Hayes bolometric correction as (1985) and adopt as a consensus value a simple average log(L/L⊙)= −0.4 [MV − V⊙ − 31.572+(BCV − BCV,⊙)] . of the three estimates, giving V⊙ = −26.76 ± 0.03, with an error that is probably realistic. The absolute visual (9) Alternately, if seeking to determine the absolute magni- magnitude of the Sun then becomes M ⊙ =4.81 ± 0.03. V, tude from knowledge of the luminosity (e.g., in eclips- Measurements of V⊙ based on absolute flux-calibrated spectra of the Sun (Colina et al. 1996; Thuillier et al. ing binaries if the temperature and absolute radius are 2004, and others) or synthetic spectra based on model known), one has atmospheres such as ATLAS9 and MARCS have also MV = −2.5 log(L/L⊙)+ V⊙ +31.572 − (BCV − BCV,⊙) . been made by many authors, and generally range from (10) −26.74 to −26.77, with minor differences depending on The last two equations involve only the difference be- the author even when using the same spectrum (see, e.g., tween two bolometric corrections, so that the zero point Bessell et al. 1998; Casagrande et al. 2006). These esti- cancels out. The apparent contradiction is thus irrele- mates agree well with the direct measurements. vant since any table of BCV may be shifted arbitrarily While the zero point of BCV is completely arbitrary, with no impact on the results from these expressions. and no particular scale has been officially endorsed by the International Astronomical Union (IAU) (but see 4. INCONSISTENCIES IN PUBLISHED TABLES OF Sect. 5), it is common for some authors of these tabu- EMPIRICAL BCV lations to define the scale by adopting a certain value for As an illustration of the confusing state of affairs BCV,⊙, sometimes for historical reasons. For example, brought about by the proliferation of zero points, and the widely used reference by Cox (2000), the successor to make readers aware of the sometimes serious inconsis- to Allen’s Astrophysical Quantities (Allen 1976, and ear- tencies lurking in these BCV sources, we examine here lier editions) indicates that it adopts BCV,⊙ = −0.08 (p. several of the widely used tables of empirical bolometric 341, but see also the next section), a value inherited from corrections and spell out their assumptions. It is not un- earlier compilations. Similar scales have been chosen in common for tabular information of this kind to be copied many other empirical tables. On the other hand, the the- over from earlier sources, but assumptions are sometimes oretical BCV values that are incorporated in the stellar changed along the way, so that each table has its own evolution models of Pietrinferni et al. (2004) are based problems: on BCV,⊙ = −0.203, while the Yi et al. (2001) isochrones • In the chapter on the Sun, the latest edition of the use either BCV,⊙ = −0.109 or BCV,⊙ = −0.08, for popular Allen’s Astrophysical Quantities (Cox 2000, p. the two different color transformation tables offered with 341) adopts an internally consistent set of solar pa- their models (Lejeune et al. 1998; Green et al. 1987). rameters given by BCV,⊙ = −0.08, Mbol,⊙ = 4.74, The more negative values above tend to come from MV,⊙ = 4.82, and V⊙ = −26.75. However, inspection the practice by many stellar modelers (starting with of the table of bolometric corrections for dwarfs in the Buser & Kurucz 1978, if not earlier) of computing theo- chapter on normal stars (p. 388), which is the one em- retical bolometric corrections from a grid of model atmo- ployed in practice, reveals that the BCV,⊙ there is −0.20 spheres for a large range of metallicities, temperatures, rather than the value advocated earlier. This implies and surface gravities, and then shifting all values by ar- V⊙ = −26.63. Therefore, the use of this table together bitrarily setting the smallest BCV to zero. This usually with Mbol,⊙ = 4.74 will introduce a systematic error of corresponds to late A-type stars of low surface gravity.3 0.13 mag, which is the same as produced by a differ- The scale adopted by Flower (1996) is such that ence of 500 K in the input temperature. In order to BCV,⊙ = −0.080. As a result, his bolometric corrections be consistent with the measured value of V⊙ = −26.76 for stars between Teff ≈ 6400 K and Teff ≈ 8500 K (spec- discussed in the previous section, the bolometric magni- tral type approximately F5 to A5) are positive, seem- tude that should be used for the Sun is Mbol,⊙ = 4.61. ingly conflicting with the idea that bolometric magni- An earlier edition of Allen’s Astrophysical Quantities has tudes ought to be brighter than V magnitudes, accord- inconsistencies of its own and adopts a rather different ing to eq.(4). Other tables with a similar zero point as BCV scale which, interestingly, does not have as seri- Flower’s share the same problem. In reality, however, ous a problem. For example, the solar parameters in the the contradiction is of no consequence because of the ar- 3rd edition by Allen (1976) are also internally consistent, bitrary nature of the zero point. Luminosities inferred and again use BCV,⊙ = −0.08, but with Mbol,⊙ = 4.75 for stars are never affected if a consistent value of Mbol,⊙ (p. 162 of that work). The BCV table on p. 206, how- is used, because the bolometric magnitudes are always ever, lists a bolometric correction for a normal star with compared to the Sun. To see this more clearly one may the solar temperature as BCV,⊙ = −0.05. Nevertheless, make use of if used in conjunction with the solar bolometric magni- tude advocated, this table implies V⊙ = −26.77, which − − Mbol Mbol,⊙ = 2.5 log(L/L⊙) (8) is much more accurate than in the later edition. • The solar values adopted by Schmidt-Kaler (1982) (p. along with eq.(4) and eq.(7) to express the luminosity − of a star in terms of its absolute visual magnitude and 451) are BCV,⊙ = 0.19, Mbol,⊙ = 4.64, MV,⊙ = 4.83 and V⊙ = −26.74, which are internally consistent. Their 3 BCV table for main-sequence stars on p. 453 gives a An example of a table of theoretical bolometric corrections of- − ten used is that of Lejeune et al. (1998), mentioned above. These slightly different value of BCV,⊙ = 0.21 for a star of so- authors initially followed the procedure just described, leading to lar temperature. This implies V⊙ = −26.72 rather than − BCV,⊙ = 0.190, but then chose to adjust the zero point to pro- their adopted value. To be consistent with V⊙ from the vide the best fit to the BCV table of Flower (1996). previous section, the bolometric magnitude to be used 4 Torres for the Sun is Mbol,⊙ =4.60. this path then it is necessary to adjust the BCV table • The extensive compilation by Lang (1992) adopts the one is using, in order to match this zero point. In general following values for the Sun (p. 103): V⊙ = −26.78, each table will require a different offset. However, most MV,⊙ = 4.82, and Mbol,⊙ = 4.75. The first two are users find it more convenient to adopt a particular BCV slightly inconsistent with the distance modulus of the table ‘as is’, in which case care must be taken to use the Sun. The last two imply BCV,⊙ = −0.07, yet the listing proper Mbol,⊙ as listed in Table 3, and not an arbitrary of bolometric corrections on p. 138 gives BCV,⊙ = −0.20 value from another source (or even from the same source for a star of solar temperature. According to eq.(7), if it is inconsistent). Yet another approach is to use eq.(9) the proper value of Mbol,⊙ to use with this table is or eq.(10) directly, as advocated by Gray (2005), which Mbol,⊙ = 4.61. Consequently, the systematic error in- dispenses with having to find a formal Mbol,⊙ value, and curred by using the Lang (1992) table in combination requires only to read off BCV for the Sun from the same with their Mbol,⊙ is 0.14 mag. table used for the star of interest. These three approaches • A BCV table still often used mainly in the binary star are of course completely equivalent. field, is that of Popper (1980). The bolometric correction Somewhat surprisingly the IAU has not issued a for- and absolute visual magnitude adopted there for the Sun mal resolution on the matter of BCV zero points, al- are BCV,⊙ = −0.14 and MV,⊙ =4.83, from which V⊙ = though two of its Commissions did agree at the Kyoto −26.74. These imply Mbol,⊙ = 4.69. To be in exact meeting of 1997 (Andersen 1999, pp. 141 and 181) on agreement with our apparent magnitude for the Sun, the a preferred scale that is equivalent to adopting a value solar bolometric magnitude should be adjusted slightly for Mbol,⊙. The scale was set by defining a star with to Mbol,⊙ =4.67. Mbol = 0.00 to have an absolute radiative luminosity of 28 • The BCV table in the textbook by Gray (2005) gives L =3.055×10 W (see also Cayrel 2002). The rationale BCV,⊙ = −0.09 for a star of solar temperature, and was that this value together with the nominal bolomet- adopts V⊙ = −26.75 (and the corresponding value of ric luminosity of the Sun adopted by the international 26 MV,⊙ =4.82). Together these imply Mbol,⊙ =4.73. For GONG project (L⊙ =3.846 × 10 W, according to the exact consistency with V⊙ from the previous section, we IAU Commission reports cited above) leads exactly to recommend using Mbol,⊙ =4.72. Mbol,⊙ =4.75, which is the bolometric magnitude for the • The work of Straiˇzys & Kuriliene (1980) contains many Sun listed in the 1976 edition of Astrophysical Quantities useful tables of average stellar properties, and adopts a (Allen 1976). This was a widely used source at the time zero point for the BCV scale that is adjusted to give (and still is, by some), so it was thought to be a logical BCV,⊙ = −0.07. These authors also adopt Mbol,⊙ = choice. Effectively, therefore, the scale is set by this value 4.72, which leads to MV,⊙ = 4.79 and V⊙ = −26.78. of Mbol,⊙. Combined with our adopted solar brightness Perfect agreement with our V⊙ requires Mbol,⊙ =4.74. of V⊙ = −26.76, it implies BCV,⊙ = −0.06. As it turns • Kenyon & Hartmann (1995) presented a table of bolo- out, however, the most recent edition of Astrophysical metric corrections that is often used in the field of pre- Quantities (Cox 2000) did not follow that recommenda- main sequence stars, and has been incorporated into tion, and adopted a slightly different zero point. So have some model isochrones for young stars such as those some other recent BCV compilations (see Table 3). by Siess et al. (2000). It is compiled from a variety of In practice the adoption of a value of BCV,⊙ or a value sources, and the zero point is such that a star of solar of Mbol,⊙ may not be the most convenient way to solve temperature has BCV,⊙ = −0.21. No value of Mbol,⊙ is the immediate problem faced by users. The first alterna- specified. tive would not be of much help to those wishing to make • Finally, as mentioned earlier, the BCV table by Flower use of an existing BCV table, and the second would force (1996) gives BCV,⊙ = −0.08 for a star of solar temper- them to adjust the table to match V⊙. To conclude, one ature, but there is no associated value of Mbol,⊙ given may argue that it would perhaps be more useful instead in the text. For consistency with V⊙ from the previous to agree on the best value for the apparent visual mag- section, the number to be used is Mbol,⊙ =4.73. nitude of the Sun, which is directly measured. Table 3 summarizes the various empirical BCV scales, along with the M ⊙ values listed by each source, as well bol, I am indebted to Phillip Flower for providing as the Mbol,⊙ recommended here to maintain consistency me with the correct coefficients for his BCV and with the adopted V⊙. The systematic error introduced when using the former instead of the latter is presented color/temperature relations, which were misprinted in in the last column. his original work (Flower 1996). They are presented here with his permission. I also thank Gene Milone for stim- ulating discussions and motivation for this paper, Todd 5. DISCUSSION Henry for alerting me to the significant errors in BCV for From a practical point of view one may approach the cool stars, and the anonymous referee for helpful com- use of published tables of empirical BCV values in several ments. Correspondence with Martin Asplund, Dainis ways, but it is essential to always maintain consistency Dravins, Arlo Landolt, Pedro Mart´ınez, Gene Milone, with the measured brightness of the sun, V⊙. Errors and Chris Sterken regarding IAU deliberations on the introduced when overlooking this requirement are seen matter of BCV is also acknowledged. This work was rather often in the literature, and are quantified in Ta- partially supported by NSF grant AST-0708229. The re- ble 3. Bessell et al. (1998) chose to define Mbol,⊙ =4.74, search has made use of NASA’s Astrophysics Data Sys- from which one derives BCV,⊙ = −0.07. If one follows tem Abstract Service.

REFERENCES Allen, C. W. 1976, Astrophysical Quantities, 3rd Ed. (London: Andersen, J. 1999, Proceedings of the Twenty-Third General The Athlone Press) Assembly, Transactions of the IAU, Vol. XXIIIB (Dordrecht: Kluwer) Bolometric corrections 5

TABLE 3 Empirical BCV scales and Mbol,⊙ values from the literature

Advocated Actual Adopted Recommended BCV,⊙ BCV,⊙ Mbol,⊙ Mbol,⊙ Error Source (mag) a (mag) b (mag) c (mag) d (mag) e Cox (2000) −0.08 −0.20 4.74 4.61 +0.13 Allen (1976) −0.08 −0.05 4.75 4.76 −0.01 Schmidt-Kaler (1982) −0.19 −0.21 4.64 4.60 +0.04 Lang (1992) −0.07 −0.20 4.75 4.61 +0.14 Popper (1980) −0.14 −0.14 4.69 4.67 +0.02 Gray (2005) ··· −0.09 4.73 4.72 +0.01 Straiˇzys & Kuriliene (1980) ··· −0.07 4.72 4.74 −0.02 Kenyon & Hartmann (1995) ··· −0.21 ··· 4.60 ··· Flower (1996) ··· −0.08 ··· 4.73 ··· a Value that each source states to have adopted as the zero point of their BCV scale. b Value read off from the relevant BCV table for each source. c Bolometric correction for the Sun adopted by each source. d Mbol,⊙ value required for consistency with V⊙ = −26.76 (Sect. 3), when using the BCV table as published. e Error incurred when using the published BCV table combined with Mbol,⊙ from the source, instead of the recommended Mbol,⊙ value in the previous column.

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