Bolometric correction

In astronomy, the bolometric correction is the correction made to the absolute magnitude of an object in order to convert an object's visible magnitude to its bolometric magnitude. It is large for stars which radiate much of their energy outside of the visible range. A uniform scale for the correction has not yet been standardized.

Description

Mathematically, such a calculation can be expressed:

$BC = M_{bol} - M_V\!\,$

The following is subset of a table from Kaler^[1] (p. 263) listing the bolometric correction for a range of stars. For the full table, see the referenced work.

Class	Main Sequence	Giants	Supergiants
O3	-4.3	-4.2	-4.0
G0	-0.10	-0.13	-0.1
G5	-0.14	-0.34	-0.20
K0	-0.24	-0.42	-0.38
K5	-0.66	-1.19	-1.00
M0	-1.21	-1.28	-1.3

The bolometric correction is large both for early type (hot) stars and for late type (cool) stars. The former because a substantial part of the produced radiation is in the ultraviolet, the latter because a large part is in the infrared. For a star like our Sun, the correction is only marginal because the Sun radiates most of its energy in the visual wavelength range. Bolometric correction is the correction made to the absolute magnitude of an object in order to convert an object's visible magnitude to its bolometric magnitude.

Alternatively, the bolometric correction can be made to absolute magnitudes based on other wavelength bands beyond the visible electromagnetic spectrum.^[2] For example, and somewhat more commonly for those cooler stars where most of the energy is emitted in the infrared wavelength range, sometimes a different value set of bolometric corrections is applied to the absolute infrared magnitude, instead of the absolute visual magnitude.

Mathematically, such a calculation could be expressed:

$BC_K = M_{bol} - M_K\!\,$^[3]

Where M_K is the absolute magnitude value and BC_K is the bolometric correction value in the K-band.^[4]

Setting the correction scale

The bolometric correction scale is set by the absolute magnitude of the Sun and an adopted bolometric magnitude for the Sun. The choice of adopted solar absolute magnitude, bolometric correction, and absolute bolometric magnitude are not arbitrary, although some classic references have tabulated mutually incompatible values for these quantities.^[5] The bolometric scale historically had varied somewhat in the literature, with the Sun's bolometric correction in V-band varying from -0.19 to -0.07 magnitude.

The International Astronomical Union voted on Resolution B2 regarding the zero points of the bolometric magnitude scale at the IAU General Assembly in Honolulu in August 2015.^[6] The XXIXth IAU General Assembly in Honolulu adopted IAU 2015 Resolution B2 on recommended zero points for the absolute and apparent bolometric magnitude scales.^[7]

Although bolometric magnitudes have been in use for over eight decades, there have been systematic differences in the absolute magnitude-luminosity scales presented in various astronomical references with no international standardization. This has led to systematic differences in bolometric correction scales. When combined with incorrect assumed absolute bolometric magnitudes for the Sun this can lead to systematic errors in estimated stellar luminosities. Many stellar properties are calculated based on stellar luminosity, such as radii, ages, etc.

IAU 2015 Resolution B2 proposed an absolute bolometric magnitude scale where $\scriptstyle M_{bol} = 0$ corresponds to luminosity 3.0128e28 Watts, with the zero point luminosity chosen such that the Sun (with nominal luminosity 3.828e26 Watts) corresponds to absolute bolometric magnitude $\scriptstyle M_{bol_{\rm Sun}} = 4.74$.^{[note 1]} Placing a radiation source (e.g. star) at the standard distance of 10 parsecs, it follows that the zero point of the apparent bolometric magnitude scale $\scriptstyle m_{bol} = 0$ corresponds to irradiance $\scriptstyle f_{o} = 2.518 021 002... e-8 \ W/m^2$, where the nominal total solar irradiance measured at 1 astronomical unit (1361 W/m²) corresponds to an apparent bolometric magnitude of the Sun of $\scriptstyle m_{bol_{\rm Sun}} = -26.832$.^{[note 1]}

A similar IAU proposal in 1999 (with a slightly different zero point, tied to an obsolete solar luminosity estimate) was adopted by IAU Commissions 25 and 36. However it never reached a General Assembly vote, and subsequently was only adopted sporadically by astronomers in subsequent literature.

See also

• Absolute Magnitude
• K band
• V band

External links

• http://www.peripatus.gen.nz/Astronomy/SteMag.html - contains table of bolometric corrections
• http://articles.adsabs.harvard.edu//full/1996ApJ...469..355F/0000360.000.html - contains detailed tables^[8] of bolometric corrections (note that these second set of tables are consistent with a bolometric magnitude of 4.73^[5] for the Sun and also be aware that there are misprint^[5] errors for a few of the figures in the tables)

Notes

1. "e" is not meant to be Euler's number here because instead the letter is being used as a shorthand to represent "multiplied by ten raised to the power of" (which could be written "x 10ⁿ") and is followed by the value of the exponent. Therefore, 3.0128e28 Watts = 3.0128 x 10²⁸ Watts, 3.828e26 Watts = 3.828 x 10²⁶ Watts and $\scriptstyle f_{o} = 2.518 021 002... e-8 \ W/m^2$ equals $\scriptstyle f_{o}\ =\ 2.518 \ 021 \ 002 \ ... \ \times\ 10^{-8} \ W\,m^{-2}$.

References

1. Kaler, James B. (1989). "Stars and their spectra: An Introduction to the Spectral Sequence": 300.
2. Bessell, M. S.; et al. (May 1998). "Model atmospheres broad-band colors, bolometric corrections and temperature calibrations for O - M stars". Astronomy and Astrophysics 333: 231–230. Bibcode:1998A&A...333..231B. Retrieved 23 August 2015.
3. Salaris, Maurizio; et al. (November 2002). "Population effects on the red giant clump absolute magnitude: the K band". Monthly Notices of the Royal Astronomical Society (John Wiley & Sons) 337 (1): 332–340. arXiv:astro-ph/0208057. Bibcode:2002MNRAS.337..332S. doi:10.1046/j.1365-8711.2002.05917.x. Retrieved 23 August 2015. Lower effective temperatures correspond to higher values of $\scriptstyle BC_K$; since $\scriptstyle M_K = M_{bol} - BC_K\!\,$, cooler RC stars tend to be brighter.
4. Buzzoni, A.; et al. (April 2010). "Bolometric correction and spectral energy distribution of cool stars in Galactic clusters". Monthly Notices of the Royal Astronomical Society (John Wiley & Sons) 403 (3): 1592–1610. arXiv:1002.1972. Bibcode:2010MNRAS.403.1592B. doi:10.1111/j.1365-2966.2009.16223.x. Retrieved 23 August 2015.
5. Torres, Guillermo (November 2010). "On the Use of Empirical Bolometric Corrections for Stars". The Astronomical Journal 140 (5): 1158–1162. arXiv:1008.3913. Bibcode:2010AJ....140.1158T. doi:10.1088/0004-6256/140/5/1158. Lay summary.
6. IAU XXIX General Assembly Draft Resolutions Announced, retrieved 2015-07-08
7. Mamajek, E. E.; et al. (2015). "IAU 2015 Resolution B2 on Recommended Zero Points for the Absolute and Apparent Bolometric Magnitude Scales". arXiv:1510.06262v2.
8. Flower, Phillip J. (September 1996), "Transformations from Theoretical Hertzsprung-Russell Diagrams to Color-Magnitude Diagrams: Effective Temperatures, B-V Colors, and Bolometric Corrections", The Astrophysical Journal 469: 355, Bibcode:1996ApJ...469..355F, doi:10.1086/177785