Bolometric correction

In astronomy, the bolometric correction is the correction made to the absolute magnitude of an object in order to convert its visible magnitude to its bolometric magnitude. It is large for stars which radiate most 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:

$${\displaystyle BC=M_{\text{bol}}-M_{v}}$$

The bolometric correction for a range of stars with different spectral types and groups is shown in the following table: ^{[1] [2] [3]}

Spectral type	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 and negative 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 the 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. ^[4] 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: ^[5]

$${\displaystyle BC_{K}=M_{\text{bol}}-M_{k}}$$

Where M_K is the absolute magnitude value and BC_K is the bolometric correction value in the K-band. ^[6]

Setting the correction scale

The bolometric correction scale is set by the absolute magnitude of the Sun and an adopted (arbitrary) absolute bolometric magnitude for the Sun. Hence, while the absolute magnitude of the Sun in different filters is a physical and not arbitrary quantity, the absolute bolometric magnitude of the Sun is arbitrary, and so the zero-point of the bolometric correction scale that follows from it. This explains why classic references have tabulated apparently mutually incompatible values for these quantities. ^[7] 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. It follows that any value for the absolute bolometric magnitude of the Sun is legitimate, on the condition that once chosen all bolometric corrections are rescaled accordingly. If not, this will induce systematic errors in the determination of stellar luminosities. ^{[7] [8]}

The XXIXth International Astronomical Union (IAU) General Assembly in Honolulu adopted in August 2015 Resolution B2 on recommended zero points for the absolute and apparent bolometric magnitude scales. ^{[9] [10]}

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 ${\displaystyle M_{\text{bol}}=0}$ corresponds to luminosity 3.0128×10²⁸  W , with the zero point luminosity chosen such that the Sun (with nominal luminosity 3.828×10²⁶ W) corresponds to absolute bolometric magnitude ${\displaystyle M_{{\text{bol}}_{\rm {Sun}}}=4.74}$. 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 ${\displaystyle m_{\text{bol}}=0}$ corresponds to irradiance ${\displaystyle f_{o}=2.518021002...\times 10^{-8}\mathrm {W/m^{2}} }$, where the nominal total solar irradiance measured at 1 astronomical unit (1361 W/m²) corresponds to an apparent bolometric magnitude of the Sun of ${\displaystyle m_{{\text{bol}}_{\rm {Sun}}}=-26.832}$.

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 the literature.

See also

• Absolute magnitude
• K band (infrared)
• V band

External links

• https://github.com/casaluca/bolometric-corrections - most up to date tables of bolometric corrections across the HR diagram and interpolation routines in different photometric filters ^{[7] [8]}
• https://web.archive.org/web/20080312151621/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 ^[11] of bolometric corrections (note that these second set of tables are consistent with a bolometric magnitude of 4.73 ^[12] for the Sun and also be aware that there are misprint ^[12] errors for a few of the figures in the tables)

References

1. Popper, Daniel M. (1980-09-01). "Stellar Masses". Annual Review of Astronomy and Astrophysics. 18 (1): 115–164. Bibcode:1980ARA&A..18..115P. doi:10.1146/annurev.aa.18.090180.000555. ISSN   0066-4146.
2. Humphreys, R. M.; McElroy, D. B. (1984). "The initial mass function for massive stars in the Galaxy and the Magellanic Clouds". The Astrophysical Journal. 284: 565–577. Bibcode:1984ApJ...284..565H. doi:10.1086/162439. ISSN   0004-637X.
3. B., Kaler, James (1989). Stars and their spectra: an introduction to the spectral sequence. Cambridge [Cambridgeshire]: Cambridge University Press. ISBN   978-0521304948. OCLC   17731797.
4. 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: 231–250. Bibcode:1998A&A...333..231B.
5. 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 . 337 (1): 332–340. arXiv: astro-ph/0208057 . Bibcode:2002MNRAS.337..332S. doi:10.1046/j.1365-8711.2002.05917.x. S2CID   17930469. Lower effective temperatures correspond to higher values of ${\displaystyle BC_{K}}$; since ${\displaystyle M_{K}=M_{bol}-BC_{K}\!\,}$, cooler RC stars tend to be brighter.
6. 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. 403 (3): 1592–1610. arXiv: 1002.1972 . Bibcode:2010MNRAS.403.1592B. doi:10.1111/j.1365-2966.2009.16223.x. S2CID   119181086.
7. Casagrande, Luca; VandenBerg, Don A. (October 2014), "Synthetic stellar photometry: general considerations and new transformations for broad-band systems", Monthly Notices of the Royal Astronomical Society , 444 (1): 392, arXiv: 1407.6095 , Bibcode:2014MNRAS.444..392C, doi:10.1093/mnras/stu1476 with up-to-date interpolation codes https://github.com/casaluca/bolometric-corrections
8. Casagrande, L; VandenBerg, Don A (2018-01-18). "Synthetic Stellar Photometry – II. Testing the bolometric flux scale and tables of bolometric corrections for the Hipparcos/Tycho, Pan-STARRS1, SkyMapper, and JWST systems". Monthly Notices of the Royal Astronomical Society. 475 (4): 5023–5040. arXiv: 1801.05508 . Bibcode:2018MNRAS.475.5023C. doi:10.1093/mnras/sty149. ISSN   0035-8711.
9. IAU XXIX General Assembly Draft Resolutions Announced , retrieved 2015-07-08
10. 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 [astro-ph.SR].
11. 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
12. 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. S2CID   119219274.