# Color index

In astronomy, the color index is a simple numerical expression that determines the color of an object, which in the case of a star gives its temperature. The smaller the color index, the more blue (or hotter) the object is. Conversely, the larger the color index, the more red (or cooler) the object is. This is a consequence of the logarithmic magnitude scale, in which brighter objects have smaller (more negative) magnitudes than dimmer ones. For comparison, the yellowish Sun has a B−V index of 0.656 ± 0.005,[2] whereas the bluish Rigel has a B−V of −0.03 (its B magnitude is 0.09 and its V magnitude is 0.12, B−V = −0.03).[3] Traditionally, the color index uses Vega as a zero point.

Sample calibration colors[1]
ClassB−VU−BV−RR−I Teff (K)
O5V−0.33−1.19−0.15−0.3242,000
B0V−0.30−1.08−0.13−0.2930,000
A0V−0.02−0.020.02−0.029,790
F0V0.300.030.300.177,300
G0V0.580.060.500.315,940
K0V0.810.450.640.425,150
M0V1.401.221.280.913,840

To measure the index, one observes the magnitude of an object successively through two different filters, such as U and B, or B and V, where U is sensitive to ultraviolet rays, B is sensitive to blue light, and V is sensitive to visible (green-yellow) light (see also: UBV system). The set of passbands or filters is called a photometric system. The difference in magnitudes found with these filters is called the U−B or B−V color index respectively.

In principle, the temperature of a star can be calculated directly from the B−V index, and there are several formulae to make this connection.[4] A good approximation can be obtained by considering stars as black bodies, using Ballesteros' formula[5] (also implemented in the PyAstronomy package for Python):[6]

${\displaystyle T=4600\,\mathrm {K} \left({\frac {1}{0.92(B-V)+1.7}}+{\frac {1}{0.92(B-V)+0.62}}\right).}$

Color indices of distant objects are usually affected by interstellar extinction, that is, they are redder than those of closer stars. The amount of reddening is characterized by color excess, defined as the difference between the observed color index and the normal color index (or intrinsic color index), the hypothetical true color index of the star, unaffected by extinction. For example, in the UBV photometric system we can write it for the B−V color:

${\displaystyle E_{B-V}=(B-V)_{\text{observed}}-(B-V)_{\text{intrinsic}}.}$

The passbands most optical astronomers use are the UBVRI filters, where the U, B, and V filters are as mentioned above, the R filter passes red light, and the I filter passes infrared light. This system of filters is sometimes called the Johnson–Cousins filter system, named after the originators of the system (see references). These filters were specified as particular combinations of glass filters and photomultiplier tubes. M. S. Bessell specified a set of filter transmissions for a flat response detector, thus quantifying the calculation of the color indices.[7] For precision, appropriate pairs of filters are chosen depending on the object's color temperature: B−V are for mid-range objects, U−V for hotter objects, and R−I for cool ones.

## References

1. Zombeck, Martin V. (1990). "Calibration of MK spectral types". Handbook of Space Astronomy and Astrophysics (2nd ed.). Cambridge University Press. p. 105. ISBN 0-521-34787-4.
2. David F. Gray (1992), The Inferred Color Index of the Sun, Publications of the Astronomical Society of the Pacific, vol. 104, no. 681, pp. 1035–1038 (November 1992).
3. "* bet Ori". SIMBAD. Centre de données astronomiques de Strasbourg.
4. Sekiguchi M. and Fukugita (2000). "A STUDY OF THE B-V COLOR-TEMPERATURE RELATION". AJ (Astrophysical Journal) 120 (2000) 1072. http://iopscience.iop.org/1538-3881/120/2/1072.
5. Ballesteros, F. J. (2012). "New insights into black bodies". EPL 97 (2012) 34008. arXiv:1201.1809.
6. BallesterosBV_T API http://www.hs.uni-hamburg.de/DE/Ins/Per/Czesla/PyA/PyA/index.html.
7. Michael S. Bessell (1990), UBVRI passbands, Publications of the Astronomical Society of the Pacific, vol. 102, Oct. 1990, p. 1181–1199.
• Query for Johnson, H. L. and Morgan, ApJ 117, 313 (1953)
• Query for Cousins, A. W. J., MNRAS 166, 711 (1974)
• Query for Cousins, A. W. J., MNASSA 33, 149 (1974)
• Query for Bessell, M. S., PASP 102, 1181 (1990)