Definitions

# mass-luminosity relation

mass-luminosity relation, in astronomy, law stating that the luminosity of a star is proportional to some power of the mass of the star. More massive stars are in general more luminous. For stars on the main sequence of the Hertzsprung-Russell diagram, it is found empirically that the luminosity varies as the 3.5 power of the mass. This means that if the mass is doubled, the luminosity increases more than tenfold. The law can be derived theoretically and was confirmed by independently measuring the masses of many visual binary stars, all at approximately the same distance. A more exact formulation of the law takes into account the chemical composition of the star. One important use of the mass-luminosity relation is in estimating the mass of a star of known luminosity that is not in a binary system.

The initial mass function (IMF) is an empirical function that describes the mass distribution (the histogram of stellar masses) of a population of stars in terms of their theoretical initial mass (the mass they were formed with). The properties and evolution of a star are closely related to its mass, so the IMF is an important diagnostic tool for astronomers studying large quantities of stars. The IMF is relatively invariant from one group of stars to another.

## Form of the IMF

The IMF is often stated in terms of a series of power laws, where $N\left(M\right)$ the number of stars of mass $M$ within a specified volume of space is proportional to $M^\left\{-alpha\right\}$ where $alpha$ is a dimensionless exponent. The IMF can be estimated from the initial luminosity function by using the mass-luminosity relation.

The exemplar form which of the IMF for stars more massive than our sun was discovered by Edwin Salpeter in 1955. His work favoured an exponent of $alpha=2.35$. This form of the IMF is called the Salpeter function or a Salpeter IMF. It shows that the number of stars in each mass range decreases rapidly with increasing mass.

Later authors extended the work below one solar mass. Glenn E. Miller and John M. Scalo suggested that the IMF "flattened" (approached $alpha=0$) below one solar mass. Pavel Kroupa kept $alpha=2.3$ above half a solar mass, but introduced $alpha=1.3$ between 0.08-0.5 solar masses and $alpha=0.3$ below 0.08 solar masses.

There are large uncertainties concerning the substellar region.

## References

• Edwin Salpeter, The luminousity function and stellar evolution, ApJ 121, 161 (1955)
• Glen Miller & John Scalo, The initial mass function and stellar birthrate in the solar neighborhood, ApJS 41, 513 (1979)
• John Scalo, The initial mass function of massive stars in galaxies. Empirical evidence, Luminous stars and associations in galaxies; Proceedings of the Symposium, Porto-Kheli, Greece, May 26-31, 1985. Dordrecht, D. Reidel Publishing Co., 1986, p. 451-466.
• Pavel Kroupa, On the variation of the initial mass function, MNRAS 322, 231 (2001) arXiv preprint
• Pavel Kroupa, The initial mass function of stars: evidence for uniformity in variable systems, Science 295, 82 (2002) arXiv preprint

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