In astrophysics, the mass–luminosity relation is an equation giving the relationship between a star's mass and its luminosity, first noted by Jakob Karl Ernst Halm.
In summary, the relations for stars with different ranges of mass are, to a good approximation, as the following:[2][4][5]
For stars with masses less than 0.43M⊙, convection is the sole energy transport process, so the relation changes significantly.
It can be shown this change is due to an increase in radiation pressure in massive stars.
Another form, valid for K-type main-sequence stars, that avoids the discontinuity in the exponent has been given by Cuntz & Wang;[6] it reads:
This relation is based on data by Mann and collaborators,[7] who used moderate-resolution spectra of nearby late-K and M dwarfs with known parallaxes and interferometrically determined radii to refine their effective temperatures and luminosities.
Those stars have also been used as a calibration sample for Kepler candidate objects.
Then, using Kepler's laws of celestial mechanics, the distance between the stars is calculated.
These masses are used to re-calculate the separation distance, and the process is repeated.
A more sophisticated calculation factors in a star's loss of mass over time.
Deriving a theoretically exact mass/luminosity relation requires finding the energy generation equation and building a thermodynamic model of the inside of a star.
[10] The derivation showed that stars can be approximately modelled as ideal gases, which was a new, somewhat radical idea at the time.
Where there is no heat convection, this dissipation happens mainly by photons diffusing.
By integrating Fick's first law over the surface of some radius r in the radiation zone (where there is negligible convection), we get the total outgoing energy flux which is equal to the luminosity by conservation of energy:
Note that this assumes that the star is not fully convective, and that all heat creating processes (nucleosynthesis) happen in the core, below the radiation zone.
These two assumptions are not correct in red giants, which do not obey the usual mass-luminosity relation.
Stars of low mass are also fully convective, hence do not obey the law.
Approximating the star by a black body, the energy density is related to the temperature by the Stefan–Boltzmann law:
Since matter is fully ionized in the star core (as well as where the temperature is of the same order of magnitude as inside the core), photons collide mainly with electrons, and so λ satisfies
is the cross section for electron-photon scattering, equal to Thomson cross-section.
The average stellar electron density is related to the star mass M and radius R
Note that this does not hold for large enough stars, where the radiation pressure is larger than the gas pressure in the radiation zone, hence the relation between temperature, mass and radius is different, as elaborated below.
One may distinguish between the cases of small and large stellar masses by deriving the above results using radiation pressure.
The consideration begins by noting the relation between the radiation pressure Prad and luminosity.
To the first approximation, stars are black body radiators with a surface area of 4πR2.
Thus, from the Stefan–Boltzmann law, the luminosity is related to the surface temperature TS, and through it to the color of the star, by
where σB is Stefan–Boltzmann constant, 5.67×10−8 W m−2 K−4 The luminosity is equal to the total energy produced by the star per unit time.
Since this energy is produced by nucleosynthesis, usually in the star core (this is not true for red giants), the core temperature is related to the luminosity by the nucleosynthesis rate per unit volume:
Additionally, S(E)/E is the reaction cross section, n is number density,
is the reduced mass for the particle collision, and A,B are the two species participating in the limiting reaction (e.g. both stand for a proton in the proton-proton chain reaction, or A a proton and B an 147N nucleus for the CNO cycle).