Mass–luminosity relation

Deriving a theoretically exact mass/luminosity relation requires finding the energy generation equation and building a thermodynamic model of the inside of a star. However, the basic relation L ∝ M³ can be derived using some basic physics and simplifying assumptions.[9] The first such derivation was performed by astrophysicist Arthur Eddington in 1924.[10] The derivation showed that stars can be approximately modelled as ideal gases, which was a new, somewhat radical idea at the time. What follows is a somewhat more modern approach based on the same principles.

An important factor controlling the luminosity of a star (energy emitted per unit time) is the rate of energy dissipation through its bulk. 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:

$${\displaystyle L=-4\pi \,r^{2}D{\frac {\partial u}{\partial r}}}$$

where D is the photons diffusion coefficient, and u is the energy density.

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:

$${\displaystyle u={\frac {4}{c}}\,\sigma _{B}\,T^{4}}$$

where

$${\displaystyle \sigma _{B}={\frac {2\pi ^{5}k_{B}^{4}}{15c^{2}h^{3}}}={\frac {\pi ^{2}k_{B}^{4}}{60\hbar ^{3}c^{2}}}.}$$

is the Stefan–Boltzmann constant, c is the speed of light, k_B is Boltzmann constant and ${\displaystyle \hbar }$ is the reduced Planck constant.

As in the theory of diffusion coefficient in gases, the diffusion coefficient D approximately satisfies:

$${\displaystyle D={\frac {1}{3}}c\,\lambda }$$

where λ is the photon mean free path.

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

$${\displaystyle \lambda ={\frac {1}{n_{e}\sigma _{e\cdot \gamma }}}}$$

Here ${\displaystyle n_{e}}$ is the electron density and:

$${\displaystyle \sigma _{e\cdot \gamma }={\frac {8\pi }{3}}\left({\frac {\alpha \hbar c}{m_{e}c^{2}}}\right)^{2}}$$

is the cross section for electron-photon scattering, equal to Thomson cross-section. α is the fine-structure constant and m_e the electron mass.

The average stellar electron density is related to the star mass M and radius R

$${\displaystyle \langle n_{e}\rangle ={\frac {M}{m_{n}4\pi R^{3}/3}}}$$

Finally, by the virial theorem, the total kinetic energy is equal to half the gravitational potential energy E_G, so if the average nuclei mass is m_n, then the average kinetic energy per nucleus satisfies:

$${\displaystyle {\frac {3}{2}}k_{B}T={\frac {1}{2}}E_{G}{\frac {m_{n}}{M}}=C{\frac {3}{10}}{\frac {GMm_{n}}{R}}}$$

where the temperature T is averaged over the star and C is a factor of order one related to the stellar structure and can be estimated from the star approximate polytropic index. 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.

Wrapping up everything, we also take r to be equal to R up to a factor, and n_e at r is replaced by its stellar average up to a factor. The combined factor is approximately 1/15 for the sun, and we get:

$${\displaystyle {\begin{aligned}L&=-4\pi \,r^{2}D{\frac {\partial u}{\partial r}}\approx 4\pi \,R^{2}D{\frac {u}{R}}\\L&\approx {\frac {1}{15}}{\frac {64\pi ^{2}}{9}}{\frac {\sigma _{B}}{\sigma _{e\cdot \gamma }}}{\frac {R^{4}T^{4}m_{n}}{M}}\\&\approx {\frac {1}{15}}{\frac {2\pi ^{3}}{9\cdot 5^{5}}}{\frac {G^{4}\,m_{e}^{2}\,m_{n}^{5}}{\alpha ^{2}\hbar ^{5}}}\,M^{3}=4\cdot {10^{26}}_{W}\,\left({\frac {M}{M_{\odot }}}\right)^{3}\\\end{aligned}}}$$

The added factor is actually dependent on M, therefore the law has an approximate ${\displaystyle M^{3.5}}$ dependence.

One may distinguish between the cases of small and large stellar masses by deriving the above results using radiation pressure. In this case, it is easier to use the optical opacity ${\displaystyle \kappa }$ and to consider the internal temperature T_I directly; more precisely, one can consider the average temperature in the radiation zone.

The consideration begins by noting the relation between the radiation pressure P_rad and luminosity. The gradient of radiation pressure is equal to the momentum transfer absorbed from the radiation, giving:

$${\displaystyle {\frac {dP_{rad}}{dr}}=-{\frac {\kappa \rho }{c}}{\frac {L}{4\pi r^{2}}},}$$

where c is the velocity of light. Here, ${\displaystyle 1/\kappa \rho =l}$; the photon mean free path.

The radiation pressure is related to the temperature by ${\displaystyle P_{rad}={\frac {4\sigma }{3c}}{T_{I}}^{4}}$, therefore

$${\displaystyle {T_{I}}^{3}{\frac {dT_{I}}{dr}}=-{\frac {3\kappa \rho }{16\sigma }}{\frac {L}{4\pi r^{2}}},}$$

from which it follows directly that

$${\displaystyle L\varpropto {T_{I}}^{4}{\frac {R}{\rho }}\varpropto {T_{I}}^{4}{\frac {R^{4}}{M}}.}$$

In the radiation zone gravity is balanced by the pressure on the gas coming from both itself (approximated by ideal gas pressure) and from the radiation. For a small enough stellar mass the latter is negligible and one arrives at

$${\displaystyle T_{I}\varpropto {\frac {M}{R}}}$$

as before. More precisely, since integration was done from 0 to R so ${\displaystyle T_{I}-T_{E}}$ on the left side, but the surface temperature T_E can be neglected with respect to the internal temperature T_I.

From this it follows directly that

$${\displaystyle L\varpropto M^{3}.}$$

For a large enough stellar mass, the radiation pressure is larger than the gas pressure in the radiation zone. Plugging in the radiation pressure, instead of the ideal gas pressure used above, yields

$${\displaystyle {T_{I}}^{4}\varpropto {\frac {M^{2}}{R^{4}}},}$$

hence

$${\displaystyle L\varpropto M.}$$

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 T_S, and through it to the color of the star, by

$${\displaystyle L=4\pi R^{2}\sigma _{B}T_{S}^{4}}$$

where σ_B 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:

$${\displaystyle L={\frac {dE}{dt}}\approx \epsilon \,{\frac {4\pi }{3}}R^{3}\,n_{A}\,n_{B}\,{\frac {4{\sqrt {2}}}{\sqrt {3m_{R}}}}\,{\sqrt {E_{0}(T)}}{\frac {S(E_{0}(T))}{kT}}e^{-{\frac {3E_{0}(T)}{kT}}}}$$

Here, ε is the total energy emitted in the chain reaction or reaction cycle. ${\textstyle E_{0}=\left({\sqrt {E_{G}}}\,kT/2\right)^{2/3}}$ is the Gamow peak energy, dependent on E_G, the Gamow factor. Additionally, S(E)/E is the reaction cross section, n is number density, ${\displaystyle m_{R}=m_{A}\cdot m_{B}/(m_{A}+m_{B})}$ 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 ¹⁴
₇N
nucleus for the CNO cycle).

Since the radius R is itself a function of the temperature and the mass, one may solve this equation to get the core temperature.