Mass–luminosity relation

In astrophysics, the mass–luminosity relation is an equation giving the relationship between a star's mass and its luminosity. The relationship is represented by the equation:

$\frac{L}{L_{\odot}} = \left(\frac{M}{M_{\odot}}\right)^a$

where L_⊙ and M_⊙ are the luminosity and mass of the Sun and 1 < a < 6. The value a = 3.5 is commonly used for main-sequence stars. This equation and the usual value of a = 3.5 only applies to main-sequence stars with masses 2M_⊙ < M < 20M_⊙ and does not apply to red giants or white dwarfs. As a star approaches the Eddington Luminosity then a = 1.

In summary, the relations for stars with different ranges of mass are, to a good approximation, as the following:

$\frac{L}{L_{\odot}} \approx .23\left(\frac{M}{M_{\odot}}\right)^{2.3} \qquad (M < .43M_{\odot})$

$\frac{L}{L_{\odot}} = \left(\frac{M}{M_{\odot}}\right)^4 \qquad\qquad (.43M_{\odot} < M < 2M_{\odot})$

$\frac{L}{L_{\odot}} \approx 1.5\left(\frac{M}{M_{\odot}}\right)^{3.5} \qquad (2M_{\odot} < M < 20M_{\odot})$

$\frac{L}{L_{\odot}} \approx 3200 \frac{M}{M_{\odot}} \qquad (M > 20M_{\odot})$

For stars with masses less than 0.43M_⊙, convection is the sole energy transport process, so the relation changes significantly. For stars with masses M > 20M_⊙ the relationship flattens out and becomes L ∝ M. It can be shown this change is due to an increase in radiation pressure in massive stars. These equations are determined empirically by determining the mass of stars in binary systems to which the distance is known via standard parallax measurements or other techniques. After enough stars are plotted, stars will form a line on a logarithmic plot and slope of the line gives the proper value of a.

The mass/luminosity relation is important because it can be used to find the distance to binary systems which are too far for normal parallax measurements, using a technique called "dynamical parallax". In this technique, the masses of the two stars in a binary system are estimated, usually as being the mass of the Sun. Then, using Kepler's laws of celestial mechanics, the distance between the stars is calculated. Once this distance is found, the distance away can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the apparent magnitudes of both stars, the luminosities can be found, and by using the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies as high as 5% can be achieved. The mass/luminosity relationship can also be used to determine the lifetime of stars by noting that lifetime is approximately proportional to M/L. One finds that more massive stars live shorter. A more sophisticated calculation factors in a star's loss of mass over time.

Derivation

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. The first such derivation was performed by astrophysicist Arthur Eddington in 1924. The derivation showed that stars can be approximately modelled as ideal gasses, which was a new, somewhat radical idea at the time. What follows is very similar to Eddington's approach, but uses a random-walk analysis and does not include the optical opacity.

To the first approximation, stars are black body radiators with a surface area of 4πR². Thus, from the Stefan–Boltzmann law, the luminosity (energy emitted per second) is obtained as

$L = 4\pi R^2\sigma T_E^4,$

where σ is Stefan's constant, 5.67 × 10⁻⁸W m⁻² K⁻⁴.

Under the condition of hydrostatic equilibrium,

$\frac{dP}{dr}=-\frac{Gm(r)\rho (r)}{r^2}$.

By integrating both sides of this relation from r = 0 to r = R, one arrives at a form of the virial theorem:

$\langle P \rangle = - \frac{1}{3}\frac{E_{GR}}{V}$.

The potential energy of a spherical mass distribution is

$E_{GR} =- \frac{3}{5}\frac{GM^2}{R}.$

This and the volume are substituted in to give

$\langle P \rangle \approx \frac{GM^2}{4\pi R^4}$.

Simplifying greatly, the ideal gas law (PV = nkT) can be used to obtain the temperature.

$\langle P \rangle = \frac{\langle \rho \rangle}{\bar m}kT,$

$kT = \frac{GM\bar m}{3R}$.

Here $\bar m$ is the average mass of gas particles within the star. Substituting this equation into the initial luminosity equation, along with

$R=\left(\frac{3}{4}\frac{1}{\rho \pi}M\right)^{\frac{1}{3}}$

yields

$L \varpropto M^{3.33}$.

A slightly more exact result can be achieved by considering that the above equation gives the average temperature based on the average pressure, but what is actually needed is the surface temperature. Because stars are much hotter in the center than near the surface, one also needs to estimate the relationship between the surface temperature and internal temperature. The center is so much hotter since energy takes a long time to escape, otherwise, thermodynamic equilibrium would be achieved quickly and the temperature would be nearly uniform throughout. A random walk analysis can be used to estimate the "delay factor", i.e. the amount of time it takes energy to escape. In reality, the mean free path, $l$ of photons within the Sun depends on the density and temperature, but here it will be approximated as a constant. After N interactions, resulting in N vector displacements in random directions, the distance traveled is:

$\mathbf{D = l_1 + l_2 + \cdots + l_n}$.

The square of the net displacement is:

$D^2 = l_1^2 + l_2^2 + \cdots + l_n^2 + 2(\mathbf{l_1\cdot l_2 + l_1\cdot l_3 + \cdots)}$.

Averaging over many random direction changes, the terms involving dot products cancel since the direction is random. Hence, for large $N$,

$D^2 = l_1^2 + l_2^2 + \cdots + l_n^2 = Nl^2.$

Thus, to escape from the Sun, on average $\frac{R^2}{l^2}$ steps are required. The time this takes is $t\approx\frac{R^2}{cl}$. In contrast, the time to escape directly out of the Sun from the center is $\frac{R}{c}$, which is a factor of $\frac{l}{R}$ shorter.

Thus, factoring this into the Stefan-Boltzmann law leads to

$T_E\approx \Big[ \frac{l}{R}\Big]^{\frac{1}{4}}T_I$.

Combining the above equations, the following result is obtained

$L \approx 4\pi R^2\sigma T_I^4\frac{l}{R}\approx \frac{(4\pi)^2}{3^5}\frac{\sigma}{k^4}G^4\bar m^4 \langle \rho \rangle l M^3$

But the mean free path is inversely proportional to the product of the cross section and the numerical density, and thus

$l \varpropto \langle\rho\rangle^{-1}$

which gives

$L \varpropto M^3.$

Distinguishing between small and large stellar masses

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 $\kappa$ 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:

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

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

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

${T_I}^3\frac{dT_I}{dr} = -\frac {3\kappa\rho}{16\sigma}\frac{L}{4\pi r^2},$

from which it follows directly that

$L \varpropto {T_I}^4 \frac {R}{\rho} \varpropto {T_I}^4 \frac {R^4}{M}$.

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

$T_I \varpropto \frac {M}{R}$

as before. More precisely, since integration was done from 0 to R so $T_I-T_E$ 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

$L \varpropto M^3.$

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

${T_I}^4\varpropto \frac{M^2}{R^4},$

hence

$L \varpropto M.$