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How to estimate the luminosity of a main sequence star

If we know the internal temperature of a main sequence star, here's the way to estimate its luminosity: calculate the total energy of the radiation field inside the star, then divide this energy by the characteristic time for the radiation to escape from the star:

L_* = E_rad/t_esc. (1)

Once again, we're going to do this in a really rough way! We assume that the star is made of pure hydrogen, and that its internal temperature, T_*, is constant and given by our previous estimate. To calculate the energy contained by the internal radiation in the star, we assume that the radiation is a blackbody at this internal temperature. If so, its total energy is given by the energy density of blackbody radiation times the volume of the star:

E_rad = aT_*⁴(4pR_*³/3). (2)

where a = 7.56 x 10^-15 ergs cm^-3 K^-4 is a fundamental constant of radiation.

Now we must calculate the escape time for the radiation. If the radiation escaped freely, the time for a photon to travel from the center of the star to its surface would be given simply by t_esc = R_*/c. But that's not what happens. The photon doesn't travel far in any direction before it is absorbed or scattered by an atom or a free electron. Then the photon is re-emitted in another random direction. So the photon escapes from the star by a process that we call random walk, or diffusion. The mean free path (average distance between scatterings) of a photon inside a star is given by the equation:

l = (n_es)^-1, (3)

where s is the cross section for absorption or scattering. Here, we have assumed that the main atomic process that hinders the escape of photons from a star is scattering by free electrons (which have density n_e), and that the relevant cross section is the Thompson cross section, s = 6.65 x 10^-25 cm². According to the theory of random walk, the typical number of scatterings that a photon will undergo in escaping the star is given by N = (R/l)², and so the average total path length, l, for a photon to escape the star is given by l = Nl, or l = R²/l. Then, the typical timescale for a photon to escape the star is given by

t_esc = R_*²/lc = n_esR_*²/c. (4)

The average number density of electrons in the star (assuming one electron per hydrogen atom) is

n_e = 3M_*/(4pR_*³m_H). (5)

Combining equations (1) - (5), we find the following expression for the stellar luminosity:

L_* = (4p/3)²(caT_*⁴R_*⁴m_H)/(M_*s) . (6)

Now, we already have an estimate for T_* (see internal temperature, equation 5). Plugging that estimate into equation (6), we find:

L_* = 2.6 x 10³⁴ ergs s^-1 (M_*/M_Sun)³. (7)

The actual luminosity of the Sun is L_Sun = 3.9 x 10³³ ergs s^-1, so equation (7) gives an estimate that is too high by a factor ~ 6.7. It's not too surprising that our estimate is off, given the roughness of our approximations. In fact, one of the main reasons that the coefficient of equation (7) is high is that other atomic processes besides electron scattering further impede the escape of photons from the star.

Just as with our estimate of the central temperatures of main sequence stars, we might guess that the correction factor for equation (7) is roughly the same for all main sequence stars. If so, we could write

L_* = 3.9 x 10³³ ergs s^-1 (M_*/M_Sun)³. (8)

Indeed, equation (8) works fairly well. It says, for example, that a 10 solar mass main sequence star will have luminosity equal to about 1000 times that of the Sun. That's not quite right. The observed mass-luminosity relationship, L_* ~ M_*^3.5, (see Lesson 4) is a bit stronger than that given by equation (8). But it is clear that equation (8) has captured the main effect.

The main effect is that the more massive main sequence stars have lower average densities than the Sun. As a result, the mean free paths of photons inside the stars are greater than those in the Sun, and so the internal radiation can escape from such stars more rapidly than it can from the Sun.