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How to estimate the internal temperature of a main sequence star

As the hypertext says, we do this by using the principle of hydrostatic equilibrium, which says that the force of attraction (due to gravity) must equal the force of repulsion (due to heat pressure).

We're going to do this in a really rough way! So, let's say that the star is made of pure hydrogen. At the center, the hydrogen atoms are fully ionized, so that there are two free particles (one proton, one electron) for every hydrogen mass. The formula for heat pressure is

P_h = 2nkT, (1)

where n is the number density of hydrogen atoms, k is Boltzmann's constant, T is the internal temperature, and the factor 2 accounts for the fact that there are two free particles per hydrogen atom. .

Now let's estimate n. To do this, we're going to assume that the star has uniform density. The total number of hydrogen atoms in the star is N = M_*/m_H, where M_* is the mass of the star and m_H is the mass of the hydrogen atom. The density of hydrogen atoms is the total number divided by the volume of the star:

n = 3M_*/(4pR³m_H), (2)

where R is the radius of the star.

Now let's estimate the inward pressure due to gravity. The force of gravitational attraction between two masses M₁ and M₂ is given by

F = GM₁M₂/R², (3)

where R is the distance between the two masses. Now, for a star, there's only one mass, M_*, and it's attracting itself. So let's simply replace both M₁ and M₂ by M_*. Moreover, there's only one characteristic distance in the picture, and that's the radius, R_* of the star itself.

Pressure is force divided by area. What area? Well, the only area in the picture is the surface area of the star, A = 4pR². So, we divide the force of gravity by A to find an expression for the pressure exerted by gravity:

P_G = GM_*²/(4pR_*⁴). (4)

Now, setting P_h = P_G_{,
we derive:}

T = GM_*m_H/(6kR_*). (5)

Let's use this equation to estimate the value of the internal temperature of the Sun. In cgs units, G = 6.67 x 10^-8, M_Sun = 2 x 10³³ g, m_H = 1.67 x 10^-24 g, k = 1.38 x 10^-16, and R_Sun = 7 x 10¹⁰ cm. Plugging these values in, we find T = 4 x 10⁶ K for the average interior temperature of the Sun.

Actually, the central temperature of the Sun is about 4 times greater, T_c = 1.5 x 10⁷ K. It's not surprising that we're off by a factor 4, given the roughness of our approximations.

To improve our estimate of the temperature inside the Sun, we need to do several things. First, we must find a more accurate formula for the pressure due to gravity. Obviously, the pressure is not uniform throughout the Sun -- it's greatest at the center. We can do this by solving a differential equation called the equation of hydrostatic equilibrium. Second, we must take into account that the density inside the Sun is not constant. The gas in the Sun is compressible, and in fact the density at the Sun's center is about 100 times greater than the average density. To do that, we must find a relationship between the interior density and pressure of the Sun -- an equation of state. The equation of state we have been using is simply that the density is uniform, and that's not a very good approximation. A better equation of state for stars is the polytropic equation of state: P = Kr^{(1 + 1/n)}, where K is a constant and n is a number called the polytropic index. For the Sun, n = 3 is a pretty good approximation (but still not exact). For most stars, n lies between 3/2 and 3.

Making these improvements, one can derive and solve a non-linear ordinary differential equation for the density structure inside a star, called the Lane-Emden equation. The solutions of the Lane-Emden equation give an equation relating the central temperature of a star to its mass and radius that is almost identical to equation (5), but with a somewhat greater numerical coefficient, depending on the value of the polytropic index.

To calculate the central temperature of a star really accurately, we can't use the polytropic equation of state. We must actually solve another equation that gives us the internal temperature of the star as a function of its radius. That equation describes how heat is transferred through the star, the equation of energy transfer. So, ultimately, to calculate the structure of a star, we must solve several simultaneous non-linear differential equations on a computer.

But we don't have to do all this to have a pretty good estimate of the central temperature of a main-sequence star. Just take equation (5), and change the numerical coefficient (increase it by a factor 4) so that the equation gives a good fit to the Sun:

T_c = 1.5 x 10⁷ K (M_*/M_Sun)(R_*/R_Sun)^-1. (6)

Equation (6) is in fact a pretty good approximation for the central temperature of any main sequence star. In our estimate of the central temperature of the Sun, we were off by a factor of about 4 because our estimates of the central density and central pressure due to gravity did not take into account the real variation of these quantities with radius. But, since all main sequence stars have very similar radial profiles of density and pressure, the correction factor that needs to be applied to equation (5) should be almost the same for all main sequence stars.

Equation (6) is no good at all for red giant stars or horizontal branch stars, because these stars have internal density profiles that are radically different from main sequence stars.