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The core temperature of the Sun

Temperatur im Inneren der Sonne  Pressure, density and temperature inside the Sun
(from Sexl, Raab, Streeruwitz: Materie in Raum und Zeit, Diesterweg/Salle/Sauerländer, 1980)

To be in a steady state the thermal gas pressure of the star must be in equilibrium with gravitation. First, we estimate the gravitational pressure in the centre of the star: The red cylinder (height R, cross-section A) has a mass of m = ρ·A·R. Assuming a constant homogeneous density ρ and applying Newton's law of gravity F = G·m·M / R2 to the surface of the star (radius R) the gravitational force F is F = G·ρ·A·R·M / R2 M = mass of the Sun G constant of gravitation
The pressure in the centre of the star is

p = F / A = G·ρ·M / R

The ratio p / ρ is given by

p / ρ = G·M / R

On the other hand, the pressure p of the star, considered as an ideal gas of N atoms of mass mA, is

p·V = N·k·T
( k = Boltzmann constant, T = abs. Temperature)

With ρ = N·mA / V we get the ratio

p / ρ = k·T / mA

To be stable the following equation must be valid:

G·M / R = k·T / mA

For the temperature T we get

T = G·mA·M / (k R)

 G = 6.7·10-11 N m2 / kg2 mA = 1.7·10-27 kg M = 2·1030 kg k = 1.4·10-23 J / K R = 7·108 m constant of gravitation mass of hydrogen atom mass of the Sun Boltzmann constant radius of the Sun

T = 2.3·107 K = 23,000,000 K

A more realistic value is 15,000,000 K (surface temperature: 5800 K)

The pressure in the centre of the Sun

p = ρ·G·M / R

with the mean density ρ = 1.4·103 kg / m3 is

p = 2.7·1014 N / m2 = 2.7·109 bar

The real value should be greater because the density increases towards the center. Another calculation for the pressure at radius r p(r) = F / A The force of gravity of an infinitesimal layer of thickness dr at radius r (with mass m) caused by the inner sphere (mass M) is dF = G m M / r2 Inserting m = ρ·V = ρ·4 pi r2 dr M = ρ 4 pi/3 r3 we get: dF = G 4 pi r2 dr ρ2 4 pi/3 r3 / r2 dp(r) = dF / A = dF / (4 pi r2) = 4 pi/3 G ρ2 r dr By integration from r to R we find: The pressure p0 at the centre of the star (r=0) is p0 = 4 pi/3 G ρ2 R2 With M = ρ·V = ρ 4 pi/3 R3 we get the same result as before: p = G·ρ·M / R

 Web Links How the Sun Shines John N. Bahcall, Institute for Advanced Study, Princeton, NJ

Last update 2020, Feb 27