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

• p = F / A = GρM / R

The ratio p / ρ is given by

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

• With ρ = Nm_A / V we get the ratio
• p / ρ = kT / m_A

To be stable the following equation must be valid:

For the temperature T we get

• G = 6.710-11 N m2 / kg2 mA = 1.710-27 kg M = 21030 kg k = 1.410-23 J / K R = 7108 m

  constant of gravitation mass of hydrogen atom mass of the Sun Boltzmann constant radius of the Sun

• T = 2.310⁷ 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

• with the mean density ρ = 1.410³ kg / m³ is
• p = 2.710¹⁴ N / m² = 2.710⁹ bar

The real value should be greater because the density increases towards the center.