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The Minimum Number of Hydrogen Atoms in a Star

The gravitation is very much weaker than the electrostatic Coulomb interaction. The latter binds electrons in atoms and joins atoms together to form molecules constituting the physical world on Earth, whereas gravity holds the solar system and the galaxy together and describes the very large scale of the universe.

To estimate roughly the the amount of hydrogen necessary to form a star we consider two protons
(mass m, charge e, distance d) in a cloud of fully ionized hydrogen:


The ratio between the electrostatic force (repulsive) and the force of gravitation (attractive) is:

The same result holds for the electrostatic and gravitational energy of two protons.

Adding protons (mass m) to the sphere (radius R) homogenously filled by hydrogen (total mass M)

the energy W of gravitation per proton mass is

W/m = G M / R

Using M R3  and R M1/3

W/m M2/3

Or using the number
N of protons (N M)

W/m N2/3

The electrostatic forces are shielded by the electrons on a small scale whereas gravitation is both long-range and unshielded. The hydrogen sphere will be bound by gravitation if the number N of atoms

N > (1.24·1036)3/2 = 1.4·1054

The corresponding total mass M=m·N = 2.3·1027 kg is about 1/1000 of the Sun's mass.

Using the density of the Sun (ρ=1.4·103 kg/m3) the radius is R = 7.3·107 m which is about 1/10 of the actual radius.

The hydrogen ball will continue to gain mass after it first formed. Hydrogen-burning will start at a limit of about 75 Jupiter masses: 75 · 1.7·1027 kg = 1.4·1029 kg = 0.07 MSun

Web Links

Star (Wikipedia)

Are They Planets or What?

Stellar Nurseries

The Jeans Mass and Gravitational Stability

Stephen Hawking's Universe


Last update 2010 Nov 13