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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)
W/m = G M / R
Using M ∼ R^{3} and R ∼ M^{1/3 }W/m ∼ M^{2/3} Or using the number N of protons (N ∼ M) W/m ∼ N^{2/3}
The
electrostatic
forces
are
shielded
by
the
electrons
on a small scale
whereas gravitation is both longrange and unshielded. The hydrogen
sphere will be bound by gravitation if the number N of atoms N > (1.24·10^{36})^{3/2}
= 1.4·10^{54} The corresponding total mass M=m·N = 2.3·10^{27} kg is about 1/1000 of the Sun's mass. Using the density of the Sun (ρ=1.4·10^{3} kg/m^{3}) the radius is R = 7.3·10^{7} m which is about 1/10 of the actual radius. The
hydrogen
ball
will
continue to gain mass after it first formed.
Hydrogenburning will start at a limit of about 75 Jupiter masses: 75 · 1.7·10^{27} kg = 1.4·10^{29} kg = 0.07 M_{Sun}
Last
update
2010
Nov
13 