The Sun has mass and radius

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Relativity & Gravitation · Newtonian gravitation as a field theory

Relativity and Gravitation, Ch Gravitational field, potential and Gauss's law for gravity

Problem

The Sun has mass

M_\odot = 1.989 \times 10^{30}\ \mathrm{kg}

and radius

R_\odot = 6.957 \times 10^{8}\ \mathrm{m}

, and is treated here as a non-rotating spherically symmetric body; for the self-energy estimate it is modelled as a uniform sphere, so

|E_{\rm grav}| = 3GM_\odot^{2}/5R_\odot

. In the Sun's core the net effect of hydrogen burning is to turn four hydrogen atoms, each of mass

1.00783\ \mathrm{u}

, into one helium atom of mass

4.00260\ \mathrm{u}

, the mass difference leaving as energy. Take

G = 6.674 \times 10^{-11}\ \mathrm{N\,m^{2}\,kg^{-2}}

c = 2.998 \times 10^{8}\ \mathrm{m\,s^{-1}}

1\ \mathrm{u} = 931.494\ \mathrm{MeV}/c^{2}

, and the gravitational potential zero at infinity. ((a)) Find the gravitational potential at the Sun's surface. ((b)) Evaluate the uniform-sphere estimate

3GM_\odot/5R_\odot c^{2}

of the Sun's gravitational mass defect as a fraction of its rest energy, and the mass in kilograms that this fraction corresponds to. ((c)) Find the energy released per helium atom formed, in MeV, and the fraction of the reacting rest mass that is converted into energy. ((d)) If the Sun turns

10\%

of its mass from hydrogen into helium in this way, what fraction of

M_\odot c^{2}

is released, and how many times larger is that than the gravitational estimate of part (b)?

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