The star Vega delivers an energy flux at the Earth and lies at a distance

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Astrophysics & Cosmology · Relativistic dynamics and fields in flat spacetime

Relativity and Gravitation, Ch Mass as energy: binding, defect and annihilation

Problem

The star Vega delivers an energy flux

f = 2.1 \times 10^{-8}\ \mathrm{W\,m^{-2}}

at the Earth and lies at a distance

d = 7.68\ \mathrm{pc}

, with

1\ \mathrm{pc} = 3.086 \times 10^{16}\ \mathrm{m}

. Assume it radiates isotropically and that nothing absorbs its light on the way. Vega burns hydrogen mainly through the CNO cycle, whose first step is the proton capture

^{12}\mathrm{C} + {}^1\mathrm{H} \to {}^{13}\mathrm{N} + \gamma

; the relevant atomic masses are

m(^{12}\mathrm{C}) = 12.000000\ \mathrm{u}

m(^{1}\mathrm{H}) = 1.007825\ \mathrm{u}

and

m(^{13}\mathrm{N}) = 13.005739\ \mathrm{u}

. Use

1\ \mathrm{u} = 1.660539 \times 10^{-27}\ \mathrm{kg} = 931.494\ \mathrm{MeV}/c^2

and

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

. ((a)) Convert the mass difference of that first CNO step into an energy in MeV. ((b)) Estimate Vega's luminosity. ((c)) Estimate the rate at which Vega loses mass by radiating, in

\mathrm{kg\,s^{-1}}

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