Take the solar luminosity and

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Astrophysics & Cosmology · Stellar structure, energy generation and evolution

Stars from the Inside, Ch The Sun as a test: calibration, sound, neutrinos and a magnetised atmosphere

Problem

Take the solar luminosity

L_\odot=3.828\times10^{26}\,\mathrm{W}

and

1\,\mathrm{AU}=1.496\times10^{11}\,\mathrm{m}

. From the pp chain's reaction sequence and Q values (an earlier result), take each completion of the chain that makes one

^4

He nucleus to release an effective energy

Q_{\rm eff}=26.20\,\mathrm{MeV}

after neutrino losses (

1\,\mathrm{MeV}=1.602\times10^{-13}\,\mathrm{J}

). About

85\%

of completions proceed through the ppI branch, which needs two

p+p\to d+e^++\nu

reactions per completion, each emitting one pp neutrino; the other

15\%

go through ppII or ppIII, which need one. ((a)) Assuming all of

L_\odot

is supplied at this effective Q value, estimate the total rate of pp-chain completions per second in the Sun. ((b)) Hence estimate the production rate of pp neutrinos per second (two per ppI completion and one per ppII or ppIII completion, neglecting the small share made by the pep reaction). ((c)) Use the inverse-square law to estimate the pp neutrino flux at Earth, in

\mathrm{cm^{-2}\,s^{-1}}

. ((d)) A gallium detector's effective capture cross-section for pp neutrinos is

c_{\rm pp}=1.17\times10^{-45}\,\mathrm{cm^2}

, and its rate in SNU is

\Phi c_{\rm pp}\times10^{36}

(with

\Phi

\mathrm{cm^{-2}\,s^{-1}}

). Compute the pp branch's predicted contribution to the gallium rate, and compare it with the measured total gallium rate of

66\,\mathrm{SNU}

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