An Introduction to Modern Astrophysics · Problem 10.25

The Sun — Carroll & Ostlie

Physics — Astrophysics · stellar-structure-equations-verification

Problem

Verify that the basic equations of stellar structure [Eqs. (10.6), (10.7), (10.36), (10.68)] are satisfied by the

1\,\mathrm{M}_\odot

StatStar model available for download from the companion website; see Appendix L. This may be done by selecting two adjacent zones and numerically computing the derivatives on the left-hand sides of the equations, for example

\frac{dP}{dr} \simeq \frac{P_{i+1} - P_i}{r_{i+1} - r_i},

and comparing your results with results obtained from the right-hand sides using average values of quantities for the two zones [e.g.,

M_r = (M_i + M_{i+1})/2

]. Perform your calculations for two adjacent shells at temperatures near

5 \times 10^6

K, and then compare your results for the left- and right-hand sides of each equation by determining relative errors. Note that the model assumes complete ionization everywhere and has the uniform composition

X = 0.7

Y = 0.292

Z = 0.008

. Your results on the left- and right-hand sides of the stellar structure equations will not agree exactly because StatStar uses a Runge--Kutta numerical algorithm that carries out intermediate steps not shown in the output file.

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