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Unformatted text preview: Physics 160: Stellar Astrophysics Homework #5 Due Tuesday November 8th at 5pm in SERF 340 Reading: Carroll & Ostlie sections 11.111.3, 12.112.3
Exercise [90 pts]:
Your homework this week is to investigate the LaneEmden equation
computationally.
(1) [50 pts] Employing a numerical integrator, compute and compare the
density profiles for polytropes n = 1.5 (degenerate star) and n = 3 (star in
radiative equilibrium). In both cases, plot the ratio of the density to the
central density (ρ/ρc) as a function of the dimensionless quantity ξ = r/λn.
Also compute the values of ξ1 and dD/dξ at ξ1, where the density goes to
zero. Be sure to show both your plot(s) and the code (e.g., C, Mathematica,
etc.) you used to calculate these profiles (see note on next page).
(2) [20 pts] For both polytropes, determine the ratio of the average density
<ρ> = 3M/4πR3 to central density ρc (remember R = λξ1 and see page 338 of
Carroll & Ostlie for an expression for M). Explain qualitatively why one
polytrope is more centrally concentrated.
(3) [20 pts] For the n = 3 case, which has an equation of state P = Kρ4/3, use
the expression for K on page 340 of Carroll & Ostlie to compute a star’s
mass for µ = 0.5 and β = 0.1, 0.5 and 0.9 (β is the ratio of gas to radiation
pressure). What do the differences in these values tell you about the
importance of radiation pressure in the interiors of low and high mass stars?
NOTE: If you are writing your own code for the numerical integration (it’s
good for the soul!), here’s a useful hint: 2nd order differential equations can
be integrated using the Euler method by solving two equations: one for the
parameter and one for its derivative. In this case, the LaneEmden equation
can be written as: d2 D 2 dD
+
= −Dn
dξ 2
ξ dξ
which can be split into two firstorder differential equations: f (ξi ) = dD
(ξi )
dξ (the derivative of D) df
2
(ξi ) = − f (ξi ) − Dn (ξi )
dξ
ξi (the LaneEmden equation) You can then iterate these as ξi+1 = ξi + h
D(ξi+1 ) = D(ξi ) + hf (ξi )
f (ξi+1 ) = f (ξi ) + h df
(ξi )
dξ until you get to the first zero point ξ1, where h is some small number (if
you’d like to use a different numerical integration scheme, you are welcome
to but be sure to explain your method).
Don’t forget your boundary values! ...
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 Fall '08
 Norman,M
 Physics, Work

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