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hwk05
Ast 4001, 2015 October 13
Homework set 5 -- Stellar interiors
Turn in your solution and explanation for problem 3 on Thursday October 22.
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1. -- A fallacy- puzzle, no calculations are needed.
Here’s a fraudulent derivation of hydrostatic equilibrium in a star. Consider a thin shell inside it.
If subscripts 1 and 2 denote the inner and outer boundaries
of the thin shell, then the inner pressure force pushing the
shell outward is area  pressure = 4  r 12 P 1 , while
the outer pressure force pushing it inward is 4  r 22 P 2 .
The net outward force due to pressure is the difference
between 4  r 12 P 1 and 4  r 22 P 2 .
In the limit as  r = r 2  r 1 becomes infinitesimally
small, this difference is
(net pressure force) =  4  { d ( r 2 P ) / d r }  r .
Meanwhile the shell is also pulled inward by gravity. The volume of the thin shell is 4  r 2  r ,
so its mass is 4 r 2  r . (Here  is mass density as usual.) If g is the local gravitational
acceleration, then gravity exerts a radial force  g m =  4 r 2  g  r on the shell.
Summing the pressure and gravity forces, we see that static equilibrium requires
4  { d ( r 2 P ) / d r }  r + 4 r 2  g  r = 0 ,
which simplifies to
d( r2 P ) / d r
  r 2  g .
This looks credible but it’s wrong ! The correct formula is d P / d r    g . Anyone
who tries to use our erroneous result will get poor models for stellar interiors.
So: Where’s the fallacy in the above derivation ?? If you don’t see it immediately, make
a serious effort to figure it out before asking someone who already knows.
Two hints:
(1) The answer isn’t some little technicality in elementary calculus, taking a limit wrong
or something like that. It’s a question of basic physics, not math.
(2) Julius Caesar would have guessed the answer from his bridge-building experience.
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4001hwk05 - p2
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2. (Practice only; useful for problem 3.) Suppose that the Sun consists of 72% hydrogen
and 28% helium by mass. Ignore heavier elements, because they have nearly the same
effect as helium for this problem. By "hydrogen" and "helium" we mean the familiar isotopes
with mass 1 and 4.
Assuming that the gas is a fully ionized plasma, calculate the gas constant  in both c.g.s.
and SI units. Aim for a precision of 1% or better. Don't forget the free electrons!
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3. (Turn in your results for this problem on Thursday October 22.) Let’s estimate the total
binding energy of the Sun, based on very little information and a few simple assumptions.
Advice: You can minimize the ugly algebra here by working with
dimensionless variable x = r / R instead of r . ( R = the Sun’s radius. )
(a) Assume that the density distribution inside the Sun is  ( r ) = { 1  r / R }  0 .
(Why? Because it's the simplest formula that has  ( R ) = 0, increases
monotonically toward the center, and gives a finite value at r = 0.)
Deduce a formula for m ( x ) where x = r / R . Evaluate the central density  0
that is consistent with the Sun’s known mass M and radius R .
(b) Do another integral to find a formula for the total gravitational energy E G , a negative
quantity, in terms of total mass M and outer radius R . Then use the Virial Theorem
to convert this to a formula for the Sun’s total energy including both gravitational and
thermal energy. Assume that it’s a perfect monatomic gas so U = 1.5 P .
(c) Integrate the hydrostatic equation to estimate P ( 0 ), the pressure at the center of the Sun.
Let’s interpret this in terms of temperature. Work out a reasonable gas constant for the
ideal gas law, P =   T . (Problem 2 above.) Use these results to estimate the
temperature at the center of the Sun in this idealized model.
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