AS3003: Stellar Physics: Stellar Structure
Problem Sheet 1
1. [1] Calculate the dynamical, Kelvin and nuclear time scales for main-sequence
stars of 1 and 20 solar masses (you may suppose that LM3.8, RM0.2). Sketch the
nuclear timescale as a function of main-sequence mass. [ 2.3 103 s, 3 107 y, 0.15 1011
y, 4.86 104 s, 7.5 104 y, 3.5 106 y]
2. [1/2] A shell of matter enclosing a mass m0 is initially at rest, with radius r0. The
conservation of energy equation may be written 1/2 [dr/dt]2 = Gm0/r – Gm0/r0.
Integrating and using substitutions x=r/r0, x=sin2, or otherwise, show that the
timescale for gravitational collapse can be written as tFF=(3/32G)1/2
3. [2] Derive the equations for convective instability
gradient under convective equilibrium
d ln T
d ln P
    ad 
P d
1
 , and the temperature
 dP 
 1

.
4. [2] Transform the stellar structure o.d.e.s and boundary conditions from Eulerian to
Lagrangian coordinates.
5. [3] Consider a solar mass star with radius 1011 m and average internal temperature
30,000 K. What would its average internal temperature be after contracting to 109 m.
[ 3,000,000 K ]
6. [3] The Virial theorem provides an estimate for the average internal temperature of
a contracting star kTGmHM2/31/3.. Electrons become degenerate at a point where
mH(mekT)3/2/h3. Derive expressions for the radius, density and temperature of a
star at this point, assuming that nuclear reactions are not ignited. What values do these
take for a solar mass star ?
[ ~1.4 105 kg m-3, ~0.22 Rsun ]
7. [3] If the internal density distribution of a sequence of chemically homogeneous
stars is given by  = (M/R3 ) F(x), where M and R are the stellar mass and radius and
F(x) is a function of radius (x=r/R), and if the equation of state is given by P  T,
show that
m = M . Fm(x),
P = (M2/R4 ) FP(x) and
T = (M/R) FT(x)
where Fm(x), FP(x) and FT(x) are also functions of radius only. If the opacity  and
nuclear energy generation rate  are given respectively by,
  T
3.5
and
  T

then show that
l rad 
M
R
5.5
0.5
Frad  x
and
l nuc 
M
2 
R
3
Fnuc  x
Hence derive the mass-radius and mass-luminosity relations for solar-type (=4) and
massive (=15) stars.
CSJ
6 February, 2016