Physics 521, Stars
Stanimir Metchev, Nov 3, 2011
Midterm
Due at 10am on Friday, Nov 4, 2011
You can attempt any combination of problems up to 50 points. You are not allowed to
collaborate with other students. Completed midterms should be handed to, or slipped
under the office door of, Prof. Jin Koda in ESS 455.
Brown dwarfs. As a protostar radiates its thermal energy it contracts and its central
density and temperature rise. If these become high enough, nuclear reactions begin to
occur rapidly, and the resulting nuclear ht source prevents further contraction on the
thermal timescale. However, another possibility is that electrons become degenerate at
the center of the protostar before the temperature becomes high enough for nuclear
reactions to occur at a significant enough rate. Then further contraction would be
prevented by the degeneracy pressure of the electrons, and the protostar would never
ignite nuclear fusion at its center. Such failed stars are called brown dwarfs. You will
calculate some approximate properties of brown dwarfs. A fact about brown dwarfs that
will be very important for you below is that they are fully and efficiently convective from
their centers to their photospheres.
1. (20 points) In ionized non-degenerate gas of cosmic abundance (i.e., near solar X, Y,
Z, µe = 1.15 for fully ionized gas) the electrons’ contribution to pressure is
comparable to that of the ions. When the electrons are degenerate, their degeneracy
pressure is much larger than the thermal corrections, but the ions continue to have
ordinary gas pressure. Show that the following expression gives the correct
approximation for the total pressure in the non-degenerate and degenerate limits:
(1)
$ # '5 3$
*2 '
–2
P " 10 & ) &1+ * +
) dyn cm ,
µ
1+
*
(
% e( %
13
where ψ ≈ 8×10–6Tµe2/3ρ–2/3 is the degeneracy parameter (ψ << 1 for degenerate
matter, ψ >> 1 for non-degenerate ideal gas electrons), and where the first term
!
comes from electron degeneracy pressure, the second from ion gas pressure, and the
third (at most a factor of 2 correction to the others) is the thermal contribution of the
electron pressure. We have neglected powers of µe in the thermal contributions.
2. (10 points) Brown dwarf interiors are in a state of degenerate coulomb plasma, with
the entropy per nucleon given by
(2)
s /k = 2.2ln(T / " 0.63 ) #11.6.
Using the first law of thermodynamics, show that for non-degenerate ionized gas, s/k
= A ln (T/ρ0.66) – B (you need not compute A or B). Thus, in both cases, show that ψ
is nearly !
constant if s is. Use fact above to argue that s is nearly constant throughout
the brown dwarf, and thus show that they are n = 3/2 polytropes.
3. (10 points) Use the relations for polytropes to show that the radius of a brown dwarf
is
#
"2 &
(3)
R = R0 %1+ " +
(,
1+ " '
$
where R0 = 2.7×109 (MSun/M)1/3µe–5/3 cm.
4. (10 points) Also, use the ratio of central to mean density for an n = 3/2 polytrope to
!
derive a relation
between ψ, the central temperature Tc, and the mass and radius:
(4)
" = 2 #10$9 Tc ( MSun / M )
4 /3
2
(R /R0 ) µ$8e / 3 .
5. (10 points) Use equations (2)–(4) to plot Tc versus R with M held fixed and R varied
from very large to very small values (as would be the case for a contracting protostar;
large and!small being relative to R0). Make plots for M = 0.04, 0.08, 0.16, and 0.32
MSun, assuming cosmic abundance of H and He. Show that Tc initially increases, but
then decreases. Show that Tc is maximized when R/R0 = 1.75 (ψ ≈ 0.55), and that
(5)
Tc,max = 8 "10 7 ( M / MSun )
4 /3
K.
6. (30 points) Use your knowledge of the temperature dependence of the p-p reaction
rates to argue that this sets a lower limit to the mass of hydrogen burning main
sequence!stars, and that protostars with masses below this critical mass will become
brown dwarfs. What is the critical mass?
Hint: a star is destined to become a brown dwarf when its fusion luminosity, at its
peak (when T = Tc,max), can not supply the energy released by gravitational
contraction to R(Tc,max) during the contraction phase. Assume that the contraction
phase lasts up to the current age of the universe. Note that this implies that the
definition of a brown dwarf changes as the universe ages. Finally, make an
appropriate assumption for the size of the nuclear burning region to get the fusion
luminosity.