Download Stellar Physics, VT 2010 Problem Set 2

Stellar Physics, VT 2010
Problem Set 2
Due date: Tue, 15 March 2011 at 13.15
1. Homology analysis of the stellar structure equations
Consider the set of equations of stellar structure (Prialnik eqs. 5.1 to 5.4) for a
main-sequence star with mass M > M and mean molecular weight µ, where
nuclear energy is released mainly by hydrogen burning via the CNO cycle and
the energy flux is dominated by radiative transport. Assume an ideal gas
equation of state – P (r) = µmk H ρ(r)T (r) – and that electron scattering is the
main contributor to the opacity (i.e. κ = κes = constant).
(a) Using the assumption of homology, derive how surface luminosity L =
F (R), stellar radius R and core temperature Tc scale with M and µ.
Assume that the star has core pressure Pc and its mean density is ρ̄.
The assumption of homology states that, as a star undergoes changes
in L, Tc , ρ̄ etc., its structure changes by the same factor at all radii.
That is, F (r) scales in the same way as L, T (r) as Tc , ρ(r) as ρ̄, r as R,
P (r) as Pc , m(r) as M , and similar for the increments dr, dm(r), dP (r),
dT (r), dF (r) (e.g. from the density equation you have dm/dr = 4πr2 ρ ⇒
M/R ∝ R2 ρ̄).
(b) Calculate the mean molecular weight µ of the following samples of gas:
(i) fully ionized hydrogen and helium in Big-Bang proportions, 92.7% H
and 7.3% He by number; (ii) fully ionized helium.
Calculate the ratio of µ in case (ii) to µ in case (i).
(c) Using the results of 1a and 1b, calculate how the radius, core temperature, luminosity and effective temperature of a star would change after a
primordial sample of hydrogen and helium has been burnt completely to
helium via the CNO cycle. Compare your results to evolutionary paths
in the H-R diagram resulting from numerical calculations, e.g. Fig. 9.9
in Prialnik (Fig. 8.8 in the previous edition). What do you conclude from
this comparison?
2. The upper end of the main sequence
(a) Consider a star in hydrostatic and radiative equilibrium. Write down
an expression for the condition when the force due to radiation pressure becomes larger than the gravitational force, leading to a violation of
hydrostatic equilibrium at the surface.
(b) From the condition above derive the upper limit LEdd (i.e. the Eddington limit) for the luminosity of a star with mass M in hydrostatic and
radiative equilibrium.
(c) Using the condition L < LEdd derive an upper limit for the mass and
luminosity of main-sequence stars. Estimate the effective temperature at
the upper end of the main sequence.
3. The energy budget of a core-collapse supernova
(a) What is the amount of gravitational energy released by the collapse of
an iron core of mass Mc = 1.5M from an initial radius of 1000 km to a
final radius of 10 km?
(b) During the collapse, about three-quarters of the iron-56 nuclei in the
core are photodisintegrated (into 13 helium nuclei and 4 neutrons each),
absorbing 124 MeV of energy for each nucleus of iron. Calculate the total
energy absorbed by this process.
(c) Following iron photodisintegration, about half of the helium-4 nuclei
also undergo photodisintegration into protons and neutrons, absorbing
28 MeV of energy for each nucleus of helium. Calculate the total energy
absorbed by this process.
(d) Another mechanism by which energy is absorbed in the collapsing core is
that of conversion of protons to neutrons by electron capture (also called
56
neutronization, e.g. e− + 56
26 Fe → 25 Mn + νe ). In this process, neutrinos
are produced, which carry away energy. Assume that each neutrino produced in such reactions carries away 10 MeV of energy, how much energy
is removed if the whole core (of 1.5M ) undergoes neutronization?
(e) Consider the combined results of 3b to 3d. This amount of energy is
removed from the star in a few seconds. How long would a 12.5M star
take to emit this much energy, radiating at its main-sequence luminosity?
(f) Estimate the energy radiated away during about one year for SN1969I,
from the light curve shown in Prialnik Fig. 10.4 (9.4 in the previous
edition).
(g) Estimate the energy carried away by the ejected envelope for a 15M
star, including the energy required for ejection and the kinetic energy
corresponding to the observed expansion velocities of 10000 km s−1 .
(h) What is the total amount of energy absorbed or carried away according
to the calculations above? How does it compare to the amount of gravitational energy released (3a)? Can you think of another mechanism which
could remove energy from the supernova core, given that just before the
final neutron star is formed, the core consists of a plasma of neutrons,
protons, electrons, neutrinos and photons, at temperatures as high as
1011 K and densities of up to 1014 kg m−3 ?
4. Convection: The physical quantities at some point inside a star are given
in the following table. Neglecting radiation pressure and assuming the mean
molecular weight is µ = 0.7, determine whether energy transport is convective
or radiative.
m(r)
F (r)
T (r)
ρ(r)
κ
0.028M 24.2L 2.2·107 K 3.1·104 kg m−3 0.04m2 kg−1
What type of star could contain such a point and where would this point be
located?