Download Answer all questions in Section A and two and only two questions in

UNIVERSITY OF SOUTHAMPTON
PHYS3010W1
SEMESTER 2 EXAMINATION 201 0/11
Stellar Evolution
Duration: 120 MINS
Answer all questions in Section A and two and only two questions in
Section B.
Section A carries 1/3 of the total marks for the exam paper and you should aim
to spend about 40 mins on it. Section 8 carries 2/3 of the total marks for the
exam paper and you should aim to spend about 80 mins on it.
A Sheet of Physical Constants will be provided with this examination paper.
An outline marking scheme is shown in brackets to the right of each question.
Only university approved calculators may be used.
Number of
Copyright 2010
© University of Southampton
Pages 9
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Section A
A1. Most of a star's life is spent on the main sequence. Why are the star's luminosity
and surface temperature roughly constant during this stage of evolution? Describe briefly the two main nuclear fusion pathways occurring in main sequence
stars, and how these determine the main energy transport mechanism within
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stellar cores.
A2. The gas density at a radius r within a star of central density Pc and outer radius
R is described by:
Derive an equation for the enclosed mass as a function of radius, m(r), and then
find an expression for the total gravitational potential energy of this star, EGR·
A3. Over what range of mass can main sequence stars exist?
Explain what
determines the upper and lower mass limits for stable stars.
A4. What are the two main energy transport processes in stars?
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Explain the
importance of the dominant energy transport mechanism for the star's structure
and observed properties, and briefly discuss which stages of evolution (and
regions of the star) are dominated by each process and why.
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AS. Type II supernovae are produced when the core of a massive star collapses.
Explain briefly the processes that lead to this collapse and describe the eventual
result. Explain what provides the energy for the supernova explosion and how
that energy is released.
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Section B
81.
(a) For a classical gas, the probability that a particle will occupy a state with
energy
Ep
is given by
1
j( Ep) = -ex_p_(E_p___J-L_)/-(-ks-T-)
where f-L is the chemical potential, ks is the Boltzmann constant, and T is
the gas temperature. If the number of quantum states available between
momenta of p and p+dp is given by:
v
g(p)dp = h3 gAnpdp
write down an integral describing the total number of particles, N, within a
classical gas of volume V.
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(b) Hence show that if the particles are non-relativistic [i.e.
Ep
= mc 2 + p 2 /(2m)]
the particle concentration n is given by the following expression:
n = exp (
(Hint: The standard integral
f-L- mc
ksT
f0
00
2
)
g
h~ (2nmksT) 1
3 2
x 2 exp(-ax2 )dx = (1/4) ~may be
useful.)
(c) The quantum concentration is defined as
_ (2nmksT)
nQh2
312
Using the expression for particle concentration obtained in part (b), show
that a non-degenerate gas will obey Maxwell-Boltzmann statistics: e.g.
TURN OVER
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4
exp [(mc 2
-
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Jl)/(ksT)] >> 1.
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(d) Hence show that the chemical potential for a species A is given by
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(e) The triple-alpha process of Helium burning involves the equilibrium reaction
4
He +4 He
r= 8 Be
Using the results of part (b), show that the relative concentrations of
8-beryllium and 4-helium nuclei (n 8 and n4 , respectively) in a gas of
temperature T are described by
n~
n4
h2
= 23/2 ( 2
k T
1rm4 B
) 3/2
exp [-Q/(ksT)]
where m4 is the mass of a 4-helium nucleus, and Q is the mass-energy
difference for the equilibrium reaction above, which has a value of 91.8 keV.
Note that the number of spin-polarization states per momentum state is 1
for both species.
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(f) Hence show that for a pure helium gas of density p = 4 x 108 kg m- 3 at
a temperature of T = 2.5 x 108 K, there will be roughly one 8-beryllium
nucleus for every 2.5 million 4-helium nuclei.
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82.
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(a) Demonstrate that the minimum rotation period of a star is given by
Tmin
R3 )
= 27r ( GM
[2]
1/2
Hence describe the most important observational evidence we have for the
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existence of neutron stars.
(b) A pulsar is observed to have a period of 89 ms, which is increasing at a rate
of
P = 1o- 13 , and has a moment of inertia of around I
that the rate of energy loss is proportional to
ww
= 1038 kg m 2 . Show
(where
w is
the angular
frequency, and wits rate of change). In comparatively slow pulsars like this
one, magnetic braking is thought to be the dominant spindown mechanism.
Can magnetic braking of this pulsar be powering the surrounding pulsar wind
nebula, which is observed to have an X-ray luminosity of
Lx = 1029 W?
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(c) The energy loss rate due to magnetic dipole radiation is proportional to w 4
(where w is the angular frequency). Hence calculate an upper limit on the
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age of the pulsar described in part (b)
(d) What is the observational evidence for the existence of stellar mass black
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holes?
(e) Interacting binary stars radiate much larger amounts of energy than can be
explained by the black-body radiation of the component stars. Describe the
process that provides the energy for this radiation in interacting binaries.
Discuss the relative efficiency of this process for white dwarfs, neutron stars
and black holes.
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TURN OVER
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83.
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(a) When a collapsing protostar becomes radiative it enters the Henyey contraction phase, during which it is in hydrostatic equilibrium. Consider the physical properties of a protostar at a fixed fractional radius, r112 = 0.5R (where
R is the star's radius). Assuming the collapse is homologous, and that the
mass enclosed within r 1; 2 remains constant, show that as the protostar collapses, the pressure P(r1; 2 ) scales with the protostar's radius according to
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(b) For the same protostar, show that the temperature and temperature gradient
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scale with radius according to
and
dT ex: R-2
dr
(c) During the Henyey contraction, the energy transport is radiative, and so the
outward energy flux at radius r 112 obeys:
where a is the radiation constant, c is the speed of light, and K(r) is the
opacity at radius r. Given that the opacity scales with radius according
to K(r 1; 2 ) ex: R 0·5 , show that the protostar's luminosity, L and radius R are
related by L ex: R0·5 at all times during the collapse phase.
(d) Hence show that during this stage of evolution the collapsing protostar will
follow a path of slope -4/5 on the theoretical Hertzsprung-Russell diagram
as it moves towards the main sequence, bearing in mind that the effective
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temperature does not scale homologously.
(e) The protostar is in virial equilibrium through its collapse.
If its volume-
averaged pressure when it reaches the main sequence is ,. . ., 1.2
x 10 14
Pa,
and its radius at this point is 1.3R0 , what must its mass be?
[2]
If the Kelvin-Helmholtz contraction stage for this star takes around 6 million
[2]
years, what is its average luminosity during this phase?
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84.
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(a) The Clayton model for stellar structure approximates the pressure gradient
in a main sequence star as:
where G is the Gravitational constant, Pc is the central density, r is radius
within the star, and lis a scale length such that l << R, where R is the outer
radius of the star. Show that in this model the pressure at a radius r is given
by:
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[Hint: the pressure at the star's surface, P(R), is zero.]
(b) The total mass of the star in this model is given by:
Using the expression for P(r) derived in part (a), show that the star's central
pressure can be approximated by:
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(c) Write down expressions for the (ideal) gas pressure and the radiation
pressure at the centre of a star in terms of the central temperature.
(d) Using the expressions from part (c), show that the central pressure of a star
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in which both gas and radiation pressure contribute is given by:
P,=
[4]
C;/t (P~Br Gr
where f3 is the fraction of the total pressure contributed by the gas (P gas =
f3Pc), ks is the Boltzmann constant,
m=
1.0 x l0- 27 kg is the mean particle
mass, and a = 7.566 x 10- 16 J m- 3 K is the radiation constant.
(e) Using the results of parts (d) and (b), determine the main sequence mass
for which the gas and radiation pressure contribute equally in the core,
assuming the Clayton model is an accurate description.
END OF PAPER
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