Download Exercise sheet 3 Theoretical Astrophysics II

SS 2014
University of Freiburg
Institute of Physics
PD Dr. Markus Roth
Kolja Glogowksi
Exercise sheet 3
Theoretical Astrophysics II
Stellar Structure and Evolution
The equations of stellar structure are given by
Gm
dp
,
(1)
= −
dm
4πr4
dr
1
=
,
(2)
dm
4πr2 ρ
F
dT
3 κ
,
(3)
= −
3
dm
16σ T (4πr2 )2
dF
= q0 ρT n ,
(4)
dm
with the boundary conditions: p(M ) = 0 and T (R) = 0, where M = m(R) denotes the total mass, R
the stellar radius, and L = F (R) defines the luminosity. For the Sun one finds: M = 1.989 · 1030 kg,
R = 6.957 · 108 m, L = 3.846 · 1026 W. In the following we use the subscript c to denote values at the
stellar center of the respective physical quantity.
Exercise 1: Star with a constant density profile
A very simple stellar model is obtained by the assumption of a constant density profile throughout the
stellar interior, i.e. ρ(r) = ρc = const.
(a) Find an expression for the mass m as a function of radius r and the parmeters R and M .
(b) Eliminate r in the equation of the hydrostatic equilibrium, use the result from (a). Integrate the
result and find an expression for the central pressure pc of the star.
(c) Show that the so found functions for the mass and pressure m(r), p(r), satisfy the virial theorem
Z M
p
−3
dm = Ω ,
(5)
ρ
0
where Ω = − 35 GM 2 /R.
Exercise 2: Star with a linear density profile
Consider a stellar model that is characterized by a linear increase of the density with radius,
r
.
(6)
ρ(r) = ρc 1 −
R
(a) Derive an expression for the density at the stellar center ρc in dependence on the parameters M and
R.
(b) Use the equation of the hydrostatic equilibrium to derive an expression for the pressure as a function
of radius r. Hint: The solution is of the form p(r) = pc × (polynomial of r/R). Derive an expression
for the central pressure pc in dependence on M and R in solar units.
(c) Now assume that the star satisfies the ideal gas law. What is the central temperature Tc ? Hint: The
equation of state of an ideal gas (ideal gas law) is given by
p=
R
ρT,
µ
(7)
where R is the ideal gas constant and µ the average molecular mass (= mass number/number of particles).
(please turn over)
Exercise 3: Main sequence stars
Main sequence stars are characterized by a long lasting phase of hydrogen burning during their evolution,
i.e. their energy is generated by the fusion of hydrogen nuclei to helium nuclei. The equations of stellar
structure (1)-(4) typically presents a complex system of nonlinear differential equations which cannot
be solved analytically. However, important insights about the properties of main sequence stars can be
obtained by the following homology considerations. For simplicity assume that the opacity κ is constant
and the stellar gas satisfies the ideal gas law (7).
(a) We define the dimensionless quantity x = m/M and separate the physical variables into a dimensionless and a dimensional part, i.e. we set r = f1 (x)R? , p = f2 (x)p? , ρ = f3 (x)ρ? , T = f4 (x)T? , and
F = f5 (x)F? .
Show that with the so defined substitutions Eq. (1) is equivalent to the following equations:
x
df2
,
=−
dx
4πf14
p? =
GM 2
.
R?4
(8)
Further derive the corresponding relations for Eq. (2)-(4) and (7).
(b) Show by means of the results in (a) that the following relations are true:
L ∝ µ4 M 3 ,
R
2(n−1)
1− 3(n+3)
L
∝ M
∝
n−1
n+3
,
4
Teff
,
(9)
(10)
(11)
where the effective temperature Teff of a star is defined by
F (R) =
L
4
= σTeff
4πR2
(12)
(black-body radiation). Calculate the slope in the Hertzsprung-Russell-Diagram (i.e. log L–log Teff Diagram) for n = 4 (p-p chain; lower main sequence) and n = 16 (CNO cycle; upper main sequence).
(c) Consider the nuclear timescale
τnuc ≈ αβM c2 /L,
(13)
where β denotes the fraction of M which is available for nuclear reactions, and α is the fraction of
βM which is actually converted in energy. One finds that noticeable changes of the star will occur,
i.e. the star will be no longer a main sequence star, if about 10% of its hydrogen is used. How long
will the Sun remain a main sequence star (for the Sun assume a mass fraction of hydrogen of 70%
and set α ≈ 0.7% for the fusion of hydrogen)? Make use of the mass-luminosity relation (9) in order
to explain why O- and B-stars are rather seldom while on the contrary K- and M-stars are frequent.
Transform Eq. (13) in solar units and show that stars of the milky way with M = 0.5M yet cannot
have left the main sequence (for the age of the milky way use 13, 6 · 109 years).
(d) Lower end of the main sequence: Stars of the lower end of the main sequence burn their hydrogen by
p-p chain reactions (n = 4). Show by means of the results in (a) and (b) that the core temperature
of such stars is given by:
Tc ∝ M 4/7 .
(14)
Assume that the lowest temperature for which hydrogen burning is possible is given by Tmin ≈ 4·106 K
(p-p chain) and derive a lower boundary for the total mass and the luminosity of a main sequence
star in solar units (Tc, ≈ 1.5 · 107 K).
Upper end of the main sequence: Hydrogen burning in stars of the upper main sequence predominantly
results from the CNO cycle (n = 16). The radiation balance at the stellar surface is only achieved if
L < LEdd , where
4πcGM
LEdd =
(15)
κ
defines the so called Eddington luminosity. Derive an upper boundary for the mass, the luminosity,
and the effective temperature in solar units (κ ≈ 0.1 m2 kg−1 ).