Download Comparing Earth, Sun and Jupiter

1. Sirius B has a temperature of 27000K, and a
luminosity of only 3% solar. How big is it?
From the Stefan Boltzmann law:
L  4R 2T 4
1/ 2
 L 
R

 
RSun  LSun 
 0.03
1/ 2
 T 


T
 Sun 
2
4.682  0.0079
The average density is

3M
3M Sun

 2.9  109 kg / m3 .
3
3
4R
4 0.0079RSun 
=16 (for oxygen), the number density is therefore:
n

mH

2.9  109 kg / m3
 1.0  1035 m 3 .
16mH
Using
2. Calculate the conditions for degeneracy pressure to
dominate:
The Fermi energy is
2  2  Z   
F 
3  

2me 
 A  mH 
2/3
The average thermal energy of an electron is 3/2 kT. Thus a gas
is degenerate when
3
kT   F
2
T
 2/3
For Z/A~0.5,
T

2/3
 2  3 2


3kme  mH
 Z 
 
 A 
2/3
 1.3 103 Km2 kg 2 / 3
E.g. in the centre of the Sun, T=1.6x107K and =1.62x105 kg/m3.
T
 2/3
 5384
Whereas in a white dwarf, T=7.6x107K and =3x109 kg/m3
so
T
 2/3
 36.5
3. Evaluate the central pressure of a degenerate star with
constant density.
dP
GM r 
G  4r 3  
4G 2 r







dr
r2
r 2  3 
3
5/3
If the pressure is given by degeneracy pressure: P  
dP
2 / 3 d


so
dr
dr
Setting the two pressure gradients equal to one another gives:
d
 2/3
  2r
dr
4 / 3
or  d  rdr so
R 
2
1 / 3
M 
 3 
R 
1 / 3
M  R 3
MV  constant
4/3
If you use the relativistic expression, P  
you get
M 
R2   2 / 3   3 
R 
2
2
2 / 3
R R M
M  constant
1 / 3
4. Calculate the cooling time of a white dwarf
The thermal energy is primarily in the nuclei, since the electrons
are degenerate.
3 
U  N nuclei  kT 
2 
 M
 
 AmH
 3 
 kT 
 2 
Equating the change in thermal energy to the luminosity:

dU
L
dt
d  M 3 
M 
T

 
kT   0.03LSun

7 
dt  AmH 2 
M Sun  2.8 10 
dT
  KT 7 / 2
dt
7/2
LSun 
2 AmH k
1

where K  3 0.03 M Sun  2.8 107 
7/2
 4.0 10 36 A
T  2.5 
5K
t  C 
2
 5

T  T0 1  KT02.5t 
 2

2.5
L 
t 
 A  T0 
 1  1.2  7 

9
L0 
 12  10 K  10 yr 
2 / 5
2.5

t 
 A  T 
 T0 1  1.2  70 

9
 12  10 K  10 yr 

2 / 5
7 / 5
5.
a. If pulsars were binary stars, what would be the
size of the semimajor axis?
a3
P 
M
2
so for a solar-mass system with P=2s, we would need
 2s 

a  1AU 
 1 year 
2/3
 0.003RSun .
Since the stars would have to be
smaller than this, the density would be >5.0x1010 kg/m3, and
it would have to be a neutron star.
b. How fast can a star rotate before it breaks up?
Equate centripetal and gravitational accelerations:
2
 max
RG
M
R2
Pmin  2
R3
GM
For white dwarfs, Pmin ~ 7s, while for neutron stars, Pmin ~
0.0005 s.
6. How much energy is output by the Crab pulsar, with
dP/dt=4.21x10-13?
The kinetic energy of a rotating sphere is K  1 I 2  2 2 I2
2
with
I
2 MR
5
2
So the rate of energy loss is:
M  1.4M Sun
For the Crab pulsar:
R  10 4 m
P  0.0333s
P  4.2110 13
dK
4 2 IP
8 2 MR 2 P



dt
P3
5P 3
, so
dK
 5.0 1031W
dt
P