Download White Dwarfs and Neutron Stars

White Dwarfs and Neutron Stars
• White dwarfs
– Degenerate gases
– Mass versus radius relation
• Neutron stars
– Mass versus radius relation
– Pulsars, magnetars, X-ray pulsars, X-ray
bursters
White dwarf
• Core of solar mass star
• Degenerate gas of
oxygen and carbon
• No energy from fusion
or gravitational
contraction
Mass versus radius relation
• For objects made of normal matter, radius
tends to increase with mass
Mass/radius relation for degenerate star
• Star mass = M, radius = R
3GM 2
• Gravitational potential energy = 
5R
• Heisenberg uncertainty: xp  h
• Electron density n  3N  M
4R3 m p R3
x  n
1 3
h
p 
 hn1 3
x
• Kinetic energy
p2

2me
M
h2 M 5 3
K  N 

mp
me m5p 3 R 2
Mass/radius relation for degenerate star
• Total energy
h2 M 5 3
GM 2
E  K U 

53 2
me m p R
R
• Find R by minimizing E
dE
h2 M 5 3
GM 2

 2 0
53 3
dR
me m p R
R
h2 M 1 3
R
Gme m 5p 3
• Radius decreases as mass increases
Mass versus radius relation
Maximum white dwarf mass
• As mass increases, electron speed  c,
kinetic energy equation  E=pc
• Electron degeneracy cannot support a
white dwarf heavier than 1.4 solar
masses, the “Chandrasekhar limit”.
12
 h3c 3 
M  # 3 4 
G m 
p 

What happens to a star more
massive than 1.4 solar masses?
1.
2.
3.
4.
There aren’t any
They shrink to zero size
They explode
They become something else
Neutron Stars
• Degenerate stars heavier than 1.4 solar
masses collapse to become neutron stars
• Formed in supernova explosions
• Electrons are not separate
– Combine with nuclei to form neutrons
• Neutron stars are degenerate gas of neutrons
Neutron energy levels
• Only two neutrons (one up,
one down) can go into each
energy level.
• In a degenerate gas, all low
energy levels are filled.
• Neutrons have energy, and
therefore are in motion and exert
pressure even if temperature is
zero.
• Neutron star are supported by
neutron degeneracy.
20
Mass v Radius
15
10
5
Neutron Stars
• Very compact – about 10 km radius
• Very dense – one teaspoon of neutron star
material weighs as much as all the buildings in
Manhattan
• Spin rapidly – as fast as 600 times per second
• High magnetic fields – compressed from
magnetic field of progenitor star
Spin up of neutron star
Collapse of star
increases both
spin and
magnetic field
Angular momentum
For single particle, angular
momentum L = mvr
Write in terms of rotation
rate , v = 2r ,
L = 2mr2  = I ·2
I = “moment of inertia”
In general, I = #MR2
M = mass of object
R = length of object
Spin up of neutron star
Angular momentum
of sphere:
4
2
L  I  2  MR 
5
Where M is mass, R is radius,  is spin rate
If the Sun (spin rate 1/25 days, radius 7108 m) were to
collapse to a neutron star with a radius of 12 km, how fast
would it be spinning?
4
4
2
2
Li  MRi  i  L f  MR f  f
5
5
Spin up of neutron star
4
4
2
2
Li  MRi  i  L f  MR f  f
5
5
 Ri
v f  i 
R
 f
2

 7 10 m 

7

1
5 1
  4.6 10 s 


1
.
6

10
s
 1.2 103 m 




8
2
Very high rotation rates can be reached simply via
conservation of angular momentum.
This is faster than any known (or possible) neutron star.
Mass and angular momentum are lost during the collapse.
Pulsars
Discovered by Jocelyn
Bell in 1967.
Her advisor, Anthony
Hewish, won the Nobel
Prize in Physics for the
discovery in 1974.
Pulsars
Energy source
is spin down of
neutron star.
Must lie along
pulsar beam to
see pulsed
signals.
Crab Pulsar
Spin down of a pulsar
1
2
Energy E  I 2
2
dE
d
2
Power P  
 4 I
dt
dt
For Crab pulsar:  = 30/s, M = 1.4 solar masses, R = 12 km,
and d /dt = – 3.910-10 s-2.
Therefore, P = 5 1031 W = 130,000 solar luminosities.
Over a year, the spin rate changes by only 0.04%.
Magnetars
Magnetic fields so
strong that they
produce
starquakes on the
neutron star
surface.
These quakes
produce huge
flashes of X-rays
and Gamma-rays.
Energy source is
magnetic field.
X-Ray Pulsars
Neutron star in binary system with a normal star
X-Ray Pulsars
High magnetic field neutron stars make regular
pulsations. Energy source is gravitational energy of
infalling matter.
X-ray Bursters
X-ray Burst
Low magnetic field neutron stars make X-ray bursts.
Source of energy is nuclear burning.
Review Questions
• What is the exclusion principle?
• Does a more massive white dwarf have a
larger or smaller radius than a less massive
one?
• What is the maximum mass of a white dwarf?
• What are some of the properties of neutron
stars?
• Why do many neutron stars spin rapidly?
• In what different forms does one find neutron
stars?