Star-forming region in Carina, NGC 3582, from Astronomy Picture of the Day: http://apod.nasa.gov/apod/ap130611.html
Star-forming region in Cassiopeia, Heart and Soul nebula, IC1805 & IC1848, from Astronomy Picture of the Day: http://apod.nasa.gov/apod/ap100601.html
Star-forming region in Cassiopeia, IC1795, from Astronomy Picture of the Day: http://apod.nasa.gov/apod/ap091210.html
Star-forming region in Carina NGC3372, from Astronomy Picture of the Day: http://apod.nasa.gov/apod/ap100226.html
Stellar Configurations
• Self gravitating
• Self-consistent solution needed
• Different processes resist collapse
8.044 L20B1
Planets
• Gravity weak because of small M
• Atomic forces provide balancing pressure
8.044 L20B2
Jupiter with moons Ganymede (upper) and Io (lower), from Astronomy Picture of the Day: http://apod.nasa.gov/apod/ap130215.html
Normal Stars
• Gravitational energy starts process
• Fusion then supplies energy
• Plasma of electrons and nuclei
• Kinetic pressure, P = nkT
• Radiation pressure, P = 1
3 u(T ), helps and dominates above about 10M
8.044 L20B3
FOUR POSSIBLE END STATES OF STARS
DISPERSED GAS
NORMAL
STAR
DWARF STAR
INCREASING
MASS
NEUTRON STAR
BLACK HOLE
8.044 L20B4
White Dwarf
• Fusion has stopped
• Collapses to a small size, nuclear spacing ∼ 1/100
that of a solid
• Electron degeneracy pressure supports it, P ∝ m1e n5/3
• White → gray → brown (dead, cold)
8.044 L20B5
Assume uniform density of α++ and e−
EK =
(α)
(e)
EK +EK
=
3N 5 e F
=
3 N h̄2
5 e 2me
3π 2(N
e/V )
2/3
small
3
V =4
πR
3
EK =
3 9π
5 2
M ≈ Nαmα = (Ne/2)mα ⇒ Ne = 2M/mα
2/3
h̄2
M
me mα
2/3
1
R2
EP =
2
3
M
−5 G R
8.044 L20B6
ET = EK + EP
ET
6
h̄
3 = 2(9π)2
M R0
5
G3m3
e mα
EK
R0
R
EP
= 0.74 × 1051 kg-m3
R0 ∝ 1/M 1/3
Stable for any M .
8.044 L20B7
X-ray image of Sirius B (brighter) and Sirius A (less bright) , from Astronomy Picture of the Day: http://apod.nasa.gov/apod/ap001006.html
Sirius B: M = 2.1 × 1030 kg
R
observed
our model
better model
5.6 × 106 m
7.1 × 106 m
8.6 × 106 m
(good)
(⇒ a problem)
8.044 L20B8
Our model of Sirius B implies
ne = 8.6 × 1029 cm−3
F = 4.7 × 10−7 ergs → 3.4 × 109K
(Tsurface ∼ 2 × 107 K)
But mec2 = 8.2 × 10−7 ergs ⇒ relativity needed
8.044 L20B9
V
Dwavevectors(k) =
(2π)3
2V
Dstates(k) =
(2π)3
2V
4 3
2N/V )1/3
πkF
⇒
k
=
(3π
F
3
(2π)3
!
N =
= c~|~k| ⇒ F = c~(3π 2N/V )1/3
8.044 L20B10



#() =

4


  2V 
3
 π k () 
 3 | {z }  (2π)3
(/c~)3
V 1
d#
= 2
D() =
d
π c~
!3
1
=
V
2
3π
!
1 3 3
c~
2
8.044 L20B11
D() = a2
N =
E =
Z F
0
Z F
0
D() d =
Z F
D() d =
0
3
a
a2 d = 1
3 F
Z F
0
4
a
a3 d = 1
4 F
= 3
4 N F
8.044 L20B12
∂E
P = −
∂V N,S
|
{z
}
!
∂F
3
=− N
4
∂V N
!
dS=0 at T =0
1
= (N/V )F ∝ (N/V )4/3
4
This pressure rises less steeply with density, (N/V )4/3,
than is the case for the non-relativistic gas, (N/V )5/3.
8.044 L20B13
For a white dwarf composed of α particles and electrons,
4
V = πR3
3
1
M ≈ Nαmα = Nemα ⇒ Ne = 2(M/mα)
2
3
3
EK = NeF = Nec~(3π 2Ne/V )1/3
4
4
!
3
M
9π M 1 1/3
= c~(
)
2
mα
2 mα R 3
3 9π
=
2 2
!1/3
M
c~
mα
!4/3
1
R
8.044 L20B14
The R dependence of the two contributions to the
total energy is straight forward: EK = a/R and
EP = −b/R where a and b are known expressions.
Then ETOTAL = (a − b)/R which is never stable.
The condition a = b is a special case, a dividing
line between collapse and infinite expansion.
8.044 L20B15
M
c~
mα
!4/3
∼ GM 2
c~
2/3
∼
M
4/3
Gmα

M ∼

3/2
c~ 
Gm2
α
mα
8.044 L20B16
The Chandrasekhar limit for the maximum possible
mass of a white dwarf is
MCh
Z
= 0.20
A
!



3/2
ch 

2
Gmp
mp
where Z/A is the average ratio of atomic number
to atomic weight of the stellar constituents. Note
that it has the same form as our expression. For
Z/A = 0.5 (α particles) this gives MCh = 1.4MSun.
8.044 L20B17
Neutron Star
• p+ + e− → n to lower coulomb energy
• Degeneracy pressure of neutrons
3 ∝ h̄6/G3m8 ⇒ R ∼ 15 km if M = 1.4M
M R0
0
n
• Nuclear forces also contribute to P
• Rotating neutron stars seen as pulsars
• Also subject to stability limit, M ∼ 2M
8.044 L20B18
http://chandra.harvard.edu/photo/2006/crab/
STELLAR CONFIGURATIONS
DISPERSED GAS
GRAVITY
NORMAL
STAR
DWARF
STAR
NEUTRON
STAR
GAS PRESSURE
(+ RADIATION )
ELECTRON
DEGENERACY
NEUTRON
DEGENERACY
+ NUCLEAR
FORCES
1 g/cm3
106 g/cm3
1014 g/cm3
BLACK
HOLE
8.044 L20B19
Artist’s conception of accretion disk around a black hole , from Astronomy Picture of the Day: http://apod.nasa.gov/apod/ap130312.html
http://commons.wikimedia.org/wiki/File:Cygnus_X-1.png
MIT OpenCourseWare
http://ocw.mit.edu
8.044 Statistical Physics I
Spring 2013
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