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 L 20 B1
Planets • Gravity weak because of small M • Atomic forces provide balancing pressure 8.044 L 20 B2
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 domi- nates above about 10 M ⊙ 8.044 L 20 B3
FOUR POSSIBLE END STATES OF STARS DISPERSED GAS DWARF STAR INCREASING NORMAL MASS STAR NEUTRON STAR BLACK HOLE 8.044 L 20 B4
White Dwarf • Fusion has stopped • Collapses to a small size, nuclear spacing ∼ 1 / 100 that of a solid • Electron degeneracy pressure supports it, P ∝ 1 m e n 5 / 3 • White → gray → brown (dead, cold) 8.044 L 20 B5
Assume uniform density of α ++ and e − h 2 � 2 / 3 E K = E ( α ) + E ( e ) � = 3 5 N e ǫ F = 3 5 N e ¯ 3 π 2 ( N e /V ) K K 2 m e � �� � small V = 4 3 πR 3 M ≈ N α m α = ( N e / 2) m α ⇒ N e = 2 M/m α � 2 / 3 1 � � 2 / 3 ¯ h 2 5 G M 2 � 9 π E K = 3 E P = − 3 M 5 2 R 2 m e m α R 8.044 L 20 B6
E T = E K + E P E T h 6 ¯ E K MR 3 0 = 2(9 π ) 2 G 3 m 3 e m 5 α R 0 R = 0 . 74 × 10 51 kg-m 3 E P R 0 ∝ 1 /M 1 / 3 Stable for any M . 8.044 L 20 B7
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 × 10 30 kg R 5 . 6 × 10 6 m observed 7 . 1 × 10 6 m our model (good) 8 . 6 × 10 6 m better model ( ⇒ a problem) 8.044 L 20 B8
Our model of Sirius B implies n e = 8 . 6 × 10 29 cm − 3 ǫ F = 4 . 7 × 10 − 7 ergs → 3 . 4 × 10 9 K ( T surface ∼ 2 × 10 7 K) But m e c 2 = 8 . 2 × 10 − 7 ergs ⇒ relativity needed 8.044 L 20 B9
V 2 V D wavevectors ( k ) = D states ( k ) = (2 π ) 3 (2 π ) 3 � 4 2 V � 3 πk 3 (2 π ) 3 ⇒ k F = (3 π 2 N/V ) 1 / 3 N = F ǫ F = c � (3 π 2 N/V ) 1 / 3 ǫ = c � | � k | ⇒ 8.044 L20B10
� 1 � 3 4 2 V 1 3 π k 3 ( ǫ ) ǫ 3 = #( ǫ ) = 3 π 2 V (2 π ) 3 c � � �� � ( ǫ/c � ) 3 � 1 � 3 D ( ǫ ) = d # dǫ = V ǫ 2 π 2 c � 8.044 L20B11
D ( ǫ ) = aǫ 2 � ǫ F � ǫ F aǫ 2 dǫ = 1 3 aǫ 3 N = D ( ǫ ) dǫ = F 0 0 � ǫ F � ǫ F aǫ 3 dǫ = 1 4 aǫ 4 E = ǫD ( ǫ ) dǫ = F 0 0 3 = 4 Nǫ F 8.044 L20B12
� ∂E � � ∂ǫ F � = − 3 P = − 4 N ∂V ∂V N,S N � �� � dS =0 at T =0 = 1 4( N/V ) ǫ F ∝ ( N/V ) 4 / 3 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 elec- trons, = 4 3 πR 3 V ≈ N α m α = 1 M 2 N e m α ⇒ N e = 2( M/m α ) E K = 3 4 N e ǫ F = 3 4 N e c � (3 π 2 N e /V ) 1 / 3 � 1 / 3 = 3 2 c � ( M � 9 π M 1 ) R 3 m α 2 m α � M � 4 / 3 1 � 1 / 3 = 3 � 9 π c � 2 2 m α R 8.044 L20B14
The R dependence of the two contributions to the total energy is straight forward: E K = a/R and E P = − b/R where a and b are known expressions. Then E TOTAL = ( 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 � 4 / 3 ∼ GM 2 c � m α c � ∼ M 2 / 3 Gm 4 / 3 α 3 / 2 c � M ∼ m α Gm 2 α 8.044 L20B16
The Chandrasekhar limit for the maximum possible mass of a white dwarf is 3 / 2 � � Z ch M Ch = 0 . 20 m p Gm 2 A p 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 M Ch = 1 . 4 M Sun . 8.044 L20B17
Neutron Star • p + + e − → n to lower coulomb energy • Degeneracy pressure of neutrons MR 3 h 6 /G 3 m 8 0 ∝ ¯ R 0 ∼ 15 km if M = 1 . 4 M ⊙ ⇒ n • Nuclear forces also contribute to P • Rotating neutron stars seen as pulsars • Also subject to stability limit, M ∼ 2 M ⊙ 8.044 L 20 B1 8
http://chandra.harvard.edu/photo/2006/crab/
STELLAR CONFIGURATIONS DISPERSED GAS GRAVITY NORMAL DWARF NEUTRON BLACK STAR STAR STAR HOLE GAS PRESSURE ELECTRON NEUTRON (+ RADIATION ) DEGENERACY DEGENERACY + NUCLEAR FORCES 10 6 g/cm 3 10 14 g/cm 3 1 g/cm 3 8.044 L 20 B1 9
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 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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