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I*M*P*R*S on ASTROPHYSICS at LMU Munich

Astrophysics Introductory Course
Lecture given by:

Ralf Bender and Roberto Saglia in collaboration with: Chris Botzler, Andre Crusius-WДtzel, Niv Drory, Georg Feulner, Armin Gabasch, Ulrich Hopp, Claudia Maraston, Michael Matthias, Jan Snigula, Daniel Thomas
Powerpoint version with the help of Hanna Kotarba

Fall 2007
IMPRS Astrophysics Introductory Course Fall 2007


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Chapter 1 Matter and Radiation

For a comprehensive treatment of this subject see: Rybicki and Lightman: Radiative Processes in Astrophysics, Wiley, New York 1979
IMPRS Astrophysics Introductory Course Fall 2007


1.1 Kinetic theory of free particles
To understand astrophysical plasmas, and especially stars, we need to know their equations of state, i.e. the relations between density , temperature T, pressure P and energy density u:
Pressure P = P ( ,T ) Equations of State Energy density u = u( ,T )

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For the classical ideal gas we have (k = Boltzmann's constant):
Pgas = nkT 3 3 Classical Ideal Gas = nkT = P 2 2

u

gas

The classical ideal gas law applies for most hydrogen burning stars (main sequence stars, see below). Generally, an equation of state for a gas can be derived with kinetic theory from the Momentum or energy distribution function of the particles. Consider a cube of volume L3 with N homogeneously distributed particles, i.e. we have the particle density n0 = N/L3. Provided the distribution function of the momenta n(p) is
IMPRS Astrophysics Introductory Course Fall 2007


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isotropic we can calculate:


no =

The pressure P on a wall is determined by the transferred momentum dp per time interval dt and per area L2:

-



n( p )d p = n( p )4 p 2 dp
3 0



P=


F 1 dp =2 L2 L dt

The momentum transferred to the wall perpendicular to the x-direction is:

dp =
which Term Term Term



2 px v x dtL2 n( p )dpx dp y dp
Term 3

z

0 - - Term 1 Term 2

is readily understood if we consider that: 1 = transferred momentum per particle 2 = all particles in this volume reach the wall during dt 3 = density of particles with momentum p

For an isotropic distribution of momenta, we can write in spherical polar coordinates:

px = p sin cos , v x = v( p ) sin cos , d 3 p = p 2 sin d d dp
IMPRS Astrophysics Introductory Course Fall 2007


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and obtain for the pressure:

i.e.

1 dp = P= 2 L dt

2 / 2



2 p v p 2 n( p ) sin 3 cos 2 d d dp

0 0 - /2

1 P = p v( p )n( p )4 p 2 dp 30



The pressure is determined by the momentum distribution function of the particles. The energy density of the particles is:


u

kin

= ( p )n( p )4 p 2 dp
0

where (p) = kinetic energy of particle with momentum p. In the non-relativistic case we have of course = p2/2m. For relativistic particles we need:
2 m0 c m0 c , ( p) = p c 1 + - p p

pv =

pc m c 1+ 0 p
2

IMPRS Astrophysics Introductory Course

Fall 2007


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1.2 Equilibrium distribution functions of fermions and bosons
All particles known obey either Bose-Einstein or Fermi-Dirac statistics. At low temperatures, the nature of the particles is important for determining their thermodynamic behaviour. At high temperatures, all ideal gases of free particles behave in the same way, i.e. like the classical ideal gas. Using the grand canonical partition function one can show that fermions and bosons have the following energy distribution functions in thermodynamical equilibrium:

dN =
= = = = +1 in denominator = -1 in denominator = = E dN dg

dg e
- + E / kT

±1

energy of the particle number of particles p in energy range (E,E + dE) number of quantum states in energy range (E,E + dE) d3xd3p/h3 (multiplicity due to particle spin) Fermions, Pauli-principle, only one particle per phase space cell h3 Bosons, no Pauli-principle "Degeneracy parameter" = (chemical potential )/(kT)
Fall 2007

IMPRS Astrophysics Introductory Course


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For all massive particles which are neither created nor destroyed, it is determined from the requirement of particle number conservation:

N=



dg e
- + E / kT

±1

= const.

This does not apply for photons in a black body environment which have = 0. Depending on we can identify: »1 -5 < < 5 « -1 highly degenerate systems medium to weakly degenerate systems non-degenerate systems

(see below for explanation).

1.2.1 Momentum distributions of non-degenerate free particles
If « -1, we have exp(- + E/kT) » 1 and consequently:

dN e e
IMPRS Astrophysics Introductory Course

- E / kT

Fall 2007


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As the kinetic energy for free particles is:
2 2 1 2 p 2 p12 + p2 + p3 E = mv = = 2 2m 2m

and the number of quantum states within (E,E + dE) is:

d 3 xd 3 p dg = h3
we have:

d 3 xd 3 p dN = h3 N=

Integrating yields:


h
3

eV ( 2 mkT

)

3/ 2

which we can use to eliminate e and to obtain the well-known Maxwell distribution:

1 N dN = V ( 2 mkT

)

3/ 2

p2 3 exp - d pdV 2mkT

This is the energy distribution function of non-degenerate free particles in thermodynamic equilibrium.
IMPRS Astrophysics Introductory Course Fall 2007


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1.2.2 Criteria for the degeneracy of a free particle gas
We first define the following two parameters to simplify the discussion: Typical de Broglie wavelength deBroglie of a particle in a thermodynamical plasma:

deBroglie =
Mean particle separation ro:

h 2 mkT

n = N / V = 1 / ro3
In the previous subsection we have shown that for large negative :
3


which can be rewritten as:

-1 :

e=



n

h

( 2 mkT


)

3/ 2

3 1 deBro = ln r3 o

glie

Therefore, « -1 implies a large separation of the particles, in which case their quantum nature is not anymore relevant and we can treat them as classical particles.
IMPRS Astrophysics Introductory Course Fall 2007


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Degenerate systems in astrophysics are mostly fermionic. Then the Pauli principle applies and the maximum phase space density is /h3, i.e. one particle per phase space cell. For fermions we have for the energy ( = p2/2m) distribution function:

f ( ) = exp
where we have used = /(kT).

- + + 1 = ex kT

-1

- p +1 kT

-1

Complete degeneracy is approached if the system is cooled to temperatures which are much smaller than the chemical potential , i.e.