Download The Maxwell-Boltzmann distribution – some useful background maths

A2 Theoretical Astrophysics
The Maxwell-Boltzmann distribution
– some useful background maths
If a classical∗ gas of freely-moving particles, each
I0 is a little more tricky. To evaluate it, first
of mass m, is in thermal equilibrium at a temper- consider the square of the integral, written as
ature T , the speeds of the particles will settle into
Z ∞
Z ∞
2
2
−ax
2
a Maxwell-Boltzmann distribution. If f (v) δv is
I0 =
e
dx ×
e−ay dy
0
the fraction of particles with speeds in the range v
Z0Z ∞
2
2
to v + δv then this distribution function is
=
e−a(x +y ) dx dy.
r 0
2 m 3/2 2
2
f (v) =
v exp −mv /2kT .
This is the volume under a two-dimensional gausπ kT
sian curve, over the positive quarter of the (x, y)
To derive this equation, and to work out the mean plane. In polar coordinates we can write this same
speed, hvi, and the mean squared speed, hv 2 i, of volume in terms of r , where r 2 = x 2 + y 2 , as
Z
the particles we need to evaluate integrals of the
1 ∞
2
2
form
I0 =
2πr e−ar dr
Z ∞
2
4 0
In =
x n e−ax dx.
π h −ar 2 i∞
0
=−
e
0
4a
We can tackle this first by integrating by parts:
π
=
.
Z ∞
4a
1
2
x n−1 (−2ax)e−ax dx
In = −
2a 0
So finally, we can present the full reduction
1 h n−1 −ax 2 i∞
formula:
=−
x e
0
2a
Z ∞
1
2
+
(n − 1)x n−2 e−ax dx
If
2a 0
Z ∞
n−1
2
In−2 .
=
In =
x n e−ax dx
2a
0
This is known as a reduction formula, and using
it we can (eventually) evaluate any integral In we
need if we know I0 and I1 .
I1 can be integrated directly:
Z ∞
2
I1 =
xe−ax dx
0
=−
=
1 h −ax 2 i∞
e
0
2a
then
1
I0 =
2
r
π
a
1
2a
n−1
In =
In−2
2a
I1 =
1
.
2a
∗ The gas is classical if quantum effects can be neglected.
Here that means that the box containing the gas is much
bigger than any particle’s de Broglie wavelength.
G.W.
2002