Download –1– Order of Magnitude Astrophysics

–1–
Order of Magnitude Astrophysics - a.k.a. Astronomy 111
Photon Opacities in Matter
If the cross section for the relevant process that scatters or absorbs radiation given by σ and
the number density of scatters is n, then the mean free path of a photon is given by l = (nσ)−1 . In
the case of radiation, it is conventional to define a quantity κ (called opacity) such that
α ≡ nσ ≡ ρκ
(1)
where ρ is the mass density of the scatterers. The optical depth of a system of size R is defined to
be
τ ≡ αR =
R
.
l
(2)
The opacity for the photons is provided mainly by three different processes: (1) scattering by free
electrons; (2) the free-free absorption of photons, and (3) the bound-free transitions induced by
matter by the photons that are passing through it.
Electron Scattering
The simplest case is the one in which the charged particle is accelerated by an electromagnetic
wave that is incident upon it. Consider a charge q placed on an electromagnetic wave of amplitude
E. The wave will induce an acceleration a ≈ (qE/m), causing the charge to radiate. The power
radiated will be
P=
2q2 a2
2q4 2
=
E .
3c3
3m2 c3
(3)
–2–
Because the incident power in the electromagnetic wave is S = cE 2 /(4π), the scattering cross
section (for electrons with m = me ) is
P 8π q2
σT ≡ =
S
3 me c2
!2
≈ 6.7 × 10−25 cm−2 ,
(4)
which is the Thomson scattering cross section. This cross section governs the basic scattering
phenomena between charged particles and radiation. The corresponding mean free path for
photons through a plasma is lT = (ne σT )−1 , and the Thomson scattering opacity, defined to be
κ ≡ (ne σT /ρ), is
ne
κT =
np
!
!
σT
= 0.4 cm2 g−1
mp
(5)
for ionized hydrogen with ne = n p .
Random Walks
A particular useful way of looking at scattering, which leads to important order-of-magnitude
estimates, is by means of random walks. It is possible to view the processes of absorption,
emission, and propagation in probabilistic terms for a single photon rather than the average
behavior of large number of photons. For example, the exponential decay of a beam of photons
has the interpretation that the probability of a photon traveling an optical depth τ before absorption
is e−τ . Similarly, when radiation is scattered isotropically we can say that a single photon has
equal probabilities of scattering into equal solid angles. In this way we can speak of a typical or
sample path of a photon, and the measured intensities can be interpreted as statistical averages
over photons moving in such paths.
Now consider a photon emitted in an infinite, homogenous scattering region. It travels a
displacement r1 before being scattered, then travels in a new direction over a displacement r2
before being scattered, and so on. The net displacement of the photon after N free paths is
R = r1 + r2 + r3 + · · · + rN
(6)
–3–
We would like to find a rough estimate of the distance | R | traveled by a typical photon. Simple
averaging of equation [6] over all sample paths will not work, because the average displacement,
being a vector, must be zero. Therefore, we first square equation [6] and then average. This yields
the mean square displacement traveled by the photon:
l∗2 ≡ hR2 i = hr21 i + hr22 i + · · · + hr2N i + +2hr1 · r2 i + 2hr1 · r3 i + · · ·+ · · ·.
(7)
Each term involvement the square of a displacement averages to the mean square of the free path
of a photon, which is denoted l2 . To within a factor of order unity, l is simply the mean free path
of a photon. The cross terms in equation [7] involve averaging the cosine of the angle between the
directions before and after scattering, and this vanishes for isotropic scattering. (It also vanishes
for any scattering with front-back symmetry, as in Thomson or Rayleigh scattering.)Therefore
l∗2 = Nl2 ,
√
l∗ = Nl.
(8)
(9)
The quantity l∗ is the root mean square net displacement of the photon, and it increases as the
square root of the number of scatterings. This result can be used to estimate the mean number of
scattering in a finite medium. Suppose a photon is generated somewhere within the medium; then
the photon will scatter until it escapes completely. For regions of large optical depth the number
of scatterings required to do this is roughly determined by setting l∗ ∼ L, the typical size of the
medium. From equation [9] we find
N≈
L2
.
l2
(10)
Since l is of the order of the mean free path, L/l is approximately the optical thickness of the
medium τ. Therefore, we have
N ≈ τ2 ,
(τ 1).
(11)
–4–
For regions of small optical thickness the mean number of scatterings is small, of order 1 − e−τ ≈ τ;
that is
N ≈ τ,
(τ 1).
(12)
For most order-of-magnitude estimates it is sufficient to use N ≈ τ2 + τ or N ≈ max(τ, τ2 ) for any
optical thickness.