Chapter 3 – Part 3
Beyond Just Elastic Particles
So far we have ignored the effects of radiation
(in particular, absorptions and emissions). Let’s
now extend to include the effects of an ambient
bath of radiation.
Suggestion: review Appendix C (and its crossreferences to Appendix D).
What Can Happen?
Absorption of Energy
Electrons bound to a nucleus can be excited to
higher orbitals (bound-bound transitions), either
radiatively (absorption of a photon) or collisionally
(some of the kinetic energy is used).
Electrons bound to a nucleus can be stripped off
completely, and the atom becomes further or
completely ionized (bound-free transitions) either
radiatively (absorption of a photon) or collisionally.
Release of Energy
Electrons in high orbitals can drop to lower ones,
with the emission of a photon, either
spontaneously (if left alone for long enough!) or by
stimulated emission (as an equivalent photon
passes by).
Electrons in higher orbitals can be collisionally deexcited (i.e. drop back down with no photon
emitted during a collision). This happens in all but
the lowest-density gases.
“Nebulium”
1864: Huggins – strong green line
1925: Bowen: Oxygen!
Can’t do this in the lab!
Free electrons can be recaptured with the
emission of a photon, and (if in higher orbitals)
can subsequently cascade down to lower
orbitals.
 Continuum and line emission
Bound-bound transitions
Statistical Equilibrium, TE, LTE
See the discussions on pp 93- 94.
Equilibrium == there’s as many “ups” as “downs,”
and no net “in” or “out.”
Can a single “T” characterize the radiation (say, TR)
and the kinetic temperature?
TE (idealized) versus LTE.
Can LTE be truly correct?
See the argument on page 95, top.
Note that the temperature gradient, the local
density, the mean-free-path of photons, and the
ensuing interaction can’t always yield LTE – think
especially of the outer parts of the sun.
(Limb darkening tells us that we see to different
depths!)
The Explanation
The Boltzmann Equation
Assume LTE. Under these circumstances, what
are the relative populations in the varied
orbitals as a function of temperature? (Play
with this! See spreadsheet.)
Note the remark about the apparent divergence
of the function at large principal quantum
numbers. This is not really a problem for the
reasons given on p 97 – think about it!
The Partition Function
Rather than intercompare two suborbitals,
compare one level to all possibilities using the
so-called partition function U.
Consider Some Values
See spreadsheet
This Explains A0 (Sirius) vs G2 (Sun)
The Low-Density Regime
In low-density gas, no LTE! Excited atoms reemit spontaneously (e.g. for Lyman alpha, rate =
6.3 x 108 sec-1 [Einstein coefficient]; lifetime ~
1.6 nanosec) rather than via frequency of
collisions (which depends on particle density).
Implication: in the low-density ISM, neutral
hydrogen will all be in the ground state.
If no LTE, what “T”??
The excitation temperature Tex is defined as that
temperature that explains the relative numbers in
the different excitation stages: e.g. 2:1, or 3:2.
It may differ from one pair of ionization stages to
another – uniqueness not guaranteed!
Note that we can have inversions, with more
particles in an excited state than in a lower state.
What About Neutral H in the ISM?
Remember it’s essentially all in the ground
state. Collisions can excite the hyperfine
state (proton and electron spin parallel;
higher energy than anti-parallel).
A spontaneous downward transition is very improbable (“forbidden”) since it
violates one of the ‘selection rules.’ (See Appendix C.) The lifetime for this rare
(not truly impossible!) occurrence is ~ 1014 sec (107 yrs). But collisions are likely
on much shorter timescales, even in the low-density ISM. So the state gets
collisionally de-excited on a timescale determined by the kinetic temperature of
the gas.
Brief Recap
• We have considered neutral H, asking what
fraction of the gas is in various stages of
excitation. That’s the Boltzmann equation.
• We could similarly look at any other species
that is not completely ionized (say He I
[neutral] or He II [one electron gone]) to see
what the various populations are in the
various excitation stages.
Ionization: The Saha Equation
What about a closely parallel
question related to the relative
numbers in different stages of
ionization? For example, in a
given hydrogen gas cloud, how
many will still be neutral H I, and
how many will be free protons (H
II), having lost their electron?
This will clearly depend on
temperature, but what else?
The Saha Equation: A Familiar Look
Note the ‘Boltzmann factor,’ with energy differences
scaling like kT. (Here ΧK is the energy required to
remove an electron from the ground state of the Kth
ionization stage.)
An Important Difference
Note that the number density of electrons enters
the equation (in the denominator). Why?
Think about the physical sense of this. It implies
that if there are more free electrons, then there
will be relatively fewer nuclei in the K+1 stage
relative to those in the Kth stage. (The electrons
can come from any species, remember!)
Hydrogen
So consider the surface of a hot star (example, p101).
We saw from example 3.6 that to pump even 10% of the
atoms of the first excited state, we need a temperature
like 32 000 K. We now see further that at that temp,
essentially all the gas is ionized.