126 The hot universe
To expressX 2 Sin terms ofXe,we use the quasi-equilibrium condition for the
intermediate 2Sreservoir; this is justified by the high rate of the reactions, shown
in Figure 3.9. For the 2Sreservoir, this condition takes the following form:
〈σv〉ep→γ 2 Snenp−〈σ〉γ 2 S→epneqγn 2 S−W 2 S→ 1 Sn 2 S= 0 , (3.193)whereneqγ is the number density of thermal photons. The relation between the cross-
sections for the direct and inverse reactions,epγ 2 S,can be found if one notes
that in a state of equilibrium these reactions compensate each other. Then, we have
〈σ〉γ 2 S→epneqγ
〈σv〉ep→γ 2 S=
neqeneqp
neq 2 S=
(
Tme
2 π) 3 / 2
exp(
−
BH
4 T
)
, (3.194)
where the Saha formula has been used to obtain the latter equality (recall that the
binding energy of 2Sstate isBH/4).With the help of this relation, we can express
X 2 S,from (3.193), as
X 2 S=
[
W 2 S
〈σv〉ep→ 2 S+
(
Tme
2 π) 3 / 2
exp(
−
BH
4 T
)]− 1
nTX^2 e, (3.195)and (3.192) becomes
dXe
dt=−W 2 S
[
W 2 S
〈σv〉ep→ 2 S+
(
Tme
2 π) 3 / 2
exp(
−
BH
4 T
)]−^1
nTX^2 e. (3.196)When the first term inside the square brackets is small compared to the second,
the electrons andexcitedhydrogen atoms are in equilibrium with each other and
with the thermal radiation. Therefore, the ratio of thee,pand 2Snumber densities
satisfies a Saha-type relation (see the second equality in (3.194)). The ionization
fraction, however, does not obey (3.189) because, as mentioned above, the ground
state is not in equilibrium with the other levels after the beginning of recombination.
The excited states are more abundant than one expects in full equilibrium and the
ionization fraction significantly exceeds that given in (3.189).
Problem 3.25The cross-section for recombination to the 2Slevel is well approx-
imated by the formula
〈σv〉ep→γ 2 S 6. 3 × 10 −^14(
BH
4 T
) 1 / 2
cm^3 s−^1. (3.197)Using this expression, verify that the two terms inside the square brackets in (3.196)
become comparable at the temperatureT2450 K.
Hence, only atT>2450 K is the reactionγ 2 Sepefficient in maintaining
chemical equilibrium between the electrons, protons and hydrogen 2Sstates. After