100
5 The Ionized Interstellar Gas
gaseous nebulae, and corresponds to Fig. 5.6. We now have N1n = Nn1 and the
equilibrium of level n is written:
Nn"n +
Nkn dν =
n−1
Nnn +
Nnk dν.
(5.35)
n =2
n">n
Photoionizations from levels n ≥ 2 are generally neglected because these levels
are much less populated than level n = 1.
In this way we arrive at an infinite (in principle) system of linear equations in n n
and thus in bn . We find a solution in Hummer & Storey [252], Storey & Hummer
[497] and references cited herein. These articles give the intensities of the Balmer
lines and also the Balmer decrement which is the intensity ratio of the different
Balmer lines to Hβ (the 4-2 line). This ratio is almost the same in cases A and B
and depends very little on the physical conditions (n e and Te ). Table 5.3 gives the
wavelength and the intensity relative to Hβ of the first lines of the Balmer (n-2),
Paschen (n-3), Brackett (n-4) and Pfund (n-5) series for n e = 104 cm−3 and Te = 104
K, for case B. See Hummer & Storey [252] and Brocklehurst [69] for higher lines,
for the lines of He ii and for other values of the physical parameters.
Table 5.3. Wavelengths in µm and intensities relative to Hβ for the most important recombination lines of hydrogen in the visible and in the infrared, for n e = 104 cm−3 and Te = 104 K,
in case B. From Hummer & Storey [252].
Balmer series
Hα
Hβ
Hγ
Hδ
H
0.6563
0.4861
0.4340
0.4101
0.3970
2.85
1.00
0.469
0.260
0.159
Brackett series
Brα
Brβ
Brγ
Brδ
Br
Paschen series
Pα
Pβ
Pγ
Pδ
P
1.875
1.282
1.094
1.005
0.9546
0.332
0.162
9.01×10−2
5.53×10−2
3.65×10−2
4.05
2.63
2.17
1.94
1.82
7.77×10−2
4.47×10−2
2.75×10−2
1.81×10−2
1.26×10−2
Pfund series
Pfα
Pfβ
Pfγ
Pfδ
Pf
7.46
4.65
3.74
3.30
3.04
2.45×10−2
1.58×10−2
1.04×10−2
7.25×10−3
5.24×10−3
The interest of the intensity ratios between recombination lines, and in particular the Balmer decrement, is that a comparison between the observed and the
theoretical ratios (Table 5.3) can be used to estimate the reddening and hence
the extinction by dust in the direction of the H ii region (cf. Caplan & Deharveng [76]). In particular, the visual extinction A V is related to the colour excess
E β−α = 2.5 log[Itheor (Hβ)/Iobs (Hβ)]/[Itheor (Hα)/Iobs (Hα)] by the relation
100
EXCITATION
from ultraviolet H, absorption measures to determine Po and nH, the
overall hydrogen density [assumed about equal to n(H I), the density of
neutral hydrogen] which multiplies yu, the collisional rate coefficient in
(Ru)& The computed value found for (k), averaged over the Werner
and Lyman bands with the interstellar radiation field in Table 5.5, is 0.11.
While G ( K ) is essentially unknown, comparison of predicted and observed
N ( J ) values suggest that G ( K ) is peaked at values of K well above 0 or 1,
as would be expected if an appreciable fraction of the energy released in
H, molecule formation [S6.2b] goes into vibrational and rotational excitation. Theoretical calculations have been made also for the relatively thick
clouds [39] in which the strong R(O), R(1), and P(1) lines are on the
square-root section of the curve of growth [S3.4c], with each &,, reduced
by the appropriate K,m factor [S5.3a] [see equations (5-43) and (5-44)].
Again, values of nH and Po were determined by fitting the observed column
densities.
The distribution of Po values found in these studies is shown in Fig. 4.5.
For comparison, the anticipated value of Po computed for mean conditions
I
1
log
I
I
{ iO'OBo (s-' I }
I
Ftsuro 4.5 ~~.b(butkn
t~ & v.lrm. Plotted in the histogram are the number of BO
values within each interval of 0.5 in log& Individual values were determined [38,
391 by applying optical pumping theory [35] to the observed H2column densities in
excited rotational levels. The expected mean value of & for interstellar starlight is
about 5 X IO-'Os-', in agreement with the observed peak at l0g(lO'~80)=0.7.
1