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Page 95
Chapter 5
Radiation
J.J. Keady and D.P. Kilcrease
5.1
5.1
Radiation Quantities and Interrelations . . . . . . . . .
5.2
Refractive Index and Average Polarizability . . . . . . 100
5.3
Absorption and Scattering by Particles . . . . . . . . . 102
5.4
Photoionization and Recombination . . . . . . . . . . 106
5.5
X-Ray Attenuation . . . . . . . . . . . . . . . . . . . . 109
5.6
Absorption of Material of Stellar Interiors . . . . . . . 110
5.7
Absorption of Material of the Solar Photosphere . . . 114
5.8
Solar Photoionization Rates . . . . . . . . . . . . . . . 114
5.9
Free–Free Absorption and Emission . . . . . . . . . . 115
5.10
Reflection from Metallic Mirrors . . . . . . . . . . . . 117
5.11
Visual Photometry . . . . . . . . . . . . . . . . . . . . . 117
95
RADIATION QUANTITIES AND INTERRELATIONS
The quantitative concepts of radiation are defined [1] in terms of I, the flux of radiation at a given point
in a given direction across a unit surface normal to that direction per unit time and per unit solid angle.
This is called specific intensity, or simply intensity.
The flux of radiation through a unit surface is the surface flux, or flux density,
F =
I cos θ dω,
4π
where θ is the angle between the ray and the outward normal and integration is in all directions.
95
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R ADIATION
The emittance is the flux of radiation emitted from a unit surface,
F =
I cos θ dω
2π
for isotropic radiation π I , where in this case the integration is over the outward hemisphere.
The radiation density is
u = (1/c)
I dω = (4π/c) I¯.
4π
The radiation quantities per unit frequency and wavelength ranges are written Iν , Iλ , Fν , etc.:
I =
Iν dν =
Iλ dλ,
c
ν2
I
=
λIλ = ν Iν ,
Iν ,
ν
c
λ2
λ2
c
dλ = − dν = − 2 dν,
c = λν.
c
ν
Iλ =
The linear absorption coefficient is κs :
d I /ds = −κs I.
The scattering coefficient σs, is similar to the absorption coefficient but applies to the radiation
scattered. It is used in the sense that κs − σs represents absorption and transference into heat.
The mass absorption coefficient is κm (the subscript is usually omitted):
d I /ds = −ρκm I,
where ρ is the density.
The atomic or particle absorption coefficient or cross section is a:
d I /ds = −N a I,
where there are N atoms or particles per unit volume and a represents the effective area over which the
incident radiation if fully absorbed.
The emission coefficient j is the radiant flux emitted per unit volume and unit solid angle. For
uniform scattering,
j = (σ/4π )
I dω,
4π
where the first term represents scattering and the integral represents incident radiation.
For scattering by electrons, atoms, molecules,
2
3
j = (σ/4π )
4 (1 + cos θ)I dω,
4π
where θ is the angle between incident and scattered light. I is assumed to be unpolarized, and the
scattered radiation when viewed at the angle θ is polarized with intensity proportional to cos2 θ in the
plane of scattering and proportional to I in the direction perpendicular to the plane of scattering.
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5.1 R ADIATION Q UANTITIES AND I NTERRELATIONS / 97
The optical thickness or depth is
τ =
κs ds =
ρκm ds.
The source function is
S = j/κs .
The intensity emitted from an absorbing medium is
I =
j exp(−τ ) ds =
S exp(−τ ) dτ.
We show two forms of the Kirchoff law:
(a) In a volume element,
jν = κs,ν Bν (T ),
where Bν (T ) is blackbody intensity at temperature T .
(b) At a surface element,
Iν = Aν Bν (T ),
where Aν is the fraction of incident radiation absorbed, i.e., 1 − Aν is the reflection coefficient and
analogous to albedo.
The atomic polarizability α is the induced dipole moment per unit electric field (ᾱ for a steady or
low-frequency field):
ᾱ = 4a03
f n /(νn /c R∞ )2
n
= 5.927 × 10−25
f n /(νn /c R∞ )2 cm3
n
= 7.138 × 10
−23
f n λ2n cm3
(λ in µm),
n
where νn /c R∞ is the frequency in rydbergs of lines connecting the ground level and f n is the
corresponding oscillator strength.
For scattering,
αs = (128π 5 /3)N (ν/c)4 α 2
= (128π 5 /3λ4 )N α 2
= 1.3057 × 1020 N α 2 /λ4
(λ in µm).
The index of refraction is n:
n − 1 = 2π N α
= 1.688 × 1020 α
at STP.
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R ADIATION
The molecular refraction is
R=
n2 − 1 M
4π
=
N0 α,
3
n2 + 2 ρ
where M is the molecular weight, ρ is the density, and N0 is the Avogadro number.
The radiation constants are
c1 = 2π hc2 = 3.741 77 × 10−5 erg cm2 s−1 ,
c2 = hc/k = 1.438 77 cm K.
The Stefan–Boltzmann constant is
σ = 2π 5 k 4 /(15c2 h 3 ) = π 4 c1 /(15c24 )
= 5.6705 × 10−5 erg cm−2 s−1 K−4 .
The blackbody emittance is
F = σ T 4.
The blackbody intensity is
B = (σ/π )T 4 = 1.804 98 × 10−5 T 4 erg cm−2 s−1 sr−1 K−4 .
The radiation density u in a cavity at temperature T is
u = aT 4 = (4σ/c)T 4 = 7.565 91 × 10−15 T 4 erg cm−3 K−4 .
In a medium of refractive index n,
B = n 2 (σ/π )T 4 ,
u = n 3 (4σ/c)T 4 ,
with similar factors applying for the Planck law with n ν and n λ .
The photon emission constant is
p = 4π ζ (3)c/c23
= 1.520 486 × 1011 photons cm−2 s−1 K−3 ,
where ζ (n) is the Riemann zeta function.
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5.1 R ADIATION Q UANTITIES AND I NTERRELATIONS / 99
The photon flux from a unit blackbody surface is
N = pT 3 .
Blackbody radiation is unpolarized, hence the intensity of radiation linearly polarized in a specific
direction will be half the value quoted in the formulas.
The Planck function in wavelength units is
(c/4)u λ = π Bλ = Fλ = 2π hc2 λ−5 /(ehc/kλT − 1)
= c1 λ−5 /(ec2 /λT − 1) (λ in cm),
where u λ , Bλ , and Fλ are the radiation density, intensity, and emittance for unit wavelength ranges.
The Planck function in frequency units is
(c/4)u ν = π Bν = Fν = 2π hν 3 c−2 /(ehν/kT − 1).
The photon distribution law is
Nλ = 2π cλ−4 /(ec2 /λT − 1),
Nν = 2π c−2 ν 2 /(ehν/kT − 1),
where Nλ and Nν are the emittance of photons per squared centimeter per second and per unit
wavelength and frequency ranges, respectively.
The Rayleigh–Jeans distribution (for the red end of the spectrum) is
Fλ = 2π ckT λ−4 = (c1 /c2 )T λ−4 ,
Fν = 2π c−2 kT ν 2 = 2π kT λ−2 .
The Wien distribution (for the violet end of the spectrum) is
Fλ = 2π hc2 λ−5 e−c2 /λT = c1 λ−5 e−c2 /λT ,
Fν = 2π hc−2 ν 3 e−hν/kT .
Wien law: The wavelength of maximum Fλ or Bλ is λmax :
T λmax = 0.201 405 2c2 = 0.289 78 cm K.
The wavelength of maximum photon emission is λm :
T λm = 0.255 057 1c2 = 0.366 97 cm K.
The frequency of maximum Fν or Bν is νm :
T c/νm = 0.354 429 0c2 = 0.509 94 cm K.
The three numerical constants above are 1/y in y = 5(1 − e−y ), y = 4(1 − e−y ), and
y = 3(1 − e−y ), respectively.
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100 / 5
5.2
R ADIATION
REFRACTIVE INDEX AND AVERAGE POLARIZABILITY
The refractive index and polarizability of atomic and molecular gases are given in Tables 5.1 and 5.2,
where n is the refractive index at STP,
n − 1 = A(1 + B/λ2 )
(λ in µm),
and ᾱ is the polarizability at low frequency.
Table 5.1. Refractive index and polarizability of atomic gases.
Atom
ᾱ [1]
(10−25
cm3 )
H
He
Li
Be
C
N
O
Ne
Na
Mg
Al
Si
P
S
6.67
2.05
243
56
17.6
11.0
8.03
3.95
236
106
83.4
53.8
36.3
29.0
n (D lines)
A
(units of
10−5 )
B
(units of
10−3 )
1.000 035 0
3.48
2.3
1.000 067 1
6.6
2.4
Atom
ᾱ [1]
(10−25
cm3 )
Cl
Ar
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
21.8
16.4
434
250
169
136
114
68
86
75
68
65
61
71
n(D lines)
A
(units of
10−5 )
B
(units of
10−3 )
1.000 283 7
27.92
5.6
Reference
1. Miller, T.M., & Bederson, B. 1977, Adv. Atom. Mol. Phys. 13, 1
Table 5.2. Refractive index of molecular gases.
Molecule
n (D lines)
A
(units of
10−5 )
B
(units of
10−3 )
Air
H2
O2
N2
H2 O
1.000 291 8
1.000 138 4
1.000 272
1.000 297
1.000 254
28.71
5.67
13.58
7.52
26.63
5.07
29.06
7.7
516 (radio freq.)
Molecule
n (D lines)
CO2
CO
NH3
NO
CH4
1.000 449 8
1.000 334
1.000 375
1.000 297
1.000 441
A
(units of
10−5 )
B
(units of
10−3 )
43.9
32.7
37.0
28.9
6.4
8.1
12.0
7.4
The refractive indices quoted in Table 5.3 are relative to air at 15◦ C. The temperatures of the media
are about 18◦ C and the temperature coefficients quoted are the change of D-line refractive index for a
1◦ C temperature rise. Manufacturers’ reports must be consulted for indices that are accurate enough
for optical design. The table also gives the spectral limits (λ in µm) within which the absorption is less
than 2.72 cm−1 (i.e., 1 cm transmission > 37%).
0.23
2.2
+0.000 005
1.91
1.722
1.683
1.666
1.657
1.652
1.648
1.643
1.626
Ord.
ray
0.23
4
+0.000 014
1.58
1.515
1.499
1.491
1.486
1.483
1.481
1.479
1.476
Extr.
ray
0.32
2.2
−0.000 001
1.557
1.531
1.522
1.517
1.513
1.511
1.507
1.496
BSC
crown
1.650
1.627
1.616
1.610
1.605
1.600
DF
flint
0.13
9.0
−0.000 01
1.495
1.455
1.442
1.437
1.434
1.432
1.430
1.429
1.424
1.398
1.303
Fluorite
CaF2
References
1. Allen, C.W. 1973, Astrophysical Quantities, 3rd ed. 1, Sec. 34; 2, Sec. 35; 3, Sec. 36
2. Garton, W.R.S. 1966, Adv. Atom. Mol. Phys. 2, 93
0.37
2.8
+0.000 003
Glass
Note
For information on atmospheric refraction, see Table 5.2.
Limits [2]
Low λ
High λ
Temp.
coef.
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.0
2
5
10
λ
(µm)
Calcite
0.17
3.6
−0.000 005
1.651
1.579
1.558
1.549
1.544
1.541
1.539
1.536
1.520
1.42
Ord.
ray
1.663
1.589
1.567
1.558
1.553
1.550
1.548
1.544
1.528
Extr.
ray
0.17
3.6
−0.000 006
Quartz
Table 5.3. Refractive indices of optical media [1, 2].
0.16
21
−0.000 003
1.550
1.489
1.471
1.463
1.458
1.455
Fused
silica
0.20
17
−0.000 04
1.792
1.602
1.568
1.552
1.543
1.538
1.535
1.532
1.526
1.519
1.494
Rock
salt
< 0.2
1.14
−0.000 08
1.423
1.358
1.343
1.336
1.332
1.330
1.328
1.325
1.315
Water
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5.2 R EFRACTIVE I NDEX AND AVERAGE P OLARIZABILITY / 101
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5.3
Page 102
R ADIATION
ABSORPTION AND SCATTERING BY PARTICLES
For scattering of free electrons, σe (Thomson scattering) is [2]
8π
σe =
3
e2
mc2
2 1−2
hν
mc2
hν
= 0.665 24 × 10−24 1 − 2 2 cm2 ,
mc
where σe is the (exponential) scattering coefficient per electron (Sec. 5.1) with the relativistic term
2hν/mc2 . At the high densities and temperatures found in stellar interiors, further corrections due to
correlations and thermal motion may be required [3, 4].
For Rayleigh scattering of atoms or molecules,
32π 3 (n − 1)2 6 + 3
3N
6 − 7
λ4
18
= 3.307 × 10 (n − 1)2 δ/λ4 N cm−1
σs =
(λ in µm),
where N is the number of atoms or molecules per unit volume, n is the refractive index of the medium,
σs is the linear scattering coefficient, and δ = (6 + 3)/(6 − 7) = depolarizing factor [5, 6]. =
0.030 for N2 and 0.054 for O2 [7].
The Rayleigh scattering cross section of an atom or a molecule is
32π 3 δ
σa =
3λ4
n−1
N
2
=
128π 5 2
δα
3λ4
= 1.306 × 1020 δα 2 /λ4 cm2
(λ in µm),
where the polarizability α = (n − 1)/(2π N ).
For atomic scattering at some distance from any absorption line,
8π
σa =
3
e2
mc2
2 2
f 12 ν 2
2 − ν2
ν12
2
,
where f 12 is the oscillator strength (1 is the ground level when excitation is low).
For the absorption of small particles (spherical) of radius a in terms of πa 2 [5], the efficiency
factors (Q = σ/πa 2 ) for extinction, scattering, absorption, and radiation pressure are Q ext , Q sca ,
Q abs , and Q pr , respectively, with
Q ext = Q sca + Q abs ,
Q pr = Q ext − cos θ Q sca ,
where cos θ is the forward asymmetry of scattering [8]. For large objects Q ext = 2.0 of which 1.0 is
intercepted and 1.0 scattered with cos θ = 1.0. The extinction coefficient k is related to the complex
dielectric constant through
= 1 ± i2 = (n ± ik)2 .
Here n is the refractive index and 1 , 2 , n, and k are real. k and n are therefore given by
k
1
= √ [(12 + 22 )1/2 ∓ 1 ]1/2 .
n
2
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5.3 A BSORPTION AND S CATTERING BY PARTICLES / 103
For measurements performed in vacuum, with 2πa/λ 1 everywhere, the measured extinction
coefficient is related to the indices of refraction via
Q abs
Q ext
2π
24nk
≈
=
.
a
a
λ n 2 − k 2 + 2 + 4n 2 k 2
See Tables 5.4–5.10, extinction efficiency factors for various compounds. The mean particle radius for
the spheroids in the amorphous carbon sample was 40 Å.
Table 5.4. Extinction efficiency factor Q ext for water droplets as a function of particle radius and wavelength [1].
λ (µm)
a = 0.3µm
1.0 µm
3.0 µm
10.0 µm
λ (µm)
a = 0.3µm
1.0 µm
3.0 µm
10.0 µm
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.80
2.00
2.10
2.20
2.30
2.50
2.60
2.70
2.75
2.80
2.90
2.95
3.00
3.10
3.20
3.30
3.40
3.50
3.60
3.75
3.83
4.00
1.39
1.07
0.80
0.58
0.44
0.35
0.28
0.23
0.18
0.15
0.099
0.067
0.054
0.043
0.035
0.022
0.012 7
0.013 6
0.047 7
0.175 4
0.417 8
0.458
0.423
0.293
0.159
0.080 6
0.042 9
0.026 4
0.018 9
0.013 0
0.011 4
0.011 4
2.79
3.37
3.76
3.90
3.89
3.60
3.41
3.08
2.83
2.56
2.07
1.65
1.45
1.25
1.09
0.69
0.33
0.189
0.297
0.940
1.65
1.74
1.76
1.79
1.93
1.52
1.21
0.944
0.763
0.569
0.497
0.428
2.47
2.13
1.95
2.37
2.68
2.80
2.28
2.04
1.86
1.83
2.09
2.87
3.16
3.51
3.67
3.66
2.61
1.58
1.61
2.58
2.45
2.44
2.49
2.67
2.74
3.14
3.53
3.97
3.99
3.84
3.76
3.58
2.07
1.99
2.05
2.14
2.18
2.05
2.25
2.25
2.01
2.27
2.11
2.13
2.42
2.46
2.28
2.34
1.96
2.26
2.34
2.23
2.21
2.21
2.22
2.24
2.26
2.33
2.42
2.17
1.97
2.38
2.66
2.80
4.50
4.66
4.80
5.00
5.26
5.50
5.80
6.00
6.05
6.40
7.00
7.50
8.00
8.50
9.00
9.50
10.00
10.50
11.00
11.50
12.00
12.50
13.00
13.50
14.00
15.00
16.00
17.50
18.00
20.00
30.00
100.00
0.018 2
0.019 7
0.018 0
0.013 1
0.009 8
0.011 1
0.026
0.076
0.083
0.030
0.022
0.021
0.020
0.020
0.020
0.021
0.025
0.031
0.044
0.063
0.083
0.091
0.103
0.107
0.114
0.123
0.104
0.086
0.080
0.061
0.037
0.014
0.342
0.319
0.291
0.242
0.189
0.162
0.177
0.39
0.419
0.215
0.145
0.120
0.106
0.097
0.091
0.087
0.094
0.112
0.155
0.220
0.286
0.317
0.360
0.376
0.400
0.434
0.371
0.309
0.289
0.223
0.129
0.048
3.02
2.88
2.76
2.56
2.22
1.93
1.50
1.95
2.02
1.95
1.47
1.19
1.02
0.886
0.734
0.596
0.509
0.487
0.547
0.706
0.884
0.992
1.13
1.20
1.29
1.44
1.38
1.30
1.29
1.23
0.528
0.149
2.46
2.26
2.21
2.02
1.85
2.02
2.65
2.40
2.38
2.30
2.73
3.09
3.24
3.30
3.21
2.95
2.50
2.08
1.78
1.83
1.97
2.09
2.20
2.26
2.32
2.42
2.51
2.64
2.71
2.92
2.75
0.758
Reference
1. Irvine, W.R., & Pollack, J.B. 1968, Icarus, 8, 324
Table 5.5. Extinction efficiency factor Q ext for ice particles as a function of particle radius and wavelength [1].
λ (µm)
a = 0.3µm
1.0 µm
3.0 µm
10.0 µm
0.95
1.00
1.20
1.50
2.00
2.35
0.572
0.49
0.295
0.158
0.063
0.029
2.92
3.80
3.35
2.52
1.52
0.94
2.41
2.66
2.28
1.82
3.10
3.77
2.02
2.08
2.15
2.24
2.35
1.96
λ (µm)
3.60
3.80
3.90
4.00
4.10
4.20
a = 0.3µm
1.0 µm
3.0 µm
10.0 µm
0.025
0.017
0.018
0.019
0.021
0.023
0.76
0.52
0.46
0.41
0.38
0.35
3.91
3.70
3.51
3.28
3.04
2.84
2.04
2.57
2.63
2.64
2.43
2.30
Sp.-V/AQuan/1999/07/30:17:53
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104 / 5
R ADIATION
Table 5.5. Continued.
λ (µm)
a = 0.3µm
1.0 µm
3.0 µm
10.0 µm
λ (µm)
a = 0.3 µm
1.0 µm
3.0 µm
10.0 µm
2.40
2.45
2.50
2.55
2.60
2.625
2.65
2.800
2.85
2.90
2.95
3.00
3.05
3.075
3.10
3.15
3.20
3.25
3.30
3.35
3.40
3.45
3.50
3.55
0.025
0.022
0.019
0.016
0.013
0.012
0.012
0.027
0.066
0.177
0.306
0.377
0.514
0.547
0.511
0.379
0.244
0.151
0.109 6
0.081 7
0.060 8
0.046
0.035
0.029
0.823
0.724
0.626
0.522
0.433
0.397
0.364
0.261
0.360
0.675
0.986
1.151
1.516
1.63
1.69
2.30
2.23
2.10
1.91
1.74
1.50
1.16
0.97
0.85
3.81
3.73
3.58
3.35
3.09
2.96
2.79
1.84
1.68
1.78
1.92
2.00
2.22
2.28
2.36
2.54
2.57
2.62
2.78
2.92
3.14
3.62
3.87
3.94
1.93
2.12
2.46
2.71
2.58
2.36
2.18
2.07
2.31
2.19
2.14
2.14
2.18
2.19
2.20
2.25
2.26
2.26
2.25
2.25
2.36
2.37
2.19
2.06
4.30
4.40
4.50
4.60
4.70
4.80
4.90
5.00
5.70
6.00
6.40
6.70
7.00
8.00
9.00
10.00
11.00
12.00
15.00
20.00
40.00
62.00
100.00
150.00
0.027
0.032
0.036
0.031
0.023
0.018
0.015
0.014
0.026
0.048
0.040
0.039
0.033
0.021
0.019
0.020
0.038
0.039
0.019
0.005
0.017
0.016
0.002 7
0.000 57
0.34
0.33
0.32
0.28
0.23
0.20
0.17
0.16
0.16
0.23
0.19
0.18
0.15
0.10
0.08
0.07
0.15
0.16
0.076
0.019
0.058
0.055
0.009
0.002
2.67
2.46
2.31
2.17
2.02
1.86
1.73
1.61
1.20
1.31
1.11
1.02
0.92
0.67
0.51
0.33
0.77
1.25
0.80
0.21
0.20
0.18
0.028
0.007
2.16
2.11
2.10
2.10
2.06
2.16
2.27
2.39
2.95
2.73
2.86
2.89
2.96
3.01
2.74
1.76
2.79
3.00
3.58
3.06
1.37
1.02
0.17
0.04
Reference
1. Irvine, W.R., & Pollack, J.B. 1968, Icarus, 8, 324
Table 5.6. Extinction efficiencies for amorphous carbon [1, 2].a
λ (µm)
Q ext /a (cm−1 )
λ (µm)
Q ext /a (cm−1 )
λ (µm)
Q ext /a (cm−1 )
0.12
0.13
0.14
0.15
0.16
0.18
0.20
0.22
0.23
0.25
5.12[5]
3.21[5]
2.02[5]
2.18[5]
1.84[5]
1.47[5]
1.43[5]
1.56[5]
1.63[5]
1.60[5]
0.27
0.30
0.50
0.70
1.00
1.60
2.00
3.13
4.00
5.00
1.49[5]
1.28[5]
7.07[4]
4.81[4]
3.52[4]
2.14[4]
1.69[4]
1.05[4]
7.96[3]
6.06[3]
7.14
10.00
15.40
20.00
30.50
50.80
70.50
101.00
205.00
289.00
4.49[3]
3.18[3]
1.74[3]
1.35[3]
8.80[2]
5.38[2]
4.04[2]
3.17[2]
1.66[2]
1.19[2]
Note
a Numbers in square brackets denote powers of 10.
References
1. Bussoletti, E. et al. 1987, A&AS, 70, 257
2. Maron, M. 1990, ApS&S, 172, 21
Sp.-V/AQuan/1999/07/30:17:53
Page 105
5.3 A BSORPTION AND S CATTERING BY PARTICLES / 105
Table 5.7. Extinction efficiency factor Q ext for graphite as a function of particle
radius and wavelength [1].a
λ (µm)
a = 0.01µm
0.1 µm
λ (µm)
a = 0.01µm
0.1 µm
0.12
0.15
0.18
0.20
0.21
0.215
0.217 5
0.22
0.225
0.23
0.24
0.26
0.28
0.30
0.33
0.365
0.486 1
0.656 2
0.80
1.00
1.40
0.425
0.244
0.512
0.995
1.51
1.25
1.22
1.17
1.01
0.88
0.63
0.40
0.31
0.25
0.20
0.160
0.093
0.058
0.043
0.030
0.017
2.58
2.32
2.49
2.83
2.92
3.03
3.06
3.08
3.11
3.13
3.12
3.02
2.98
3.01
3.09
3.11
3.34
2.65
1.88
1.09
0.45
2.00
3.00
4.00
5.00
6.00
9.00
10.00
11.00
11.52
11.54
12.00
20.00
40.00
60.00
80.00
100.0
200.0
400.0
700.0
1 000.0
2 000.0
1.00[−2]
5.37[−3]
3.44[−3]
2.43[−3]
1.83[−3]
4.65[−6]
9.00[−4]
8.21[−4]
2.98[−3]
8.82[−4]
7.60[−4]
6.91[−4]
6.51[−4]
4.28[−4]
2.77[−4]
1.89[−4]
5.24[−5]
1.40[−5]
4.71[−6]
2.33[−6]
5.87[−7]
0.19
0.084
0.048
0.032
0.024
1.23[−2]
1.07[−2]
9.61[−3]
3.13[−2]
1.01[−2]
8.78[−3]
7.45[−3]
6.87[−3]
4.41[−3]
2.80[−3]
1.90[−3]
5.13[−4]
1.29[−4]
4.24[−5]
2.08[−5]
5.22[−5]
Note
a Numbers in square brackets denote powers of 10.
Reference
1. Draine, B.L. 1985, ApJS, 57, 587
Table 5.8. Extinction efficiencies for silicon carbide [1].a
λ (µm)
0.10
0.20
0.40
0.78
0.99
1.96
3.09
3.97
4.69
6.02
7.10
8.39
Q ext /a (cm−1 )
3.46[6]
7.65[4]
1.29[4]
8.85[4]
8.13[3]
7.01[3]
4.15[3]
2.55[3]
1.66[3]
6.97[2]
4.83[2]
4.03[2]
λ (µm)
9.12
9.90
10.14
10.37
10.61
10.86
11.11
11.37
11.63
11.90
12.17
12.45
Q ext /a (cm−1 )
λ (µm)
Q ext /a (cm−1 )
4.24[2]
9.65[2]
1.66[3]
3.06[3]
5.76[3]
9.35[3]
1.37[4]
1.45[4]
1.04[4]
7.30[3]
4.84[3]
3.31[3]
12.74
13.04
13.34
13.65
13.96
14.29
15.30
22.00
36.90
52.00
103.3
205.3
2.53[3]
1.94[3]
1.50[3]
1.18[3]
9.69[2]
7.96[2]
5.76[2]
3.23[2]
1.57[2]
9.74[1]
3.79[1]
1.23[1]
Note
a Numbers in square brackets denote powers of 10.
Reference
1. Pégourié, B. 1988, A& A, 194, 335
Sp.-V/AQuan/1999/07/30:17:53
106 / 5
Page 106
R ADIATION
Table 5.9. Extinction efficiency factors Q ext for silicate as a function of particle
radius and wavelength [1].a
λ (µm)
0.12
0.14
0.16
0.20
0.23
0.30
0.40
0.55
a = 0.01µm
0.1 µm
0.943
0.51
0.154
0.032
0.014 6
1.09[−2]
7.99[−3]
5.76[−3]
2.56
2.62
2.67
3.13
3.98
3.41
2.26
0.78
λ (µm)
1.00
1.65
2.00
2.60
3.00
4.00
5.00
6.00
a = 0.01µm
0.1 µm
3.22[−3]
4.19[−4]
2.25[−4]
9.40[−5]
5.75[−5]
2.06[−5]
8.64[−4]
9.95[−4]
0.11
3.18[−2]
2.31[−2]
1.67[−2]
1.46[−2]
1.14[−2]
1.01[−2]
1.00[−2]
Note
a Numbers in square brackets denote powers of 10.
Reference
1. Draine, B.L. 1985, ApJS, 57, 587
Table 5.10. Extinction efficiencies for silicate for λ > 6 µm [1].a
λ (µm)
7.0
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
Q ext /a (cm−1 )
λ (µm)
Q ext /a (cm−1 )
λ (µm)
1.04[3]
3.26[3]
6.75[3]
1.20[4]
1.32[4]
1.20[4]
1.05[4]
8.38[3]
6.80[3]
5.58[3]
13.0
14.0
15.0
16.0
17.0
18.0
19.0
20.0
23.0
25.0
3.74[3]
2.79[3]
3.08[3]
3.70[3]
4.34[3]
4.70[3]
4.62[3]
4.19[3]
3.06[3]
2.57[3]
27.5
30.0
40.0
50.0
80.0
100.0
200.0
500.0
1 000.0
2 000.0
Q ext /a (cm−1 )
2.10[3]
1.75[3]
9.80[2]
6.15[2]
2.26[2]
1.41[2]
3.40[1]
5.38
1.34
3.36[−1]
Note
a Numbers in square brackets denote powers of 10.
Reference
1. Draine, B.L. 1985, ApJS, 57, 587
5.4
5.4.1
PHOTOIONIZATION AND RECOMBINATION
Photoionization Fit Parameters for Ground States
The following parameters are taken from [9] and are used in the formula for the photoionization cross
section:
a E = aT × 10−18 [R(E T /E)s + (1 − R)(E T /E)s+1 ] cm2 ,
E > ET ,
where E T is the threshold energy in eV and aT is the threshold cross section, divided by 1.0 ×
10−18 cm2 . The fitting coefficients R and s are found in Table 5.11.
Sp.-V/AQuan/1999/07/30:17:53
Page 107
5.4 P HOTOIONIZATION AND R ECOMBINATION / 107
Table 5.11. Photoionization cross-section fits [1].
Parent
Resulting ion
ET
aT
R
s
H(2 S)
H+ (1 S)
13.6
24.6
54.4
11.3
24.4
47.9
64.5
14.5
29.6
47.5
77.5
97.9
13.6
16.9
18.6
35.2
54.95
77.4
113.9
138.1
21.6
41.1
42.3
47.99
63.74
68.8
71.5
97.2
126.5
138.1
157.96
6.30
7.83
1.58
12.2
4.60
1.60
0.68
11.4
6.65
2.06
1.08
0.48
2.94
3.85
2.26
7.32
3.65
1.27
0.78
0.36
5.35
4.16
2.71
0.52
1.80
2.50
1.48
3.11
1.40
0.36
0.49
1.34
1.66
1.34
3.32
1.95
2.60
1.00
4.29
2.86
1.63
2.60
1.00
2.66
4.38
4.31
3.84
2.01
0.83
2.60
1.00
3.77
2.72
2.15
2.13
2.28
2.35
2.23
1.96
1.47
1.00
1.15
2.99
2.05
2.99
2.00
3.00
3.00
2.00
2.00
3.00
3.00
3.00
2.00
1.00
1.50
1.50
2.50
3.00
3.00
2.00
2.10
1.00
1.50
1.50
1.50
2.00
2.50
2.50
3.00
3.00
2.10
3.00
He(1 S)
He+ (2 S)
C(3 P)
C+ (2 P)
C2+ (1 S)
C3+ (2 S)
N(4 S)
N+ (3 P)
N2+ (2 P)
N3+ (1 S)
N4+ (2 S)
O(3 P)
O(3 P)
O(3 P)
O+ (4 S)
O2+ (3 P)
O3+ (2 P)
O4+ (1 S)
O5+ (2 S)
Ne(1 S)
Ne+ (2 P)
Ne+ (2 P)
Ne+ (2 P)
Ne2+ (3 P)
Ne2+ (3 P)
Ne2+ (3 P)
Ne3+ (4 S)
Ne4+ (3 P)
Ne5+ (2 S)
Ne5+ (2 P)
He+ (2 S)
He2+ (1 S)
C+ (2 P)
C2+ (1 S)
C3+ (2 S)
C4+ (1 S)
N+ (3 P)
N2+ (2 P)
N3+ (1 S)
N4+ (2 S)
N5+ (1 S)
O+ (4 S)
O+ (2 D)
O+ (2 P)
O2+ (3 P)
O3+ (2 P)
O4+ (1 S)
O5+ (2 S)
O6+ (1 S)
Ne+ (2 P)
Ne2+ (3 P)
Ne2+ (1 D)
Ne2+ (1 S)
Ne3+ (4 S)
Ne3+ (2 D)
Ne3+ (2 P)
Ne4+ (3 P)
Ne5+ (2 P)
Ne6+ (1 S)
Ne6+ (1 S)
Reference
1. Osterbrock, D.E. 1974, Astrophysics of Gaseous Nebulae (Freeman, San Francisco)
5.4.2
Photoionization of Light Hydrogenic Ions
A semiempirical expression [10] for the photoionization cross section per K -shell electron for
hydrogenic light elements is given by
29 π 2 a02 −E 1 4
3 γ (γ − 2)
1+β
1
exp(−4η arccot η)
3
aν =
(ηβγ ) 1 +
ln
1−
,
2
5
2
hν
4 γ +1
1−β
1 − exp(−2π η)
3α Z
2βγ
where η = [−E 1 /(hν + E 1 )]1/2 . E 1 is the negative binding energy of the 1s electron, v is the electron
velocity, and
v
[(hν + E 1 )2 + 2(hν + E 1 )mc2 ]1/2
∼ [2(hν + E 1 )/mc2 ]1/2 ∼ α Z /η,
=
c
hν + E 1 + mc2
γ = (1 − β 2 )−1/2 = 1 + (hν + E 1 )/mc2 ,
α 1/137.036.
β=
Sp.-V/AQuan/1999/07/30:17:53
108 / 5
5.4.3
Page 108
R ADIATION
Radiative Recombination
Given an absorption coefficient aν for a particular level, microscopic reversibility demands that the
recombination cross section into that same level be given by
σ (v) =
gi (hν)2
aν .
gi+1 (mcv)2
This is the Milne relation. Here gi is the statistical weight for the particular level or term i in the
recombined ion, gi+1 is that of the original ion.
Using the Milne relation and the above analytic form for the photoionization cross section, the
recombination rate coefficient can be expressed as [9]
4 gi m 3/2 E T /kT E T3
αi (T ) = √
RE
e
a
/kT
+
(1
−
R)E
/kT
,
(E
(E
)
)
T
T
T
s−2
s−3
m 3 c2
π gi+1 2kT
where En (x) is the exponential integral function. If s is noninteger, the relation En (x) → x n−1 (1 −
n, x) can be used, where (a, x) is the incomplete gamma function.
Recombination into excited states is generally at least as, if not more, important than recombination
into the ground state. If the excited state can be approximated as hydrogenic (often a good
approximation), then the
cumulative recombination coefficient α(n) for principal quantum number
n and higher is α(n) = ∞
n =n αn and is given in Table 5.12 as a function of temperature for the first
four values of n for Z = 1 [9]. For an arbitrary ionic charge Z , α(n; Z , T ) = Z α(n; 1, T /Z 2 ).
Table 5.12. Recombination coefficients α(n) in cm3 s−1 for hydrogen [1].
n
1250 K
2 500 K
5 000 K
10 000 K
20 000 K
1
2
3
4
1.74[−12]
1.28[−12]
1.03[−12]
8.65[−13]
1.10[−12]
7.72[−13]
5.99[−13]
4.86[−13]
6.82[−13]
4.54[−13]
3.37[−13]
2.64[−13]
4.18[−13]
2.60[−13]
1.83[−13]
1.37[−13]
2.51[−13]
1.43[−13]
9.50[−14]
6.83[−14]
Reference
1. Osterbrock, D.E. 1974, Astrophysics of Gaseous Nebulae (Freeman, San
Francisco)
Recombination into the nth hydrogenic level can be written as (taking s = 3 and R = 1)
4 gi m 3/2 E T3
αn = √
aT e E T /kT E1 (E T /kT ).
m 3 c2
π gi+1 2kT
Note that exp(x)E1 (x) ≈ 1/x for x > 5.
The threshold photoionization cross section aT can be obtained from the table of photoionization
cross-section fitting parameters or from the Kramers–Gaunt formula:
ng
8h 3 gn
aT (Kramers & Gaunt) = √
= 7.907 × 10−18 2 cm2 ,
2
2
2
2
Z
3 3π m ce Z
where g is the Gaunt factor [11] given in Table 5.13.
Sp.-V/AQuan/1999/07/30:17:53
Page 109
5.5 X-R AY ATTENUATION / 109
Table 5.13. Bound–free Gaunt factors for the hydrogen atom [1].
Configuration
g at absorption edge
g level average
1s
2s
2p
3s
3p
3d
4s
4p
4d
4f
5
6
7
0.80
0.96
0.88
1.14
1.14
0.73
1.3
1.3
0.80
0.89
0.92
0.94
0.95
0.96
0.97
Reference
1. Gaunt, J.A. 1930, Philos. Trans. 229, 163
5.4.4
Dielectronic Recombination
For dielectric recombination into ion X + , with excited state X +∗ and charge Z , we have the Burgess
formula [12] for the recombination coefficient αd ,
αd = 3.0 × 10−3 T −3/2 f A(x)B(Z ) exp[−χC(Z )/T ] cm3 s−1 ,
where f is the oscillator strength for the transition X + → X +∗ with T in K and
x = 2[E(X +∗ ) − E(x + )]/[(Z + 1)E0 ],
E0 ≡ 27.2 eV,
1/2
2
A(x) = x /(1 + 0.105x + 0.015x ),
x > 0.05,
1/2
B(Z ) = Z (Z + 1)5 /(Z 2 + 13.4)
,
Z ≤ 20,
xC(Z )/T <
C(Z ) = 1.58 × 105 (Z + 1)/ 1 + 0.015Z 3 /(Z + 1)2 ,
∼ 5.0.
5.5
X-RAY ATTENUATION
The smoothed fits in Table 5.14 provide an approximate representation (∼ 10% or better) to both the
Henke experimental data [13] and to relativistic calculations [14] for the photoionization cross section
σ (1 barn = 10−24 cm2 ). The photon energy E is measured in keV, and
log10 [σ (barn/atom)] =
n
m=0
am (log10 E)m .
Sp.-V/AQuan/1999/07/30:17:53
110 / 5
Page 110
R ADIATION
Table 5.14. Cold material X-ray attenuation (total cross section) fits.a
H
He
C
N
O
Na
Mg
Al
Si
S
Ar
E range (keV)
n
a0
a1
0.1089−8.0470
8.0470−44.77
0.1086−8.0470
0.0305−0.2885
0.2888−30.000
0.0105−0.4027
0.4031−32.20
0.0305−0.5374
0.5380−30.320
0.0415−0.0708
0.0724−1.079
1.0800−9.886
0.0305−0.0574
0.0576−0.0824
0.0824−1.3113
1.3130−30.921
0.0305−0.0813
0.0815−0.1144
0.1144−1.5680
1.5690−30.000
0.0305−0.1087
0.1089−0.1300
0.1303−1.8470
1.849−30.800
0.0394−0.1740
0.1742−0.1932
0.1996−2.4772
2.479−31.970
0.0504−0.1085
0.1085−0.2497
0.2653−3.2002
3.2033−32.200
5
2
5
3
4
4
4
4
5
5
5
3
3
4
5
4
3
4
5
4
4
4
4
4
4
3
4
4
4
3
4
3
1.000 266
−0.105 7762
2.596 995
3.113 512
4.641 890
3.653 780
4.880 570
3.851 303
5.087 381
2.522 877[2]
4.372 625
5.551 365
1.819 299
0.759 7808
4.556 145
5.664 605
1.516 313
4.047 249[2]
4.713 295
5.732 064
4.653 385
1.704 974[3]
4.842 629
5.818 893
4.862 989
−3.287 565[2]
5.105 629
5.957 286
0.103 8804
3.766 572
5.325 805
6.236 304
−2.874 876
−8.314 851[−2]
−3.278 818
−3.093 789
−2.929 378
−2.036 035
−2.726 891
−2.158 767
−2.624 299
7.578 575[2]
−2.803 046
−2.508 414
−3.725 268
2.040 594[2]
−2.709 325
−2.340 560
−5.356 061
1.227 899[3]
−2.629 794
−2.167 852
1.585 146
5.668 094[3]
−2.691 588
−2.084 606
1.407 408
−9.238 824[2]
−2.701 358
−1.944 676
0.158 0149
−3.816 305
−2.604 915
−2.427 534
a2
1.337 216
−5.823 671[−2]
−0.505 4382
−0.391 9827
0.592 8674
−0.317 1900
0.745 1610
−0.341 0862
8.701 549[2]
−0.783 3634
−0.295 8825
−0.968 6183
2.000 915[2]
−0.624 9299
−0.642 0745
−1.787 405
1.261 952[3]
−0.421 7032
−0.759 3566
4.064 835
6.305 044[3]
0.127 3064
−0.807 0467
4.580 753
−6.359 921[2]
−0.2269 985
−0.839 6531
0.180 2517
−1.506 623
−0.411 9090
−0.185 3513
a3
a4
0.690 5931
−0.273 4547
0.637 2270
0.344 4271
0.364 0990
0.429 1417
6.076 411[−2]
0.548 5890
−0.197 2559
4.394 846[2]
−1.137 645
65.192 85
−1.125 062
0.235 7794
4.319 780[2]
−1.136 888
0.254 8878
1.700 534
2.337 491[3]
0.821 4728
0.261 1724
2.192 223
0.244 2756
82.195 58
−0.794 4890
−0.961 1755
−1.261 846
0.697 1264
0.246 5677
6.544 871
0.483 1536
Note
a Numbers in square brackets denote powers of 10.
See Sec. 5.4 for a semiempirical equation for hydrogen and hydrogenic ion cross sections.
5.6
ABSORPTION OF MATERIAL OF STELLAR INTERIORS
The opacity of stellar interiors is usually expressed as the Rosseland mean of the mass absorption
coefficient κ̄. Tabulations are available [15, 16] for a wide range of compositions expressed by X , Y ,
and Z . Tables 5.15–5.17 give log10 κ̄ in cm2 g−1 as a function of log10 ρ where density ρ is in g cm−3
and temperature T in units of 10−6 K. These tables are based on interpolations of data in [15].
Sp.-V/AQuan/1999/07/30:17:53
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5.6 A BSORPTION OF M ATERIAL OF S TELLAR I NTERIORS / 111
Table 5.15. Hydrogen Rosseland mean opacity log10 κ̄ [1].
Ta
log ρ = −10.0
−9.0
−8.0
−7.0
−6.0
0.006
0.007
0.008
0.009
0.010
0.011
0.012
0.014
0.016
0.018
0.020
0.025
0.030
0.035
0.040
0.045
0.050
0.055
0.060
0.070
0.080
0.090
0.100
0.120
0.150
0.200
0.250
0.300
0.400
0.500
0.600
0.800
1.000
1.200
1.500
2.000
2.500
3.000
4.000
5.000
6.000
8.000
10.00
15.00
20.00
30.00
40.00
60.00
80.00
100.0
−1.67
−0.66
0.18
0.71
0.85
0.76
0.60
0.31
0.12
−0.01
−0.08
−0.15
−0.17
−0.21
−0.27
−0.32
−0.35
−0.37
−0.38
−0.40
−0.40
−0.40
−0.40
−1.41
−0.49
0.33
1.01
1.42
1.56
1.50
1.19
0.89
0.68
0.52
0.30
0.21
0.15
0.07
−0.02
−0.11
−0.18
−0.24
−0.31
−0.35
−0.37
−0.38
−0.39
−0.40
−0.40
−1.00
−0.19
0.54
1.20
1.71
2.05
2.21
2.16
1.91
1.68
1.48
1.17
1.00
0.89
0.78
0.65
0.50
0.35
0.21
0.00
−0.14
−0.22
−0.28
−0.34
−0.37
−0.39
−0.40
−0.40
−0.40
−0.61
0.16
0.82
1.40
1.89
2.27
2.57
2.86
2.86
2.73
2.56
2.25
2.06
1.92
1.78
1.62
1.43
1.21
1.01
0.66
0.39
0.19
0.05
−0.13
−0.26
−0.35
−0.38
−0.39
−0.40
−0.40
−0.40
−0.40
−0.40
−0.13
0.57
1.15
1.65
2.09
2.46
2.77
3.22
3.48
3.57
3.55
3.37
3.19
3.04
2.88
2.67
2.44
2.20
1.97
1.56
1.22
0.94
0.71
0.37
0.07
−0.17
−0.28
−0.33
−0.37
−0.39
−0.39
−0.40
−0.40
−0.40
−0.40
−0.40
−0.40
−0.41
−0.41
−0.41
−0.41
−0.41
−0.40
−0.40
−0.41
−0.41
−0.41
−0.42
−0.42
−0.38
−0.39
−0.40
−0.41
−0.41
−0.42
−0.42
−0.43
−0.44
−0.28
−0.33
−0.36
−0.38
−0.40
−0.41
−0.42
−0.43
−0.44
−0.46
−0.47
−0.49
−5.0
2.36
2.68
2.98
3.47
3.82
4.07
4.23
4.35
4.29
4.15
3.96
3.72
3.46
3.20
2.96
2.53
2.17
1.86
1.60
1.17
0.72
0.26
0.02
−0.13
−0.27
−0.33
−0.36
−0.39
−0.39
−0.40
−0.40
−0.40
−0.40
−0.40
0.00
−0.14
−0.22
−0.31
−0.35
−0.39
−0.41
−0.43
−0.44
−0.46
−0.47
−0.49
Note
a Units of 106 K.
Reference
1. Iglesias, C.A., Rogers, F.J., & Wilson, B.G. 1992, ApJ, 397, 717
−4.0
−3.0
−2.0
−1.0
0.0
1.45
1.10
0.62
0.32
0.13
−0.10
−0.21
−0.27
−0.33
−0.37
−0.38
−0.39
1.07
0.79
0.40
0.17
0.02
−0.13
−0.25
−0.31
−0.34
0.85
0.60
0.34
0.07
−0.08
−0.17
1.06
0.68
0.41
0.23
0.49
0.25
−0.07
−0.22
−0.35
−0.40
−0.44
−0.47
−0.49
0.43
0.14
−0.16
−0.29
−0.41
−0.45
−0.48
−0.01
−0.29
−0.42
−0.48
−0.17
−0.35
4.96
5.10
5.07
4.90
2.56
2.11
1.60
1.01
0.61
0.34
0.03
−0.14
−0.22
−0.32
−0.35
−0.37
−0.39
−0.40
−0.40
−0.40
0.60
0.34
0.16
−0.05
−0.17
−0.31
−0.37
−0.41
−0.43
−0.46
−0.47
−0.49
Sp.-V/AQuan/1999/07/30:17:53
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Page 112
R ADIATION
Table 5.16. Helium Rosseland mean opacity log10 κ̄ [1].
Ta
log ρ = −10.0
−9.0
−8.0
0.006
0.007
0.008
0.009
0.010
0.011
0.012
0.014
0.016
0.018
0.020
0.025
0.030
0.035
0.040
0.045
0.050
0.055
0.060
0.070
0.080
0.090
0.100
0.120
0.150
0.200
0.250
0.300
0.400
0.500
0.600
0.800
1.000
1.200
1.500
2.000
2.500
3.000
4.000
5.000
6.000
8.000
10.00
15.00
20.00
30.00
40.00
60.00
80.00
100.0
−5.79
−4.95
−4.13
−3.36
−2.71
−2.04
−1.49
−0.82
−0.74
−0.83
−0.88
−0.93
−0.73
−0.53
−0.60
−0.64
−0.66
−0.68
−0.67
−0.67
−0.67
−0.67
−0.68
−5.81
−5.14
−4.43
−3.73
−2.99
−2.31
−1.67
−0.72
−0.33
−0.37
−0.51
−0.72
−0.64
−0.28
−0.20
−0.34
−0.45
−0.52
−0.57
−0.61
−0.62
−0.62
−0.62
−0.65
−0.68
−0.69
−5.71
−5.13
−4.52
−3.94
−3.24
−2.43
−1.76
−0.68
−0.01
0.24
0.19
−0.17
−0.27
0.02
0.39
0.43
0.17
−0.03
−0.16
−0.33
−0.42
−0.46
−0.47
−0.49
−0.56
−0.66
−0.69
−0.69
−0.70
−0.70
−0.70
−0.70
−0.70
−0.71
−0.71
−0.70
−0.70
−0.70
−0.71
−0.71
−0.71
−0.72
−7.0
−3.15
−2.41
−1.73
−0.65
0.13
0.62
0.86
0.72
0.46
0.52
0.85
1.17
1.19
0.95
0.69
0.35
0.15
0.02
−0.06
−0.15
−0.23
−0.45
−0.59
−0.65
−0.69
−0.69
−0.70
−0.70
−0.70
−0.66
−0.68
−0.69
−0.70
−0.71
−0.71
−0.72
−0.73
−0.74
−6.0
1.49
1.36
1.24
1.34
1.63
1.90
1.98
1.83
1.40
1.11
0.93
0.80
0.63
0.47
0.12
−0.21
−0.42
−0.60
−0.66
−0.68
−0.69
−0.70
−0.70
−0.70
−0.70
−0.53
−0.60
−0.64
−0.68
−0.69
−0.71
−0.72
−0.73
−0.74
−0.76
−0.77
−0.79
−5.0
−4.0
2.39
2.57
2.69
2.57
2.26
2.03
1.89
1.68
1.48
1.03
0.52
0.13
−0.30
−0.49
−0.58
−0.66
−0.68
−0.69
−0.70
−0.70
−0.70
−0.70
−0.19
−0.36
−0.46
−0.57
−0.63
−0.69
−0.71
−0.72
−0.74
−0.75
−0.77
−0.79
3.04
2.83
2.58
2.04
1.45
0.97
0.32
−0.06
−0.28
−0.50
−0.60
−0.64
−0.67
−0.69
−0.70
−0.70
0.43
0.15
−0.05
−0.29
−0.43
−0.59
−0.66
−0.71
−0.73
−0.75
−0.77
−0.79
Note
a Units of 106 K.
Reference
1. Iglesias, C.A., Rogers, F.J., & Wilson, B.G. 1992, ApJ, 397, 717
−3.0
−2.0
−1.0
0.0
2.41
1.91
1.18
0.68
0.32
−0.10
−0.33
−0.46
−0.57
−0.64
−0.67
−0.69
1.58
1.17
0.60
0.22
−0.03
−0.27
−0.47
−0.56
−0.62
1.02
0.68
0.32
−0.05
−0.26
−0.40
0.65
0.34
0.09
0.29
0.04
−0.31
−0.48
−0.63
−0.69
−0.74
−0.77
−0.79
0.24
−0.07
−0.39
−0.54
−0.68
−0.74
−0.77
−0.22
−0.51
−0.64
−0.72
−0.40
−0.57
Sp.-V/AQuan/1999/07/30:17:53
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5.6 A BSORPTION OF M ATERIAL OF S TELLAR I NTERIORS / 113
Table 5.17. Solar composition (X = 0.73, Z = 0.018) Rosseland mean opacity log10 κ̄ [1].
Ta
log ρ = −10.0
−9.0
−8.0
−7.0
−6.0
0.006
0.007
0.008
0.009
0.010
0.011
0.012
0.014
0.016
0.018
0.020
0.025
0.030
0.035
0.040
0.045
0.050
0.055
0.060
0.070
0.080
0.090
0.100
0.120
0.150
0.200
0.250
0.300
0.400
0.500
0.600
0.800
1.000
1.200
1.500
2.000
2.500
3.000
4.000
5.000
6.000
8.000
10.00
15.00
20.00
30.00
40.00
60.00
80.00
100.0
−1.77
−0.78
0.05
0.55
0.67
0.61
0.47
0.24
0.13
0.04
−0.04
−0.13
−0.12
−0.14
−0.21
−0.26
−0.31
−0.34
−0.35
−0.36
−0.37
−0.36
−0.33
−1.53
−0.62
0.20
0.88
1.27
1.41
1.35
1.06
0.82
0.66
0.54
0.33
0.25
0.25
0.20
0.09
0.00
−0.09
−0.15
−0.22
−0.25
−0.27
−0.26
−0.19
−0.04
−0.20
−1.10
−0.32
0.42
1.08
1.60
1.94
2.08
2.00
1.76
1.56
1.39
1.14
0.97
0.92
0.90
0.81
0.66
0.50
0.38
0.18
0.06
0.00
−0.04
−0.05
0.09
−0.01
−0.24
−0.38
−0.46
−0.62
0.07
0.72
1.30
1.80
2.20
2.48
2.74
2.71
2.56
2.40
2.15
1.97
1.85
1.80
1.74
1.62
1.43
1.25
0.95
0.74
0.60
0.50
0.38
0.38
0.33
0.02
−0.19
−0.40
−0.44
−0.45
−0.44
−0.45
0.00
0.52
1.08
1.59
2.03
2.41
2.71
3.14
3.36
3.42
3.37
3.20
3.06
2.94
2.83
2.73
2.62
2.47
2.29
1.96
1.70
1.51
1.36
1.18
0.99
0.84
0.53
0.24
−0.18
−0.37
−0.40
−0.44
−0.44
−0.44
−0.44
−0.45
−0.46
−0.46
−0.47
−0.47
−0.47
−0.47
−0.43
−0.45
−0.46
−0.47
−0.47
−0.48
−0.48
−0.26
−0.38
−0.43
−0.46
−0.46
−0.47
−0.48
−0.49
−0.50
0.29
−0.03
−0.22
−0.38
−0.43
−0.46
−0.48
−0.49
−0.50
−0.52
−0.54
−0.56
−5.0
2.33
2.65
2.95
3.40
3.71
3.94
4.08
4.16
4.11
4.03
3.92
3.78
3.63
3.47
3.32
3.02
2.76
2.56
2.39
2.15
1.90
1.56
1.25
0.97
0.48
0.05
−0.20
−0.34
−0.38
−0.39
−0.40
−0.45
−0.46
−0.47
0.82
0.50
0.23
−0.11
−0.27
−0.39
−0.44
−0.49
−0.50
−0.52
−0.54
−0.56
Note
a Units of 106 K.
Reference
1. Iglesias, C.A., Rogers, F.J., & Wilson, B.G. 1992, ApJ, 397, 717
−4.0
−3.0
−2.0
−1.0
0.0
3.66
3.12
2.74
2.45
2.03
1.76
1.49
0.93
0.52
0.27
0.05
−0.11
−0.25
−0.36
2.52
2.26
2.06
1.74
1.43
1.15
0.83
0.48
0.26
0.05
2.23
2.03
1.87
1.65
1.29
0.98
0.73
2.10
1.78
1.50
1.24
0.72
0.48
0.11
−0.09
−0.32
−0.42
−0.50
−0.53
−0.55
0.53
0.22
−0.11
−0.28
−0.44
−0.50
−0.54
−0.30
−0.44
−0.51
−0.18
−0.37
4.80
4.93
4.95
4.88
3.93
3.75
3.58
3.44
3.17
2.85
2.42
2.05
1.77
1.34
0.94
0.53
0.05
−0.15
−0.25
−0.29
−0.35
−0.42
−0.44
1.19
0.88
0.64
0.29
0.06
−0.21
−0.32
−0.44
−0.49
−0.52
−0.54
−0.56
Sp.-V/AQuan/1999/07/30:17:53
114 / 5
Page 114
R ADIATION
The opacity due to electron scattering alone for a completely ionized plasma, with hydrogen mass
fraction X , is given by κ̄e = 0.200(1 + X ) [17].
5.7
ABSORPTION OF MATERIAL OF THE SOLAR PHOTOSPHERE
The Rosseland mean opacity for the solar photosphere including diatomic species is given by
Table 5.18, as log10 κ̄ with κ̄ in cm2 g−1 . The assumed microturbulent velocity is 2 km/s.
Table 5.18. Solar photospheric Rosseland mean opacity log10 κ̄ [1].
T (10−6 K)
log ρ = −10.0
−9.0
−8.0
−7.0
−6.0
−5.0
−4.0
0.0021
0.0030
0.0040
0.0050
0.0062
0.0071
0.0081
0.0093
0.0100
0.0126
0.0158
0.0200
0.0316
0.0398
0.0501
0.0631
0.0708
0.0794
0.0891
0.1000
−5.31
−4.12
−3.42
−2.83
−1.60
−0.70
0.15
0.61
0.66
0.35
0.07
−0.08
−0.17
−0.23
−0.32
−0.36
−0.35
−0.36
−0.37
−0.37
−5.27
−3.66
−2.72
−2.42
−1.34
−0.53
0.33
1.05
1.27
1.24
0.78
0.49
0.20
0.16
−0.03
−0.21
−0.26
−0.28
−0.30
−0.31
−5.16
−3.04
−2.00
−1.74
−0.92
−0.21
0.54
1.29
1.61
2.05
1.73
1.34
0.87
0.83
0.62
0.26
0.11
0.01
−0.06
−0.11
−4.95
−2.49
−1.38
−0.92
−0.43
0.19
0.83
1.48
1.79
2.56
2.64
2.35
1.83
1.71
1.54
1.11
0.87
0.66
0.50
0.38
−4.61
−2.14
−0.84
−0.14
0.17
0.65
1.20
1.76
2.03
2.81
3.28
3.29
2.92
2.71
2.50
2.09
1.84
1.61
1.39
1.21
−4.03
−1.94
−0.38
0.47
0.85
1.18
1.63
2.11
2.33
3.04
3.63
4.01
3.98
3.80
−3.12
−1.72
−0.11
0.86
1.43
1.72
2.10
Reference
1. Kurucz, R.L. 1992, Rev. Mexicana Astron. Af., 23, 181
5.8
SOLAR PHOTOIONIZATION RATES
The solar photoionization rates are calculated as [18]
∞
R=
σν (ν)Fν dν,
ν0
where σν (ν) is the photoionization cross section having threshold ν0 and Fν is the solar flux.
The unattenuated photoionization rate for the quiet and active Sun for some monatomic species, at
heliocentric distance 1 AU, is shown in Table 5.19.
Table 5.19. Solar photoionization rate R in s−1 [1].a
Species
H−
H
He
Quiet Sun
14.
7.3[−8]
5.2[−8]
Active Sun
Species
Quiet Sun
Active Sun
14.
1.9[−7]
2.2[−7]
O (1 S)
2.0[−7]
2.1[−7]
1.6[−5]
7.5[−7]
9.5[−7]
1.7[−5]
F
Na (expt.)
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5.9 F REE –F REE A BSORPTION AND E MISSION / 115
Table 5.19. (Continued.)
Species
Quiet Sun
Active Sun
C (3 P)
4.1[−7]
3.6[−6]
4.3[−6]
1.9[−7]
2.1[−7]
1.8[−7]
1.0[−6]
1.0[−5]
1.2[−5]
6.3[−7]
8.5[−7]
7.4[−7]
C (1 D)
C (1 S)
N
O (3 P)
O (1 D)
Species
Na (theor.)
S(3 P)
S(1 D)
S(1 S)
Cl
K
Quiet Sun
Active Sun
5.9[−6]
1.1[−6]
1.1[−6]
1.0[−6]
5.7[−7]
2.2[−5]
6.6[−6]
2.6[−6]
2.6[−6]
2.5[−6]
1.5[−6]
2.3[−5]
Note
a Numbers in square brackets denote powers of 10.
Reference
1. Huebner, W.F., Keady, J.J., & Lyon, S.P. 1992, AP&SS, 195, 1
5.9
FREE–FREE ABSORPTION AND EMISSION
The free–free linear absorption coefficient [19, 20] is
g
4π Z 2 e6
κs = √
· 3 Ne Ni (κ in exp cm−1 )
2
3 3 hcm v ν
= 1.802 × 1014 (Z 2 g/ν 3 υ)Ne Ni (υ in cm/s)
= 6.686 × 10−18 Z 2 gλ3 Ne Ni /υ
(λ in cm),
where υ is the electron velocity, g is the Gaunt factor representing the departure from Kramers’s theory,
Z is the ionic charge, and Ne and Ni are the electronic and ionic densities in cm−3 . The mean 1/υ is
(2m/π kT )1/2 , whence
κs = 3.692 × 108 Z 2 gT −1/2 ν −3 Ne Ni
= 1.370 × 10−23 Z 2 λ3 g Ne Ni /T 1/2
(λ in cm).
The effective linear absorption coefficient κ after allowance for stimulated emission is
κ = 3.692 × 108 [1 − exp(−hν/kT )]Z 2 gT −1/2 ν −3 Ne Ni .
For small hν/kT (= 1.438/λT ), e.g., for radio waves,
κ =
e6
Z 2g
8 π 1/2
N e Ni
3/2
3 6
c(mkT )
ν2
(κ in exp cm−1 )
= 0.017 8Z 2 gν −2 T −3/2 Ne Ni
= 1.98 × 10−23 Z 2 gλ2 Ne Ni T −3/2
(λ in cm).
The Gaunt factor for radio waves is [19, 21]
g = 10.6 + 1.90 log10 T − 1.26 log10 Z ν.
Gaunt-factor calculations incorporating relativistic effects and electron degeneracy can be found
in [22, 23]. The temperature parameter γ 2 is defined by
γ 2 = Z 2j Ry/kT,
= Z 2j
Zj = 1
1.579 × 105 K
T
(for H),
(T in K).
Zj = 2
(for He),
Sp.-V/AQuan/1999/07/30:17:53
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R ADIATION
For −3 ≤ log10 γ 2 ≤ 2.0 we list in Table 5.20 approximate Gaunt factors for both hydrogen and
helium, where η (defined as the chemical potential divided by kT ) is the degeneracy parameter and
u ≡ hν/kT .
Table 5.20. Relativistic thermally averaged free–free Gaunt factors.
log u
η = −6.0
−2.0
0.0
1.0
2.0
3.0
−4.0
−3.5
−3.0
−2.0
−1.4
−1.0
−0.7
−0.2
0.0
5.5
4.89
4.26
3.03
2.37
1.95
1.69
1.30
1.17
5.13
4.55
3.97
2.84
2.22
1.85
1.59
1.25
1.13
3.77
3.35
2.95
2.14
1.69
1.42
1.25
1.03
0.96
2.77
2.48
2.18
1.61
1.28
1.10
0.96
0.83
0.80
1.94
1.74
1.54
1.15
0.93
0.80
0.72
0.63
0.62
1.36
1.22
1.09
0.82
0.67
0.58
0.52
0.47
0.46
These smoothed numbers are least accurate (∼ 10–20 %) for large u and small log10 γ 2 (≈ −3),
improving to a few percent for smaller u and larger log10 γ 2 . For larger u, Gaunt factors for hydrogen
and helium can differ significantly [22, 23].
When degeneracy is not important (η ≤ −4, say),
√

 3 ln 4 , = 1.781 for u 1,
π
u
g∼
=

 −0.4
u
for u ≈ 1.
The free–free emission (bremsstrahlung) per unit solid angle, volume, time, and frequency range is
jν = κν Bν (black body)
16 π 1/2 e6 Z 2 m 1/2
hν
=
N e Ni
g exp −
3 6
kT
c3 m 2 kT
c 2
= 5.444 × 10−39 Z 2 g exp −
T −1/2 Ne Ni erg cm−3 s−1 sr−1 Hz−1
λT
(T in K, N in cm−3 ).
The free–free emission from a cosmic plasma is
c 2
jν = 6.2 × 10−39 g exp −
T −1/2 Ne2 d V erg s−1 sr−1 Hz−1 (T in K, Ne in cm−3 ),
λT
2
where Ne d V (integrated over volume) is called the emission measure.
The integrated free–free emission is
64π π 1/2 e6 Z 2 kT 1/2
4π
jν dν =
g N e Ni
3
6
hc3 m m
= 1.426 × 10−27 Z 2 T 1/2 g Ne Ni erg cm−3 s−1 .
For a completely ionized plasma with solar abundance [24], i Ni Ne Z i2 ∼
= 1.4Ne2 , and thus
4π
jν dν = 2.0 × 10−27 gT 1/2 Ne2 d V erg s−1 .
Sp.-V/AQuan/1999/07/30:17:53
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5.11 V ISUAL P HOTOMETRY / 117
Free–free absorption from neutral atoms: For highly polarizable target atoms, an approximation for
the thermally averaged free–free absorption coefficient valid for long wavelength and low temperature
is [25]
k = 1.62 × 10−19 Ne Na α 1/2 λ3 T (1 − e−hc/λkT ) cm−1 ,
where Ne (Na ) are the number of free electrons (neutral atoms) per cm3 , α is the polarizability (cm3 ),
λ is the wavelength (cm), and T is the temperature (K).
The validity criterion is 0.633/α < ε < 2/α 1/3 , where the free-electron kinetic energy ε is
measured in Rydbergs and α is in units of a03 .
5.10
REFLECTION FROM METALLIC MIRRORS
In Table 5.21, no attempt has been made to differentiate between different methods of deposition [26].
Table 5.21. Reflection from metallic mirrors.
λ
(µm)
Ag
(%)
Al
(%)
0.20
0.22
0.24
0.26
0.28
20
25
27
27
23
72
78
81
82
82
0.30
0.32
0.34
0.36
0.38
12
7
63
77
82
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.80
1.00
2.0
5.0
10.0
Speculum
(%)
Hg
(%)
Ni
(%)
Cu
(%)
Au
(%)
Si
(%)
Pt
(%)
Steel
(%)
W
(%)
26
33
38
58
61
35
40
42
40
39
34
34
31
29
28
18
27
32
34
34
68
68
68
68
67
20
29
35
37
38
24
27
30
33
36
15
16
18
20
21
82
82
83
83
84
44
48
51
54
56
64
67
69
71
73
39
41
43
45
47
29
30
32
34
36
35
33
33
33
34
65
61
56
50
41
39
40
42
43
45
39
41
44
46
49
23
25
27
30
34
85
90
91
92
93
94
95
85
86
87
88
89
88
87
58
61
63
65
66
67
68
74
74
73
73
74
74
75
50
57
61
63
65
67
69
38
42
47
60
74
82
85
34
37
51
77
84
89
93
35
30
30
30
30
30
30
48
56
59
60
61
63
66
51
55
57
57
56
55
56
38
45
49
52
51
52
53
97
98
98
99
99
85
93
96
97
98
70
72
82
89
92
70
73
82
89
92
70
73
84
92
96
89
92
96
98
99
95
97
98
99
99
29
28
28
28
28
70
74
81
91
95
59
63
77
90
93
56
60
87
95
98
Reflections in the EUV [27, 28] are strongly dependent on the details of deposition, the age of the
surface, and the reflection angle. No summary can be made.
5.11
VISUAL PHOTOMETRY
Units of visual photometry are given in Chapters 2 and 15. For values of the relative visibility factor K λ
for normal brightness (about 5×10−4 stilb or greater), the photopic curve (international) (cone vision
Sp.-V/AQuan/1999/07/30:17:53
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118 / 5
R ADIATION
at fovea) is given in Table 5.22 [29]. This and the following table are actually one-dimensional tables
so that the last column and first row entry of the first table applies to a wavelength of 3 900 Å.
Table 5.22. Relative visibility factors.
λ (Å)
0
100
200
300
400
500
600
700
3000
4000
5000
6000
7000
0.000 4
0.323
0.631
0.004 1
0.001 2
0.503
0.503
0.002 1
0.004 0
0.710
0.381
0.001 05
0.011 6
0.862
0.265
0.000 52
0.023
0.954
0.175
0.000 25
0.038
0.995
0.107
0.000 12
0.060
0.995
0.061
0.000 06
0.091
0.952
0.032
0.000 03
800
900
0.000 04
0.139
0.870
0.017
0.000 12
0.208
0.757
0.008 2
The equivalent width of the K λ curve is K λ dλ = 1 068 Å.
The mechanical equivalent of light (experimental) [29] is
K λ lumens ≡ 0.001 47 W.
The luminous energy (in lumergs) is 680 K λ eλ dλ, where eλ dλ is the element of energy in joules.
1 lumen (5 550 Å radiation) = 4.11 × 1015 photons s−1 ,
1 nanolambert (5 550 Å radiation) = 1.31 × 106 photons s−1 cm−2 sr−1 .
The number of lumens L entering a telescope of diameter D in cm for a star of visual apparent
magnitude V near the zenith (clear conditions) is
log10 L = 2 log10 D − 0.4V − 9.86.
For relative visibility for dark-adapted eyes (about 10−7 stilb or less), the scotopic curve (rod
vision) is given in Table 5.23.
Table 5.23. Relative visibility factors.
λ (Å)
0
100
200
300
400
500
600
700
800
900
4000
5000
6000
7000
0.018 5
0.900
0.049 0
0.000 11
0.040
0.985
0.030 0
0.076
0.960
0.017 5
0.132
0.840
0.010 0
0.212
0.680
0.005 8
0.302
0.500
0.003 2
0.406
0.350
0.001 7
0.520
0.228
0.000 87
0.650
0.140
0.000 44
0.770
0.083
0.000 21
For dark-adapted eyes,
1 lumen at 5 100 Å (scotopic) ≡ 0.000 58 W.
The quantum thresholds for a single scintillation with most favorable conditions for human eye are
4 quanta in 0.15 s (absorbed) and 60 quanta in 0.15 s (incident).
The threshold intensity for a large steady source [30] is 1.4 × 10−10 stilb.
The size of the retinal image for 1 arc is 4.9 µm.
The eye resolving power 1 5 µm at fovea.
Sp.-V/AQuan/1999/07/30:17:53
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5.11 V ISUAL P HOTOMETRY / 119
Density of rods and cones in the retina [29]:
Rods 30 × 106 rods/sr = 2.7 rods/(minutes of arc)2 ,
Cones 1.2 × 106 cones/sr = 0.1 cones/(minutes of arc)2 .
The density of cones in the fovea 50 × 106 cones/sr.
The equivalent diameter of the fovea region containing no rods [31] is 1◦ 40 .
The diameters of individual cones are 2 µm ≡ 25 (variable).
The diameters of individual rods are 1 µm ≡ 12 .
The approximate brightness of common objects is given in Table 5.24 [32].
Table 5.24. Object and brightness (stilb).
Candle
Acetylene (Kodak burner)
Welsbach (high-pressure) mantle
Tungsten lamp filament
Sodium vapor lamp
Mercury vapor lamp (high pressure)
Arc crater (plain carbon)
Clear blue sky
Overcast sky
Zenith Sun
0.6
10.8
25
800
70
150
16 000
0.2–0.6
0.3–0.7
165 000
Table 5.25 gives approximate albedos for common objects.
Table 5.25. Approximate albedos [1, 2].
White cartridge paper
Magnesium oxide (or carbonate)
Black cloth
Black velvet
0.80
0.98
0.012
0.004
References
1. Walsh, J.W.T. Photometry, 3rd ed. (Dover,
New York), p. 529
2. Houston, R.A. 1924, Treatise on Light
(Longmans, London)
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7. Penndorf, R. 1957, J. Opt. Soc. Am., 47, 176
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10. Huebner, W.F. 1986, Physics of the Sun, Vol. I, edited
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11. Gaunt, J.A. 1930, Philos. Trans., 229, 163
12. Burgess, A. 1965, ApJ, 141, 1588
13. Henke, B.L. et al. 1982, At. Data Nucl. Data Tables,
27, 1
14. Saloman, E.B., & Hubbell, J.H. 1986, X-Ray Attenuation Coefficients (Total Cross Sections): Comparison
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19.
20.
21.
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