Curve of Growth
The curve of growth describes how the line strength increases as the optical
depth increases. An appropriate measure of line strength is the equivalent
width: the area of a rectangle with the same area as the line pro¤le and height
equal to the continuum.
We begin with a purely absorbing, simple slab model with radiation incident
on one side. Then from the solution to the transfer equation, we have the
intensity emerging:
Iº = I0 e¡¿ n
where the optical depth is
¿n =
Z
nx ·0 Áv d` ' nx L· 0Áv
where nx is the density of atoms of element x, L is the thickness of the slab,
· 0 is the absorption coe¢cent
per atom at line center, and Áº is the line pro¤le
R
function. The factor nx d`R´ N x ' nxL is called the columnR density The
area under the curve Iº (º ) is Iº dº and the area of the line is (I0 ¡ Iº ) dº
The equivalent width of the line is de¤ned to be
¶
Z µ
Z 1
¡
¢
Iº
W =
1¡
dº =
1 ¡ e¡N x ·0Áv dº
I0
0
Of course this is an idealization. In practice the limits are frequencies far
enough from the line center that we are clearly in the continuum.
In the classical oscillator model,
´
¼e2 ³
·0 =
f 1 ¡ e¡hº 0=kT
mc
where the oscillator strength f takes quantum e¤ects into account and º 0 is the
frequency at line center.
The line pro¤le function depends on the broadening mechanisms in e¤ect.
1. We always have Doppler broadening due to the velocity distribution of
the atoms. In LTE,
" µ
¶2 #
1
º ¡ º0
Áº; do p ple r = p
exp ¡
¼¢º D
¢º D
1
and the line breadth factor is
·
¸1=2
º 0 2kT
¢º D =
+ (turbulent velocity)2
c
M
2. Natural broadening is due to radiation reaction (my notes, R&L pg 287,
Shore pg 197-8)
°
Áº =
2
4¼ (º ¡ º 0) 2 + (°=2)2
This is sometimes called the Lorentz pro¤le. We can include the e¤ect of
collisions by adding a term °C to the natural broadening damping factor
°:
0.1
Weak lines
N x ·0 Á0 ¿ 1
Then
W
Z
1
¢
1 ¡ e¡N x ·0Á v dº '
0
Z 1
= N x ·0
Áv dº = N x· 0
=
¡
Z
1
0
N x· 0 Áv dº
0
The result is independent of the shape of Áº and thus of the type of broadening. The equivalent width is linearly proportional to the column density.
0.2
Strong lines
¿ 0 = N x· 0 Á 0 À 1
Notice that N x· 0 Áº cannot be À 1 everywhere, since Áº ! 0 far from the line
center. Let the frequency where ¿ º ' 1 be º 0 § ¢: Then
W '
Z
º 0¡¢
0
¡
¢
1 ¡ e¡N x ·0Áv dº+
Z
º 0+¢
º 0¡¢
¡
¢
1 ¡ e¡N x ·0Áv dº+
Z
1
º 0+¢
¡
¢
1 ¡ e¡N x ·0Á v dº
In the middle factor, e¡ ¿ º is ¿ 1: The two other terms are each of order
Z 1
N x ·0
Áv dº ¿ Nx · 0
º 0+¢
and we neglect them. Thus
W ' 2¢
2
0.2.1
Moderately strong lines
In these lines N x ·0 Áº becomes <1 while still in the Doppler core. So
N x ·0 p
1
¼¢º D
Nx · 0 Áv (º 0 + ¢) =
" µ
¶ 2#
¢
exp ¡
=
¢º D
¢
=
1
1
¢º D
s
ln
µ
N x ·0
p
¼¢º D
¶
where the square root is . 1 for self consistency, and it follows that
s µ
¶
Nx · 0
W ' 2¢º D ln p
¼¢º D
Thus the equivalent width increases logarithmically with N:
Check that the terms we neglected are small:
W n eg lec te d
¿
Wk ep t
N x· 0
e
r ³
´ . 2 »1
x ·0
2¢º D ln pN¼¢º
D
So the approximations are consistent.
0.2.2
Very strong lines
Here N x· 0 Áº does not become <1 until the wings of the line. Thus the transition occurs at ¢ where
N x· 0 °
2 = 1
2
4¼ ¢2 + (°=2)
Uusally ° ¿ 2; so we may approximate:
p
N x· 0 °
¢=
2¼
and then
W '
p
Nx · 0 °
¼
and increases as the square root of N :
Check as in the previous case. The neglected term is:
Z 1
Z
N x· 0 °
Nx · 0 ° 1
dx
dx =
2
2
2
2
2
2
2
4¼
¢ 4¼ x + (°=2)
¢ x + ° =16¼
µ
¶
Nx · 0 ° 4¼ ¼
¡ 1 4¼¢
=
¡
tan
4¼ 2 °
2
°
p
Nx · 0 °
N x ·0 °
»
»
4¼ 2 ¢
2¼
3
Thus we get the curve of growth:
let x = N¢ºx ·D0 and y = W=¢º D then
The transition to the saturated regime occurs sooner (lower W ) for lower
temperatures because ¢º D is smaller. The damped (square root) portion is
higher is the damping paramater ° is larger. The line pro¤le becomes squarer
in the saturated portion abd devlops broad wings in the damping portion. (See
¤gure on next page.)
To use the curve of growth one compares the theoretical curve with an emipirical curve contructed from the data. The column density and temperature can
then be deduced. One can use line multiplets of the same atom, or di¤erent
trancitions (but beware here if ° is dominated by natural rather than collisional
broadening.)
Herezs some data for hydrogen from
http://nedwww.ipac.caltech.edu/level5/Charlton/Charlton1_1.html
p
b is the Doppler parameter ¢º D in velocity units: b = 2kT =m and m is
the atomic mass.
4
Herezs another one from http://www-int.stsci.edu/~leonidas/voss03/
5
6
Q0420-388 z=3.12
Lyα
α Forest
+
metals
Ly-limit
Damped Lyα
α
QSO Lyα
α em.
7