Stellar Absorption and Emission Processes Revisited (2)

In approaching stellar line formation we will be looking at a number of related equations, including:

1)      The Boltzmann equation

2)     The Saha equation

3)     Combined Boltzmann and Saha equations

 

These will enable us to form a picture of spectral line formation which can then be generalized for different atoms and energy transitions.  We start then with the Boltzmann equation, which we already introduced in the previous chapter:

 

N₂ / N₁   =     [g₂ / g₁ ]   exp (- E2 – E1) / kT

 

That is, for the atoms of a given element in a specified state of ionization, the ratio of the number of atoms N₂ with energy E2, to the number of atoms N₁   with energy E1, in different states of ionization is given by the above formula. The same form of the equation can also be used to find the ratio of probabilities, i.e. that the system will be found in any of the  g₂   degenerate states with energy E2 to the probability that the system is in any of the g₁   degenerate states E1, viz.

 

P(E2) / P(E1)   =     [g₂ / g₁ ]   exp (- E2 – E1) / kT

 

Thus, the Boltzmann equation can be posed in two forms.  In statistical mechanics we could have also seen the partition function:

 

Z  =   å _j   exp ( - e _j )/ t

 

Which is just the summation over the Boltzmann factor (exp ( - e _j )/ t ) for all states j for which the number of particles (N) is constant. We will find it useful to rewrite it:

 

Z = g₁ +  å^¥ _{j = 2}   g _j  exp (- E _j – E1) / kT

 

Of interest now are the relative numbers of atoms in ionization stage i, which is written:

 

N _e N _{i + 1} / N _i  =  2 Z _{i + 1} / Z _i (2 p m _e kT/ h ²) ^1.5  e ^{- c i/ kT}

 
This is the Saha equation, named after the Indian astrophysicist who first derived it.  Here,  N _e  is the number of free electrons per unit volume and c _i  is the ionization potential of the ith ionization stage. Thus, the equation relates the number of atoms in two successive  ionization stages to the quantities that are relevant. As per our introduction to quantum mechanics, the factor ‘2’ in the equation refers to the two possible spins of the free hydrogen election with spin quantum number:

m _s =  +½.

 

 Recall that for thermodynamic equilibrium, the rate of ionization cannot exceed the rate of recombination[1].  In other words, the rate at which atoms in the ith stage are ionized (i.e. to the i +1st stage) must equal the rate at wich ions in that i +1st stage are recombining with free electrons to form ions in the ith stage. The latter depends on N _e N _{i + 1}   and the former on N _i . Hence, Saha’s equation simply expresses the fact these two processes must occur at the same rate.

One can also rewrite the equation in a more manageable logarithmic form if one substitutes the numerical constants:

log(N _e N _{i + 1} /N _i)  = 

 

15.38 + log (2 Z _{i + 1} / Z _i ) + 1.5 log T – 5040 c _i/ T

 

The units here are important to note, and are consistent with the ionization potential being measured in electron volts (eV). Therefore N _e  must be in particles per cubic centimeter.

 

    Yet another way to express the Saha equation is via the electron pressure, P _e . This acknowledges that each separate species of particle makes its own contribution to the total gas pressure. The free electrons in a gas therefore produce a pressure given by: P _e = N _e kT.

 

Then we may write another log form of the Saha equation:

 

log(P _e N _{i + 1} /N _i)  = 

 

-0.48  + log (2 Z _{i + 1} / Z _i ) + 2.5 log T – 5040 c _i/ T

 

There are also two distinct processes by which lines can be formed:

 

1)      Bound-bound transitions

2)     Bound-free transitions

In the first case, the photon goes from one bound atom to another. Also, in case (1) the photon has a good chance of being scattered, i.e. emitted in the same downward transition.  This may be at the same frequency as the absorption, in which case we say the scattering is coherent, or not, in which case it is non-coherent.

Details of bound-bound transitions differ in significant ways from bound-free transitions. In the first case, the transitions are also affected by a broadening function which is not so important for continuous emission. If we write out the equation for absorption in more detail we get:

a _u  =  [1 - e ^{-  h u o / kT}] (p e²/ mc) f f _u

which yields units in  cm² / atoms at lower level. Two other absorption derivative values are possible from the preceding:

 

i)                   The absorption coefficient per unit length (cm^-1)

ii)                 The mass absorption coefficient  k _u .

 The value for (i) is just a _u  multiplied by the number of absorbing atoms per unit volume. The value for  k _u  is just  multiplied by the number of absorbing atoms.  

The value for a _o   is just:  a _o =  a _u  /  f _u.  

Sometimes referred to as a “fudge factor”, f is known as the oscillator strength or f-value of the line. It is basically the  laboratory.

The broadening function  f _u  is:

f _u  =   1/ Öp   [exp (u  -u _o)  /D u _D ]²  D u_D

Which can also be rewritten as:   f _u  du  =

1/ Öp   [exp (u  - u _o)  /D u_D ]²  du / D u_D

This would be the probability that the absorbed photon lies between u  and  u +  du, assuming equal intensities for all frequencies. Thus the integral:

ò f _u  du  =  1

 

Where du is over all frequencies.  The value of f _u  is larger near  u _o  the frequency of the line center, as may be deduced from the line profile diagram below:

Absorption line profile showing the core and "wings"

Note here that  D u_D  is the Doppler half-width of the line. As can be seen on inspection, f _u is very large at line center and falls off in the “wings”, i.e. at larger and smaller frequencies.

 The three important types of line broadening are doppler effect, natural and pressure broadening. We will confine our attention to the first type which is given by the broadening probability equation, provided the velocity is Maxwellian and that the frequency at line center  u _o is also observed for some u . The Maxwellian will display the distribution of velocities as shown below where the central line defines the most probable.

Maxwellian showing distribution of velocity with proportion of particles.

For the Sun, solar physics, it is also necessary to consider the Zeeman effect, a broadening due to strong magnetic fields such as in sunspots. An example of this applied to a sunspot is depicted below:

The left image shows the line-centered sunspot for which the Zeeman effect in classic "triplet" form (right image) is detected and measured. The greater the spectral line splitting the greater the magnitude of the associated magnetic field.  George Ellery Hale, who discovered the effect, posed the quantitative relationship in terms of the original wavelength   l_o  (undisturbed line) and the spread of wavelengths, D  l:


D  l =     (l_o)²  e H/ 4 π  m_e c²  


Where H is the intensity of the sunspot magnetic field in gauss (to be found), e is the electron charge in e.s.u.,  m_e  is the mass of the electron in grams, and c the velocity of light in cm/ sec.

 

Problems:

1) For the temperature and conditions of problem (1) of the previous set, find the ratio of the probability that the system will be found in any of the eight degenerate states of energy level E2, to the probability that the system will be in any of the two degenerate states of energy level E1.

2) An H-alpha line undergoes triplet splitting in the vicinity of a sunspot. The undisturbed line is measured at:   l_o  =    6.62 x 10  ^-5 cm. The line shift on either side is: + 0.0025 Å.   Use this information to find the strength of the sunspot magnetic field: a) in gauss, b) in Tesla.

Looking at Stellar Emission and Absorption (3)

The study of spectral lines is facilitated by using what is called the “equivalent width”. This provides the total strength of a line, yielding the same area of the line (for a rectangular facsimile) provided the depth is complete from the continuum to zero brightness.  

    We can express the equivalent width in two ways, based on frequency of wavelength, or wavelength itself, e.g.

W =   ò ^¥₀  (I _c – I _u /  I _c )du =  ò ^¥₀  (F _c – F _l / F _c ) dl

   The left side defines W in terms of the intensity from the continuous spectrum outside the spectral line where the quantity (I _c – I _u / I _c ) is referred to as the “depth of the line”, the analogous quantity to (F _c – F _l / F _c ) on the right side where we have radiant flux units. Technically, the integral should be taken only from one side of the line to the other but the limits can be as shown provided I _c    (or  F _c)  is kept constant in the neighborhood of the line.

      The Curve of Growth:

Having obtained the equivalent width, the next logical step is to generate the “curve of growth”.  Recall from the previous section that W, the equivalent width, represents the line strength. Then the curve of growth describes how the latter increases as the optical depth, τ  increases.

To fix ideas, we use a model slab of  finite thickness ds and over which the optical depth increases by d τ.  The incident intensity is  I _c  for frequencies in the neighborhood of the line,  and the intensity emerging from the opposite side is    I _u  which we seek to find.

Starting with  the original basic transfer equation based on wavelength:

dI(l)/ds = -k(l)  I(l) + k(l)  S(l)

We can derive the frequency form:

d I_u / dτ _u  =  -  I_u – S _u

We have (from the original equation) after adjusting for frequency:

d I_u / ds   =  -k _u I_u  +   k _u S _u

Where:   k _u   =   d τ _u / ds

So: 

d I_u / ds   =  -( d τ _u / ds) I_u  +   (d τ _u / ds) S _u

d I_u =   - d τ _u  I_u  +   d τ _u  S _u  =   - d τ _u   (I_u – S _u)

whence:

d I_u / dτ _u  =  -  I_u – S _u

For which the solution is found (on integration):

I_u =  I _c  exp (-τ _u)

Where:  τ _u   = N L a _u  =  N L a _o  f _u

Where ‘N’ is the number of absorbing atoms per unit volume and we already saw that:

a _o =  a _u  /  f _u = a _u [1 - e ^{-  h u o / kT}] (p e²/ mc) f

In terms of this new information, the equivalent width of the line can then be written as:

W _u =  ò ^¥₀ [1  -   exp (- N L a _o  f _u) ] du

For small x we may use the approximation that:

exp (-x) much greater than   1 – x

So rewrite the equation for  W _u:

W _u »   ò ^¥₀ [1  -  (1 - N L a _o  f _u)  du

»   ò ^¥₀   N L a _o  f _u  du » N L a _o  ò^¥₀  f _u  du

So we find W _u  depends only on the form for the broadening function.  For very weak lines, for example:

ò ^¥₀  f _u  du  much greater than   1 so that W _u » N L a _o 

Or simply proportional to N, the number of absorbing atoms. For “strong” lines the absorption near the center is very large so we can expect: N L a _o  f _u    >> 1 and with sufficient working this can be shown to give the corresponding equivalent width: W _u  = 2  D u _D

And for the moderately strong lines: 

W _u  =  2  D u _D {ln  (N L a _o/Öp  D u _D )} ^½

While very strong lines yield:

W _u  =   1/p  (N L a _o  g)^½

Where g is the damping constant.

Assembling all the diverse W _u    and plotting log W _u vs.  log (N L a _o ) one gets the curve of growth shown below: 

Here, each contribution can be separately considered, in terms of the “linear”, “saturated” and “damping” segments. The first is for the weakest lines such that W _u  is simply proportional to the number of absorbing atoms. But as more atoms are added this approximation becomes invalid and so we are in the saturated part. In effect,  the near zero slope shows the degree to which the equivalent width increases at an almost uniform rate. Finally, we have the damping segment wherein the line under consideration develops “damping wings” because the number of absorbing atoms is so large. (Bear in mind again that  a _o  is proportional to the transition probability)

Problems:

1)Consider the ionization of helium, with three stages of ionization for which the partition functions are:    Z _I =    Z ₃    =   1,  and Z ₂ =    2

With ionization energies:

 c  ₁ =   24.58 eV    and c  ₂ =   13.6 eV  

For a temperature of T = 5040 K write out two versions of the Saha equation for:

log( P _e N ₂ /N ₁)   and log( P _e N ₃ /N ₂)  

And hence, compute:   (P _e N ₂ /N ₁)   and (P _e N ₃ /N ₂)  

2)The diagram below was sketched by a first year astrophysics student. It represents part of the emission spectrum of atomic hydrogen.

 It contains a series of lines, and the wavelengths of some (in nm) are marked. '
There are no lines in the series located at less than 91.2 nm

(a)   In which region of the E-M spectrum would these lines occur?

(b) Obtain a relation between E and l   and thence find the photon energies equivalent for all the wavelengths marked.

(c)Use this information to map a partial energy level diagram for hydrogen. Show and label clearly the electronic transitions responsible for the emission lines labeled.

3)a) A stellar atmosphere is composed of pure hydrogen, at a temperature of 9600 K and with electron pressure, P _e    =  200 dyne cm ^-2 .  Use the Saha equation to compute the fraction of atoms

that are ionized in relation to the total, i.e. find:   N _II /( N _I   + N _II )  

The partition functions are:  Z _II =   1 and  Z _I =   2

Hint: Check to see whether (E ₂  -  E ₁ ) >> kT with

k =  8.6174 x 10 ^-5 eV/K

So the Boltzman factor  exp (- E2 – E1) / kT << 1

b)Repeat the exercise again for a temperature of 5040 K and 20,000 K and thence account for why the Balmer lines attain maximum intensity at a temperature of 9520 K.

(4) Calculate how far you could see through the Earth’s atmosphere if it had the same opacity as the solar atmosphere, where:

k _l   =   0.264 cm ² /g.

Take the density of Earth’s atmosphere as:

   r   =   0.0012 g cm ^-3

The density of the solar atmosphere is taken as :

r _s   =  2.5 x 10  ^-7 g cm ^-3

And we know the intensity declines over a characteristic distance (length scale) such that:

ℓ  =   1/ k _l r