Bound-free opacity takes place when a photon interacts with an electron which is bound to an atom. Unbound electrons have energy E > E¥ while bound electrons have E £ E¥. The energy of a level n is given by En
| (1) |
From an astrophysical point of view, H is the most important atom since it makes up the major fraction of matter both in stars and in the insterstellar gas. In this case the energy levels are arranged so that
| (2) |
where \cal R is the Rydberg constant. In the case n = 1 we get the binding energy,
| (3) |
and hence \cal R = 13.6 eV.
Important upper wavelengths produced by capture of an electron into the three lowest levels of the H atom are shown in Table 7.1, and are called the Lyman, Balmer and Paschen series. They are in the ultra-violet (UV), optical and infra-red (IR) regions repsectively.
The most important of these for historical reasons is in the optical region, the Balmer series. The first few Balmer lines are shown in Table 7.2.
These lines are very clear in hot stars of spectral type from O to F and strongest at about spectral type A (see figure 7.2). We'll look at the reasons for this later on in the lecture.
We consider the two possible processes of electron capture and electron photoionisation. In both cases the energy is conserved:
| (4) |
where the electron has velocity v. The major difference between the bound-free and free-free processes is that we now need to take into account the quantisation of the energy levels of the atom.
It turns out that most of the electron's kinetic energy is radiated away as it approaches the ion, as in the free-free case, and so the form of the opacity for the bound-free case is quite similar to free-free emission. The opacity has the form
| (5) |
In a stellar context, in order to compute the opacity of the plasma, we need to know the relative amounts of the various ions, which may be parameterised conveniently by X, Y and Z, and what ``state'' of ionisation they are in i.e. in which energy levels the electrons are sitting.
Lower | Series | Symbol | Wavelength | Energy |
level | name | l | eV | |
n = 1 | Lyman | La | 1216 Å | 0 |
n = 2 | Balmer | Ha | 6563 Å | 10.19 |
n = 3 | Paschen | Pa | 18750 Å | 12.07 |
Name | transition | wavelength l |
Ha | n = 3 ® 2 | 6563 Å |
Hb | n = 4 ® 2 | 4861 Å |
Hg | n = 5 ® 2 | 4340 Å |
Hd | n = 6 ® 2 | 4101 Å |
H¥ | n = ¥® 2 | 3647 Å |
In order to determine the amount of Bound-Free opacity we need to know the state of the gas - i.e. which ions are present, in what quantities and the state (energy level) of the electrons. This is determined by the Boltzmann equation
| (6) |
where NA and NB are number densities of atoms in state A and state B, EA and EB are the energies of the states, and gA and gB are the statistical weights of the levels (i.e. the number of states permitted at that energy level).
The Boltzmann equation allows us to compute the ratio of atoms of a certain species, such as Hydrogen, in various possible energy states as a function of temperature.
For Hydrogen gn = 2 n2, which is the number of spin and angular momentum states for each energy level En. For a given energy level n, the quantum number l can take values from 0 to n-1, while for a given l the quantum number m takes values from -l to l. This makes n2 states, and the electrons themselves can be either spin-up or spin-down, so that the total number of states is 2n2.
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The previous problem shows that there is a very small energy per electron, much less than the energy required to get from the ground state to the first excitation level (see Table 7.1). This indicates that the H is not strongly ionised. The amount of inonisation can be computed from the Boltzmann equation.
This shows that at solar surface temperatures a very small fraction of Hydrogen is in the excited state. |
For atoms of different species, one makes use of the Saha equation:
| (8) |
where N is the number of atoms, Ni is the number of ions, ne is the number density of electrons and c is the energy of the level.
Note that the Saha equation is the equivalent of the Boltzmann equation applied to ionisation states, rather than energy states within the same ionisation state.
Expressing the electron number density in cm-3, temperature T in K, and c in eV, Saha's equation takes the form
| (9) |
where U is called the partition function and is given by
| (10) |
By starting from the ground state (I = 0) one can thus determine the occupancy of the ionised states (I = 1, 2, etc). For example, for the ionisation of Hydrogen we get
| (11) |
and for the first ionisation of Helium we get
| (12) |
where ne is in cm-3.
and
Show that for Hydrogen
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A similar analysis for Calcium shows that for first ionised electron for which c = 6.113 eV, we have
| (16) |
and for the second ionised state (c = 11.871 eV)
| (17) |
Practically all the Ca is in the singly ionised state. It is for this reason that the Calcium lines (and many other low ionisation energy ``metals'' such as Na, Mg, Cl, Cr, Mg, Fe etc) show prominent lines in the Solar spectrum. These lines are called the Fraunhofer lines. Some prominent lines are shown in Figure 7.1 for a G5 dwarf star, which is quite similar to the Sun.
The strength of the main lines in stellar spectra varies strongly with spectral type, or in other words, temperature. The spectral types at which various lines peak in their strength, and the range of spectral types in which they are strong, is shown in Table 7.1.
The problem above suggests that in hotter stars than the Sun, Hydrogen lines should become much more prominent. Figure 7.2 shows the spectrum of an A0 dwarf star, for which the effective surface temperature is about 10,000 K, showing that the H lines are in fact much more prominent. The next two problems examine this in mode detail by computation of the strength of the Hydrogen line as a function of temperature.
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Species | Peak | Range |
H I | A0 | O - K |
He I | B5 | B0 - A0 |
He II | O | O - B |
Fe II | F0 | A - M |
Ca II | K5 | A0 - M5 |
Sr II | K5 | F0 - M5 |
TiO | M | M0 - M9 |
From the Saha Equation, a list of the relative abundances of different elements in the Sun and the energy levels of these elements, we can calculate the degrees of ionisation of all atomic species. This gives us the density of the species as a function of the electron density ne and the temperature T. Assuming that the Sun is electrically neutral, means that the source of the electrons is just the ionised atoms themselves, allowing us to determine ne. In the atmosphere of the Sun, where the lines are actually formed, one can construct a model of the temperature, density and pressure as a function of height (or optical depth), and from the line strength and electron density derive the amount of each element or abundance present in the Sun. This calculation is in principal straight forward but in practice it is a lot of work. Table 7.4 shows the relative abundances of the elements in the Solar atmosphere. This relative pattern of abundances is very typical of stars near the Sun (i.e. the so-called Population I stars).
Element | Abundance | Element | Abundance |
H | 3.2×1010 | Sc | 35 |
He | 2.1×1010 | Ti | 2.8 ×103 |
Li | 49.5 | V | 262 |
Be | 0.81 | Cr | 1.3 ×104 |
B | 350 | Mn | 9.3 ×103 |
C | 1.2 ×107 | Fe | 8.3 ×105 |
N | 3.7 ×106 | Co | 2.2 ×103 |
O | 2.2 ×107 | Ni | 4.8 ×104 |
F | 2.5 ×103 | Cu | 540 |
Ne | 3.4 ×106 | Zn | 1.2 ×103 |
Na | 6.0 ×104 | Ga | 48 |
Mg | 1.1 ×106 | Ge | 115 |
Al | 8.5 ×104 | As | 6.6 |
Si | 1.0 ×106 | Se | 67.2 |
P | 9.6 ×103 | Br | 13.5 |
S | 5.0 ×105 | Kr | 46.8 |
Cl | 5.7 ×103 | Rb | 5.9 |
Ar | 1.2 ×105 | Sr | 26.9 |
K | 4.2 ×103 | Y | 4.8 |
Ca | 7.2 ×104 | Zr | 28 |
Following on from section 7.1, we have the cross-section for photon emission in the bound-free case in terms of the electron velocity v for a state of a Hydrogen-like atom n.
| (18) |
The reverse of this process is photo-ionisation, or the ejection of an electron by the incoming photon. In thermodynamic equilibrium, the capture and photo-ionisation processes must balance. Furthermore, the electron velocity distribution is Maxwellian. These taken together can be used to show that the cross-section of level n is
| (19) |
where gbf is the bound-free Gaunt factor, and similarly to the free-free case, is a slowly varying function of order unity taking into account corrections for quantum effects.
Photons involved in this process must have an energy greater than the binding energy of the level n considered. Hence, for hn< En, the cross-section is identically zero. In terms of the Hydrogen-like atom, we may express this as
| (20) |
or that the energy level must satisfy
| (21) |
Given the population of excited states in the star, the total opacity kbf due to bound-free absorption can be computed.
| (22) |
where Nn is the number density of atoms in state n and r is the (mass) density. Substituting this expression into the form for the Rosseland mean opacity, one derives
| (23) |
where k0 is
| (24) |
where t is a correction factor (called the guillotine factor) which takes into account low temperature effects of the number of electrons in bound states per atom not considered here.
In a plasma of electrons and ions, the electrons have much greater typical velocities than the ions do, so that the electrons are able to transfer heat through the system. Opacity associated with electron conduction is in general much smaller than other sources of opacity, and only becomes significant at quite high densities or temperatures, or when the number of electron states available starts to become limited (i.e. the plasma starts to become degenerate).
Consider electrons of mean velocity [`v] transfering energy outward along a direction x, and that the electrons have a mean free path le. The electron flux is ne [`v] where ne is the electron density. The conducted flux Fe is then
| (25) |
where e is the energy of the electrons, and ¶e/¶x is the gradient of the energy over the distance traveled.
If we consider non-relativistic electrons, the energy is e = 3kT/2, so that
| (26) |
We now define the thermal conductivity lc of a medium in terms of the temperature gradient along x
| (27) |
and so the conducted flux is
| (28) |
where le is a scale length for thermal conductivity.
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The electrons are scattered by the ions. We may relate the kinetic energy me[`v]2/2 of the electrons to the size of the scattering region around each ion r0 by equating with the potential energy of the ion Ze2/r0,
| (30) |
so that the cross-section for scattering, s = pr02 is
| (31) |
The electron density in an ionised gas of density r, and Hydrogen mass fraction X is
| (32) |
and the ion density ni is given by
| (33) |
where A is the atomic weight. Using the relationship between mean free path and cross-section and the above expressions for ne and ni, one can show that
| (34) |
This can be expressed as an opacity for conduction kc, by defining
| (35) |
so that
| (36) |
where r is in g cm-3.
In general, putting together all the sources of opacity is quite complicated, particularly in the outer stellar layers where the frequency dependence must be carefully considered. In stellar interiors, it is usually adequate to use frequency averaged opacity, such as the Rosseland mean. Also, in stellar interiors, the mostly ionised state of the gas means that the main sources of opacity are due to free-free, bound-free and electron scattering, for all of which relatively simple expressions can be computed. A summary of these is shown in Table 7.5.
Opacity Source | Expression |
e- scattering | ke = 0.2(1+X) |
free-free | kff = 3.9 ×1022 (1+X) (1-Z) [`g]ffrT-[7/2] |
bound-free | kbf = 4.3 ×1025 Z (1+X) [([`g]bf)/t] rT-[7/2] |
e- conduction | kc = 5 ×103[(Z2/A)/(1+X)][([Ö(T/107)])/(0.1r)] |