Does anybody know why Thomson scattering by free electrons is apparently not a significant broadening mechanism for the Fraunhofer absorption lines in the solar spectrum?

The electron speed for a temperature of 6000 K is about a factor 1.4*10^-3 of the speed of light, so a spectral line at 6000 A should be Doppler-broadened to a half-width of about 10 A by Thomson scattering if there is a sufficiently high electron density. Now the accepted particle density of the photosphere is 10^23/m^3 (which also would be the electron density as the gas is almost fully ionized) and considering the scale height of the photosphere of about 150 km=1.5*10^5 m, the column density of electrons amounts to 1.5*10^28/m^2 . The Thomson scattering cross section is 6.7*10^-29 m^2 and if one multiplies these two numbers one gets an optical depth with regard to Thomson scattering of about 1. The scattering should therefore be significant and the lines broadened to a half width of around 10 A. This seems to be in contradiction to observations which show a line width of less than 1 A.

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I posted this question also in a different forum and got the reply that Thomson scattering is insignificant because, contrary to my above assumption, the electron density is about 3 orders of magnitude smaller than the neutral density.

However, with a neutral density of 10^23/m^3, there should be again an excessive line broadening as the optical depth for resonance scattering (which produces the Fraunhofer lines in the first place) becomes very large. Although most of the atoms would be in the ground state for a temperature of 6000 K (which would be irrelevant for transitions in the visible part of the spectrum), there will be a large number of atoms in the metastable 2s state from which the Balmer series can be excited: assuming a plasma density of around 10^20/m^3 (0.1% of the neutral density) and a recombination coefficient of 10^-18 m^3/sec (10^-12 cm^3/sec) the recombination rate would be 10^40*10^-18 = 10^22/m^3/sec . Now the 2s level of hydrogen is metastable with a lifetime of 1/7 sec. If one multiplies this with the recombination rate, one gets a 2s density of about 10^21/m^3 which would yield an optical depth of around 10^10 for the Balmer series lines (10^21*10^-16*10^5 =10^10; where 10^-16 m^2 is the resonance scattering cross section in the line and 10^5 m the scale height of the photosphere). This still would result in an equivalent width of about 200 A (which is not observed for the lines). So it appears that the genereal assumption of the neutral density exceeding the plasma density is inconsistent

Also, with a ground state density of 10^23/m^3 and a 2s density of 10^21/m^3, all the ultraviolet radiation shortward of 911 A and 3660 A respectively would be absorbed as the optical depth with regard to photoionization would be about 10^6 and 10^4 respectively (assuming a photoionization cross section of 10^-22 m^2 and a photospheric scale height of 10^5 m). All the ultraviolet continuum would literally be absorbed in the bottom few meters of the photosphere and one would not observe any UV continuum at all (although relatively weak, the UV continuum is in fact still significant; the Lyman-alpha photon flux for instance is less than 0.1% of the total EUV flux). Also, with all the UV radiation would be absorbed in the bottom layer of the photosphere, how could there be any ionization at all in the rest of the photosphere?
I don't know at the moment what the lifetime of the 3s level of hydrogen is, but if it is any longer than about 10^-5 sec, even the continuum in the visible region of the spectrum would be absorbed by the photosphere (which obviously is not the case).

This was my post above. I wasn't logged in when I posted it.

P.S.: In the meanwhile I found out that the lifetime of the 3s level is of the order of 10^-6 sec, which, according to the above assumed values, would only yield an optical depth of 0.1 with regard to photoabsorption in the visible region of the spectrum, so this is not much of an issue here.
However, the discrepancies in the ultraviolet and for the width of the Fraunhofer lines remain.

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I noticed that the consideration I submitted above as 'Guest' neglects both the effects of collisions and photoionization on the atomic level density, so here a revised treatment that takes these issues into account. Although it does not refer to the line widths explicitly anymore, it still shows that the presently accepted values for the photosphere are inconsistent:


The ionization energy for n=1 is 13.6 eV, but a temperature of 6000 K corresponds only to 0.5 eV, i.e. only electrons in the high energy 'tail' of the Boltzmann distribution could lead to collisional ionization. Even an energy of 13.6 eV won't be enough as the cross section at the threshold energy is practically zero (only an exact head-on collision could transfer the full energy and the cross section for this is strictly speaking zero). From measurements it is apparent that the collisional ionization cross section reaches its maximum value only at an energy of twice the threshold energy, increasing roughly linearly from the threshold. This means that the combined Boltzmann and cross section correction factor has to be taken of the form exp(-13.6*(1+d)/0.5)*d where d is a certain percentage that maximizes this value. This factor is of the form d*(exp(-a*d+b )) and if you differentiate this and set the result to zero you find that it has a maximum at d=1/a, which, taking the explicit numbers here, results in d=3.7*10^-2 and with this the correction factor has a value of 2*10^-14. Multiplying this to a total electron density of 10^20/m^3, one has only an effective density of 2*10^6/m^3 capable of ionizing the ground state. With the corresponding Coulomb collision cross section of 10^-21 m^2 and velocity of 2*10^6 m/sec, the resultant collision frequency is 2*10^6 * 10^-21 * 2*10^6 = 4*10^-9/sec. i.e. the lifetime of the ground state of atoms in the photosphere would be about 10^8 sec.

On the other hand, the EUV photon flux shortward of 912 A is about 10^15 photons/m^2/sec at 1AU from the sun. Taking the inverse square law for the flux into account, this means that the ionizing photon flux in the solar atmosphere is about 5*10^19 photons/m^2/sec. Multiplying this to a photoionizatin cross section of 10^-22 m^2 yields a photoionization frequency of 5*10^-3 i.e. the lifetime with regard to photoionization is about 100 sec. This is 6 orders of magnitudes less thanfor the collisional ionization i.e. the latter would be completely irrelevant for the ground state at this temperature.



If one does the same calculations for n=2, one finds now a reduction factor exp(-3.4*(1+0.15)/0.5)*0.15 = 6*10^-5 i.e. an effective density of 6*10^15/m^3. The Coulomb collision cross section for this energy is 10^-20 m^2 and the velocity is 10^6 m/sec i.e. the collisional ionization frequency is 6*10^15 * 10^-20 * 10^6 = 60/sec i.e. a lifetime of a about 1/100 of a second.

On the other hand, the photon flux shortward of the Balmer edge is about 2 orders of magnitude larger than the flux shortward of the Lyman edge i.e. about 5*10^21 photons/m^2/sec. The photoionization cross section from n=2 is about a factor 5 larger than from the ground state i.e. 5*10^-22 m^2. This results therefore in a photoionization frequency of 2.5/sec or a lifetime of 0.4 sec. As this is longer than the collisional lifetime, photoionization is therefore insignificant for n=2 (assuming an electron temperature of 6000 K) but it is definitely the determining factor for the ground state.



Let's reverse the argumentation now and assume the n=2 density known through the linewidth measurements of the Fraunhofer lines (it should be emphasized though that this assumption neglects the possibility of other line broadening mechanism than by radiative transfer):
from the above values one has a lifetime of the ground level of 100 sec (determined by photoionization) and a
collisionally determined lifetime of the metastable 2s level of 1/100 sec (the natural lifetime of the 2s level is 1/7 sec and therefore not relevant here). On the other hand the recombination rate for an electron temperature of 6000 K is about a factor 5 larger for n=2 than for the ground state. Taking these figures together, one finds that the ground state should have a density only about a factor 10^3 higher than n=2 i.e. about 10^20/m^3 if the n=2 density is 10^17/m^3 . This is a full 3 orders of magnitudes below the generally assumed total density and shows therewith that the present theory regarding the physical state of the photosphere is very much inconsistent.

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Hi Thomas,

Sorry for there being not much response here. I suspect that there is an interesting subject in what you're writing about, but it is a kind of dry subject at the moment, and one that most of us don't start off knowing much about (there ARE a few spectroscopists here). I will try to take some time over the weekend to read and understand this line of questions.

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Thomson scattering by free electrons does broaden the spectral lines, but by too much and produces overlapping and a continuous spectrum as a result.

This is because the energy levels of bound-free, free-bound, and free-free electrons are not quantized.