 Great, is the volume okay? All right, so we'll continue where we left off yesterday. And we had gotten to this point where we talked about some radiation transport equation and then absorption and emission coefficients. And finally come up with the expressions for the emission coefficients, emission and absorption coefficients in terms of the basic atomic transitions in terms of the cross sections and the populations. So just a brief mention that what you'll see mentioned in the literature is average opacities or absorption coefficients. The opacity is conventionally defined as the absorption coefficient per unit mass per mass density. And there are two average opacities which are in common use in the literature. There's the Roslin mean opacity, which is an inverse sum over the opacity averaged by the derivative of the plane function. And this, because it's an inverse average, emphasizes low opacities. And so this is an appropriate one for measuring transmission. When you have high optical depth and high absorption coefficients, it's easiest for the photons to transport in these gaps here where the opacity is lowest and they tend to get trapped where the opacity is high. So this gives more or less a measure of how easily the photons can transport through the plasma. On the other hand, the plane mean opacity, which is an average, a straightforward average over a plane function emphasizes absorption. And so this opacity is more representative of how the radiation couples to the material, how the energy is transferred back and forth. And of course, both of these are strictly applicable only in LTE because they're using a plane function. So let's look at a couple of examples of absorption coefficients and emission coefficients. The easiest one to do is hydrogen. And so this looks like exactly what you would think. The bound bound transitions, the line transitions have this current discrete energies. There's a line shape which you've heard about for the last couple of hours, which is not being resolved here on this plot. But basically the opacity are large where the line shape is large and the opacity is quite small in between. Between there is the bound free opacity. You see opacity edges here for each transition, ionization out of a bound state. There is an edge and below that, the opacity or the absorption from that transition falls off approximately is one over new cube. But there are many different transitions here. Each one representing ionization out of a different bound state. The largest of which corresponds to N equals one in the ground state of neutral hydrogen. So this is an example of a continuum emission. And then there's also the free free, the Bremstahl emission, which is featureless and just falls off basically as a power law. So this is what the absorption coefficient looks like. Notice that there's several orders of magnitude difference here. It's actually not too different in LTE, not LTE, at least in this case. So over here, amongst the lines, you can barely tell any difference. Over here, around the continuum and the edge, there is a difference and that's due to the different populations of the ground state of neutral hydrogen in the LTE and non-LTE case. So now if we look at the emissivity, we see again, characteristic, we can see underneath here that there is a continuum, part of which is due to Bremstahl. Part of which is due to bound free absorptions. You can see the edges coming in here. And then there's the line emission. Over if we look at the source function, which is the ratio of the emissivity to the absorption coefficient, that looks like this. And now we see a large difference between LTE and non-LTE. The non-LTE emissivity doesn't look too different from the LTE, but the ratio does look quite a bit different. So here where the lines occur, where there's a lot of opacity, a lot of absorption, comes up close to a playing function. Over here, where we saw the difference between LTE and non-LTE in the continuum and near the first ground state, first bound state, there is a large difference in the source functions. So last time we did an example to illustrate some of the effects of optical depth of a uniform sphere of krypton. And we did that in LTE so that the absorption and emission coefficients did not depend on the radiation density. So now we can go back and do the same thing, except we can do it in non-LTE where those coefficients do depend on the radiation. So to do this kind of a calculation at each point on this sphere, and this has been discretized to do this, the radiation density is calculated, the rate coefficients, the radiative rate coefficients are calculated in the presence of the radiation field, and then the collisional radiative models used to calculate the populations, which then is used to calculate the absorption and emission coefficients, and that is done in a self-consistent manner so that the resulting radiation field is accurately reflected in the absorption and emission coefficients. So once again, this is the small sphere, 0.001 centimeters, and I've reproduced the LTE results and I'll put the corresponding non-LTE results on here. And again, this is a small sphere, there's not much optical depth. Notice that the optical depth here is different between the non-LTE and LTE cases, and it's different by quite a bit, by at least an order of magnitude here, and that's because of the different populations, non-LTE and LTE. So the LTE case will be finalized much more than the non-LTE case here. So the opacity is quite different and the emissivities are quite different here. Again, this is a small sphere, so the flux that you get coming out now, which is what a detector would see, is nowhere near the playing function, and it shows characteristics of the atomic transitions. And if we ramp this up by two orders of magnitude in size, now we're getting some more serious optical depths here, the large transitions have optical depths approaching 100. And now we notice that the optical depths are coming closer together here, they're not different by an order of magnitude anymore, maybe it's half an order of magnitude. And some of the emission characteristics are similar and some are different. We have strong lines here, they differ by a factor of pu in the strong lines, they differ by an order of magnitude or more in some of the places with less absorption. And while in the LTE case, there's been saturation at the block body limit up here where there's a large opacity in the in the continuum and in the K-shell transitions, that's not close to the black body limit for the non-LTE case. And if we go two more orders of magnitude in size, so now the optical depths again are becoming very large and now we can barely distinguish the optical depths in the non-LTE case and the LTE case, but we can certainly distinguish the emission and the flux that we get. So this illustrates actually one important point is that even if the opacities look the same, the emissivities might not. And that's because the opacities are reflective mostly of lower state populations, primarily ground state populations and lower level populations, while the emissivities are characteristic of excited states. And so you'll see much more of the effect of the populations in the higher living states and emissivity than any absorption coefficient. So again, this is again very optically thick now. Optical depths up near 10 to the fourth and almost everything is up 10 or more, but yet in the non-LTE case, we're still not close to a black body function, to the black body curve, even out here in the continuum, whereas the LTE case is nearly fully plunky in. So we can take another look at these calculations in terms of looking at the radiation, which is now the radiation density inside the sphere at different positions in the sphere. And now for all three cases of the sphere, this is a normalized position from the center of the sphere to the outside of the sphere. And what these curves show is the radiation temperature, radiation temperature being defined as the energy density in the radiation, sigma T to the fourth equals the energy density. So that's defining the radiation temperature, although it doesn't tell you what the spectral content is. So again, for small spheres with low optical depths, the radiation temperature is actually quite low. So remember that the material temperature here is a couple hundred volts. The radiation temperature is down here at 50 volts. So the energy density in the radiation is, well, that's a factor of four, four to the fourth below what it would be if it was equilibrated. And the radiation densities are not too different, but the spectra would look quite different. Now, as we increase the optical depth, increase the depth of the sphere, the radiation builds up inside the sphere. So this is an effective radiation trapping. And when we get very large optical depths, the radiation temperature is nearly the same for the LTE and non-LTE cases, at least in the interior of the sphere. So one lesson we can take away from this, well, we've already talked about the fact that true black body emission requires very large optical depths. Another thing is that large optical depths produce radiation trapping, and the high radiation fields then act to produce conditions which are closer to LTE. They might not reach LTE, but they give you something which is closer to LTE than a non-LTE case with no radiation field. Now, another important feature here is that something's different on the boundaries. The interior of the sphere for these high optical depths is pretty uniform. Now, remember, it's said to be uniform in density and temperature, but not in radiation. However, at the boundary, it is not uniform anymore. So the presence of boundaries in radiation transport introduces non-uniformity. So if somebody tells you, I've got a uniform plasma here, but it's optically thick. You know immediately that that's not actually accurate because at the boundary, and the boundary is what you're viewing, it is not uniform because the radiation is trapped inside and escapes freely on the outside. Now, this is true even for constant density and temperature, but in practice, in a laboratory plasma, the radiation field at boundary also changes the temperature. And so if you've got a plasma which is being heated up or cooling off, it will cool more quickly at the boundary because the radiation can escape and it will in the interior. So it's very difficult to get an optically thick uniform plasma, basically impossible. So that's a major point that you have to be aware of is the conditions do not remain uniform in the presence of radiation in boundaries. So let's look at something similar to this again, but now we'll do a single transition, a single line, and probably the simplest possible case, which is hydrogen, limon, alpha, where we really need just three levels to describe this in a collisional radiative model. We need neutral hydrogen in the ground state in the first excited state because we're looking at the transition between those and we need the ionized state. So here's a simple case again. Now this is mandated to be a uniform plasma, temperature of one EV density of 10 to the 14th cubic centimeter, which means that there's a modical optical depth. And when I say optical depth here, what I mean is the optical depth you would measure at line center going from one side of the plasma to the other because going through the plasma at anywhere else or in any different direction or in any different frequency will give you a different optical depth. And what we want to do is look at the radiation coming out of here. What will we view from this? And we'll do this at two different viewing angles, one at 90 degrees to the plasma to the slab and one at 10 degrees. So this one has a much longer path length going through. So as you might think, then if we look quite through the slab, we will see some effects from the optical depth. And here I've got the, yes, the optical depth here as a function of energy with respect to the line center. And it does come up to just above about five and a half in the center. So there's a large optical depth, five and a half in the center as we go out in the wings, there's less and less optical depth. So by the time we get out here in a wing 0.001 away, the optical depth is quite a bit less than one. And so we would expect to see something different at these frequencies. So here at the center, line center, we're seeing the maximum amount of radiation. As we go out on the wings, we see through a smaller optical depth. And if you remember the expression we got for the exiting intensity from this slab when we did the characteristic solution, what we're seeing is the intensity from positions in the slab, each one attenuated by the optical depth to the boundary. And we're basically just seeing radiation from the outermost optical depth. So the radiation we see is very similar from the center out to about this portion of energy space, which corresponds to about here to a few optical depths. As we get out further, we see less and less radiation. So we see something which is pretty flat at the top and falling off on the outside. Now, if we take a different view through the plasma, the 10 degrees, we've got a much longer path length. And so this flat top persists further out. So we get a larger optical depth further out into the wings of the line. So actually this is one case where we can say that what we actually see here will actually depend quite sensitively on what the shape of this line wing is. So the line profiles do make a difference here. But now this was done with, again, I was mandating a uniform plasma and then in particular, I was mandating uniform populations. Now what happens if we let the radiation field, which is built up inside the slab, affect the populations? So now we see something quite different. So now this is pleasant, it's got uniform temperature and density, but not population. So now, again, at each point in the plasma, the collisional radius model is calculated using the photon field for this transition at that point. So now we see a couple of different effects. We see this radiation trapping effect. So the radiation field is building up here. I don't have a plot here quite the radiation field, but we can infer that from this plot, which is the upper state population. So the upper state population now in the center of the slab is several times higher than it was in the uniform case with no radiation pumping. And that's just because of the radiative excitation. And again, it's non-uniform now. It falls off in the center of the slab to the boundaries. So this becomes a very non-uniform plasma. Now the exiting intensity from the uniform case, that's this case that we just talked about. Now we've got much more upper state population so we're getting much more emission in the line. So we expect to see more radiation coming out, but we also see another effect here, which is the shape here. So we do see more radiation coming out, but at the center here, we see something, not a flat line, not a flat top profile as we saw here, but we see an inverted profile. And for the case of 10 degrees where you've got a larger optical depth, what we see is the same effect, but magnified. So what's going on here is, again, we're seeing only into about one optical depth at any given frequency. So if we go to line center where the optical depth is the largest, then we're only seeing in a small portion of the way of the slab. So what we're viewing is this part of the plasma. If we go further into the plasma, or if we go further away from line center where the optical depth decreases to about one, then we can view most of the plasma. So we're sampling this portion of the plasma. So we're seeing the regions with a higher population and a higher mission. And then finally, as we go out into the wings now, further out into the wings, then one optical depth spans over the entire plasma and we start to see a drop-off. And this becomes a very strong effect here. So this phenomenon is known as line reversal and it's caused by the radiation trapping. The increased population here is called radiation pumping. And again, it becomes very non-uniform. And so this effect, in an optically thick plasma, this effect will happen to different extents in each transition. So now we finally get to ask, okay, how do we actually calculate what the radiation field is? And along with the radiation field, we need to calculate the rest of the system. So we've got a coupled system. So question is, what do we actually mean when we say we're doing radiation transport? Now, the simplest form of radiation transport, we talked about maybe a backlighting example yesterday, is you put radiation from one side, it goes through the plasma, it doesn't perturb the plasma at all, and so we just see the effects of absorption. That's a pretty simple form of radiation transport because it doesn't come into the collisional radiator equation, it doesn't perturb, change the energy balance at all, but that's not the most common case. Usually, what we need to do with radiation transport is we need to calculate the radiation field and the effects of that radiation field on the material. And there are two sets of equations that are used for this in different areas of plasma physics. For LTE, energy transport. If it's in LTE, then you know what the stage of the material is from the temperature, and so the question is, what's the temperature of the material? And that depends on what the energy balance in the plasma is. And in this case, the radiation is usually carrying a fair amount of energy, and so we couple the radiation transport equation to an energy equation. So this energy equation here is, here's an absorption coefficient. Again, this is the angle averaged intensity, so looking at radiation from all different directions and the source function. So this equation is just saying that the rate of change of the material energy is just the sum of the, is just the absorption of radiation from, it's just the absorption of energy from the radiation. And then this absorption coefficient times the source function at the emission. So this is the integrated emission. So this is energy absorbed and energy radiated. And in this case, if it's LTE, we also know what the source function is. It's just the Planck function. So there's a set of equations, a set of methods used to solve this sort of a system of equation. So the material and radiation coupling is less direct than in the non-LTE collisional radiative sense. It's indirect because the radiation acts to change the material energy. The energy changes the temperature. All the frequencies are involved, independent of what the spectrum is. Now what we're usually more concerned about is the case we use for non-LTE spectroscopy, which is where we couple the radiation transport equation to the collisional radiative equations. And Yuri talked a lot about this yesterday. So we've got our set of rate equations here. And now the rates, at least the radiative rates, depend on the radiation field. Now we talked a little bit about the characteristic equation yesterday and said, okay, well we can integrate this equation if we know what the source function is. And there were a couple subtleties hidden in that. I said the hard part is determining what the source function is. Well, let me point out one of the subtleties there is that this intensity here that comes into the coupling is the radiation intensity averaged or integrated over all directions. So if we want to figure out what the radiation field is at any point in the plasma, we need to have the radiation heading in all different directions, which means that we can't just do radiation transport one direction, but we need to do it in all directions. As a matter of fact, when we do this in one dimension or two dimensions, the radiation transport is actually always must be done in three dimensions. And we can use the dimensionality to reduce the size of the system, but the transport is always done in three dimensions. Now in the rate equations, we'll see in just a moment that the source function for a single transition, a single line transition, actually can come down to a very simple expression, which is a constant times something proportional to the radiation field integrated over that line profile. We'll talk about that in just a bit. So here we have a more direct coupling of the radiation to the material. Not all the frequencies are involved in any given transition here. It's just a narrow range of transitions. And so the solution methods tend to be very different than they are for the LTE case. We don't need to spend too much time on line profiles because you're getting just two nice lectures on that. But we'll just point out that the line profiles are determined by the multiple effects, the natural broadening, the collisional broadening, which is actually the lifetime, the effects of the lifetime on the state due to the other transitions, Doppler broadening due to the material temperature, and then stark effect due to the plasma fields. And we get, usually we're dealing with a void profile because we always have the collisional and the natural broadening and some form of Doppler broadening. And then if the stark effect is important, then that makes the line shape more complicated than this. There's one other effect that we've alluded to but are not going to spend much time on, which is that there's a separate line profile for emission and absorption. And the emission profile can be determined by multiple effects. So if any talked about the fact that the Doppler effect can change the line profile. So a photon being absorbed by an atom by a transition at one frequency can then be emitted in a different direction. Now, in the frame of the material, that will usually come out of the same frequency. In the frame of the observer, however, there's a Doppler shift involved there. And that comes into the line profile. So the photo excitation, that's the photo excitation plus coherent scattering. There's also photo excitation plus elastic scattering where you get this excited state which is that a particular energy which might not be specifically at the center of line energy. And it's undergoing these weak elastic collisions which will change its energy somewhat. And so that affects the line profile. We've got the Doppler broadening, there's some other things which come into it. And this can be described by redistribution function. Now, this actually gets into a lot of detail, some of which are dependent on details of the plasma. The nice thing about this is that in most cases, the emission profile turns out to be very close to the absorption profile. And that is largely on account of the Doppler broadening. I can give you references to this if you want to go through it all. But we usually deal with the case of complete redistribution where we assume that the two profiles are the same. If there's only Doppler broadening, then we can write down what the redistribution profile looks like. And if we integrate this over Maxwell and we get something which is not too different again from the absorption profile. So in practice, this ends up being maybe a five or 10%. It's stronger for large optical depth because then the form of the emission profile and the form of the wings matters more. But it's rare to find a case where this matters a lot. So here's that hydrogen lineman alpha case again, now done with partial redistribution where I have calculated what the emission profile is. Again, it's the same conditions. And now I'm specifying the collisionality of this class by giving the Voight parameter. So I've calculated the collisional rates and this Voight parameter gives us the ratio of the collisional broadening to the Doppler broadening. And now again, we've got three different cases here. So let's look at the fractional population which tells us pretty much what the radiation field is in that line. The uniform case, which is not physical. The one we did before, which was complete redistribution where the two profiles emission and absorption profile equal. And now we've got something which is a little bit different which is partial redistribution. So there's a difference here of maybe 10% in the populations and the intensity that we see coming out or the flux coming out that a detector would see is again different by about the same amount. And this is actually a fairly large difference due to partial redistribution. There are specific cases where it becomes important where you care very much about the details of line profile. But most of the plasmus patroscably we do, this is something to be aware of because you don't want to be claiming too much accuracy in your identifications and profiles if you have an accountant for this effect but usually it's not a big deal. Now, there are other effects which are more important. So let's talk about velocities, material velocity. So far we've been talking about essentially matter at rest. So the ions, the atoms themselves are not at rest. They've got a Maxwellian distribution of velocities which gives us Doppler broadening of the lines but one part of the plasma is not moving with respect to the other. However, in most laboratory produced plasmas, there are different parts of the plasma are moving with respect to the other. And then the discussion that we've done so far applies only in one single reference frame. If we go from one frame to another, then we have Doppler shifts again, proportional to the velocity of the material. Now in the reference frame of the material, the emission is usually isotropic unless there's an applied magnetic field or something. So but that's true in the frame of the material. If we look in the frame of the laboratory, which is usually what we're concerned about, then the emission actually becomes non isotropic as well. So there's two ways of handling this. If you've got velocity gradients present, if there's material motion in there, then you either have to take the radio transfer equation and transform it from the laboratory frame into a co-moving frame, which there's several treatises on how to do this. It's a fairly complicated thing and it really complicates the numerics. Or you can transform the material properties into the laboratory frame. So if the velocities are not too large, then this is somewhat easier to do. So this option A is complicated, transforming the radio transfer equation. Option B is actually pretty simple, but then you need to take the kind of non isotropic absorption and emission coefficients. Now what sort of velocity gradients, what sort of velocity changes do we need to worry about? Because usually the material isn't moving very fast, but the overseas is fairly small. But if we're shifting the energy of a photon, we only need to shift it from one part of the line profile to another. Or if we shift it, shifting it completely out of the profile is actually a fairly simple thing because the lines tend to be very narrow. So a V over C of 10 to the minus three is quite large for photons emitted in lines. V over C of 10 to the minus four is even significant. So let's look at an example of this. So here's an aluminum sphere with a uniform expansion velocity. So here's my sphere. What I want to do then is put a detector out here and say, okay, what kind of flux do I see? What's the spectrum of the flux? I've got some parameters here, 500 DV, 10 to the minus three gram per cubic centimeter. And actually this is an annulus, 0.1 centimeter thick of a one centimeter radius. Now optical depths here. So this optical depth chart will tell us a few interesting things. One, it will give us some idea of the thickness, the width of the lines in energy space. So this covers about a kilovolt of energy. The lines, the strong lines here are very narrow. Also the optical depths get to be fairly large. So in the absence of any Doppler shifts, we expect a lot of radiation trapping in these lines. Now, if we look at the fluxes that we get out for different values of V over C. And I've got this plot blown up on the next slide so we'll talk about it a bit here and then look more details. I've looked at three values of V over C here or four values from zero up to 0.03. So a moderately large velocity but our laboratory plasmas do get there. Now what do we expect to see? Well, from plasma headed toward us, we expect to see photons blue shifted. So we'll see photons coming out at high energies. If we can see through this plasma if it's optically thin enough, we'll see photons going the other direction. So we'll see red shifts. And of course, going around the sphere, the velocities are headed in all different directions here. The velocities going out here, we would only see a second order shift. So that's negligible. So we're going to see photons coming out at all different frequencies here. So again, if we look at this, the blue line is what you would see if there were absolute, no velocities. So we see here, if you can make out these blue lines, we see spectra here. We see fluxes from limon alpha and helium alpha which are those optically thick lines. And they look pretty much like the emission coefficients modified by optical depth effects. When we start putting the velocity gradients on here, so if we go to the extreme one here, V over C of 0.3, now we see emission from the helium alpha which has been shifted out here quite a ways. We see, these are the blue ships from the material coming towards us. We also see red ships from the material headed away from us. And as a matter of fact, the emission from these features actually blends together in a sense because the blue ships and the red ships can overlap here. For smaller velocities, we see that happening but not quite as much. So it's only for the large velocity 0.03 which again is about the width of this line feature compared to its energy. And we get the blending, otherwise they remain, the features remain the state, but they don't look like the emission profiles anymore. So having large velocity gradients can really complicate the interpretation of a spectrum. Yes? J-Byr is an angular integral and yes, we are dealing with that because in the frame of the material, we integrate over the intensity to get the angle average intensity. So the total radiation density which is pumping that transition. Of course, that is coming because the radiation is non-local, that transition will be seeing radiation which was emitted. So the line central will be seeing radiation emitted at line center only from material which is traveling at the same velocity. From other velocities, it will be seeing radiation which has been redshifted or blueshifted. Okay, so now let's look at a bit more detail about what happens with the radiative transitions and in particular, again the simplest system we can have a two level atom. So we now put together a collisional radiative equation for this two level system and that collisional radiative system is just the single equation which is saying that the excitations out of the ground state, collisional excitations and photo excitations is equal to the de-excitation coming out of the upper state which is spontaneous emission, stimulated emission and collisional de-excitation where again this J bar is the integral of the angle average intensity over the line shape. And from detailed balance, we know the relation between the coefficients here and the collisional upward and downward excitations. And so if we derive the populations here and derive the absorption and emission coefficients, then we can derive the source function here. And the source function turns out to have this very simple form. It's a constant times J bar, this integral of the intensity plus something times the plane function. And the plane function comes in here from these collisional terms because the collisions are thermal. So they know about the Maxwellian distribution. Now this epsilon here, so the magnitude here, the epsilon is the collisional divided by the radiative rate. So since the radiative transitions are so strong unless you've got lots of electrons and collisions, this tends to be a small number. And so that the source function for this single level system or this single transition system is close to actually J bar, a little bit less than J bar and a small edmixture of the plane function put in. So the important thing here is that the source function, and remember this is the source function across the entire line profile is actually independent of frequency. This is in the approximation that the line is narrow and it's linear in the radiation field. So the most successful solution methods exploit this dependence. So here's a most common solution technique for this for a single line. Here's this characteristic system now that we wrote down yesterday. Again, this is just headed in a single, along a single characteristic. So we would have to have a series of these, but we can write the system and we can formally integrate the system and get a solution is that the intensity is given by some operator acting on the source function, which we can write down explicitly here. So this operator, the lambda operator only depends on the population. And we can get, if we solve the system, we get a numerical system, we only need to solve it for one quantity J bar because the J bar itself, if we then put it back into the system, will give us the rates and it will give us the populations. So this full system becomes nonlinear, but you can write down explicitly what this system is and come up with means to solve it. Fourthly, the dependence of the operator itself is usually pretty small. So we usually collect that as a first pass. So just some algebra here, some operator algebra to write down, okay, here's what the intensity looks like to get J bar here. We take the intensity, we integrate it over the line profile. We put in what the source function is, integrate it over angles as well. And we come up with the system here, which tells us what J bar is in terms of this lambda operator. Now notice, nothing here depends on J bar since there's only a very weak dependence of the lambda operator on radiation intensity itself. So this gives us an explicit system and so we can solve this directly for J bar. And in 1D, this becomes very efficient. However, this system couples together all the frequencies of the system, it couples together all the spatial points of the system. So it's, in practice, it becomes a very large system. However, there are a number of efficient frequency efficient solution techniques which have been derived for this, which approximate key parts of this operator lambda bar. Now notice, the other thing about this, this integrated operator lambda bar includes terms which look like one minus e to the minus tau. So this one minus lambda, we're going to take this and this is an inverse operator here, this amplifies the radiation field. So this explicitly exhibits the radiation trapping effect. So this is a very nice formal method for getting a solution. But then you need to go and build a numerical method for this. And then in a multi-level atom, you can do a generalization of this approach. For multiple lines, the source function looks like this. So we have to take now the emissivity for each transition. So the total emissivity divided by the total absorption coefficient. But various means of dealing with this have been proposed, have been used in the past. The first one was the ETLA extended to level atom where the assumption is made that in each transition, it's really only the piece dealing with that transition, which is the most important and everything else can be considered constant. And then you get again a source function, which is some constant times J bar for that system, plus another constant. You can also, you can do a better job on this if you linearize this with respect to a given transition or with respect to a sum of transitions. And again, you come up with form of the source function, which is a constant plus a linear term. And then you solve it as before. And actually, you can even do a partial redistribution case for this, even though now there's two different J bars, one to the absorption profile and one to the emission profile. So the other piece to do this is you need a numerical method for the transport operator itself. And there's two things that you need to worry about here. One is that the formal transport solution, which is just what the transport operator itself gives you has to be accurate when it has to be conservative. Has to be conservative, so to be energy conservative, that's what you're concerned about or photon conservative where that's appropriate. Has to be non-negative and it has to be second order accurate so that you reach the right limit in the limiting case of large optical depths. Plus, since remember, we're actually doing a time-dependent system here and we're taking kind of retarded times if we're doing a characteristic approach, it has to be causal. Now it turns out that obeying all these constraints has been rather difficult and it took probably a couple of decades for good numerical methods to be derived which obey all these constraints. But we do now have several options for that. Now in addition to that, once we've got a transport method, a transport operator, then we need some method to converge the solution of the full system which is the coupled systems that we've been talking about. So there's a few classes of that. I mean, you can try to do a full system solution but the size of these systems grows very quickly with either with spatial extent or number of transitions. So very few people try that. There's also an acceleration solution and I'll show you one of those. Or you can try to take pieces of the transport operator and put it into explicitly into say your collisional radiative equation and I'll show you something about there. But now when we talk about doing a radiation transport solution, then there's actually, then the two pieces that we need is we need some formal transport solution which is the transport operator and we need some method of solving a coupled set of equations. And the simplest thing that people use, the first thing they used was just iteration. So we call this source iteration or lambda iteration depending on what field you're coming from which is you take the radiation field you have at the current time, you evaluate the source function by doing the collisional radiative operator, collisional radiative system, you solve the radiation transport equation with that source function, you use the intensities to evaluate the temperatures and the populations again. So you solve a new collisional radiative system here, get new populations, get new opacities, emissivities, you put it back in, get a new source function and just iterate this. And that's a very simple way to do it. It's very simple to implement. It doesn't matter what your transport method was is you can implement this with your transport method. Now the disadvantage of this is it doesn't converge very well. For optically thin systems, it works just fine. But the number of iterations that this usually takes is proportional to the maximum optical depth or maximum optical depth squared. And since it's easy to get strong transitions with optical depth in the hundreds or thousands, this takes a lot of iterations. So actually if you implement one of these and you've got any significant optical depth in your system, you'll usually see this converge in a sense after a number of iterations, but the convergence could be false convergence. So here's another example, here's actually an example we've already done, which is this hydrogen, lime and alpha. Okay, now I told you that I solved this system self-consistently before, but I didn't tell you how I did it. So now we'll try it with source iteration. And so what I've done here started from the, started from the uniform case, evaluated the populations, the source functions, solved the radiation transport equation, gotten the radiation intensity, put that back in and iterated it. And each iteration I get specific intensity, which looks like this. So these are successive iterations and you can see it slowly building up to what I'm calling the self-consistence solution. And again, we can see the effects of radiation trapping here slowly building up. So basically the effects of a source iteration, they go for a maximum of one optical depth at a time. And so that gives you a minimum number of iterations of equal to the optical depth, but in practice in most situations, it's more like a random walk here. And so it takes the square of that number. So again, this is roughly 10 iterations here and it's coming close to the self-consistence solution, but it's not there yet. However, using a solution technique, which accounts for the fact that the source function actually is just a linear function of J bar and accounting for that, then a single iteration will get you the self-consistence solution because you can cast it as a linear system. So you have to be careful of this. The source iteration or just the formal iteration process can lead to fault conversions. Now, this also becomes a problem if you try to use Monte Carlo as your solution method because it's straightforward Monte Carlo simulation, which is you sample the emission distribution in space to create photons, which actually stand for a whole bunch of photons. You track the photons until they either escape the system or they do an interaction. So they can interact, they can be absorbed in the transition and be remitted in another direction. And finally, they'll be destroyed by a collision. So this is very easy to implement. You can do it for energy, any geometry. You don't have to worry about what the discretization is. You can put lots of details in there. It does have a problem with statistical noise, but we've got very fast computers, large memories now, so we can do lots of particles. However, we still have this problem of the inner devaluation of the coupled system. That with increasing optical depth, the convergence is very, very slow. So there is a semi implicit form of Monte Carlo, which transforms some of the absorption and emission events into an effective scattering, which gives you faster convergence here. And there actually is a form of implicit Monte Carlo, symbolic implicit Monte Carlo, which provides you a fully implicit solution. But again, it gives you an operator, which is much like the one I showed formally before, which gives you an operator, which couples all the frequencies together and all the spatial positions. So it ends up being a very large operator. In an effect, what you've done is use Monte Carlo as a very expensive method to construct this operator. So you can go to some other transport technique, and discrete ordinances is the one that I tend to use most. So the form of solution method is the discretized radiation transport equation. Now the angle dependent transport equation, not just the characteristic equation. So you get a bunch of integral differential equations, couple differential equations. So this does quite well with optically thin and optically thick regions. If you get a good modern spatial discretization, you get the optically thick limit correct. And it's a deterministic method, so you can actually iterate the solution of this. So you get a self consistent solution out. Now this has disadvantages as well, because we're choosing a set of directions. It means that we need to resolve all the angular features. And this becomes more and more difficult as the relative spatial length scales become more disparate in the system. So we get effects at different angles, which are called gray effects due to the preferred directions that we've chosen. So if you doing this in one D is actually fairly easy and you can convert it quite well. In 2D and 3D, it depends on exactly what your geometry is because you could need an enormous number of angles. And remember that we're not discretizing, we're discretizing the system in space and up to three dimensions. We're discretizing it in phase space as well. We're discretizing it in momentum, so we're discretizing it in frequency and in angles, and in time as well. So it's a seven dimensional discretization. However, these days with current solution techniques in the large computers, we can actually do this pretty efficiently up to several thousand frequencies, say on a three dimensional mesh. Now there are ways of accelerating this. Fortunately for the non-LTE case, there is a fairly simple method to iterate to accelerate this kind of solution. And that takes account of the fact that in the single transition single line case, the fact that the source function is proportional to the radiation field net line is equivalent to saying that the most important thing, the most important physical effect in there is the scattering of the photons in frequency space as opposed to in spatial, as opposed to in position space. And so if we take account of the frequency scattering and we can do that on a very local basis, then we don't need to worry so much about the coupling of different spatial points. And this actually comes out, it gives a very efficient acceleration method so that it's actually not too difficult to do a full non-LTE mind transport solution in 3D using a method like this. Let me talk just briefly about one more method which is commonly used in the spectroscopy community which is escape factors. So the escape factor approach is something different. It's taking part of the radiation transport operator and putting it into the collisional radiative equations. And what this escape factor is being used to do is it's being used to eliminate the radiation field from the collisional radiative equations. So the escape, I'll tell you in just a moment how to calculate the escape factor but you take the net rate. So this is the stimulated rate with upward and downward and cast them in terms of a fraction of the spontaneous rate. So the advantage here is it's very fast. There's no transport solution. So you can put it into your code with no problem at all. The disadvantage is that, well there's no transport solution so it's missing all those details. The escape factors depend very strongly on the line profiles and on the system geometry and basically the solution you get from an escape factor is what you get. There's no real way to iteratively improve that solution. So this can save you a lot of time in coming close to a detailed solution but it's not going to get you there. Basically the escape factor is built off a single flight escape probability looking at every frequency. What's the optical depth to the boundary and calculating an escape factor in this way by averaging e to the minus optical depth over the line profile. And there actually is some, the radical justification for this is on average, average meaning a spatial average weighted by mission. This is exactly the correct thing to do but this is very dependent on the line profile and the geometry. And so there are many different escape factors in the literature and it's complicated by overlapping lines, top shifts, et cetera, et cetera. So you can use these but use them very carefully. So each of these solution techniques is actually a very large topic in its own right which we certainly don't have time to go through but I do have some references to show you here and some references to recommend. So the classic reference in this field has been this second edition of stellar atmospheres by Demetri Mahalas. I'm happy to say that even though this is out of print, this next reference here by Hubeini and Mahalas is basically it's the third edition of stellar atmospheres and this book has a lot of information on collisional radiative equations, how you calculate the rates, how you do the radiation transport in particular some of these escape factors stuff and put this together. It's actually quite weak on the formal radiation transport methods but in the rest it's very good and added bonuses this is available in paperback so it's quite inexpensive. Now to get a good overview of what radiation transport is about in various effects, I highly recommend this next book, Radiation Hydrogenomics by John Caster. I think it's also available in paperback and there are some more detailed references down here, kinetic theory of particles in photons. The theoretical foundations of non-LT plasma spectroscopy is very useful if you want to get into details of some of these topics and it's out of print but you can find copies. Radiation trapping in atomic vapors talks about a lot of different methods of approaching radiation, non-LT radiation transport and radiation trapping. Radiation processes in plasmas talks about plasma effects and radiation transport which is a topic that we haven't touched at all. And then currently there aren't very many good references on formal methods, discretization methods doing radiation transport but the most successful methods have actually been developed not in the field of photon radiation transport but neutron transport and so this book here, this last book down here will give you a good feel for what this field is like. So I apologize for running over time and delaying lunch but I think that's as far as I've gotten. These lectures will be available, I'll tell you in the lab this afternoon how you can get a hold of a copy of these lectures including the references.