# Plasma luminosity

Staff Emeritus

## Main Question or Discussion Point

I'm interested in the general topic of the luminosity of plasmas. I would expect that this topic would come up most often in astrophysics, so that's where I'm posting the question as to get the best shot at getting an answer.

I'm interested in applying the information in a rather unusual way, though. Basically, I want to be able to guess within 1-2 orders of magnitude the amount of heat radiated by hypothetical ultra-high temperature, thrust, and ISP rocket exhausts.

The most reasonable exhausts would probably
a) pure hydrogen
b) "fusion exhaust" (the exact composition could vary, though all variants would include helium).

So if I knew some thermodynamic quantities like pressure and temperature, is there any reasonably simple way to roughly (1-2 orders of magnitude) figure out the rate at which the plasma would lose energy?

Black body laws gives me radiated power as the fourth power of the temperature, but I'm pretty sure this is not a sufficiently good approximation, that real plasmas will radiate a lot less.

If the radiated power went as the 4th power temperature , and since ISP should scale as sqrt(temperature) for a simple rocket, every doubling of the ISP would increase the cooling problem by a factor of 2^8 = 256.

However, while the black-body approximation is simple, I don't think it's reasonable.

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SpaceTiger
Staff Emeritus
Gold Member
What are the typical densities and temperatures? Is the initial heating effectively instantaneous?

Astronuc
Staff Emeritus
The 'luminosity' or more exactly 'energy loss' is a technical challenge for controlled fusion systems. There are correlations for particular contributions from recombination, brehmsstrahlung, and cyclotron radiation. The cyclotron radiation is a consequence of the confining magnetic field.

I would recommend looking at the problem parametrically in terms of plasma temperature and pressure, from which one would determine density. Also, one has to look at the ion species, because Z affects the energy losses from recombination and brehmsstrahlung, as the energy states of an atom/ion depend on Z (obviously).

The stongest superconducting magnets generate a field of about B = 20T, but one could go higher just to evaluate the plasma pressure. The magnetic pressure is given by P = B2/2$\mu_o$.

A typical plasma density would be on the order of 1020 ions/m3 and electrons would have similar density. One could try higher densities subject to the constraints of plasma temperature and pressure.

Staff Emeritus
I'm estimating that pressures would be in the 3000 psi range (i.e about 20 Mpa in MKS units), and that temperatures would range from 4,000K on up - way up. The higher the better, the question is how high can we get it without melting anything.

Heating would be done by anti-matter injection, I really don't have much of a handle on how that would actually work in practice. I suppose that the magnetic bottle would have to contain the charged pions from the anti-matter decay, while they transfer their energy to the plasma - the uncharged pions would escape, causing major radiation issues in the process :-(.

The pressure estimates were made by simply matching the chamber pressure of a SSME.

Temperature estimates were made by oversimplifying eq 12 in
http://members.aol.com/ricnakk/th_nozz.html [Broken]

i.e Pe=0, k=1.6, M=.001 (ionized H) or Pe=0, k=1.4, M=.002 (non-ionized H).

After correcting some initial numerical errors, this gives a very modest temperature to match the space-shuttle performance (only 700 degrees for non-ionized H - this is possible only because of the low molecular weight of the exhaust).

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Astronuc
Staff Emeritus
Well, at those temperatures, it won't really be a plasma, and the H gas densities will be much higher.

The ionization energy of H is 13.6 eV and with 1 eV = 11605 K, this is equivalent to ~158000 K. Fusion plasmas are on the order of 10 keV and higher or 116 million K, hence the low particle density.

Anti-matter, eh? Figuring the pion and muon scattering should be interesting, and the fact that (IIRC) half the reactions may produce neutral pions.

The other feat in this idea is getting the anti-matter into the chamber of high pressure H gas. The anti-matter has to pass through a magnetically confining vacuum chamber, so as to prevent annihilation before the propulsion chamber.

Staff Emeritus
As I was writing this out, I realized that there was a significant gain in ISP to be made just by switching to a lower molecular weight exhaust. This is actually somewhat old news, the nuclear thermal fission rockets like Nerva used this idea.

But if we push the idea further, and start raising the "combustion" chamber temperature of the rocket - chemical combustion runs at maybe 3,000K, so say we go to 30,000K, or 300,000K, and 3,000,000K or however high we can get - when do we start to run into the cooling problems?

This is rather a subjective question, of course, but if we are using superconductors to build a magnetic bottle (and a magnetic nozzle), keeping them cold next to a radiating plasma is going to start to become difficult and eventually practically impossible. I want to get some sort of handle on when "eventually" occurs.

Probably the 4 points above would give me a rough idea of what I want to know - what the total radiation flux would be from hydrogen at 3000, 30,000, 300,000 and 3,000,000K at a constant pressure of 20 Mpa.

The thrust of the rocket should be set by the chamber pressure, multiplied by some effective area. So we can imagine keeping the pressure at current values (20 Mpa or so) to keep thrust at current values, and ask what happens to the radiated energy as we raise the chamber temperature. Black body radiation would give a 4th power law vs temperature, but I don't think that's going to be within an few order of magnitudes of the correct answer unfortunately.

[add]It could also be useful to know what happens if we raise or lower the pressure by a factor of 10, i.e. what happens at 2 Mpa and 200 Mpa, just to get some clue as to whether we want big, low-pressure bottles or smaller, higher-pressure ones.

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Why do you think the blackbody curve would be insufficient?

The Sun's surface (the only high temperature plasma I can say much about) is pretty close to a blackbody (closer than 1 order of magnitude). Just looking at the last zone of some simulation data I have, the difference between $$A \sigma T^4$$ and the actual luminosity is just a factor of two. If I try to be accurate and more properly use the temperature at optical depth $$\tau=1$$ then the difference is only 23% (with the Stefan Blotzmann Law being higher than the luminosity). And thats is with a very rough determination of the temperature at $$\tau=1$$(more technically error in the determination of the radius for which $$\tau=1$$). If I allow for just 10% error in the temperature I used (compared to the 'real' temperature at $$\tau=1$$) then the temperature giving perfect agreement between Stefan-Boltzmann and the real luminosity is within reach.

The one possible complaint I can think of is that the Stefan-Blotzmann law doesn't take into account that only discrete sets of energies can be emitted by the gas, but thats just an issue of changing an integral over all possible energies into a sum over all allowed energies in the derivation of the law. But the discrepancy is already less than than the level of accuracy you want, so I don't see what the issue is.

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Staff Emeritus
Let's look at the laboratory standard for black body radiation per the Wiki

n the laboratory, the closest thing to black-body radiation is the radiation from a small hole entrance to a larger cavity. Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped and is almost certain to be absorbed by the walls in the process, regardless of what they are made of or the wavelength of the radiation (as long as it is small compared to the hole). The hole then is a close approximation of a theoretical black body and if the cavity is heated, the spectrum of the hole's radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous, and will not depend on the material in the cavity (compare with emission spectrum). By a theorem proved by Kirchhoff, this curve depends only on the temperature of the cavity walls.
So if the light in the rocket exhaust had to interact with many, many atoms of the same temperature before we saw it, I'd agree it would have to be close to a black body of that temperature.

The Sun is pretty close to a black body just because it's so thick - gamma rays in the interior part of it interact with a heck of a lot of atoms on the way out.

Unfortunately, I don't think the rocket exhaust qualifies for the defintion of a black body. This was actually pointed out to me by someone else when I floated the T^4 argument. I tend to believe the criticisms, though, because I don't think the necessary precondition of "enough bounces" has been met for the black body approximation to work.

While I've gotten criticism of the T^4 argument (which I believed), I haven't gotten any insight into the correct way to approach the problem.

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Astronuc
Staff Emeritus
I'll attempt to answer some questions.

Using just Stefan-Boltzmann would not work because at low temperatures, conduction and convection also transmit thermal energy, and at much higher energies there are effects like brehmsstrahlung and cyclotron radiation. When one discusses a ionized gas or plasma, one does not use Stefan-Boltzmann, but rather other factors like recombination rates, brehmsstrahlung and cyclotron radiation. On has to consider the significance of a 'blackbody'. At 5800 K or thereabouts, the sun's surface is relatively cold, as compared to the core (~13.6 million K) or corona (~5 million K). At 5800 K, one actually finds neutral atoms, briefly, since gamma radiation readily ionizes them.

I believe at low temperatures, SB would underestimate power loss (radiant losses) because conduction and convection are competing, and at very high temperatures SB would overestimate radiant loss (I need to think about the latter).

In engineering, a factor of 2 or even 20% undercertainty is large.

The problem is challenging for several reasons.

As for material behavior, probably 1800 K is where one starts to worry about cooling, i.e. that's essentially the max temperature for long term operation of a solid. Remember the shuttle motor, which is operating at about 3573 K (http://en.wikipedia.org/wiki/Space_Shuttle_Main_Engine) only operates for about 2.5 minutes, and the nozzle is cooled IIRC but he liquid H. Long term operation is limited by metal creep which is limited to about 1% or so in order to avoid creep rupture.

If one is using a propulsive device for hours, days, weeks, months, then erosion and creep become critical technical limits.

An antimatter system is very complicated because of the pions (particularly neutrals) and muons, and gamma-radiation. The gammas pass right through the propellant and into the solid structure. Besides the structure, there is the matter of thermal/radiation burden on any superconducting material. If the temperature rises above a critical point, one losses superconductivity, the magnetic field drops and the plasma escapes magnetic confinement.

Determining the equations of state for this problem is probably a PhD topic. I don't think anyone has developed a complete physics package, which would provide an adequate solution. A key factor is the energy deposition, i.e. anti-matter/matter reaction source as a function of position, and linear energy transfer methods would seem appropriate.

The engineering part enters the problem in the design and construction of the propulsion system. One particluarly difficult matter is the chamber in which the anti-matter will be combined with matter. The entry port has to allow the anti-matter to pass through to the matter, but not allow the matter to enter the anti-matter, i.e. a one-way stream. Otherwise, one has a window (solid/matter) which would slowly be eroded away by interactions with anti-matter.

Thank you, pervect, for a most interesting problem.

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pervect said:
Let's look at the laboratory standard for black body radiation per the Wiki

So if the light in the rocket exhaust had to interact with many, many atoms of the same temperature before we saw it, I'd agree it would have to be close to a black body of that temperature.

The Sun is pretty close to a black body just because it's so thick - gamma rays in the interior part of it interact with a heck of a lot of atoms on the way out.

Unfortunately, I don't think the rocket exhaust qualifies for the defintion of a black body. This was actually pointed out to me by someone else when I floated the T^4 argument. I tend to believe the criticisms, though, because I don't think the necessary precondition of "enough bounces" has been met for the black body approximation to work.

While I've gotten criticism of the T^4 argument (which I believed), I haven't gotten any insight into the correct way to approach the problem.
So essentially, you're arguing that the problem lies in the plasma(emitted photons included) being in a non-equilibrium state? This sounds valid, I just want to be clear.

Astronuc said:
I'll attempt to answer some questions.

Using just Stefan-Boltzmann would not work because at low temperatures, conduction and convection also transmit thermal energy, and at much higher energies there are effects like brehmsstrahlung and cyclotron radiation. When one discusses a ionized gas or plasma, one does not use Stefan-Boltzmann, but rather other factors like recombination rates, brehmsstrahlung and cyclotron radiation. On has to consider the significance of a 'blackbody'. At 5800 K or thereabouts, the sun's surface is relatively cold, as compared to the core (~13.6 million K) or corona (~5 million K). At 5800 K, one actually finds neutral atoms, briefly, since gamma radiation readily ionizes them.

I believe at low temperatures, SB would underestimate power loss (radiant losses) because conduction and convection are competing, and at very high temperatures SB would overestimate radiant loss (I need to think about the latter).

In engineering, a factor of 2 or even 20% undercertainty is large.
Yes, but pervect was asking for order of magnitude accuracy. Not high precision, and I still think SB will be within that range. If you want 'good' accuracy, then yes, its probably not satisfactory. However, using SB deep in the solar interior (under the convection zone) works for determining the radiative temperature gradient. And temperatures there are a few million Kelvin. However, you also have a system that is in local thermodynamic equilibrium, which may well not be the case for this confined plasma.

Convection shouldn't be an issue in in considering the way the plasma heats the things around it, convection simply transports heat within the plasma. Assuming a vaccuum between the confined plasma and the surrounding enclosure, radiation heating should be the only concern. Convection within the plasma would affect just the spatial distribution of the heat within the plasma (which would affect the details of how it radiates the heat away yes, but its still all being radiated).

Astronuc
Staff Emeritus
Convection shouldn't be an issue in in considering the way the plasma heats the things around it, convection simply transports heat within the plasma. Assuming a vaccuum between the confined plasma and the surrounding enclosure, radiation heating should be the only concern. Convection within the plasma would affect just the spatial distribution of the heat within the plasma (which would affect the details of how it radiates the heat away yes, but its still all being radiated).
When I referred to convection, I am thinking of 'forced convection' as a heat transfer method from propellant to nozzle. This would be at low temperatures <3000 K where there is essentially no plasma, i.e. very low ion fraction and the conduction/convection (basically heat transfer by atomic collision and conduction electrons) of heat is much greater than radiative heat transfer. One needs a certain degree of ionization in order to use magnetic confinement. The ionization potential of hydrogen is 13.6 eV (~ 158,000 K), and even there one has neutrals that would leak out of magnetic confinement and transfer heat to the containment system.

If one wants to really model this problem over a range of temperatures from 3000K to 3 billion K, then one has to describe all the phenomenon. It is not trivial, because if it was, someone would have done it already, and I am pretty sure it hasn't been done.

The problem would be simplified if it had a cut-off at say 1 keV (11.6 million degrees), or perhaps 136 eV (1.58 million K). The actual cutoff depends on the fraction of neutrals that can be tolerated, and I imagine that number is very low, because the steady-state ion density imposes a certain temperature (energy) distribution in the system.

I suspect that the heat transfer in the propellant will be primarily conduction/diffusion (with virtually no convection - at least normal to the flow) because of the size.

Astronuc said:
When I referred to convection, I am thinking of 'forced convection' as a heat transfer method from propellant to nozzle. This would be at low temperatures <3000 K where there is essentially no plasma, i.e. very low ion fraction and the conduction/convection (basically heat transfer by atomic collision and conduction electrons) of heat is much greater than radiative heat transfer. One needs a certain degree of ionization in order to use magnetic confinement. The ionization potential of hydrogen is 13.6 eV (~ 158,000 K), and even there one has neutrals that would leak out of magnetic confinement and transfer heat to the containment system.
Ok, I see what you mean. I was visualizing this as a fully ionized plasma, magnetically confined, with essentially a vaccuum between the plasma and the chamber walls, but I suppose for engineering purposes one cannot assume that.

If one wants to really model this problem over a range of temperatures from 3000K to 3 billion K, then one has to describe all the phenomenon. It is not trivial, because if it was, someone would have done it already, and I am pretty sure it hasn't been done.
Certainly not, I'm simply trying to work through defining the actual problem in my own head, thats all. I don't normally deal with plasmas that are not in local thermodynamic equilibrium (Where the kinetic temperature and the radiative temperature are the same, i.e. its a blackbody radiator), so I need to work through which assumptions are not applicable to this problem. It is a very interesting problem.

The problem would be simplified if it had a cut-off at say 1 keV (11.6 million degrees), or perhaps 136 eV (1.58 million K). The actual cutoff depends on the fraction of neutrals that can be tolerated, and I imagine that number is very low, because the steady-state ion density imposes a certain temperature (energy) distribution in the system.

I suspect that the heat transfer in the propellant will be primarily conduction/diffusion (with virtually no convection - at least normal to the flow) because of the size.

Astronuc
Staff Emeritus
Staff Emeritus
franznietzsche said:
Yes, but pervect was asking for order of magnitude accuracy. Not high precision, and I still think SB will be within that range. If you want 'good' accuracy, then yes, its probably not satisfactory.
If you can give referenes or arguments to support that statement, I'd be satisfied. However, I am not currently convinced that that this is true.

Of course, part of the reason I'm asking the question is to get more insight into the physics.

Consider hydrogen gas (for definiteness) at a low temperature at reasonable pressure levels. An ideal black body would absorb all incoming radiation - cold hydrogen at the pressures we are talking about would be quite transparent to optical frequencies, absorbing almost no radiation at those frequencies.

Therfore it seems problematical to me to assume that the hydrogen at the pressures we are talking about is a "black body". It is not meeting the technical conditions that a black body is supposed to have - furthermore, it is not even close to meeting those conditions, being virtually transparent.

[add]. I'm fairly sure that emissivity should be equal to absortivity, from thermodynamic arguments - the emissivity being an extra multiplicative factor in the Steffan-Boltzman law. We have seen that the absoptivity of hydrogen is quite low at optical frequencies, (not 1 as it would be for a black body, nor even .3 to be within an order of magnitude of a black body). As the absoprtivity is so low, I would expect that the emissivity of hydrogen would be likewise low.

Looking at the details of the black body formula, it looks like we want to multiply the density of states at a given frequency by hv/(e^hv/kt - 1)

http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html

The black-body law comes from a v^2 density of states.

Thus I suppose we should be asking what the density of states in the gas / plasma is. Unfortunately I still don't know how to go about computing or even roughly estimating that.

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Astronuc
Staff Emeritus
Just a concept on a chamber for high temperature rocket motor.

http://www.afrlhorizons.com/Briefs/Jun05/ML0403.html [Broken]
Integrated High-Payoff Rocket Propulsion Technology Program Materials Working Group

For any rocket motor, a detailed analysis of performance involves the solution of a basic set of equations: mass (continuity), momentum, and energy.

A more basis steady-state analysis can be accomplished, but one must define the distribution of energy generation and steady-state heat transport - which involves conduction, convection and radiation.

For a matter-antimatter system, the energy distribution is complicated because energy is transmitted by charged particles and gamma radiation, in addition to more conventional mechanisms.

The radiative transport and reaction rates (and energy source) depend on the local particle densities, so one must know the density profile of all species involved.

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Staff Emeritus
I'm still trying to understand how to do the simplest, most basic, steady state analysis.

The references you posted in another thread were a great help - unfortunately, I'm having some troubles getting Cygwin up and running (which I need to get Xstar up and running). At least they give me some idea of how complex the problem really is.

As far as serious, detailed proposals for antimatter propoulsion go, the ones that I think are most plausible for "near term" (i.e. maybe sometime in my lifetime) use antimatter to induce fission, fission fragments, fusion, or both. Examples include the Penn State ICAN-II or AIMStar proposals

http://www.engr.psu.edu/antimatter/documents.html [Broken]
(has both, and some other papers on antimatter)

Some of the fission-fragment designs are also interesting, for instance

http://www.niac.usra.edu/files/studies/abstracts/850Howe.pdf

(the above is not particulary detailed, alas).

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Astronuc
Staff Emeritus
I have some texts, which I will check, and one is on hydrogen.

I think the best way to address the problem is look at some reasonable steady-state densities and energy production rates (based on particles densities and reaction cross-sections), then applying pressure constraints, determine the corresponding temperature, from which one can work out the heat/energy transfer mechanisms.

Otherwise, one works backwards from a confinement system, lets say some ring structure of tungsten other high temperature structure, and figure what kind of cooling system is required to keep it at some appropriate temperature (~1800 K) at a particular heat flux. With that heat flux, determine the energy in the propellant.

Either way, one must know the heat transfer properties.

If the propellant is a plasma, then it can be magnetically confined. Heat will be radiated out of the plasma, but not necessarily at BB radiation. IIRC, BB radiation applies to atoms (i.e. bound states) - not ionized nuclei and free electrons.

If the propellant is cool (cold) then it may not be ionized, and therefore it would contact the confinement structure and one has then to determine conduction/convection heat transfer.

Let me try developing some numbers, particularly on the number of neutrals and ions as a function of temperature, and some equations.

Staff Emeritus
I'm finding xstar's online documentation of the physics
to be useful, but a bit terse to follow.

For example:

http://heasarc.gsfc.nasa.gov/xstar/docs/html/node106.html

What is a "LTE ion density"? (TE = Thermodynamic equilbirium, perhaps?)

Are eq 3 and 4 the "Milne relationship?

It the variable $\varepsilon$ = h*frequency? What is $\varepsilon_{th}$?

Also, is it correct to say that for line emission,
opacity = $\kappa = n_i \sigma_{pi}$

n_i being the number density of an ion, and $\sigma_{pi}$ being the photo-ionization cross section

and is it correct to say that

optical depth = $\tau = \int \kappa dR$

Astronuc
Staff Emeritus
Pervect, indeed the documentation is terse, and it seems to reflect the expectation that the user is familiar with the calculations, i.e. it's not newbie friendly.

Anyway - LTE = Local thermodynamic equilibrium
http://en.wikipedia.org/wiki/Thermal_equilibrium

I believe $\varepsilon_{th}$ is an energy threshold or thermal energy, perhaps. Perhaps is does coincide with photon energy, but I am thinking of an ionization energy perhaps.

n_i being the number density of an ion, and being the photo-ionization cross section
this is correct.
----------------------------------
http://heasarc.gsfc.nasa.gov/xstar/docs/html/node105.html
We treat heating and cooling by calculating the rate of removal or addition of energy to local radiation field associated with each of the processes affecting level populations. Heating therefore includes photoionization heating and Compton heating. The cooling term includes radiative recombination, bremsstrahlung, and radiative deexcitation of bound levels. Cooling due to recombination and radiative deexcitation is included only for the escaping fraction, as described elsewhere in this section.
I think reference Osterbrock, D., 1974, Astrophysics of Gaseous Nebulae San Francisco: Feeman) may cover this.
------------------------------------

Meanwhile - http://www.atom.physto.se/~dnikolic/DraganNikolicThesis_WEB.pdf [Broken]
Autoionizing states and their relevance in electron-ion recombination

-------------------------------------

Something else to investigate
http://cxc.harvard.edu/atomdb/physics/plasma/plasma.html

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Chronos
Gold Member
High energy reactions are user-unfriendly. Even your garden variety fusion reactor would cook the humans before they manage to escape the solar system. Unless, of course, you shield them. Unfortunately, the shielding required is very massive. I recall playing around with a very cleverly designed water shield one time. We dang near made it to the Oort cloud before getting deep fried. We were very thirsty by then, so the crew did not mind dying all the much.

Astronuc
Staff Emeritus
$$T\,=\,Wb / m^2\,=\,V-s/m^2$$

$$V\,=\,W/A\,=\,J/C$$

$$Wb\,=\,V-s$$

$$H\,=\,V-s/A\.=\.J/A^2\,=\,Wb/A$$

$$P\,=\,N/m^2\,=\,J/m^3$$

from

$$P\,=\,\frac{B^2}{2\mu_0},\, \,with\,\mu_0\,=\,4\pi\,\times{10}^{-7}\,H/m$$

$$\frac{T^2}{H/m}\,=\,\frac{(\frac{V-s}{m})^2}{H/m}\,=\,\frac{(\frac{V-s}{m})^2}{(V-s)/(A-m)}\,=\,\frac{V-s}{m^3}\,A\,=\,\frac{J/C-s}{m^3}(C/s)\,=\,J/m^3\,=\,Pa$$

With B = 20 T, P = 159 MPa, but this is about the maximum field.

With B = 15 T, then P ~90 MPa.

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