As many people will have read there was a glitch in the surface temperature record reporting for October. For many Russian stations (and some others), September temperatures were apparently copied over into October, giving an erroneous positive anomaly. The error appears to have been made somewhere between the reporting by the National Weather Services and NOAA’s collation of the GHCN database. GISS, which produces one of the more visible analyses of this raw data, processed the input data as normal and ended up with an October anomaly that was too high. That analysis has now been pulled (in under 24 hours) while they await a correction of input data from NOAA (Update: now (partially) completed).
There were 90 stations for which October numbers equalled September numbers in the corrupted GHCN file for 2008 (out of 908). This compares with an average of about 16 stations each year in the last decade (some earlier years have bigger counts, but none as big as this month, and are much less as a percentage of stations). These other cases seem to be mostly legitimate tropical stations where there isn’t much of a seasonal cycle. That makes it a little tricky to automatically scan for this problem, but putting in a check for the total number or percentage is probably sensible going forward.
It’s clearly true that the more eyes there are looking, the faster errors get noticed and fixed. The cottage industry that has sprung up to examine the daily sea ice numbers or the monthly analyses of surface and satellite temperatures, has certainly increased the number of eyes and that is generally for the good. Whether it’s a discovery of an odd shift in the annual cycle in the UAH MSU-LT data, or this flub in the GHCN data, or the USHCN/GHCN merge issue last year, the extra attention has led to improvements in many products. Nothing of any consequence has changed in terms of our understanding of climate change, but a few more i’s have been dotted and t’s crossed.
But unlike in other fields of citizen-science (astronomy or phenology spring to mind), the motivation for the temperature observers is heavily weighted towards wanting to find something wrong. As we discussed last year, there is a strong yearning among some to want to wake up tomorrow and find that the globe hasn’t been warming, that the sea ice hasn’t melted, that the glaciers have not receded and that indeed, CO2 is not a greenhouse gas. Thus when mistakes occur (and with science being a human endeavour, they always will) the exuberance of the response can be breathtaking – and quite telling.
A few examples from the comments at Watt’s blog will suffice to give you a flavour of the conspiratorial thinking: “I believe they had two sets of data: One would be released if Republicans won, and another if Democrats won.”, “could this be a sneaky way to set up the BO presidency with an urgent need to regulate CO2?”, “There are a great many of us who will under no circumstance allow the oppression of government rule to pervade over our freedom—-PERIOD!!!!!!” (exclamation marks reduced enormously), “these people are blinded by their own bias”, “this sort of scientific fraud”, “Climate science on the warmer side has degenerated to competitive lying”, etc… (To be fair, there were people who made sensible comments as well).
The amount of simply made up stuff is also impressive – the GISS press release declaring the October the ‘warmest ever’? Imaginary (GISS only puts out press releases on the temperature analysis at the end of the year). The headlines trumpeting this result? Non-existent. One clearly sees the relief that finally the grand conspiracy has been rumbled, that the mainstream media will get it’s comeuppance, and that surely now, the powers that be will listen to those voices that had been crying in the wilderness.
Alas! none of this will come to pass. In this case, someone’s programming error will be fixed and nothing will change except for the reporting of a single month’s anomaly. No heads will roll, no congressional investigations will be launched, no politicians (with one possible exception) will take note. This will undoubtedly be disappointing to many, but they should comfort themselves with the thought that the chances of this error happening again has now been diminished. Which is good, right?
In contrast to this molehill, there is an excellent story about how the scientific community really deals with serious mismatches between theory, models and data. That piece concerns the ‘ocean cooling’ story that was all the rage a year or two ago. An initial analysis of a new data source (the Argo float network) had revealed a dramatic short term cooling of the oceans over only 3 years. The problem was that this didn’t match the sea level data, nor theoretical expectations. Nonetheless, the paper was published (somewhat undermining claims that the peer-review system is irretrievably biased) to great acclaim in sections of the blogosphere, and to more muted puzzlement elsewhere. With the community’s attention focused on this issue, it wasn’t however long before problems turned up in the Argo floats themselves, but also in some of the other measurement devices – particularly XBTs. It took a couple of years for these things to fully work themselves out, but the most recent analyses show far fewer of the artifacts that had plagued the ocean heat content analyses in the past. A classic example in fact, of science moving forward on the back of apparent mismatches. Unfortunately, the resolution ended up favoring the models over the initial data reports, and so the whole story is horribly disappointing to some.
Which brings me to my last point, the role of models. It is clear that many of the temperature watchers are doing so in order to show that the IPCC-class models are wrong in their projections. However, the direct approach of downloading those models, running them and looking for flaws is clearly either too onerous or too boring. Even downloading the output (from here or here) is eschewed in favour of firing off Freedom of Information Act requests for data already publicly available – very odd. For another example, despite a few comments about the lack of sufficient comments in the GISS ModelE code (a complaint I also often make), I am unaware of anyone actually independently finding any errors in the publicly available Feb 2004 version (and I know there are a few). Instead, the anti-model crowd focuses on the minor issues that crop up every now and again in real-time data processing hoping that, by proxy, they’ll find a problem with the models.
I say good luck to them. They’ll need it.
>Why does it warm night-time temperatures more than daytime temperatures?
Ray, I have stopped using this argument since I read this in the AR4:
“The global average DTR has stopped decreasing. A
decrease in DTR of approximately 0.1°C per decade was
reported in the TAR for the period 1950 to 1993. Updated
observations reveal that DTR [Diurnal Temperature Range]
has not changed from 1979 to 2004 as both day- and night time temperature have risen
at about the same rate. The trends are highly variable from
one region to another. {3.2}”
AR4 Technical Summary: TS.3.1.1 Global Average Temperatures
This is now becoming dated, but anyhow….
Martin (471), “rigorous” was a poor term for my argument as some of the assumed mathematics are quite detailed. But, if one can get out of the doctrinaire belief syndrome, the fact that molecular absorption, emission and energy transfer is arguably one of the least understood physical process in climatology (I don’t mean at the basic level but at the detailed atomic level) does not mesh well with “precision”. I’m glad you see the obvious that (5.35)ln(C/C_o) is not set in stone as many others have effectively claimed.
This is over-simplified, I admit, but you’re suggesting I use the output to prove the input function of the model.
Hank (475), well, all a matter of degree, and I don’t disagree in any substance what you say here. But my point/suggestion/question is that there MIGHT be a pony in there (or might not), not IS; versus the dogmatic assertions that there is NO pony, NADA, ZERO, NEVER HAS BEEN, NEVER WILL BE, ABSOLUTLY IMPOSSIBLE, UNASSAILABLY PROVEN — all for the sacrosanct (5.35)ln(C/C_o). I can’t argue for IS for the reasons you and Ray cite.
Phil, when did calculus first refute algebra?? What does the rate of forcing change with concentration have to do with anything? Depending on the constants, the forcing log function is larger (and so is its differential, BTW!) than a linear function up to a certain concentration. THEN the log function is less.
501: So if you admit that it has stopped, what caused it in the first place if not CO2? After all, an action that didn’t happen cannot stop. So you are saying it happened before.
So what did it?
Hank, “thicker”? No, more dense. There were models and some evidence from satellite 3D stereographs showing the sunspot was higher than the surrounding gas. STEREO A/B may have something newer either showing it as a miscalculation or definitively true but I couldn’t find anything.
And how would I know a link to something I don’t know exists? The only way I could follow your request is if I knew I was lying.
So that’s a non-starter.
Snorbert how does the icy water near the pole get to 10C?
Sheesh Mark (503), why troll so hard to start a fight? To answer your question: “I dunno, you tell me.” I respect Ray’s comments however and I would like to know more about the topic. In general I like to stick to the statements made in the AR4. Besides 25 years is getting close to “climate”.
BTW, “501” is the name of my pants. You can call me “arch”.
Rod asks: “What does the rate of forcing change with concentration have to do with anything?”
Well, Rod, what we’re talking about is climate CHANGE, so it is precisely the change in forcing that we are interested in.
Arch, I stand corrected. I’d read this, but it slipped my mind. Thanks. Any interpretation why?
> how would I know a link to something
> I don’t know exists?
Teh Google Scholar. Useful for checking whether what we believe can be cited to a current scientific source
Mark, I didn’t think that vibration relaxation (emission) was the same process that generates blackbody emission ala Stefan’s law, though there seems to be some confusion here, not the least in my mind. I don’t think vibration emission is the same as stellar blackbody emission; is there some combo radiation taking place on stellar surfaces?
Ray, I don’t materially disagree with your 466 post (other than that pesky “log increases slower than linear”, which does depend on the constants and ratios…); it’s pretty much what I’ve been questioning. But, I’m not sure of your justification for “only a few percentage points change”; what is a few? two? 25? Why not 5.35 to 2.0? (at least at higher concentrations…)?
Ray (707), “the change in forcing” is not the same thing as the rate of the change in forcing.. (moving from the ridiculous to the sublime) ;-)
Rod, we know that logarithmic forcing works over a broad range of concentrations. As you move outside of that range, you would expect to see gradual changes. If we were to characterize it, we might expect things to follow logx(1+ax+bx^2…). (Really, this is just a sort of Taylor series expansion) The thing is for the log forcing to have worked at all, a and b have to be small.
And the change in forcing is proportional to the rate of change. Also, as to blackbody emission–any mode that is thermally excited contributes to it. So, if the vibrational mode is excited, it’s part of the blackbody spectrum. Again, I strongly recommend the treatment in Landau and Lifshitz, Statistical Mechanics.
Ray, but vibration modes are not (classically) thermally excited. [It’s deja vu all over again ;-) ] On the other hand radiation is radiation: if its emission cooled the emitting surface ala blackbody, or if it did not cool the emitting material ala vibration relaxation, I don’t suppose the radiation field much cares… I’m thinking aloud here; make any sense?
I have yet to see Ray make any serious error, people should listen to him.
Re #502
“Phil, when did calculus first refute algebra??”
When the algebra is being misused!
“What does the rate of forcing change with concentration have to do with anything?”
Actually it’s the topic of this thread!
At very low concentration the forcing changes linearly wrt concentration i.e. constant slope, after a certain threshold the slope starts to decrease following a ln curve, i.e. the forcing is less than it would have been if the forcing had continued the linear response.
“Depending on the constants, the forcing log function is larger (and so is its differential, BTW!) than a linear function up to a certain concentration. THEN the log function is less.”
It can’t happen that way!
RodB: No, you aren’t making much sense.
The kinetic energy of any molecule in an ideal gas is a spectrum of velocities (and velocity=enedgy). The IR band is of a similar value of absorbtion energy as the top end of the kinetic energy within that gas. Since the atmosphere at STP is VERY concentrated, the number of collisions per unit time is VERY high. So the chance of a collision that is of the right energy to excite the vibrational state is pretty high too.
Classical mechanics.
Ray #512. I think you meant “we need x to be small”.
A pendulum under simple harmonic oscillation has the potential energy of the raised weight linearly dependent on angle. It is true only for small enough angles because the expansion of a Sin (X) is approximately X when X is very small.
A pendulum that rises higher no longer obeys the simple harmonic motion.
RodB 510: It is. Ask a stellar physicist. ANY absorbtion spectra is part of the emission spectra. It doesn’t care whether it’s because aliens put a plastic wrapper around the star or a vibrational state.
Maybe your “problem” is that you think there is such a thing as a black body.
There isn’t.
The sun is a fraction of a % different from a blackbody radiation curve.
But you can tell whether the star is a helium rich or helium poor star by taking the peak emission and relating it to sigma T^4 to get the temperature of the star and checking it against the specral type.
If they were blackbodies they would tell you nothing since there would be no difference.
Arch #506: I’ll tell you what did: CO2 greenhouse gas forcing caused it. Now, if you don’t want to be told that or don’t believe it, why the clucking bell did you ask me to answer?
YOU’RE the one who thinks it could be something else.
So, what is it?
RodB 502. So if there only MIGHT be a pony in there, there is no pony. It’s a good working assumption. Or do you take your burgers to the local university to have its composition tested to check for pony meat because there MIGHT be a pony in it?
Rod, think about it. If you heat a gas, how are you going to keep the vibrational modes from being thermally excited? Try as you might, you cannot repeal equipartition. Start with a gas in thermal equilibrium. You have a Maxwell-Boltzmann distibution in energy–and that has to include ALL energies of the gas molecules. You are in equilibrium, so energy emitted via radiation has to be equal to energy absorbed via the same state transition. Now lets say we add some energy via radiation to a vibrational state of the gas. We’ve disturbed thermal equilibrium and equipartition. Yes, you’ll have more radiative decays, but you’ll also have more collisional relaxations until equipartition is again restored. What about if we heat the gas? Well, now we’ve changed the velocity distribution of molecules, so that more molecules have enough energy to excite the vibrational mode via collisions. The result: more radiative decays and more photons until again we have equipartition and equilibrium.
Mark, actually the point I was making is that a and b must be small, since deviations from log forcing are not evident over the range of observations we have. However, your point is well taken that if x becomes too large, this approximation will also fail.
Ray (508), I have no idea. I was hoping you knew something that I didn’t that would call that finding into question. I’m sorry if my original post sounded more hostile than intended.
I agree with jcbmack (514). ;-)
Re 519: Mark, I didn’t see any indication in his posts that Arch *does* think that the forcing was not C02. As he wrote to you, “I like to stick to the statements made in the AR4.” And as we know, the AR4 did have some reasonably strong statements on CO2 attribution! And in fact, Arch’s post to Ray was pointing to an update found in AR4. Looking up some older posts, I find that Arch refers to AR4 quite a bit, and clearly knows it pretty well.
I enjoy your posts, but I think here you are inferring an agenda that Arch doesn’t actually have.
Mark (519) I’m sorry if I confused you. My intended point was that there are lots of good arguments to use but that one is not very strong if you argue from the perspective of the AR4.
Thanks Kevin (524). I am not a scientist but I did take the time to read much of the report. Some of it I understand, some of it I can even recall.
I don’t always word things as clearly as I think I have however.
Most of the time I just lurk and learn here. I’ll go back now.
Oh! Woe! Phil, it is a strong doctrine indeed that says 2 + 2 can not equal four. Tell you what: 1) graph Y = (6)ln(X) and Y = X – 1 (the negative 1 to normalize the graphs, which start at X=1). 2) Open eyes. 3) see which Y is larger… up to some value of X. It still ain’t that hard. Are you maybe trying to refute something I didn’t say?? I’m evidently missing it.
Mark, I’m totally missing your point. I didn’t and don’t disagree with what you say in #516. I understand vibration modes can be excited via molecular collision (decreasing the average kinetic energy and classic temperature of the gas) as can vibration modes relax via collision. But we’re talking about radiative excitation and relaxation. :-?
Mark, you ought to know, but I think the spectral stellar lines relate to (mostly) electron energy levels (similar to vibration discrete energy levels) and are in addition to, not the same as, blackbody type radiation spectra. It would seem that the peak blackbody radiation would only by sheer coincidence indicate the elemental composition. Blackbody type radiation is dependent only on temperature and is completely independent of composition.
Re #521
Ray almost but not quite, or maybe you made it so brief that a key point was missed?
Any state, say the first vibrationally excited state, will achieve equilibrium as you say, but that does not mean radiative excitation=radiative emission.
dni/dt= (rad in) + (collisions in)- (rad out)-(collisions out)=0
In the lower atmosphere collisions are taking place ~100,000 times/radiative lifetime for CO2 so to a first approximation we can neglect that sink term. That means that the temperature of the system will increase until the source terms balance the sink term. So if you increase absorbance by increasing CO2 the temperature goes up to achieve LTE. Higher in the atmosphere when collisions become less frequent the losses by emission become significant.
Ray, I agree with your thoughts in #521, but maybe disagree with a couple of nuances. 1) I think the precise Maxwell-Boltzmann distribution is of the kinetic (translation) energies only and does not include electronic, rotation, vibration, etc. internal energies. 2) equipartition and excitation/relaxation are two different processes, though highly related. But I don’t see any fundemental disagreement here. Am I overlooking something?
Rod, think about it. You are talking about blackbody radiation as if it were some separate physical phenomenon and not the product of a photon gas coming into equilibrium with surrounding matter. No matter how much you torture a molecle or atom, it will not radiate except as a transition between its energy levels (yes, you can distort the bands, but they are STILL the bands).
So ask yourself:
1)How does the photon gas come into equilibrium with itself? (Remember photons do not interact with each other.)
2)How do the photon gas and surrounding material come into equilibrium with each other?
Phil, Once you are in true equilibrium, (no excess or deficit of photons, boltzmann distribution, etc.), you’ll emit as many photons as you absorb. Yes, collisional relaxation is more likely, but (at equilibrium) so is collisional excitation. In the atmosphere you have a net flux of IR photons from the warmer surface to the cooler atmosphere, so there’s a heating of the atmosphere. The higher collisional relaxation rate is how the energy gets from the radiation field into the kinetic energy of the molecules.
Rod, No, the M-B distribution refers to the total energy of the molecule for all modes that are thermally active at the temperature (and for the # of molecules in the gas) concerned. You cannot build a wall between different modes. Suppose we start with a gas where all the energy is in kinetic energy of the center of mass–no vibration, electronic excitation, etc. Over time, you will have collisions which excite vibrational and even electronic excitations (as long as there are molecules with sufficient kinetic energy.) The number of such excitations depends on the number of molecules with sufficient kinetic energy to bring about such excitations–so they too, will follow an M-B distribution. Now suppose the gas is isolated–no energy in or out. You start out with all energy being kinetic–and following an M-B distribution. Later, you’ve got the same total energy, but a different kinetic energy distribution. See, for things to make sense, you have to deal with total energy.
Equipartition is a characteristic of equilibrium–excitation/relaxation processes are how you bring equipartition about.
RodB 527.
How can that be when there are spectral lines from the sun (for example) in a region where there are no electron shells to cause them to happen?
Because they are constrained rovibrational states.
Arch 525. So we have:
Clear signal that CO2 caused most of the warming until 2003 ish.
An unclear signal that the warming trend has reduced.
How does that undo all the heating pre 2003?
It can’t.
So, CO2 was the best responsible candidate for the heating.
Now, for the recent trend, it isn’t signifiant yet, so we don’t know if there’s anything to query. So the original assumption stands: CO2 wot done it.
OK.
Barton Paul Levenson,
Your estimates leave 19 W/m^2 to be accumulated by the ocean. The sum of your heat inputs is 492 W/m^2, and the sum of your heat outputs is just 473 W/m^2. I think that the error is in your calculation of the rate of heat lost from the ocean surface to outer space. Your estimate is 371 W/m^2, I think that it should be 390 W/m^2.
Although I do think that during times of high sun spot activity, the oceans sequester heat, I do not believe that the rate is as high as your numbers imply. At your rate of 19 W/m^2 the 100 meter column of water would have to rise at a rate of 2.6 degrees C per year. I do not believe that is happening. I think that the thermal imbalance is much smaller, perhaps a few tenths of a W/m^2 and that it explains the rate of ocean temperature rise observed.
If we net out the atmosphere/ocean exchange, i.e., the ocean loses 390 W/m^2 and gains 324 W/m^2 for a net change of 66 W/m^2 and then the sum of the three outputs (C&C, Rad. and Evap.) will be 168 W/m^2, and will be equal to the solar input. Evaporation losses, 78 W/m^2 would be 46% of the total insolation rate, 168 W/m^2.
However, I think that your estimate of the return radiation from the atmosphere to the ocean is too low. The atmosphere would have to be about 15 degrees cooler than the ocean to attain that low a rate of energy return. I don’t believe that the worldwide atmosphere/ocean temperature difference is that high.
Ray, I can simply accelerate a bipolar or ionized molecule translationally and generate a radiation field without changing any of the internal quantized energy levels. Can I not? How does an iron rod radiate?
Ray, I guess you ought to know, but I’ve never seen a Maxwell distribution discussed or derived with anything other than velocity/kinetic energy, other than an insignificant inclusion of potential energy, talking of 3 dimensions of velocity space and 3 dimensions of “space” space. I am aware of M-B being used to derive specific heat constants which does include internal energy modes. I agree you have to deal with and account for all energy and energy transfers; but I don’t see how that is in conflict with the above. In any case, I can’t see anything of importance than I disagree with — or have disagreed with. What am I missing? Where is our disagreement?
Mark, sure. I said mostly electron energy level shell changes, but rovibrational states change similarly and (can) appear in spectroanalysis. It is just not the same as blackbody type radiation/emission.
Rod, Of course an accelerated charged particle will emit electromagnetic energy. So now, I ask where the energy comes from to accelerate the particle? Where did the energy come from to ionize the molecule? If not from the gas itself, then you are adding energy to the gas, and it is not in equilibrium. If it is from the gas, then either the gas is very hot, or you have very few molecules with enough energy to ionize a molecule in a collision. Acceleration of charged particles is important in stars and other plasmas. Yet, we see blackbody curves even at cryogenic temperatures (the Universe being an example). Energy has to come from somewhere.
Dear Ray and Jim,
I did some reading related to Jim’s topic from
Why don’t op-eds get fact checked? On Realclimate
I posted it here, since it there is already a discussion about this.
For the microscopic interpretation of the radiation transfer equation I found in Sir John T. Houghton’s book: „The Physics of Atmospheres“, the following central equation on page 74:
….after some expression he writes the following:
Beginning of quote: “Substituting in (5.31) for n1 and n2 from (5.28) and (5.30) results in a expression for the Source function Js:
Js=(Iv+((c*b12*A21)⁄(4*π*a21*B12)*a21/A21))/(1+a21/A21)
In the limit where local thermodynamic equilibrium applies collisonal activation and deactivation dominates so a21/A21 converges vs. infinity and Js, the source function converges vs Bv, the Planck function. The quantity in brackets ((c*b12*A21)⁄(4*π*a21*B12)) must therefore be the Planckfunction Bv.” End of quote.
Note that I describe the parameter in the above equation with slightly different symbols compared to John Houghton:
Iv describes the average intensity integrated over solid angle and frequency interval.
The concentration of the absorber is described with c.
Capital A21 is the einstein coefficient for spontaneous emission. With a21 he describes the einstein coefficient for deactivation by collision. Capital B12 is the einstein coefficient for absorption. With b21 he describes the einstein coefficient for activation by collision. With n1 and n2 he describes ground state and excited state. With Js he denotes the source function.
For the limit of Local Thermodynamic Equilibrium (LTE), collisions dominate.
John T. Houghton argues: the source function converges versus the planck function, denoted by Bv, if the quotient a21/A21 converges towards infinity.
Houghton identifies ((c*b12*A21)⁄(4*π*a21*B12)) with the planck function.
My math says however, if a21/A21 converges towards infinity the quantity in brackets ((c*b12*A21)⁄(4*π*a21*B12)) converges towards zero and the source function also converges to zero. This is exactly Jim’s argument. However, John T. Houghton’s argument points to Ray’s position.
Can you guys point out to me what I miss?
Best regards
Guenter
RodB 536, you are SO in denial.
How do you get the KE of an ideal gas of a diatomic molecule?
Atom: 5/3 X
Diatom: 7/3 X
Why the extra +2 in the numerator?
Because there are two more degrees of freedom for a diatom: spin and vibration.
Now if the heat capacity goes up, that must mean heat (energy) is being taken up by those two extra degrees of freedom.
So these modes must be as important to the KE content of an ideal gas at a given temperature as any of the three degrees of movement in three dimensional space.
Insignificant? Only if the movement of the gas through space is insignificant…
Rod 535: “How does an iron rod radiate?”
Black body basic. Plus, given there’s always a magnetic field, radio wave frequency radiation. If it is hot enough, it will produce radiation that excludes somewhat the radiation in the absorbtion spectra of neutral iron.
However I need to know a question: What atmosphere is made of iron (which is metallic as opposed to neutral and a solid as opposed to a gas)?
Re #531
Ray Ladbury Says:
4 December 2008 at 12:36 PM
“Phil, Once you are in true equilibrium, (no excess or deficit of photons, boltzmann distribution, etc.), you’ll emit as many photons as you absorb. Yes, collisional relaxation is more likely, but (at equilibrium) so is collisional excitation. In the atmosphere you have a net flux of IR photons from the warmer surface to the cooler atmosphere, so there’s a heating of the atmosphere. The higher collisional relaxation rate is how the energy gets from the radiation field into the kinetic energy of the molecules.”
Fine except for the “you’ll emit as many photons as you absorb”, that can’t happen (if it did there’d be no change in temperature). Look up fluorescence quenching for example, try OH a case where I’ve done the measurements and published, as the pressure is increased the fluorescence goes down due to collisional quenching.
Guenter,
Fortunately, Google books allows people to view this online, so that we can see the equations clearly. Houghton makes a physical argument, the mathematics of which are given previously in the discussion of local thermodynamic equilibrium. As phi approaches infinity, the distributions of energies is the Boltzmann distribution, which gives rise to the blackbody radiation Planck function. Mathematically, as phi is infinity, the first term is (I/(1+infinity)=0. Therefore the second term must be the Planck function B. The second therm is phi/(1+phi) times the bracketed function in question. phi/(1+phi) = 1 as phi approaches infinity, so the bracketed term must be B.
Mark, so when some of the added energy goes into vibration/rotation modes you have to add more energy to the molecule to get the same increase in KE that you got if no energy went into the internal modes. What I implied. I am in denial about what, exactly??
Mark, I checked google and Wiki and couldn’t find an atmosphere with pure iron in its composition. Golly! And your point is…??
Guenter, can you clarify what you mean by “converges”. Do you mean “goes to zero”? I think you also have to consider what is happening with b21 and B21, and I think you can use the LTE to derive relations between the factors.
I found this–a little qualitative, but it points in right direction:
http://www.astro.indiana.edu/~classweb/a540_s05/notes/ch06_1.ppt
The course looked to be pretty good itself:
http://www.astro.indiana.edu/~classweb/a540_s05/schedule.htm
This one was interesting, if not totally germane to your question:
http://www.strw.leidenuniv.nl/~brown/college_sterren/BlackbodyThermodynamics.pdf
t_p_hamilton #544
I understand that, but by close inspection, calculating myself, I realized that the bracketed function contains an additional 1/phi which should go to zero right?
Ray,
Yes with converges I meant goes to zero. Of course one should consider what is happening with b21 and B21.
But then pulling a21/A21 in the nominator out of the bracket is highly questionable mathematically speaking, wouldn’t you agree?
http://scholar.google.com/scholar?q=neutron+star+iron+atmosphere