# Earth black body temperature wrong?

by Andre
Tags: black, body, earth, temperature
PF Gold
P: 5,459
I see I can upload zips

So here is the excel sheet to scrutinize.
Attached Files
 blackbody.zip (13.6 KB, 7 views)
 P: 355 Dont you have to modify albedo as well ? The constant albedo is consistent only with the circular area of intercepted radiation. Im still working and will be for a longtime , but I will have a crack at this when im off.
 P: 355 Andre, your intercepting area and emitting area are not exact of what it should be, that introduces error. Also as I said above the assumption of constant albedo with sperical earth is very arbitrary one could argue and the temperature difference is not significant as you presume it to be given all that. And a one technicall, why is the height in emitting area not corrected with the cosine lat? Is that not saying that the distance between 1deg lat at lets say 70-71 is the same as at the equator?
 PF Gold P: 5,459 no, the effective height of the slice intercepting the incoming parallel insolation is corrected for the lattitude with the cosine, whereas the emitting surface area is indeed the complete surface, so the correction is not applied there. I made an error not showing that in the simple drawing that I attached. I agree that the one degree slices introduce an error, which diminishes when it is replaced by the integral. I doubt if the error is large enough to explain the difference. Somebody interested to do the math? Also not sure about the albedo problems. Is anything changing in (bond)albedo between the two methods?
 P: 355 I wrote a program with double precission of those cosine factors and i got 252.71 :D . I was thinking that it could be quite managable to do the integration using calculus technique of rotating curves around y axis. (I will do but when I get to the book, its been a while). Albedo depends on the angle of incomming radiation. There is whole "science" to it and im not sure what the best approach would be. I do understand that when intercepting earth is presented as a solid circle, the assumption of constant albedo is not that bad. However, once you consider spericity which introduces difference in Temp. the immediate response of a sceptic (my role in this thread) would be the changing albedo. I have to think what it all means for a while it definitelly very interesting.
 P: 355 Andre, im uneasy about the emitting area. The schollary formula implies 4times bigger emitting area. Your formula has 10times biger emitting area (on average). [from 3times bigger to 180time biger at the poles]
 PF Gold P: 5,459 You are right, if you check the ratio of the sums of the area we get 3.97 instead of 4. I rechecked my original spreadsheet, that was right, this one is wrong. So I decreased the step size to 0,1 degree with the original spreadsheet and made some refinements like using the average cosine of the area (area ratio now checks 5 decimals accurate) to get a black body temp of 251.78K. It's getting colder. For Mars I get 207.78K degrees now. I'll upload the spreadsheet later.
PF Gold
P: 5,459
Attached Files
 black-body.zip (85.4 KB, 9 views)
 P: 355 Also, the solar flux is now hitting spherical surface, ther should be correction for cos(zenith angle) [this will further cool us]. Albedo must be changed as well but thats the biggest uncertainty how. Its certainly not constant, it needs to be weighted since you get different fluxes and incident angles over different areas. This will indeed produce great discrepancies. on different note: Why do you put so much significance to this formula BB earth approximation? Its just approximation for introductory text into subject. It is nowhere used for modeling or research. (If it is dont read that reasearch). Its very crude approximtion which only demonstrates concept. There is much better approximation using a homogeneous atmospheric layer above the surface and bellow TOA. That one produces 286K which is close to 288K. But any real GCM of atmosphere includes full radiative transfer which are nowdays upgradet to full blown non-LTE rad. transfer codes. Im involved with group that develops one for Mars atmosphere. (Note that temperature of Mars is not know with high accuracy. There is plenty of data that needs to be interpreted using non-LTE rad. transfer which is still being developed without the crude approximation that comes from LTE.)
PF Gold
P: 5,459
 Quote by sneez Also, the solar flux is now hitting spherical surface, ther should be correction for cos(zenith angle) [this will further cool us].
The cosine is there, basically the calculations are truly reversable now, ensuring radiative balance

 Albedo must be changed as well but thats the biggest uncertainty how. Its certainly not constant, it needs to be weighted since you get different fluxes and incident angles over different areas. This will indeed produce great discrepancies.
Definitely. It's also possible that we have circular reasoning here, the albedo being calculated from the average temperature and flux.

 on different note: Why do you put so much significance to this formula BB earth approximation? Its just approximation for introductory text into subject.
if there is a principle error in textbooks, I think that is serious and significant enough to address.

 It is nowhere used for modeling or research. (If it is dont read that reasearch).
I could imagine that the difference between black body temperature and actual average plays a role in parametrization of radiation models like MODTRAN.

BTW Also running the thread here:
http://www.ukweatherworld.co.uk/foru...osts=8&start=1
P: 355
 The cosine is there, basically the calculations are truly reversable now, ensuring radiative balance
I thought that cosine is for changing of height with increasing lat. In other words 1 deg lat at equator is (60nautical miles) is not the same at 45 deg lat. The cosine solar zenith angle I do not see.

 Definitely. It's also possible that we have circular reasoning here, the albedo being calculated from the average temperature and flux.
Its possible but its derived from measurements so i guess its not that circular. Also its very CRUDE approximation.

 if there is a principle error in textbooks, I think that is serious and significant enough to address.
NO, the schollary formula is "correct". Its much deeper than what meets the eye and its direct implication of mathematical rules and conservation of energy. (while im not strick on ether one philosophically, for the sake of a computational approximation its correct).

 I could imagine that the difference between black body temperature and actual average plays a role in parametrization of radiation models like MODTRAN.
I dont really know MODTRAN. But bb temperature computed from that formula does not enter any radiative mode or circulation modell I know. Im working with state of the art non-LTE radiative model that is implemented in already some GCMs and there is no such parametrization. The models move away from parametrization (the good ones) and implement more and more sophisticated rad. transfer models.

You are making mistake to see your computation and the scholary one as two different computation models. They both wrong when it comes down to it. The fact is that the scholary formula is accepted as ONLY very crude approximation using the simpliest of parameters (flat disk, constant albedo, sol flux const, etc). Its correct in that respect. Your approach does the same thing differently. Thats currently my opinion, I did llok up some of my books and went over the deeper part of the BB temp, and must say that its correct given the assumptions it list.
PF Gold
P: 5,459
 Quote by sneez I thought that cosine is for changing of height with increasing lat. In other words 1 deg lat at equator is (60nautical miles) is not the same at 45 deg lat. The cosine solar zenith angle I do not see.
.

Are you sure you're? I'm from the very old fashioned protocols that advises to generate your own cosine tables if you use it more than once, to increase speed.

Note that the B column in the new spread sheet is generating the cosines and that the C and D column is all about jugling with b cells hence cosines. The C column even with a square because both the lenght and height are cosine functions.

Note also that field B911 checks the ratio of the sum of the two areas, intercepting and emitting, to be 4,000002, where it must be 4 exactly. Talk about coarse approximation.

 NO, the schollary formula is "correct".
The scholar formula is correct if the black body is also a perfect heat conductor, dividing the heat evenly over the surface between intercepting and emitting radiation. As soon as there are different temperatures, the fourth root kicks in to destroy the assumed linearity.

 Its much deeper than what meets the eye
It would be more elegant to present that deeper stuff.

 and its direct implication of mathematical rules and conservation of energy.
As I can see the mathematical rules are violated here by assuming T[sub]k[/sup] to be a linear constant whereas in reality it's the sum of the different not lineair ti for i=1 to 0,5 pi

There is no volation of the conservation of energy. You start just as easily with the temperature to get the same radiation sum back.

 You are making mistake to see your computation and the scholary one as two different computation models. They both wrong when it comes down to it. The fact is that the scholary formula is accepted as ONLY very crude approximation using the simpliest of parameters (flat disk, constant albedo, sol flux const, etc). Its correct in that respect. Your approach does the same thing differently.
That's true, I merely changed the order of the operators to be correct, generate temperatures first before averaging them instead of averaging the flux first before converting them to temperature.

 I did llok up some of my books and went over the deeper part of the BB temp, and must say that its correct given the assumptions it list.
Please don't hide behind books and explain.
P: 355
 Are you sure you're? I'm from the very old fashioned protocols that advises to generate your own cosine tables if you use it more than once, to increase speed. Note that the B column in the new spread sheet is generating the cosines and that the C and D column is all about jugling with b cells hence cosines. The C column even with a square because both the lenght and height are cosine functions. Note also that field B911 checks the ratio of the sum of the two areas, intercepting and emitting, to be 4,000002, where it must be 4 exactly. Talk about coarse approximation.
There is misunderstanding here. Since we have sperical intercepting area this formula I would think does not apply anymore: (scholary version for simplicity):
pi*R^2 (1-A)*Flux = RHS, instead we need: pi*R^2(1-A)Flux*cos(sol zenith angle)

this is on top of that cosine(lat) you have ther already which corrects the height. However, I did not check your new spread sheet. Let me know if thats corrected there.

 The scholar formula is correct if the black body is also a perfect heat conductor, dividing the heat evenly over the surface between intercepting and emitting radiation. As soon as there are different temperatures, the fourth root kicks in to destroy the assumed linearity.
Lets make something clear. Black body is by definition perfect heat absorber, conductor and emitter. The intensity radiation from is determined by its Temperature which does not vary from place to place in that body, its in complete THERMODYNAMIC eqilibrium. The entire body has one temperature.

Now what you talking about is , if and how can radiation at top of atmosphere be approximated as black body. Thats what we are discussing. What you describe is not balck body anymore, since it does not have uniform temperature. (it has average temperature but thats not strict BB). So be aware of this divergence in those 2.

 It would be more elegant to present that deeper stuff. Please don't hide behind books and explain.
SOrry andre, im in the middle of new discovery (yey) and also very busy with reasearch at this point. I was refering to Grant Petty book, introduction to atmos. radiation. Very clear exposition of the issue and not just brushing over it. It has even better model there.

 That's true, I merely changed the order of the operators to be correct, generate temperatures first before averaging them instead of averaging the flux first before converting them to temperature.
which is fine and interesting change, however, it is not strict black body plus another problem with albedo and complication with sphericity is introduced and not a significant results are made. This model is just illustrative one.
PF Gold
P: 5,459
 Quote by sneez Lets make something clear. Black body is by definition perfect heat absorber, conductor and emitter. The intensity radiation from is determined by its Temperature which does not vary from place to place in that body, its in complete THERMODYNAMIC eqilibrium. The entire body has one temperature.
That's indeed a question of definition that precludes the simplification on Earth/Mars that the difference between black body temperature and actual global average temperature is the greenhouse effect.

Now, Planets are more near to perfect heat insolators than heat conductors and the sun emitting from a single source instead of onmiversal from all directions. Because that's what my little model is assuming. This makes quite a difference between a theoretical black body and a practical black body to figure out what the basic temperature of a planet should be.

Note that assuming no heat transport over the surface that my little world is also in thermal equilibrium. After all even after billion years of dynamic proximations of sort of thermal equilibrium, our planets still have polar ice caps