Earth black body temperature wrong?

[Andre]

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.

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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 length 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 into 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 Tk[/sup] to be a linear constant whereas in reality it's the sum of the different not lineair t_i 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.

 

[sneez]

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 length 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 that's 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 into 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 that's 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, I am in the middle of new discovery (yey) and also very busy with reasearch at this point. I was referring 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.

 

[Andre]

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

 

[sneez]

agreed, but than do not say things like black body temperature is wrong, because its not. you just developed different model which presumably allows you to form ice caps at the poles, while having averaged temperature closed to black body. I applaud to you ,honestly, for the thought, but on the other hand the scholary formula does not aim at that, and from what I undestand it was not your intentional aim either.

The scholary formula really applies to top of atmosphere equilibrium not surface temperature. It can be said that it represents sufrace temperature since the emitting atmosphere is much colder usuallly than the surface.

another point, don't call your model "practical black body", that term will cause confusion for ppl who are not familiar with your model. Black body is black body with clear definition. and scholary formula already exists for it. Your formula is not for black body, it assumes that slices of Earth act as black body but have different temperature, which when averaged give you close to black body temperature. (yes, they still are at almost equilibrium but not with each other but with top of atmosphere radiative field). This fact could be advantage and disadvantage depending.

is the temperature your model computes for, let say, poles approximate to measured mean temperature at poles? and mid lats? (this will make your model more credible in this sense, of cause if you add those 33K or more in your case)

PS: how is it with the cosine zenith angle ?

 

[Andre]

PS: how is it with the cosine zenith angle ?

To the best of my knowledge it has been incorporated in the square in the C column. otherwise we have a misunderstanding here.

 

[ykkhor]

What is the reason we assume black body when solving radiation problem?

 

[Xnn]

A black body is the ideal radiator. Of course, the Earth is not really a black body and a correction factor needs to be applied; it's called emissitivity.

BTW, this is a very old thread and I can not vouch that it's correct. I also don't see much of a discussion of emissitivity either.