A model of lunar temperature

Posted: April 2, 2012 by tchannon in Astrophysics, methodology

Figure 1

This is presenting a model of lunar equatorial temperature using SPICE electronic simulation, which is an exact dual of thermodynamics, different units. Why do this? Very simple, what if? For simple problems paper and pencil, calculate is fine, we design like that anyway but putting the whole things together quickly and without damaging anything is confirmation and much more as you are about to find out. Sometimes too there is no analytic answer or is so involved, forget it.

The following plot demonstrates an adequate fit with provisional data from the Diviner lunar temperature mapping project.

There are problems and a mysterious issue over law which I hope can be discussed and answers found, improving the model and our understanding of radiative law.


Figure 2

Here is the difference, the error.


Figure 3

About me

A degree of trust is needed in what follows otherwise the text will need to be very long but I am going to attempt to make this accessible to a fairly wide technical audience. If I talk down, sorry.

I am a veteran equipment designer, rather wider in breadth than usual.

The technology

Most electronic or electrical circuit simulation uses some variation on something called SPICE. In essence it is a lumped element method and akin to finite element in eg. structural engineering. The parameters of elements are entered into a maths matrix, which is then solved (such as Gauss) and produces a result.

In practice it is just a little more complex. (yes I have written such a thing, know hands on)

We are approximating and I am finding a way of achieving the wanted result given the available tools and so on. There will be many alternative ways of doing the same thing.


It may come as a surprise to learn that the same basic laws are common between many different fields. Hydraulics, mechanics, electronics, thermodynamics and so on, they share, are duals. The units are different.

I am using the exact dual between electrical and heat.


Only need a few

  • Thermal flow == Current
  • Temperature == Voltage
  • Thermal capacity == Electrical capacitor
  • Thermal resistance == Electrical resistance
  • Heat source == Voltage source
  • Thermal flux source == Constant current source


We are irradiating the top of a 1 sqm column which reaches down into the moon. Sideways heat flow is neglible.

Mental adjustment, can’t rotate the moon so we flash the sun on and off as would be seen by the 1 sqm. This follows a half sine wave, with the rest darkness. The diviner plot shows a time-scale 0 to 24, so for convenience we use 0 to 24 seconds, scale doesn’t matter, everything scales.

Solar irradiance generator

The is relatively complicated for non-obvious reasons to do with flexibility and suitability for parametric sweeps later on. (unimportant)


Figure 4

V1 is the primary signal generator, a sine wave of amplitude 2.0, which is +-1, green trace in follow plot. A controlled switch S1 only passes through the positive half of the sine wave. R1 is incidental (no effect), necessary to avoid an open node when the switch is off.


Figure 5

ARB2 is a maths function which was found necessary to correct the small error in the original simulation, why is a question I need discussing. It raises the sine to power 1.3, value found by trial and error. Aside: the error was matched to an exact sine.

The output of the function is the red trace in the plot to the right.

G1 might be more difficult to understand. This is a voltage to constant current device, or temperature to thermal flux in dual terms. The current source is V2, set to some very high temperature and will supply as much flux as wanted without changing.


Figure 6

G1 has a single parameter, which is the transfer factor. This is set to 1360 turning an input of 1 into 1360 out, which for a constant current, which is a thermal flux, the effect is as much force as necessary, there will be 1360. In this case (list above) 1360 amps is 1360 watts/sqm, see plot right.

That is the solar generator, what is being blasted at 1 sqm of the lunar surface.

The Moon itself

The body, rock of the moon, regolith behaves as a simple conducive column, how it should be and what the simulation implies is so.

I’ll come back to this later including with what simulation can do, we will look inside the moon.

The lunar surface


Figure 7

The infamous Stefan Boltzmann equation is applied as a skulduggery function to the incoming solar flux and the result is taken off to the interior of the moon, heat is conducted away or perhaps from inside the moon.

G1 was mentioned previously and is the source of solar flux peaking at 1360 watt/sqm, moon body is via R4.

ARB1 is a trick. This is a maths function with it’s two inputs connected together. It is a constant current sink, dualled thermal flux sink. Current goes into the simulation ground, where for the purposes here is absolute zero, deep space temperature. I’m not bothering to make that 2.7K.

The current depends on the input voltage or temperature, with the maths function, you guessed, SB. Trivially simple.

A reminder.


Figure 8

What does it do? You tell me.


Figure 9

The “temperature” is the red trace here. Probably within 1%, not actually worked it out.

That sine power 1.3 worries me, why? I assume it is to do with the double curvature, yet it is not sine squared.


Figure 10

I can do parametrics swept by cosine at 5′ intervals to 5′ the pole, probably wrong and shown as a possibility if further progress seems worthwhile.

Lunar body

The simple way to model a thermal conduction is by an RC attenuation (there are a few papers showing this and validating the idea, no references this is many hours of work anyway).


Figure 11

In this case we don’t have to worry about heat leaking sideways etc. is negligable, is next to almost the same.

The values used were arrived at by trial and error. This is what causes the exponential curve darkside.

I’ve left some wires going no-where, did have fixed simulation probes attached.

Lets look inside the moon, only simulation can do this. Click for full size. (ignore the time-scale, yes it goes to 600+, about time to stabilise simulation)


Figure 12

The temperature vs. depth changes shape, and delays in time to a sine, if it went deeper still it would be a flat line, which is the true body temperature of the moon for this latitude. Being a sphere it isn’t that simple and I don’t want to think about that just yet.

If this seems an unexpected result I point out it is similar to earth soil temperature data.

Numeric results

As it stands right now.

Lunar surface:

  • 392.8K peak
  • 93.99K minimum
  • 214.7K mean (common average)

Is that the correct mean? This is a good question which has been asked and disputed on the blog. (I’ve said nothing but smiled)

  • 222.3K body peak
  • 206.5K body minimum
  • 214.4K calculated mean

The two means are the same, therefore the common average is correct.

Want a closer look?


Figure 13

[UPDATE 2012-04-02 14:30]

In a comment Hans Jelbring questions why the lunar surface only drops to 93K during the lunar night, on the implication it ought to be colder.

A few seconds work provides what I hope is an intuitive answer, a plot of lunar surface thermal flux.

In the simulator plotting the current, dual thermal flux left hand end of R4 shows the thermal flux in Watt sqm. Plotting the surface temperature concurrently gives a timing reference for the human.


Figure 14

Red trace (green is temperature K) is surface thermal flux which is much small than might be imagined. The majority during the daytime is emitted by the Stefan Boltzmann surface. This is I think a critical problem in understanding what SB means.

  • 13.48 watts max surface emission at 6 hours, start of lunar night
  • 12.77 watts max surface absorption at 21 hours
  • 4.4 watts emission at lunar dawn (read off magnified plot)
  • 0.2 watt mean, which is simulation error, not fully stabilised.

Note too, I can easily provide copious numeric output. Have to be XLS for WordPress.


I can supply more information etc., ask if you want. I might answer.

Any suggestions please on why the sine power 1.3

Simulation software is an old copy of Simetrix, this is commercial, an engineers choice, about ergonomics and so on. Unfortunately I have no justification or income warranting an update, this stuff is $$$ seriously so.

This has taken a lot of effort to put together so I hope this acts as an independent confirmation from total different methods. It’s also worth keeping in mind the Diviner data is in error and my result is in error, all we can do is our best. Painstaking detail is often necessary.

Posted by Tim Channon, co-moderator

Been cross posted to my own blog but I expect discussion will be here.

  1. gallopingcamel says:


    I used PSPICE to analyze 100 MW PFNs (Pulse Forming Networks) for S-Band klystrons. With that kind of power one needs to design the network with great care.

    It never occurred to me that you could use circuit analysis to solve energy balance problems.

    My copy of PSPICE is ten years old so its graphical output is cruder than yours. Where can I find a more up to date copy so that I can attempt to duplicate your achievements? (Just for the fun of it).

    My PC prefers Debian Linux but I can run many Windows executables using WINE. For example, I can run Photoshop but not Dreamweaver or Quicken (I use GNUcash instead).

    I have an amazingly fast (Russian) 6,000 point Finite Element Analysis program for magnetic field and heat transfer problems if you are interested. It runs from the Windows command line.

  2. Truthseeker says:


    Someone has turned the Geek dial up to 11.

  3. Truthseeker says:

    Seriously though, the error rate is so small it could be measurement error from the observations unless I am reading the plots incorrectly …

  4. tallbloke says:

    Tim, this is very impressive work, which I don’t understand yet, so I’ll be asking some questions. 🙂

    In the mean time, I’ll just add a couple of pointers to possibly relevant snippets.

    1) Kramm et al presented a plot of subsurface lunar temperature. It is included in a (laughably wrong) rebuttal attempted by Eli Rabett:

    2) I came across a paper which found a 35V negative charge on the dark side of the Moon.
    If this modulates throughout the Moon’s orbit as the Moon’s dark side interacts with Earth’s magnetosphere and the Solar wind, would that affect things in terms of the emissivity of the regolith? Might that have anything to do with your puzzle:

    “That sine power 1.3 worries me, why? I assume it is to do with the double curvature, yet it is not sine squared.”

    Here’s the abstract:

    Observations of electron distributions above the shadowed surface of the Moon show energy-dependent loss cones which indicate reflection by both magnetic and electric fields. At the same time, low energy (<~100 eV) field aligned upward-going electron beams are observed. Together, these observations imply average night-side potential differences between the surface and the Lunar Prospector (LP) spacecraft of ~-35 V. The lunar surface may be at an even higher negative potential relative to the ambient plasma, since LP will likely also charge negative. The potential difference is consistent with simple current balance models which include secondary emission. No clear dependence is found on surface terrane type and age, or on ambient electron density and temperature. Instead, the potential difference is found to depend strongly on the angle from the subsolar point and the angle between the magnetic field and the normal to the lunar surface.

  5. wayne says:

    Tim, out of this world! Don’t you just love it when other branches parallel classic physics, it’s all mass and energy at the core, right? Well I do, and you just hit the ball out of the park! Anyone know an easy way to convert the J’s of electron’s to photon’s? That would ice the cake.

  6. Hans says:


    You have done a very nice approach to quantitatively decide temperature when varying some parameters. As an example it is easy for you to directly see how the distance to sun would affect lunar equatorial temperature.

    It is well known that by using the simpliest electrotrotechnical components almost any (mostly cyclic) process can be modelled quite well. The diod on top of that makes it easier to produce jumps in the output.
    The reason is that the derivation and integration function is modelled by coils and a capacitors.
    In short, almost any process can be modelled by using the electronic curcuit language. As such it is a very effective way to simulate physical situations without need to tuch reality (and avoid more expensive investigations). In a way it is also an alternative to mathematical formula, another language that can be quite useful.

    The danger is of course that you cannot simulate an outcome that depends on a physical process you don´t know exist. This is often the case relating to complex systems in nature. During certain
    conditions a secondary process is suddenly becoming the dominant one. If it happens seldom it is hard to model. Prime examples could be constructing a model to predict El Nino, solar cycles, the year without a summer 1815, the US blizzard 1899 etc.

    The Diviner data set seems so clean and nice. Still, it hides one distinct problem that has to be solved besides a discussion of the SB law validity (It is quiote a good approximation though). Why is the minimum equatorial temperature as high as 93 K? Is there any way to know what physical process(es) is causing the minimum temperature to stay that high? IMO there is and I also think that Tim´s impressive model cannot give the answer to this question. He needs more information before finishing his model and add one or two important processes before letting it out on the market for sale.

  7. davidmhoffer says:

    I remember sitting in on a mechanical engineering class many years… ok, a few decades… ago. I was floored to see that the equations they used to describe things like springs, friction, and so on were the exact same equations that electrical engineers used to describe things like capacitors and resistors and so on. It is no suprise therefore to see that one can build a model of surface temperatures using modeling tools written for electrical components. The mathematical relationships between variables being governed by the same relative functions, it would actually be a surprise if one could NOT do this.

    What I do not understand is why you are worried about the sine power 1.3? You’ve built a model that seems at first glance to be accurate well within the error of the measurement instrumentation and methodology. Sort of like Boltzmann worrying that 5.67 * 10^-8 seems like an awfull small number and so must be questionable.

    I’d be more interested in seeing time and effort being put into extending your model. Could you use it to describe, for example, a layer of gas exposed to IR? There’s plenty of spectroscopy work out there specific to various gasses. If you could mimic those…. and then from there mimic mixed gasses…. and from there mimic mixed gasses layered on top of… oh, I don’t know… how about a rocky surface covered 2/3 in water?

  8. Roy Martin says:

    Tim, re:

    “That sine power 1.3 worries me, why? I assume it is to do with the double curvature, yet it is not sine squared.”
    “I can do parametrics swept by cosine at 5′ intervals to 5′ the pole, probably wrong and shown as a possibility if further progress seems worthwhile.”

    The meridional distribution of temperatures is not a cosine curve either. If you scale and plot temperatures at latitude degrees 0, 60, 75 and 90 from Fig.3 in N & V’s Reply to comments Part 1, for different times of day, you will find that they have the same type of lattitudinal distribution as we see in the diurnal daytime rise and fall. Very uncosine like too. Something to do with matt surface?

  9. Tenuk says:

    Nice work ,Tim, with a result close to the Diviner model, which is based on actual temperature observations. It also illustrates how thermal energy and the EM charge field behave in similar ways.

    Regarding 1.3 ‘tweak’, could this be needed as a correction to the different speeds of energy transfer in the system – photons go speed of light, electrons a bit slower, conduction competitively very slow?

    “I can do parametrics swept by cosine at 5′ intervals to 5′ the pole, probably wrong and shown as a possibility if further progress seems worthwhile.”

    Is there a function in to simulate an analogue tuner to integrate from equator to poles, instead of using parametric sweeps by latitude?

    “Is that the correct mean? This is a good question which has been asked and disputed on the blog. (I’ve said nothing but smiled)”

    Good question, Tim – how much information does the correct arithmetic mean of our telephone numbers convey???

  10. Harriet Harridan says:

    Nice work Tim!


    “My copy of PSPICE is ten years old so its graphical output is cruder than yours. Where can I find a more up to date copy so that I can attempt to duplicate your achievements? (Just for the fun of it). My PC prefers Debian Linux”

    Have you seen: http://ngspice.sourceforge.net/

  11. Brian H says:

    Geez, Tim, when you go into full tech-y mode, your grammar goes all Yoda! When I read, “For simple problems paper and pencil, calculate is fine, we design like that anyway but putting the whole things together quickly…. Sometimes too there is no analytic answer or is so involved, forget it,” I knew we were in for it.

    BTW, is this a new data descriptor? “Sideways heat flow is neglible …”

    Your “it’s” score this time is again 50:50; zero + or – correlation with correct usage. 😉

    This line seems illogical:
    “The two means are the same, therefore the common average is correct.”

    The average of 0 and 200 is 100. The average of 90 and 110 is 100.
    So what?

  12. B_Happy says:

    So this method, which contains some experimental data, produces a mean of 214K

    The back-of-the-envelope average pretending it is a black body produces 255K, which is wrong.

    And, the averaging method of Nikolov and Zeller which assumes the temperature is 3K where the sun does not reach, produces 154K, which is even more wrong.

    Is that a fair summary?

    [Reply] N&Z have an update coming out this week which includes a term for heat retention in the regolith which resolves the disparity for the global mean.

  13. tallbloke says:

    A thought that occurs to me is whether the accuracy of the DIVINER data might be a useful aid in determining the solar ‘constant’. Tim used a value of 1360W/m^2 and gets a mean temp of ~214.5K

    Ned is telling us the latest estimate from the DIVINER team is 192-197K.

    What would the value of the solar constant in Tim’s model be if his result matched the middle of this range?

    The solar constant dropped 5W/m^2 or so when the TIM instrument had the aperture optimised for excluding stray reflection of incoming solar irradiance. Perhaps it should drop a llittle more.

    If this idea is right, then ongoing Lunar measurement might offer a better method for measuring variation in solar irradiance than pointing a sensor straight at the sun. This idea is somewhat analogous to the projection of the solar image onto a sheet of paper to facilitate sunspot observation, rather than squinting through a telescope directly at the Sun, and going blind like Galileo did (or was it the cataracts?). Sensors pointed directly at the Sun degrade. Calibration of successive instruments has been a big problem in trying to get a continuous and valid record.

    It would also be interesting to see what cyclical variation was due to the Moon’s declination cycle, as that might tell us something about modulation of the solar irradiance by the Earth’s magnetosphere too.

  14. tallbloke says:

    Picking up on Roy Martin’s point, a further thought on the 1.3 error. Is it due to the emissivity of the regolith changing with temperature? My intuitive guess is that this would be a bigger factor than any electrical effect due to magnetosphere-heliosphere interaction.

    Also, the Moon is not completely without an atmosphere. Could there be a bit of water condensing in the night side which affects the curve as the surface warms at dawn? Could it be due to the dew. 🙂

    Probably not, as such an effect would be very short lived, not evenly spread over the lunar morning.

  15. Vuk says:

    “That sine power 1.3 worries me, why? I assume it is to do with the double curvature, yet it is not sine squared.”

    How about instead 1.3 using e/2 (e ≈ 2.718) where e is base of the natural logarithm ?

  16. mkelly says:

    The analysation/instruction of the flow of heat is often by the similarity to resistors in circuits.

  17. Sparks says:

    That is very interesting, I like how this is evolving! I’m a bit rushed today but I want to get back to this and bounce some Ideas around later.

    davidmhoffer says:
    April 2, 2012 at 8:23 am
    “I was floored to see that the equations they used to describe things like springs, friction, and so on were the exact same equations that electrical engineers used to describe things like capacitors and resistors and so on.”

    How cool is it when you realize for the first time how it all works like that, electrical-mechanical laws evolved from the same original known laws of physics.

  18. gallopingcamel says:

    Harriet Harridan, April 2, 2012 at 8:52 am,

    Thanks for that link. From the “README” it should work well on my system. It has been a while so it may take me a day or two to get up to speed!

  19. tchannon says:

    [reply] In response to Hans I have updated the post with a new section about lunar surface flux / heat flow.
    This unfortunately due to vagaries of HTML / WordPress editor has changed the post formatting, like fighting a joke monster.

  20. tallbloke says:

    From Gerhard Kramm et al: http://arxiv.org/ftp/arxiv/papers/0904/0904.2767.pdf

    Figure 5: Moon’s disk temperature at 2.77cm wavelength versus moon phase angle φ during two complete cycles from twice new moon via full moon to new moon again (adopted from Monstein.)

    with a comment below of

    Our Moon nearly satisfies the requirement of a planet without an atmosphere. It is well known that the Moon has no uniform temperature. There is not only a variation of the temperature from the lunar day to the lunar night, but also from the Moon equator to its poles.

    Using Eqs. (1.4) would provide T 270 K e ≈ when the albedo, 0.12 M α = , and the emissivity, 1 M ε = (black body), are considered. However, as illustrated in Figure 5, the mean disk temperature of the Moon observed at 2.77cm wavelength by Monstein (2001) is much lower than this equilibrium temperature

  21. tallbloke says:

    Looks to me like 214K is on the money at the moment, but who knows what Ned will come back with in his and Karl’s update. Nothing much changed with the equatorial mean temp I hope.

    What the heck is “the mean disk temperature” anyway? 🙂

  22. tchannon says:

    “How about instead 1.3 using e/2 (e ≈ 2.718) where e is base of the natural logarithm ?”

    I wish. Worse, not hugely.

    Something which crossed my mind is how does surface roughness affect the law?

    Don’t want to clutter up the comments, could send you compare data via email.

    A bit of history and maybe I need to show the evidence along the way, maybe later.

    Originally I used plain sine() and whilst close there is a definite systematic error but not at peak. I investigated by extracting the error curve and thought that look like a sine fragment. I threw this at a data analyser software (good for making tea, doing all manner of things) and sure enough it ate it for breakfast, matches an exact single sine, with Diviner data noise r2=0.97

    On thinking I changed the way the generator is done to allow variable shape. I tried the obvious, a double sine (sine squared). Worse.

    I hit on trying a sine power and that came in nicely.

    Note: the Diviner data is extracted from the published plot but is obviously very close to the underlying data. (decided to published the software used to help others, see my blog)

  23. tallbloke says:

    Tim, couple of quick points.

    1)Tenuc said something about a ~1.4 power that appears frequently as a non-linear factor in nature recently.

    2) Did you not see the comment where Ned offered to send you a copy of the Diviner data? Email him.

  24. Hans says:

    Thanks for the link to the article.
    This lunar temperature curve is a red herring for several reasons. What do you say about its phase and what does tim say?

  25. Hans says:

    tallbloke says: April 2, 2012 at 3:11 pm

    What the heck is “the mean disk temperature” anyway?

    A very good question. And what is a “disk temperature at 2.77cm wavelength”?

  26. tchannon says:

    Rog, yes I know about that, and yes.

    Switching horse on the data would take time (timebase has to match with other things) and since it overlays rather well, no rush to do that.

  27. vukcevic says:

    Hi Tim
    I am far to busy with many other projects, anyway not familiar with all ins and outs, but if I was starting from scratch I would use two equations, one for daily insolation
    and other for the accumulated (average) energy discharge
    as illustrated here:

    p.s. have all sorts of problems with wordpress

  28. tchannon says:

    “Is that a fair summary?”

    No. I am only doing this for a 1 metre wide swath around the girth.

    Extending this to whole globe to poles is a to-do, maybe.

    I already have a large array of data from which I could compute a mean, don’t trust it.

    What the hell. Probably slightly high, computes to 202.7K

    This assumes I am using the correct math, yada, yoda.
    Is the weighted mean according to area.

    That 1.3 factor if it is about a spherical shape might make a mess.

  29. Hans says:


    I can inform you that your simulation (figure 12) is almost exactly what was simulated by some scientists in the 80th based on measured temperaures from Apollo sites. I will not reveal the source just now since I am writing myself about lunar temperature variations.

    Your surface simulation is close to identical.
    Your green curve fits a depth of 20 mm, your yellow green fits 50 mm depth and your light bloue one is somewhat less than 100 mm.

    This type of simulations were known by NASA before 1975. Quite interesting.

  30. tallbloke says:

    “computes to 202.7K”

    Wow! Within a couple of percent of the DIVINER upper end estimate.

    Nice going Tim!

  31. tchannon says:

    Anyone interested in trying this for themselves,

    A possibility is LTSpice, (Windows, also specifically mentioned Linux Wine), free and funded by Linear Technology a well known semiconductor manufacturer. Don’t know it but pulled a copy, small, installed fine, runs. Need to avoid all their product specific stuff… not going to give models from the competition, but apparently will load external models. Has to because designs are usual a mish-mash from different people.
    The docs mention behavioural devices but this might get awkward. With sneak could use semiconductors to help out.

    Little alternative I can see quickly, evaluation version of various packages, most are silly constrained. (don’t know about Simetrix, actually based in the town here, was a one man operation but he sold out to a multinational)

    ng-spice has been mentioned, can do it but you are on your own typing in and analysing results.

  32. David Appell says:

    tallbloke says:
    Wow! Within a couple of percent of the DIVINER upper end estimate.

    Of course, as I’ve been trying to point out here for a few days now, standard physics gives exactly the right average, and predicts the entire dayside curve as well, especially including its peak. (Radiative considerations alone can’t fix the nightside temperature — thermal conductance of the regolith must be included.)

    average equatorial T = (2.7/2*pi)B +Tn/2 = 212 K

    where B=[S(1-alpha)/sigma}^(1/4), S is the Earth’s solar constant, and Tn is the nightside temperature of about 95 K. The number 2.7 comes from the Sweger integral.

  33. David Appell says:

    Tim gets a “wow!” for coming within a few percent of the upper end estimate, but standard physics doesn’t get one for predicting exactly the right average and, to boot, the entire curve?

    [Reply] read it all again David. Tim is almost spot on with the equatorial mean, as good as ‘standard physics’ and within a couple of percent of the global mean as estimated from DIVINER data. Meanwhile, ‘standard physics’ has to borrow results from contributors to this blog to get close to that without resorting to fudges involving ‘internal fluxes’… 🙂

  34. tallbloke says:

    tchannon says:
    April 2, 2012 at 3:23 pm

    “How about instead 1.3 using e/2 (e ≈ 2.718) where e is base of the natural logarithm ?”

    I wish. Worse, not hugely.

    Try 1.236

  35. David Appell says:

    [Reply] read it all again David. Tim is almost spot on with the equatorial mean, as good as ‘standard physics’ and within a couple of percent of the global mean as estimated from DIVINER data. Meanwhile, ‘standard physics’ has to borrow results from contributors to this blog to get close to that without resorting to fudges involving ‘internal fluxes’…

    That’s really unfair. I simply referred to the “Sweger integral” for convenient of citation — anyone can do the numerical integral for themselves, as I did too to verify it.

    Also, standard physics is “spot on” with the global mean (212 K), not off by a few percent as is Tim.

    It’s also “spot on” with the shape of the curve, and the maximum dayside temperature (382 K).

    I simply do not see what is lacking in the calculation of standard physics. What is it?

    Did you note that Vasavada et al (1999) use the same standard physics, with thermal conductance physics added it, to calculate the Mercury and moon temperatures?

    [Reply] You get 212K, DIVINER gets 213K Tim gts 214K. Where is this “few percent”?
    Also, it’s not just Sweger’s integration you borrowed, but the 95K night side temperature. Where did you get that from if not this blog? Vasavada et al are forced to resort to ‘internal fluxes’ to make up the deficiencies.

  36. tchannon says:

    Parameter stepped from 1.2 through 1.4

    Parameter at 1.3 left as wide black line. Diviner data blue wide,
    (aside: emailed Ned, he is going to supply the data when he gets a moment)

    Pretty close to 1.30

    Forget to mention r2=0.999772
    We are into funny farm stuff.

  37. tchannon says:

    Suspicion based on brain emergence.

    As you approach the pole time slows down… surface speed slows therefore irradiance duration per unit area is longer.

    Tried changing the simulation timebase to see what happens. Effect is diurnal temperature variation increases as expected. Subsurface variation goes deeper.

    This is unconfirmed. Don’t think it matters much on overall temperature.

  38. tallbloke says:

    Forget to mention r2=0.999772
    We are into funny farm stuff.

    Heh. Most impressive Tim. I’ll leave you with it for the night. Leave Mr Appell for me to deal with.

    BTW, the moon is rigid. no slowdown near the poles. 14 days sun, 14 days dark, same as the equator.
    Less velocity, less distance, same time.

  39. tchannon says:

    Something weird about 1.3
    It is the square root of 1.69

    So? Something new on me. It is a “real root”

    sqrt(1.69) = sqrt(169/100) = sqrt(169)/sqrt(100)

    169 and 100 are perfect squares

    Therefore 169 = 13×13 and 100 = 10×10

    and 13/10 = 1.3

    Right, okay. Wind is up the yard arm. Worrying given the question is associated with area and odd maths stuff.

  40. tchannon says:

    By J. C. JAEGER*
    [Manuscript received November 10, 1952]


  41. Tenuk says:

    Another though, Tim, is that the moon isn’t a perfect sphere. The feldspathic crust of the moon on the side facing Earth has been almost obliterated in places down to the basalt.

    This will have an effect on the albedo and rate of conductance of heat through the crust. significant to the 1.3 issue, I don’t know?

  42. Tenuk says:


    Thermal conductivity depending on rock type W/(m·K)

    Feldspar 2.04 to 2.72 – Avg 2.38
    Basalt 1.70 to 2.00 – Avg 1.85

    2.38/1.85 = 1.29

    Seems 1.3 is a popular number!

    N.B. Can’t vouch for numbers, Wiki and other dubious sources used… 🙂

  43. Bryan says:


    Great post!

    A possible addition to your thermo-electronic analogues?

    Its seems to me that Latent Heat of Vapourisation of Water acts like an Inductor (L).
    In that it opposes a rise in temperature caused by Suns irradiation and also opposes a drop in temperature caused by night.

    Not much use in your Luner model but very important in an Earth analogue.

    It also makes possible a tuned LCR circuit where the energy circulating in the system can be greater than the make up energy from the Sun

  44. Sparks says:

    Tim Channon, Roger.

    I’ll write a windows app for this if you supply a formula, I’ll do the Programing etc.. (I’ve been thinking about writing a DX-Central type of app for a while .)

    For example converting this Wind Chill formula to pascal;

    Wind Chill = 35.74 + 0.6215T – 35.75(V^0.16) + 0.4275T(V^0.16)
    T = degrees Fahrenheit
    V = wind speed MPH

    I can produce a function like this

    WC,T, V :integer;

    WC := round(35.74 +0.6215*T – 35.75*power(V,0.16) + 0.4275*T*Power(V,0.16));

    We can then output the results in a real time readout and maybe use sliders or live data as an input similar to a virtual volt tester that demonstrates ohms law.

    Ohms law example;


    Which would look like this converted to Pascale;
    v, p, r,i : real;
    r:=(v / i);
    edit3.Text:= floattostr(r);
    p:= (I * V);
    edit4.Text:= floattostr(p);

    We can then produce an app that looks like this (for example);

    And possibly input live data like solar flux, Sunspot number etc..

  45. It is probably not relevant here but in forced convection heat transfer from a single sphere in a large (infinite) fluid, experimentation has found the empirical relation that the Nusselt number is a function of the Reynolds number and Prandtl number thus
    Nu =2.0 +0.6* (Re)^1/3 * (Pr)^1/3.
    From the Colburn analogies for momentum transfer, heat transfer and mass transfer the following relation can be found
    h/k = Cp (Sc/Pr)^2/3
    where h is the heat transfer coefficient, k is the mass transfer coefficient, Cp is the heat capacity at constant pressure, Sc is the Schmidt number (which Gavin Schmidt does not understand) and Pr is the Prandtl number
    Maybe the electrical current and radiation heat flux can be treated in the same way as convective heat flux, mass flux and momentum fluxes by using appropriate dimensionless numbers.

  46. tchannon says:

    That must do something, there is though a layer of dust, outbound accumulated prayers or swear words from earth, need to do a dna check on which it is.
    Wonder if anything shows in the Diviner data.

    I’ve not thought about that one, but I am not surprised.
    This model is trivially small, I’ve built models of very large and novel stuff which has ended up in commercial product literally on air broadcast. Investigating overload and other things, real spooky when a screwdriver across pins does the same thing.

    Moving this to earth might shed some light on a few processes, whole thing, no way.

    Although I can’t remember how to do it, was a contract where I recorded the real signals from an automotive engine control system and used those as the stimulation input. Worked spooky stuff again.

  47. tallbloke says:

    Is this the beginning of a plan for a modular climate model?

    Start with an N&Z style minimum framework of a spinning sphere with orbital parameters and solar illumination – the grey body.

    Then add the atmospheric mass and the ocean, plus the basic layout of landmass and some physical rules and params to get the ocean circulation going.

    Then we start adding climate processes and biomechanics as bolt on modules….

    The Talkshop Terrestrial Testbed – T4 🙂

  48. tallbloke says:

    Tim, do you remember I sent you an email at some point last year when we were discussing Milankovitch cycles? There’s a guy who wrote a matlab program to calculate insolation at any point on th planet for any time over the last and next 3M years or so. Run a search on your inbox for me please.

    We could maybe use that as a starting point. The annual north-south shifting insolation is the pump which primes the oceanic circulation.

  49. gallopingcamel says:

    Tried ngSPICE but my existing PSPICE is easier to use owing to its slick graphical (schematic) input and large library of sources, components and tools.

    The PSPICE post processor gives pretty good graphics but not as pretty what Tim is using. I will try LTspice next.

  50. Sparks says:

    My original thought was, we have a circuit, Let’s plug it in and test it!

  51. Tenuk says:

    Hi Tim – Not sure dust is a big problem as it covers the whole surface and would still have the same relative difference in properties as each type of the solid rock it came from, but at a lower energy density. Here are a couple of pictures, which show the huge dark low ‘seas’ of basalt on the near side facing Earth and the much lighter feldspar on the higher mountainous far side.

    Definitely a big difference (1.3?) in albedo between near and far sides.

  52. tchannon says:

    I’m being a bit slow, that’s a nice idea on the thermal time constant.

    I did try using delay lines, don’t work but are known problematic. A good thing was coming across an article in EE Times about a better way to do lossy lines.

  53. Roy Martin says:

    Simulating the lunar temperature in this way is most interesting, and has led to a quite informative passage of thoughts about the moon.

    Another observation on the 1.3 issue:

    It struck me that the daytime part of the temp. curve looked like an ellipse. On overlaying it turned out that an ellipse with a minor axis of twelve hours set at 80K, and the semi major axis from 80K to 389K is visually an exact fit to the curve of the measured data.

    The area of the semi ellipse is 1.24 times that of a half sine wave with twelve hour half period set on the ellipse minor axis and to the 389K peak. Not quite the 1.3 figure you are looking for, but the close match to the ellipse again highlights the deviation from a basic sine wave shape.

  54. tallbloke says:

    Roy, is it exactly 1.24 or is it 1.236?

    I have a specific reason for the precision of the question.


    BTW, did you mean 12 hours or 14 days?

  55. Roy Martin says:

    Actually 1.2337. ??

    12 hours, as in Tim’s Fig.2 in the post.

  56. Jose Suro says:

    Maybe a relevant paper that could interest N&Z (lunar regolith considerations). The numerical simulation in particular (section 5):



  57. tallbloke says:

    Sorry, 12 hours, as you were.

    Hmm, 1.2337 is very close to 2*phi

    Need to think, this could be a bit of a Eureka moment… when my poor battered brain gets into gear. 🙂

  58. tallbloke says:

    Hose Suro: Welcome and thanks for your input. That is a very interesting and thorough paper.

  59. tallbloke says:

    If the Moon were orbiting Earth at a slower rate, then the curve would become ‘flat topped’ wouldn’t it? i.e. the regolith would reach a temperature which enabled it to radiate at the rate it was being heated by the incident radiation, so causing the temperature curve to level out.

    Is that right?

  60. tchannon says:

    Yesterday I looked at that leading to head scratching, the shape seems to be very close to a hemicircle. Something odd is trying to tell me something.

    There is no question the fundamental method is exact, equivalence are not a whim, are actual, as is the usage of SB. The difference though between a sine and with the sine power 1.3 is small.

    I’ve used no factor at all for albedo because it has no effect on SB.

    The basic assumption is that planet rotation which causes the 1m sq to tilt from flat can be simulated by sine modulation of the radiation source might be subtly flawed. If it is I am sure this can be fixed.

    How about this: we are dealing with a rough surface.

    Therefore at an angle parts of the surface are plane to the illumination and full temperature, parts are unilluminated.

    I think this might lead to several effects, most critically the fully illuminated emit at the full SB temperature, not the lower Beer-Lambert temperature, therefore the whole region of the surface is colder than might otherwise be expected. (if you get what I mean)

    There is also less illuminated area for conduction into the surface but I suspect that tends to cancel.

    And to help here is an XLS (97/xp/2000, 75k) of this data with plot and image (might not survive).

  61. tchannon says:

    Just occurred to me there is a proof for this.

    If there is a technically good image of the moon, (photographically flat field) is the radial reflectance falloff as per a theoretic flat surface or not?

  62. tallbloke says:

    OK, now I’m confused. What’s the difference between the ‘pure sine wave’ in that plot and the ‘sine wave’ in fig 5 please.

  63. tallbloke says:

    Here’s an image of the moon I took last year

  64. tallbloke says:

    The falloff won’t be the same as a flat surface because of the roughened sphericity of the grains of dust.

  65. tchannon says:

    Fig 5 and 6 show the sine and then the sine^1.3

    Those drive the flux generator.

    The flux generator then drives the SB flux sink and the plot just above is the resultant SB temperature at the flux sink as the source and sink fight each other.

  66. tchannon says:

    Oh dear, here we go again, it’s a mess out there with lots of contradictory statements, ie. quite likely no-one knows. There are various buzz terms but probably limb darkening, more often applied to bodies with an atmosphere.

    An opinion which looks right is the moon is not lambertian but some declare it is. (guess the kind of people)


    Using Rog’s lunar image, seems to fit cos()^1.3 but not so simple to be sure. Won’t go into the details except show some evidence.

    Some lovely software lets me take a profile line. Can’t screen capture fully so I have faked enough here. Select rectangular and then use one or other diagonal, take that to image. (with measurements)
    Difficulty is the lunar surface is uneven, tried a lot of different ploys to try and get a good line, this seems the cleanest.

    Overlay, seems closest to sine()^1.3

  67. tallbloke says:

    Tim, yes, but you’ve chosen to go to the Earth-shadow there, and the earth’s shadow casts a penumbra. What do you get to the other limb?

  68. Joseph says:


    Great work Tim!

    “Wow, yes, for goodness’ sake. He’s doing the exact same things I am doing with my math. Everything he’s modeled there I can justify theoretically via the math and physics, Tim Channon is not just making something up, he’s modelling the real situation, intelligently. I should get the new paper out asap which might help what he’s doing there.”

  69. Stephen Wilde says:

    Good work, chaps.

    Get the Moon nailed and a lot else follows.

  70. tchannon says:

    A further problem occurs to me, gamma. Cameras are bent and vary horribly. Would need test results.

  71. tchannon says:

    Rog, you think I have the wrong edge?
    Perfectly possible, was a guess.

    Flat line.

    Ideas anyone?

  72. tallbloke says:

    Tim, that definitely the Earth’s penumbral shadow cutting across the Moon on that side. The other side (7oclock clockwise round to 2oclock) is the limb of the moon against space. And brightness doesn’t fall off very much at all. I think that’s due to the shape of the particles reflecting the light. But I could be wrong.

  73. wayne says:

    “Tim, that definitely the Earth’s penumbral shadow cutting across the Moon on that side.”

    TB, have seen your misassumption there? Tim has is right. And Tim, you have done one fine work on these questions of the moon’s relation to radiation and temperature.

    It is apparent, especially on Earth, that is thermal inertia is the key. The moon is never less than 100K average and when looking at a one moon-day cycle the radiation only has to handle the difference between the highest daytime and lowest nighttime differences. On Earth this aspect is even more magnified for the mean diurnal range is a mere ~10K according to NASA. That means it takes much less energy than if you are trying to account the difference between the highest mean and absolute zero. Thermal inertia makes this possible in reality and most of climate sciences assumptions are flatly wrong. Venus is the ultimate example of this with a diurnal range of nearly zero.

  74. tallbloke says:

    Wayne, I took the photo myself. I was stood out there that night. It wasn’t an eclipse, just a normal moon phase. Penumbra is the wrong word, but there’s an edge effect from the Earth’s shadow

    Look at this image of the full moon. It always keeps the same face to us. You tell me which side the Earth’s shadow falls across in my photo. (my image is upside down, it was a reflecting telescope)

  75. tchannon says:

    Lovely image Rog, and the missus just has a look too, likes it. There is a fascination with partial illumination seen live, the edges of craters.

  76. tchannon says:

    Oh dear, on recomputing taking into account the 1.3 factor both ways I am getting a different mean. 199.26

    In this case I have included a band at 89 degrees, doubt it matters.

    What I’ve done here is do a parametric sweep from 0 to 85 degrees at 5 degree steps, plus 89 degrees, changing the irradiance factor by sine()^1.3 on the basis the factor applies to X and Y

    I’ve then computed the equal area mean, which is the weighted mean but without the 1.3 factor.

    Heaven knows if this is valid.

    Anyone able to check my maths?
    XLS is here

    There is a large array which is imported data, longer than needed. 600 to 630

    To the right of that is copied data (reason, can paste over original) for the period of interest 600 to 623 (see earlier plots)

    Bottom right of that is the answer. Been over and over after making stupid finger trouble mistakes.

  77. wayne says:

    Roger, now that you mention earth’s shadow again, maybe I am misunderstanding you. I just said that half-kidding. When you say ‘edge effect’ from the Earth’s shadow on the moon, the only thing I can think of is a partial eclipse. Or are you speaking of earth shine. And wasn’t Tim trying to get a series of numbers to plot the curve of how much solar radiation drops off at the horizons as seen from the sun, the edge effect?

  78. tallbloke says:

    Wayne: it was brain fade on my part, too many threads at once. It wasn’t an eclipse, and so it isn’t Earth’s shadow. So Tim is quite correct and I get the dunce’s hat. 🙂

    In theory there is Earthshine on the portion unlit by the Sun. However, when we compare the tiny wattage with the solar glare, it is negligible.

    Tim: 199.26 is bang on the money with the DIVINER estimate. Time to stop and let it all sink in. I think your methodology is correct, but why take the word of a man who can’t remember what causes the phases of the Moon? 😦 🙂

  79. Tenuk says:

    Excellent result, Tim, and this seems to nail the mysterious 1.3 without having to delve into the difference in thermal conductance between feldspar and basalt.

    Would it be difficult to do the parametric sweeps at <5 degree steps,as this could perhaps make a difference to the final result.

  80. tchannon says:

    I will probably update the article, or I suppose it could be republished, with content pulled out of contents.

    This needs crediting
    MeeSoft, software by Michael Vinther

    Profile Line tool

  81. tchannon says:

    Rog. Happens to me too.

  82. tchannon says:

    Could but that would take days because it exceeds the triviality limit. It would need internal scripting and external programming.

    Easier would be trying to estimate the effect.

    Quick as flash, try this, take the XLS just above and delete (or set to zero) every other weighting factor starting at 5 degrees, above the right hand matrix. Read off the new mean temperature figure. (works because J * 0 =0 and S = S + J, values fall out of calculation)

    Call it 200K (just under). Betcha the error is larger than that anyway for other reasons.

  83. tchannon says:

    For the current equatorial simulation conditions.

    TSI 1360 W/sqm T = 214.7K
    TSI 1362 W/sqm T = 214.8K

  84. Tenuk says:

    tchannon says:
    April 5, 2012 at 3:44 pm
    “Tenuk – Could but that would take days because it exceeds the triviality limit. It would need internal scripting and external programming.”

    Thanks Tim. Definitely not worth the time, life’s too short.

  85. tchannon says:

    A provisional switch to earth has produced results where I am nervous of showing anything.

    Lunar model.

    Change R and C only, the parameters of the body surface.
    Increase simulation time from 600 to 6,000 to allow longer settling time.

    Global mean temperature 14C
    Equator mean 25C
    Range on equator 20C
    Mean at 65N -1C

    Surface thermal conductance for this 0.015K / Watt which I make 66 Watts / K sqm (Moon 0.1 about right for insulation, particles under dessicated high vacuum)

    Keep clearly in mind there is no atmosphere and this is ground temperature.

  86. SergeiMK says:


    Of interest the LTSPICE programme has the option of outputting a WAV file so you can here the simulations. – you obviously need to change the time (to bring the frequency in audio range) and amplitude (to prevent saturation).

    Just feed the cycles into a summing circuit and monitor the output

    Contact me if you want more info.

  87. […] average surface temperature for the Moon’s equatorial band of 214.4K. Tim Channon then built a very neat model to replicate the Vavasada result. Ned Nikolov showed that the way that the average surface […]

  88. […] model by Tim Channon that accurately reproduces the Diviner data can be found on Tallbloke’s […]