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The full problem statement is:

Measured from the time when the first rays of sunshine appear above the horizon until the moment when the sun is fully visible, sunrise lasts 2.1 minutes. Based on this information, and assuming that the earth and the sun are “black bodies”, can you estimate the temperature of the sun?

I found this problem in a statistical physics course: link.

I am not able to see how I could possibly connect the given information to the temperature of the Sun.

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2 Answers 2

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The sunrise time gives you an estimate for the angular size of the Sun, roughly it's diameter divided by the distance to Earth.

This comes from dividing the sunrise time by 24 hours, assuming this is the fraction of $2\pi$ radians occupied by the Sun's disk, that the Sun rises perpendicularly to the horizon and ignoring complications due to the Earth's atmosphere.

Assuming a balance between flux received and flux radiated, it is easy to show that the equilibrium temperature of the Earth depends on the effective (blackbody) temperature of the Sun multiplied by the square root of its angular size. You possibly ought to include the albedo of the Earth here.

Details:

Angular size in radians is given by $2\pi\tau$, where $\tau$ is the sunrise duration expressed as a fraction of a day and this is equal to the angular diameter of the Sun: $$2\pi \tau = \frac{2R}{D}\ , $$ where $R$ is the Sun's radius and $D$ the Earth-Sun distance.

Then the flux balance between the flux intercepted by the Earth and the flux radiated by the Earth, assuming both Sun and Earth radiate as blackbodies (and ignoring any albedo) $$\left(\frac{4\pi R^2 \sigma T_{\rm Sun}^4}{4 \pi D^2}\right) \pi R_E^2 = 4\pi R_{E}^2 \sigma T_{E}^4\ , $$ where $R_{E}$ and $T_{E}$ are the radius and temperature of the Earth, and $\sigma$ is the Stefan-Boltzmann constant.

Cancelling and rearranging: $$T_{\rm Sun}^4 = 4\frac{D^2}{R^2}T_E^4 = \frac{4T_E^4}{\pi^2 \tau^2}\ .$$ Hence $$ T_{\rm Sun} = \sqrt{\frac{2}{\pi \tau}}T_E\ . $$

If we assume $T_E\sim 280$ K (just based on our experience of temperatures on Earth) and $\tau = 2.1/(24\times 60)$, then the answer follows as $T_{\rm Sun} \simeq 5850$ K, and is surprisingly accurate. But then I guess the sunrise time of 2.1 minutes was chosen to give a sensible answer given a simple treatment.

However, you could easily make your answer more sophisticated. If the Sun isn't rising perpendicularly to the horizon, or refractive effects make it become visible before it has geometrically cleared the limb of the Earth, then the angular diameter calculated above would be an upper limit to the true angular diameter, leading to a lower limit to the solar temperature. The answer only depends on the square root of this parameter.

Exactly what you choose to be the equilibrium temperature is also up for debate. The value I chose above gives roughly the "right answer", but is probably a bit lower and can be calculated in increasingly sophisticated ways but would require additional information beyond simple observation and everyday experience.

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    $\begingroup$ I'd like some extra details. This sounds like an interesting answer. $\endgroup$
    – Tripasect
    Aug 2, 2023 at 0:04
  • $\begingroup$ Isn't the equilibrium temperature at 260K (we need to ignore the greenhouse effect for the blackbody, and that is the result when doing the forward calculation with the Stefan-Boltzmann-law) ? And wouldn't we need this information as a given? (earth-sun distance is something reasonable well known, but would technically also be needed). Lastly, the question appears to assume a given latitude and time of year. Bad question overall. $\endgroup$
    – Chieron
    Aug 2, 2023 at 7:44
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    $\begingroup$ @Chieron no, the Earth-Sun distance is not required for this question to be answered reasonably. We estimate the equilibrium temperature of the Earth from our experience, which suggests a temperature of about 280K. That is a bit too high - and affects the answer by about 10%. Latitude etc., yes, the angular diameter would be an upper limit. Good question overall. $\endgroup$
    – ProfRob
    Aug 2, 2023 at 8:43
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    $\begingroup$ Does refraction angle of light hitting the atmosphere depend strongly enough on blackbody temperature of the sun for that to affect when the sun first becomes visible? I guess it would have an almost equal effect on when the sun fully clears the horizon, though, so sunrise duration would depend much less strongly on it. $\endgroup$ Aug 2, 2023 at 9:51
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    $\begingroup$ @PeterCordes I really don't know the full details of that process - because the Sun emits a broad spectrum and refraction is wavelength dependent, it gets complicated quite quickly. I think people are over-analysing this. The intent of the question is clear and the choice of 2.1 minutes is designed for a simple assumption to give the correct answer. $\endgroup$
    – ProfRob
    Aug 2, 2023 at 10:23
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No, definitely not. The Sunrise duration depends on the angle of inclination between the Ecliptic and the horizon, which varies depending on the latitude of the position on Earth. In addition, the duration will be modified by refraction depending on local atmospheric conditions.

In the Polar regions there are months when the Sun remains above or below the horizon (i.e. Sunrise duration is zero) and extended periods where it skims the horizon for an entire day (i.e. Sunrise duration is 24 hours). In addition, it is plausible to assume that when the Sun is skimming the horizon there are extended periods when its centre is significantly below the horizon but it is visible due to refraction. This does not imply that the temperature of the Sun is varying day-by-day, or for that matter that the temperature of the Sun varies for different observers.

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    $\begingroup$ Nevertheless, the question has clearly been set so that the duration is what you would expect with none of these effects in operation. $\endgroup$
    – ProfRob
    Aug 2, 2023 at 6:50
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    $\begingroup$ @ProfRob The question is meaningless without being told the location and date. $\endgroup$ Aug 2, 2023 at 7:30
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    $\begingroup$ Fair enough, maybe that's the answer that would have got the marks (with some arguments, as you have presented). Possibly a better answer though would be to say that 2.1 minutes gives an upper limit to the angular diameter and therefore a lower limit to the solar temperature. $\endgroup$
    – ProfRob
    Aug 2, 2023 at 8:28
  • $\begingroup$ @ProfRob That's a possibility, but since one has to allow a couple of minutes imprecision when predicting the time the Sun will first become visible or will appear to have cleared the horizon, it's also reasonable to assume that the time between those points can be changed to some extent due to refraction. Observational exercise for students: to what extent is the Sun "pear shaped" as it rises and sets due to refraction effects? :-) $\endgroup$ Aug 2, 2023 at 9:39

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