Temperature evolution on Mercury
Orbital dynamicsWhile Mercury seems an otherwise boring planet, it's orbital dynamics is rather interesing. The surface temperature evolution is driven by a 3:2 spin orbit resonance in combination with the moderate eccentricity of the orbit. This means that the planet rotates slowly - three times around its axis every two orbits. This in turn implies that if at periapsis one spot on the surface has the Sun at the zenith, an orbit later the spot at the opposite side of the planet will have the Sun at the zenith, and the next orbit will again see the first spot in this special role.
Why is this important? Because the orbital eccentricity causes the apparent movement of the Sun to vary from periapsis to Apoapsis. The speed at which the Sun moves across the sky is always the combination of the planet's (unchanging) rotation and the (changing) orbital velocity. For Earth, the rotation (24 h) is much faster than the orbital period (365 d) and moreover the orbit has a low eccentricity, so none of this is an issue. For Mercury on the other hand, this matters a lot. If one plots the motion of the Sun across the sky (here as 'day fraction' beween 0 and 1) against the time, the result is not a straight line as it would be to very good approximation for Earth - instead, the motion holds and even slightly reverses twice every 'day':
This reversal happens every time the planet reaches periapsis and the orbital motion is fast - as it moves out to apoapsis, it slows again and the rotation around the axis dominates the solar motion.
Total irradiationFor the total irradiation summed over two orbits, this has interesting consequences. Take the subsolar point at periapsis (i.e. the point at which the Sun is in the zenith). Since Mercury is at periapsis, the distance to the Sun is smallest, so the radiative flux arriving is largest. At the same time, the apparent motion of the Sun in the sky holds for a couple of days, which means this surface point gets exposed to a high radiation flux over an unusually long period of time. And an orbit later, the same happens for the point 180 degrees around the planet.
When one plots the total irradiation received as a function of latitude and longitude, it becomes clearly apparent that there are two points on the equator which receive maximal energy while the points 90 degrees off in longitude receive less than half of that value. Superimposed is the gemeotrical reduction of the irradiation by the radiation incident angle at high latitudes.
Temperature evolutionMercury does not have an atmosphere, which means there is no significant heat transport and the temperature of each surface element is the result of a radiative calculation only, where the radiation flux from the Sun heats the surface, the IR emission from the surface cools it, and both fluxes feed into the heat capacity of a surface layer which participates in this diurnal cycle.
Using Mercury's mean albdo of 0.068 and guesstimating values for the thermal inertia of the surface rock, the simulation produces the following results for the temperature distribution at periapsis:
The existence of a 'hot spot' underneath the sun where temperatures reach to ~680 K is clearly apparent. Since the surface does not cool instantly, the simulation leads to an asymmetric temperature distribution around the hot spot. During the night, temperatures drop down to ~ 200 K.
Running the simulation to the apoapsis finds the hotspot displaced by 90 degrees in longitude as expected. It is significantly cooler, just about 580 K, and the average temperature across the planet is also lowered.
Max. daytime and min. nighttime temperatures are in decent agreement with what has been measured on Mercury.
Definition fileThe basic worldbuilder definition file which has been used to create the above simulation is as follows:
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