As mentioned before, the thermal inertia is a somewhat important ingredient. Now, it isn't hard to find out the heat capacity of rock - but the main question is how deep into the ground rock (water,...) participates in the thermal evolution.

Since fundamentally heat is transported inside the ground with a diffusion equation, it is at least clear that the size scale has to go like the square root of the timescale (because that is a general characteristic of diffusion equations). So the problem boils down to fixing a size scale for one Earth day, and then using the scaling with time to apply the value to other timescales.

Of course, for the Stormhold project, there is yet another complication, because the 'day' (the sidereal rotation) isn't even the relevant timescale for thermal evolution - because the day merely compensates the orbital rotation. Instead, the 'year'ly cariation, i.e. the motion from apoapsis to periapsis, is the relevant scale.

Anyway, using a basic size scale of 30 cm rock influenced by thermal variations during a day, scaling this with a Mercury day and appliying it to Mercury in its orbit gives these temperature fields at periapsis

and at apoapsis

which is halfway realistic.

Applying the same values to Earth on the other hand gives too harsh contrasts between high and low latitudes. The reason is of course the atmosphere - it not only does Greenhouse effect, but also transport.

Generally, atmospheres are complicated beasts, because while they influence the thermal balance, the thermal balance also influences the atmosphere. For instance, at Earth gravity and temperature one can't have a hydrogen atmosphere component - it would boil off into space, thermal speeds of molecules are frequently above escape velocity. So the atmosphere must be heavy enough to be bound at the given temperature.

Then there's the Greenhouse effect (we'll cover that in more detail at some point) - heavily driven by water vapour in the atmosphere. But if the atmosphere gets warmer, it can hold more water, so the Greenhose effect increases, so it can hold yet more water... There's equilibrium points that are earth-like, where the overall role of the Greenhouse effect on thermal balance is limited - but there's also the Venus-like equilibrium where the process continues until all water is in the atmosphere which is heated to unbearable temperatures.

There's snow and albedo - the colder the air is, the more snow stays on the ground. Snow has a high albedo, so it cools the planet, so more snow remains, so the albedo increases yet further. Again this has an Earth-like equilibrium where the seasonal variations are enough to undo it, and a runaway scenario called 'snowball earth' - that seems to have happened more than once in the past (it really depends on which latitudes the continents drift...)

The code generally doesn't check whether the solution is self-consistent, that is something the user has to do, so usually with an exoplanet it's down to iterating the conditions a few times (for Earth, we of course know orbital parameters and the amount of Greenhouse effect and so on...)

Heat transport on Earth is pretty complicated, it involves convection cells (Hadley cells) of limited extent because Coriolis force prohibits large-distance migration of airmasses, sea currents dependent both on underwater topology and Coriolis forces... It's... pretty hard to even halfway simulate that in any meaningful detail without a supercomputing facility.

So the transport code does something rather simple - it does transport on average. If there's a temperature differential between two cells, it permits heat transport between them proportional to the difference and a selectable transport coefficient. So we don't have detailed transport, but we do have transport 'of the magnitude of Earth' or 'a tenth of Earth' to get the right ballpark.

By selecting both thermal inertia and atmosphere transport, one can achieve plausible average daily temperature profiles on Earth at different locations and seasons.

So... using an earth-sized albedo, a somewhat smaller than earth-size Greenhouse effect, earth-sized transport with air and sea currents and then iterating a bit with the details of the orbital parameters gives these solutions for the star and the planet:

Mime

------------

Mass [m_sun]: 0.4

Surface T [K]: 3700

Luminosity [L_sun]: 0.0279554

Radius [Mkm]: 0.283751

Mean density [g/cm^3]: 8.3118

Spectral fraction IR 0.818121

Spectral fraction vis 0.170524

Spectral fraction UV 0.0113552

Stormhold

------

Mass [m_earth]: 1.05

Radius [R_earth]: 1.01

Mean density [g/cm^3]: 5.61902

Surface gravity [g]: 1.03034

Semimajor axis [Mkm]: 28.4236

Eccentricity 0.1

Periapsis [Mkm]: 25.5812

Apoapsis [Mkm]: 31.266

Period [days]: 47.8297

Rotation period [d]: 234648

Sid. rot. period [d]: 47.82

Inclination [deg] 3

Thermal properties

------------------

Max. irrad. [W/m^2]: 1307.44

Min. irrad. [W/m^2]: 875.229

Albedo: 0.3

T_rad periapsis [K]: 280.776

T_rad apoapsis [K]: 254.005

and these temperature profiles across the surface at periapsis

and apoapsis

so there is quite some temperature difference left between day and night side, but there clearly is a ring of habitable temperatures that is much larger than in the atmosphere-less case.