Airless Body Thermal Model

Picture the Moon sliced like a loaf of bread, one column running straight down from the surface to the very centre. That column is chopped into a stack of boxes — thin ones near the surface, fatter ones deeper down. Each box is treated as one uniform blob: one temperature, one composition, no structure of its own.

Every box only ever talks to its immediate neighbours — mainly the box above and the box below it, plus a far weaker sideways exchange with the neighbouring latitude's box at the same depth (the north–south flow mentioned above). It gains or loses heat by conduction to and from those neighbours — and, only at the very top of the stack, gains heat from sunlight by day and loses heat to space as thermal radiation, day and night.

Zoom out and the same rule applies all the way down, box after box, growing thicker with depth, until a single ball-shaped cell at the very centre stands in for the whole core (see "More on the numerical scheme" below for why).

None of this happens in one leap. The simulation advances in small steps of time — a few simulated minutes at once — and at each step works out, for every box, how much heat it gained or lost since the last step, then nudges that box's temperature by the resulting amount before moving on. Thousands of these small steps, one after another, is what turns "boxes exchanging heat with their neighbours" into a full day/night cycle — and enough repeated cycles is what settles into the steady state plotted throughout this page.

Regolith is the loose layer of dust, broken rock, and rubble that blankets the Moon's solid bedrock — built up over billions of years as meteorite impacts, large and small, kept pulverizing the surface. It isn't "soil" in the Earth sense: no water, no organic matter, just fragmented mineral grains.

Those grains barely touch each other, especially with no air pressure to pack them together, so heat has very few solid paths to conduct through — a big part of why regolith conducts heat so poorly (see "More on conductive heat transfer" below).

Sunlight arrives as parallel rays. Straight overhead, that bundle of rays lands on the smallest possible patch of ground — maximum energy per square metre. Tilt the same bundle to a shallower angle and it spreads over a wider patch, so the same total energy gets diluted over more area. This is Lambert's cosine law: the sunlight a patch of ground actually receives scales with cos(angle from directly overhead) — full strength with the Sun straight up, tapering smoothly to zero right at sunrise and sunset, where the rays graze the surface edge-on.

That angle isn't just "time of day" — it comes from three things combined: latitude, the Sun's declination (how far north or south of the equator it's shining, cycling with the orbit and the axial tilt), and the hour angle (how far the rotation has carried this point since local noon). Put those three together and you get how high the Sun sits in the sky, for any latitude, at any time of day, at any point in the year — and that's exactly what multiplies against the raw solar constant to give the sunlight a given box actually receives right now.

It's this smooth cosine taper, not a day/night on-off switch, that shapes sunrise and sunset in the simulated temperature curve.

In this model the declination cycles over a ~365-day year — the year of a body orbiting the Sun at the Moon's distance — and stays on that cycle no matter what you set the rotation period to: spinning the body faster changes its day length, not its orbit. The real Moon's seasonal cycle is a little shorter (about 347 days) because its spin axis is coupled to its orbit around the Earth; with a tilt of only 1.54°, the difference between those two cycles is far smaller than anything else this model resolves.

The model's radiation law is εσT⁴ — the energy a surface radiates per square metre, where ε (emissivity) is how efficiently it radiates compared to a perfect emitter and σ is a universal constant (Stefan–Boltzmann). That is really shorthand for the general law: net radiative exchange between two surfaces goes as the difference of their temperatures raised to the fourth power, εσ(T_surface⁴ − T_surroundings⁴). For the Moon's surface radiating out into space, "surroundings" is the cosmic microwave background at about 2.7 K — not a literal 0 K. Throughout, the model uses ε = 0.95 — a representative value for silicate regolith — with no slider: it applies identically to every run.

That background term turns out to be negligible here: 2.7⁴ ≈ 53, versus around 65,000,000 for a surface at 90 K (90⁴) — roughly a million times smaller. Dropping it, as this model does, isn't a simplification that changes anything measurable; it's far below the precision of everything else in the calculation.

Conduction follows Fourier's law: heat flux is proportional to the local temperature gradient, −k·dT/dz, where k is thermal conductivity. By day, the hot surface conducts heat downward into the cooler regolith beneath it; by night, that stored heat conducts back up as the surface radiates and cools.

How much lag this creates depends entirely on k. The Moon's loose surface regolith conducts extremely poorly — the layered model's surface default here, 0.0007 W/(m·K), is close to 3,000 times smaller than its deep default, which represents solid compacted rock beneath the regolith (measurements of deeper regolith itself give values much closer to the surface number). That's why so little of the day's heat reaches even a few centimeters down before nightfall.

Conduction sets how fast heat moves between boxes; specific heat sets how much a box's temperature changes per joule it gains or loses. For regolith, specific heat is itself strongly temperature-dependent: Apollo sample calorimetry (Hemingway et al., 1973) shows it rising from roughly 250 J/(kg·K) near 90 K to roughly 850 near 350 K — a cold box needs only about a third as much heat per kelvin as a hot one.

The model therefore re-computes each regolith box's specific heat from that box's current temperature at every step (via a published polynomial fit to those measurements), identically under both conductivity models — this isn't a choice you make, it applies to every run. In the layered model, the compacted material beneath the regolith keeps a single fixed value instead.

Depth cells are geometrically graded (thin near the surface, where the diurnal thermal wave actually lives, coarser with depth) and switch to mass-conserving spacing near the center, ending in a single innermost cell that represents a small ball rather than a vanishing wedge — avoiding the coordinate singularity at r=0 without needing to resolve it exactly. The same grid is used for every run, whatever the parameters — so when two runs differ, the difference comes from the physics you changed, never from a change of numerical resolution. (One disclosed exception: the cheese material uses a coarser surface cell matched to its much deeper day/night wave — cheese conducts so much better that resolving it on regolith's millimetre grid would make runs ~20× longer to capture detail thousands of times smaller than any cheese-vs-regolith difference. All cheese runs share that one grid too.)

Reaching a periodic steady state by pure explicit time-marching can take many simulated rotation periods, especially for the real deep interior. This uses Aitken-extrapolation acceleration: run a few real periods, extrapolate each cell's asymptotic value from the trend, jump most of the way there, then run more real periods to "heal" whatever the jump got wrong — same equations and domain throughout, just a faster way of finding that same model's steady state.

Because the surface and the deep interior settle at wildly different speeds. Only the top few centimetres actually feel the day/night cycle within a single rotation (see "More on conductive heat transfer" above for why conduction here is so slow). Everything below that thin skin barely warms or cools on any single pass — heat trickles down only a little further each rotation, and whatever's already down there keeps conducting back up and down long after the rotation that put it there has ended.

That means a box a metre down isn't just reflecting "today" — it's a running memory of many previous rotations layered on top of each other. Simulating a single rotation from a guessed starting temperature captures none of that memory. Only after enough repeated rotations does every box's day-to-day swing stop changing from one rotation to the next; that repeating pattern is a periodic steady state, and it's the only point at which the simulated temperatures actually mean anything (see "More on the numerical scheme" above for how this model reaches that state faster than by brute-force repetition alone).

Two consequences of this are visible while the simulation runs. First, run length: the simulation advances in physics steps of a fixed few simulated seconds each — set by the box sizes, which are identical for every run, not by the day length — so a rotation that contains more simulated time contains more steps, and one 708-hour lunar rotation simply holds more simulated time to work through than one 24-hour rotation. (Fast rotations are also never allowed to get crude: every rotation is resolved by at least a few thousand steps, however short its day.) A quick run is not a shortcut; each rotation just contains less simulated time. Second, the number of rotations is not fixed: the run does at least 12, then keeps adding more (up to 36) while the equator's coldest nightly temperature is still changing from one rotation to the next. The "settling in" chart below shows exactly the quantity that test watches — and if a run stops at the cap before flattening out, that chart will show it.

Lunar regolith conducts so poorly that heat barely moves a few centimetres vertically over an entire rotation (see "More on conductive heat transfer" above) — let alone sideways, between one longitude and its neighbour, a surface distance measured in hundreds of kilometres. Whatever's happening at one longitude simply can't conduct its way over to affect the next one before the whole pattern has already reversed with the next sunrise. East-west conduction is negligible enough to ignore entirely.

That has a convenient consequence: every longitude experiences exactly the same day/night cycle as every other longitude at the same latitude, just shifted in time by however far apart they are in rotation phase. Once a single simulated column settles into its repeating daily cycle, every other longitude's history is already sitting inside that same cycle — just read starting from a different point in the rotation. The globe elsewhere on this page is built exactly that way: one column's temperature history, replayed at every longitude with a different time-shift, rather than an actual simulation of each one separately.

It isn't that internal heat is assumed to be negligible — this version simply doesn't model it, on purpose. The deep boundary at the very centre carries no imposed heat flow of its own; it's given a starting temperature and then left to evolve purely by conduction from everything above it, the same rule every other box follows.

That's a deliberate choice about what this particular model is testing: how much of the Moon's surface temperature is explained by sunlight, conduction, and radiation alone, with nothing else added. Whether folding in a small steady geothermal flux at depth — the kind Apollo's own heat-flow probes measured at two landing sites — would noticeably change the answer is a genuinely open, separate question. Testing it means adding a flux term to this same validated model, not assuming the answer either way in advance.

One consequence worth knowing: regolith conducts so slowly that below roughly the top metre, no run of any practical length can move the temperature much in either direction. The deep interior therefore effectively keeps whatever temperature the simulation starts every box at (220 K here). That starting value is itself an assumption — it quietly stands in for everything below the sunlight's reach, including any internal heat — and it is one more reason not to over-read a close match in the comparison section, if you get one.