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Back to Math And Terraforming

It is hard to predict the exact climate patterns of a new terraformed world. Just keep in mind the thousands of scenarios scientists make for Earth's climate changes caused by manmade emissions of greenhouse gasses. Although all these models show a global warming, rising in sea levels and weather instability, no model predicted exactly what happened in the last years and exactly what will happen in the nearby future.

Scientists from NASA and ESA have made simulations for virtual planets and for terraformed planets, but nobody knows exactly how the weather patterns will be. We don't have the supercomputers and simulations NASA has, but still we have some simple formulas that will give you an idea of what to expect.

Average temperature Edit

The initial average temperature on a planet or moon might be between absolute zero to over 2000 degrees. However, once terraforming occurs, we will want something close to what we have here on Earth. Earth's average temperature is of 15 degrees Celsius or 288 degrees Kelvin. The average temperature on a terraformed planet would be 15 or close to 15 degrees. Sometimes, terraformers and technicians might argue on a higher or lower value. For example, on a Desert planet, since most of the water is expected to condense close to the poles, we would like to have acceptable temperatures there and we will let the equator scourged by heats exceeding 100 C. On the opposite, on an Oceanic Planet, if we want to use ice as the basement for Artificial Continents, then we would focus on an average temperature very close to oceans' freezing point.

Variations Edit

Temperature variations are found between day-night cycles, between seasons, between latitudes and between altitudes. The factors that influence all these changes are the Solar Constant (amount of light and heat received from the parent star), day length, year length, orbital ellipse, axial tilt, atmospheric composition, planet size and many other factors.

Majority of these factors can be put into an equation and you can have an idea of how the climate patterns will look like. I made the formulas myself and I have improved them for years. They are close to reality, but not as exact as the data you would gather from a supercomputer with a high-performance climate simulation. The formulas were first made for an old computer program, Quattro Pro. Later, I upgraded them for Microsoft Excel, which is currently the most common program used for mathematics.

Formulas Edit

For other details, see Temperature.

It all starts with the solar constant or Ks, which is the amount of energy received by the planet or moon. For Earth, Ks is 1.98.

Local solar constants Edit

Local solar constant (LKs) is the amount of energy received for each latitude. For a planet with no axial tilt or for a planet that is at equinox, the local solar constant can be estimated by the following formula:

LKs = Ks*(90-deg)/45

where deg is the latitude in degrees, Ks is the solar constant and LKs is the local solar constant for each latitude. Please note that LKs will have the same value as Ks for a latitude of 45 degrees, at the equator it will be double and at the poles it will be zero. The formula reflects the angle in which the sun illuminates our planet.

If the axis is tilted, we have to do some correction. Suppose our planet is tilted by 50 degrees and you measure at South solstice. You have to do two corrections. Firstly, the sun will illuminate the planet under a tilted angle. Secondly, the day length is increasing in South and beyond a latitude of 50 degrees, you will have a polar day.

LKS = Ks*(90-deg)/45*((90-cor)/90)


  • LKS is the local solar constant for tilted angles
  • deg is the maximum solar angle.
    • deg = latitude + axial tilt for Northern latitudes between equator and the Arctic circle.
    • deg = 90 for latitudes beyond the arctic circle.
    • deg = axial tilt - latitude for latitudes between the equator and the Antarctic circle.
    • deg = latitude - axial tilt for latitudes beyond the Antarctic Circle.
  • cor is the correction for day length.
    • cor = 90 for latitudes beyond the Arctic Circle.
    • cor = axial tilt + latitude for altitudes between equator and the Arctic Circle.
    • cor = axial tilt - latitude for altitudes between the equator and the Antarctic Circle.
    • cor = 0 for latitudes beyond the Antarctic Circle.

This way, you will get an idea about the amount of heat the planet receives at each latitude. You can make variation models, using different axial tilts. If you want to do an estimation for a moment between equinox and solstice, use an intermediary value for axial tilt. If you do this, you will have to move the Arctic Circle and the Antarctic Circle further North respectively South, to where polar day and polar night will start. For example, if a planet has its axis tilted of 40 degrees and you want to know how it would look halfway between equinox and solstice, do the calculations for an axial tilt of half (20 degrees).

If you have a planet that moves on an ellipse, you have to do the math at closest and furthest approach, since you will have different solar constants. The planet will move faster when close to the star and slower when further away. Also, if the planet has a tilted axis, things will be even more complicated.

Sometimes, by computing all these data, you will get a large table of values. This is the way it is done. For the entire planetary year, you have to find out the local solar constants. However, once you've done it, you will have the first part needed.

If the planet has an elliptical orbit, you also need to find out the planet's solar constant for each column in the table, as a reference point.

Planetary constraints Edit

Before we can transform the LKS values into actual temperatures, there still is one step. This step is to fix planetary constraints. Why this? Because the atmosphere absorbs part of the heat during daytime and radiates it back during nighttime. Atmospheric currents also allow air to mix. This brings heat to the poles and gently cools the equatorial regions. If our planet is tidal locked, air currents will keep some heat to the night side, also cooling the day side.

We see on Earth that even in Antarctica, during polar night, temperature does not drop to absolute zero. It reaches values of -70 C.

The most extreme temperatures will be reached on a tidal locked planet. This kind of planet allows us to see the planetary maximum and minimum constrain. For this, we will set the North pole at center of the night side and South pole at center of the day side. This would look like an extreme polar day and polar night. The formulas will be:

Var = 1/((1+Ks*0.8205)*speed*(1/(diam/12.756))*atm)

Here, we have:

  • Var is the temperature variation (a factor to be used in other formulas later).
  • Ks is the solar constant.
  • speed is the atmosphere speed (the speed with which air circles between poles or between day and night hemispheres, for tidal locked planets; Earth = 1).
  • diam is the planet's diameter in thousand km.
  • atm is the atmosphere mass per square meter (Earth's is 1 by default). Sometimes, you can use density, but it is better to look at formulas shown at Atmosphere.

The next step, using the variation, you can get the minimum and maximum temperatures on the planet:

Tmin = Tavg-(Tavg*var)

Tmax = Tavg+(Tavg*var*0.75)


  • Tmin is the possible minimum temperature
  • Tmax is the possible maximum temperature
  • Tavg is the average planetary temperature (in degrees Kelvin)
  • var was defined above.

The values you will get are in degrees Kelvin, so they can be transformed into Celsius.

For Earth, you will get an absolute minimum of -94 and an absolute maximum of +97 C.

Another important value that we need to get is the temperature variation factor:

Tvar = var*Tavg.

The temperature variation factor is important and will be further used in the next formulas.

Converting solar constants into crude temperatures Edit

The Temperature formula, based on solar constant, is:

Tv = LOG10(1+Ks*(1e+100))/((4.6e+22)*0.58674^(LOG10(1+Ks*(1e+100))))

where Tv is the void temperature (as measured in cosmos, by an object painted 25% gray, exposed to the solar constant Ks). For every local constant from the table, we must apply the formula, to get an estimated temperature.

When applied the temperature formula, we get what I usually call space temperature map. Values are extreme and in polar night areas it is as cold as absolute zero. In fact, what we now have is what the temperatures would be without any atmospheric protection, if the planet was painted 25% gray and if the soil would radiate no heat during night.

The next step is the crude temperature map. For this, use the following formula:

Tlc = var*Tl/Tv+(Tmed-var)

Here, we have:

  • Tlc = local average temperature
  • var = temperature variation factor (as described above)
  • Tl = local void temperature (from space temperature map)
  • Tv = the void temperature (as measured for the planetary solar constant)
  • Tmed = the average temperature resulted after terraforming.

The system gives values relatively close to the truth, except for the poles and parts of the planet under polar night, where temperatures are always at the minimum.

Inertia Edit

The results are close to the true values, but there is one more factor we have to include: thermal inertia. This actions in two ways: by keeping some heat from summer to winter (this is why winter solstice is closer to the beginning of winter) and by keeping some cold from winter (again, this is why the summer solstice is closer to the beginning of summer, not in the middle of it). Also, the inertia keeps some heat from the day to the night.

There are many parameters that influence the inertia. And when you do the best calculations possible, the air currents shift and you get what never expected: a Siberian wind blows in Europe bringing extreme cold temperatures that otherwise could not be explained (or hot air blows from the Middle East and Sahara and overheats everything). So, it is hard, if not impossible, to predict what climate will be linked to a limited area, because air, as moves, changes slowly its temperature.

What can be said for sure is that on a planet with short seasons (for example rotating closer and to a dimmer star), the seasons will be shorter and softer. It will not be enough time in summer to heat-up, while in winter there will not be enough time to cool down.

Day - night cycle Edit

On any planet, temperatures change during the day-night cycle. These temperature variations are influenced by many factors, like rains, clouds and altitude. The factors that we can include into a mathematic equation are: day length and greenhouse effect.

For day length, the maximum and minimum values are in case of a tidal locked planet and are described above, at planetary constraints.

The formula is:

Dv = var*(1-1/((day*0.1001)+1))


  • Dv is the daily temperature variation
  • var is the temperature variation factor (described a few sections above)
  • day is the day length (in Earth days).

This formula gives you the average difference between day and night (max and min) temperatures, on a clear sky.

The formula is calibrated for Earth, where temperature changes are usually around 10 degrees. A longer day will result in higher day-night temperature fluctuations, but never above a certain limit. A shorter day will result in much smoother temperatures.

Winds and their influence Edit

The models listed here have one problem: that the winds change automatically when temperature changes. In case of an inner planet or an Earth-like planet, if the axis is very tilted, winds will speed-up and will try to mix the atmosphere. The increase in wind speed is important and results in a substantial reduction of the temperature variation factor.

In case of outer planets, the opposite happens. Winds there are expected to be extremely slow, because temperature changes are too small. On Earth, high altitude currents move with speeds that sometimes pass over 100 km/h. On a terraformed Titan, it will be a big surprise to see winds stronger then 5 km/h. As a result, there will be some temperature changes during Titan's long polar day and polar night, but not that much. I estimated that on Titan difference between poles cannot be over 7 degrees even at solstice.

In the end Edit

An interesting fact is that outer planets will have very small temperature differences. Their temperature variation factor is very small, sometimes below one degree. During daytime, the tiny amount of heat from the Sun gently heats them, while in night time, this heat is trapped by the greenhouse gasses. Also, their seasons will be very soft. A terraformed Triton or titan will keep its temperature for long and will certainly not freeze during a long polar night.

Also, in case of inner planets, things are interesting. The math shows that they will have much higher temperature limits. We don't know for sure. In case of an inner planet, terraformers will use anti-greenhouse gasses or space shields to reflect part of the light, while in night time, they will allow heat to escape into cosmos. Probably these planets will behave similar to Earth, but also there is a chance they will be more sensible to climate changes.

The formulas predict that a small planet will have smaller temperature variations then a large one. A Super Earth might be one of the most hazardous places to be, because air currents will travel much more and will not bring heat to the poles and cool air to the equator as on a smaller planet.

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