Back to Math And Terraforming
There are a few planetary parameters that are easy to work with. Using Microsoft Excel, we can see the behavior of any theoretical planet. The following is an example I use. Some formulas are adapted from books and sites, while some are my invention.
In this table, I left row A for column names, while row B is for the parent star. There, we should write the Solar Constant and the solar mass. If you want to work with a system with multiple stars, see additional instructions below.
Single Sun System Edit
- The first column is for the planet's Name.
- Distance to the star is to be placed in the second column, in million km. One AU has 149.5 million km. All formulas are calibrated for this and not for AU.
- Solar Constant is calculated in 3rd column. So, on cell C3, write star's solar constant (for Sun, it is 44235). Then, in cell C33, type =C$2/(B3*B3) and copy this for following cells in the column. For Solar System, for Earth, you should get a value close to 1.98. This is the star's global energy output at that planet's orbit.
- Visual Constant is a term invented by me. Some stars have less blue or less red luminosity levels. Plants need both blue and red light. It is displayed on column 4. Type here the lowest amount of star radiation (blue or red) on cell B4. Then, on cell C4, type the formula: =D$2/(B3*B3). Earth plants can survive as far as the orbit of Neptune with the amount of light they receive. There, the solar constant drops to 0.02. If there is less light, in red or blue wavelength, less then 0.2, plants cannot survive and you will need an extra source of light.
- Mass (Earth = 1) is displayed on column 5. If you don't know the mass, you can detect it from diameter and density, using the following formula: =(F3/12.756)^3*(H3/5.5). Copy the formula across the column to where you need it.
- Diameter (in thousands km) is displayed on column 6. If you don't know it, you can determine it from mass and density, using the following formula: =((E3/(F3/5.5))^0.3333333333)*12.756. Copy the formula across the column to where you need it.
- Gravity is calculated in column 7 (Earth = 1). The formula is: =E3/(F3/12.756)^2.
- Density is written in column 8. The formula is =(E3/(F3/12.756)^3)*5.5 and is calculated using mass and diameter. You can use a pre-defined density, to get mass in relation with diameter or to find diameter in relation with mass.
- Rotation is listed on column 9. There is no formula to calculate rotation, it is all up to your imagination.
- Revolution (year length) is listed on column 10. The formula is =(B3^2/(0.00000050300751*E$2))^0.5. On cell E2, you have to write star's mass (Earth = 1). The resulting value for revolution period is in Earth's days.
- Revolution (years) is listed on column 11. The formula is =J$2/365.25.
- Stellar Gravity is listed on column 12. The formula is =(E$2/B3^2)/14.89916, where on E2 you have the star's mass (Earth = 1). For Earth, you will get a value of 1. A strong stellar gravity means strong tides and that the planet should probably be tidal locked. A weak gravity means that the planet is experiencing no tides.
- Hill Sphere is listed on column 13. This is the boundary where planet's gravity equals star's gravity. Satellites should be placed at least half of that distance. Hill Sphere (in million km) is calculated by the formula: =B3*(E3/(3*E$2))^0.33333333. Multiply the value by 500 and if the number you get is equal with the diameter, it means that your planet is below Roche limit (star's gravity will tear it into a ring). The value should be at least twice that.
- Temperature is listed on column 14. The formula is =LOG10(1+C3*1E+100)/(4.6E+22*POWER(0.58674,LOG10(1+C3*1E+100))). This formula is based on solar constant and nothing else. The value you get is the temperature that an object with 25% gray will get exposed to the solar radiation. I use to call this void temperature because it is not influenced by atmosphere in any way. This formula is my own and I worked a few weeks to get it right. I figured it out from many temperature examples given and is given in Kelvin degrees. There is a special article, Temperature.
- Temperature (Celsius) is listed on column 15. The formula is =N3-273.15. For Earth, you should get something like +93.
- Surface. It is listed in column 16 and has the formula =F3^2*PI(). Results are in millions of square km.
- Population Limit is listed on column 17 and is a term created by myself. See Population Limit for more details. The formula is:
- For objects with a higher visual constant then 1.98: =(F3^2/12.756^2)/(D3/1.98)*5000
- For objects with a lower visual constant then 1.98: =(F57^2/12.756^2)*D57/1.98*5000
Binary & Trinary Stars Edit
In this case, you might want planets around each star and distant planets orbiting both stars. So, you will have for each star one table and then another one with combined mass & luminosity.
Usually, in case of satellites, the same table can be used. The only difference is that the solar and visual constants will be those of the star, as determined for the planet. In rare cases, extra infrared light from the planet can increase the solar constant and temperature.
Elliptic Orbits Edit
In this case, you need to do more math. Basically, you will have to do one table for aphelion and another one for perihelion, to get a better image of what's going on.
Simplified Formulas Edit
- KS - star's own solar constant (for Sol, 44253)
- KV - star's own visual constant (lowest amount of light in a visible spectra)
- d - distance (millions km)
- M - star's mass (unit: Earth = 1)
- Solar Constant Ks = KS/d^2 (for Earth, 1.98)
- Visual Constant Kv = KV/d^2 (for Earth, 1.98)
- Mass m = dm^3*(ds/5.5) (unit: Earth = 1)
- Diameter dm = ((m/(ds/5.5))^0.3333333333)*12.756 (thousands km)
- Gravity: G = m/(dm/12.756)^2 (unit: Earth = 1)
- Density: ds = (E3/(F3/12.756)^3)*5.5 (kilograms per liter)
- Revolution period: Rv = (d^2/(0.00000050300751*M))^0.5 (days)
- Stellar Gravity: Gst = (M/m^2)/14.89916 (for Sun - Earth = 1)
- Hill Sphere: Hs = d*(m/(3*M))^0.33333333 (millions km)
- Temperature: t = LOG10(1+Ks*(1e+100))/((4.6e+22)*0.58674^(LOG10(1+Ks*(1e+100)))) (degrees Kelvin) See Temperature for more.
- Surface: S = dm^2*PI() (millions square km)
- Population limit:
- P = (F3^2/12.756^2)/(D3/1.98)*5000 if Kv>1.98 (millions people)
- P = (F57^2/12.756^2)*D57/1.98*5000 if Kv<1.98 (million people)
Additional Parameters Edit
There are a few other things to take into equation when we talk about planets.
Flattering occurs when a planet is rotating along its own axis. It is influenced by the rotation speed and depends on many parameters. Its basic formula is
Fl = (a-b)/a
where a is equatorial diameter and b is polar diameter.
Escape Velocity is the speed needed by a spaceship to break away from the surface. Its formula is:
Ev = ((1601*m)/diam)^0.5
where m is planet's mass (Earth = 1) and diam is planet's diameter (in thousands km). Ev is returned in km/s.
Launch Windows Edit
Launch windows often occur and are vital for Trade Routes. The idea is that two planets or two moons are aligned in conjunction or in opposition or aligned for a good flight path at nearly the same amounts of time.
For two satellites orbiting one planet or two planets orbiting a star, on orbits close to circular, the formula is:
Lw(p) = a+((a^2/b)+(a^3/b^2)+(a^4/b^3)+(a^5/b^4)+(a^6/b^5)+(a^7/b^6)+(a^8/b^7)+(a^9/b^8)+(a^10/b^9))
Lw(r) = a+((a^2/b)-(a^3/b^2)+(a^4/b^3)+(a^5/b^4)+(a^6/b^5)+(a^7/b^6)+(a^8/b^7)+(a^9/b^8)+(a^10/b^9))
where Lw(p) is the frequency of a launch window for two prograde or two retrograde planets' Lw(r) is the frequency of a launch window for one prograde and one retrograde planet, a is the orbital period (revolution period) for the first planet and b is the orbital period (revolution period) for the second planet. Always, the planet that orbits faster must be a and the planet that orbits slower must be b.
If you try to find the launch windows between a planet and one of its moons, the formula is the same. Only that, this time, a is the rotation period of the planet and b is the orbital period of the moon. If the moon orbits faster then the planet rotation, a becomes the orbital period of the moon and b the rotation period of the planet.
However, in practice, there are other things to count: elliptical orbits and gravity assists. Celestial bodies moving on highly elliptical orbits require far more complex formulas. Also, additional, more easy routes can occur when another planet (or another moon) is in the right place for a gravity assist.
The formulas can help you find how often the best flight windows will occur. However, a spaceship can travel at anytime between two planets, but with much higher costs.