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A sextant

Back to Math And Terraforming

This page is made to help you see what kind of conditions settlers would encounter on a planet orbiting a specified type of star.

Light Spectra[]

Before everything, keep in mind that stars are different and there is NO WAY a star will behave just as expected by our math. If at possible you can get some fresh data about your star from a database, better than trying yourself to calculate. The best example I can give you is our own Sun, which emits in some spectra about 10% more light than it should be as a G2 main sequence star. On the other hand, datasheets contain telescope measurements from Earth, which are affected by interstellar environment and might not show exactly how the star behaves.

Solar Constant[]

The Solar Constant, Ks, is the energy output of a star. For our sun, Sol, it is 1. The formula for this is:

Ks = (((T)^4)*8.972008E-16)*(r^2), where Ks is the solar constant, T is the star's temperature in degrees Kelvin and r is the star's radius in Solar radii.

Finding the main spectra[]

The luminosity of a star is similar to a black-body radiation spectra. However, the formulas shown there are hard to apply, since you will need to work with a lot of difficult numbers, like Planck constant and Boltzmann constant. If you use those formulas, you will get very complicated numbers, sometimes with up to 15 zeros and hard to compare. I worked and simplified the formulas to get results that are more easier to digest.

This is the way to determine which is the main spectra (wavelength) of a star's energy output:

Lw = (0.002897/(T))*1096400000 where T is temperature in degrees K. For the Sun, it is 549.7, which corresponds to yellow-green light. Values below 350 correspond to infrared, of 350-400 correspond to red light, 600-700 to blue and violet light and above 700 to ultraviolet.

Amount of light for a specified wavelength[]

The formula is:

L(w) = 1.35E-44*(((W*1000000000000)^3)/(2.1828^(0.000000000048*(W*1000000000000)/T)-1))*(r^2)

Here, L(w) is the amount of light for a specified wavelength, W is the wavelength in nanometers, T is star's temperature in degrees K and r is star's radius in Solar radii.

This formula will give you a number that doesn't mean anything. You have to use the formula for the Sun and then to your star, using the same wavelength. Comparing the results you will get somewhere. If you want some fast results, follow the following formulas:

  • Lw(100) = 1.35E-44*(((100*1000000000000)^3)/(2.1828^(0.000000000048*(100*1000000000000)/T)-1))*(r^2)*67.6024 - to get near infrared values
  • Lw(400) = 1.35E-44*(((400*1000000000000)^3)/(2.1828^(0.000000000048*(400*1000000000000)/T)-1))*(r^2)*14.3312 - to get red light values
  • Lw(520) = 1.35E-44*(((520*1000000000000)^3)/(2.1828^(0.000000000048*(520*1000000000000)/T)-1))*(r^2)*14.82431 - to get yellow light values
  • Lw(700) = 1.35E-44*(((700*1000000000000)^3)/(2.1828^(0.000000000048*(700*1000000000000)/T)-1))*(r^2)*20.00458 - to get blue light values
  • Lw(2000) = 1.35E-44*(((2000*1000000000000)^3)/(2.1828^(0.000000000048*(2000*1000000000000)/T)-1))*(r^2)*3973.8 - to get near ultraviolet values
  • Lw(10000) = 1.35E-44*(((10000*1000000000000)^3)/(2.1828^(0.000000000048*(10000*1000000000000)/T)-1))*(r^2)*(1.078496E+24) - to get far ultraviolet values.

These formulas are adapted to give you a value of 1 for the Sun. This way you can easily compare a star's light output and colour to the Sun.

To calculate the amount of each light received by a planet, use the following formula:

Lw(local) = Lw/(d^2) where Lw(local) is the light received by a planet, Lw is the light of the specified lightweight and d is distance to the planet in AU.

The formula is made so that it gives you for the Sun and the Earth values of 1. For habitability, the limits are as follows:

  • Red light: most plants cannot survive if it falls below 0.001.
  • Blue light: most plants cannot survive if it falls below 0.001.
  • Near ultraviolet light: most organisms will not survive if it is above 100.
  • Far ultraviolet light: most organisms will not survive if it is above 10^5.

Temperature - Luminosity - Radius[]

There are 3 equations that can easily find basic properties of a star.

If you know temperature (T in degrees Kelvin) and radius (d in solar radii) of a star, then you can get the Solar Constant (Ks):

Ks = (d^2)*(T^4)*(8.97E-16)

Again, if you know radius and solar constant, you can get the temperature:

T = (Ks/((d^2)*(8.97E-16)))^0.25

Finally, if you know temperature and solar constant, you can find the radius:

d = (Ks/((t^4)*(8.97E-16)))^0.5

Spectral Class[]

This is the hardest part of all, because spectral classes are a complete mess. They follow prototype stars, which sometimes change. Also, if you compare catalogues from the 60's, from the 80's, from the 2000's and from today, you will see differences. The most updated list for Main Sequence Stars, is at this address:

www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt

If it helps someone's work, I made some calculations using the values from there and put them in the following articles:

For stellar remnants and atypical stars:

Stellar Wind[]

There are some useful formulas for Stellar Wind, please go to the page to see them.

All-In-One Chart[]

Here is how to create a sheet in Excel that allows you to quickly get the parameters of stars:

Suppose on rows 1 and 2 you write what each column means, the first parameters will start from row 3. The formulas I give you are for row 3. You can copy them and paste to the next rows.

  1. A3: Name - write the name of the star.
  2. B3: Temperature - write the temperature of the star.
  3. C3: Mass - write the mass of the star in Solar masses.
  4. D3: Diameter - write the radius of the star in Solar radii or the diameter in Solar diameters.
  5. E3: Bolometic magnitude - usually given from charts.
  6. F3: Visual magnitude - usually given from charts.
  7. G3: Main wavelength =(0.002897/(B3))*1096400000 - the wavelength most light is emitted.
  8. H3: Gross solar constant =((B3)^4)*8.972008E-16 - solar constant if the star had same radius with the Sun
  9. I3: Frequency =(1/G3)*302344.9 - frequency of the wavelength most light is emitted.
  10. J3: Main wavelength gross energy =1.35E-44*(((I3*1000000000000)^3)/(2.1828^(0.000000000048*(I3*1000000000000)/B3)-1))*15.3146 - energy output for the wavelength assuming the star has the same radius as the Sun.
  11. K3: Infrared light gross output =1.35E-44*(((100*1000000000000)^3)/(2.1828^(0.000000000048*(100*1000000000000)/$B3)-1))*67.6024 - energy output at wavelength of 100 (infrared) assuming the star as the same radius as the Sun. For the Sun, result will be 1.
  12. L3: Red light gross output =1.35E-44*(((400*1000000000000)^3)/(2.1828^(0.000000000048*(400*1000000000000)/$B3)-1))*14.3312 - energy output at wavelength of 400 (red) assuming the star as the same radius as the Sun. For the Sun, result will be 1.
  13. M3: Visible light gross output =1.35E-44*(((520*1000000000000)^3)/(2.1828^(0.000000000048*(520*1000000000000)/$B3)-1))*14.82431 - energy output at wavelength of 520 (yellow) assuming the star as the same radius as the Sun. For the Sun, result will be 1.
  14. N3: Blue light gross output =1.35E-44*(((700*1000000000000)^3)/(2.1828^(0.000000000048*(700*1000000000000)/$B3)-1))*20.00458 - energy output at wavelength of 700 (blue-violet) assuming the star as the same radius as the Sun. For the Sun, result will be 1.
  15. O3: Ultraviolet light gross output =1.35E-44*(((2000*1000000000000)^3)/(2.1828^(0.000000000048*(2000*1000000000000)/$B3)-1))*3973.8 - energy output at wavelength of 2000 (ultraviolet) assuming the star as the same radius as the Sun. For the Sun, result will be 1.
  16. P3: Far ultraviolet light gross output =1.35E-44*(((10000*1000000000000)^3)/(2.1828^(0.000000000048*(10000*1000000000000)/$B3)-1))*1.078496E+24 - energy output at wavelength of 10000 (far ultraviolet) assuming the star as the same radius as the Sun. For the Sun, result will be 1.
  17. Q3: Solar constant =J3*($D3*$D3) - total energy output of that star, for the Sun result will be 1.
  18. R3: Infrared output =K3*($D3*$D3) - infrared light output at wavelength of 100, for the Sun result will be 1.
  19. S3: Red light output =L3*($D3*$D3) - red light output at wavelength of 400, for the Sun result will be 1.
  20. T3: Yellow light output =M3*($D3*$D3) - yellow light output at wavelength of 520, for the Sun result will be 1.
  21. U3: Blue light output =N3*($D3*$D3) - blue-violet light output at wavelength of 700, for the Sun result will be 1.
  22. V3: Ultraviolet output =O3*($D3*$D3) - ultraviolet light output at wavelength of 2000, for the Sun result will be 1.
  23. W3: Far ultraviolet output =P3*($D3*$D3) - far ultraviolet light output at wavelength of 10000, for the Sun result will be 1.
  24. X3: Distance where Ks = 20 =(Q3/20)^0.5 - distance in AU where solar constant = 20 (inner edge where terraforming might be possible).
  25. Y3: Distance where Ks = 1 =(Q3*1)^0.5 - distance in AU where solar constant = 1 (orbit of an Earth-Like Planet).
  26. Z3: Distance where Ks = 1/1000 =(Q3*1000)^0.5 - distance in AU where solar constant = 1/1000 (outer edge where terraforming might be possible).
  27. AA3: Outer limit for red light =(S3*1000)^0.5 - outer limit in AU where red light is enough for plants to survive.
  28. AB3: Outer limit for blue light =(U3*1000)^0.5 - outer limit in AU where blue light is enough for plants to survive.
  29. AC3: Inner limit for ultraviolet light =(V3/100)^0.5 - inner limit in AU where ultraviolet light is low enough for most Earth-like life forms.
  30. AD3: Inner limit for far ultraviolet light =(W3/1000000000000)^0.5 - inner limit in AU where far ultraviolet light is low enough for most Earth-like life forms.
  31. AE3: Roche limit =(C3/78486792.13)^0.5 - outer distance in AU where an Earth-like planet could orbit without breaking into a ring.
  32. AF3: Inner limit for habitable zone =MAX(X3, AC3, AD3, AE3) - inner limit in AU where terraforming allows and plant life is possible.
  33. AG3: Middle of habitable zone =Y3 - orbit in AU of an Earth-Like Planet based on temperature and solar constant.
  34. AH3: Outer limit for habitable zone =MIN(Z3, AA3, AB3) - outer limit in AU where terraforming allows and plant life is possible.
  35. AI3: Tidal forces at minimal orbit =$C3/AF3/AF3 - tidal stress from the host star at the inner edge of habitable zone. For Earth's orbit in the Solar System, value = 1.
  36. AJ3: Tidal forces at optimal orbit =$C3/AG3/AG3 - tidal stress from the host star at the orbit of an Earth-like planet. For Earth's orbit in the Solar System, value = 1.
  37. AK3: Tidal forces at maximal orbit =$C3/AH3/AH3 - tidal stress from the host star at the outer edge of habitable zone. For Earth's orbit in the Solar System, value = 1.
  38. AL3: Minimal orbit period =((AF3)^3/(1*($C3)))^0.5*365 - time to orbit in days at the inner edge of habitable zone.
  39. AM3: Optimal orbit period =((AG3)^3/(1*($C3)))^0.5*365 - time to orbit in days for an Earth-like planet.
  40. AN3: Maximal orbit period =((AH3)^3/(1*($C3)))^0.5*365 - time to orbit in days at the outer edge of habitable zone.
  41. AO3: Angular Size - minimal distance =($D3*1396.96)/(AF3*149.5) - how big the star will look like from the inner edge of habitable zone.
  42. AP3: Angular Size - optimal distance =($D3*1396.96)/(AG3*149.5) - how big the star will look like from an Earth-like planet.
  43. AQ3: Angular Size - maximal distance =($D3*1396.96)/(AH3*149.5) - how big the star will look like from the outer edge of habitable zone.
  44. AR3: Stellar wind mass =(10^(-12.76+1.3*LOG10(Q3)))*5754400000000 - mass of particles ejected in the Stellar Wind, for the Sun, value = 1.
  45. AS3: Stellar wind force =(10^(-12.76+1.3*LOG10(Q3)))*5754400000000*((B3/5778)^4) - energy of the stellar wind, for the Sun, value = 1.
  46. AT3: Minimum stellar wind =$AS3/AF3/AF3 - stellar wind at the inner edge of habitable zone, where force solar wind at Earth's orbit is 1.
  47. AU3: Optimal stellar wind =$AS3/AF3/AF3 - stellar wind at the orbit of a habitable zone planet, where force solar wind at Earth's orbit is 1.
  48. AV3: Maximum stellar wind =$AS3/AF3/AF3 - stellar wind at the outer edge of habitable zone, where force solar wind at Earth's orbit is 1.
  49. AW3: Heliosphere boundary =(AS3/0.0001)^0.5 - assuming a similar interstellar environment with the area around Solar System, the radius of heliosphere in AU.

These formulas are very useful, for example if you want to study a planet orbiting Brown Dwarfs or a moon of a rogue planet, since almost all their light is in infrared. Also, you can get an image of the pale amount of visible light these bodies give to their planets.

For White Dwarfs[]

There is a clear relation between mass and radius of a White Dwarfs. However, the formulas available are very complicated. Instead, I prefer to write a table with the approximate values for mass and diameter:

Mass vs diameter relation
Mass (Solar) Diameter (Solar) Diameter (km)
0.0323 0.04 55900
0.036 0.037 52300
0.043 0.035 48900
0.053 0.032 45200
0.068 0.03 41900
0.106 0.028 38500
0.121 0.025 34900
0.165 0.022 31400
0.225 0.02 27900
0.25 0.0192 26800
0.33 0.0171 23900
0.43 0.015 21000
0.5 0.0139 19400
0.62 0.0122 17000
0.75 0.0107 15000
0.82 0.01 14000
0.91 0.0090 12600
1 0.0081 11300
1.11 0.0067 9340
1.25 0.00515 7200
1.26 0.005 6980
1.38 0.0029 3990

Downloads[]

You can download Excel simulations from these links:

  • Climate simulation: [1]
  • Planet simulation: [2]
  • Solar System simulation: [3]
  • Star simulation: [4].

These files are stored on Mega, the successor of Megaupload.

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