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Planetary distances and year lengths
May. 14th, 2011 02:16 pmIn previous entries I've dealt with the age and metallicity of various nearby (i.e. within 100 light years) stars, since these appear to be the main factors in determining whether a star is a good candidate to have a habitable planet.
But let's say you've weighed these factors and decided on a particular star as the primary for your fictional planet. You're now going to want to figure out some basic data, such as how far the planet is from the star and how long its year is.
Here are a couple of formulas that will let you make these calculations.
1. How far should your planet be from the star?
Well, if humans are going to live there without needing special protection all the time, it needs to be somewhere in the habitable zone. If you want a hot planet, you'll put it closer to the inner edge of this zone; if you want a cold one, you'll put it near the outer edge, and if you want one that's in the middle, well, you know what to do.
To figure out where the boundaries of the habitable zone lie, start with the visual luminosity (L) of the star. Luminosity means "how bright the star is relative to Sol," so a star with a luminosity of 0.8 would be 80% as bright as Sol. The luminosity for stars within 75 light years or so should be readily available on such sources as the Internet Stellar Database or Solstation.com. (If you happen to know a star's mass but can't find out its luminosity, you can get a rough approximation by raising the star's mass to the power of 3.5.)
All right, now that we've got the star's luminosity, let's do a little basic math to find the boundaries of the habitable zone.
Sol's habitable zone is generally believed to be from 0.7 to 1.5 AU. For any other star that might have a terrestrial planet:
The inner edge in AU is the square root of L x 0.7
The Earth twin distance in AU is the square root of L
The outer edge in AU is the square root of L x 1.5
(There are more involved formulas you can use to determine these values, but this method will do for rough-and-ready calculations.)
2. Now that you know the planet's distance from its primary, how long would its year be?
Start with the distance of the planet in AU.
Cube this number.
The planet's year (in Earth years) will equal the square root of the distance cubed. (Google "Kepler's Third Law" for more information on why this is so.)
Let's go through this exercise for the three stars of the Alpha Centauri system.
Proxima Centauri is a red dwarf. Its luminosity is a mere 0.0017 -- that is, it's 0.17 percent that of Sol.
The square root of 0.0017 is 0.041231.
Thus, the inner edge of the habitable zone is 0.7 of 0.041231, or 0.029 AU.
The Earth twin distance is equal to the square root of L, so a planet with Earthlike temperatures would likely be located about 0.041 AU from Proxima.
And the outer edge of the habitable zone is 0.062 AU out from the star.
Remember, however, that these numbers are just approximations, and you would have to take other factors into consideration, such as the fact that Proxima is a flare star, in deciding what distance your planet would be... or if you think Proxima could have a habitable planet at all.
Let's say there is a planet at 0.041 AU. The cube of this number is 0.00007, and the square root of 0.00007 is roughly 0.008. So this planet's year would be 0.008 years long, or 3.056 days. (Of course, the planet would not be likely to rotate on its axis in one Earth day -- the figure in Earth days is just to give a better idea of how long the year is.)
Alpha Centauri A is much more Sol-like, with a luminosity of 1.1.
The square root of 1.1 is 1.048809.
So the inner edge of the habitable zone is at 0.734 AU; Earth twin distance is 1.049 AU; and the outer edge is at 1.573 AU.
Finally, let's take Alpha Centauri B, an orange dwarf with a luminosity of 0.5.
Inner edge of the habitable zone is at 0.495 AU; Earth twin distance is 0.7 AU; and the outer edge is at 1.061 AU. The Earth twin's year would be 0.59 Earth years, or 217 Earth days, long.
These calculations are only meant to apply to planets that are at least somewhat Earthlike. They would not apply to planets that harbored liquid water under a thick crust of ice, as may be the case with one or more of Jupiter's moons, or planets that were self-warming because of radioactivity, or... well, think up your own offbeat scenarios.
But let's say you've weighed these factors and decided on a particular star as the primary for your fictional planet. You're now going to want to figure out some basic data, such as how far the planet is from the star and how long its year is.
Here are a couple of formulas that will let you make these calculations.
1. How far should your planet be from the star?
Well, if humans are going to live there without needing special protection all the time, it needs to be somewhere in the habitable zone. If you want a hot planet, you'll put it closer to the inner edge of this zone; if you want a cold one, you'll put it near the outer edge, and if you want one that's in the middle, well, you know what to do.
To figure out where the boundaries of the habitable zone lie, start with the visual luminosity (L) of the star. Luminosity means "how bright the star is relative to Sol," so a star with a luminosity of 0.8 would be 80% as bright as Sol. The luminosity for stars within 75 light years or so should be readily available on such sources as the Internet Stellar Database or Solstation.com. (If you happen to know a star's mass but can't find out its luminosity, you can get a rough approximation by raising the star's mass to the power of 3.5.)
All right, now that we've got the star's luminosity, let's do a little basic math to find the boundaries of the habitable zone.
Sol's habitable zone is generally believed to be from 0.7 to 1.5 AU. For any other star that might have a terrestrial planet:
The inner edge in AU is the square root of L x 0.7
The Earth twin distance in AU is the square root of L
The outer edge in AU is the square root of L x 1.5
(There are more involved formulas you can use to determine these values, but this method will do for rough-and-ready calculations.)
2. Now that you know the planet's distance from its primary, how long would its year be?
Start with the distance of the planet in AU.
Cube this number.
The planet's year (in Earth years) will equal the square root of the distance cubed. (Google "Kepler's Third Law" for more information on why this is so.)
Let's go through this exercise for the three stars of the Alpha Centauri system.
Proxima Centauri is a red dwarf. Its luminosity is a mere 0.0017 -- that is, it's 0.17 percent that of Sol.
The square root of 0.0017 is 0.041231.
Thus, the inner edge of the habitable zone is 0.7 of 0.041231, or 0.029 AU.
The Earth twin distance is equal to the square root of L, so a planet with Earthlike temperatures would likely be located about 0.041 AU from Proxima.
And the outer edge of the habitable zone is 0.062 AU out from the star.
Remember, however, that these numbers are just approximations, and you would have to take other factors into consideration, such as the fact that Proxima is a flare star, in deciding what distance your planet would be... or if you think Proxima could have a habitable planet at all.
Let's say there is a planet at 0.041 AU. The cube of this number is 0.00007, and the square root of 0.00007 is roughly 0.008. So this planet's year would be 0.008 years long, or 3.056 days. (Of course, the planet would not be likely to rotate on its axis in one Earth day -- the figure in Earth days is just to give a better idea of how long the year is.)
Alpha Centauri A is much more Sol-like, with a luminosity of 1.1.
The square root of 1.1 is 1.048809.
So the inner edge of the habitable zone is at 0.734 AU; Earth twin distance is 1.049 AU; and the outer edge is at 1.573 AU.
Finally, let's take Alpha Centauri B, an orange dwarf with a luminosity of 0.5.
Inner edge of the habitable zone is at 0.495 AU; Earth twin distance is 0.7 AU; and the outer edge is at 1.061 AU. The Earth twin's year would be 0.59 Earth years, or 217 Earth days, long.
These calculations are only meant to apply to planets that are at least somewhat Earthlike. They would not apply to planets that harbored liquid water under a thick crust of ice, as may be the case with one or more of Jupiter's moons, or planets that were self-warming because of radioactivity, or... well, think up your own offbeat scenarios.