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Orbital data

rancke

Absent Friend
Would someone with a world generation program or a spreadsheet do me the kindness of figuring out the orbital distance and period of a world orbiting in the life zone of a K6 V star?

I used to do this sort of calculations by hand, but it has been so long that I'm rather rusty.


Hans
 
I get:
PO = 124.1662989 days
PR = 34.3845 hours

Mind you, there's randomness involved in the PR.

EDIT: Oops, I forgot the orbital distance: 0.7AU, 104.7million km
(LBB6 and WBH)
 
I get:
PO = 124.1662989 days
PR = 34.3845 hours

Mind you, there's randomness involved in the PR.

EDIT: Oops, I forgot the orbital distance: 0.7AU, 104.7million km
(LBB6 and WBH)

Thank you very much.

PR is the planetary rotation, right? The canonical data implies something close to that of Earth (24 hour give or take).


Hans
 
Yes, PR is the rotation - there is a random element to that in the generation, so it would be different if I reopened my spreadsheet and re-calculated it. 24 hours would be well within the range of normal results.
(PO is the orbital period.)
 
Yes, PR is the rotation - there is a random element to that in the generation, so it would be different if I reopened my spreadsheet and re-calculated it. 24 hours would be well within the range of normal results.
Altohugh in theory differing rotation periods is an excellent way to remind the players that they're not in Kansas, in practice they're an umitigated pain. I gave Regina a rotation 91 minutes longer than 24 hours and went to great lengths to work out the ramifications ("8½ hours sleep, 8½ hours work, 8½ hours leisure", as the union slogan has it ;)), including the interaction of the Reginan day and calendar with the Galactic Standard day and the Imperial calendar. But both I and my players found it a lot of bother to keep track of for far too little gain.


Hans
 
Taking a K6 V star to be 0.64 solar masses and to have a luminosity of 0.137 (very middle-of-the-road values):

The orbital distance at which a planet receives the same insolation as Earth is 0.37 AU.

The orbital period of the planet will be 0.308 years. It's proportional to the 3/2 power of the semi-major axis divided by the square root of the star's mass.

Realistically, the planet's rotation period is almost certainly going to equal to the length of its year (i.e., it will be tidally locked to its star if it's older than a few hundred million years).

As a rule of thumb, planets in habitable-zone orbits are expected to be tidally locked to any star of 0.7 solar masses or less, because of the necessary proximity to the star. You can thwart the tidal lock with an eccentric orbit and the right rotation-to-revolution ratio (see Mercury for an example), but this is only going to make conditions on the planet less conductive to complex life than a simple tidal lock.
 
Realistically, the planet's rotation period is almost certainly going to equal to the length of its year (i.e., it will be tidally locked to its star if it's older than a few hundred million years).

As a rule of thumb, planets in habitable-zone orbits are expected to be tidally locked to any star of 0.7 solar masses or less, because of the necessary proximity to the star. You can thwart the tidal lock with an eccentric orbit and the right rotation-to-revolution ratio (see Mercury for an example), but this is only going to make conditions on the planet less conductive to complex life than a simple tidal lock.


I'd say we have no real idea if that makes it harder for life in a habitable zone. It is a guess based on zero case studies.
 
I'd say we have no real idea if that makes it harder for life in a habitable zone. It is a guess based on zero case studies.

I'd have to disagree with that. Yes, it's a guess, but it's by no means a stab in the dark. We can make some reasonably educated guesses at this point, simply by modeling what surface conditions are likely to be like and comparing that to the general environmental parameters required by carbon-and-water-based life (and even if you have some more exotic biochemistry in mind, similar arguments apply).

A tidally-locked planet has stable temperature zones. A planet on an eccentric orbit with a 3:2 lock doesn't. It has long, long, long days and nights (for the example planet above, one complete solar day as observed from the surface would last over 200 Earth days). So, every point on the resonance-locked planet bakes for 100 days, and then freezes for another hundred. I.e., it presents an environment that combines the worst features of both the Sahara and the Antarctic, fit only for small, hardy extremophiles that are equally at home in hot sand and glaciers. Maybe you might get something as complex as fish in the oceans, which offer a sort of thermal flywheel to damp the extremes of temperature, but the land surface is going to be subject to too much thermal fluctuation for higher organisms to endure, and will probably be scoured by some pretty ferocious winds as well. As a biologist, I'd expect the equivalents of bacteria and algae to make a living in sheltered spots it, and not much more. If you're talking multicellular animal life suited to that environment, then microscopic nematodes and tardigrades could handle the conditions too-- if they could find other organisms to eat.

The truly tidally-locked planet, on the other hand, is a comparatively serene place according to the mathematical models, with definite and stable bands of temperature/precipitation, and moderate winds. They're arranged quite differently than they are on rapidly rotating planets, but that shouldn't be a problem. Life should face no special challenges leaving the oceans and making a living on land. It's easy to picture complex ecological communities evolving.

If theory isn't convincing, there's the empirical observation that a strong inverse correlation exists on Earth between the extremity of day/night and seasonal environmental variations and the amount of biomass a particular terrestrial habitat can support. A planet with 5000-hour days is going to be strongly biased towards the zero end of the biomass-per-square-meter scale for land life.
 
Turns out that this particular world has both a biosphere and a rotation period close to (but not exactly) 24 hours.


Hans
 
I'd have to disagree with that. Yes, it's a guess, but it's by no means a stab in the dark. We can make some reasonably educated guesses at this point, simply by modeling what surface conditions are likely to be like and comparing that to the general environmental parameters required by carbon-and-water-based life (and even if you have some more exotic biochemistry in mind, similar arguments apply).

Way incorrect. With us being ON Earth, senors all over the place we can't even model our own climate with any real accuracy. So no, we can only make uneducated guesses.
 
Taking a K6 V star to be 0.64 solar masses and to have a luminosity of 0.137 (very middle-of-the-road values):

The orbital distance at which a planet receives the same insolation as Earth is 0.37 AU.

The orbital period of the planet will be 0.308 years. It's proportional to the 3/2 power of the semi-major axis divided by the square root of the star's mass.

This gives me a problem. Fritz says the orbital period is 124.166 days and the orbital distance is 0.8 AU. You say 112.497 (0.308 * 365.25) days and 0.37 AU. Now I don't know what to believe.

EDIT:

I dug out my copy of First In and got to work.

The star is a K6 V. From the tables in FI I got a temperature of 4,300K and a luminosity of 0.133, a mass of 0.60 and a
diameter of 0.0062. (Solar jump limit = 0.62).

From BtC i got an average surface temperature of 289 K ('Cool') which, with an albedo of 0.30 and a greenhouse
effect of 0.15 gave me a blackbody temperature of 275. That in turn gave me an orbital radius of 0.3727 AU.

From the orbital radius and the mass of the star I get an orbital period of 0.29374 year or 107.29 days.


Hans
 
Last edited:
This gives me a problem. Fritz says the orbital period is 124.166 days and the orbital distance is 0.8 AU. You say 112.497 (0.308 * 365.25) days and 0.37 AU. Now I don't know what to believe.


Hans

You'll have to ask Fritz how he arrived at his figures.

As for me, I'm using mass and luminosity data from K.R. Lang's "Astrophysical Data: Stars and Planets" (a standard reference). The numbers are for a typical K6 V star.

I'm using bolometric luminosity, the total energy output over all wavelengths, which is necessary if you want to figure the equilibrium temperature and the habitable zone correctly. If bolometric luminosity is expressed relative to Sol, then the distance in AU that a planet must be from the star in order to receive exactly the same amount of energy from its star as Earth does from Sol is directly proportional to the square root of the star's luminosity. I used this particular figure because you did not specify what you meant by 'habitable zone', so I just calculated the distance at which a planet would receive exactly the same amount of insolation as Earth.

The math for determining the year length has been explained in a previous post. I don't think I've made any errors in the calculations, but you can double-check them yourself. You have all numbers and formulas.

I'll be out of town and off the grid for two weeks starting tomorrow morning (frantically packing camping stuff now), so I probably won't be checking back until August 2nd or so. Happy world-building!
 
I used this particular figure because you did not specify what you meant by 'habitable zone', so I just calculated the distance at which a planet would receive exactly the same amount of insolation as Earth.
Sure, I had decided to forget about accuracy and just get a ballpark figure. But when I overcame my indolence and did the calculations, I thought I might as well include the small details too.

The math for determining the year length has been explained in a previous post. I don't think I've made any errors in the calculations, but you can double-check them yourself. You have all numbers and formulas.
Yes, and thank you for your help. Right now I'm wishing someone would check my calculations for errors.

I'll be out of town and off the grid for two weeks starting tomorrow morning (frantically packing camping stuff now), so I probably won't be checking back until August 2nd or so. Happy world-building!
Thank you and have a good holiday.


Hans
 
Yes, and thank you for your help. Right now I'm wishing someone would check my calculations for errors.


Hans

You're welcome. Remember, real stars do not have mass and luminosity values that lie on a precise, infinitely thin curve. There's a fair amount of slop in the numbers, depending on factors such as age and composition. For a given spectral class, mass may vary by at least a few percent from the canonical value, and luminosity can vary even more widely. Pick a value that isn't too outrageous and run with it.
 
You're welcome. Remember, real stars do not have mass and luminosity values that lie on a precise, infinitely thin curve. There's a fair amount of slop in the numbers, depending on factors such as age and composition. For a given spectral class, mass may vary by at least a few percent from the canonical value, and luminosity can vary even more widely. Pick a value that isn't too outrageous and run with it.

Thanks for reminding me. FI mentions that luminosity, in particular, can vary significantly. Since I want the world closer to the solar jump limit, I think I'll increase the luminosity a bit.


Hans
 
As to technique, Hans, I used the tables and formulas in World Builder's Handbook (Orbital distance is from LBB6). The distance is for the orbit in the habitable zone, and the mass based on interpolation of the stellar data tables. I then used the formula to calculate the orbital period. (I actually plugged the star info into my spreadsheet, and it did the rest.)
 
As to technique, Hans, I used the tables and formulas in World Builder's Handbook (Orbital distance is from LBB6). The distance is for the orbit in the habitable zone, and the mass based on interpolation of the stellar data tables. I then used the formula to calculate the orbital period. (I actually plugged the star info into my spreadsheet, and it did the rest.)

Thanks for trying, Fritz. That would have been good enough for my original intent, to get a ballpark figure, but now that I've actually dusted off my old books, I'm going the extra mile.

As an aside, the rigid fixed orbital distances from LBB6 and its descendents is one of the reasons I prefer First In and its much more flexible orbital distance creation.


Hans
 
Thanks for trying, Fritz. That would have been good enough for my original intent, to get a ballpark figure, but now that I've actually dusted off my old books, I'm going the extra mile.

As an aside, the rigid fixed orbital distances from LBB6 and its descendents is one of the reasons I prefer First In and its much more flexible orbital distance creation.


Hans

Hi all. I'm back from vacation.

Hans, if you are interested in planetary orbit spacing and want a fresh perspective, you would do well to look at Rubčić and Rubčić 2010: http://fizika.phy.hr/fizika_a/av10/a19p133.pdf

They are proposing a quantization law for planetary orbits. So far as I can tell they've been universally ignored by the astronomical community. I can see why: I'm just a biologist, and I can see things in it that make me think their reasoning is flawed. Nevertheless, it is remarkably useful for gaming purposes. It is possible to derive from their paper a simple formula for orbital spacing, and you can find a reasonably good fit using their equations for most known real-universe planetary systems. It's more complex than the Titius-Bode equation, but not daunting. It lends itself well to randomly generating realistic-looking planetary systems.

Here's the Reader's Digest version: Every planetary system has an integer packing constant K, the inverse square of which describes the "tightness" (higher numbers mean closer spacing). Actual orbits will fall at or near the square of another series of integers. The basic spacing for the system is directly proportional to the star's mass.

Taking Sol as an example:

The fundamental orbit size is 1.545 * M(star) AU (which is just 1.545 AU for Sol).
Sol's empirically observed packing constant K is 6. The inverse square is 1/36.
Orbits scale as (1.545/36) times N^2, where N is the integer number of the orbit.
Mercury is at N=3, Venus is at N=4, Earth is at N=5. Orbits 1 and 2 are empty. Some of the higher numbers are skipped as you move out past the asteroid belt. As an empirical observation, this is because the ratio of any two adjacent real-world orbits O(n+1)/O(n) almost always falls between the square root of 2 and 2 (probably for reasons of dynamic stability); you can see from that that certain sequences of integers will be disallowed in the Rubčić series, and that for numbers above N=7 there are multiple solutions that fall within these boundaries (note that this property makes it especially easy to generate natural-looking variety in randomly generated planetary systems if you want to go this route).

Earth's theoretical orbit is at (1.545/36) * 25 AU, or 1.073 AU. Actual orbits vary from the theoretical ones about as much as the Titius-Bode orbits do, usually within 5% but occasionally up to 10%.

It's not as cookie-cutter as Titius-Bode, and it gives good-looking results. For gaming purposes, it's easy to simulate a system using random numbers and you can vary the packing constant to account for all those epistellar jovian planets (packing constants in the range of 10 to 15 or even higher in extreme cases).
 
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