Drakaster - a sextuple star system

[Dracontes]

Screenshot


STC file
Just right click, "Save target as", and substitute filetype "txt" for "stc" in the prompt, or if you want to look at the code right click, "Open in New Window".

It's not a terribly original name for a stellar system but I thought it better to worry with the science first and let the creative writing for last, and I was right, thank spreadsheets!

Anyhow, this sextuple stellar system will have planets, 'bout twenty of them, and since I am a Biology and Geology buff I can't but have a good number of the terrestrial types teeming with life.
However since I dread doing textures for them with a mouse they'll be featureless spheres for a good time to come.

There are a few problems:
- Discovering better formulae to calculate stellar characteristics from star mass.
- Relating spectral classification to calculated characteristics in a more precise manner than educated guesswork.
- Find out if the "(1/5)*(separation), 5*(separation)" rule for binary systems with planets would be better substituted by the "half of Hill sphere radius for binary component, half of Hill sphere radius for barycenter" rule (if that makes any sense).

Well, enjoy!

[Malenfant]

EDIT: Link is working now . I'll check it tonight... I've made a sextuple system myself, I know it's hard getting the barycentres right...


There's a few of us here who are dedicated worldbuilders... I've been posting my attempts at realistic trinary and quadruple systems.
http://www.celestiaproject.net/forum/viewtopic.php?t=10198
http://www.celestiaproject.net/forum/viewtopic.php?t=10054

I've been using this paper to figure out the exact distances at which you can get planets: http://arxiv.org/abs/astro-ph/9809315

It's better than a straight 1/5th or 5x rule (which is wrong anyway), as it really depends on the masses of the stars and the eccentricity of their orbits.

The other problems are less easy to solve. I've been using the Geneva Stellar Evolution Grids (which I think are getting on a bit now, but they're good enough for my purposes), which can be found at http://obswww.unige.ch/~mowlavi/evol/stev_database.html (the raw data that can be imported into Excel is there somewhere, but it takes a hell of a lot of processing).

Once you have the Log T and Log L values for the stars, you just do "10^" of those to get the temperature (in K) and lumuninosity (in Sols). The radius (in metres) for stars with more than 0.3 solar masses can be found by:

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((SQRT(L*3.846E+26))/((T^2)*SQRT(4*PI()*0.0000000567)))


where L is in Sols, and T is in K.

Habitable zone (in AU) is just the SQRT of the luminosity (in Sols).

Snow Line is where the blackbody temperature is 175K:

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SQRT((L*3.846E+26)/(16*PI()*0.0000000567*(175^4))))


Here the L is in Sols.


That should help a bit

[Dracontes]

Thanks very muchly for the links and formulas That Snow Line formula is one I'm quite keen on
Actually this is the second time I've done this. Drakaster's previous incarnation went defunct this Summer on account of a computer malfunction and subsequent fixing. But a brainchild never dies...

It's better than a straight 1/5th or 5x rule (which is wrong anyway), as it really depends on the masses of the stars and the eccentricity of their orbits.

If thats it, then I think Hill sphere radii can be a good approximation. In fact I've recently tightened the system and I've paid much more attention to the Hill sphere radii for the components than to the previous rule of thumb.

Habitable zone (in AU) is just the SQRT of the luminosity (in Sols).

I've been using this webpage I serendipitously found on my hard drive and it states at some point that:

Now, for Earthlike planets I[nsolation] must be close to 1; according to Brian Davis, "recent work suggests very conservatively 1.1 > I > 0.53".

I assumed that the values were in insolation (Earth units) and thought my way to this formula:

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SQRT(L/I)

For each of the limit insolations. Not sure if it is correct but it made sense at the time.

[Malenfant]

Hm, I had a look at the stc with 1.5.0 and for some reason it doesn't look right somehow. Can't put my finger on what exactly is the problem though. I look at the things you've done, and look at my quad system and I see the same things there that I didn't notice before (eg the barycentres on orbits on their own, like your a-b-c). But something's bugging me still about your ones.

Maybe it's the fact that your orbits are all very circular, just makes it look odd to my eyes? I'll try to check the numbers when I have time (probably not for a couple of days though).

[Dracontes]

From the little write-up I made to set my goals for this system's design, I had it as originating from a massive, dense protostellar nebula that collapsed into these stars more or less where they are today. I don't have any pretentions to the likelihood of that happening, I just need the inkling that it's a possibility.
I devised it that way, circular orbits, to get the stars closer to each other so they are all important in the sky of any planet of the multiple system. An artistical justification if anything else But given some leeway I can bump up eccentricities a bit.

Might I point out it's perhaps the hierarchy of the system? The B component may be out of place; I don't think it's at all costumary to have a wide binary (C) orbiting a close ternary (A-B) or even to have a close ternary at all. Though, with this being science fiction I do like to take some risks.

I've had a few problems with the paper's equations (which I presume were the least squares approximations):
- Do they mean by "binary semi-major axis" half or the total mean object separation? If they mean the semi-major axis of the stars' orbits for satellite-type stable orbits that would give rather spurious numbers that don't check well with the objects' Hill sphere radii.
- The expression for planet-type stable orbits is rather baffling. First the binary semi-major axis coeficient is nowhere to be seen and when I get the equation on the spread sheet it gives every other cell negative numbers and rather unconsistant.
However, I'll be sure to have a spreadsheet available online tomorrow, if you'd like.

Provisional solutions:
- For S-type orbits, I reasoned half the Hill sphere radii plus 5-10% would be a good choice; it's the one I've been using.
- For P-type orbits I thought that the main reason a planet feels perturbations from each component of the binary is that the Hill spheres (note that the radii are rather stable) sweep an area around the barycenter as the stars orbit. So, I added the mean object separation multiplied by 1+e plus each object's Hill sphere radius and I reckon that gives a pretty good safety margin.
The numbers do check up with those I calculated, albeit fudgedly, with the paper formulae; I do still have to think this through but with effort I'll get it plausible.

[Malenfant]

I've had a few problems with the paper's equations (which I presume were the least squares approximations):
- Do they mean by "binary semi-major axis" half or the total mean object separation? If they mean the semi-major axis of the stars' orbits for satellite-type stable orbits that would give rather spurious numbers that don't check well with the objects' Hill sphere radii.

Well, if you're doing what I'm doing then you should be calculating the critical values beyond which you can't find planets (ie the S and P orbits) - those are given as distance of each star from the barycentre - that's what it means by "critical semimajor axis". The bianry semimajor axis seems to be the total separation between the stars.

- The expression for planet-type stable orbits is rather baffling. First the binary semi-major axis coeficient is nowhere to be seen and when I get the equation on the spread sheet it gives every other cell negative numbers and rather unconsistant.

Basically, equation (1) gives you the S orbits around each star, and equation (3) gives you the P orbit around both. I think they've been a bit confusing about the phraseing though - if you take eqn (1) and don't multiply it by the binary semimajor axis (ie the separation), then you have the ratio of the S orbit to the total separation. Equation (3) gives you the same thing as it stands, except for the P orbit.

So when you multiply that by the binary separation (ab) you're actually getting the orbital distance in AU of those orbits. Without that, you're just getting the ratio of the S and P orbits to that separation.

For example, if the masses are the same, then the equations will give you a ratio of 0.274 for the S-orbits and 2.3875 for the P-orbit. If your binary separation is 10 AU, then that means that the S-orbits will be at 2.74 AU around each star, and the P-orbit will be at 23.875 AU around both.

Provisional solutions:
- For S-type orbits, I reasoned half the Hill sphere radii plus 5-10% would be a good choice; it's the one I've been using.
- For P-type orbits I thought that the main reason a planet feels perturbations from each component of the binary is that the Hill spheres (note that the radii are rather stable) sweep an area around the barycenter as the stars orbit. So, I added the mean object separation multiplied by 1+e plus each object's Hill sphere radius and I reckon that gives a pretty good safety margin.
The numbers do check up with those I calculated, albeit fudgedly, with the paper formulae; I do still have to think this through but with effort I'll get it plausible.

I wouldn't fudge it like this if I were you, especially since you have the equations to calculate it properly. Figure 1 in the paper also shows that it diverges quite a bit from the hill sphere at mass ratio near 1.

[Dracontes]

Thanks for the pointers, Malenfant
I've just finished the spreadsheet using those formulas, this time correctly, even going to the trouble of including the error margins. Numbers crunched, it seems I won't have to change things too much. The only hassle will be splicing these calculations into my Drakaster System workbook
Though of course, you should take what is below with a grain of salt.

SSC File
The planets are created! You should however download the STC file again as I tweaked it a bit and created a new barycenter.

Screenshots


Overview of the stellar system in question. The wide binary C is closer to us, the ternary A-B is in the mid distance while D is the farthest away.


The D planetary system seen from about 10?? above the star's equatorial plane. I have yet to randomize the angular orbital elements so it has this rather becoming if unnatural gradient to the orbits.


The C binary. I'm not too sure about the three thightly packed orbits in Drakaster Ca's habitable zone but as a worldbuilder I've got to have life teeming or else.


The A-B ternary. Yes, those bright dots around the A components are planets orbiting each of the stars; think Galilaean moons. Heck, I've even put them into a Laplace resonance!


The pi?©ce de resistance. A binary planet orbiting a binary star. Land on Ib and watch the sequential eclipses of Aa and Ab by Ia, or vice-versa.

Next stop, the moons... I've got my work cut out for me

[Dracontes]

This time I fancied zipping the *.stc and *.ssc inside a folder and preparing a download page at my incipient site would be a better idea. So to spare people time, I mean.

I added moons, corrected the radii of planets relating to more accurate estimates of their density. Also, for your viewing pleasure, all bodies are clad in default textures I'm sure most of you should have.

Now for the quandaries:

I used to determine orbital stability of planetary and moon orbits the same equation as I used for the stars themselves. I'm not too sure if it's the best approach though it follows through logically, thinking of mutually orbiting bodies as mass points. You are free to show me otherwise.

I'm wondering about Drakaster D VIII's three outermost moons' periods being greater than the parent planet's period. While they are inside the inner half of the Hill radius of Drakaster D VIII there is something counterintuitive about that setup being stable in the long run.

Are motion resonances a must have in Jovian moon systems? How far must a moon be from its parent planet to avoid being tidally locked.

I'm interested in developing each planet in detail while I finish up with the celestial mechanics calculations. I have already found the formula for atmospheric gas escape and I'm wondering if anyone here remembers the link of a worldbuiding-related site that had a form to calculate system characteristics and went to the detail of palcing estimate on mountain height and plate tectonics duration. I would be quite keen on those equations... Though I'm not afraid to search for them

I'll be sure to posit more questions and doubts soon enough.