Orbital periods in binary systems?

[Evil Dr Ganymede]

They can't reach maximum elongation at the same time, because they're constrained to be on opposite sides of the centre of mass - when one reaches max elongation, the other is either well past it or still approaching it. You've worked out sin(0.5/2.5), which is correct for the max elongation of one star, but their farthest separation is going to be closer to 2*tan(0.5/2.5) - ie, when they're in the 90-degree position.

That's when they should be farthest apart from eachother, yes - and it does look like that happens when they are in the 90 degrees position (I swung above the system at that point to take a look).

But each star should apparently still be a maximum of 11.536 degrees from the CoM when it's at 'the 78 degree position' (0 degrees being the line from the CoM to the planet), but that doesn't seem to be happening at all - it looks like they're at the widest separation from the CoM at 90 degrees, and the separation is too small. It doesn't sound like my calculation is in error (for once!), and I can't see anything wrong in the ssc file... the orbits aren't eccentric, the stars' periods and distances are exactly the same and the orientation is correct. I'm using the FOV measurements at the bottom right of the screen to tell me the angular diameter of my field of view - am I misinterpreting that, or could it be wrong?

I dunno if this will give you an idea what I'm doing... Star2 is on the right there at the edge of the top view (the view from the planet). The bottom view is the view from above the system, but you can't really see exactly where the stars and planet are.

[granthutchison]

My original message has been deleted in an effort to sort out an editing mess ... The correct message appears in quotes below. I seem to have been logged-off accidentally during a session, and so I ended up quoting my own message as a guest instead of editing it as myself

Grant

[Guest]

But each star should apparently still be a maximum of 11.536 degrees from the CoM when it's at 'the 78 degree position' (0 degrees being the line from the CoM to the planet), but that doesn't seem to be happening at all - it looks like they're at the widest separation from the CoM at 90 degrees

The difference in elongation between max elongation and quadrature is only 0.2 degrees, so it might not be easily detectable by eyeball - essentially, the star will hang about in the vicinity of its max elongation for 20-30 degrees of its orbit around max elongation.

I'm using the FOV measurements at the bottom right of the screen to tell me the angular diameter of my field of view - am I misinterpreting that, or could it be wrong?

Ah-ha! I think a significant part of your problem is that the FOV figure Celestia gives relates to the vertical dimension of the screen, not the horizontal as you're using to make your measurement - so you'll be out by a factor of ~4/3. See this thread for confirmation of this (to me) rather counterintuitive approach. (You can quickly check and make sure that this is still the way Celestia behaves by zooming in on the Moon or the Sun from the surface of the Earth, and checking that it fills the screen vertically at an FOV of around half a degree.)

Grant

[Evil Dr Ganymede]

The difference in elongation between max elongation and quadrature is only 0.2 degrees, so it might not be easily detectable by eyeball - essentially, the star will hang about in the vicinity of its max elongation for 20-30 degrees of its orbit around max elongation.

This is true.

Ah-ha! I think a significant part of your problem is that the FOV figure Celestia gives relates to the vertical dimension of the screen, not the horizontal as you're using to make your measurement - so you'll be out by a factor of ~4/3. See this thread for confirmation of this (to me) rather counterintuitive approach.

Argh, that's annoying! So my numbers are correct after all, phew... I just basically rotated the field of view so that it's 90 degrees (so the CoM is at the top edge of the screen, and the stars appear to be bouncing down from that when you speed up the field of view), and set the FOV to be about 11.5 degrees and it looks like it is correct - the star reaches the bottom of the FOV and then returns to the CoM.

[granthutchison]

Another paper you might find interesting is Habitable planet formation in binary star systems, if you have access to Icarus (if not, I can e-mail it to you). It examines the limited case of a one solar mass star, but with quite a wide range of binary companion parameters. They found that for planetesimal accretion to take place, the semimajor axis of the companion's orbit was limited by:

a(comp) > 16/(1-e) * m^0.31 * a^0.8

where a(comp) is the semimajor axis of the companion orbit, in AU, e is the eccentricity of the companion orbit, m is the mass of the companion in solar masses, and a is the semimajor axis at which planetesimal accretion is to take place, in AU. The 1/(1-e) relationship simply fixes the closest approach of the companion - it ensures that the periastron is kept beyond a certain limit. The equation implies that a one-solar mass companion cannot have a periastron less than 16AU, if a planet is to form in the 1AU habitable zone - a pretty major restriction if true.

For an orbit around the centre of mass of a close binary pair, they don't give an analoguous formula, but they do give some useful examples. For a pair of (main sequence) solar-mass stars, the habitable distance is 1.4AU (inverse square law and twice-solar luminosity). A planet cannot form at this distance if the central suns are farther apart than 0.11AU. If the pair consists of a 1 solar mass star and a 0.5 solar mass star, the habitable zone is close to 1AU, and the pair cannot be separated by more than 0.1AU if planets are to form at that distance. (You can expect solar-mass stars with this sort of separation to have circular orbits because of tidal effects, so eccentricity wasn't factored into their calcs.)

Grant

[Evil Dr Ganymede]

Another paper you might find interesting is Habitable planet formation in binary star systems, if you have access to Icarus (if not, I can e-mail it to you).

Thanks, I can access that... (or at least, I'll be able to when my net connection stops being so darn flaky).

Right, got it now. Ta, that looks interesting, I'll read it later on today.
BTW, did I mention this paper by Quintana et al. about planet formation in the Alpha centauri system before? That's kinda nifty.

[granthutchison]

BTW, did I mention this paper by Quintana et al. about planet formation in the Alpha centauri system before?

Thanks!

Grant

[Evil Dr Ganymede]

Oops - I did mention it before. That'll teach me to be too lazy to go back one page .

[granthutchison]

And for completeness here's another messy expression for the angle over which the eclipse is total/annular (second contact to third contact):

theta = 2*asin{[r1*a2+r2*a1]/[a*sqrt((a1+a2)^2+(r2-r1)^2)]} + 2*atan{(r2/r1)/(a1+a2)}

a1, a2, r1, r2 and a have the same meanings as before. Star 1 is the star being eclipsed.
If theta is negative, this indicates that the eclipsed star has a larger disc than the transiting star - it's an annular eclipse. If theta is positive, the transiting star completely eclipses the other star.
Divide the absolute value of theta by omega, calculated as before, to get the eclipse duration.

(I've tested the formula in Celestia for various mass ratios and radii, and it has performed very well indeed, so far. )

Grant

[revent]

I'll get back to you on the eclipse thing ... seems interesting. But planetary temperature wouldn't necessarily plummet - there's a fair old thermal inertia to something the size of a planet, and we don't all freeze to death during the night.

My thermo prof would rap you on the knuckles, Grant.

There is no such thing as thermal inertia. If you stop heating an object, it begins to cool immediately (actually, it continues to radiate heat at it's previous rate). What people call 'thermal inertia' is just the effect of being attached to a large heat sink, the Earth in this case.

Thermal inertia was a big pet peeve of several instructors I've had.

[granthutchison]

If you stop heating an object, it begins to cool immediately ...

For sure. But the analogy of "thermal inertia" concerns the rate at which it cools - I wasn't suggesting that the planet didn't change in temperature, just that it would cool relatively slowly, so the temperature doesn't really plummet. The idea refers to the behaviour of a body with high inertia, which changes velocity relatively slowly when subjected to a force.
I'm aware that some people dislike the term (even to the extent of writing fulminating letters to learned journals when others use it). But it does, as you say, seems to have the status of a "pet peeve", like splitting infinitives, which serves no useful purpose and is on slightly shakey logical ground. "Thermal inertia" can be quite rigorously defined in strict analogy to conventional inertia, it's used regularly and productively by atmospheric physicists, and I'd staunchly maintain that it's a useful and intuitive concept - at least in the sense I used it above.

Grant

(Slight edit for clarity after rereading.)