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PostPosted: Sun Jun 17, 2018 4:31 pm 
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It's an older G/? binary that I thought would be an interesting example for EDG's "how-to-binary" post.

Normal BBT is 278.66 x (L^0.25/D^0.5

For binaries, the esteemed doctor advises doing this:

"Flux" = (278.66^4) * (L/D²)

I'm not entirely sure how to apply it, so let's take Delta T as an example.

Spectral type is G0V paired with something... G9V to K4V

Masses 1.0 and 0.8, age estimated 8.5+ Gyr, which would be closer to the end of the line for a G0 star.

Luminosity not calculated on Wikipedia. Beta Canum is a G0 with lum of 1.15. Let's say as an older, more cantankerous star, it's hotter, maybe 1.5?

Its companion, let's say for ease of calculation it's a K0-ish star with L of 0.5.

The two stars are pretty tight, an 0.1 AU distance.

I'm not quite sure how to apply the flux formula. Example pretty please?


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PostPosted: Mon Jun 18, 2018 1:54 am 
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ok, so let's get some numbers first:

Delta Trianguli A: Age 8.5 Ga, Mass 1.0 Sols, Luminosity 1.5 Sols, Radius 1.2 Sols (G0 V).
Delta Trianguli B: Age 8.5 Ga, Mass 0.8 Sols, Luminosity 0.34 Sols, Radius 0.76 Sols (K2 V). (these are from my own stellar evolution tables)
Orbital separation = 0.11 AU, eccentricity = 0.02. ( http://www.solstation.com/stars2/del-tri2.htm ). Star1 = 0.0489 AU from barycentre, Star2 = 0.061 AU from barycentre.

Let's say your planet orbits the binary barycentre at a distance of 1.1 AU. It'd orbit the barycentre with a period of 316.67 days.

Since the planet orbits the pair, you need to calculate the temperature when the stars are at 90° (quadrature) to the planet and when the stars are in a line with the planet.
Let's assume for simplicity's sake that the planet doesn't move (the stars will be orbiting each other much more quickly, once every 10 days).

The planet starts with the stars lined up with it, with the primary closer. The planet is 1.051 AU from the G0 V, and 1.16 AU from the K2 V (which is eclipsed by the G0 V anyway). So it's just getting energy from the G0V, so it's basically a straightforward calculation with the first equation (L = 1.5, D = 1.051) so its blackbody temperature is about 300 K.

Five days later, the stars are in a line again but are reversed - the companion is the closer star to the planet. So here we use the first equation again (L = 0.34, D = 1.0389) and find that the planet's blackbody temperature is about 209 K. That's a big drop, but keep in mind that these eclipses are very short (not entirely sure how to calculate it, but it wouldn't be long - maybe an hour or so?). But also this assumes that the second star completely eclipses the first one, which in this case doesn't actually happen (if you calculate the angular diameter of each star, at this point the more distant primary would be 0.56 degrees and the closer companion would be 0.39 degrees, so some of the primary's radiation is still getting through and the temperature would be higher).

Between those times is when you'd use the second Flux equation. At quadrature (where the stars are at 90° to the planet), both stars are 1.011 AU from the planet. So the Flux from the primary star is (278.66^4) * (L/D²) = 601072003 * (1.5/1.212390123) = 7436616222. For the companion star, the Flux = 6010720039 * (0.34/1.213734568) = 1683765848. Add those together and you get 9120382070. Now the combined blackbody temperature would be (9120382070 ^ 0.25 = ) about 309 K.

The full temperature graph would look something like this:
Attachment:
graph.PNG
graph.PNG [ 5.97 KiB | Viewed 60 times ]


Does that make any more sense? You really need a spreadsheet to calculate all this as you need to know where the stars are in their orbits etc. Not sure I can get into too much detail on it as I'm pretty rusty on all this now.

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PostPosted: Mon Jun 18, 2018 10:44 am 
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EDG wrote:
Delta Trianguli A: Age 8.5 Ga, Mass 1.0 Sols, Luminosity 1.5 Sols, Radius 1.2 Sols (G0 V).
Delta Trianguli B: Age 8.5 Ga, Mass 0.8 Sols, Luminosity 0.34 Sols, Radius 0.76 Sols (K2 V). (these are from my own stellar evolution tables)
Orbital separation = 0.11 AU, eccentricity = 0.02. ( http://www.solstation.com/stars2/del-tri2.htm ). Star1 = 0.0489 AU from barycentre, Star2 = 0.061 AU from barycentre.

Let's say your planet orbits the binary barycentre at a distance of 1.1 AU. It'd orbit the barycentre with a period of 316.67 days.

Since the planet orbits the pair, you need to calculate the temperature when the stars are at 90° (quadrature) to the planet and when the stars are in a line with the planet.
Let's assume for simplicity's sake that the planet doesn't move (the stars will be orbiting each other much more quickly, once every 10 days).


I think that's overkill. I don't think that Matt is asking about calculating the effects of stellar transits and ever-changing orbital geometry: He's only after the annual-average temperature. This is the sort of calculation in which it is customary to ignore the eccentricity of planetary orbits. It's just a question of how to sum the heating effects of two stars.

Flux = (L₁/D₁² + L₂/D₂²)

But for a planet in a nearly-circular orbit around the common centre of the two stars, with the planet's orbit large compared with the stars' mutual orbit, the average D₁ = D₂, so write

Flux = (L₁ + L₂)/D²

BBT = 278.66 × ∜Flux

BBT = 278.66 × (∜(L₁ + L₂)/√D)

With L₁ = 1.5 and L₂ = 0.34

BBT = 278.66 × (∜1.84)/√D

= 278.66 × 1.1647/√D

= 324.5 / √D kelvins

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PostPosted: Mon Jun 18, 2018 10:29 pm 
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Yeah, the average and extremes were what I was looking for. Thanks.

Pretty wild that the temperature would change by that much.

Imagine the strange beliefs that might arise from watching it happen.

Especially if the orbits are slightly off so it doesn't happen regularly. Maybe it's seen as a struggle, or a mating ritual, or one swallows the other and then it's reborn. A lot of interesting stories could come of it.


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PostPosted: Mon Jun 18, 2018 10:58 pm 
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there's probably something that can be done with sins and cosines to more easily calculate the temperature when the stars are at different angles relative to the planet, but I don't have the brains anymore to figure that out so I just brute force it in the spreadsheet I have :p. (my spreadsheets are turning into black boxes...!)

Also while the blackbody temperature might drop a lot during the eclipses, keep in mind that they're only eclipsed for an hour or so, and the planetary environmental systems generally have a much slower response time than that - so while it may get colder for that duration i don't think it's likely that the atmosphere's suddenly going to freeze out or massive hurricanes appear instantly or whatever.

Also, I have no idea if I'm right about all this :P. The logic seems to make sense at least so I presume this is broadly how it would work. AFAIK nobody's done a detailed paper about how temperatures would vary on a planet orbiting a binary system.

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PostPosted: Wed Jun 20, 2018 6:15 pm 
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Now I'm all set for my character to get mad at his uncle and pout while he watches the suns set.


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PostPosted: Thu Jun 21, 2018 11:05 am 
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EDG wrote:
Also while the blackbody temperature might drop a lot during the eclipses, keep in mind that they're only eclipsed for an hour or so, and the planetary environmental systems generally have a much slower response time than that - so while it may get colder for that duration i don't think it's likely that the atmosphere's suddenly going to freeze out or massive hurricanes appear instantly or whatever.


I was really surprised at how chilly it got during last year's total solar eclipse (which I witnessed in Donnely, Idaho).

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