phoenixshade
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Proud to be an Ape
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Greetings, one and all, after a long absence. I'm seeing what I can do with this problem. I set up the quaternary the oldfashioned way: I did the math. The first step was to consider one tight binary. First, we look at the universal formula for gravity, F = Gm_{1}m_{2}/r^{2}. Since m_{1} and m_{2} are equal, this becomes F = Gm^{2}/r^{2}. Next, we set this equal to centripetal force, F = mv^{2}/R. This is a different R, equal to half the distance between the stars (since they orbit their mutual center), so substituting R=0.5r into the equation gives F = mv^{2}/0.5r, or F = 2mv^{2}/r. Now we can set this equal to gravity and solve for our initial v: Gm^{2}/r^{2} = 2mv^{2}/r. This reduces to Gm/r = 2v^{2}, or v = √(Gm/2r). Plugging in the numbers gives v = 111.8197 km/s. Next, we repeat the process, treating each binary as a single object of mass 2m located at its barycenter. By the same calculations, we get v = 58.17067 km/s. Dropping into GravSim, I created the four stars with appropriate mass, then adjusted them via Edit Objects (state vector). This was problematic at first; I had to manually edit the .gsim to make all four stars "Floating." Otherwise, it was forcing Star Ab to use Star Aa as a Reference Object, making its velocity relative to Star Aa instead of equal and opposite... equivalent to setting its initial velocity to zero. (Hey Tony, any chance we can get "Floating" in the Reference Object dropbox when editing state vectors??) To make everything nice and easy, the first tight binaries start on the yaxis (x=0), each 0.0205 au from the origin. Both stars of the second binary start on the xaxis (y=0), with xvalues of 0.2825 and 0.3235 au (0.303 +/ 0.0205). Getting the tight binaries to orbit is now easy: for the first pair, all the initial velocity is in the x direction; one is positive 111.8197 and the other negative. For the second pair, the velocity is all in the y direction. Now, to get the pairs to orbit each other, we could give each pair a push of 58.17067 km/s in opposite y directions, but rather than doing that, I gave all of the relative y velocity to the first pair by doubling the value. (This way, I didn't have to mess with the already assigned y velocities for the second tight binary.) Since their relative motion is the same, the resulting orbits are identical. In all cases, I assigned positive or negative values to produce anticlockwise rotation. Here's the resulting .gsim. You might have to click the "floating/absolute" button on the graphics options... otherwise it tracks on one of the stars and things look a little weird. Working on planets, but what distance did you have in mind to represent the HZ? Based on their mass, I'd estimate that each of these four stars have about twice our sun's energy output, or about 8 times solar output in total. I believe that, based on the inverse square law, that would put the HZ around sqrt(8) or about 2.83 au, so that's my starting point for planet 2. Thoughts? By the way, if you want to change anything, I recommend you do it with state vectors. Trying to mess with orbital parameters WILL result in chaos (and not the fun kind). Trust me.
