We can establish a formula that describes the relationship between a and M+m in a system that has this coincidence where 1 light-annum = 1 parallax distance measurement.
Let’s continue from the above and define a few more terms. I think I did all the algebra right; could someone please check me?
c = speed of light
s = angular measures in a full circle (Babylonian arcseconds = 3606060 = 129600)
S = Parallax distance measurement, AKA “local parsec”
S = a * π * s
L = P * c ( 1 “light annum” is the distance light travels during 1 orbital period.)
L = S ( This is our coincidence we’re solving for.)
I’m gonna throw out the m term for the time being, since it’s usually negligible. Since M and m are always considered together, we can put m back in at the end if we want.
Now, solve for a and M.
a * π * s = 2 * π * sqrt(a[sup]3[/sup]/(GM)) * c
a * s = 2 * sqrt(a[sup]3[/sup]/(GM)) * c
π falls out right away.
s = 2c * sqrt( a / (GM) )
s[sup]2[/sup] = 4c[sup]2[/sup] * a / (GM)
a = GM * s[sup]2[/sup] / (4c[sup]2[/sup])
M = a * 4c[sup]2[/sup] / (G * s[sup]2[/sup])
Did I do that right?
Now, here’s the real question: Assuming the star is a main-sequence dwarf, what’s the range in values for M that result in the planet orbiting within the life zone?