What's a possible way of finding time t for an overlap of three objects going around a circular orbit around a common fixed center. So like the solar system in 2D, How can I find the time for when the angle between three planets so that they are perfectly in line with each other and create a eclipse? I can do it for two using angular motion but am confused about how to do it with two. Any help will be appreciated
2026-03-25 06:08:54.1774418934
Orbital Motion overlap
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Suppose $[x_i, y_i] = [r_i \cos(\alpha_i t + \beta_i), r_i \sin(\alpha_i t + \beta_i)] $ are the positions of your three objects at time $t$, $i = 1,2,3$. These are collinear iff $$ (x_2 - x_1)(y_3 - y_1) - (y_2 - y_1)(x_3 - x_1) = 0$$ which simplifies to $$ r_1 r_3 \sin((\alpha_1 - \alpha_3) t + \beta_1 - \beta_3) + r_1 r_2 \sin((\alpha_2 - \alpha_1) t + \beta_2 - \beta_1) + r_2 r_3 \sin((\alpha_3 - \alpha_2) t + \beta_3 - \beta_2) = 0$$
In general, this can't be solved in "closed form", but numerical methods can be used.