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Approximations are made: Earth's orbital velocity remains the same: Angle between the Earth and the Earth's perihelion $\theta$ is increasing constantly. Eccentricity is small enough, that ellipse can be approximated to be $r=a(1-e \cos\theta)$. Earth is at its perihelion on 4th of January, and its eccentricity is 0.0167, so the given formula can be ...

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$$1- 0.01672*\cos(0.9856*(\text{day}-4))$$ This is an approximate expression. Term by term, $1$ The mean distance between the Earth and the Sun is about one astronomical unit. $0.01672$ This is the eccentricity of the Earth's about about the Sun. $\cos$ This is of course the cosine function, but with argument in degrees rather than radians. $0.9856$ This ...

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A presentation on the SETI Weekly Seminar series (available on Youtube) points out that tidal locking (e.g. expected of a planet in the habitable zone of a red dwarf) can involve higher muliples than same-face-shows, and in fact an eccentric orbit favors an odd half multiple (3:2 like Mercury). There is also orbital inclination to consider. The pattern of ...

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By the sounds of it you have made a mistake with the units. In fact, you should not be using SI units at all in your simulation; astronomical values in SI units vary by such huge orders of magnitude that they are often a source of floating point errors that can destroy trajectories. You should instead use the astronomical system of units. Specifically, ...

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