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Anomalies for Planetary Motion

The Kepler motion of a planet (brown dot) around a star (yellow dot) follows an ellipse. Given a semimajor axis and eccentricity , the current position of the planet at time since the perihelion passage is given by the true anomaly . To determine this, we first have to calculate the eccentric anomaly from the mean anomaly .

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Kepler's third law gives a relation between the semimajor axis and the period of a planet, , where is the gravitational constant and the solar mass is set equal to 1. The mean anomaly is defined as , where is the time since the perihelion passage of the planet. To obtain the eccentric anomaly, we have to solve the Kepler equation . The true anomaly can be determined via . The distance between the star and the planet is given by .
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