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Kepler's Mysterium Cosmographicum

Solving Kepler's equation of elliptical motion

Movie of elliptical motion

Plots of elliptical Kepler motion

The circumference of an ellipse

Kepler's third law

Mars

Kepler Applet

This applet is simulating the elliptical motion of a celestial body.

 Use the third scrollbar (blue) to increase uniformly the true anomaly angle phi. Use the fourth scrollbar (red) to increase uniformly the time (mean anomaly M). Select from the menu for details, such as eccentric anomaly. radius, velocity. F focus P periapsis a = OP semimajor axis e = OF/a = c/a eccentricity Q position of the planet at time t (Q=P for t=0) R point on the blue auxiliary circle with radius a r = FQ radius vektor E = angle(ROP) eccentric anomaly φ = angle(QFP) true anomaly T orbital period M = t/T*360° mean anomaly The mean anomaly M is the angular distance from perihelion which a (fictitious) planet would have if it moved on the circle of radius a with a constant angular velocity and with the same orbital period T as the real planet moving on the ellipse. By definition, M increases linearly (uniformly) with time.   Choose Show/Hide Velocity v and Show/Hide Vel. Circle: Applet for Mars As the planet P is moving around the ellipse, the end of the velocity vector OP' will move around the gray circle (center M), and FP is always pependicular to to MP'. The data window of the applet: E - e*sin(E) = M(t) 