Planetary Orbit by Kepler

[b]What is this applet about?[/b] This applet simulates the motion of a planet in its orbit around the Sun. The planetary orbit is an ellipse (Kepler's first law), with the Sun in one of the focal points. How the planet moves alongs this orbit, is described by Kepler second law. This applet calculates the planetary motion the same way as Kepler did in 1605. [b]The planetary orbit[/b] According to Kepler's first law the planetary orbit is an ellipse with the Sun in one of the focal points. This applet lets you construct your own planetary orbit by moving the sliders [b]a[/b] and [b]e[/b]. The value of [b]a[/b] is called the [i]semi long axis[/i] of the ellipse. This equals the mean distance from the planet to the Sun. This value is expressed in [i]Astronomical Units[/i]. 1 AU is approximately 150 million kilometer. The point of the orbit closest to the Sun is called [i]perihelion[/i], the most distant point is the [i]aphelion[/i]. [b]The period of one revolution around the Sun[/b] The period T of the planet is calculated with Kepler's third law: [math]\frac{ T^2 }{ a^3 } = constant[/math]. [i]T[/i] is expressed in years and [i]a[/i] in AU. In this applet T is expressed in days by multiplication with 365.25: [math]T = \sqrt{a^3} * 365.25[/math]. [b]True anomaly[/b] This applet starts with the planet in its perihelion at [i]t=0[/i]. The planet moves anti-clockwise, and arrives at the perihelion again at [i]t=T[/i]. In this applet you can see the slider [b]t[/b], with a value ranging from 0 to T. At a certain time t the planet has moved over an angle [math]\theta[/math] with reference to the Sun. This angle is called the [i]true anomaly[/i]. [b]Mean anomaly[/b] The mean anomaly [i]M[/i] is calculated from the average angular velocity: [math]M = \frac{ 2 * \pi * t }{ T }[/math]. [b]Excentric anomaly[/b] Kepler plotted a circle around the elliptic orbit, with centre O. The planet's position is the projected onto the circle in point [i]P[/i]. The excentric anomaly [i]E[/i] is the angle between line [i]OP[/i] and the line from O to the perihelion. [b]Kepler's Equation[/b] The relation between mean anomaly M and excentric anomaly E is given by Kepler's equation: [math]M = E - e * \sin{ E }[/math] This equation can be solved by Geogebra by intersecting a function [math]f: y = e * \sin{ E }[/math] with a straight line [math]y = E - M[/math]. The value of E is calculated as a function of time t. [b]Finding the true anomaly[/b] As the entire diagram is constructed with Geogebra in the way Kepler did this 4 centuries ago, the true anomaly can be measured from the diagram using Geogebra's angle tool.