Note that is in the plane of motion; the plane of motion is the plane whose normal is and . Taking the dot product with r, or where is the angle between and now. r Kepler’s First Law Case 1. = 0. e Then The orbit is a circle. Now let Then Convert to Cartesian coordinates: We now obtain Case 2: If = 1, the orbit is a parabola. e Also if > 1, then we get Case 3: If > 1, the orbit is a hyperbola. e e Now assume . Complete the square to get Let , . Then distance from the center to the ellipse to the focus is where . c
Note that is in the plane of motion; the plane of motion is the plane whose normal is and . Taking the dot product with r, or where is the angle between and r now. Kepler’s First Law Case 1. e = 0. Then The orbit is a circle. Now let Then Convert to Cartesian coordinates: We now obtain Case 2: If e = 1, the orbit is a parabola. Also if e > 1, then we get Case 3: If e > 1, the orbit is a hyperbola. Now assume . Complete the square to get Let , . Then distance from the center to the ellipse to the focus is c where .