Equation of the Orbit and Keplers First

In order to find the geometric shape of the orbit, we now derive the equation of the orbit. Since e is a constant vector lying in the orbital plane, we choose it as the reference direction. We denote the angle between the radius vector r and e by f. The angle f is called the true anomaly. (There is nothing false or anomalous in this and other anomalies we shall meet later. Angles measured from the perihelion point are called anomalies to distinguish them from longitudes measured from some other reference point, usually the vernal equinox.) Using the properties of the scalar product we get r ■ e = re cos f .

This is the general equation of a conic section in polar coordinates (Fig. 6.4; see Appendix A.2 for a brief summary of conic sections). The magnitude of e gives the eccentricity of the conic:

0 < e < 1 ellipse , e = 1 parabola, e > 1 hyperbola.

Inspecting (6.14), we find that r attains its minimum when f = 0, i. e. in the direction of the vector e. Thus, e indeed points to the direction of the perihelion.

Starting with Newton's laws, we have thus managed to prove Kepler's first law:

The orbit of a planet is an ellipse, one focus of which is in the Sun.

Without any extra effort, we have shown that also other conic sections, the parabola and hyperbola, are possible orbits.

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