Its mentioned in several books that a satellite launched with a velocity less than the escape velocity and other than the critical velocity will follow an elliptical orbit. However I can't find a derivation of its equation of trajectory.

Question

Its mentioned in several books that a satellite launched with a velocity less than the escape velocity and other than the critical velocity will follow an elliptical orbit. However I can&#039;t find a derivation of its equation of trajectory.

Bridget Vang · Accepted Answer

This problem is most easily done in polar coordinates. In polar coordinates, the Newtonian gravity is given by      g    =  −            G      M              r      2                          r        ^            Applying Newton&#039;s Second Law in polar coordinates gives two differential equations: one radial and one angular:                    (1)                                          r            ¨                          −        r                                            θ              ˙                                2                =        −                              G            M                                r            2                                                  (2)                    r                              θ            ¨                          +        2                              r            ˙                                                θ            ˙                          =        0            Multiplying (2) by r, we observe that       r    2              θ      ˙        =  h is a constant. This is really just a result of conservation of angular momentum, as there is no force in the angular direction. h is nothing more than the specific angular momentum of the orbiting body.By substituting   u  =  1      /    r into (1) and (2) and solving, it can be shown that                    d        2            u              d              θ        2              +  u  =            G      M              h      2      This can then be solved to obtain  r  =            h      2              G      M      (      1      +      e      cos      ⁡      (      θ      −              θ        0            )      )      where e is the eccentricity.This is the equation we were looking for: a conic section.   e  =  0 gives a circle,   0  &amp;lt;  e  &amp;lt;  1 gives an ellipse,   e  =  1 gives a parabola, and   e  &amp;gt;  1 gives a hyperbola.

Proof for elliptical orbits

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Answer & Explanation

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