# mechanical similarity and planetary orbits

14 Nov 2012What would be Kepler’s law of planetary motion (the cube of the orbit’s major axis is proportional to the square of the orbital period) if the Newtonian gravitational potential was \(\propto 1/r^2\) instead of \(\propto 1/r\)? Can you generalise Kepler’s law to arbitrary integer power laws for the potential, \(\propto 1/r^n\)? Would the two other laws by Kepler be affected (elliptical orbits and angular momentum conservation)?

bonus question: What’s \(\int_0^\infty\mathrm{d}x\:x^3\exp(-\alpha x^2)\)?