# A first integral of motion for logarithmic spiral trajectories

### Abstract

A first integral of motion similar to the Laplace-Runge-Lenz vector is obtained for logarithmic spirals trajectories (LST). The LST is a special solution to the equation of motion of a planar solar sail, where the radial and tangential velocities of the spacecraft are related by a constant. By removing extraneous solutions in the resulting orbit equation, we obtained the requirement that the integral of motion must be identically zero for an LST orbit to exist. Minimum-travel time logarithmic spiral trajectories have been obtained, consistent with what was obtained in the literature.

*Proceedings of the Samahang Pisika ng Pilipinas*

**37**, SPP-2019-1D-06 (2019). URL: https://paperview.spp-online.org/proceedings/article/view/SPP-2019-1D-06.