The Dark Matter in the Solar System

Problem 1

Estimate upper bound of the relative perturbation of the Earth orbit due to presence of the dark matter in the Solar system. Assume that the upper limit on the dark matter in the Solar system lies on the level $\rho_{DM}\approx1.5\cdot10^{-12} {\it g/cm}^3$ and it remains constant at the distance discussed.

solution

\[\Delta=\frac{\delta F}{F_{SE}}\times1.5\cdot10^{-12}\] Here $\delta F$ is correction to the gravitational force $F_{SE}$ acting upon the Earth from the Sun. This correction is found easily by means of the Gauss theorem.

Problem 2

Find the velocity change of a spaceship rotating around the Earth with period $T$ as the result of dark matter particles scattering on the nuclei of particles that compose the spaceship.

solution

Problem 3

Assume that due to interaction with dark matter particles the spaceship's velocity changed by $\Delta v$ in one period. Given its mass is $m$ and it moves around the Earth on a circular orbit, estimate the dark matter density in the Earth's neighborhood.

solution