Question

Assume dark matter exists throughout space with a uniform density of $6.00\times {10}^{28}kg/m^3$.

(a) Find the amount of such dark matter inside a sphere centered on the Sun, having the Earth’s orbit as its equator.
(b) Explain whether the gravitational field of this dark matter would have a measurable effect on the Earth’s revolution.

Answer 1

(a) The volume of the sphere bounded by the Earth’s orbit is
$V=\frac{4}{3}\pi r^3=\frac{4}{3}\pi (1.496\times {10}^{11}m{)}^3=1.40\times {10}^{34}{m}^3$
$m=\rho V=(6\times {10}^{-28}kg/m^3)(1.40\times {10}^{34}{m}^3)=8.41\times {10}^{6}kg$
(b) By Gauss’s law, the dark matter would create a gravitational field acting on the Earth to accelerate it toward the Sun. It would
shorten the duration of the year in the same way that $8.41\times {10}^{6}kg$ of extra material in the Sun would. This has the fractional effect of
$\frac{8.41\times {10}^{6}kg}{1.99\times {10}^{30}kg}=4.23\times {10}^{-24}$ of the mass of the Sun.
No. It is only the fraction $4.23\times {10}^{-24}$ of the mass of the Sun.