Thursday, October 20, 2011

Back of Napkin Dark Matter Calculations

The volume of the solar system inside Pluto's orbit is ~10^39 m^3, while the volume of the galaxy inside a radius of 10 kpc is ~10^62 m^3. Assume for the sake of argument that the density of dark matter in the galaxy is ~.01 Msun/pc^3. It isn't a constant density, but this will illustrate the point. Then the total mass of dark matter inside Pluto's orbit is ~10^-13 Msun, which is completely negligible. On the other hand, the total mass of dark matter inside a radius of 10 kpc is >10^10 Msun, which is comparable to the mass of stars.

From here.

The Milky Way has about 400 billion (i.e. 4*10^11) stars.

The Earth has a mass ca. 3*10^-6 MSun, and the Moon has a mass ca. 3*10^-8 MSun. Thus, the total mass of all of the dark matter inside Pluto's orbit, if it exists, is on the order of about 1/100,000th the mass of the Moon (or a Moon rock material body with a radius of about 100 miles which is about the size of a big asteroid), but distributed somewhat evenly across the entire solar system.

Since people feel only the net pull of gravity from all directions and the force of gravity declines as 1/r^2, only the somewhat local inhomogenities in a distribution of dark matter are observable to someone inside that distribution of dark matter. So, gravitational impact of dark matter in the vicinity of the solar system on someone in the solar system is much, much smaller than the gravitational impact of a single large asteroid within the solar system.


Mitchell said...

But what is the flux of dark matter through the napkin, as a function of its angle with respect to the galactic plane?

Andrew Oh-Willeke said...

Ha, ha! I'll leave that as an exercise (knowing, of course, that the problem doesn't contain enough information to infer the answer).