Question

Can anyone help me with the solution to one of the Fermi problems (No. 4) in the link (http://ocw.mit.edu/courses/physics/8-01sc-physics-i-classical-mechanics-fall-2010/problem-solving-and-estimation/MIT8_01SC_problems02.pdf)
to calculate the average density of matter in the galaxy?

Answer 1

Since density = mass / volume, I think we can start with getting the mass and the volume of the Milky Way Galaxy.

Volume: http://en.wikipedia.org/wiki/Orders_of_magnitude_(volume)
Mass: http://en.wikipedia.org/wiki/Milky_Way

I'm citing Wikipedia but these pages are citing other relevant sources so you can check them out if you need verification. And this research provided me with the values:

Volume: 10^63 cubic meter
Mass: 1.3 x 10^12 solar mass

Now, a solar mass is around 2.0 x 10^30 kg. Let's convert it to kg and we get:

1.3 x 10^12 x 2.0 x 10^30 = 2.6 x 10^42 kg

Alright now we can solve for the actual density. We simply divide the mass by the volume, and we get:

2.6 x 10^42 / 10^63 = 2.6 x 10^-21 kg / cubic meter

Which is... pretty close to the one in the solutions (if you count three zeros as close enough). The solution describes a more complex process, including summing up all the masses of the components of Milky Way, and simplifying the density to proton level. Please note that we're just making an estimation here and the fact that we're not entirely correct is not important. :)