Blog #8: WS3.1, #7

7. The mass of the Milky Way, measured using stars and probes of the gas in the Galaxy, yield a Milky Way mass of around \(10^{10}-10^{11}M_\odot\). The puzzle of the apparent mass discrepancy with the actual mass has confounded scientists since the 1930s with no clear resolution. They dubbed this mysterious and invisible mass 'dark matter.' Many theories attempt to explain the nature of dark matter. A notable one was the MACHO -- MAssive Compact Halo Object. The idea is that the invisible mass of the Galaxy is entirely due to a great number of non self-luminous individual astrophysical bodies such as brown dwarfs (tiny, failed stars), white dwarfs (the core of a star left over after the star dies), very low-mass red dwarfs (small, faint stars), or even black holes! All of these objects are very hard to detect by conventional techniques. In this problem, we'll test the validity of the MACHO theory of dark matter. 
a) If MACHOs are composed entirely of brown dwarfs, and a typical brown dwarf has mass 0.05 \(M_\odot\), what must the number density of brown dwarfs in the Galaxy be to account for the mass deficit between the stellar and total mass of the Galaxy? Assume the MACHOs are distributed uniformly in a sphere with radius 10kpc.

Comparing the stellar mass of the Galaxy with the total mass of the Galaxy
\(M_{stellar} = 10^{10}M_\odot\)
\(M_{tot} = 3*10^{12}M_\odot\)
shows a deficit of \(2.99*10^{12}M_\odot\), which needs to be made up with brown dwarfs.
\(\frac{2.99*10^{12}M_\odot}{\frac{4}{3} \pi (10kpc)^{3}} \frac{BD}{0.05M_\odot} = 1.43*10^{10}BD/kpc^3 \rightarrow 14BD/pc^3\)

b) What does the number density imply about the distance to the nearest MACHO? 

Equating a volume of a sphere unknown radius to the volume that should contain one brown dwarf: \(\frac{4}{3}\pi r^3 = \frac{1}{14}pc^3\)
\(r = (\frac{3}{56\pi})^{1/3} = 0.38pc\) is the implied distance to the nearest MACHO.
0.38pc is a much smaller distance than our distance to the nearest brown dwarf in reality, Luhman 16, which is 2pc away. Therefore, this theory is probably incorrect.

c) If MACHOs were made of nachos, how many nachos would be required to populate the galaxy for dark matter to be a solved problem? What would be our distance to the nearest nacho? 

We'll assume an average nacho mass of 10g, and we need to use these nachos make up the deficit of \(2.99*10^{12}M_\odot\) or \(2.99*10^{45}g\).
\(\frac{2.99*10^{45}g}{10g} = 2.99*10^{44}\) nachos would be needed
\(\frac{2.99*10^{44}nachos}{\frac{4}{3}\pi (10kpc)^3} = 7.14*10^{40}nachos/kpc^3 \rightarrow 7.14*10^{31}nachos/pc^3\)
\(\frac{4}{3}\pi r^3 = \frac{1}{7.14*10^{31}}pc^3\)
\(r=(\frac{1}{7.14*10^{31}}\frac{3}{4\pi})^{1/3} = 1.5*10^{-11}pc\) would be the distance to the nearest nacho...close, but not nearly close enough for Chris Pratt.


via GIPHY