Question

LP 5. (20 pts) It is known that most of the mass of galaxies is some exotic material that doesn't interact with light or normal matter. Let's assume that this "dark matter" is due to a new elementary particle---alaskonium. Like Alaska in the winter, alaskonium in the galaxy is cold and dark. We will assume that alaskonium is a fermion and that it is so cold that it is degenerate; a gravitationally bound, nonrelativistic, degenerate gas. Take the total mass and radius of the gas to be that of a typical galaxy, M- 1010 solar masses and R- 105 light years. a) What is the rest mass (energy) of the alaskonium particle? Give answer in eV. b) What is the Fermi energy? c Is our assumption that alaskonium is nonrelativistic validated? What are the implications if it is not?

Answer 1

1. E =mc²

1010 x 2 x 1030(3 x 108)2

= 18 x 10⁵⁶ J

= 1.12 x 10⁷⁶ eV

2.$E_F = \bar{h}^2 \pi ^2n^2 /2mR^2$

Ег = 6.58 × 10-163.142 × 32/2(10102 × 230)(10° × 9.46 × 1015)

= 583.88 x 10^-16/37.84 x 10⁶⁰

=15.43 x 10^-76 J

3. NO, we cant consider alaskonium as non-relativistic. since this means the particle is stationary and not moving. But, any particle in space cant be considered as stationary. It must be moving relative to something. Thus the energy of the particle changes.