Book: An Introduction to Modern Astrophysics, 2nd Edition, by Carroll and Ostlie.???????

4. (a) We have discussed the radius of a black hole and the radius of a neutron star. The formula for the radius of a neutron star is rather ugly: (9) 3(187r)2/3 h? (1)\"3 RNs= 10 G Mil/S" E (b) Show that the ratio of the radius of a neutron star to that of a black hole (the Schwarzchild radius) is proportional to (note that m H is the mass of the hydrogen atom and therefore a constant): A\" ENS/356,. or MNS ' Mg}, (10) (c) Let me do the arithmetic for you. In mks units (meters) I nd RNS =15,000(M/M@)'1/3 m (11) and Rsch = 3000(M/M9) m (12) Assuming the neutron star is 2M9, at what black hole mass are the radii of the neutron star and black hole equal? 5. The COBE satellite showed that the Universe is almost a perfect blackbody at T : 2.7255K. The blackbody curve, rewritten to calculate the number of photons, not the energy, is 8 h 3d e(f)df = %% (13) where EU) is the number of photons in the frequency range I > f + df. Integrating this formula (which is a challenge!) over all frequencies gives the formula for the number density of photons as "v 2 3T3 (14) where )3 is 2.029 x 107m3 K'3. (a) Calculate the photon density nT (# photons m'3) of the present-day Universe. (b) For this problem, let us agree on a common value of the critical density of the Universe (the amount needed to stop expansion) of 7.8 x 1040.] m'a. Only 4% of this is in normal matter like baryons (protons, neutrons). Assuming the normal matter is all protonsz, what is the baryon density n3 (# baryons 1113)? To do this, you need to calculate the equivalent energy of a proton in joules by using E = ch2 where any is the mass of a proton