If the image looks blurry, please click on the image to enlarge it to the best resolution.

**Question 1**

Assuming that
the present density of baryonic matter is ρ_{b0} = 4.17 x 10^{-28}
kg m^{-3} , what was the density of baryonic matter at the time of Big
Bang nucelosynthesis (when T ̴ 10^{10} K)? Assume
the present temperature, T_{0} to be 2.7 K.

**Answers**

**Question 2**

On the night of
January 21st, 2019, there was a total lunar eclipse during a supermoon. At the
time, the moon was close to perigee, at a distance of 351837 km from the earth,
which was 1.4721 x 10^{8} km from the sun. The gamma (γ) of a lunar
eclipse refers to the closest distance between the center of the moon and the
center of the shadow, expressed as a fraction of the earth’s radius. For this
eclipse, γ = 0.3684. Given this information, find the closest estimate for the
duration of totality of the eclipse.

**Answers**

**Question 3**

You are in the
northern hemisphere and are observing rise of star A with declination δ = -8^{o}
, and at the same time a star B with declination δ = + 16^{o} is
setting. What will happen first: next setting of the star A or rising of the
star B?

**Answers**

Consider a star with mass M and radius R. The star’s density varies as a function of radius r according to the equation

, where ρ_{center
}is the density at the center of the star. Derive an expression for dP/dr
in terms of G, M, R, and r, where P is the pressure at a given radius r.

**Answers**

**Question 5**

Cygnus X-1/HDE
226868 is a binary system consisting of a black hole Cygnus X-1 and blue
supergiant HDE 226868. The mass of HDE 226868 is 30M_{ʘ} and the period of the
binary system is 5.6 days. Radial velocity data reveals that the orbital
velocity of HDE 226868 is 116.68 km/s at apoapse and 123.03 km/s at periapse.

*(a) Determine
the eccentricity of the orbit of HDE 226868. *

*(b) Determine
the length of the semimajor axis of the orbit of HDE 226868.*

*(c) Determine
the mass of Cygnus X-1, to at least 3 significant figures.*

The peak
blackbody temperature of an accretion disk occurs at a distance of r_{peak}
and a temperature of T_{peak}. One can determine the peak blackbody
temperature by assuming that it corresponds to the peak in the x-ray spectrum.
Due to relativistic effects, the actual peak blackbody temperature T_{peak}
is related to the peak color temperature T_{color} derived from
observed spectral data by T_{color} = f_{GR}f_{col}T_{peak},
where f_{GR} ≈
0.510 and f_{col} ≈
1.7. Three x-ray spectra of Cygnus X-1 are shown in Figure below.

Figure Three x-ray spectra from Cygnus X-1. From Gou et al. (2011).

*(d) Using spectrum SP2, determine the peak blackbody temperature T _{peak}
of the accretion disk around Cygnus X-1.*

The total
luminosity of the blackbody component of the accretion disk can be estimated by
L_{disk} ≈ 4πσr^{2}_{peak}T^{4}_{peak}
(Makishima et al. 1986). The radius rlast of the innermost edge of the
accretion disk is related to the radius r_{peak} of the peak blackbody
temperature by r_{peak} = ηr_{last}, where η = 0.63. In 1996,
the blackbody luminosity of the accretion disk around Cygnus X-1 was estimated
to be 2.2 x 10^{37} ergs/s.

*(e) Determine
the radius rlast of the innermost edge of the accretion disk around Cygnus X-1.*

Assume that the innermost edge of the accretion disk is located at the innermost stable circular orbit (ISCO), whose radius risco is a function of the spin of the black hole. The relationship between risco and a˚, the spin parameter of the black hole, can be estimated by:

(f) Determine the spin parameter a˚ of Cygnus X-1.

**Answers**