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**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.

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**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.

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**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?

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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.

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**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.

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