Prediction International Astronomy Olympiad 2022 (Problem and Solution) Edition 1

Here we provide predictions about the International Astronomy Olympiad in 2022 Edition 1. We have prepared this question for all students who are pursuing the astronomy olympiad. This question can be used not only at the international level, but also at the local and national level.

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Question 1

Assuming that the present density of baryonic matter is ρ_b0 = 4.17 x 10^-28kg m^-3 , what was the density of baryonic matter at the time of Big Bang nucelosynthesis (when T  ̴ 10¹⁰ K)? Assume the present temperature, T₀ 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 10⁸ 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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Question 4

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_peakand 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_peakis related to the peak color temperature T_color derived from observed spectral data by T_color = f_GRf_colT_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_peakof the accretion disk around Cygnus X-1.

The total luminosity of the blackbody component of the accretion disk can be estimated by L_disk ≈ 4πσr²_peakT⁴_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 10³⁷ 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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