Prediction International Astronomy Olympiad 2022 (Problem and Solution) Edition 2

Here we provide predictions about the International Astronomy Olympiad in 2022 Edition 2. 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.

Question 1

Figure below shows a full-phase light curve (“phase curve") of the exoplanet HD 189733b taken by the Spitzer space telescope. Use this figure to answer the following questions. The star HD 189733 has an effective temperature of 4785 K and a radius of 0.805 Solar radii.

Part A : Use the depth of the planet’s transit to estimate the radius of HD 189733b, in Jupiter radii.
Part B : Use the depth of the eclipse of the planet by the host star to estimate the ratio of the flux of the planet HD 189733b to that of the host star HD 189733.
Part C : HD 189733b is so close-in to its host star that it is expected to be tidally locked. Use the phase curve to estimate the ratio of the dayside flux emitted by the planet to the nightside flux emitted by the planet.
Part D : This phase curve also noticeably has a phase curve offset, that is, the maximum in planet and star flux does not occur exactly at secondary eclipse. What process that occurs in a planetary atmosphere could cause such a phase curve offset?

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Question 2

Mass-Radius Relation Stellar physics often involves guessing the equation of state for stars, which is typically a relation between the pressure P and the density ρ. A family of such guesses are known as polytopes and go as follows

P = K ρ^γ

where K is a constant and the exponent γ is fixed to match a certain pressure and core temperature of a star. Given this, show that one can obtain a crude power-law scaling between the mass M of a polytopic star and its radius R of the form M α R^α.Find the exponent α for polytopic stars (justify all steps in your argument). Also, indicate the exponent γ for which the mass is independent of the radius R. Bonus: Why is this case interesting?

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Question 3

In a rather weird universe, the gravitational constant G varies as a function of the scale factor a(t).

Consider the model f(a) = e^(pa-1) where b = 2.09.

Part A : Assuming that the universe is flat, dark energy is absent, and the only constituent is matter, estimate the present age of this weird universe according to this model. Assume that the Friedmann equation: still holds in this setting.

Part B : What is the behaviour of the age of the universe t as the scale factor a(t) → ∞? Note that all parameters with subscript 0 indicate their present value. Take the value of Hubble’s constant as H₀ = 67.8 kms^-1Mpc^-1. 

Hint: You might need the following integrals

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Question 4

Part A : Find the shortest distance from Boston (42.3601⁰N; 71.0589⁰ W) to Beijing (39.9042⁰ N; 116.4074⁰E) traveling along the Earth’s surface. Assume that the Earth is a uniform sphere of radius 6371 km.
Part B : What fraction of the path lies within the Arctic circle (north of 66.5608⁰ N)?

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Question 5

In this problem, we will try to understand the relationship between magnetic moments and angular momenta, first for charged particles and how this can be extended to planetary objects

Part A : Consider a charge e and mass m moving in circular orbit of radius r with constant speed v. Write down the angular momentum L of the charge and magnetic moment µ of the effective current loop. Recall that the magnetic moment of a current loop with current I and radius r is given as µ = I A where A is the area of the loop.
Part B : Use the above results to find a relationship between the magnetic moment µ and angular momentum L in terms of intrinsic properties of the particle (charge,mass).
Part C : The relationship from part (2) can be expressed as µ = γ L. γ is usually referred to as the classical gyromagnetic ratio of a particle. Evaluate the classical gyromagnetic ratio for an electron and for a neutron in SI units.
Part D : For extended objects such as planets, the magnetic dipole moment is not directly accessible whereas the surface magnetic field can be measured. Assuming a magnetic dipole of magnetic moment µ located at the center of a sphere of radius r, write down the expression for the surface magnetic field B_surf and the surface magnetic moment defined as M_surf = B_surf r³. You may use the value of the angular dependence at the magnetic equator for the following parts.
Part E : Assuming a gyromagnetic relationship exists between magnetic moment µ and angular momentum L of an extended object, write down the relationship between the surface magnetic moment M_surfand angular momentum L as M_surf  = κ L. You will observe that κ depends only on fundamental constants and intrinsic properties of the extended object.
Part F : The surface magnetic moments for Mercury and Sun are 5 x 10¹² T m³ and 3 x 10²³ T m³respectively. Assuming the bodies are perfect spheres, evaluate the constant κ for Mercury and the Sun. Comment on values obtained and if they fit into the model developed in parts (3) and (4).
Part G : The surface magnetic moments M_surf and angular momenta L of various solar system bodies are plotted in the figure below. Justify that the data implies M_surf   ̴ L^α and calculate the constant α. What is the expected value of α from the model developed in parts (3) and (4)?
Part H : Certain bodies such as Venus, Mars and the Moon are remarkably separated from the trend observed for other bodies. What can you say about magnetism in these bodies when compared to the others?

Figure Surface magnetic moment vs angular momentum for solar system objects. Figure taken fromVallee, Fundamentals of Cosmic Physics, Vol. 19, pp 319-422, 1998.

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