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?
Answers
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 ργ
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 Rα.
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?
Answers
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
Answers
Question 4
Part A : Find the shortest distance from Boston (42.3601
0
N; 71.0589
0 W) to Beijing (39.9042
0 N; 116.4074
0
E) 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 66.5608
0 N)?
Answers
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
Bsurf and the surface magnetic moment
defined as
Msurf =
Bsurf 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
Msurf
and angular momentum
L as
Msurf =
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 10
12 T
m3 and 3 x 10
23 T
m3
respectively. 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 Msurf and
angular momenta L of various solar system bodies are plotted in the
figure below. Justify that the data implies Msurf ̴ Lα 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 from
Vallee,
Fundamentals of Cosmic Physics, Vol. 19, pp 319-422, 1998.
Answers
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