Download Coursework 6 File

Our Universe
Coursework 6
Due in on Wednesday of week 8 at 16:00
Exercise class question - not to be handed in
The purpose of this question is to illustrate the fact that the tidal acceleration exerted by a central
star acting to tidally deform an orbiting planet can also act to limit the size of stable orbits for a
satellite gravitationally bound to the planet.
1. A planet of mass Mp forms around a star with mass M∗ with separation r, and an orbiting
satellite forms around the planet with semimajor axis asat . By considering the tidal acceleration exerted by the central star on the planet–satellite system (that acts to pull them apart),
and the gravitational acceleration exerted by the planet on the satellite, derive an expression
for the critical value of the satellite semimajor axis around the planet, acrit , above which the
satellite becomes unbound from the planet.
2. A planet of mass Mp = 1.9 × 1027 kg orbits at distance r = 7.8 × 1011 m from its central star
that has mass M∗ = 2 × 1030 kg. It has a low-mass satellite orbiting stably with semimajor
axis asat = 6 × 109 m. The planet undergoes slow inward migration toward the star, and
during this process the satellite maintains the same semimajor axis. Determine how close
the planet can get to the star before the satellite becomes unbound. Express your answers in
metres and in AU (1 AU = 1.5 × 1011 m).
Please note that all questions below should be handed in for assessment
* Homework question 1
The luminosity of the Sun is L = 3.8 × 1026 W, but when it evolves to the red-giant phase it will
reach a value approximately 1000 times larger than the present-day value. Mars orbits the Sun at
a distance amars ≈ 1.5 AU (where 1 AU = 1.5 × 1011 m). The radius of Mars is Rmars = 3400 km.
The Stefan-Boltzmann constant σ = 5.7 × 10−8 W m−2 K−4 , and the mass of the hydrogen atom
mH = 1.67 × 10−27 kg.
(Hint: You may find coursework 3 useful in answering this question.)
1. Calculate the radiative flux at the position of Mars due to the Sun when it is a red giant.
2. Calculate the total amount of radiative energy that is intercepted by Mars every second from
the red-giant Sun. You need to consider the cross-sectional area of Mars in order to answer
this question.
3. Using the simplifying assumption that Mars is a sphere of constant surface temperature T
that emits radiation as a perfect black-body, obtain an expression for the total radiant energy
emitted by Mars per second.
4. By assuming that Mars is in thermal equilibrium, with the rate of emission obtained in part (3)
being balanced by the rate of energy incident obtained in part (2), estimate the temperature
T of Mars due to solar irradiation during the red giant phase.
5. By considering the mean velocity of atoms and molecules in a gas at temperature T , and
how this relates to the escape velocity from a planet of mass Mp and radius Rp , obtain an
expression for the mass of an atom or molecule in the planet atmosphere that is just able to
remain bound to a planet.
6. Explain why the approximate expression that you have derived in part (5) is unsufficient to
make an accurate determination of whether or not a substantial abundance of a particular
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atomic or molecular species can remain bound to a planet over long time periods. What
simple refinement should be made to the expression you derived in part (5) ?
7. What is the fate of carbox dioxide (CO2 ) in the atmosphere of Mars during the red giant phase.
The mass of Mars is Mmars = 6.4 × 1023 kg and Boltzmann’s constant is k = 1.38 × 10−23 m2
kg s−2 K−1 .
Homework question 2
Explain briefly how Venus evolved from being a planet with a significant inventory of water in its
atmosphere and on its surface in the past, to become the hot dry world that we observe today.
Homework question 3
Explain briefly how Mars evolved from being a warmer planet with a denser atmosphere in the
past, to the cooler planet hosting a thin atmosphere that we see today.
Homework question 4
1. Derive the following relation between the mass of an orbiting planet, Mp , and the magnitude
of the observed radial velocity of the host star, vobs , as it orbits the common centre of mass
of the star+planet system:
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M∗ P
Mp sin (i) = vobs
,
2πG
where M∗ is the mass of the star, P is the orbital period, G is the gravitational constant, and
i is the angle of inclination between the line of sight and the normal to the planet-star orbital
plane.
2. During monitoring of the spectrum of a sun-like star it is noticed that the spectral absorption
lines shift periodically in wavelength with a period equal to 4.23 days. More specifically,
a spectral line located at wavelength λ = 650 nm shifts periodically by an amount equal to
∆λ = ±2.17×10−4 nm. The sun-like star has a mass M∗ = 2×1030 kg and a radius 7×108 m.
Assuming that the periodic shifting of spectral lines is due to an unseen orbiting companion,
calculate its mass based on the line-of-sight component of its orbital velocity (express your
answer in kilograms and in Jupiter masses (1 MJup = 1.9 × 1027 kg).
3. Photometric monitoring of the star in part (2) indicates that its observed luminosity decreases
by 1% for a few hours before returning to its original value every 4.23 days. What further
physical properties of the companion may be deduced by combining this observation with
those described in part (2) ?
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