Download Astronomy 111 Review Problems Solutions

Astronomy 111 Review Problems
Solutions
Problem 1:
Venus has an equatorial radius of 6052 km. Its semi-major axis is 0.72 AU. The Sun has
a radius of
cm.
a) During a Venus transit (such as occurred June 8 2004 and June 5 2012), what is
the angular diameter of Venus in arcseconds?
Note the distance between Earth and Venus is used here d=(1 – 0.72)AU.
b) What minimum diameter telescope would you need to be able resolve the Venus
transit at visible wavelengths?
The diffraction limit gives diameter D = λ/θ ~ 500nm/(60” x 5e-6 radians/”) ~
0.2cm. Your eye can resolve it (but it’s a bad idea to look at the Sun).
c) During a Venus transit, what fraction of the Sun’s light is blocked by Venus?
The angular diameter of the Sun is.
However we need to compare solid angles (areas) to estimate the fraction of the Sun
blocked. The angular area is
. So the fraction of light blocked is
. We now insert our value calculated in part a) and the value for the Sun
calculated above using radians in both cases.
d) Consider a transit on a distant star similar to the Sun. The transit is caused by a
Venus sized extra solar planet. The viewer is very distant from the star. What
change in the observed magnitude of the star does this transit cause? Does the
magnitude of the star increase or decrease during the transit?
Note that we need to consider a distant viewer. In this case, the angular diameters
will not depend on the distance of the planet from the star, only on the areas of the
star and planet. The fraction of light blocked is then
. Magnitudes are defined as
. So the change in magnitude would be
. The star becomes dimmer so the magnitude of the
star increases during the transit.
Problem 2:
At a pressure of 1 atmosphere, water freezes at 273K and boils at 373K. We say that a
planet is in the habitable zone if its equilibrium temperature is within the range allowing
liquid water. Assume that the planet has an albedo similar to that of Earth. The
equilibrium temperature of the Earth (at 1AU from a solar type star and with Earth’s
albedo) is 263 K.
a) How does the equilibrium temperature of a planet depend upon the planet’s
distance from the star R and the luminosity of the star, L? Write your answer in
the following form:
and find the exponents
and the constant X.
The amount of light emitted is equal to that absorbed so
that
we rewrite this as
. We find
.
b) What are the inner and outer radii (in AU from the star) of the habitable zone
near a star that is 100 as luminous as the Sun (such as a red giant)? The Sun will
become a red giant in a few billion years. Speculate on which moons and
satellites in our solar system might become nice places to live during this time.
We need to solve two equations
equation for freezing. We find
and a similar
And
. When the Sun becomes a red giant the
moons of Saturn and Uranus might become nice places to live.
c) Consider the possibility that the planet is not in a circular orbit. What is the
maximum eccentricity for the planet’s orbit that would allow the planet to remain
in the habitable zone during its entire orbit.
The boiling temperature of water is 1.37 times the freezing temperature. This
means the outer boundary (in radius) of the habitable zone is 1.87 that of the
inner boundary (1.87 is the square of 1.37) and as we found in the previous
problems
. A planet in an elliptical orbit has radius of periapse
and apoapse of
. We need to solve the following
.
Solving for the eccentricity we find
. This
would be the maximum eccentricity allowing the planet to remain in the
habitable zone during its entire orbit.
Problem 3:
A dust particle is in a nearly circular orbit about a star of mass M, and has a ratio of
radiation pressure to gravitational force of β. The particle experiences forces due to
gravity from the central star and radiation pressure from the central star. Ignore
drag forces such as Poynting Robertson drag.
a) As a function of distance r from the star, what is the period, P, of the
particle’s orbit about the star?
b) The particle is trapped in the 2:1 mean motion resonance outside a planet
with semi-major axis ap. Because of radiation pressure, the dust particle does
not have the same semi-major axis that a larger object in this resonance
would have. What is the semi-major axis of the dust particle in terms of the
planet’s semi-major axis and β?
Problem 4:
Pluto has a mass of 132×1023g and Charon has a mass of 15 ×1023g. Charon is in
orbit about Pluto with a semi-major axis of 19.6×103 km. Pluto and Charon are in an
orbit around the Sun with a semi-major axis of 39.5AU and eccentricity of e=0.25.
a) At what distance (in km) from Pluto is the Center of Mass of the Pluto/Charon
system?
b) How far away (in AU) from the Earth would Pluto and Charon be at
perihelion? Assume that they are at opposition.
c) Pluto is observed to have visual magnitude of mv=7.5. Charon is 1.9 magnitudes
fainter than Pluto. What visual magnitude are the two bodies observed together?
Problem 5:
The scale height of a planetary atmosphere is approximately given by
where
k is Bolzmann’s constant, T is the temperature, g is the surface gravitational acceleration
and m is the mean molecular mass.
a) How does the scale height of a planetary atmosphere depend on (scale with) the
mass and radius of the planet?
The only variable that depends on the mass and radius of the planet is the gravitational
acceleration,
. The scale height is inversely proportional to g, so
b) Assume that the temperature of the atmosphere is set by an equilibrium between
the radiation absorbed from the Sun and that emitted by the planet. How does
the scale height depend on the distance of the planet from the Sun?
The only variable in
equilibrium temperature T.
that depends on distance to the Sun, d, is the
.
From this we see that
. Since
we find that
.
c) By what factor would the scale height of Pluto’s atmosphere change between
perihelion and aphelion?
The atmosphere of Pluto should shrink as it gets further from the Sun. The drop in
temperature also means that some of the gases that contribute to the atmosphere at
perihelion condense onto the surface as ices.
d) Consider a planet with mass M which has a satellite with mass Ms. The satellite
has radius Rs and is on an eccentric orbit with semi-major axis a and eccentricity
e. How much larger is the tidal force on the satellite at periapse compared to that
at apoapse?
The tidal force at the surface of the satellite is
between the satellite and the planet. At apoapse
The tidal force at periapse is
where d is the distance
and at periapse
times larger than that at apoapse.
Problem 8:
The focal length of the 24” Mees Telescope is 324 inches or 823cm. The ST9 camera
pixels are 20x20microns (µm) large. How many arcseconds on the sky is one pixel
large?
8.23m/20e-6m=2.4e-6 which is about 0.5“
.