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 . We find

that we rewrite this as .

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 and a similar

equation for freezing. We find

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 that depends on distance to the Sun, d, is the equilibrium temperature T.

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 where d is the distance

between the satellite and the planet. At apoapse and at periapse .

The tidal force at periapse is 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“