Astronomy
Kepler's Laws of Planetary Motion
 The orbit of every planet is an ellipse with the sun at a focus
 A line joining a planet and the sun sweeps out equal areas during equal intervals of time
 The square of the orbital period of a planet is proportional to the cube of the semimajor axis of its orbit.
Hubble's Law
The redshift in the light from a distant galaxy is proportional to its distance.
The light coming to us from distance galaxies has its wavelength increased due to the Doppler Effect. This phenomenon is called redshift. Virtually all the galaxies (except a few very close to us) show redshift. You can calculate the speed of a galaxy from the redshift z in this equation:
Hubble analyzed this observation and concluded that those galaxies are moving away from us and also moving away from each other, i.e., the universe is expanding. He derived the following relationship between the velocity of the galaxy and its distance from us:
This is called Hubble's Constant
Problem 1
According to Hubble's Law, what is the relationship between the distance D and speed v of a galaxy?
Solution 1
According to Hubble's Law, z is directly proportional to D, and v is directly proportional to z. Hence v is directly poroportional to D
Problem 2
Solution 2
AnswerProblem 3
At the present time, the temperature of the universe (i.e. the microwave radiation background) is about 3K. When the temperature was 12K, typical objects in the universe, such as galaxies, were:
 4 times as distant as they are today
 2 times as distant as they are today
 1/4 as distanct as they are today
 1/2 as distanct as they are today
 separated by about the same distances as they are today
Solution 3
CProblem 4
If the Sun were suddenly replaced by a black hole of the same mass, it would have a Schwarzschild radius of 3,000 m. What effect, if any, would this change have on the orbits of the planets?
 The planets would oscillate about their former elliptical orbits.
 The planets would move in spiral orbits.
 The orbits would remain unchanged.
 The orbits would precess much more rapidly.
 The planets would move directly toward the Sun
Solution 4
Answer
Problem 5
A satellite of mass m orbits a planet of mass M in a circular orbit of radius R. What is the time required for one revolution?
Solution 5
It's an intuitive question. The answer is an ellipseProblem 6
The primary source of the Sun's energy is a series of thermonuclear reactions in which the energy produced is c^{2} times the mass difference between
 2 hydrogen atoms and 1 helium atom
 4 hydrogen atoms and 1 helium atom
 2 hydrogen atoms plus 2 helium atoms and 1 carbon atom
 3 helium atoms and 1 carbon atom
 6 hydrogen atoms and 2 helium atoms
Solution 6
BProblem 7
A satellite orbits the Earth in a circular orbit. An astronaut on board perturbs the orbit slightly by briefly firing a control jet aimed toward the Earth's center. Afterward, what shape is the satellite's path? Is it an ellipse, a hyperbola, a bigger circle, a spiral, or does it show many radial oscillations per revolution?
Solution 7
It's an intuitive question. The answer is an ellipseProblem 8
The magnitude of the Earth's gravitational force on a point mass is F(r), where r is the distance from the Earth's center to the point mass. Assume the Earth is a homogenous sphere of radius R.
Suppose there is a very small shaft in the Earth such that the point mass can be placed at a radius of R/2. What is
?
Solution 8
It's an intuitive question. The answer is an ellipseProblem 9
Which of the following is most nearly the mass of the Earth? (the radius of the Earth is about 6.4E6 m.)
Solution 9
Answer
Problem 10
Suppose that the graivtational force law between 2 massive objects were
where ε is a small positive number. Which of the following statements would be FALSE?
 A single planet could move in a stational circular orbit about the Sun
 The total mechanical energy of the planetSun system would be conserved
 A single planet could move in a stationary noncircular elliptical orbit around the Sun
 The angular momentum of a single planet moving about the Sun would be conserved
 The periods of planets in circular orbits would be propotional to the (3 + ε)/2 power of their respective orbital radii
Solution 10
Stable noncircular orbits can only occur for the simple harmonic potential and the inversesquare law force. The answer is C
Problem 11
A black hole is an object whose gravitational field is so strong that even light cannot escape. To what approximate radius would Earth (mass 5.98E24 kg) have to be compressed in order to become a black hole?
Solution 11
The Schwarzschild radius (aka gravitational radius) is the radius of a sphere such that, if all the mass of an object is compressed within that sphere, the escape speed from the surface of the sphere would equal the speed of light. A black hole is an object that is smaller than its Schwarzchild radius. The radius is:
Kepler's Laws
 The orbit of every planet is an ellipse with the sun at a focus
 A line adjoining a planet and the sun sweeps out equal areas during equal intervals of time
 The square of the orbital period of a planet (T^{2}) is directly proportional to the cube of the semimajor axis of its orbit (a^{3})
Circular Orbits

Gravitational pull is the centripetal force on the satellite
F = G \frac{mM}{r^2} = \frac{mv^2}{r}because centripetal acceleration isa = \frac{v^2}{r}

The speed of a satellite in circle orbit about a body of mass M
v = \sqrt{\frac{GM}{r}}

The time period (T) of the satellite is proportional to r^{3/2} (Kepler's Third Law)
T^2 = \frac{4 \pi^2}{GM} r^3
Problem 12
An astronomy observes a very small moon orbiting a planet and measure the moon's minimum and maximum distances from the planet's center and the moon's maximum orbital speed. Which of the following CANNOT be calculated from these measurements? Mass of the moon
 Mass of the planet
 Minimum speed of the moon
 Period of the orbit
 Semimajor axis of orbit
Solution 12
The mass of the moonProblem 13
The period of a hypothetical earth satellite orbiting at sea level would be 80 minutes. In terms of the earth's radius R_{e}, what is the radius of a synchronus satellit orbit (period 24 hours)? 3 R_{e}
 7 R_{e}
 18 R_{e}
 320 R_{e}
 5800 R_{e}