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Kepler's Laws of Planetary Motion

1. The orbit of every planet is an ellipse with the sun at a focus
2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time
3. The square of the orbital period of a planet is proportional to the cube of the semi-major 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:

z + 1 = \frac{\lambda_0}{\lambda_e} = \sqrt{\frac{1 + \frac{v}{c}}{ 1 - \frac{v}{c} }}

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:

v = H_0 D
H_0 = 70.1 \pm 1.3 \mathrm {(km/s)/Mpc}

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 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:

1. 1/2 as distanct as they are today
2. 4 times as distant as they are today
3. 2 times as distant as they are today
4. 1/4 as distanct as they are today
5. separated by about the same distances as they are today

D

Problem 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?

1. The orbits would remain unchanged.
2. The planets would oscillate about their former elliptical orbits.
3. The planets would move in spiral orbits.
4. The orbits would precess much more rapidly.
5. The planets would move directly toward the Sun

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 ellipse

Problem 6

The primary source of the Sun's energy is a series of thermonuclear reactions in which the energy produced is c2 times the mass difference between

1. 2 hydrogen atoms plus 2 helium atoms and 1 carbon atom
2. 3 helium atoms and 1 carbon atom
3. 4 hydrogen atoms and 1 helium atom
4. 2 hydrogen atoms and 1 helium atom
5. 6 hydrogen atoms and 2 helium atoms

C

Problem 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 ellipse

Problem 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

\frac{F(R)}{F(R/2)}

?

Solution 8

It's an intuitive question. The answer is an ellipse

Problem 9

Which of the following is most nearly the mass of the Earth? (the radius of the Earth is about 6.4E6 m.)

Problem 10

Suppose that the graivtational force law between 2 massive objects were

F_{12} = r\hat frac{G m_1 m_2}{r_{12}^{2 + \epsilon}}

where ε is a small positive number. Which of the following statements would be FALSE?

1. A single planet could move in a stational circular orbit about the Sun
2. The angular momentum of a single planet moving about the Sun would be conserved
3. The periods of planets in circular orbits would be propotional to the (3 + ε)/2 power of their respective orbital radii
4. A single planet could move in a stationary noncircular elliptical orbit around the Sun
5. The total mechanical energy of the planet-Sun system would be conserved

Solution 10

Stable non-circular orbits can only occur for the simple harmonic potential and the inverse-square law force. The answer is D

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:

r_s = \frac{2 G m}{c^2}
= \frac{2 (6.67 \times 10^{-11}) (5.98 \times 10^24)}{(3.00 \times 10^8)^2}
= \frac{2 (6.67 \times 10^{-11}) (5.98 \times 10^24)}{(3.00 \times 10^8)^2}

Kepler's Laws

1. The orbit of every planet is an ellipse with the sun at a focus
2. A line adjoining a planet and the sun sweeps out equal areas during equal intervals of time
3. The square of the orbital period of a planet (T2) is directly proportional to the cube of the semi-major axis of its orbit (a3)

Circular Orbits

1. Gravitational pull is the centripetal force on the satellite
F = G \frac{mM}{r^2} = \frac{mv^2}{r}
because centripetal acceleration is
a = \frac{v^2}{r}
2. The speed of a satellite in circle orbit about a body of mass M
v = \sqrt{\frac{GM}{r}}
3. The time period (T) of the satellite is proportional to r3/2 (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?
1. Mass of the moon
2. Mass of the planet
3. Minimum speed of the moon
4. Period of the orbit
5. Semimajor axis of orbit

Solution 12

The mass of the moon

Problem 13

The period of a hypothetical earth satellite orbiting at sea level would be 80 minutes. In terms of the earth's radius Re, what is the radius of a synchronus satellit orbit (period 24 hours)?
1. 3 Re
2. 7 Re
3. 18 Re
4. 320 Re
5. 5800 Re

Kepler's 3rd Law