# Astronomy 300: Homework Assignment #3

Due: Monday February 15

1. Consider a planet in a circular orbit around a star with mass M. Use Newton's Second Law and Newton's Law of Gravity to derive the planet's velocity v as a function of its orbital distance r.

2. Mass of the Milky Way This relationship holds for any object in circular motion around a mass, including the Sun's orbital motion around the center of our galaxy. In your last homework assignment you calculated the velocity of Sun's orbital motion. Take the Sun's orbital vellocity to be 220 km/s, and assume its orbit to be circular. If the sun is 8 kiloparces from the center of the Galaxy, what is the mass of the Galaxy (interior to the Sun's orbit)?

3. Compute the gravitational force of:

A.)The Sun on the Earth

B.)The Earth on the Moon

C.)The Earth on you.

D.)A proton on the Earth's surface.

E.)A proton on another proton located 0.1 cm away.

Which, if any, of the these numbers would you consider to be so small as never have important physical effects?

Reading: C&O, Ch 3, Croswell, Ch 3&4

4. Parallax of Polaris: Given that the North Star's distance is 430 light years, compute the Parallax of Polaris. (Make sure to state the units you are using.) Explain how this number was a hindrance to Kepler and others trying to prove the Heliocentric Model in the 1500s (when the best positional measurements, those of Tycho Brahe, had errors of a few minutes of arc.)

5. C&O, Chapter 3, Problem 3.2

6. A.)Given the Sun's Luminosity, LSUN, the inverse square law, and Earth's radius RE, derive an expression for the total Power, P, intercepted by the Earth. What fraction of the Sun's radiant energy strikes the Earth?

Of the energy striking the Earth, a fraction a, called the albedo, is reflected back to space by clouds, water, etc. So, if P, above, is the total solar power received by Earth, then P * (1-a) is the energy absorbed by the Earth each second.

B.) Approximating Earth's albedo as: a = 0.3, present an expression for the total solar energy absorbed by Earth per second. Evaluate units of Watts.

7. This energy, once absorbed, is re-radiated from the Earth's surface, in all directions, approximately like a black body. (Much of this energy is in the infrared, but here we consider total energy at all wavelengths) Write an expression for the Luminosity of the Earth as a function of Earth's temperature and radius.

8. Assuming that the Earth is in equilibrium, equate the energy Earth absorbs from the Sun to the energy Earth emits into space and solve for the equilibrium temperature of the Earth. Express your answer in K, C and F.

9. How does your answer compare with Earth's average surface temperature of 290K? Attempt to expain any discrepancy.

10. Harlow Shapley is credited with being the first to correctly determine the structure of the Milky Way. What was the primary method he used to determine the distances to the nearest globular clusters, and how was it calibrated? How was he then able to extrapolate to these results to more distant clusters?