1.    a) Imagine that a planet the same size as Jupiter was orbiting the nearest star to the Sun, Alpha Centauri A, which is about 300,000 AU away from us. What is the approximate angular diameter of this "Jupiter II" as seen from Earth, in seconds of arc?

b) What is the average angular diameter of the "real" Jupiter, as seen from Earth, an average of 5 AU away? 

HINT: Both of these are straightforward applications of the Angular Diameter formula. Remember that the units for actual diameter (that's diameter, NOT radius!) and distance must agree with each other, so you may have to do some converting.

c) Consider that even our most powerful telescopes can typically only see objects more that 1 second of arc across -- everything else looks like a dot, or can't be seen at all. Also consider that Alpha Centauri A is the closest star to the Sun, by a large margin. What then are the chances of us being able to directly observe other planets from Earth, if they exist? Justify your answer.

2.    a) A certain comet has a semi-major axis of 60,000 AU. What is the orbital period of this comet, in years?

HINT: Bear in mind that the comet is traveling around the Sun. What simple equation relates the average distance between an object and the Sun to the amount of time it takes that object to orbit the Sun once?

    b) The eccentricity of the comet's orbit is such that its aphelion distance is 100,000 AU. At this point, the comet lies directly between the Sun and the Sun's closest neighbor, the star Alpha Centauri A. Compare the force of gravity between the comet and the Sun to the force of gravity between the comet Alpha Centauri A, given that Alpha Centauri A is about 300,000 AU from the Sun, and is nearly identical to our Sun. Draw a diagram of the situation to help you.

HINT: This is a straightforward application of the gravity equation, as illustrated in the gravity handout. Also, check out the answers to Problem 3 on Homework 1. The procedure is the same, just the numbers are different!

ANOTHER HINT: Be careful of the values you use for distance! It REALLY helps to draw a diagram!

3.     a) Many believe that most of the liquid water on the surface of the Earth came from comets crashing into Earth. Given that there are currently 2 X 10²⁴ grams of water on the Earth, and that the average comet has a mass of 10¹⁶ grams, how many comets would have to hit Earth to fill up the oceans?

b) If this "comet rain" occurred during the first half a billion years of the Earth's history, how many comets hit the Earth in a given year?

HINT: Neither of the parts of Problem 3 involve a "formula" per se. They just involve a little common sense, and some division. If you get an answer less than 1 for part b) (And why would I be mentioning it if you're not supposed to?) then flip flop the number and give your answer in "years between comets," since, for example, "half a comet per year" makes no sense.

Updated 7/6/06

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