Physics 9 The Solar System with Prof. D. Tytler Oct. 2000

Homework 4, due Nov 2 in class

We encourage you to work together, to discuss the topics and solutions, but your answers must use all of your own words, calculations and diagrams.

You must not hand in work copied from someone else.

Please write down the code numbers of all in your study group.

These problems all require thought.

  1. Sketch the orbit of an Oort cloud comet coming into the inner solar system for the first time. The comet may or may not return to the Oort cloud (ignore the caption below the figure).

  2. Give two reasons why this comet might not return to the Oort cloud.

    3. Big planets like Jupiter often shift the orbit of comets coming through the inner solar system such that the comets could end up on an orbit confined to the inner solar system. Other comets do survive the solar heating heat. They might hit a planet or the sun, and they could be broken apart by tidal forces.

  3. Imagine that an Oort cloud comet was first detected when it was crossing Jupiter’s orbit. Assuming that its speed is the same as Jupiter’s, how long would it take to reach the Earth’s orbit, if it were travelling directly towards the sun? Show the steps towards the answer.

    4. Basically we need to find out how long it would take Jupiter to go 4AU (the distance between Earth’s orbit and Jupiter’s). We have to use the relationship between distance, velocity and time:

    v = d / t

    First calculate Jupiter’s velocity, v = circumference of orbit / orbital period. The calculation is easiest if you work with AU and years: v = 2x3.14x5.2 AU/11.9 years = 2.7 AU/year. Then find out how long it would take to go 4AU: t = 4AU / v (If you do it you find t = 1.5yrs)

  4. Will the comet’s speed vary along its orbit? If yes, give the reason(s).

    5. The comet will accelerate, moving faster as it approaches the sun, then slowing down again, as it moves away form the Sun. This is Kepler’s Second Law. There is a second much smaller non-gravitational effect. The comet will emit gas as it nears the sun, and this gas acts like a rocket engine, changing the comets orbit a bit.

  5. Estimate the probability that this comet will hit the Earth. Get an answer accurate to the nearest order of magnitude (e.g. 10-1, 10-2, 10-3 etc.). Ignore the motion of the Earth and the Earth’s gravity. The gravity will bend the comet’s orbit a bit, such that some comets which would have just missed the Earth’s surface, by < 1 Earth radius, will now impact.

    6. Probability = (number of orbits which the Earth is hit) /(number of all possible orbits)

    It is hard to count orbits, but we can instead use area. Consider throwing a dart at a dartboard. Assume you can hit the board, but only just, and that you do not have the skill to hit a given number. The dart then hits somewhere on the board, by chance. The probability of hitting a given part of the dartboard is then the area of that part, divided by the area of the board. For the comet, let us take the area of the board to be the area of the Earth’s orbit. The area of the part of the board is the area of the Earth. Here the areas are cross-section areas, given by pi r2, where r is the radius. They are not the surface areas. Note that the comet can approach the inner solar system at any angle: it is not confined to the plane of the ecliptic.

    P = (pi r2) / (pi R2), where r is the radius of the Earth and R is the distance from the Earth to the Sun. The result is about 10-9, or one chance in a billion. One major uncertainty is that we do not know that the comet’s orbit will pass within 1 AU of the sun. In practice, we would replace the R in the above by the uncertainty in our best estimate of where the comet was heading. As the comet moves along its orbit, we can better predict where it will go, and R drops. In nearly every case, there comes a time when we learn that the comet will probably miss the Earth.

    Note that this is the most general way of answering the question. If you answered it assuming the comet was in the ecliptic, the probability is just the diameter of the Earth divided by the circumference of its orbit, or (2 r)/(2 pi R), which is about 10-5. Thus you can see how fortunate it is that the Oort cloud comets aren’t all in the plane of the ecliptic.

  6. Discuss whether asteroids have atmospheres.

    7. The gravitational field of an asteroid is not large enough to hold an atmosphere.

  7. The temperature of an asteroid depends on its distance from the sun and its reflectivity. Why are brighter asteroids colder?

    8. An asteroid with a higher reflectivity will appear brighter. But this also means that less light will heat the surface of the asteroid, resulting in a lower temperature than an asteroid with lower reflectivity at the same distance from the sun.

  8. Astronomers estimate that 40% of asteroids which cross the Earth’s orbit are extinct comets. Why are they extinct, where did they come from, and what are their orbits now?

These asteroids were Oort Cloud comets, long ago. Some Oort comets move onto orbits which take them near to the sun many times, such as Halley’s comet. They become extinct comets when they have exhausted their volatiles.