### Figuring for Yourself

Refer to Example 13.1. How would the calculation change if a typical comet in the Oort cloud is only 1 km in diameter?

Refer to Example 13.1. How would the calculation change if a typical comet in the Oort cloud is larger—say, 50 km in diameter?

The calculation in Example 13.1 refers to the known Oort cloud, the source for most of the comets we see. If, as some astronomers suspect, there are 10 times this many cometary objects in the solar system, how does the total mass of cometary matter compare with the mass of Jupiter?

If the Oort cloud contains 10^{12} comets, and ten new comets are discovered coming close to the Sun each year, what percentage of the comets have been “used up” since the beginning of the solar system?

The mass of the asteroids is found mostly in the larger asteroids, so to estimate the total mass we need to consider only the larger objects. Suppose the three largest asteroids—Ceres (1000 km in diameter), Pallas (500 km in diameter), and Vesta (500 km in diameter)—account for half the total mass. Assume that each of these three asteroids has a density of 3 × 10^{3} g/cm^{3} and calculate their total mass. Multiply your result by 2 to obtain an estimate for the mass of the total asteroid belt. How does this compare with the mass of the Oort cloud?

Make a similar estimate for the mass of the Kuiper belt. The three largest objects are Pluto, Eris, and Makemake (each roughly 2000 km). In addition, assume there are eight objects (including Haumea, Orcus, Quaoar, Ixion, Varuna, and Charon, and objects that have not been named yet) with diameters of about 1000 km. Assume that all objects have Pluto’s density of 2 × 10^{3} g/cm^{3}. Calculate twice the mass of the largest 13 objects and compare it to the mass of the main asteroid belt.

What is the period of revolution about the Sun for an asteroid with a semi-major axis of 3 AU in the middle of the asteroid belt?

What is the period of revolution for a comet with aphelion at 5 AU and perihelion at the orbit of Earth?