Chapter 7, Problem 60PQ

You are a planetary scientist studying the atmosphere of Jupiter through a large telescope when you observe an asteroid approaching the planet. This asteroid is large, so you know it is held together by gravity rather than the cohesive forces that hold a large rock together. If the asteroid gets too close to Jupiter, the massive tidal forces will tear it apart, scattering small particles that will add to the ring system. You have calculated the closest distance the asteroid will come to Jupiter. How do you know if the asteroid will survive?

a. A measure of the cohesive gravitational force holding such an asteroid together is the gravitational field on the surface due to the mass of the asteroid. This field is independent of the distance of the asteroid from Jupiter. Calculate the gravitational field at the surface of the asteroid due only to the mass of the asteroid. Assume the asteroid has a diameter of 10,000 km and a density of 1300 kg/m³.

b. Tidal forces from Jupiter tend to disrupt the asteroid by pulling it apart. The tidal forces depend on the distance between Jupiter and the asteroid. There is a distance between Jupiter and the asteroid known as the Roche limit where the tidal forces are balanced by the asteroid’s own cohesive gravitational force. If the asteroid is within the Roche limit, it will be torn apart. Figure P7.60 shows Jupiter’s gravitational field as a function of distance from its center. By looking at this graph, can you determine an approximate value for the Roche limit for this asteroid in the vicinity of this planet?

c. What will happen to the Roche limit if we consider an asteroid of lower density?

FIGURE P7.60