ASTR 100 (McGaugh)
Homework #4 solutions

Chapter 7, Problem # 5

Europa contains a large percentage of ice rather than rock. Our Moon is primarily made of rock. Thus, since ice is less dense than rock, Europa has a lower density than our Moon. Europa and our Moon are almost the same size. Since the mass of an object is equal to its volume (size) times its density, the mass of Europa is less than the mass of our Moon. Finally, the force of gravity acting on you on any moon's surface is given by:

From this equation, we see than since the mass of Europa is less than the mass of our Moon, and the radius of Europa is approximately equal to the radius of our Moon, the force of gravity on Europa would be less than the force of gravity on our Moon.

Chapter 7, Problem # 6

It would be more dangerous to fly through Saturn's rings than through Jupiter's ring because Jupiter's ring is made of microscopic dust particles, whereas Saturn's rings are made of larger icy particles which range from golf ball size up to the size of a house. Thus, a collision with a Saturnian ring particle would potentially be much more damaging than a collision with a Jovian ring particle. Also, Saturn's ring system is more substantial than Jupiter's (ie. it is a larger and denser system with a greater total mass) so there would be a greater chance of hitting something during a flight through Saturn's rings than through Jupiters ring.

Chapter 8, Problem # 2

Kepler's laws are derived from Newton's universal law of gravitation and this law applies to all objects. Thus, Kepler's third law, which describes the motion of an object in orbit around the Sun, applies to any object which orbits the Sun, whether it is called a planet or a comet.

Since Kepler's third law applies to comets, we can use it to determine the period of a typical comet in the Oort cloud, given that the comet's semimajor axis is 100,000 AU:

Kepler's third law: a3 = P2 (where a is given in AU and P is given in years)

rearranging the equation, we get: P = a3/2

plugging in a for the comet yields: P = (105)3/2 = 3.16 × 107 years

So a typical comet from the Oort cloud returns to the inner solar system about once every 3.16 × 107 years.

Chapter 8, Problem # 6

a) The warning time would be the time it took the asteroid to travel the 15 million kilometers to the Earth. We assume that the asteroid is moving at a speed of 15 km/sec.

velocity = distance/time

time = distance/velocity

time = 15,000,000 km / (15 km/sec)

time = 1,000,000 sec × (1 day / 86400 sec)

time = 11.6 days

b) A kilometer-sized asteroid impact is equivalent to the explosion of a bomb with a yield of hundreds to thousands of megatons (ie. several extremely large nuclear explosions). Thus the shock wave from the impact would cause devastation over a large area. Also, the impact would raise a large dust cloud which could block out the Sun for weeks or months.

c) If an asteroid (assumed to be traveling at approximately 15 km/sec as in part a) were discovered when it was 384,000 km from the earth, the warning time would be (as in part a):

time = distance/velocity

time = 384,000 km / (15 km/sec)

time = 25600 sec × (1 hr / 3600 sec)

time = about 7 hours

d) Assuming that the asteroid is not substantially slowed down by the earth's atmosphere, we can determine the time it would take the asteroid to pass through the 100 km thickness of the atmosphere using the same method as was used in parts a) and c):

time = distance/velocity

time = 100 km / (15 km/sec)

time = 6.7 seconds

Chapter 9, Problem # 3

The basic model for the formation of our solar system (contraction of a revolving gas cloud) leads necessarily to a flat disk structure with all the planets moving in the save direction about the Sun. Thus, if we discovered a system around another star in which the planets orbited in randomly inclined orbits, both prograde and retrograde, a different formation process must have occurred or additional processes beyond those which formed our galaxy must have come into play to create the planetary orbits observed in this new system. For example, the planets of that system may have formed very close to one another so that gravitational interactions between the planets were a significant factor in the formation of the system and caused the orbits of the planets to change from their original planar prograde motion.

Angular momentum would have to be conserved in the new planetary system, just as it is in our own Solar System, in accordance with the law of conservation of angular momentum.