203
Problem 12.
12 S. The figure shows an image from the Galileo space probe taken
during its August 1993 flyby of the asteroid Ida. Astronomers were
surprised when Galileo detected a smaller object orbiting Ida. This smaller
object, the only known satellite of an asteroid in our solar system, was
christened Dactyl, after the mythical creatures who lived on Mount Ida,
and who protected the infant Zeus. For scale, Ida is about the size and
shape of Orange County, and Dactyl the size of a college campus. Galileo
was unfortunately unable to measure the time, T, required for Dactyl to
orbit Ida. If it had, astronomers would have been able to make the first
accurate determination of the mass and density of an asteroid. Find an
equation for the density,
.
, of Ida in terms of Ida’s known volume, V, the
known radius, r, of Dactyl’s orbit, and the lamentably unknown variable T.
(This is the same technique that was used successfully for determining the
masses and densities of the planets that have moons.)
13
.
. If a bullet is shot straight up at a high enough velocity, it will never
return to the earth. This is known as the escape velocity. We will discuss
escape velocity using the concept of energy in the next book of the series,
but it can also be gotten at using straightforward calculus. In this problem,
you will analyze the motion of an object of mass m whose initial velocity is
exactly equal to escape velocity. We assume that it is starting from the
surface of a spherically symmetric planet of mass M and radius b. The trick
is to guess at the general form of the solution, and then determine the
solution in more detail. Assume (as is true) that the solution is of the form
r = kt
p
, where r is the object’s distance from the center of the planet at
time t, and k and p are constants. (a) Find the acceleration, and use
Newton’s second law and Newton’s law of gravity to determine k and p.
You should find that the result is independent of m. (b) What happens to
the velocity as t approaches infinity. (c) Determine escape velocity from
the Earth’s surface.
14. Astronomers have recently observed stars orbiting at very high speeds
around an unknown object near the center of our galaxy. For stars orbiting
at distances of about 10
14
m from the object, the orbital velocities are
about 10
6
m/s. Assuming the orbits are circular, estimate the mass of the
object, in units of the mass of the sun, 2x10
30
kg. If the object was a
tightly packed cluster of normal stars, it should be a very bright source of
light. Since no visible light is detected coming from it, it is instead be-
lieved to be a supermassive black hole.
15 S. Astronomers have detected a solar system consisting of three planets
orbiting the star Upsilon Andromedae. The planets have been named b, c,
and d. Planet b’s average distance from the star is 0.059 A.U., and planet
c’s average distance is 0.83 A.U., where an astronomical unit or A.U. is
defined as the distance from the Earth to the sun. For technical reasons, it
is possible to determine the ratios of the planets’ masses, but their masses
cannot presently be determined in absolute units. Planet c’s mass is 3.0
times that of planet b. Compare the star’s average gravitational force on
planet c with its average force on planet b. [Based on a problem by Arnold
Arons.]
Homework Problems