CHAPTER 5 | GRAVITY 83
that follows the inverse square law, then the planet must follow
an elliptical path.
Even though Kepler correctly identifi ed the shape of the
planets’ orbits, he still wondered why the planets keep moving
along these orbits, and now you know the answer. Th ey move
because there is nothing to slow them down. Newton’s fi rst law
says that a body in motion stays in motion unless acted on by
some force. In the absence of friction, the planets must continue
to move.
Kepler’s second law says that a planet moves faster when it is
near the sun and slower when it is farther away. Once again,
Newton’s discoveries explain why. Imagine you are in an elliptical
orbit around the sun. After you round the most distant part of
the ellipse, aphelion, you begin to move back closer to the sun,
and the sun’s gravity pulls you slightly forward in your orbit. You
pick up speed as you fall closer to the sun, so, of course, you go
faster as you approach the sun. After you round the closest point
to the sun, perihelion, you begin to move away from the sun, and
the sun’s gravity pulls slightly backward on you, slowing you
down as you climb away from the sun. If you were in a circular
orbit, the sun’s gravity would always pull perpendicular to your
motion, and you would not speed up or slow down. So Kepler’s
second law makes sense when you analyze it in terms of forces
and motions.
Physicists think of Kepler’s second law in a more elegant
way. Earlier you saw that a body moving on a frictionless surface
will continue to move in a straight line until it is acted on by
some force; that is, the object has momentum. In a similar way,
an object set rotating on a frictionless surface will continue rotat-
ing until something acts to speed it up or slow it down. Such an
object has angular momentum, a measure of the rotation of the
body about some point. A planet circling the sun in an orbit has
a given amount of angular momentum; and, with no outside
infl uences to alter its motion, it must conserve its angular
momentum. Th at is, its angular momentum must remain con-
stant. Mathematically, a planet’s angular momentum is the prod-
uct of its mass, velocity, and distance from the sun. Th is explains
why a planet must speed up as it comes closer to the sun along
an elliptical orbit. Because its angular momentum is conserved,
as its distance from the sun decreases, its velocity must increase.
Conversely, the planet’s velocity must decrease as its distance
from the sun increases.
Th e conservation of angular momentum is actually a com-
mon human experience. Skaters spinning slowly can draw their
arms and legs closer to their axis of rotation and, through conser-
vation of angular momentum, spin faster (■ Figure 5-7). To slow
their rotation, they can extend their arms again. Similarly, divers
can spin rapidly in the tuck position and then slow their rotation
by stretching into the extended position.
Kepler’s third law says that a planet’s orbital period depends
on its distance from the sun. Th at law is also explained by a con-
servation law, but in this case it is the law of conservation ofCalculating Escape Velocity
If you launch a rocket upward, it will consume its fuel in a few
moments and reach its maximum speed. From that point on, it
will coast upward. How fast must a rocket travel to coast away
from Earth and escape? Of course, no matter how far it travels,
it can never escape from Earth’s gravity. Th e eff ects of Earth’s
gravity extend to infi nity. It is possible, however, for a rocket to
travel so fast initially that gravity can never slow it to a stop. Th en
the rocket could leave Earth.
Escape velocity is the velocity required to escape from the
surface of an astronomical body. Here you are interested in
escaping from Earth or a planet; in later chapters you will con-
sider the escape velocity from stars, galaxies, and even black
holes.
Th e escape velocity, Ve, is given by a simple formula:
Ve (^) √
2 ___GMr
Again, G is the gravitational constant 6.67 10 ^11 m^3 /s^2 kg, M
is the mass of the astronomical body in kilograms, and r is its
radius in meters. (Notice that this formula is very similar to the
formula for circular velocity; in fact the escape velocity formula
is √
2 times the circular velocity formula.)
You can fi nd the escape velocity from Earth by again using
its mass, 5.97 1024 kg, and plugging in its radius, 6.38 106
m. Th en the escape velocity is:
Vc
√
2 6.67 10
(^11) 5.97 1024
6.38 106
√
7.96 ^10
14
6.38 106
(^) √
1.25 108 11,200 m/s 11.2 km/s
Th is is equal to about 24,600 miles per hour.
Notice from the formula that the escape velocity from a
body depends on both its mass and radius. A massive body might
have a low escape velocity if it has a very large radius. You will
meet such objects when you consider giant stars. On the other
hand, a rather low-mass body could have a very large escape
velocity if it had a very small radius, a condition you will meet
when you study black holes.
Once Newton understood gravity and motion, he could do
what Kepler had failed to do—he could explain why the planets
obey Kepler’s laws of planetary motion.
Kepler’s Laws Reexamined
Now that you understand Newton’s laws, gravity, and orbital
motion, you can look at Kepler’s laws of planetary motion in a
new and more sophisticated way.
Kepler’s fi rst law says that the orbits of the planets are ellipses
with the sun at one focus. Th e orbits of the planets are ellipses
because gravity follows the inverse square law. In one of his most
famous mathematical proofs, Newton showed that if a planet
moves in a closed orbit under the infl uence of an attractive force