82 PART 1^ |^ EXPLORING THE SKY
right distance from Earth it could be a very useful geosyn-
chronous satellite.
Notice that objects orbiting each other actually revolve
around their center of mass.
Finally, notice the diff erence between closed orbits and open
orbits. If you want to leave Earth never to return, you must
accelerate your spaceship at least to escape velocity, Ve, so it
will follow an open orbit.Orbital Velocity
If you were about to ride a rocket into orbit, you would have to
answer a critical question: “How fast must I go to stay in orbit?”
An object’s circular velocity is the lateral velocity it must have to
remain in a circular orbit. If you assume that the mass of your
spaceship is small compared with the mass of Earth, then the
circular velocity is:Vc (^) √
GM____r
In this formula, M is the mass of the central body (Earth in this
case) in kilograms, r is the radius of the orbit in meters, and G is
the gravitational constant, 6.67 10 ^11 m^3 /s^2 kg. Th is formula is
all you need to calculate how fast an object must travel to stay in
a circular orbit.
For example, how fast does the moon travel in its orbit?
Earth’s mass is 5.97 1024 kg, and the radius of the moon’s orbit
is 3.84 108 m. Th en the moon’s velocity is:
Vc
√
6.67 ^10
(^11) 5.98 1024
3.84 108
√
39.9 ^10
13
3.84 108
√
1.04 106 1020 m/s 1.02 km/s
Th is calculation shows that the moon travels 1.02 km along its
orbit each second. Th at is the circular velocity at the distance of
the moon.
A satellite just above Earth’s atmosphere is only about 200
km above Earth’s surface, or 6578 km from Earth’s center, so
Earth’s gravity is much stronger, and the satellite must travel
much faster to stay in a circular orbit. You can use the formula
above to fi nd that the circular velocity just above Earth’s atmo-
sphere is about 7780 m/s, or 7.7 km/s. Th is is about 17,400
miles per hour, which shows why putting satellites into Earth
orbit takes such large rockets. Not only must the rocket lift the
satellite above Earth’s atmosphere, but the rocket’s trajectory
must then curve over and accelerate the satellite horizontally to
circular velocity.
A Common Misconception holds that there is no
gravity in space. You can see that space is fi lled with gravitational
forces from Earth, the sun, and all other objects in the universe.
An astronaut who appears weightless in space is actually falling
along a path at the urging of the combined gravitational fi elds in
the rest of the universe. Just above Earth’s atmosphere, the orbital
motion of the astronaut is dominated by Earth’s gravity.
2
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at the particle’s location. Th e resulting force is directed toward
the center of the fi eld.
Th e fi eld is an elegant way to describe gravity, but it still does
not say what gravity is. Later in this chapter, when you learn
about Einstein’s theory of curved space-time, you will get a better
idea of what gravity really is.
Orbital Motion
and TidesOrbital motion and tides are two diff erent kinds of gravita-
tional phenomena. As you think about the orbital motion of the
moon and planets, you need to think about how gravity pulls on
an object. When you think about tides, you must think about
how gravity pulls on diff erent parts of an object. Analyzing these
two kinds of phenomena will give you a deeper insight into how
gravity works.
Orbits
Newton was the fi rst person to realize that objects in orbit are
falling. You can explore Newton’s insight by analyzing the
motion of objects orbiting Earth. Carefully read Orbiting
Earth on pages 84–85 and notice three important concepts and
six new terms.
An object orbiting Earth is actually falling (being acceler-
ated) toward Earth’s center. Th e object continuously misses
Earth because of its orbital velocity. To follow a circular
orbit, the object must move at circular velocity, and at the5-2
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SCIENTIFIC ARGUMENT
What do the words universal and mutual mean when you say
“universal mutual gravitation”?
Scientists often build arguments that proceed step by step, and
this is a good example. Newton argued that the force that makes
an apple accelerate downward is the same as the force that accel-
erates the moon and holds it in its orbit. The third law of motion
says that forces always occur in pairs, so if Earth attracts the
moon, then the moon must attract Earth. That is, gravitation is
mutual between any two objects.
Furthermore, if Earth’s gravity attracts the apple and the moon,
then it must attract the sun, and the third law says that the sun
must attract Earth. But if the sun attracts Earth, then it must
also attract the other planets and even distant stars, which, in
turn, must attract the sun and each other. Step by step, Newton’s
third law of motion leads to the conclusion that gravitation must
apply to all masses in the universe. That is, gravitation must be
universal.
Aristotle explained gravity in a totally different way. Why
couldn’t Aristotle’s explanation of a falling apple on Earth
account for a hammer falling on the surface of the moon?