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Work and Mechanics
Lesson 38: Celestial Orbits
Newton’s Cannonball
Newton’s cannonball is an ingenious thought experiment that provides an intuitive way of
understanding why one body orbits about another. Consider a cannonball that is fired
horizontally from the top of a mountain. Here, we assume that the mountain is so tall that it
extends above the atmosphere, eliminating the wind resistance that would slow the
cannonball. If the cannonball is fired horizontally, it will curve toward the center of the
Earth due to gravitational attraction, as shown.
1
F
2
F
Consider what happens when the cannonball is fired at increasing velocities, as shown
below. When the velocity is low, the cannonball drops quickly to the Earth (trajectory A).
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Work and Mechanics
Lesson 38: Celestial Orbits
As the velocity increases, the cannonball lands further away from the cannon (trajectories
B and C). When the cannonball is fired at just the right velocity, it falls toward the Earth so
that its elevation above sea level remains constant; this produces a circular orbit
(trajectory D), and the cannonball will return to the exact position from which it was fired.
If the cannonball is fired at an even higher velocity, it will trace out an elliptical orbit
(trajectory E), returning again to the position from which it was fired.
D
E
A
B
C
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Work and Mechanics
Lesson 38: Celestial Orbits
Elliptical Orbits
Since a body must be traveling at exactly the right velocity for a circular orbit to occur,
perfectly circular orbits are not common in nature. The planets in our solar system all have
slightly elliptical orbits around the sun.
CLASS PROBLEM: Let’s perform a thought experiment of our own. Consider a comet
in an elongated elliptical orbit about the Sun.
3
2
1
(a) Will the force acting on the comet be higher at position 1, 2 or 3? Why?
(b) Will the velocity of the comet be higher at position 1, 2 or 3? Why? HINT:
Draw the force vector of the Sun acting on the comet on the diagram above
and consider the acceleration due to this force (Newton’s 2nd Law).
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Work and Mechanics
Lesson 38: Celestial Orbits
Effect of Center of Mass on Orbits
We tend to assume that the Earth traces out a perfect ellipse as it orbits our “stationary”
Sun. We also tend to think of the Moon tracing out a perfect ellipse around the center of the
Earth. Newton’s Universal Law of Gravitation shows that the force exerted by the Earth on
the Moon is equal to the force exerted by the Moon on the Earth; we computed this force to
be 2.58(10)20 N in an earlier class. The force exerted by the Moon on the Earth will cause
the Earth to accelerate toward the Moon according to Newton’s 2nd Law (F = m•a). So, the
center of the Earth does not follow a perfectly elliptical path around the Sun. Instead, it is
tugged back and forth across this elliptical path by the Moon. That is, the Earth experiences
a small counter orbit with the Moon due to the force of the Moon acting on the Earth.
When two isolated bodies interact due to gravitational attraction, they each orbit about the
center of mass of the two bodies. The center of mass of a pair of bodies lies on a line
somewhere between the centers of the two bodies. If two bodies have the same mass, then
the center of mass is half way between the bodies. If one body is much more massive than
the other body, then the center of mass lies close to the center of the more massive body as
shown by the example below.
EXAMPLE: If one body has a mass of 90 kg and the other body has a mass of 10 kg, then determine
the center of mass of the two-body system when they are separated by 10 m. That is, compute the
distance r1 from the center of the more massive body to the center of mass of the pair.
r1
m2 = 10
kg
m1 = 90 kg
center of mass of the pair of
bodies
a = 10 m
𝑟1 = 𝑎 ∙
𝑚2
10𝑘𝑔
= 10𝑚 ∙
= 1𝑚
𝑚1 + 𝑚2
90𝑘𝑔 + 10𝑘𝑔
NOTE: If these two bodies were orbiting about one another, then the less massive body would
orbit about the center of mass of the pair, NOT the center of the more massive body. Likewise, the
more massive body would orbit about the center of mass of the pair. The mean radius of the more
massive body’s orbit would be 1m while the mean radius of the less massive body’s orbit would be
9m.
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Work and Mechanics
Lesson 38: Celestial Orbits
Now, let’s apply this idea to the Earth / Moon pair. The Moon has a mass that is 1.23% that
of Earth. Using 384,400 km as the average distance between the Earth and the Moon shows
that the center of mass of the pair is 4,670km from the center of the Earth. Since the radius
of the Earth is 6,380 km, the Earth experiences a significant “wobble” as it traces out its
elliptical orbit around the sun.
Historical Background and Perspective
Johannes Kepler presented Kepler’s Laws of Planetary Motion in 1609 and 1619. Kepler’s
Laws provided equations describing the elliptical geometry of orbits, the speed of a planet
at various orbital positions, and the orbital period (the time for one complete revolution
about the sun). In 1687, Isaac Newton published “the Principia” that that derived Kepler’s
Laws using his Laws of Motion (1st, 2nd and 3rd laws) and his Law of Universal Gravitation.
In 1915, Albert Einstein showed that gravity was due to curvature of space-time, removing
the assumption of Newton that gravity propagated instantaneously. While Einstein’s
discovery, which was a consequence of his General Theory of Relativity, has been
confirmed so far using our most accurate measurement devices, Newton’s Laws are
accurate enough for most situations and are much easier to use.
Einstein’s General Theory of Relativity was a giant leap in our understanding of the
universe. However, it does not describe the interactions between the fundamental
particles, some yet to be discovered, that define the universe and gravity itself. General
Relativity is not unified with the existing theory of quantum mechanics. Scientists are
searching for a “theory of everything” that fully explains all physical phenomena. Perhaps
ongoing theoretical physics research along with experiments at the Large Hadron Collider
and elsewhere will provide clues to how everything fits together, satisfying our curiosity
and providing the scientific basis for future technologies.
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Lesson 38: Celestial Orbits
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