Proof of the Law of Ellipses
Let v = r′(t) be the velocity vector. Then r″ = v′ and Eq. (1) may be written
REMINDER
Eq. (1) states:
Where r(t) = ∥r(t)∥.
On the other hand, by the Chain Rule and the relation r(t)2θ′(t) = J of Eq. (3),
Together with Eq. (6), this yields 
, or
This is a first-
where c is an arbitrary constant vector.
We are still free to rotate our coordinate system in the plane of motion, so we may assume that c points along the y-axis. We can then write c = 〈0, (k/J)e〉 for some constant e. We finish the proof by computing J = r × v:
Direct calculation yields
so our equation becomes 
. Since k is a unit vector,
Solving for r, we obtain the polar equation of a conic section of eccentricity e (an ellipse, parabola, or hyperbola):
REMINDER
The equation of a conic section in polar coordinates is discussed in Section 11.5.
This result shows that if a planet travels around the sun in a bounded orbit, then the orbit must be an ellipse. There are also “open orbits” that are either parabolic and hyperbolic. They describe comets that pass by the sun and then continue into space, never to return. In our derivation, we assumed implicitly that J ≠ 0. If J = 0, then θ′(t) = 0. In this case, the orbit is a straight line, and the planet falls directly into the sun.
Kepler’s Third Law is verified in Exercises 23 and 24.
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CONCEPTUAL INSIGHT
We exploited the fact that J is constant to prove the law of ellipses without ever finding a formula for the position vector r(t) of the planet as a function of time t. In fact, r(t) cannot be expressed in terms of elementary functions. This illustrates an important principle: Sometimes it is possible to describe solutions of a differential equation even if we cannot write them down explicitly.
The Hubble Space Telescope produced this image of the Antenna galaxies, a pair of spiral galaxies that began to collide hundreds of millions of years ago.
HISTORICAL PERSPECTIVE
The astronomers of the ancient world (Babylon, Egypt, and Greece) mapped out the nighttime sky with impressive accuracy, but their models of planetary motion were based on the erroneous assumption that the planets revolve around the earth. Although the Greek astronomer Aristarchus (310–
Constants:
Gravitational constant:
G ≈ 6.673 × 10−11 m3 kg−1 s−2
Mass of the sun:
M ≈ 1.989 × 1030 kg
k = GM ≈ 1.327 × 1020
The German astronomer Johannes Kepler was the son of a mercenary soldier who apparently left his family when Johannes was 5 and may have died at war. He was raised by his mother in his grandfather’s inn. Kepler’s mathematical brilliance earned him a scholarship at the University of Tübingen and at age of 29, he went to work for the Danish astronomer Tycho Brahe, who had compiled the most complete and accurate data on planetary motion then available. When Brahe died in 1601, Kepler succeeded him as “Imperial Mathematician” to the Holy Roman Emperor, and in 1609 he formulated the first two of his laws of planetary motion in a work entitled Astronomia Nova (New Astronomy).
In the centuries since Kepler’s death, as observational data improved, astronomers found that the planetary orbits are not exactly elliptical. Furthermore, the perihelion (the point on the orbit closest to the sun) shifts slowly over time (Figure 6). Most of these deviations can be explained by the mutual pull of the planets, but the perihelion shift of Mercury is larger than can be accounted for by Newton’s Laws. On November 18, 1915, Albert Einstein made a discovery about which he later wrote to a friend, “I was beside myself with ecstasy for days.” He had been working for a decade on his famous General Theory of Relativity, a theory that would replace Newton’s law of gravitation with a new set of much more complicated equations called the Einstein Field Equations. On that 18th of November, Einstein showed that Mercury’s perihelion shift was accurately explained by his new theory. At the time, this was the only substantial piece of evidence that the General Theory of Relativity was correct.
