Prof. Dale E. Gary
Origins of Modern Astronomy
Kepler's Laws of Planetary Motion
When Tycho lay on his deathbed, he begged is protege, Johannes Kepler (1571-1630) to take his careful measurements of the positions of the planets and prove that the Tychonic theory was correct. However, Kepler was a proponent of the Copernican theory, and he was a careful mathematician who simply wanted to discover the true laws of planetary motion. With Tycho's measurements, Kepler tried to fit a number of different curves to the positions of the planets.
To get a flavor for how he did this in the case of Mercury, he took the measurements of Mercury's greatest elongation, as shown in the following movie (made for elongations in 2001 and 2002) and determined the location of the planet relative to the Sun and Earth. The path of multiple elongations maps out a curve, which he tried to fit.
He tried fitting circles, with centers offset from the Sun as needed to fit the points, but Tycho's measurements were much too precise to allow such a solution. He tried ovals and other curves of various kinds, and only after many long years of checking and rechecking his numbers did he come up with his famous three laws of planetary motion:
- Kepler's 1st Law: Planets orbit the Sun in ellipses with the Sun at one focus.
- Kepler's 2nd Law: An imaginary line from the Sun to the planet sweeps out equal areas in equal times.
- Kepler's 3rd Law: The square of a planet's orbital period is proportional to the cube of the orbit's semi-major axis (or P2 = ka3).
Let's examine these a bit more, and look at some implications:
Kepler's 1st Law: An ellipse has a long axis (called the major axis). It also has two foci (focuses). One focus is where the Sun is located, the other focus is empty. As the two foci are brought together, the ellipse looks more and more like a circle. In fact, a circle is just a special case of an ellipse with the two foci at the same place (the center of the circle), in which case the major axis is the diameter of the circle. Half the major axis is called the semi-major axis (a in the upper figure, r in the lower figure).
Kepler's 2nd Law: Combined with the first law, this law states how fast a given planet moves in its orbit. A planet moves faster when it is near the Sun (perihelion), and slower when it is farther from the Sun (aphelion). But it says that in a very quantitative way.
Kepler's 3rd Law: This law states how one planet's speed (or period) relates to another. The two planet orbits on the left have the same period, because they have the same semi-major axis.
It is important to realize that the 2nd and 3rd laws are quantitative, meaning that one can calculate the actual distances (semi-major axes) and speeds, etc., using them. For the case of planets going around the Sun, for example, with P in years and a in Astronomical Units (AU), the third law becomes just: P2 = a3. This is trivial for Earth -- P = 1 year and semi-major axis a = 1 AU, but we can also measure the period of Venus (P = 0.615 years), and get its distance from the Sun (a = 0.723 AU). We will see later what the constant k is in the formula P2 = ka3, so that Kepler's 3rd law can be extended to other orbits, such as the Moon around the Earth.
Lecture Question #1
Kepler's Laws allow us to construct a scale model of the solar system, knowing the exact shapes and relative sizes of the planetary orbits. But it does not allow us to determine the absolute size of any orbit. All the distances are relative to the Astronomical Unit (distance of the Earth from the Sun). In fact, the actual distance to the Sun was not known until a transit of Venus in 1761 offered a chance to get an accurate triangulation. Nowadays we can measure planetary distances very accurately using radar signals.
Venus crossing the disk of the Sun on 2004 Jun 08
(photo taken by Dale Gary)
Venus just touching the limb of the Sun on 2004 Jun 08
(photo taken by Dale Gary)
Isaac Newton (1642-1727) took the ideas of Kepler and Galileo, and put them together into three Laws of Motion and the Universal Law of Gravitation. These laws, plus a little calculus, are sufficient to explain and quantify virtually all mechanical phenomena we see on Earth and throughout the universe. Newton's 1st law, the law of inertia, states that something in motion will continue to move at the same speed in a straight line unless acted on by an outside force. This was a key realization, also noted by Galileo, that finally could account for the motions of planets. One of the big questions had been, what keeps the planets moving forever in their orbits. The answer is inertia -- they are moving so they keep moving. But they are NOT moving in a straight line, and Newton realized that that must mean there is a force acting on them. But what was this force, and how could it work at such a distance?
Lecture Question #2
Another help was his 2nd law, which quantifies the relationship between force and acceleration and introduces the concept of mass. He determined that a body of half the weight would be accelerated twice as much by a given force, i.e. force equals mass x acceleration: F = ma. But equally important was his 3rd law, which states that for every force there is an equal and opposite reaction force. That means that when a falling object, say a stone or an apple, is accelerated toward the Earth due to the force of gravity by the Earth, the object also exerts exactly the same force on the Earth (so the Earth is accelerated toward the object, albeit by an imperceptible amount).
He noted that falling objects accelerate with constant acceleration toward the center of the Earth, and postulated that the Moon is also doing the same -- that is, the Moon is continually falling. It is only the side-ways motion of the Moon that keeps it from hitting the Earth. But to check that idea, he had to know quantitatively what the magnitude of the force should be at the distance to the Moon. He knew it should become smaller, but exactly how?
The Sun's inward pull of gravity on a planet competes with the planet's tendency to continue moving in a straight line. These two effects combine, causing the planet to move smoothly along an intermediate path, which continuously "falls around" the Sun. This unending tug-of-war between the Sun's gravity and the planet's inertia results in a stable orbit. (Fig. 1.24, Chaisson & McMillan, Copyright Prentice Hall, 2004)
So Newton basically guessed that the fall-off in the force should go as the square of the distance (a so-called inverse-square-law), and by his third law the force between the Earth and the Moon should be proportional to the product of the masses. Using this guess, he calculated how far the Moon would fall in 1 second, and using his knowledge of the distance to the Moon and the period of the Moon's orbit, he found that the amount agreed exactly with his law. This was the Universal Law of Gravitation (F = - GMm/r2), which states that the force between two masses is proportional to the product of the masses and inversely proportional to the square of the distance between them. G is called the gravitational constant. The Universal Law of Gravitation is the reason things fall to Earth, and it is also the reason the Moon circles the Earth and the planets circle the Sun.
It turns out that using Newton's Laws and some other laws of physics it is possible to theoretically understand all of Kepler's laws. In particular, the constant k can now be given as k = 4p2/[G(M+m)], where M is the mass of the more massive body and m is the mass of the less massive body. This allows us to use the law for any two orbiting bodies. Without it, we could not send a spacecraft to Mars, or predict the orbit of the Space Shuttle around Earth. The overall equation for Kepler's third law is:
P2 = (4p2/[G(M+m)]) a3.
Lecture Question #3
Lecture Question #4
Figure 1.25 Orbits (a) The orbits of two bodies (stars, for example) with equal masses, under the influence of their mutual gravity, are identical ellipses with a common focus. That focus is not at the center of either star but instead at their center of mass, located midway between them. The pairs of numbers indicate the positions of the two bodies at three times. (Note that the line joining the bodies at any given time always passes through the common focus.) (b) The orbits of two bodies when one body is twice as massive as the other. The elliptical orbits again have a common focus (at the center of mass of the two-body system), and the two ellipses have the same eccentricity, but the more massive body moves more slowly and in a smaller orbit. The larger ellipse is twice the size of the smaller one. (c) In the extreme case of a hypothetical planet orbiting the Sun, the common focus of the two orbits lies inside the Sun.
(Chaisson & McMillan, Copyright Prentice Hall, 2004)