29:50 Modern Astronomy
Fall 1999
Lecture 32 ...November 10, 1999
Orbits in the Solar System

Back to the Solar System I want to do some preparatory discussion before the Leonids next week so you will know what you are looking at. To understand what is going on with meteors, you need to know about orbits of objects in the solar system, and what comets are.

Orbits in the Solar System and Kepler's Laws So far in this class I have used the term orbit fairly liberally, and I hope by this time you have an intuitive sense of its significance. We are now at the point where we will give a more rigorous definition begin by discussing some of the rules or ``laws'' obeyed by orbits, and then proceed later to describe why this is so. The rules of orbits are called Kepler's Laws.

To begin with, an orbit is the path followed by a celestial object in space. The central problem of orbits in astronomy is to answer the question where something is at a given time. In the case of the solar system, we usually deal with one object moving around a much more massive object, such as the planets around the sun, the moon around the Earth, etc. In this case it is quite accurate to speak of the orbit of one object (say a planet) around another object (say the Sun).

Now let's consider the case of the major planets. Even in antiquity people had ideas about which ones were further and which ones were closer to the Sun. Mercury moves the fastest, Jupiter takes 12 years to complete an orbit, and Saturn takes 29. This corresponds to increasing distance as well. Look at Appendix 4 of the book, which gives the orbital properties of the planets
Transparency giving orbital data on the planets.

Kepler's Laws: The Laws governing planetary motion were discovered by Johannes Kepler, an astronomer living around the year 1600. These laws govern not only the major planets, but also comets, asteroids or minor planets, and meteors.
Kepler's First Law: The orbits of planets are ellipses (not circles) with the Sun at one focus.
This brings up two questions. First, what is an ellipse, a secondly, what is a focus.

Figure 4-13 from your book shows a representation of an ellipse. It looks like a squashed circle. The long axis is referred to as the major axis and the short axis the minor axis. Half of the major axis is the semi-major axis.
Figure 4-13 from book.

The closer the foci move to each other, the more closely circular the ellipse is. Kepler's First Law says that the planets move along such a figure, with the sun at one of the foci. There is nothing at the other focus.

The further apart the foci are, the more egg shaped the ellipse is. We quantify this by the eccentricity, which is the ratio of the distance between the foci to the major axis.

There are a number of ways of describing an ellipse mathematically. The simplest is that every point on an ellipse (i.e. every point in the orbit of a planet) is such that the sum of the distances from the point to the two foci is the same. This is illustrated nicely in Figure 4-14 of the textbook.
Figure 4-14 from book.
If we define an (x,y) coordinate system, then an ellipse is defined by

If A=B we have the equation for a circle.

The larger the eccentricity, the greater the difference between the closest approach of the planet to the sun and its most distant approach, called perihelion and aphelion. If you look at Appendix 4, you will see that the eccentricity of the Earth's orbit around the sun 0.017; this means that the orbit is quite close to a circle, and the variations in the Earth's distance from the Sun during the course of the year are not great. By contrast, the eccentricity of Mars' orbit is 0.093, and that of Mercury is 0.206. If the changes for the Earth were this great, life on Earth might be very difficult.

The most extreme cases are comets; Comet Hale-Bopp had an eccentricity of 0.995, so its aphelion was far out in space.

Kepler's Second Law: Kepler's Second Law concerns how fast planets move in their orbits. Essentially it says that objects move slowly when near aphelion, and faster near perihelion. Kepler's Second Law is expressed as the ``Equal Area Law'', which states that a line from the Sun to the planet sweeps out equal areas in equal amounts of time. This is illustrated in Figure 4-15 from the book.
Figure 4-15.

This figure shows a rather eccentric planetary orbit around the sun. Each of the tick marks occurs at equal intervals of time, say a month or year apart. We can think of the area enclosed by the orbit for each of these intervals. Kepler's Second Law says that the area if the same time intervals are the same. The sophisticated reason for Kepler's 2nd Law is the conservation of angular momentum. Kepler's 2nd Law is the reason why a comet such as Halley's comet (comets typically have highly eccentric orbits) spends so virtually all of its time far from the Sun, then moves through the inner solar system in a very short period of time.

Kepler's Third Law As mentioned above, the further an orbiting object is from the Sun, the longer it takes to complete its orbit. Kepler deduced that this could be expressed in algebraic terms as a relationship between the semimajor axis and the period:

This Law is very precisely adhered to in the Solar system, and is a big hint as to the nature of the force holding the Solar System together. Kepler didn't know enough physics to figure out why the 3rd Law holds, but Newton about a century later showed why this is so.

Let's work out some examples to show it works. The orbital period of Jupiter is 11.86 years. How far is it from the Sun (i.e. what is its semimajor axis?). We have
If P=11.86, years.
so, according to Kepler's Law .
Taking the 1/3 root, a=5.20 astronomical units.

This is indeed the semimajor axis of Jupiter's orbit. Point to transparency with planetary orbital data.
Let's do another example to show that this wasn't a special case for Jupiter. Let's work out the planet Neptune. The distance from the Sun to Neptune (strictly speaking, the semimajor axis of its orbit) is a=30 astronomical units. Kepler's Law
if a=30,
so , taking the square root, P=164 years

which again is the correct answer.

Figure from Zeilek showing adherence of planetary orbits to Kepler's 3rd Law.

I will come up with some problems dealing with Kepler's Laws.

Kepler's 3rd Law allows an interesting conclusion about the nature of the motion of the planets in their orbits. Are the longer orbital periods just due to the facts that the planets further out have a longer way to go, or is it due to the planets also moving slower? Let's work it out with some algebra.

Let's stick to circular orbits (ellipses with eccentricity 0)for simplicity. The circumference of the circle, or distance a planet would have to travel is with R being the radius of the circle. Let the planet be moving at a speed V (km/sec), then the period is the time it takes to complete one loop of the track, or . so . If the planets were all moving at the same speed V, we would have . Instead we have which is a more extreme dependence.

In words the periods at greater distances are longer than can be accounted for just by the fact that the orbital circumferences are larger, or that the planets have a greater distance to travel. They are also moving more slowly at greater distances.

Let me finish with a few words on the historical and cultural significance of Kepler's Laws. These were a big advance in our knowledge of the sky, and clearly demonstrated the mathematical structure of the astronomical universe which had been more or less an act of faith on the part of Archytas of Tarentum. These laws are still used today in the calculation of spacecraft trajectories and the prediction of cometary orbits.

However, Kepler did not know why this was the case, why we have instead of . This epiphany came in the work of Isaac Newton who lived about a hundred years after Kepler, and illuminated an incredible amount about the physical universe. With Newton's work came the beginning of dynamics and the beginning of the understanding of the underlying laws of the universe.

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