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3.6 Elliptical Orbits

Kepler's first law of planetary motion says that each planet orbits the Sun on an elliptical path, with the Sun at one focus. What does this mean? You can draw an ellipse in this simple way: Take a piece of string about six to ten inches long and tie it in a loop. Put two thumbtacks in a piece of cardboard. Loop the string around the tacks, put a pencil against the string, and pull it taut. Then trace all the way around the tacks, keeping the loop of string taut. The result will be an ellipse. Each tack occupies a special point in the ellipse called a focus (the plural is "foci"). 

A clean, scalable example of an ellipse with an inscribed circle. Click here for original source URL.

So according to Kepler's first law, if the ellipse you drew represents the orbit of a planet, the Sun would be located at one of the tacks. What lies at the other focus? Nothing — it's just a mathematical concept. If you move the two foci closer together, the ellipse becomes more like a circle. We can continue this progression until the two foci meet at the same point. The ellipse is then a perfect circle. In other words, a circle is just a special case of an ellipse. The Greeks favored circular orbits because a circle is the most simple and symmetrical shape. Now we have another way to think about orbits: They're a whole family of ellipses — just think of circles that are successively more "squashed" — and the circle is just one example of these ellipses. The ellipse is the more general figure. The generality of Kepler's description is often the hallmark of a good scientific idea. 

Ellipse, showing semi-latus rectum. Click here for original source URL.

A circle can be described by just one number: the radius. But it takes two numbers to describe an ellipse. The widest diameter of an ellipse is called the major axis and half of this distance is the semimajor axis (symbolized by the letter ""a""). The semimajor axis is not only the distance from the center of the ellipse to one end; it is also equal to the average distance of a planet from the Sun. It is therefore analogous to the radius of a circular orbit. The word used to describe the flattening or non-circularity of an ellipse is eccentricity (symbolized by the letter ""e""). Mathematically, it is the ratio of the distance between the foci to the major axis (2a). A circle has zero eccentricity; the distance between the foci is zero. Most planets are in nearly circular orbits with very low eccentricity. Very flattened or elongated orbits have high eccentricity. Many comets move in very eccentric orbits, but they still follow Kepler's first law: their orbits are elliptical, with the Sun at one focus. 

Elliptical motion offers a good example of the progress of science as more accurate data is gathered. If you only had measurements of the motions of planets accurate to 10% you would find that the orbits of most planets are consistent with circular motion. But departures from uniform motion are seen when the data's accuracy reaches 1%. The effects are quite subtle, which is why Tycho's excellent records were required to make the breakthrough. For example, in Earth's elliptical orbit, the Earth-Sun distance varies only 3.4% between when Earth is at the closest point to the Sun (called perihelion) and when it's at the farthest point (called aphelion). In other words, the ellipse of the Earth's orbit has a distance between the foci that is only 3.4% of the semi major axis — a very slightly squashed circle. 

The focus of the orbit isn't exactly where the sun it, it is at the center of mass between the sun and the planets. An example of such motion is shown here. . Click here for original source URL.

If you measured orbits with even greater accuracy, say 0.1%, you'd find that there are tiny discrepancies. In other words, although planets move in nearly elliptical orbits, they do not move in exactly elliptical orbits. The reason is that they are pulled slightly out of their ellipses by the gravitational tug of other planets. Scientists realized this as early as the 1700s. Also, at a higher level of precision, it is not quite correct to say that the Sun lies exactly at a focus. Rather, the focus of the orbit is the center of mass of the Sun and the planet. To find where the center of mass would be, imagine a balancing beam, and put the Sun at one end and a planet at the other. The center of mass would be the balance point between the two. The Sun is so massive that the center of mass for each planet is inside the Sun but not quite at the center. By taking these slight departures from perfect ellipses into consideration the orbits of the planets can be more completely described. In this way, data of higher and higher accuracy can lead to a whole new level of understanding.