Why are ellipses important in astronomy

These two points inside the ellipse are called its foci singular: focus , a word invented for this purpose by Kepler. This property suggests a simple way to draw an ellipse Figure 3. We wrap the ends of a loop of string around two tacks pushed through a sheet of paper into a drawing board, so that the string is slack.

If we push a pencil against the string, making the string taut, and then slide the pencil against the string all around the tacks, the curve that results is an ellipse. At any point where the pencil may be, the sum of the distances from the pencil to the two tacks is a constant length—the length of the string. The tacks are at the two foci of the ellipse. The widest diameter of the ellipse is called its major axis.

Half this distance—that is, the distance from the center of the ellipse to one end—is the semimajor axis , which is usually used to specify the size of the ellipse. Figure 3: Drawing an Ellipse. Each tack represents a focus of the ellipse, with one of the tacks being the Sun. Stretch the string tight using a pencil, and then move the pencil around the tacks. The length of the string remains the same, so that the sum of the distances from any point on the ellipse to the foci is always constant.

The distance 2a is called the major axis of the ellipse. The shape roundness of an ellipse depends on how close together the two foci are, compared with the major axis. The ratio of the distance between the foci to the length of the major axis is called the eccentricity of the ellipse. If the foci or tacks are moved to the same location, then the distance between the foci would be zero.

This means that the eccentricity is zero and the ellipse is just a circle; thus, a circle can be called an ellipse of zero eccentricity. In a circle, the semimajor axis would be the radius. Next, we can make ellipses of various elongations or extended lengths by varying the spacing of the tacks as long as they are not farther apart than the length of the string.

The greater the eccentricity, the more elongated is the ellipse, up to a maximum eccentricity of 1. The size and shape of an ellipse are completely specified by its semimajor axis and its eccentricity. The eccentricity of the orbit of Mars is only about 0. Kepler generalized this result in his first law and said that the orbits of all the planets are ellipses. Here was a decisive moment in the history of human thought: it was not necessary to have only circles in order to have an accepTable cosmos.

The universe could be a bit more complex than the Greek philosophers had wanted it to be. He expressed the precise form of this relationship by imagining that the Sun and Mars are connected by a straight, elastic line. When Mars is closer to the Sun positions 1 and 2 in Figure 4 , the elastic line is not stretched as much, and the planet moves rapidly.

Farther from the Sun, as in positions 3 and 4, the line is stretched a lot, and the planet does not move so fast. As Mars travels in its elliptical orbit around the Sun, the elastic line sweeps out areas of the ellipse as it moves the colored regions in our figure. Note that if you follow the Starry Night instructions on the previous page to observe the orbits of Earth and Mars from above, you can also see the shapes of these orbits and how circular they appear.

The image below links to an animation that demonstrates that when a planet is near aphelion the point furthest from the Sun, labeled with a B on the screen grab below the line drawn between the Sun and the planet traces out a long, skinny sector between points A and B. When the planet is close to perihelion the point closest to the Sun, labeled with a C on the screen grab below , the line drawn between the Sun and the planet traces out a shorter, fatter sector between points C and D.

These slices that alternate gray and blue were drawn in such a way that the area inside each sector is the same. That is, the sector between C and D on the right contains the same amount of area as the sector between A and B on the left.

Since the areas of these two sectors are identical, then Kepler's second law says that the time it takes the planet to travel between A and B and also between C and D must be the same. If you look at the distance along the ellipse between A and B, it is shorter than the distance between C and D. Since velocity is distance divided by time, and since the distance between A and B is shorter than the distance between C and D, when you divide those distances by the same amount of time you find that:.

The orbits of most planets are almost circular, with eccentricities near 0. In this case, the changes in their speed are not too large over the course of their orbit. For those of you who teach physics, you might note that really, Kepler's second law is just another way of stating that angular momentum is conserved. That is, when the planet is near perihelion, the distance between the Sun and the planet is smaller, so it must increase its tangential velocity to conserve angular momentum, and similarly, when it is near aphelion when their separation is larger, its tangential velocity must decrease so that the total orbital angular momentum is the same as it was at perihelion.

This is usually referred to as the period of an orbit. Kepler noted that the closer a planet was to the Sun, the faster it orbited the Sun. He was the first scientist to study the planets from the perspective that the Sun influenced their orbits.

What this means mathematically is that if the square of the period of an object doubles, then the cube of its semimajor axis must also double.

The shape of the ellipse is described by its eccentricity. The larger the semi-major axis relative to the semi-minor axis, the more eccentric the ellipse is said to be.

The eccentricity is defined as:. This gives an interpretation of the eccentricity as the position of the foci as a fraction of the semi-major axis.