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Physics: High School18 chapters | 211 lessons

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Lesson Transcript

Instructor:
*David Wood*

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this lesson, you should be able to explain what eccentricity is and calculate the eccentricity of an orbit given relevant distance measurements. A short quiz will follow.

When you think about eccentricity, perhaps you think of your crazy aunt who climbed on the roof during Thanksgiving waving a pair of pantyhose. But did you know that planets can be eccentric, too? In physics, **eccentricity** is a measure of how non-circular the orbit of a body is. A particularly eccentric orbit is one that isn't anything close to being circular.

An eccentricity of zero is the definition of a circular orbit. When the eccentricity increases above this, but hasn't reached a value of 1, the orbit is elliptical. At an eccentricity of exactly 1, the object is on a parabolic trajectory, and an eccentricity of greater than 1 makes it a hyperbolic trajectory. These names come from parameters of conic sections since every orbit is one type of conic section or another.

But are eccentric orbits common in the universe? What about in our solar system?

The planets are generally not especially eccentric. The Earth has one of the least eccentric orbits, at 0.017, though Venus and Neptune are even more circular. In fact, almost all the planets are below 0.1 eccentricity. Only one planet in our solar system is particularly eccentric: Mercury with an eccentricity of 0.21. The dwarf planet Pluto also has an eccentricity of 0.25. Pluto's orbit is especially wild because it's also at an angle compared to the plane the other planets orbit in.

But how do we determine these numbers? How are they calculated?

A more specific definition of eccentricity says that **eccentricity** is half the distance between the foci, divided by half the length of the major axis. The major axis is shown on this diagram and these are the two foci. The eccentricity of this orbit is wildly exaggerated so the two foci are nicely far apart. In a circular orbit, the two foci are both at the same point, right in the middle of the circle.

Here is the definition of eccentricity in equation form: *e* = *c* / *a*. So all you have to do is plug the numbers into the equation and solve. Since this is a ratio, you can use whatever unit you wish for distance, the standard SI unit of meters, or non-scientific units like miles, or astronomical distance measurements like astronomical units. Whatever you happen to have on hand.

Okay, let's go through an example. Let's say you have a planet which we'll call Planet X, and that planet is orbiting its star in an eccentric orbit. At its closest approach it is 2 astronomical units away from the star and at its furthest approach, its 3.2 astronomical units away from the star. What is the eccentricity of the orbit?

The issue with this question is that you're not given numbers to plug into the equation directly. You really have to draw a diagram to figure out what to plug in. Here is a diagram of an orbit and here are the distances we're given: 2 AU on closest approach and 3.2 AU on furthest. We need to figure out half the distance between the foci, marked here, and half the length of the major axis, marked here.

The major axis is clearly a total of 2 + 3.2, which equals 5.2 astronomical units (AU). Half of that is 2.6 AU, so we have our value for *a*. To get the value of *c*, realize that the distance at closest approach, 2 AU, is also mirrored on the opposite side. So the distance between the foci must be the total length of the major axis, 5.2, minus 2 AU on one side, minus another 2 AU for the other side. So the foci must be 1.2 AU apart; clearly this diagram is not to scale. Half of that distance is 0.6 AU, so that's our value of *c*.

Plug those into the eccentricity equation, and we find that the eccentricity is equal to 0.6 divided by 2.6, which gives us an eccentricity of 0.23. Actually pretty similar to Pluto.

And that's it; we're done.

**Eccentricity** is a measure of how non-circular the orbit of a body is. A particularly eccentric orbit is one that isn't anything close to being circular. An eccentricity of zero is a circular orbit, between zero and 1 is elliptical, equal to one is parabolic, and greater than 1 is hyperbolic.

The planets are generally not especially eccentric. The Earth has one of the least eccentric orbits, at 0.017, and Mercury is the most eccentric planet with an eccentricity of 0.21 (not counting dwarf planets like Pluto).

A more specific definition of eccentricity says that **eccentricity** is half the distance between the foci, divided by half the length of the major axis. This can be expressed by this equation: *e* = *c* / *a*. The major axis is shown on this diagram and these are the two foci. Since this is a ratio, you can use whatever unit you wish for distance. All you have to do is plug the numbers into the equation and solve.

After finishing this lesson, you should be ready to:

- Define eccentricity
- Explain the meaning of the numerical values for eccentricity
- Recall the equation for finding the eccentricity of an orbit and explain how to use it

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Physics: High School18 chapters | 211 lessons

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