Use Kepler's third law to determine the period of a comet with an orbit extending from 1 AU to 5

Use Kepler's third law to determine the period of a comet...

Key Concepts

Kepler's Third Law

Kepler's third law of planetary motion establishes that the square of the orbital period of an object is directly proportional to the cube of the semi?major axis of its orbit. This relationship is fundamental in celestial mechanics for relating the time an object takes to make one complete orbit to the size of its orbit.

Semi-Major Axis

The semi-major axis is the longest radius of an ellipse and represents half the longest diameter of the orbital path. In orbital mechanics, it is the principal parameter used in Kepler's third law to calculate the orbital period and overall size of the orbit.

Orbital Period

The orbital period is the time taken by an astronomical object to complete one full orbit around another body. It is determined by the size of the orbit, typically described by the semi-major axis, and follows the proportionality described in Kepler's third law.

Elliptical Orbits

Elliptical orbits describe the shape of the path followed by objects under gravitational influence, where the distance between the orbiting body and the central body varies over time. Despite this variation, the semi-major axis remains the key measure of the orbit's size, which is crucial for calculating the orbital period using Kepler's third law.

Transcript

00:02 So we'll start off with kepler's third law, which basically says that the period of an orbit squared is proportional to the semi -major axis cubed of that orbit.

00:18 And the constant of proportionality works for the orbit around any massive body.

00:24 We'll call that big m.

00:26 So that constant is 4 pi squared divided by universal constant g.

00:33 Times the mass of the body around which the orbit is occurring.

00:40 So p is period and a is semi -major axis, and also interpreted to be the average distance of the orbiting object from the main mass m.

00:59 So we're going to take a look at using this law to do a little bit of rocket science, i suppose.

01:06 So the idea is to put an object in orbit around the earth.

01:11 So mass of the earth is the parent body.

01:15 And we're going to take a little spacecraft that starts at parogy.

01:22 So a parogy is a certain distance from the center of the earth.

01:29 Here we'll call that 6 .67 times 10 to the 6 meters.

01:39 So this is not going to be a scale drawing.

01:42 We're just showing a sketch to give the idea.

01:45 And that is going to go in an elliptical orbit with the endpoint.

01:53 I'm probably not going to draw a very good ellipse, but the endpoint is the moon, the position of the moon.

02:05 And so the appalian distance is from the center of the earth to the moon.

02:14 Which is 3 .85, roughly 3 .84 times 10 to the 8th meters.

02:29 So we could ask how long does this journey take using this elliptical orbit? and we can see that essentially we've gone through one -half of a period.

02:43 So if we found the period, we could determine how long the trip would take...

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