*This post is a reprint of a previous post that shows how to calculate the perimeter of an ellipse.*

**Issue:**In a previous post I calculated the speed of the earth under the assumption that the earth’s orbit was circular.

The earth’s orbit is elliptical, not circular. The
link below provides a description of the actual orbit of the earth around the
sun. At its nearest point the earth is

Describes the orbit of the earth around the sun

According to this web site at its nearest point the earth is
147 million kilometers away from the sun and at its furthest point the earth is
152 million kilometers away from the sun.

Find an equation for an ellipse that approximates the
earth’s orbit around the sun.

What is the perimeter of an object that travels around this
ellipse?

How do estimates of the earth’s speed and distance traveled
based on the elliptical-orbit estimate differ from the earth’s distance
traveled and speed based on the circular orbit assumption.

**Answer**: The equation of an ellipse with the center at (0,0) is of the form

X

^{2}/a^{2 }+ Y^{2}/b^{2 }= 1
It has been a really long time since I thought about
ellipses. (The web site below has a nice
explanation.)

If we set a=147 and b=152 we get an ellipse where the
furthest point is 147 kilometers and the nearest point is 152 kilometers.

The formulas for the perimeter of an ellipse are actually
very difficult. See the link below for
an explanation.

The approximation for the ellipse perimeter that we use is

PER = pi x [ 3 x
(a+b) – ((3a+b)(a+3b)}

^{0.5}]
Plugging in values of a=147 and b=152 we find that the
earth’s annual orbit around the sun is approximately 939.4 million
kilometers. Our estimate of the earth’s
orbit assuming a circle (radius 149.7 kilometers) is 940.4 kilometers.

The circular orbit and elliptical orbit estimates are in
fact quite similar.

Please feel free to check my work, to comment and to provide
suggestions for more posts.

David

Isn't 152Km larger than 147Km so the furthest and nearest is switched?

ReplyDeleteI want to know this answer too!

ReplyDelete