The earth's actual orbit

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

http://www.dailymathproblem.com/2019/08/speed-of-object-going-around-sun.html

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

http://www.universetoday.com/14483/orbit-of-earth/

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

X²/a² + Y²/b² = 1

It has been a really long time since I thought about ellipses.  (The web site below has a nice explanation.)

http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php

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.

http://www.mathsisfun.com/geometry/ellipse-perimeter.html

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.