Written by Sunny Yoon

Here is a picture of the earth orbiting around the sun. In order to plot the orbit of the earth, we need to use heliocentric (Sun-centered) longitude of the earth. However, all measurements give us geocentric (Earth-centered) longitude of the sun.

Here are the instructions on how to plot Earth's orbit.

1) Determine a coordinate system. Determine a starting point and the direction.

2) Find the measurements of geocentric longitude of the sun.

3) Convert the geocentric longitude to heliocentric longitude.

4) Plot Earth's orbit.

1) Determine a coordinate system. Determine a starting point and the direction.

Use 0 degrees (positive x-axis) as the direction from Earth to the Sun on the vernal equinox. The vernal equinox is when the sun crosses the celestial equator and when the length of day and night are approximately equal. There is also the autumnal equinox.

So in this picture, 0 degrees represent the direction to the Sun at the vernal equinox while 180 degrees represent the direction to the Sun at the autumnal equinox.

2) Find the measurements of geocentric longitude of the sun.

http://www.archaeoastronomy.com/2010.shtml

I went to the web site above to the dates of the vernal equinox and autumnal equinox. This web site will give you the dates starting from year 2000 until 2020. The vernal equinox for 2010 is March 20th and the autumnal equinox for 2010 is September 22nd.

Afterwards, I downloaded the The Astronomical Almanac in order to find the dates, geocentric longitude of the Sun, and the apparent size of the Sun for the current year. The Astronomical Almanac is almanac published by the United States Naval Observatory and Her Majesty's Nautical Almanac Office, containing solar system ephemeris and catalogs of selected stellar and extragalactic objects.

OR

I'm going to use the data given by Mission Mathematics II: Grade 9 -12 on pg. 79. If you prefer to use actual data, you can look them up through internet.

 Date Geocentric Longitude of the Sun (degrees) Apparent Size of the Sun (cm) March 21 0 April 6 15.7 48.7 May 6 45.0 48.7 June 5 73.9 48.5 July 5 102.5 48.1 August 5 132.1 48.6 September 4 162.0 49.0 October 4 191.3 49.5 November 3 220.1 49.7 December 4 250.4 49.9 January 4 283.2 50.0 February 4 314.7 49.6 March 7 346.0 49.5

3) Convert the geocentric longitude to heliocentric longitude.

Let's look at this picture again. In this picture, 0 degrees represents the direction to the Sun at the vernal equinox and the Sun is moving counterclockwise direction from the reference point. We need to change the picture so the Sun will be at the center, that is, looking back at Earth from the Sun.

In heliocentric coordinates, we are looking back at Earth from the Sun so the vernal equinox is at 180 degrees and the autumnal equinox is at 0 degrees and the Earth is moving clockwise.

 Date Geocentric Longitude of the Sun (degrees) Heliocentric Longitude (degrees) Apparent Size of the Sun (cm) March 21 0 180 April 6 15.7 195.7 48.7 May 6 45.0 225.0 48.7 June 5 73.9 253.9 48.5 July 5 102.5 282.5 48.1 August 5 132.1 312.1 48.6 September 4 162.0 342.0 49.0 October 4 191.3 11.3 49.5 November 3 220.1 40.1 49.7 December 4 250.4 70.4 49.9 January 4 283.2 103.2 50.0 February 4 314.7 134.7 49.6 March 7 346.0 166.0 49.5

I added 180 degrees from March 21 to September 4 and I subtracted 180 degrees from October 4 to March 7.

What is the mathematical reasoning behind this?

When you are changing from geocentric longitude to heliocentric longitude, it provides an opportunity to talk about reference point. When you look into the illustration of a geocentric model, the earth is in the center and the sun is moving counterclockwise from the right. In this model, earth is the reference point as the sun is changing. When you look into the illustration of a heliocentric model, the reference point has changed. Since the Sun is the new reference point, when it becomes the vernal equinox, earth is on the left of the sun.

Also, think about the direction on how the earth is portrayed moving from the sun. When you're standing on the earth, the sun moves in a counterclockwise direction. Or, more specifically, from 0 degrees to 180 degrees. However, when you're standing on the sun, the earth is moving clockwise from left to right from 180 degrees to 360 degrees. Therefore, when you're converting the longitude, you add 180 degrees starting from the vernal equinox until the autumnal equinox. After the autumnal equinox, you subtract 180 degrees.

4) Plot Earth's orbit.

Here is a picture I made using GSP.

I plotted the position of the earth using dilation and rotation in GSP.

Using (1,0) as my starting point, I started with the longitude on October 4th. Heliocentric longitude is 11.3 degrees, so I rotated point (1,0) 11.3 degrees counterclockwise. After the rotation, I dilated the point 4.95 which is 49.5 divided by 10.

After that, I repeatedly rotated (1,0) to the heliocentric longitude given at a specific date and dilated the point one-tenth the size given in the table.

The picture above shows how close Earth's orbit is to a circle rather than an ellipical orbit we're used to see. The eccentricity of the Earth's orbit is currently about 0.0167, meaning that the Earth's orbit is nearly circular.