Illuminations: Mars Earth Orbit - Model

# Mars Earth Orbit - Model

This model uses circular orbits to approximate the paths of Mars and Earth. It uses 3:2 as the ratio of their orbital radii (the actual is about 1.524:1), and it uses a ratio of 2:1 as the ratio of the time it takes them to complete one revolution about the sun (the actual is about 1.881:1). It also assumes that the orbits of both planets are circular with the Sun as the center, although this is not the case; in actuality, the orbits are ellipses, and the Sun is one of the foci.

However, these approximations are close to the actual data, and they allow for the development of a polar equation (5 ‑ 4cosθ) that produces a perfect limaçon. A representation that uses a better approximation of the actual data will not result in such a nice curve.

### Instructions

 The Orbit buttons allow for various explorations when the planets are in motion. You can also move the Earth and Mars by clicking and dragging them. The Show EM and Show E'M' buttons show the relative distance and orientation between Earth and Mars as they make their orbits about the Sun. When a portion of the path has been drawn, a red X will appear in the lower right corner; press this button to clear the graphic.

### Exploration

 People used to believe that the Earth was the center of the Universe. Instead of thinking that the Earth was one of several planets that revolved around the Sun, they believed that the Sun and the other planets revolved around the Earth. Relative to the Earth, what would the path of Mars look like? Press the Put Earth in Orbit button. The graphic to the right will show the relative path of Mars if only the Earth were in orbit. Press the Show EM and Show E'M' buttons to see a segment connecting Earth and Mars; notice that the length and angle of these segments are identical, which explains how the path is being drawn. Press the Put Earth in Orbit button again to stop the orbit of the Earth. Press the Put Mars in Orbit button. The graphic will show the relative path of Mars if only Mars were in orbit. Press the Put Mars in Orbit button again to stop the orbit of Mars. Press the red X in the lower right corner to clear the graphic. Now, think about this: What would the path of Mars look like, relative to Earth, if both planets were in motion? To test your conjecture, press the Put Both in Orbit button. What does the path look like? Was your prediction correct? Reset the graphic and press the Put Both in Orbit, One Revolution by Mars button to see just one loop.