Computing distance as a function of time

[tony873004]

I want to compute the position of a planet from its star as a function of time.
Here is an illustration describing the problem: http://orbitsimulator.com/PF/pft.GIF

Each of these 5 planets has a semi-major axis of 1 AU and a period of 1 year. The case of the planet in a circular orbit is easy. It's distance doesn't change, so for this planet, d(t)=sma, where sma is the semi-major axis of the planet.

But the other planets have eccentric orbits, causing their distances to vary. They travel fastest when near the star.

I made a plot of their distances vs time using a numerical method. But I'd like to what analytic formula could give me these distances as well. Here's the graph: http://orbitsimulator.com/PF/pft2.GIF
The y-axis is in meters, the x-axis in days. The names of the lines reveal the eccentricity of the planet (p8 is 0.8).

The graph for p2 looks like a sin wave, but the higher the eccentricity, the pointier the bottom of the sin wave. Is there a formula to describe these curves?

Thanks!

 

[DaveC426913]

I'm imagining converting them to polar coordinates might yield some insight.