Computing distance as a function of time

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To compute the distance of a planet from its star as a function of time, the semi-major axis (sma) is straightforward for circular orbits, where distance remains constant at 1 AU. For planets with eccentric orbits, distances vary significantly, especially as they travel faster when closer to the star. A numerical method has been used to plot these distances over time, revealing sinusoidal patterns that become sharper with increased eccentricity. The discussion seeks an analytic formula to describe these distance variations, with a suggestion that converting to polar coordinates may provide further insights. Understanding these dynamics is crucial for accurately modeling planetary orbits.
tony873004
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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!
 
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I'm imagining converting them to polar coordinates might yield some insight.
 

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