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As you may know, as eccentricity gets smaller, the location of apoapsis/periapsis becomes less well-defined. Rather equivalently, I've noticed that if eccentricity is < 4e-7 when converting from Keplerian elements to Cartesian and back again, significant errors start to creep in.

I'd like to know how small of an eccentricity my code should realistically be expected to handle. I know these issues are handled well by equinoctial elements, but would rather avoid that complication if possible.

Wikipedia says, "Neptune's largest moon Triton has an eccentricity of 1.6 × 10−5, the smallest eccentricity of any known body in the Solar System."

That's very fine and interesting, but what about artificial satellites? What is the smallest eccentricity achieved by one of those?

GPS Pilot
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How about zero?

There are artificial satellites which are placed in circular orbits, particularly geostationary orbits where the satellite stays over the same spot on the earth's surface.

https://en.wikipedia.org/wiki/Geosynchronous_orbit

user16622
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    No satellite has exactly zero eccentricity (or exactly zero inclination). Geostationary orbits have small eccentricity and small inclination, but not zero. – GPS Pilot Apr 26 '16 at 01:27
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    @GPSPilot: There will be deviation from perfect circle of the orbit caused by non-uniformity of Earth's gravitational field and influence of other bodies (Moon in particular), but they hardly qualify as eccentricity, as the deviation from the circle is no longer the shape of an ellipse, and will vary from one orbital period to another, as the influence shifts; trying to describe it in terms of ellipse would be a misguided endeavor. – SF. Apr 26 '16 at 09:43
  • Nothing like this is exactly zero. Even geo-stationary satellites drift a little, which is how eccentricity would manifest itself when viewed from earth. Such satellite require occasional corrections. – Olin Lathrop Apr 26 '16 at 11:32