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The eccentric anomaly,$E$, is defined via the equation,

$$ r = a(1-e \cos E) \tag{1} $$ for radius $r$, semi-major axis $a$ and eccentricity $e$.

It is ALSO defined in terms of the Kepler Equation as,

$$ E - e \sin E = \frac{2 \pi t}{T} \tag{2} $$ for time $t$, orbital period $T$.

Now I have some data $r(t)$, where I know the orbital parameters $a,e,T$.

From my data, if I calculate $E$ from the first equation, I can calculate the LHS of Eq. 2.

I can also calculate the RHS of Eq. 2 directly from my data.

However when I do this LHS $\ne$ RHS: The RHS increases ~linearly, whilst the LHS is ~ oscillatory. Given the form of these two equations, this is what I would expect, but how can I get the two to agree?

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$E$ is monotonic like $t$. It isn’t limited to be between $0$ and $2\pi$.

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  • $\begingroup$ So does that mean it is not possible to calculate $E$ from equation 1, given $r(t)$? $\endgroup$
    – user1887919
    Commented Apr 1, 2020 at 19:15
  • $\begingroup$ (1) determines $E$ modulo $2\pi$. It doesn’t know how many times you’ve gone around the ellipse. $\endgroup$
    – G. Smith
    Commented Apr 1, 2020 at 19:18
  • $\begingroup$ What are you actually trying to do? If you already know $r(t)$, why would you care about the eccentric anomaly? Usually the point of using these two equations is to calculate $r(t)$, but I assume you have done that instead by numerical integration of the equations of motion. $\endgroup$
    – G. Smith
    Commented Apr 1, 2020 at 19:21

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