Consider a probe approaching a planet for a hyperbolic flyby manoeuvre;

The eccentricity of the hyperbolic trajectory can be calculated using the following formula:

$$e = 1 + \frac{r_pv_\infty^2}{\mu_1}$$

In the event that eccentricity and periapsis radius are both unknown, however, this formula is insufficient. Does there exist an alternative formula that would allow the eccentricity and the periapsis radius to be worked out separately, or by use of simultaneous equations?

Assume for now that all other necessary values are known, excluding $e$ and $r_p$.

I suspect it could be possible using a known aiming radius $∆$. Where:

$$ \Delta= r_p\sqrt{1+\frac{2\mu_2}{r_pv_\infty^{2} }}, $$

But I unfortunately can't quite get my head around the maths.

  • $\begingroup$ What are the knowns? $\endgroup$ Aug 6, 2015 at 13:21

1 Answer 1


You have answered your own question. It appears that you know $\mu$ and $v_\infty$. Solve your $\Delta$ equation for $r_p$. It's just a quadratic (pick the positive solution). Then from $r_p$, you can get $e$ from your other formula.

  • $\begingroup$ Am I also correct in assuming that $r_p$ refers to the distance between the probe and the centre of the planet? ie. The planet's mean radius + height of flyby at closest point. $\endgroup$ Aug 9, 2015 at 12:35
  • $\begingroup$ Yes, that is the definition of periapsis. Though to be precise, it is the altitude of the closest approach plus the body's actual (not mean) surface radius under that point. So it is simply the distance from the closest approach point to the center of mass of the body. $\endgroup$
    – Mark Adler
    Aug 9, 2015 at 14:08

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