Calculating the periapsis radius of a flyby manoeuvre without knowledge of the eccentricity of hyperbolic trajectory

Question

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

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

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

• $v_\infty$ refers to the probe's hyperbolic excess velocity.
• $\mu_1$ refers to the standard gravitational parameter $GM$ of the primary.
• $r_p$ refers to the periapsis radius.

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.

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.