What is the characteristic time to reach a hyperbolic excess velocity within a sphere of influence?

Question

When departing a planet using a hyperbolic trajectory, one should ideally reach the sphere of influence boundary with an excess velocity needed for interplanetary transfer. Of course, the true excess velocity is only reached after infinite time has passed. But how much time does it require for the spacecraft's velocity to decay to within a reasonable threshold relative error of the excess velocity (within its sphere of influence)? What percent error should one expect when reaching the boundary of a planet's sphere of influence?

Answer 1

tl;dr: For a trip from Earth to Mars, I get a velocity 1% greater than $v_{\infty}$ at point about 70% to the way to the Earth's SOI.

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The question is about departing a planet and proceeding to interplanetary transfer, and since the term "sphere of influence" is invoked, presumably the patched conics approximation is assumed, so we can think of the Earth as the only source of gravity, and work in the rest frame of the Earth.

The vis-viva equation gives everything we need.

$$v^2=GM_E\left(\frac{2}{r}-\frac{1}{a}\right).$$

If we start from a circular orbit in LEO with a radius $a_0$, the velocity is

$$v_{LEO}=\sqrt{\frac{GM_E}{a_0}.}$$

After a propulsive $\Delta v$ the velocity is

$$v_0 = v_{LEO} + \Delta v.$$

Rearrange the vis-viva to solve for the new semi-major axis of the hyperbola:

$$a_{hyp}=1/\left(\frac{2}{a_0}-\frac{v_0^2}{GM_E}\right).$$

For a hyperbolic orbit, the semi-major axis is negative.

Now you can get the velocity (speed really) at any distance r:

$$v^2=GM_E\left(\frac{2}{r}-\frac{1}{a_{hyp}}\right),$$

$$v(r)=\sqrt{GM_E\left(\frac{2}{r}-\frac{1}{a_{hyp}}\right)},$$

and taking the limit of $r \rightarrow \infty$ we get $v_{\infty}$

$$v_{\infty}= \sqrt{\frac{GM_E}{-a_{hyp}}}.$$

Incidentally, if you want to calculate kinetic and potential energy of a spacecraft with mass $m$ at any distance $r$ you can just use:

$$T(r)=\frac{1}{2}mv^2(r),$$

$$U(r)=\frac{-GM_E m}{r},$$

and the sum of the two should stay constant. That's actually the soul of the vis-viva equation!

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OK, now let's put in some numbers;

$GM_E$ = 3.986E+14 m^3/s^2

$GM_S$ = 1.327E+20 m^3/s^2

$a_0$ = 6,378,000 + 250,000 meters

$\Delta v$ = 5,7000 m/s

$AU$ = 1.5E+11 meters

$R_{SOI} = AU\left(\frac{GM_E}{GM_S} \right)^{2/5}$ ~ 9.27E+08 meters

I get $v_{\infty}$ of about 7,795 m/s, and a velocity of 1% above that at a distance of 655,000 km from earth, or about 70% of the way to the edge of the sphere.

Answer 2

Zero.

The actual excess velocity is reached right upon leaving the planet's sphere of influence. Since then the velocity starts changing (in an arbitrary direction) primarily according to influence of the body whose sphere of influence the craft entered.

Of course that's a simplification - in fact both (as well as all other) bodies influence the velocity at all times, it's just that the strongest influence comes from the body whose SOI you're in, but for purpose of this explanation, let's assume Patched Conics instead of N-body model; the difference is moot for this purpose.

Since you're always in some SOI, it's pointless to think of "true excess velocity" relative to one SOI after traveling ways through another SOI. You have the (one, actual, true) excess velocity upon reaching the edge, then you just have the velocity in the other SOI, and it may differ a lot, and you may still get the relative velocity to the first body if you're inclined to, but it won't be any special "true" one, it will just be a different one.

Say, you escape Moon SOI on the Free Return trajectory. What will be the "true excess velocity" if you're on an Earth reentry trajectory? What velocity are we supposed to calculate the error relative to?

And even if you go into deep space, eject out of Solar System towards Proxima Centauri, you'll remain in Sun's SOI until roughly halfway through, when you'll enter Proxima's SOI - and at that point you'll cease to lose velocity as you'll start getting pulled towards Proxima.