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Imagine a mission which requires a retrograde heliocentric orbit. How could this be achieved? a direct launch would require ridiculous delta-v.

There was a similar delta-v problem placing Ulysses in a high inclination heliocentric orbit. The problem was solved by using a Jupiter gravity assist.

Could a retrograde orbit be achieved with a second Jupiter fly-by a half a Jovian year later?

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Edit: I anticipate someone will point out that a retrograde heliocentric orbit is as stupid as driving the wrong way around the track at an Indy 500. I thought the orbit would be great for doing an asteroid survey. The orbit would be inclined, to avoid the worst of the traffic, with the nodes in the ecliptic

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Edit ... @SE - Stop firing the good guys suggested several Jupiter passes to reduce the delta-v required on the first leg.

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Yes, your proposed manoeuvre works out geometrically.

To meet up with Jupiter on the other side of the Sun, the probe must be travelling in a roughly circular orbit to arrive at the same time as Jupiter. That means travelling at roughly the same velocity as Jupiter after the first encounter:

velocity diagram

That's an exit velocity of $\sqrt{2} \cdot v_J$ relative to Jupiter.

The good news: It's possible to to turn 90º with such a high relative velocity. The perijove is at 2.2 Jupiter radii, which is close but possible.

The not so good news: $v_{in} = v_{out}$ The relative entry velocity also has to be $\sqrt{2} \cdot v_J$. How are you going to achieve that? If a perfect Hohmann transfer is used, orbiting prograde and slower than Jupiter at perihelion, the relative velocity has to be $v < v_J$, at least 40% too slow. The original orbit would have to be massively out of plane, already retrograde, or have a high radial velocity component at Jupiter distance for such a high relative entry velocity to be possible. The first two options are clearly impractical.

Having a large radial component at the flyby means lifting the perihelion significantly, to 11.9 AU.

high perihelion

That is going to cost you an additional 1.25km/s of delta-v at launch compared to a regular Jupiter Hohmann transfer. Doable, but not cheap. But certainly better than a direct retrograde solar orbit.

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  • $\begingroup$ ... excellent answer. The need for spacecraft speed at first Jupiter encounter ruins my strategy to accomplish it economically. There is also the flip problem: how to scrub off that speed after the second Jupiter encounter. How would you suggest a heliocentric retrograde orbit be accomplished? $\endgroup$
    – Woody
    Dec 29, 2022 at 5:29
  • $\begingroup$ @Woody Perhaps just like this, just adding in a few years of Venus-Earth ping pong to make up for the difference. Or 6 years extra by doing 3 Jupiter flybys with 60º inclination increments. $\endgroup$ Dec 29, 2022 at 16:07
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Jupiter's orbital velocity ranges from 12.4 km/s to 13.7 km/s. To get heliocentric retrograde from a single Jupiter flyby you would need to steer your flyby departing asymptote opposite of Jupiter's motion w.r.t. the Sun (w/ $v_{\infty}$ greater than Jupiter's orbital velocity).

In 2023's Jupiter-direct launch window, trajectories exist with conceivably feasible launch C3 requirements:

2023 Jupiter-direct porkchop plot

(Personal work, filtered for $C3 < 200$ $km^2/s^2$ & $v_{\infty,J} > 12.4$ $km/s$)

individual trajectories

(Personal work, individual trajectories from above)

The remaining question is if it's possible to deflect the trajectory that much at Jupiter. At an unrealistic flyby radius of 1 $R_J$ deflection angles are over 100°, and at a more reasonable 10 $R_J$ deflection angles are a considerable ~60°:

Desmos graph of flyby deflection angle, degrees

(Personal work, flyby deflection angle in degrees (y-axis), flyby $v_{\infty}$ (x-axis), Desmos graph link)

Using the required deflection angle (from the incoming $\vec{v}_{\infty}$ determined by the Earth-Jupiter transfer orbit) and the $v_{\infty}$ magnitude to calculate the flyby close approach distance we get this:

$$r_p = \frac{\mu}{{v_{\infty}}^2} \left( \frac{1}{sin(\frac{\delta}{2})} -1 \right)$$

required flyby close approach plot

(Personal work, individual trajectories)

This shows that a single, very close, flyby of Jupiter can put a spacecraft into a retrograde orbit of the Sun.

Here is an animation of what an example trajectory from this batch looks like:

(Personal work)

Further Details of this specific trajectory:

  • Launch Date: 23-Jul-2023
  • Flyby Date: 09-Jul-2024 (352 day transfer time-of-flight)
  • Launch C3: 188 $km^2/s^2$, Flyby $v_{\infty}$: 19.5 km/s
  • Flyby Deflection: 77.8°, Flyby Close Approach: 196,500 km (2.75 $R_J$)
  • Post Flyby Heliocentric Orbit:
    • SMA: 2.8 AU
    • Eccentricity: 0.79
    • Inclination: 178.7° (retrograde)
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I don't have an explicit answer, but just the observation that orbits are defined most basically by their energy and their angular momentum. If you want to change the energy it's most effective if the delta-V is applied when the velocity is highest (or the radius is smallest). That is the Oberth effect. Conversely, if you interested in changing the angular momentum, then this is best done when the radius is largest (and the velocity is smallest).

To get a retrograde orbit, you want to reverse the angular momentum. One way to achieve this would be start an elliptical transfer orbit towards Jupiter. You want the timing such that the spacecraft at aphelion passes just in front of Jupiter and gets kicked backwards reversing its angular momentum.

However, as @SE pointed out in the comment, it is impossible to actually reverse the motion since any change is with respect to Jupiter's large prograde velocity. However, Jupiter does have several massive moons. The two inner Galilean moons, Io and Europa, have orbital velocities higher than Jupiter's velocity around the sun (i.e. on the sunny side of their orbit, their motion is retrograde with respect to the sun).

So although Jupiter alone may not be able to give a retrograde orbit, perhaps the Jovian system with its relatively massive moons may provide a mechanism.

retrograde solar orbit using Io

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    $\begingroup$ This does not work. You would have to leave the Jupiter system with a higher velocity (v > v_J) than you entered with (v < v_J), which does not conserve momentum. $\endgroup$ Dec 25, 2022 at 20:12
  • $\begingroup$ @SE-stopfiringthegoodguys agreed! $\endgroup$
    – Roger Wood
    Dec 26, 2022 at 1:41
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    $\begingroup$ @RogerWood ... great idea, To work, the flyby would need to generate an acute Asymptote Angle, significantly smaller than 90*. This requires a small Impact Factor (0.5 or less) . This, in turn, requires a very low approach speed or a very dense moon. If the attempted Impact Factor is too low, lithobraking terminates the maneuver. The devil is in the details. Would a very low periapsis combined with an Oberth maneuver pull it off? $\endgroup$
    – Woody
    Dec 29, 2022 at 5:45
  • $\begingroup$ @SE-stopfiringthegoodguys - I'm not sure why you think that's the case? You can certainly use a gravity assist to increase the spacecraft's velocity; that's one of their main uses. Momentum is conserved when this happens because the spaceship is also having an (infinitesimally small) effect on the orbit of the planet, reducing its momentum by an equal amount. en.wikipedia.org/wiki/Gravity_assist $\endgroup$
    – Salda007
    Jan 13, 2023 at 9:10
  • $\begingroup$ @Salda007 I'm perfectly aware. What this does is changing the relative velocity to the flyby body, which is not possible. $\endgroup$ Jan 13, 2023 at 9:32
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Although it has yet to be demonstrated doing such a thing, it should be practical to use a solar sail to do the job. Here's a visualization of the mission trajectory from Solar Sail Trajectory Optimization for the Solar Polar Imager (SPI) Mission:

Visualization of solar polar orbiter mission trajectory, showing solar sail temperature, orbital radius and inclination. Orbit changes from 1AU to .48AU and inclination from 7° to 75°, with the solar sail reaching a peak temperature at aphelion of 520K

a merely highly inclined 0.5 AU radius orbit isn't quite what you asked for, but seems like if it were possible to stick something into a solar-polar orbit in 5 years it shouldn't be beyond the realms of possibility to go all the way over into a retrograde orbit and spiral back out again in a reasonable period of time, if you really wanted.

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