# Travel time from lunar orbit (EML2) to Jupiter using skyhook only?

I am doing background research for a science fiction story, and hoping someone can help with this navigational question.

Voyager 2 reached Jupiter from Earth in about 2 years. However (if I understand correctly) it used a powerful boost from its propulsion system to accelerate it along the desired trajectory after it cleared Earth's gravity well.

If there were a skyhook in a halo orbit around Earth-Moon L2, above the dark side of the Moon - could a probe launched from the surface of the Moon use only the skyhook to accelerate it onto the proper trajectory to reach Jupiter?

If this is possible, how long might such a probe spend in transit?

• Would a Venus gravity assist (or even a couple) be allowable. At the cost or a couple of extra years in transit, most recent probes get to Jupiter by a series of Venus and Earth gravity assists. This dramatically reduces the requirements for the initial propulsion. Oct 20, 2020 at 8:25

# Transfer

Regardless of how escape from the Earth system is achieved, orbital mechanics pose some restrictions on travel time.

The minimum velocity transfer possible is an elliptical transfer orbit touching Earth's orbit when closest to the Sun, and Jupiter's orbit when farthest from the Sun. The transfer time is then 940 to 1060 days, depending on where in its orbit Jupiter is.

"reaching" Jupiter is easy, "staying" there is difficult. Flying past Jupiter quicker than the minimum velocity transfer is possible, but the relative velocity to the Jovian system grows a lot the more you cut down on transfer time.

If the goal is to fly past the planet, like Voyager did, this doesn't matter. But if you want to get into orbit, you don't want any transfer that's quicker than a thousand days, unless your spacecraft has:

• Large propulsive capabilities
• A very capable heat shield for Jupiter aerobraking
• Or some skyhook system in the Jovian system waiting for it

# Skyhook

Sadly, this may not be the proper location of a skyhook. The minimal velocity after Earth escape for a Jupiter transfer is 8790 m/s. EML2 isn't very deep in Earth's gravity well, so the skyhook must provide at least 7900 m/s of $$\Delta v$$ on top of its own orbital velocity.

non-rotating case

If centred at EML2, it has to be at least seven times the distance to the Moon in length, and in that case there's not a lot of extra effort anchoring it to the Moon to turn it into a Lunar space elevator. And in that case, it's not an EML2 skyhook anymore.

A non-rotating lunar skyhook is still plausible for reaching Jupiter, but it must be placed in low lunar orbit (it would still have to extend over 7000 km vertically, and be barely within the limitations of current materials).

rotating skyhook

7900 m/s tip velocity is a lot.

Consider for instance the acceleration experienced at the tip:

$$a = \frac{v_{tip}^2}{r}$$

Even if it's a thousand kilometres in radius, the probe is still experiencing over 6 Gs!

A subtler issue is the strength of the tether. The integrated acceleration it has to withstand is independent of tether radius, and proportional to the square of the tip velocity:

$$\int_0^r a(r) dr = \frac{v_{tip}^2}{2}$$

The cross section ratio between the centre of the tether and the tip is then:

$$e^{\frac{\rho v^2}{2T}}$$

Where $$\rho$$ is the tether density and $$T$$ its tensile strength. Even with our best currently available materials, that's a cross section ratio of about 4000.

Conclusion

An EML2 skyhook does not fulfil the minimum requirements of sending a probe to Jupiter on its own. This is not to say that this can't be done with a skyhook, but EML2 is clearly not the right place for it.

In general, tethers generally scale badly with high velocity requirements, proportional to $$e^{v^2} v^2$$. That's a much worse asymptotic behaviour than the $$e^v$$ of the rocket equation, which is already bad enough. For low to medium velocity requirements, they are excellent, for Jupiter transfer not so much.

• On the bright side, since the OP stated "SciFi," he may have some supermaterials to build his sky hook out of. Oct 20, 2020 at 11:58
• You're figuring a minimum-energy transfer. Nothing (other than tether strength) precludes throwing it faster. Oct 21, 2020 at 3:24
• @LorenPechtel Yes, I decided to stop at the first roadblock. Oct 21, 2020 at 6:45
• Thank you! I really appreciate this in depth discussion of the mechanics involved. It sounds like what may work the best is booster rockets on the launch side, with skyhooks coming into play to "catch" probes rather than "throw" them. Oct 21, 2020 at 14:30
• @CrystalEidson You have the same problems with "catch". It's more like skyhooks have a "ceiling", above which you have to use rockets. Oct 21, 2020 at 14:42