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.
