Benefit of sling shot effect with a space elevator

Question

The upper end of a space elevator is moving considerably faster than orbital speed at that distance from Earth. As a result anything released from the top of elevator would get thrown away from the Earth instead of just entering into high Earth orbit. How much would being released from the end of a space elevator at twice geosynchronous radius reduce travel times or the amount of rocket thrust needed during Earth departure for interplanetary probes.

I'm assuming the cable is capable of damping out the oscillations caused by releasing the payload. Doing so is necessary to release a payload at any altitude other than geosynchronous altitude; due to the amount of activity currently occurring at significantly lower altitudes being able to do so seems to be a necessary requirement for constructing an elevator in the first place.

Answer 1

Your total energy is ${v^2\over 2}-{\mu\over r}$. If that's negative, you're still in orbit. If positive, you've escaped.

At the point of release, $r$ is twice geosynchronous radius, $2\times 42164\,\mathrm{km}$, and $v$ is twice geosynchronous velocity, $2\times 3.075\,\mathrm{km\over s}$. The $\mu$ for Earth is $398600\,\mathrm{km^3\over s^2}$. The result is $14.18\,\mathrm{km^2\over s^2}$.

You have escaped.

The result is equal to your $v_\infty^2\over 2$, so your $C_3$, which is $v_\infty^2$, is $28.36\,\mathrm{km^2\over s^2}$.

Not enough to get you directly to Jupiter, but you'll get well past Mars' orbit or well inside Venus' orbit, depending on your preference. A judicious set of Venus and Earth flybys could then get you to Jupiter. From there you could go pretty much anywhere, including potentially escaping the solar system. Also a Venus flyby could get you to Mercury.

You will be constrained on the departure plane, but perhaps a lunar flyby could help get you going in the direction you want.

The $\Delta V$ required to get to that $C_3$ from low-Earth orbit is about $4.5\,\mathrm{km\over s}$. Or from GEO, about $3.3\,\mathrm{km\over s}$ (by lowering periapsis to $150\,\mathrm{km}$ altitude and escaping from there). So it buys you a lot. Though that's one hell of a cable you'll have to build.

Answer 2

How much would being released from the end of a space elevator at twice geosynchronous radius reduce travel times or the amount of rocket thrust needed during Earth departure for interplanetary probes?

There are a lot of variables in this question and the answer, even given the clarity of the length of the elevator. The major variable being the reduction of non-defined values for times or thrust. So I will just focus on the velocities and destinations obtainable at different release elevations, with no added fuel costs.

Where Earth geosynchronous orbits, whether circular or elliptical, have a semi-major axis of 42,164 km (26,199 mi), Twice would equal 84,328 km (52,398 mi).

I personally don't have the math to define the exact velocity imparted at 84,328 km, and my research did not find a calculated value for this. But values for just above and below this are available.

Wikipedia lists a few "free" destinations at release heights just above GSO. An object attached to a space elevator at a radius of approximately 53,100 km will be at escape velocity when released. Transfer orbits to the L1 and L2 Lagrangian points can be attained by release at 50,630 and 51,240 km, respectively, and transfer to lunar orbit from 50,960 km

For destinations beyond the earth moon system, the calculations get a bit more complex, as you need to also consider the sun. In the same manor the elevator gives velocity in relation to it's height above earth, its position in relationship to the sun also effects relative velocity.

An elevator cable of somewhat over 100, 000 km in length should suffice as a sling to launch spacecraft to Jupiter (at the outer end) and Mercury (at the inner end). Reaching Jupiter is critical, because we can take advantage of Jupiter’s gravity assist to send spacecraft further outward or even beyond the solar system.

Aravind, P. K. (2007). "The physics of the space elevator". American Journal of Physics (American Association of Physics Teachers)

As we see velocity or obtainable destinations is simply a matter of height. Or said another way, the sky really is not the limit, the higher you go the faster and the farther you can fly.