As I understand, to reach a Lagrange point the spacecraft would need to slow down. Also, can spacecraft passing nearby Lagrange points get captured within the point?
$\begingroup$ No exact answers here, but there may be some helpful information: Did DSCOVR travel “along the stable manifold of it's future SE L1 Halo orbit” to get there? and Do Lissajous orbits have stable/unstable manifolds? and Rendezvouses in halo or lissajous orbits $\endgroup$– uhohFeb 24, 2021 at 7:51
$\begingroup$ There are stable and unstable LPs. If your relative speed is low enough, then yes you can be "captured" at a stable LP. $\endgroup$– Carl WitthoftFeb 24, 2021 at 12:41
$\begingroup$ @CarlWitthoft The sable Lagrange points currently are not of interest. The current points of interest are the unstable L1 and L2 Lagrange points of the Sun and the Earth, and of the Earth and the Moon. $\endgroup$– David HammenFeb 25, 2021 at 13:58
$\begingroup$ @DavidHammen ok, but of interest to whom? The OP didn't specify $\endgroup$– Carl WitthoftFeb 25, 2021 at 14:15
$\begingroup$ @CarlWitthoft Of interest to current spacecraft designers. The OP appears to be subject the widespread misconception ("how to spacecraft reach Lagrange points") that spacecraft are at those Lagrange points. This is not the case. They instead go into pseudo orbits about those Lagrange points, and this most likely would also be the case with the L4 and L5 Lagrange points. $\endgroup$– David HammenFeb 25, 2021 at 15:36
How do spacecraft reach Lagrange points?
First things first: Spacecraft don't go to the various Lagrange points of interest. They instead go into pseudo orbits about those Lagrange points.
How do spacecraft reach Lagrange points?
Either fuel efficiently, but slowly, or by brute force.
Consider the Earth-Moon Near Rectilinear Halo Orbits (NRHOs) that are of high interest to NASA as of late. There are paths that can bring a spacecraft to an NRHO with rather small propellant expenditures. Unfortunately, it can take a long time (several weeks to months) to follow such trajectories. These long duration paths might be used to deliver automated vehicles to NRHO.
But that long transfer time won't be good if crew are aboard the vehicle. That's when brute force is needed.
$\begingroup$ How do the spacecrafts enter orbit around the lagrange point then? I know its not a "point" rather just a large area of space, but how would a spacecraft enter a pseudo orbit in that viscinity? $\endgroup$ Mar 17, 2021 at 9:26
As I understand, to reach a Lagrange point the spacecraft would need to slow down.
If it's launched from the Earth it does not need to use propellant to slow down. NASA explains this in its article about the James Webb Space Telescope which will orbit the Earth-Moon L2 point.
It will take roughly 30 days for Webb to reach the start of its orbit at L2, but it will take less than a day to get far away from Earth and much of the way there. Getting Webb to its orbit around L2 is like reaching the top of a hill by pedaling a bicycle vigorously only at the very beginning of the climb, generating enough energy and speed to spend most of the way coasting up the hill so as to slow to a stop and barely arrive at the top.
After being launched into low Earth orbit by the first stage of an Ariane V, the second stage will "pedal vigorously" to send the JWST "uphill" towards L2. Just 30 minutes after launch JWST will separate from the second stage and, aside from small mid-course corrections, coast for the next 30 days until it slips into orbit around L2 1,400,000 km away. There's a nice animation of the flight. After 1 day it is 200,000 km from Earth. Day 3 it is 400,000 km passing the Moon. Day 5, 600,000 km. On day 6 it will pass the halfway mark with 24 days still to go.
L2 is metastable and JWST will have to use some propellant to maintain its orbit. Once its propellant is exhausted in (hopefully) 10 years it will drift away from L2 into its own orbit around the Sun.
Can spacecraft passing nearby Lagrange points get captured within the point?
Yes, particularly the stable L4 and L5 points. This is how we get trojan asteroids. Though being "within a point" is a contradiction. Instead they orbit around the point in a sort of kidney bean shape.
as a body moves away from the exact Lagrange position, Coriolis acceleration curves the trajectory into a path around (rather than away from) the point.
Supplemental answer to @Schwern's answer
The GIF below shows SOHO traveling out to it's halo orbit around Sun-Earth L1. JWST will do something that looks similar, but towards Sun-Earth L2.
Why does the trajectory nicely dovetail into the halo orbit?
This is really interesting. If you first put the spacecraft in its halo orbit, but just a tiny bit too close to the Earth, say a few kilometers compared to a 200,000 or 400,000 km diameter orbit, then it would start drifting towards the Earth as it orbited with an exponentially increasing distance from it's ideal halo orbit. That deviation would double every few weeks (the whole orbit is about 6 months) so after one orbit it would be 10,000 km closer to the Earth, and on the second orbit it would spiral completely away from the halo in a path that looks just like this!
The cool thing is that for two body and for circular restricted three-body problems you can run them backwards and forwards in time equally well.
So if you put a spacecraft on a trajectory near Earth that matches this escape trajectory, it will naturally follow this path and wander up to the halo and nicely fall into place. After one or two orbits it will snuggle up to it's halo and just need small station keeping maneuvers to stay there.
This is what the NASA folks are taking about in the block quotes in the other answer.
The items below are from this answer.
This is plotted data from Horizons from Is this what station keeping maneuvers look like, or just glitches in data? (SOHO via Horizons) using a script like this: https://pastebin.com/7XULFDea written when I was just starting to learn Python. Still images combined into a GIF using ImageJ. Data from https://ssd.jpl.nasa.gov/horizons small black dots are 1 day intervals, red dot is Sun-Earth L1 and blue blob represents the various places Earth is relative to L1.
1$\begingroup$ The statement is even very strong: Time reversibility implies unstable L-points must have nice dovetail transfers. $\endgroup$ Feb 27, 2021 at 18:33