Will the lunar gateway be visible for ground based (amateur) telescopes?
Yes! It will be a bit of a challenge but certainly doable.
The Lunar Gateway will be much further away - about a thousand times
Yes that's roughly right.
From Wikipedia's International Space Station; Sightings from Earth:
Because of the size of its reflective surface area, the ISS is the brightest artificial object in the sky (excluding other satellite flares), with an approximate maximum magnitude of −4 when in sunlight and overhead (similar to Venus), and a maximum angular size of 63 arcseconds.
Five magnitudes is a factor of 100, 7.5 a factor of 1000, so the ISS at the Moon might be as bright as +3.5.
Gateway will be about the same distance from the Sun (thus illuminated similarly) but a lot smaller, so except for flares when the flat solar panels might be perfectly aligned to reflect sunlight towards Earth, I'll optimistically ballpark estimate a maximum brightness of +6 or 10x less.
Your milage may vary as the shapes and surface reflectivity's will be different, but certainly in terms of brightness a small telescope should certainly be able to make it visible if
- already pointed in the correct place at the moment and the positions of nearby circa +6 magnitude stars are charted to serve as a finder chart
- the blinding glare of the illuminated part of the Moon doesn't overpower your field of view. (keep your optics clean, use a steady tripod and high enough magnification to exclude much or all of the illuminated part of the Moon)
Wikipedia's Lunar Gateway; Orbit and Operations says that Gateway's near rectilinear halo orbit will bring it as far as 10° south of the Moon, which solves the glare problem at least.
This is a bit like looking for Uranus (about +5.5 mag) in that you'll need a chart
but different because
- It's brightness will fluctuate a lot depending on its orientation, presenting different surfaces at different angles
- It will move a lot faster relative to the stars, meaning if you find it one night, it won't be there the next night.
You'll need a finder chart similar to those for NEOs or comets that pass close to Earth, with little tick marks along the path with time and date labels. Or you can use something like JPL's Horizons to get a table of RA/Dec positions calculated for your position on Earth and either a telescope with a calibrated digital drive, or you can plot your own tick marks on a fixed star chart.
will it still be possible to track it with (amateur) telescopes (considering both visual observation and imaging)?
Yes. Once you have the relevant start chart you can make visual annotations with pencil or pen, noting the time in UTC and keeping track of your geographic coordinates. Certainly a modest astrophotographic setup, even a good DSLR with a dark sky and fast lens will be able to do it.
In fact with an orbital period of only 7 days it will usually be moving much faster than the Moon moves against the stars, so if you take a series of exposures over one night you'll see big changes in position, and if you do it through a small telescope tracking the stars you'll see a slightly curved line as the object moves relative to the stars even during a single exposure.
The tracking part will be a FUN project! But here's an update:
Thanks to @DanHanson's comment for pointing out that while tracking the gateway will be a bit like a "fast moving" star-like object somewhere between Uranus and Neptune in brightness, getting anything like a resolved image beyond single pixel would be very challenging for existing instrumentation.
First let's look at "prior art" about resolving ~10 to 20 meter sized things or disturbances at the Moon's distance of greater than 360,000 km in Astronomy SE:
and Space SE:
Next a bit of math; 20 meters divided by 360,000,000 meters is 6×10-8 or about 0.01 arseconds ("10 mas" for astrometers)
If we use the canonical Rayleigh criterion of 1.22 $\lambda/D$ and a 500 nm wavelength, we might start seeing something with a diameter (or at least a baseline) above 9 meters.
Let's go directly to our canonical very large telescope, known officially as The Very Large Telescope. This is a cluster of
4 x 8.2-metre Unit Telescopes (UT) (and) 4 x 1.8-metre moveable Auxiliary Telescopes (AT)
that are optically joined via a large optical interferometer (and using adaptive optics) that apparently nobody in Astronomy SE knows how it works, nor can even cite a paper despite a tweet and two bounties! (okay I'll add a third, +400 bounty now)
Wikipedia give's its resolution as
which means if you were able to compete with astronomers around the world doing real science and get observing time to gratuitously look at a spaceship, you'd get perhaps a 10 pixel blob.
Enter the military intrigue aspect
There is an evolving interest in looking at spacecraft, not for science but for security, so there are several efforts to boost ground-based interferometric capabilities both in visible light and via radar.
Let's revisit @Natsfan's answer to Will the Magdalena Ridge Optical Interferometer be able to image extended objects like the surface of the Moon? There's too much helpful information and quotes there to block and double-block quote, but the gist is that imaging spacecraft via optical interferometry is precisely one of the driving uses for the development this instrument (and perhaps other instruments we can't read about so easily).
There's also upcoming space radar systems using flexible clusters of dishes which include beam shaping by coherent transmission of radar pulses from multiple dishes with tens of kilometer baselines. See for example military applications discussed in
Magdalena Ridge Observatory Interferometer computer graphic overlay of the BCF building and the ten telescopes Source