Will the Artemis Gateway Lunar Space Station be visible from Earth by amateur observers?
If they can see +15 magnitude stars routinely, they'll probably be able to see Gateway.
Answers to Good source for the relationship between absolute magnitude, diameter, and albedo? give a simple expression for absolute magnitude for solar system (reflective) bodies which is defined as the apparent magnitude of the object at 1 AU from the Sun and 1 AU from the viewer. From @MikeG's well-sourced answer:
$$ H = C - 5 \log_{10} D - 2.5 \log_{10} p_V$$
where $H$ is absolute magnitude, $p_V$ is albedo, and $C$ = 15.618
and D is the spherical diffuse object's diameter in kilometers.
I'm going to model the gateway as a 5 meter diameter spherical cow and ignore any potential flares from any solar panels, heat radiators, windshields, or other flat surfaces for now.
With an average albedo of 0.5, that gives us an absolute magnitude $H$ of +27.9. Now Gateway will presumably stay at 1 AU from the sun, but remain a heck of a lot closer than 1 AU from us. We can scale using inverse square and get the visual magnitude as follows:
$$V = H + 5 \log_{10} \left(\frac{r}{1 \ \text{AU}}\right)$$
With a lunar distance $r$ of 400,000 km or 0.00267 AU, that gives an apparent spherical cow magnitude of +15.
If we can comfortably see +5 magnitude with a 6 mm night-adapted pupil, then to go another 10 magnitudes requires about a factor of 100 in diameter, or a 60 cm (24 inch) aperture.
Maybe Gateway will be brighter, maybe you can see +6 or +6.5 stars, but it's going to be a 30 to 60 cm (12 to 24 inch) amateur telescope if you want to see it in the eyepiece.
An excellent follow-up question would be how far might Gateway move relative to the background stars in a 10 minute exposure!
If you are an astrophotographer then a small telescope is all you'll need if you have a good camera and good astrophotography technique.
Per sources in my answer to Is it possible to capture geostationary satellites with DSLR? a big communications satellite in GEO can brighten by 5 or 6 magnitudes during a flare.
Certainly Gateway (like any space station) will be quite power-hungry like a global communications satellite, so I think we can expect flares -- if they do happen say during a full Moon when anything near the Moon that's pointing at the Sun will also point roughly towards Earth -- will be fun to watch for.
Could we track it as it goes across the terminator and is still lit by the Sun?
Now that's an interesting question!
1 arcsecond is sort of a nominal best resolution from the surface of the Earth for a telescope without dynamic wavefront correction. The technique of lucky imaging might help here.
The disk of the Moon has an area of about $2.5\times 10^6$ arcseconds. Its maximum (full) and minimum (new) magnitudes are -12.9 and -2.5. To get the area brightness in funky sounding but traditional units of magnitude per square arcsecond, we add $2.5 \log_{10}(2.5\times 10^6) \approx +16$ giving +3.1 and +13.5 mag/arcsec.
This means that the light reflected from the tiny, deep sub-pixel spacecraft will barely brighten a pixel in an image of even the unlit area of the Moon.
(Unless of course there is a flare!)
Deep sub-pixel means it will not significantly darken a pixel in an image of the bright, lit side of the Moon.
Therefore you'll probably need to wait until Gateway is away from the disk of the Moon and against dark space to see it.
Images of a satellites in Earth orbit crossing the Sun look the way they do because they are resolved images
or much closer and brighter than Gateway
Further reading:
