If we were in a spaceship and somehow appeared in an unknown galaxy and solar system, what real-life instruments and techniques could be used to orient ourselves?
In both cases, you're pretty screwed, a bit less so if you're in another solar system.
Looking at star positions or galaxy positions could work, but only if you can identify the galaxies you're looking at. But I imagine you'd need quite a bit of time on a supercomputer to work out your position.
If you're in another solar system, I'd look for pulsars.
How quickly would it be possible to measure our position relative to the other bodies and trajectories and velocities? Would this process take a long time even with our current technology? I know with the physics we know today we can calculate most things from formulas but that's using data that has been accumulated over centuries.
With the entire astronomical infrastructure of Earth, it'd still probably take months, if not days to get a good position fix. Decades to get velocity.
So if our ship appeared somewhere completely random, how quickly do you think we could build a decent understanding of the solar system and our relative orientation to it?
Well, the solar system itself is pretty easy. Given the infrastructure, you'd have a decent accuracy within a year or so.
Of course, keep in mind that the probes we send to outer planets in our own solar system still refine the accuracy of orbits even today.
And if possible, what instruments would we use?
Visual and radio telescopes would come in handy, if you were looking to compare star patterns, then wide-angle telescopes and super computers. Another handy tool to use would be spectroscopes. You could probably identify certain stars by their light spectrum.
Of course, something like a paralax measuring space telescope would be nice...
If you're in or near a star system, finding the planets is "easy" with a good enough telescope scanning your surroundings. You should also be able to estimate their orbits and masses with a few days' observation.
Finding the planets is not as easy. Something big and shiny like Venus, Jupiter and Saturn are easy to detect, but you can't guarantee that the star will be bright and the planets large and reflective. You'd need a infrastructure for fast sky survey. Our quickest surveys are capable of scanning the entire sky in a matter of days, but you'd have to do it repeatedly to detect a planet. It'd take a few months of data to get accurate orbits.
Stars would be more difficult, but again not impossible with a good telescope. After you've determined your own orbit, you'd probably be looking for stellar parallax, which could take a while longer, at least a few orbits around the star for reliable measurements.
That would help you determine the distance to the nearest stars, but that doesn't help you much. Our best attempts can only measure to about 100 light years out.
It's like waking up in an unknown city somewhere on Earth. You know the distance to the nearest buildings, but that doesn't help you get home. You need a database of buildings both near and far or even better, landmarks like tall iconic mountains, rivers,...
