system
Given the following post, respond as if you're commenting on Reddit:
human
How do we realistically measure distances to stars outside of the milky way and how do we prove universal expansion?

I understand the parallax method, but how can that be trusted when we don't know the speeds of the objects we measure or the speeds of the objects we are measuring against. I also know the speed we orbit our Galaxy, and the speed we orbit the sun. I dismiss the speed of Earth's rotation for the sake of the question and I also dismiss the fact that orbits aren't round but rather more like a spirograph. How do we measure distance to a star with any kind of accuracy whatsoever, when were moving so fast inside of two different systems, and we don't know the speed the Galaxy it's in is moving, or it's rotational speed, no do we know the speed that star is orbiting it's Galaxy? Please help me understand this.
gpt
You do actually see stars move along in a twirly "spiral" pattern. The Earth's orbit around the Sun makes the stars appear to move in ellipses, while the star's motion relative to the Sun makes the stars appear to move in a straight line at a roughly constant speed. Over the time periods we're looking at, the stars orbits really are basically straight lines - the curvature as they move around the galaxy is very small if you're only looking over years or decades. We call the straight motion the "proper motion" of the star, while the ellipses are just the parallax caused by the orbit of the Earth. You add this proper motion to the parallax, and you get a twirly type pattern across the sky. 

Depending on how fast the star is moving relative to us and how close it is, these two effects will be different sizes. But we can measure all that and figure out how fast it's going and how far away it is at the same time.

For parallax, we don't need to know about how fast our galaxy is moving, because we're all in the same galaxy. We can only do parallax for quite nearby stars.

To build up for longer distances, we use a "cosmic distance ladder". We can use parallax to find out the distances to variable stars called Cepheids. These variable stars are special, in that their mean brightness depends on how fast they flash. But we don't see the *intrinsic* brightness of a star - we just see how bright it appears at this distance. But if we use parallax to find the distance, we can then work out the *intrinsic* brightness of a star. This tells us the relationship between how fast they flash and how bright they are, intrinsically. And this gives us the next step in the ladder: we can look at how fast a Cepheid is flashing to figure out how bright it is intrinsically, and compare that with how bright it appears from here, to tell us how distant it is. This is what we call a "standard candle". We do the same thing with supernovae for very large distances.

But often we can't use supernovae or cepheids directly, because there isn't usually a convenient one in a galaxy. But we can use the galaxies that *do* have supernovae or cepheids to calibrate our *redshift* scale. The more distant a galaxy is, the more its light is redshifted by the expansion of the universe. If we know the distances of some galaxies directly from cepheids and supernovae, we can use this to work out exactly how distant a galaxy is to get redshifted by so much.

Galaxies are moving around relative to each other, but they are going slowly enough that the expansion of the universe dominates beyond a certain distance. But if you look at nearby galaxies, you don't see uniform expansion - in fact, the Andromeda galaxy is coming towards us - so we can't use those to measure the large scale expansion of the universe. We have to look at larger scales.