We’ve come up with a few different coordinate systems for the Universe, but none of them are really “universal”. By that, I mean that you wouldn’t be able to use them no matter where you are in the universe - most of them only work conveniently from Earth.
Starting small, we’ve got one coordinate system to describe our solar system, with our sun at the center. This coordinate system is easiest to visualize as if you had a birds eye view of our solar system, watching all the planets go around the sun. Every coordinate system needs three coordinates to describe a location - normally we think of up/down, right/left, and forward/back. The three coordinates here are distance outwards from the sun (in any direction), the position of the object in a circle around the sun at that distance (usually measured as an angle from a line drawn from the position of the Earth in September to the sun), and distance away from the plane described by the orbits of the planets. Perhaps unsurprisingly, this is only really a useful coordinate system for objects in our solar system.
We also have a galactic coordinate system, which is not centered on the center of the galaxy as you might expect, but instead, it’s still centered on the location of the sun. The second coordinate here is the distance “up” out of the disk of the Milky Way, and the third is distance from the sun once again. The disk of the Milky Way is not in line with the disk of our solar system, which you can spot if you go outside in a dark sky. The Milky Way usually goes North-South in the sky, and all the planets (and the moon) will always be found in a line that goes East-West. This coordinate system is generally considered to be confusing to work with, and is mostly not used, unless you’re trying to map the Milky Way itself.
By and large, what astronomers use to describe the locations of things in the sky is actually a projection of the Earth’s latitude and longitude lines onto the sky. Take the line between the north and south poles, and extend them out into the sky - those are the northern and southern celestial poles. (The north celestial pole points almost exactly to the North Star.) The equator, expanded outwards in a plane from the surface of the earth, describes the Celestial Equator. The units we then use to describe the positions of objects are right ascension, declination, and distance (if we have it). Declination is similar to latitude - it describes the angle above or below the celestial equator. Right Ascension is like longitude, but instead of being measured in degrees, is measured in hours, minutes, and seconds. We do this because we know how fast the earth rotates - 360 degrees in 24 hours. This means that the earth rotates through 15 degrees in an hour, and we can easily tell how long we need to wait for another object to be overhead. If an object at the 0 hour coordinate is overhead right now, objects at the 3 hour line will be overhead in three hours. This coordinate doesn’t tell you anything about the absolute position of an object in the Universe, but it’s very good at describing where that object appears to be placed in the sky.
Using the earth as the zero point starts to make more sense once you start looking at things that are very far away, because at that point, you start looking increasingly far back in time. Looking very far away in the universe is looking backwards through shells of time, and since we are observing from Earth, those shells are by definition centered on the Earth - the objects that we see as five billion years old are being viewed at that age because it took light that long to get here. The speed of light is the same in every direction, so you’ll see the same age in every direction at that distance, from the perspective of the Earth. This would be true of anyone observing from anywhere in the universe, but they might see a slightly different set of galaxies.
Because we only have coordinate systems that are centered on the earth or the sun, if you were elsewhere in the galaxy, or elsewhere in the universe, you’d have to constantly convert your location into where you would appear to be (from the perspective of the earth or the sun) if you wanted to use one of these systems. I suspect that whenever we do start exploring outside our solar system, we will come up with another coordinate system that’s convenient for keeping track of our spacecraft, but we’ll keep all the old ones we’ve come up with “for historical reasons”.
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