We can calculate the expected wavelength of light emitted from a star, so we can calculate the amount by which the wavelength has changed because the star is moving away. That can be used to let us calculate the speed at which the star is moving away from us. If we know how far away that star is, we can calculate the speed of the expansion per unit of distance.
How do we get the expected wavelength in the first place, if the only way if measuring it is by observing the changed wavelength?
The answer to that is a bit less ELI5, but there are more equations that use the fact that the wavelength of light emitted by a star is inversely proportional to the surface temperature, and we know the constant that relates them because it’s the same for every star. We then use a few other equations connecting the temperature, radius and the intensity of the light that reaches us to calculate the emitted wavelength from all of that.
my best attempt at ELI15: hot things glow because their electrons get energized, jump up to a new orbit level, then fall back down - and shoot off a photon when they do. The wavelength of photon shot out is equal to the energy difference between the orbit levels. These orbit levels are particular and finite, so a given atom shoots off a finite number of wavelengths of light.
Hydrogen is very simple, with one electron. It only ever shoots out light in only like 5 wavelengths, which each correspond to its one electron jumping up to each possible orbital. Here's an image of what that looks like, through a special prism. Larger atoms have way more electrons and way more ways for electrons to jump up and down, so it gets complicated very fast, but every element has a signature set of wavelengths that it shoots out when heated.
This means that a) the Sun is not giving us a complete, smooth spectrum of light, but a ton of different specific wavelengths that our eyes interpret as essentially smooth, and b) we can tell what elements are in a star by measuring the specific wavelengths of light from that star. If the light passes through the atmosphere of a moon or planet, we can see what lines are absorbed to learn about the atmosphere there.
Wr/t expansion: hydrogen is most of every star, and a star moving very fast away from us "stretches" light it emits (not exactly like the doppler effect, but close enough for ELI5), shifting those signature wavelengths longer, towards red. The level of "redshift" can be measured very precisely, and it tells us how fast something is moving away from us. These measurements line up very closely with the few other ways we have to estimate distance - more distance = more redshift and moving away from us faster. The best explanation for this is that space is expanding, everywhere, and carrying other galaxies away from each other (this is also the best explanation for a bunch of other things, which is a good sign it's right). Small differences in redshift also let us measure how fast a galaxy is rotating, or how fast stars are orbiting a black hole, without having to wait 100,000 years to watch them move.
For example, hydrogen emits a very specific frequency of light - it's also the most common element in stars. We can also reproduce this on Earth.
there are several ways, and unfortunately they give different answers. anywhere between 65Km/s/Mpc to 75Km/s/Mpc, but with accuracy of about +-1Km/s/Mpc, so there is a bit of a problem.
The method I know is that as light moves through space that is expanding, its frequency changes. And as light moves through a space containing hydrogen atoms (including the stars that created the light), specific frequencies are absorbed. These frequencies are very specific, and very well known. So, if you look at a distant galaxy that should be emitting white light, you look for absorbed frequencies. These will have shifted based on how fast the universe is expanding. the bigger dips will be from the star its self, but there will be smaller dips from interstellar hydrogen atoms. This lets you measure how fast that galaxy is moving away, and the rate at which the universe is expanding between here and there.
The book, ‘The First Three Minutes’ speaks a lot to this. Essentially it’s similar to Doppler effect but with light, radiation, and radio waves. We measure these frequencies and compare the data against other known constants
It is measured through red-shift. This is like the doppler effect (where sounds get higher in pitch coming towards you, and lower going away). However, it's the shift in light. If you know what wavelength some light is supposed to be, but you see it is shifted to longer wavelengths, you know it is going away from you. Red light is longer wavelengths, which is where the name comes from, but the light isn't necessarily red. You also need to know how far away the thing you're looking at is. If you get the red shift and distance for lots of things, you can do math to figure our how fast things are moving away per how far away they are. aka how fast the universe is exapnding
Universal expansion doesn't have a speed so much as a rate (or frequency). It isn't measured in miles per hour or m/s, but in "per second." Or more conventionally, km/s / Mpc (kilometers per second, per Mega-parsec).
The current value for the "Hubble parameter" (or Hubble not-quite-a-constant) is around 70 km/s/Mpc. Meaning that if you have two things that are 1 Mega-parsec apart, the distance between them will be increasing at a speed of 70 km/s.
In SI base units this works out as about 2.3 x 10-18 /s. This is a rate, not a speed as there is no distance. Given any (huge, inter-galactic) distance, a second later that distance will have grown by a factor of about 2.3 x 10-18. The "speed" depends on the current distance. Objects that are further apart are moving away from each other faster than objects that are closer together.
Measuring this is tricky. Just looking at the Wikipedia page you can see a whole range of measurements, many disagreeing with each other.
In simple terms, you measure the Hubble Parameter by finding something in the sky that is a long way away, where you know how far away it is, and you measure how fast it is moving away from you (using red shift). You then divide the speed by the distance, to get the Hubble Parameter. And then repeat for as many objects as you can find, hopefully getting the same number for each.
And then you do a whole bunch of maths to correct for various things, including the finer details of whatever cosmology model you are using.
This article has some more details on how the Hubble Parameter is measured - looking at how we figure out how far away things are, and how we measure how fast they are moving away from us.
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My understanding is that even in an expanding universe, light still obeys its speed limit and therefore they can still use the speed of light to measure how it's expanding.
Scientists divide how fast things are moving away from us by how far away they are.
Certain types of celestial bodies are really useful for this. We know from observation that certain types of supernovae all have a very similar brightness and frequency of light. But brightness lessens predictably with distance, so we can then tell based off the difference what that brightness should be at the source versus the observed brightness we see via instruments how far away they are. Further, since light from a source which is moving away from you appears slightly redshifted, we can tell based on how redshifted the supernova is how fast it is moving away from us.
Now, it's possible that any given supernova's red shift could be explained by it simply moving away from us. And some of them will be moving away from us less and some more. But basically ALL of them are moving away from us, and importantly, the ones that are further away are moving away from us faster. So scientists gather up data on ALL the things they can determine the distance and relative velocity for, and they divide the relative velocity by the distance for each, and they average it out, and you get a pretty good measurement for what the actual expansion rate is.