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Star Stories
Tales from the night sky of Tenerife, brought to you by the guides of the stars!
Astro Physics and Cosmology,
with Stargazing Tenerife team, your guide to the dark skie of Tenerife.

Today for us hardened observational cosmologists I am talking about the second fundamental principle of cosmology.  Last month we brushed on this subject and today I thought we could look in more depth at it.  This comes, like so much of astronomy, the study of spectra.

Spectrum is what happens when we spread the colours of light out into the colours of a rainbow and assign each colour with a wavelength.  Remembering blue light and ultraviolet has short wavelengths, so we’re going to have green, yellow, orange, red, infrared, as you get in the longer and longer wavelengths and that’s the wavelength of light that you’re going to measure.  We measured it in angstroms, which is 10 to the minus 10 meters per angstrom.  So, they’re very, very short wavelengths.

If we had plotted the amount of energy at each wavelength, you could see, the wavelengths have a lot of energy, and for the wavelengths down the end don’t have very much energy coming out of that wavelength.  So when you see a spectrum like this, it’s actually a mixture of two things. Part of it is because we’re looking at a galaxy, if it were full of stars and the stars were hot.  Hot things glow, just like if you really turn, for example, your stove top up really, really hot, it’ll start glowing red. That’s because it’s hot. But then there are little narrow lines, absorption lines, and sometimes there’s emission lines, and that’s due to the atomic transitions of each element. They have these places of energy motion between the electrons, they’re different levels, and those show up at places of colour. So for example, hydrogen has very specific lines, for example, at 6,563 angstroms as one of the big lines of hydrogen, H-alpha. But then things like sodium, magnesium, various molecules, calcium, all have different things.  This is very familiar to any astronomer, and we would look at a spectrum like this and read off which elements are there. Now, that’s all well and good. But the strange thing is that when you look at a faraway galaxy, you see a different spectrum from a nearby one.

“Observations indicate that the universe is expanding at an ever increasing rate.  It will expand forever, getting emptier and darker.”

-Stephen Hawkin

A far away galaxy would look the same but shifted, red-shifted.  Moved to a longer wavelength. What you find is that every dip, or peak, or bump in the spectrum, every emission or absorption line, has had its wavelength increased by a constant ratio and that ratio is called the red shift and written for some unknown reason as Z rather than R.  It’s given by the shift in wavelength divided by the wavelength you’d expect in a nearby galaxy or in a laboratory. So, what’s going on here? Well, we know that waves get shifted by the Doppler shift.  So, we hear that in sound when a police car goes by you. The sound waves get shifted, and compressed, and stretched, and the pitch changes. Light’s a wave, so that same effect’s going to happen.  This mean that you can use the red shift to measure velocity.  When we see a red shift, it’s like the objects are moving away from us.  For the first order, that velocity can be calculated as the ratio of the shift of light is equal to the ratio to the speed of light.  There’s an approximation valid for speeds much less than the speed of light, but the galaxies we are talking about are like that.  What it’s telling you is, if the lines been shifted by a 1% longer wavelength, means whatever has produced them has been moving away from us at 1% of the speed of light. That’s a red shift of 0.01 and so a galaxy that’s been shifted by this much though, that’s a big shift. So that means it’s moving, apparently, at a good fraction of the speed of light. Now we’re going to see that maybe it’s not actual motion, but it’s sort of apparently moving.

Now it turns out that almost every galaxy is moving away from us. There are one or two very nearby galaxies that aren’t, but almost every galaxy is moving away from us. And if you plot how fast they’re moving away from us against distance, you get a plot. What you would see is how fast they’re moving away from us.  30,000 kilometers per second is 10% of the speed of light, if it’s a velocity and then we measure distance.  You can see there’s a one-to-one correspondence.  The further away you go, the faster the motion.  This is known as the Hubble law, and its parameterised as a velocity is proportional to a constant, Hubble’s constant, which is about 70 kilometers per second per megaparsec, times the distance.  So, 70 kilometers per second per megaparsec means if I’m a megaparsec away, I expect to be moving with a velocity of 70 kilometers per second away from us. Now everything is moving away from us.  This is something that Edwin Hubble discovered in 1929. Now if everything’s moving away from us, that seems to violate the Copernican principle, e.g. we’re no place special. It seems like we are a special place. We’re this place that everything’s going away from. We’re a very unpopular place in the universe.  Let’s say you’re living on some distant galaxy a billion light years away.  The universe, as we’ve just said, may look uniform.  But they’d be able to tell, hey, everything’s moving away from that point. That point must be the worst place in the universe or something, but anyway, special. But it turns out that, actually, that’s not quite the case. I’m going to show you a little calculation here. this uses the vector form of the Hubble law equation. What it’s telling us, this is us. We’re in the outskirts of this galaxy over here. We got two other galaxies I call galaxy A and galaxy B away from us and what we’ve said is the velocity is equal to a constant times the distance.  These have vectors, which means that the velocity is in the same direction as the distance, and proportional to it. So, the vectors, remember, are how far and the direction, typically, is how we can think of them. If Galaxy A is relatively near, it’s got a relatively small velocity, and the velocity’s in the same direction as the displacement vector.

Galaxy B could be further away, so it’s got a bigger velocity, and once again, it’s in the same direction.  This means everything’s moving away from us.  The further away they are, the faster they’re moving.  What we are stating here is Hubble’s law, saying the direction and the distance is the same.  It just says that direction, we have one answer. That direction, we have another answer.  But where this vector thing comes in very useful is if you now ask, OK, let’s say you’ve got aliens living on galaxy A what would they see?  Now if they look back at us, we would appear to moving away from them at equal but opposite speed. But how about galaxy B? What would they see for this other galaxy?  So the first thing they can ask is, what’s the distance, the vector displacement, from galaxy A to galaxy B?  That’s, what’s the vector from there to there?  That’s simply xb minus xa, the vector sum. This is vector arithmetic.  You go minus xa plus xb, and that’s how far away galaxy B looks from galaxy A.  But then we look at the velocity.  The velocity of galaxy B with respect to velocity A.  Once again, the same thing applies. You’ve got the change in relative velocity is equal to velocity B minus velocity A.  So, if you go back, it’s got this velocity minus this velocity is going to tell how that thing appears to be moving from its point of view.  So, if you do that calculation, we know that the relative velocity is velocity B vector minus velocity A.  But we also know, that velocity B is H0xb, and velocity A is H0xa.

If we take the H0s outside, it’s just H0 times xb minus xa, which we just said is delta x.  So it’s the distance times the Hubble constant.  So, the aliens on galaxy A are seeing that the velocity of galaxy B, and relative to them, is just equal to the Hubble constant times the distance, relative to them. Just what we see on Earth.  That means what we’ve just shown, using mathematics, and we’ve literally proved, is that if we see this Hubble law, everyone sees exactly the same Hubble law. They see the same thing to us and to this other galaxy. We all see the same thing.  In fact, this is perfectly consistent with our homogeneous uniform universe, the Copernican principle.  Because not only do we see galaxies uniformly, but wherever you are, you see everything moving away.  So how can you work out where the real center is?  Well, I guess we’re going to have to think about general relativity but it strikes me that if we think of the universe, it literally seems to be, the further away, the faster the motion.  So that means the universe is expanding.  So, let’s do a balloon analogy.  Let’s just think of the universe as being the surface of the balloon.  I’m going to put little dots on it, and as I blow the balloon up, every dot, which is a galaxy, is moving away from every other dot. The galaxies are expanding away from each other and the further away you are, the faster the motion will be as I blow the balloon up.  So I sort of get Hubble’s law. But where is the center? It’s the center of the balloon. And what was that? That was when I started blowing the balloon up.  The center would be like the moral equivalent to the Big Bang, when I started blowing the universe up.  Because we can’t really tell where the center is, because you can never actually measure if you’re moving, or you can measure something relative to something else.  Another analogy, my favourite, would be like baking up bread. Let’s say you’d had a raisin loaf and you bake it in the oven.  We’ll have rasins as galaxies, and the dough is transparent. As it gets bigger and bigger, it carries them all apart and so every raisin will think, hey, every other raisin’s moving away from us.  Unless they actually look outside the loaf into of oven, they can’t really tell which one’s moving.

It’s like with the balloon.  If they look outside their sheet of balloon, they can see where the center of the balloon is.  But we’re talking about a three-dimensional universe.  We can’t see outside it.  If all the galaxies are all moving away from each other then they are all moving away from us, and they’re getting smaller, because they’re getting further and further away.  So, we’re literally embedded in this universe.  If we ask ourselves, well, where’s the common center, the common center is right at the beginning of when we started when we are all crammed into the same spot. From where we’re sitting here, looks like everything’s moving away from us.  But if we were actually on a different galaxy, we would see exactly the same thing. So they can’t really tell. So those are our two clues. We seem to have a universe that looks the same from any point of view and it’s one that’s, everything’s moving away from every point.

I hope that this has helped explain in more detail the expansion of the universe with out your head having exploded. I look forward to writing again next week when we will have more star stories from the guides to the dark skies of Tenerife!

So until next time, this is Stargazing Tenerife team signing out, live long and prosper!

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