"Let us go rambling about the Universe." This is the invitation that American astronomer Harlow Shapley gave to an audience in Washington DC in 1920. He was taking part in the so-called Great Debate with fellow scientist Heber Curtis on the scale of the Universe.
Shapley believed that our Milky Way galaxy was 300,000 light years across. That is actually three times too big according to the latest thinking, but his measurements were pretty good for the time. In particular, he calculated broadly correct proportional distances within the Milky Way - the position of our Sun relative to the centre of the galaxy, for instance.
In the early 20th Century, though, 300,000 light years seemed to many of Shapley's contemporaries an almost absurdly large figure. And the idea that other Milky Way-like spiral galaxies - which could be seen with telescopes - were equally large was outlandish.
Indeed, Shapley himself believed the Milky Way must be exceptional. "Even if the spirals are stellar, they are not comparable in size with our stellar system," he told his listeners.
Curtis disagreed. He thought, correctly, that there were many other galaxies as big as our own spread throughout the Universe. But, interestingly, his starting point was a belief that the Milky Way was far smaller than Shapley had calculated. According to the calculations that Curtis used, the Milky Way was just 30,000 light years in diameter - or, about three times too small going on modern measurements.
Three times too big; three times too small - when we are talking about such enormous distances it is understandable that astronomers debating almost a century ago could get their figures a little bit wrong.
Today we are fairly confident that the Milky Way is probably about 100,000 light years across. The observable Universe is, of course, much larger. According to current thinking it is about 93 billion light years in diameter. How can we be so sure? And how did we ever come up with such measurements from right here on Earth?
Ever since Copernicus argued that the Earth was not the centre of the Solar System, it seems we have always found it difficult to rewrite our preconceptions of what the Universe is - and especially, how big it may be. Even today, as we will see, we are gathering new evidence to suggest the whole Universe may be much bigger than some have recently thought.
Caitlin Casey, an astronomer at the University of Texas at Austin, studies the Universe as we know it. As she points out, astronomers have developed an ingenious array of tools and measuring systems to calculate not just the distance from Earth to other bodies in our Solar System, but the spans between galaxies and the journey to the edge of the observable Universe itself.
According to the calculations that Curtis used, the Milky Way was just 30,000 light years in diameter
The steps to measuring all these things are known as the "cosmic distance ladder". The first rung of the ladder is easy enough for us to get onto and these days it relies on modern technology.
"We can just bounce radio waves off of neighbouring planets in the Solar System, like Venus and Mars, and measure the time it takes for those waves to come back to Earth," says Casey. "That gives us a very precise measurement."
Big radio telescopes like Arecibo in Puerto Rico can do this sort of work - but they can also do even more than that. Arecibo, for instance, can detect asteroids flying around the Solar System and even produce images of them based on how radio waves reflect off the asteroid's surface.
But using radio waves to measure distances beyond our Solar System is not practical. The next rung on the cosmic distance ladder is something known as parallax measurement.
This is also something we do all the time without realising. Humans, like many animals, intuitively recognise the distance between themselves and objects, thanks to the fact that we have two eyes.
If you hold an object in front of you - say your hand - and look at it with one eye open, then switch to using only the other eye, you will see your hand appears to shift sideways slightly. This is called parallax. The difference between those two observations can be used to work out the distance to the object in question.
At this distance, we are still nowhere near the edge of our own galaxy
Our brains do it naturally with the information from both our eyes, and astronomers do exactly the same thing with nearby stars, except they use different sensors: telescopes.
Imagine having two eyes floating in space, either side of our Sun. Thanks to the Earth's orbit, that is exactly what we do have, and we can view stars' shift relative to objects in the background by this method.
"We take a measurement of where stars are in the sky, say, in January and we wait six months and measure those same stars in July, when we're on the opposite side of the Sun," says Casey.
However, there is a point at which objects are so far away - about 100 light years - that the observed shift is too small to provide a useful calculation. At this distance, we are still nowhere near the edge of our own galaxy.
Main sequence stars, when used for this analysis, are considered one type of "standard candle"
The next step up is a technique called "main sequence fitting". It relies on our knowledge of how stars of a certain size - known as main sequence stars - evolve over time.
For one thing, they change colour, gradually becoming redder with age. By measuring their colour and brightness accurately, and then comparing this to what is known about the distance of closer main sequence stars measurable by parallax, we can estimate the positions of these more distant stars.
The principle that backs these calculations is that which states that stars of the same mass and age would appear equally bright were they the same distance from us. Since they are often not, we can use the difference in those measurements to work out how far away they actually are.
Main sequence stars, when used for this analysis, are considered one type of "standard candle" - meaning a body whose magnitude (or brightness) we can calculate mathematically. These candles are dotted around space, lighting the Universe in predictable ways. But main sequence stars are not the only examples.
This understanding of how brightness relates to distance is pretty fundamental to working out the distance to even farther objects - like stars in other galaxies. Main sequence fitting will not work there, though, because the light from those stars - which are millions of light years away if not more - is hard to analyse with accuracy.
By observing how bright it actually appears to us, they can calculate its distance
But way back in 1908, a scientist called Henrietta Swan Leavitt at Harvard came up with a fantastic discovery that has helped us measure such colossal distances. Swan Leavitt realised that there was a special class of stars called Cepheid variables.
"She made this observation that a certain type of star varies its brightness over time, and the variation in the brightness, the pulsations of these stars, relates directly to how bright they are intrinsically," says Casey.
In other words, a brighter Cepheid will "pulsate" more slowly (over the course of many days, in fact) than a dimmer Cepheid. Because astronomers can measure the pulse of a Cepheid relatively easily, they can predict how bright the star is. Then, by observing how bright it actually appears to us, they can calculate its distance.
This is similar in principle to the main-sequence fitting approach, in that brightness is again the key. But the key point is that distance can be measured in different ways. And the more ways of measuring distances we have, the better we can understand the true scale of our cosmic backyard.
It was the detection of such stars in our own galaxy that convinced Harlow Shapley of its great size.
In the early 1920s, Edwin Hubble detected Cepheid variables in the nearby Andromeda galaxy and discerned that it was just under a million light years away.
There is one more feature of the Universe that can help us to measure really extreme distances
Today, our best estimate is that the galaxy is actually 2.54 million light years away. But that does not shame Hubble's measurement. In fact, we are still trying to work out a best estimate for the distance to Andromeda. The 2.54 million light years figure is actually an average of several recent calculations.
This is the point at which the sheer scale of the Universe, even now, continues to boggle our minds. We can make very good estimates, but in truth it is extremely difficult to measure distances between galaxies with fine accuracy. The Universe really is that big. And it does not stop there.
Hubble also measured the brightness of exploding white dwarf stars - Type 1A supernovas. These can be seen in quite distant galaxies, billions of light years away.
Because the brightness of these explosions is calculable, we can determine how far away they are, just like we can with Cepheid variables. The Type 1A supernovas and Cepheid variables, then, are both additional examples of what astronomers call standard candles.
But there is one more feature of the Universe that can help us to measure really extreme distances. It is called redshift.
If an ambulance or police car blaring a siren has ever passed you in the street, you will be familiar with the Doppler Effect. As the ambulance approaches you the siren seems high in pitch and then, as it passes you and moves away, it falls again.
As the Universe is expanding, each galaxy is moving away from the others
The same thing happens with light waves, on a much finer scale. We can detect the change by analysing the spectrum of light from distant bodies. This spectrum will have dark lines in it because some specific colours are absorbed by elements in and around the light source - the surface of stars, for example.
The further away objects are from us, the further towards the red end of the spectrum those lines will be shifted. That is not just because the objects are far away, but because they are actually moving further away from us over time, thanks to the Universe's expansion. And seeing redshift in the light from distant galaxies is one way of proving that the Universe is, indeed, expanding.
It is like putting dots on the surface of a balloon - each representing a galaxy - and then blowing up the balloon, says Kartik Sheth, a programme scientist at NASA. As the balloon expands the distance between the dots on its surface grows. "As the Universe is expanding, each galaxy is moving away from the others."
"Basically, a wave would normally just be whatever frequency it was emitted at, but now you're stretching space-time itself so therefore the wave looks longer."
Light has reached us from galaxies that are 13.8 billion light years old
The faster that galaxy is moving from us, the further away it must be - and the more redshifted its light will be when we analyse it back here on Earth. Again, it was Edwin Hubble who discovered that there was a proportional relationship between his Cepheids in distant galaxies and how much the light from those galaxies was redshifted.
Now comes the big key to our puzzle. The most redshifted light we can detect in the observable Universe suggests that light has reached us from galaxies that are 13.8 billion light years old.
Because this is the oldest light we have detected, that also gives us a measurement for the age of the Universe itself.
But over the last 13.8 billion light years, the Universe has been continually expanding - and at first it did so very rapidly. Taking that into account, astronomers have worked out that the galaxies right on the edge of the observable Universe, whose light has taken 13.8 billion years to reach us, must now be 46.5 billion light years away.
One possibility is that, somewhere, a few of our calculations are not quite right
That is our best measurement for the radius of the observable Universe. Doubling it, of course, gives the diameter: 93 billion light years.
This figure rests on many other measurements and bits of science, and it is the culmination of centuries of work. But, as Casey notes, it is still a little rough.
For one thing, given the complexity of some of the oldest galaxies we can detect, it is not clear how they were able to form so quickly after the Big Bang. One possibility is that, somewhere, a few of our calculations are not quite right.
"If one of the rungs of the cosmic distance ladder is off by 10%, then everything's off by 10%, because they rely on each other," says Casey.
The whole Universe is roughly 250 times as large as the observable Universe
And where things get really complex is when we try to think about the Universe beyond that which is observable. The "whole" Universe, as it were. Depending on which theory of the shape of the Universe you prefer, the whole Universe could actually be finite or infinite.
Recently, Mihran Vardanyan and colleagues at the University of Oxford in the UK analysed known data about objects in the observable Universe, to see if they could work out anything about the shape of the whole Universe.
The result, after using computer algorithms to look for meaningful patterns in the data, was a new estimate. The whole Universe is roughly 250 times as large as the observable Universe.
But estimates and models aside, we just do not know. Still, the observable Universe alone should be big enough for most people. Indeed, for scientists like Casey and Sheth, it remains a constant source of fascination.
We're not even at the centre of our Solar System or at the centre of our galaxy
"Everything that we've learned about the Universe - how big it is, all the amazing objects that are in it - we do that simply by collecting these photons of light that have travelled millions and millions of light years only to come and die on our detectors, our cameras or radio telescopes," says Sheth.
"It's rather humbling," says Casey. "Astronomy has taught us that we're not the centre of the Universe, we're not even at the centre of our Solar System or at the centre of our galaxy."
One day, we might travel physically much further into the Universe around us than we have so far dreamed. For now, we can only look. But just looking can let us ramble pretty far.
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