In previous lectures, we've not hesitated to say that a certain star is 100 light-years away, or a particular galaxy is 3 billion light-years away. Yet you may be wondering how it is we know the distances to such remote objects? It's one thing to watch your odometer click around as you drive across the country, but imagine looking at the night sky. How do you figure out the distance to those tiny points of light? Yet knowing these distances is crucial to our cosmological story. If you think the stars lay at the same distances as the sun and moon, as many pre-scientific cultures did, then the earth becomes a dominant, possibly primary cosmic player.

Yet instead, if stars are like the sun, yet are just very faint because they're far away, then we live in a radically different kind of universe, one in which the earth shrinks into insignificance in the vastness of it all. So this is the topic for the lecture, how distances are measured. As we'll learn, knowing distances doesn't just tell us the universe is really big, but they do much more. They allow us to ascertain the intrinsic properties of objects, their true sizes and powers. They allow us to know how far back in time we're looking, which is extremely important in cosmology for constructing the history of the universe. The next lecture will show that measuring accurate distances is essential in figuring out the properties of the universe as a whole, its expansion rate, its age, its geometry, and its composition.

Let's start with a very important point. There is no single method for measuring distances to astronomical objects. Some methods work for nearby objects, while others work only for more distant objects. The distances at which the various methods work, overlap. So nearby methods are used to calibrate more distant ones, which in turn are used to calibrate yet more distant methods, and so on, in what is called the distance ladder. Given out limited time, we'll talk about just 5 of the most important steps on the ladder, starting nearby and ending with the most distance galaxies.

Now the first two steps in the distance ladder, make use of a standard surveying technique called triangulation, which astronomers often call parallax. So lets take a quick look at that now. Imagine a person looking at a foreground object against a more distant background. You can actually try doing this using your finger or a pencil as the object, looking alternatively with your left eye and the your right. Your finger appears to jump across the background, where closer objects jump more, further objects jump less.

Of course, our brains automatically use the difference between the images of the left and right eye, to intuit the object's distance. Yet we want to more general method that works for any two viewpoints. The amount of jump is given by this angle, the same as the apex angle. You can see we have a long, skinny triangle, with a base, a distance, and a little angle. As you may recall from high school geometry, knowing any two of these, gives you the third. So for a skinny triangle, we have:

distance = base x 57.3 / angle(degrees)

Let's make a mental check. Smaller angles yield longer distances, which seems about right. For a skinny triangle, with a one degree apex, and a one inch base:

distance = 1 x 57.3 / 1

The length is about 57.3 inches, or 5 feet, which also seems about rights. For reasons that will become apparent shortly, astronomers often work with extremely tiny angles. So lets just look at those briefly now. We're all familiar with degrees, 90 degrees in a right angle, so one degree is already pretty small. Now if you need much smaller angles, then an arc-minute is a sixtieth of a degree, and an arc-second is a sixtieth of an arc-minute, which is a 3600th of a degree.

So now back to our slim triangle with a one inch base. If we now make the apex angle 3600 times smaller, so it's now just one arc-second, then the length increases to:


distance = 1 x 57.3 / (1/3600)
distance = 1 x 57.3 x 3600
distance = 206,000 inches or 3.25 miles.

So a one inch base at 3.25 miles is a really very skinny triangle!

We're now ready for our first astronomical triangulation method. Remember that the distance ladder starts close to home, so the fir enormously important step on that ladder is to measure the size of the earth's orbit around the sun. The first attempt to do this, using triangulation, was by the great Italian French astronomer Giovanni Cassini 1625-1712, observed the position of Mars from Paris, while his assistant, Jean Richer 1630-1696, simultaneously did it from the coast of South America, 4000 miles away.

When they compared the two positions, they found a small sift of about 20 arc-seconds, from which they could calculate a distance to Mars. Then using that distance, they calculated the earth-sun distance, at only about 10% off the true value. Now since then, there have been many attempts at using triangulation to measure the earth-sun distance. Yet now they're really only of historical interest, because modern measurements use huge radio-telescopes to bounce radar pulses off Venus, timing the reflected signal, and knowing the relative distance to Venus and the sun, then gives an earth-sun distance to an accuracy of about 30 yards!

Now the true earth-sun distance, actually its average value, since the earth's orbit is slightly elliptical, is called an astronomical unit, or AU. It's very close to about 150 million km, or 93 million miles. By human standards, it is of course a huge distance. If one step represents the US, New York to San Francisco, then the sun is 20 miles away.

So now that we have a securely measured orbit for the earth, we're ready to move on to our next step. in our ladder, and reapply the triangulation method to measure the distance to stars, by using the earth's orbit as the base for our skinny triangles. Now this is an almost identical illustration for our first one of human binocular vision. Yet it replaces the human head with the earth's orbit. The position of a nearby star is observed twice, six months apart from opposite sides of the orbits.

Sure enough, nearby stars show an observable jump, against the background of much more distant stars. Now with a base of two AUs, and a measured jump angle, the distance to these nearby stars can now be calculated using our handy triangle equation.

Now as usual, although this is simple in principle, in practice it's extremely difficult because the jump angles are extremely small. In fact, despite enormous efforts, it wasn't until 1838 that the great German mathematician and astronomer Friedrich Bessell 1784-1846, measured parallel shifts.

The nearest stars have shifts of about one arc-second, with smaller shifts for more distant stars. Now this is equivalent to trying to see the jump of an object 10 miles away by closing your left and then your right. It's no wonder it took great care and excellent instruments to see thee tiny shifts.

So what distances do we find for the nearby stars? Using our small angle equation, and apex angle of one arc-second tells us that the distance is about 260,000 times the base. Now lets use light travel times to estimate this.

Our orbit base is twice 8 light-minutes, or 16 light-minutes. So the distance is about 206,000 times that, for about 3 million light-minutes. This is equal to six light-years, and nearby stars are a few light-years away. Yet in human terms, this is an unimaginable distance. Returning to our one-pace crosses the US, with the sun 20 miles away, to get to the nearest star we'd need to walk 30 times further than the moon!

Now at this point we'll introduce and then immediately ignore a famous unit of distance that astronomers often use, called the parsec. The word comes from a contraction of parallax and arc-second. It's simply the length of a slim triangle whose base is the radius of the earth's orbit, one AU, and whose apex angle is one arc-second. So from what we've just learned, you can see that a parsec is simply 206,000 AUs, or 3.26 light-years, a little more than 30 trillion km.

Now although astronomers tend to use parsecs, kiloparsecs, even megaparsecs, we'll continue to use light-years instead. They conveniently give us the look-back time for objects in years, ideal if your subject happens to be cosmology. Yet if you ever need to convert parsecs to light-years, it's very easy. You just multiply by three and then add a bit.

Despite the incredibly tiny parallax angles, modern technology has made huge advances. Ground-based observations suffer from atmospheric blurring, which limits the measurements to about a hundredth of an arc-second. Yet above the atmosphere, you can achieve enormous improvements.

So since the 1990s the European Hipparcos satellite measured parallel shifts for 120,000 stars with a precision of a thousandth of an arc-second. The successor satellite called Gaia, due for launch by 2011, will achieve 20 millionths of an arc-second precision. That's the width of a human hair from 500 miles away! It will give distances to millions of stars across the entire galaxy, a truly wonderful project if successful.

Now before moving on, we'll make one historical point about the parallel shifts of stars. Even the ancient Greeks knew that if the earth went around the sun, then there should be shifts in the star positions. In the great debate of whether the earth went around the sun, or vice-versa, Aristotle argued that it must be stationary or we would notice the stars shifting back and forth. For him, the alternative possibility, that the stars are so far away we can't see the shifts, which is the actual truth of the matter, was philosophically unacceptable to him. It would have implied an unreasonably big cosmos. Yet this wasn't the first nor last time that nature far outstripped our timid notions of what is or is not reasonable!

Returning to our distance ladder, you can see here the current limits of trigonometric parallax. It reaches only a small part of our galaxy. So we're going to need different methods to go further.

By far, the most versatile method is to derive distances for objects with known luminosity. This illustrates the method. In a row of similar candles, the further ones look fainter.

As long as we know the true power of the candle, and we can measure how bright it appears to be, then we can figure out its distance. Of course the key to this matter, is to find objects whose intrinsic brightness, what astronomers call luminosity, is already known. Basically we need to know the intrinsic brightness of a candle.

Now this diagram spells out the method a bit more precisely. Imagine an object, star or galaxy, it rally doesn't matter, with a known luminosity L. The units of L are watts, just like the labels on light bulbs. As the light from that object moves out, it becomes spread over ever-larger spheres. Now at a distance (d), the area of a sphere (A) from high school geometry is 4πd².

Now obviously if we happen to be on that sphere, those L watts have been spread evenly over the A square meters, giving the famous inverse square law for apparent brightness (b).

b = L/A (Luminosity / Area)

b = L/4πd² (watts/m²)

Now measuring (b) at the telescope is relatively easy. So as long as we know L, we can easily calculate the distance (d) using our simple equation.

Now the last 3 rungs of our distance ladder rely on this method, and each use a type of object with known luminosity, certain kinds of stars, certain kinds of galaxies, and certain kinds of supernova explosions. So lets look at these in turn.

To a casual observer, the stars in the sky seem not only fixed in position, but also fixed in brightness. Yet this is not always the case. If you look carefully night after night, a few stars change in brightness, becoming brighter and then dimmer in a regular, periodic way. One of the most famous of these stars is called Delta Cepheus. Using normal binoculars you can watch this star getting slowly brighter and dimmer relative to its neighbors, repeating its cycle about every 5.3 days as it first swells and then shrinks.

By 1912, a Harvard observatory astronomer Henrietta Leavitt 1868-1921, observed a very interesting property of this kind of "Cepheid variable" star, as we now call this class of star. The more intrinsically luminous Cepheids, pulsate slowly, while the less luminous Cepheids, pulsate quickly. This sort of makes reasonable sense, where big stars breath more slowly than smaller stars, a bit like elephants breathing more slowly than people, who breath more slowly than mice.

It turns out that Cepheids have a period/luminosity relation, like the one shown here. Of course you still need to calibrate the relation by using parallax to measure the distances to a few Cepheids, yet in fact that was one of the primary motivations for the Hipparcos satellite.

Now this kind of relation is extremely useful to astronomers, because once you've measured the period of the star's variation, you can use the period/luminosity relation to read off its true luminosity. Then by using its apparent brightness in the sky, you can quickly find the distance using our simple inverse square law equation. So this is how Cepheids are used to find the distances to the nearest galaxies.

Fortunately, Cepheids are really quite powerful stars, between 100 and 10,000 times the power of the sun. So they can be seen quite far away as individual stars in nearby galaxies. This was in fact one of the primary tasks of the Hubble Space Telescope, to find Cepheids in other galaxies and measure their distances. This HST "Key Project" was led by Wendy Freedman and was compiled in 1999, giving distances to 800 Cepheids in 18 galaxies out as far as 65 million light-years.

Now these distances to the 18 galaxies can in turn be used to calibrate the next step in the distance ladder. Yet before we look at that next step, we'll visit a wonderful piece of astronomical history in which Cepheids played a key role. Prior to the 1920s, it wasn't known whether the spiral nebula which had been discovered in the mid 19th century, were simply single stars in formation within our own Milky Way system, or entire island universes, other galaxies, in their own right, external to the Milky Way.

The issue was so vexed, that by 1920, Harlow Shapley 1885-1972 and Herber Curtis 1872-1942 debated the question publicly in Washington DC, with no firm conclusions. Then by 1923, Edwin Hubble used the newly built 100" telescope on Mt. Wilson to find and measure Cepheids in the great Andromeda Spiral Nebula. He found a distance of a million light-years, immediately placing it well-outside the Milky Way. So that single observation confirmed nebulae as separate galaxies. By implication, so were all the other spiral nebulae, and suddenly the universe was recognized to be an enormously bigger place than it had been previously thought.

Let's now go to the fourth step on the distance ladder, which is going to get us out to about half a billion light years. The step uses a relation similar to the one we just used or Cepheids, yet this time it applies to entire Spiral galaxies. It was discovered in the 1970s by Brent Tully b.1943 and Rick Fisher b.1943, taking the form of a tight correlation between the intrinsic luminosity of a spiral galaxy and its speed of rotation. Basically, big luminous spiral rotate fast, while the small low-luminosity spirals rotate more slowly.

You can probably guess how this is used to measure distances, since you can take a galaxy and measure its rotation speed, then use the Tully Fisher relation to read off the galaxy's intrinsic luminosity, then compare that the to luminosity of that galaxy's apparent brightness, and then figure out the distance using our handy equation b=L/4πd².

Now at this point you may well be wondering how can we measure the rotation speed of spiral galaxies? Yet we'll postpone that discussion until lecture 9, yet briefly, astronomers can relatively easily measure the speeds of things by use of the Doppler effect. Again, it's something we'll meet in coming lectures.

So using the Tully-Fisher method has measured fairly accurate distances to thousands of galaxies, out as far as maybe a few hundred million light-years, allowing many truly impressive studies of the structure and motion of the nearby universe. This is something we'll come back to by lecture 22.

However, in the cosmic scheme of things, these galaxies are still relatively nearby, perhaps a few percent of the way to our visible limit of 14 billion light-years. If we're going to measure distances to really far away galaxies, lets say ten billion light-years, then we need to find something exceedingly luminous, yet also with known luminosity. Well in the last 10 years, a great deal of effort has finally yielded such a class of object, called white-dwarf supernovae, technically known as supernovae type Ia.

They provide the fifth and final rung on our distance ladder. There are two qualities that make these supernovae ideal for distance measurements. They have almost exactly the same luminosity when they explode and they are extremely bright, so they can be seen very far away, out to 10 billion light-years. The reason the explosions are so uniform, is that they all involve exactly the same type of thermonuclear bomb.

As this artist's picture suggests, the supernovae is thought to come from a rare kind of binary star in which the outer atmosphere of one normal star, is pulled over onto its companion white dwarf star, which is just a dense, burned out ball of carbon and oxygen, roughly the size of the earth, yet containing about as much mass as the sun.

As more mass piles on, the temperature at the interior of the white-dwarf, reaches a critical point at which thermonuclear fusion rips through the star and in a few seconds the entire star explodes, creating an extremely bright fireball that can outshine an entire galaxy for several days.

A simple animation illustrates a supernova event where a star rapidly appears and then slowly fades. More precisely, here's a so-called light-curve which shows the supernova's rapid appearance and then gradual fading over about 60 days. Here's a lovely example of a supernovae in a nearby galaxy. Of course these nearby supernovae in galaxies with known distances are very important because you can calculate their true luminosity and so calibrate the method.

The most challenging task, yet the more important for cosmology, is to find extremely distant supernovae. For this, large telescopes are used on several consecutive nights, to take deep images that contain thousands of faint galaxies. Each image is subtracted from the previous night's image so that any new object is immediately visible in the difference image. Often a spectrum is then taken to confirm the nature of the supernova, and its slow fading is followed over eh next few weeks using HST.

Some of these supernovae are enormously distant, 5-8 billion light-years away, with look-back times half the age of the universe. As we'll see in lecture 9, it's these extremely distant supernovae, at the very end of the distance ladder, which gave the first evidence for an accelerating expansion, which then uncovered the distance to dark energy.

Throughout this lecture, we've needed to simplify the story of how distances are measured. The true history is much more nuanced and troubled, taking over 100 years to develop. In fact, until 15 years ago, the cumulative uncertainty was 50%, not very good.

Yet in the last few years, it has seen enormous progress. The Hipparcos satellite measured parallaxes of nearby stars including many Cepheids. The HST has used Cepheids to measure distances to dozens of nearby galaxies. The Tully Fisher and other methods have been greatly improved. and the intrinsic brightness of white-dwarf supernovae have finally been calibrated. So today, the combined uncertainty in the distance scale is more like 5-10%, a huge improvement.

In future lectures, the importance of reliable distance estimates will return time and time again. So although this lecture hasn't included much exotic material from the cosmological story itself, we've been gathering some hard, but necessary material that needs to be in place before we can move forward with that story.

Now we'll end this lecture with a new method currently being explored which promises to bypass the distance ladder completely, and yield a simple trigonometric distance out to several billion light-years. The method has been developed using this extremely interesting nearby galaxy. Right at the center is a small, one light-year sized disk of gas that's orbiting around a massive black hole.

The method works by imagining a blobby ring of gas moving rapidly in a circular orbit. First we measure the speed of the blob in one spot moving away or towards us, to perhaps be 1000 km/s. Next we follow the position of another blob in the ring for a year, moving across our line of sight, and notice that it moves by perhaps 10 micro-arc seconds across the sky, ten millionths of an arc second. Since these two blobs have the same speed, then a year moving at 1000 km/s, which is about 30 billion km, is the same as 10 micro-arc seconds as seen from the earth.

So we have an exceedingly skinny triangle with a base of 30 billion km, and an angle of ten micro-arc seconds, bingo! We then have the distance to the galaxy. Now you may be wondering how on earth can we measure those things? Yet it's possible using powerful radio telescopes. The method hasn't been fully developed yet, but in time it's hoped that it may ultimately yield distances out to 8 billion light-years to roughly 2 percent accuracy. This would be a great improvement and simplification over the conventional distance ladder.

Now measuring an accurate distance to a small number of rather unusual types of galaxies, isn't itself particularly important. Yet what is important, is that it allows you to measure the expansion of the universe with high precision. That's our next topic for the next lecture.