This work was created by Dr Jamie Love and Creative Commons Licence licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Distances in Space

by Dr Jamie Love Creative Commons Licence 1997 - 2011

Stars and galaxies are very, very far away.

How do you know that? How can you tell the distance to a star or galaxy?

Well, it depends on the star or galaxy you are trying to measure. Astronomers use different techniques depending on how far away the object is. Stars that are nearby are measured using a different method from those that are far away, in another galaxy, for example. Let's start with the method used to measure the distance to our nearest stellar neighbors. It is easy to understand because it's all math but let me give you an idea of the process without the math.

Hold your finger out at arm's length and look at it with one eye closed. Notice how it is positioned with respect to the background. Now, without moving your head or your finger, look at it with the other eye. You will notice that the image of your finger has moved with respect to the background. That is the parallax effect. Your brain uses parallax all the time to estimate distances to objects, but the separation between your eyes limits your ability to estimate distances to less than 10 meters. When the base line is the distance between your eyes we call this binocular parallax.
You estimate the distances to more distant objects by noticing how small they appear. You know how big they are supposed to be so you adjust the thought of that image in your mind to estimate its probable distance. That's NOT parallax, but it's useful and not too different from how astronomers estimate more distant objects in the universe. But I'm getting ahead of myself. Let's start with parallax.

Parallax can be used to measure the distances to far off objects if you make careful observations and understand trigonometry. Surveyors use it all the time. They measure a "baseline" distance between two points and then observe the angle of the "target" (say, a mountain peak) from the two points at each end of the baseline. Because they know the length of the baseline, surveyors can calculate the distance to the target using trigonometry.

The first distance to a star, other than the Sun, was calculated in 1838 by the German astronomer Friedrich Bessel. He used parallax to measure the distance to 61 Cygni (better know as star number 61 in the constellation of CYGNUS). He knew that he needed the longest baseline possible in order to produce the most accurate measurement so he used the diameter of the Earth's orbit! This is called annual parallax.

Using the stars in the background to guide him, Bessel carefully measured the angle to 61 Cygni and then exactly half a year later he measured the angle again.
He knew that the distance to the Sun is (about) 150 million kilometers, so the diameter of the Earth's orbit would be 300 million kilometers.
(I've made some assumptions here to keep the math "easy". Kepler is probably spinning in his grave! )

Bessel then calculated the very tiny angle formed by joining the lines from 61 Cygni (T, the "target") to the baseline points (A and B). That is, he carefully measured the angle TAS and TBS and from that he could calculate the angle ATB.
ATB is the angle of (annular) parallax.

He used the angle of parallax of 61 Cygni (ATB) and the baseline (AB) to calculate (using trigonometry) that the distance to that star was 106 trillion kilometers! That's 106,000,000,000,000 kilometers.

These vast distances are impossible to grasp and astronomers rarely think of these distances in kilometers.

A light-year is the distance light travels in one year and is roughly 9.5 trillion kilometers. So 61 Cygni is about 11.2 light-years away.

The mathematically minded might want a more complete explanation (otherwise ignore all this stuff in this box).

After calculating the angle of parallax, ATB, you divide it in half, using the Sun to get two equal angles, ATS and BTS. Notice that the angles AST and BST are right triangles (90o). To arrive at the distance ST you work with one of these right triangles. They are identical triangles so it doesn't matter which one you choose. I'll use AST.

The angle SAT will be very, very close to 90o (not at all like in the drawing). The distance SA is one Astronomical Unit (AU) or 150 million kilometers. Because you know the angle SAT and the length SA, you have all the values you need to calculate the rest of this right triangle.
ST is the distance to the star, from our star (the Sun) so let's just focus on that part of the triangle using the angle SAT.
The tangent of SAT is equal to the opposite side (ST) divided by the adjacent side (SA) or simply
tangent of SAT = ST/SA and solving for ST we find it equals the tangent of SAT times SA.
The angle SAT will be very, very large and its tangent will be almost infinite (!) but not quite. Bessel calculated the angle SAT to be 89.9999182o and the tangent of that angle is 706,666. When you multiply 706,666 by 150 million kilometers (the distance SA) you get 106,000,000,000,000 kilometers as the distance ST!

There are other ways to do this problem.
You could just as well have used the very small angle ATS to make the calculation. Now we have the arrangement that the tangent of ATS equals SA divided by ST and solving for ST we find that it equals SA divided by the tangent of ATS.
Bessel calculated ATS is 0.0000818 of a degree and the tangent of that small angle is the small value of 0.0000014277. When you divide the distance to the Sun (150 million kilometers) by that very small number (0.0000014277) you find that ST equals (about) 106 trillion kilometers. (Just like before.)

Why use ST? The distance from me, on the Earth, to that star is the distance AT (or BT), not ST!

OK, you're right but the difference is pretty small. I would be off by about one Astronomical Unit (150 million kilometers) and that isn't much compared to the 106 trillion kilometers! And you'd have to work with sin and cosine!
Besides, and more to the point, it makes sense to measure the distances between stars, including our Sun. After all, as the year goes by the distance from that star to Earth will vary by as much as two AUs (as we orbit the Sun - think about it).

Parallax is so important to astronomers they often use a unit of distance based upon it. An object with a parallax of one second of arc as viewed from the baseline of the radius of the Earth's orbit (therefore, its annular parallax) is calculated to be at a distance of one "parsec". One parsec works out to 3.25 light-years (about 31 trillion kilometers).
By coincidence, stars in our neighborhood are about one or two parsecs apart so parsecs are a convenient way to measure interstellar travel. Starfleet uses it all the time!

The parallax method is a great way to measure distances, without having to travel there, and it is based on pure math and careful observations so there is no need to make any assumptions about the "target". Unfortunately, it only works well for measurements up to a few hundred light-years but beyond that distance the angle of parallax becomes too small to measured (with current technology).

So, how do we know the distances to more distant objects like the Andromeda Galaxy?

Standard candles!

"Standard candles" is the name used by astronomers for a source of light whose absolute magnitude they know with certainty. Absolute magnitude is the brightness of a star at a fixed distance, agreed to be 10 parsecs or 32.5 light-years. If you know a star's absolute magnitude you can measure its apparent magnitude (the amount of light as seen from the Earth) and calculate how far away it is.

This idea is really very simple and you can see the effect at night along a series of streetlights or a row of candles. All the lights have the same light output - assuming they use the same light bulbs or candles. The closest light appears the brightest, the next closest light seems slightly dimmer, and so on to the last light which will seem the dimmest of all. This can be expressed mathematically (of course ) and used to determine precisely the distance to other lights assuming you know how bright the lights are in the first place.
In math terms we use an equation called the "inverse square rule" and this rule has applications in all areas of physics.

The inverse square law states that the magnitude of a physical quantity is "proportional to the reciprocal of the square of the distance from that physical property". It is used throughout physics for such things as Newton's law of gravitation (hinted to you in Kepler's third law), Coulomb's law of electrostatic force (relevant to the magnetic fields of planets, but we won't go into it) and the propagation of light (our subject here).

Light (and gravity, and electrostatic forces) spreads out evenly in all directions if unhindered by other objects such as, in the case of light, mirrors, lens or dust. Its intensity diminishes the further it travels. That's what we mean by "inverse". The further away it is, the weaker it gets. That makes sense. If you want to read by candlelight move the candle closer to you. By shortening its distance you increase its relative brightness - but its absolute brightness is unchanged.

Imagine this inverse part of the rule as a kind of "dilution" of the energy. This energy radiates away in countless lines as if creating a sphere. You might understand that the surface of a sphere is a two dimensional property so it's measured in "squared units" like square meters or square kilometers. (I'm talking about the SURFACE of the sphere NOT its volume.) However, the distance from the point source to the edge of a sphere is measured in "linear units" of simply meters or kilometers. This is where the "square" part comes in. You might think that a candle two meters away would give you half the illumination as a candle one meter away but you would be wrong! The candle that is two meters away will provide only a quarter of the illumination it would provide if only one meter away. That's the "inverse square" effect.

Let's say we have a candle that produces X amount of light at a distance of one meter.
At two meters it will provide only 1/4 of X (not 1/2 of X). That's because we square the distance (22 = 4) and use that number to divide into X. The division is the "inverse" part.
At three meters that candle provides only 1/9 as much light to you. That's because the inverse square law tells us to inverse the distance (1/3) and square it (1/32 = 1/9).
At four meters that candle provides only 1/42 or 1/16 as much illumination to you.
If you have access to a light meter (on a camera) and a meter stick (or yard stick if you live in the Stone Age) you can do your own experiment to prove that light obeys the inverse square law. (I won't go into how to set up that experiment. You can figure it out. )

So how do astronomers use the inverse square law?

Well, the trick is to be able to observe and measure the light output from a star whose absolute magnitude you know. That's where the idea of a standard candle comes from. When astronomers speak of a "standard candle" they are talking about a type of star (or other source of light) whose absolute magnitude is known.

Unfortunately, there are no perfect standard candles!
Astronomers use a variety of techniques to try to figure out the absolute magnitude of a star based upon such features as its spectrum, mass or motion through space. But all of these techniques have their difficulties.
However, the most successful standard candles are "Cepheid variables".

You already know about some variable stars. You know that binaries may eclipse each other causing them to change in brightness in a regular clock like manner. You also know that some nova have a recurrent brightness due to the constant infalling of materials drawn from a companion star (usually a red giant) onto a white dwarf. There are other kinds of variable stars but Cepheids variables are particularly important to astronomers.

So what are they and how did they get that name. Did a man named "Cepheid" discover them?

No. (But that's a good guess! )
In the 18th century a young deaf-mute astronomer named John Goodricke noticed that delta-Cephei varied in brightness. Over the course of exactly 5.37 days its magnitude dropped from 3.5 to 4.4 and back again. You can even make the same observations he did! You should have no trouble finding delta-Cephei because it's opposite the bright star Alderamin that forms the southern base of CEPHEUS.

Next to delta()-Cephei are two other stars that together form a small triangle.
Epsilon()-Cephei has a constant magnitude of 4.19 and zeta()-Cephei has a constant magnitude of 3.55.

If you watch these three stars over the course of several days you will see delta()-Cephei quickly brightens to about the same magnitude as zeta()-Cephei then slowly fades over a couple days to become almost as dim as epsilon()-Cephei.

Many amateur astronomers enjoy watching this site.
[Note: if you use a telescope or even binoculars to make these observations, you may notice that delta-Cephei has a small companion of magnitude 7.5. This won't affect your observations of delta-Cephei's variation, but you might want to be aware that delta-Cephei is actually a binary separated by 41 arc seconds.]

So did Goodricke use Cepheid's to determine distance?

No, Goodricke merely identified delta-Cephei as a star which varied regularly in brightness. Two centuries have gone by and we now know that delta-Cephei represents a class of "pulsating stars" that have come to be called "Cepheid variables". Cepheid variables, or simply Cepheids, are yellow supergiants that have exhausted their supply of hydrogen and helium so they can be easily identified by color and, more importantly, a close look at their spectrum will show no sign of hydrogen or helium. This total lack of hydrogen and helium causes the star to "pulse". Complex nuclear reactions send waves of energy into the center of the star and when they all met (in the core) they bounce back again to the surface. A large Cepheid obviously has a greater distance from the surface to the center so its period of pulsation is longer than that of a smaller Cepheid. These pulses cause the absolute magnitude, and thus the relative magnitude, to fluctuate at a rate that depends up the size of the star. This link between the period of each pulse and the star's size can then be used to calculate the star's maximum luminosity. This allows astronomers to use Cepheids as standard candles.

Don't get confused here. Cepheids change in brightness and that is what causes the periods. You may think it's odd to use a star with changing brightness as a standard candle, but it makes sense if everyone agrees to use the maximum brightness of the variable as the "set point".

In 1912 a clerk at Harvard College Observatory named Henrietta Leavitt, who had accumulated data on over two thousand variable stars, published a paper called "Periods of 25 Variable Stars in the Small Magellanic Cloud". Commenting on the data she wrote, "A remarkable relation between the brightness of these variables and the length of their periods will be noticed." She continued, "Since the variables are probably at nearly the same distance from the Earth [all in the Small Magellanic Cloud], their periods are apparently associated with their actual emission of light, as determined by their mass, density, and surface brightness." She concluded, "It is to be hoped, also, that the parallaxes of some variables of this type may be measured." because doing so would give us standard candles from which we can assign distances.

Sadly, the annular parallax method I described earlier cannot measure the distances to these unusual stars because they are so far away. The baseline drawn by the Earth's orbit is only twice its orbital radius or 2 AUs. That is not long enough.

A year after Leavitt's paper, Ejnar Hertzsprung (yes the "H" in the H-R diagrams) used a method called "statistical parallax" to try to calculate the distance to the Small Magellanic Cloud. Statistical parallax uses the Solar System's motion, relative to the rest of space, to create a much longer baseline. The Sun, and all of the rest of our Solar System (including Earth), is moving through space at a constant velocity. Unlike annular parallax which returns to its starting point each year, this "space based" baseline gets longer and longer with each year. There were plenty of old photographic plates covering many years so by adding those years up you can arrive at a very long baseline for statistical parallax.

Hang on. How could Hertzsprung know the speed of the Solar System through space?

Good question. The problem is that all the stars are in motion and have their own velocities. That's where the statistics comes in. Hertzsprung used statistics to remove the effects of their individual motions by assuming that their average velocity would be zero. That is, if all the stars are moving in all possible directions at all possible speeds, averaging their motion would eliminate the variations and allow him to plot the motion of the Solar System against what he calculated as a zero velocity space. (If you don't understand that - don't worry about it.)
So in 1913 Hertzsprung used statistical parallax to calculate the distance to the Small Magellanic Cloud. However, he made so many mistakes that it was embarrassing! He did not have enough data to accurately calculate the velocity of the Sun relative to the rest of the Galaxy so his "statistical baseline" was wrong. That made the calculated distance too short by a factor of about 5 or 6. Of course, you can't blame a man for trying, even with bad data. However, Hertzsprung made a simple math error - he dropped a zero - that caused him to be off by ten fold! These two errors, due to bad data and bad math, caused him to conclude that the Small Magellanic Cloud was only 3,000 light-years away. (It's about 170,000 light-years away.)
Oh, well - not bad for a first try. Hertzsprung's 1913 paper showed the method to calculate the distance to the Small Magellanic Cloud, even though he used it incorrectly. Importantly, his 1913 paper described these variable stars as like delta-Cephei and coined the word "Cepheid" as a general term.

In 1918 Harlow Shapley repeated Hertzsprung's work but did it correctly. He used better data to create his statistical baseline and watched his zeros to make a correct estimate to the Small Magellanic Cloud. With that data Shapley was able to correlate the absolute luminosity of Cepheids at much greater distances and used the Small Magellanic Cloud to calibrate them. Shapely's 1918 paper gave astronomers the standard candles that allowed us to more our thinking about space from Astronomical Units to millions of light-years!

In 1923 Edwin Hubble used the most powerful telescope at the time (the Hooker telescope on Mount Wilson) to observe Cepheids in several spiral galaxies including the Andromeda Galaxy. By measuring the periods of each Cepheid he was able to figure out their absolute magnitude and from that he was able to calculate their distances. The world of astronomy was shocked when Hubble told them that his calculations, based upon Cepheids, indicated that the Andromeda Galaxy was nearly a million light-years away. Until that time there were many astronomers who believed that "galaxies" were nothing more than glowing dust in our own neighborhood. Hubble's work convinced other astronomers that galaxies were very distant and very large groups of very many stars! The distances Hubble calculated proved we live in a much larger universe than previously thought.

Hey, I thought the Andromeda Galaxy was about two million light-years away!

I was just getting to that!

In 1952 an astronomer named Walter Bade used the newest, largest telescope (the Palomar telescope) to discover that there were two types of Cepheids. Delta-Cephei is a Population I star and there are many like it in the Galaxy's arms. Bade showed there is another kind of Cepheid found in the halo of galaxies and (as you know from earlier lessons) these are Population II stars. He discovered that these Population II Cepheids, which he chose to call W Virginis variables, were much less luminous than the Population I Cepheids ("proper Cepheids"). That means a proper Cepheid and a W Virginis with exactly the same period (of brightening and dimming) do NOT have the same maximum absolute magnitude - the W Virginis is dimmer.

Hubble had used Population II "Cepheids" in his surveys because they are in a galaxy's halo making them much easier to single out than Population I stars hidden in the disk. Bade explained that Hubble's calculations were all underestimated because Hubble used W Virginis variables by mistake! The absolute magnitudes of the stars Hubble observed were lower than Hubble had calculated because Hubble was assuming the Population II stars (W Virginis) in distant galaxies had the same period/luminosity properties as the Population I stars (the proper Cepheids) from which he had based his calculation.

Thanks to Bade we now have TWO variable stars as standard candles: Population I Cepheids and Population II W Virginis stars. Also, thanks to Bade, we now know that the Andromeda Galaxy is over two million light-years away, more than twice as far as Hubble had estimated.

Astronomers have learned to use pulsating stars like Cepheids and W Virginis as standard candles. By knowing the distance to one standard candle we can assume that other stars associated with it (in a binary, in the same cluster, or in the same galaxy) are at a similar distance. Astronomers are always on the look out for better standard candles. Research often finds something unexpected, like there being two different types of Cepheids, and a new standard candle method might come along at any time and allow us to make better calculations.

Sadly, even Cepheid standard candles are not good enough for the galaxies much farther away. Instead we try to use supernovas as standard candles. Supernovas can be seen in galaxies at the very edge of our universe. (We will discuss the edge of our universe next month.) We have a pretty good idea of what the absolute magnitude should be for a supernova especially when we use a spectrum to define exactly what kind of supernova it is. However, some astronomers are not confident about those calculated values so one day, in the future, you may be told that the use of supernova as standard candles was "rubbish"! On the other hand, until we come up with something better, supernovas will have to be our standard candles for the far off reaches of the universe.

And we really need a good standard candle for those far off places because using supernovas is not a very useful approach. Supernovas are infrequent. A very large galaxy might have an average of one supernova per century! That's a long time to wait.

Once astronomers determine the distance to a galaxy, using Cepheids or supernova, they can calculate its dimensions (from its angular size) and compute its luminosity, not merely the luminosity of its standard candle, using the inverse square law in reverse. As you know from our lesson on strange galaxies, some galaxies are extremely bright. But, we must accept that these "far off" (distant) values, based on far off distances, are still written in pencil - not ink. If someone like Bade comes along and discoveries that we have been doing the numbers wrong then we will have to rewrite the books!

On the other hand, you should understand that parallax measurements are "pure" and are the best way to measure the distance to nearby stars (a few hundred light-years away). You also can use parallax to measure motion of the nearby stars, but it takes some time to do it.

Stars move?

Yes, they move very fast but the actual motion appears very small because they are so far away. Star motion is explained in your next lesson.




This work was created by Dr Jamie Love and Creative Commons Licence licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.