Chapter 1: The Distance Scale of the Universe

Cosmological distances are so large that it is inconvenient to use normal units, such as cm, km, miles, etc., so another fiducial set of units has been devised. The distance from the earth to the sun, approximately 1.5 × 10¹³ cm, is called an Astronomical Unit (AU). Solar system distances are typically measured in AU’s. However, for objects outside our solar system, another unit is more appropriate. Since relativity implies that the speed of light is constant for all inertial (non-accelerating) observers, the distance light travels in a fixed time is a constant for all observers. For example, it takes light approximately eight minutes to travel from the sun to the earth, so that we can characterize the earth-sun distance as eight light-minutes.

By contrast, the distance to the nearest star, Alpha Centauri (actually a double star) is approximately 4.3 light years. This is typical of the average distance between neighbors among the approximately 100 billion stars in our galaxy. The distance to the center of our galaxy is roughly 30,000 light years, while the distance across the whole galaxy is approximately 100,000 light years. The average distance between galaxies is about one million light years. There are roughly 100 billion galaxies in the observable universe.

Another, often-used unit of distance is the parsec. One parsec is approximately 3.26 light-years. This unit of distance is related to the measure of parallax that we shall describe shortly.

Because these distances are so large, as we look out across our own galaxy, and out to other galaxies we are also looking backward in time—the light from a galaxy one million light years away has taken one million years to travel to us. Thus, as we explore the most distant objects with telescopes we are also performing a kind of cosmic archaeology. In 1987 we witnessed a spectacular supernova explosion on the outskirts of our galaxy, which we called Supernova 1987a. The star that exploded was located approximately 150,000 light years away, so that the actual event we witnessed in February 1987 occurred some 150,000 years earlier at the site of the explosion.

When we wish to measure our height or the length of a piece of wood that we need to cut, we do it directly by putting it up against a standard ruler — a meter stick or a tape measure. Even when we drive our cars our odometers measure the distance that we have travelled by counting the number of times that our tires have gone around and multiplying by the circumference of the tire. Because they are so large, cosmological distances cannot be measured directly, and can only be indirectly inferred. A Cosmic Distance Ladder has been developed to try to reach from the nearest scales to the farthest cosmological ones. This sequence of distance estimators is based on standard candles and standard rulers. The exercises developed here are aimed at giving students some experience with these techniques.

During a thunderstorm it is often fun to figure out how far away the lightning and thunder originate. After a flash of lightning count the seconds until you hear the thunder; divide by three to find the distance to the lightning strike in kilometers. Because the light from the lightning travels to us at 300,000 km/s it reaches us in much less than a second. The thunder on the other hand travels at the speed of sound in air that is about 1/3 km/s. By timing the arrival of the thunder one can deduce the distance to the place at which it is emitted. Astronomers use a similar technique to measure distances within the solar system. The AU is measured by radar ranging of the planets. Since the speed of light is known to be constant, by bouncing radar beams off Mars and Venus and measuring how long it takes until they return we can determine that the semi-major axis of the earth’s orbit is a=1.49597892(1) × 10¹³ cm.

Hold your arm straight out in front of your left eye with your index finger up. Close your left eye and notice where the finger appears with respect to objects in the background. Now open your left eye and close your right one. Notice that the finger appears to have moved to the right. This effect is called parallax. Try it again with your hand closer to your face, notice that your finger appears to move more. The angle, q, by which the object appears to move is simply related to the distance, d, of the object and the separation, s, between the two observers (your right and your left eyes):

Your brain uses the parallax effect to give you stereoscopic vision — depth perception. By observing stars at different times of the year, astronomers use the motion of the earth around the sun to obtain the same parallax effect. Since the separation, s, between two observations six months apart is 2 AU, astronomers can derive the distance to the star in AU. By definition, an object one parsec away will appear to move back and forth on the sky by one arc-second (1/3600th of a degree) every year as the earth goes around the sun in its orbit.

Angular displacements below a few percent of an arcsecond cannot be measured from the ground. This limits parallax measurements to stars within about a hundred parsecs from the sun. Recently the Hipparcos satellite measured the parallaxes to thousands of stars with a resolution of approximately 0.001 arcseconds, extending the reach of stellar parallaxes to about a thousand parsecs. Even this, however, allows us to measure distances only within our corner of the Milky Way galaxy. Although satellites are being designed which will measure parallaxes to a micro-arcsecond using interferometric techniques (in particular the Space Interferometry Mission), this will extend our reach only to the nearby Local Group of galaxies. To determine the distance to more distant objects we must use another technique. Think about lighting a candle in a dark room. If you stand very near the candle it will appear quite bright. As you move further away it will appear dimmer and dimmer. The candle has not changed. Its luminosity, L (total energy emitted per unit second) is nearly constant, but the area, A, of the sphere that the candle illuminates is proportional to the square of the distance, r. The apparent brightness, B, of the candle (the amount of energy per second per unit area), is the luminosity divided by the area illuminated, thus B decreases as the square of the distance of the observer from the source (candle):

Turning this discussion around, if we know the luminosity of a distant star, and measure the apparent brightness, then we can infer the distance to the star. The problem lies in determining the luminosity of a distant star.

If all stars were exactly the same luminosity, then we would be in business; stars would make excellent standard candles — find a nearby one, such as the sun, measure its luminosity, and you would know the luminosity of all stars. Unfortunately, in the same way that not all candles or all light bulbs are equally luminous, we know that not all stars have the same luminosity. Stars vary from class M dwarf stars that are hundreds of times less luminous than our sun, to the most brilliant of supergiants that are 10,000 times more luminous. However, if we narrow the criteria somewhat then stars make excellent standard candles. For example, stars with similar spectra have nearly the same luminosity. This is because both the luminosity and the spectrum of a star depend only on the mass of the star, its composition and its age. By measuring the spectra of many nearby stars, we can classify all stars by the shape of their spectra into spectral types. Measuring parallax distances for these stars, we can determine the luminosity of stars of each spectral type. Then by measuring the spectrum of a distant star and matching it to the spectrum of a nearby star with a known distance, we can determine the star’s luminosity. Since we know the actual luminosity of the star, we can infer its distance. Most usefully we can determine the distances to certain clusters of stars in our Galaxy. By measuring the distance to one or two stars in the cluster we can know the distance to the whole cluster. This will prove essential to building the next rung on the distance ladder.

Within our own local neighborhood there are only a few thousand stars whose parallax we can measure. None of those stars are of a type luminous enough to be visible if they were moved much outside our own galaxy. To measure distances to other galaxies we therefore require brighter standard candles. There are several particularly good candidates.

The first is a very luminous class of variable stars called Cepheids. These stars oscillate in brightness very regularly over periods ranging from 3 to 100 days, depending on the mass of the star. Why are they such good standard candles? Because they are very bright, and therefore are visible very far away, and because they have very characteristic time signatures that are not easily confused. Moreover by observing the Cepheids located in clusters whose distance can be determined, it has been shown that all Cepheids of a given mass have almost exactly the same period and luminosity — the more massive the Cepheid, the longer the period (just like more massive bells have lower tones) and the more luminous the star. It is relatively easy to measure the period of a Cepheid, and Cepheids are luminous enough to be seen in nearby galaxies as far away as 15 or 20 million parsecs. These have recently been observed using the Hubble space telescope in the Virgo cluster of galaxies in order to determine the distance to this system.

To go further out than the nearby galaxies, we must use certain empirical relations that allow us to use classes of galaxies as standard candles. The most accurate of these is the Tully-Fisher relation and is based on the observation that the luminosity of a spiral galaxy is related to the velocity at which it rotates. This is an empirical relation that is not well understood but seems to hold quite well for all sorts of spiral galaxies. How does one measure the rotation velocity of a distant galaxy? This can be done by using the Doppler effect. When a source of radiation is moving toward you the frequency of the radiation increases slightly — the light becomes bluer. When a source is moving away from you the frequency is lowered, the light becomes redder. This is the effect that makes train whistles sound high pitched as they approach you and then suddenly drop in pitch as they pass you. It is also how police radar measures the speed of passing motorists. The rotation velocity of a distant galaxy is determined by looking for Doppler shifting of features in its spectrum. Once we know the velocity, we can use the Tully Fisher relation to determine its intrinsic luminosity and therefore determine its distance. Since these velocity measurements can be made on galaxies out to 100 million parsecs it can be used to extend the distance ladder out a factor of five past where the Cepheids can no longer be seen.

The third standard candles that have begun to be used are Type la supernovae. These supernovae occur in systems composed of two stars, one of which is a very compact dense type of star called a white dwarf. As the two stars orbit around each other, the white dwarf sucks material off its companion. When the white dwarf reaches a certain critical mass it can no longer support itself against gravity and it collapses catastrophically releasing a tremendous amount of radiation. Because the critical mass is relatively independent of the history of the star, all these events are relatively similar. In particular the peak luminosity has been found to be a very good standard candle. Type la supernovae can be seen as far away as five billion light years using the newest ground and space based telescopes.

A second technique for determining distances is the use of standard rulers. Think of looking at a friend whose height you know to be to be 2 m. If you stand 10 m away from your friend they will occupy approximately 11 degrees in your field of view. If you stand 20 m away, they will occupy only 5.7 degrees. In general an object of length, l, a distance, d, away will subtend an angle θ given by:

For objects that are very far away (d>>l), this reduces to θ/radians = l/d. Thus if all objects of a certain type had a given size l, then by measuring the angle, θ, that they subtend on the sky, we could figure out how far away they are.

There are several complications with this procedure however. In the first place, finding good standard rulers is even more difficult than finding good standard candles. At relatively short distances the expanding gas-shell which results from a supernova can be used. This is not because all such shells are the same size (after all the shell is expanding), but because we can measure the expansion velocity of the shell. By measuring this velocity over time, we can tell how large the gas shell is.

The only things that are large enough to be resolved at the largest distances are galaxies that tend to be quite varied in their properties although some attempts have been made to use them as standard rulers. In particular, with very long baseline interferometry, high resolution studies of the cores of galaxies suggest that characteristic radio structures on scales of about 50 -100 parsecs might be constant from galaxy to galaxy, and at least two recent studies have attempted to utilize such structures as standard rulers.

A problem with using all standard distance measures at very large distances is that the curvature of space alters the definitions of the quantities, so that the interpretation of observations then becomes cosmology dependent. In particular, if the universe has any spatial curvature, the standard flat space argument used above to relate angular size and distance can change dramatically.

Equipment: Two people (A and B), 1 meter stick, one protractor, tape

• a)

  A and B stand on opposite sides of the room facing each other and mark their positions with tape on the floor.

• b)

  A moves 1.0 m to the left. using the protractor A determines how many degrees to the right B has appeared to move.

• c)

  A moves 2.0 m to the right (1.0 m to the right of his/her starting position) and determines how many degrees to the left of his/her original position B appears.

• d)

  A and B infer the distance between them.

• e)

  A and B measure the distance between them using the meter stick and compare to the result obtained in (d).

• f)

  Relate this to earth-star distance measured at two different times of year.

Equipment: Two people (A and B), light source, light sensor (e.g., Vernier’s LS-DIN light sensor for calculator based laboratory (CBL)), meter stick, tape, graph paper.

• a)

  Darken the room as completely as possible.

• b)

  A and B stand 1.0 m apart facing each other.

• c)

  B shines the light at A.

• d)

  Using the light sensor, A measures the light intensity.

• e)

  B moves an unknown distance away and marks his/her spot with tape.

• f)

  B shines the light at A; A takes another measurement.

• g)

  Infer the distance from A to B.

• h)

  Measure the distance from A to B using the meter stick and compare to the result obtained in (g).

• i)

  Repeat (c)-(h) several times at increasing distances, recording the distance inferred from the sensor readings, and the distance measured directly using the meter stick.

• j)

  Plot the inferred distances versus the directly measured distances, and consider why the technique may fail at larger distances.

Equipment: Two people (A and B), one ordinary (round) foam ball with oscillator insert, one foam football with oscillator insert. The oscillator insert consists of one oscillator, one battery holder, one switch, and batteries. (For example, Radio Shack: piezo buzzer, catalogue number 273-060A; AA battery holder, 270-401 A; push-on/push-off switch, 275-011A.)

• a)

  Activate the oscillator in the round ball.

• b)

  Stand a comfortable throwing distance away from each other (4 to 5 meters).

• c)

  Throw the ball from A to B and B to A.

• d)

  Note the change in pitch as the ball moves toward/away from you.

• e)

  Repeat (a)-(d) with the football, throwing spirals and listening for the changes in pitch as the ball turns. (This closely resembles the observations of light from rotating galaxies).

Bonus: (stopwatch, meter stick and third person required)

• a)

  Using meter stick, measure the distance between A and B.

• b)

  Using stopwatch, time how long it takes ball to travel from A to B.

• c)

  Repeat (b) several times and average times.

• d)

  Calculate the average velocity of ball using results from (a) and (c).

• e)

  Compare the result in (d), to the velocity of sound (340 m/s at sea level).

• f)

  The (“Doppler-shifted”) frequency is given by

  $f=f(11−vball/vsound)$
  v_ball is positive if the ball is moving toward you. Calculate the percentage change in the frequency of the sound heard when the ball is moving compared to when it is at rest.

• g)

  Compare the result in (f) to a semitone, the difference between adjacent musical notes in a chromatic scale (C and C#, C# and D, …) which differ by about 6%.

Equipment: Two people (A and B), meter stick, one stick 10-50 cm long, protractor, graph paper.

• a)

  Measure the length of the stick.

• b)

  B stands across the room from A, holding the stick in front of him/her.

• c)

  Using the protractor, A determines the angle subtended by the stick.

• d)

  Infer the distance from A to B.

• e)

  Measure the distance from A to B using the meter stick. Compare the results from (d) and (e).

• f)

  Repeat steps (b) through (e) for several different separations of A and B, recording each time both the distance inferred from the angular measurement and the distance measured directly using the meter stick.

• g)

  Plot the inferred distances against the directly measured distances and consider why the technique may fail at larger distances.