Proficiency Step #6---Cosmic Scale & Distances
When we, here on our planet Earth, think about an inch, it doesn’t really seem like much. And 12 inches, while
considerably larger, is still only about the length of…well…a foot. Three feet (36 inches) is a yard, and 16
yards is a rod, and 100 yards is a football field. Five thousand, two hundred eighty feet is a mile---well now,
this is beginning to get significant. We have all used these measurements often in our lives---and we are very
familiar with them. We have no trouble envisioning them and can even feel fairly comfortable with greater
distances---perhaps a hundred miles, or even a thousand miles! But when we start to think about larger
distances, our ability to envision how far things are gets increasingly difficult to comprehend---and the
distances get VERY, VERY big. So how can we as human beings relegated to planet Earth study and
comprehend the unimaginable distances of the universe? It becomes necessary to “add to” our repertoire of
descriptive terms dealing with distances. We have inches, feet, rods, miles, etc. (or millimeters, centimeters,
meters, etc.)---and we use them all the time, but now we are forced to add more terms that describe even
greater distances---AU, light-years, parsecs, etc. It can be quite intimidating to imagine the immense distances
involved with our study of the universe, but with a little knowledge and practice, it will become more and more
comfortable---just like inches and feet and miles.
It is usually a good idea to use distance measurements that are “related” to the topic of study for the distant
object. For example, we wouldn’t want to say a football field is 10,800 inches long (although it is)---it just isn’t
easy to “envision” that distance that way. On the other hand, we also wouldn’t want to say that it is 1/17.6th of
a mile! We try to make the distance understandable using measurements that “make sense” for the distance
being discussed. We therefore use inches when measuring a board to cut with our saw. We use miles for how
far it is to Disney World. In space, therefore, we will use miles at first---to get to the moon, let’s say, but then
we will switch to AU (astronomical units) to get around the planets---and go from there.
Scale Before Distance:
The pictures below begin to illustrate the ever-increasing scale of objects in space. These are the objects
themselves---the distance between the objects gets much,
this lesson.
much, much larger, as you will see throughout
Some Common Distance Units:
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Astronomical Unit (AU): the average separation of the earth and the sun (1.5 x 1011m—or about 93
million miles).
Light Year: the distance that light travels in one year (1016m—or about 6 trillion miles).
Parsec (pc): 3.26 light years (or 3 x 1016m).
Kiloparsec (kpc): 1000 parsecs.
Megaparsec (Mpc): 1,000,000 parsecs.
Some Representative Distances:
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The Solar System is approximately 100 Astronomical Units in diameter.
The nearest star (other than the sun) is 4.2 light years away.
Our Galaxy (the Milky Way) is about 100,000 light years in diameter.
Diameter of local cluster of galaxies: about 1 Megaparsec.
Distance to M87 in the Virgo cluster: 50 million light years.
Distance to most distant object seen in the universe: about 12 billion light years (12 x 109 light years).
If the Sun is a small ball bearing 3mm or 0.1” in diameter, then the Earth is a small spec at a foot away.
Jupiter is a small pinhead at 2 yards, and the nearest star is in La Crosse. The Milky Way stretches
past the Moon, so is not very dense at all. The Universe is 13.7 Billion years old; hence the maximum
viewable distance is 13.7 billion l-y, practical limit 12 billion l-y.
Because the distances to stars are so large, it is useful to introduce some large units of distance measure. The
most common are the light year and the parsec.
The Light Year Distance Unit:
In popular discussions, large distances (such as the ones between stars) are often given in units of light years
(which we shall abbreviate l-y). A light year (which is a unit of distance, NOT time!) is the distance that light
travels in a single year. Since light is very fast, the light year is a very large distance. From the knowledge that
light travels at a speed of 3 x 108 m/s in a vacuum, we can determine the length of a light year in meters simply
by multiplying by the number of seconds in a year (a tropical year has 3.156 x 107 seconds in it). The result is
that a light year is equal to 9.46 x 1015 meters, or 9.46 x 1012 kilometers---or about 6 trillion miles. (Since it is
not common for most people to be familiar with scientific notation, it is common practice to simply use the light
year as the principal unit of astronomical measure.)
A light year is often a convenient unit in discussing distances between stars, because the average separation
between stars in a galaxy is typically of that order of magnitude. For example, the nearest star to our Sun is
Proxima Centauri, which is about 4.2 light years away. Here is a summary of some important distances
expressed in light years:
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Light Year: the distance that light travels (through a vacuum) in one year (1016 m).
The nearest star (other than the Sun) is 4.2 light years away.
Our galaxy (the Milky Way) is about 100,000 light years in diameter.
Parallax And The Parsec Distance Unit:
Parallax is a method of measuring distance that uses basic trigonometry to calculate how far a distant star is.
The method is useful for stars that are relatively close, but with the great distances in space the parallax angle
soon becomes too small to be useful. It does however give rise to the parsec distance unit.
The parallax angle p is illustrated in the following figure.
The parallax angle p
Since we know the distance from the earth to the sun (1 AU), and can measure the angle 2p, we can then
calculate the distance to the star.
Unfortunately, the parallax angle is very small because of the great distances to stars. Thus, only for the more
nearby stars can it be measured reliably. Roughly speaking, ground-based telescopes can only measure
parallax reliably for stars that are within a few hundred light years from us. Telescopes above the atmosphere
such as the Hubble Telescope can measure smaller parallax shifts and thus larger distance, but even in that
case the most distant objects for which distance can be determined by parallax is only a few thousand light
years away. For more distant objects the parallax is simply too small to measure and we must turn to other
less direct methods to determine distance.
The method of parallax gives rise to a natural distance unit that astronomers call the parsec (which we shall
abbreviate as pc). The parsec is defined to be the distance at which a star would have a parallax angle p equal
to one second of arc. From basic trigonometry we find that this distance is equal to 206,265 astronomical units
(where the astronomical unit is the average separation of the Earth and the Sun) or 3.26 light years. One also
commonly uses the kiloparsec (kpc) and the megaparsec (Mpc) as a distance unit, which correspond to 1000
and 1,000,000 parsecs, respectively. Although the light year is often found in popular level discussions,
astronomers probably use the parsec, kiloparsec, and megaparsec more commonly as units of large distance.
It is useful to remember the following average distance scales:
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The average separation between stars in a galaxy like our own is of an order of parsecs.
The diameter of a galaxy like our own is typically of an order of 100 kpc.
The separation between galaxies in a cluster of galaxies like our own local cluster is typically several
Mpc.
The separation between clusters of galaxies is typically of an order of 10 Mpc.
The most distant galaxies observed are thousands of Mpc away from us.
Cepheid Variables:
The relationship between a Cepheid variable's luminosity and variability period is quite precise, and Cepheids
have been used as standard candles (a unit of luminosity measure) for almost a century. This connection was
discovered in 1908 by Henrietta Leavitt (later Henrietta L. Swan. She measured the brightness of hundreds of
Cepheid variables and discovered a distinct period-luminosity relationship. A three-day period Cepheid has a
luminosity of about 800 times that of the Sun. A thirty-day period Cepheid is 10,000 times as bright as the Sun.
The scale has been calibrated using nearby Cepheid stars, for which the distance was already known. This
high luminosity, and the precision with which their distance can be estimated, makes Cepheid stars the ideal
standard candle to measure the vast distances to clusters and external galaxies. Of course, a small error will
be present because we do not know the precise location of the Cepheid variable within the cluster or galaxy.
This error is typically small enough to be irrelevant in these kinds of measurements.
The Measurement of Distances: Standard Candles:
At very large distances such as those to galaxies beyond the local group or the local supergroups,
astronomers can no longer use methods such as trigonometric parallax or Cepheid variables that we have
discussed before because the parallax shift becomes too small, and because at sufficiently large distances we
can no longer even see individual stars in galaxies.
At those distances, astronomers turn to a series of methods that use standard candles: objects (besides
Cepheids) whose absolute magnitude is thought to be well known. Then, we can compare the relative intensity
of light observed from the object with that expected based on its assumed absolute magnitude. The inverse
square law for light intensity can then be used to infer the distance.
Example of a Standard Candle: Type 1a Supernovae:
One example of a standard candle is a type 1a supernova. Astronomers have reason to believe that the peak
light output from such a supernova is always approximately equivalent to an absolute blue sensitive magnitude
of -19.6. Thus, if we observe a type 1a supernova in a distant galaxy and measure the peak light output, we
can use the inverse square law to infer its distance and therefore the distance of its parent galaxy.
Because type 1a supernovae are so bright, it is possible to see them at very large distances. Cepheid
variables, which are giant stars, can be seen at distances out to about 10-20 Mpc (Mega-parsec); supernovae
are about 14 magnitudes brighter than Cepheid variables, which means that they can be ‘seen’ (real large
telescope!) a billion l-y away, which is a significant fraction of the known Universe (13.7 billion l-y).
A Comparison of Methods for the Virgo Cluster:
The following table lists a variety of methods for determining the distance to the Virgo Cluster of galaxies. The
second column gives the distance as calculated by the various objects used as standard candles. Note that
although the distances vary somewhat with the method used, the approximate distance is fairly uniform.
Distance Methods Applied to the Virgo Cluster
Method
Uncertainty
(Mag)
Distance
(Mpc)
Uncertainty
(Mpc)
Range
(Mpc)
Cepheids
0.16
14.9
1.2
20
Novae
0.40
21.1
3.9
20
Plan. Nebulae
0.16
15.4
1.1
30
Glob. Clusters
0.40
18.8
3.8
50
S. Bright.
Fluct.
0.16
15.9
0.9
50
Tulley-Fisher
0.28
15.8
1.5
>100
D-Sigma
0.50
16.8
2.4
>100
Supernova
(1a)
0.53
19.4
5.0
>1000
Source: An Introduction to Modern Astrophysics, B. W. Carrol and D. A. Ostlie (Addison-Wesley, 1996)
We shall not discuss the details of how all these methods work, but we note that there is reasonably good
agreement on the distance to the Virgo Cluster (the average among the different techniques is approximately
15 Mpc). This gives us some confidence that these methods can be used to measure large distances. The last
column (Range) gives the largest distance at which these methods can be used. We see that distances in
excess of 1000 Mpc may be measured using standard candles (luminosity/distance relationship).
Redshift & Hubble’s Law:
The picture to the left shows the absorption lines in the optical spectrum of a supercluster
of distant galaxies (right side), as compared to absorption lines in the optical spectrum of
the Sun (left side). Arrows indicate how the wavelength (or in this case the position of the
absorption lines) shifts up towards the red end of the scale (and beyond). This “redshift”
shows that the cluster is moving away from us as evidenced by the doppler effect (where
wavelengths increase as the speed of moving objects increases). The amount of the
redshift is proportional to the speed of the receding object---and indirectly proportional to
the distance of the object. This relationship (Hubble’s Law) can be used to determine the
very great distance of very far away objects such as galaxy clusters. It also put Albert
Einstein on the right track to formulate his expanding universe theory (He personally made
a trip to thank Edwin Hubble).
There are many methods to calculate cosmic distances. Most of these methods will only work for a certain
range of distances due to the very great distances we must deal with in the universe. Some are better suited
for a range of distances than others. Because of this phenomenon, it is standard procedure for cosmologists
to “switch” from one method to another as distances increase. This ever-changing group of methods is called
the cosmic distance ladder. It probably isn’t necessary for typical amateur astronomers to become proficient
with all of these measurement methods, but it is useful to have a brief understanding of what they are and why
they are used. This lesson---and the resources below should fulfill this need.
Suggested Sources to learn more:
www.wyalusing.org (Click on “StarGazing” for lots of StarSplitters and Astronomy information)
http://www.astro.ucla.edu/~wright/distance.htm (a list of 26 different methods for determining cosmic
distances—with concise explanations)
http://en.wikipedia.org/wiki/Cosmic_distance_ladder (an explanation of the various methods used to calculate
cosmic distances using different methods for everincreasing distances)
http://coolcosmos.ipac.caltech.edu/cosmic_classroom/cosmic_reference/redshift.html (an excellent simplified
explanation of redshift)
Google terms: Cosmic Distance Ladder, Hubble’s Law, Cosmic Distance Scale, Cepheid Variables, Redshift,
Light Year, Parallax, Standard Candles.
Please watch the excellent Filippenko “lessons” on the VHS or DVD media in the StarSplitters’ library.