Steven Dutch, Natural and Applied Sciences, University
of Wisconsin - Green Bay

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It is no accident that we call enormous numbers "astronomical". Astronomers deal with quantities so vast, distance, mass, and energy, that they tax our imaginations and mathematical skills. But such vast quantities also provide some of the most exciting ideas the human mind can encounter.

We can determine the distances to objects in the Solar System, and to nearby stars, by *Triangulation*.
If we observe an object from two stations a known distance apart, we can find the distance
to the object. For objects in the Solar System, we can measure the position of a the
object from different sides of the Earth. Observations of Venus crossing or *transiting*
the face of the Sun, or asteroids passing very close to the Earth, were once used. Once we
know the actual distance of any object, Kepler's Third Law allows us to determine the
distances of every object in the Solar System. [Nowadays we can actually bounce radar
signals off nearby planets and measure the distance more accurately this way]

The stars are so far away that they look the same all over the Earth. The farthest
apart we can ever make observations at present is from opposite sides of the Earth's
orbit, on opposite sides of an ellipse 186 million miles in diameter. When we photograph
the stars six months apart, some of them shift very slightly because of the Earth's
motion. You can demonstrate this shift, or *parallax*, easily. Hold up your finger
at arm's length and alternately open and close each eye. Your finger will appear to move
relative to objects in the background. For the stars, the *parallactic shift* is
extremely tiny but it can be measured. The distances to the stars turn out to be so great
that we need a new unit of distance: the

light year

. The light year is not a unit of time; it is the distance light travels in a year, or about 6 trillion miles (10 trillion km)

Measuring parallax involves measuring very tiny angles. A degree can be divided into 60
min*utes* and each minute into 60 *seconds*. A quarter spans one degree at a
distance of 1.5 meters. The Sun and Moon span about half a degree in the sky. A quarter
spans one minute at a distance 60 times greater, or 90 meters. Venus at its nearest and
Jupiter both span about a minute of arc in the sky. A quarter spans one second at a
distance of 5 kilometers. The moons of Jupiter appear about one second of arc across.
Astronomers routinely measure the positions of objects to about 0.1 second of arc and the
Hubble Space Telescope, above the distortions produced by Earth's atmosphere, can see
details that span a few hundredths of a second of arc.

Astronomers also use another distance term, the *parsec*. A parsec is the
distance at which a star would have a parallax of one second of arc, or 3.26 light years.
Parsecs are convenient because the distance in parsecs is simply one over the parallax,
making it easy to convert the two units.

Hipparcos of Nicaea lived in the second century B.C. and left us the first star
catalog. On August 8, 1989, a satellite named in his honor, the **Hi**gh **P**recision
**Par**allax **Co**llecting **S**atellite, was launched by the European Space
Agency. Despite an engine malfunction that left the satellite in a less desirable orbit
than planned, the spacecraft gathered data on stellar positions and magnitudes until March
1993. The data were processed and finally released in catalog form in the summer of 1997.
The data are of unprecedented accuracy and have completely revolutionized the
determination of stellar distances.

Hipparcos worked by observing stars through two telescopes aimed 58 degrees apart. The light from the two telescopes was merged into a detector with a fine grid of wires. As the satellite rotated, different stars passed through the field of view of each telescope and blinked on and off as the stars passed across the grid of wires. These observations allowed extremely accurate relative positions of the stars to be determined. The relative positions of all the stars could then be combined into an extremely accurate catalog of star positions across the entire sky.

For 118,000 selected stars, Hipparcos measured their parallax accurate to .001 second of arc. That's the apparent diameter of a quarter at a distance of 5000 kilometers, or putting a quarter in New York and viewing it from San Francisco. It's also the amount the hair on a person a meter away appears to grow in one second. A secondary mission named Tycho measured another million stars to an accuracy of "only" 0.01 second.

Distances in older star catalogs look so authoritative that many people think the distances to the stars are known very exactly, but this is not true. For parallax measurements made from Earth:

- The nearest star is some 4.2 light years away, and that distance was known to within an accuracy of better than 0.1 light year.
- For a star measured as 20 light years away, the distance estimate is accurate to within a light year or so.
- For stars 50 light years away, the margin of error is perhaps 5 to 10 light years.
- Beyond 100 light years, the parallax is so tiny it cannot be measured accurately, and astronomers resort to more indirect methods to find the distances to the stars. We will examine some of these methods later.

The Hipparcos catalog of star distances has made older distance estimates obsolete and all new astronomical publications will use the Hipparcos figures. Even more accurate observational systems are in the planning stages. However, until there is a distance measuring spacecraft on continuous duty, capable of being aimed at new targets on demand, there will still be earth-based parallax measurements and these will be subject to the same limitations as the older measurements

The following table shows how Hipparcos has improved our knowledge of stellar
distances:

Accuracy Level | Earth-Based Data | Hipparcos Data | ||
---|---|---|---|---|

1 percent | 50 stars | 10 light years | 400 stars | 30 light years |

5 percent | 100 stars | 20 light years | 7,000 stars | 150 light years |

10 percent | 1000 stars | 50 light years | 28,000 stars | 300 light years |

The earth-based data are approximate. Much of the improvement in Hipparcos is due to its
being completely automatic, whereas earth-based data were combined by laborious
measurements of individual stars. Thus, earth-based data were incomplete and of variable
accuracy.

Films like Star Wars create the impression that travelling from star to star is only a bit more complex than driving down to the corner store for a loaf of bread. In fact, distances in the Universe are so vast it is hard to comprehend them. Everyday speeds will not begin to cover the distances in the Universe. For example, walking nonstop at a brisk pace, a person could cover about 75 miles (120 km) in a 24-hour day. If he or she could keep up the pace indefinitely, it would take about 40 days to cross the 3000-mile (5000 km) width of the United States, and about 11 months to travel the 25,000 (40,000 km) miles around the Earth. It would take about 8 1/2 years to travel the 235,000 miles (380,000 km) to the Moon, and 95 years to cover the distance to Venus at its closest.

In an automobile travelling a steady 55 miles (90 km) per hour, it takes about 2 1/2 days to cross the United States and about 19 days to circle the Earth. It will still take six months to travel the distance to the Moon. Venus, however, is 100 times farther away at its closest, and would be a 54-year trip. We are not even out of the inner Solar System yet. At 55 miles an hour, just under half a million miles (800,000 km) a year, the Sun is two centuries away. If we had started out from the Sun at that speed in 1 A.D. we would now have covered about 960 million miles (1.54 billion km) and would be just beyond Saturn, only a third of the way out of the Solar System.

In a jetliner, at 550 miles (900 km) an hour, we cover distance 10 times as fast as in a car. The United States is 5-1/2 hours wide, and two days will take us around the world. The Moon is three weeks away, Venus over 5 years, the Sun 20. Pluto, the outermost planet, is still 750 years away. The nearest star is still so far away as to be unimaginable: 52 million years.

At 18,000 miles (29,000 km) an hour, the Space Shuttle crosses the Unites States in 10 minutes and circles the Earth in an hour and a half. In 13 hours it covers the distance to the Moon, but even so would take two months to cover the distance to Venus and seven to cover the distance to the Sun. Pluto is still over two years away. The nearest star is still 160,000 years away.

Light, at 186,300 miles (298,000 km) per second, could circle the Earth seven times in one second and reach the Moon in 1-1/2 seconds. The Sun is about 8-1/2 minutes away at the speed of light. It takes light about 5-1/2 hours to reach Pluto from the Sun, but 4.3 years to reach the next star. % Scale Models of the Universe

It is impossible to make a physical model that shows Man, the planets, and the stars on the same scale. If we make the Earth a quarter (one inch or 2.5 cm in diameter), the Moon becomes a pea 29 inches (74 cm) away. The Sun is 9 feet (2.7 m) across and 1000 feet (300 m) away. Pluto is 7 miles (11 km) away, and the nearest star is still off the real Earth: 49,000 miles (79,000 km) away.

If we let the Sun be a quarter, the Earth is a speck 1/100 inch (1/40 cm) in diameter and ten feet (3 m) away. Pluto is more than a football field away: 350 feet (110 m). And at last we can begin to show stars in our scale model. The nearest one is about 500 miles (900 km) away, and it, too, is the size of a quarter, with a pea-sized companion star about 200 feet (60 m) away. Placing a single coin in each State capital covers the U.S. with coins more densely than space is filled with stars.

Most of the stars we see with the naked eye at night are within a few hundred light years. A few are as far away as 2000 light years, only about 1/50 the diameter of our Galaxy. We can only use parallax to determine the distances of stars within about 50 light years, a realm that bears about the same relationship to our Galaxy as a period bears to the size of this page. How can astronomers measure the distances of distant stars?

There are many ways to explain how we know the arrangement of the universe, but perhaps the clearest analogy is the City Lights Analogy. Imagine being on the roof of a tall building with no way to leave. At night you can see lights in all directions. How can you map your "universe"? For nearby lights, you can use triangulation. By observing from different locations on the roof, and measuring changes in the relative positions of the lights, you can determine how far away the lights are. Once you know how far away the lights are, you can determine their true brightness. In nearby towns, the lights are all too far away to measure their distances by triangulation, but you can recognize lights of the same types as those you see nearby. Since you know how bright these nearby lights are, you can calculate how far off the lights in the town are. For distant towns, you cannot even see individual lights, but you know how far away the nearby towns are, and how much total light the towns emit, so you can use this information to estimate how far the distant towns are. At even greater distances, only clusters of towns are visible, and finally only great urban complexes, but at each stage you can use what you already know to discaover facts about the more distant universe. We build our picture of the Universe in much the same way.

Within the range we can measure parallax accurately, there are thousands of stars of many spectral types. With the apparent brightness and distances of these stars known, it is possible to determine their absolute magnitudes. Distant stars of the same spectral types will probably have the same absolute magnitudes, enabling us to estimate the distance to the unknown star. For isolated stars, this is virtually the only method of determining distance. When we say that the bright star Deneb is 1600 light years away, for example, the distance is an estimate based on its brightness and spectral type. The distance could easily be 25% larger or smaller. Deneb is still too far away for even Hipparcos to measure accurately.

Another technique for finding the distances of faraway stars involves star clusters. On photographs of clusters taken many years apart, the stars move. Some stars will move at higher speeds than others, but the velocity will show a characteristic statistical pattern. We can also measure the Doppler shift of stars in the cluster to find out how fast the stars are moving toward or away from us. The star cluster has no connection to Earth at all, so the velocity pattern of stars along our line of sight should be the same as that across our line of sight. Radial velocity can be determined in kilometers per second. By assuming the range of velocities across our line of sight is the same, we can determine what angle in the sky corresponds to a known distance and thus determine the distance to the cluster.

The cluster method works out to a thousand light years or so. Fortunately, within that
distance are stars that give us a yardstick to distant galaxies. These are the *Cepheid
Variables*. Cepheids are named for a star in the constellation Cepheus, the first star
of this type discovered, but the most famous Cepheid, and also the nearest, is Polaris,
the Pole Star. Polaris is about 300 light years away and varies in brightness too slightly
to be obvious to the unaided eye. Before Hipparcos, the distance to Cepheids had to be
determined indirectly by the cluster method, but now the distances to several have been
determined directly.

Cepheid variables are yellow-white giant stars that pulsate and vary in brightness, but in a regular way. The brighter the absolute magnitude of a Cepheid, the faster it pulsates. It is easy to measure the period of a Cepheid variable, and armed with this information, we can determine the absolute brightness of the star. Instead of using distance and apparent magnitude to find absolute magnitude, we compare the absolute and apparent brightness of the star to determine its distance.

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*Created 26 March 1998, Last Update 2 April 1998*

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