Interstellar Distance Measurement Help (page 2)
Interstellar Distance Measurement—The Astronomical Unit
The distances to stars in our part of the Milky Way galaxy can be measured in a manner similar to the way surveyors measure terrestrial distances. The radius of the earth’s orbit around the sun is used as the base line.
The Astronomical Unit
Astronomers often measure and express interplanetary distances in terms of the astronomical unit (AU). The AU is equal to the average distance of the earth from the sun, and is agreed on formally as 1.49597870 × 10 8 kilometers (this is sometimes rounded off to a figure of 150 million kilometers). The distances to other stars and galaxies can be expressed in astronomical units, but the numbers are large.
The Light Year
Astronomers have invented the light year, the distance light travels in one year, to assist in defining interstellar distances so the numbers are reasonable. One light year is the distance a ray of light travels through space in one earth year. You can figure out how far this is by calculation. Light travels approximately 3.00 × 10 5 kilometers in one second. There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and about 365.25 days in a year. So a light year is roughly 9.5 × 10 12 kilometers.
Let’s think on a cosmic scale. The nearest star to our Solar System is a little more than four light years away. The Milky Way, our galaxy, is one hundred thousand (10 5 ) light years across. The Andromeda galaxy is a little more than two million (2.2 × 10 6 ) light years away from our Solar System. Using powerful telescopes, astronomers can peer out to distances of several billion light years (where one billion is defined as 10 9 or one thousand million).
The light year is an interesting unit for expressing the distances to stars and galaxies, but when measurements must be made, it is not the most convenient unit.
The true distances to the stars were unknown until the advent of the telescope, with which it became possible to measure extremely small angles. To determine the distances to the stars, astronomers use triangulation, the same way surveyors measure distances on the earth.
Figure 8-5 shows how distances to the stars can be measured. This scheme works only for “nearby” stars. Most stars are too far away to produce measurable parallax against a background of much more distant objects, even when they are observed from the earth at different times of the year as it orbits the sun. In Fig. 8-5, the size of the earth’s orbit is exaggerated for clarity. The star appears to be in slightly different positions, relative to a background of much more distant objects, at the two observation points shown. The displacement is maximum when the line segment connecting the star and the sun is perpendicular to the line segment connecting the sun with the earth.
Suppose a star thus oriented, and at a certain distance from our Solar System, is displaced by one second of arc when viewed on two occasions, three months apart in time, as shown in Fig. 8-5. When that is the case, the distance between our Solar System and the star is called a parsec (a contraction of “parallax second”). The word “parsec” is abbreviated pc; 1 pc is equivalent to approximately 3.262 light years or 2.063 × 10 5 AU.
Sometimes units of kiloparsecs (kpc) and megaparsecs (Mpc) are used to express great distances in the universe. In this scheme, 1 kpc = 1000 pc = 2.063 × 10 8 AU, and 1 Mpc = 10 6 pc = 2.063 × 10 11 AU. Units such as the kiloparsec and the megaparsec make intergalactic distances credible.
The nearest visible object outside our Solar System is the Alpha Centauri star system, which is 1.4 pc away. There are numerous stars within 20 to 30 pc of our sun. The Milky Way is 30 kpc in diameter. The Andromeda galaxy is 670 kpc away. And on it goes, out to the limit of the observable universe, somewhere around 3 × 10 9 pc, or 3000 Mpc.
A Point Of Confusion
The parsec can be a confusing unit. If the distance to a star is doubled, then the parallax observed between two observation points, as shown in Fig. 8-5, is cut in half. That does not mean that the number of parsecs to the star is cut in half; it means the number of parsecs is doubled. If taken literally, the expression “parallax second” is a misleading way of expressing the distances to stars, because the smaller the number of parallax seconds, the larger the number of parsecs.
To avoid this confusion, it’s best to remember that the parsec is a fixed unit, based on the distance to an object that generates a parallax of one arc second as viewed from two points 1 AU apart. If stadimetry were used in an attempt to measure the distance to a rod 1 AU long and oriented at a right angle to the line of observation (or a person 1 AU tall as shown in Fig. 8-4), then that object would subtend an angle of one arc second as viewed by the observer.
Interstellar Distance Measurement Practice Problems
Suppose we want to determine the distance to a star. We measure the parallax relative to the background of distant galaxies; that background can be considered infinitely far away. We choose the times for our observations so that the earth lies directly between the sun and the star at the time of the first measurement, and a line segment connecting the sun with the star is perpendicular to the line segment connecting the sun with the earth at the time of the second measurement (Fig. 8-6). Suppose the parallax thus determined is 5.0000 seconds of arc (0° 0′ 5.0000″). What is the distance to the star in astronomical units?
First, consider that the star’s distance is essentially the same throughout the earth’s revolution around the sun, because the star is many astronomical units away from the sun. We want to find the length of the line segment connecting the sun with the star. This line segment is perpendicular to the line segment connecting the earth with the sun at the time of the second observation. We therefore have a right triangle, and can use trigonometry to find the distance to the star in astronomical units.
The measure of the parallax in Fig. 8-6 is 5.0000 seconds of arc. We divide this by exactly 3600 to get the number of degrees; let’s call it (5/3600)° and consider it exact for now. (We’ll round the answer off at the end of the calculation.) Let d be the distance from the sun to the star in astronomical units. Then, using the right triangle model:
1/ d = tan(5/3600)°
1/ d = 2.4240684 × 10 –5
d = 41,252.96 AU
This rounds to 4.1253 × 10 4 AU because we are justified in going to five significant figures. If you have a good calculator, you can carry out the calculations in sequence without having to write anything down. The display will fill up with a lot of superfluous digits, but you can and should round the answer at the end of the calculation process.
Practice problems for these concepts can be found at: Surveying, Navigation, and Astronomy Practice Test
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