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Interstellar Distance Measurement Help (page 2)

By — McGraw-Hill Professional
Updated on Oct 3, 2011

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

Practice 1

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?

Solution 1

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.

Surveying, Navigation, and Astronomy Interstellar Distance Measurement The Parsec

Fig. 8–6. Illustration for Solution 1.

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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