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What Are Astronomical Units, Light-Years, Parsecs And Megaparsecs?

By Thomas Keyes
Dec. 9, 2005

In measuring the distance to a neighboring star, astronomers sight the star at two points in time six months apart, measuring with extreme precision the angular difference between the lines of sight. These two points in time must be such that a line drawn between to the two corresponding positions of the Earth and passing through the Sun is perpendicular to a line from the star to the Sun. This angle is divided by 2, and the quotient is called the ‘parallax’. This division by 2 leaves us with a right triangle, the shortest leg being the distance from the Earth to the Sun. The parallax is invariably less than one second of arc. For those who are new to this sort of thing, a circle is divided into 360 degrees. Therefore, a right angle is 90°. A degree is divided into 60 minutes (60’), and a minute is divided into sixty seconds (60”). Consequently, a degree has 3600 seconds and a second equals .0002777777778 degrees.

An Astronomical Unit (AU) is the mean distance from the Earth to the Sun, expressed as 1.4959 x 10^11 meters, that is, 149,590,000,000 meters, or 149,590,000 kilometers, or 92,950,000 miles. This is the short side or opposite side of the triangle that we have constructed here. To get the long side or adjacent side, we merely multiply the cotangent of the parallax of the angle by 1 AU.

Suppose we have a parallax of exactly one second, which in fact never happens, since all the stars are farther. On any calculator with trigonometric functions, we keypunch ‘tan (1 /3600)’ or ‘tan .000277777778’ and we get 4.848136811^-6, which means .00000484136811. This is the tangent. To get the cotangent, we merely punch the key ‘x^-l’ (usually -1 is a superscript and the caret (^) does not appear) followed by that number. On some calculators the key may read ‘1/x’. Alternatively, we may merely punch ‘1 / .00000484136811’. The answer comes out 206,264.8062. This is the cotangent of the parallax, which is merely multiplied by the value of 1 AU. The resulting number is called a ‘parsec’ (the distance of a star having a parallax of one second), and amounts to about 3.0855^16 meters, that is, 30,855,000,000,000,000 meters, or 30,855,000,000,000 kilometers, or about 19,173,000,000,000 miles.

A megaparsec is 1,000,000 parsecs, or 3.0855^22 meters, or 30,855,000,000,000,000,000,000 meters, or 30,855,000,000,000,000,000 kilometers, or 19,173,000,000,000,000,000 miles.

Currently the farthest known stars are supposedly 1300 megaparsecs away, or 4.0111^25 meters, or 40,011,100,000,000,000,000,000,000 (40 septillion) meters, or 40,011,100,000,000,000,000,000 (40 sextillion) kilometers, or 24,924,000,000,000,000,000,000 miles.

A light-year is the distance that light traveling at the speed of 299,792,458 meters per second, or 299,792.458 kilometers per second, or 186,282.3971 miles a second, would go in a year of 365.25 days of 86,400 seconds each. This equals 9.4607^15, that is, 9,460,700,000,000,000 meters, or 9,460,700,000,000 kilometers, or 5,879,000,000,000 miles. So a parsec is about 3.26 light-years.

In actuality, the nearest stars have parallaxes of less than one second, but the calculation is similar. Suppose the parallax is .5 seconds, which is .5 x .0002777777778, or .00013888888889. The tangent is 2.424068^-6, that is, .000002424068, and the cotangent is 412, 529.6. Multiplying this by 1 AU, we get a distance of 6.1710^16 meters, that is two parsecs.

Determining the distance to a star by triangulation, as above, is applicable only to the nearest stars. Beyond that it becomes impractiable to measure the parallax, so distance is estimated by using the red shift, but I’ll get to that in another article maybe some day.

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About the author Thomas Keyes: I have written two books: A SOJOURN IN ASIA (non-fiction) and A TALE OF UNG (fiction), neither published so far.

I have studied languages for years and traveled extensively on five continents.

Email: udikeyes@yahoo.com


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