How to Calculate Parsecs

Length of a parsec

A parsec (short for parallax second) is a unit of length of about 3.26 light years that is commonly used by astronomers. The use of the parsec as a unit measure follows naturally from the first method used to measure interstellar distances. The distance to a star was measured using parallax in 1838 and the term "parsec" first appeared in a scientific paper in 1913. The parsec can be calculated directly from its definition used in basic trigonometry.

Things You'll Need

• Calculator with scientific functions

Instructions

1. Define the parsec. Let ABC be a right triangle with the vertices A, B, and C such that A is the sun, B is the earth and C is a point in space. The angle B is the right angle and angle C has a measure of 1 second. The length of side BC is equal to 1 parsec.

2. Determine the measurements of the triangle. There are 60 seconds in a minute and 60 minutes in a degree. Therefore, there are 60 x 60 = 3600 seconds in a degree. Thus, angle C measures 1/3600 degree. The length of side AB is the distance from the earth to the sun, which is 1 astronomical unit (AU) by definition.

3. Find the length of side BC, which is the distance from the earth to the point in space described in step 1. From trigonometry, we have BC/AB = cot(C) where cot is the cotangent function. Since AB = 1 AU, we can say that BC = cot(C) where length BC will be measured in AU. Therefore, we have BC = cot(C) = cot(1/3600) = 206,265 AU. Thus, BC = 206,265 AU, so a parsec is therefore equal to 206,265 AU.

4. Calculate the distance BC into miles to get the number of miles in a parsec. BC = 206,265 AU and there are about 93,000,000 miles in an AU, so a parsec is 206,265 x 93,000,000 = 1.92 x 10^13 miles.

5. Convert 1.92 x 10^13 miles to light years. Light travels 186,282 miles/second, so a parsec is (1.92 x 10^13) / (186,282) = 1.03 x 10^8 seconds. There are about 60 x 60 x 24 x 365 = 31,536,000 seconds in a year so a parsec is about (1.03 x 10^8) / (31,536,000) = 3.26 light years.

References

• PhysLink.com