Distance by Parallax 

Useful Equations:

Solution:

Parallax is the angle in arcseconds that a star would appear to be displaced on the sky by an observer moving one astronomical unit.

One radian equals approximately 206,265 seconds of arc. Therefore, one second of arc equals

Since 1 pc = 206,265 seconds of arc, this formula reduces to dd = 1/alpha as the distance in parsecs, where one parsec is defined as the distance at which one A.U. would subtend an angle of one second of arc.

In the triple-star α Centauri system, Proxima Centauri (α Centauri C) is the nearest star to the Sun. Its parallax is 0.77233 arcsec. Its distance, therefore, is

or approximately 1.29 parsecs. This corresponds to

or approximately 3.98 * 10¹⁶ metres.

The right ascension (α) of Proxima Centauri for epoch J2000.0 is 14^h 29^m 42.95^s, and its declination (δ) is -62^o 40' 46.1". The brightest star in the α Centauri system. α Centauri (A), has coordinates (α, δ) of 14^h 39^m 36.50^s, -60^o 50' 02.3". Create a function that converts hours, minutes, and seconds of arc to degrees:

The two stars are separated on the sky by

in right ascension, and

in declination. The declination of α Centauri (A) in radians is

and the cosine of this angle is

The separation of the two stars in the sky is approximately 2.239 degrees. (More accurate measurements give 2.18 degrees.)

or 0.039 radians. Their physical separation is

five hundredths of a parsec, or 10,436 A.U. (The accepted value is about 13,000 A.U.)

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Reference

Cox, A. (Ed.). (2000). Allen's Astrophysical Quantities. (4th ed.). New York: Springer.