... Castor may deserve its Alpha designation for other reasons.
This month, the constellation Gemini is well positioned in the evening sky. Castor and Pollux, the Gemini twins of mythology, are represented by the two brightest stars on this constellation with Castor shining to the north of Pollux. Castor, also known as Alpha Geminorum, may be considered it's first star since alpha is the first letter of the greek alphabet.
Castor has been designated Alpha Gemonorum since the star maps of Bayer's Uranometria in 1603. Bayer's system ranks the brighter stars in each constellation in order of brightness, hence Alpha Geminorum is the brightest star in Gemini.
Except it isn't.
The other twin of Gemini, Pollux, is designated Beta Geminorum, but Pollux is the 17th brightest star in the sky (magnitude 1.16) while Castor is the 23rd brightest (magnitude 1.59). As noted in Burnham's Celestial Handbook: "The star, although designated Beta, is actually brighter than Alpha Geminorum and it has been suggested that one of these stars has changed in luminosity in the last few centuries."
Regardless of its brightness, Castor may deserve its alpha designation for other reasons. It may be considered the finest double star to view in the northern skies. The two stars visible in amateur telescopes, Castor A and B, are magnitude 2.0 and 2.8. Their visual separation has been increasing since periastron in 1968 and 1969 of 1.8 arcseconds, and currently appears to be about 4 arcseconds.
These two stars orbit each other with an average physical separation of 90 astronomical units (AUs) which is also the approximate diameter of our solar system. I'm always delighted to realise that our entire solar system could fit neatly between Castor A & B.
A lot of motion has been recorded since these stars positions were first measured in 1718. We have not yet observed one complete revolution of these stars, but one estimate of the period of revolution is around 400 years. From Burnham's Celestial Handbook: "In 1803, Sir William Herschel announced that the components form a system in which the two stars are gravitationally connected and revolve about each other in space. Castor was the first true physical binary to be recognized, and the first object beyond our own Solar System in which the force of gravitation was shown to be operating, as it does in the planetary system."
So Castor may be considered the first binary star due to it's historical significance, it's pleasing visual appearance in small telescopes, and it's unusual Bayer's designation.
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