Caption: Trigonometry, parallax, and parsecs illustrated.

From the trigonometry shown in the diagram, we get the stellar parallax formula

r(AU) d(parsecs) = _____________ , θ(arcseconds)where θ is the observed parallax measured in the unit the arcsecond ('') = 1/3600°, r is the baseline distance measured in the unit the astronomical unit (AU) = 1.49597870700*10**11 m (exact), d is the distance to be determined in the unit the parsec (pc) = 648000/π AU (exact), and the formula is a small-angle approximation formula that is valid for the specified units.

Note:

1 parsec = 648000/π AU = 206264.806 ... AU = (3.085 677 581 491 367 ...)*10**16 m = 3.261563777 ... lyr ≅ 3.26 lyr ≅ 3 lyr(see Wikipedia: parsec (pc) = 648000/π AU (exact); Wikipedia: astronomical unit (AU) = 1.49597870700*10**11 m (exact), Wikipedia: light-year (ly) = 9.460730472580800*10**15 m (exact)).

Parsecs originally became the distance units of choice in astronomy because they are the distance units the above formula gives:

1 AU / 1 arcsecond = 1 parsec.The parsec is, in fact, a good natural unit for interstellar distances since nearest neighbor stars are typically of order a parsec apart. The light-year is a secondary natural unit for interstellar distances. In fact, the light-year would have been a better choice for the primary natural unit since it gives the lookback time immediately: an astronomical object X light-years away is seen as it was X years ago (more exactly X Julian years ago; the Julian year = 365.25 days exactly by definition). But the dead hand of the past weighs on us.

Credit/Permission: ©
David Jeffery,
2004 / Own work.

Image link: Itself.

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