Caption: Stellar parallax (i.e., parallax) and the definition of the parsec.
Features:
This motion is stellar parallax (in one meaning of the term): i.e., the shift in angular position of the star due to the motion of the Earth.
Just for historical reasons, it is the semi-major axis and NOT the major axis which is used to define the stellar parallax.
Thus it is the semi-angle subtended by the major axis that is the stellar parallax (in the second meaning of the term).
tan(θ) = b/r ,where tan is tangent function, θ is the semi-angle, b is baseline (which for stellar parallax is the astronomical unit (AU), the mean Earth-Sun and also the semi-major axis of the Earth's orbit), and r is the distance to the star.
r = b/tan(θ) = (b/θ)*[1 - (1/3)θ**2 + ...] ,where we have expanded tan(θ) in the tangent function power Series, and used the geometric series to inverse the tangent function power Series in order to create a power series for 1/tan(θ). The θ must be in radians for the power series.
In fact, stellar parallaxes are always so small that the power series expansion can be truncated to 1 with negligible error---the error is negligible because θ**2 << 1. This is the small angle approximation for tangent function.
Thus, to 1st-order in small θ, we have the distance formula
r = b/θ .
r = b_AU * (1.49597870700*10**11 m ) /[ θ_arcsecond * (π rad/180°) * (1°/60') * (1'/60'') ] = (b_AU/θ_arcsecond) * (3.08567758 ... *10**16 m) .We define the parsec (pc) by
1 parsec = 3.08567758 ... *10**16 m ≅ 3.086*10**16 m. = 206264.806 ... AU = 3.261563777 ... light-years (ly) = 1/(0.30660139 ... ) ly .
The name parsecs is a contraction of parallax and arcsecond (see Wikipedia: Parsec) and just means the distance implied by a stellar parallax of 1 arcsecond.
r_pc = b_AU/θ_arcsecond ,where r_pc is distance in parsecs.
Also the public understands light-years and has mostly never heard of parsecs.
But for historical reasons, we're stuck with the parsec (1 pc ≅ 3.26 ly).