Caption: Stellar parallax (i.e., parallax) and the definition of the parsec.

Features:

1. The orbital motion of the Earth around the Sun causes an apparent motion of a star in an ellipse on the celestial sphere.

   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.

2. Stellar parallax can be measured relative to background stars which are so remote that they exhibit no measurable parallax themselves.

3. The angle subtended by the semi-major axis of the ellipse is the stellar parallax (in a second meaning of the term).

   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).

4. The distance to the star follows from trigonometry:

    
     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.

5. Inversing the above formula, we get:

             
     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  .
             

6. Say we write b in terms of meters and do a conversion to astronomical units for b and a conversion to arcseconds for θ. We can then write

     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.

7. Thus we obtain the usual stellar parallax formula

     r_pc = b_AU/θ_arcsecond  , 
             

   where r_pc is distance in parsecs.

8. The parsec is the natural unit preferred by astronomers for interstellar distances since it is of order of distance between nearest neighbor stars.

9. The light-year (ly) would probably have been the better as a natural unit since distance in light-years converts to lookback time in years using a factor of 1 for times short compared to cosmic time. This is a very convenient conversion.

   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).