Caption: An explication of stellar parallax (i.e., the parallax of stars) and the definitions of the parsec and the light-year.

Features:

1. The orbital motion of the Earth (i.e., the Earth's orbit) around the Sun causes an apparent motion of a star in an ellipse on the celestial sphere.

   Recall that in astro jargon, "apparent" does NOT mean false: it means as seen from the Earth.

   The observed 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: i.e., the background stars identify the celestial sphere.

3. The angle subtended by the semi-major axis of the ellipse of the observed motion 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 with a stellar parallax measurement follows from trigonometry:

     tan(θ) = b/r  , 

   where tan is the 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 * (1'/60'') * (1°/60') * (π rad/180°) ]
       = (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
              = 3.26156377 ... light-years (ly)  = 1/(0.306601393 ... ) 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. Using parsecs as our unit of distance, we obtain the usual stellar parallax formula

     r_pc = b_AU/θ_arcsecond  ,  

   where r_pc is distance in parsecs.

8. The parsec is a natural unit for interstellar distances since it is of order of the distance between nearest neighbor stars (more precisely nearest neighbor systems consisting of single stars or star systems).

9. Now the definition of the light-year (ly) is the distance light travels in 1 Julian year (Jyr) at the vacuum light speed:

     1 ly = (365.25 days)*(86400 s/day)*(2.99792458*10**8 m/s) 
          = 0.94607304725808*10**16 m = 0.306601393 ... pc 

   which is exact by definition since our all the factors are exact by definition.

10. The light-year (like the parsec though is only about 1/3 of a parsec) is a natural unit for interstellar distances since nearest neighbor interstellar distances are of order 1 ly.

    It probably would have been better if astronomers had chosen to use the light-year as the primary natural unit for interstellar distances since it is easy to understand and it gives lookback times in years (more precisely Julian years) instantly for spatially static astronomical systems. The procedure is simple: lookback time t(years) = r(ly)*c(ly/Jyr) = r(ly)*(1 ly/Jyr) since the vacuum light speed c = 1 ly/Jyr. This is a very convenient procedure.

    In fact, almost all astronomical systems are approximately spatially static for lookback times short compared to cosmic time since the Big Bang which is the age of the universe = 13.799(21) Gyr current value.

    One major reason for preferring the light-year is that the public understands light-years and has mostly never heard of parsecs.

    But because the distances to relatively nearby stars are measured by stellar parallax, the natural unit directly applicable to stellar parallax measurements (i.e., the parsec (pc)) was chosen as the primary natural unit for interstellar distances. This historical choice CANNOT be changed---we are ruled by the dead hand of the past in astronomy: the oldest exact empirical science.

11. To summarize our results and for general reference, see the insert Larger-Scale Astronomical Distance Natural Units AKA Characteristic Size Scales below (local link / general link: astronomical_distances_larger.html).