= Stellar parallax
{title2=$p,\ \varpi$}
{wiki}
= Annual stellar parallax
{synonym}
Stellar parallax is the apparent change in a <star>'s viewing direction caused by the observer's changing position. Nearby <stars> shift more against a distant reference background. In an ideal circular <Earth> orbit, the conventional annual <stellar parallax> $p$ uses a baseline of one <astronomical unit>, approximately the <Earth>-<Sun> distance. Two opposite observing positions, about six months apart, are separated by approximately two <astronomical units>. Their maximum angular separation is $2p$, so $p$ is half that excursion, not the full January-to-July shift. The observed excursion depends on the projected baseline and observing dates. https://www.esa.int/ESA_Multimedia/Images/2013/06/Measuring_stellar_distances_by_parallax[ESA: Six-month parallax geometry]{ref}.
For the perpendicular baseline in the diagram, a <right triangle> gives
$$
\tan p=\frac{1\,\mathrm{au}}{d},\qquad d=\frac{1\,\mathrm{au}}{\tan p}\simeq\frac{1\,\mathrm{au}}{p},
$$
where $p$ in the final expression is in <radians>. Combining the <small-angle approximation> with the <parsec> convention gives
$$
\frac{d}{1\,\mathrm{pc}}\simeq\frac{1}{p/(1\,\mathrm{arcsec})}.
$$
Thus $p=0.05$ <arcseconds> implies $d\simeq20$ <parsecs>, while the ideal six-month excursion is $0.10$ <arcseconds>. Geometry supplies this distance without assuming a <luminosity> or a particular <stellar atmosphere>.
\Image[/stellar-parallax-geometry.png]
{title=One-AU parallax angle and an ideal annual signal; geometry schematic, not to scale}
In practical <astrometry>, many observations separate the annual <stellar parallax> signal from <proper motion>, the <star>'s own changing sky position. The Gaia DR3 catalogue reports <stellar parallax> and its uncertainty in milliarcseconds: $50\,\mathrm{mas}=0.05$ <arcseconds>. A catalogue angle is a fitted measurement with <measurement error>. https://gea.esac.esa.int/archive/documentation/GDR3/Gaia_archive/chap_datamodel/sec_dm_main_source_catalogue/ssec_dm_gaia_source.html[ESA Gaia DR3: Parallax, errors, and proper motion columns]{ref}.
For a positive <stellar parallax> with small <standard deviation> $\sigma_p$, first-order propagation gives $\sigma_d/d\simeq\sigma_p/p$. Thus $p=50\,\mathrm{mas}$ with $\sigma_p=1\,\mathrm{mas}$ gives $d\simeq20$ <parsecs> with $\sigma_d\simeq0.4$ <parsecs>, assuming systematic offsets are negligible. This linear estimate requires $\sigma_p\ll p$; a reciprocal is nonlinear and does not generally preserve symmetric uncertainty intervals.
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