= Astrometry
{wiki}
Astrometry measures the <positions> and motions of celestial objects. Repeated <position> measurements can distinguish <stellar parallax>, caused by the observer's motion, from <proper motion> of the source across the sky.
An <astrometry> measurement needs a date and a <reference frame>, as well as an <angle>. In an <equatorial coordinate system>, <right ascension> and <declination> label two viewing directions. A catalogue's reference date states when its quoted <position> applies; comparing entries at different dates without accounting for <proper motion> can make the same <star> appear misplaced. The observational problem is to infer a consistent history from successive incoming <light> directions, while accounting for the observer's changing <position> and <measurement error>.
To see the separation clearly, consider just one angular coordinate $\theta$, with $t$ the numerical elapsed <time> in years. Assume a simplified <circular orbit> of the <Earth>, a fixed source <Euclidean distance>, constant <proper motion> in this coordinate and no other <accelerations>. Choose the direction and the starting date so that the annual <stellar parallax> projection is the <cosine> term
$$
\theta(t)=\theta_0+\mu t+p\cos(2\pi t).
$$
Here $\theta_0$ is the coordinate without the annual <stellar parallax> offset at the reference date; the observed $\theta(0)$ is $\theta_0+p$. The <proper motion> coefficient $\mu$ is quoted in <arcseconds> per year, and $p$ is the annual <stellar parallax> in <arcseconds>. This is an illustrative choice of a full-<amplitude> projection. General sky directions have different projected <amplitudes> and <phases> in their two coordinates; their apparent paths need not be <circles>. Real <astrometry> uses the observer's calculated <position> rather than imposing this single <cosine>. https://gea.esac.esa.int/archive/documentation/GDR3/Data_processing/chap_cu3ast/sec_cu3ast_cali/ssec_cu3ast_cali_source.html[ESA Gaia DR3: Astrometric source model]{ref}.
Set $\theta_0=0$, $\mu=100\,\mathrm{mas\,yr^{-1}}$ and $p=50\,\mathrm{mas}$, where $1\,\mathrm{mas}$ is one thousandth of an <arcsecond>. The model gives $\theta(0)=50$, $\theta(0.5)=0$ and $\theta(1)=150\,\mathrm{mas}$. Same-season observations one year apart differ by $100\,\mathrm{mas}$ because the annual <stellar parallax> contribution repeats. Subtracting that fitted straight trend leaves an annual signal between $-50$ and $+50\,\mathrm{mas}$. The existing <stellar parallax> distance relation then gives $20$ <parsecs>.
\Image[/proper-motion-parallax-separation.png]
{title=Separating a 100-mas-per-year proper-motion trend from a 50-mas annual parallax projection}
Observing only at the same annual <phase> would recover the <proper motion> trend but leave $p$ indistinguishable from a constant offset. Sampling several seasons as well as several years separates the repeating signal from the accumulating <displacement>. The two signatures answer different questions using the same measurements.
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