= Proper motion
{title2=$\mu$}
{wiki}
Proper motion describes the change in a celestial object's direction on the sky due to its motion relative to the observer. Its <angular velocity> differs from the annual <stellar parallax> signal caused by the observer's orbit.
Thus <proper motion> is an <angular velocity>, rather than a <speed> in kilometres per second. It describes the relative transverse motion of source and observer in a specified <reference frame>. A <star> moving towards us can have little <proper motion>, while a nearer <star> can show larger <proper motion> than a faster but more distant one. In the example above, the observed coordinate actually decreases over the first half-year, although $\mu$ is positive: the annual <stellar parallax> excursion temporarily outweighs the steady drift. A short observed <displacement> alone therefore need not estimate <proper motion> well.
Real catalogues give two <proper motion> components. For <right ascension> $\alpha$ and <declination> $\delta$, the <angular velocities> in perpendicular directions on the local sky are
$$
\mu_{\alpha *}=\cos\delta\,\frac{d\alpha}{dt},\qquad
\mu_\delta=\frac{d\delta}{dt},\qquad
\mu_{\rm tot}=\sqrt{\mu_{\alpha *}^2+\mu_\delta^2}.
$$
The <cosine> factor accounts for the shorter east-west arc associated with a given <right ascension> increment away from the <celestial equator>. ESA's Gaia DR3 fields `pmra` and `pmdec` already contain these two components; `pm` is their total <proper motion>. Do not multiply `pmra` by another <cosine>. Components of $60$ and $80\,\mathrm{mas\,yr^{-1}}$, for example, give total <proper motion> $100\,\mathrm{mas\,yr^{-1}}$. 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: Proper-motion columns and conventions]{ref}.
For nearby <stars> in the ordinary nonrelativistic approximation, <Euclidean distance> $d$ converts total <proper motion> into transverse <speed> $v_\perp$:
$$
\frac{v_\perp}{1\,\mathrm{km\,s^{-1}}}
=4.74047\,\frac{\mu_{\rm tot}}{1\,\mathrm{arcsec\,yr^{-1}}}\,\frac{d}{1\,\mathrm{pc}}.
$$
The <small-angle approximation> gives transverse <speed> as <Euclidean distance> times <angular velocity> in <radians> per unit <time>. With <arcseconds> and <parsecs>, the <parsec> convention turns their product into <astronomical units> per year. One <astronomical unit> per Julian year of $365.25$ days is approximately $4.74047\,\mathrm{km\,s^{-1}}$. The coefficient is a unit conversion, rather than a property of a particular <star>.
Taking the model's $100\,\mathrm{mas\,yr^{-1}}=0.100\,\mathrm{arcsec\,yr^{-1}}$ as the total <proper motion>, with the other component zero, gives $v_\perp=4.74047\times0.100\times20\simeq9.48\,\mathrm{km\,s^{-1}}$. At the same total <proper motion>, <Euclidean distances> of $10$, $20$ and $100$ <parsecs> instead give transverse <speeds> of $4.74$, $9.48$ and $47.40\,\mathrm{km\,s^{-1}}$, respectively. Their <stellar parallax> values are $100$, $50$ and $10\,\mathrm{mas}$. The same annual angular <displacement> can therefore describe quite different journeys through space.
\Image[/angular-rate-versus-speed.png]
{title=The same 100-mas-per-year proper motion gives larger transverse speed at greater distance}
The transverse <speed> still omits the line-of-sight <velocity>, which can be measured through the <Doppler effect> in <spectral lines>. Combining these measurements permits a three-dimensional <velocity> estimate in a consistent <reference frame>. https://www.esa.int/Science_Exploration/Space_Science/Gaia/How_does_Gaia_study_the_Milky_Way[ESA: Combining proper motion, distance and radial velocity]{ref}.
Historically, <Edmond Halley>'s 1718 paper compared contemporary and ancient stellar directions and argued that some supposedly fixed <stars> had moved. Its title, "Considerations on the change of the latitudes of some of the principal fixt stars", captures the conceptual surprise: long-separated measurements could reveal motion invisible in an ordinary night's watching. Modern <astrometry> retains that insight, using repeated precise measurements to turn the apparently static sky into a record of changing <positions>. https://archive.org/details/paper-doi-10_1098_rstl_1717_0025[Halley: Original Royal Society paper, pages 736-738]{ref}. https://www.esa.int/Science_Exploration/Space_Science/Gaia/How_does_Gaia_study_the_Milky_Way[ESA: Halley's 1718 comparison]{ref}.
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