Proper motion

Definition

Let $\alpha(t)$ and $\delta(t)$ be the barycentric right ascension and declination of a star in a fixed reference frame. Here parallax, aberration, precession and nutation are removed. The proper motion of the star is the pair $\mu_{\alpha*} = \frac{d\alpha}{dt}\cos\delta, \qquad \mu_\delta = \frac{d\delta}{dt}.$ Its total proper motion is $\mu = \sqrt{\mu_{\alpha*}^2 + \mu_\delta^2}$ .

Remarks

The factor $\cos\delta$ makes $\mu_{\alpha*}$ an angle on the sky; a change in right ascension near the pole is a small angle. So $\mu$ is the angular speed of the star's direction on the celestial sphere. With $\mu$ in arcseconds per year and the distance $d$ in parsecs, the tangential velocity is $v_t \approx 4.74\, \mu\, d$ km/s. The factor $4.74$ is one astronomical unit per year in km/s. Example: Barnard's Star has $\mu \approx 10.4''$ per year at $d \approx 1.83$ pc, so $v_t \approx 90$ km/s. Non-example: the yearly back-and-forth ellipse traced by a nearby star is parallax, not proper motion; it is periodic, while proper motion is a steady drift. The velocity along the line of sight is the radial velocity, measured by the Doppler effect, not by positions.

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Declination concept
Right ascension concept
Proper motion definition

Tag	`M28Y9SHA1D`
URI	`m2://M28Y9SHA1D`
Kind	definition
Writing	W0 stub
MSC	85A04
Defines	Proper motion
Uses	Right ascension, Declination
Changed	3 November 2026

Proper motion

Definition

Remarks

Reading path

Further reading

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