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# Magnitude of proper motion (μα*)

## Description

Proper motion is the astronomical measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the center of mass of the Solar System, compared to the abstract background of the more distant stars.

The components for proper motion in the equatorial coordinate system (of a given epoch, often J2000.0) are given in the direction of right ascension (μα) and of declination (μδ). Their combined value is computed as the total proper motion (μ). It has dimensions of angle per time, typically arcseconds per year or milliarcseconds per year. Knowledge of the proper motion, distance, and radial velocity allows calculations of true stellar motion or velocity in space in respect to the Sun, and by coordinate transformation, the motion in respect to the Milky Way.

Proper motion is not entirely “proper” (that is, intrinsic to the celestial body or star), because it includes a component due to the motion of the Solar System itself.

The proper motion is a two-dimensional vector (because it excludes the component in the direction of the line of sight) and is thus defined by two quantities: its position angle and its magnitude. The first quantity indicates the direction of the proper motion on the celestial sphere (with 0 degrees meaning the motion is due north, 90 degrees meaning the motion is due east, and so on), and the second quantity is the motion’s magnitude, typically expressed in arcseconds per year (symbol arcsec/yr, as/yr) or milliarcsecond per year (mas/yr).

The magnitude of the proper motion μ is given by the Pythagorean theorem. In this equation, right ascension has been converted by cosδ,thus the result is designated μα*. For example the proper motion results in right ascension in the Hipparcos Catalogue (HIP) have already been converted. Hence, the individual proper motions in right ascension and declination are made equivalent for straightforward calculations of various other stellar motions.

Related formulas

## Variables

 μ magnitude (dimensionless) μ_δ proper motion (declination) (dimensionless) μa proper motion (rigth ascension converted by cosδ) (dimensionless)