Proper Motion - Introduction

Introduction

Over the course of centuries, stars appear to maintain nearly fixed positions with respect to each other, so that they form the same constellations over historical time. Ursa Major, for example, looks nearly the same now as it did hundreds of years ago. However, precise long-term observations show that the constellations change shape, albeit very slowly, and that each star has an independent motion.

This motion is caused by the true movement of the stars relative to the Sun and solar system through space. The Sun travels in a nearly circular orbit (the solar circle) about the center of the Milky Way at a speed of about 220 km/s at a radius of 8 ± 0.65 kpc from the center, which can be taken as the rate of rotation of the Milky Way itself at this radius.

The proper motion is measured by two quantities: the position angle and the proper motion itself. 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 gives the motion's magnitude, in seconds of arc per year.

Proper motion may also be given by the angular changes per year in the right ascension (μ_α) and declination (μ_δ). On the celestial sphere, positions are located by latitude and longitude. The coordinate δ corresponds to latitude. The coordinate α corresponds to longitude measured from the vernal equinox V, the point on the sky where the Sun crosses the celestial equator on near March 21.

The components of proper motion by convention are arrived at as follows. Suppose in a year an object moves from coordinates (α, δ) to coordinates (α₁, δ₁), with angles measured in seconds of arc. Then the changes of angle in seconds of arc per year are:

The magnitude of the proper motion μ is given by vector addition of its components:

where δ is the declination. The factor in cos δ accounts for the fact that the radius from the axis of the sphere to its surface varies as cos δ, becoming, for example, zero at the pole. Thus, the component of velocity parallel to the equator corresponding to a given angular change in α is smaller the further north the object's location. The change μ_α, which must be multiplied by cos δ to become a component of the proper motion, is sometimes called the "proper motion in right ascension", and μ_δ the "proper motion in declination".

The position angle θ is related to these components by:

Barnard's star has the largest proper motion of all stars, moving at 10.3 seconds of arc per year. Large proper motion is usually a strong indication that a star is relatively close to the Sun. This is indeed the case for Barnard's Star which, at a distance of about 6 light-years, is, after the Sun and the Alpha Centauri system, the nearest known star to Earth (yet, being a red dwarf, too faint to see without a telescope or powerful binoculars, with an apparent magnitude of 9.54).

In 1992, Rho Aquilae became the first star to have its name invalidated by moving to a neighbouring constellation - it is now a star of the constellation Delphinus. This will next happen to Gamma Caeli, which is due to become a star of the constellation Columba in the year 2400.

A proper motion of 1 arcsec per year at a distance of 1 light-year corresponds to a relative transverse speed of 1.45 km/s. For Barnard's star this works out to 90 km/s; including the radial velocity of 111 km/s (which is at right angles to the transverse velocity) gives a true motion of 142 km/s. True or absolute motion is more difficult to measure than the proper motion, as the true transverse velocity involves the product of the proper motion times the distance; that is, true velocity measurements depend on distance measurements, which are difficult in general. Currently, the nearby star with the largest true velocity (relative to the Sun) is Wolf 424 which moves at 555 km/s (or 1/540 of the speed of light).