The proper motion of a star is its apparent angular movement per year on the celestial sphere. It is a combination of its actual motion through space and its motion relative to the solar system. Most stars are so distant that the proper motion is almost negligible.

The full space motion of a star is the combination of this proper motion (which is a quantity measured by the Hipparcos satellite) with the "radial velocity" of the star, along the line of sight. The radial velocity of stars or galaxies, are usually measured from the Doppler shifts in their spectra: this quantity was not measured by Hipparcos. Stars don’t have a fix position in space, they move around so over the time the sky changes as well. Short term calculations apply when the star moves only a small distance or angle. For most of the stars, considering their slow motion, the short term refers to something less than 10000 years.

In order to start computing the proper motion of a star, one should consider the following elements from the Hipparcos catalogue:

• α - as the right ascension (in decimal hours)
• δ - as the declination (in decimal degrees)
• μ_α - the proper motion in right ascension (in arc seconds per year)
• μ_δ - the proper motion in declination (in arc seconds per year)

For short term calculations, it will be assumed that the proper motion in both right ascension and declination are constant. Most of the time, considering the small angular motion, this is a fair enough assumption. Then the new values of right ascension and declination are found as:

• α_t = α₀ + μ_α • t
• δ_t = δ₀ + μ_δ • t

The only thing that should be carried on is to keep the original units used when reading the star’s data from the catalogue.

This simple approach works well for time changes up to about 10000 years past or future compared with the time when the catalogue data have been produced. However the calculation losses the accuracy for longer times, since the star’s distance from Earth changes appreciably and so its proper motion changes, invalidating this simple approach. As a general rule, if the star moves by more than about 30 degrees- roughly the length of the Big Dipper or the height of Orion, these calculations are braked down. After a few hundred thousand years, about any star will have moved enough to invalidate this simple calculation. Therefore a more complex model for calculating the proper motion can be put in the scene.

In order to accurately calculate the stellar motion, the star’s motion in three dimensions should be known. The three conventional space velocity components are:

• υ_R the star’s radial velocity
• μ_α the proper motion in right ascension
• μ_δ the proper motion in declination
• d the star’s distance in parsecs

If the radial velocity is missing, a null value can be used in all the following calculations. If the catalogue gives the distances in light years, the distance should be divided by 3262 in order to get parsecs. If the catalogue gives a parallax pa, it can be converted to distance by using the parallax formula pa=1/d (parallaxes less than about 0.01 arc seconds in older catalogues or 0.001 arc seconds in the Hipparcos catalogue are usually considered poor, and should be used carefully)...

It should be considered that both components of proper motion are in the same units (seconds of arc per year) - if not they should be converted accordingly. Then they should be converted from these angular units, to linear velocities- the two components of the transverse velocity υ_T of the star. To make these velocities consistent with the radial velocity , they should be converted into km/s.

• υ_TRA = μ_α (arcsec/year) • d • 4740 (km/s)
• υ_TDEC = μ_δ (arcsec/year) • d • 4740 (km/s)
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These velocities are used to calculate the change in position of a star over time. However these velocities as calculated are hard to work with. First their orientation in space will vary from star to star. υ_TDEC, for instance, points towards the celestial poles if the star has a declination of zero, but points 90 degrees away from the poles if the star is at the pole. It is generally easier to transform these velocities to Cartesian velocities – with components along some consistent set of axes. If we use the coordinate system as:

• +x axis towards δ = 0 degrees, α = 6 hours (vernal equinox)
• +y axis towards δ = 0 degrees, α = 6 hours
• +z axis towards δ = 90 degrees (north celestial pole)

The three Cartesian velocities can be calculated in terms of the three velocities υ_R , υ_TRA and υ_TDEC :

The Cartesian velocities retain the original unit (that is km/s), but for interstellar motions it is more appropriate to express distances in terms of parsecs and times in terms of years rather than seconds. Then the natural speed unit will turn to be expressed in parsecs per year. A conversion from km/s to pc/year is obtained by dividing by 977780.

If we express the three Cartesian coordinates for a star at time t = 0 (present time):

We can calculate the new positions at time t considering that for the time frames we are interested in (a few thousand years to a million years) the stellar motion is pretty much linear - stars are too far apart for the gravity to curve their paths appreciably.

Now it can go back to equatorial coordinates (right ascension and declination) using the following formulas:

The result can be converted from degrees to hours by dividing the result to 15.

Over a few thousand years, a star’s motion generally doesn’t change its brightness very much, but as these calculations are highly accurate over millions of years, it can be seen that many stars will get measurably brighter or fainter. If we introduce in the following equations the apparent magnitude V as seen from Earth:

The calculation is identical to the one for calculating the brightness of a star as seen from a different reference point in space- a given change in distance to a star, whether it’s observer that moves or the star, yields the same effect on brightness.

The following objects are the ten highest proper motion stars contained in the Hipparcos Catalogue (the 61 Cygni binary is seen at this resolution only as a single object):