Proper motion

gravitational lensing

or otherwise then μ = ${\displaystyle {\frac {v_{t}}{d}}}$ where ${\displaystyle v_{t}}$ is the distance (usually expressed as annual velocity) transverse (tangential or perpendicular) to line of sight from the Sun. The angle is shaded light blue from the Sun to the object's start point and its year later position as if it had no radial velocity.
In this diagram the radial velocity happens to be one of the Sun and object parting, so is positive.

Proper motion is the

Over the course of centuries, stars appear to maintain nearly fixed positions with respect to each other, so that they form the same

This motion is caused by the movement of the stars relative to the

Proper motion may alternatively be defined by the angular changes per year in the star's right ascension (μ_α) and declination (μ_δ) with respect to a constant epoch.

The

The magnitude of the proper motion μ is given by the Pythagorean theorem:^[10]

${\displaystyle \mu ^{2}={\mu _{\delta }}^{2}+{\mu _{\alpha }}^{2}\cdot \cos ^{2}\delta \ ,}$

technically abbreviated:

${\displaystyle \mu ^{2}={\mu _{\delta }}^{2}+{\mu _{\alpha \ast }}^{2}\ .}$

where δ is the declination. The factor in cos²δ accounts for the widening of the lines (hours) of right ascension away from the poles, cosδ, being zero for a hypothetical object fixed at a celestial pole in declination. Thus, a co-efficient is given to negate the misleadingly greater east or west velocity (angular change in α) in hours of Right Ascension the further it is towards the imaginary infinite poles, above and below the earth's axis of rotation, in the sky. The change μ_α, which must be multiplied by cosδ 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".^[11]

If the proper motion in right ascension has been converted by cosδ, the result is designated μ_α*. For example, the proper motion results in right ascension in the

The position angle θ is related to these components by:[2]^[13]

${\displaystyle \mu \sin \theta =\mu _{\alpha }\cos \delta =\mu _{\alpha \ast }\ ,}$

${\displaystyle \mu \cos \theta =\mu _{\delta }\ .}$

Motions in equatorial coordinates can be converted to motions in

In 1992 Rho Aquilae became the first star to have its Bayer designation invalidated by moving to a neighbouring constellation – it is now in Delphinus.^[16]

Usefulness in astronomy

Stars with large proper motions tend to be nearby; most stars are far enough away that their proper motions are very small, on the order of a few thousandths of an arcsecond per year. It is possible to construct nearly complete samples of high proper motion stars by comparing photographic sky survey images taken many years apart. The

Proper motions of the galaxies in the Local Group are discussed in detail in Röser.^[19] In 2005, the first measurement was made of the proper motion of the Triangulum Galaxy M33, the third largest and only ordinary spiral galaxy in the Local Group, located 0.860 ± 0.028 Mpc beyond the Milky Way.^[20] The motion of the Andromeda Galaxy was measured in 2012, and an Andromeda–Milky Way collision is predicted in about 4.5 billion years.^[21] Proper motion of the NGC 4258 (M106) galaxy in the M106 group of galaxies was used in 1999 to find an accurate distance to this object.^[22] Measurements were made of the radial motion of objects in that galaxy moving directly toward and away from Earth, and assuming this same motion to apply to objects with only a proper motion, the observed proper motion predicts a distance to the galaxy of 7.2±0.5 Mpc.^[23]

History

Proper motion was suspected by early astronomers (according to

The lesser meaning of "proper" used is arguably dated English (but neither historic, nor obsolete when used as a postpositive, as in "the city proper") meaning "belonging to" or "own". "Improper motion" would refer to perceived motion that is nothing to do with an object's inherent course, such as due to Earth's axial precession, and minor deviations, nutations well within the 26,000-year cycle.

Stars with high proper motion

The following are the stars with highest proper motion from the Hipparcos catalog.^[26] It does not include stars such as Teegarden's Star, which are too faint for that catalog. A more complete list of stellar objects can be made by doing a criterion query at the SIMBAD astronomical database.

Highest proper motion stars^[27]

#	Star	Proper motion		Radial velocity (km/s)	Parallax (arc seconds)	Distance in parsecs ${\displaystyle \left({\frac {1}{\text{parallax}}}\right)}$
		μα · cos δ (mas/yr)	μδ (mas/yr)
1	Barnard's Star	−798.58	10328.12	−110.51	0.54831	1.82
2	Kapteyn's Star	6505.08	−5730.84	+245.19	0.25566	3.91
3	Groombridge 1830	4003.98	−5813.62	−98.35	0.10999	9.09
4	Lacaille 9352	6768.20	1327.52	+8.81	0.30526	3.28
5	Gliese 1 (CD −37 15492) (GJ 1)	5634.68	−2337.71	+25.38	0.23042	4.34
6	HIP 67593	2118.73[28]	5397.57[28]	−4.4	0.18776	5.33
7	61 Cygni A & B	4133.05	3201.78	−65.74	0.286	3.50
8	Lalande 21185	−580.27	−4765.85	−84.69	0.39264	2.55
9	Epsilon Indi	3960.93	−2539.23	−40.00	0.27606	3.62

The figure for HIP 67593 is almost certainly an error, probably because the star has a relatively nearby brighter visual binary companion; the movement between the DSS2 and SDSS9 images is less than it. Gaia measured a much smaller proper motion for its Data Release 2, yet a 15-fold parallax between it and its likely common-proper-motion companion HIP 67594. Reconciling its distance and motion will have to wait for Data Release 3 expected to analyse well very high proper motion objects.

See also

• Astronomical coordinate systems
• Galaxy rotation curve
• Leonard–Merritt mass estimator
• Milky Way
• Peculiar velocity
• Radial velocity
• Relative velocity
• Solar apex
• Space velocity (astronomy)
• Stellar kinematics
• Very-long-baseline interferometry

References