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ASTR 5610, Majewski [SPRING 2018]. Lecture Notes

ASTR 5610 (Majewski) Lecture Notes




Stellar (trigonometric) parallax is the apparent shift in the position of a nearby star because of the orbital motion of the Earth about the Sun.


The space velocity of a star:



Obviously, the line-of-sight (radial) velocity for Galactic stars can be obtained by the Doppler shift:

VR = c (λ - λ0) / λ0

where λ is the observed wavelength of a particular spectral line and λ0 is the rest frame wavelength of the line.

At the telescope we actually measure geocentric radial velocities, which are not "standard", since there is a significant imposed velocity variation due to the relative position of the source and the mean motion of the Earth at the time of the observation.

Heliocentric radial velocities are reported because they correct out the following components of Earth's motion projected on the line of sight:

In terms of understanding Galactic dynamics, it often more useful to interpret radial velocities that also take out the motion of the Sun projected on the line of sight.

The GSR velocities of stars should be interpreted as the velocity that a stationary observer in the Galactic rest frame would see at the position of the Sun.


Transverse velocities, VT, cannot be measured directly.

Only the angular change, the proper motion, can be observed.

The proper motion, μ, has units of angle change per unit time.

By the above equation, we see that a proper motion can be large if:

Typical star velocities with respect to the Sun are 10's of km/s.

But typical stars are far, so proper motions are small!

Some famous catalogs in astronomy containing high proper motion stars:

These catalogs contain lots of interesting stars, because, as mentioned above, there are two ways to get high μα VT / d:

  1. large VT = "high velocity stars" - stars with large VT w.r.t. solar motion, stars not moving like the Sun, so typically not a disk star. Halo stars!
  2. small d = very nearby stars (e.g. Barnard's), often hard to find in any other way except large proper motion.
Other presently popular proper motion catalogs are:

Knowing whether your target has a sizable proper motion is important because outdated coordinates will not point you in the correct place.

Note that a proper motion is a motion in both right ascension and declination.

Generally we actually derive relative proper motions of stars with respect to other reference stars nearby.


Now that the definitions of parallax, radial velocity and proper motion have been given, it is possible to use their combination to derive (U,V,W) space velocities with some appropriate coordinate transformations.

Johnson & Soderblom (1987, AJ, 93, 864) have summarized the relevant equations in a right-handed coordinate system, and the following equations and discussion are taken from their paper:

If you start with the following data, including proper motions in equatorial coordinates:

...and consider the following definitions of the transformation angles:

The first consideration is the conversion from equatorial to Galactic coordinates, completed by a matrix operation:

(To convert to a left-handed coordinate system, invert the signs in the top row of T.)

One can also propagate the errors using the standard equation and assuming the errors are uncorrelated:



The measurement of both parallaxes and proper motions starts with simply measuring positions of stars at one epoch.

To see the changes in position that result from parallax and proper motion requires first measuring accurate relative positions between targets and nearby reference stars....

... much more difficult is to get absolute positions/parallaxes/proper motions:

To address this problem, catalogs of fundamental stars with "absolute positions and proper motions" are set up by astrometrists to establish a global reference frame for more localized positional measures.

Many of these reference catalogs are based on meridian circle measurements.

All of these reference systems are references to the extension of the Earth's equator on the sky:

The International Celestial Reference System (ICRS), an astrometric system referenced to distant extragalactic sources, has been established to fix this problem.

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All material copyright © 2003,2006,2008,2010,2012,2014,2016,2018 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 551 and Astronomy 5610 at the University of Virginia.