PARALLAX, PROPER MOTION, RADIAL VELOCITY AND SPACE VELOCITY
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 parallax, π, is the difference between the geocentric and
heliocentric positions of the star
The effect is small: π < 1" for all stars
tan(π) ~ π (radians) = r / d
To convert to arcsec: π (arcsec) = 206265 π (radians)
If r = 1 Astronomical Unit (A.U.) = 150,000,000 km:
We define 1 parsec (pc) = 206265 A.U. = 3x1013 km =
3.26 light years (l.y.) Then:
Nearest star: Proxima Centauri
1.31 pc = 4 l.y.
π = 0.76" = size of dime at d = 6 km.
Thus, parallaxes are hard to measure (not seen until 1838 by F.W. Bessel,
who determined the parallax of 61 Cygni at 0.29 arcsec --
final proof of heliocentric solar system):
Best π from ground (e.g., McCormick
refractor) π ~ 0.002" errors
Thus d ~ 100 pc stars (Vyssotsky) have ~20% distance errors, which
yields 40% errors in brightness.
This is a radius around the Sun that is rather limiting in terms of
the range of types of targets.
Many of the ground-based parallaxes have been collected into
the Yale Parallax Catalogue, also known as General Catalogue
of Trigonometric Parallaxes (van Altena et al. 1995).
Space astrometry offers a number of advantages.
No atmospheric refraction.
Low gravity -- reduced mechanical flexure.
Space astrometry missions:
Best current space π:
High Precision Parallax Collecting Satellite (HIPPARCOS)
π ~ 0.001" for 120,000 stars, complete to V ~ 8, but
stretching to V ~ 12.
--> star of ~200 pc has 20% distance errors - some halo stars,
some variable star standard candles
Still rather limiting volume: Little representation by halo stars and
other rare astrophysically interesting species (e.g., Cepheids).
Note program using HST Fine Guidance Sensors (interferometers)
on small number
of selected, astrophysically interesting targets.
~ 0.4 mas parallaxes to V=15.
A check on HIPPARCOS.
ESA's Gaia satellite
(scanning, all-sky mission launched end of 2013):
π ~ 10-5 arcsec to V ~ 15.
The Astrometric instrument (ASTRO) is devoted to star angular position
measurements, providing the five astrometric parameters:
Star position (2 angles)
Proper motion (2 time derivatives of position)
ASTRO is functionally equivalent to the main instrument used on the Hipparcos
The Photometric instrument provides very low resolution
(30 to 270 Angstroms/pixel)
star spectra (sufficient to judge the spectral energy distribution, "SED")
to derive estimates of stellar parameters (like temperature, metallicity)
in the band 320-1000 nm and the ASTRO chromaticity calibration.
The Radial Velocity Spectrometer (RVS) provides radial velocity and
medium resolution (R ~ 11,500) spectral data in the narrow band 847-874 nm,
for stars to about 16th magnitude (~150 million stars) and astrophysical
information (reddening, atmospheric parameters, rotational velocities)
for stars to 12th mag (~5 million stars), and elemental abundances to about 11th mag
(~2 million stars).
measure the positions of ~1 billion stars both in our Galaxy and other members
of the Local Group, with an accuracy down to 24 microarcseconds for stars to V = 15
and to 0.5 milliarcsec for stars to V = 20;
perform spectral and photometric measurements of these objects;
derive space velocities of the Galaxy's constituent stars using the stellar
distances and motions;
create a three-dimensional structural map of the Galaxy.
The first GAIA data release has already happened! It includes proper motions
as determined by comparing GAIA positions against those of the Tycho instrument on the
Hipparcos satellite ("TGAS" = "Tycho-Gaia Astrometric Sollution").
Gaia Data Release 2 (DR2) with proper motions from all Gaia data is expected in a few months.
Gaia will soon have a major impact on this field.
Historical note: The U.S. has attempted several of its own astrometric missions in the
past 2 decades, including the U.S. Naval Observatory efforts FAME, OBSS, and even a more
recent version (J-MAPS) -- but none have come to fruition.
Also of historical note was NASA's
Space Interferometry Mission (SIM), which went through various incarnations (e.g., SIMLite,
SIM PlanetQuest) and advanced quite
far in planning until cancelled in 2010. It was a pointed interferometer in space, with
π ~ 10-6 arcsec to V ~ 20. (Like reading a newspaper
headline at lunar distance!)
d ~ 1000 kpc - entire galaxy at least, and even bright stars at M31-like distances.
Would have been able to find Earth-like planets around the nearest stars based on measuring the reflex
motions of the central stars (a primary motivation for the mission).
The actual yearly reflex motion of a star on the sky from
parallactic motion depends on the position of the star
relative to the ecliptic:
One actually measures the relative parallax of a nearby star
with respect to a set of presumably more distant reference stars in the
same part of the sky.
These background sources move on parallactic ellipses similar
to those of the target star, but hopefully much smaller.
These motions have to be corrected out
based on an estimate of
the distances of the stars, which are not known a priori
and must be deduced statistically based on a Galactic model of
mean distances with magnitude or some other method (e.g.,
photometric or spectroscopic parallaxes).
The result of these corrections is an absolute parallax
for the foreground star.
Parallax work from the ground is best done near/at twilight
(morning, evening). Why??
The space velocity of a star:
VS = space velocity (total velocity of a star)
VT = transverse velocity (velocity perpendicular to
line of sight -- obtained by knowing proper motion, μ, and distance, d )
VR = radial velocity (velocity parallel to line of sight = Doppler
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.
This is the formula in the non-relativistic regime.
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:
Earth orbital velocity (maximum 30 km/s correction).
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 Local Standard of Rest radial velocity corrects the heliocentric
radial velocity to that one would see in the LSR rest frame after removing the
Sun's peculiar motion.
In a right-handed coordinate system, and adopting the "basic
solar motion", we have:
The Galactic Standard of Rest radial velocity further corrects the velocity
for the LSR velocity projected on the line of sight.
Assuming ΘLSR = 220 km/s:
(It should be noted that the vGSR velocity is sometimes
called "vLSR", meaning "corrected for LSR velocity.)
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
This is a natural system in which to explore motions of stellar populations
in the Galaxy without having to worry about the (l,b ) direction of
the stars and the Sun's motion projected on this line of sight.
Transverse velocities, VT, cannot be measured directly.
Only the angular change, the proper motion, can be observed.
To convert from the proper motion to the transverse velocity, one needs
to know the distance, d, to the star.
When one works out the math (problem for the student) one finds that:
The proper motion, μ, has units
of angle change per unit time.
Common units for proper motions (which are typically very small) are:
By the above equation, we see that a proper motion can be large if:
the star has small distance, d
the star has a large transverse velocity with respect to
Typical star velocities with respect to the Sun are 10's of km/s.
Highest GSR speed in Galaxy (determined by the
Galactic escape velocity) should be ~475 km/s.
But note recent interesting discovery of hypervelocity stars,
which appear to be ejected from the center of the Milky Way (red lines
in cartoon below):
Note that because the origin point of the stars is assumed to be known, if
we can measure the motion of the stars now we can see the curvature
of their orbit induced by any non-spherical distribution of mass in the
Milky Way (yellow lines in cartoon above).
This is a new way to measure the shape of the dark matter in the Milky Way
(see Gnedin et al. 2006), if we can measure the proper motions accurately
(e.g., with something of the precision that had been intended for SIM PlanetQuest).
If Sun revolves around Galaxy about
11 km/s in advance of the Local Standard of Rest...
What does the above fact say about the orbit of the
Sun and where the Sun is in that orbit??
The apex of the solar motion,
the direction in the
sky that the Sun seems to be headed (and the direction
from which comes the mean motion of nearby stars)
is at Galactic coordinates (l,b)=(56, 22)o,
which should be compared to the nominal LSR direction
The Sun's motion imposes
a solar reflex motion on the proper motions (and this,
of course, can be removed to give Galactic standards of rest proper
How might one determine the solar motion
with respect to the LSR?
How does this problem compare to the idea of
the radiant of a meteor shower?
In which direction should the mean radial velocities
of stars be highest? Lowest?
In which direction should the mean proper motions
of stars be zero?
The techniques of measuring secular parallaxes and statistical parallaxes
make use of these global solutions against surbveys of radial velocities and proper motions
(see Mihalas & Binney or your textbook).
But typical stars are far, so proper motions are small!
For naked eye stars, typically < 0.1" per year
But, over time, this amount of motion adds up:
Only a few hundred stars have proper motions > 1.0" per year
The largest known proper motion is that of the nearby Barnard's star:
μ = 10.25" per year
Barnard's Star has a large motion because it is the 4th closest star to us at
only 1.8 pc.
This is an animated GIF composed of four images of
Barnard's star taken 5, 2 and 1 year apart. First in 1991 on 4th June,
5min exposure on hypered Tech Pan 2415 film, second in 1996 on 24th June,
30s exposure using an SXL8 camera (kindly loaned at the time by
Eric Strach) and the last two in 1998 and 1999 using the SX mono camera.
Images were exported as JPG files, aligned and made into the animated GIF
A crude pixel to pixel measurement of the motion over 5 years gave just
over 10" per annum, very close to the correct proper motion of 10.36" per
annum. From www.fornax.pwp.blueyonder.co.uk/images.html.
Some famous catalogs in astronomy containing high proper motion stars:
1976: Luyten Half-Second Catalogue (LHS) -- all stars μ > 0.5" per annum
1955-1961: Luyten Two-Tenths Catalogue (LTT) -- μ > 0.2" per annum
1971: GICLAS Catalogue/LPMS (Lowell Proper Motion Survey)
-- μ > 0.2" per annum
These catalogs contain lots of interesting stars, because, as mentioned above,
there are two ways to get high μα
VT / d:
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!
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:
HIPPARCOS had precise positions, but only an ~3 year mission
proper motions did not benefit from long time baseline.
(~120,000 stars to V ~ 9) to 1 mas/yr
Tycho Catalog on board HIPPARCOS mission
(1 million stars to V ~ 11) to 20 mas/yr.
New reduction, Tycho 2 Catalog proper motions, also known as
Astrographic Catalogue 2000 (AC2000), combined positions from
Tycho and older Astrographic Catalogue (a program
using the Carte du Ciel photographic survey of 1895-1920,
median epoch 1904) to obtain
proper motions with average error 2.4 mas/yr.
(2.5 million stars to V ~ 11.5)
Northern Proper Motion Survey (NPM1/NPM2):
Lick Observatory photographic program of northern sky
proper motions for northern 2/3 of sky (total of about 300,000 stars
for 8 < B < 18 referenced to galaxies with errors 5 mas/yr).
(Not all stars in the magnitude limit, but Input Catalog of Special Stars
and a random sampling of other "anonymous" stars.)
Southern Proper Motion Survey (SPM): Southern complement of NPM
by Yale/San Juan (1 million stars with 5 < V < 18.5 referenced to galaxies
with precision of 4.0 mas/yr for well-measured stars).
USNO CCD Astrograph Catalog (UCAC): CCD astrometry on > 50 million
9 < R < 16 with 20-70 mas accuracies.
A catalog of proper motions combining the USNO-B (see below) and Sloan DSS positions
by Munn et al. (2004).
The future of astrometry is fairly bright, especially with the success of Gaia, and
with several large scale surveys underway
or planned, as well as a number of facilities that will be available for undertaking
ground-based astrometry of high precision (note that this table contained information
on several missions, like J-MAPS and SIM-Lite, that have been cancelled):
From Majewski 2010, in "IAU262: Stellar Populations: Planning for the Next Decade",
eds. G. Bruzual & S. Charlot.
Knowing whether your target has a sizable proper motion is important
because outdated coordinates will not point you in the correct place.
Thus, in addition to giving the equinox of your coordinates --
which tells you what precessional year your coordinate system corresponds
to -- for large proper motion stars you have also to give the epoch
of the coordinates of the star, which tells you in what year the star
was at any specific coordinates.
PLEASE know the difference between epoch and equinox of coordinates!
For example, what would it mean to give the position of a
star at epoch 1975 in equinox 2000 coordinates?
If you know the proper motion of the star for one year, you can correct
the coordinates to the position the star has in any other year.
Note that a proper motion is a motion in both right ascension and
Thus, we always have to give information on both the size of the
proper motion, and the direction of the motion.
The direction of motion is called the position angle,
θ of the motion, and it is the angle between the
direction of the NCP and the direction of motion of the star.
We define θ = 0o as motion
due North and θ = 90o
as motion due East.
Thus, proper motions are given as the pair of values
Alternatively, instead of giving total motion and position angle
(a "radial coordinate system"), one could break up the motion into
components of proper motion in the right ascension and declination
θ = position angle of star = angle between direction
to NCP and the direction of motion of the star, and μα
is the annual rate of change in right ascension (can be given in seconds of
time per year, but above shown in arcseconds per year).
Note that the cosδ term is needed to
account for the convergence of
meridians toward the NCP and SCP. (The cosδ is small when
δ is large.)
Generally we actually derive relative proper motions of stars with respect
to other reference stars nearby.
Relative proper motions must then be corrected to absolute
proper motions by correcting for the average proper motions of the
reference stars (obtained by a model, or by transfer to some established
fundamental reference system).
Alternatively, one can tie to some extragalactic sources
Compact sources that are reliably centroided -- e.g., QSOs or compact
galaxies -- are preferred.
For example, the SIM proper motions were to be put on an absolute
reference frame via nearby stars that are part of a global
Astrometric Grid of reference stars, whose motions themselves will be
measured with respect to quasars.
The magnitude limit one needs to achieve to see these extragalactic
sources depends on the field of view of the image.
The density of sources to R ~ 21 are:
all galaxies: ~3000 deg-2 -- but only
about 10% of these are probably appropriately compact.
quasars: ~100 deg-2
Absolute proper motions are essential for a proper interpretation
of space velocities of stars.
SPACE VELOCITY (REPRISE)
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
...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
One can also propagate the errors using the standard equation and assuming
the errors are uncorrelated:
ASTROMETRIC REFERENCE SYSTEMS
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 easier to measure relative positions/parallaxes/proper motions
between stars with small angular separation.
Differential effects of atmospheric refraction, telescope flexure,
telescope guiding, etc. minimized on smallest angular scales in single
... much more difficult is to get absolute positions/parallaxes/proper motions:
All of the above effects are more insidious over larger angles, and
not trivial to account for mean position/parallax/proper motion of the
entire field of study.
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
Every catalogue of such stars establishes an astrometric system.
Several of these have been set up and continue to be updated, since
every reference frame set up deteriorates with time (as reference stars move from
their nominal positions with proper motions that can differ from their
imperfectly measured values).
Some examples of reference catalogs are:
Astrographic Catalogue Reference Stars (ACRS) -- 320,000
reference stars to V ~ 10.5 with 0.23" positions.
Fundamental Katalog (FK5) -- about 4500 stars to V ~ 9
to 0.15" precision.
International Reference Stars (IRS) -- 36,000 reference stars
with 7 < V < 9 to 0.22".
PPM -- ~380,000 reference stars at 0.30" precision in north
and 0.16" in the south.
A useful summary of astrometric catalogs and which are recommended for various
uses has been made
U.S. Naval Observatory.
Note that some catalogs are based on other ones (i.e., intended as extensions
of a previous astrometric system).
A problem you can see with the above reference catalogs is their
bright magnitude limits. This can be a serious problem for new digital
surveys, where such stars can be very saturated.
A commonly used positional catalog for use at faint magnitudes
(to V ~ 21), the
USNO-B, has been set up by the U.S. Naval Observatory by scanning Schmidt
plates from the Palomar Observatory Sky Surveys and the ESO Schmidt survey
of the southern sky.
The positional accuracy of USNO-B is 0.2 arcsec.
The catalogue has over 1 billion objects.
The USNO-B finds numerous uses in calibrating positions
of sources at faint magnitudes (e.g., astrometry needed
for fiber/multislit spectroscopy).
Many of these reference catalogs are based on meridian circle measurements.
These are telescopes mounted so that that have
motion only along the local meridian.
The position of a star is precisely measured by the time of its
transit through the meridian.
The Carlsberg Automated Meridian Circle at La Palma.
These tend to be small aperture refractors, and so generally
look at brighter stars that tend to have larger proper motions.
All of these reference systems are references to the extension of the Earth's
equator on the sky:
Thus they are affected by precession, nutation, and
many other higher order Earth motions which are unknown to certain degrees,
and which mean that the reference system is constantly changing.
A summary of these kinds of effects is given
Imperfect knowledge of the Earth's motion means that things like
the (constantly changing) position of the vernal equinox are not
defined well enough for high precision astrometric work.
In addition, systematic errors in stellar proper motions can
result in time-dependent warps and spurious rotations of the reference frame.
The International Celestial Reference System (ICRS), an astrometric system
referenced to distant extragalactic sources, has been established to fix this problem.
The ICRS is defined with respect to the solar system barycenter at a
specific epoch (e.g., 1 January 2000) and has "fixed" axes.
Precise positional measurements use Very Long Baseline Interferometry
(VLBI) radio observations of mostly several hundred quasars to
establish an "inertial" reference frame.
These define the ICRS pole and right ascension origin to 20 microarcseconds.
The ICRS defining objects (top) and extended list of secondary reference objects (bottom).
From Kovalevsky & Seidelmann Fundamentals of Astrometry.
Creation of the ICRS is coordinated by the IAU.
The VLBI work has shown deficiencies in models of Earth's motion
(e.g., the precession rate had been overestimated by ~ 0.3 arcsec/century)
and, e.g., the FK5 system.
The bridge ("frame-tie")
from the radio definition of the ICRS to the optical is
an issue of great concern.
The ICRS defining objects have 14 < V < 23, with majority 18 < V < 19.
Not easy to observe optically.
The radio stability of the QSOs is also of great concern (see example below) -- especially
as efforts move to defining the system on microarcsecond levels.
The evolution in shape of a potential (but rejected) ICRS radio source.
From Kovalevsky & Seidelmann Fundamentals of Astrometry.
Gaia will help with some of this, given its broad magnitude coverage, although with
poorer precision at the magnitudes of QSOs.