Astrometry: Revealing the Other
Two Dimensions of Velocity
Space
H.A. McAlister
Center for High Angular Resolution Astronomy
Georgia State University
Astrometry: What is It?
•
Astrometry deals with the measurement of the positions and
motions of astronomical objects on the celestial sphere. Two
main subfields are:
•
•
Spherical Astrometry – Determines an inertial reference
frame, traditionally using meridian circles, within which
stellar proper motions and parallaxes can be measured.
Also known as fundamental or global astrometry.
•
Plane Astrometry – Operates over a restricted field of view,
using, through much of the 20th century, long-focus
refractors at plate scales of 15 to 20 arcsec/mm, with the
goal of accurately measuring stellar parallax. Also known
as long-focus or small-angle astrometry.
Astrometry relies on specialized instrumentation and
observational and analysis techniques. It is fundamental to all
other fields of astronomy.
Astrometry: What are its tools?
•
Spherical Astrometry:
•
•
•
•
•
•
Meridian circles & transits
Astrolabes
Space (eg. Hipparcos)
Phased optical interferometry
Very long baseline radio interferometry
Plane Astrometry:
•
•
•
•
•
•
•
•
Photography (until relatively recently)
Visual Micrometry (until relatively recently)
CCD imaging
Scanning photometry
Speckle interferometry (O/IR)
Michelson interferometry (O/IR)
Radio interferometry
Space (imaging & interferometry)
Some 19th & 20th C. Astrometric Instruments
Greenwich Meridian Circle House
USNO 6-in Transit Circle 1898-1995
Photo from National Maritime Museum, Greenwich
W. Finsen’s Eyepiece
Interferometer
c. 1960
Worley’s recording
filar micrometer at
USNO
U.S. Naval Observatory Photo
U.S. Naval Observatory Photo
Hipparcos Test 1988
ESA Photo
Spherical Astrometry
What does it measure?
Spherical Astrometry attempts to measure stellar & planetary
positions and motions in an inertial reference frame. This is
very challenging and must account for effects of:
•
•
•
•
•
Proper motion
Parallax
Galactic rotation
Precession
Nutation
•
•
•
•
•
Aberration
Polar wandering
Refraction
Solar motion
Earth dynamics (including time)
Oh, yeah, and whatever the heck systematics are in
your instrumental and reduction methodology!
A truly inertial reference frame has been achieved through the
tie-in to extragalactic radio sources using VLBI.
Sound difficult?
It is!
Plane Astrometry
What does it measure?
Plane Astrometry attempts to measure small angle effects
relative to a background of one or more reference stars. These
effects include:
•
•
•
•
•
•
Proper motion
Parallax (relative and absolute)
Secular acceleration
Sub-motions arising from unseen companions
Binary star relative separation and orientation
Stellar diameters
And, here again, whatever the heck systematics are
in your instrumental and reduction methodology!
Also sound difficult?
It is!
Astrometry
What does it give us?
Spherical Astrometry gives us:
•
•
•
•
•
Galactic kinematics and dynamics
Solar system dynamics
Time
Periodic and secular changes in Earth orientation
Navigation (mostly historical except during WWIII)
Plane Astrometry gives us, when combined with other
techniques, stellar:
•
•
•
•
•
Luminosities
Masses
Diameters (and shapes)
Temperatures
Multiplicity frequencies
•
•
•
•
Space velocities
Distribution in solar
neighborhood
Unseen companions
Calibration of cosmic
distance scale
And, yet again, the systematics in your instrumental and
reduction methodology must be understood!
Sound useful?
It sure is!
“Astrometry is the metrological basis of astronomy”
from J. Kovalevsky, Modern Astrometry, p.8.
Astrometry for Dummies
This says it all
(Circulated c. 1974 by Ron Probst at UVa)
Okay, here’s a basic: Trigonometric Parallax
A star is observed against a background of
distant stars at 6-month intervals
t1
Trig Parallax =
Earth’s orbit
Nearby Star
t2
Very Distant
Background
Stars
2x
t2
t1
The nearby star appears to shift back and forth with
respect to the background stars
Okay, here’s a basic: Trigonometric Parallax
A star is observed against a background of
distant stars at 6-month intervals
t1
Trig Parallax =
Earth’s orbit
More
Distant Star
t2
Very Distant
Background
Stars
2x
t2
t1
A somewhat more distant star presents a smaller back and
forth motion with respect to the background stars
The Challenge of Astrometry
•
Astrometry constantly pushes the limits imposed by the
current status of its instrumentation and reduction &
analysis techniques.
•
Historically, breakthroughs in astrometry are almost always
the result of discrete advances in instrumentation followed
by refinements made by clever people.
•
Ultimately, astrometry is all about understanding and
probing the distinction between precision and accuracy.
Gain in Accuracy
Adapted from Parallax by A. W. Hirshfield
Observer
Technique
Date
Limiting
Accuracy
Hipparchus
Tycho
Flamsteed
Bradley
Bessel
Schlesinger et al.
USNO et al.
Various
USNO et al.
HIPPARCOS
HST
Various
GAIA (ESA)
OBSS (USNO/NASA)
SIM (JPL/NASA)
visual sextant
visual quadrant
mural quadrant
improved quadrant
visual heliometer
early photography
later photography
speckle interferometry
CCD imagine
space craft
FGS
long-baseline interfer.
space craft
space craft
space interferometry
150 BC
1600
1700
1750
1835
1920
1970
1990
2000
1990
2000
2000
2010
?
2009
5 arcmin
1 arcmin
10 arcsec
0.5 arcsec
0.1 arcsec
0.05 arcsec
5 mas
3 mas
1 mas
1 mas
0.5 mas
100 µas
10 µas
10 µas
1 µas
Gain
1
5
Telescope
30
600
3x103
Photography
6x103
6x104
1x105
3x105
3x105
6x105
3x106
3x107
3x107
3x108
Jesse Ramsden
Some results from astrometry 18001800-1900
Dividing machines
1801 : First asteroid,
asteroid, Ceres, discov.
discov. by Guiseppe Piazzi
1838 : Parallax of stars by Bessel,
Bessel, Henderson, Struve
1850 : 20 parallaxes in a catalogue by Peters
Lathes to make round
mechanical parts, screws
1837 : From 390 proper motions Argelander : solar apex
1846 : Neptune predicted and discovered
Celestial mechanics flourishes
AltAlt-azimuth circle 1789 for
Piazzi’s Palermo Observatory
1860 - : Bonner + Cordoba Surveys 1,000,000 stars
1890 - : Photography : parallaxes and sky surveys
13
Astrometry of small angles 1/2
Astrometry of small angles 2/2
i.e. within the telescope field of view
i.e. within the telescope field of view
1611 – 1660 : Estimation: diameters, relative positions,
Galilei et al.
1659 – 1990 : Wire micrometer by Huygens
Gascoigne 1640, published later
B
A
From 1750 also heliometer =
. C
.
divided--lens micrometer
divided
A
for the same purposes
.
Science :
..
Diameters of planets
Relative positions
Double stars : stellar masses
Relative parallaxes 18381838-1900
14
1890 - 1990 : Photography : same science as micrometer :
Diameters of planets
Relative positions of stars and solar system objects
Double stars : stellar masses
Relative parallaxes
1990 - : 1,000,000,000 stars, e.g. US Naval Observatory Catalogues
1920 - : Interferometry : stellar diameters, double stars
1990 - : Space astrometry : Hipparcos, Hubble
1980 - : CCD astrometry :
on ground : 100,000,000 stars to 17th mag + solar system
from space : Hubble relative parallaxes 0.001”
Principle of the
wire micrometer
15
Some results from astrometry 19001900-2000
By 1900 : 539 stars 0.01”/a motions Decl. > -10 deg
16
Copenhagen meridian circle
Photoelectric astrometry begins in 1925
1905 : Hertzsprung discovers dwarfs/giants using motions for distances
100 stars 0.04” relative parallaxes
By 1950 : 33,342 stars 0.01”/a motions,
5822 stars 0.01” relative parallaxes
500 stars with <10% error on distances
! 1970 - : Radio astrometry : accurate absolute positions,
reference system by quasars, Earth rotation
1996 : Hipparcos satellite : accurate large & small angles
120,000 stars 0.001”/a motions (N & S)
120,000 stars 0.001” absolute parallaxes
22,000 stars with <10% error on distances
2000 : Tycho
Tycho--2 : 2.500,000 stars 0.002”/a motions
USNO : 1,000,000,000 stars to 20th mag
17
Courtesy: Steno Museum, Aarhus
Bengt Strömgren 1925 and 1933
Experiments with
photoelectric recording of transits
18
3
Slits + Photon counting vs. Time
=> Astrometry + Photometry
Carlsberg automatic meridian circle
on La Palma from 1984
Ideas 1960
y
y
t2
t1
x1
x2
Photoelectric
during 14 years
star
x
Light intensity
= Photons
per second
x ~ time
t1
t2
Then from 1998
CCD micrometer :
20,000,000 star
observations
per year
0.1” per obs.
time
+ switching
mirror mirror
B. Strömgren 1933: slits
+ switching
Atomic bombs 1957 : Counting techniques
E. Høg 1960 : Slits + counting >>> implementation on meridian circles
P. Lacroute 1967: Go to space!
E. Høg 1975: Design of Hipparcos
19
Telescope of Hipparcos
Hipparcos and Tycho 19751975-2000
Focal plane of Hipparcos –
Tycho
20
Star mapper
grid
Mission concept 1975
Mission approval Feb. 1980
Tycho proposal April 1981
Observing 1989 - 93
Catalogues 1996
Tycho--2 Catalogue in 2000
Tycho
2.500,000 stars
500 citations until 2008
Schmidt type
system
D = 0.29 m
F = 1.4 m
Two fields on the sky
Modulating grid
Observing 1989 - 93
22
21
Telescope and payload of Gaia
Astrometric Accuracy versus Time
Launch 2012
Two SiC primary mirrors
Rotation axis (6 h)
1.45 × 0.50 m2 at 106.5°
106.5°
Basic angle
monitoring system
1000
Hipparchus/Ptolemy - 1000 stars
T he Landgrave of Hesse - 1000 stars
100
SiC toroidal
structure
(optical bench)
Tycho Brahe - 1000
Flamsteed - 3000
10
arcsec 1
Bradley - aberration
0.1
0.001
Superposition of
two Fields of View
0.0001
0.00001
Erik Høg
1995/2008
Argelander - 34,000
FK5 - 1500
Bessel - 1 star
Jenkins - 6000
0.01
Combined
focal plane
(CCDs)
F = 35 m
Lalande - 50,000
Positions
Parallaxes =
Small angles
All parameters
Hipparcos - 120,000
ROEMER - 4 5 million
Proposal 1992
PPM - 379,000
TychoTycho-2
- 2.5 million
U SNO - 360
Gaia
- 1200 million
Gaia 23 million
SIM - 10,000
150 BC … 1600
1800
2000
Year
Two anastigmatic offoff-axis telescopes
Figure courtesy EADSEADS-Astrium
23
4
Astrometry Through the Ages VII.
The Race for Stellar Parallax I.
Friedrich Wilhelm Bessel (1784-1846)
•
Quit school at 14 to become an accountant and spent his “leisure” time
studying navigation and astronomy.
•
Calculated orbit of Comet Halley in 1804.
•
At Konigsberg Observatory, Bessel undertook measuring positions of
50,000 stars and determined parallax of 61 Cygni in 1838.
•
Discovered the submotion of Sirius.
Friedrich Georg Wilhelm von Struve (1793-1864)
•
From 1820-1839 he was director of the Dorpat Observatory in
Tartu, Estonia, where he measured the parallax of Vega in 1839.
•
Founded Pulkovo Observatory near St. Petersburg in 1839.
•
First “modern” observer of double stars and published his great
Stellarum Duplicium et multiplicium Mensurae Micrometricae in
1837.
Thomas Henderson (1798-1844)
•
A Scottish lawyer by trade, his amateur astronomy successes earned
him the first directorship at the Cape Observatory in S. Africa.
•
He spent 13 months there during 1832-33, a period of time he loathed,
and obtained many stellar position measurements.
•
Calculated the parallax of Centauri in 1838 during his retirement,
but Bessel and Struve had already beaten him in the literature.
Astrometry Through the Ages VIII.
The Race for Stellar Parallax II.
Who did the best job?
Bessel (61 Cyg)
0.314
HIPPARCOS (61 Cyg)
0.292
% error
7.5
Struve (Vega)
0.261
HIPPARCOS (Vega)
0.129
% error
102
Henderson ( Cen) HIPPARCOS (Vega)
1.0
0.742
% error
35
So, Bessel wins not only in timing but in quality!
“It is the greatest and most glorious accomplishment which
practical astronomy has ever witnessed.”
Sir John Herschel upon Bessel’s award of the Gold Medal of the Royal
Astronomical Society
By 1904, parallaxes of only 72 stars were listed in Newcomb’s The
Stars, with vastly discordant results. The field had hit a wall and
required a new approach.
Astrometry in Transition
19th and Early 20th Century Parallax Technique
Bessel used the 6-in Koenigsburg heliometer
(built by Fraunhofer) to visually measure
the offset of 61 Cyg and a reference star.
Schlesinger used the 30-in Thaw refractor at
Allegheny Observatory to embark on a
revolutionary program of photographic
astrometry.
The Modern Era I.
Photography Enters the Fray
“Measures of stellar distances presented difficulties so great that even today we
possess reliable knowledge on the approximate distances of of not over a
hundred stars. At no point in astronomical science is fuller knowledge more
desirable, more pressingly urgent, than in the subject of stellar distances.”
W.W. Campbell
1910
Frank Schlesinger (1871-1943)
•
Started career as a surveyor then entered Columbia to study
astronomy, earning his PhD in 1898 after experimenting with
“plate constant solutions” to measuring stellar positions on
photograph plates.
•
Hired by George Ellery Hale to work with the Yerkes refractor
in 1902, then became director of Allegheny Observatory in
1905, and finally director at Yale University Observatory in
1920.
•
Introduced the “method of dependences” and invented the
rotating sector.
•
Established precepts that dominated long-focus astrometry
until the advent of the CCD.
•
Founded the Yale southern observatory in S. Africa.
•
Produced two parallax catalogs as well as The Bright Star
Catalogue.
The Modern Era II.
Schlesinger’s Technique
Standard Coordinates and “Dependences”:
•
Compensate for plate-to-plate differences in origin, tilt, and scale using measurements of
reference star positions on a series of photographic plates. For example, the simplest such
equations of condition in the “standard coordinates” xs and ys are:
axxs + bxys + cx = xs – xobs
and
ayxs + byys + cy = ys – yobs
for which a, b and c are the plate constants and easily determined by least squares.
•
Additional terms can be added to compensate for coma, magnitude, color, etc. effects
resulting from telescope aberrations and the non-linear photographic detection process.
•
Schlesinger introduced a mathematical simplification to reduce computation effort
(remember this was when “computers” were humans with adding machines and tables of
logarithms!) incorporating “dependences” that also led to weights for the reference stars.
Left: Gaertner single-axis measuring
machine, designed by Schlesinger, purchased
for McCormick Observatory by S.A.
Mitchell in 1916 for $650. Its precision was
±1µm. Right: McCormick advanced its
computing capability in 1936 with the
purchase of a Marchant Calculating Engine.
From the “McCormick Museum” at
www.astro.virginia.edu
The Modern Era III.
The Original Parallax Club (~1900-1920)
Observatory
Telescope
PI
Allegheny
30-in refractor
F. Schlesinger
Dearborn
18.5-in refractor
P. Fox
Greenwich
26-in refractor
F. Dyson
McCormick
26-in refractor
S.A. Mitchell
Mt. Wilson
60-in reflector
A. van Maanen
Sproul
24-in refractor
J.A. Miller
In 1924, Schlesinger’s first General Catalogue of Trigonometric
Parallaxes included results for 1,870 stars. That number climbed to
4,260 stars.
The 1995 “Fourth Catalogue” by van Altena, Lee & Hoffleit contains
15,994 parallaxes for 8,112 stars and showed that the most recent
parallaxes had standard errors of ±0.004" in comparison with the
“Third Catalogue” value of ±0.016 ".
The Modern Era IV.
First “Discovery” of Extrasolar Planets
Peter van de Kamp (1901-1995)
•
Trained at Groningen and Lick, he worked with
Mitchell at UVa before becoming director of the Sproul
Observatory at Swarthmore College in 1937.
•
Sproul developed into a major center for parallax and
proper motion determinations as well as for astrometric
studies of binary stars.
•
With its record proper motion, Barnard’s star was
placed on the Sproul program to determine the star’s
secular acceleration.
•
Based on 2,316 plates from the 24-in Sproul refractor
taken during 1938-1961, van de Kamp in 1963
announced that Barnard’s star exhibited a 1µm submotion due to a 1.6 MJup planet in an eccentric orbit with
an orbital period of 24 years.
•
In 1974, van de Kamp announced that the perturbation
was due to two sub-Jupiter mass planets in circular
orbits.
•
Never confirmed by observers at Van Vleck, Allegheny,
McCormick and the U.S. Naval Observatories.
•
Van de Kamp always maintained the reality of his
discovery, citing the long time-span of his data base. The Chattanooga Times,
22 April 1963
The Modern Era V.
Reflectors Rejoin and Outshine the Old Refractors
Kaj Strand (1907-2000)
•
Trained at Copenhagen, he subsequently worked at
Leiden and then at Sproul on photographic
observations of double stars.
•
Following WWI, he held positions at Yerkes and
Dearborn before joining the U.S. Naval Observatory
in 1958, becoming its Scientific Director in 1963.
•
Responsible for the development of the USNO 61-in
astrometric reflector, located in Flagstaff.
•
Led development of an automatic measuring engine
SAMM.
•
USNO still produces the best ground-based
parallaxes, with accuracies of ±1.0 mas.
U.S. Naval Observatory photos
Trigonometric Parallax
- The stellar parallax is the apparent motion of a star due to our changing perspective as the
Earth orbits the Sun.
- parsec: the distance at which 1 AU subtends an angle of 1 arcsec.
Relative parallax - with
respect to background stars
which actually do move.
d( pc) =
1
p(")
Absolute parallax - with
respect to a truly fixed
frame in space; usually a
statistical correction is
applied to relative
parallaxes.
Trigonometric Parallax
Measured against a reference frame made of
more distant stars, the target star describes
an ellipse, the semi-major axis of which is
the parallax angle (p or π ), and the semiminor axis is π cos β, where β is the ecliptic
latitude. The ellipse is the projection of the
Earth s orbit onto the sky.
Parallax determination: at least three sets of
observations, because of the proper motion
of the star.
Van de Kamp
Parallax Measurements: The First Determinations
All known stars have parallaxes less than 1 arcsec. This number is
beyond the precision that can be achieved in the 18th century.
Tycho Brahe (1546-1601) - observations at a precision of 15-35 .
Proxima Cen - 0.772 - largest known parallax (Hipparcos value)
1838 - F. W. Bessel - 61 Cygni, 0.31 +- 0.02 ( modern = 0.287 )
1840 - F. G. W. Struve for Vega (α Lyrae), 0.26 (modern = 0.129 )
1839 - T. Henderson for α Centauri (thought to be Proxima!), 1.16 +0.11 (modern = 0.742 )
1912 - Some 244 stars had measured parallaxes. Most measurements
were done with micrometers, meridian transits, and few by
photography.
Parallax Measurements: The Photographic Era
Observatory
Telescope*
Percentage (%)**
Yale (Johannesburg, South Africa)
26-in f/16.6
15.5
McCormick (Charlottesville, VA)
26-in, f/15
15.4
Allegheny (Riverview Park, PA)
30-in, f/18.4
15.1
Royal Obs. Cape of Good Hope (now SAAO)
24-in, f/11
13.9
Spoul (Swarthmore, PA)
24-in. f/17.9
10.4
USNO (Flagstaff, AZ)
61-in, f/10 reflector 6.6
Royal Obs. Greenwich
26-in, f/10.2
6.1
Van Vleck (Middletown, CT)
20-in, f/16.5
4.7
Yerkes (Williams Bay, WI)
40-in, f/18.9
3.6
Mt. Wilson (San Gabriel Mountains, CA)
60-in, f/20 reflector 3.5
* All are refractors unless specified otherwise
** by 1992; other programs, with lower percentages are not listed
Source: nchalada.org/archive/NCHALADA_LVIII.html
Accuracy: ~ 0.010 = 10 mas
Parallax Measurements: The Modern Era
Catalog
Date
#stars
σ(mas)
Comments
YPC
1995
8112
±15 mas
Cat. of all π through 1995
USNO pg
To 1992
~1000
±2.5 mas Photographic parallaxes
USNO ccd
From 92
~150
±0.5 mas CCD parallaxes
Nstars &
GB
Current
100?
± 2 mas
Southern π programs
Hipparcos
1997
105
±1 mas
First modern survey
HST FGS
1995-2010 100?
?
±0.5 mas A few important stars
SIM
2016?
103
±4 µas
Critical targets & exoplanets
Gaia
2016?
109
±10µas
Ultimate modern survey
van Altena - MSW2005
Parallax Precision and the Volume Sampled
Photographic era: the accuracy is 10 mas -> 100 pc;
Stars at 10 pc: have distances of 10 % of the distance accuracy
Stars at 25 pc: have distances of 25 % of the distance accuracy
By doubling the accuracy of the parallax, the distance reachable doubles, while the
volume reachable increases by a factor of eight.
Parallax Size to Various Objects
•  Nearest star (Proxima Cen) 0.77 arcsec
•  Brightest Star (Sirius)
0.38 arcsec
•  Galactic Center (8.5 kpc)
0.000118 arcsec
118 µas
•  Far edge of Galactic disk (~20 kpc) 50 µas
•  Nearest spiral galaxy (Andromeda Galaxy)
1.3 µas
Future Measurements of Parallaxes: SIM and GAIA
1%
10%
SIM
2.5 kpc 25 kpc
GAIA
0.4 kpc 4 kpc
Hipparcos 0.01 kpc 0.1 kpc
SIM !
25 kpc!
(10%)!
SIM!
2.5 kpc!
(1%)!
You are here
SIM(planetquest.jpl.nasa.gov)
Proper Motions: Barnard s Star
Van de Kamp
Proper Motions
- reflect the intrinsic motions of stars as these orbit around the Galactic center.
-  include: star s motion, Sun s motion, and the distance between the star and the Sun.
-  they are an angular measurement on the sky, i.e., perpendicular to the line of sight;
that s why they are also called tangential motions/tangential velocities. Units are arcsec/
year, or mas/yr (arcsec/century).
-  largest proper motion known is that of Barnard s star 10.3 /yr; typical ~ 0.1 /yr
-  relative proper motions; wrt a non-inertial reference frame (e. g., other more distant
stars)
-  absolute proper motions; wrt to an inertial reference frame (galaxies, QSOs)
V2 = VT2 + VR2
VT (km /s)
µ("/ yr) =
4.74d( pc)
Proper Motions
dα
dt
dδ
µδ =
dt
µα =
µα - is measured in seconds of time per year (or
century); it is measured along a small circle;
therefore, in order to convert it to a velocity, and
have the same rate of change as µδ , it has to be
projected onto a great circle, and transformed to
arcsec.
µδ - is measured in arcsec per year (or century); or mas/
yr; it is measured along a great circle.
Proper Motions
µ 2 = (µα cos δ ) 2 + µδ2
Proper Motions - Some Well-known Catalogs
High proper-motion star catalogs
> Luyten Half-Second (LHS) - all stars µ > 0.5 /yr
> Luyten Two-Tenth (LTT) - all stars µ > 0.2 /year
> Lowell Proper Motion Survey/Giclas Catalog - µ > 0.2 /yr
High Precision and/or Faint Catalogs
Ø  HIPPARCOS - 1989-1993; 120,000 stars to V ~ 9, precision ~1 mas/yr
Ø  Tycho (on board HIPPARCOS mission) - 1 million stars to V ~ 11, precision 20 mas/
yr (superseded by Tycho2).
Ø  Tycho2 (Tycho + other older catalogs time baseline ~90 years) - 2.5 million stars to
V ~ 11.5, precision 2.4-3 mas/yr
Ø  Lick Northern Proper Motion Survey (NPM) - ~ 450,000 objects to V ~ 18, precision
~5 mas/yr
Ø  Yale/San Juan Southern Proper Motion Survey (SPM); 10 million objects to V ~ 18,
precision 3-4 mas/yr.
For this, it is convenient to use a
local system of celestial coordinates
centered at a certain point A ( 0, 0).
The equatorial coordinates
of a point in the vicinity of A are
,
0 +
0 +
The image of this region of the celestial sphere
on an ideal focal surface is planar
one has to transform the differential coordinates
and
into linear coordinates.
It is done by a
conic projection
from the center
of the unit
celestial sphere on A.
Ax, Ay are tangents to the declination small circle
=> increasing right ascensions,
along the celestial meridian, the positive direction =>N
This local system of coordinates = standard coordinates
The transformation
differential coordinates => standard coordinates
gnomonic or central projection
Transforming Measured to
Standard Coordinates:
Models for wide-field astrographs
and simplifications for long-focus telescopes
- T. M. Girard (Yale Univ.)
MSW 2005
1
References
A. König, 1962, “Astronomical Techniques”, Edited by W. A. Hiltner, [University
of Chicago Press] (Chapter 20)
P. van de Kamp, 1962, “Astronomical Techniques”, Edited by W. A. Hiltner,
[University of Chicago Press] (Chapter 21)
P. van de Kamp, 1967, “Principles of Astrometry”, [Freeman & Company],
(Chapters 5 and 6)
L. Taff, 1981, “Computational Spherical Astronomy”, [Wiley-Interscience]
personal class notes, Astro 575a, 1987 (taught by W. van Altena)
MSW 2005
2
Wide-Field vs. Long-Focus Telescopes
Wide-field
Long-focus
field of view
2°
10°
< 2°
focal length
2
4m
> 10 m
f/4
f/10
f-ratio
scale
uses
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50
100 /mm
positions,
absolute proper motions
f/15
10
f/20
20 /mm
parallaxes,
binary-star motion,
relative proper motions
3
A Typical Astrometric Reduction
The goal is the determination of celestial coordinates (
stars of interest on a plate or other detector.
) for a star or
1. Extract reference stars from a suitable reference catalog.
2. Identify and measure target stars and reference stars on the plate.
3. Transform reference-star coordinates to standard coordinates.
4. Determine the plate model (e.g., polynomial coefficients) that
transforms the measured x,y’s to standard coordinates. Use the
reference stars, knowing their measures and catalog coordinates,
to determine the model.
5. Apply the model to the target stars.
6. Transform the newly-determined standard coordinates into
celestial coordinates.
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4
Relation Between Equatorial and Standard Coordinates
Standard coordinates
(aka tangential coordinates, aka ideal coordinates)
A. The coordinate system lies in a plane tangent to the
celestial sphere, with the tangent point T at the
origin, (0,0).
B. The “y” axis, , is tangent to the declination circle that
passes through T, (+ toward NCP).
C. The “x” axis, , is perpendicular to , (+ toward
increasing R.A.)
D. The unit of length is the radius of the celestial sphere
or that of its image - the focal length. (In
practice, arcseconds are commonly used.)
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5
Equatorial & Standard Coordinates (cont.)
tan
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tan sin
tan cos
6
Equatorial & Standard Coordinates (cont.)
Tangent point is at
Star is at
o, o
o
o
Spherical triangle formed by star S,
tangent point T, and north celestial
pole NCP.
cos
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sin sin
o
cos cos
sin sin
cos sin
sin cos
sin cos
o
o
cos
cos sin
o
cos
7
Equatorial & Standard Coordinates (cont.)
cos sin
sin sin o cos cos
o
cos
Standard from Equatorial:
sin cos
sin sin
tan
cos
cos sin
cos cos
o
o
sin
o
cos
o cos
o
o
Equatorial from Standard:
sin
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sin
cos
o
1
2
o
2
8
Corrections to Measured Coordinates
A.
Correct for known “measuring machine” errors – repeatable
deviations (offsets and rotation) from an ideal Cartesian system.
Direct and reverse measures can be used to calibrate such effects.
B.
Correct for instrumental errors – plate scale, orientation, zero-point,
plate tilt, higher-order plate constants, magnitude equation, color
equation, etc.
(To be discussed.)
C.
Correct for “spherical” errors – refraction, stellar aberration,
precession, nutation.
(To be discussed.)
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9
“Spherical” Errors: Atmospheric Refraction
z
tan ,
'
o
tan 2
where is the true zenith distance,
i.e., the arclength ZS
and
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is the parallactic angle.
10
“Spherical” Errors: Atmospheric Refraction (cont.)
Refraction varies by observing site, and with atmospheric pressure and
temperature.
17 P
( P, TF )
58.2"
o,
o
460 TF
See R. C. Stone 1996, PASP 108, 1051 for an accurate method of
determining refraction, based on a relatively simple model.
Importantly, refraction also varies with wavelength!
Differential Color Refraction (DCR) can introduce color equation, an
unwelcome correlation between stellar color and measured position.
See R. C. Stone 2002, PASP 114, 1070 for a discussion of DCR and a
detailed model for its determination.
(In practice, it is sometimes incorporated into the plate model.)
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11
“Spherical” Errors: Stellar Aberration
If
is the angle between the star and the apex
of the Earth’s motion,
v
sin ,
c
v
c
where
Thus, differentially across a field of size
(
Note: For
)
20.5"
cos
20.5"
,
)2
(
2
sin
...
= 5°, the maximum quadratic effect has amplitude ~100 mas.
In practice, this would be absorbed by general quadratic terms in the plate
model, which would almost certainly be present for such a large field.
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12
“Spherical” Errors: Precession & Nutation
As both precession and nutation represent simple rotations, these
are almost never applied explicitly.
They are effectively absorbed by the rotation terms in the plate
model, which are always present.
Note: The equinox of the reference system is therefore, in a
practical sense, arbitrary. It is typically chosen to be that of the
reference catalog for convenience. Of course, the tangent point
must be specified in whatever equinox is chosen.
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13
The Plate Model
Often, a polynomial model is used to represent the transformation from
measured coordinates, (x,y), to standard coordinates,
a1 x a2 y a3 a4 x 2
a5 xy a6 y 2
a7 x 3 a8 x 2 y a9 xy 2
a10 y 3
a11m a12CI ...
b1 y b2 x b3 b4 y 2 b5 yx b 6 x 2 b7 y 3 b8 y 2 x b9 yx 2 b10 x 3
b11m b12CI ...
Note: The various terms can be identified with common corrections...
For each reference star, i, calculate deviation,
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i
ref i
i
ref i
i
Minimize
2.
( a k , xi , y i , mi , CI i )
i
(bk , xi , y i , mi , CI i )
14
The Plate Model (cont.)
Correction
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- solution
- solution
scale
x
y
orientation
y
x
zero point
constant
constant
plate tilt
x*(px+qy)
y*(px+qy)
cubic distortion
x*(x2+y2)
y*(x2+y2)
magnitude equation
m, (m2, m3...)
m, (m2, m3...)
coma
x*m
y*m
color equation
CI
CI
color magnification
x*CI
y*CI
higher order terms...
...
...
15
The Plate Model (cont.)
Some Helpful Hints
The simplest possible form should be used. The modeling error is thus
kept minimal. Reference stars are usually at a premium!
(Rule of thumb: Nref > 3*Nterms.)
Pre-correct measures for known (spherical) errors.
Update reference star catalog positions to epoch of plate material, i.e.,
apply proper motions when available.
Uniform distribution of reference stars is best. Avoid extrapolation.
Iterate to exclude outliers, but trim with care.
Plot residuals versus everything you can think of!
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16
When The Plate Model Is Just Not Enough
“Stacked” differences wrt an
external catalog can uncover
residual systematics.
A comparison between preliminary
SPM3 positions (derived using a plate
model with cubic field terms) and the
UCAC.
The resulting “mask” was used to
adjust the SPM3 data, field by field.
(See Girard et al. 2004, AJ 127, 3060)
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17
Magnitude Equation – The Astrometrist’s Bane
Magnitude equation = bias in the measured “center” of an image
that is correlated with its apparent brightness.
It can be particularly acute on photographic plates, caused by the nonlinearity of the detector combined with an asymmetric profile, (due to
guiding errors, optical aberrations, etc.)
• Difficult to calibrate and correct internally
• Reference stars usually have insufficient magnitude range
• Beware of “cosmic” correlations in proper motions
• In clusters, magnitude and color are highly correlated
NOTE: Charge Transfer Efficiency (CTE) effects can induce a similar bias in CCD centers.
(More often, the CTE effect mimics the classical coma term, i.e., x*m).
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18
Magnitude Equation – The Astrometrist’s Bane (cont.)
A comparison of proper motions derived from
uncorrected SPM blue-plate pairs and yellow-plate pairs
indicate a significant magnitude equation is present.
Using the grating images to correct each plate’s
individual magnitude equation, the proper motions are
largely free of bias.
SPM (and NPM) plates use
objective gratings, producing
diffraction image pairs which
can be compared to the
central-order image to deduce
the form of the magnitude
equation.
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19
Long-Focus Telescope Astrometry
Traditional Simplifications - due to scale & small field of view
Plate tilt
often negligible, but should be checked
Distortion
can be significant for reflectors; usually can be
ignored for refractors - constant over the FOV
Refraction
usually ignored unless at large zenith angle, or for
plate sets with a large variation in HA
Aberration
small, ignored
Magnitude equation
usually present! Can be minimized by
using a limited magnitude range.
Color equation
DCR will be present. Careful not to confuse with
magnitude equation for cluster fields.
Color magnification
Coma
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generally not a problem over the FOV
images are often affected, but variation across FOV is
slight and can be neglected in general (but check)
20
Long-Focus Telescope Astrometry (cont.)
A Parallax and Binary-Motion Example: Mass of Procyon A & B
(Girard et al. 1999, AJ 119, 2428)
Overview:
The plate material consisted of 250
(primarily) long-focus plates, containing
>600 exposures and spanning 83 years.
Magnitude-reduction methods were used
during the exposures to bring Procyon’s
magnitude close to that of the reference
stars.
Linear transformations between plates,
putting all onto the same standard
coordinate system.
Astrometric orbit and parallax were found.
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21
Long-Focus Telescope Astrometry (cont.)
A Relative Proper-Motion Example: Open Cluster NGC 3680
(Kozhurina-Platais et al. 1995, AJ 109, 672)
Overview:
The plate material consisted of 12 Yale-Columbia
26-in. refractor plates, spanning 37 years.
Explicit refraction correction was needed as plates
were taken at two observatories, Johannesburg
and Mt. Stromlo.
Plates exhibited magnitude and color equation
which affected the derived relative proper motions.
The cluster’s red giants were displaced in the
proper-motion VPD.
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22
Long-Focus Astrometry (NGC 3680 Example cont.)
Before
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After
23
In lieu of a summary slide...
Some Helpful Hints
The simplest possible form should be used. The modeling error is thus
kept minimal. Reference stars are usually at a premium!
(Rule of thumb: Nref > 3*Nterms.)
Pre-correct measures for known (spherical) errors.
Update reference star catalog positions to epoch of plate material, i.e.,
apply proper motions when available.
Uniform distribution of reference stars is best. Avoid extrapolation.
Iterate to exclude outliers, but trim with care.
Plot residuals versus everything you can think of!
MSW 2005
24
Astrometry Through the Ages I.
Hellenic Beginnings
Hipparchus ( ~190-120 BC)
•
•
•
Born in Nicaea (Turkey), but probably
worked in Rhodes and Alexandria
Only one minor, original work survives,
and he is mostly known from the writings
of others, particularly from Ptolemy’s
Almagest.
Accomplishments include:
•
Early work in trigonometry
•
Introduced 360º circle into Greece
•
Discovered precession from
comparison of historical measures of
length of the year
•
46 "/yr (Hipparchus)
•
36 "/yr (Ptolemy, 300 years
later)
•
50.26 "/yr (Modern value)
•
Developed a “lunar theory”
•
Determined eclipse period
•
Calculated lunar distance
•
Produced a star catalog of ~850 stars
Farnese Atlas: Dr. Gerry Picus,
courtesy Griffith Observatory
Astrometry Through the Ages II.
Greco-Roman Culmination
Claudius Ptolemy (~85-165 AD)
•
•
•
Observed from Alexandria between 127-141 AD
Greatest work: The Mathematical Compilation
The Greatest Compilation
Al-majisti
Almagest (Greek to Arabic to Latin).
The Almagest presents:
•
Lunar and solar theories
•
Eclipses
•
Motions of the “fixed” stars
•
Precession
•
Planetary theory
•
Incorporating epicycles and
eccentrics.
•
One of the most successful theories
of all time.
•
A catalog of 1,000 stars
•
Some successors (Tycho, Delambre,
Newton) accused Ptolemy of stealing
his catalog from Hipparchus and
precessing it by 300 years.
Astrometry Through the Ages III.
Islamic Contributions
Ulugh Beg (1393-1449)
•
•
•
•
Grandson of Tamarlane who was appointed
ruler of Samarkand (in present day
Uzbekistan) by his father in 1409.
Built a madrasah (center of Islamic higher
learning) with a great observatory 50 m in
diameter and 35 m high featuring a large
marble sextant.
Ulugh Beg’s accomplishments:
•
Determined the length of the year to
within 51 sec of actual value.
•
Created the first major new star catalog
since Ptolemy, the Zij-I Sultani,
containing 992 stars.
•
Advanced trigonometry and calculated
value of sin 1° accurate to one part in
1015.
Forcibly “retired” by his son who had him
murdered in 1449.
Archnet.org
Astrometry Through the Ages IV.
The Copernican Revolution
The Usual Suspects:
•
Nicolaus Copernicus (1473-1543) – Challenged the
prevailing thought that dated to Aristotle while still
keeping the practical aspects of Ptolemaic modeling.
•
Tycho Brahe (1546-1601) – The greatest pretelescopic observer (and overall fascinating guy).
•
Johannes Kepler (1571-1630) – Tycho’s brilliant
assistant who outshone his master intellectually.
•
Galileo Galilei (1564-1642) – Sought to measure
stellar parallax and suggested that “double stars”
might present that possibility.
•
Isaac Newton (1642-1727) – Not usually thought of
as an astrometrist, but he revolutionized the
theoretical basis of the field.
Astrometry Through the Ages V.
Taking Catalogs to their Next Level
John Flamsteed (1646-1719)
•
First Astronomer Royal and builder of Greenwich Observatory.
•
Attempted to provide Newton with observations toward
understanding the orbit of the Moon, but never quite got Newton
what he needed. They grew to dislike one another enormously.
•
Published the catalog Historia Coelestis Britannica in 1725 with
3,000 star positions of unprecedented accuracy.
•
Feuded extensively with Newton’s protégé Edmond Halley, who
ironically succeeded him as Astronomer Royal.
•
“… an intemperate man even by the standards of an intemperate
age.” Dictionary of Scientific Biography (New York, 1970-90).
Edmond Halley (1656-1742)
•
Born into a prosperous family, Halley’s father provided him with
instruments and introductions.
•
Worked as an assistant during his brief undergraduate career to
Flamsteed, but dropped out of Queen’s College Oxford.
•
Sailed to St. Helena to attempt the first southern star catalog
•
Observed the transit of Mercury on 7 Nov 1677.
•
Published his catalog of 341 southern stars in 1678.
•
Anticipated Newton’s proof of the inverse square law of attraction.
•
Black balled by Flamsteed from the Savilian Chair at Oxford.
•
Discovered stellar proper motion by comparison with Ptolemy.
•
Instrumental in Newton publishing The Principia.
Astrometry Through the Ages VI.
More Newtonian Era Advances
Jean-Dominique Cassini (1625-1712)
•
Began career in his native Italy (Giovanni Domenico Cassini) and
originally accepted the geocentric theory.
•
At Bologna, he measured rotation periods of Jupiter and Mars and
first suggested (then rejected) the finite speed of light.
•
Louis XIV invited him to oversee the construction of the Paris
Observatory. Cassini became a French citizen in 1673.
•
Paris accomplishments:
•
Four new moons of Saturn
•
Famous gap in Saturn’s rings; suggested true nature of rings
•
Paris meridian
•
First accurate value of solar parallax (from observations of
Mars made at Cayenne and Paris).
•
Rejected Newtonian theory and believed Earth prolate.
James Bradley (1693-1762)
•
Originally trained for the Church but resigned to take the Savilian
Chair in Astronomy at Oxford in 1721.
•
Announced the discovery of the abberation of star light in 1729.
•
Announced the discovery of nutation after following the moon for
an entire Saros cycle.
•
Became the third Astronomer Royal in 1742 and modernized the
equipment at Greenwich.
Double Stars – Beginnings I.
•
First reference ever to duplicity is that of C. Ptolemy (2nd C
AD) who used the term
to describe and Sgr
(which is optical). Ptolemy did not mention the other obvious
case of Alcor and Mizar.
•
Oldest recognized system discovered
telescopically is the “Trapezium” in
Orion in 1619 by Johannes Cysat of
Ingolstadt (1587 - 1657).
•
First physical binary resolved by the telescope was Mizar,
generally credited to Giovanni Ricciolo (1598 – 1671) in about
1650. It now appears that Galileo had actually resolved the
star as early as 1617.
Galileo was interested in double stars as a means for proving
the heliocentric theory. He assumed systems were optical in
nature.
•
Double Stars – Beginnings II.
•
By end of 17th C, Centauri, Cruxis, Geminorum and
Virginis had all been discovered. They were still generally
considered to be chance alignments, although Lambert
mentioned the possibility of physical association in 1761.
•
In 1667, G. Montanari noticed brightness change in Algol
(ras Al gul) although the Arabic name probably resulted
from that phenomenon. In 1783, J. Goodricke attributed the
variation as being due to either large spots or an eclipse by
a giant planet (which was not proven until 1889).
•
First evidence for physical nature was presented by J.
Mitchell in 1767 and C. Mayer in 1779 whose catalogs of
observations showed relative motion. Others, for example
Bode, held onto the accidental alignment hypothesis and
advanced the utility of double stars for parallax and proper
motion studies.
Double Stars – Beginnings III.
First great observer of double stars was
Wilhelm (William) Herschel (1738-1822).
Between 1779 and 1784, he made his
first series of observations of ~700 objects. On
9 June 1803, he presented the paper “Account
of the Changes that have happened, during the
last Twenty-five Years, in the relative Situation
of Double-Stars; with an Investigation of the
Cause to which they are owing.” He used
Gem (Castor) to argue persuasively for orbital
rather than any other kind of motion.
Herschel’s 20 ft telescope, now on display at the
Air and Space Museum in Washington
Double Stars – 19th Century I.
Friedrich Georg Wilhelm Struve (17931864) published Mensurae Micrometicae
in 1837 as the first systematic systematic
program of discovery and synoptic
observation to deduce orbital motion.
Introduced the use of ( , ) as standard
measured quantities for binary stars.
E
N
Used the 9-inch Fraunhofer
refractor at the Dorpat
Observatory in Russia
(Estonia) to discover 3,134
pairs.
Nine-inch refractor by Fraunhofer on display at the Deutsches
Museum in Bonn (identitical to that used by Struve)
Double Stars – 19th Century II.
John Frederick William Herschel (1792-1871)
observed double stars from the Cape of
Good Hope, South Africa, during 18341838.
Southern double star work would continue well
into the 20th C by Innes, Van den Bos and
Finsen at Johannesburg and Rossiter at
Bloemfontein.
Alvin Clark resolved Sirius in 1862 (with an
18-inch refractor originally ordered
by the University of Mississippi but
diverted to Dearborn Observatory of
Northwestern University because of
the Civil War).
Lick Observatory 3-m
telescope image of Sirius
A and B
Chandra Image of Sirius
B and A
Double Stars – “Modern” Era I.
Sherburne Wesley Burnham (1838-1921) was a
court reporter with a keen interest in double
stars and observed at Dearborn, Washburn,
Lick and Yerkes, eventually discovering
1,336 doubles. He specialized in discovering
“close” pairs ( as small as 0.2 arcsec and
large m pairs. In 1906, he published A
General Catalogue of Double Stars Within
120o of the North Pole, with 13, 665 star.
(Typically referred to as the “BDS” catalog.)
Robert Grant Aitken (1864-1951) joined Lick
Observatory in 1895 and served as
Director during 1930-35. He initiated a
survey for duplicity of all stars north of
= -22o that resulted in 4,400 new pairs,
3,100 of which he discovered. In 1932, he
published A New General Catalogue of
Double Stars Within 120o of the North Pole
(the “ADS”) with 17,180 entries. His book
The Binary Stars (1935 with several Dover
reprints) is a classic of the field.
Double Stars – “Modern” Era II.
Charles Worley (1935-1997) – Surveyed nearby faint
dwarfs for companions in 1960 at Lick and went on
to observe close pairs at the 26-inch USNO refractor
until his death. He transferred the Lick “Index
Catalogue” to the USNO and initiated the
“Washington Double Star Catalogue” (WDS) which
continues today under the direction of Brian Mason
and Bill Hartkopf. Worley retired his micrometer
and initiated a speckle interferometry program at the
USNO in the 1990’s.
Worley with GSU speckle camera at
Lowell 26-inch refractor
Other Productive Techniques:
Photography – Some 100,000 (many are spurious or optical)
CPM stars – Willem Luyten discovered ~2,000 common proper motion pairs from the
Palomar Sky Survey. These are probably the most common type of binary, but orbital
periods are too long to measure.
Speckle Interferometry – First systematically introduced in 1975. Approximately 35,000
measures to date (majority from CHARA and USNO) and ~300 new, bright and close
pairs.
HIPPARCOS – 51,500 measures
Interferometry
Michelson Perfects Extremely Narrow Angle Astrometry
Albert Michelson (1852-1931)
•
Born in Prussia, his family emmigrated to the
U.S. when he was two. He would become the first
American scientist to win the Nobel Prize (in
1907 “for his optical precision instruments and the
spectroscopic and metrological investigations
carried out with their aid.”)
•
While he did not invent interferometry, his
experiments at Mt. Wilson on the then new 100inch telescope showed great promise for
measuring stellar diameters and resolving close
binary star systems.
Michelson’s 20-ft Interferometer on the Mt. Wilson 100-inch
A.A. Michelson & F.G. Pease, ApJ, 53, 249, 1921
Michelson’s 20 ft Interferometer
On Display in CHARA’s Mt. Wilson Exhibit Hall
Pease’s 50-ft Interferometer on Mt. Wilson
An Instrument Ahead of its Time and Technology
Adapted from a diagram by Peter Lawson, JPL
Speaking of Mt. Wilson ...
By the way, watch where you step!
The Discovery of Speckle Interferometry
Labeyrie’s Brilliant Insight into “Seeing”
Antoine Labeyrie first proposed the technique of
speckle interferometry as a graduate student in
Meudon before going to Stonybrook on a postdoc.
On the occasion of Labeyrie receiving the
Franklin Medal in Philadelphia (April 2002).
Deane Peterson (SUNY Stonybrook) and
yours truly lectured at a special symposium
in Labeyrie’s honor.
Binary Star Speckle Interferometry
Star A
Star B
Atmospheric
Turbulent
Layers
Telescope
Entrance
Pupil
ADS 11483
Telescope
Focal
Plane
G2V+G2V, 1985.51, Sep = 1.74 arcsec
GSU Speckle Camera @ CFH Telescope
Enhanced Measurement Accuracy
Visual Binary
Peg: P = 11.6 yr, a = 0.25 arcsec
0.1 arcsec
Micrometer Observations
Speckle Observations
An Interferometric Renaissance
Hanbury Brown & Labeyrie Reawaken Interferometry
The Intensity Interferometer was
developed by Robert Hanbury
Brown (1916-2002) and his
colleagues and operated at
Narrabri, NSW, Australia during
1963-1972 with the sole purpose of
accurately measuring the angular
diameters of 32 bright, southern
stars.
Built by Labeyrie and his collaborators in
the mid-1980’s in the south of France, the
Grand Interféromètre à 2 Télescopes
incorporates Labeyrie’s novel ideas for
telescope design, path length compensation,
and beam combination.
Other Optical Interferometers I.
The NRL/USNO Mark III Set the Modern Standard
The Mark III Interferometer, built by Michael Shao, Mark Colavita and their collaborators,
was operated on Mt. Wilson from 1986 to 1992. It was the third of a series of technological
stepping stones and successfully demonstrated in a productive scientific program what has
become the modern standard for interferometric subsystems. The Mark III is the direct ancestor
to the Palomar Testbed Interferometer, the Keck Interferometer and the Space Interferometry
Mission.
Other Optical Interferometers II.
The USNO NPOI as an Astrometric Instrument
U.S. Naval Observatory photo
Operated by the U.S. Naval Observatory in collaboration with Lowell Observatory on Anderson
Mesa in northern Arizona, the Navy Prototype Optical Interferometer has the dual role of
serving positional astronomy using its “astrometric subarray” surrounded by the longer baseline
imaging array.
Other Optical Interferometers III.
These Things are Complicated!
Photo by Steve Golden, GSU/CHARA
An interior view of the delay line laboratory of the CHARA Array, operated on Mt. Wilson,
California, by Georgia State University with support from the National Science Foundation, hints at
the terrific overhead imposed on interferometers by the requirements of fringe detection.
Four Centuries of Progress
From Tycho to SIM
60 arcsec precision
1 micro-arcsec precision
Some Useful References
Fundamentals of Astrometry J. Kovalevsky & P.K. Seidelmann, Cambridge, 2004.
Observing and Measuring Visual Double Stars R. Argyle, Springer, 2004.
Modern Astrometry 2nd Ed., J. Kovalevsky, Springer, 2002.
Parallax A.W. Hirshfield, Freeman, 2001.
Astrometry of Fundamental Calalogues H. Walter & O. Sovers, Springer, 2000.
Relativity in Astrometry, Celestial Mechanics and Geodesy M.H. Soffel, Springer, 1989.
Astrometry W. van Altena, Ann. Rev. Astr. Astp., 21, 1983.
Vectorial Astrometry C.A. Murray, Hilger, 1983.
Stellar Paths P. van de Kamp, Reidel, 1981.
Double Stars W.D. Heintz, Reidel, 1978.
Astronomy of Star Positions H. Eichhorn, Ungar, 1974.
Principles of Astrometry P. van de Kamp, Freeman, 1967.
A Compendium of Spherical Astronomy S. Newcomb, Dover reprint, 1906.
see also The Double Star Library at ad.usno.navy.mil/wds/dsl.html