Transcript A Lecture by Dana Casetti-Dinescu of Yale University

Celestial Coordinate Systems
Horizon Coordinates
h - altitude: +-90 deg
A - azimuth (0-360 deg, from N
through E, on the horizon)
z - zenith distance; 90 deg - h
(refraction, airmass)
Kaler
Equatorial Coordinates
RA: 0 - 24 h (increases
eastward from the Vernal
Equinox)
Dec: +- 90 deg
H - hour angle: negative - east of
the meridian, positive - west of
the meridian.
Tsid = RA + H
Scott Birney
Ecliptic Coordinates
Scott Birney
- ecliptic longitude (0-360deg, increases eastwards from the Vernal
equinox)
- ecliptic latitude (+-90 deg)
- Earth’s axial tilt = 23.5 deg
Galactic Coordinates
l - galactic longitude (0-360 deg,
increases toward galactic rotation
from the galactic center
b - galactic latitude, +- 90 deg
The Galactic plane is inclined at an
angle of 62.6 deg to the celestial
equator.
RA (J2000)
Scott Birney
Dec
__________________________________
NGP:
192.859
27.128
GC:
266.404
-28.936
Galactic Coordinates (cont.)
l = 0 - Galactic center
l = 90 - in the direction of Galactic rotation
l = 180 - anticenter
l = 270 - antirotation
II
III
www.thinkastronomy.com
IV
I
l = 0 - 90
first quadrant
l = 90 - 180
second quadrant
l = 180 - 270 third quadrant
l = 270 - 230 fourth quadrant
Galactic Coordinates: Position and Velocity Components
The cylindrical system
R,  z
W (Z)
R,  - positive away from the GC
 - positive toward Galactic rotation
z, W(Z) - positive toward the NGP
Note: this is a left-handed coordinate
system; right-handed 
W
The Cartesian system: defined with respect to the Local Standard of Rest (LSR)
X, Y, Z
U, V, W
Z, W
X, U - positive away from the GC
Y, V - positive toward Gal. rotation
X, U
Z, W - positive toward NGP
Left-handed system; right-handed: U= -U
X = d cos l cos b
Y = d sin l cos b
Z = d sin b
d - distance to the Sun
Y, V
Coordinate Transformations
1) Spherical Trigonometry: Transformations Between Different Celestial
Coordinate Systems
Law of cosines:
cos a = cos b cos c + sin b sin c cos A
Law of sines:
sin a
sin b
sin c
------ = ------ = -------
Kaler
sin A sin B
sin C
And:
cos A = - cos B cos C + sin B sin C cos a
Spherical Trigonometry: Transformations Between Different Celestial
Coordinate Systems
- Application: Equatorial <--> Galactic (BM - p. 31)
Useful angles:
G , G - eq. coordinates of
the North Gal. Pole (G)
longitude of the North
Celestial Pole (P) (122.932,
defined as 123.0 for RA,Dec.
at B1950)
Coordinate Transformations
2) Euler Angles: Transformations of Vectors (Position, Velocity) From One
Coordinate System to Another
The three basic rotations about x, y, z axes by a total amount of  are equivalent
to the multiplication of the matrices: (e.g., Kovalewski & Seidelman )
1
0
0 


R1  0 cos  sin  


0 sin  cos  
cos  0 sin  


R2   0
1
0 


sin  0 cos  
cos  sin  0


R3  sin  cos  0


0
1
 0
x'
x 
 
 
y'

R
R
R
  3 2 1y 
 
 
z'
 
z 
Read Johnson and Soderblom
(1987) for an application to
positions and velocities
determined from proper motions,
RVs and parallax.
From Celestial Coordinates to Coordinates in the Focal Plane:
The Gnomonic Projection
Girard - MSW2005
Standard Coordinates
Standard Coordinates
Girard MSW2005
Standard Coordinates
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  ), and the semiminor axis is  cos , where  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 ( Lyrae), 0.26” (modern = 0.129”)
1839 - T. Henderson for  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
s(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
19952010?
100?
±0.5
mas
A few important stars
SIM
2016?
103
±4 µas
Critical targets & exoplanets
±10µas
“Ultimate” modern survey
9
Gaia
van Altena 2016?
- MSW2005 10
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 mas
• Far edge of Galactic disk (~20 kpc)
50 mas
• Nearest spiral galaxy (Andromeda Galaxy)
1.3 mas
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
m(" / yr) 
VT (km /s)
4.74d( pc)

Proper Motions
d
m 
dt
d
m 
dt
m - 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 m , it has to be
projected onto a great circle, and transformed to
arcsec.
m - is measured in arcsec per year (or century); or
mas/yr; it is measured along a great circle.
Proper Motions
m 2  (m cos ) 2  m2
Proper Motions - Some Well-known Catalogs
High proper-motion star catalogs
> Luyten Half-Second (LHS) - all stars m > 0.5”/yr
> Luyten Two-Tenth (LTT) - all stars m > 0.2”/year
> Lowell Proper Motion Survey/Giclas Catalog - m > 0.2”/yr
High Precision and/or Faint Catalogs
 HIPPARCOS - 1989-1993; 120,000 stars to V ~ 9, precision ~1 mas/yr
 Tycho (on board HIPPARCOS mission) - 1 million stars to V ~ 11, precision 20
mas/yr (superseded by Tycho2).
 Tycho2 (Tycho + other older catalogs time baseline ~90 years) - 2.5 million stars to V
~ 11.5, precision 2.4-3 mas/yr
 Lick Northern Proper Motion Survey (NPM) - ~ 450,000 objects to V ~ 18, precision
~5 mas/yr
 Yale/San Juan Southern Proper Motion Survey (SPM); 10 million objects to V ~ 18,
precision 3-4 mas/yr.
Precession
The system of equatorial coordinates is not
inertial, because the NCP and the vernal
equinox (VE) move (mainly due to the
precession of the Earth).
- It amounts to 50.25”/year (or a period of
25,800 years); the VE moves westward.
- Tropical year: 365.2422 days (Sun moves
from one VE to the next; shorter by 20
minutes than the sidereal year.
- Sidereal year: 365.2564 day (Sun returns
to the same position in the sky as given by
stars).
- Therefore the equatorial coordinates are
given for a certain equinox (e.g. 1950, or
2000); for high proper-motion stars,
coordinates are also given for a certain
epoch.
Van de Kamp
Astrometric Systems (Reference Frames)
A catalog of objects with absolute positions and proper motions: i.e., with respect to an
inertial reference frame define an astrometric system. This system should have no
rotation in time.
1) The dynamical definition: - with respect to an ideal dynamical celestial reference
frame, stars move so that they have no acceleration. The choice of this system is the
Solar System as a whole. Stars in this system have positions determined with respect to
observed positions of planets. Observations made with meridian circles contribute to the
establishment of this type of reference frame (FK3, FK4, FK5 systems).
2) The kinematic definition: - an ideal kinematic celestial frame assumes that there exists
in the Universe a class of objects which have no global systemic motion and therefore are
not rotating in the mean. These are chosen to be quasars and other extragalactic radio
sources (with precise positions from VLBI). This system is the International Celestial
Reference System (ICRS, Arial et al. 1995).