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)
h - altitude: +-90 deg
A - azimuth (0-360 deg, from N through E, on the horizon)
z - zenith distance; 90 deg - h
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
- Earth’s axial tilt = 23.5 deg
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) Dec
NGP: 192.859 27.128
GC: 266.404 -28.936
l = 0 - Galactic center
l = 90 - in the direction of Galactic rotation
l = 180 - anticenter
l = 270 - antirotation
l = 0 - 90 first quadrant
l = 90 - 180 second quadrant
l = 180 - 270 third quadrant
l = 270 - 230 fourth quadrant
The cylindrical system
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
The Cartesian system: defined with respect to the Local Standard of Rest (LSR)
X, Y, Z
U, V, W
X, U - positive away from the GC
Y, V - positive toward Gal. rotation
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
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
------ = ------ = -------
sin A sin B sin C
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)
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)
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 )
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
Relative parallax - with respect to background stars which actually do move.
Absolute parallax - with respect to a truly fixed frame in space; usually a statistical correction is applied to relative parallaxes.
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 semi-minor 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
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.
* All are refractors unless specified otherwise
** by 1992; other programs, with lower percentages are not listed
Accuracy: ~ 0.010” = 10 mas
van Altena - MSW2005
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
You are here
SIM 2.5 kpc 25 kpc
GAIA 0.4 kpc 4 kpc
Hipparcos 0.01 kpc 0.1 kpc
Van de Kamp
V2 = VT2 + VR2
- 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 , it has to be projected onto a great circle, and transformed to arcsec.
- is measured in arcsec per year (or century); or mas/yr; it is measured along a great circle.
High proper-motion star catalogs
> Luyten Half-Second (LHS) - all stars > 0.5”/yr
> Luyten Two-Tenth (LTT) - all stars > 0.2”/year
> Lowell Proper Motion Survey/Giclas Catalog - > 0.2”/yr
Van de Kamp
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).