1. Celestial Coordinate Systems Horizon Coordinates Kaler h - altitude: +-90 deg A - azimuth (0-360 deg, from N through E, on the horizon) z - zenith distance; 90 deg - h (refraction, airmass)

2. 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. T sid = RA + H Scott Birney

3. Ecliptic Coordinates  - ecliptic longitude (0-360deg, increases eastwards from the Vernal equinox)  - ecliptic latitude (+-90 deg)  - Earth’s axial tilt = 23.5 deg Scott Birney

4. Galactic Coordinates Scott Birney 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

5. l = 0 - Galactic center l = 90 - in the direction of Galactic rotation l = 180 - anticenter l = 270 - antirotation Galactic Coordinates (cont.) www.thinkastronomy.com l = 0 - 90 first quadrant l = 90 - 180 second quadrant l = 180 - 270 third quadrant l = 270 - 230 fourth quadrant IIV II III

6. 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

7. X = d cos l cos b Y = d sin l cos b Z = d sin b d - distance to the Sun 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 Y, V Z, W X, U

8. Coordinate Transformations 1) Spherical Trigonometry: Transformations Between Different Celestial Coordinate Systems Kaler 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 And: cos A = - cos B cos C + sin B sin C cos a

9. 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)

10. 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 ) Read Johnson and Soderblom (1987) for an application to positions and velocities determined from proper motions, RVs and parallax.

11. From Celestial Coordinates to Coordinates in the Focal Plane: The Gnomonic Projection Girard - MSW2005

12. Standard Coordinates

13. Girard MSW2005 Standard Coordinates

15. 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. Absolute parallax - with respect to a truly fixed frame in space; usually a statistical correction is applied to relative parallaxes.

16. 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 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

17. 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 ” ) 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) 1912 - Some 244 stars had measured parallaxes. Most measurements were done with micrometers, meridian transits, and few by photography. Parallax Measurements: The First Determinations

18. Parallax Measurements: The Photographic Era ObservatoryTelescope * Percentage (%) ** Yale (Johannesburg, South Africa)26-in f/16.615.5 McCormick (Charlottesville, VA)26-in, f/1515.4 Allegheny (Riverview Park, PA)30-in, f/18.415.1 Royal Obs. Cape of Good Hope (now SAAO)24-in, f/1113.9 Spoul (Swarthmore, PA)24-in. f/17.910.4 USNO (Flagstaff, AZ)61-in, f/10 reflector6.6 Royal Obs. Greenwich26-in, f/10.26.1 Van Vleck (Middletown, CT)20-in, f/16.54.7 Yerkes (Williams Bay, WI)40-in, f/18.93.6 Mt. Wilson (San Gabriel Mountains, CA)60-in, f/20 reflector3.5 * 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

19. CatalogDate#stars  mas) Comments YPC19958112±15 masCat. of all π through 1995 USNO pgTo 1992~1000±2.5 masPhotographic parallaxes USNO ccdFrom ‘92~150±0.5 masCCD parallaxes Nstars & GB Current100?± 2 masSouthern π programs Hipparcos199710 5 ±1 masFirst modern survey HST FGS1995- 2010? 100?±0.5 masA few important stars SIM2016?10 3 ±4 µasCritical targets & exoplanets Gaia2016?10 9 ±10µas“Ultimate” modern survey van Altena - MSW2005 Parallax Measurements: The Modern Era

20. 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. Nearest star (Proxima Cen) 0.77 arcsec Brightest Star (Sirius) 0.38 arcsec Galactic Center (8.5 kpc) 0.000118 arcsec 118  as Far edge of Galactic disk (~20 kpc) 50  as Nearest spiral galaxy (Andromeda Galaxy)1.3  as Parallax Size to Various Objects

21. Future Measurements of Parallaxes: SIM and GAIA SIM 25 kpc (10%) SIM 2.5 kpc (1%) You are here SIM(planetquest.jpl.nasa.gov) 1% 10% SIM 2.5 kpc 25 kpc GAIA 0.4 kpc 4 kpc Hipparcos 0.01 kpc 0.1 kpc

22. Proper Motions: Barnard’s Star Van de Kamp

23. 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) V 2 = V T 2 + V R 2

24. Proper Motions   - 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.

25. Proper Motions

26. Proper Motions - Some Well-known Catalogs 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 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.

27. 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

28. 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 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: 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).