Celestial Coordinate SystemsCelestial 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)

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

- ecliptic longitude (0-360deg, increases eastwards from the Vernal equinox)

- ecliptic latitude (+-90 deg)

- Earth’s axial tilt = 23.5 deg

Scott Birney

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

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

IIIII

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

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= -UY, V

Z, W

X, U

Coordinate TransformationsCoordinate 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

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 TransformationsCoordinate Transformations

2) Euler Angles: Transformations of Vectors (Position, Velocity) From One Coordinate System to AnotherThe 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 )

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R1 =

1 0 0

0 cosθ sinθ

0 −sinθ cosθ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

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R2 =

cosω 0 sinω

0 1 0

−sinω 0 cosω

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

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R3 =

cosφ sinφ 0

−sinφ cosφ 0

0 0 1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

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x '

y '

z'

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟= R3R2R1

x

y

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: From Celestial Coordinates to Coordinates in the Focal Plane: The Gnomonic ProjectionThe Gnomonic Projection

Girard - MSW2005

Standard Coordinates

Girard MSW2005

Standard Coordinates

Standard Coordinates

Trigonometric ParallaxTrigonometric 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.

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d(pc) = 1p(")

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d(pc) = 1p(")

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.

Trigonometric ParallaxTrigonometric 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

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

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

Catalog Date #stars 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 Measurements: The Modern Era

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

Future Measurements of Parallaxes: SIM and GAIA

SIM 25 kpc(10%)

SIM2.5 kpc(1%)

You are hereSIM(planetquest.jpl.nasa.gov)

1% 10%1% 10%SIM 2.5 kpc 25 kpcSIM 2.5 kpc 25 kpcGAIA 0.4 kpc 4 kpcGAIA 0.4 kpc 4 kpcHipparcos 0.01 kpc 0.1 kpcHipparcos 0.01 kpc 0.1 kpc

Proper Motions: Barnard’s StarProper Motions: Barnard’s Star

Van de Kamp

Proper MotionsProper 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)

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("/ yr) =VT (km /s)

4.74d(pc)

V2 = VT2 + VR

2

Proper MotionsProper Motions

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=dα

dt

μδ =dδ

dt

- 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.

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2 = (μα cosδ)2 + μδ2

Proper MotionsProper Motions

Proper Motions - Some Well-known CatalogsProper 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.

PrecessionPrecession

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 definition1) 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).