How to Get Equator Coordinates of Objects from Observation Zhenghong TANG Yale Astrometry Summer Workshop July, 2005

2 1. Two basic ways to determine positions: --- absolute determination --- relative determination 2. Measurement coordinates (X,Y)  equator coordinates (α,δ) 3. Astrometric Calibration Regions (ACRs) 4. Block Adjustment of overlapping observations Contents

3 1.1 Absolute determination Basic idea: Measuring the positions of objects directly in the equator coordinate system. Measuring the positions of objects directly in the equator coordinate system.  The observational quantities are related to the equator coordinate system directly;  ’absolute’: Never uses any known star at any time.

4 1.1 Absolute determination Typical astrometric instrument : meridian circle Micrometer, clock and angle reading equipment α=t δ=φ-z P: North polar of equator system Z: Zenith point

5 1.1 Absolute determination

6 Advantage: Any instrument can be used to construct an independent star catalogue. Before 1980, the fundamental catalogues were realized by it. FK3, FK4, FK5

7 1.1 Absolute determination Disadvantages: Low efficiency  only one object in each observation Influence of abnormal atmosphere refraction. Instability of the instruments and clocks (mechanical and thermal distortion)  Uncertainty > 0”.15

8 1.2 Relative determination Basic idea: Determining positions of objects with the help of reference stars, whose positions are known.  doesn’t care much about the real form of all influences;  the influences on the known objects and neighboring unknown objects are similar.

9 1.2 Relative determination typical instrument: Astrograph Receiver:  Photographic plate (before 1980)  CCD (after 1980) ► higher quantum efficiency ► higher linearity ► greater convenience

Relative determination Disadvantages: Precondition: reference catalogue with known positions and proper motions Precision of the results will rely on the quality of reference catalogue. Errors in positions of reference stars will appear in the positions of objects.

Relative determination NameDateNstars  (p)(mas ）  pm( mas/yr ） SAO ACRS PPM HIPPARCOS Tycho ACT ~ 2.5 TYCHO ~ 2.5 UCAC ~701~6 Table. Some astrometric reference catalogues

Relative determination Advantages:  higher efficiency (10,000 stars)  higher precision (0”.01~0”.05) (Hipparcos catalogue, CCD)

13 2. How to get (α,δ) from (X,Y) Basic information of an observation:  Focal length: F  Field of view  Pixel size  Observational date: T  Center position of the observation (α0,δ0)  Imaging model

14 2. (X,Y)  (α,δ) Basic assumption: There exists a uniform model that transforms celestial equator coordinates to measurement coordinates for all stars in the field of plate/CCD. Notes: Suppose (X,Y) of all objects have been obtained from the plate/CCD.

15 2 (X,Y)  (α,δ) The procedure can be divided to four steps: First Step: Compute (ζ,η) of reference stars from the reference catalogue * Correcting proper motions from T0 to T, T0: catalogue time, T: observational date

16 2 (X,Y)  (α,δ) Transformation between spherical coordinates (α,δ) and standard coordinates (ζ,η). i.e. from spherical surface  focal plane Gnomonic projection:

17 2. (x,y)  (α,δ) Second Step: Solve parameters of the model that describes the relationship between (ζ,η) and (x,y)  Observational equation (i=1,Nref) ζ(i)=aX(i) +bY(i) +c η(i)=a’X(i)+b’Y(i)+c’  Normal equation [X]a +[Y]b +c =[ζ] [X]a’+[Y]b’+c’=[η]  Least square solution

18 2. (x,y)  (α,δ) Third Step: Compute the standard coordinates of unknown objects based on their measurement coordinates and the model parameters. ζ(j)=aX(j) +bY(j) +c η(j)=a’X(j)+b’Y(j)+c’ (j=1,Nobj)

19 2. (x,y)  (α,δ) Fourth Step: Compute the spherical equator coordinates of unknown objects from their standard coordinates.

20 2. (x,y)  (α,δ) The whole procedure: 1) (α,δ) ref  (ζ,η) ref 2) (ζ,η) ref + (x,y) ref  (a,b,c,d,e,f,…) 3) (a,b,c,d,e,f,…)+ (x, y) obj  (ζ,η) obj 4) (ζ,η) obj  (α,δ) obj

21 2. (x,y)  (α,δ) Note: When there are some factors which don’t have similar influences on reference stars and objects, they should be corrected additionally. Example: satellite observation Z1: appearing zenith distance, Z2: real zenith distance if S is far, Z3: real zenith distance if S is nearby, like satellite of the Earth (When OS  ∞, Z2=Z3)

22 3. Astrometric Calibration Regions Why need ACRs:  Testing imaging characteristics of telescopes and receivers, i.e. selecting the suitable model for a certain telescope.  Looking for systematic errors of other catalogues.

23 3. Astrometric Calibration Regions (ACRs) Some sky areas with plenty of stars distributed evenly, whose positions and proper motions are known very well. Also called Astrometric Standard Regions.  Pleiades, Praesepe (~ 500 stars)  SDSS ACRs (16, 7º.6×3º.2 ， 1999, 765~10,772 stars/per square degree)

24 NparModel forms 4ζ=aX+bY+c η=aY-bX+d 6ζ=aX +bY +c η=a’X+b’Y+c’ 7ζ=aX +bY+c+pX 2 +qXY+rX(X 2 +Y 2 ) η=aY － bX+d+pXY+qY 2 +rY(X 2 +Y 2 ) 8ζ=aX +bY +c +pX 2 +qXY η=a’X+b’Y+c’ +pXY+qY 2 9ζ=aX +bY+c +pX 2 +qXY+rX(X 2 +Y 2 ) η=a’X+b’Y+c’ +pXY+qY 2 +rY(X 2 +Y 2 ) 3. Astrometric Calibration Regions Different telescopes/receivers, different models Table. Models of different forms

25 NparModel forms 10ζ=aX +bY+c +pX 2 +qXY +rY 2 η=aY － bX+d+p’X 2 +q’XY+r’Y 2 12ζ=aX +bY+c +pX 2 +qXY +rY 2 η=a’X+b’Y+c’+p’X 2 +q’XY+r’Y 2 13ζ=aX +bY+c +pX 2 +qXY +rY 2 +sX(X 2 +Y 2 ) η=a’X+b’Y+c’+p’X 2 +q’XY+r’Y 2 +sY(X 2 +Y 2 ) 14ζ=aX +bY+c +pX 2 +qXY +rY 2 +sX(X 2 +Y 2 ) η=a’X+b’Y+c’+p’X 2 +q’XY+r’Y 2 +s’Y(X 2 +Y 2 ) 3. Astrometric Calibration Regions Table Models of different forms (continued) Note: For magnitude terms: adding M, MX, MY

26 3. Astrometric Calibration Regions How to select suitable model?  With the help of ACRs, solve the parameters of different models, and compare the residual of the least square solution. The model with highest precision and fewest parameters is the suitable one.

27 3. Astrometric Calibration Regions Real example 1: 60/90cm Schmidt in Xinglong Station of NAOC CCD: 2K*2K FOV: 1º*1º

28 3. Astrometric Calibration Regions Real example 2: 2.16m Telescope in Xinglong Station of NAOC CCD: 2K*2K, FOV: 11’*11’

29 3. Astrometric Calibration Regions The density and faintness of ACRs are important. It usually takes long time to construct good ACRs For other passbands, like X-ray and Gamma-ray, the procedure of the relative determination is similar. Good ACRs in these passbands are also needed.

30 4. Block Adjustment of overlapping observations Why?  The precision of the relative determination relies on the quality of reference catalogue, more exactly, on the quality of reference stars around the objects.  Errors exist in the catalogue positions and measurement coordinates of the reference stars

31 4. Block Adjustment Why? (continued)  Small size of CCD ( ~5 cm) ► small FOV ~15’ for f=10m ► Covering few reference stars ► uneven distribution

32 4. Block Adjustment Results with systematic errors like translation, rotation and distortion when reference stars are not good:

33 4. Block Adjustment How? The start point of BA: one star has only one position at a time (absolute restriction). 1 2 × × ×× × × × ×

34 4. Block Adjustment How? (continued) Compared with single CCD adjustment, BA has many more equations provided by common stars, besides the equations of reference stars. All equations are solved by least square method to obtain the parameters of all CCD frames. Compute (α,δ) of objects.

35 Advantages:  Enlarging FOV  Increasing reference stars  Reducing the importance of reference stars  Detecting and removing possible systematic errors in reference stars. 4. Block Adjustment

36 Simulation: 4*4+3*3, noise in (X,Y) and (α,δ) 15+4 reference stars, each frame has one reference star, except the center one has four reference stars. 4. Block Adjustment

37 Case 1: When reference stars covering small area: Real line  BA Dash line  SA 4. Block Adjustment

38 4. Block Adjustment Case 2: When one reference star has big error in α: Real line  BA Dash line  SA

39 4. Block Adjustment Case 3: When some reference stars have systematic errors in α : Real line  BA Dash line  SA

40 4. Block Adjustment Application: (Long focal length + high precision)  Linkage of radio-optical reference frames

41 Application:  Open clusters of big area  Constructing ACRs  Detecting systematic errors in catalogues 4. Block Adjustment

42 Thanks!