Name Date Nstars (p)(mas）  pm(mas/yr） SAO 1966 260 000 1000 10 ACRS 1991 320 000 200 5 PPM 1991 469 000 200 4 HIPPARCOS 1997 120 000 0.8 0.9 Tycho1 1997 1 060 000 40 40 ACT 1997 989 000 40 ~ 2.5 TYCHO-2 1999 2 500 000 25 ~ 2.5 UCAC-2 2003 48 330 000 22~70 1~6 1.2 Relative determination Table. Some astrometric reference catalogues

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

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

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.

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

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

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

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)

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

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

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)

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.

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)

3. Astrometric Calibration RegionsDifferent telescopes/receivers, different models Table. Models of different forms

3. Astrometric Calibration Regions Table Models of different forms (continued) Note: For magnitude terms: adding M, MX, MY

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.

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

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

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.

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

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

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

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

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.

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

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

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

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

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

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

Thanks!

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