How to Get Equator Coordinates How to Get Equator Coordinates of Objects from Observationof Objects from Observation

Zhenghong TANGZhenghong TANG

Yale Astrometry Summer WorkshopYale Astrometry Summer WorkshopJuly, 2005July, 2005

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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

ContentsContents

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1.1 Absolute determination1.1 Absolute determination

Basic idea: Basic idea: Measuring the positions of objects directly in Measuring the positions of objects directly in the equator coordinate system.the equator coordinate system.

The observational quantities are related to the The observational quantities are related to the equator coordinate system directly;equator coordinate system directly;

’’absolute’: Never uses any known star at any absolute’: Never uses any known star at any time.time.

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1.1 Absolute determination1.1 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

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1.1 Absolute determination1.1 Absolute determination

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1.1 Absolute determination1.1 Absolute determination

Advantage:

Any instrument can be used to construct an independent star catalogue.

Before 1980, the fundamental catalogues were realized by it. FK3, FK4, FK5

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1.1 Absolute determination1.1 Absolute determination

Disadvantages:Low efficiency

only one object in each observationInfluence of abnormal atmosphere refraction .Instability of the instruments and clocks

(mechanical and thermal distortion)

Uncertainty > 0”.15

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1.2 Relative determination1.2 Relative determination

Basic idea: Determining positions of objects Basic idea: Determining positions of objects with the help of reference stars, whose with the help of reference stars, whose positions are known.positions are known.

doesn’t care much about the real form of doesn’t care much about the real form of all influences;all influences;

the influences on the known objects and the influences on the known objects and neighboring unknown objects are similar.neighboring unknown objects are similar.

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1.2 Relative determination1.2 Relative determination

typical instrument: Astrograph

Receiver: Photographic plate (before 1980)

CCD (after 1980) ► higher quantum efficiency ► higher linearity ► greater convenience

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1.2 Relative determination1.2 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.

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1.2 Relative determination1.2 Relative determination

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

Table. Some astrometric reference catalogues

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1.2 Relative determination1.2 Relative determination

Advantages:

higher efficiency (10,000 stars)

higher precision (0”.01~0”.05)

(Hipparcos catalogue, CCD)

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2. How to get (α,δ) from (X,Y)2. How to get (α,δ) from (X,Y)

Basic information of an observation:

Focal length: F Field of view Pixel size Observational date: TCenter position of the observation (α0,δ0) Imaging model

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2. (X,Y) 2. (X,Y) (α,δ) (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.

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2 (X,Y) 2 (X,Y) (α,δ) (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

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2 (X,Y) 2 (X,Y) (α,δ) (α,δ)

Transformation between spherical coordinates (α,δ) and standard coordinates (ζ,η).

i.e. from spherical surface focal plane

Gnomonic projection:

)cos(osossinsin

)cos(ossinsincos

)cos(osossinsin

)sin(cos

000

000

000

0

cc

c

cc

＝

＝

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2. (x,y) 2. (x,y) (α,δ) (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

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2. (x,y) 2. (x,y) (α,δ) (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)

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2. (x,y) 2. (x,y) (α,δ) (α,δ)

Fourth Step: Compute the spherical equator coordinates of

unknown objects from their standard coordinates.

00

000

000

cossin

sincos)cos(cot

cossin)sin(cot

s

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2. (x,y) 2. (x,y) (α,δ) (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

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2. (x,y) 2. (x,y) (α,δ) (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)

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3. Astrometric Calibration Regions3. 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.

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3. Astrometric Calibration Regions 3. Astrometric Calibration Regions (ACRs)(ACRs)

Some sky areas with plenty of stars distributed Some sky areas with plenty of stars distributed evenly, whose positions and proper motions are evenly, whose positions and proper motions are known very well. Also called Astrometric known very well. Also called Astrometric Standard Regions.Standard Regions.

Pleiades, Praesepe (~ 500 stars) Pleiades, Praesepe (~ 500 stars)

SDSS ACRs (16, 7º.6×3º.2 SDSS ACRs (16, 7º.6×3º.2 ，， 1999,1999, 765~10,772 stars/per square degree) 765~10,772 stars/per square degree)

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Npar Model forms

4 ζ=aX+bY+c

η=aY-bX+d

6 ζ=aX +bY +c

η=a’X+b’Y+c’

7 ζ=aX +bY+c+pX2+qXY+rX(X2+Y2)

η=aY － bX+d+pXY+qY2+rY(X2+Y2)

8 ζ=aX +bY +c +pX2+qXY

η=a’X+b’Y+c’ +pXY+qY2

9 ζ=aX +bY+c +pX2 +qXY+rX(X2+Y2)

η=a’X+b’Y+c’ +pXY+qY2 +rY(X2+Y2)

3. Astrometric Calibration Regions3. Astrometric Calibration RegionsDifferent telescopes/receivers, different modelsDifferent telescopes/receivers, different models

Table. Models of different forms

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Npar Model forms

10 ζ=aX +bY+c +pX2 +qXY +rY2

η=aY － bX+d+p’X2 +q’XY+r’Y2

12 ζ=aX +bY+c +pX2 +qXY +rY2

η=a’X+b’Y+c’+p’X2 +q’XY+r’Y2

13 ζ=aX +bY+c +pX2 +qXY +rY2+sX(X2+Y2)

η=a’X+b’Y+c’+p’X2 +q’XY+r’Y2+sY(X2+Y2)

14 ζ=aX +bY+c +pX2 +qXY +rY2+sX(X2+Y2)

η=a’X+b’Y+c’+p’X2 +q’XY+r’Y2+s’Y(X2+Y2)

3. Astrometric Calibration Regions3. Astrometric Calibration Regions

Table Models of different forms (continued)

Note: For magnitude terms: adding M, MX, MY

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3. Astrometric Calibration Regions3. 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.

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3. Astrometric Calibration Regions3. Astrometric Calibration Regions

Real example 1:

60/90cm Schmidt in Xinglong Station of NAOC

CCD: 2K*2KFOV: 1º*1º

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3. Astrometric Calibration Regions3. Astrometric Calibration Regions

Real example 2:

2.16m Telescope in Xinglong Station of NAOC

CCD: 2K*2K, FOV: 11’*11’

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3. Astrometric Calibration Regions3. 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.

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4. Block Adjustment of overlapping 4. Block Adjustment of overlapping observationsobservations

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

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4. Block Adjustment4. Block Adjustment

Why? (continued)

Small size of CCD ( ~5 cm) ► small FOV ~15’ for f=10m ► Covering few reference stars ► uneven distribution

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4. Block Adjustment4. Block AdjustmentResults with systematic errors like translation, rotation and distortion when reference stars are not good:

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4. Block Adjustment4. Block Adjustment

How?

The start point of BA: one star has only one position at a time (absolute restriction).

1

2

× × × ×× × × × × ×

),,;,(

),,;,(

1

1

n

n

aayxI

aayxH

）（）＝（ tt 21

),,,(

),,,(

00

00

＝＝

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4. Block Adjustment4. 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.

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Advantages:

Enlarging FOV

Increasing reference stars

Reducing the importance of reference stars

Detecting and removing possible

systematic errors in reference stars.

4. Block Adjustment4. Block Adjustment

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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 Adjustment4. Block Adjustment

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Case 1:

When reference stars covering small area:

Real line BA

Dash line SA

4. Block Adjustment4. Block Adjustment

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4. Block Adjustment4. Block Adjustment

Case 2:

When one reference star has big error in α:

Real line BA

Dash line SA

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4. Block Adjustment4. Block Adjustment

Case 3:

When some reference stars have systematic errors in α :

Real line BA

Dash line SA

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4. Block Adjustment4. Block AdjustmentApplication: (Long focal length + high precision)

Linkage of radio-optical reference frames

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Application:

Open clusters of big area

Constructing ACRs

Detecting systematic errors in catalogues

4. Block Adjustment4. Block Adjustment

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Thanks!