How to Get Equator Coordinates of Objects from Observation Zhenghong TANG Yale Astrometry Summer Wor

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How to Get Equator Coordinates of Objects from
ObservationZhenghong TANGYale Astrometry
Summer WorkshopJuly, 2005
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Contents

• 1. Two basic ways to determine positions
• --- absolute determination
• --- relative determination
• 2. Measurement coordinates (X,Y)
• ? equator coordinates (a,d)
• 3. Astrometric Calibration Regions (ACRs)
• 4. Block Adjustment of overlapping observations

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1.1 Absolute determination
Basic idea 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.
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1.1 Absolute determination

• Typical astrometric instrument meridian circle
• Micrometer, clock and angle reading equipment

at df-z P North polar of equator
system Z Zenith point
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1.1 Absolute determination
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1.1 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 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 gt 0.15

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1.2 Relative determination
Basic idea Determining positions of objects with
the help of reference stars, whose positions are
known.? doesnt care much about the real form
of all influences?the influences on the known
objects and neighboring unknown objects are
similar.
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1.2 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 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 determination
Table. Some astrometric reference catalogues
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1.2 Relative determination

• Advantages
• ? higher efficiency (10,000 stars)
• ? higher precision (0.010.05)
• (Hipparcos catalogue, CCD)

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2. How to get (a,d) from (X,Y)

• Basic information of an observation
• ? Focal length F
• ? Field of view
• ? Pixel size
• ? Observational date T
• Center position of the
• observation (a0,d0)
• ? Imaging model

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2. (X,Y) ? (a,d)

• 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) ? (a,d)

• The procedure can be divided to four steps
• First Step
• Compute (?,?) 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) ? (a,d)

• Transformation between spherical coordinates
  (a,d) and standard coordinates (?,?).
• i.e. from spherical surface ? focal plane
• Gnomonic projection

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2. (x,y) ? (a,d)

• Second Step
• Solve parameters of the model that describes
  the relationship between (?,?) and (x,y)
• ? Observational equation (i1,Nref)
• ?(i)aX(i) bY(i) c
• ?(i)aX(i)bY(i)c
• ? Normal equation
• Xa Yb c ?
• XaYbc?
• ? Least square solution

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2. (x,y) ? (a,d)

• Third Step
• Compute the standard coordinates of
  unknown objects based on their measurement
  coordinates and the model parameters.
• ?(j)aX(j) bY(j) c
• ?(j)aX(j)bY(j)c
• (j1,Nobj)

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2. (x,y) ? (a,d)

• Fourth Step
• Compute the spherical equator coordinates
  of unknown objects from their standard
  coordinates.

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2. (x,y) ? (a,d)

• The whole procedure
• 1) (a,d)ref ?(?,?)ref
• 2) (?,?)ref (x,y)ref ? (a,b,c,d,e,f,)
• 3) (a,b,c,d,e,f,) (x, y)obj ?(?,?)obj
• 4) (?,?)obj ? (a,d)obj

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2. (x,y) ? (a,d)

• Note
• When there are some factors which dont 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? 8, Z2Z3)

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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.

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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.63º.2,1999, 76510,772
stars/per square degree)
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3. Astrometric Calibration RegionsDifferent
telescopes/receivers, different models
Table. Models of different forms
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3. Astrometric Calibration Regions
Table Models of different forms (continued)
Note For magnitude terms adding M, MX, MY
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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.

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3. Astrometric Calibration Regions
Real example 1 60/90cm Schmidt in Xinglong
Station of NAOC CCD 2K2K FOV 1º1º
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3. Astrometric Calibration Regions
Real example 2 2.16m Telescope in Xinglong
Station of NAOC CCD 2K2K, FOV 1111
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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.

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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

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4. Block Adjustment

• Why? (continued)
• ? Small size of CCD ( 5 cm)
• ? small FOV 15 for f10m
• ? Covering few reference stars
• ? uneven distribution

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

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

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2

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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 (a,d) of objects.

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4. Block Adjustment

• Advantages
• ? Enlarging FOV
• ? Increasing reference stars
• ? Reducing the importance of reference stars
• ? Detecting and removing possible
• systematic errors in reference stars.

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4. Block Adjustment

• Simulation
• 4433, noise in (X,Y) and (a,d)
• 154 reference stars,
• each frame has one reference star, except the
  center one has four reference stars.

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4. Block Adjustment

• Case 1
• When reference stars covering small area
• Real line ? BA
• Dash line? SA

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4. Block Adjustment
Case 2 When one reference star has big error
in a Real line ? BA Dash line? SA
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4. Block Adjustment
Case 3 When some reference stars have
systematic errors in a Real line ? BA Dash
line? SA
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4. Block Adjustment

• Application (Long focal length high precision)
• ?Linkage of radio-optical reference frames

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4. Block Adjustment

• Application
• ?Open clusters of big area
• ?Constructing ACRs
• ?Detecting systematic errors in catalogues

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• Thanks!