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Pulkovo Method for an Accuracy Estimation

For reliable estimation of the accuracy of asteroid CCD observations we usually consider from ten or more various NUMBERED MINOR PLANETS (NMPs) observed during several nigths per month (more then 50 positions). We are sure that the errors of theory of NMP motion are more less then observational errors of their modern CCD observations. We have our EPOS Software package for exact asteroid ephemeris calculations with a full account for perturbation. It is important also that one night of observations ought to have three or more positions of the NMP for calculation of the "internal" accuracy (a symbol "int" in the Tables) and we use several close nights of observations of the same NMP for deriving the "external" accuracy (a symbol "ext" in the Tables).

The main idea of our method of estimation of an accuracy of asteroid observations is to use these (O-C) deviations when we calculate the mean value of (O-C) and its mean error for the considered NMP. The (O-C) values may be very different for various NMPs, but their mean error shows a reliability of "Atmosphere + Telescope + CCD matrix + Catalogue" system at the mean observational conditions. They can be considered as an indicator of accuracy of the CCD positional observations with given MPC code. We get a set of such mean errors derived from all NMPs observed at the given observatory during investigated observational period and then we calculate a total average value for this set of the "Mean error of single observation". This average value is an accuracy of the investigated CCD observations. Of course, we must have a lot of NMP observations in the set for each observatory under consideration.

Examples
Table 1.
Observations with normal values of the "Mean error of single observation"
-----------------------------------------------------------------------------
 NMP Code   Dynamical Time        RA      (O-C)       DEC    (O-C)    V    z
-----------------------------------------------------------------------------
3907  761  1998 08 24.383673  3 17 44.739  0.28 +30 46 00.84 -0.08  15.8  18
3907  761  1998 08 24.409133  3 17 46.152  0.28 +30 46 10.85  0.12  15.8  11
3907  761  1998 08 25.392523  3 18 40.107  0.33 +30 52 27.69  0.04  15.4  15

   Total mean value of    3 observations   0.297+-0.017       0.027+-0.058
   Mean error of single observation        0.029              0.101
                                           =====              =====
COMMENT. There are two neighbour nights of CCD observations. For any NUMBERED minor planet the (O-C) values must be zero if the errors of observations and the errors of its theory are absent. The errors of the theory for usual NUMBERED minor planet during several successive nights are very small, more less then observational errors in this conditions. Therefore, the range of these values due to observational errors may be a good characteristic of the accuracy of observational data. The real meanings of (O-C) may be different but we are investigating namely the range of these values during several close nights, or "Mean error of single observation" in this and other Tables.
Table 2.
Observations with the large values of the "Mean error of single observation"
-----------------------------------------------------------------------------
 NMP  Code  Dynamical Time        RA      (O-C)       DEC    (O-C)    V    z
-----------------------------------------------------------------------------
3955  761  1998 10 16.163935  0 52 43.326  2.34 + 3 26 20.94  1.62  15.4  27
3955  761  1998 10 16.194495  0 52 41.796  2.68 + 3 26 17.64  1.68  15.2  24
3955  761  1998 10 17.165035  0 51 52.843  5.76 + 3 24 29.44 -0.82  15.5  26
3955  761  1998 10 17.181125  0 51 52.129  7.22 + 3 24 25.54 -3.00  15.2  25

  Total mean value of    4 observations    4.500             -0.130
  Mean error of single observation         2.378              2.240
COMMENT. The large values of (O-C)s may be due to errors of the theory of NMP motion (old orbital unimprovement elements). We take into account not values but their range during one night observations. There are very large (O-C) ranges of RA & DEC for two close nights. Observations are only responsible for these (O-C) changings (from 2.3 in the first night to 7.2 arcsec in the next night for RA and from 1.6 to -3.0 arcsec for DEC).
See below the same asteroid observations made by other observer.
 3955  699  1998 12 15.074492  0 32 36.715   0.73  + 4 27 42.71   0.67 16.3  19
 3955  699  1998 12 15.094822  0 32 37.118   1.12  + 4 27 47.90   0.51       17
 3955  699  1998 12 15.115212  0 32 37.521   1.49  + 4 27 53.90   1.13       17
-----------------------------------------------------------------------
NORMAL PLACE 3 obs. 15.094842  0 32 37.1184  1.113 + 4 27 48.172  0.770 
                                          +/-0.219             +/-0.186 
Mean error of single observation             0.380                0.322
RA&DEC velocities and (O-C)      19.791     18.65     274.96     11.25
                               (per day)            (per day)
Relative errors of velocities (%)            6.7                  4.3                                            
COMMENT. For each set of spheric coordinates (RA & DEC) obtained during one night we use a linear approximation to calculate a normal place with its accuracy and the first derivatives (dRA/dt & dDEC/dt), i.e the velocities of asteroid's motion (in time seconds per day for RA and arcseconds per day for DEC). The relative errors of these values may additionally characterise an accuracy of CCD observations.
Table 3.  (O-C)s for the CCD observations made with MPC code 719
--------------------------------------------------------------------------------
  NMP  MPC   Dynamical Time        RA       (O-C)       DEC    (O-C)   V    z
       code       (DT)        (geocentr.)          (geocentr.)
--------------------------------------------------------------------------------
 1615  719  1999 01 23.295214  5 37 23.987  1.01  +21 19 44.10  2.29  15.2  34
 1615  719  1999 01 23.359444  5 37 22.481 -0.21  +21 19 45.93  1.48  15.5  53

 4150  719  1999 01 23.295214  5 36 58.694  0.79  +21 20 26.01  0.45  16.0  34
 4150  719  1999 01 23.359444  5 36 56.569  0.38  +21 20 32.70  1.67  17.1  54

 5945  719  1999 01 23.315254  8 20 36.813  0.34  +27 37 45.73  0.30  16.5   6
 5945  719  1999 01 23.400954  8 20 30.592  0.61  +27 38 03.31  0.79  17.1  30
COMMENT. The observations with two positions per night do not give us to make a numerical estimation of CCD observations accuracy. The coincidence of the first and the second (O-C)s may show a quality of observations.
Table 4. Examples of typical errors of CCD observations  (I - IV)
------------------------------------------------------------------------------
   NMP  Code       DT           Geocentric positions and their (O-C)  Mgn  Z
------------------------------------------------------------------------------
I.  CRUDE TIME ERROR (*):
 31759 683   2001 12 18.369964  8 18 49.392 -0.24  + 3 28 44.53  1.01 18.2 32
 31759 683   2001 12 18.381134  8 18 49.021 -1.33  + 3 28 43.62 -0.94 18.5 30
 31759 683   2001 12 18.394534  8 18 48.365 -5.82* + 3 28 43.52 -2.29*18.4 29
NORMAL PLACE  3 obs. 18.381877  8 18 48.926 -2.463 + 3 28 43.89 -0.740
                                           +-1.708             +-0.958
Mean error of single observation             2.958**             1.659**
RA&DEC velocities and (O-C)      -42.016  -231.16     -39.91  -133.10
Relative errors of velocities (%)           57.9               142.8

COMMENT. The third position of the NMP 31759 have a crude error of UT registration


II.SYSTEMATIC ERROR IN NORMAL PLACES DUE TO CATALOG POSITIONS OF REFERENCE STARS:
 31759 704   2001 12 19.380534  8 18 21.828  0.98  + 3 30 24.25  0.11 18.0 30
 31759 704   2001 12 19.396184  8 18 21.386  0.95  + 3 30 26.35  0.55 18.4 30
 31759 704   2001 12 19.411874  8 18 20.924  0.65  + 3 30 27.15 -0.32 18.0 30
 31759 704   2001 12 19.427554  8 18 20.481  0.64  + 3 30 28.76 -0.39 18.4 31
 31759 704   2001 12 19.443894  8 18 20.010  0.48  + 3 30 30.86 -0.03 18.3 33
NORMAL PLACE  5 obs. 19.412008  8 18 20.926 +0.740!+ 3 30 27.48 -0.016!!
                                           +-0.097             +-0.169
Mean error of single observation             0.216               0.377
RA&DEC velocities and (O-C)      -28.724    -8.25      98.89    -7.62
Relative errors of velocities (%)            2.0                 7.2

 31759 608   2001 12 22.471034  8 16 47.719 -0.68  + 3 36 55.92  1.36 18.7 25
 31759 608   2001 12 22.481174  8 16 47.399 -0.52  + 3 36 57.32  1.27 18.6 23
 31759 608   2001 12 22.491744  8 16 47.050 -0.57  + 3 36 58.51  0.91 18.7 20
NORMAL PLACE  3 obs. 22.481317  8 16 47.389 -0.590!+ 3 36 57.25 +1.180!!
                                           +-0.047             +-0.137
Mean error of single observation             0.082               0.238
RA&DEC velocities and (O-C)      -32.341     5.07     124.93   -21.85
Relative errors of velocities (%)            1.0                14.9

COMMENT. Symbols (!) and (!!) may point out the errors connected with various reference star systems taken into consideration by astrometric reduction in two different observatories.


III. VARIOUS ERRORS OF OBSERVATIONS made by automatic telescope (MPC code 704) in comparison with positions obtained by MPC code 644 :
 19713 704   2002 01 11.352004  9 36 47.506 -6.76  +10 49 43.97  5.80 18.2 26
 19713 704   2002 01 11.367104  9 36 47.117 -6.02  +10 49 46.06  5.06 18.3 24
 19713 704   2002 01 11.382204  9 36 46.679 -6.00  +10 49 49.76  5.92 18.9 22
 19713 704   2002 01 11.397304  9 36 46.150 -7.30  +10 49 52.56  5.88 18.1 22
NORMAL PLACE  4 obs. 11.374654  9 36 46.863 -6.520 +10 49 48.09  5.665
                                           +-0.314             +-0.203
Mean error of single observation             0.629               0.406
RA&DEC velocities and (O-C)      -29.840   -10.64     194.99     7.32
Relative errors of velocities (%)            2.4                 3.9

 19713 704   2002 01 13.383044  9 35 46.539  0.71  +10 56 12.76 -1.24 17.8 22
 19713 704   2002 01 13.398254  9 35 46.011 -0.03  +10 56 13.46 -3.61 18.9 23
 19713 704   2002 01 13.413444  9 35 45.542  0.11  +10 56 15.37 -4.78 19.4 24
NORMAL PLACE  3 obs. 13.398247  9 35 46.030  0.263 +10 56 13.87 -3.210
                                           +-0.227             +-1.041
Mean error of single observation             0.393               1.804
RA&DEC velocities and (O-C)      -32.778   -20.14      85.88   116.42
Relative errors of velocities (%)            4.3                57.5

 19713 704   2002 01 19.415624  9 32 17.813 -0.23  +11 18 35.70 -1.24 18.3 27
 19713 704   2002 01 19.430774  9 32 17.294  0.51  +11 18 39.13 -1.49 18.3 31
 19713 704   2002 01 19.445934  9 32 16.654 -0.52  +11 18 43.05 -1.23 18.9 34
NORMAL PLACE  3 obs. 19.430777  9 32 17.254 -0.080 +11 18 39.29 -1.320
                                           +-0.307             +-0.085
Mean error of single observation             0.531               0.147
RA&DEC velocities and (O-C)      -38.245    -9.95     242.56     0.50
Relative errors of velocities (%)            1.8                 0.2

 19713 644   2002 01 26.408114  9 27 34.965  0.19  +11 49 03.45 -0.12 17.5 24
 19713 644   2002 01 26.418664  9 27 34.501  0.03  +11 49 06.16 -0.35 17.3 26
 19713 644   2002 01 26.429074  9 27 34.076  0.36  +11 49 09.08 -0.33 17.3 28
NORMAL PLACE  3 obs. 26.418617  9 27 34.514  0.193 +11 49 06.23 -0.267
                                           +-0.095             +-0.074
Mean error of single observation             0.165               0.127
RA&DEC velocities and (O-C)      -42.454     8.47     268.49   -10.15
Relative errors of velocities (%)            1.3                 3.6
IV. AN EXAMPLE OF POSSIBLE INFLUENCE OF ATMOSPHERIC TURBULENCE:

 19480 644   2002 01 13.534954 10 17 33.722  0.19  - 0 25 09.12 -0.51 17.1 46
                     13.536094       33.684 -0.12         07.32  1.25 17.1 46
 19480 644   2002 01 13.545524 10 17 33.558  0.15  - 0 25 09.02 -0.71 17.1 48
                     13.546674       33.520 -0.16         07.02  1.26 17.0 49
 19480 644   2002 01 13.556384 10 17 33.403  0.33  - 0 25 08.52 -0.52 17.1 51
                     13.557534       33.355 -0.14         06.72  1.25 17.0 51
NORMAL PLACE  6 obs. 13.546194 10 17 33.540  0.042 - 0 25 07.95  0.337
                                           +-0.085             +-0.411
Mean error of single observation             0.208               1.007
RA&DEC velocities and (O-C)      -15.204     1.33      34.76     6.50
Relative errors of velocities (%)            0.6                23.0
COMMENT. A changing mark from (+) to (-) may be explaned by very short time interval between two neighbourhood exposures, near 0.001 day or 1.4 min, if atmospheric turbulence was large.
Table 5. (O-C) residuals for CCD observations made by Blanco Telescope in 2000
------------------------------------------------------------------------------
   NMP  Code       DT           Geocentric positions and their (O-C)  Mgn  Z
------------------------------------------------------------------------------
   355  807  2000 08 25.061526 21 13 18.279 -0.15  -19 41 42.75 -0.23 15.1 59
   355  807  2000 08 25.124136 21 13 14.932 -0.63  -19 41 50.33 -0.06      50
   355  807  2000 08 26.043236 21 12 26.676 -0.36  -19 43 41.43 -0.07 15.6 62
   355  807  2000 08 26.110266 21 12 23.192 -0.23  -19 43 49.25  0.01      51
   Total mean value of    4 observations    -0.343+-.105        -0.088+-.051
   Mean error of single observation          0.210               0.101

  2884  807  2000 07 31.337196 22 38 57.746 -0.07  -11 27 37.89  0.33 16.1 45
  2884  807  2000 07 31.417086 22 38 55.368  0.16  -11 27 53.99  0.38      62
  2884  807  2000 08 27.133756 22 20 41.339 -0.59  -13 16 20.86 -0.03 16.2 48
  2884  807  2000 08 27.205096 22 20 37.967 -0.62  -13 16 38.90 -0.06      43
   Total mean value of    4 observations    -0.280+-.194         0.155+-.116
   Mean error of single observation          0.387               0.232

  4434  807  2000 08 25.310056 23 42 57.209  0.18  - 1 36 09.34 -0.39 15.6 36
  4434  807  2000 08 25.367456 23 42 55.384  0.12  - 1 36 34.27 -0.30      49
  4434  807  2000 08 26.315236 23 42 24.641 -0.08  - 1 43 32.64 -0.42 16.1 37
  4434  807  2000 08 26.387316 23 42 22.290  0.45  - 1 44 04.39  0.03      55
   Total mean value of    4 observations     0.167+-.109        -0.270+-.103
   Mean error of single observation          0.219               0.206

  5219  807  2000 08 26.178286 22 14 05.649 -0.02  -14 03 05.14  0.19 16.9 44
  5219  807  2000 08 26.232106 22 14 03.224 -0.13  -14 03 18.06  0.19      45
  5219  807  2000 08 27.124556 22 13 23.170 -0.32  -14 06 51.30  0.02 16.6 49
  5219  807  2000 08 27.194696 22 13 20.017 -0.48  -14 07 07.93  0.05      44
   Total mean value of    4 observations    -0.237+-.102         0.113+-.045
   Mean error of single observation          0.204               0.090

 10298  807  2000 08 26.984296 19 48 59.496  0.20  -21 43 14.48 -0.22 17.1 63
 10298  807  2000 08 27.049606 19 48 58.031  0.29  -21 43 21.38 -0.17      53
 10298  807  2000 08 27.986296 19 48 37.694  0.21  -21 44 58.64 -0.11 17.2 62
 10298  807  2000 08 27.998716 19 48 37.435  0.21  -21 44 59.87 -0.08      60
   Total mean value of    4 observations     0.227+-.021        -0.145+-.031
   Mean error of single observation          0.042               0.062

 15954  807  2000 07 29.236676 21 25 31.931  0.03  -19 58 27.83 -0.46 16.3 49
 15954  807  2000 07 29.349856 21 25 26.565  0.04  -19 59 11.31 -0.42      63
 15954  807  2000 08 01.279616 21 23 04.788 -0.35  -20 17 54.89 -0.54 16.2 53
 15954  807  2000 08 01.356226 21 23 01.020 -0.31  -20 18 24.11 -0.51      67
   Total mean value of    4 observations    -0.147+-.106        -0.482+-.027
   Mean error of single observation          0.211               0.053

 16422  807  2000 07 30.257726 21 47 17.021  0.19  -10 47 42.75 -0.04 17.6 40
 16422  807  2000 07 30.367456 21 47 11.849  0.27  -10 48 23.81  0.00      57
 16422  807  2000 08 28.043556 21 21 23.581  0.15  -14 11 52.44 -0.13 17.9 58
 16422  807  2000 08 28.104826 21 21 20.450  0.09  -14 12 17.99 -0.27      47
   Total mean value of    4 observations     0.175+-.038        -0.110+-.060
   Mean error of single observation          0.075               0.120

 16655  807  2000 07 29.285356 22 17 44.429  0.06  -13 44 15.25 -0.11 17.7 43
 16655  807  2000 07 29.393976 22 17 40.363 -0.05  -13 44 54.39  0.04      60
 16655  807  2000 08 01.320356 22 15 44.976 -0.39  -14 03 01.62 -0.28 17.7 48
 16655  807  2000 08 01.395346 22 15 41.893 -0.07  -14 03 29.85  0.01      63
   Total mean value of    4 observations    -0.113+-.097        -0.085+-.073
   Mean error of single observation          0.194               0.145

 17041  807  2000 07 29.269376 21 58 20.474  0.23  -17 30 40.32 -0.12 17.5 47
 17041  807  2000 07 29.378016 21 58 15.242  0.34  -17 31 19.95 -0.20      62
 17041  807  2000 08 28.048186 21 29 53.848  0.34  -20 20 28.36 -0.12 17.7 63
 17041  807  2000 08 28.109696 21 29 50.407  0.10  -20 20 44.42 -0.31      53
   Total mean value of    4 observations     0.253+-.057        -0.188+-.045
   Mean error of single observation          0.114               0.090

 18163  807  2000 08 26.033866 20 33 41.654 -0.04  -18 59 43.06 -0.09 18.0 58
 18163  807  2000 08 26.099186 20 33 38.819  0.06  -19 00 01.14 -0.15      49
 18163  807  2000 08 26.995616 20 33 00.348 -0.44  -19 04 05.46  0.02 17.9 65
 18163  807  2000 08 27.061566 20 32 57.591 -0.03  -19 04 23.42 -0.15      53
   Total mean value of    4 observations    -0.112+-.111        -0.092+-.040
   Mean error of single observation          0.223               0.080


     Table 6. Accuracy of CCD observations made with MPC code 807
              (Pulkovo method for estimation of external accuracy
                       with the use of Table 5 data)

       ------------------------------------------------------
         NMP    j   Date 2000    Mgn    Z         Errors
       ------------------------------------------------------

        15954   4   29, 01 08    16.2   58     0.211"  0.053"
        16655   4   29, 01 08    17.7   56     0.194   0.145
        16422   4   30, 28 08    17.8   50     0.075   0.120
         2884   4   31, 27 08    16.2   50     0.387   0.232
          355   4   25, 26 08    15.3   56     0.210   0.101
         4434   4   25, 26 08    15.8   46     0.219   0.206
         5219   4   26, 27 08    16.8   45     0.204   0.090
        18163   4   26, 27 08    18.0   55     0.223   0.080
        10298   4   27, 28 08    17.2   60     0.042   0.062
        17041   4   29, 28 08    17.6   56     0.114   0.090
       ======================================================
   Total   10  40    9 nights                  0.188"  0.118"
                                            +/-0.030   0.019
COMMENT. It's a good accuracy for mass professional CCD observations. Our estimation contains an average influence of the system "Atmosphere + Telescope + CCD matrix + Image processing + Astrometric reduction + Star Catalog" in considered period of observations.

Copyright by Izmailov I. S., Bykov O. P., Kastel G. R.
Calculated by Epos software.