MEMORANDUM

OPTIMIZATION OF THE MOON PERTURBED ORBIT

Astro Space Center

of the P.N. Lebedev Physical Institute, Moscow, Russia,


December 1999

For the optimization of scientific program of RADIOASTRON mission with application of the different angular resolutions (small for the near the Earth diameter baseline projections, medium for near the 28 hours orbit altitudes , and high up to the Moon distances and higher) was fined a new class orbits [1]. One example of such obit is described it the table I.

Table.I

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   N   t,days    Rp       Ra       T      RAn      Dn      RAp     Dp

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    1     .000   10.574  336.221  8.317  173.85   36.69  236.50  -31.65
         4.154   10.102  332.548  8.168  170.31   32.03  237.42  -31.86
    2    8.562   10.454  331.464  8.142  170.14   31.81  237.60  -31.72
        12.280   10.261  332.719  8.180  169.33   30.60  237.83  -31.78
    3   16.843   10.422  333.024  8.197  168.06   29.51  237.79  -31.48
        20.510   10.439  332.556  8.181  167.83   29.22  237.88  -31.39
    4   24.705   10.671  331.859  8.164  167.95   29.33  238.00  -31.27
        28.674   10.592  332.609  8.188  167.57   28.59  238.21  -31.31
    5   33.239   14.123  335.016  8.401  168.23   29.73  238.80  -30.21
        37.036   14.394  336.119  8.451  168.72   30.51  238.64  -30.23
    6   41.626   14.586  336.674  8.478  167.77   29.66  238.62  -29.94
        45.549   14.565  334.159  8.386  167.56   29.35  238.80  -29.76
    7   50.035   15.120  335.515  8.455  168.16   29.95  239.11  -29.53
        54.024   15.260  335.543  8.462  168.48   30.51  239.05  -29.44
    8   58.324   20.348  336.057  8.665  163.88   25.47  238.95  -28.41
        62.872   29.446  334.758  8.951  167.31   32.27  237.76  -27.91
    9   67.549   29.782  334.624  8.958  166.97   32.02  237.80  -27.71
        71.853   29.703  332.196  8.866  166.96   32.05  238.01  -27.41
   10   76.447   30.674  333.137  8.936  167.67   32.70  238.42  -27.17
        80.802   31.092  333.182  8.954  168.06   33.61  238.22  -27.05
   11   85.314   35.502  332.812  9.103  164.44   29.65  238.18  -26.20
        90.557   53.811  328.158  9.614  166.52   36.38  238.22  -23.09
   12   95.609   54.055  327.503  9.598  166.25   36.25  238.14  -22.97
       100.170   53.886  326.016  9.536  166.26   36.50  238.30  -22.62
   13  105.230   54.930  325.334  9.549  166.80   37.01  238.48  -22.64
       109.668   55.453  326.779  9.624  166.91   37.57  238.10  -22.74
   14  114.606   67.259  329.012 10.159  167.40   38.64  239.56  -20.97
       119.755   69.771  332.077 10.374  168.04   40.74  238.49  -21.23
   15  125.302   69.874  332.470 10.393  167.57   40.55  238.42  -20.97
       130.131   69.846  331.735 10.364  167.52   40.84  238.44  -20.72
   16  135.601   69.744  329.695 10.281  167.39   40.79  237.99  -21.06
       140.336   68.712  331.796 10.322  167.32   40.51  237.95  -21.23
   17  145.692   68.691  329.368 10.228  167.07   40.30  237.53  -21.52
       150.467   67.807  331.009 10.257  166.95   39.85  237.55  -21.70
   18  156.091   68.310  332.206 10.322  166.63   39.70  237.79  -21.25
       160.813   68.284  332.681 10.340  166.58   39.90  237.67  -21.19
   19  166.020   67.372  331.636 10.264  164.24   37.70  237.12  -20.86
       171.089   53.249  324.109  9.440  162.77   35.55  240.06  -17.12
   20  175.911   52.591  323.309  9.386  162.07   34.79  239.77  -17.05
       180.480   51.413  322.476  9.310  161.98   34.12  240.20  -16.76
   21  185.169   51.857  323.933  9.381  161.94   34.11  240.46  -16.37
       189.858   51.537  324.523  9.392  161.92   34.24  240.46  -16.27
   22  195.135   45.543  330.264  9.382  158.81   30.13  242.20  -11.23
       199.823   40.620  330.453  9.205  159.42   35.25  242.25  -10.01
   23  204.490   40.151  331.106  9.212  158.95   34.52  242.17   -9.75
       209.075   38.953  329.343  9.102  158.92   34.15  242.57   -9.27
   24  213.755   39.267  331.207  9.183  159.14   34.55  242.90   -8.97
       218.280   38.910  331.653  9.186  159.12   34.90  242.90   -8.82
   25  223.767   31.619  352.291  9.687  160.07   42.12  244.74   -5.86
       228.358   31.194  353.980  9.735  160.00   44.37  244.43   -5.66
   26  233.321   30.796  355.763  9.788  159.81   44.16  244.53   -5.41
       238.196   29.913  353.301  9.661  159.70   44.65  244.83   -4.90
   27  243.526   30.524  354.156  9.716  160.20   45.78  245.19   -4.86
       247.910   30.272  355.034  9.740  160.15   46.26  245.18   -4.74
   28  253.146   33.736  363.687 10.203  162.36   50.45  246.98   -4.43
       258.081   33.514  364.944 10.243  162.31   50.94  246.89   -4.38
   29  263.704   33.818  367.377 10.349  162.52   51.63  247.25   -4.16
       268.463   33.901  365.624 10.284  162.22   52.73  247.23   -3.78
   30  273.941   34.013  363.436 10.204  162.39   53.05  247.12   -3.95
       278.598   31.825  363.277 10.114  162.18   53.38  247.47   -3.49
   31  283.720   32.007  360.677 10.021  162.37   53.52  247.44   -3.63
       288.605   31.443  360.230  9.982  162.30   53.58  247.58   -3.48
   32  293.876   32.203  361.327 10.054  162.68   54.45  247.87   -3.43
       298.654   33.029  361.953 10.109  162.40   55.49  247.52   -3.35
   33  303.894   25.878  360.076  9.765  161.12   53.60  246.37   -3.49
       308.444   21.757  360.367  9.620  160.12   59.74  246.19   -2.29
   34  313.351   21.060  358.018  9.505  159.70   59.00  245.85   -2.31
       317.942   20.621  357.624  9.474  159.59   59.05  245.96   -2.17
   35  322.872   20.916  357.438  9.478  159.81   59.58  246.04   -2.21
       327.409   21.344  359.798  9.583  159.69   60.10  245.75   -2.26
   36  332.773   13.159  365.356  9.484  163.22   70.80  244.49   -3.02
       337.019   14.084  369.850  9.688  163.99   75.33  243.81   -2.65
   37  341.882   13.007  368.377  9.592  162.88   74.56  243.45   -2.59
       346.630   12.433  368.858  9.588  162.74   74.69  243.66   -2.47
   38  351.581   12.376  367.536  9.536  163.14   75.04  243.68   -2.52
       356.150   12.134  371.354  9.671  163.17   75.03  243.70   -2.52
   39  361.362   13.198  381.237 10.088  201.02   84.76  244.39   -3.81
       366.200   13.200  386.140 10.277  207.55   85.32  244.23   -3.75
   40  371.665   12.127  387.552 10.290  211.66   85.54  244.10   -3.77
       376.549   11.739  387.683 10.280  220.84   86.03  244.34   -3.64
   41  382.083   11.316  385.525 10.181  228.93   86.16  244.28   -3.71
       386.704    9.609  386.926 10.169  247.15   86.54  245.13   -3.46
   42  391.815    9.600  384.281 10.067  264.73   86.10  245.35   -3.68
       396.763    8.641  387.834 10.167  270.59   85.99  245.84   -3.64
   43  401.935    8.706  387.411 10.153  286.84   84.97  246.09   -3.81
       406.977    8.930  390.331 10.274  298.28   84.03  246.19   -3.68
   44  412.451    6.935  385.178  9.999  312.48   80.55  245.75   -3.76
       417.000    7.570  386.314 10.067  332.46   60.38  245.82   -1.91
   45  422.324    7.408  378.715  9.771  332.41   58.91  245.80   -2.05
       426.808    7.296  383.407  9.945  332.84   57.51  246.07   -2.06
   46  432.183    8.287  377.399  9.754  332.73   57.11  246.43   -2.39
       436.586    8.883  381.508  9.933  332.69   57.14  246.29   -2.32
   47  441.915   11.853  378.571  9.935  332.62   40.95  245.47   -3.28
       446.468   15.442  381.430 10.182  332.14   42.05  244.52   -2.63
   48  451.969   15.677  380.708 10.163  332.04   41.42  244.48   -2.76
       456.671   16.253  378.156 10.087  331.95   41.55  244.44   -2.80
   49  461.853   17.376  376.438 10.064  332.01   42.42  244.65   -2.89
       466.705   17.846  377.963 10.141  331.96   42.76  244.54   -2.79
   50  471.867   20.634  377.675 10.237  331.76   42.64  244.88   -3.39
       476.853   20.992  380.267 10.351  331.67   42.92  244.66   -3.21
   51  482.521   21.513  381.015 10.400  331.53   42.64  244.65   -3.38
       487.268   22.446  380.929 10.433  331.26   43.20  244.36   -3.30
   52  492.620   21.753  377.119 10.259  331.18   42.31  243.92   -3.00
       497.462   19.666  369.683  9.894  332.78   37.55  245.17   -3.10
   53  502.791   19.024  366.303  9.741  332.74   36.81  244.90   -2.88
       507.167   18.483  366.665  9.734  332.96   36.08  245.09   -2.92
   54  512.479   18.876  367.105  9.766  332.86   35.77  245.09   -3.09
       516.930   19.235  369.136  9.857  332.73   36.06  244.90   -2.98
   55  521.982   18.535  367.008  9.749  332.43   29.49  244.12   -2.99
       526.878   28.229  358.760  9.804  332.80   23.50  243.63   -1.92
   56  532.000   27.413  357.447  9.723  332.54   21.86  243.27   -1.83
       536.612   26.866  355.723  9.637  332.79   20.77  243.57   -2.05
   57  541.925   27.510  356.158  9.678  332.74   20.49  243.60   -2.29
       546.291   27.489  357.555  9.730  332.82   20.25  243.64   -2.23
   58  551.380   33.919  359.175 10.037  332.27   17.61  243.35   -3.40
       556.280   37.197  361.167 10.239  331.80   19.12  242.81   -2.91
   59  561.629   37.340  362.459 10.295  331.52   17.71  242.58   -3.32
       566.623   37.223  359.249 10.166  331.62   17.18  242.79   -3.75
   60  571.910   38.040  358.817 10.181  331.61   17.01  242.82   -3.94
       576.788   37.173  357.734 10.106  332.03   16.03  243.25   -4.23
   61  582.134   38.957  357.295 10.158  332.04   16.04  243.37   -4.61
       586.900   38.558  357.001 10.131  332.26   15.51  243.58   -4.76
   62  592.113   40.063  359.724 10.294  332.16   15.28  243.67   -5.53
       597.272   40.727  357.527 10.235  331.95   15.71  243.60   -5.85
   63  602.569   41.286  356.208 10.206  331.89   14.12  243.38   -5.89
       607.520   37.638  339.502  9.432  333.17    9.15  244.61   -8.87
   64  612.321   37.951  338.381  9.402  333.17    8.93  244.59   -8.92
       616.928   37.598  336.315  9.311  333.29    8.64  244.72   -9.31
   65  621.664   38.890  337.780  9.414  333.29    9.28  244.92   -9.86
       626.361   39.462  337.465  9.424  332.90   10.13  244.69   -9.94
   66  631.213   41.323  336.766  9.468  333.11    6.74  244.34  -10.35
       636.190   42.455  324.170  9.040  332.15    6.30  243.82  -14.79
   67  641.041   42.486  322.963  8.997  332.15    6.09  243.75  -14.73
       645.212   42.119  321.649  8.935  332.13    6.09  243.77  -15.06
   68  649.775   43.003  322.110  8.984  332.08    6.81  243.96  -15.36
       654.183   43.419  322.874  9.028  331.65    7.50  243.71  -15.25
   69  658.991   44.756  322.589  9.067  332.35    3.22  243.29  -16.24
       663.461   40.032  316.184  8.658  330.96    4.78  242.68  -19.74
   70  668.037   39.821  314.531  8.590  331.00    4.41  242.57  -19.52
       672.067   39.424  314.098  8.560  331.09    4.26  242.62  -19.70
   71  676.349   39.898  314.454  8.590  331.03    4.65  242.71  -19.88
       680.632   40.159  315.452  8.636  330.68    5.12  242.53  -19.74
   72  685.266   39.441  314.083  8.560  331.17    2.82  242.17  -19.63
       689.200   35.524  310.406  8.286  331.67    1.65  242.31  -21.23
   73  693.398   35.511  308.709  8.224  331.75    1.41  242.29  -21.01
       697.394   35.144  309.236  8.230  332.11     .90  242.46  -21.03
   74  701.966   35.380  310.042  8.268  332.09     .74  242.38  -21.26
       705.683   35.331  310.241  8.273  331.95     .89  242.29  -21.29
   75  710.028   35.213  308.913  8.221  332.07     .33  242.19  -21.06
       713.868   34.243  308.998  8.189  332.68    -.54  242.47  -21.31
   76  717.983   35.520  309.770  8.263  332.53     .30  242.65  -21.72
       722.080   35.386  311.368  8.315  332.66     .17  242.73  -21.54
   77  726.585   35.398  312.124  8.343  332.76    -.72  242.48  -21.78
       730.461   35.012  311.080  8.292  332.85    -.89  242.49  -22.09
   78  734.697   35.228  311.020  8.297  332.87    -.94  242.49  -22.16
       738.732   34.911  311.765  8.313  333.14   -1.26  242.62  -22.20
   79  743.352   39.080  318.087  8.693  332.14     .89  242.55  -24.50
       747.477   38.656  318.843  8.705  331.33    2.05  242.27  -24.61
   80  752.030   38.469  319.382  8.718  331.51    1.08  242.01  -24.83
       756.219   37.617  317.434  8.616  331.75     .73  242.10  -25.41
   81  760.604   38.055  318.689  8.677  331.70     .99  242.18  -25.77
       764.897   37.758  319.012  8.678  331.66    1.06  242.18  -25.89
   82  769.686   37.481  328.791  9.027  330.72    1.60  241.62  -29.27
       774.191   35.693  328.611  8.955  328.68    4.25  241.12  -29.83
   83  778.789   35.441  329.291  8.970  328.83    3.71  240.98  -30.04
       783.195   34.398  327.046  8.849  328.90    3.59  241.03  -30.68
   84  787.677   34.900  328.858  8.934  328.65    4.37  241.29  -31.09
       792.141   34.719  329.124  8.938  328.22    4.83  241.16  -31.21
   85  797.180   28.488  340.666  9.134  326.27    6.96  240.87  -33.28
       801.602   27.539  340.859  9.106  323.62    9.82  240.22  -33.57
   86  806.609   27.237  341.153  9.106  323.62    9.75  240.20  -33.69
       810.722   26.599  339.053  9.004  323.30   10.00  240.17  -34.14
   87  815.725   27.306  340.119  9.070  322.63   11.38  240.54  -34.36
       819.800   27.487  340.889  9.105  322.00   11.95  240.30  -34.32
   88  825.118   22.261  353.981  9.398  316.87   19.01  240.83  -35.01
       829.409   22.897  355.457  9.478  314.09   21.40  240.00  -34.96
   89  834.189   22.603  355.038  9.451  314.08   21.32  239.89  -34.92
       838.877   22.389  353.182  9.373  313.44   21.78  239.80  -35.19
   90  843.606   23.126  352.691  9.382  312.93   22.65  240.04  -35.19
       848.224   23.572  354.573  9.470  312.36   23.11  239.76  -35.02
   91  853.316   26.335  365.833 10.001  307.98   29.35  241.66  -35.54
       858.199   27.642  369.448 10.190  306.45   30.38  240.85  -35.17
   92  863.498   26.931  369.918 10.181  306.56   30.08  240.60  -35.12
       868.400   27.052  369.049 10.152  305.96   30.39  240.43  -35.23
   93  873.533   26.795  366.494 10.044  306.08   30.19  240.11  -34.99
       878.375   25.555  368.136 10.060  306.24   30.04  240.21  -35.09
   94  883.712   25.760  367.414 10.040  306.22   30.23  240.29  -35.00
       888.358   24.960  369.472 10.088  306.71   29.89  240.40  -34.95
   95  893.720   24.503  372.326 10.180  306.76   29.72  240.39  -35.07
       898.560   24.599  372.917 10.207  306.40   29.93  240.23  -35.07
   96  903.712   20.731  371.508 10.004  307.51   28.22  239.24  -34.60
       908.538   15.351  366.898  9.624  303.70   30.74  239.42  -36.13
   97  913.637   14.359  366.311  9.565  304.43   29.89  239.14  -36.03
       918.108   13.267  364.594  9.459  305.07   29.41  239.57  -36.34
   98  923.132   13.227  368.013  9.586  304.70   29.82  239.76  -36.46
       927.717   13.150  368.430  9.599  304.34   30.07  239.74  -36.52
   99  932.655    7.403  371.822  9.510  299.42   34.33  238.80  -35.70
       937.386    7.935  367.885  9.383  283.49   43.59  237.84  -36.29
  100  942.178    7.298  370.749  9.466  283.20   43.73  237.68  -36.22
       946.889    6.688  366.415  9.281  282.05   44.15  238.07  -36.54
  101  951.734    7.089  369.917  9.427  280.46   45.03  238.45  -36.59
       956.336    7.180  367.250  9.331  279.35   45.42  238.48  -36.69
  102  961.279    8.210  378.343  9.787  252.10   53.96  239.19  -35.35
       966.174   10.414  378.917  9.893  248.49   54.20  238.44  -35.38
  103  971.463   10.213  380.193  9.934  246.32   54.48  238.54  -35.27
       976.170   10.346  375.844  9.774  244.62   54.47  238.73  -35.39
  104  981.529   11.351  376.188  9.825  244.36   54.49  239.17  -35.40
       985.980   11.323  375.585  9.801  243.17   54.40  239.31  -35.54
  105  991.247   14.017  378.754 10.024  240.72   54.73  240.49  -35.27
       995.953   14.548  379.743 10.083  240.47   54.71  240.25  -35.29
  106 1001.352   15.061  381.597 10.174  239.24   54.81  240.53  -35.18
      1006.180   16.083  379.190 10.120  239.14   54.83  240.33  -35.17
  107 1011.653   16.038  376.224 10.005  237.99   54.85  240.06  -35.13
      1016.134   14.431  370.082  9.710  231.96   53.95  241.03  -35.70
  108 1021.217   14.845  367.833  9.641  231.97   53.94  241.07  -35.71
      1025.742   14.750  367.908  9.640  231.44   53.85  241.10  -35.77
  109 1030.927   15.457  369.262  9.718  231.20   53.88  241.36  -35.69
      1035.469   16.464  369.714  9.773  231.94   54.04  240.92  -35.62
  110 1040.290   14.776  366.952  9.605  225.97   53.83  240.08  -35.34
      1045.152   15.720  354.492  9.173  208.18   48.97  240.67  -36.28
  111 1049.842   15.289  352.194  9.072  207.38   48.78  240.34  -36.31
      1054.205   14.913  351.986  9.050  206.53   48.40  240.49  -36.36
  112 1059.233   15.283  352.989  9.101  206.42   48.43  240.63  -36.26
      1063.304   15.675  354.368  9.167  207.00   48.74  240.33  -36.24
  113 1067.983   14.585  352.801  9.069  199.48   46.55  239.60  -35.92
      1072.510   17.297  342.262  8.780  190.60   41.61  240.11  -36.17
  114 1077.204   16.784  340.569  8.700  189.29   41.01  239.80  -36.18
      1081.199   16.291  340.416  8.676  188.29   40.14  240.11  -36.24
  115 1085.808   16.521  341.775  8.734  187.68   39.91  240.23  -36.01
      1089.944   16.630  342.530  8.766  187.77   40.04  240.15  -36.00
  116 1094.622   16.135  341.162  8.698  183.91   37.90  239.68  -35.86
      1095.000   16.073  341.140  8.694  182.99   37.17  239.73  -35.88
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Here N and t are number of orbit (started from the first apogee) and time of perigees crossing after launch in October 14, 2000. Rp and Ra are perigee and apogee radii of the orbit (in ths. km). T is orbital period (in days). RAn and Dn is right ascension and declination (in degrees) of the normal to the orbit plane, RAp and Dp - the same for perigees direction.

The orbit described above allows us to implement both small and large baselines of observations practically for all sources of celestial sphere and to ensure an capability of receiving high-quality images as well as definition of coordinates and own movements of sources to within micro second of an arch.

We divided the celestial sphere on 1250 parts and calculated Bmin(i,j) and Bmax(i,j),
where j - the part number j = 1,2,...1250,
i - number of a half year after Radioastron launch, i=1,2,...6,
Bmin(i,j) - minimal baseline projection,
Bmax(i,j) - maximal baseline projection.

The restriction of the communication with the Earth was taken into account - an angle between "center of the Earth - SRT" vector and source direction not greater than 90 degrees.

Figures fig.1a - fig.6b show colored distribution of the minimal and maximal baseline projections for different half years.

Figures fig.7a - fig.7b shows colored distribution of the minimal and maximal baseline projections for the second half year for 28-hours orbit. For all other half years the figure look the same.

References: [1] Project of the high orbit model, in Russion, ASC and Lavochkin Association Report, November 1999, Moscow.


Figures: