Up: Mutual positions of the

Subsections

# 5 Results of the reduction of photometric observations

## 5.1 The notation

We recorded observing times on the UTC scale and converted them into the TT scale prior to reduction. To this end, we used the following relation:

The results of the reduction of photometric observations are given in Tables 3 and 4, and in Figs. 1-31. We describe now the notation that we used to present the results of this paper.

t0 is the time of observation, MJD (MJD = JD - 2400000.5) with fractions in the TTtime scale. This quantity is not necessary for subsequent determination of the elements of the satellite orbit, however, it can be used to control and identify the data.For each observed event the time t0 was chosen arbitrary within the period of phenomenon.

, are the times as explained above, MJD with fractions in the TT time scale.

, are two components (in kilometers) of the mutual position vector of satellites calculated using formulas (3) as the main results of the reduction of photometric observations.

, are the standard errors of , as inferred from the reduction of photometric observations (in kilometers). These errors can be used in subsequent computations to determine the weights in the least squares method.

ax, ay, az, bx, by, bz: the dimensionless coefficient required for application of the results obtained (see formula (2)). The value of azis always equal to zero.

The results presented also include corrections Dx, Dy, which incorporate the contributions due to observational errors and the errors of the theory. These corrections characterize the agreement between theory and observations (O-C).

The accuracy of the relative satellite positions inferred from photometric observations of mutual events is characterized by standard errors , expressed as linear quantities. It is interesting to compare these errors with the accuracy of ground-based angle measurements expressed in arcseconds. The observational errors of the two different types can be co-ordinated by taking into account the angles at which intervals and - the projections of the satellite - satellite vector on the sky plane located at a topocentric distance to the active satellite - are seen from the Earth. This is the case when mutual occultations of satellites are considered. When analyzing the mutual eclipses the sky plane is perpendicular to the heliocentric direction of the eclipsed satellite. In this case the angles corresponding to intervals and are heliocentric angles. The accuracy of the determination of these angles is then similar to that of ground-based photographic angle measurements. Therefore along the standard errors , we also give the corresponding angular errors computed using the following formulas:

where r is the topocentric or heliocentric distance of the active satellite depending on the type of the observed event. Standard errors given in Table 4 are in arcsec.

We number the Galilean satellites in accordance with the generally accepted numbering system: Io - 1, Europa - 2, Ganymede - 3, and Callisto - 4.

We identify mutual phenomena of the Galilean satellites by assigning to them the codes consisting of date (yymmdd) and the following values separated by dash: (number of active satellite), P - type of mutual phenomenon, (number of passive satellite). P is O for mutual occultation and P is E for mutual eclipse. For example, identifier 970803-4E1 refers to Io's eclipse by Callisto on August 3, 1997.

Each event could have been observed at several observatories yielding several light curves. To designate the particular light curve, we add the observer's code to the event identifier (see Table 1). This conventional code identifies the observatory, observers, and the equipment involved in each particular observation. Thus, if phenomenon 970803-4E1 was observed at the Pulkovo observatory, the results should be coded as 970803-4E1-k. The results of observations of the same phenomenon made at the Nauchny observatory by observer Irsmambetova T.R. using photoelectric photometer will be referred to via identifier 970803-4E1-t.

 Observation ax ay bx by bz 970413-1O2-d 50550.990029 50550.989991 379 -964 0.647097 0.762407 0.213239 -0.180988 0.960090 970422-4O3-d 50560.963826 50560.963768 -218 540 0.626288 0.779592 0.212176 -0.170452 0.962251 970622-1E2-a 50621.854428 50621.854459 531 -1396 0.700797 0.713360 0.216057 -0.212252 0.953031 970706-3E4-v 50635.913714 50635.913775 -674 2058 0.685900 0.727696 0.216626 -0.204184 0.954663 970715-1O3-a 50644.767217 50644.767163 1020 -2570 0.605868 0.795565 0.214879 -0.163643 0.962833 970718-3E2-a 50647.766400 50647.766442 -1108 3155 0.670902 0.741545 0.216719 -0.196073 0.956341 970718-3O2-a 50647.843641 50647.843601 491 -1443 0.609167 0.793042 0.215321 -0.165397 0.962435 970718-3O2-g 50647.843647 50647.843607 531 -1551 0.609165 0.793043 0.215322 -0.165397 0.962434 970719-3O1-a 50648.851532 50648.851479 400 -1055 0.611784 0.791025 0.215666 -0.166798 0.962116 970724-1E4-a 50653.741856 50653.741934 818 -2368 0.665072 0.746779 0.216751 -0.193036 0.956951 970725-1E4-a 50654.846134 50654.846210 -676 1823 0.663051 0.748574 0.216693 -0.191937 0.957186 970725-1E4-t 50654.846106 50654.846182 -533 1678 0.663051 0.748574 0.216693 -0.191936 0.957186 970725-1E4-g 50654.846302 50654.846378 -649 1880 0.663051 0.748574 0.216693 -0.191936 0.957186 970725-3E2-a 50654.916027 50654.916066 -1037 2996 0.662727 0.748861 0.216675 -0.191753 0.957227 970725-3E2-k 50654.915886 50654.915926 -1021 2998 0.662727 0.748861 0.216675 -0.191753 0.957227 970725-3E2-g 50654.915801 50654.915841 -997 2991 0.662727 0.748861 0.216675 -0.191753 0.957227 970725-3O2-k 50654.969840 50654.969801 556 -1432 0.620168 0.784469 0.216654 -0.171277 0.961106 970725-3O2-t 50654.970338 50654.970300 515 -1521 0.620169 0.784468 0.216653 -0.171277 0.961106 970731-4E3-e 50660.991690 50660.991752 414 -1170 0.655323 0.755349 0.216531 -0.187857 0.958031 970801-4E2-g 50661.818732 50661.818817 1076 -3235 0.654879 0.755734 0.216545 -0.187647 0.958069 970801-4E2-l 50661.818615 50661.818700 1012 -2951 0.654879 0.755733 0.216545 -0.187647 0.958069 970801-4E3-k 50660.991652 50660.991714 423 -1195 0.655323 0.755349 0.216531 -0.187857 0.958031 970803-4E1-k 50662.980645 50662.980730 -325 773 0.654328 0.756211 0.216578 -0.187399 0.958110 970803-4E1-l 50662.980643 50662.980728 -134 372 0.654328 0.756211 0.216578 -0.187399 0.958110 970803-4E1-t 50662.980706 50662.980791 -311 886 0.654328 0.756211 0.216578 -0.187399 0.958110 970830-3E2-a 50690.754789 50690.754816 -1214 3592 0.620757 0.784003 0.214981 -0.170217 0.961670 970830-3O2-a 50690.646325 50690.646296 -614 1892 0.678793 0.734330 0.220080 -0.203436 0.954033 970903-1E3-a 50694.620004 50694.620040 -79 194 0.617212 0.786797 0.214789 -0.168494 0.962016 970906-3O2-g 50697.809825 50697.809798 -746 2432 0.688229 0.725493 0.220037 -0.208735 0.952897 970910-1O3-a 50701.655382 50701.655345 -222 646 0.693845 0.720125 0.219966 -0.211938 0.952206 970914-3E1-e 50705.716657 50705.716714 323 -526 0.603294 0.797519 0.213624 -0.161598 0.963458 970915-3E2-a 50706.608493 50706.608526 -173 -445 0.603013 0.797731 0.213642 -0.161495 0.963471 970918-1O3-g 50709.687811 50709.687766 -967 2883 0.700427 0.713724 0.219700 -0.215607 0.951444 970918-1E3-e 50709.791892 50709.791940 -166 467 0.598091 0.801428 0.213127 -0.159053 0.963991 970918-1E3-t 50709.791845 50709.791893 -138 526 0.598091 0.801428 0.213127 -0.159053 0.963991 970921-3E1-l 50712.831184 50712.831240 462 -1201 0.594581 0.804035 0.212791 -0.157358 0.964343 970922-3E2-g 50713.768904 50713.768940 928 -2302 0.594290 0.804251 0.212808 -0.157252 0.964357 971007-4O1-t 50728.790911 50728.790826 1185 -3373 0.708427 0.705784 0.219125 -0.219946 0.950583 971109-4O2-t 50761.672069 50761.672005 1390 -2436 0.686812 0.726835 0.219121 -0.207056 0.953474 971109-4O2-w 50761.672881 50761.672816 918 -2967 0.686812 0.726835 0.219122 -0.207056 0.953474 971110-3E1-t 50762.710973 50762.711012 1139 -2885 0.531928 0.846789 0.204298 -0.128334 0.970460 971118-3O1-g 50770.562083 50770.562049 -1155 3335 0.674907 0.737903 0.219004 -0.200308 0.954942

 Observation Fig t0 Dx, km Dy, km , km , km E(S) Q C 970413-1O2-d 1 50551.021346 571 56 24 27 0.006 0.007 K V 6 970422-4O3-d 2 50560.994237 299 -573 34 78 0.009 0.020 KPL P 4 970622-1E2-a 3 50621.879701 291 222 44 32 0.012 0.008 KQ P 1 970706-3E4-v 4 50635.938173 -959 -156 107 99 0.029 0.027 K V 7 970715-1O3-a 5 50644.791125 218 -419 63 42 0.021 0.014 KQ P 2 970718-3E2-a 6 50647.790257 104 -15 93 46 0.025 0.012 KQ P 1 970718-3O2-a 7 50647.867410 -74 327 7 7 0.002 0.002 KQ P 2 970718-3O2-g 7 50647.867415 -31 223 12 13 0.004 0.004 K V 6 970719-3O1-a 8 50648.875257 -80 88 19 23 0.006 0.008 KQ P 2 970724-1E4-a 9 50653.765616 944 -2155 165 106 0.045 0.029 KQ P 1 970725-1E4-a 10 50654.869863 -118 -175 73 196 0.020 0.054 K V 5 970725-1E4-t 10 50654.869835 -1 -330 28 27 0.007 0.007 K V 6 970725-1E4-g 10 50654.870031 91 -54 31 28 0.008 0.008 K V 6 970725-3E2-a 11 50654.939655 142 116 96 51 0.026 0.014 KQ P 1 970725-3E2-k 11 50654.939515 20 71 105 58 0.029 0.016 K P 1 970725-3E2-g 11 50654.939429 -41 36 71 36 0.019 0.010 K V 6 970725-3O2-k 12 50654.993386 -341 -255 77 96 0.026 0.032 K V 5 970725-3O2-t 12 50654.993885 87 -179 6 16 0.002 0.006 KPL P 4 970731-4E3-e 13 50661.015210 152 112 12 15 0.003 0.004 KQ P 2 970801-4E2-g 14 50661.842276 371 101 151 92 0.041 0.025 K V 6 970801-4E2-l 14 50661.842159 157 333 57 39 0.015 0.011 K V 6 970801-4E3-k 15 50661.015172 138 80 52 55 0.014 0.015 KQ P 1 970803-4E1-k 16 50663.004171 55 302 4 6 0.001 0.002 KQ P 1 970803-4E1-l 16 50663.004169 242 -101 31 70 0.008 0.019 K V 6 970803-4E1-t 16 50663.004232 172 426 163 451 0.044 0.123 KPL P 4 970830-3E2-a 17 50690.778555 -1517 -495 138 60 0.038 0.016 KQ P 1 970830-3O2-a 18 50690.670037 128 -47 7 5 0.002 0.002 KQ P 2 970903-1E3-a 19 50694.643972 128 -402 53 141 0.014 0.038 KQ P 1 970906-3O2-g 20 50697.833822 272 -454 24 18 0.008 0.006 K V 6 970910-1O3-a 21 50701.679597 65 -81 16 27 0.005 0.009 KQ P 2 970914-3E1-e 22 50705.741160 -88 18 13 24 0.004 0.006 KQ P 2 970915-3E2-a 23 50706.633011 -54 569 71 160 0.019 0.044 KQ P 1 970918-1O3-g 24 50709.712467 160 75 63 39 0.020 0.012 K V 6 970918-1E3-e 25 50709.816646 -23 267 11 21 0.003 0.006 KQ P 2 970918-1E3-t 25 50709.816599 -71 300 22 46 0.006 0.013 K V 6 970921-3E1-l 26 50712.856114 821 598 16 19 0.004 0.005 K V 6 970922-3E2-g 27 50713.793868 189 -282 49 106 0.014 0.029 KPL P 3 971007-4O1-t 28 50728.816855 -248 -37 37 43 0.011 0.013 KPL P 4 971109-4O2-t 29 50761.700869 516 -373 10 7 0.003 0.002 K V 6 971109-4O2-w 29 50761.701680 397 -796 31 47 0.009 0.013 KPL P 8 971110-3E1-t 30 50762.739970 1425 2503 262 555 0.072 0.152 KPL P 4 971118-3O1-g 31 50770.591704 245 248 26 16 0.007 0.004 K V 6

## 5.2 Description of the results

Each line in Tables 3 and 4 corresponds to observation of one event by one observer, i.e., refers to one light curve.

Table 3 contains the data required to refine the elements of satellite orbits. Table 4 gives the parameters that allow the accuracy and reliability of results to be assessed. To establish a correspondence between the lines in two tables that refer to the results of the same observation, we give the observation identifier in the first column of each table.

The results of reduction of observations and depend substantially on time t0, which was set equal to one of the observing times near the light minimum of the passive satellite. The sets of observing times for the same event differ from one observatory to another. Therefore and inferred from observations made at different observatories cannot be compared to each other. By contrast, we assume Dx and Dy to be constant throughout the particular phenomenon as is explained above allowing us to compare Dx and Dy values inferred from observations performed at different observatories. The discrepancies between values inferred from observations made at different observatories are therefore due to observational errors, thereby providing an external estimate for the latter. Parameters an characterize the internal observational errors.

The accuracy of the photometric observations performed and the quality of the data obtained can be assessed from graphs illustrating the agreement between the theory and observations (Figs. 1-31). The dots or other symbols show the satellite flux, Si, corresponding to the measured photometric count Ei and parameters Dx, Dy, K, Q, and L, P obtained from the process of the reduction of observations using formula

The model of mutual occultation or eclipse of the satellites is used to compute and plot theoretical satellite fluxes, , for each measurement time. Theoretical values on graphs are connected by a continuous curve. Each graph (the curve and the dots) represents the results of a single observation of one event at one observatory and is identified as explained above, e.g., 970803-4E1-t in the case of Io's eclipse by Callisto on August 3, 1997 observed by Irsmambetova T.R. at the Nauchny observatory. We plot in the same figure the results of all observations of the same event made simultaneously at several observatories. Therefore some of the figures show multiple satellite light curves thereby allowing the results obtained at different observatories to be compared and the systematic errors made during observations to be revealed.

Some light curves were obtained with CCD camera. In this case zero value of satellite flux S corresponds to zero value of E. This make it possible to put . No observation allowed us to determine all the parameters K, Q, P, L together empirically. The more detailed comments to each light curve reduced are in the following subsection.

For each observation Table 4 contains the reference (Fig) to the corresponding figure, the reference (C) to the special comment, and the list (E(S)) of the parameters from the set K, Q, P, L which were really determined. To each observation we assigned a quality index Q. The value P (perfect) of the quality index means that the list of the parameters determined fully corresponds to the method of observation. The value V (vague) shows that the only parameter K could be determined and that real error of the values , may be more important than the estimations , .

## 5.3 Comments and remarks on the observations

Each of following comments is attached to some light curve according to the reference C in Table 4.

1.
Measurements of the flux from occulted and occulting satellites or the flux from eclipsed satellites relative to the flux from the reference satellite were made by the CCD detector so the variations of the air mass transparency were taken into account. In this case the parameter Q may be put at zero or determined from observations;

2.
Measurements of the flux from the occulted and occulting satellites or from the eclipsed satellite were made by the CCD detector. The parameters K, Q were successfully determined as necessary;

3.
Measurements of the flux from occulted or eclipsed satellite relative to the flux from the reference satellite were made by the photometer so the variations of the air mass transparency were taken into account. The parameters K, P, L were successfully determined as necessary;

4.
Measurements of the direct flux from the occulted and occulting satellites or from the eclipsed satellite were made through the diaphragm by the photometer and four parameters K, Q, P, L are to be determined. However the data allowed the only parameters K, P, L to be successfully determined so the variations of the air mass transparency were only partially taken into account;

5.
Measurements of the flux from the occulted and occulting satellites or from the eclipsed satellite were made by the CCD receiver. The parameters K, Q are to be determined. However the data do not allow us to determine the parameter Q and only parameter K was determined. The real errors may be more important than the estimates and ;

6.
Measurements of the flux by the photometer were made. The data do not allow us to determine the parameters Q, P, L and only parameter K was determined. The real errors may be more important than the estimates and ;

7.
Visual measurements of the flux from the eclipsed satellite were made and the parameters K, P, L were determined;

8.
Visual measurements of the flux from the occulted and occulting satellites were made and only parameter K was determined. The real errors may be more important than the estimates and .

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