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:

$$TT = UTC + 63.184 \;\; {\rm s} .$$

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.

$t_{\rm a}$, $t_{\rm p}$ are the times as explained above, MJD with fractions in the TT time scale.

$X^{\rm (o)}$, $Y^{\rm (o)}$ 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.

$\sigma X$, $\sigma Y$are the standard errors of $X^{\rm (o)}$, $Y^{\rm (o)}$ 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.

a_x, a_y, a_z, b_x, b_y, b_z: the dimensionless coefficient required for application of the results obtained (see formula (2)). The value of a_zis always equal to zero.

The results presented also include corrections D_x, D_y, 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 $\sigma X$, $\sigma Y$ 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 $X_{\rm a}$ and $Y_{\rm a}$ - 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 $X_{\rm a}$ and $Y_{\rm a}$ 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 $\sigma X$, $\sigma Y$we also give the corresponding angular errors $\sigma_{\rm a} X, \sigma_{\rm a} Y$computed using the following formulas:

$$\sigma_{\rm a} X = {\rm arctg} \left( \frac{\sigma X}{r} \rig... ...gma_{\rm a} Y = {\rm arctg} \left( \frac{\sigma Y}{r} \right),$$

where r is the topocentric or heliocentric distance of the active satellite depending on the type of the observed event. Standard errors $\sigma_{\rm a} X, \sigma_{\rm a} Y$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: $n_{\rm a}$ (number of active satellite), P - type of mutual phenomenon, $n_{\rm p}$ (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.

 

 

Table 3: Relative positions of the Galilean satellites derived from the photometric observations

Observation	$t_{\rm a}$	$t_{\rm p}$	$X^{\rm (o)}$	$Y^{\rm (o)}$	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

   

Table 4: The fit of the theory to the photometric observations and precision estimates

Observation	Fig	t0	Dx, km	Dy, km	$\sigma X$, km	$\sigma Y$, km	$\sigma_{\rm a} X, ^{\prime\prime}$	$\sigma_{\rm a} Y, ^{\prime\prime}$	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 $X^{\rm (o)}$ and $Y^{\rm (o)}$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 $X^{\rm (o)}$ and $Y^{\rm (o)}$ inferred from observations made at different observatories cannot be compared to each other. By contrast, we assume D_x and D_y to be constant throughout the particular phenomenon as is explained above allowing us to compare D_x and D_y 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 $\sigma X$ an $\sigma Y$ 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, S_i, corresponding to the measured photometric count E_i and parameters D_x, D_y, K, Q, and L, P obtained from the process of the reduction of observations using formula

$$S_i=\frac{E_i - P - L \; (t_i - t_{\rm b})}{K + Q \; (t_i - t_{\rm b}) } .$$

The model of mutual occultation or eclipse of the satellites is used to compute and plot theoretical satellite fluxes, $S(X_{\rm t} + D_x, Y_{\rm t} + D_y)$, 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 $P = 0, \; L=0$. 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 $X^{\rm (o)}$, $Y^{\rm (o)}$ may be more important than the estimations $\sigma X$, $\sigma Y$.

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 $\sigma X$ and $\sigma Y$;

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 $\sigma X$ and $\sigma Y$;

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 $\sigma X$ and $\sigma Y$.