- Email: [email protected]
Astrometric results of observations of mutual occultations and eclipses of the Galilean satellites of Jupiter in 2002–2003
ARTICLE IN PRESS
Planetary and Space Science 56 (2008) 1785–1790 www.elsevier.com/locate/pss
Astrometric results of observations of mutual occultations and eclipses of the Galilean satellites of Jupiter in 2002–2003 Nikolay Emelyanova,b, a
Sternberg Astronomical Institute, Moscow, Russia Institut de me´canique ce´leste et de calcul des e´phe´me´rides – Observatoire de Paris, UMR 8028 du CNRS, Paris, France
b
Accepted 5 February 2008 Available online 22 July 2008
Abstract We suggest a new approach and develop an original method for deriving astrometric data from the photometry of mutual occultations and eclipses of planetary satellites. We decide to model not the relative apparent motion of one satellite with respect to another satellite but the deflection of the observed relative motion with respect to the theoretical motion implied by appropriate ephemerides. We have attempted to reduce the results of photometric observations of the Gallilean satellites during their mutual occultations and eclipses in 2002–2003. The data of observation for 319 light curves of 106 mutual events were received from the observers. The reliable 245 light curves were processed with our method. Eighty six apparent relative positions have been obtained. Systematic errors arise inevitably while deriving astrometric data. Most of them are due to factors that are unrelated to the methods for deriving astrometric data. The systematic errors are more likely due to incorrect excluding the effect of background on photometric counts. In the case of mutual occultations, the flux drop is determined to a considerable degree by the ratio of the mean albedos of the two satellites. Some mutual event observations revealed wrong adopted values of the mean albedos. r 2008 Elsevier Ltd. All rights reserved. Keywords: Occultations; Eclipses; Planets and satellites; Jupiter satellites
1. Introduction The photometry of mutual occultations and eclipses of natural planetary satellites can be used to derive very accurate astrometric data. This can be achieved by analyzing the light curves of the satellites observed during these mutual events. The final goal of observations is to refine the models of motion for the natural satellites. We propose the most accurate photometric model of mutual events to date based on all available data about the satellites, and developed the corresponding method for extracting astrometric data. We use our new method to reduce the series of observations of mutual occultations and eclipses of the Gallilean satellites in 2002–2003 made at different observatories. Corresponding author at: Institut de me´canique ce´leste et de calcul des e´phe´me´rides – Observatoire de Paris, UMR 8028 du CNRS, Paris, France. E-mail address: [email protected]
0032-0633/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2008.02.017
The purpose of this paper is to analyze the most important sources of the systematic errors. 2. Observations Many mutual events involving the Galilean satellites of Jupiter have been observed. In 2002–2003, the Institut de Mecanique Celeste et de Calcul des Ephemerides (IMCCE) organized and coordinated an international campaign to observe these rare events. The goal was to observe as many events as possible. Several independent observations of each event are needed to eliminate observational errors. This campaign allowed the international network of 37 sites to collect 319 lightcurves of 106 mutual events. Only 23 events were simultaneously observed from at least five observatories and 47 events were observed from only one site of the network. Light curves were derived from relative photometry. We preferred to use for our reduction the raw data provided by
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N. Emelyanov / Planetary and Space Science 56 (2008) 1785–1790
IMCCE. Thus no light curve had been pre-processed or altered by anyone before our treatment. 3. Extracting astrometric data from the photometry of mutual events Observations of both mutual occultations and eclipses of natural planetary satellites consist of satellite flux measurements during a single event. Extracting astrometric data from such photometry is an important but difficult task. The methods for solving this problem are now at new stage of development. Several groups of authors have been working extensively on the development of such methods. Aksnes and Franklin (1976), Aksnes et al. (1984), Franklin et al. (1991), and Kaas et al. (1999) developed a method for systematically deriving and publishing the differences in the topocentric coordinates of the Galilean satellites based on the photometry of their mutual occultations and eclipses. This method was significantly improved by Vasundhara (2002), Vasundhara et al. (2003), and Noyelles et al. (2003). In some cases the apparent motion of one satellite relative to another is very complex. A very accurate geometric and photometric model of mutual occultations and eclipses of satellites is much needed. 4. Method for deriving astrometric data In our papers (Emel’yanov, 1995; Emelianov, 2003) a new approach was suggested and an original method for deriving astrometric data from the photometry of mutual occultations and eclipses of planetary satellites was developed. The main idea is not to model the apparent relative motion of one satellite with respect to the other one but instead the departure of the observed relative motion from the theoretical motion provided by relevant ephemeris. The measured flux EðtÞ during an event at a given time t may be expressed by EðtÞ ¼ K SðX ðtÞ; Y ðtÞÞ
(1)
where X ðtÞ and Y ðtÞ are the projections of the differences of planetocentric Cartesian coordinates of the two satellites onto the tangent plane of the event. In the case of mutual occultations, this plane coincides with the plane passing through the occulted satellite perpendicular to the line of sight of the observer. In the case of a mutual eclipse, the plane of the event passes through the eclipsed satellite perpendicular to the line connecting the satellite with the center of the Sun. The origin of coordinates is placed at the center of the passive (occulted or eclipsed) satellite. The occulting or eclipsing satellite is referred to as the active satellite. The function Sðx; yÞ describes a model of the phenomenon. It is supposed Sðx; yÞ ¼ 1 off event. The parameter K is a scale factor for the light drop during the event and it is equal to the total flux outside the event.
Given appropriate theories of the motion of planets and satellites, one can compute the theoretical values of functions X ðtÞ, Y ðtÞ, i.e., X th ðti Þ, Y th ðti Þ for the time ti ði ¼ 1; 2; . . . ; mÞ of each photometric measurement. Here m is the number of photometric counts during a single event. The real values of X ðti Þ and Y ðti Þ differ from X th ðti Þ and Y th ðti Þ by corrections Dx , Dy . Our proposed method consists of solving conditional equations E i ¼ K SðX th ðti Þ þ Dx ; Y th ðti Þ þ Dy Þ ði ¼ 1; 2; . . . ; mÞ (2) for parameters Dx , Dy , and K. Here E i is the photometric recorded at time ti . We linearize conditional equations with respect to parameters Dx , Dy and then solve them using the least-square method. We computed the theoretical values X th ðti Þ and Y th ðti Þ and corrections Dx and Dy using the theory of Galilean satellite motion developed by Lainey et al. (2004). The parameter K is determined from all the photometric measurements before, after and during the event. This fact makes it possible to use observations when there is no measurement before and after the event. We have some examples of such observations. In these cases the parameters Dx , Dy , and K were successfully determined from observations. The function Sðx; yÞ is calculated as an integral of the flux from each point of satellite over the hemisphere facing the Earth. For each point we consider wavelengthdependent reflective properties of the satellites, various laws of light scattering by a rough surface, variation of reflective properties over the satellite surface, wavelengthdependent solar limb darkening. We consider also a wavelength-dependent sensitivity of the detector. Given the topocentric (or heliocentric in the case of mutual eclipse) distances of the active and passive satellites, one can compute the corresponding projections X 00 and Y 00 of the angular separation between the satellites onto the celestial parallel and meridian, respectively. In a similar way we designated D00x , D00y , the angular values, corresponding to the corrections Dx , Dy . Let us consider an example. We have received photometric observations of the occultation of Io by Europa made on December 20, 2002 at Cluj-Napoca observatory. Fig. 1 shows scaled values of the combined flux of the occulted and occulting satellites (circles). Using the theory of the Galilean satellites by Lainey we calculated the apparent motion of Europa with respect to Io. It is evident that this event cannot be represented by rectilinear and uniform apparent motion of one satellite relative to another as it is plotted in Fig. 2. Then using our method we fitted of the parameters K, Dx , and Dy to that observation. The resultant light curve is shown in Fig. 1 (line). By this means corrections to the theoretical relative apparent motion of the satellites were derived (see Fig. 3).
ARTICLE IN PRESS N. Emelyanov / Planetary and Space Science 56 (2008) 1785–1790
Fig. 1. Photometric observations of the occultation of Io by Europa made December 20, 2002 at Cluj-Napoca observatory (circles) and the fit of the parameters K, Dx , and Dy to these observations (points).
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Fig. 3. The derived corrections to the theoretical relative apparent motion of the satellites.
Second, to determine the value of K one can use photometric measurements made before, after and during the event. This fact makes it possible to use observations when there is no measurement before and after the event. We have some examples of such observations. 5. Persistent issues
Fig. 2. The apparent motion of Europa with respect to Io as calculated using the theory of the Galilean satellites by Lainey.
As an astrometric result of the observation one may take any corrected relative position of satellites X ðt Þ ¼ X th ðt Þ þ Dx ; Y ðt Þ ¼ Y th ðt Þ þ Dy together with associated time t inside the interval of the event. This method has the following advantages: First, the deflection of the observed relative motion of satellites with respect to the theoretical motion can be described by two parameters Dx , Dy in most cases. There is no need for an impact parameter and the method may be applied to arbitrary apparent relative satellite motion even somewhat unusual.
Systematic errors arise inevitably when astrometric data are derived from the photometry of mutual occultations and eclipses of planetary satellites. Most of them are due to factors that are unrelated to the methods for deriving astrometric data. These problems come when we use our new method as well as using other methods. Let us consider the most important sources of the systematic errors. During the observation the detector is illuminated in addition to the satellite light by the sky background and the light scattered by the parts of the instrument. Moreover, the photometric count also includes the signal produced by the detector itself. Hereafter, we refer to the part of the photometric count that is not due to the satellite flux as the background, and denote it as P. In such a case, we have to solve the conditional equations E i ¼ K SðX th ðti Þ þ Dx ; Y th ðti Þ þ Dy Þ þ P
ði ¼ 1; 2; . . . ; mÞ
(3) where the parameter P as well as the parameters Dx , Dy , and K are unknown. Our analysis leads us to the following two important conclusions. First, joint least-square determination of four parameters Dx , Dy , K, and P may result in highly correlated parameter errors, and the differences between
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Fig. 4. The results of photometric observation of the mutual occultation of Io by allisto on February 3, 2003 made at the observatories Kazakstan (K), Cluj-Napoca (C) and Pulkovo (P) and the fit of the theoretical light curves.
the derived parameters and their real values may exceed the random errors substantially. That is why it is better not to treat parameter P (background) as an unknown, but eliminate the effect of the background on the result of measurement beforehand and set P ¼ 0. Second, if we set P ¼ 0 while incorrectly eliminating background so that this parameter actually differs from zero, an unaccounted systematic error appears in parameter Dy . To illustrate these circumstances we have several examples. Let us consider one of them. The mutual occultation of Io by Callisto on February 3, 2003 was observed at three observatories: Kazakstan, Cluj-Napoca, and Pulkovo (Devyatkin). The results of observations as well as results of the fit of the theoretical light curves are shown in Fig. 4. Significant differences in flux drop between the results of three observatories are evident. The only explication is that the effects of the background on the result of measurement were not eliminated correctly. As a consequence the astrometric results differ significantly also. This is shown in Table 1. The systematic errors may be estimated equal to 0.5 arsec. Our analysis also leads us to conclude that, in the case of mutual occultations of planetary satellites, the flux drop is determined to a considerable degree by the ratio of the mean albedos of the two satellites. Small errors in the adopted albedos translate into errors in the derived coordinate differences of the satellites. If a uniform scattering law is assumed the value of SðtÞ can be approximated by this equation SðtÞ ¼
1 þ ðp2 r22 =p1 r21 Þk2 ðdðtÞÞ 1 þ ðp2 r22 =p1 r21 Þ
(4)
where r1 and r2 are the radius of the occulting and occulted satellites, and p1 and p2 , their geometric albedos. Function
Table 1 Corrections to the theoretical apparent position of Callisto relative to Io during its mutual occultation obtained from the photometric observations made at three observatories in February 03, 2003 Observatory
Dx (km)
Dy (km)
D00x (arcsec)
D00y (arcsec)
Kazakstan Cluj-Napoca Pulkovo (Devyatkin)
158.3 245.2 147.8
143.5 391.3 1621.7
0.050 0.078 0.047
0.046 0.125 0.518
k2 ðdðtÞÞ is a fraction of unocculted part of the disk of the occulted satellite. The function k2 ðdðtÞÞ depends on the distance between the apparent centers of satellites dðtÞ and k2 ðdðtÞÞ ¼ 1 outside the event. As dðtÞ depends on the satellites coordinates we have from the same observation different astrometric results for different assumed values of the ratio of the mean albedos of the two satellites. Some mutual event observations fall to be inconsistent with assumed values of the mean albedos. Let us consider some examples. The occultation of Europa by Callisto on March 9, 2003 was observed at Terskol and Pulkovo. The theoretical light curves were fitted to the observations using Hapke law (Hapke, 1981) with the parameters from Domingue and Verbiscer (1997) as it is shown in Fig. 5. For both satellites we have realized the three parameter function (3P-HG) and taken the set of parameters from Domingue and Verbiscer (1997) for 0:55 mm spectral band as the observations were made with V filter. During the event in question almost a half of leading and a half of trailing hemisphere of each satellite were visible from the Earth. Thus the mean values of the corresponding parameters were taken in consideration. It is obvious that the ratio p2 =p1 of the albedos of the two satellites is not assumed correctly. In Table 2, we collected the values of
ARTICLE IN PRESS N. Emelyanov / Planetary and Space Science 56 (2008) 1785–1790
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Source
p2 =p1
Hapke law with the parameters from Domingue and Verbiscer (1997) From Morrison and Morrison (1977) Photometric observation of the event 2003.03.09 4o2 at Pulkovo (Devyatkin) Photometric observation of the event 2003.03.09 4o2 at Terskol
2:67
accurate enough. In the cases where the apparent disk of one of the satellites overlaps fully with that of the other satellite, the solution becomes uncertain in one of the relative coordinates. The above effects also occur in the case of the mutual eclipses of satellites. We have several mutual phenomena with these circumstances. The brightness of each point on the disk can be computed using the appropriate scattering law and an available satellite map. It is thus possible to compute the mean brightness of the disk as a function of the angle of rotation of the satellite. On the other hand, this function can be determined from the ground-based photometry of a particular satellite viewed as a point light source. The two sources mentioned above were compared to each other to test the reliability of the data (Emelyanov and Gilbert, 2006). The modeled and observed dependences for Io and Europa differ insignificantly. However, the differences for Ganymede and Callisto are greater. The available satellite maps do not permit yet confident allowance for the variation of reflective properties over the satellite surface. These maps are actually intended for other purposes, and they describe the brightness distribution across the surface only approximately.
4:43 X3:74
6. Some results of the reduction of observations
X3:29
In spite of the problems mentioned above we have attempted to reduce the results of photometric observations of the Gallilean satellites during their mutual occultations and eclipses in 2002–2003 performed at various observatories worldwide. At this first step, the variation of reflective properties over the satellite surface was not taken into account. The data of observation for 319 light curves were received from the observers. Only 245 light curves proved to be reliable. They were processed with our method. A special processing is needed for 44 light curves and they remain to be processed. The data of 30 light curves cannot be processed for various reasons: poor data received (10), no event detected (7), errors revealed in the data (6), insufficient explications on the data (7). Relative apparent satellite positions obtained for 41 time instants from the observations of mutual eclipses and for 45 time instants from the observations of mutual occultations. In total, relative apparent satellite positions obtained for 86 time instants. In order to evaluate the obtained astrometric results we used following estimates. The least-squares method gives us precision estimates for the parameters Dx , Dy derived from the observed light curves. These estimates follow the random errors of the photometry and present an internal precision of the astrometric results. R.m.s. values of these estimates for all the processed light curves are given in Table 3 as total random errors. Furthermore for each event observed at N (N41) observatories simultaneously we calculated mean values of Dx , Dy designated as Dx , Dy . Then the differences
Fig. 5. The occultation of Europa by Callisto on March 9, 2003 observed at Terskol and Pulkovo and the fit of the theoretical light curves.
Table 2 The ratio p2 =p1 of the albedos of the two satellites from different sources
p1 is the albedo of Callisto and p2 is of Europa one.
r ¼ p2 =p1 from different sources. In this event, p1 is the albedo of Callisto and p2 is of Europa one. Using (4) and the function k2 ðdðtÞÞ we concluded that changing p2 =p1 from 2.67 to 3.74 may change the differences ðJ4 J2Þ in the topocentric coordinates derived from the photometry of the mutual occultations by 0.15 arcsec. These circumstances cause significant systematic errors in the astrometric results derived from the photometric observations of mutual events of natural satellites. Considering r ¼ p2 =p1 as a free parameter to be determined from an observed light curve was proposed by Vasundhara et al. (2003). These authors have mentioned that r should absorb the uncertainty in the sky measurements. Our attempts considering r as a free parameter resulted in highly correlated parameter errors between Dy and r. Another source of possible error is that any observed light curve of the satellites has two solutions for the corresponding differences of apparent coordinates of the satellites. The occulting satellite may pass above or below the occulted satellite at the same distance from the latter producing the same dips in the total flux. Of the two solutions we may choose the one that agrees better with the available theory of motion. However, this approach may result in selecting the wrong solution if the theory is not
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Table 3 Precision evaluation of the obtained astrometric results. Type of estimates
In X 00 (arcsec)
In Y 00 (arcsec)
Total random errors Total systematic errors Total r.m.s. of O–C
0.039 0.058 0.097
0.027 0.085 0.124
Dx Dx , Dy Dy were calculated for each observatory for a given event. Finally we calculated r.m.s. of all these differences through all events and all observatories. We consider these residuals as systematic errors arising from incorrectly eliminated background in the photometric results and from wrong values of satellites albedos used for processing the mutual occultation observations. These estimates are designated in Table 3 as total systematic errors. The corrections Dx and Dy remain constant during the event and characterize the discrepancy between the theory and observations. These values obtained from observations made at different observatories can be compared to each other. The resulting discrepancies are due to observational biases. Total r.m.s. of all Dx and Dy calculated through all events and all observatories are designated in Table 3 as total r.m.s. of O–C. We plan in the future to reduce all the data collected in IMCCE. The final astrometric results will be published. 7. Conclusions As of now, the differences of topocentric or heliocentric coordinates of satellites pairs were derived for 86 time instants from a series of photometric observations of the Galilean satellites made during their mutual occultations and eclipses in 2002–2003. Preliminary analysis shows that the systematic errors of the astrometric results have about the same magnitude as the effects of neglecting a convenient scattering law and the nonuniformity of the reflective properties of the satellite surfaces. The systematic errors of the photometry of satellites during their mutual occultations and eclipses are most likely due to incorrect exclusion of the effect of background (P) on photometric counts.
The available satellite maps do not permit yet to make confident allowance for the variation of reflective properties over the satellite surface.
Acknowledgment This work was supported by the Russian Foundation for Basic Research (Project no. 06-02-16966-a).
References Aksnes, K., Franklin, F., 1976. Mutual phenomena of the Galilean satellites in 1973. III—Final results from 91 light curves. Astron. J. 81, 464–481. Aksnes, K., Franklin, F., Millis, R., Birch, P., Blanco, C., Catalano, S., Piironen, J., 1984. Mutual phenomena of the Galilean and Saturnian satellites in 1973 and 1979/1980. Astron. J. 89, 280–288. Domingue, D., Verbiscer, A., 1997. Re-analysis of the solar phase curves of the icy galilean satellites. Icarus 128, 49–74. Emelianov, N.V., 2003. A method for reducing photometric observations of mutual occultations and eclipses of planetary satellites. Solar Syst. Res. 37, 314–325. Emel’yanov, N.V., 1995. Features of mutual occultations and eclipses in the system of Saturn’s satellites. Astron. Rep. 39, 539–542. Emelyanov, N.V., Gilbert, R., 2006. Astrometric results of observations of mutual occultations and eclipses of the Galilean satellites of Jupiter in 2003. Astron. Astrophys. 453, 1141–1149. Franklin, F.A., Africano, J., Allen, W., Aksnes, K., Birch, P., 1991. An analysis of the 1985 observations of mutual phenomena of the Galilean satellites. Astron. J. 102, 806–815. Hapke, B., 1981. Bidirectional reflectance spectroscopy. 1. Theory. J. Geophys. Res. 86, 3039–3054. Kaas, A.A., Aksnes, K., Franklin, F., Lieske, J., 1999. Astrometry from mutual phenomena of the galilean satellites in 1990–1992. Astron. J. 117, 1933–1941. Lainey, V., Arlot, J.E., Vienne, A., 2004. New accurate ephemerides for the Galilean satellites of Jupiter. II. Fitting the observations. Astron. Astrophys. 427, 371–376. Morrison, D., Morrison, N.D., 1977. Photometry of the Gallilean satellites. In: Planetary Satellites. University of Arizona Press, Tucson, p. 363. Noyelles, B., Vienne, A., Descamps, P., 2003. Astrometric reduction of lightcurves observed during the PHESAT95 campaign of Saturnian satellites. Astron. Astropyhs. 401, 1159–1175. Vasundhara, R., 2002. Astrometry from CCD photometry of mutual events of Jovian satellites from VBO during 1997. Astron. Astropyhs. 389, 325–333. Vasundhara, R., Arlot, J.-E., Lainey, V., Thuillot, W., 2003. Astrometry from mutual events of Jovian satellites in 1997. Astron. Astrophys. 410, 337–341.
Planetary and Space Science 56 (2008) 1785–1790 www.elsevier.com/locate/pss
Astrometric results of observations of mutual occultations and eclipses of the Galilean satellites of Jupiter in 2002–2003 Nikolay Emelyanova,b, a
Sternberg Astronomical Institute, Moscow, Russia Institut de me´canique ce´leste et de calcul des e´phe´me´rides – Observatoire de Paris, UMR 8028 du CNRS, Paris, France
b
Accepted 5 February 2008 Available online 22 July 2008
Abstract We suggest a new approach and develop an original method for deriving astrometric data from the photometry of mutual occultations and eclipses of planetary satellites. We decide to model not the relative apparent motion of one satellite with respect to another satellite but the deflection of the observed relative motion with respect to the theoretical motion implied by appropriate ephemerides. We have attempted to reduce the results of photometric observations of the Gallilean satellites during their mutual occultations and eclipses in 2002–2003. The data of observation for 319 light curves of 106 mutual events were received from the observers. The reliable 245 light curves were processed with our method. Eighty six apparent relative positions have been obtained. Systematic errors arise inevitably while deriving astrometric data. Most of them are due to factors that are unrelated to the methods for deriving astrometric data. The systematic errors are more likely due to incorrect excluding the effect of background on photometric counts. In the case of mutual occultations, the flux drop is determined to a considerable degree by the ratio of the mean albedos of the two satellites. Some mutual event observations revealed wrong adopted values of the mean albedos. r 2008 Elsevier Ltd. All rights reserved. Keywords: Occultations; Eclipses; Planets and satellites; Jupiter satellites
1. Introduction The photometry of mutual occultations and eclipses of natural planetary satellites can be used to derive very accurate astrometric data. This can be achieved by analyzing the light curves of the satellites observed during these mutual events. The final goal of observations is to refine the models of motion for the natural satellites. We propose the most accurate photometric model of mutual events to date based on all available data about the satellites, and developed the corresponding method for extracting astrometric data. We use our new method to reduce the series of observations of mutual occultations and eclipses of the Gallilean satellites in 2002–2003 made at different observatories. Corresponding author at: Institut de me´canique ce´leste et de calcul des e´phe´me´rides – Observatoire de Paris, UMR 8028 du CNRS, Paris, France. E-mail address: [email protected]
0032-0633/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2008.02.017
The purpose of this paper is to analyze the most important sources of the systematic errors. 2. Observations Many mutual events involving the Galilean satellites of Jupiter have been observed. In 2002–2003, the Institut de Mecanique Celeste et de Calcul des Ephemerides (IMCCE) organized and coordinated an international campaign to observe these rare events. The goal was to observe as many events as possible. Several independent observations of each event are needed to eliminate observational errors. This campaign allowed the international network of 37 sites to collect 319 lightcurves of 106 mutual events. Only 23 events were simultaneously observed from at least five observatories and 47 events were observed from only one site of the network. Light curves were derived from relative photometry. We preferred to use for our reduction the raw data provided by
ARTICLE IN PRESS 1786
N. Emelyanov / Planetary and Space Science 56 (2008) 1785–1790
IMCCE. Thus no light curve had been pre-processed or altered by anyone before our treatment. 3. Extracting astrometric data from the photometry of mutual events Observations of both mutual occultations and eclipses of natural planetary satellites consist of satellite flux measurements during a single event. Extracting astrometric data from such photometry is an important but difficult task. The methods for solving this problem are now at new stage of development. Several groups of authors have been working extensively on the development of such methods. Aksnes and Franklin (1976), Aksnes et al. (1984), Franklin et al. (1991), and Kaas et al. (1999) developed a method for systematically deriving and publishing the differences in the topocentric coordinates of the Galilean satellites based on the photometry of their mutual occultations and eclipses. This method was significantly improved by Vasundhara (2002), Vasundhara et al. (2003), and Noyelles et al. (2003). In some cases the apparent motion of one satellite relative to another is very complex. A very accurate geometric and photometric model of mutual occultations and eclipses of satellites is much needed. 4. Method for deriving astrometric data In our papers (Emel’yanov, 1995; Emelianov, 2003) a new approach was suggested and an original method for deriving astrometric data from the photometry of mutual occultations and eclipses of planetary satellites was developed. The main idea is not to model the apparent relative motion of one satellite with respect to the other one but instead the departure of the observed relative motion from the theoretical motion provided by relevant ephemeris. The measured flux EðtÞ during an event at a given time t may be expressed by EðtÞ ¼ K SðX ðtÞ; Y ðtÞÞ
(1)
where X ðtÞ and Y ðtÞ are the projections of the differences of planetocentric Cartesian coordinates of the two satellites onto the tangent plane of the event. In the case of mutual occultations, this plane coincides with the plane passing through the occulted satellite perpendicular to the line of sight of the observer. In the case of a mutual eclipse, the plane of the event passes through the eclipsed satellite perpendicular to the line connecting the satellite with the center of the Sun. The origin of coordinates is placed at the center of the passive (occulted or eclipsed) satellite. The occulting or eclipsing satellite is referred to as the active satellite. The function Sðx; yÞ describes a model of the phenomenon. It is supposed Sðx; yÞ ¼ 1 off event. The parameter K is a scale factor for the light drop during the event and it is equal to the total flux outside the event.
Given appropriate theories of the motion of planets and satellites, one can compute the theoretical values of functions X ðtÞ, Y ðtÞ, i.e., X th ðti Þ, Y th ðti Þ for the time ti ði ¼ 1; 2; . . . ; mÞ of each photometric measurement. Here m is the number of photometric counts during a single event. The real values of X ðti Þ and Y ðti Þ differ from X th ðti Þ and Y th ðti Þ by corrections Dx , Dy . Our proposed method consists of solving conditional equations E i ¼ K SðX th ðti Þ þ Dx ; Y th ðti Þ þ Dy Þ ði ¼ 1; 2; . . . ; mÞ (2) for parameters Dx , Dy , and K. Here E i is the photometric recorded at time ti . We linearize conditional equations with respect to parameters Dx , Dy and then solve them using the least-square method. We computed the theoretical values X th ðti Þ and Y th ðti Þ and corrections Dx and Dy using the theory of Galilean satellite motion developed by Lainey et al. (2004). The parameter K is determined from all the photometric measurements before, after and during the event. This fact makes it possible to use observations when there is no measurement before and after the event. We have some examples of such observations. In these cases the parameters Dx , Dy , and K were successfully determined from observations. The function Sðx; yÞ is calculated as an integral of the flux from each point of satellite over the hemisphere facing the Earth. For each point we consider wavelengthdependent reflective properties of the satellites, various laws of light scattering by a rough surface, variation of reflective properties over the satellite surface, wavelengthdependent solar limb darkening. We consider also a wavelength-dependent sensitivity of the detector. Given the topocentric (or heliocentric in the case of mutual eclipse) distances of the active and passive satellites, one can compute the corresponding projections X 00 and Y 00 of the angular separation between the satellites onto the celestial parallel and meridian, respectively. In a similar way we designated D00x , D00y , the angular values, corresponding to the corrections Dx , Dy . Let us consider an example. We have received photometric observations of the occultation of Io by Europa made on December 20, 2002 at Cluj-Napoca observatory. Fig. 1 shows scaled values of the combined flux of the occulted and occulting satellites (circles). Using the theory of the Galilean satellites by Lainey we calculated the apparent motion of Europa with respect to Io. It is evident that this event cannot be represented by rectilinear and uniform apparent motion of one satellite relative to another as it is plotted in Fig. 2. Then using our method we fitted of the parameters K, Dx , and Dy to that observation. The resultant light curve is shown in Fig. 1 (line). By this means corrections to the theoretical relative apparent motion of the satellites were derived (see Fig. 3).
ARTICLE IN PRESS N. Emelyanov / Planetary and Space Science 56 (2008) 1785–1790
Fig. 1. Photometric observations of the occultation of Io by Europa made December 20, 2002 at Cluj-Napoca observatory (circles) and the fit of the parameters K, Dx , and Dy to these observations (points).
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Fig. 3. The derived corrections to the theoretical relative apparent motion of the satellites.
Second, to determine the value of K one can use photometric measurements made before, after and during the event. This fact makes it possible to use observations when there is no measurement before and after the event. We have some examples of such observations. 5. Persistent issues
Fig. 2. The apparent motion of Europa with respect to Io as calculated using the theory of the Galilean satellites by Lainey.
As an astrometric result of the observation one may take any corrected relative position of satellites X ðt Þ ¼ X th ðt Þ þ Dx ; Y ðt Þ ¼ Y th ðt Þ þ Dy together with associated time t inside the interval of the event. This method has the following advantages: First, the deflection of the observed relative motion of satellites with respect to the theoretical motion can be described by two parameters Dx , Dy in most cases. There is no need for an impact parameter and the method may be applied to arbitrary apparent relative satellite motion even somewhat unusual.
Systematic errors arise inevitably when astrometric data are derived from the photometry of mutual occultations and eclipses of planetary satellites. Most of them are due to factors that are unrelated to the methods for deriving astrometric data. These problems come when we use our new method as well as using other methods. Let us consider the most important sources of the systematic errors. During the observation the detector is illuminated in addition to the satellite light by the sky background and the light scattered by the parts of the instrument. Moreover, the photometric count also includes the signal produced by the detector itself. Hereafter, we refer to the part of the photometric count that is not due to the satellite flux as the background, and denote it as P. In such a case, we have to solve the conditional equations E i ¼ K SðX th ðti Þ þ Dx ; Y th ðti Þ þ Dy Þ þ P
ði ¼ 1; 2; . . . ; mÞ
(3) where the parameter P as well as the parameters Dx , Dy , and K are unknown. Our analysis leads us to the following two important conclusions. First, joint least-square determination of four parameters Dx , Dy , K, and P may result in highly correlated parameter errors, and the differences between
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Fig. 4. The results of photometric observation of the mutual occultation of Io by allisto on February 3, 2003 made at the observatories Kazakstan (K), Cluj-Napoca (C) and Pulkovo (P) and the fit of the theoretical light curves.
the derived parameters and their real values may exceed the random errors substantially. That is why it is better not to treat parameter P (background) as an unknown, but eliminate the effect of the background on the result of measurement beforehand and set P ¼ 0. Second, if we set P ¼ 0 while incorrectly eliminating background so that this parameter actually differs from zero, an unaccounted systematic error appears in parameter Dy . To illustrate these circumstances we have several examples. Let us consider one of them. The mutual occultation of Io by Callisto on February 3, 2003 was observed at three observatories: Kazakstan, Cluj-Napoca, and Pulkovo (Devyatkin). The results of observations as well as results of the fit of the theoretical light curves are shown in Fig. 4. Significant differences in flux drop between the results of three observatories are evident. The only explication is that the effects of the background on the result of measurement were not eliminated correctly. As a consequence the astrometric results differ significantly also. This is shown in Table 1. The systematic errors may be estimated equal to 0.5 arsec. Our analysis also leads us to conclude that, in the case of mutual occultations of planetary satellites, the flux drop is determined to a considerable degree by the ratio of the mean albedos of the two satellites. Small errors in the adopted albedos translate into errors in the derived coordinate differences of the satellites. If a uniform scattering law is assumed the value of SðtÞ can be approximated by this equation SðtÞ ¼
1 þ ðp2 r22 =p1 r21 Þk2 ðdðtÞÞ 1 þ ðp2 r22 =p1 r21 Þ
(4)
where r1 and r2 are the radius of the occulting and occulted satellites, and p1 and p2 , their geometric albedos. Function
Table 1 Corrections to the theoretical apparent position of Callisto relative to Io during its mutual occultation obtained from the photometric observations made at three observatories in February 03, 2003 Observatory
Dx (km)
Dy (km)
D00x (arcsec)
D00y (arcsec)
Kazakstan Cluj-Napoca Pulkovo (Devyatkin)
158.3 245.2 147.8
143.5 391.3 1621.7
0.050 0.078 0.047
0.046 0.125 0.518
k2 ðdðtÞÞ is a fraction of unocculted part of the disk of the occulted satellite. The function k2 ðdðtÞÞ depends on the distance between the apparent centers of satellites dðtÞ and k2 ðdðtÞÞ ¼ 1 outside the event. As dðtÞ depends on the satellites coordinates we have from the same observation different astrometric results for different assumed values of the ratio of the mean albedos of the two satellites. Some mutual event observations fall to be inconsistent with assumed values of the mean albedos. Let us consider some examples. The occultation of Europa by Callisto on March 9, 2003 was observed at Terskol and Pulkovo. The theoretical light curves were fitted to the observations using Hapke law (Hapke, 1981) with the parameters from Domingue and Verbiscer (1997) as it is shown in Fig. 5. For both satellites we have realized the three parameter function (3P-HG) and taken the set of parameters from Domingue and Verbiscer (1997) for 0:55 mm spectral band as the observations were made with V filter. During the event in question almost a half of leading and a half of trailing hemisphere of each satellite were visible from the Earth. Thus the mean values of the corresponding parameters were taken in consideration. It is obvious that the ratio p2 =p1 of the albedos of the two satellites is not assumed correctly. In Table 2, we collected the values of
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Source
p2 =p1
Hapke law with the parameters from Domingue and Verbiscer (1997) From Morrison and Morrison (1977) Photometric observation of the event 2003.03.09 4o2 at Pulkovo (Devyatkin) Photometric observation of the event 2003.03.09 4o2 at Terskol
2:67
accurate enough. In the cases where the apparent disk of one of the satellites overlaps fully with that of the other satellite, the solution becomes uncertain in one of the relative coordinates. The above effects also occur in the case of the mutual eclipses of satellites. We have several mutual phenomena with these circumstances. The brightness of each point on the disk can be computed using the appropriate scattering law and an available satellite map. It is thus possible to compute the mean brightness of the disk as a function of the angle of rotation of the satellite. On the other hand, this function can be determined from the ground-based photometry of a particular satellite viewed as a point light source. The two sources mentioned above were compared to each other to test the reliability of the data (Emelyanov and Gilbert, 2006). The modeled and observed dependences for Io and Europa differ insignificantly. However, the differences for Ganymede and Callisto are greater. The available satellite maps do not permit yet confident allowance for the variation of reflective properties over the satellite surface. These maps are actually intended for other purposes, and they describe the brightness distribution across the surface only approximately.
4:43 X3:74
6. Some results of the reduction of observations
X3:29
In spite of the problems mentioned above we have attempted to reduce the results of photometric observations of the Gallilean satellites during their mutual occultations and eclipses in 2002–2003 performed at various observatories worldwide. At this first step, the variation of reflective properties over the satellite surface was not taken into account. The data of observation for 319 light curves were received from the observers. Only 245 light curves proved to be reliable. They were processed with our method. A special processing is needed for 44 light curves and they remain to be processed. The data of 30 light curves cannot be processed for various reasons: poor data received (10), no event detected (7), errors revealed in the data (6), insufficient explications on the data (7). Relative apparent satellite positions obtained for 41 time instants from the observations of mutual eclipses and for 45 time instants from the observations of mutual occultations. In total, relative apparent satellite positions obtained for 86 time instants. In order to evaluate the obtained astrometric results we used following estimates. The least-squares method gives us precision estimates for the parameters Dx , Dy derived from the observed light curves. These estimates follow the random errors of the photometry and present an internal precision of the astrometric results. R.m.s. values of these estimates for all the processed light curves are given in Table 3 as total random errors. Furthermore for each event observed at N (N41) observatories simultaneously we calculated mean values of Dx , Dy designated as Dx , Dy . Then the differences
Fig. 5. The occultation of Europa by Callisto on March 9, 2003 observed at Terskol and Pulkovo and the fit of the theoretical light curves.
Table 2 The ratio p2 =p1 of the albedos of the two satellites from different sources
p1 is the albedo of Callisto and p2 is of Europa one.
r ¼ p2 =p1 from different sources. In this event, p1 is the albedo of Callisto and p2 is of Europa one. Using (4) and the function k2 ðdðtÞÞ we concluded that changing p2 =p1 from 2.67 to 3.74 may change the differences ðJ4 J2Þ in the topocentric coordinates derived from the photometry of the mutual occultations by 0.15 arcsec. These circumstances cause significant systematic errors in the astrometric results derived from the photometric observations of mutual events of natural satellites. Considering r ¼ p2 =p1 as a free parameter to be determined from an observed light curve was proposed by Vasundhara et al. (2003). These authors have mentioned that r should absorb the uncertainty in the sky measurements. Our attempts considering r as a free parameter resulted in highly correlated parameter errors between Dy and r. Another source of possible error is that any observed light curve of the satellites has two solutions for the corresponding differences of apparent coordinates of the satellites. The occulting satellite may pass above or below the occulted satellite at the same distance from the latter producing the same dips in the total flux. Of the two solutions we may choose the one that agrees better with the available theory of motion. However, this approach may result in selecting the wrong solution if the theory is not
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Table 3 Precision evaluation of the obtained astrometric results. Type of estimates
In X 00 (arcsec)
In Y 00 (arcsec)
Total random errors Total systematic errors Total r.m.s. of O–C
0.039 0.058 0.097
0.027 0.085 0.124
Dx Dx , Dy Dy were calculated for each observatory for a given event. Finally we calculated r.m.s. of all these differences through all events and all observatories. We consider these residuals as systematic errors arising from incorrectly eliminated background in the photometric results and from wrong values of satellites albedos used for processing the mutual occultation observations. These estimates are designated in Table 3 as total systematic errors. The corrections Dx and Dy remain constant during the event and characterize the discrepancy between the theory and observations. These values obtained from observations made at different observatories can be compared to each other. The resulting discrepancies are due to observational biases. Total r.m.s. of all Dx and Dy calculated through all events and all observatories are designated in Table 3 as total r.m.s. of O–C. We plan in the future to reduce all the data collected in IMCCE. The final astrometric results will be published. 7. Conclusions As of now, the differences of topocentric or heliocentric coordinates of satellites pairs were derived for 86 time instants from a series of photometric observations of the Galilean satellites made during their mutual occultations and eclipses in 2002–2003. Preliminary analysis shows that the systematic errors of the astrometric results have about the same magnitude as the effects of neglecting a convenient scattering law and the nonuniformity of the reflective properties of the satellite surfaces. The systematic errors of the photometry of satellites during their mutual occultations and eclipses are most likely due to incorrect exclusion of the effect of background (P) on photometric counts.
The available satellite maps do not permit yet to make confident allowance for the variation of reflective properties over the satellite surface.
Acknowledgment This work was supported by the Russian Foundation for Basic Research (Project no. 06-02-16966-a).
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