Astrometric results of the mutual events between the Saturn's satellites observed at Yunnan Observatory

Planetary and Space Science 76 (2013) 83–86

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Astrometric results of the mutual events between the Saturn’s satellites observed at Yunnan Observatory Xi-liang Zhang a,b,n, Zhong Liu a, Zheng-hong Tang b a b

National Astronomical Observatories/Yunnan Observatory, Joint Laboratory for Optical Astronomy, Chinese Academy of Sciences, Kunming, PR China Shanghai Observatory, Chinese Academy of Sciences, Shanghai, PR China

a r t i c l e i n f o

abstract

Article history: Received 31 October 2012 Received in revised form 7 December 2012 Accepted 12 December 2012 Available online 22 December 2012

The mutual eclipse of S4 Dione by S3 Tethys (Dec 23, 2009) and mutual occultation of S3 Tethys by S4 Dione (Apr 24, 2010) were observed at Yunnan Observatory during the international campaign in 2009–2010. The aim of this paper is to calculate the astrometric data of the involved Saturnian satellites during the mutual events, using a same dynamical model as proposed by Assafin et al. (2009) for the mutual occultation and a same one as proposed by Zhang and Liu (2011) for the mutual eclipse, and taking the scattering properties of the Saturnian satellites surfaces into account. Compared with the theory TASS 1.7 (Vienne and Duriez, 1995), the O–C(x) and O–C(y) are 10.1 mas, 3.4 mas for the mutual eclipse S4 by S3 and 28.4 mas,  1.3 mas for the mutual occultation S3 by S4, and with respect to the theory by Lainey et al. (2011), the results are 1.0 mas,  5.0 mas and  0.2 mas, 2.5 mas respectively. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Mutual events Saturnian satellites Dynamics

1. Introduction Photometric observations of the mutual occultation and eclipse between the Saturnian satellites are more rare and difficult than that of the Galilean satellites to obtain, because of the farther distance and smaller diameter of the Saturnian satellites, but it also can provide very accurate and invaluable astrometric data for the study of the dynamics of the Saturnian satellites. The photometry of the mutual events between the natural satellites has been verified to be a most effective and accurate ground-based means of obtaining astrometric data of the natural satellites. The Sun and the Earth traverse the common orbital plane of the Saturnian satellites, that is the equatorial plane of the Saturn, every 15 years, so the observable period of the mutual events of the Saturnian satellites is 15 years. In 2009–2010, two mutual occultations and two mutual eclipses between the Saturnian satellites that had been predicted by Arlot and Thuillot (2008), were observed at Yunnan Observatory, but only two light curves were got. The aim of this paper is to calculate several physical and dynamical quantities of the Saturnian satellites by fitting the light curves we observed. In the following sections, we provide the detailed descriptions of the

n Corresponding author at: National Astronomical Observatories/Yunnan Observatory, Joint Laboratory for Optical Astronomy, Chinese Academy of Sciences, Kunming, PR China. Tel.: þ86 871 3920040. E-mail address: [email protected] (X.-l. Zhang).

0032-0633/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.pss.2012.12.004

observations, reductions, analysis and fitting of the light curves, and the astrometric results of these two mutual events.

2. CCD photometric observations 2.1. Observation We used the 1-m and 60-cm telescopes at Yunnan Observatory (1021470 :3E, 25110 :5N, altitude¼2000 m, IAU code 286), with the attached DW436 2048  2048 CCDs, to observe the mutual eclipse and the mutual occultation of the Saturnian satellites on Dec 23, 2009 and Apr 24, 2010 respectively. The effective field of view of the 1-m telescope at the Cassegrain focus is about 7  7 square arc min, and that of the 60-cm telescope is 12  12 square arc min. During the observations, a Johnson R filter was used. The frequency of the readout we chose is 1 ms=pixel, and then one data point was got for almost every 7–8 s. Each data point corresponds to one flux information of the satellites involved. Table 1 shows the properties of both the 1-m telescope and the 60-cm telescope at Yunnan Observatory. 2.2. Reduction During processing the images, each image was bias-corrected and flat-fielded, and then the curves of flux variations relative to UT time were obtained for each event. Each image observed during the mutual events corresponds to one flux data point of the satellites involved. Table 2 shows the details of our observations and reductions, where S3, S4, and S6 are the Saturnian

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Table 1 Properties of the 1-m and 60-cm telescopes at Yunnan Observatory, with the attached CCDs. Telescope F-length (mm)

CCD FOV Size of pixel

1-m 60-cm

13:5 mm  13:5 mm 2048  2048 000 :21 70  70 120  120 13:5 mm  13:5 mm 2048  2048 000 :35

13 000 7500

Size of CCD

Size/ pixel

Fig. 2. Geometry of a mutual eclipse of S2 by S1 with the radius R1 o R2 .

Table 2 Observational details of the mutual events observed at Yunnan Observatory in 2009 and 2010. Date ymd

Type

Ref

Telescope

Exposure (s)

2009 12 23 2010 04 24

S3eS4 S4oS3

S6 S6

60-cm 1-m

1 1.5

Fig. 3. Dynamical model of the mutual event corresponding to Fig. 1.

Fig. 1. Geometry of a partial mutual event where the disks S1 and S2 of radius R2 o R1 partially intercept each other with an area A, S1 and S2 representing the two Saturnian satellites involved.

satellites Tethys, Dione, and Titan, respectively, ‘S4oS3’ means the event of Dione occults Tethys, ‘S3eS4’ means the event of Tethys eclipses Dione. ‘Ref’ indicates the satellite used as a reference for the photometry, which in this paper is Titan for the events between Tethys and Dione. ‘Telescope’ indicates the apertures of the two different telescopes used whose complete descriptions are provided in Table 1. ‘Exposure’ gives the exposure time of each event.

value of albedo ratio of the occulting to the occulted satellites is a constant during fitting the mutual occultation because of their short duration, the corresponding formula used to perform the fitting of the light curve is given as below. Eq. (1) shows the expression of flux variation of the satellites involved, which has been normalized to 1 before and after mutual occultation, where F 1o2 represents the flux of the satellites involved during the mutual occultation, F 1 þ 2 , that of before and after the mutual event, and R1, k1 and R2, k2, the radius and albedos of S1 and S2, respectively, such that

F occ ¼

1 1 R21 ða1  sin 2a1 Þ þ R22 ða2  sin 2a2 Þ F 1o2 2 2 ¼ 1 k1 2 F1 þ 2 2 pR þ pR2 k2 1

where, 2

cos ai ¼ 2

R2i R2j þ d 2Ri d

2

d ¼ d0 þv2 ðtt 0 Þ2 , 3. Analysis and adjustment As mentioned above, mutual events depend on the relative positions of the Sun, satellites, and the Earth (observer). Figs. 1 and 2 show the geometrical projection of the two satellites involved during the mutual events, as seen from the center of the Earth for the mutual occultation and seen from the center of the Sun for the mutual eclipse respectively. 3.1. Dynamical model 3.1.1. Modeling the occultation The model proposed by Assafin et al. (2009) is used to fit the observed light curve of the mutual occultation, with an assumption being made, what is that the occulted satellite is assumed to be unmovable, while the occulting one has a linear uniform motion relative to the occulted one (as shown in Fig. 3). During a mutual occultation, the flux variation of the satellites involved is mainly determined by the occulted area A (as shown in Fig. 1) of S2 by S1 and the different surface properties of them. In spite of the surfaces of satellites being nonuniform, when the

ð1Þ

ð2Þ ð3Þ

i¼1 or 2 and j ¼2 or 1. Eq. (3) corresponds to Fig. 3, in which d and d0 are the relative distance and the impact parameter, v is the velocity of the occulting/eclipsing satellite from to the occulted/eclipsed one, t and t0 are the time of observation and the mid-time respectively. 3.1.2. Modeling the eclipse Modeling a mutual eclipse is more complicated than that of a mutual occultation because of the existence of the penumbra zone, caused by the limb darkening effect of the Sun. The same model and formulas as introduced by Zhang and Liu (2011) were adopted to fit the mutual eclipse of Dione by Tethys. Fig. 4 shows the dynamical model corresponding to Fig. 2. 3.2. Photometric model In order to get higher accuracy astrometric data of the Saturnian satellites, the effect of the scattering properties of the surfaces of the Saturnian satellites is considered during the fitting of the light curves, so following photometric function (shown as Eq. (4)) is proposed, in this paper, to calculate the quantities of the

X.-L. Zhang et al. / Planetary and Space Science 76 (2013) 83–86

impact parameters and mid-times.

q¼

I m0 ¼A f ðaÞ þð1AÞm0 F m þ m0

ð4Þ

p¼

D¼

FðaÞpq

ð5Þ

D

2 Af ð0Þ ð1AÞ þ 3 2

ð6Þ

Fig. 4. Dynamical model of the mutual eclipse corresponding to Fig. 2.

Table 3 Values of the photometric parameters of the Saturnian satellites S3 Tethys and S4 Dione. Satellite

A

f(0)

b

S3Tethys S4Dione

0.7 1.0

1.45 1.0

0.016 0.023

2 ð1AÞ½sin a þðpaÞ cos a 3p

ð7Þ

FðaÞ ¼ 100:4ba

where, f ðaÞ ¼

85

ð8Þ

Af ð0Þ  a a a 1sin tan ln cot 2 2 2 4

ð9Þ

I and F represent the incident solar flux and reflected light per surface unit of the Saturnian satellites, m0 and m are the cosines of the incident and emission angles, and f ðaÞ and b are the surface phase function and phase coefficient (Buratti, 1984; Buratti and Veverka, 1984), with the values given in Table 3. Since the dynamical and photometric models had been given above, the surfaces of the satellites were divided into many area units and the flux of each area unit was calculated, combined with the photometric model (Eq. (4)). The loss of the flux of the satellites during the mutual events was then derived through integrating all the occulted/eclipsed area units, determined by the dynamical model. The algorithm of Gauss–Newton iteration (Teunissen, 1990) was chosen to carry out the least-squares fit to the light curves, since the dynamical and photometric models and corresponding formulas had been determined, with the initial values of the dynamical parameters deriving from the MULTISAT ephemerides available at WWW.IMCCE.FR/SAT using the available ephemerides (Emelianov and Arlot, 2008). The radii of the Satellites S3 Tethys and S4 Dione we chose in this paper are 531.0 km and 561.4 km (Thomas, 2010) respectively.

4. Results and discussion Several parameters of the Saturnian satellites were derived for each event, as shown in Table 4, in which the first to third

Table 4 Astrometric results. Date ymd

Type

Midtime hms

Impact ð00 Þ

PA (1)

2009 12 23

S3eS4

18 53 28.97

0.07594

168.7

2010 04 24

S4oS3

15 31 29.46

0.07127

356.4

Da cos d

Dd

ð00 Þ

ð00 Þ

0.01488 7 0.0078  0.00448 7 0.0021

 0.07447 7 0.0078 0.07113 7 0.0021

Flux drop (%) 15.76 30.88

Fig. 5. Observed and fitted light curves of involved satellites. The dots and bold lines represent the observed and fitted flux variations of involved satellites normalized to 1 before and after the event, respectively. The x-axis corresponds to the date (in hours) and the y-axis to the relative flux.

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Table 5 O–C(x) and O–C(y) with respect to theory TASS 1.7 and the theory Lainey et al. (2011). Event

2009 12 23 3e4 2010 04 24 4o3

TASS 1.7

Lainey et al.

O–C(x) ð00 Þ

O–C(y) ð00 Þ

O–C(x) ð00 Þ

O–C(y) ð00 Þ

0.0101 0.0284

0.0034  0.0013

0.0010  0.0002

 0.0050 0.0025

5. Conclusion The results of our observations presented in this paper demonstrate a comparable accuracy with those of Arlot et al. (2012) and such accuracy is not far from the one of the observations by the space probe Cassini, it will be helpful to improve the dynamical models of the Saturnian satellites. The observations of such events should be organized during the next occurrence.

Acknowledgments columns denote the observed dates, types, and observed midtimes of each event, the fourth and fifth columns are the impact parameters we observed and the position angles of the occulting/ eclipsing satellite relative to the occulted/eclipsed one, and the following two columns give the observed values of Da cos d and Dd. The decline in flux is presented in the last column. The observed and fitted light curves, indicated by dots and bold line respectively, are plotted in Fig. 5, with the flux of the satellites involved being normalized to 1 before and after the events. It appears that the light fluxes are slightly higher after the events than before because the involved satellites being closer to the bright ring of Saturn at the end of the events. We compare our results with the theoretical values of two different dynamical theories, and get their residuals denoted as O–C(x) and O–C(y). As shown in Table 5, O–C(x) and O–C(y) are 10.1 mas, 3.4 mas for the mutual eclipse S4 by S3 and 28.4 mas, 1.3 mas for the mutual occultation S3 by S4 compared with the theory TASS 1.7 (Vienne and Duriez, 1995), and the results are 1.0 mas,  5.0 mas and 0.2 mas, 2.5 mas with respect to the theory Lainey et al. (2011), respectively. We can find that the deviations in Da cos d and Dd got with Lainey’s theory are smaller than those of TASS 1.7 theory.

We would like to thank the staff of 1-m and 60-cm telescopes team of Yunnan Observatory for their help and support for our work. This work is partly supported by Chinese Natural Science Foundation (no. 11203070) and the Project of MOST (2011CB811400). References Arlot, J.-E., Emelianov, N.V., Lainey, V., et al., 2012. Astronomy & Astrophysics 544, A29. Arlot, J.-E., Thuillot, W., 2008. Astronomy & Astrophysics 485, 293. Assafin, M., Viera-Martins, R., Braga-Ribas, F., et al., 2009. Astrophysical Journal 137, 4046. Buratti, B.J., 1984. Icarus 59, 392. Buratti, B.J., Veverka, J., 1984. Icarus 58, 254. Emelianov, N.V., Arlot, J.-E., 2008. Astronomy & Astrophysics 487, 759. Lainey, V., Karatekin, O., Desmars, J., et al., 2011. Astrophysical Journal 752, 14. Teunissen, P.J.G., 1990. Manuscripta Geodaetica 15, 137. Thomas, P.C., 2010. Icarus 208, 395. Vienne, A., Duriez, L., 1995. Astronomy & Astrophysics 297, 588. Zhang, X.L., Liu, Z., 2011. Research in Astronomy and Astrophysics 11, 1243.