Mutual Occultations and Eclipses of Galilean Satellites of Jupiter in 2002–2003

Information on the approach of a new epoch of mutual occultations and eclipses in the system of Galilean satellites of Jupiter is presented. The importance of observations of such phenomena is highly emphasized.     

Determination of parameters of Long-Term variability of the X-ray pulsar LMC X-4

We have investigated the temporal variability of the X-ray flux measured from the high-mass X-ray binary LMCX-4 on time scales from several tens of days to tens of years, i.e., exceeding considerably the orbital period (~1.408 days). In particular, we have investigated the 30-day cycle of modulation of the X-ray emission from the source (superorbital or precessional variability) and refined the orbital period and its first derivative. We show that the precession period in the time interval 1991–2015 is near its equilibrium value P _sup = 30.370 days, while the observed historical changes in the phase of this variability can be interpreted in terms of the “red noise” model. We have obtained an analytical law from which the precession phase can be determined to within 5% in the entire time interval under consideration. Using archival data from several astrophysical observatories, we have found 43 X-ray eclipses in LMC X-4 that, together with the nine eclipses mentioned previously in the literature, have allowed the parameters of the model describing the evolution of the orbital period to be determined. As a result, the rate of change in the orbital period ? _orb/P _orb = (1.21 ± 0.07) × 10^?6 yr^?1 has been shown to be higher than has been expected previously.

     

Special program of observations of Jovian and Saturnian satellites for 2009

Noteworthy phenomena, viz., mutual occultations and eclipses in the system of Jupiter’s Galilean satellites and in the system of Saturn’s principal satellites, will occur in 2009. The relatively simple photometry of these phenomena makes it possible to obtain positional data at a higher accuracy than can be achieved in regular astrometric observations. The visibility conditions for the satellites are described here and observational recommendations are given. The ephemerides of these phenomena are available via the Internet from the MULTI-SAT ephemerides server at http:/www.sai.msu.ru/neb/nss/index.htm.     

Fourier analysis of the light curves of eclipsing variables,V

The methods of analysis of the light changes of eclipsing variables in the frequency-domain, developed in our previous papers (Kopal 1975a, b, c, d) for an interpretation of mutual eclipses in systems consisting of spherical stars, have now been extended to analyse the light variations — between minima as well as within eclipses — ofclose binaries whose components are distorted by axial rotation and mutual tidal action. Following a brief introduction (Section 1) in which the need of this new approach will be expounded, in Sections 2 and 3 we shall deduce the theoretical changes of close eclipsing systems between minima (Section 2) as well as within eclipses (Section 3), which in Sections 4 and 5 will be analysed in the frequency-domain; and explicit formulae obtained which should enable us to separate the photometric proximity and eclipse effects directly from the observed data as they stand-without the need of any preliminary ‘rectification’. Section 6 will contain the explicit forms of the expressions for photometric perturbations in the frequency-domain, due to rotational and tidal distortion of both stars; and the concluding Section 7 will then be concerned with practical aspects of the application of these new methods to an analysis of the observed light changes of close eclipsing systems — in which the proximity and eclipse effects cannot be distinguished from each other by mere inspection.     

Mutual phenomena of Jovian satellites

Results of the observations of mutual eclipses of Galilean satellites observed from the Vainu Bappu Observatory during 1985 are presented. Theoretical models assuming a uniform disc, Lambert’s law and Lommel-Seeliger’s law describing the scattering characteristics of the surface of the eclipsed satellite were used to fit the observations. Light curves of the 1E2 event on 1985 September 24 and the 3E1 event on 1985 October 24 observed from VBO and published light curves of the 1E2 event on 1985 September 14, the 3E1 event on 1985 September 26 and the 2E1 event on 1985 October 28 (Arlotet al 1989) were fitted with theoretical light curves using Marquardt’s algorithm. The best fitting was obtained using Lommel-Seeliger’s law to describe the scattering over the surface of Io and Europa. During the fitting, a parameter^δxshift which shifts the theoretical light curve along the direction of relative motion of the eclipsed satellite with respect to the shadow centre, on the sky plane (as seen from the Sun) was determined along with the impact parameter. In absence of other sources like prominent surface features or non perfect sky conditions which could lead to asymmetric light curves,^δxshift would be a measure of the phase correction (Aksnes, Franklin & Magnusson 1986) with an accuracy as that of the midtime. Heliocentric Δα cos (δ) and gDδ at mid times derived from fitted impact parameters are reported     

Fourier analysis of the light curves of eclipsing variables,XVIII

The aim of the present paper has been to establish explicit expressions for the photometric perturbations in the light changes of close eclipsing systems, arising from the mutual distortion of the components, for any type of eclipses — be these occultations or transits; partial, total, or annular — and exhibiting arbitrary distribution of brightness (limb- or gravity-darkening) over the apparent disc of the eclipsed star.These perturbations have been expressed in terms of certain general types of series that can be easily programmed for automatic computation. They represent a generalization of results previously obtained by Kopal (1975) or Livaniou (1977, 1978) in so far as the expansions derived in this paper hold good for any real (not necessarily integral) value ofm>0. As such, they can be used to free from the photometric proximity effects within eclipses the empirical momentsA _2m of the light curves of non-integral orders, and the task performed within seconds of real time on high-speed automatic computers now available. Closed-form expressions for such perturbations, obtaining in the case of total eclipses, are given correctly to terms of first order in quantities which represent the distortion of each component.     

Astrometric results of observations of mutual occultations and eclipses of the Galilean satellites of Jupiter in 2002-2003

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.     

Mutual occultations and eclipses of the Galilean satellites of Jupiter in 2002–2003: final astrometric results

The photometry of mutual occultations and eclipses of natural planetary satellites can be used to infer very accurate astrometric data. This can be achieved by processing the light curves of the satellites observed during international campaigns of photometric observations of these mutual events.
This work focuses on processing the complete data base of photometric observations of the mutual occultations and eclipses of the Galilean satellites made during the international campaign in 2002–2003. The final goal is to derive new accurate astrometric data.
We propose the most accurate photometric model of mutual events based on all the data available to date about the satellites, and develop the corresponding method for extracting astrometric data. This method is applied to derive astrometric data from photometric observations of mutual occultations and eclipses of the Galilean satellites.
We process the 371 light curves obtained during the international campaign of photometric observations of the Galilean satellites in 2002–2003. As compared with the theory, the rms 'O-C' residuals with respect to theory is equal to 0.055 and 0.064 arcsec in right ascension and declination, respectively, for the 274 best observations. Topocentric or heliocentric angular differences for satellite pairs are obtained for 119 time instants during the time period from 2002 October 10 to 2003 July 17.     

Computation of the elements of eclipsing binary systems in the frequency-domain

The aim of the present paper will be to translate the essential parts of the theory of Fourier analysis of the light changes of eclipsing variables into more practical terms; and describe procedures (illustrated by numerical examples) which should enable their users to obtain the desired results with maximum accuracy and minimum loss of information by processes which can be fully automated.In order to unfold in steps how this can be done, the scope of the present paper-the first of two-will be restricted to an exposition of the analysis of light changes caused by eclipses of spherical stars; while between minima due to this cause the light of the system should remain sensibly constant. An extension of our analysis to incorporate photometric effects arising from mutual distortion of the components of close eclipsing systems between minima as well as within eclipses is being postponed for the second communication.In developing this subject we shall single out for the user's attention only those parts of the whole theory which are of direct relevance to practical work. Their justification can be largely found in sources already published; and new developments essential for our work, not yet made public, will be relegated to several Appendices at the end of the text, in order not to render its text too discursive and deflect the reader's attention from the main theme of its narrative.After a brief outline of the subject given in Section 1, Section 2 will introduce the reader to practical aspects of the Fourier analysis of the light curves; and Section 3 will be devoted to its use to determine the numerical values of the momentsA _2m of the light curves which constitute the cornerstones for all subsequent work. Section 4 will describe an algebraization of the process of determination of the elements for the case of total (annular) eclipses; while Section 5 will do the same for partial eclipses. The concluding Section 6 will be devoted to an error analysis of our problem, and to an outline of the way by which the errors of the individual observations will compound to the uncertainty of the final results. Lastly, Appendices 1–5 concluding the paper will contain additional details of some aspects of our work, or proofs of new processes made use of to obtain our results, whose earlier inclusion would have made the main text too discursive.     

A study of the distortion wave in the RS CVn eclipsing binary SV Camelopardalis

The light outside the eclipses of the totally eclipsing RS CVn binary SV Camelopardalis (SV Cam) is Fourier analysed and the amplitudes of the distortion waves have been derived. The distribution of the percentage contributions of these amplitudes inV, B andU colours with respect to the luminosities of the binary components indicates that the hotter component is the source of the distortion waves. These distortion waves, attributed to star spots, are modelled according to Budding (1977) and spot parameters like longitude, latitude, temperature and size are obtained. From this study it is noticed that while symmetric waves with two minima could be fitted satisfactorily, asymmetric waves with more than two minima could not be fitted well. From the longitudes of the minima of the best fitted curves, migration periods of four spot groups are determined. Assuming synchronism between rotation and orbital periods, the rotation periods of the four spot groups are derived from their migration periods. The period of rotation of one of the spot groups having direct motion is found to be 0^d.5934209 while the periods of the other three spot groups having retrograde motion are 0^d.5926588, 0^d.592607 and 0^d.5924688. As the latitudes of these spots are known from modelling parameters, the latitude having a rotation period equal to that of the orbital period (co-rotating latitude) is found to be about 30°     

Fourier analysis of the light curves of eclipsing variables,II

The aim of the present paper will be to extend the Fourier methods of analysis of the light curves of eclipsing binaries, outlined in our previous communication (Kopal, 1975) in connection with systems whose components would appear as uniformly bright discs, to systems whose components exhibit discs characterized by an arbitrary radially-symmetrical distribution of brightness —i.e., an arbitrary law of darkening towards the limb — be it linear or nonlinear.In Section 2 which follows a few brief introductory remarks, fundamental equations will be set up which govern the light changes arising from the mutual eclipses of limb-darkened stars — be such eclipses total, partial or annular; and Section 3 will contain a closed algebraic solution for the elements of the occulation eclipses terminating in total phase. Such a solution proves to be no more complicated than it turned out to be for uniformly bright discs in our previous paper; and calls for no special functions for the purpose — as will be put in proper perspective in the concluding Section 4.The cases of transit eclipses terminating in an annular phase, of partial eclipses of occulation or transit type, will be similarly dealt with by Fourier methods in the next paper of the present series.     

Fourier analysis of the light curves of eclipsing variables,XI

The aim of the present paper will be to introduce a new definition of the loss of light suffered by mutual eclipses of the components of close binary systems: namely, as across-correlation of two apertures representing the eclipsing and eclipsed discs.The advantages of such a strategy over the more conventional (geometrical) approach are (a) greater symmetry of the respective expressions; (b) greater affinity of expressions arising from distortion with those expressing the light changes due to eclipses of spherical stars; and (c) greater freedom in dealing with the effects of particular distribution of brightness over the disc of the star undergoing eclipse (generalized limb-darkening), as well as of possible semi-transparency of the eclipsing component (Wolf-Rayet stars!). In point of fact, none of these tasks could be handled with equal ease by any other technique; nor could the corresponding loss of light be so automated by any other approach.In Section 2 which follows brief introductory remarks we shall evaluate the loss of light arising from distribution of brightness within the aperture undergoing eclipse, and appropriate opacity of the occulting disc. In Section 3 we shall take advantage of these new forms of our results to deduce a number of new properties of the eclipse functions — both algebraic and differential — which have so far escaped attention and which are of considerable practical interest. Lastly, in Section 4 we shall generalize the same concepts to the modification of the light changes caused by the departures of the respective apertures from circular forms.It will be shown that all these phenomena can be most naturally described in terms of Hankel transforms of the products of two Bessel functions with orders depending on the physical characteristics (distribution of brightness; opacity) of the two components; while the geometry of the system (i.e., the fractional radiir _1,2 of the two stars; or the inclinationi of their orbit) enter only through their arguments. Such formulation of our problem should bring a theory of the light changes of eclipsing variables in much closer contact with the adjacent parts of physical optics.     

Mutual phenomena of Jupiter's satellites

Observations of an occultation of Europa by Io are fitted by a model light curve. The model has five free parameters, namely the radius of Europa, the impact parameter, the brightness ratio of the satellites, the time of midevent and the mean relative velocity. The model assumes a fixed value for the radius of Io and for the solar phase angle α, and that Europa has a uniform surface brightness. The OC residuals of the best fitting light curve are very small (～0.002 mag) and of a purely random nature; there is no evidence of albedo features. Taking α = 0 does not affect significantly the quality of the fit. Six mutual eclipses were also observed, and their times of minima agree well with the predictions of Aksnes Icarus21 (1974). For two events these predictions differ by about 20 min from those of Brinkmann and Millis Sky & Telescope45 (1973).     

Fourier analysis of the light curves of eclipsing variables,IV

The methods of analysis of the light changes of eclipsing variables in the frequency domain, developed in our previous papers (Kopal, 1975b, c) for total or annular eclipses of arbitrarily limbdarkened stars, have now been extended to the case of partial eclipses of occultation as well as transit type. In Section 2 which follows brief introductory remarks the even Fourier sine coefficients are formulated — in the guise of the momentsA _2m of the light curve — in terms of the elements of the eclipse; and their use for a solution for the elements is detailed in Section 3. A brief appendix containing certain auxiliary tables to facilitate this task concludes the paper. An extension of the same method to an analysis of the light changes exhibited by close eclipsing systems — in which the photometric proximity effects arising from mutual distortion can no longer be ignored — will be given in the subsequent paper of this series.     

Astrometric results of observations at Russian observatories of mutual occultations and eclipses of Jupiter’s Galilean satellites in 2009

In 2009, in five Russian observatories photometric observations of Jupiter’s Galilean satellites during their mutual occultations and eclipses were carried out. Based on these observations, an original method was used to ascertain astrometric results such as the difference between the coordinates of pairs of satellites. Fifty-three phenomena were successfully observed. A total of 94 light curves of satellites were measured. The error in the coordinates of satellites due to random errors in photometry, calculated on all data obtained, was 0.041″ in right ascension and 0.046″ in declination. The discrepancies between the theory and observations in these coordinates was found to be 0.060″ and 0.057″, respectively. The results were uploaded to the common database for all observations of natural satellites of planets at the Natural Satellites Data Center (NSDC), which is available online at http://www.sai.msu.ru/neb/nss/index.htm. For the first time in the practice of photometric observations of satellites in epochs of mutual occultations and eclipses a new method of observation was tested, which eliminates from astrometric results the major systematic errors caused by an inaccurate account of the background level. The tests were conducted in the Terskol Observatory and the observatory of the Crimean laboratory of the Sternberg State Astronomical Institute of the Moscow State University. The application of the new method showed that the elimination of the background level at these observatories was carried out correctly.     

Mutual occultations and eclipses of the Galilean satellites of Jupiter in 1997: Astrometric results of observations

In 2006, a complete database of the international campaign on photometric observations of the Galilean satellites of Jupiter in the 1997 epoch of mutual occultations and eclipses was published. Only two thirds of the observations were considered by other authors beforehand. In this study, we have processed the whole observational database with an original technique in order to obtain the astrometric data. We determined 301 relative positions of the satellites from photometric observations performed at 50 observatories around the world. The results are put into a common database of all observations of the natural planetary satellites called the Natural Satellites Data Center (NSDC) available on the Internet site http://www.sai.msu.ru/neb/nss/index.htm. The influence of random and systematic errors on the accuracy of determining the coordinates of satellites has been analyzed. It has been shown that the largest systematic errors are caused by inaccurate elimination of the background of the photometric measurements and by the erroneous data on the albedo of satellites. The actual accuracy of astrometric results is 0.05″ and 0.07″ in right ascension and declination, respectively. New recommendations for photometric observations of satellites during the considered phenomena have been developed in order to avoid the systematic errors.     

Mutual phenomena of Saturn's satellites in 1979–1980

Using two sets of orbital elements and the radii of the Saturnian satellites 1 (Mimas) through 7 (Hyperion), we find that from October 1979 until August 1980 nearly 300 mutual eclipses and occultations involving these bodies will occur. To allow for the expected errors in the satellite ephemerides, we repeat these calculations in order to obtain the additional events that occur when all satellite radii (save Titan's) are increased by 1000 km. A third listing predicts eclipses of satellites by (the shadow of) the ring. Photometric observations of a large number of these events will add much precise information to our knowledge of the Saturnian system at a critical time.     

Fourier analysis of the light curves of eclipsing variables,VIII

The main aim of this paper will be to develop explicit form of the moments of the light curvesA _2m(r ₁,r ₂,i) required for the solution for the geometrical elementsr _1,2 andi of eclipsing systems exhibiting annular eclipses (Sections 2 and 3), as well as partial eclipses (Section 4).In the concluding Section 5 we shall demonstrate that — regardless of the type of eclipse and distribution of brightness on the apparent disc of the eclipsed star, or indeed of the shape of the eclipsing as well as eclipsed components — the momentsA _2m satisfy certain simple functional equations — a fact which relates them to other classes of functions previously studied in applied mathematics.     

Composite spectra

Composite-spectrum binary stars generally consist of a late-type giant and a main-sequence star of type A or B. Their spectra are therefore rather confusing; but by a technique of digital subtraction of the spectra of appropriate single late-type giants, composite spectra can be split into their individual components. In favourable cases the radial velocities of both components can be measured and the mass ratio determined. The procedures are illustrated by reference to HR 6902, a fifth-magnitude composite-spectrum binary. Its components are shown to have spectral types of G9 II and B8 V, with a mass ratio of 1.31, and its orbit is determined. There is some evidence that the system shows eclipses. If it does, the masses of the components are 3.9 and 3.0M⊙ respectively, and HR 6902 becomes the sixth known member of the important class ofζ Aur binaries.     

Binaries with Total Eclipses in the LMC: Potential Targets for Spectroscopy

35 Eclipsing binaries presenting unambiguous total eclipses were selected from a subsample of the list of Wyrzykowski et al. (2003). The photometric elements are given for the I curve in DiA photometry, as well as approximate teff and masses of the components. The interest of these systems is stressed in view of future spectroscopic observations.