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Physics ± Uspekhi 50 (6) 637 ± 647 (2007) INSTRUMENTS AND METHODS OF INVESTIGATION

#2007 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences PACS numbers: 95.55.Qf, 95.85.Kr, 96.30.Ys

Spectral and spectral-frequency methods of investigating atmosphereless bodies of the Solar system
V V Busarev, V V Prokof'eva-Mikhailovskaya, V V Bochkov
DOI: 10.1070/PU2007v050n06ABEH006333

Contents
1. 2. 3. 4. 5. 6. Introduction On the features of taking planetary reflectance spectra Spectral-frequency method Study of the asteroid 21 Lutetia by the spectral-frequency method Color spots on the surface of the asteroid 4 Vesta Conclusions References 637 638 640 641 644 646 647

Abstract. A method of reflectance spectrophotometry of atmosphereless bodies of the Solar system, its specificity, and the means of eliminating basic spectral noise are considered. As a development, joining the method of reflectance spectrophotometry with the frequency analysis of observational data series is proposed. The combined spectral-frequency method allows identification of formations with distinctive spectral features, and estimations of their sizes and distribution on the surface of atmospherelss celestial bodies. As applied to investigations of asteroids 21 Lutetia and 4 Vesta, the spectral-frequency method has given us the possibility of obtaining fundamentally new information about minor planets.

1. Introduction
Lately, the significant increase in the observational capabilities of modern astronomy and, in particular, in planetary astronomy, has led to discoveries of many new objects. For example, the total number of minor planets in the Solar system already approaches 150,000 [1]. However, it should be noted that for almost all new minor planets and satellites only orbital elements are known and estimates of sizes available. Due to a large number of new objects, their physical and chemical ± mineralogical characteristics will remain unexplored for a long time. Meanwhile, this information is extremely important for solving several cosmogonical
V V Busarev Sternberg Astronomical Institute, Moscow State University, Universitetskii prosp. 13, 119992 Moscow, Russian Federation Tel. (7-495) 939 10 29 E-mail: busarev@sai.msu.ru V V Prokof'eva-Mikhailovskaya, V V Bochkov Research Institute `Crimean Astrophysical Observatory', p/o Nauchnyi, 334413 Crimea, Ukraine Tel. (380) 65 54 71 124 E-mail: prok@crao.crimea.ua; bochkov@crao.crimea.ua Received 22 January 2007 Uspekhi Fizicheskikh Nauk 177 (6) 663 ± 675 (2007) Translated by K A Posnov; edited by A Radzig

problems related to the evolution of both minor and major planets of the Solar system. Moreover, solving these problems is relevant not only for our planetary system, but also for the many exoplanetary systems discovered over the last 15 ± 20 years [2]. This means that wide use and development of ground-based distant (astrophysical) methods represents a realistic way of rapidly exploring these objects. Large ground-based telescopes with a mirror diameter of the order of 4 ± 10 m would enable the most effective astrophysical studies of minor planets and planetary satellites. However, the actual number of operating telescopes of this class is still small and the observational time on them is strongly limited. So, at the moment it is virtually impossible to count on obtaining a lot of observational data on new planetary objects with these telescopes. The same reasoning applies to observations from space telescopes (Hubble, Spitzer, etc.) and space missions. For these reasons, one should seek the possibility of using for systematic planetary studies intermediate and small-class telescopes (with diameters in the range 0.5 ± 3 m) equipped with high-sensitivity detectors. The large number and easy availability of these instruments make the solution of the problem considered realistic in, say, two ± three decades. This time could be made shorter if a larger number of specialized observatories for regular planetary observation are constructed. It should be emphasized that the employment of ground-based telescopes to explore the composition and other surface characteristics of solid bodies in the Solar system can be effective only if they lack gas envelopes distorting the reflected solar light. Successful studies of atmosphereless planets with small telescopes can be carried out using effective methods and highly sensitive detectors. The principal optical methods include, first of all, photometry, spectrometry, and polarimetry of celestial objects. These methods are well elaborated and widely applied in astrophysics, so it is not necessary to describe them in detail. They are also applied to study the structure and characteristics of the surface material of planets. In this paper, we show how a combination of the two above-mentioned methods and the frequency analysis method (FA) enables us to obtain new information on solid


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celestial bodies. These observational methods are based on the property of a cold atmosphereless body to reflect incident solar radiation which can be measured on the detector. Due to a strong roughness of the surface of solid celestial bodies (down to the microlevel where the size of roughnesses is comparable to the optical wavelength), the term `reflection of a body' implies that the solar light, reflected by the surface of a body, includes both mirror-reflected and scattered components. It should be stressed that the scattered component of interest for us includes the radiation scattered by both the surface (determined by the scattering indicatrix) and the internal volume of the material. It is this latter component of the radiation reflected by a celestial body, which is called the diffusive component, that gives information on the composition of its matter. If the surface of a body has a spot-like structure or some relief details which have different spectral characteristics or scattering indicatrixes, then the rotation of the body will modulate the radiation flux (spectral or integral) reflected towards the observer. This information is contained in the observational data, and the question arises as to how to extract it correctly. During one complete revolution of a celestial body about its axis of rotation, the observer can register all the changes in the radiation field reflected by the body in the form of brightness variations. The obtained brightness curves (sometimes termed as light curves) give information on details on the surface of the investigated planet in the form of higher harmonics of its rotational frequency. By determining such frequencies across the observed phase interval of the light curve of the planet, one can estimate the characteristic size of inhomogeneities on the corresponding fraction of its surface. A variant of such FA was applied earlier to the set of photometrical data obtained for the asteroid 1620 Geographer [3]. Its relief sizes were estimated using photometrical observations obtained in 1994 with a time resolution of 0.4 min and 1 min. FA of these data revealed the presence of several periods repeatedly found in three data sets: 5.4, 7.5, and 15 min (the second harmonics of the previous value). In the primary maximum, a long period of around 40 min was present. The amplitudes were found to be small (about 0m.02), but were detected quite reliably. The difference in the period values in different maxima of the asteroid light curves suggested different sizes of the details on different sides of the body. For example, the largest and smallest details had a size of 1 ± 1.2 km and 150 ± 200 m, respectively. The estimates of the surface details of the asteroid 1620 Geographer, obtained by us from FA of photometrical data, were found to be fully consistent with the results of radar studies by Ostro et al. [4]. This enabled us to extend the application of the method under consideration to the area of spectrophotometric and spectral studies of asteroids.

Thus, consider the case (by the way, the most typical for ground-based astronomical observations) where a body is sufficiently small or remote to appear as a point-like source. In addition, we shall assume that the body has no atmosphere. Clearly, these bodies include asteroids, Edgeworth ± Kuiper bodies, and atmosphereless satellites of major planets. Then, from measurements in some spectral range of the investigated planet and a standard star, the monochromatic illuminance Ep l produced by the planet at the conventional upper boundary of the terrestrial atmosphere can be calculated using the differential photometry method: Ep l Ess l Ip l pl Iss l


þdM

Y

1

where Ess l (erg cmþ2 s A) is the monochromatic illuminance set up by the standard star at the upper boundary of the t errest ri al at m os pher e, ta ken, f or exam ple, f rom some publication; Ip l and Iss l are the counts of the light intensity from the planet and the standard star, respectively, measured by a spectral detector (CCD or any other) and corrected for the sky background; rl is the atmospheric spectral transparence function calculated for a given observational night, and dM Mp þ Mss is the air mass difference corresponding to the planet and the standard star at the moment of observations. On the other hand, according to paper [5], the monochromatic illuminance produced by the planet at the upper boundary of the terrestrial atmosphere (for the normal incidence of the light rays) can be expressed through the monochromatic illuminance El of the investigated planet due to the Sun: Ep l pGl F aY l El r2 Y D2 2

where Gl is the geometric albedo accounting for the integral physical ± chemical properties of the observed hemisphere of the planet, F aY l is the phase function of the planet (for a 0, one has F aY l 1, r is the radius of the planet, and D is its distance to the Earth. But since the unknown quantity El can be expressed according to the inverse square law through the known (from the literature) solar illuminance E0 l at the upper boundary of the terrestrial atmosphere as E l E0 l
2 R0 R2

(R0 is the distance from the Earth to the Sun, R is the distance from the planet to the Sun), formula (2) can be recasted in the form Ep l pGl F aY l E0 l
2 r 2 R0 X R 2D 2

2. On the features of taking planetary reflectance spectra
L e t u s f i rs t c o n si d e r i n m o re d e t a il th e fe at u re s o f t h e measurement and calculation method for the reflectance spectrum of a planet. This method is somewhat different from the differential spectral measurements well-known in astrophysics. In addition, specialists in the close fields of astrophysics do not always fully understand the physical sense of the term `the planetary reflectance spectrum' and its relation to the chemical-mineral composition of the matter.

3

Denoting raY l pGl F aY l, which is called the coefficient (or factor) of the mean spectral brightness of the observed hemisphere of the planet (more precisely, its projection on the observer's plane of the sky), and equating the right sides of expressions (3) and (1) yield the formula for calculation of r aY l in absolute units: raY l kEss l Ip l pl E0 l Iss l
þdM

Y

4


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2 where k D 2 R 2 ar 2 R0 is a dimensionless factor. Clearly, r aY l is also dimensionless. If the brightness factor r aY l is considered in some spectral range, its spectral distribution represents the reflectance spectrum. When the phase angle of a planet is close to zero, i.e., a % 0, the reflectance spectrum constitutes the spectral dependence of the geometrical albedo of the observed hemisphere of the planet. Equation (4) explains the physical sense of the reflectance spectrum that is obtained by dividing the spectral energy distribution of the planet by the same characteristic of the solar radiation incident on it. The reflectance spectrum of a planet (at least in the near UV, optical, and near IR bands where the proper thermal radiation of the planet is negligible) is the spectral dependence of the diffusive component of the radiation flux that passed through the matter and brings information on its physical-chemical properties. The present-day practice of planetary spectral studies widely uses the approximate but more simple and convenient method of measuring planetary reflectance spectra, which does not require the detection of the spectral energy distribution of the solar radiation at the moment of observations. Using a solar-like G0V star (for example, from the list in Ref. [6]) as the standard star, one can put Ess l % E0 l into expression (4) and recast it in the form

r aY l

kIp l plþ Iss l

dM

X

5

The reflectance spectra of a planet calculated with formula (5) approximate with good accuracy real ones when using the best solar-spectrum analogs. The accuracy of planetary reflectance spectra obtained in this way can be checked by comparing them with similar spectra calculated using different solar analogs or solar spectral data. However, despite obvious advantages, such a simplified method has significant shortcomings due most frequently to a poor quality of the resulting reflectance spectra, i.e., a high noise level. Consider in more detail possible solutions to this problem. Clearly, a high final signal-to-noise level is required to obtain high-quality planetary reflectance spectra. At the stage of recording of a spectrum, the level of statistical noise can be reduced simply by increasing the exposure time. But the subsequent procedure of dividing spectral data Ip laIss l to obtain the reflectance spectra according to formula (4) or (5) introduces an additional noise component (ANC) into the final result. For example, a low spectral resolution of the detector can be one of the simplest causes of appearing an ANC. In this case, the ANC can result from a poor matching of the spectral scales of the planet and the standard star or the star-analog even if they are correctly calibrated. This error cannot be less than the spectral resolution of the telescope + spectrograph + detector system. If the half-width of some absorption band in the resulting reflectance spectrum turns out to be comparable to the instrumental spectral resolution, its identification is virtually impossible. Increasing the spectral resolution of the detector system enables one to significantly remove this `source' of the ANC in the planetary reflectance spectra. Let us consider some other causes of appearing the ANC and try to outline ways of removing or constraining it. The stars themselves taking the part of solar-spectrum analogs serve as the most significant source of the ANC in the planetary reflectance spectra. As is well known from

theoretical and practical astrophysics, in spite of the enormous number and variety of stars there are no ideal solar twins among them because of different values of their physical parameters (the effective temperature, gravity acceleration in the atmosphere, color index, etc.) or chemical composition. Thorough all-sky searches have resulted so far in a list containing only a dozen stars (of 4th ± 6th magnitudes and spectral classes G0V ± G3V) which can be considered as good solar-spectrum analogs [7]. A longer list of stars that are considered solar-analog candidates is also being compiled. Relative spectral differences between some stars-analogs and the Sun were analyzed earlier [5]. Paper [5] showed that these differences are most pronounced in the vicinity of the Balmer jump (0.38 ± 0.40 mm), as well as near the strongest absorption lines (H and K (CaII), Ha , Hb , Hg , and Hd ). The differences appear not only at wavelengths corresponding to the centers of these lines, but also in the line wings which become more extended with increasing line intensity (in particular, in early-spectral-type stars with increasing Teff or hydrogen content). It also turned out that more shallow spectral inhomogeneities, which appear due to metal lines in spectra of stars-analogs of later spectral classes, can give rise to a noise background in almost the entire visible region of planetary reflectance spectra, when such stars-analogs are used. Thus, when dividing the observed spectrum of a planet (reflecting solar light) to the solar-analog stellar spectrum, the consecutive comparison of the star-analog spectrum and solar spectrum is done. Here, all spectral differences between them can be obtained in the form of a strong ANC which cannot be fully removed. The use of good stars-analogs in this case is the only way to constrain the ANC. Another significant source of the ANC that is worth considering is the terrestrial atmosphere. As is well known, on the one hand, when solar light passes through the Earth's atmosphere, water vapor, oxygen, carbon dioxide, nitrogen, and ozone give rise to the appearance of intensive telluric lines and bands (see, for example, Ref. [8]). On the other hand, the tiny water droplets and dust in the clouds (even light), as well as atmospheric turbulence cells, cause scattering and fluctuation of light passing through them. Although these scattering phenomena are not spectral-selective, they lead to distortions of the wave front of the radiation flux, which are transformed to its monochromatic components upon spectral decomposition. So, when comparing (dividing) the intensities of radiation fluxes from a planet and a standard star, which passed different optical paths in the terrestrial atmosphere, random fluctuations of their monochromatic components lead to the appearance of the ANC in the resulting planetary reflectance spectrum. Ground-based observations under good sky conditions, as a rule, provide high-quality lownoise spectra of celestial objects. However, for most astronomical observatories (with several exceptions) such atmospheric conditions occur quite rarely and for a limited time period. So looking for new methods for minimizing the ANC in the planetary reflectance spectra obtained from the ground under nonideal sky conditions is a very topical task. Smoothing is quite an effective way of suppressing the ANC in the planetary reflectance spectra. There are several methods, e.g., frequency filtering, running-mean, etc., which improve the quality of the reflectance spectra and help to identify real spectral features. For example, Fig. 1a depicts the reflectance spectrum of the asteroid 21 Lutetia, calculated using the asteroid and solar-analog stellar spectra (HD117176) [March 5, 2006; 1.25-m telescope of the Stern-


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Normalized reÉectance spectrum

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 0.35 1.6 1.4

a

0.45

0.55 0.65 0.75 Wavelength, mm

0.85

0.95 b

1.2 1.0 0.8 0.6 0.4 0.2 0 0.35 0.45 0.55 0.65 0.75 Wavelength, mm 0.85 0.95

Figure 1. (a) Example of the noisy reflectance spectrum of the asteroid 21 Lutetia. The reflectance spectrum was calculated using the spectra of the asteroid (exposure time 300 s) and the solar analog HD117176 (exposure time 10s) obtained on March 5, 2006 (UT 01h31m and 03h26m, respectively) on the 1.25-m telescope of the Crimean Laboratory of SAI with a CCD-spectrograph with a spectral resolution of 0.0009 mm in the 0.40 ± 0.96-mm range; the reflectance spectrum was normalized at the wavelength 0.55 mm. (b) The same spectrum after running-mean smoothing over 5 points.

berg Astronomical Institute (SAI) equipped with spectrograph and CCD matrix ST-6] taken under the presence of thin cirrus clouds. Figure 1b shows the same spectrum after smoothing (over 5 points) with the running-mean method. It is seen from Fig. 1b that after the smoothing the asteroid reflectance spectrum (especially near the blue and red edges) retains the noise structure, although with a significantly smaller amplitude. Below we shall consider special frequency methods that also facilitate identifying and extracting useful information in the reflectance spectra of atmosphereless planets and satellites.

3. Spectral-frequency method
The proposed spectral-frequency method of studies is based on the recording of a sufficiently large number of spectra of a certain atmosphereless solid body and their subsequent FA. As mentioned above, the goal of the method is, first, to d is c o ve r s o m e s t ru c tur e s o n th e pl a n et ar y s ur f ac e a n d, second, to associate with them the observed variations of the p l a ne ta r y s p ec tr a l c ha r ac t er i s t i c s , w hi c h c or r e s p on d t o changes in the physical ± chemical parameters of the matter. In the considered FA method, the rotation of a rigid celestial body a ro und i ts axis i s ess enti al l y invol ved . T hus, the repetition of or change in a certain spectral feature (for example, an absorption band) in consecutively taken reflectance spectra of the body studied with a period equal to the rotation period or some fraction of it can serve as proof of its reliability. The measurement of the amplitude of such

periodic variations in 2 ± 3 adjacent regions in the reflectance spectrum enables a more complete and reliable description of the spectral shape and its variations and, hence, obtaining more detailed information on the structure inhomogeneities or the composition of the material of this celestial body. Thus, the determination of the frequency ( or frequencies) of periodic spectral variations gives the possibility of estimating the sizes and the surface distribution of spectral-contrast spots of the planet's material which has some physical ± chemical peculiarities. Clearly, the necessary condition for the frequency analysis of the spectral data is the measurement of their absolute values. This means that the data obtained at different times and under different conditions (apparatus, atmospheric, and astronomical) must be reduced to a single system with the aid of their calibration, taking outside the terrestrial atmosphere, the reference to stellar standards, and the recalculation to the standard distance of the object from the Earth and the Sun. After such a correction of the original data, in the corresponding spectra of the object studied some characteristic feature can be found (for example, an absorption band or a spectral segment with the particular slope). To describe this feature one, two, or several synthetic photometric bands are calculated. In particular, to characterize an absorption band, it is sufficient to determine its equivalent width; to describe the reflectance spectrum slope in a fixed spectral interval, it is sufficient to calculate the color index, i.e., the ratio of stellar magnitudes of the object in photometric bands chosen at the boundary of this interval. Modern spectrophotometric equipment provides a good accuracy of measurements (for high signal-to-noise ratios ï better than 1%) and high time resolution (up to tenths of a second). During photometric observations in a limited spectral band, radiation fluxes from the solid object studied (a planet) and from neighboring stars are simultaneously detected to control changes in atmospheric extinction (transparency) at night. In the absolute photometry method, the photometric calibration of all observational data is also carried out using an artificial photometric standard [3], and the Earth's atmospheric extinction is determined at each individual night. Such measurements can be made in any spectral band, but the V band providing a sufficiently high signal-to-noise ratio seems to be preferential. FA is performed for the absolute stellar magnitudes of a solid celestial body reduced to a unit distance from the Earth and the Sun and to zero irradiation and observation phase angles. It should be noted that interesting results can also be obtained from FA of the color indices of the object, which can be obtained by dividing its brightnesses in the neighboring spectral bands. This requires simultaneous or quasisimultaneous photometry of the body studied in these spectral bands. Such a color index is free, on the one hand, from the integral changes of the object's brightness related to the observed rotation-induced variations in its shape but, on the other hand, gives information on the distribution of color spots over the body's surface. For example, color spots on different components of a binary object give rise to frequencies in its integral color indices corresponding to the angular rotation velocities of the components, but do not produce the frequency pertaining to their orbital motion. In order to extract the full information on a celestial object by means of FA of its observational data, the spectroscopy in a wide spectral range (for example, in the entire visible and/or near infrared band) is prefered to broad-band photometry in

Normalized reÉectance spectrum


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one or several spectral bands. Such spectra, as a rule, contain the necessary additional information that can be used for the subsequent choice of narrower photometrical bands. Besides, the synthetic photometric magnitudes of the object in these bands can be evaluated from its known spectral data. These photometric magnitudes can be calculated either in the standard photometric system (for example, the Johnson one), or taking into account the location of some interesting spectral features. Two approaches to data processing for FA are possible. One is based on the experience of constructing light curves with trial periods and elaborating automatic means of finding the most probable period that minimizes dispersion of data points relative to the mean curve. Different methods of FA rely on different means of estimating the dispersion of points, for example, the Lafler ± Kinman method [9] and the Jurkevich method [10]. It should be noted that these methods of FA are applicable to light curves of any form and to series of data with extended breaks. Anyway, to provide a high reliability of finding the true period, a large number of observational points are required. The second approach is based on the Fourier analysis. It is more rigorous mathematically, but its applicability is limited for series with breaks. In 1975, Deeming [11] proposed a method enabling removing frequencies modulated by the series on-off ratio. Recently, the Breger method [12] also based on the Fourier analysis has been widely applied to study brilliance variations in stars and rotating solid bodies. The Breger method is commonly used for preliminary FA of data series. Note that the last two methods can be applicable for objects with sine-like light curves; in the case of light curves with two maxima, they find the second harmonics of the true period. FA of photometrical data of asteroids obtained at the Crimean Astrophysical Observatory has been performed using a software package enabling frequency searches in light curves of arbitrary shapes. The main program (Period) of data processing was written by M Yu Klepikov and completed by K V Prokof'eva. Here, calculations are simultaneously carried out by three different methods: Lafler ± Kinman, Jurkevich, and Deeming [9 ± 11]. The program allows the user to quickly look at the convolution of data with a given period and to find the power of a polynomial to approximate slow changes in the light curve. Subtracting this polynomial from the corresponding photometrical data removes the variations in brilliance with a chosen period, its harmonics, and conjugate periods. In this way, the data whitening is performed, which is required for further searches for hidden periods of smaller amplitudes [13]. If a multiperiodicity is present, the data whitening, as a rule, is carried out in order of decreasing amplitudes of periodic variations. The complication of the frequency spectrum of astronomical object brilliance in the presence of the series of observations made with extended time breaks is well known in astrophysics as the frequency substitution phenomenon [13]. Frequencies that appear as a result of the interaction of the true frequency of variations in brilliance of the object studied with on-off ones are often called artifacts. The amplitudes of peaks corresponding to these frequencies are comparable to those from the true frequency. In addition, in the frequency spectrum so-called combination or conjugate frequencies that arise when summing the signal with frequencies of the medium the signal passes through are observed [14]. They represent the sum and the difference of the main frequency of

the oscillation of the signal from the object studied and the medium frequencies. In FA of astrophysical data, the role of the `medium' can be played by the on-off ratio of the series analyzed. It carries frequencies determined by the observational conditions ï the length of the series or its parts, diurnal, monthly disruptions, and other causes. Note that the use of some a priori information, for example, the allowance for two maxima in the light curves of a celestial body, facilitates the estimation of reliability of periods revealed by FA. Several features are used, as a rule, to find real variations in brilliance of an astronomical object. The detected frequency is considered to be real under the following conditions: in the power spectrum there are harmonics of this frequency and the frequencies of multiple periods, as well as combination frequencies located symmetrically relative to the detected frequency; phase diagrams (convolutions or light curves) constructed with the studied period have no troughs due to the on-off ratio of observations and show the presence of two maxima and two minima of nearly equal amplitudes; phase dependences of the light curves constructed from different series of observations coincide; the peak corresponding to the frequency of a detected period is present in the phase diagrams obtained by different methods, and the periodograms of the model constructed by replacing the observed stellar magnitudes of the object in the series under study by random values, demonstrate no features at the would-be true frequency.

4. Study of the asteroid 21 Lutetia by the spectral-frequency method
The asteroid 21 Lutetia will be one of the most interesting objects to be studied by the Rosetta space mission, which should approach the asteroid in 2010. It belongs to the spectral type M accor