Fourier analysis of the lightcurves of 53 known pole asteroids

Plrrrw/.

Pergamon

si.. Vol. 43. No. 2. pp. 77 85. IYYh Copyright (‘ I9Yh Else\,& Science Ltd Printed In Great Britain. All rights reserved 003J Oh3i~% SI 5.(10+0.00 sptrcv

0032-0633(95)00073-7

Fourier analysis of the lightcurves of 53 known pole asteroids M.T.

Capria,’ M. C. De Sanctis,’ M. A. Barucci’ and M. Fulchignoni’

IAS Reparto di Planetologia. V. le dell‘Universili. I I. 00185 Rorna. Ital> Istituto A~tronomico. Umvrrsitli La Sapienza. via Lancisi. 29. 00161 Rnmx Obsetwtoirr de Paris, 92 I95 Meudon Principal Cedex. France Recei\td

28 h~~\cmhcr

lYY5: revised 31 January

lY95: accepted

Abstract. The Fourier coefficients of 254 composite lightcurves of 53 asteroids are analysed, continuing a previous work (Barucci et al., Icarus 78, 31 l-322, 1989), with the aim of inferring asteroidal shapes and possible albedo variegations. Coefficients are obtained only from good quality photometric data of asteroids for which a reliable estimate of the orientation of the spin axis is available. For each of the 53 asteroids an indicative shape is derived, on the basis of the coefficients resulting from the Fourier analysis. The shapes determined for 243 Ida and 951 Gaspra are confirmed by the Galileo images. Most of the analysed asteroids show a quasi-regular shape, probably due to a bias on the diameters of the sample ; the remaining part, having smaller diameters, seems to show an irregular shape. The obtained results cannot be interpreted in a statistical sense but they can constitute a promising starting point for the understanding of the evolution of asteroidal population.

Irol>

5 May lYY5

tridimensional shape of an asteroid. Since then. man) observations and cxperimcntal theoretical advances. simulations have changed the situation (Cellino 1st (I/.. 1987. l9XYa ; Barucci et al., IYY?: Barucci and Fulchignoni. IY86). In I988 we started the Fourier analysis of the liphtcurves existing in the APC database (Lagerkiist ct ~1.. 19X7, 1988) and published tirst results in Barucci (‘I II/. (1989). Now we restrict our analysis to asteroids with known rotation pole coordinates. The aim of the work is to infer the approximated shape of each object and the albedo variegations, if any, using the method described later on. The obtained results give rough information on the shape and albedo variegations of the single object but the analysis of all the data in statistical terms could gi\,e valuable information on the behaviour 01’the population of large asteroids. Moreover. the resulting Fourier coefhcients for each asteroid can he used to define the “zero level” model to start the iterative numerical procedure of inversion of the asteroid lightcurve which allows us to obtain a more detailed model through the best tit of the whole set ol‘a\ailable information.

Introduction The shape is one of the more interesting properties ot an asteroid. due to the possibilities it offers to infer the geological and collisional history of the body and. on a statistical basis. to get more insight in the evolutionary history and origin of the whole population. Several methods have been developed to derive the shape of asteroids (Magnusson c’t trl.. 1989 : Ostro ct r/l.. 198X) ; among them the lightcurve inversion technique has given good results when direct observations are not available. Russell was the first to analyse theoretically the problem of liphtcurve inversion (Russell, 1906) and his conclusions were quite pessimistic : he demonstrated that lightcur\,e data alone are not sufficient to determine the

Application of Fourier analysis to asteroid lightcurves Our analysis was performed only on high quality lightcurbes coming from asteroids for which the pole coordinates are known. We used. mainly. ;t list of asteroid poles given by Magnusson (1993). The lightcur\,es came mostly from the APCs (Lagcrkvibt ct d.. lUX7. IYXX, 1997) : some recent lightcurves of133 Ida and 951 Gaspra were taken from Binzel et trl. ( lYY3). Gonano-Beurcr ct d. (1997). Blanco et trl. (1991). anti Wisniewaki c’t ol. (19Y?). In Table I the fundamental parameters 01 the 53 asteroids analysed can be seen : number and name. diameter. period. family type and laxonomic class. We used only single hghtcurves spanning a full rotation cycle without significant gaps or. when possible. more

78

M. T. Capria rt d. : Fourier

Table 1. Physical parameters LY Asteroids

1 Ceres 2 Pallas 3 Juno 4 Vesta 6 Hebe 7 Iris 9 Metis 12 Victoria I5 Eunomia 16 Psyche 19 Fortuna 20 Massalia 21 Lutetia 22 Kalliope 39 Amphitrite 3 1 Euphrosyne 37 Fides 39 Laetitia 41 Daphne 43 Ariadne 44 Nysa 45 Eugenia 52 Europa 55 Pandora 63 Ausonia 64 Angelina 87 Sylvia 88 Thisbe 107 Camilla 125 Liberatrix 129 Antigone 130 Elektra 30 1 Penelope 2 16 Kleopatra 343 Ida 250 Bettina 349 Dembowska 354 Eleonora 423 Diotima 433 Eros 45 1 Patientia 511 Davida 532 Herculina 584 Semiramis 624 Hektor 694 Ekard 704 Interamnia 951 Gaspra 1580 Betulia 1620 Geographos 1685 Toro IS67 Apollo 3908 1980 PA

of the observed

asteroids

Period’ th)

Family’

(km) 848.4 498. I 233.9 468.3 185.2 199.8 179.0 112.8 255.3 253.2 ‘21.0 145.5 95.8 181.0 213.3 355 9 A__. 108.3 149.6 174.0 65.9 70.6 114.6 302.5 66.7 103.1 58.0 260.9 200.6 737 6 A.__. 43.6 113.0 18?.? 65.4 135.1 30.0 79.2 139.8 155.7 208.8 13.0 234.9 336.1 717 -7 _A_._ 54.0 233.0 90.8 3 16.6 16.0 5.0 2.0 5.0 7.0 1.0

9.075 7.81 I 7.110 5.340 7.274 7.138 5.079 8.659 6.083 4.196 7.443 8.098 Y. 167 4.147 5.390 5.531 7.333 5.138 5.988 5.761 6.42 I 5.699 5.631 4.804 9.298 8.752 5.184 6.041 4.844 3.Y68 4.957 5.‘35 ‘__ 3.747 5.385 4.633 5.055 4.701 4.177 4.612 5.270 9.777 5.130 9.405 5.069 6.92 1 5.922 8.717 7.042 6.130 5.__. “3 _ 10.196 3.065 4.426

67 139

type

169

170 171 140 158 162 158

142 67 I85 24 133

165

132

124 200 Am

163 T

Am AP AP AP Am

Taxonomi?’ class G B S V s s S S S M G S M M S c S S c S E F/C F:C M S E P F,‘C c M M G M M S M R S c S c:u C S S D CP F S c S z V

analysis of the lightcurves

criteria. The data handling successive steps :

of 53 known pole asteroids

procedure

is composed

of two

(I) all the lightcurves, chosen with the given criteria. are composed using the method explained in Harris et ~1. (1989); (2) the Fourier coefficients of the I’ functions, the squares of the lightcurve intensity. are computed. Intensities are used because, under the assumptions of uniform brightness. geometric scattering and triaxial ellipsoidal shape. the Fourier components are more easily separated into meaningful quantities if magnitudes are converted to intensities and then squared. The lightcurve flux Zcan be considered proportional to the cross-sectional area A of a triaxial ellipsoid with semiaxes N 3 h 3 C. The square of this area can be written, dropping a factor of TI: A’ = (u/7 sin (j)2 + (~*cos0cos

$)‘+

(h~,cos (Isin rl/)’

= (ahsin0)‘+0.5(~cosB)‘(a’+h’) +0.5(ccosH)2(~~‘-h’)cos3$

(1)

where H is the latitude of the sub-Earth point and $ the rotational phase measured from maximum light. The Fourier series for I’ can be written as I’ = q + $J [a,,cos (n$)+h,, ,>=I For a triaxial

ellipsoid

sin (lr$)).

(2)

we have

(10 = ‘(rrbsinB)‘+((,cosf~)‘(cr’+h~)

(3)

II, =o

(4)

LIl = O.5(c~cosu)~(a~ -/I’)

(5)

II > 2. II,! = h,, = 0. The only coefficients containing information about a triaxial ellipsoid under the above assumptions are N(, and the amplitude of the second component of the Fourier expansion of I’ : c2 = (~1;f/G)’ ‘. Even order terms greater than 3 cannot arise from a triaxial ellipsoid shape: higher order shape parameters will contribute to even terms higher than 2. Odd terms will only arise from albedo features or perhaps scattering phenomena. Hereafter the following definitions apply : l first-, second-. third- and fourth-order harmonics are the first (n = 1). second (17 = 2). third (!I = 3) and fourth (n = 4) components of the Fourier expansion of I’ : l the amplitude of the rzth Fourier coefficient is L’,,= (0,; + h,’ ) ’ 2 : l c’,, = N,,,:?.

.’Tedesco rt d. ( 1992) ; Tholen ( 1991). ’ Lagerkvist or al. ( 1989, 1992). ’ Williams ( 1989). ” Tholen ( 1989).

Results of Fourier analysis applied to lightcurves

taken within a range of phase angle < 10 that together yield a lightcurve with full coverage and no gaps. Obviously, it was not possible for all the asteroids with known pole coordinates to find lightcurves following these

In the previous investigation (Barucci it N/.. 1989) we pointed out that it is possible. comparing the plots of the Fourier coefficients describing the lightcurves of an asteroid at different aspect angles. to infer the general shape and some characteristics of the surface : in order to

lightcurves

understand the information contained in the coefficients of the Fourier expansion of asteroid lightcurves. we obtained. with laboratory simulations and analytical calculations, a sample of synthetic lightcurves. In our pre\ious work it is explained how, plotting the ;Implitude of the Fourier components describing synthetic IightcurLrs obtained from a set of laboratory models, tlifferent in shape and albedo markings. some regions in each plot can be identitied \vhich arc indicative of the \hape and or albedo variegations of the model. In this work wc have taken into account 53 asteroids. $jut of those with kno\vn pole coordinates (Binzel cztt/i., I993 : Drummond and Wisniewski. 1990 : Magnusson, IYXY. 199.3: Maenusson c’t t/l.. 1992: Michalowski, lY9.1). having two or m;re lightcurves suitable for Fourier analy\lS.

The coctlicienls obtained from this analysis are summarircd in Table 7. For each asteroid only the coefficients relative to one lightcur\,e per opposition are reported. c’, and C, > 0.03. it could be some overlapping ol‘astcroids presenting irregularities in the shape or albcdo features. Irregular objects arc thohe with the higher LIIUCS of the amplitude of Ihe third cocllicien~ (particularI) those with I’? > (‘, and (‘3 > 0.031. In Fig. 2 \omc cxamplcs of Fourier coefficient trends. that arc reprcsentati\e ofdilTercnt types of objects, can bc SWII : 2 t’allx I‘or those asteroids modelled as spheroids. 216 Klcopatra as representative of ellipsoidal objects. 39 ,4mphrilritc for the irregularly shaped asteroids. 4 Vesta for xtercjidh

with albedn

marks.

This time MC ha\:c analysed 253 composite lightcurve\ of 53 astcroldh to &rive a rough shape and some charactcristics of each object. Among these asteroids, I8 objects ha\,c been tiehcrihed in the previous paper. For these IX

asteroids. wc have now analysed a larger number of lightcurves with the aim of inferring tt more detailed model ot the objects. We have taken particular care 01‘ 23.3 Ida and 951 Gaspra lightcurve analysis. to compare the shape determined by the Fourier method with the real shape 01‘ these asteroids.

I C‘C,I.CJ.S. The Fourier analysis ot‘rhe lightcurves gives ;I set ol‘coetlicients that fall close to the origin. It should bc noted that the lightcur\e amplitudes arc vcr! small and, consequentI). the values obtained by Fourier ;lnalysis should be uncertain. The largest amplitude is shown b> the tirst component and suggests ;I spherical shape with some minor albedo variegations. -7 P~/ltr.\-. The Fourier expansion 01‘ the lighlcurvcs shows that all the componenls lie cluitc clo\c to the origin. The values of the coefticients art‘ indicati\,e ot‘ ;I quasispherical object \vith some Abedo murk\. 3 ./~r~o. The analysis of the data gi\,c\ wlues ot‘tlle tirst coefhcient higher than the second one in \omc oppositionh. while c’; is Lery low. .4 model ol‘.luno could bc an ablate spheroid with albedo markh. I I +.stu. The data obtained during lY517. 1959. I Yhl. 1073. 1977. 1978 and 19X5 ha\re been ud to dcri\c the Fourier coetticients. The very mall \ alucs 01‘ c J and the larger values of the lirst coef%icnts suggest an txluilibrium spheroidal shape with an albcdo hpcjt on cmisphcrical scale. This model agrees with t how sugsestcd h! Ccllino c’t rrl. ( 19X7. I YXYb) and Drummond (‘I i/l. ( I 08X). 6 Ildx,. The Fourier analysis ot‘ the data obtained at aspects ranging between 60 and 80 gives Gniilar \;ilues of the tirst and second coclficient\. A mndcl of :I spheroid with some albedo variegation5 could c\;plain the Gmilal small Lalues for both these coctficicnl4. 7 11.i.v.The trend of Fourier coctticicnt\ t’or increasing aspect angles. and the high vaiucs ot‘ c , . arc Indicati\,c ()I‘ albcdo variegations on a rough ellipsoid:il shape. Polarimetric studies seem to contirm thi\ niodcl (Broglia and

Manara. IYYO). Y .Ilc/i.\. The hta ot’ nine lightcur\cx ~&en in wu oppositions ( IY-IY. 19%. 1964. lY7-l. lY75, lY7CI. I YXh) refer to apcct angle values bet\\een 45 and ‘)I) t or intcrmediate I alueh of the aspect the F.ouricr c~~ctticion~x lit quite near to the origin, while I‘or increasing aspect angle\ the (“ and (‘, values become imporrant. 4 tria\;ial ellipmid \vith mnc albcdo marks or irrqul;lritia ~_~~ulcihe an explanation ot‘ these rchults. 12 I .ic,/orill. The dab calnc t‘rom I %S and I Y7 I cjppositionh. Thcsc lightcurves show tc>uricr coetticient \ alue\ indicati\c 01‘ an ellipsoidal shape, hill the data al-c rlc>t enough to determine a more detailed model. 13 E~/ro/r~i~. All the analyscd data (IIIIW lightcuncs of 1950. IY51. lY55. 1959. IYW. lYS7. lW5 oppositions) refer to nearly equatorial view\. A quite regular cllipx~idat shape is sugges;tcd by the high ~alutlz oft,,. u hich dominate

the complete 4c1 of Ihe Fourier Ih P.\~I.c~/~c. Sixteen lightcurves ing between

30 and 90

e\;pan~ion.

ot‘ninc oppositions. rangof a3pcct an&. \hom (‘, larger

80

M. T. Capria rt al. : Fourier analysis of the lightcurves

Table 2. Asteroid

Fourier

Asteroid

Cl

1 2

3

4

6

9

12 15

16

19

20

21

22

29

0.019 0.024 0.013 0.053 0.040 0.096 0.086 0.047 0.095 0.085 0.099 0.095 0.134 0.037 0.074 0.016 0.054 0.056 0.082 0.061 0.043 0.069 0.055 0.043 0.095 0.080 0.075 0.025 0.120 0.033 0.007 0.023 0.021 0.028 0.009 0.019 0.023 0.033 0.090 0.080 0.070 0.100 0.077 0.133 0.137 0.140 0.144 0.061 0.088 0.058 0.104 0.043 0.019 0.018 0.059 0.099 0.065 0.026 0.029 0.032 0.043 0.022 0.036 0.062 0.063 0.022

Table 2. Cotttitzurd

coefficients

0.014 0.022 0.012 0.03 1 0.077 0.040 0.080 0.127 0.022 0.018 0.017 0.011 0.020 0.039 0.029 0.031 0.050 0.074 0.109 0.156 0.168 0.040 0.073 0.057 0.122 0.272 0.245 0.196 0.259 0.140 0.204 0.342 0.321 0.349 0.363 0.404 0.443 0.400 0.055 0.044 0.038 0.020 0.042 0.089 0.078 0.181 0.190 0.185 0.170 0.161 0.137 0.136 0.128 0.123 0.05X 0.053 0.187 0.095 0.100 0.107 0.218 0.140 0.025 0.024 0.012 0.054

of 53 known pole asteroids

0.001 0.002 0.015 0.009 0.026 0.006 0.006 0.018 0.014 0.012 0.013 0.014 0.009 0.012 0.007 0.01 I 0.009 0.005 0.013 0.022 0.020 0.03 I 0.026 0.025 0.026 0.007 0.039 0.052 0.037 0.008 0.038 0.032 0.008 0.002 0.035 0.047 0.053 0.016 0.020 0.015 0.007 0.023 0.013 0.016 0.013 0.023 0.023 0.024 0.013 0.020 0.033 0.030 0.007 0.030 0.020 0.010 0.016 0.019 0.006 0.019 0.002 0.003 0.008 0.033 0.059 0.048

(‘4

t

Asteroid

“I

0.003 0.003 0.011 0.006 0.005 0.002 0.011 0.013 0.005 0.004 0.006 0.003 0.010 0.003 0.002 0.015 0.020 0.050 0.036 0.034 0.036 0.054 O.OIO 0.004 0.018 0.023 0.013 0.013 0.027 0.014 0.018 0.062 0.038 0.010 0.058 0.073 0.083 0.104 0.009 0.01 I 0.014 0.010 0.009 0.010 0.015 0.054 0.057 0.03X 0.03 I 0.042 0.017 0.013 0.008 0.003 0.027 0.019 0.00 1 0.011 0.011 0.013 0.042 0.013 0.003 0.009 0.017 0.009

40 44 40 56 71 56 63 63 60 61 64 77 81 86 88 58 66 72 79 76 87 88 48 49 56 69 70 84 87 22 46 73 76 76 78 82 84 85 35 35 37 38 40 54 56 83 X7 63 78 80 80 80 81 89 55 57 88 45 46 46 48 54 24 34 47 70

31

0.030 0.033 0.064 0.074 0.076 0.077 0.092 0.069 0.006 0.057 0.050 0.109 0.010 0.108 0.053 0.061 0.055 0.047 0.035 0.038 0.018 0.061 0.027 0.057 0.026 0.083 0.057 0.077 0.044 0.071 0.018 0.045 0.188 0.015 0.138 0.059 0.026 0.088 0.022 0.025 0.013 0.021 0.044 0.018 0.013 0.067 0.025 0.035 0.203 0.033 0.030 0.092 0.059 0.057 0.042 0.045 0.062 0.076 0.116 0.0’1 0.033 0.028 0.026 0.019 0.03 I

37 39

41

43 44

45

52 55 63

64

87

88 107

135 129

130

201 216

(‘4

0.042 0.032 0.048 0.08 1 0.016 0.154 0.109 0. I50 0.398 0.448 0.255 0.396 0.386 0.131 0.093 0.270 0.078 0.272 0.164 0.237 0.235 0.325 0.351 0.329 0.269 0.389 0.034 0.038 0.211 0.023 0.209 0.237 0.639 0.403 0.904 0.057 0.009 0.08 1 0.322 0.329 0.360 0.092 0.093 0.264 0.321 0.329 0.337 0.373 0.382 0.236 0.147 0.IY.i 0.777 mm0.340 0.265 0.254 0.417 0.091 0.391 0.106 0.139 0.151 0.181 0.253 1.1 I5

0.002 0.009 0.010 0.016 0.007 0.027 0.01 I 0.033 0.030 0.019 0.044 0.089 0.023 0.046 0.029 0.040 0.018 0.068 0.026 0.03 I 0.026 0.019 0.032 0.043 0.027 0.040 0.008 0.004 0.035 0.006 0.011 0.050 0.100 0.035 0.058 0.098 0.006 0.020 0.026 0.047 0.010 0.007 0.009 0.033 0.036 0.043 0.034 0.007 0.165 0.028 0.019 0.008 0.019 0.05 I 0.02Y 0.034 0.065 0.041 0.06 1 0.008 0.01 I 0.013 0.016 0.025 0.046

0.010 0.001 0.011 0.021 0.006 0.014 0.008 0.019 0.055 0. I30 0.050 0.1 IX 0.121 0.012 0.056 0.069 0.01 I 0.041 0.018 0.004 0.036 0.074 0.097 0.07 I 0.025 0.089 0.003 0.003 0.016 0.00’ 0.016 0.023 0.164 0.03 I 0.341 0.049 0.002 0.009 0.044 0.06X 0.073 0.036 0.040 0.042 0.04 I 0.058 0.059 0.081 0.161 0.090 0.013 0.038 0.036 0.05x 0.034 0.063 0.0X4 0.019 0.043 0.00X 0.006 0.024 0.038 0.0’) I O.SY3

49 54 75 17 13 x5 52 62 69 70 71 X7 87 55 77 89 31 60 52 60 62 67 77 83 X5 XX 31 33 77 77 51 77 63 Xl 83 29 33 55 73 7X X5 66 86 69 17 77 81 X2 83 8X 45 59 61 66 6X 71 87 33 85 2s 35 36 37 59 X5

M. T. Capria VI rrl. : Fourier

analysis of the lightcurves

XI

of 53 hnown pole asteroids

(a) 0.X

.Isteroid 143

250 34Y

354

123 333

451 511

532 584

(’ 0 105

0.141 0.051 0.084 0.066 0.050 0.066 O.OY6 0.1 IO 0.057 0.040 0.003 0.049 0.054 0.036 0. 167 0.07 I

0.035 0.050 0.034 0.026 0.034 0.012 0.020 O.OlY 0.043 0.03: 0.05 I

0.0’) 674

0.01 3 0.032 0.074 (I.05 I

6Y4

0.00s (I.1I3

704

(l.OlZ (I.01 I 0.1 I? (I.013 (1.03Y (1.041 0.193 0.037 0.0x4 0.083 0.(17X

951

1580 1620 1685 1862 3908

(‘T

0.323 0.270 0.650 0.588 0.16 I 0.31 I 0.016 0.288 0.301 0.094 0.747 0. I94 0.203 0.066 0.235 I .462 0.960 0.01-l 0.013 0.032 0.02’ 0.070 O.OYt( 0. I?6 0. I55 0. I’7 0. I74 0. I.12 0. I x4

0.08X 0.300 0.715 I.015 0. I70 0.330 0.0’2 0.055 0.1’0 0.371 0.382 0.22Y ’ ‘36 _.__ 0.643 0.176 0.266 0.077

(‘3

(‘4

0.018 0.015 0.060 0.019 0.056 0.027 0.006 0.026 0.029 0.025 0.013 0.014 0.041 0.074 0.026 0.403 0.156 0.026 0.030 0.014 0.016 0.005 0.010 0.014 0.035 0.015 0.049 0.015 0.090 0.006 0.026 0.076 0.062 0.059 0.054 0.023 0.02 I 0.03 I

0.02 I 0.026 0.064 0.032 0.039 0.093 0.005 0.057 0.042 0.023 0.084 0.056 0.046 0.025 0.038 0.634 0.398 0.00s 0.015 0.003 0.008 0.008 0.008 0.01 I 0.012 0.046 0.035 0.032 0.032 0.003 0.039 0.027 0.496 0.013 0.027 0.008 0.029 0.010 0.024 0.028 0.075 1.185 0.128 0.015 0.03 I

0.020 0.016 0.169 0.3Y I 0.036 0.070 0.058 0.058

0.101

55 55 83 x9 77 85 31 X2 84 43 70 71 86 74 35 85 x5 84 63 36 35 54 57 62 65 74 37 41 85 3’ G 71 XY 60 77 36 71 51 67 7’ 51 x7 x3 4(I 42 3’)

than the others. except the 1974 and 1984 oppositions. A good model could be an ellipsoid with strong presence of albedo features, confirmed by polarimetric observations (Broglia and Mnnara. 1992) and in agreement with the model of Lupishko PI II/. ( 1983). /Y Fwtw7rr. The Fourier analysis of the data gives high values of c’, (c, > 0.04) but smaller than the values obtained for (‘I, These results could give an indication on the shape of I9 Fortuna: an ellipsoid with some albedo marks or surface peculiarity. 20 ~4a.s.~/irr. The analysis of data from four oppositions (1955, 1963. 1979, 198?), obtained at nearly equatorial views. shows similar values of c2. The 195.5 lightcurves are described by a high c,, suggesting albedo marks : an

06

C!

0.4

(b) 08

0.6

c1

.

0.4

.

cl.6

.

. CJ

.

0.4

.

Fig. 1. Plots of the amplitudes of the first (a). third (b) and fokth (c) Fourier components versus the amplitudes of the second ones for the asteroid lightcurves. Because: of the very high values of the coetiicients. the data of 1620 Geoyxphos arc excluded

ellipsoid with some albedo features on the asteroid surface could be a good model of Massalia. Z/ Ottetia. The analysis of the lightcurve taken at 90 of aspect angle suggests the shape of a regular ellipsoid, but the high values of C, at intermediate aspect angles indicate the presence of some albedo features. Lutetia seems to have an ellipsoidal shape with some albedo variegations. which become visible at intermediate values ot the aspect. 22 Kdliope. All the data. obtained in five oppositions. refer to intermediate similar aspect angles. The small

M. T. Capria et ul. : Fourier analysis of the lightcurves (4 0.157

‘) 4

0.10

:.:

c,

,’

(b) 0.15

,1,

0.10 c3

0.05,

*

29

I

.2

1~

0.00 0.0

216 1

0.1

-0.2

-Pi

0.3

c2

Cc) 0.15

-,

O.lOi c4 0.05 29 216 4Y * 2 0.00 *s b ..-I_ 0.0' Yr0.2

0.3

.i-

C?

Fig. 2. The arrows

show the ranges

of the first (a). third

(b)

and fourth Cc) Fourier components versus the second one, for increasing values of the aspect angle, of four representative asteroids

values of the coef5cients are indicative of an object not very elongated as an oblate spheroid. .?9 Anzphitt-ire. The five lightcurves analysed show Fourier components that lie near the origin. The relatively high values of C, and cj, decreasing for increasing aspect angles, indicate an irregular, not elongated object (as an ovoid), with possible albedo variegations. 31 Ezrphros~ne. The Fourier analysis of the lightcurves obtained at small aspect angles gives small values of c1 and c,. At large aspect angles. the values of C, increase. exceeding in value the other coefficients; this behaviour suggests a spheroidal shape with some albedo variegations as a possible model of Euphrosyne.

of 53 known pole asteroids

37 Ficics. The analysed lightcurves refer to small aspect angles and show the first Fourier component larger than the second component in one opposition. The values of (‘i and c1 are quite small. A spheroid with a visible albedo mark could be a model of 37 Fides. 3Y Lttc~titiu. On the basis of the analysis of nine lightcurves of eight oppositions this asteroid might be modelled as an ellipsoidal object with some albedo variegations and,or minor irregularities. The c’, has high values for intermediate aspect angles, suggesting albedo features, while the values of the third coeficient are higher for high aspect angles, indicating irregularity in the shape. 41 DLI~Iw. The behaviour of c,. decreasing for increasing values of the aspect. is indicative of large irregularities in the shape. The Fourier analysis suggests an elongated nonsymmetrical object with some irregularities of the surface. 43 .ilriarh. On the basis of the Fourier analysis of six lightcurves ( 1965. I984 oppositions). this asteroid seems to be a nonsymmetric elongated object with irregularities on the surface. The decreasing values of (‘, and the increasing values of c7 and c’?. for increasing aspect angles. are indicative of irregular shape. 44 NJ,.w. The data from nine lightcurves have been analysed. The behaviour of C, and I’? is indicative of an ellipsoidal object with some albedo features which arc described by the amplitudes of the first and third component. 4.5 Euyniu. The analysis of the lightcurves shows relatively high values of c’, at intermediate aspect angles; the trend of c2, for increasing aspect, suggests an ellipsoidal model with albedo features visible at intermediate values of aspect angles. 52 Ettroyxt. The data show similar values of aspect angles (70 ) and low values of the second Fourier component. The values of c, larger than c3 and the very small (‘1 and C’~suggest a model of a regular object as. for example, a spheroid with albedo marks. 55 Pmtiortr. The Fourier expansion of five lightcurves (taken during 1977 and 1984 oppositions) shows trends of I~, and c2 which are indicative of an ellipsoidal shape on which albedo marks are superimposed. h3 Atuo~tict. The data. obtained during 1980. I98 I and 198.5. refer to intermediate and high aspect angles. Ausonia seems to be an elongated ellipsoid, as indicated by the large amplitude of the second Fourier coefficient. with albedo variegations of the asteroid surface. suggested by the high values of the tirst coetlicient. 63 ,~II,~Y(~/IIILI. The lightcurves analysed have been obtained at intermediate values of aspect angles. The C, and the 1’: values are indicative of a quasispherical object with some albedo variegations probably added to irregularities of the surface. &‘7 S\,ll.i~. The data show large values of C, corresponding at high aspect angles. The second coetficicnt dominates the Fourier expansion. even if the third component is not very low. A model of Sylvia could be an elongated ellipsoid with some minor surface irregularities. 8K Tltishc. The analysed lightcurves show similar values of c1 for increasing aspect angles. The values of the first. second and third coefficients obtained at equatorial view suggest that an oblate spheroid could fit these data.

M. T. Capria PI tri. : Fourier analysis of the lightcurves of 53 known pole a+zrolcts 107 C’trr~illtr. Ten lightcurves taken in five oppositions ( 19XI. I981, 1983, 19X4. 1985 ). referring to aspect angles between 60 and 90 . have been analysed. The small values of the tirst component, with the exception of the 1981 lightcurves. and the high c1 suggest a quite elongate eltipxoidal shape with an albedo mark visible in 1981 opposition. I23 Lih~rrr~~i.\-. The analysis of Liberatrix data shows ;~n anomulouz trend of the Fourier components for small increases of aspect angles. All the components decrease Ihr increasing aspect. suggesting a very irregular non\ymmetric shape for this asteroid. 1.29 .-lr~t&o~c. The data from four oppositions (1976. 1981. 1983. lY85) have been analvsed. The values of the \et of Fourier components are indicative of an elongated object, white the high value of C, and the presence of ,‘J suggest an ellipsoidal shape with irregularities and;or ;ilbedo fcaturcs. 130 E/r~/;/r~r. The lightcurvca anatysed were obtained at .Ispect anglea ranging between 70 and 90 The data show increasing ~lucs of c’; for increasing aspect angles. A quite elongated eltipsoldat shape dominates the trend of rhe lightcurvcs, c\en if the values of (‘, and c3 are indicati\,e of some small albedo variegations or irregularities. Xl Prtrelopc~. The analysed data refer to aspect angles ranging between 30 and 90 The behaviour of c2. increasing for increasing aspect angles, suggests an ellipsoidal shape, white the high values of the first and the third iomponents suggest some atbedo variegations and ‘or irregularities of the surface. _‘/d Klrt~~~trr~. The data of 1977. 1982. 1983, 1984. 19X5 give high amplitudes of the second Fourier component at high aspect angles. The small values of C, and ~7~suggest an elongated quite regular ellipsoid as a possible model of Kleopatra. 2.50 B~//i/rrr. The observations obtained during tw’o oppositions ( IWO, 19X3) both refer to high aspect angle values. The Fourier C, has values slightly larger than those due to scattering effct only. suggesting atbebo variegations . m,hitc the (‘? is indicative of an ellipsoidal J1npe. _;3’1 DcJ/II/v)I~.~X~/. The analysis of six tightcurves ( 1965. 1977. 19X4 oppositions) show5 values of c, quite high ; the behaviour 01‘ c’, and C’ indicates an elongated ellipsoid. maybe with one minor asymmetry (relatively high third coefficient) and some atbedo marks. 3.V HOOIIIWC The data of ten tightcurves (taken during four oppositions) have been analysed. The complex trend of c’, for increasing aspect angles and the relatively high \~alucs of c’, arc indicative of an irregular nonsymmetric shape of thib asteroid. 423 L)iori~/tr. On the basis of the analysis of lightcurvea taken in I Y82. referring to aspect angles of about 70 . this object sccm~ 10 be ;I not very elongated ellipsoid with some alhedo marks. as suggested, respectively. by the small \,nlues of c2 and c’ , , slightly higher than those typical of uniform albcdo models. 4j.g EJ-o.v. The 1:ourier expansion of nine lightcur\es gi\,es vcr! Iargc amplitudes of the coefficients. The trends. for incrensintr ~ aspects. of these components suggest a ver! elongated shape with large asymmetries and strong irregularities.

cY:

451 P~ticnti~. This asteroid seems to be described by ;I spheroidal model with an atbedo spot, as follows from the analysis of the Fourier coefficients obtained for the 1974 and 1979 lightcurves. These data she\\ values of c and C, larger than C” values. -51I Doritltr. The analysis of eight lightcurves (taken during 1952. 1958, 1962. 1982. I YX4 and 1086) shows a (‘, descending trend and c2 increasing trend for increasing aspect angles. This behaviour i, similar to that of ;I truncated ellipsoid model. which seem4 to lit the 5 I I Davida data. 53.2 H~/~c~~/i~tr. On the basix of the analtsi:, of I‘OUI lightcurves this asteroid seems to bc an ellipsoid not \‘erh elongated, as suggested by retativcly lo\\, \aluea of (‘> at high aspect angles. with sonic albdo features on the surface. indicated by the large tir\t Fourier component. 5S4 Se~~btr~~i.~.The analysed data show high values of the third component, indicati\,c 01‘ an irregular shape. A model of Semiramib could he an a\ymmctric irregular object inscribed in an ellipsoid. h-7-! Hrl\/o~. The Fourier expansion 01‘ the data obtained in four oppositions ( 1957. 1965. I Y6?. I Y6X) ih dominated by the second component. The trend of C‘ and c’,. fat increasing aspects. indicates a~1 dongated ellipsoidal bhapc of this asteroid. 6%’ EX-OI.~/.The analysis of three lightcur\,es shows relatively high values of the first and second coel~icients. The bchaviour of all the set of Fourier components suggests ;I model of an ellipsoid with a deformation (or albedo hpot) visible at ncarl!, equatori;ll \ ICU4. 70-1 I~/c~~tr/~r~irr.These data refer to I969 and I Y7-l opp~~ sition and shob IOU value\ 01‘ C’ for increasing aspect anglca. suggesting a quasi-spheroidal shape 01‘ Interamnia. /iSO B~t/r/i~. This sstero~d <~CN \ unusual light cur\xs with three mnsima and three minima. The ,inal\s~s of lY76 data gives high flues of the lirst and the tliird Fourier components (c,: larger than c’,). Betu)ia could have an irregular shape. \crh dilti-rcn( t‘rom clas\ic~al cqulibrium shapes. In.20 (;(‘o!/I.L//~/~~.\.The lightcurve analysed have been obtained at \rery large phase angles. All the Fourier components hai,e high values, but the ~cond component dominates the others. A \‘er!’ elongated object \\ith some albcdo variegations and ‘or irregularities could hc ;t model of Geographos. 11’Si To/.c~. The data, referring to ~111aspect angle 01‘ about 70 . show high values of all f-ourier components. An elongated object with an albedo mark and or :I large irregularity \isihle when the aspect ih intermediate can explain this result. ISh.2 .-l/~//o. The data 01‘ 19x0 and IW2 have been analysed. The values of the second component. referring to intermediate aspects. and the relati\cl\; high values of and c2. suggest ~1 quite dongatd ohiect \\,ith WJW Zib e d 0 variegations and irrcgulariti~s of Ihe asteroid surface. _?(IOSIYKO P.l. The analyscd lightcurvc~ show rclati\,elh high \xlues of c’~ and c’?. The behaviour of the Fourier asymmetric objccr. components suggests an clongatcd with the presence of irregularitie\ on the \urt’ace.

84

M. T. Capria et rrl. : Fourier

analysis of the lightcurves

of 53 known pole asteroids

Shapes of.243 Idu and 951 Gaspra \

The Galiko spacecraft carried out the first encounter with minor planets, making a flyby of the asteroids 951 Gaspra and 243 Ida. During the encounters Gulileo obtained images and measurements of the shapes and characteristics of these two objects (Chapman, 1994). We have analysed the ground-based lightcurves of these asteroids to compare the inferred rough shapes with the Gulileo images. ,343 Ida. The Fourier analysis of the lightcurves, spanning from 1984 to 1993 oppositions, gives a set of coefficients which are indicative of a very irregularly shaped object. The lightcurves show decreasing values of the first component and increasing values of cm2for increasing aspect angles. This behaviour suggests the presence of large irregularities in an object having elongated shape (large amplitude of c2). The high values of c, ( >0.04) could be due to albedo marks on the surface of Ida, as indicated by spectral differences. 951 Gaspru. The Fourier expansion of the lightcurves obtained during 1988, 1990 and 1991 oppositions shows in the c,-cl plane a trend similar to that obtained from the synthetic lightcurves of a truncated ellipsoid model (Barucci ef ul., 1989). For increasing values of the aspect angles there is a decrease of the first coefficient and an increase of the second one. The values of (‘i and of (‘J are relatively low. From this analysis it seems that a good model for 951 Gaspra could be a nonsymmetrical object, as a truncated ellipsoid, with no presence of large albedo features. The analysis of Gufileo’s images of these asteroids shows that the derived models are quite realistic, confirming that some indications on the shape and albedo variegation can be obtained using the Fourier coefficients method.

Concluding remarks The result of our analysis can be summarized as follows : 22.6% of the considered asteroids have a quasi-spherical shape. and most of them show evidence of some albedo variegations. One object (64 Angelina) seems to have a slightly irregular spheroidal shape, and constitute a transition element between spheroids and irregulary shaped asteroids, which constitute the 28.3% of our sample. The remaining 47.2% belongs to the large group of the ellipsoidally shaped asteroids : five members of this group. mostly the smallest, show in the Fourier coefficients the signature of some small/medium surface irregularities. and we consider these asteroids as the transition group between ellipsoids and irregular objects. In Fig. 3 a pie chart summarizes these results. Thirty out of the 53 analysed asteroids have a diameter larger than 125 km, the value adopted by Farinella rt L/I. (1981) as the lower diameter value for objects having an equilibrium shape. Nine of them have a diameter smaller than 50 km, a value adopted by Farinella et ul. (1981) and Fulchignoni rt a/. (1995) as the upper limit of the population of “small” asteroids, and 14 have intermediate diameters. Of course. due to the fact that asteroid diam-

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objects

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Fig. 3. Pie chart of the distribution of asteroid shapes : ellipsoids 37.80/o, irregular ellipsoids 9.4%. irregular objects 28.39io. irregular spheroids 1.9%. and spheroids 22.6%

eters are known with a very poor precision. this is just a qualitative description: all the limits (large. small, etc.) are purely indicative. It is not surprising that most (71.7%) of the asteroids of our sample are interpreted on the basis of the adopted analysis of the Fourier coefficients as regularly shaped bodies. varying from spheroids to elongated ellipsoids with some large albedo marks on l/3 of them. The objects (17% of the total sample) having diameters in the range between 50 and 125 km. belonging to the regular objects group, show high values of the odd Fourier coefficients indicating the presence of some irregularity on their shape. A total of 28.3% of our sample seems to be formed by irregularly shaped objects: half of these asteroids are much smaller than 50 km and probably consist of fragments produced by collisions (as in the case of Ida) ; the remnant half are larger (D > 50 km) and their irregularity may be explained by variations in the surface curvature (truncated shapes. as the case ofGaspra) due to the effects of the fragmentation. Surface albedo variegations are less frequent in this group. The adopted sample of asteroids is biased because it contains mostly large objects, those of which the pole coordinates are available. More than 2:‘3 of the analysed asteroids have a quasi-regular shape. because of their dimensions, but the remaining of the sample. which includes all the objects with small diameter. is characterized by values of Fourier coefficients that indicate irregularities in the shape. These irregularities can be the results of the collisional processes experienced by the objects during their evolution which are modifying their original shapes, through a total fragmentation (smaller objects) or a major reshaping. The obtained results cannot be interpreted in statistical sense. but are very interesting. The suggested analysis can be adopted as a tool to investigate the evolution of the asteroid population.

.4~.kno~l,l~r!yef1lrrlt. We are very grateful to A. Cellino for his comments and suggestions which helped us improve the text.

M. T. C‘apria (pi (ri. : Fourier

anulpsis of the lightcurves

of 53 known pole astrroic!\

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X5

vations of asteroids 3. 24. 60. 261 and X63. IWW\ 77, I71 186, 198Y. Lagerkvist. C. I., Barucci, M. .4., Capria, M. T., Guerriero, L., Fulchignoni, M., Perozzi, E. and Zappali, V.. Asteroid photometric catalogue. Consiglio Narionale delle Ricerche Ed.. Roma. 1987. Lagerkvist, C. I., Barucci, M. A., Capria, M. T., Fulchignoni, M., Magnusson. P. and Zappal& V., .Asteroid photometric catalogue. tirst update. Consigho Nazlonals deile Ricerche Ed.. Roma. IYXX. Lagerkvist, C. 1.. Harris, A. W. and Zappal& \ ., Asteroid hghtcurve parameters. in .4strroiJs /I (edited by R. P. Blnzel. T. Gehrels and M. S. Matthews). pp. I Ifi? I 179 lim\crsity 01‘ Arizona Press. Tucson. Arizona. IYSY Lagerkvist. C. I., Barucci, M. A., Capria, M. T., Dahlgren, M., Erikson. A., Fulchignoni, M. and Magnusson, P., Asteroid photometric catalogue. second update. I’ppsala Ilniversitet. Cppsala. I YY_7, Lupishko, D. F.. Akimov, L. A. and Belskaya, 1. 5.. On photometric hetrrogeneitl of asrcroid surtax in .4 \/f~i&. C‘O/>W!.\..2/[~for~.\ (edited by <‘. I. Laperk\lst and II. Rickmxi), pp. h!- 70. Uppsala Unlvrrsitet. Uppsala. 19x3. Magnusson. P., Pole determinations ot‘;~~roid~. in .~l.\~~oit/.~/I (edited hy R. P. Binzel. T. G&r& ami M. S. Matthews). pp. I I XI)- I IW I!ni\er5ity of Arirona Pre$\. Tucson. 2rimna.

1w’). Magnusson, P., Private communication. lYY3. Magnusson. P., Barucci, M. A., Drummond, J. D.. Lumme, K., Ostro, S. J.. Surdej, J., Taylor, R. c’. and Zappali, V., Determination of pole orientations and zhapeb of asteroids. in .-l.\/c~ic/.\ /I (edited by, R. Binzel. T. Gehrel:, and M. S. Matthews). pp. 66~Y7. Umversit! of Arirona PI-~\. Tucson. Arizona. I WY, Magnusson, P., Barucci, M. A., Binzel. R. P.. Blanco, c‘.. Di Martino, hl., Goldader, J. D., Gonano-Beurer, M., Harris, A. W., Michalowski, T., Mottola, S., Tholen, D. J. and Wisnieaski, W. Z.. Asteroid 951 Gaspra : pre-Galileo physical model. fWIc.s 97. I’4 12’). I YY2. Michalowski, T.. Poles. shapes. sense5 01‘ rotalion and sidereal periods of‘xsteroids. f~~tr~~r.s 106. 563 572. lYY3 Ostro, S. J., (‘onnelly, R. and Dorogy, M.. C‘onves-profile lnvcrsion 01’‘Istcroid lightcurves : theor! and applications. /WKIJ.\ 75, 30- 63. 1088. Russell, H. N., On the light variations ol‘asteroid~ and qteltites. .-l.rt,op/l~l~a. ./. 24, l-1 8. 1906. Tedesco. E. F., Veeder, G. J.. Fowler. .J. W. and C’hillemi, .J. R., The IR.4S Minor Planet Surve\. Philtip Lahor;ktory. Massachuzetts. IYY’. Tholen, D. J., Asteroid tauonomic classltications. in .-I.\~~~~itl.\/I (edIted h> R. Binzel. T. Gehrels and M. S. Matthews). pp. I I3Y-- I I SO l’nicrrsity of Arirona Prra. Tucson. Arizona. IUSY. Tholen, D. J.. Ephemeris Program ‘&Ephem”. I YY?. Williams, G. J.. Asteroid fanil?; identification\ and propal element>. in .4.\/~oitl.s II (edited by R. Binzel. T Gehrcls and M S. Matthews). pp. 107.7 ~IOXY I ‘nl\ckty of‘ Arizona Press. Tucson. Arizona. I9XY. Wisniewski, W. Z., Barucci, M. A., Fulchignoni, .%I., De Sanctis, 11. C.. Dotto, E.. Rotundi, A., Binzel. R. P., Madras, C. D., Green. S. F., Kelly, M. L., Newman. P. J., Harris, A. W., Joung, J. W., Blanco, C.. Di Martino, M., Ferreri, W.. Gonano-Beurer, M., Mottola. S., Tholen, D. J., Goldader, J. D., Coradini, M. and Magnusson, P., Ground-based photometry of :tsteroid 951 Gaspra. I0~rr.s 101. 2 I3 121. lY91.