Lightcurves and poles of seven asteroids

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Planetary and Space Science 53 (2005) 1147–1165 www.elsevier.com/locate/pss

Lightcurves and poles of seven asteroids M.J. Lo´pez-Gonza´lez, E. Rodrı´ guez Instituto de Astrofı´sica de Andalucı´a, CSIC, P.O. Box 3004, E-18080 Granada, Spain Received 10 November 2003; received in revised form 5 April 2005; accepted 18 April 2005 Available online 17 June 2005

Abstract The Asteroids 5 Astraea, 19 Fortuna, 51 Nemausa, 68 Leto, 138 Tolosa, 196 Philomela and 409 Aspasia have been studied using Stro¨mgren photometry observations made in 1997. Simultaneous lightcurves in the uvby Stro¨mgren filters of synodic periods of 0.70004
0.00020, 0.31013
0.00003, 0.32400
0.00020, 0.61910
0.00100, 0.42087
0.00009, 0.34750
0.00020 and 0.37576
0.00060 days, and amplitudes, in the y filter, of 0.m 16
0.m 05, 0.m 30
0.m 02, 0.m 15
0.m 04, 0.m 29
0.m 02, 0.m 43
0.m 03, 0.m 30
0.m 04 and 0.m 15
0.m 06 have been found for 5 Astraea, 19 Fortuna, 51 Nemausa, 68 Leto, 138 Tolosa, 196 Philomela and 409 Aspasia, respectively. Additional observations of 138 Tolosa and 196 Philomela during February 2000, show lightcurve amplitudes equal or greater than 0.m 15 for 138 Tolosa and of 0.m 45 for 196 Philomela in 2000 opposition. Solutions for the sense of rotation, sidereal period, pole orientation and shape properties have been proposed for the first time for 138 Tolosa and improved solutions have been obtained for 5 Astraea, 19 Fortuna, 51 Nemausa, 196 Philomela and 409 Aspasia. r 2005 Elsevier Ltd. All rights reserved. Keywords: Asteroids; Photometry; Rotation; Lightcurves

1. Introduction Lightcurves from different aspects as an asteroid orbits provide information about spin period, axis orientation and body shape. In this paper we continue our work of adding to the available asteroid photometric data and obtaining or improving the rotational properties of selected asteroids (Lo´pez-Gonza´lez and Rodrı´ guez, 1999, 2000). Lightcurves of 5 Astraea, 19 Fortuna, 51 Nemausa, 68 Leto, 138 Tolosa, 196 Philomela and 409 Aspasia from 1997 observations at the observatory of Sierra Nevada, Spain, are presented. Additionally two nights of observations of 138 Tolosa and 196 Philomela obtained in February 2000 are also presented. Simultaneous uvby photometry was obtained from each asteroid and used to derive Stro¨mgren and Johnson

Corresponding author. Fax: +34 958 814530.

E-mail address: [email protected] (M.J. Lo´pez-Gonza´lez). 0032-0633/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2005.04.010

zero-phase angle reduced magnitudes and colour indices. Rotational and shape parameters for these asteroids have also been obtained and compared with previous results. The lightcurves presented in this work will be added to the asteroid photometric data to improve future work on the photometric and rotational properties of these asteroids.

2. Observations The observations were carried out on different nights from July to October on 1997, and an additional two nights of February 2000. The observations were performed with the 90 cm telescope at Sierra Nevada Observatory, Spain. This telescope is equipped with a six channel uvbyb photometer for simultaneous measurements in uvby or in the H b channels (Nielsen, 1983). Only uvby measurements were made for the presented observations.

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Table 1 Aspect data and comparison stars Date (0 UT)

Long (deg) (1950)

Lat (deg) (1950)

Phase (deg)

r (AU)

D (AU)

C1 (SAO)

C2 (SAO)

5 Astraea 25 7 1997 26 7 1997 27 7 1997 29 7 1997 30 7 1997 31 7 1997 02 8 1997 04 8 1997 05 8 1997 08 9 1997

333.69 333.54 333.38 333.05 332.88 332.70 332.34 331.96 331.76 324.34

0.33 0.36 0.38 0.43 0.45 0.47 0.52 0.57 0.59 1.36

10.18 9.86 9.54 8.88 8.55 8.21 7.51 6.80 6.44 6.62

2.15948 2.15200 2.14475 2.13095 2.12441 2.11812 2.10628 2.09548 2.09046 2.08750

3.06675 3.06656 3.06635 3.06593 3.06570 3.06548 3.06500 3.06450 3.06425 3.05197

146135 (V ¼ 6:m 2, F8) ’’ ’’ ’’ ’’ ’’ ’’ ’’ ’’ 164693 (V ¼ 7:m 9, G0)

146111 (V ¼ 7:m 6, G0) ’’ ’’ ’’ ’’ ’’ ’’ ’’ ’’ 164779 (V ¼ 7:m 5, G5)

19 25 26 27 29 30 31 02 04 05 08 09 10

Fortuna 7 1997 7 1997 7 1997 7 1997 7 1997 7 1997 8 1997 8 1997 8 1997 9 1997 9 1997 9 1997

336.25 336.16 336.05 335.82 335.70 335.57 335.28 334.97 334.80 327.29 327.08 326.88

2.46 2.47 2.47 2.49 2.50 2.50 2.51 2.52 2.53 2.35 2.33 2.32

14.86 14.47 14.07 13.25 12.83 12.40 11.52 10.61 10.15 7.98 8.49 9.01

1.35630 1.34819 1.34026 1.32497 1.31762 1.31047 1.29678 1.28394 1.27784 1.21207 1.21441 1.21698

2.26521 2.26354 2.26187 2.25854 2.25689 2.25523 2.25193 2.24864 2.24700 2.19372 2.19224 2.19076

146135 (V ¼ 6:m 2, F8) ’’ ’’ ’’ ’’ ’’ ’’ ’’ ’’ 164693 (V ¼ 7:m 9, G0) ’’ ’’

146111 (V ¼ 7:m 6, G0) ’’ ’’ ’’ ’’ ’’ ’’ ’’ ’’ 164779 (V ¼ 7:m 5, G5) ’’ ’’

51 26 27 29 30 31 02 04 05

Nemausa 7 1997 7 1997 7 1997 7 1997 7 1997 8 1997 8 1997 8 1997

280.08 279.89 279.53 279.35 279.19 278.88 278.59 278.46

15.17 15.08 14.91 14.82 14.74 14.55 14.36 14.26

10.92 11.26 11.94 12.28 12.62 13.30 13.96 14.29

1.46288 1.46817 1.47940 1.48535 1.49151 1.50447 1.51825 1.52544

2.41347 2.41415 2.41552 2.41620 2.41688 2.41823 2.41958 2.42025

142443 (V ¼ 7:m 9, G0) ’’ ’’ ’’ ’’ ’’ ’’ ’’

142326 (V ¼ 6:m 5, K0) ’’ ’’ ’’ ’’ ’’ ’’ ’’

68 02 03 07 08 09 10 11 12

Leto 10 1997 10 1997 10 1997 10 1997 10 1997 10 1997 10 1997 10 1997

44.70 44.58 44.05 43.90 43.75 43.59 43.42 43.25

3.02 2.97 2.77 2.71 2.66 2.60 2.55 2.49

14.66 14.26 12.58 12.15 11.71 11.27 10.82 10.36

1.48043 1.47478 1.45417 1.44954 1.44513 1.44094 1.43698 1.43325

2.36045 2.36178 2.36717 2.36853 2.36990 2.37128 2.37267 2.37406

093169 (V ¼ 7:m 8, G5) ’’ ’’ ’’ ’’ ’’ ’’ ’’

093085 (V ¼ 7:m 6, F5) ’’ ’’ ’’ ’’ ’’ ’’ ’’

138 Tolosa 30 7 1997 31 7 1997 04 8 1997 05 8 1997 08 9 1997 02 2 2000 03 2 2000

332.93 332.79 332.16 331.99 324.95 199.46 199.54

6.00 6.03 6.13 6.15 5.98 3.27 3.28

13.15 12.66 10.68 10.17 9.89 19.84 19.71

1.09406 1.08976 1.07452 1.07123 1.09313 2.15482 2.14080

2.04965 2.04986 2.05080 2.05107 2.06676 2.68740 2.68609

146135 (V ¼ 6:m 2, F8) ’’ ’’ ’’ 164693 (V ¼ 7:m 9, G0) 139256 (V ¼ 7:m 8, G0) ’’

146111 (V ¼ 7:m 6, G0) ’’ ’’ ’’ 164779 (V ¼ 7:m 5, G5) 139271 (V ¼ 8:m 3, F2) ’’

44.20 44.08

6.26 6.26

10.93 10.64

2.27915 2.27143

3.14071 3.14094

093169 (V ¼ 7:m 8, G5) ’’

093085 (V ¼ 7:m 6, F5) ’’

196 Philomela 02 10 1997 03 10 1997

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Table 1 (continued ) Date (0 UT)

Long (deg) (1950)

Lat (deg) (1950)

Phase (deg)

r (AU)

D (AU)

C1 (SAO)

C2 (SAO)

07 08 09 10 11 12 02 03

43.55 43.40 43.25 43.10 42.94 42.78 197.72 197.77

6.25 6.24 6.23 6.23 6.22 6.21 8.28 8.31

9.44 9.13 8.81 8.49 8.17 7.85 16.78 16.66

2.24284 2.23628 2.22996 2.22389 2.21807 2.21251 2.59327 2.58003

3.14186 3.14209 3.14232 3.14255 3.14278 3.14301 3.12329 3.12305

’’ ’’ ’’ ’’ ’’ ’’ 139256 (V ¼ 7:m 8, G0) ’’

’’ ’’ ’’ ’’ ’’ ’’ 139271 (V ¼ 8:m 3, F2) ’’

274.67 274.52 274.23 274.10 273.98 273.75 273.54 273.45

12.70 12.71 12.72 12.72 12.72 12.72 12.70 12.70

12.03 12.39 13.10 13.45 13.79 14.46 15.11 15.43

1.52648 1.53295 1.54651 1.55359 1.56088 1.57602 1.59193 1.60015

2.45748 2.45806 2.45923 2.45981 2.46040 2.46158 2.46277 2.46337

142443 (V ¼ 7:m 9, G0) ’’ ’’ ’’ ’’ ’’ ’’ ’’

142326 (V ¼ 6:m 5, K0) ’’ ’’ ’’ ’’ ’’ ’’ ’’

10 1997 10 1997 10 1997 10 1997 10 1997 10 1997 2 2000 2 2000

409 Aspasia 26 7 1997 27 7 1997 29 7 1997 30 7 1997 31 7 1997 02 8 1997 04 8 1997 05 8 1997

In order to perform differential photometry, different sets of comparison stars were used for each selected asteroid. The comparison stars were chosen taking into account their spectral type (close to solar) and close to the asteroids for better reduction of the data avoiding extinction problems. Each set of comparison stars contains one main comparison star, C 1 , and at least one check star, C 2 . Table 1 shows the comparison stars identifications together their visual magnitudes and spectral types. During the observations reported here, neither of the comparison stars showed any sign of variability at the level of
0.m 005. To transform our data into the standard uvby system we have used the same procedure described in (Rodrı´ guez et al., 1997). After differential magnitudes in the standard system were obtained for the asteroids with respect to the main comparison star (C 1 of each group) we correct them to unit distance from the sun and the earth and perform light-time corrections on all the observations. Then, these differential magnitudes are transformed to absolute magnitudes using the absolute values of the respective main comparison star, C 1 as listed in the Hauck and Mermilliod (1998) catalogue.

3. Analysis The aspect data for each asteroid, the longitude and latitude relative to the ecliptic, the solar phase angle, the geocentric, r, and heliocentric, D, distances for every night of observation, are listed in Table 1.

The internal errors of the observations of these selected asteroids are within the error limits of 68 Leto, which present the smallest observational errors, and of 5 Astraea, which present the largest observational errors. These are in the u filter from 0.m 009 for 68 Leto to 0:m 018 for 5 Astraea; in the v filter from 0.m 004 for 68 Leto to 0.m 008 for 5 Astraea; in the b filter from 0.m 003 for 68 Leto to 0.m 006 for 5 Astraea; and in the y filter from 0.m 004 for 68 Leto to 0.m 006 for 5 Astraea. This means internal observational errors smaller than 0.m 02 in the u filter and smaller than 0.m 01 in the vby filters. 3.1. Photometry These observations were carried out during different nights from July to October on 1997 and two nights of February 2000. Table 2 shows the average phase angle of the observations, a, and the range of phase angle covered, for each asteroid. The mean reduced magnitudes observed for each uvby Stro¨mgren filter, uvbyð1; aÞ, and the observed Stro¨mgren colour indices, b-y and u-b, are also listed. For 1997 observations, we have applied a linear phase correction to transform the observed reduced magnitudes, uvby (1,a), to zero-phase angle magnitudes, uvby (1,0). The zero-phase angle magnitudes are calculated from uvbyð1; aÞ ¼ uvbyð1; 0Þ þ bm a, where bm is the mean linear phase coefficient, that is more suitable for the observed magnitudes in all the Stro¨mgren filters. The uvby (1,0) Stro¨mgren values are converted into the UBV Johnson system using the equations of Warren and Hesser (1977) to transform the Stro¨mgren b-y and u-b colour indices to Johnson B-V and U-B colour

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1150 Table 2 Photometry uvbyð1; aÞ (mag)

uvbyð1; 0Þ (mag)

UBVð1; 0Þ (mag)

TRIAD (mag)

5 Astraea y 7:32
0:08 b 7:88
0.09 v 8:62
0.09 u 9:72
0.15 b-y 0.56
0.04 u-b 1.84
0.11

(a ¼ 7:0 97 y 7:13
0:07 b 7.69
0.07 v 8.44
0.08 u 9.53
0.15


1:0 91) V ð1;0Þ 7.13
0.07 Bð1;0Þ 8.04
0.14 U ð1;0Þ 8.40
0.22

V ð1;0Þ 7.43 Bð1;0Þ 8.26

B-V 0.90
0.07 U-B 0.35
0.08

B-V 0.83 U-B 0.42

51 Nemausa y 8.15
0.07 b 8.64
0.07 v 9.40
0.07 u 10.63
0.07 b-y 0.49
0.02 u-b 1.99
0.06

(a ¼ 12:0 89 y 7.64
0.05 b 8.13
0.04 v 8.89
0.04 u 10.12
0.07


1.0 70) V ð1;0Þ 7.64
0.05 Bð1;0Þ 8.43
0.07 U ð1;0Þ 8.90
0.10

V ð1;0Þ 7.67 Bð1;0Þ 8.45

B-V 0.79
0.03 U-B 0.46
0.04

B-V 0.78 U-B 0.46

138 Tolosa y 9.49
0.11 b 10.13
0.12 v 10.91
0.13 u 12.05
0.15 b-y 0.64
0.03 u-b 1.91
0.10

(a ¼11.0 16 y 9.25
0.11 b 9.89
0.12 v 10.66
0.13 u 11.80
0.16

196 Philomela y 7.09
0.09 b 7.66
0.09 v 8.42
0.09 u 9.60
0.10 b-y 0.56
0.02 u-b 1.93
0.04

(a ¼9.0 05 y 6.75
0.09 b 7.31
0.09 v 8.07
0.09 u 9.20
0.10

409 Aspasia y 8.45
0.10 b 8.93
0.09 v 9.58
0.09 u 10.68
0.11 b-y 0.48
0.03 u-b 1.76
0.07

(a ¼14.0 02 y 7.75
0.06 b 8.24
0.06 v8.89
0.05 u 11.82
0.14

a


1.0 53) V ð1;0Þ 9.25
0.11 Bð1;0Þ 10.29
0.15 U ð1;0Þ 10.70
0.22

V ð1;0Þ 9.01 Bð1;0Þ 9.89

B-V 1.04
0.05 U-B 0.41
0.07

B-V 0.88 U-B 0.49


1.0 58) V ð1;0Þ 6.75
0.09 Bð1;0Þ 7.70
0.10 U ð1;0Þ 8.10
0.10

V ð1;0Þ 6.81 Bð1;0Þ 7.65

B-V 0.90
0.03 U-B 0.42
0.03

B-V 0.84 U-B 0.44


1.0 69) V ð1;0Þ 7.75
0.06 Bð1;0Þ 8.50
0.10 U ð1;0Þ 8.80
0.20

V ð1;0Þ 7.75 Bð1;0Þ 8.47

B-V 0.77
0.05 U-B 0.30
0.05

B-V 0.72 U-B 0.34

uvbyð1; aÞ (mag)

uvbyð1; 0Þ (mag)

UBV ð1; 0Þ (mag)

TRIAD (mag)

19 Fortuna y 7:85
0:12 b 8:31
0.13 v 9:01
0.13 u 10:18
0.13 b-y 0.46
0.01 u-b 1.86
0.04

(a ¼ 11:0 44 y 7:30
0:06 b 7.76
0.07 v 8.46
0.07 u 9.63
0.07


3:0 05) V ð1;0Þ 7:30
0:06 Bð1;0Þ 8.04
0.08 U ð1;0Þ 8.41
0.11

V ð1;0Þ 7.42 Bð1;0Þ 8.16

B-V 0.73
0.02 U-B 0.37
0.03

B-V 0.74 U-B 0.38

68 Leto y 7.64
0.10 b 8.20
0.10 v 8.95
0.10 u 10.20
0.11 b-y 0.55
0.01 u-b 2.00
0.02

(a ¼12.0 06 y 7.24
0.08 b 7.79
0.08 v 8.55
0.08 u 9.79
0.08


2.0 21) V ð1;0Þ 7.24
0.08 Bð1;0Þ 8.12
0.09 U ð1;0Þ 8.59
0.10

V ð1;0Þ 7.15 Bð1;0Þ 7.99

B-V 0.88
0.02 U-B 0.47
0.02

B-V 0.84 U-B 0.49

138 Tolosaa y 9.70
0.05 b 10.20
0.10 v 11.10
0.10 u 12.30
0.10 b-y 0.54
0.07 u-b 2.10
0.20

(a ¼19.0 75

196 Philomelaa y 7.56
0.24 b 8.08
0.16 v 8.89
0.18 u 10.15
0.24 b-y 0.52
0.03 u-b 2.10
0.11

(a ¼16.0 7


0.0 05) V ð1;0Þ 9.01 Bð1;0Þ 9.89

B-V 0.90
0.10 U-B 0.50
0.10

B-V 0.88 U-B 0.49


0.0 05) V ð1;0Þ 6.81 Bð1;0Þ 7.65

B-V 0.84
0.05 U-B 0.52
0.08

B-V 0.84 U-B 0.44

Photometry from February 2000 observations.

indices. The Johnson magnitudes, UBV ð1; 0Þ, and colour indices obtained are contained in Table 2. 3.2. Synodic periods and amplitudes Rotational synodic periods for each asteroid have been calculated by frequency analysis of our data using the method described in Rodrı´ guez et al. (1998). Synodic periods of 0.70004
0.00020, 0.31013
0.00003, 0.32400
0.00020, 0.61910
0.00100, 0.42087
0.00009, 0.34750
0.00020 and 0.37576
0.00060 days are

obtained for 5 Astraea, 19 Fortuna, 51 Nemausa, 68 Leto, 138 Tolosa, 196 Philomela and 409 Aspasia, respectively. Figs. 1–7 show the composite lightcurves derived using these synodic periods for each asteroid. The (1,0) magnitudes derived in the different Stro¨mgren filters and the b-y, v-b and u-b colour indices versus the rotational phase are also shown in these figures. The composite lightcurves for each asteroid show different characteristics. Most of them present two regular maxima and minima of differing depth and amplitude per rotational

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Fig. 1. Lightcurves and colour indices of 5 Astraea in rotational phase. The 0 phase time corresponds at JD 2450655.4851 corrected for light-time.

cycle, however three maxima and minima are clearly distinguished in 51 Nemausa, while 409 Aspasia shows an overall ‘‘single maximum and minimum’’ lightcurve. The maximum amplitude of the lightcurves together with the synodic periods of the selected asteroids are listed in Table 3. The maximum amplitudes in the y filter are 0.16
0.05, 0.30
0.02, 0.15
0.04, 0.29
0.02, 0.43
0.03, 0.30
0.04 and 0.15
0.06 magnitudes for 5 Astraea, 19 Fortuna, 51 Nemausa, 68 Leto, 138 Tolosa, 196 Philomela and 409 Aspasia, respectively. The uncertainty in the amplitude determinations is of similar value for the vby filters while is of slightly greater value in the u filter. Extreme cases are those of 5 Astraea and 409 Aspasia where due to the scatter of the data in the u lightcurves the values of the amplitude of the lightcurves obtained are comparable with the values of the uncertainty in the amplitude determinations. There are not significant variations in the amplitude of the

lightcurves in the different uvby filters, being the differences in the amplitude in the uvby filters smaller than the error values of the amplitude determinations (see Table 3). 3.3. Poles and shapes Poles and shapes for these asteroids have been determined using the Epoch/Amplitude method (e.g., Taylor, 1979; Magnusson, 1986; Magnusson et al., 1989; Michalowski and Velichko, 1990; De Angelis, 1993). The Epoch and amplitude equations were resolved following Lo´pez-Gonza´lez and Rodrı´ guez (1999). Thus solutions were found for the sidereal period of rotation of the asteroid, T sid , the pole coordinates of the asteroid, lp and bp , and the ratios of the symmetry axes of the asteroid (considered as a triaxial ellipsoid of axis aXbXc rotating about their shortest axis), a=b and b=c.

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Fig. 2. Lightcurves and colour indices of 19 Fortuna in rotational phase. The 0 phase time corresponds at JD 2450655.4803.

Lightcurve data reported in the literature together with lightcurves obtained here for these asteroids have been used in the analysis. Many of the lightcurves used in this work can be found in the Asteroid Photometric Catalogue (Lagerkvist et al., 1987a, 1989, 1992). In Table 4 we have listed the final solution obtained for T sid ; lp ; bp ; a=b and b=c for each asteroid.

4. Results 4.1. 5 Astraea 5 Astraea is a S-type asteroid (Tholen, 1989) with a diameter of 125 km (Tedesco, 1989). 5 Astraea has been observed in eight oppositions between 1958 and 1997 (Gehrels and Owings, 1962; Chang and Chang, 1962; Taylor, 1978; Pavlovski et al., 1981; Melillo, 1987;

Weidenschilling et al., 1990; Harris et al., 1999; Riccioli et al., 2001; Shevchenko et al., 2002). Observations of 5 Astraea in 1997 opposition are considered here. We find a synodic period for 5 Astraea of 0.d 70004
0.d 00020 (16.h 801) The composite lightcurves obtained for this asteroid show two maxima and two minima per rotational cycle in all the uvby filters. The colour indices do not show any variation greater than the scatter of the data during the rotational phase of this asteroid (see Fig. 1). 5 Astraea was observed independently in its 1997 opposition by Riccioli et al. (2001) and Shevchenko et al. (2002), and in both cases partial lightcurves were reported. Shevchenko et al. (2002) reported a value of the amplitude in the V filter of 0.m 16 while Riccioli et al. (2001) reported a value greater than 0.m 17 in the V filter. From our lightcurve we find a maximum amplitude of 0.m 16 in the y filter with an uncertainty of
0.m 05 in the determination of the amplitude. The amplitude values of

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Fig. 3. Lightcurves and colour indices of 51 Nemausa in rotational phase. The 0 phase time corresponds at JD 2450656.3739 corrected for light-time.

the lightcurves in the other filters seem to be rather larger than in the y filter, but these differences in the amplitude values are smaller than the uncertainty in the amplitude determinations and thus may not be significant. For a linear phase angle correction, a mean linear phase coefficient, bm , of 0.024
0.007 mag/degree is calculated by averaging the linear phase coefficients obtained in each of the Stro¨mgren filters. The mean values of V ð1; 0Þ ¼ 7:m 13, B-V ¼ 0:m 90 and U-B ¼ 0:m 35 found, using a linear phase angle correction, agree with the values reported in the TRIAD file of Bowell et al. (1979). The results obtained here for the sidereal period, pole and shape of 5 Astraea are T sid ¼ 0:d 7000262 being lp ¼ 115 , bp ¼ 55 (or lp ¼ 310 , bp ¼ 55 ), a=b ¼ 1:35 and b=c ¼ 1:5, having 5 Astraea a prograde sense of rotation. The values obtained for the sidereal period and pole for 5 Astraea agrees with earlier determinations

(Erikson and Magnusson, 1993; De Angelis, 1995). The greater values of a=b and b=c obtained, also agree with the results calculated recently by Blanco et al. (2000). Fig. 8 shows the observed amplitudes together with the numerical amplitudes at zero phase angle calculated with the solution values of a=b and b=c versus the aspect angle. We have plotted the amplitudes observed, AðaÞ, and corrected, Að0Þ, using the relation given by Zappala et al. (1990), Að0Þ ¼ AðaÞ=ð1 þ maÞ, with a m coefficient of 0.007. 4.2. 19 Fortuna 19 Fortuna is a G-type asteroid (Tholen, 1989) with a diameter of 225 km (Blanco et al., 1996). This asteroid has been observed in more than 10 oppositions between 1963 and 1998 (Yang et al., 1965; van HoutenGroeneveld et al., 1979; Lupishko et al., 1981; Lagerk-

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M.J. Lo´pez-Gonza´lez, E. Rodrı´guez / Planetary and Space Science 53 (2005) 1147–1165

Fig. 4. Lightcurves and colour indices of 68 Leto in rotational phase. The 0 phase time corresponds at JD 2450724.5769 corrected for light-time.

vist et al., 1987b; Weidenschilling et al., 1987, 1990; Harris et al., 1992, 1999; Hainaut-Rouelle et al., 1995; Blanco et al., 1996; Denchev, 2000). Here we present the lightcurves of 19 Fortuna in 1997 opposition. We find a synodic period of 0.d 31013
0.d 00003 (7.h 443) from our data. The composite lightcurves obtained for 19 Fortuna show two unequal maxima and minima per rotational cycle in all the uvby filters. The colour indices do not show any variation during the rotational phase of this asteroid (see Fig. 2). The largest amplitude in the vby filters is 0.m 30
0.m 02 and 0.m 27
0.m 06 in the u filter. A mean linear phase coefficient of 0:048
0:003 mag/ degree is obtained from all the Stro¨mgren filters, for a linear phase angle correction. The mean values of V ð1; 0Þ ¼ 7:m 30, B-V ¼ 0:m 73 and U-B ¼ 0:m 37 found, using a linear phase angle correction, agree with the ones reported by the TRIAD file.

The results obtained here for the pole and shape of 19 Fortuna show this asteroid as a prograde rotator with T sid ¼ 0:d 3101345, lp ¼ 60 and bp ¼ 63 or (lp ¼ 230 and bp ¼ 63 ) and values for a=b of 1.25 and for b=c of 1.3. The solution for the sidereal period of 19 Fortuna corresponds with earlier determinations (Drummond et al., 1988, 1991; De Angelis, 1995; Blanco and Riccioli, 1998; Torppa et al., 2003). The pole solution for 19 Fortuna showed values for bp greater than those proposed in earlier determinations and a value of lp in agreement with Drummond et al. (1988, 1991) and Blanco and Riccioli (1998), but smaller than the 98 found by De Angelis (1995) and Torppa et al. (2003). The ambiguity in the solution of the longitude of the asteroid pole is owing to not very large amplitude differences (see Fig. 9) in the lightcurves obtained at different oppositions. The solution of lp ¼ 60 would produce aspect angles close to 90 at asteroid longitudes

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Fig. 5. Lightcurves and colour indices of 138 Tolosa in rotational phase. The 0 phase time corresponds at JD 2450660.4948 corrected for light-time.

of about 60
90 and at those geometric longitudes are when maxima amplitudes would be waited for the 19 Fortuna lightcurves. The solution of lp ¼ 98 would produce aspect angles close to 90 at asteroid longitudes of about 98
90 . It would be interesting to observe 19 Fortuna at these longitudes and to determine with precision the amplitudes of the lightcurves in order to resolve the ambiguity over the value for the longitude of the pole of this asteroid. Nevertheless with the available data up to now the solution with less residuals and ‘‘best fit’’ times of maximum and the amplitudes of the lightcurves available is proposed in Table 4. In addition, the shape solution agrees with previous solution values for a=b (Drummond et al., 1988, 1991; De Angelis, 1995; Torppa et al., 2003), however the value of b=c seems to be a little greater than the average values of previous determinations.

Fig. 9 shows the observed amplitudes of 19 Fortuna, AðaÞ, together with the numerical amplitudes at zero phase angle calculated with the solution values of a=b and b=c versus the aspect angle. The plots show the amplitudes observed corrected, Að0Þ, by using, Að0Þ ¼ AðaÞ=ð1 þ maÞ, with a m coefficient of 0.01. 4.3. 51 Nemausa 51 Nemausa is a CU-type asteroid (Tholen, 1989) with a diameter of 153 km (Tedesco, 1989). This asteroid has been observed in eight oppositions between 1963 and 1994 (Chang and Chang, 1963; Wamsteker and Sather, 1974; Gammelgaard and Kristensen, 1983, 1986; Kristensen and Gammelgaard, 1985, 1993, 1997; Di Martino et al., 1987; Weidenschilling et al., 1990; Kristensen, 1991, 1992; Belskaya and Dovgopol, 1992; Dovgopol and Lisina, 1992; Gammelgaard, 1998). Here

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Fig. 6. Lightcurves and colour indices of 196 Philomela in rotational phase. The 0 phase time corresponds at JD 2450724.5738 corrected for light-time.

we show lightcurves of 51 Nemausa in 1997 opposition. We find a synodic period of 0.d 32400 (7.h 776) from our data. The composite lightcurves obtained for 51 Nemausa show three maxima and minima per rotational cycle in all the uvby filters (see Fig. 3). The largest amplitude is 0.m 15
0.m 04 in the y filter, 0.m 13
0.m 05 in the b filter, 0.m 12
0.m 05 in the v filter and 0.m 16
0.m 08 in the u filter. However the possible differences in the amplitude values in the uvby filters are within the limits of the uncertainty in the amplitude determinations. From our data no clear variation in the colour indices during the rotational phase of this asteroid was detected, contrary to the variations in the colour and albedo detected by Kristensen and Gammelgaard (1993) and Gammelgaard (1998) in the lightcurves of 51 Nemausa. A mean linear phase coefficient of 0:04 mag/degree is used to apply a linear phase angle correction to our data in all the Stro¨mgren filters. Mean values of

V ð1; 0Þ ¼ 7:m 64, B-V ¼ 0:m 79 and U-B ¼ 0:m 46 are obtained. The results obtained here for the pole of 51 Nemausa show this asteroid as a retrograde rotator with T sid ¼ 0:d 3243683, lp ¼ 34 and bp ¼ 60 or (lp ¼ 175 and bp ¼ 65 ). The lightcurves of 51 Nemausa in its different oppositions show together with the two maxima and minima typical of an ellipsoidal body one or two more maximum and minimum. An ellipsoidal body shape cannot explain these features. An irregular pyramid should better fit the body shape of this asteroid. However, under the assumption of ellipsoidal shape, values for a=b of 1.12 and for b=c of 1.5 are obtained here. Fig. 10 shows the observed amplitudes of 51 Nemausa, AðaÞ, together with the calculated amplitudes at zero phase angle. The observed amplitudes corrected by using, Að0Þ ¼ AðaÞ=ð1 þ maÞ, with a m coefficient of 0.015, are also shown.

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Fig. 7. Lightcurves and colour indices of 409 Aspasia in rotational phase. The 0 phase time corresponds at JD 2450656.3935 corrected for light-time.

4.4. 68 Leto 68 Leto is a S-type asteroid (Tholen, 1989) with a diameter of 127 km (Tedesco, 1989). This asteroid has been observed in one opposition in 1978 (Surdej and Schober, 1980; Harris and Young, 1980). A synodic period of 14.h 8497 h was derived from the 1978 lightcurves (Surdej and Schober, 1980). The lightcurve of 68 Leto in 1978 opposition showed three maxima per rotational cycle with an amplitude of 0.16 mag. Fig. 4 shows the lightcurves of 68 Leto obtained in 1997 opposition. We find a synodic period of 0.d 61910 (14.h 858) from our data. The composite lightcurves obtained for 68 Leto show only two maxima and minima per rotational cycle in all the uvby filters. No significative difference is detected in the amplitude of the lightcurves obtained in the uvby filters. The largest amplitude is 0.m 29
0.m 02 in the y filter, 0.m 28
0.m 02 in the vb filters and 0.m 29
0.m 04 in

the u filter. No variation in the colour indices was detected. A linear phase angle correction was applied to our data using a mean linear phase coefficient of 0:035 mag/degree in all the filters. Mean values of V ð1; 0Þ ¼ 7:m 24, B-V ¼ 0:m 88 and U-B ¼ 0:m 47 were obtained. We have times of maximum and amplitude data for only two oppositions, and therefore it was not possible to find a solution for sidereal period, pole and shape of this asteroid. More observations of 68 Leto at different aspect angles are needed to resolve a solution for the pole and shape of 68 Leto. 4.5. 138 Tolosa This object is a S-type asteroid (Tholen, 1989) with a diameter of 47.5 km (Tedesco, 1989). This asteroid has

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Table 3 Synodic periods and maximum amplitude of the lightcurves Asteroid

Synodic period (days)

Amax (mag)

5 Astraea

0.70004
0.00020

Ay ¼ 0:16
0:05 Ab ¼ 0:18
0:10 Av ¼ 0:20
0:10 Au ¼ 0:30
0:30

19 Fortuna

0.31013
0.00003

Ay ¼ 0:30
0:02 Ab ¼ 0:30
0:02 Av ¼ 0:30
0:02 Au ¼ 0:27
0:06

51 Nemausa

0.32400
0.00020

Ay ¼ 0:15
0:04 Ab ¼ 0:13
0:05 Av ¼ 0:12
0:05 Au ¼ 0:16
0:08

68 Leto

0.61910
0.00100

Ay ¼ 0:29
0:02 Ab ¼ 0:28
0:02 Av ¼ 0:28
0:02 Au ¼ 0:29
0:04

138 Tolosa

0.42087
0.00009

Ay ¼ 0:43
0:03 Ab ¼ 0:45
0:07 Av ¼ 0:46
0:10 Au ¼ 0:40
0:20

196 Philomela

0.34750
0.00020

Ay ¼ 0:30
0:04 Ab ¼ 0:30
0:05 Av ¼ 0:30
0:06 Au ¼ 0:31
0:06

409 Aspasia

0.37576
0.00060

Ay ¼ 0:15
0:06 Ab ¼ 0:16
0:05 Av ¼ 0:12
0:05 Au ¼ 0:20
0:20

been observed in three oppositions during 1978 and 1995 (Vesely and Taylor, 1985; Harris, 1996; Armstrong et al., 1996). Vesely and Taylor (1985) obtained a partial lightcurve of 138 Tolosa, during its 1978 opposition, showing an amplitude greater than 0.28 mag. Harris (1996) from observations of 138 Tolosa in 1984 found a synodic period of 0:d 42096 (10:h 103) and an amplitude of 0.4 mag. From our observations in 1997 we measured a synodic period for 138 Tolosa of 0:d 42087 (10:h 101). The composite lightcurves obtained for 138 Tolosa show two unequal maxima and two minima per rotational cycle (see Fig. 5). There is a gap of 10% in the coverage of the rotational phase of the lightcurve of this asteroid. However the data of the individual nights overlap. The largest amplitude in the y filter is of 0.m 43
0.m 03. The amplitudes in the other uvb are slightly different by about
0.m 03 but these differences are smaller than the uncertainty in the amplitude determinations. The colour indices do not show any significant difference during the rotational phase of this asteroid. A mean linear phase coefficient, bm , of 0:022 mag/ degree is applied to make phase angle correction to our data. We find mean values of V ð1; 0Þ ¼ 9:m 25, B-V ¼ 1:m 04 and U-B ¼ 0:m 41. Our observations of 138 Tolosa in February 2000, do not cover the whole rotational cycle of 138 Tolosa (only 40%) but we adopt the synodic period found in 1997, the composite lightcurves of 138 Tolosa in 2000, show an amplitude equal or greater than 0.m 15 and colour indices of B-V ¼ 0:m 9 and U-B ¼ 0:m 5. With the lightcurves data of 1978, 1984, 1995, 1997 and 2000, we have obtained a tentative sidereal period, pole and shape of 138 Tolosa. We have derived the values of T sid ¼ 0:d 4207348, lp ¼ 40 and bp ¼ 30 (or lp ¼ 230 and bp ¼ 30 ) and values for a=b of 1.45 and for b=c of 1.01 (being a probable prograde rotator).

Table 4 Rotational properties Asteroid

Sense

T sid

lp

bp

lp

bp

5 Astraea

P

19 Fortuna

P

51 Nemausa

R

138 Tolosa

P

196 Philomela

P

409 Aspasia

P

0.d 7000262
0.d 0000002 0.d 3101345
0.d 0000002 0.d 3243683
0.d 0000001 0.d 4207348
0.d 0000002 0.d 3468707
0.d 0000002 0.d 3757102
0.d 0000002

115
5 60
10 34
5 40
5 92
3 65
10

55
5 63
5 60
10 30
10 36
3 55
10

310
5 230
10 175
10 230
5 283
3 205
5

55
5 63
5 65
10 30
10 31
3 45
5

a/b

b/c

1.35
0.04 1.25
0.02 1.12
0.02 1.45
0.05 1.45
0.02 1.11
0.02

1.50
0.20 1.30
0.40 1.50
0.20 1.01
0.30 1.15
0.10 1.30
0.20

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Fig. 8. Lightcurve amplitude versus aspect angle. Solid line: numerical reproduction obtained considering 5 Astraea as a triaxial ellipsoid with a=b ¼ 1:35 and b=c ¼ 1:5. Points: observed amplitudes, AðaÞ. Circles: corrected amplitudes, Að0Þ, by Að0Þ ¼ AðaÞ=ð1 þ maÞ using a m coefficient of 0.007.

The Fig. 11 shows the observed amplitudes together with the calculated amplitudes at zero phase angle. For 138 Tolosa our information is limited. From the partial lightcurve in 1978 oppositions we can assume that the amplitude was greater than 0.m 28, from the partial lightcurve in the 2000 opposition we know that the amplitude has to be equal to or greater than 0.m 15, for the 1995 lightcurve we adopted an amplitude of approximately 0.m 30 from the composite lightcurve of Armstrong et al. (1996). More observations of 138 Tolosa at different aspect angles will help to improve on the proposed solution. It will be interesting to observe 138 Tolosa in April 2004 and October 2005 when

minima amplitudes are waited for the lightcurves in according to our proposed solution.

4.6. 196 Philomela 196 Philomela is a S-type asteroid (Tholen, 1989) with a diameter of 146 km (Tedesco, 1989). This asteroid has been observed in four oppositions (Yang et al., 1965; Zappala et al., 1983; Erikson et al., 1991; Licandro et al., 1994; Kryszczynska et al., 1996). A synodic period of 0:d 347625 (8:h 343) has been determined (Kryszczynska et al., 1996).

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Fig. 9. Lightcurve amplitude versus aspect angle. Solid line: numerical reproduction obtained considering 19 Fortuna as a triaxial ellipsoid with a=b ¼ 1:25 and b=c ¼ 1:3. Points: observed amplitudes, AðaÞ. Circles: corrected amplitudes by Að0Þ ¼ AðaÞ=ð1 þ maÞ using a m coefficient of 0.01.

From our observations in 1997 of 196 Philomela we find a synodic period of 0.d 34750 (8.h 340). The composite lightcurves obtained for 1997 show only two maxima and minima per rotational cycle in all the uvby filters (see Fig. 6). The largest amplitude in the vby filters is of 0.m 30 (with an uncertainty of
0.m 04 in the y filter,
0.m 05 in the b filter and
0.m 06 in the v filter). The observations of 196 Philomela during 2000 only gave a partial lightcurve. This partial lightcurve shows that the amplitude in the uvby filters is at least equal or greater than 0.m 45 (see Fig. 6). From 1997 observations a mean linear phase coefficient, bm , of 0:04 mag/degree was obtained for

all the Stro¨mgren filters. We find mean values of V ð1; 0Þ ¼ 6:m 75, B-V ¼ 0:m 90 and U-B ¼ 0:m 42 using a linear phase angle correction. From our observations in February 2000, colour indices of B-V ¼ 0:m 84 and U-B ¼ 0:m 52 are obtained, in agreement with those found from our observations in 1997. From 196 Philomela lightcurves of 1964, 1981, 1989, 1994, 1997 and 2000 we found this asteroid to be a prograde rotator with T sid ¼ 0:d 3468707, lp ¼ 92 and bp ¼ 36 or (lp ¼ 283 and bp ¼ 31 ) and values for a=b of 1.45 and for b=c of 1.15. The sidereal period obtained for 196 Philomela is smaller than that derived by

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Fig. 10. Lightcurve amplitude versus aspect angle. Solid line: numerical reproduction obtained considering 51 Nemausa as a triaxial ellipsoid with a=b ¼ 1:12 and b=c ¼ 1:5. Points: observed amplitudes, AðaÞ. Circles: corrected amplitudes, Að0Þ ¼ AðaÞ=ð1 þ maÞ, using a m coefficient of 0.015.

Kryszczynska et al. (1996), however it is the one that produces the minimum residuals with respect to the times of maximum of 196 Philomela from 1964 to 2000 opposition. More times of maximum for 196 Philomela and lightcurve amplitudes will help to resolve the ambiguity over the sidereal period. The proposed values of bp are a little greater than those proposed in earlier determinations. The value for a=b is within the range of values of previous determinations (Michalowski, 1993; Licandro et al., 1994; De Angelis, 1995; Kryszczynska et al., 1996). Fig. 12 shows the observed amplitudes of 196 Philomela together with the amplitudes calculated at zero phase angle.

4.7. 409 Aspasia 409 Aspasia is a CX-type asteroid (Tholen, 1989) with a diameter of 168 km (Tedesco, 1989). This asteroid has been observed in five oppositions (Lagerkvist, 1981; Hanslmeier, 1982; Di Martino and Cacciatori, 1984; Hainaut-Rouelle et al., 1995; Piironen et al., 1998). A synodic period of 9:h 02 has been determined (Piironen et al., 1998). From our observations of 409 Aspasia in 1997 opposition a synodic period of 0.d 37576 (9.h 018) was obtained. The composite lightcurves obtained for 1997 show amplitude values of 0.m 15
0.m 06 in the y filter, 0.m 16
0.m 05 in the b filter, 0.m 12
0.m 05 in the v filter,

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Fig. 11. Lightcurve amplitude versus aspect angle. Solid line: numerical reproduction obtained considering 138 Tolosa as a triaxial ellipsoid with a=b ¼ 1:45 and b=c ¼ 1:01. Points: observed amplitudes. Crosses: partial amplitudes (from partial lightcurves). Circles: corrected amplitudes, Að0Þ ¼ AðaÞ=ð1 þ maÞ, using a m coefficient of 0.004.

and 0.m 20
0.m 20 in the u filter (see Fig. 7). The scatter of the data in the u lightcurve produces an uncertainty in the u lightcurve amplitude of similar magnitude to the amplitude value. The differences found in the amplitude derived in the uvby lightcurves are within the limits of the uncertainty in the amplitude determinations and thus may not be significant. We find mean values of V ð1; 0Þ ¼ 7:m 75, B-V ¼ 0:m 77 and U-B ¼ 0:m 30 using a linear phase angle correction (bm ¼ 0:05 mag/degree). Using 409 Aspasia lightcurves from 1980, 1981, 1982, 1993, 1996 and our lightcurves from 1997 we found this asteroid as a more probable prograde rotator with T sid ¼ 0:d 3757102, lp ¼ 65 and

bp ¼ 55 or (lp ¼ 205 and bp ¼ 45 ) and values for a=b of 1.11 and for b=c of 1.3. The proposed values are in agreement with those proposed by Blanco and Riccioli (1998) although here slightly larger values are proposed for bp and b=c. Fig. 13 shows the observed amplitudes of 409 Aspasia together with the calculated amplitudes.

5. Conclusions Lightcurves of the asteroids 5 Astraea, 19 Fortuna, 51 Nemausa, 68 Leto, 138 Tolosa, 196 Philomela and 409

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Fig. 12. Lightcurve amplitude versus aspect angle. Solid line: numerical reproduction obtained considering 196 Philomela as a triaxial ellipsoid with a=b ¼ 1:45 and b=c ¼ 1:15. Points: observed amplitudes. Circles: corrected amplitudes, Að0Þ ¼ AðaÞ=ð1 þ maÞ, using a m coefficient of 0.007.

Aspasia obtained from July to October 1997 using uvby Stro¨mgren photometry, are shown. Partial lightcurves of 138 Tolosa and 196 Philomela from observations in February 2000 are included. The lightcurves obtained for each of these asteroids show two maxima and two minima per cycle, apart from the lightcurve of 51 Nemausa with 3 maxima and 3 minima per rotation cycle and the lightcurve of 409 Aspasia with different maxima and minima superimposed on a single maximum and minimum lightcurve. The absolute magnitudes and colour indices obtained agree well with previous measurements. No significative variations are found in the colour indices during the rotational phases

of the selected asteroids. Values of sidereal periods, poles and shapes parameter are proposed.

Acknowledgements This research was partially supported by the Direccio´n General de Investigacio´n (DGI) under projects AYA2000-1559 and AYA2003-04651, the Comisio´n Interministerial de Ciencia y Tecnologı´ a under project REN 2001-3249 and the Junta de Andalucı´ a. This research has made use of the Asteroid Photometric Catalogue database. We very gratefully

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Fig. 13. Lightcurve amplitude versus aspect angle. Solid line: numerical reproduction obtained considering 409 Aspasia as a triaxial ellipsoid with a=b ¼ 1:11 and b=c ¼ 1:3. Points: observed amplitudes. Circles: corrected amplitudes, Að0Þ ¼ AðaÞ=ð1 þ maÞ, using a m coefficient of 0.03.

acknowledge the staff of Sierra Nevada Observatory for their help during the run of observations. Acknowledgements are also especially made to M.C. Romero for making available many papers used in this investigation and to V.G. Brown for proofreading. We wish to thank the referees, T. Michalowski and C. Blanco, for their useful comments and suggestions.

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