4 Results

4.1 10 Hygiea

10 Hygiea is a C-type asteroid (Tholen 1989) with a diameter of 429 km (Tedesco 1989). 10 Hygiea has been observed in eight oppositions between 1953 and 1991 (Groeneveld & Kuiper 1954; Vesely & Taylor 1985; Lagerkvist et al. 1987b, 1989, 1992c; Michalowski et al. 1991), and now in 1996. 10 Hygiea has been also observed by ISO satellite (Dotto et al. 1998, 1999) and imaged with the Hubble Space Telescope (Storrs et al. 1998, 1999). No companions to 10 Hygiea have been detected down to a limit of seven magnitudes.

From our data we find a synodic period for 10 Hygiea of 27.63 $\pm$ 0.02 hours. This synodic period is consistent with the sidereal period derived by Michalowski et al. (1991) from all the observations of 10 Hygiea between 1953 and 1989, and also with the observations in 1991 (Erikson & Magnusson 1993). Figure 1 shows the composite lightcurves of 10 Hygiea derived using this synodic period. Although there is a gap in the coverage of the rotational phase of the lightcurve of this asteroid at the time of one of the minima, the composite lightcurves obtained for this asteroid show two maxima and two minima per rotational cycle in all the uvby filters. The two maxima are of different magnitude by $0\hbox{$.\!\!^{\rm m}$ }06$. The maximum amplitude is of $0\hbox{$.\!\!^{\rm m}$ }28$ in all the uvby filters.

The b-y colour index does not show any variation during the rotational phase of this asteroid (see Fig. 1). The v-b and u-b colour indices seem to show a variation during the rotational phase of this asteroid. These v-b and u-b colour curves seem to be anticorrelated with the uvby lightcurves, although the lack of data that overlap during fractional parts of the rotational phase of this asteroid does not let us to conclude that these detected variations in the v-b and and u-b colour curves are anticorrelated with the uvby lightcurves. Recently, rotational spectral variability of 10 Hygiea, in the 0.7 $\mu$m region, has been detected by Howell et al. (1999).

The mean values of $B-V=0\hbox{$.\!\!^{\rm m}$ }66$ and $U-B=0\hbox{$.\!\!^{\rm m}$ }31$ found for this asteroid, agree with the values, $B-V=0\hbox{$.\!\!^{\rm m}$ }69$ and $U-B=0\hbox{$.\!\!^{\rm m}$ }31$, reported by Bowell et al. (1979).

The results obtained here for the sidereal period, pole and shape of 10 Hygiea are $Tsid=1\hbox{$.\!\!^{\rm d}$ }150967$ being $\lambda_{\rm p}=120^\circ$, $\beta_{\rm p}=34^\circ$ (or $\lambda_{\rm p}=295^\circ$, $\beta_{\rm p}=43^\circ$), a/b= 1.31, b/c= 1.20 and $\beta_{\rm A}= 0.001$, having a retrograde sense of rotation, in agreement with the later determinations (Michalowski et al. 1991; Michalowski 1993; Erikson & Magnusson 1993).

The observed amplitudes together with the theoretical amplitudes, at zero-phase angle, obtained with the solution values of a/b and b/c versus the aspect angle are plotted in Fig. 2.

4.2 241 Germania

241 Germania is a CP-type asteroid (Tholen 1989) with a diameter of 169 km (Tedesco 1989). This asteroid has been observed in two oppositions in 1990 and 1991 (Lagerkvist et al. 1992b; Shevchenko et al. 1992) and now in 1996. Lagerkvist et al. (1992b) observed 241 Germania during five nights in 1990 and the most likely solution that they found for the synodic period for 241 Germania was about 15.2 hours. Shevchenko et al. (1992) observed 241 Germania during four nights in October-November 1991. They found, for 241 Germania, a synodic period of 8.998 hours from their observations. We find a synodic period of 15.51 $\pm$ 0.01 hours from our data. This is 18.6 minutes larger than the one proposed by Lagerkvist et al. (1992b). We tried to make composite lightcurves of our observations using these two previously suggested synodic periods, but none of them were consistent with our observations. The composite lightcurves derived for 241 Germania, from our observations, using a synodic period of 15.51 hours are shown in Fig. 3. We also tried to make composite lightcurves of 1990 and 1991 observations of 241 Germania using the synodic period obtained from our observations. Table 3 contains the date of available observations of 241 Germania together with its ecliptic longitude and latitude, the solar phase angle and the mean reduced magnitude observed, $\overline{V}(1,\alpha)$, during each night. Figure 4 shows the composite lightcurves derived, using a synodic period of 15.51 hours, from 1990, 1991 and 1996 observations. We have not applied any phase correction to the reduced magnitudes, $V(1,\alpha)$, however for the nights 13 $^{{\rm rd}}$ and 16 $^{{\rm th}}$ of October 1991, where only relative magnitudes are available, we have shifted these relative magnitudes by additional constants of $10\hbox{$.\!\!^{\rm m}$ }5$ and $12\hbox{$.\!\!^{\rm m}$ }9$ for the nights 13 $^{{\rm rd}}$ and 16 $^{{\rm th}}$ of October, respectively. The different value of the solar phase angle on November 3, 1991 ( $5\hbox{$.\!\!^\circ$ }01$) and on October 12, 1991 ( $12\hbox{$.\!\!^\circ$ }69$) explains the difference in the magnitudes observed those days. The composite lightcurves obtained from 1990 and 1991 observations seem to be well defined with the data points in their rotational phases. This gives us confidence about the synodic period derived for 241 Germania.

   

Table 3: Observations of 241 Germania

Date	Long	Lat	Phase	$\overline{V}(1,\alpha)$
(0 UT)	(1950)	(1950)	(deg)	(mag)
241 Germania
22 08 1990	301.40	5.38	9.59	8.25 $\pm$ 0.02
23 08 1990	301.27	5.39	9.93	8.64 $\pm$ 0.10
24 08 1990	301.13	5.40	10.28	8.24 $\pm$ 0.03
26 08 1990	300.87	5.41	10.96	8.34 $\pm$ 0.07
29 08 1990	300.52	5.42	11.95	8.39 $\pm$ 0.05
12 10 1991	56.42	5.70	12.69	8.36 $\pm$ 0.04
13 10 1991	56.34	5.70	12.45	-2.23 $\pm$ 0.02
16 10 1991	55.96	5.71	11.43	-4.60 $\pm$ 0.03
03 11 1991	52.96	5.59	5.01	7.90 $\pm$ 0.03
04 10 1996	18.94	8.25	4.23	8.03 $\pm$ 0.04
07 10 1996	18.32	8.24	3.43	7.95 $\pm$ 0.05
08 10 1996	18.11	8.23	3.23	7.96 $\pm$ 0.04

The composite lightcurves obtained here for 241 Germania show two maxima of similar magnitude and two unequal minima per rotational cycle in all the uvby filters. The largest amplitude in all the uvby filters is of $0\hbox{$.\!\!^{\rm m}$ }17$. The colour indices do not show any significant variation during the rotational phase of this asteroid (see Fig. 3).

The mean values of $B-V=0\hbox{$.\!\!^{\rm m}$ }64$ and $U-B=0\hbox{$.\!\!^{\rm m}$ }26$ found for this asteroid, are in good agreement with the values ( $B-V=0\hbox{$.\!\!^{\rm m}$ }69$ and $U-B=0\hbox{$.\!\!^{\rm m}$ }30$) reported by Bowell et al. (1979).

We have used all the available observations of 241 Germania, from 3 oppositions, to obtain information about its rotational and shape parameters. The most probable solution obtained for the sidereal period, pole and shape is $Tsid=0\hbox{$.\!\!^{\rm d}$ }646605$, $\lambda_{\rm p}=94^\circ$ and $\beta_{\rm p}=44^\circ$ or ( $\lambda_{\rm p}=256^\circ$ and $\beta_{\rm p}=42^\circ$) and values for a/b of 1.20, for b/c of 1.5 and for $\beta _{\rm A}$ of 0.002, having a prograde sense of rotation (a retrograde solution with $Tsid=0\hbox{$.\!\!^{\rm d}$ }647033$, might also be possible). No previous solutions have been determined for this asteroid. The prograde solution has slightly smaller residuals. We choose the prograde solution as the most probable.

The observed amplitudes together with the theoretical amplitudes, at zero-phase angle, obtained with these solution values of a/b and b/c versus the aspect angle are plotted in Fig. 5. The agreement is very good, however more lightcurves of 241 Germania would improve the determination of its rotational and shape parameters. According to the poles presented here, during the next oppositions in May/June 2000 ( $\lambda=245^0$) and in December 2002 ( $\lambda=80^0$), the lightcurves of 241 Germania will have small amplitudes, while during the oppositions in September 2001 ( $\lambda=342^0$) and in the February/March 2004 ( $\lambda=150^0$), the aspect of 241 Germania will be close to equatorial, and then, the lightcurves will have larger amplitudes. Those observations would help to improve the results presented here.

Figure 5: Amplitude obtained considering 241 Germania as a triaxial ellipsoid with a/b=1.2 and b/c=1.5 Amplitudes observed and corrected by a phase factor $\beta _{\rm A}$ = 0.002

4.3 509 Iolanda

This object is of S-type (Tholen 1989) with a diameter of 59 km (Tedesco 1989). There are no available lightcurves for this asteroid in previous oppositions.

We find a synodic period for 509 Iolanda of 12.72 $\pm$ 0.02 hours. The composite lightcurves obtained for 509 Iolanda show two maxima and two minima per rotational cycle (see Fig. 6). There are gaps in the coverage of the lightcurve of this asteroid, and only one maximum and minimum are completely defined. The largest amplitude observed in all the uvby filters is greater than $0\hbox{$.\!\!^{\rm m}$ }45$.

Figure 6: Lightcurves and colour indices of 509 Iolanda in rotational phase. The 0 phase time corresponds to JD 2450361.3211 corrected for light-time

The colour indices obtained for this asteroid show large dispersions and no significant variations are detected in the colour indices along the rotational phase. We find a mean value of $U-B=0\hbox{$.\!\!^{\rm m}$ }43$ colour index that agrees with the value, $U-B=0\hbox{$.\!\!^{\rm m}$ }41$, reported by Bowell et al. (1979). Although the mean value of $B-V=0\hbox{$.\!\!^{\rm m}$ }70$ obtained is smaller than the value, $B-V=0\hbox{$.\!\!^{\rm m}$ }83$, reported by Bowell et al. (1979), the difference between the value obtained and the one reported is within the large error bars of the determination and thus may not be significant.

There are not enough lightcurves to deduce the sidereal period, pole and shape of 509 Iolanda. For this asteroid, lightcurves taken at different ecliptic longitudes are needed to determine its rotational and shape parameters.