3 Analysis

In Table 1 the aspect data for each asteroid, the longitude and latitude relative to the ecliptic, the solar phase angle, the geocentric, r, and heliocentric, $\Delta$, distances, are listed for every day of observation.

   

Table 1: Aspect data

Date	Long	Lat	Phase	r	$\Delta$
(0 UT)	(1950)	(1950)	(deg)	(AU)	(AU)
10 Hygiea
22 09 1996	20.85	5.34	6.60	2.43850	3.38599
04 10 1996	18.63	5.42	2.89	2.40571	3.39552
07 10 1996	18.04	5.43	2.10	2.40397	3.39785
08 10 1996	17.84	5.43	1.89	2.40398	3.39862
241 Germania
04 10 1996	18.94	8.25	4.23	1.80108	2.78756
07 10 1996	18.32	8.24	3.43	1.79872	2.78908
08 10 1996	18.11	8.23	3.23	1.79848	2.78959
509 Iolanda
04 10 1996	21.52	8.22	4.90	1.81454	2.79611
07 10 1996	20.88	7.99	3.90	1.80876	2.79640
08 10 1996	20.66	7.91	3.60	1.80740	2.79650

3.1 Photometry

These observations were carried out during three nights of October 1996 when the asteroids were very close to the opposition. 10 Hygiea was observed one additional night of September when the solar phase angle of this asteroid was 4$^\circ$ greater than its average phase angle corresponding to October observations. The absolute magnitudes obtained during September for 10 Hygiea are about $0\hbox{$.\!\!^{\rm m}$ }28$ greater than those obtained during October. In Table 2 the mean reduced magnitudes observed for each uvby Strömgren filter, $\overline{uvby}(1,\alpha)$, the observed Strömgren colour indices, b-y and u-b, and the average phase angle of October observations, $\overline{\alpha}$, for each asteroid, are presented.

   

Table 2: Photometry

$\overline{uvby}(1,\alpha)$	$\overline{UBV}(1,\alpha)$	$\overline{uvby}(1,\alpha)$	$\overline{UBV}(1,\alpha)$	$\overline{uvby}(1,\alpha)$	$\overline{UBV}(1,\alpha)$
(mag)	(mag)	(mag)	(mag)	(mag)	(mag)
10 Hygiea	( $\overline{\alpha}= 2\hbox{$.\!\!^\circ$ }2$ $\pm$ $0\hbox{$.\!\!^\circ$ }5$)	241 Germania	( $\overline{\alpha}= 3\hbox{$.\!\!^\circ$ }5$ $\pm$ $0\hbox{$.\!\!^\circ$ }5$)	509 Iolanda	( $\overline{\alpha}= 4\hbox{$.\!\!^\circ$ }0$ $\pm$ $0\hbox{$.\!\!^\circ$ }7$)
y 5.64 $\pm$ 0.08	V 5.64 $\pm$ 0.08	y 7.98 $\pm$ 0.05	V 7.98 $\pm$ 0.05	y 9.04 $\pm$ 0.13	V 9.04 $\pm$ 0.13
b 6.05 $\pm$ 0.08	B 6.30 $\pm$ 0.10	b 8.38 $\pm$ 0.06	B 8.62 $\pm$ 0.09	b 9.48 $\pm$ 0.12	B 9.74 $\pm$ 0.24
v 6.71 $\pm$ 0.08	U 6.61 $\pm$ 0.15	v 9.03 $\pm$ 0.06	U 8.88 $\pm$ 0.13	v 10.22 $\pm$ 0.15	U 10.17 $\pm$ 0.46
u 7.84 $\pm$ 0.07		u 10.09 $\pm$ 0.09		u 11.42 $\pm$ 0.30
b-y 0.41 $\pm$ 0.01	B-V 0.66 $\pm$ 0.02	b-y 0.40 $\pm$ 0.02	B-V 0.64 $\pm$ 0.04	b-y 0.44 $\pm$ 0.07	B-V 0.70 $\pm$ 0.12
u-b 1.78 $\pm$ 0.05	U-B 0.31 $\pm$ 0.04	u-b 1.71 $\pm$ 0.06	U-B 0.26 $\pm$ 0.04	u-b 1.94 $\pm$ 0.30	U-B 0.43 $\pm$ 0.22

These Strömgren values are transformed into the UBV Johnson system by using the equations of Warren & Hesser (1977) to transform the Strömgren b-y and u-b colour indices to Johnson B-V and U-B colour indices. The Johnson magnitudes and colour indices obtained, $\overline{UBV}(1,\alpha)$, are also shown in Table 2.

Rotational synodic periods for each asteroid have been obtained performing analysis of frequencies on our data using the method described in Rodríguez et al. (1998). Synodic periods of $1\hbox{$.\!\!^{\rm d}$ }1511$, $0\hbox{$.\!\!^{\rm d}$ }6464$ and $0\hbox{$.\!\!^{\rm d}$ }5300$, and estimated errors, after optimizing these periods using a nonlinear least squares fit, of $0\hbox{$.\!\!^{\rm d}$ }0007$, $0\hbox{$.\!\!^{\rm d}$ }0006$ and $0\hbox{$.\!\!^{\rm d}$ }0010$ are obtained for 10 Hygiea, 241 Germania and 509 Iolanda, respectively. Then, synodic periods of 27.63 $\pm$ 0.02 hours for 10 Hygiea, 15.51 $\pm$ 0.01 hours for 241 Germania and 12.72 $\pm$ 0.02 hours for 509 Iolanda are derived from these observations.

Figure 1: Lightcurves and colour indices of 10 Hygiea in rotational phase. The 0 phase time corresponds to JD 2450349.4941 corrected for light-time

Figures 1, 3, and 6 show the composite lightcurves derived using these synodic periods for each asteroid. The uvby $(1,\alpha)$ magnitudes and the b-y, v-b and u-b colour indices versus the rotational phase are shown in these figures. The lightcurves obtained for each asteroid show regular shapes with two maxima and two minima per rotational cycle. We have not applied any phase correction to these magnitudes, however the magnitudes measured for 10 Hygiea on September 22, when the solar phase angle was greater, have been shifted by an additional constant of $-0\hbox{$.\!\!^{\rm m}$ }28$ in all the uvby filters.

Figure 2: Amplitude obtained considering 10 Hygiea as a triaxial ellipsoid with a/b=1.31 and b/c=1.2. Amplitudes observed and corrected by a phase factor $\beta _{\rm A}$ =0.001

Figure 3: Lightcurves and colour indices of 241 Germania in rotational phase. The 0 phase time corresponds to JD 2450361.3180 corrected for light-time

Figure 4: Lightcurves and colour indices of 241 Germania in rotational phase. The 0 phase time corresponds to JD 2448125.5747 corrected for light-time

3.2 Poles and shapes

Poles and shapes for these asteroids have been determined using the Epoch/Amplitude method (see Taylor 1979; Magnusson 1986; Magnusson et al. 1989; Michalowski & Velichko 1990; De Angelis 1993). We used the method to resolve the Epoch and Amplitude equations as it was explained in López-González & Rodríguez (1999). In this way we obtain the most likely solution for the sidereal period of rotation of the asteroid, Tsid, the pole coordinates of the asteroid, $\lambda_{\rm p}$ and $\beta_{\rm p}$, the ratios of the symmetry axes of the asteroid (considered as a triaxial ellipsoid of axis $a\ge b \ge c$ rotating about their shortest axis), a/b and b/c, and a phase coefficient that takes into account the phase angle effects on the amplitude of the lightcurves, $\beta _{\rm A}$.

Lightcurve data reported in the literature together with lightcurves obtained here for these asteroids have been used in the analysis. Most of the lightcurves used in this work can be found in the Asteroid Photometric Catalogue by Lagerkvist et al. (1987a, 1988, 1992a).