4 Results

4.1 40 Harmonia

40 Harmonia is a S-type asteroid (Tholen [1989]) with a diameter of 111 km (Tedesco [1989]). From our data we find a synodic period for 40 Harmonia of $8^{\rm h}54^{\rm m}36^{\rm s}$. The composite lightcurves obtained for 40 Harmonia show a regular shape in all the uvby filters, with two maxima and two minima per rotational cycle. The maximum amplitude in all the uvby filters is of $0\hbox{$.\!\!^{\rm m}$ }15$ and the amplitude averaged in the rotational cycle is of $0\hbox{$.\!\!^{\rm m}$ }13$. No significative difference in the lightcurve amplitude is found in the lightcurves measured during October, $\overline{\alpha}=12\hbox{$.\!\!^\circ$ }001$, with respect to those measured during September, $\overline{\alpha}=4\hbox{$.\!\!^\circ$ }025$. 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).

Using a linear phase angle correction, a mean linear phase coefficient, $\beta_{\rm m}$, of $0.029 \pm 0.004$ mag/degree is obtained by averaging the linear phase coefficients obtained in each of the Strömgren filters. The mean values of $V(1,0)=7\hbox{$.\!\!^{\rm m}$ }369$, $B-V=0\hbox{$.\!\!^{\rm m}$ }862$ and $U-B=0\hbox{$.\!\!^{\rm m}$ }398$ found using a linear phase angle correction, agree with the mean values of $V(0)=7\hbox{$.\!\!^{\rm m}$ }043$, $B-V=0\hbox{$.\!\!^{\rm m}$ }867$ and $U-B=0\hbox{$.\!\!^{\rm m}$ }382$ found by using the radiative transfer theory of Lumme & Bowell ([1981b]) and with the values reported in the TRIAD file of Bowell et al. ([1979]). The difference found in the magnitudes deduced for both methods, $V(1,0)-V(0^\circ)$, of $0\hbox{$.\!\!^{\rm m}$ }33$ is as expected for all the asteroids (Bowell & Lumme [1979]). The multiple scattering factor obtained from the u Strömgren filter, Q_u, is smaller than that obtained from the other Strömgren filters, that have very similar values. However the difference found for the Q_u factor is within the error bars of the determination and thus may not be significant. The average value found in all the Strömgren filters of the multiple scattering factor, Q_m, of $0.141 \pm 0.021$ is in agreement with the mean value of an asteroid of type S (Bowell & Lumme [1979]).

Tancredi & Gallardo ([1991]) determined for this asteroid values of $\lambda_{\rm p}$ of $15^\circ-25^\circ$ and $\beta_{\rm p}$ of $20^\circ-60^\circ$ (or $\lambda_{\rm p}=195^\circ-210^\circ$ , $\beta_{\rm p}=20^\circ-70^\circ$) and a value for a/b of 1.27-1.35. Michalowski (1993) obtained a prograde sense of rotation with a $Tsid=0\hbox{$.\!\!^{\rm d}$ }3712522$ and values $\lambda_{\rm p}=208^\circ$, $\beta_{\rm p}=21^\circ$, a/b= 1.27 and b/c= 2.07.

Here we obtain the best fit considering 40 Harmonia as a prograde rotator obtaining $Tsid=0\hbox{$.\!\!^{\rm d}$ }3711872$ being $\lambda_{\rm p}=22^\circ$, $\beta_{\rm p}=28^\circ$ (or $\lambda_{\rm p}=203^\circ$, $\beta_{\rm p}=38^\circ$), a/b= 1.31, b/c= 1 and $\beta_{\rm A}=1~10^{-5}$. A solution with a value of $Tsid=0\hbox{$.\!\!^{\rm d}$ }3712535$ is also obtained but this solution has slightly greater residuals.

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. 4. The agreement obtained is surprisingly good, however more lightcurves of 40 Harmonia would improve the determination of its sidereal period and rotational parameters.

Figure 4: Amplitude obtained considering 40 Harmonia as a triaxial ellipsoid with a/b=1.31 and b/c=1. Amplitudes observed and corrected by a phase factor $\beta _{\rm A}=0.00001$

4.2 45 Eugenia

45 Eugenia is a 214 km FC-type asteroid (Tedesco [1989]; Tholen [1989]). This asteroid has been observed during eight oppositions between 1969 and 1988 and now in 1997. We find a synodic period of $5^{\rm h}42^{\rm m}00^{\rm s}$ from our data. The composite lightcurves obtained for 45 Eugenia show regular shapes in all the uvby filters, with two maxima and two minima per rotational cycle. The scatter of the data is greater in the u and y filters. The maximum amplitude in all the uvby filters is of $0\hbox{$.\!\!^{\rm m}$ }12$ and the amplitude averaged in the rotational cycle is of $0\hbox{$.\!\!^{\rm m}$ }11$. No significant differences in the amplitude can be deduced from our data for October observations, $\overline{\alpha}=9\hbox{$.\!\!^\circ$ }83$, with respect to September observations, $\overline{\alpha}=1\hbox{$.\!\!^\circ$ }046$. The colour indices during the rotational phase of this asteroid show large dispersions. The b-y colour index seems to present maximum values at half of the rotational phase of this asteroid while the u-b colour index seems to show a variation anticorrelated with that of the b-y curve. However these variations are within the scatter of the data and may be of no significance (see Fig. 2).

Using a linear phase angle correction a mean linear phase coefficient of $0.030 \pm 0.010$ mag/degree is obtained from all the Strömgren filters. The mean values of $V(1,0)=7\hbox{$.\!\!^{\rm m}$ }815$, $B-V=0\hbox{$.\!\!^{\rm m}$ }659$ and $U-B=0\hbox{$.\!\!^{\rm m}$ }$ 207 found using a linear phase angle correction and the ones of $V(0)=7\hbox{$.\!\!^{\rm m}$ }504$, $B-V=0\hbox{$.\!\!^{\rm m}$ }689$ and $U-B=0\hbox{$.\!\!^{\rm m}$ }201$ found following Lumme & Bowell ([1981b]) theory are in very good agreement. However, the value of U-B obtained here is smaller than the one reported by the TRIAD file. The multiple scattering factors, $Q_{{\rm filters}}$, obtained from the different Strömgren filters do not vary greatly. The average value found in all the Strömgren filters, Q_m, is of $0.159 \pm 0.023$. These values are greater than the mean values reported for C type asteroids, and also for U type asteroids (Bowell & Lumme [1979]; Lumme & Bowell [1981b]).

Previous solutions show 45 Eugenia as a retrograde rotator. Here we obtain the best fit with $Tsid=0\hbox{$.\!\!^{\rm d}$ }2374644$ and $\lambda_{\rm p}=313^\circ$ and $\beta_{\rm p}=41^\circ$ or ( $\lambda_{\rm p}=106^\circ$ and $\beta_{\rm p}=42^\circ$) and values for a/b of 1.33, b/c of 1.4 and $\beta _{\rm A}$ of 3 10^-3. This solution is in agreement with previous determined values for $\lambda_{\rm p}$ and a/b, however the value of $\beta_{\rm p}$ seems to be greater than the average value of the previous determinations (although Drummond et al. [1988], [1991], reported values of $\beta_{\rm p}=44^\circ$).

Figure 5: Amplitude obtained considering 45 Eugenia as a triaxial ellipsoid with a/b=1.33 and b/c=1.4. Amplitudes observed and corrected by a phase factor $\beta _{\rm A}=0.003$

In Fig. 5 the observed amplitudes together with the theoretical, at zero phase angle, obtained with these values of a/b and b/c are plotted versus the aspect angle. This figure shows very dispersed amplitude values, for aspect angles from 40$^\circ$to 60$^\circ$, from 120$^\circ$to 150$^\circ$ and close to 100$^\circ$. More lightcurves covering the gaps in the aspect angle would help us to discern the rotational parameters of 45 Eugenia.

4.3 52 Europa

This object is a CF-type asteroid (Tholen [1989]) with a diameter of 312 km (Tedesco [1989]). Zappala et al. ([1983]) observed 52 Europa in January 1983 and obtained a synodic period of $5\hbox{$.\!\!^{\rm h}$ }631$ and an amplitude of $0\hbox{$.\!\!^{\rm m}$ }10$. From our data we find a synodic period of $5^{\rm h}37^{\rm m}46^{\rm s}$. The composite lightcurves obtained for 52 Europa show regular shapes in all the uvby filters, with two maxima and two minima per rotational cycle (see Fig. 3). The maximum amplitude in all the uvby filters is of $0\hbox{$.\!\!^{\rm m}$ }20$ and the amplitude averaged in the rotational cycle is of $0\hbox{$.\!\!^{\rm m}$ }14$. No significant differences in the lightcurve amplitude can be deduced from our data at $\overline{\alpha}=7\hbox{$.\!\!^\circ$ }75$ (October observations) with respect to those at $\overline{\alpha}=2\hbox{$.\!\!^\circ$ }77$ (September observations) and the colour indices seem to be constant during the rotational phase of the asteroid.

A mean linear phase coefficient, $\beta_{\rm m}$, of 0.040 $\pm$ 0.012 mag/degree is obtained from all the Strömgren filters. This coefficient is obtained for an average phase angle of $7\hbox{$.\!\!^\circ$ }75$ when a possible greater phase correction than the one deduced at greater phase angles is to be expected. We find mean values of $V(1,0)=6\hbox{$.\!\!^{\rm m}$ }640$, $B-V=0\hbox{$.\!\!^{\rm m}$ }673$ and $U-B=0\hbox{$.\!\!^{\rm m}$ }267$ using a linear phase angle correction that agree with the mean values of $V(0)=6\hbox{$.\!\!^{\rm m}$ }382$, $B-V=0\hbox{$.\!\!^{\rm m}$ }704$ and $U-B=0\hbox{$.\!\!^{\rm m}$ }273$ obtained by using Lumme & Bowell ([1981b]) theory. The difference $V(1,0)-V(0)=0\hbox{$.\!\!^{\rm m}$ }26$ is slightly smaller than the expected but could be explained by considering that the magnitude, V(1,0), has been obtained with a linear phase correction to values obtained at phase angles between 7$^\circ$ and 9$^\circ$ and, as was commented before, the linear phase coefficient used could be a little greater than that obtained at greater phase angles, producing a consecutive decrease in the extrapolation to the V(1,0) magnitude. The values obtained here for the magnitude and the colour indices agree with the values reported in the TRIAD file.

There are slight differences in the values of the multiple scattering factors, $Q_{{\rm filters}}$, deduced using the different Strömgren filters. The value obtained in the y filter is the smallest one, increasing for the b, v and u filters. This value, $Q_y=0.093 \pm 0.026$, is within the range of the mean values of Q_V reported for a C-type asteroids (Bowell & Lumme [1979]).

Scaltriti & Zappala ([1977]), Zappala et al. ([1983]) and Barucci et al. ([1986]) observed 52 Europa in December 1976, January 1983 and April 1984 finding amplitudes in their lightcurves of 0 $.\!\!^{\rm m}$09, 0 $.\!\!^{\rm m}$10 and $0\hbox{$.\!\!^{\rm m}$ }08$, respectively. Dotto et al. ([1995]) observed 52 Europa during August-September 1986 and obtained a lightcurve of $0\hbox{$.\!\!^{\rm m}$ }23$ of amplitude. They found 52 Europa as a retrograde rotator with $Tsid=0\hbox{$.\!\!^{\rm d}$ }2346504$ and $\lambda_{\rm p}=70^\circ$, $\beta_{\rm p}=40^\circ$ (or $\lambda_{\rm p}=260^\circ$, $\beta_{\rm p}=55^\circ$) and values for a/b of 1.21 and for b/c of 1.30. Michalowski et al. ([1995]) observed 52 Europa in 1992 and 1994 obtaining lightcurves with $0\hbox{$.\!\!^{\rm m}$ }10$ and $0\hbox{$.\!\!^{\rm m}$ }11$ of amplitude, respectively. They obtained 52 Europa as a retrograde rotator with $Tsid= 0\hbox{$.\!\!^{\rm d}$ }2347019$ and $\lambda_{\rm p}=77^\circ$, $\beta_{\rm p}=18^\circ$ (or $\lambda_{\rm p}=264^\circ$, $\beta_{\rm p}=32^\circ$) and values for a/b of 1.20 and for b/c of 1.17.

Here we obtain the best fit when 52 Europa is considered as a prograde rotator (when 52 Europa is considered as a retrograde rotator the residuals coming from the Epoch method increase substantially) obtaining $Tsid=0\hbox{$.\!\!^{\rm d}$ }2345855$ and $\lambda_{\rm p}=261^\circ$, $\beta_{\rm p}=60^\circ$ (or $\lambda_{\rm p}=63^\circ$, $\beta_{\rm p}=46^\circ$) and values for a/b of 1.19, for b/c of 2.2 and for $\beta_{\rm A}=5~10^{-3}$. These results are in agreement with previous determinations for $\lambda_{\rm p}$ and $\beta_{\rm p}$ obtained by Dotto et al. ([1995]). However the values of $\beta_{\rm p}$ obtained by Michalowski et al. ([1995]) are smaller. The value obtained for a/b is in agreement with previous determinations while the value found for b/c is greater than previous determinations. The values of the observed amplitudes and the theoretical ones calculated with this solution at zero phase angle are plotted in Fig. 6. The agreement obtained is very good. Again new lightcurves for increasing the ecliptic longitude coverage of this asteroid would help to improve future work on the rotational properties of 52 Europa.

Figure 6: Amplitude obtained considering 52 Europa as a triaxial ellipsoid with a/b=1.19 and b/c=2.2. Amplitudes observed and corrected by a phase factor $\beta _{\rm A}$ = 0.005