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)
40 Harmonia
08 09 1997	352.11	-7.65	4.92	1.16620	2.16443
09 09 1997	351.86	-7.67	4.56	1.16501	2.16431
10 09 1997	351.61	-7.68	4.23	1.16406	2.16419
02 10 1997	346.36	-7.59	10.43	1.20539	2.16231
03 10 1997	346.17	-7.56	10.91	1.21001	2.16225
07 10 1997	345.47	-7.45	12.76	1.23066	2.16208
45 Eugenia
08 09 1997	348.58	-3.53	1.89	1.78740	2.79222
09 09 1997	348.36	-3.57	1.60	1.78771	2.79301
10 09 1997	348.13	-3.60	1.40	1.78831	2.79380
02 10 1997	343.59	-4.19	8.63	1.87130	2.81091
03 10 1997	343.43	-4.21	8.99	1.87811	2.81167
07 10 1997	342.81	-4.27	10.40	1.90779	2.81470
52 Europa
08 09 1997	352.22	-7.19	3.22	2.23452	3.23064
09 09 1997	352.03	-7.22	2.98	2.23227	3.22979
10 09 1997	351.83	-7.25	2.76	2.23030	3.22893
02 10 1997	347.54	-7.60	6.69	2.25939	3.20973
03 10 1997	347.37	-7.61	7.00	2.26392	3.20885
07 10 1997	346.71	-7.61	8.25	2.28467	3.20528

These observations were carried out during different nights of September and October when each asteroid had different phase angles, thus different magnitudes were observed. The mean reduced magnitudes observed for each uvby Strömgren filter, $\overline{uvby}(1,\alpha)$ , the average phase angle of the observations, $\overline{\alpha}$ , and the range of phase angle covered, during September and October observations, are presented in Table 2. The observed Strömgren colour indices, b-y and u-b, are also listed.

**Table 2:** Photometry

$\overline{uvby}(1,\alpha)$	$\overline{uvby}(1,\alpha)$	$\overline{uvby}(1,0)$	$\overline{UBV}(1,0)$	Q factors	$\overline{uvby}(0)$	$\overline{UBV}(0)$	$\Delta_{((0,1)-(0))}$	TRIAD
(mag)	(mag)	(mag)	(mag)		(mag)	(mag)	(mag)	(mag)
40 Harmonia
$\overline{\alpha}=4\hbox{$.\!\!^\circ$ }25$	$\overline{\alpha}=12\hbox{$.\!\!^\circ$ }01$	$\beta_{\rm m}$ = 0.029
$0\hbox{$.\!\!^\circ$ }31$	$1\hbox{$.\!\!^\circ$ }21$	0.004
y 7.402	y 7.705	y_(1,0)7.369	V_(1,0)7.369	Q_y 0.145	y₍₀₎7.043	V₍₀₎7.043	$\Delta V$ 0.33	V₍₀₎7.14
0.050	0.060	0.048	0.048	0.021	0.045	0.045	0.09
b 7.939	b 8.239	b_(1,0)7.903	B_(1,0)8.231	Q_b 0.148	b₍₀₎7.581	B₍₀₎7.910	$\Delta B$ 0.32
0.050	0.061	0.048	0.059	0.021	0.045	0.195	0.25
v 8.683	v 8.984	v_(1,0)8.648	U_(1,0)8.629	Q_v 0.148	v₍₀₎8.325	U₍₀₎8.292	$\Delta U$ 0.34
0.050	0.061	0.048	0.074	0.021	0.045	0.265	0.35
u 9.831	u 10.146	u_(1,0)9.810		Q_u 0.122	u₍₀₎9.460
0.061	0.065	0.051		0.022	0.051
b-y 0.536	b-y 0.535		B-V 0.862		b-y 0.538	B-V 0.867	$\Delta_{B-V}-0.005$	B-V 0.85
0.007	0.007		0.011		0.090	0.150
u-b 1.892	u-b 1.902		U-B 0.398	Q_m 0.141	u-b 1.879	U-B 0.382	$\Delta_{U-B}$ 0.016	U-B 0.43
0.021	0.020		0.015	0.021	0.096	0.070
45 Eugenia
$\overline{\alpha}=1\hbox{$.\!\!^\circ$ }46$	$\overline{\alpha}=9\hbox{$.\!\!^\circ$ }83$	$\beta_{\rm m}$ 0.030
$0\hbox{$.\!\!^\circ$ }14$	$0\hbox{$.\!\!^\circ$ }92$	0.010
y 7.661	y 8.098	y_(1,0)7.815	V_(1,0)7.815	Q_y 0.142	y₍₀₎7.504	V₍₀₎7.504	$\Delta V$ 0.31	V₍₀₎7.27
0.043	0.054	0.052	0.052	0.020	0.042	0.042	0.09
b 8.086	b 8.216	b_(1,0)8.216	B_(1,0)8.455	Q_b 0.174	b₍₀₎7.935	B₍₀₎8.193	$\Delta B$ 0.26
0.046	0.044	0.044	0.120	0.020	0.039	0.177	0.30
v 8.688	v 9.112	v_(1,0)8.828	U_(1,0)8.665	Q_v 0.157	v₍₀₎8.532	U₍₀₎8.394	$\Delta U$ 0.27
0.049	0.051	0.044	0.161	0.019	0.039	0.279	0.44
u 9.721	u 10.142	u_(1,0)9.856		Q_u 0.162	u₍₀₎9.5650
0.082	0.041	0.066		0.031	0.063
b-y 0.425	b-y 0.401		B-V 0.659		b-y 0.431	B-V 0.689	$\Delta B-V-0.030$	B-V 0.66
0.020	0.041		0.051		0.081	0.135
u-b 1.634	u-b 1.643		U-B 0.207	Q_m 0.159	u-b 1.630	U-B 0.201	$\Delta U-B$ 0.006	U-B 0.27
0.055	0.056		0.041	0.023	0.102	0.074
52 Europa
$\overline{\alpha}=2\hbox{$.\!\!^\circ$ }77$	$\overline{\alpha}=7\hbox{$.\!\!^\circ$ }75$	$\beta_{\rm m}$ 0.040
$0\hbox{$.\!\!^\circ$ }21$	$0\hbox{$.\!\!^\circ$ }81$	0.012
y 6.665	y 6.935	y_(1,0)6.640	V_(1,0)6.640	Q_y 0.093	y₍₀₎6.382	V₍₀₎6.382	$\Delta V$ 0.26	V₍₀₎6.25
0.058	0.060	0.052	0.052	0.026	0.071	0.071	0.12
b 7.094	b 7.348	b_(1,0)7.053	B_(1,0)7.313	Q_b 0.127	b₍₀₎6.822	B₍₀₎7.086	$\Delta B$ 0.23
0.059	0.061	0.053	0.081	0.035	0.072	0.214	0.29
v 7.732	v 7.980	v_(1,0)7.685	U_(1,0)7.580	Q_v 0.138	v₍₀₎7.463	U₍₀₎7.359	$\Delta U$ 0.22
0.058	0.063	0.054	0.107	0.035	0.073	0.335	0.44
u 8.819	u 9.064	u_(1,0)8.769		Q_u 0.143	u₍₀₎8.551
0.088	0.066	0.061		0.046	0.094
b-y 0.429	b-y 0.413		B-V 0.673		b-y 0.440	B-V 0.704	$\Delta B-V-0.031$	B-V 0.66
0.021	0.014		0.029		0.143	0.239
u-b 1.725	u-b 1.717		U-B 0.267	Q_m 0.125	u-b 1.729	U-B 0.273	$\Delta U-B-0.006$	U-B 0.33
0.062	0.036		0.036	0.036	0.166	0.121

We have applied a linear phase correction to transform the observed uvby reduced magnitudes to zero-phase angle magnitudes, uvby (1,0). No significative differences are found in the individual linear phase coefficients, $\beta_{{\rm filter}}$ , obtained using the different Strömgren filters. The slight differences found are, in all the cases, within the error bars of the determinations, but these slight differences could influence the colour indices when they are transformed to zero-phase angle. In order to preserve the colour indices measured, a mean linear phase coefficient, $\beta_{\rm m}$ , that is more suitable for the observed magnitudes at phase angles greater than 7 $^\circ$ in all the Strömgren filters, is used to transform to zero-phase angle magnitudes. The magnitudes uvby (1,0) are calculated as $uvby(1,\alpha)= uvby(1,0) + \beta_{\rm m}\alpha$ . This relation has been applied to the measurements taken during October when the solar phase angles were greater than 7 $^\circ$ for all the asteroids.

Lumme & Bowell ([1981a], [1981b]) showed that the light observed at phase angle $\alpha$ relative to the luminosity at zero phase is decomposed into a single scattering part and a multiple scattering part, and the magnitudes at phase angle $\alpha$ can be expressed as $V(\alpha) = V(0^\circ)-2.5~{\rm Log}((1-Q)\phi(\alpha) +Q)$ . Here, $\phi(\alpha)$ is the corresponding phase function for single scattered light and Q is the multiple scattering factor. Lumme & Bowell ([1981b]) found that $\phi(\alpha)$ can be expressed as $\phi(\alpha) = 1 - {\rm sin}\alpha/(0.124 + 1.407~{\rm sin}\alpha -0.758~ {\rm sin}^2 \alpha)$ in the phase range $0^\circ \le \alpha \le 25^\circ$ . We have used all the data measured from September to October, for each asteroid, to find the values of $Q_{{\rm filter}}$ and $V_{{\rm filter}}(0)$ which produce the best fit, for each filter, to the Lumme & Bowell ([1981b]) relation.

The Strömgren magnitudes and colour indices obtained by linear analysis, $\overline{uvby}(1,0)$ , and those obtained by following the radiative transfer theory of Lumme & Bowell ([1981b]), $\overline{uvby}(0)$ , are shown in Table 2. These Strömgren values are transformed into the UBV Johnson system by using the transformation 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. As the "y'' standard magnitude obtained using the "y'' Strömgren filter is equivalent to the "V'' magnitude of the "V'' Johnson filter, knowing the B-V and U-B colour indices, the B and U magnitudes are calculated directly. The Johnson magnitudes and colour indices obtained are also shown in Table 2, together with the differences found between the Johnson magnitudes calculated using a linear phase correction, $\overline{UBV}(1,0)$ , and those obtained by using the Lumme and Bowell relation $\overline{UBV}(0)$ , (values $\Delta_{((0,1)-(0))}$ in Table 2).

In order to obtain the corresponding rotational synodic periods for each asteroid, analysis of frequencies were carried out on our data using the method described in Rodríguez et al. ([1998]). Synodic periods of $0\hbox{$.\!\!^{\rm d}$ }37125$ , $0\hbox{$.\!\!^{\rm d}$ }23750$ and $0\hbox{$.\!\!^{\rm d}$ }23456$ are obtained for 40 Harmonia, 45 Eugenia and 52 Europa, respectively.

Figures 1 to 3 show the composite lightcurves derived using these synodic periods for each asteroid. The (1,0) magnitudes derived in the different Strömgren filters and the b-y, v-b and u-b colour indices versus the rotational phase are also shown in these figures. The Strömgren magnitudes observed during September need a greater phase angle correction than that applied to those observed during October when the phase angles are greater than 7 $^\circ$ . Therefore September magnitudes are shifted by additional constants of $+0\hbox{$.\!\!^{\rm m}$ }07$ , $+0\hbox{$.\!\!^{\rm m}$ }18$ and $+0\hbox{$.\!\!^{\rm m}$ }05$ , in all the uvby filters, for 40 Harmonia, 45 Eugenia and 52 Europa, respectively. The composite lightcurves obtained for each asteroid show very regular shapes with two maxima and two minima per rotation cycle.

3.2 Poles and shapes

Poles and shapes for these asteroids have been determined using the Epoch/Amplitude method (see Taylor [1979]; Magnusson [1986] and Magnusson et al. [1989]). 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. ([1987], [1988], [1992]). The epochs of maximum brightness and the lightcurve amplitudes used for each asteroid together with the references for the lightcurves used are listed in Table 3.

**Table 3:** Epochs of maximum brightness and Amplitudes observed
Date	JD	Ecliptic	Ecliptic	Phase	Amp.	Reference
		Long	Lat	(deg)	(mag)
		(1950)	(1950)
40 Harmonia
14 1 1958	2436217.8864	146.93	4.37	13.48	--	Gehrels and Owings (1962)
29 1 1958	2436232.7303	143.89	4.99	6.65	0.23	Gehrels and Owings (1962)
29 1 1958	2436232.9298	143.84	5.00	6.55	--	Gehrels and Owings (1962)
14 2 1975	2442457.5334	127.61	4.73	7.25	0.28	Lagerkvist (1978)
6 10 1983	2445614.2994	291.30	-3.62	26.81	0.30	McCheyne et al. (1985)
7 10 1983	2445615.3196	291.56	-3.62	26.86	--	McCheyne et al. (1985)
8 10 1983	2445616.3273	291.82	-3.61	26.91	--	McCheyne et al. (1985)
8 5 1986	2446558.7162	233.41	5.12	3.59	0.15	Gallardo and Tancredi (1987)
9 9 1997	2450701.4624	351.62	-7.68	4.24	--	This work
10 9 1997	2450701.6148	351.58	-7.68	4.20	0.15	This work
11 9 1997	2450702.5703	351.34	-7.70	3.95	--	This work
2 10 1997	2450724.4850	346.17	-7.56	10.90	0.15	This work
3 10 1997	2450725.3888	346.00	-7.54	11.33	--	This work
45 Eugenia
11 6 1969	2440383.6909	217.87	9.96	16.05	--	Taylor et al. (1988)
4 5 1978	2443632.6831	226.59	10.77	4.56	0.29	Debehogne and Zappala (1980)
1 6 1978	2443660.6991	221.04	10.50	11.95	--	Harris and Young (1979)
13 1 1982	2444982.9384	172.49	0.95	18.66	0.17	Weidenschilling et al. (1987)
13 3 1982	2445041.5846	164.76	3.62	2.97	--	Debehogne et al. (1983)
21 5 1982	2445110.6812	161.27	4.68	22.75	0.20	Weidenschilling et al. (1987)
14 7 1982		176.10	4.60	21.40	0.18	Weidenschilling et al. (1987)
21 5 1983	2445476.0576	298.15	6.82	19.90	0.14	Weidenschilling et al. (1987)
30 6 1983	2445515.9700	294.80	6.81	7.17	0.11	Weidenschilling et al. (1987)
11 10 1983	2445618.7777	290.34	1.91	22.04	0.15	Weidenschilling et al. (1987)
11 11 1983	2445649.6401	298.90	0.89	20.45	0.15	Weidenschilling et al. (1987)
29 9 1984	2445972.7803	35.09	-8.14	9.97	--	Taylor et al. (1988)
24 10 1984	2445998.1970	29.82	-8.87	3.03	--	Taylor et al. (1988)
18 11 1984		24.90	-8.80	10.2	0.36	Weidenschilling et al. (1987)
27 11 1984	2446032.1503	23.74	-8.52	13.01	--	Taylor et al. (1988)
17 1 1985	2446082.7266	26.52	-6.86	19.51	0.41	Weidenschilling et al. (1987)
20 10 1985	2446359.0875	119.07	-5.22	20.27	0.15	Weidenschilling et al. (1987)
17 1 1986	2446447.8954	116.55	-5.23	1.83	0.09	Weidenschilling et al. (1987)
16 6 1987	2446962.6469	223.69	10.06	15.59	--	Lebofsky et al. (1988)
9 9 1997	2450700.6000	348.34	-3.57	1.58	0.12	This work
2 10 1997	2450724.4542	343.44	-4.21	8.97	0.12	This work
3 10 1997	2450725.4125	343.28	-4.22	9.32	--	This work
52 Europa
17 11 1976	2443100.3604	64.73	-10.51	4.88	--	Scaltriti and Zappala (1977)
18 11 1976	2443100.6041	64.68	-10.51	4.82	--	Scaltriti and Zappala (1977)
19 11 1976	2443102.4801	64.29	-10.50	4.35	--	Scaltriti and Zappala (1977)
11 12 1976	2443124.2884	59.81	-9.93	7.52	0.09	Scaltriti and Zappala (1977)
12 12 1976	2443124.5264	59.76	-9.92	7.60	--	Scaltriti and Zappala (1977)
24 1 1983	2445359.3366	119.91	-1.52	1.49	0.10	Zappala et al. (1983)
24 1 1983	2445359.4646	119.91	-1.52	1.49	--	Zappala et al. (1983)
25 1 1983	2445359.5729	119.86	-1.51	1.58	--	Zappala et al. (1983)
25 4 1984	2445815.5260	211.19	10.83	3.58	0.08	Barucci et al. (1986)
2 9 1986	2446675.5158	347.15	-6.32	3.26	0.23	Dotto et al. (1995)
2 9 1986	2446675.6309	347.12	-6.33	3.22	--	Dotto et al. (1995)
2 11 1992	2448928.7170	26.44	-10.90	5.53	0.10	Michalowski et al. (1995)
8 2 1994	2449391.7604	124.43	0.16	4.94	0.11	Michalowski et al. (1995)
8 2 1994	2449391.8824	124.41	0.16	4.99	--	Michalowski et al. (1995)
9 9 1997	2450700.6467	351.99	-7.22	2.95	0.20	This work
11 9 1997	2450702.5240	351.62	-7.27	2.57	--	This work
2 10 1997	2450724.3380	347.39	-7.61	6.95	0.20	This work
3 10 1997	2450725.3984	347.21	-7.61	7.29	--	This work
7 10 1997	2450729.3837	346.58	-7.61	8.52	--	This work

We use the method proposed by Michalowski & Velichko ([1990]) that use the Epoch and Amplitude equations to build expressions, $f_{\rm l} (Tsid,\lambda_{\rm p},\beta_{\rm p}, b/a,b/c,\beta_{\rm A})=0$ for l=1,...(k+m), where k is the number of pairs of lightcurves used for the Epoch equations, m is the number of lightcurves used for the Amplitude equations, Tsid is the sidereal period of rotation of the asteroid, $\lambda_{\rm p}$ and $\beta_{\rm p}$ are the pole coordinates of the asteroid, $a\ge b \ge c$ are the symmetry axes of the asteroid (considered as a triaxial ellipsoid of axis $a\ge b \ge c$ rotating about their shortest axis) and $\beta _{\rm A}$ is a phase coefficient taking into account the phase angle effects on the amplitude of the lightcurves. Following the work of De Angelis ([1993]), we have used the same procedure of standardization of the variables $f_{\rm l}$ , dividing each $f_{\rm l}$ by the standard deviation $\sigma$ of all $f_{\rm l}$ of the same group of equations, and then, these expressions for $f_{\rm l}$ have been used as elements of a sum of squares to be minimized by a least-square fit.

First of all we look for a mean synodic period that suits all the epochs of maximum light for each asteroid. There are cases where this mean synodic period is not well defined, as it is possible to choose between different mean synodic periods. For these cases we look for different solutions for each of the possible mean synodic periods by using a grid of $\lambda_{\rm p}$ and $\beta_{\rm p}$ as trial poles making a least-square fit for each trial pole, first for prograde and then for retrograde solutions. The solutions with least residuals are considered as the most probable solutions. In this way we obtain a more likely solution for the corresponding values of $Tsid, \lambda_{\rm p}, \beta_{\rm p}, a/b, b/c$ and $\beta _{\rm A}$ simultaneously.

In Table 4 we have listed the final solution obtained for $Tsid, \lambda_{\rm p}, \beta_{\rm p}, a/b, b/c$ and $\beta _{\rm A}$ for each asteroid, together with the mean synodic period used. Previous solutions reported by earlier authors are also shown in Table 4.

**Table 4:** Rotational properties
Reference	Mean Tsyn	Sense	Tsid	$\lambda_{\rm p}$	$\beta_{\rm p}$	$\lambda_{\rm p}$	$\beta_{\rm p}$	a/b	b/c	$\beta _{\rm A}$

40 Harmonia
Tancredi & Gallardo (1991)				20 $^\circ$	40 $^\circ$	203 $^\circ$	45 $^\circ$	1.31	1
Michalowski (1993)		P	$0\hbox{$.\!\!^{\rm d}$ }3712522$			208 $^\circ$	21 $^\circ$	1.24	2.07
This work	$0\hbox{$.\!\!^{\rm d}$ }3712973$	P	$0\hbox{$.\!\!^{\rm d}$ }3711872$	22 $^\circ$	28 $^\circ$	203 $^\circ$	38 $^\circ$	1.31	1	0.00001
This work	$0\hbox{$.\!\!^{\rm d}$ }3713638$	P	$0\hbox{$.\!\!^{\rm d}$ }3712535$	12 $^\circ$	34 $^\circ$	201 $^\circ$	41 $^\circ$	1.31	1	0.00001

45 Eugenia
Taylor et al. (1988)		R	$0\hbox{$.\!\!^{\rm d}$ }2374645$	106 $^\circ$	26 $^\circ$	295 $^\circ$	34 $^\circ$
Drummond et al. (1988)		R	$0\hbox{$.\!\!^{\rm d}$ }2374646$			307 $^\circ$	44 $^\circ$	1.33	1.65
Magnusson (1990)		R	$0\hbox{$.\!\!^{\rm d}$ }2374646$	116 $^\circ$	26 $^\circ$	305 $^\circ$	35 $^\circ$	1.36	1.48
Lumme et al. (1990)		R	$0\hbox{$.\!\!^{\rm d}$ }2374646$	128 $^\circ$	16 $^\circ$	313 $^\circ$	25 $^\circ$	1.3	1.4
Drummond et al. (1991)		R	$0\hbox{$.\!\!^{\rm d}$ }2374646$			307 $^\circ$	44 $^\circ$	1.33	1.65
De Angelis (1995)		R	$0\hbox{$.\!\!^{\rm d}$ }2374650$			289 $^\circ$	27 $^\circ$	1.33	1.23	0.0054
This work	$0\hbox{$.\!\!^{\rm d}$ }2374297$	R	$0\hbox{$.\!\!^{\rm d}$ }2374644$	106 $^\circ$	42 $^\circ$	313 $^\circ$	41 $^\circ$	1.33	1.4	0.0030

52 Europa
Barucci et al. (1986)				0 $^\circ$	37 $^\circ$	203 $^\circ$	38 $^\circ$	1.12	1
Dotto et al. (1995)		R	$0\hbox{$.\!\!^{\rm d}$ }2346504$	70 $^\circ$	40 $^\circ$	260 $^\circ$	55 $^\circ$	1.21	1.30
Michalowski et al. (1995)		R	$0\hbox{$.\!\!^{\rm d}$ }2347019$	77 $^\circ$	18 $^\circ$	264 $^\circ$	32 $^\circ$	1.20	1.17
This work	$0\hbox{$.\!\!^{\rm d}$ }2346131$	P	$0\hbox{$.\!\!^{\rm d}$ }2345855$	63 $^\circ$	46 $^\circ$	261 $^\circ$	60 $^\circ$	1.19	2.2	0.0050

3 Analysis

3.1 Photometry

3.2 Poles and shapes