2 Observations and modelling methods

We carried out $R_{\rm C}$ filtered CCD observations at Piszkésteto Station of Konkoly Observatory on ten nights from October, 1998 to January, 1999. The data were obtained using the 60/90/180 cm Schmidt-telescope equipped with a Photometrics AT200 CCD camera ( $1536\times 1024$ KAF 1600 MCII coated CCD chip). The projected sky area is $29'\times 18'$ which corresponds to an angular resolution of 1 $.\!\!^{\prime\prime}$1/pixel.

The exposure times were limited by two factors: firstly, the asteroids were not allowed to move more than the FWHM of the stellar profiles (varying from night to night) and secondly, the signal-to-noise (SN) ratio had to be at least 10. This latter parameter was estimated by comparing the peak pixel values with the sky background during the observations. The journal of observations is summarized in Table 1.

   

Table 1: The journal of observations. (r - geocentric distance; $\Delta$ - heliocentric distance; $\lambda$ - ecliptic longitude; $\beta$ - ecliptic latitude; $\alpha$ - solar phase angle; aspect data are referred to 2000.0)

Date	RA	Decl.	r(AU)	$\Delta$(AU)	$\lambda$	$\beta$	$\alpha$
683 Lanzia
1998 12 14/15	00 12.78	+19 58.9	3.25	2.82	20	18	17
1998 12 16/17	00 13.78	+19 49.5	3.25	2.84	20	18	17
725 Amanda
1999 01 26/27	06 31.61	+27 13.5	2.34	1.44	110	24	12
852 Wladilena
1998 12 12/13	11 40.37	+27 28.2	2.98	2.71	163	23	19
1998 12 14/15	11 41.67	+27 31.9	2.98	2.68	163	23	19
1998 12 16/17	11 42.89	+27 36.2	2.98	2.65	163	23	19
1999 01 24/25	11 47.52	+30 52.9	2.95	2.18	170	19	14
1627 Ivar
1998 12 14/15	05 03.41	+10 30.4	2.22	1.26	76	-12	6
1998 12 15/16	05 02.00	+10 32.6	2.23	1.26	76	-12	6
1998 12 16/17	05 00.62	+10 34.9	2.23	1.27	76	-12	7
1999 01 22/23	04 30.95	+13 09.2	2.35	1.65	85	-13	20
1998 PG
1998 10 23/24	23 47.69	+09 15.0	1.23	0.26	2	9	25
1998 10 26/27	23 55.11	+08 26.9	1.23	0.27	2	7	25
1998 10 27/28	23 57.63	+08 11.6	1.23	0.27	2	7	26

The image reduction was done with standard IRAF routines. The relatively high electronic noises and low angular resolution did not permit the use of psf-photometry and that is why a simple aperture photometry was performed with the IRAF task noao.digiphot.apphot.qphot. Unfortunately other filters were not available during the observing run and consequently we could obtain only instrumental differential R magnitudes in respect to closely separated comparison stars. The precision was estimated with the rms scatter of the comp.-check magnitudes (tipically 0.01-0.03 mag).

We have also investigated the possible colour effects in neglecting standard photometric transformations. We made an R filtered 60-seconds CCD image of open cluster M67 on December 14, 1998. This cluster contains a widely used sequence of photometric standard stars (Schild 1983). We determined the instrumental magnitude differences in respect to star No. 81 in Schild (1983), which various colour indices are close to zero ( (B-V)=-0.098, $(V-R)_{\rm C}=-0.047$ mag). The studied standards were stars Nos. 106, 108, 117, 124, 127, 128, 129, 130, 134 and 135, following Schild's notation. We plotted the resulting differences ( $\Delta R_{{\rm ins}}-\Delta R_{{\rm std}}$) vs. (B-V) and $(V-R)_{\rm C}$ in Fig. 1. For a wide colour range they do not differ more than 0.1 mag, while the colour dependence is quite weak. Therefore, the obtained instrumental R-amplitudes of minor planet lightcurves are very close to the standard ones, allowing reliable comparison with other measurements.

Figure 1: The colour dependence of instrumental minus standard magnitude differences for selected photometric standard stars in M67

The presented magnitudes throughout the paper are based on magnitudes of the comparison stars taken from the Guide Star Catalogue (GSC) (Table 2). Therefore, their absolute values are fairly uncertain (at level of $\pm 0.2-0.3$ mag). Fortunately it does not affect the other photometric parameters needed in the minor planet studies, such as the amplitude, time of extrema, or photometric period. The final step in the data reduction was the correction for the light time. Composite diagrams were calculated using APC11 by Jokiel (1990) and are also light time corrected. Times of zero phase are included in the individual remarks.

   

Table 2: The comparison stars. The typical uncertainty in the magnitude values is as large as $\pm 0.2-0.3$ mag

Date	Comp.	m(GSC)
683 Lanzia
1998 12 14	GSC 1182 337	15.3
1998 12 16	GSC 1182 85	14.4
725 Amanda
1999 01 26	GSC 1887 1325	12.3
852 Wladilena
1998 12 12	GSC 1984 2286	12.7
1998 12 14	GSC 1984 2516	12.0
1998 12 16	GSC 1984 2496	13.8
1999 01 24	GSC 2524 1778	12.6
1627 Ivar
1998 12 12	GSC 702 759	12.6
1998 12 14	GSC 689 1331	12.8
1998 12 16	GSC 689 2101	12.6
1999 01 22	GSC 681 519	13.7
1998 PG
1998 10 23	GSC 1170 1119	14.2
1998 10 24	GSC 1171 632	14.3
1998 10 26	GSC 1171 1424	14.5

Two methods were applied for modelling. The first is the well-known amplitude-method described, e.g., by Magnusson (1989) and Michaowski (1993). For this the amplitude information is used to determine the spin vector and the shape. An important point is that the observed $A(\alpha)$amplitudes at solar phase $\alpha$ should be reduced to zero phase ( $A(0^\circ)$), if possible, by a simple linear transformation in form of $A(\alpha)=A(0^\circ)(1+m \alpha)$. m is a parameter, which has to be determined individually and that can be difficult, or even impossible if there are insufficient observations (Zappala et al. 1990).

The other possibility is to examine the times of light extrema ("epoch-methods'', "E-methods''). In this paper a modified version was used, which gives the sense of the rotation unambiguously. The pole coordinates can be also estimated independently. Further details can be found in Szabó et al. (1999) and Szabó et al. (in preparation), here we give only a brief description.

The initial idea is that the prograde and retrograde rotation can be distinguished by following the virtual shifts of moments of light extrema (e.g. times of minima). From a geocentric point of view, a full revolution around the Earth causes one extra rotational cycle to be added (retrograde rotation) or subtracted (prograde rotation) to the observed number of rotational cycles during that period. The virtual shifts increase or decrease monotonically and their cumulative change is exactly one period over one revolution. Therefore, plotting the observed minus calculated (O-C) times of minima versus the geocentric longitude, we get a monotone function ascending or descending by the value of the period. The definition of the observed O-C is as follows:

$${\rm O-C}=T_{\rm min}-(E_{\rm0}+N\cdot P_{\rm sid})\equiv \left\langle {\Delta T \over P_{\rm sid}} \right\rangle P_{\rm sid}$$ (1)

where $T_{\rm min}$ means the observed time of minimum, $E_{\rm0}$is the epoch, while the N integer number denotes the cycles (e.g. the number of rotation) of $P_{{\rm sid}}$ period between the observed extremum and the epoch. ${\Delta T}$ means the time interval between $T_{\rm min}$ and $E_{\rm0}$, and $\langle \rangle$ denotes fractional part. The theoretical O-C curve depends on the pole coordinates:

$${\rm O-C}'\!:~=~{{\rm O-C} \over P_{\rm sid}} = {1 \over \pi... ...an}(\Lambda-\lambda_{\rm p}) \over {\rm sin}(\beta_{\rm p}-B)}$$ (2)

where $\Lambda$ and B denote geocentric longitude and latitude; ${\lambda _{\rm p}}$ and ${\beta _{\rm p}}$ are the pole coordinates.

The main difference between the classical E-methods and this O-C' method is that time dependence is transformed into the geocentric longitude domain. Because of the system's basic symmetries, the O-C' diagrams are calculated for a half revolution and with the half sidereal period. The fitting procedure consists of altering $P_{{\rm sid}}$ until the observed times of minima do not give a monotone O-C' diagram showing an increase or decrease of exactly 1. Fitting a theoretical curve (Eq. 2) to the observed points, the pole coordinates can be also estimated.