3 Pole and shape of the observed asteroids

 

 

Table 2: Asteroid parameters

Asteroid	$D({\rm km})$	albedo	Type	H(90,0)	G	m
225 Henrietta	120	0.040	F	8.57	(0.04)	0.031
360 Carlova	116	0.054	C	8.13	(0.04)	(0.010)
416 Vaticana	85	0.169	S	7.84	0.30	0.026
516 Amherstia	73	0.163	M	8.09	(0.21)	0.038
1223 Neckar			S	10.61	(0.23)	0.017

 

 

Table 3: Spin and shape models

Sidereal	Sense of	Pole 1		Pole 2		a/b	b/c	Method	Reference
period (days)	rotation	$\lambda_{{\rm p}}$	$\beta_{{\rm p}}$	$\lambda_{{\rm p}}$	$\beta_{{\rm p}}$
225 Henrietta
		$135^\circ$	$13^\circ$			1.23	1.08	EAM	Present work
		$\pm4^\circ$	$\pm6^\circ$			$\pm0.03$	$\pm0.05$
360 Carlova
		$108^\circ$	$+51^\circ$	$337^\circ$	$+47^\circ$	1.57	1.00	EA	Dotto et al. (1995)
0.2578997	P	$105^\circ$	$+47^\circ$			1.42	1.52	EAM	Present work
$\pm0.0000001$		$\pm9^\circ$	$\pm10^\circ$			$\pm0.02$	$\pm0.11$
416 Vaticana
0.2238486	P	$132^\circ$	$+58^\circ$			1.55	1.20	EAM	Present work
$\pm0.0000001$		$\pm4^\circ$	$\pm 3^\circ$			$\pm 0.01$	$\pm0.02$
0.2238486	P			$310^\circ$	$+22^\circ$	1.45	1.17	EAM	Present work
$\pm 0.0000002$				$\pm 7^\circ$	$\pm6^\circ$	$\pm0.02$	$\pm 0.06$
516 Amherstia
		$75^\circ$	$+63^\circ$	$256^\circ$	$+55^\circ$	1.83	1.85	EA	De Angelis (1995)
		$76^\circ$	$+30^\circ$			1.53	1.23	EAM	Micha owski (1996)
0.3116334	P	$75^\circ$	$+17^\circ$			1.36	1.82	EAM	Present work
$\pm 0.0000003$		$\pm 5^\circ$	$\pm 3^\circ$			$\pm 0.01$	$\pm 0.04$
0.3116332	R	$255^\circ$	$-15^\circ$			1.36	1.81	EAM	Present work
$\pm 0.0000003$		$\pm 5^\circ$	$\pm4^\circ$			$\pm 0.01$	$\pm 0.04$
1223 Neckar
0.3232105	P	$70^\circ$	$+45^\circ$	$255^\circ$	$+42^\circ$	1.47	1.28	EAM	Present work
$\pm 0.0000004$		$\pm 8^\circ$	$\pm6^\circ$	$\pm 7^\circ$	$\pm6^\circ$	$\pm 0.04$	$\pm0.05$

The spin vectors, sidreal periods, and triaxial ellipsoid models for the observed asteroids were determined by the method described by Michaowski (1993). In this method the magnitudes, amplitudes, and epochs of maxima are considered. The results were obtained by building a set of nonlinear equations whose solution was found by least square fitting. The observed amplitudes and magnitudes of the brightness maxima were reduced to zero phase angle by using the amplitude-phase (Zappala et al. 1990) and the HG-magnitude system (Bowell et al. 1989), respectively.

As described by Michaowski (1993), the method had two-fold ambiguity in the location of an asteroid pole when the sense of rotation was fixed. This problem was also discussed earlier in a review paper by Magnusson et al. (1989). They stated that by combining the solutions from amplitude-magnitude and epoch methods, it was sometimes possible to obtain a unique solution even though each individual method had failed to achive this. However, they did not discuss when such situations were possible.

The method used in the present paper combines these methods (instead of solutions alone) in one process of calculation as mentioned above. From the results obtained here (see below) and in other papers (Michaowski 1993, 1996; Michaowski et al. 1995; Kryszczynska et al. 1996) we can also state that in some cases it is possible to obtain a unique solution for the pole location. This possibility can sometimes occur when an asteroid is observed during a few oppositions and the ecliptic latitudes are within a wide range of values (usually from about $-20^{\circ}$ to $+20^{\circ}$). An example of such an asteroid in the present work is 360 Carlova (see below). We have also noticed that for an asteroid which has always been observed in low ecliptic latitudes (e.g. 1223 Neckar from the present study), we have obtained two solutions for its pole coordinates and the two-fold ambiguity is not broken.

The basic parameters of the asteroids are summarized in Table 2. Their IRAS diameters (D) and albedos are taken from The Small Bodies Node of the NASA Planetary Data System (http://pdssbn.astro.umd.edu/), while the taxonomic types are from Tholen (1989). The maximum brightness of the asteroid obtained for aspect $90^{\circ}$ and zero solar angle H(90,0) is shown in the next column. The last two columns display the G and mvalues obtained during reduction the magnitudes and amplitudes to zero phase angle, respectively. If the existing data were insufficient for such reduction, the average value for a given taxonomic type was taken: from Harris (1989) for G parameter and from Zappala et al. (1990) for m. The assumed values are given in parentheses. This table is not complete as the asteroid 1223 Neckar was not observed by the IRASsatellite.

Table 3 contains the spin and shape models for the asteroids studied in the present paper. This table shows the sidereal periods, the senses of rotation (P - prograde, R - retrograde), the ecliptic coordinates (equinox 2000) of the north poles, and the ratios a/b, b/c of triaxial ellipsoid models. The available results obtained by other authors are given for comparison. The methods, used for calculation, are also given (E - Epochs, A - Amplitude, M - Magnitude). If no previous results are listed in Table 3, it means the results from the present work are the first published ones for a given asteroid.

3.1 225 Henrietta

No model has been previously reported for this asteroid. We have used the data from four oppositions: 1982, 1983, 1987, and 1995. During these apparitions the ecliptic latitude of Henrietta ranged from $-11^{\circ}$ to $+29^{\circ}$. The results are presented in Table 3. The available data are insufficient for obtaining a unique solution for the sidereal period and sense of rotation. New observations are needed to improve the presented preliminary results.

3.2 360 Carlova

Dotto et al. (1995) used the EA method and data from three oppositions (1979, 1984, 1986) and obtained a model of this asteroid (see Table 3). We were able to calculate the model of Carlova using the lightcurves from six apparitions: 1979, 1984, 1986, 1996, 1997, 1998 (ecliptic latitude from $-15^\circ$to $+16^{\circ}$). The results are presented in Table 3. We determined the sidereal period and the prograde rotation of this asteroid. The coordinates of the north pole are close to the first solution of Dotto et al. (1995) and a/b is a little smaller than that obtained by these authors. The difference in the ratios b/c obtained by Dotto et al. (1995) and in the present study is much greater. Dotto et al. (1995) used only the EA method and the A approch is not so good for b/c determination. The magnitude (M) method is a much better indicator for b/c and such a method has also been used in the present work (see Michaowski 1993 for details).

3.3 416 Vaticana

There is no previously published model for this asteroid. We have lightcurves from 1985, 1989, 1995-96, and 1998. The unique value of the sidereal period and prograde sense of rotation have been obtained (see Table 3). The ecliptic latitudes of Vaticana during these four apparitions varied from $-21^{\circ}$to $+18^{\circ}$ but the two-fold ambiguity still exists (see table). It is probably due to the small number of observed oppositions and the future observations should help to resolve this ambiguity.

3.4 516 Amherstia

Using the lightcurves from three oppositions (1978, 1985, 1989), De Angelis (1995) and Michaowski (1996) obtained slightly different parameters, especially for the triaxial ellipsoid model (see Table 3). This problem was analyzed earlier by Michaowski (1996).

The data from five oppositions (1978, 1985, 1989, 1995, 1996) allowed us to determine a model of this asteroid (Table 3). Unfortunately, we have obtained two similar solutions for both prograde and retrograde senses of rotation (the ecliptic coordinates for these solutions indicate two poles of the same axis of rotation). It probably means that Amherstia has a spin vector located in the ecliptic plane. For such asteroids the sense of rotation is undefined.

3.5 1223 Neckar

For this asteroid, the data obtained during seven apparitions (1977, 1983, 1987, 1989, 1990, 1993, 1995-96) are available. The model is given in Table 3. As the ecliptic latitude of Neckar is always close to zero (from $-3^{\circ}$ to $+4^{\circ}$ during previous oppositions), two solutions for the north pole have been obtained. The available lightcurves have allowed us to obtain the unique value for the sidereal period and the prograde sense of rotation. The results presented here are the first ones published for 1223 Neckar.

Acknowledgements

This work was partially supported by the Polish KBN Grants 2 P03D 024 09 and 2 P03D 007 18.