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Astron. Astrophys. 352, 508­516 (1999)

ASTRONOMY AND ASTROPHYSICS

Circumstellar disc of Pictoris: constraints on grain properties from polarization
¨ N.V. Voshchinnikov1 and E. Krugel2
1 2

Sobolev Astronomical Institute, St. Petersburg University, Bibliotechnaya pl. 2, St. Petersburg-Peterhof, 198904 Russia ¨ ¨ Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121 Bonn, Germany

Received 7 May 1999 / Accepted 14 September 1999

Abstract. We model the positional dependence of the optical scatterer and observer is essential. Whereas it is ill determined polarization (BVRI­bands) in the circumstellar disc of Pic- in reflection nebulae and allows only a very rough derivation toris as observed by Gledhill et al. (1991) and Wolstencroft et of the grain properties, the geometrical configuration in the ciral. (1995). The particles are spherical, have a size distribution cumstellar disc of Pic is clear and simple. In the case of Pic n(a) a-q and their number density decreases with distance one observes: from the star as r-s . We consider both compact and porous i) no circumstellar extinction (Crifo et al. 1997); grains. Varying the grain size and the exponent, s, of the density ii) very weak polarization of the star itself (Tinbergen 1979); distribution as well as the refractive index of the grain material m, we find that the measured polarization, colours and bright- iii) scattered light from the nearly edge­on disc (Smith & Terrile 1984; Kalas & Jewitt 1995; Mouillet et al. 1997) and its ness distribution in the disc are best reproduced by a model in polarization (Gledhill et al. 1991; Wolstencroft et al. 1995); which the grains are larger than interstellar grains (the minimum iv) an infrared excess (Aumann et al. 1984) which extends up grain size amin = 0.15 µm). The value of the maximum size is to 1300µm (Chini et al. 1991); there are also mid IR images ill determined because it has little influence and was taken to (Lagage & Pantin 1994; Pantin et al. 1997). be amax = 100 µm. The best­fit exponents of the power laws are q = 3.2 and s = 3. The grains have in the R­band a reMany numerical calculations have been performed to exfractive index mR = 1.152 - 0.005i. Such a value is roughly plain the IR emission of Pic (see detailed discussion in Li & appropriate for porous grains where half of the volume is ice Greenberg 1998). Depending on the wavelength range considand the other half vacuum, or where 24% of the volume consists ered, the observations were reproduced using compact or fluffy of silicate and the remaining 76% of vacuum. grains with sizes ranging from smaller 1 µm to up to 1 mm. With respect to modelling the scattering and polarization of Key words: stars: individual: Pic ­ stars: circumstellar matter light from the disc of Pic, Artymowicz et al. (1989), Kalas & ­ ISM: dust, extinction ­ polarization Jewitt (1995, 1996) and Pantin et al. (1997) considered scattering only at one wavelength. They assumed either isotropic or anisotropic scattering without computing the asymmetry factor g . Scarrott et al. (1992), on the other hand, applied Mie 1. Introduction theory and treated multi­wavelength scattering, however, the In the search for extrasolar planetary systems, Pictoris and polarization was only calculated at the then available R­band. other Vega­type stars have become very popular and our present Artymowicz (1997) reproduced the observation in the V­band knowledge is summarized in several reviews (Backman & employing the empirical phase and polarization function of zoParesce 1993; Vidal­Madjar & Ferlet 1994; Artymowicz 1994, diacal and cometary dust. In this paper, we model scattering and polarization at all ob1996, 1997). In the last review, Artymowicz (1997) analyzed three principal components of the Pictoris system: the star it- served wavelengths with particle cross sections computed from self, the circumstellar dust and the circumstellar gas. Although Mie theory. As a result, we are able to constrain the properties of there is an improved determination of the distance towards Pic the grains and to exclude certain choices of dust models which from the Hipparcos satellite (Crifo et al. 1997), the main con- were hitherto thought possible. clusions of the previous analyses remain in force. In general, information on dust comes from measurements 2. Observations of polarization and colours of extinction, polarization, scattered light and thermal emission. For scattering, the geometrical relation between source, Imaging polarimetry of the Pic disc was performed by Gledhill et al. (1991) in the R waveband and by Wolstencroft et al. (1995) in the B, V, R and I wavebands. The polarization patterns Send offprint requests to: Nikolai V. Voshchinnikov (nvv@aispbu.spb.su) are centro­symmetric and indicate that Pic itself is the illu-


¨ N.V. Voshchinnikov & E. Krugel: Modelling of Pic polarization

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Fig. 2. Observed colours along the disc of Pic.

Fig. 1. Results of the multi­wavelength measurements of the polarization along the disc of Pic.

minating source. The observational data are collected in Fig. 1 and suggest the asymmetry in the polarizing behaviour between the SW and NE sides, especially in the I­band. The wavelength dependence of polarization P () is typical of reflection nebulae and characterized by the increase of the polarization degree with wavelength. The increase is rather weak in the NE side and well pronounced in the SW side.

The polarization vectors are oriented perpendicular to the disc (Gledhill et al. 1991; Wolstencroft 1998). Tinbergen (1979) included Pic in the list of zero polarization standard stars. Using a large diaphragm which included the disc he found that the mean degree of polarization over the wavelength range 0.4­0.7 µm is P = 0.020 ± 0.008%. This value is compatible with the absence of material in front of the star under the assumption of a maximum polarization efficiency (P/AV = 3%) 0.006 mag. The small polarization observed and implies AV by Tinbergen (1979) should be due to dust scattering in the disc. In ordinary reflection nebulae, the colour of the scattered light is usually bluer than the illuminating star at small offsets and redder at large distances (Voshchinnikov 1985). Unfortunately, the disc colours of Pic were observed only out to 12 offset (Paresce & Burrows 1987; Lecavelier des Etangs et al. 1993). The data of Fig. 2 indicate that the disc has the same colour as the star or is slightly redder. The colours do not depend on position, the only exception being the innermost point ( = 2. 5). It falls into the central gap which is presently the subject of various speculations.


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We thus conclude that the properties of the scattered light are not peculiar. We therefore believe that the usual approach to the study of reflection nebulae can also be applied to Pic. 3. Modelling 3.1. Polarization mechanism and disc model The variation of the position angle and the degree of polarization in the disc of Pic speaks in favour of light scattering by spheres or arbitrarily oriented non­spherical particles. The disc is seen nearly edge­on. As it is optically thin, we can choose a model of single scattering in an optically thin medium. In this case, the radiation escaping from the disc at the angular distance from the star is the integral along the line of sight over some range of scattering angles (1 , 180 - 1 ), where 1 = arcsin
max

(1)

and max denotes the maximum angular size of the disc. We will adopt max = 45. 0 (Kalas & Jewitt 1995). With growing , the range of scattering angles, which is always centered at 90 , becomes narrower. A slightly tilted disc orientation (10 ) would not change the picture of light scattering significantly. 3.2. Step 1: Polarization diagrams vs scattering angle As a first step in the modelling, we compute the polarization and scattered intensity of an elementary volume. In Mie scattering by spherical particles, such polarization at a given wavelength depends on the refractive index of the particle, m = n - k i, and its radius a, P (m, a, , ) = i1 (m, a, , ) - i2 (m, a, , ) . i1 (m, a, , ) + i2 (m, a, , ) (2)

Here i1 and i2 are the dimensionless intensities which determine the radiation scattered perpendicular and parallel to the scattering plane, respectively (van de Hulst 1957; Bohren & Huffman 1983). Fig. 3 shows the polarization diagrams at = 0.70 µm (R­ band) for which the observational database is largest (Fig. 1). The particles considered are a mixture of silicate and ice. They could be porous, i.e. contain some volume fraction of vacuum. Porous grains have been discussed many times for the disc of Pic (see Artymowicz 1997; Li & Greenberg 1998). The refractive indices used are specified in Table 1. Note that we chose the volume fractions of the grain constituents in such a way that several of the refractive indices in Table 1 are identical, although they refer to grains of different chemical composition. The cross sections of grains with a silicate core and an ice mantle were ¨ computed from Guttler's (1952) theory for two­layered spheres. It is important to note that all polarization diagrams of Fig. 3 are similar and independent of the particle composition and structure. Small grains belong to the Rayleigh domain where the polarization has the well known bell­like shape with the maximum at = 90 . Polarization diagrams for very large

Fig. 3. The degree of linear polarization for various kinds of particles. Their composition and refractive indices are given in Table 1. The peak of the curve for a = 0.1 µm is always slightly shifted to larger values of scattering angle relative to those for a = 0.01 µm.


¨ N.V. Voshchinnikov & E. Krugel: Modelling of Pic polarization Table 1. Grain refractive indices mR at = 0.70 µm Refractive index 1.715 - 0.030i 1.508 - 0.020i 1.310 - 0.010i 1.152 - 0.005i 1.715 - 0.030i/1.310 - 0.010i Particle composition compact silicate grains composite grains: mixture of 50% silicate + 50% ice porous grains: mixture of 72% silicate + 28% vacuum compact grains of dirty ice porous grains: mixture of 45% silicate + 55% vacuum porous grains: mixture of 50% ice + 50% vacuum porous grains: mixture of 24% silicate + 76% vacuum core­mantle grains: silicate core + ice mantle

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Optical constants for silicate are from Draine (1985), for dirty ice from Greenberg (1968). The optical constants for composite and porous grains were obtained from the Bruggeman rule (Bohren & Huffman 1983).

et al. 1997). Moreover, Lumme et al. (1997) demonstrated that Mie theory of spheres reproduces the polarization diagrams for complex particles rather well, except sometimes for forward and backward scattering. The polarization measurements in Pic were made at offsets from = 8. 5 to 31. 2 (see Fig. 1). According to Eq. (1), the scattering angles vary in the limits from (11 - 169 ) to (44 - 136 ). With these limits, we conclude with the aid of Fig. 3: the polarimetric observations of Pic cannot be explained by light scattering of very small (< 0.1µm) or very large (> 10µm) particles only because they produce the polarization too high for the scattering angle range. 3.3. Step 2: Polarization diagrams vs particle size As the second step of the modelling procedure we include averaging along the line of sight (scattering angles) at fixed angular distance . We adopt that the number density of dust grains in the disc has a power­law distribution nd (r) = 0, n
0 r r0 -s

r < r0 , , r0 r < rout .

(3)

Fig. 4. a Upper panel: polarization vs particle size at the angular distance = 10. 5 for various exponents, s, of the dust density distribution. The grains are compact silicates with mR as indicated. The horizontal dashed lines show the observed polarization PR in the NE and SW sides. b Lower panel: as a, but now for a fixed density distribution (s = 2) and particles of various refractive indices.

Here, r0 is the radius of the central hole where the density is equal to n0 and rout is the outer radius of the disc. The latter is given by rout = Dmax = 19.28 · 45 870 AU, assuming a distance D = 19.28 pc (Crifo et al. 1997). The central cavity reaches out to < 0 6. 0 (Artymowicz et al. 1989).1 With these numbers, r0 120 AU and the ratio of the inner to outer radius equals 0 /max = r0 /rout 0.13. The polarization degree can be expressed as P (m, a, , , s) = I
j

spheres are also simple. They resemble smooth curves with the maximum at 70 . In both cases, the polarization does not change sign and reaches 100%. For particles of intermediate size, polarization reverses sign repeatedly and a ripple­like structure is seen. This behaviour of the curves P () in Fig. 3 reflects the general principles of light scattering by small particles and does not principally change for non­spherical bodies, like spheroids, cylinders, bispheres, or fluffy aggregates arbitrarily aligned in space (Mishchenko et al. 1996; Kolokolova et al. 1997; Lumme

I1 - I2 , I1 + I2
1

(4) (5)

=

K
s+1

- 1

ij (m, a, , ) sins d ,

The adaptive optics observations made by Mouillet et al. (1997) show that the scattered light is present at angular distances up to 1. 5. Perhaps, this is the light from the outer disc scattered in almost forward and backward directions. However, because the polarization observations are made outside of this radius, this does not impact on the result, given in the single scattering case.

1


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Fig. 5. As Fig. 4b, but with polarization plotted vs the parameter |mR - 1|a.

Fig. 6. Polarization vs amin at the angular distance = 10. 5. The upper size limit equals amax = 100 µm and the exponent of the size distribution q = 3.5.

Fig. 7. As Fig. 6, but with polarization vs the parameter |mR - 1|a

min

.

Fig. 8. Polarization in the disc of Pic as a function of the angular offset for the R­band. Solid lines present models with different refractive indices (see Table 1). In all calculations, amax = 100 µm and q = 3.5. The other model parameters (amin and s) are shown in the panels.

where the intensities Ij (j = 1, 2) depend on m, a, , , s, and K is a constant. If < 6. 0 and sin > 0.1 · max , the line of 13 sight intersects the hole. Examples of polarization diagrams are shown in Fig. 4 at the offset = 10. 5 for different types of particles and various values of s. The polarization degree observed in the R­band at this offset angle in the NE and SW extensions is shown by horizon-

tal dashed lines. From this figure we again conclude that grains of intermediate size have to be included into consideration to explain the observed polarization. Interestingly, the polarization observed at given offset may be explained for any density distribution of particles: as evident from Fig. 4a (the parameter


¨ N.V. Voshchinnikov & E. Krugel: Modelling of Pic polarization Table 2. Observational and theoretical colour excesses in the disc of Pic at = 10. 5. Colour excess (B­V) -disc (V­R) -disc (V­I) -disc Observations -0m 21 ± 0m 20 . . -0m 17 ± 0m 20 . . 0m 01 ± 0m 20 . . Model 1 -0m 217 . -0m 200 . -0m 313 . Model 2 -0m 213 . -0m 226 . -0m 344 . m m m m m , , , , , s s s s s = = = = = 2 2 3 3 3 ( ( ( ( ( Model 3 -0m 221 . -0m 228 . -0m 354 . = = = = = 3. 3. 4. 4. 4. 4, 4, 2, 4, 4, = = = = = Model 4 -0m 228 . -0m 252 . -0m 400 . 0. 0. 0. 0. 0. 67, 66, 67, 67, 89, g g g g g = = = = = 0. 0. 0. 0. 0. Model 5 0m 034 . 0m 061 . 0m 105 . 80). 86). 91). 96). 82).

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Model 1: mR = 1.715 - 0.030i, amin = 0.7 µ Model 2: mR = 1.508 - 0.020i, amin = 1.2 µ Model 3: mR = 1.310 - 0.010i, amin = 2.4 µ Model 4: mR = 1.152 - 0.005i, amin = 5.5 µ Model 5: mR = 1.152 - 0.005i, amin = 0.2 µ All models have amax = 100 µm and q = 3.5. ­ exponent in the brightness distribution, ­

single scattering albedo, g ­ asymmetry factor.

s determines only the polarization level at low and large size limits). The curves plotted in Fig. 4b look even more similar if we use the product |m - 1|a as an argument. That this is so is well known for the extinction efficiency (e.g. Greenberg 1968). The corresponding polarization diagrams are shown in Fig. 5. Thus, considering light scattering in the continuum, the effects of refractive index and particle size cannot be separated. In other words, we can only estimate |m - 1|a, the product of refractive index times particle size, from observations at one wavelength. In more realistic models, we must also average over the size distribution nd (a). Afterwards the spikes in polarization seen in Fig. 3 disappear. We use the power­law nd (a) a-q (6) with minimum and maximum radii amin and amax , respectively. Figs. 6 and 7 are analogous to Fig. 4 and 5, but include an average over the size distribution. The upper radius and the exponent are kept fixed (amax = 100 µm,q = 3.5) and amin is being varied. The changes of amin are most important with respect to polarization. The increase or decrease of the maximum size has no influence if |m - 1|amax > 10. So if very large particles are present in the disc of Pic, they are not visible as scatterers at waveband R. Changes in q do not affect the overall picture and will be discussed in Sect. 3.6. Figs. 6 and 7 contain plots for all refractive indices listed in Table 1. The figures show that in order to explain the observed polarization one needs to keep amin in the range 0.6 µm < amin < 10 µm. However, very porous grains with small values of amin may, in principle, be used to model the observations as well. 3.4. Step 3: Polarization vs angular distance from star In the next step, we model the angular distribution of polarization using the minimum cut­off in the size distribution as estimated from Fig. 6. The curves in Fig. 8 demonstrate that the same observations may be satisfactorily fit with particles of different composition by changing only the parameters amin and s.2 Note that the variations of s and amin influence the polarActually, the choice of the parameter s is determined by the radial brightness distribution of the scattered light, I (). For the outer parts
2

ization mainly at small and large offset angles. (We adopted amax = 100 µm and q = 3.5.) Therefore, the polarimetric observations of the extended sources (nebulae, circumstellar shells, galaxies) at one wavelength do not allow to determine the chemical composition of particles. Numerous other sets of refractive indices and size parameters may also explain the observations. As an example, let us consider the model of Scarrott et al. (1992). They reproduced the angular dependence of the scattered light and polarization in the R­band observed by Gledhill et al. (1991) with silicate particles of mR = 1.65 - 0.05i and amin = 0.01 µm, amax = 3 µm, q = 4.0, s = 2.75. However, the disc colours in their model are too red compared . to those observed (from -0m 72 in (B­V) -disc to -1m 24 in . (V­I) -disc ). 3.5. Step 4: Polarization vs wavelength With the parameters found from modelling of the polarization in the R­band (see Fig. 8), we calculated the wavelength dependence of polarization at the offset distance = 10. 5. We also checked the colour excesses given by the corresponding model (see Table 2) and compared them with observations. The results are shown in Fig. 9 and Table 2. It is clear that despite our thorough procedure and the successful fits of the changes in polarization and brightness in the R­band with angle (see Fig. 8 and Table 2), the dependence P () cannot be well explained for any refractive index. From Table 2 it also follows that the colour index V­I is crucial for the choice of the model. The colours in the models 1 ­ 4 are too red and these models must be rejected. In principle, the correct polarization behaviour (i.e. its growth with wavelength) can be reproduced for all refractive indices of Table 1 if the minimum size amin is lowered. But even if we were to fit the dependence P (), very small particles would yield unacceptably blue colours. So in order to explain all colour and polarization observations we need to reduce the lower limit of the size distribution, amin , and the refractive index m.
of the disc of Pic ( > 6. 0), I () - with = 3.6­4.3 (Kalas & Jewitt 1995). The values of for the models considered are given in Table 2.


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¨ N.V. Voshchinnikov & E. Krugel: Modelling of Pic polarization

Fig. 9. The wavelength dependence of polarization in the disc of Pic at the angular distance = 10. 5. The observations in the SW and NE sides and their errors are plotted by dashed lines. The parameters of the model are the same as in Fig. 8.

Fig. 10. The polarization observed in the disc of Pic. The solid lines correspond to the theoretical models for which mR = 1.152 - 0.005i, amin = 0.15 µm, amax = 100 µm, q = 3.2 and s = 3.

3.6. Step 5: Polarization and colours: Whole picture Finally, we suggest a fit to the angular dependence of the polarization (in all wavebands) and the colour excesses (see Figs. 10 and 11). In order to improve the fits, we varied the parameters amin and q around the values given in Table 2 for model 5. Un-

fortunately, there remains some problem for the position with the minimum distance = 2. 5. The colour index B­V can be modeled only if the grains have very narrow size distribution around a 15 µm, but with such grains the polarization at observed offsets becomes too large (see Fig. 4).


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g = 0.79 . Still better fits could be obtained assuming different dust properties on either side of the disc. Previous estimates of the grain albedo in the disc of Pic based on visible/IR models gave = 0.5 ± 0.2 (Artymowicz 1997), which is smaller than our value for very porous grains. But Artymowicz et al. (1989) also derived 0.9. As usual, the albedo of the grains was estimated independently of the scattering phase function. Evidently, the latter cannot be isotropic in the visual because so far all suggested grain sizes imply forward scattering. The value of our model (g 0.8) is larger than previous ones. Using the Heyney­ Greenstein phase function, Kalas & Jewitt (1995) found that the brightness distribution observed in the disc of Pic in the R­band may be reproduced with 0.3 |g | 0.5. Note that mirror particles (with g < 0) used by Kalas & Jewitt (1995, 1996) are astronomical nonsense because no grain material (with the exception of very small iron particles) gives at visual wavelengths a negative asymmetry parameter. Even pure conductors (m = )have g -0.2 only (van de Hulst 1957). In our favourite model (see Figs. 10, 11) we only specify the refractive index (see Table 1). It was taken arbitrarily and is not motivated from the physical point of view. From other side, the porous particles may be easily kept in the shell because the radiation pressure acting on them is smaller: we found that for silicate grains of radii a > 2.0 µm with 76% porosity (Table 1) the radiation pressure force is smaller than gravitational force. Very porous aggregates were adopted by Li & Greenberg (1998) to explain the IR observations. Mie theory can, of course, not be applied in such cases. Although Li & Greenberg used a standard mixing rule to treat the fluffy aggregates, the underlying theory has not been developed for particles with sizes larger than the wavelength nor for computing the scattering of light. We modeled scattering and polarization in the disc of Pic at all wavelengths where it has been observed. Because the properties of the scattered light are not peculiar, we could compute particle cross sections from Mie theory. Our models exclude the possibility that the grains are compact spheres, or that they are all very small or all very large grains. Instead, the particles must be rather porous (degree of porosity > 50 %) and have a lower cut­off in the size distribution as small as amin 0.2 µm. This limit is larger than the mean size of interstellar grains. Our model can be further verified by computing the resulting IR fluxes and check how well they agree with observations.
Acknowledgements. The authors are thankful to Vladimir Il'in, Nicolas Mauron and an anonymous referee for helpful comments. NVV ¨ wishes to thank for hospitality Max­Planck­Institut fur Radioastronomie where this work was begun. The work was partly supported by grants of the program "Astronomy" of the government of the Russian Federation, the program "Universities of Russia ­ Fundamental Researches" (grant N 2154) and the Volkswagen Foundation.

Fig. 11. The colour excesses observed in the disc of Pic. The theoretical model has the same parameters as in Fig. 10.

4. Concluding remarks All models presented here were constructed with a small number of parameters which are rather well determined. Our favourite model presented in Figs. 10, 11 has parameters: mR amin amax q s = 1.152 - 0.005i , = 0.15 µm , = 100 µm , = 3.2 , = 3.

The model fits reasonably well the observed angular dependence of polarization (in four wavebands), three colour excesses and the radial brightness distribution. The calculated values of the exponent in the brightness distribution, single scattering albedo and asymmetry factor in the R­band are: = 4.4 , = 0.88 ,


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¨ N.V. Voshchinnikov & E. Krugel: Modelling of Pic polarization Kolokolova L., Jockers K., Chernova G., Kiselev N., 1997, Icarus 126, 351 Lagage P.O., Pantin E., 1994, Nat 369, 628 Lecavelier des Etangs A., Perrin G., Ferlet R., et al., 1993, A&A 274, 877 Li A., Greenberg J.M., 1998, A&A 331, 291 Lumme K., Rahola J., Hovenier J.W., 1997, Icarus 126, 455 Mishchenko M.I., Travis L.D., Mackowski D.W., 1996, JQSRT 55, 535 Mouillet D., Lagrange A.M., Beuzit J.L., Renaud N., 1997, A&A 324, 1083 Pantin E., Lagage P.O., Artymowicz P., 1997, A&A 327, 1123 Paresce F., Burrows C., 1987, ApJ 319, L23 Scarrott S.M., Draper P.W., Gledhill T.M., 1992, In: Gondhalekar P.M. (ed.) Dusty discs. p. 8 Smith B.A., Terrile R.J., 1984, Sci 226, 1421 Tinbergen J., 1979, A&AS 35, 325 Vidal­Madjar A., Ferlet R., 1994, In: Ferlet R., Vidal­Madjar A. (eds.) Circumstellar dust disks and planet formation. p. 7 van de Hulst H.C., 1957, Light scattering by small particles. John Wiley, New York Voshchinnikov N.V., 1985, In: Morozhenko A.V. (ed.) Photometric and polarimetric investigations of the celestial bodies. Kiev, p. 111 Wolstencroft R.M., 1998, private communication Wolstencroft R.M., Scarrott S.M., Gledhill T.M., 1995, Ap&SS 224, 395

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