2 Observations and data reduction

Figure 1: Typical model-predicted Mira star spectrum $L_{\nu }$ (ergs-1Hz-1) (see BSW96) together with the transmission curves of our six interference filters (center wavelength (nm)/bandwidth (nm) 656/10, 673/8, 699/6, 754/6, 781/14 and 1045/9). The displayed spectrum is from the P model at phase 0.5 in cycle 1 (see Table 6).

 

 

Table 1: Observational parameters.

Data set	Epoche	Visual cycle	Exposure time	Photons/frame	Frames	FOV	Pixel size
		and phase	per frame
656/10	1996.255	0+0.20	5 ms	53800	516	2 $.\!\!^{\prime\prime}$40	4.69 mas
673/8	"	"	10 ms	23400	2310	"	"
699/6	"	"	5 ms	71100	130	"	"
754/6	"	"	"	94500	200	"	"
781/14	"	"	"	120600	1940	"	"
1045/9	"	"	70 ms	$\sim$4 Mio	260	3 $.\!\!^{\prime\prime}$72	14.53 mas

2.1 Observational parameters

The R Leo speckle interferograms were obtained with the Russian 6 m telescope at the Special Astrophysical Observatory on April 4, 1996 (see Table 1). The data were recorded through narrow-band interference filters with center wavelength (nm)/bandwidth (nm) of 656/10, 673/8, 699/6, 754/6, 781/14 and 1045/9 (filter width of the 6 filters at 10% transmission level: 12 nm, 9 nm, 8 nm, 6 nm, 16 nm, 13 nm; 1% level: 16 nm, 11 nm, 10 nm, 9 nm, 21 nm, 19 nm, respectively). Figure 1 shows the transmission curves of the filters. Due to the narrow bandwidth and the nearly rectangle-shaped transmission curve of our filters, specific regions of the molecular band structure of the Mira star spectrum can be selected which is important for sound physical interpretation (cf. Hofmann & Scholz 1998 = HS98; Hofmann et al. 1998 = HSW98). With these narrow-band filters quasi-monochromatic radii of R Leo can be measured in the strong TiO absorption band at 673 nm, at the moderate TiO absorption bands at 656 nm, 699 nm and 781 nm, in the weak TiO absorption band at 754 nm, and in the continuum at 1045 nm, suited for the comparison with predictions of Mira star models.

The observational parameters are listed in Table 1. Seeing was approximately 1 $.\!\!^{\prime\prime}$6. The plate scale error is $\pm$1.5% and the error of detector orientation $\pm$0.7$^\circ$ (derived from speckle observations of calibration binaries). The optical speckle interferograms were recorded with the speckle camera described by Baier & Weigelt (1983). The detector used was an image intensifier (gain 500000, quantum efficiency: 9% at 600 nm, 8% at 700 nm, and less than 1% at 900 nm) coupled optically to a fast CCD camera (512² pixels/frame, frame rate 4 framess^-1, digital correlated double sampling). The near-infrared speckle raw data (1045 nm continuum) were recorded with our NICMOS-3 camera.

Figure 2: Diffraction-limited bispectrum speckle interferometry images of R Leo at 656 nm, 673 nm, 699 nm, 754 nm, 781 nm and 1045 nm, and for comparison of the unresolved star HIC 49637 at 673 nm (the filter widths are described in Sect. 2.1). In each panel the contour levels are plotted from 7 to 98% of peak intensity in steps of 7%. North is at the top and east to the left.

Figure 3: Azimuthally averaged visibilities (diamonds) of R Leo and fitted visibilities of the artificial spherical symmetric UD, FDD and Gaussian CLV functions. From top left to right bottom: R Leo at 656 nm, 673 nm, 699 nm, 754 nm, 782 nm and 1045 nm. The solid line corresponds to the best fitting Gaussian CLV function, the dashed line to the best-fitting UD CLV function, and the small dashed line to the best-fitting FDD CLV function. The visibility data are plotted up to the telescope cut-off frequency (44.1, 43.2, 41.6, 38.6, 37.2, and 27.8 cycles/arcsec for 656 nm, 673 nm, 699 nm, 754 nm, 782 nm, and 1045 nm, respectively).

2.2 Diffraction-limited images and visibilities

Diffraction-limited images were reconstructed from the speckle interferograms by the bispectrum speckle interferometry method (Weigelt 1977; Lohmann et al. 1983; Hofmann et al. 1995b). The visibilities of R Leo were determined with the speckle interferometry method (Labeyrie 1970). The speckle transfer function was derived from speckle interferograms of unresolved stars (HIC 49637, HIC 49669). The correct speckle transfer function was determined by comparison of the object-independent spectral ratio function (von der Lühe 1984) of object and reference star. The bispectrum of each frame consisted of $\sim$37 million elements.

Figure 2 presents the reconstructed diffraction-limited R Leo images and for comparison the 673 nm reconstruction of the unresolved star HIC 49637. These diffraction-limited images of R Leo (April 1996 at cycle+phase of 0+0.20) show no significant asymmetry in all six filters, i.e. in the continuum at 1.04 $\mu$m and in the TiO absorption bands showing the upper atmosphere. Note, however, the weak asymmetry reported by Lattanzi et al. (1997; November 1995 at cycle+phase of -1+0.71) and Tuthill et al. (1999; January 1992 at cycle+phase of -5+0.27, June 1993 at cycle+phase of -4+0.88). In contrast to R Leo, R Cas shows a strong asymmetry of its shape in all TiO absorption band filters (see Hofmann et al. 2000a).

 

 

Table 2: Disk parameters (diameter; FWHM) derived from fits of artificial spherical symmetric CLVs (UD, FDD; Gaussian) to the azimuthally averaged visibilities of R Leo.

Data set	Diameter/
	FWHM (mas)
UD
656/10	60.6 $\pm$ 3.0
673/8	75.6 $\pm$ 3.7
699/6	52.5 $\pm$ 2.5
754/6	48.7 $\pm$ 2.3
781/14	55.0 $\pm$ 2.7
1045/9	37.9 $\pm$ 4.0
Gaussian
656/10	38.1 $\pm$ 1.8
673/8	47.6 $\pm$ 2.3
699/6	32.7 $\pm$ 1.6
754/6	30.5 $\pm$ 1.5
781/14	34.6 $\pm$ 1.6
1045/9	23.6 $\pm$ 2.6
FDD
656/10	68.3 $\pm$ 3.2
673/8	85.3 $\pm$ 4.0
699/6	59.0 $\pm$ 2.8
754/6	54.8 $\pm$ 2.6
781/14	62.1 $\pm$ 2.9
1045/9	42.7 $\pm$ 4.6

Figure 3 shows the reconstructed diffraction-limited visibilities from which we derived the disk parameters of R Leo by fitting the following artificial (i.e. non-physical) center-to-limb variations (=CLV) of emitted intensity: uniform disk (UD), fully darkened disk (FDD) and Gaussian function (Gauss). Inspection of the 2-dimensional visibilities of R Leo in all six wavelength bands used yields axis ratios of the stellar disk ranging between 0.93 and 0.98. However, these axis ratios do not indicate any significant asymmetry of the shape of R Leo as the uncertainty of the axis ratios is approximately 10%. Hence, the disk parameters were derived from the azimuthally averaged 2-dimensional visibilities by fitting spherical symmetric CLVs.

 

 

Table 3: Linear UD and FDD radii and Gaussian HWHM (in solar radii) based on the HIPPARCOS parallax of R Leo.

Data set	UD	Gaussian	FDD
656/10	660$\pm$142	415$\pm$89	744$\pm$160
673/8	823$\pm$177	518$\pm$112	929$\pm$200
699/6	572$\pm$123	356$\pm$77	642$\pm$138
754/6	530$\pm$114	332$\pm$72	597$\pm$128
781/14	599$\pm$129	377$\pm$81	676$\pm$145
1045/9	413$\pm$97	257$\pm$61	465$\pm$110

In Fig. 3 the azimuthally averaged visibilities at 656 nm, 673 nm, 699 nm, 754 nm, 782 nm and 1045 nm together with the visibilities of the fitted spherical symmetric uniform disk and Gaussian functions are shown. Note, that the Gaussian function fits the reconstructed visibilities at optical wavelengths much better than FDD and UD, and that all three artificial CLVs fit the 1045 nm visibility equally well. Table 2 lists the fitted disk parameters of R Leo. Table 3 contains the diameters of Table 2 converted to linear radii (in solar radii $R_{\odot}$) using R Leo's HIPPARCOS parallax of 9.87 mas $\pm$ 2.07 mas (ESA 1997, Whitelock & Feast 2000). Note, that Gaussian FWHM values are not directly comparable to diameter quantities.

 

 

Table 4: Ratios of the R Leo diameters at different wavelengths (filters 1045 nm/9 nm, 781 nm/14 nm, 754 nm/6 nm, 699 nm/6 nm, 673 nm/8 nm and 656 nm/10 nm filter). The diameters used are derived from fits of the spherical symmetric fully darkened disk-model (FDD) to the reconstructedvisibilities.

Feature	FDD
	diameter ratio
(754/6)   / (1045/9)	1.28 $\pm$ 0.15
(699/6)   / (1045/9)	1.38 $\pm$ 0.17
(781/14) / (1045/9)	1.46 $\pm$ 0.18
(656/10) / (1045/9)	1.60 $\pm$ 0.19
(673/8)   / (1045/9)	2.00 $\pm$ 0.24
(699/6)   / (754/6)	1.08 $\pm$ 0.08
(781/14) / (754/6)	1.14 $\pm$ 0.08
(656/10) / (754/6)	1.25 $\pm$ 0.09
(673/8)   / (754/6)	1.56 $\pm$ 0.11
(781/14) / (699/6)	1.06 $\pm$ 0.07
(656/10) / (699/6)	1.16 $\pm$ 0.08
(673/8)   / (699/6)	1.45 $\pm$ 0.10
(656/10) / (781/14)	1.09 $\pm$ 0.08
(673/8)   / (781/14)	1.36 $\pm$ 0.10
(673/8)   / (656/10)	1.25 $\pm$ 0.09

Table 4 presents the ratios of the R Leo diameters at different wavelengths. We only list the FDD diameter ratios since the FDD approximation provides reasonable fits to the CLV in near-continuum filters and many other filters (HS98, HSW98), and since UD and Gaussian fits yield almost identical ratios. Note, however, that real diameter ratios can be different from those based on these artificial CLVs because, in particular, the shape of the physical CLV may be angle-dependent (e.g. occurrence of hot or cool spots). The UD diameter is approximately two times larger in the TiO absorption band head at 673 nm than in the 1045 nm continuum. The UD diameters at 656 nm, 699 nm and 781 nm (moderate TiO absorption band) are about 1.60, 1.38 and 1.46 times larger than at 1045 nm, respectively. The UD diameter at the weak TiO absorption band (754 nm) is about 1.28 times larger than at 1045 nm.