3 Results of refraction calculations and their analysis

For our calculations we have adopted the following values of meteorological conditions at the observing site: atmospheric temperature $7^{\circ}$C, pressure 745 mm Hg, dew point $40^{\circ}$F. In the present study all calculations are performed at the zenith distance of $60^{\circ }$. Results may be compared with other data presented for the same zenith distance in other similar studies (e.g. Schildknecht [1994]).

We have calculated the mean refraction with Eq. (2) for stellar spectral energy distributions of every spectral type separately for three representative luminosity classes: dwarfs (luminosity class V), giants (luminosity class III) and supergiants (luminosity class I) as well as at different degrees of interstellar reddening (E(B-V)= 0.0, 0.5, 1.0, 2.0, 3.0, 4.0).

In Fig. 1 the dependence of the calculated refraction on MK spectral types is given for dwarfs. In the paper of Schildknecht ([1994]) these data were calculated for the same zenith distance and are given in tabulated form, therefore the direct comparison is possible. As the data of Schildknecht are given differentially, we have added our data at G0 to his zero-point. The dependencies are in good agreement, taking into account that the different spectral energy distributions, CCD detectors and telescope optics were involved. But our dependence in Fig. 1 shows the gentlest sloping for late type stars while in the dependence of Schildknecht this is the case for early type stars. In their analysis of the dependence of differential refraction on stellar temperatures Gubler & Tytler ([1998]) have found that for cool stars the dependence shows the gentlest sloping, in agreement with our data. Schildknecht ([1994]) as well as Gubler & Tytler ([1998]) have not discussed any luminosity and interstellar reddening effects.

Figure 1: The dependence of the calculated refraction on MK spectral types for dwarfs. The continuous spectral codes for discrete spectral types were introduced as follows: B0 = 2.0, A0 = 3.0, F0 = 4.0, G0 = 5.0, K0 = 6.0 and M0 = 7.0. The reddenings (thin lines) are shown for B0, G0 and M0 dwarfs at different E(B-V). The dependence from Schildknecht ([1994]) is imposed by the dashed line. One desirable error in refraction (0.05 arcsec) is indicated. All calculations have been made at the zenith distance of $60^{\circ }$

Accurate estimates of observational errors in determining star positions have been made by Stone et al. ([1996]) with the FASTT telescope. After careful reductions and analysis the accuracy of about 0.15 arcsec has been achieved in their paper (0.14 and 0.17 arcsec, respectively in right ascension and declination for a single observation). It is desirable to calculate refraction with better accuracy (by about one order) than the accuracy of observed stellar positions.

The errors in calculated refraction may arise from incomplete knowledge of stellar and ground-based atmospheric parameters as well as inherent limitations of the refraction model. To every stellar spectral energy distribution used in the analysis we have attributed the $T_{\rm {eff}}$ values with the use of the calibration "Spectral type versus $T_{\rm {eff}}$'' of Schmidt-Kaler ([1982]). The dependence of the calculated refraction on $T_{\rm {eff}}$ for dwarfs is given in Fig. 2. We see that the slope of this dependence is changing with temperature. We have divided the data in four groups according to effective temperatures ( 30000-10500 K, 9500-8700 K, 8200-4900 K, 4700-4100 K and 3800-2900 K, which correspond to spectral types B, A0-A3, A5-K2, K3-K7 and M, respectively). The slopes of the dependence within each group are approximately linear. From these slopes we have estimated how accurate the $T_{\rm {eff}}$ values should be to achieve the desirable accuracies of 0.02, 0.05 and 0.1 arcsec in calculated refraction, respectively. These estimates for the accuracy required for $T_{\rm {eff}}$ (based in Fig. 2) are presented in Table 1. The respective estimates for the accuracy required for spectral types (based in Fig. 1) are presented in Table 2.

Figure 2: The dependence of the calculated refraction on $T_{\rm {eff}}$ for dwarfs. The reddenings (thin lines) are shown for B0, B5 and G0 dwarfs ( $T_{\rm {eff}}$ = 3000, 5860 and 3580 K, respectively) at different E(B-V). One desirable error in refraction (0.05 arcsec) is indicated. All calculations have been made at the zenith distance of $60^{\circ }$

 

 

Table 1: Accuracies in $T_{\rm {eff}}$ [K] required to achieve the desirable accuracies per parameter in calculated refraction for dwarfs

Accuracy in	Accuracy in $T_{\rm {eff}}$
refraction [as]	B	A0-A3	A5-K2	K3-K7	M
0.02	4550	690	300	140	80
0.05	11370	1730	740	340	200
0.10	22740	3470	1480	680	410

 

 

Table 2: Accuracies in spectral subtypes required to achieve desirable accuracies per parameter in calculated refraction for dwarfs

Accuracy in	Accuracy in spectral subclasses
refraction [as]	B-G2	G3-K0	K2-K7	M
0.02	2.1	3.0	0.8	0.6
0.05	5.1	7.4	1.9	1.6
0.10	10.3	14.8	3.9	3.2

The most stringent requirement to accuracy of $T_{\rm {eff}}$ refers to K - M dwarfs. We conclude from the first columns of Tables 1 and 2 that we should have the accuracies of 80-140 K in $T_{\rm {eff}}$ or 0.6 - 0.8in spectral subtype respectively in order to achieve an accuracy of 0.02 arcsec in calculated refraction. The corresponding analysis for other luminosity classes (giants and supergiants) leads us to about the same conclusions.

The results of Tables 1 and 2 should be compared with the real accuracies of spectral classification achieved with available spectral classification methods. A system of spectral classification for K - M stars developed by Malyuto et al. ([1997]) is based on spectral indices measured with different spectral libraries of about 10 Å resolution (data from photoelectric scanners, diode-array and CCD detectors are involved). The accuracy of classification is 0.6 of spectral subtype which roughly corresponds to 100 K according to the calibration "Spectral type versus $T_{\rm {eff}}$'' of Schmidt-Kaler ([1982]). This accuracy estimate includes the uncertainty of the original MK classifications from the literature and the accuracy of the method should be better. A more informative approach to classification exhausting the information contained in the spectrum (so called perturbation method) has been developed by Cayrel et al. ([1991]) and others (Soubiran et al. [1998]; Katz et al. [1998]) for F5 - K7 stars having 5 Å resolution CCD spectra, which provide the accuracy of 145 K in $T_{\rm {eff}}$, the inhomogeneity of the parameters from the literature is included, too. These two approaches have been successfully applied to stars up to $V=13^{\rm m}$ and $15^{\rm m}$, respectively. As to photometric classifications, the Vilnius system (Straizys [1992]) provides the respective classification accuracy of 0.8 of spectral subtype or about 150 K for K - M stars (the uncertainty of the original MK classifications are included too). Straizys et al. ([1998]) have calculated the accuracies of some photometric systems from the analysis of the Kurucz model atmospheres for stars earlier M0. The Vilnius system provides the best accuracy (better than 100 K for K stars, the corresponding errors of spectral types are from 0.1 to 0.5 of spectral subtypes). From the comparison of these temperature accuracy estimates with the data of Tables 1 and 2 for K - M stars we conclude that the accuracy of 0.02 arcsec per $T_{\rm {eff}}$ parameter in refraction is achievable with the available spectral and photometric classification systems.

If we agree to be within the larger error budget of 0.05 arcsec per temperature parameter in calculated refraction, we may respectively weaken our requirements to the accuracies of spectral types (it allows to reach fainter stars). For example, judging on Table 2, accuracy of 2 spectral subtypes could then be sufficient. If we increase this error budget to 0.1 arcsec, accuracy of 4 spectral subtypes could be sufficient. In this aspect it is to the point to mention some classification systems providing relatively low classification accuracy (these systems are applicable to faint stars). Some of them are based on photographic objective prism spectra of very low resolution (e.g. Upgren [1962], dispersion of 580 Å/mm, what corresponds to 9 Å resolution, classification accuracy is about 2 spectral subtypes, the limiting magnitude is about $13^{\rm m}$ with the Hamburg 80 cm Schmidt telescope). Seitter ([1975]) has developed spectral classification systems based on photographic objective prism spectra of different low dispersions (645 and 1280 Å/mm, what correspond to 10 and 20 Å resolutions, respectively). Observations have been performed at the Hoher List Observatory of Bonn University, diameter of correcting plate of the Schmidt telescope is 34 cm. The classification accuracies are 1.6 and 3 spectral subtypes, respectively. As an example of very low resolution spectral classification studies, we mention a paper of Nandy et al. ([1977]) where a description of an objective prism for the 1.2 m Schmidt telescope is given, dispersion is 2500 Å/mm (about 40 Å resolution), what provides photographic spectral classification with an accuracy of 5 spectral subtypes, the limiting magnitude of $20^{\rm m}$ may be even achieved.

Variations of calculated refraction with the spectral luminosity classes should be analysed, too. In Fig. 3 the calculated refraction is given as a function of spectral type for luminosity classes I, III and V (supergiants, giants and dwarfs), respectively. The luminosity effects in refraction seem to be significant, differences between luminosity classes reach 0.06 arcsec for K - M stars. Judging on positions of luminosity class lines in Fig. 3 we see that we should only distinguish supergiants from stars of other luminosity classes (giants and dwarfs) for stars of spectral types earlier G0; and we should separate supergiants, giants and dwarfs for stars G5 and later to provide the accuracy of 0.02 arcsec per luminosity class parameter in calculated refraction. There are many spectral classification methods aimed at study of spacial distributions of stars in galactic investigations and therefore providing segregation of stars into luminosity classes. The above mentioned classification system of Malyuto et al. ([1997]) easily segregates K - M dwarfs, giants and supergiants. The segregation of stars into luminosity classes is possible in the conventional MK system (Johnson & Morgan [1953]) and other similar ones (e.g. Upgren [1963]; Seitter [1975]; Houk [1994]) as well as with the use of photometric methods (e.g. Straizys [1992]). We conclude that the accuracy of 0.02 arcsec per luminosity class parameter in calculated refraction is achievable. If we are satisfied with the accuracy of 0.05 arcsec per luminosity class parameter in calculated refraction or lower, we would need spectral types only (luminosity effects become unsignificant).

Figure 3: The dependencies of the calculated refraction on spectral types for different luminosity classes. The solid line corresponds to luminosity class V, the short-dashed line to luminosity class III, the long-dashed line to luminosity class I. One desirable error in refraction (0.05 arcsec) is indicated. All calculations have been made at the zenith distance of $60^{\circ }$

If astrometric observations near the galactic plane are performed, interstellar reddening effects should be taken into account. In Fig. 1 and Fig. 2 the calculated reddening effects are shown for representative spectral types and $T_{\rm {eff}}$, respectively, at different degrees of interstellar reddening. Stellar temperature and reddening effects in refraction are of the same order; for reddened stars the temperature effects are smaller than for unreddened stars. Table 3 contains the accuracies in interstellar reddening required to achieve the indicated accuracies in calculated refraction at different degrees of interstellar reddening for B0, G0 and M0 dwarfs. The data show that the most stringent requirement to accuracy in reddening is for earlier type stars. We see from the first column of Table 3 that we should achieve an accuracy of 0.05 in E(B-V) for B0V star in order to provide an accuracy of 0.02 arcsec per parameter in refraction. The same conclusions have been made for other luminosity classes (giants and supergiants).

 

 

Table 3: Accuracies in E(B-V) required to achieve the desirable accuracies per parameter in calculated refraction for dwarfs

Accuracy in	E(B-V)
refraction	0.5	1	2	3	4
[as]	Accuracy in E(B-V)
0.02	0.05	0.06	0.11	0.21	0.38
0.05	0.13	0.16	0.28	0.53	0.95	B0
0.10	0.26	0.32	0.56	1.06	1.90
0.02	0.07	0.09	0.17	0.31	0.51
0.05	0.18	0.24	0.42	0.77	1.28	G0
0.10	0.36	0.48	0.84	1.54	2.56
0.02	0.11	0.14	0.26	0.44	0.67
0.05	0.28	0.36	0.64	1.1	1.68	M0
0.10	0.56	0.72	1.28	2.2	3.36

Recently Weaver & Torres-Dodgen ([1997]) has described a classification system (based on 15 Å resolution spectra obtained with silicon-based detector) which provides an accuracy of 0.4 spectral subtype and 0.05 in E(B-V) for A-stars. In more general case the parameter E(B-V) is defined from the formulae

 

E(B-V)=(B-V)-(B-V)₀, (3)

where B-V is the observed index for a concrete star, (B-V)₀ is the intrinsic index for respective spectral type and luminosity class in the MK system. Schmidt-Kaler ([1982]) has estimated the intrinsic scatter for a given type being from $\pm 0.02$ to $\pm 0.06$ in (B-V)₀, the typical accuracy of the observed (B-V) index is about 0.01. We see from the equation 3 that, as a rule, accuracy of 0.05 in E(B-V) may be realized if reliable MK spectral types and luminosity classes are available (examples of the respective spectral classification systems were given above). We conclude that the accuracy of 0.02 arcsec per reddening parameter in calculated refraction is achievable.

Combining the above results for stellar parameters used in refraction calculations (for spectral type, luminosity class, degree of interstellar reddening) we conclude that there are about the same achievable accuracies in refraction (about 0.02 arcsec per each stellar parameter). Therefore, the combined error in calculated refraction due to uncertainties in three stellar parameters discussed above is about 0.03 - 0.04 arcsec (it is by a half order better than the error for the best available observed stellar positions).

Till now all authors had ignored the possible dependencies of atmospheric refraction on stellar metal abundance (it is the third main physical parameter in spectral classification, side by side with spectral types and luminosity classes). To estimate it, we have added the available spectral energy distributions for selected metal-deficient stars (giants and dwarfs), as given by Sviderskiene ([1992]). The data were corrected for interstellar reddening. The values of the main physical parameters for these stars are taken from the catalogue of [Fe/H] determinations (Cayrel de Strobel et al. [1997]). For our estimates only the stars with the lowest metal abundance ( $-1.4>{\rm [Fe/H]}>-2.9$) are discussed, the known spectral binaries are excluded. There are 9 metal-deficient G-K giants and 11 metal-deficient F-G dwarfs in our list. Because of the uncertainty of spectral types for such stars we have calculated and discussed the dependence of the calculated refraction only on $T_{\rm {eff}}$ (Fig. 4). We conclude that the effect of metal abundance on refraction is small and reaches only 0.02 - 0.03 arcsec in these extreme cases. The metal-deficient dwarfs (F-G subdwarfs) have slightly larger refraction than the dwarfs of normal metal abundance with the same temperature.

Figure 4: The dependence of the calculated refraction on $T_{\rm {eff}}$ for metal-deficient stars ( $-1.4>{\rm [Fe/H]}>-2.9$). The designations used: plus signs - metal-deficient G-K giants, crosses - metal-deficient F-G dwarfs. The lines are the same as in Fig. 3 (correspond to stars of normal metal abundance) but the calibration "Spectral type versus $T_{\rm {eff}}$'' of Schmidt-Kaler ([1982]) is involved. One desirable error in refraction (0.05 arcsec) is indicated. All calculations have been made at the zenith distance of $60^{\circ }$

We have also analysed the dependencies of calculated refraction on atmospheric parameters (ground-based temperature, pressure, dew point, subsequently). From the slopes of these dependencies we have estimated how high the errors per every parameter in calculated refraction should be to achieve desirable accuracies of 0.01, 0.02, 0.05 and 0.1 arcsec per input parameter in calculated refraction. The results are given in Table 4 and should be compared with the accuracies of measuring devices. According to Stone ([1996]), the atmospheric pressure measured with Setra digital barometers may be calibrated to $\pm 0.1$ mm, the dew point sensors used by Stone ([1996]) are accurate to $\pm 2^\circ$F (about $\pm 1^\circ$C). From the comparison of the just mentioned accuracies with the data of Table 4 we conclude that the accuracies of about 0.01 - 0.02 arcsec per every parameter in calculated refraction are achievable with these devices (the highest accuracy is achievable per dew point parameter).

 

 

Table 4: Accuracies in atmospheric parameters necessary to reach the desirable accuracies per parameter in calculatedrefraction

Accuracy in	Accuracy in atmospheric parameters
Refraction	Temperature	Pressure	Dew point
[as]	[degC]	[mm]	[degC]
0.01	0.03	0.07	1.1
0.02	0.06	0.15	2.3
0.05	0.14	0.37	5.7
0.10	0.29	0.75	11.5

But the temperature case is more complicated. Although the resistance temperature detector may be calibrated to $\pm 0.05^\circ$C (Stone [1996]), "in general, the accuracy of measurements of the temperature cannot be forced to lower than $1^\circ$C. The reason is not to be found in the measuring instruments but in local temperature gradients inside the dome and telescope itself'' (Ploner [1996]). The problem has been carefully studied by Stone et al. ([1996]), their observational stellar positions are corrected for so called room and tube refractions (with the use of some temperature probes giving the temperature distribution within the dome at any particular time, air circulation system is installed). The room refraction is found to be the most serious source of errors in calculated refraction (typically the error is about 0.03 arcsec but can exceed 0.1 arcsec on some nights). The refraction model used in the present paper doesn't take into account the room refraction. However, the respective temperature errors in calculated refraction may be avoided or radically reduced if we consider differential refraction only (see the next section).