4 Differential calculations for atmospheric refraction

If two or more stars are observed simultaneously within a frame of a CCD detector, differential measurements of stellar positions become possible. In differential calculations for refraction the respective uncertainties in atmospheric parameters become unimportant because ground-based atmospheric temperature, pressure and dew point are the same for every object at the time of the observation. We may expect that any changes in these parameters (as well as in temperature distribution within the dome) are small during an exposure. Then we should take into account only the color effects in calculated refraction due to differences in spectral types of stars and to different degrees of interstellar reddening. Judging on the results of the previous section, we conclude that the total error in calculated differential refraction should be within 0.04 arcsec if the best available spectral classification systems are applied.

In the subsequent analysis of differential refraction we follow mainly the pattern used by Gubler & Tytler ([1998]) although the aim, method and details of the analysis are different. Regardless of which approach is used, the following quantities must be specified in order to perform the calculations:

$$\begin{array}{lp{0.8\linewidth}} T_1 & effective temperature... ...at the time of the observation, respectively. \\ \end{array}$$

We present the results of calculations for one example: the differential refraction $\Delta R_{\rm m}$ between stars 1 and 2 for $z_1=60^{\circ }$ and $\Delta z_0=12$ arcmin has been calculated (taking into account that the frame of our CCD detector corresponds to the area of $12\times 18$ arcmin on the sky). We use the same approach to calculations as in the previous section with the same typical set of ground-based atmospheric parameters. A knowledge of the exact values of atmospheric parameters is not crucial in this differential case: as an example, if we change the ground-based temperature (the most uncertain atmospheric parameter) by $2^{\circ}$C, the respective calculated differential refraction are changed less than by 0.005 arcsec in the most extreme cases (if one goes from spectral type B to M at a zenith distance of $60^{\circ }$). Keeping in mind the planned differential measurements of positions of solar system bodies relative to positions of reference stars, we take star 1 as a solar type star (G2V), star 2 is G2V, B0V, M2V, B0I, M2I at different degrees of interstellar reddening (0.0, 0.5, 1.0, 2.0, 3.0 and 4.0) subsequently. The calculated differential positions are given in Table 5.

 

 

Table 5: Differential positions $\Delta R_{\rm m}$ (star 2 minus star 1) given in arcsec. Star 1 is G2V observed at $z_1=60^{\circ }$, the calculated refraction is 98.977 arcsec. In this example star 2 is observed at $z_2=60\hbox{$.\!\!^\circ$ }2$. Its E(B-V) and spectral types are as given in this table

E(B-V)	Differential positions $\Delta R_{\rm m}$
for star 2	G2V	B0V	M2V	B0I	M2I
0.0	0.796	1.099	0.520	1.074	0.489
0.5	0.647	0.899	0.427	0.877	0.405
1.0	0.526	0.726	0.356	0.709	0.340
2.0	0.365	0.481	0.263	0.471	0.255
3.0	0.277	0.347	0.211	0.342	0.206
4.0	0.227	0.275	0.178	0.272	0.175

Figure 5: The filter transmission curves together with the quantum efficiency (the lower line) of a CCD detector used in our study

Figure 6: The dependencies of the calculated refraction on B-V indices for different luminosity classes. The thin line is the reddening line for B5 dwarf at different E(B-V). Other designations are the same as in Fig. 3. All calculations have been made at the zenith distance of $60^{\circ }$

Figure 7: The dependencies of the calculated refraction on V-R indices for different luminosity classes. The designations are the same as in Fig. 6. All calculations have been made at the zenith distance of $60^{\circ }$

Figure 8: The dependencies of the calculated refraction on V-I indices for different luminosity classes. The designations are the same as in Fig. 6. All calculations have been made at the zenith distance of $60^{\circ }$

Three effects are combined together in this table: one depends on separation of the stars along the zenith direction (separation effect) a second is influenced by the difference in spectral types of the stars (stellar temperature effect) and a third is influenced by degrees of interstellar reddening (interstellar reddening effect). The pure separation effect at different reddenings is given in the second column where star 2 is of the same spectral type (G2V) as star 1. The total effect is largest for B0V (of order 1 arcsec) and decreasing with increasing the reddening. The appropriate observations for some pairs of stars with very differing spectral types taken from the Hipparcos catalogue are planned to compare calculated and true refraction.