In order to find some typical estimates of such instrumental polarization, we shall consider the case of 2.3 m Vainu Bappu telescope of Indian Institute of Astrophysics, Kavalur. This telescope has beam sizes f/3.23 and f/13 at the prime and Cassegrain focii. The diameters of the primary and secondary mirrors are 2.32 and 0.63 m. The hyperbolic secondary has an eccentricity of 2.7776.
We consider a bunch of unpolarized parallel rays incident on the primary of
such a telescope and calculate the instrumental polarization at the prime focus.
The bunch of parallel rays are coming from a direction making a field angle
with the telescope axis. We consider different values of within
range 0-300 arcsec and corresponding instrumental polarization values
as calculated are reproduced in Table 1 (click here). At 300 arcsec field angle, the
amount of instrumental polarization is of the order of 
. This
value is too low and no present day astronomical polarimeter can measure
such a small value.
| FLD (sec) | POL (unpol) | Pol |
|
| (for 100% pol. str.) | |
| 0 | 0.000000 | 99.999886 |
| 20 | 0.000001 | 99.984192 |
| 40 | 0.000002 | 99.937111 |
| 60 | 0.000005 | 99.858650 |
| 80 | 0.000007 | 99.748764 |
| 100 | 0.000010 | 99.607513 |
| 120 | 0.000015 | 99.434906 |
| 140 | 0.000020 | 99.230873 |
| 160 | 0.000026 | 98.995458 |
| 180 | 0.000033 | 98.728645 |
| 200 | 0.000041 | 98.430466 |
| 220 | 0.000049 | 98.100891 |
| 240 | 0.000059 | 97.739960 |
| 260 | 0.000069 | 97.347595 |
| 280 | 0.000079 | 96.923820 |
| 300 | 0.000092 | 96.468719 |
We consider the above rays coming from a star instead of being unpolarized,
are now 
polarized with the polarization vector making an angle 45
degrees with the reference X-axis. Here we want to see the extent of instrumental
polarization effect. At zero field angle the depolarized value of stellar polarization
is 
. Also as we can see from Table 1 (click here), the degree of polarization
reduces to 96.5% from at a field angle 300 arcsec. However,
the polarization vector remains fixed at 45 degrees (not shown in
Table 1 (click here)).
It can be seen from the detailed calculations described in previous sections, that such instrumental polarization values increase with the fastness of prime focus beam and obviously with the field angle.
Now we consider the instrumental polarization observed at the Cassegrain
focus for the above two cases, (i) unpolarized light and (ii)
polarized light incident on the primary. In the first case we can
see from Table 2 (click here) that the polarization values are
definitely higher compared to the prime focus. At 90 arcsec one gets a
value of 
polarization. With a high precision polarimeter one can
make attempts to measure such a value. In the second case, we observe that
at the Cassegrain focus there is
considerable depolarization. The depolarized value of polarization at zero
field angle is 
, slightly higher than the corresponding value
at prime focus. At a field angle of 90 arcsec, as can
be seen from Table 2 (click here), a polarized star will show 
polarization.
| FLD (sec) | POL (unpol) | Pol |
|
| (for 100% pol. str.) | |
| 0 | 0.000000 | 99.998334 |
| 10 | 0.001774 | 99.998329 |
| 20 | 0.003552 | 99.998322 |
| 30 | 0.005328 | 99.998314 |
| 40 | 0.007114 | 99.998299 |
| 50 | 0.008901 | 99.998283 |
| 60 | 0.010693 | 99.998253 |
| 70 | 0.012491 | 99.998230 |
| 80 | 0.014294 | 99.998207 |
| 90 | 0.016104 | 99.998184 |
The instrumental polarization at the Cassegrain focus depends upon the diameters of two mirrors, the corresponding beam sizes and eccentricity of the hyperboloid. For unpolarized light we normally expect a higher instrumental polarization value at the Cassegrain focus as compared to the prime focus. In case of primary mirror the light is incident symmetrically over the entire mirror surface for any field angle. However, for secondary mirror the rays are incident asymmetrically and this asymmetry increases with the field angle. This should cause higher values of instrumental polarization for the Cassegrain focus. In our calculations we have considered only those rays which are reflected from the periphery of the primary mirror. However, in actual case we should consider reflections from the entire surface of the primary mirror (down upto the Cassegrain hole) and then integrate the Stokes parameter values. The inner region of the mirror will exhibit a lower instrumental polarization value as the beam becomes slower there. Therefore the instrumental polarization values that we have calculated can be considered only as the upper limit of such effects.
It has been already discussed in Sect. 1 (also Sen & Tandon 1994), that
a typical present day polarimeter can measure linear polarization with an
accuracy 
(barring the case of very bright objects like
moon, planets etc.). Under such a condition one may question our attempts to
quote up to six places after the decimal, the percent polarization values.
This we have done to understand the nature of field angle dependence of such
polarization. Higher accuracy in polarization measurements, helps one to
understand the ongoing astrophysical processes in a better way. For example,
way back in 1974, Clarke & Mclean (1974) had discussed about the polarization
occurring within the stellar line profile and recommended the need for measuring
polarization with a detectability 
. One also requires high
accuracy on the estimated p values in order to study the nature of its wavelength
dependence, where the polarization is caused by synchrotron emission or dust
scattering.
During imaging polarimetry, while we try to improve polarimetric accuracies with better telescope aperture and instrument, we should also keep in mind the limitation put by the telescope optics itself.
Acknowledgements
We are thankful to Prof. S. Bujarbarua, Director, Centre of Plasma Physics, for his encouragement to this work and Prof. S.N. Tandon, Inter University Centre of Astronomy and Astrophysics, for useful discussions on the problem.