Right picture: Palomar Observatory image made by R. Minkowski in 1960, giving the optical identification of the central galaxy in the cluster 3C295. Angular size of box = 15 arc min. (source for 3C295 info: Chandra Photo Album)
Without going into details, just a little bit of background. Most astronomical sources are large in physical size, even though small in angular size, and radiation is emitted by a large number of statistically independent sources- atoms, molecules, or electrons. The resulting signals are noise-like, that is, the electric fields are Gaussian random variables with spectra that depend on the details of the emission mechanisms. Laboratory laser and maser oscillators are usually coherent because of the cavity in which they oscillate. Astronomical masers have never been found to have non-Gaussian statistics. For similar reasons, many radio-astronomical sources are unpolarized; that is, signals in one polarization are statistically independent of signals in the orthogonal polarization. But some sources, especially those involving magnetic fields that extend over a significant part of the spatial extent of the source, can be polarized, and studying such polarization sometimes leads to significant insights.
3C295 is such an unpolarized source. However, observing this source with, for example, the Westerbork Synthesis Radio Telescope (WSRT) will show the presence of polarization. As the atmosphere cannot be the source of this polarization, this polarization must have an instrumental origin. In September 1989, Frank Robijn and I, wrote a report on the internal polarization of the WSRT as part of our curriculum.
Observations where made with respect to 3C295 according to
Δα = r sinλ,
Δδ = r cosλ
where λ is the position angle and r the angular distance from the center. So Δα and Δδ correspond to the offset of the source relative to the telescope pointing. Thus a grid was defined of 97 observation points in which the values of the Stokes parameters I, Q, U and V were determined. From this the polarization fraction
π = (Q2 + U2)1/2I -1
and the polarization angle
θ = ½arctan(U/Q)
have been determined. Plotting the data, after reduction, for U/I and Q/I, suggested a depedance of the form sin(2λ).
Modeling the observations
The simplest function, suggested by the results, to fit the data seemed to be
S = Ao + A2sin(2λ + φ)
where S respresents one of the Stokes parameters Q and U, and λ is the position angle. The amplitude and phase then follow from elementary Fourier analysis. Though the amplitude calculated seemed to represent the data rather well and turned out to be a regular function of r, the phase appeared to be determined badly in a large number of cases. The obvious thing to do is then to write out the relation given above, thus giving
S = Ao + A2sin(2λ) + B2cos(2λ)
because in determining the phase, one first has to derive A2 and B2, and then calculate the phase from these numbers. The error in the phase is now determined by the combined errors in A2 and B2. An expert in the field of polarization suggested including a term proportional to sin(λ), leading to
S = A1sin(λ) + B1cos(λ) + A2sin(2λ) + B2cos(2λ)
Since the results clearly suggested the presence of a 'constant' term, we also tried to fit
S = Ao + A1sin(λ) + B1cos(λ) + A2sin(2λ) + B2cos(2λ)
As it turned out, this last function respresented the observations really well. If we allow for explicit r-dependence, we want the coefficients to be functions of r in the following way
Ao = Ao(r)
A1 = a1A1(r)
B1 = b1A1(r)
A2 = a2A2(r)
B2 = b2A2(r)
because the parameter values found suggested constant ratios A1/B1 and A2/B2. In order to derive ai, bi and Ai(r), the following procedure was chosen. Because the absolute error in determining the coefficients for the fit in the position angle is the same at every radius, those coefficients were simply summed over all radii. From the ratios of the summed coefficients ai and bi were determined, adding a further constraint that |ai| + |bi| = 1 (i = 1,2).
Then by adding the absolute values of the coefficients per radius, the Ai(r) is determined. The data indicated that the assumption of a constant phase was indeed justified. On theoretical grounds one would expect that the amplitudes Ai behave like
Ai = Cisin2(Dir)
Chandra Photo Album)