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Analysis of linear polarization basing on the single Stokes parameter
G. V. Lipunova June 4, 2015
Abstract One MASTER telescop e is equipp ed with two orthogonal p olarizers. It happ ens sometimes that a single telescop e observes a transient source. We investgate what information ab out the source's linear p olarization PL can b e learned from the single Stokes' parameter. For the case of zero Stokes parameter, dep endences of the 1 and 2 PL upp er limits on the Stokes parameter's uncertainty D are found. Different values of observed Stokes parameter corresp ond to different dep endences max(PL ) vs. D. They can b e calculated by the method prop osed.

1

Limit on the degree of the linear p olarization

When observing with only two perpendicular polaroids, just one Stokes parameter can be inferred. This Stokes parameter is the lower limit on the degree of the linear polarization PL . The polarization angle cannot be defined. Let I1 , I2 be observable fluxes in two perpendicular polaroids. We derive value I1 - I2 D= . I1 + I2 If I is the total flux (the value proportional to counts) from the source, then I1 = I PL cos2 and one Stokes parameter is D = PL cos 2 . If errors I of I1 and I2 are normally distributed, I1 = I2 , then approximately 1 I D = D 2I (But see Simmons J. F. L. & Stewart B. G. 1985, A&A, 142, 100). I2 = I PL sin2

2

Allowed values of PL from D, zero noise

If a source with the degree of linear polarization PL is observed at some polarization angle , the noise-free Stokes parameter is as follows: D() = PL в cos(2 ) 1


We can find the range of allowable angles that corresponds to some variation

Figure 1: Dimensionless Stokes parameter versus polarization angle. The grey band corresponds to an observed value, Do ± D = 0.1 ± 0.02. Three vertical lines correspond to three values of angle, from left to right: o - 1 , o , and o + 2 ­ see relations (1). of D : Do = PL в cos(2 o ) Do + D = PL в cos(2 (o - 1 )) Do - D = PL в cos(2 (o + 2 )) (1) Probability for the polarizer's direction to occur in the corresponding range over angles is (1 + 2 ) 2/ (2) as can be seen from Fig. 1. For a set of values PL , Do , and D, system (1) can be solved to find o , 1 , and 2 . Values (2) are shown by the green lines in Figs.2­8, designated as `Probability'. If Do + D > PL , 1 or 2 are cut accordingly to provide cos = 1. Thus, for values PL = 0 non-zero D cannot be obtained in the case of zero noise. This is a shortcoming of such deterministic method. To address possible errors of observed D, we perform simulations using the Monte-Carlo method.

3

Application of the Bayes' theorem

Let X and Y are the continous random variables. Consider events X = x and Y = y , where x and y are some numerical values. Consider the the Bayes'

2


theorem, formulated in terms of the probability densities f fX (x|Y = y ) = fY (y |X = x) fX (x) fY (y )

X

and fY , (3)

Let X be all possible values of the degree of linear polarization: X = PL and [0, 1]. Then x = PL is the degree of the linear polarization of the source. Let Y be all possible values of the dimesionless Stokes parameter Y = D, which can be observed, and Y [-1, 1]. For the specific data, we derive the dimesionless Stokes parameter Do . According to (3), the probability that the source has the linear degree of polarization PL if we observe value Do is: f
PL (PL |D = Do ) = fD (D = Do |PL = PL ) в f fD (Do ) PL (PL )

(4)

It is be more practical to find the following probability: f
PL (PL PL |D = Do ) = fD (D = Do |PL PL ) в f fD (Do ) PL (PL PL )

(5)

It is assumed that the probability of a source to have specific PL is uniform:

f

PL

(PL ) = 1 .

(6)

In its turn, the probability density of observing certain Do from a source with some polarization less than PL is the limit of the ratio of probability dP to measure D inside some small interval [Do - dD..Do + dD] to the size of the interval: fD (D = Do |PL Similarly, fD (Do ) = dP (D [Do - dD..Do + dD]) 2 dD (8)
PL

)=

dP (D [Do - dD..Do + dD] PL PL )

2 dD

(7)

3.1

Monte-Carlo simulations

Performing Monte-Carlo simulations, we can derive probability density (7) as follows. Let us generate n sources with identical PL and different polarization angles , uniformly distributed over interval [0.. ]. Consequently, we obtain diffrent values of D = PL cos 2, which would be observed from such sources in the case of zero noise. To take into account the noise (experimental random errors) we shift each D by a random value distributed as N (0, D) (normal distribution with zero mean and standard deviation D) and obtain Dsh . Hence we presume that the absolute error of observed D is distributed normally. We vary value PL from 0 to 1 at equal steps, each time generating n points, and count the number of the following events: Ntot -- total number of points. NA -- when PL PL . NB -- when |Dsh - Do | dD. nB (PL ) -- when |Dsh - Do | dD for each value PL . 3


Figure 2: Do = 0.4, D = 0.025, dD = 0.025.
NB A -- when |Dsh - Do | dD and PL PL at the Value dD can be chosen arbitrarily but it should be The simulation consists of 501 steps over PL with dD = D or dD = 0.5 в D. The total numer of events We understand that P (PL PL ) PL = NA /N

same time. of order of D. parameters n = 2000, is Ntot = 1002000.

tot

P (A)

(9) (10) (11)

fD (Do ) = NB /N

tot

P (B ) /N
A

fD (D = Do |PL PL ) = N

BA

P (B |A)

Furthermore, the probability that a source with PL gives observed D in the interval [Do - dD...Do + dD] is P (PL , Do , D, dD) = nB /n . (12)

Evidently, the last function depends on the value of dD. This function, designated as 'Probability', is shown by black curves in Figs. 2-8 and is to be compared with the result obtained by (2) shown by the green lines. Thus, the value, which we seek, f can be found as P (A|B ) =
PL (PL PL |D = Do )

P (B |A) в P (A) NB A . = P (B ) NB the maximum se. The result be also calcula its maximum

(13) value, giving is shown in Figs. 2-9 ted as the cumulative value.

The last value should be normalized by fPL (PL 1|D = Do ) = 1 at the extreme ca in the correspondingly named panels. It can sum of (2) (green lines), also normalized by

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Figure 3: Do = 0.4, D = 0.005, dD = 0.005.

Figure 4: Do = 0, D = 0.005, dD = 0.005.

Figure 5: Do = 0, D = 0.005, dD = 0.0025.

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Figure 6: Previous results for the probability calculated by (2). Values D are shown for the corresponding curves. The 2%-curve agrees with the green curves in Figs. 7 and 8 (the left upper panel). Figure is from Gorbovskoy et al. (2012)

3.2

Comparison to previous results

In Gorbovskoy et al (2012) a result using formula (2) for Do = 0 was reported, which we reproduce here. Fig. 6 shows the probability of the degree of linear polarization to be greater than the value on the horizontal axis. The case of observed Do = 0 is considered (zero Stokes' parameter). Different curves are plotted for different D. In Figs. 7 and 8 we show that those results are consistent with the present ones (the green curves in panels for P (P > PL |Dobs )).

4

Upp er limits on PL for different D and different confidence levels

The practical interest is to provide a limit on PL -value of a source basing on the observed Do . In Fig. 10, results for Do = 0 using (2) are presented. We calculate the same dependences via Monte-Carlo simulations described above (Fig. 11). The upper limits can be calculated for other values of Do . An example, corresponding to the case of GRB 140801, is plotted in Fig. 9.

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Figure 7: Do = 0, D = 0.02, D = 0.02.

7


Figure 8: Do = 0, D = 0.02, D = 0.01.

8


Figure 9: Do = 0.024, D = 0.025, D = 0.025. 1- upper limit on PL is 24% and 2- upper limit on PL is 81%. These are slightly higher than the limits for Do = 0 and D = 0.025, 21% and 80%, which can be found from Fig. 11. The case of GRB 140801.

9


Figure 10: For observed Do = 0, the value on the vertical axis represents the upper limit on then degree of linear polarization versus D rel . Three curves correspond to different confidence levels, designated for each curve.

10


Figure 11: Results of Monte-Carlo simulations for Do = 0, dD = D (black) and dD = 0.5 D (red). The value on the vertical axis represents the upper limit on then degree of linear polarization versus D. Two curves correspond to different confidence levels, designated for each curve.

11