Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.asc.rssi.ru/RadioAstron/publications/articles/cr_2015,53,199.pdf Äàòà èçìåíåíèÿ: Tue Jun 9 10:31:32 2015 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:56:28 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: arp 220

ISSN 0010 9525, Cosmic Research, 2015, Vol. 53, No. 3, pp. 199­208. © Pleiades Publishing, Ltd., 2015. Original Russian Text © I.N. Pashchenko, Yu.Yu. Kovalev, P.A. Voitsik, 2015, published in Kosmicheskie Issledovaniya, 2015, Vol. 53, No. 3, pp. 214­224.

First Estimate of the Value of the Instrumental Polarization of the RadioAstron Space Radio Telescope Using the Results of an Early Scientific Program for Observing Active Galactic Nuclei
I. N. Pashchenko, Yu. Yu. Kovalev, and P. A. Voitsik
Astro Space Center, Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia e mail: in4pashchenko@gmail.com
Received December 16, 2013

Abstract--To interpret radio interferometric observations sensitive to linear polarization it is necessary to take into account the effect of instrumental or "parasitic" polarization. In the case of ground­space very long baseline interferometry, this procedure involves a session of polarization sensitive mapping with the suf ficient number of ground stations and sufficiently small projections of ground­space baselines. In the paper, the problem of estimating the value of the instrumental polarization of the Spectr R space radio telescope of the RadioAstron project using the results of the early scientific program for observing active galactic nuclei is addressed. We use a statistical approach to estimate the value of the instrumental polarization associated with the analysis of already obtained to present time intermediate results of the review of brightness temperatures. Estimates obtained for the frequency bands C and L (95% probability intervals [0.0646, 0.1267] and [0.0945, 0.1736] for 6 and 18 cm, respectively) suggest that the instrumental polarization of the radio telescope does not exceed the values typical for the ground VLBI stations. DOI: 10.1134/S0010952515030053

1. INTRODUCTION One of the main scientific goals of the RadioAstron project [1] is to perform ground­space radio interfer ometric observations of active galactic nuclei (AGN) with ultrahigh angular resolution, which are sensitive including to the linear polarization of the radio emis sion. Information about the polarization of the radio emission of magnetized plasma relativistic ejections and its frequency dependence is, in fact, a unique opportunity to study magnetic fields, elemental com position, and the energy spectrum of ejection matter particles. This is connected with the dependence of transfer coefficients of the polarized emission on these parameters [2, 3]. According to existing concepts, magnetic fields play an important role in the forma tion, collimation, and acceleration of jets in many space objects such as quasars, microquasars, and young stellar objects [4]. However, the problem of ana lyzing the results of polarization sensitive observations using the method of very long baseline interferometry (VLBI) is complicated by the presence of instrumental effects, one of which is so called polarization "leak age." Taking into account the weakness of the signal of the linear polarization of active galactic nuclei ejec tions (the degree of linear polarization in optically dense regions is typically only a few percent), ignoring this effect can lead to incorrect conclusions about the properties of the investigated objects. The unac counted for (or mistakenly accounted for) effect of the instrumental polarization in magnitude and its

location in the region of the maximum full intensity can coincide with the expected signal [5], which, in this case, makes interpretation extremely difficult. It should be noted that the methods of estimates of the value of the instrumental polarization in the arse nal of ground VLBI are also applicable, as the experi ence of the VSOP project shows [6], in the case of ground­space VLBI. However, the accuracy of esti mates obtained by these methods in connection with small experience of using the ground­space VLBI for polarization measurements have not been sufficiently studied. In particular, this applies to the case, in which in this frequency band the orbital radio telescope is able to detect only one orthogonal polarization com ponent (as, for example, the HALCA Japanese satellite of the VSOP project [6] or Spektr R in the frequency band of 6 cm). We performed a series of experiments with the observational data of the MOJAVE project [7] obtained using the VLBI network of the Very Long Baseline Array (VLBA) and taken by us from the stan dard archive, simulating the situation arising when calibrating the instrumental polarization of a space radio telescope (SRT). Within these experiments we have been considered the influence of two factors the oretically serving as a source of uncertainty when obtaining estimates of leakage, namely: the absence of intermediate sized baselines with ground radio tele scopes and recording only one of the orthogonal (in this case, circular) polarization components. The results of the performed experiments indicate that, in the case of the influence of both factors, the estimation

199


200

PASHCHENKO et al.

accuracy of the instrumental polarization can be reduced by an order of magnitude. In this paper, we solve the problem of obtaining a first estimate of the SRT instrumental polarization using the data of early scientific program (ESP) for observing AGNs, namely, the review of brightness temperatures. 2. LINEAR MODEL OF THE INSTRUMENTAL POLARIZATION The signals induced in both polarizers of the radio telescope antenna, each of which is intended to receive one of the two orthogonal polarizations (in our case, the left and right circular polarizations) are actually the sum of two components: the response to the nominal and the response to orthogonal to it polarization: V R = G R ( E R exp ( ­ i ) + D R E L exp ( + i ) ) , V L = G L ( E L exp ( ­ i ) + D L E R exp ( ­ i ) ) , where ER and EL are electric fields of the right and left orthogonal components of circular polarization of the radiation field, GR and GL are time dependent complex antenna gain coefficients (GiR(t) = giR(t)exp(iiR(t)) for an antenna with the number i), DR and DL are complex coefficients (also designated as "D member") that characterize the response to radiation of orthogonal polarization and are a quantitative measure of the effect of "leakage" and is the parallax angle of the polarizer, which characterizes its orientation relative to the observed source. It is important that the parallax angle changes continuously when accompanying the source by the antenna having altazimuth mounting, but remains constant in the case of an antenna with an equatorial mounting. This makes it possible, in many cases, to separate the signal of the parasitic polarization and the source signal. It can be shown that the cross correlations of responses of a pair of antennas are associated as fol lows with the Stokes parameters (visibility functions or amplitude of corresponding interferometric lobes) of emission of investigated source I 12, Q12, U 12, V12 and instrumental coefficients G1R, G1L, G2R, G2R and D1R, D1L, D2R, D2R [8]:

L 1 L * V L 1 V *2 2 L ~ = G 1 L G *L [ ( ~ 12 ­ V 12 ) exp ( i ( + 1 ­ 2 ) ) I 2 ~ + D 1 L D *L ( ~ 12 + V 12 ) exp ( i ( ­ 1 + 2 ) ) 2I ~ 12 ~ 12 + D 1 L P * exp ( i ( ­ 1 ­ 2 ) ) +D *L P * exp ( i ( + 1 + 2 ) ) . 2 R 1 L * V R 1 V *2 2 L ~ = G 1 R G *L P 12 exp ( i ( ­ 1 ­ 2 ) ) 2 ~ 12 + D 1 R D *L P * exp ( i ( + 1 + 2 ) ) 2 ~ + D 1 R ( ~ 12 ­ V 12 ) exp ( i ( + 1 ­ 2 ) ) I ~ + D *R ( ~ 12 + V 12 ) exp ( i ( ­ 1 + 2 ) ) , 2I L 1 R * V L 1 V *2 2 R ~ = G 1 L G *R [ P 21 exp ( i ( + 1 + 2 ) ) 2 ~ + D 1 L D *R P 12 exp ( i ( ­ 1 ­ 2 ) ) 2 ~ + D 1 L ( ~ 12 + V 12 ) exp ( i ( ­ 1 + 2 ) ) I ~ + D *R ( ~ 12 ­ V 12 ) exp ( i ( + 1 ­ 2 ) ) . 2I ~ ~ ~ Here, P 12 = Q 12 + iU 12 = E 1 R E *L is the complex 2 linear source polarization. The tilde over the Stokes parameter designates the value in the spatial frequency plane, but not in the image plane, and the asterisk des ignates the operation of complex conjugation. How ever, in practice, linearized ratios are often used (tak ing into account the small value of the degree of linear and especially circular source polarization, as well as D members, about a few percent): R 1 R * = G 1 R G *R ~ 12 exp ( i ( ­ 1 + 2 ) ) , 2 2I L 1 L * = G 1 L G *L ~ 12 exp ( i ( + 1 ­ 2 ) ) , 2 2I ~ R 1 L * = G 1 R G *L [ P 12 exp ( i ( ­ 1 ­ 2 ) ) 2 2
+ D 1 R ~ 12 exp ( i ( + 1 ­ 2 ) ) + D *L ~ 12 exp ( i ( ­ 1 + 2 ) ) ] . I 2I

R 1 R * V R 1 V *2 2 R ~ = G 1 R G *R ( ~ 12 + V 12 ) exp ( i ( ­ 1 + 2 ) ) I 2 ~ + D 1 R D *R ( ~ 12 ­ V 12 ) exp ( i ( ­ 1 + 2 ) ) 2I ~ 21 ~ +D 1 R P * exp ( i ( ­ 1 + 2 ) ) +D *R P 12 exp ( i ( ­ 1 ­ 2 ) ) , 2

~ L 1 R * = G 1 L G *R [ P 21 exp ( i ( + 1 + 2 ) ) 2 2
+ D 1 L ~ 12 exp ( i ( ­ 1 + 2 ) ) + D *L ~ 12 exp ( i ( + 1 ­ 2 ) ) ] . I 2I 3. FORMULATION OF A PROBABILITY MODEL Let us pass to the formulation of a probability model that connects the observational data and unknown
COSMIC RESEARCH Vol. 53 No. 3 2015


FIRST ESTIMATE OF THE VALUE OF THE INSTRUMENTAL POLARIZATION

201

instrumental coefficients. For this, it is necessary to construct a joint distribution of the observational data and unknown parameters, including both instrumental effects and the parameters of the observed source. Using the constructed model, we can estimate the probability density of the unknown parameters, using the Bayesian approach. We consider separately observational data and model parameters. 3.1. Used observational data. The main observa tional mode in the RadioAstron project within ESP for observing AGN is sufficiently short time observa tional sessions with different a value and projection direction of a ground­space baseline. The obtained values for the visibility function after amplitude cali bration or their upper limits are to be used for estimat ing the correlated flux and, therefore, the source brightness temperature [9]. The intermediate result of a review of brightness temperatures is the set of values of the signal/noise (S/N) ratios for the measured inter ferometric responses for various observational ses sions. It follows from section 1 that in each lobe mea * surement of cross correlation (i.e., R1L* or LR2 ) for a 2 1 baseline with SRT, there is a contribution from the instrumental polarizations for both baseline antennas. Taking into account the proposed constancy of coeffi cients of the SRT instrumental polarization, the inter mediate results of the current review of brightness temperatures can be used for estimating these values. It is possible to significantly reduce the number of unknown parameters in the model correlations con necting measurements in the review correlations with instrumental coefficients, forming the following ratios of cross to parallel correlations [8]: R1 L* / R1 R* 2 2 = 1+ e r2
i
2

ence between the corresponding phases, 12 is the dif ference of the parallax angles for both antennas, ~ M 12 is complex degree of the linear polarization of the observed source. Here, m is the degree and is the position angle of the linear source polarization. We will assume that the sensitivities of both channels (left and right circular polarizations) of both radio tele scopes formed the baseline are equal, which, strictly speaking, cannot be true. Then, the ratio of the ampli tudes of interferometric lobes with cross to parallel correlations can be replaced by the signal/noise ratio for lobes of the corresponding correlations, i.e., to put R 1 L * / R 1 R * = (S/NRL)/(S/NRR). Thus, using the 2 2 ratio of the measured data, we eliminate the need for amplitude calibration. The model ratio connecting unknown parameters r, M, DGRT, i, and DRA with the observed data ((S/N)+/(S/N)|| is as follows: ( S / N )+ / ( S / N ) = r M exp ( i 1 ) + D
GRT

(2) exp ( ( i 2 ) + D RA exp ( i 3 ) ) ,

||

~ (M 12 exp ( ­ 2 i 2 ) + D 1 R exp ( + 2 i 12 ) + D *L ) , 2 L 1 R * / R 1 R * = 1 exp ( ­ 2 i 1 ) 2 2 r1

~ â M * exp ( + 2 i 2 ) + D 1 L exp ( ­ 2 i 12 ) + D *R , 2 21 1 ~ L 1 R * / R 1 R * = exp ( ­ i 1 ) M * exp ( ­ 2 i 1 ) 2 2 21 r2 +D
1L

(1)

+ D *R exp ( + 2 i 12 ) , 2

~ L 1 R * / L 1 L * = r 2 exp ( ­ i 2 ) M * exp ( + 2 i 2 ) 2 2 21 + D 1 L exp ( ­ 2 i 12 ) + D *R , 2 where r i = g 1 R / g iL is the ratio of amplitudes of gain coefficients for the left and right channels of the antenna with the number i, i = iR ­iL is the differ
COSMIC RESEARCH Vol. 53 No. 3 2015

where (S/N)+ is the signal/noise ratio for the cross cor relation lobe (or upper limit for it), (S/N)|| is the same, but for the parallel correlation, r is the ratio of the amplitudes of the gain coefficients for the ground antenna, M is the degree of the linear polarization of the source, DGRT is the amplitude of the coefficient of the instrumental polarization for the ground antenna, DRA is the same for RadioAstron, i are phases (i = 1, 2, 3). Table 1 shows the distribution and their parameters. Using the intermediate results of ESP for observing AGN, namely, the correlated data of the review of brightness temperatures, we estimated the signal/noise ratio for lobes of cross correlations in the PIMA soft ware package [10]. Further, we considered only scans with the detected signal (interferometric lobe) in one of the parallel correlation having the signal/noise ratio S/N > 30. For each session, consisting of several 10 min scans, the signal/noise ratios for the lobes of cross and parallel correlations were averaged over the scans (in the case of the detected signal, the criterion of which was the ratio of S/N > 5.7). In the presence in the session of only upper limits, the lowest upper limit was chosen to be used in the future. Table 2 presents a summary of the data thus obtained. We have not estimated the leakage in the K band in connection with the insufficient number of interferometric lobes with necessary S/N detected in this band. 3.2. Unknown parameters. In the Bayesian approach, all unknown parameters r, M, DGRT, i of the ratio (2) used by us, except SRT leakage ampli tude, which is interesting for us, are usually included in the model as so called nuisance parameters. These are the parameters required to construct the model, over which integrations are performed in the end. In our case, in particular, taking into account the incon stancy of these parameters on time scales much


202 Table 1 Physical value Ratio of the leakage amplitude in the right and left bands Amplitude of the complex degree of the source polarization Amplitude of the leakage coeffi cient of the ground antenna Phase members

PASHCHENKO et al.

Designation r M D
GRT

Used distribution Lognormal logN(m, s) Generalized (or four parametric) beta distribution B(Mmin, Mmax, a, b) Generalized (or four parametric) beta distribution B(Dmin, Dmax, a, b) Uniform U(min, max)

Parameters = 0, = 0.25 Mmin = 0.0, Mmax = 0.20, a = 2, b = 3 Dmin = 0.01, Dmax = 0.2, a = 5, b = 5 min = ­, max =

i, i = 1, 2, 3

shorter than the ESP duration, this approach would lead to a significant increase in the number of model parameters. Thus, coefficients of the instrumental polarization of the ground antennas usually vary with changes in the configuration of polarizers/receivers, for example, in connection with the next stage of their service. In the presence of N estimates of the ratios of S/N lobes of the cross and parallel correlations (including upper limits for these ratios), the model will contain 5N + 1 nuisance parameter and the interesting for us amplitude of the DRA coefficient of the SRT instrumental polarization. Basically, because of the possibility to indicate a priori sufficiently close con straints on the values of nuisance parameters, the esti mate of interesting for us amplitudes of the SRT leak age coefficients could perform in such manner. How ever, we used a different approach, in which the nuisance parameters were changed by the distributions of the corresponding values. The parameters of these distributions were fixed in accordance with the limita tions discussed below. 3.3. The choice of parameters for the model distri butions. Fixed parameters of the model distributions of r, M, DGRT, i were chosen on the basis of the limi tations following from both the theoretical analysis and the previous experimental results. Thus, the amplitudes of coefficients of the instrumental polar ization of the VLBI ground stations are usually of about 5% or less in the case of the VLBI network of VLBA, and up to 15­20% for some radio telescopes of the European VLBI Network, EVN. Taking into account this moment, it is possible to limit the ampli tudes of the DGRT leakage coefficients for ground sta tions by the value of 20%. The situation with the value of M degree of the linear source polarization is less certain. On the scales of the VLBI resolution, there are very few unpolarized sources. As the experience of the
Table 2 Form of data Number of detected lobes Number of upper limits Band C 15 11 Band L 9

VSOP space interferometer project shows, the "VLBI core" observed from the Earth1 is the result of blend ing the "real" optically thick "core" and more polar ized close VLBI component resolved during VLBI observations with ground­space baselines [11]. Since for the majority of observations according to EST the implemented resolution corresponds to baselines in several times larger than the HALCA satellite orbit of the VSOP project [11], we can assume that the total detectable on the baselines with SRT flux corresponds to an optically thick "core." Taking into account the small values of the degree of the linear polarization of optically thick synchrotron source [2], it is possible to limit the distribution of the degree of the linear polar ization by the value of 5%. Further, taking into account the randomness of the phase of the complex degree of the polarization of each observed source depending on the structure of the magnetic field for ejection from a particular source and a variety of other factors, as well as the constantly varying parallax angle of Spektr R due to the SRT turns, we can assume that all phases i in the model ratios are random values selected from the distribution uniform on the interval [­, ]. Strictly speaking, the phase of the coefficient of the SRT instrumental polarization as the amplitude is expected to be constant, which, in fact, allows us to perform such statistical analysis. However, taking into account that in the RL/LL ratio (see (1)), which is applicable to the frequency band of 6 cm, the SRT leakage coefficient enters with phase factor 12 varying with time, the phase distribution 3 of corresponding summand in (2) can also be considered homogeneous. In the case of observations in the band of 18 cm, we used only the model ratios for cross and parallel corre lations, which contain the SRT leakage coefficient also with the "random" phase summand. It should be noted that at the selected parameters of the model distributions of DGRT and M, the result of the DRA estimate corresponds also to the possibility, in which the DGRT parameters actually are the M param eters, i.e., the so called nonidentification of these parameters is shown. This is expressed in the presence of several peaks in the likelihood function. However,
1

2

The VLBI core is designated optically thick base of usually sin gle sided ejections observed in the radio images obtained by the VLBI method. COSMIC RESEARCH Vol. 53 No. 3 2015


FIRST ESTIMATE OF THE VALUE OF THE INSTRUMENTAL POLARIZATION

203

these parameters are not evaluated, and integration is carried out over them. For given amplitude DRA of the SRT leakage coeffi cient and parameters of the model distributions (see Table 2), the predicted data (the values of the sig nal/noise ratios for lobes of cross to parallel correla tions) are a sample from the distribution of R(DRA, ) obtained in accordance with ratio (2). Then, the like lihood function of DRA, parameters for obtaining the data y (i.e., the probability of the data y obtained in ESP to be selected from the distribution of R(DRA, )) is given by the expression:
N

L ( y ( D RA, ) ) =


i=1

det

N

p ( R = Ri )


j=1

lim

F ( R < Rj ) ,

where p(R) and F(R) are the probability density and the distribution function (integrated probability den sity) estimated from the predicted distribution of R(DRA, ) and the observed data y, Ndet is the number of detections for lobes of cross correlations, Nlim is the number of upper limits per the amplitude for lobes of cross correlations. Fixing the parameters of the model distributions of nuisance parameters based on a pri ori information, we, thus, exclude them from the model parameters. That is, the likelihood can be writ ten as L ( y ( D RA, ) ) , where I is the used model (with all parameters). Thus, for estimating the required amplitude we come to the following probability model: P ( D RA, y, I ) = L ( y D RA, I ) Pr ( D RA, I ) Pr ( I ) , where Pr(DRA, I) is a priori probability distribution of the SRT leakage amplitude for the model I, but L ( y D RA, I ) is the likelihood presented above. Pr(I) is an a priori probability for the I model. 4. ESTIMATE OF LEAKAGE AMPLITUDE According to Bayes' theorem [13], the a posteriori probability density of the SRT leakage amplitude is given by the ratio P(D
RA

y, I ) =

P ( D RA, y, I ) L ( y D RA, I ) Pr ( D RA, I ) = , P ( y, I ) P ( y, I )

where P(y, I) is the normalization constant, the so called evidence or the model likelihood. The evidence is the probability of the observed variables (i.e., the data) after marginalization (integration) of all model parameters and is the main instrument for the Baye sian comparison of models. For further analysis as non informative a prior distribution Pr(DRA, I), we selected a distribution uniform over the interval [0, 1]. Wherever necessary, the estimate of the corresponding densities is performed by the nonparametric Parzen window method with Gaussian core and the width of the window determined in accordance with the Scott rule [14].
COSMIC RESEARCH Vol. 53 No. 3 2015

To construct a sample from a posteriori distribution P ( D RA y, I ) we used the Markov Chained Monte Carlo, MCMC, or rather, its affine invariant imple mentation [15, 16]. Assembly chains were initialized in the areas of the DRA parameter selected on the basis of a preliminary study of the likelihood L ( y D RA, I ) . However, preliminarily a "burn" was carried out, after reset of the results of which the simulation was per formed, so that the exact values of the chain initializa tion should not influence the result. To reduce the effect of the correlation between neighboring chain parameters on the obtained estimates we used only every tenth value of the obtained chain. It should be noted that in order to obtain estimates of the SRT leakage amplitudes we could also use an estimate of the maximum likelihood. However, taking into account that, in the future, we will plan to esti mate the polarization properties of the observed sources using the leakage estimates obtained when mapping, it seems reasonable to use the mechanism of Bayesian analysis, which allows us to take into account a priori information on unknown parameters and study the hierarchical models. Besides which, the sample obtained by MCMC from the a posteriori probability distribution of the leakage amplitude allows us sufficiently simply to perform a predictive analysis of the used model for the adequacy of the description of the observable data (see below). The a posteriori probability distribution of the leak age amplitudes for both frequency bands was also esti mated by direct numerical integration. Indeed, in the one dimensional case the application of the MCMC method for constructing the distribution density may seem excessive, but to check the model, in any case, it is necessary to obtain a sample of corresponding poste riori density. In addition, the used MCMC algorithm does not require adjusting any parameters, which essentially simplifies its use. In both cases, direct inte gration gave a similar result. Figures 1a and 1b show histograms of a posteriori density distribution of the DRA amplitude of leakage coefficients for the frequency bands (6 cm) (the coefficient of leakage of the right polarization to the left DL) and L (18 cm) (the coefficient of leakage of the left polarization to the right DR) with the limits of 95% probability intervals (dashed lines). When construct ing histograms of densities, a method [20] was used that maximizes the a posteriori probability of the num ber of cells of the histogram in the case of the piecewise constant density model implemented in [21]. Table 3 shows the corresponding average distributions, as well as their 95% probability intervals. 5. DISCUSSION It should be noted that, in the solution to the prob lem, there are unaccounted for sources of uncertainty. Thus, the probability model used is based on the linear model of leakage. However, we obtained a result in


204 (a) 30 25 20 15

PASHCHENKO et al. () 7 6 5 4 3 C

A posteriori probability density

10 2 Number of sessions 5 0 0.02 1 0 (b) 7 6 5 4 15 10 5 0 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 Band L DR
Fig. 1. Histogram of a posteriori distribution P ( D RA y ) of the amplitude of the coefficient of the instrumental polar ization DL for the C and L frequency bands. To construct histograms 5000 points were used, which were obtained from P ( D RA y ) by the Monte Carlo Markov chains.

0.04

0.06

0.08 0.10 Band C DL (b)

0.12

0.14

0.16

25 20

L

3 2 1 0 2 4 10 6 8 Baseline projections, diameters of the Earth

Fig. 2. Histogram of distribution of lengths of baseline pro jections used in observational sessions in the C and L bands. Bold vertical line corresponds to apogee of orbit for the HALCA space radio telescope of the VSOP project.

magnitude greater than the value after which when estimating leakage by the standard for VLBI methods terms of the second order of smallness are added in the linear model. But, since we are interested in how we cannot better estimate the effect of leakage in order to interpret a particular polarization experiment, but
Table 3. Parameters of obtained samples from a posteriori distributions of the amplitudes of coefficients of the instru mental polarization Parameter Band C Band L Average value 0.0968 0.1328 95% probability interval [0.0646, 0.1267] [0.0945, 0.1736]

rather estimate the value of the effect, using the linear model seems justified. Moreover, when using a nonlin ear model of leakage, a sufficiently simple (in view of the number of parameters) formulation of the proba bilistic model is found to be impossible. We plan to return to the formulation of a more complete model after finishing the review of the AGN brightness tem peratures and determining the value of leakage by the standard for VLBI methods in order to, for example, perform a statistical study of the polarization proper ties of observed objects on the resolution scales of cor responding ground­space baselines with SRT. Further, parameters probability parameters estimating in the implemented approach, the fixed of the distributions used to construct the model, in fact, are the structural model and must be separately determined, before the required leakage amplitude DRA. We
Vol. 53 No. 3 2015

COSMIC RESEARCH


FIRST ESTIMATE OF THE VALUE OF THE INSTRUMENTAL POLARIZATION Table 4. Values of the evidence for considered models. The C band Boundaries of distributions B(2, 3) [0.00, 0.05] [0.00, 0.10] [0.00, 0.20] [0.05, 0.20] [0.10, 0.20] ln Z ± ln Z 16.3678 15.9771 15.5265 15.3821 12.3782 15.5640 ± ± ± ± ± ± 0.1794 0.1694 0.1565 0.2159 0.2230 0.1569 ln Z ­ ln Z 3.9896 3.5989 3.1483 3.0040 0 3.1858
min

205

B

i min

see Fig. 3

54.0332 36.5580 23.2964 20.1660 -- 24.1870

fixed their values on the basis of the limitations follow ing from both theoretical considerations (in the case of the maximum value of the degree of polarization) and previous experimental results (the results of the VSOP project and available information on the leakage amplitudes of ground stations). However, if the distri bution of the amplitudes of leakage coefficients DGRT of ground stations can be made sure, estimation the distribution of the degree of polarization M of observed sources cannot be considered final. The fact is that it is based on the data from ground based VLBI observations and the data of ground space interferom eter of the VSOP project. Data used in the paper were obtained with an angular resolution several times bet ter than that achieved in the VSOP project (see Fig. 2). In addition, we use the idea of a "VLBI core" as the basis of an optically thick ejection base that is the model assumption, which, although is supported by
35 30 25 Probability density 20 15 10 5 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Degree of the polarization
Fig. 3. Histogram of distribution of observed degree of lin ear polarization in sources investigated in this work according to the MOJAVE review at 15 GHz. It was used 1000 subsamples with size matched with the size of the sample used in this work (only detected signal) in the C band. Each subsample included the polarization degree of the MOJAVE archive for the last 3 years for each source selected randomly in proportion to the number of sessions of particular source used in an analysis. Solid line shows probability density obtained by corer method of estimation with Gaussian core (see text). COSMIC RESEARCH Vol. 53 No. 3 2015

the ground based data, cannot be consistent with observational data for ultra high resolution of the ground­space VLBI with the RadioAstron SRT. We could continue the analysis in this direction, for example, considering the parameters of the distribu tions of the degree of the polarization of the observed sources as "latent" parameters and giving for them any a priori distributions, to estimate together with the SRT leakage amplitude. Or choosing the parameters of model distributions (or even the type of distribu tions) maximizing the evidence of the model to esti mate the a posteriori distributions P ( D RA y, I ) , using the model already obtained on the basis of the selected distributions. This analysis was performed. We used a generalized beta distribution with different upper and lower limits for the simulation of possible situations described as a standard optically thick "core," bright strongly polarized component with the degree of polarization of up to 20%, as well as the intermediate cases. In addition, we use an empirical distribution obtained on the basis of the review of the MOJAVE data [7] for the degree of the polarization of the sources of observed sample for the last 3 years (see Fig. 3). Although the MOJAVE data were obtained for the VLBI network of VLBA at a frequency of 15 GHz, using the empirical distribution can be considered rea sonable, taking into account the fact that the charac teristic size of cells of heterogeneity of the Faraday screen estimated according to the ground VLBI is not much less than synthesized at this resolution [17]. Therefore, with the RadioAstron resolution the Fara day depolarization on the external screen should be small.2 An estimate of the evidence of each of the models was the byproduct of the MCMC sampling by the method of thermodynamic integration using the algorithm of parallel hardening. Tables 4 and 5 show the obtained values of the evi dence Z for the investigated set of models for the C and L band, respectively. The columns contain: boundaries of the beta distribution used in each of the models B(2, 3) being the distribution of the degree of the polarization of the observed sources; the natural
2

However, it should be remembered for one of the results dis cussed in section 3.2 of the VSOP project [10], according to which the VLBI core observed from Earth when mapping with the ground­space interferometer reveals close to the "true" VLBI core a bright strongly polarized component.


206 25 20 15 10 5 ()

PASHCHENKO et al. Table 5. Values of the evidence for considered models. The L band Boundaries of distributions B(2, 3) [0.00, 0.05] [0.00, 0.10] [0.00, 0.20] [0.05, 0.20] [0.10, 0.20] see Fig. 3 ln Z ± ln Z 6.9359 7.1373 7.3599 7.5868 7.0668 7.1217 ± ± ± ± ± ± 0.1578 0.1483 0.1126 0.1169 0.1278 0.1456 ln Z ­ ln Z 0 0.2014 0.4240 0.6509 0.1309
min

B

i min

-- 1.2232 1.5281 1.9172 1.1399

0.1858

1.2041

0 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Band C DL (b) 9 8 Probability density 7 6 5 4 3 2 1 0 25 20 15 10 5 0.1 0.2 0.3 0.4 Band C DL (c) 0.5 0.6

logarithm lnZ of the evidence of the models and its error dln Z; removing each of the models from the model with a minimum evidence in the logarithmic scale; the Bayesian factor Bi min (the evidence ratio) of the models with respect to the model with the smallest evidence. As can be seen, the too small volume of the sample of the observational data used cannot confidently select the best model (models) from a set of models on the basis of ratios of the evidence of the models (the so called "Bayesian factors") [18]. An exception, per haps, is the evidence against the model described as "highly polarized component." The same results were obtained by direct numerical integration, as well as calculating the evidence of the Laplace approximation (fitting the likelihood function peak by the Gaussian function). Again, we are planning to return to the sta tistical study of the polarization of nuclei for ejections after the review of brightness temperatures and deter mination of the value of SRT leakage by methods already standard for VLBI. To check the adequacy of the description of the observed data of the used probability model (including the linear approximation of leakage, the selected parameters of model distributions, a prior distribution of the leakage amplitude) we performed an a posteriori predicative check of the model as follows [19]. Using a sample from the a posteriori distribution of the leakage amplitude DRA, i obtained during MCMC and obtained in accordance with the generating model of the distri bution of corresponding relations for the correlations P ( D RA y, I ) a set of the hypothetical replications for the data y ire p l was composed. Further, the distribution of these statistics for the data replications was com posed as data average, maximum, and minimum val ues. Then, it was checked how the "implemented" replications (i.e., the observational data) correspond to the constructed distributions of statistics. Corre sponding distributions and observational data, as well as 5 and 95% boundaries, are shown in Figs. 4 and 5 for the average, minimum, and maximum values for the C and L bands, respectively. As can be seen from the fig ures, the "implemented" data are consistent with the distribution of the statistics for the replication data, except for the minimum value in the C band. Thus, in this band, the probability model used has difficulties
COSMIC RESEARCH Vol. 53 No. 3 2015

0

0.02

0.04

0.08 0.06 Band C DL

0.10

0.12

Fig. 4. Histogram of distributions of average (a), maxi mum (b), and minimum (c) values for 5000 sets of hypo thetical "replication" data obtained on the basis used for analysis probability model and a sample from a posteriori distribution of the amplitude of leakage coefficient for the C frequency band. Dotted lines indicate the 5 and 95% lev els. Solid line shows observational data.


FIRST ESTIMATE OF THE VALUE OF THE INSTRUMENTAL POLARIZATION () 16 14 12 10 8 6 4 2 0 0.10 0.15 0.20 Band L DR 0.25 0.30

207

Fig. 5. Histogram of distributions of the average (a), maxi mum (b), and minimum (c) values for 5000 sets of the hypo thetical "replication" data obtained on the basis used for the analysis probability model and sample from a posteriori dis tribution of amplitude of leakage coefficient for the L fre quency band. Dotted lines indicate the 5 and 95% levels. Solid line shows observational data.

with an adequate description of the minimum values of the observational data. This can be connected with both the simplified linear model of leakage used by us and with used model distributions. CONCLUSIONS We carried out a statistical analysis of the ESP results for observing AGN, by which was shown that the effect of "leakage" of the polarization of the RadioAstron SRT does not exceed the values typical for some ground VLBI stations (95% probability inter val [0.0646, 0.1267] and [0.0945, 0.1736] for 6 and 18 cm, respectively). It allows us to hope for successful implementation as already carried out in the frame work of ESP, and planned within Key Scientific Pro grams experiments on polarization sensitive mapping. Clarifying the corresponding estimates of the leakage coefficients by the standard for VLBI methods, when performing polarization sensitive mapping, allows us to use the data of the review of AGN brightness tem peratures for statistical studies of the degree of the polarization of nuclei of compact radio sources at the ultra high resolution in the different frequency bands. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research (project no. 13 02 12103), as well as Program for fundamental researches of RAS OFN 17 "Active processes in galactic and extragalac tic objects". The RadioAstron project was performed by Astro Space Center of Lebedev Physical Institute and Lavochkin Association according to the contract with the Russian Space Agency together with many scientific and technical organizations in Russia and other countries. In this work, we used data from the MOJAVE database supported by the MOJAVE team. REFERENCES
1. Kardashev, N.S., Khartov, V.V., Abramov, V.V., et al., RadioAstron--telescope with dimensions of 300 000 km: Basic parameters and first results of observations, Astron. Zh., 2013, vol. 90, p. 179. 2. Jones, T.W. and O'Dell, S.L., Transfer of polarized radiation in self absorbed synchrotron sources. II. Treatment of inhomogeneous media and calculation of emergent polarization, Astrophys. J., 1977, vol. 215, p. 236. 3. Vitrishchak, M.V., Circular polarization of radio emis sion of active galactic nuclei on parsec scale, Cand. Sci.

8 Probability density 7 6 5 4 3 2 1 0 0.1 0.2 0.3

(b)

0.4 Band L D

0.5
R

0.6

0.7

(c) 16 14 12 10 8 6 4 2 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Band L DR
Vol. 53 No. 3 2015

COSMIC RESEARCH


208

PASHCHENKO et al. (Phys. Math.) Dissertation, Moscow, Shternberg Astron. Inst., Moscow State Univ., 2008. Pudritz, R.E., Hardcastle, M.J., and Gabuzda, D.C., Magnetic fields in astrophysical jets: From launch to termination, Space Sci. Rev., 2012, vol. 169, p. 27. Leppanen, K.J., Zensus, J.A., and Diamond, P.J., Lin ear polarization imaging with very long baseline inter ferometry at high frequencies, Astron. J., 1995, vol. 110, p. 2479. Kemball, A., Faltters, C., Gabuzda, D., et al., VSOP polarization observing at 1.6 GHz and 5 GHz, Publ. Astron. Soc. Japan, 2000, vol. 52, p. 1055. Lister, M.L., Aller, H.D., Aller, M.F., et al., MOJAVE: Monitoring of jets in active galactic nuclei with VLBA experiments. V. Multi epoch VLBA images, Astron. J., 2009, vol. 137, p. 3718. Roberts, D.H., Wardle, J.F.C., and Brown, L.F., Linear polarization radio imaging at milliarcsecond resolu tion, Astrophys. J., 1994, vol. 427, p. 718. Sokolovsky, K.V., Radioastron early science program space VLBI AGN survey: Strategy and first results, Proc. of the 11th EVN Symposium, Bordeaux (France), October 2012, arXiv: 1303.5451. Petrov, L., Kovalev, Y.Y., and Fomalont, E.B., The Very Long Baseline Array Galactic Plane Survey--VGaPS, Astron. J., 2011, vol. 142, p. 23. Gabuzda, D.C., What we've learned from VSOP polar ization observations, in Radio Astronomy at the Fringe, vol. 300 of ASP Conference Proceedings, Zensus, J.A., Cohen, M.H., and Ros, E., Eds., 2003, p. 57. Hirabayashi, H., Fomalont, E.B., Horiuchi, S., Lovel, J.E.J., et al., The VSOP 5 GHz AGN survey. I. Compilation and observations, Publ. Astron. Soc. Japan, 2000, vol. 52, p. 997. Gilks, W.R., Richardson, S., and Spiegelhalter, D.J., Markov Chain Monte Carlo in Practice, Chapman & Hall/CRC Interdisciplinary Statistics, 1995. Scott, D.W., Multivariate Density Estimation: Theory, Practice, and Visualization, New York, Chichester: John Wiley & Sons, 1992. Goodman, J. and Weare, J., Ensemble samplers with affine invariance, Applied Mathematics and Computa tional Science, 2010, vol. 5, p. 65. Foreman Mackey, D., Hogg, D.W., Lang, D., and Goodman, J., Emcee: The MCMC hammer, Publica tions of the Astronomical Society of the Pacific, 2013, vol. 125, p. 306. Hovatta, T., Lister, M.L., Aller, M.F., et al., MOJAVE: Monitoring of jets in active galactic nuclei with VLBA experiments. VIII. Faraday rotation in parsec scale AGN jets, Astron. J., 2012, vol. 144, p. 34. Raftery, A.E., Markov Chain Monte Carlo in Practice, Chapman & Hall/CRC Interdisciplinary Statistics, 1995. Gelman, A. and Meng, X. L., Markov Chain Monte Carlo in Practice, Chapman & Hall/CRC Interdiscipli nary Statistics, 1995. Knuth, K.H., Optimal Data Based Binning for Histo grams. ArXiv:physics/ 0605197. Vanderplas, J., Connolly, A.J., Ivezic, Z., and Gray, A., Introduction to astroML: Machine learning for astro physics, Proc. of Conference on Intelligent Data Under standing (CIDU), 2012, p. 47.

4.

13. 14. 15. 16.

5.

6.

7.

17.

8.

18. 19. 20. 21.

9.

10.

11.

12.

Translated by N. Topchiev

COSMIC RESEARCH

Vol. 53

No. 3

2015