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The Statistics of Radio Astronomical Polarimetry: Bright Sources And Draft version November 7, 2018 A Preprint typeset using L TEX style AASTeX6 v. 1.0 THE STATISTICS OF RADIO ASTRONOMICAL POLARIMETRY: BRIGHT SOURCES AND HIGH TIME RESOLUTION W. van Straten Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia ABSTRACT A four-dimensional statistical description of electromagnetic radiation is developed and applied to the analysis of radio pulsar polarization. The new formalism provides an elementary statistical explana- tion of the modal broadening phenomenon in single pulse observations. It is also used to argue that the degree of polarization of giant pulses has been poorly defined in past studies. Single and giant pulse polarimetry typically involves sources with large flux densities and observations with high time resolution, factors that necessitate consideration of source-intrinsic noise and small-number statis- tics. Self noise is shown to fully explain the excess polarization dispersion previously noted in single pulse observations of bright pulsars, obviating the need for additional randomly polarized radiation. Rather, these observations are more simply interpreted as an incoherent sum of covariant, orthog- onal, partially polarized modes. Based on this premise, the four-dimensional covariance matrix of the Stokes parameters may be used to derive mode-separated pulse profiles without any assumptions about the intrinsic degrees of mode polarization. Finally, utilizing the small-number statistics of the Stokes parameters, it is established that the degree of polarization of an unresolved pulse is funda- mentally undefined; therefore, previous claims of highly polarized giant pulses are unsubstantiated. Unpublished supplementary material is appended after the bibliography. Keywords: methods: data analysis — methods: statistical — polarization — pulsars: general — techniques: polarimetric 1. INTRODUCTION the position angle distribution was interpreted as ev- Radio pulsars exhibit dramatic fluctuations in total idence of a superposed randomized emission compo- and polarized flux densities on a diverse range of lon- nent. McKinnon & Stinebring (1998) revisited the issue gitudinal, temporal and spectral scales. As a func- with a statistical model that described correlated in- tion of pulse phase, both average profiles and sub-pulses tensity fluctuations of completely polarized orthogonal make sudden transitions between orthogonally polarized modes. Although the predicted position angle distri- modes (e.g. Taylor et al. 1971; Manchester et al. 1975). butions were qualitatively similar to the observations, arXiv:0812.3461v3 [astro-ph] 2 Aug 2017 The radiation at a single pulse phase often appears as an the measured histograms were wider than those pro- incoherent superposition of modes, both orthogonal and duced by a numerical simulation of modal broadening non-orthogonal (e.g. Backer & Rankin 1980; McKinnon that included instrumental noise. Source-intrinsic noise 2003a). Evidence has also been presented for variations was later used to explain the excess polarization scat- that may indicate stochastic generalized Faraday rota- ter of bright pulses in an analysis of mode-separated tion in the pulsar magnetosphere (Edwards & Stappers profiles (McKinnon & Stinebring 2000); however, it was 2004). given no further consideration in all subsequent statisti- In an extensive study of single-pulse polarization cal treatments by McKinnon (2002, 2003b, 2004, 2006). fluctuations at 1400 MHz, Stinebring et al. (1984) ob- These begin with the reasonable assumptions that served that histograms of the linear polarization po- the instantaneous signal-to-noise ratio S/N is low and sition angle were broader than could be explained by a sufficiently large number of samples have been av- instrumental noise alone. This modal broadening of eraged, such that the stochastic noise in all four Stokes parameters can be treated as uncorrelated and normally distributed. These assumptions form part [email protected] of the three-dimensional eigenvalue analyses of the Stokes polarization vector by McKinnon (2004) and 2 W. van Straten Edwards & Stappers (2004). As in Stinebring et al. derlying stochastic process. Relevant theoretical results (1984), these studies concluded that modal broadening are drawn from various sources, ranging from studies of is due to the incoherent addition of randomly polar- the scattering of monochromatic light in optical fibres ized radiation that is intrinsic to the pulsar. The ba- (e.g. Steeger et al. 1984) to the classification of synthetic sic premises of these experiments are valid for the vast aperture radar images (e.g. Touzi & Lopes 1996). majority of pulsar observations; however, they become Following a brief review of polarization algebra in 2, § untenable for the bright sources on which these stud- the joint probability density functions of the Stokes pa- ies focus. That is, when the instantaneous S/N & 1, rameters at different resolutions are derived in 3, where § the Stokes polarization vector can no longer be treated the results are compared and contrasted with previous in isolation and the correlated self noise intrinsic to all works. The formalism is related to the study of radio four stochastic Stokes parameters must be accounted. pulsar polarization in 4, where the effects of ampli- § These considerations are particularly relevant to the tude modulation and wave coherence on the degrees of study of giant pulses. Those from the Crab pulsar reach freedom of the pulsar signal are discussed. Finally, the brightness temperatures in excess of 1037 Kelvin and re- results are utilized to reexamine past statistical analy- main unresolved in observations with nanosecond resolu- ses of orthogonally polarized modes, randomly polarized tion (Hankins et al. 2003). Consequently, previous anal- radiation, and giant pulse polarimetry. It is concluded yses of giant pulses have typically presented polarization that randomly polarized radiation is unnecessary and data at the sampling resolution of the instrument; that the degree of polarization of giant pulses must be more is, where the time-bandwidth product is of the order rigorously defined. Potential applications of the four- of unity. For example, Heiles et al. (1970) studied gi- dimensional statistics of polarized noise are proposed in ant pulses from the Crab pulsar using an instrument 5. § with bandwidth, δν = 8.3 kHz, and presented plots of the Stokes parameters at a resolution of τ = 120µs. 2. POLARIZATION ALGEBRA Cognard et al. (1996) plotted the total intensity and cir- This section reviews the relevant algebra of both Jones cular polarization of the strongest giant pulse and in- and Mueller representations of polarimetric transforma- terpulse observed from PSR B1937+21 with τ = 1.2µs tions. Unless otherwise noted, the notation and termi- and δν = 500 kHz. Using a baseband recorder with nology synthesizes that of the similar approaches to po- δν = 500 MHz, Hankins et al. (2003) plotted the inten- larization algebra presented by Cloude (1986), Britton sities of left and right circularly polarized radiation from (2000), and Hamaker (2000). Crab giant pulses at a time resolution of 2 ns. Each of 2.1. Jones Calculus these studies concluded that giant pulses are highly po- larized. However, in each experiment, τδν 1; at this The polarization of electromagnetic radiation is de- ∼ resolution, every discrete sample of the electric field is scribed by the second-order statistics of the transverse completely polarized, regardless of the intrinsic degree electric field vector, e, as represented by the complex of polarization of the source. That is, the instantaneous 2 2 coherency matrix, ρ ee† (Born & Wolf 1980). × ≡ h i degree of polarization is fundamentally undefined. Here, the angular brackets denote an ensemble average, To study the polarization of intense sources of impul- e† is the Hermitian transpose of e, and an outer product sive radiation at high time resolution, it is necessary to is implied by the treatment of e as a column vector. A consider averages over small numbers of samples and useful geometric relationship between the complex two- source-intrinsic noise statistics. These limitations are dimensional space of the coherency matrix and the real given careful attention in the statistical theory of po- four-dimensional space of the Stokes parameters is ex- larized shot noise (Cordes 1976). This seminal work pressed by the following pair of equations: enables characterization of the timescales of microstruc- ρ = Sk σk/2 (1) ture polarization fluctuations via the auto-correlation S = Tr(σ ρ). (2) functions of the total intensity and degrees of linear and k k circular polarization (e.g. Cordes & Hankins 1977). It Here, Sk are the four Stokes parameters, Einstein no- has also been extended to study the degree of polariza- tation is used to imply a sum over repeated indeces, tion via the cross-correlation as a function of time lag 0 k 3, σ is the 2 2 identity matrix, σ − are ≤ ≤ 0 × 1 3 between the instantaneous total intensity spectra of gi- the Pauli matrices, and Tr is the matrix trace operator. ant pulses (Cordes et al. 2004). The Stokes four-vector is composed of the the total in- This paper presents a complimentary approach based tensity S0 and the polarization vector, S = (S1,S2,S3). on the four-dimensional joint distribution and covari- Equation (1) expresses the coherency matrix as a linear ance matrix of the Stokes parameters, with emphasis combination of Hermitian basis matrices;
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