Correlation of Diffuse Emissions as a Function of Baseline Length R.H. Tillman∗ February 17, 2014

Contents 1 Introduction

2

2 Theory

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3 Isotropic Antenna in a Uniform Sky

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∗ Virginia

Tech, email: [email protected]

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1

Introduction

This memo came about to investigate the non-zero phase shift seen in correlations from two colinear dipoles spaced very close together, where a zero phase shift was expected. The formulation expands on that of LWA Memo 142 [1], and much of the same notation is used. We develop a closed form expression for the “correlation temperature” between two isotropic antennas in a uniform sky, which shows a sinc amplitude variation and no phase shift. This is likely due in part to the lack of any antenna pattern, however the addition of the antenna pattern will not likely disturb the results found here too a large extent. Therefore, the cause of the non-zero phase shift seen in the afore mentioned data is likely RFI. These results are none-the-less interesting. The rest of this report is structured as follows. Section 2 develops a theoretical framework for this discussion. Section 3 applies this model to the simplest application imaginable, an isotropic antenna in a uniform sky.

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Theory

Consider a point in the sky, ψ, with temperature T (ψ). From the Rayleigh-Jeans Law the associated flux density from this point is S(ψ) =

2k T (ψ) λ2

W m−2 Hz−1

(1)

where k is Boltzmann’s constant. Assuming a plane wave is radiated from this point, the amplitude of the electric field is p |Ei (ψ)| = 2ηS(ψ) V m−1 Hz−1/2 (2) where η = 120π is the wave impedance of free space. Assuming the wave is un-polarized, its amplitude is equally divided between the two eµ spherical coordinate base vectors1 . By selecting the origin as the phase reference, an antenna at position pn receives an electric field of √ 2kη p i En (ψ) = eµ fµ (ψ) T (ψ)ejβˆr(ψ)·pn (3) λ where fµ (ψ) is a zero-mean complex-valued Gaussian-distributed random variable with unit variance, β = 2π/λ is the wave number, and ˆ r(ψ) is the radial unit vector directed at the source. The open circuit voltage induced by the source at ψ is related to the incident electric field by the antenna’s vector electric length ln vn (ψ) = Ein (ψ) · ln (ψ) The total open circuit voltage, accounting for the entire sky, is then ZZ Vn = Ein (ψ) · ln (ψ)dΩ

(4)

(5)

Ω 1 Index notation and the Einstein summation convention are used when abstract vector notation no longer applies. Latin indices on vector valued functions indicate a summation in three dimensions, while Greek indices a summation in two. Throughout this memo eµ represents the two spherical surface base vectors, with e1 = θ, the altitude angle, and e2 = φ, the azimuthal angle measured East (ˆ x) to West (ˆ y).

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where dΩ = sin θdφdθ is the element of solid angle. The correlation between antenna m and n is therefore, ignoring mutual coupling, ρmn = Vm (f )Vn∗ (f ) ZZ ZZ = Eim (ψ) · lm (ψ)dΩ Ein (ζ) · ln (ζ)dΩ Ω

(6)

Ω

which only impacts the fµ (ψ), such that fµ (ψ)fµ∗ (ζ) = 1 if and only if ψ = ζ, and otherwise fµ (ψ)fµ∗ (ζ) = 0. Thus we may write ZZ ρmn = (Eim (ψ) · lm (ψ))(Ein (ψ) · ln (ψ))dΩ Ω ZZ (7) 2kη (eµ · lm ) (eµ · l∗n ) T (ψ)ejβˆr(ψ)·(pm −pn ) dΩ = 2 λ Ω which is the spacial-continuum form of Equation 13 in [1]. The correlation may be expressed in terms of temperature by considering the Thevenin equivalent circuit model for a receiving antenna [1]. The antenna temperature is found from the autocorrelation of the antenna voltage by ρnn TA = (8) 4kRA where RA is the real part of the antenna’s self-impedance. This can be generalized to a “correlation temperature” between two antennas with the self-impedance, (mn)

TA

=

ρmn 4kRA

(9)

These expression are completely general for any type of antenna, and for any baseline orientation. We now look at the very specific case of isotropic antennas under a uniform sky.

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Isotropic Antenna in a Uniform Sky

Considering a sky with a uniform brightness distribution T (ψ) = T0 . In addition, assume two identical isotropic antennas that receive all polarizations equally and with unit amplitude, so that eµ · lm = 1. Under these (admittedly gross) assumptions, (7) reduces to ZZ 2kη ejβˆr(ψ)·(pm −pn ) dΩ (10) ρmn = 2 T0 λ Ω Now, assuming an East-West baseline of length B, (pm − pn ) = x ˆB such that the term in the exponential is jβB sin θ cos φ. All together we have Z Z 2kη ρmn = 2 T0 sin θ ejβB sin θ cos φ dφdθ (11) λ θ φ which is ripe for integral tables. For the azimuthal integral we use the identity [2] Z π 1 e−j(z sin v−nv) dv Jn (z) = 2π −π 3

(12)

where Jn (z) is the Bessel function of the first kind. The expression is now reduced to Z 4πkη T ρmn = sin θJ0 (βB sin θ) dθ 0 λ2 θ for which we have the identity [3] Z π sin(2µθ)J2ν (2z sin θ)dθ = π sin(µπ)Jν+µ (z)Jν−µ (z)

(13)

(14)

0

and we obtain

4π 2 kη T0 J1/2 (βB/2)J−1/2 (βB/2) (15) λ2 which is a completely closed form expression for the correlated voltage between two isotropic antennas with a uniform sky distribution as a function of the baseline length. Equation 15 may be simplified by applying properties of the Bessel function [3] ρmn =

4π 2 kη T0 J1/2 (βB/2)J−1/2 (βB/2) λ2 4π 2 kη 4 = T0 sin(βB/2) cos(βB/2) λ2 πβB (16) 8πkη sin(βB) T = 0 λ2 βB 8πkη = T0 sinc(2B/λ) λ2 where sinc x = sin(πx)/πx. Software packages understandably do not understand that the pole in sin x/x at x = 0 is removable, but at least MATLAB/Octave have a built in sinc function, hence its use in (16). We now apply (16) to a 47 MHz (λ = 6.38 m) system with a uniform sky temperature of 3000 K, roughly the sky temperature at this frequency [4]. Equation 8 was used to find the self impedance of the antenna, and was found to be 60.5 Ω. Fig. 1 shows the correlation temperature as a function of baseline length. Interestingly enough, despite the lack of any antenna response to speak of, this curve significantly resembles the x baseline curve in Fig. 1 of [1], in both shape and the location of zeros. This seems to provide validation to our the previous analysis, as the results in [1] came about from numerical integration. ρmn =

References [1] S. W. Ellingson, “Sky Noise-Induced Spacial Correlation,” LWA Memo 142, Oct. 2008. [Online]. Available: http://www.ece.vt.edu/swe/lwa/ [2] N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, 1996. [3] F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions, 1st ed. New York, NY, USA: Cambridge University Press, 2010. [4] H. V. Cane, “Spectra of the non-thermal radio radiation from the galactic polar regions,” MNRAS, vol. 189, pp. 465–478, Nov. 1979.

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Figure 1: Correlation temperature between two isotropic antennas. Uniform T0 = 3000 K sky temperature distribution.

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