Fringes and Fringe Rates R.H. Tillman∗ October 9, 2013

Contents 1 Introduction

2

2 Theory

2

2.1

Transformation of Coordinate Planes . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2.2

Fringe Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2.3

Multiple Point Source Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

3 Examples

4

3.1

Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

3.2

Outrigger Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3.3

Data Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Document History

∗ Virginia

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Tech, email: [email protected]

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1

Introduction

This memo serves two primary purposes: • First, to concisely derive the expected fringe rate for a source at a given Right Ascension and Declination given a two element baseline at an earth relative latitude. This is done from the perspective of an engineer with little experience of astronomy, spherical, radio, or other wise. • To debug the results seen from the LWA-1 dataset 056540 000061571, which should contain the transit of both Cyg A and Cas A. Once these two goals have been established, this memo will also serve as a demonstration of the newly installed outrigger antennas at LWA-1.

2

Theory

This section derives the expected fringe rate for a given object, given the objects position and the interferometer’s location and geometry. We begin by presenting the various coordinate systems and how they transform between one another. Then we use these transformations to find the fringe rate. We conclude with a general model to yield the fringe spectrum.

2.1

Transformation of Coordinate Planes

We begin at the local coordinates, that of the interferometer in Cartesian coordinates (x, y, z), and will transform those coordinates to the (u, v, w) plane. The local origin is at the first antenna, with latitude L . The base vectors are oriented with x ˆ toward the east, yˆ toward North, and zˆ toward the zenith, and the components are normalized by wavelength. The (u, v, w) coordinates track the object, and have base vectors oriented with w ˆ toward the source, vˆ toward the celestial pole, and u ˆ = vˆ × w. ˆ Rather than directly directly transforming to the (u, v, w) plane, we form an intermediate transformation to another Cartesian system (X, Y, Z), whose origin is in the center the celestial sphere. The base vectors are identical to a local system located on the equator. The transformation from the local system to this systems is [1]     X x  Y  = z sin L + y cos L  (1) Z z cos L − y sin L

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Now, given the Right Ascension α and Declination δ of the object1 , the transformation from the intermediate system to (u, v, w) is.      u cos H 0 sin H X  v  =  sin δ sin H cos δ − sin δ cos H   Y  (2) w − cos δ sin H sin δ cos δ cos H Z where H is the hour angle of the object, and for a given Local Sidereal Time t, is computed as H =t−α

2.2

(3)

Fringe Rate

With the geometry described in the previous section, the fringe rate is the time variation of the component that tracks the source, namely [1] dw dw dH = dt dH dt d (− cos δ sin HX + sin δY + cos δ cos Hz) ωE = dH = −ωE (cos HX + sin HZ) cos δ

(4)

= −ωE u cos δ = −ωE (x cos H − y sin L sin H + z cos L sin H) cos δ where dH/dt = ωE = 2π/(24 · 3600) ≈ 7.27 · 10−5 rad/s is the earth’s rotation rate. Thus, the last equation in (4) provides an expression for the fringe rate in the terms of the desired quantities2 . Further incite comes from the derivative of the fringe rate. d2 w v 2 = ωE dt2 tan δ

(5)

Equation (5) will seldom be zero can be used to determine whether the fringe rate should increase or decrease with time. This can be used to validate simulations and measurements.

2.3

Multiple Point Source Model

Given the above formulation, the response of the two element correlator from a point source, with amplitude |S| and (α, δ) as before, is X(t) = |S|ej2π(dw/dt)t 1A

(6)

handy resource for finding (α, δ) for a specific object is available at http://ned.ipac.caltech.edu/. note that, for an E-W baseline at transit (H = 0), this formulation is identical to the one at http://fringes. org/xgal_fr.html. 2 We

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From superposition, the output given N point sources is x(t) =

N X

|Sn |ej2π(dwn /dt)t

(7)

n=1

where each Sn and wn corresponds to source n at (αn , δn ). The fringe rate spectrum is then X(f ) = F {x(t)} =

N X

  dw |Sn |δ 2πf − dt n=1

(8)

So each point source will generate an impulse in the fringe rate spectrum that, for sufficiently spaced sources, should be easily identifiable.

3

Examples

This section applies to model to practical observations, and demonstrates the theory using the LWA-1 radio telescope [2]. First, we verify the model derived in the previous section against an observation where fringes were observed.

3.1

Model Verification

LWA Memo 184 demonstrates, among many things, the idea of using the fringe spectrum to identify discrete point sources [3]. Figure 1 shows the fringe rates that are expected in the dataset. The expected fringe rates are nearly identical to those in LWA Memo 184 Fig. 3, thus the model appears to be reliable.

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Figure 1: Expected fringe rate for the data presented in LWA Memo 184. Compared to Fig. 3 in that memo, these modeled fringe rates well match those seen using actual TBN data.

3.2

Outrigger Analysis

Now, we look at an example using LWA-1’s outrigger antennas. The dataset (056540 000061571) was taken on September 5, 2013 from 04:00-08:00 UTC, during which time both Cyg A and Cas A transit occurred, and is TBN at center frequency of 74.03 MHz. The outriggers locations, relative to the LWA-1 core, are shown in Fig. ??. We take the E-W polarizations, and use a baseline from stands 248 from the LWA-1 core to the outrigger stand 259, with the origin at stand 248. This provides an almost pure E-W baseline, with (x, y, z) = (388.1, 27.0, −0.2) m= (95.8, 6.7, −0.1) m/λ. The measured fringe pattern for this baseline is shown in Fig. 2. The system was modeled using the derivations in the previous section, with just two sources; Cyg A and Cas A. The expected fringe pattern is generated in the time domain, is zero padded to obtain a power of 2 number of samples, and is windowed by a triangular function to mitigate effects due to the finite amount of data. The FFT is then taken of the processed time series to find the fringe spectrum. The dataset is processed in like manner. Figure 3 shows the modeled fringe rate spectrum for an hour of time beginning at Cyg A’s transit, t = αCygA . Two very distinct spikes are seen, the one on the left for Cyg A and the one on the right for Cas A. The spikes are separated even further than in Fig. 1, as the baseline is longer. Figures 4 is the same as Figs. 3 but for the measured dataset. An integration time of 10 s was used. Three spikes are seen, corresponding to, from positive fringe rate to negative, Cyg A, Cas A, 5

Figure 2: Magnitude and phase of the correlation for the dataset and baseline in question. A strong fringe pattern is observed. and a DC term from to the diffuse galactic background3 . The spikes corresponding to Cyg A and Cas A are located at fringe rates that agree with the model. Additionally, the relative amplitudes of the spikes are similar to the model. The model was then applied over time, resulting in Fig. 5. Each spectra represents an hour, and the time resolution is a half hour. Cyg A’s fringe rate is decreasing in magnitude to 0, while Cas A’s is increasing. This makes sense, as during this time interval Cyg A is past transit and is setting to the west, while Cas A is rising, and transits near the end of the time interval. Identical processing was performed on the dataset, and the results are shown in Fig. 6. The fringes corresponding to Cyg A and Cas A follow the model with time.

3 When correlating two outriggers together, the DC term is usually significantly large than the terms from either source. It is speculated that this is due to cross talk in the instrument, for example if the outriggers enter analog receivers or ADCs on shared boards.

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Figure 3: Modeled fringe rate of Cyg A and Cas A around Cyg A’s transit. The spike on the left is from Cas A, and the one on the right, Cyg A.

Figure 4: Fringe rate spectrum for the first hour of the 056540 000061571 observation.

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Figure 5: Modeled fringe rate spectrograph of Cyg A and Cas A from Cyg A’s transit to Cas A’s transit. The top line is from Cyg A, and the bottom, Cas A.

Figure 6: Fringe rate spectrograph for the 056540 000061571 observation.

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3.3

Data Verification

This section is now outdated since the error was found to be due to the antenna identification.

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Document History • Version 3 (October 7, 2013) – Found error in stand identification software, set straight by J. Dowell. Found fringes! Strong DC signal still seen, when correlating outriggers against each other, possibly due to cross talk in the instrument? When correlating an outrigger against a core antenna, the DC term is substantially reduced. Removed Section 3.3. • Version 2 (September 30, 2013) – Corrected error in original modeled code, Figs. 3 and 5 updated. Nothing was wrong with the math, there was simply a coding mistake. – Added Sections 3.1 and 3.3. • Version 1 (September 25, 2013) – First Version.

References [1] A. Thompson, J. Moran, and G. Swenson JR., Interferometry and Synthesis in Radio Astronomy, 1st ed. Wiley, 1986. [2] S. Ellingson, G. Taylor, J. Craig, J. Hartman, J. Dowell, C. Wolfe, T. Clarke, B. Hicks, N. Kassim, P. Ray, L. Rickard, F. Schinzel, and K. Weiler, “The LWA1 radio telescope,” Antennas and Propagation, IEEE Transactions on, vol. 61, no. 5, pp. 2540–2549, 2013. [3] S. Ellingson, “Fun with TBN,” LWA Memo 184, http://www.ece.vt.edu/swe/lwa/

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Sept. 2011. [Online]. Available: