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• Outline
• Glossary
• 4. The Visibility Space
  • Previous: 4.4 The Visibility Function
  • Next: 4.5.2 UV Coverage: Improving Your Coverage

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4.5.1 UV coverage : UV tracks

The objective of $\S$ 4.5.1 ⤵ and $\S$ 4.5.2 ➞ is to give you a glimpse into the process of aperture synthesis. TLG:GM: Check if the italic words are in the glossary. An interferometer measures components of the Fourier Transform of the sky by sampling the visibility function, $\mathcal{V}$. This collection of samples lives in ($u$, $v$, $w$) space, and are often projected onto the so-called $uv$-plane. In $\S$ 4.5.1 ⤵, we will focus on the way the visibility function is sampled. This sampling is a function of the interferometer's configuration, the direction of the source and the observation time. In $\S$ 4.5.2 ➞, we will see how this sampling can be improved by using certain observing techniques.

4.5.1.1 The projected baseline with time: the $uv$ track

A projected baseline is obtained via a baseline and a direction in the sky. It corresponds to the baseline as seen from the source. The projected baseline is associated with the measurement of a spatial frequency of the source. TLG:RC: Rewrite previous sentence. As the Earth rotates, the projected baseline and its corresponding spatial frequency (defined by the baseline's ($u$, $v$)-coordinates) vary slowly in time, generating a path in the $uv$-plane.

We will now generate test cases to see what locus the path takes, and how it can be predicted depending on the baseline's geometry.

4.5.1.1.1 Baseline projection as seen from the source

Let's generate one baseline from two antennas Ant$_1$ and Ant$_2$.

Let's express the corresponding physical baseline in ENU coordinates.

Let's place the interferometer at a latitude $L_a=+45^\circ00'00''$.

Figure 4.5.1: A baseline located at +45$^\circ$ as seen from the sky. This plot is interactive and can be rotated in 3D to see different baseline projections, depending on the position of the source w.r.t. the physical baseline.

On the interactive plot above, we represent a baseline located at +45$^\circ$, aligned with the local south-west/north-east as seen from the celestial sphere. TLG:RC: Rewrite previous sentence. By rotating the sphere westward, you can simulate the variation of the projected baseline as seen from a source in apparent motion on the celestial sphere.

4.5.1.1.2 Coordinates of the baseline in the ($u$,$v$,$w$) plane

We will now simulate an observation to study how a projected baseline will change with time. We will position this baseline at a South African latitude. We first need the expression of the physical baseline in a convenient reference frame, attached to the source in the sky.

In $\S$ 4.2 ➞, we linked the equatorial coordinates of the baseline to the ($u$,$v$,$w$) coordinates through the transformation matrix: \begin{equation} \begin{pmatrix} u\ v\ w

\end{pmatrix}

\frac{1}{\lambda} \begin{pmatrix} \sin H_0 & \cos H_0 & 0\\ -\sin \delta_0 \cos H_0 & \sin\delta_0\sin H_0 & \cos\delta_0\\ \cos \delta_0 \cos H_0 & -\cos\delta_0\sin H_0 & \sin\delta_0\\ \end{pmatrix} \begin{pmatrix} X\\ Y\\ Z \end{pmatrix} \end{equation}

\begin{equation} \begin{bmatrix} X\\ Y\\ Z \end{bmatrix} =|\mathbf{b}| \begin{bmatrix} \cos L_a \sin \mathcal{E} - \sin L_a \cos \mathcal{E} \cos \mathcal{A}\nonumber\\ \cos \mathcal{E} \sin \mathcal{A} \nonumber\\ \sin L_a \sin \mathcal{E} + \cos L_a \cos \mathcal{E} \cos \mathcal{A}\\ \end{bmatrix} \end{equation}

Equation 4.5.1

This expression of $\mathcal{b}$ is a function of ($\mathcal{A}$,$\mathcal{E}$) in the equatorial ($X$,$Y$,$Z$) systems. TLG:RC: Rewrite previous sentence as the meaning is unclear. Notice the unbold $b$.

4.5.1.1.2 Observation parameters

Let's define an arbitrary set of observation parameters to mimic a real observation.

• Latitude of the baseline: $L_a=-30^\circ43'17.34''$
• Declination of the observation: $\delta=-74^\circ39'37.481''$
• Duration of the observation: $\Delta \text{HA}=[-4^\text{h},4^\text{h}]$
• Time steps: 600
• Frequency: 1420 MHz

4.5.1.1.3 Computing of the projected baselines in ($u$,$v$,$w$) coordinates as a function of time

As seen previously, we convert the baseline coordinates using the previous matrix transformation.

As the $u$, $v$, $w$ coordinates explicitly depend on $H$, we must evaluate them for each observational time step. We will use the equations defined in $\S$ 4.2.2 ➞:

• $\lambda u = X \sin H + Y \cos H$
• $\lambda v= -X \sin \delta \cos H + Y \sin\delta\sin H + Z \cos\delta$
• $\lambda w= X \cos \delta \cos H -Y \cos\delta\sin H + Z \sin\delta$

We now have everything that describes the $uvw$-track of the baseline (over an 8-hour observational period). It is hard to predict which locus the $uvw$ track traverses given only the three mathematical equations from above. Let's plot it in $uvw$ space and its projection in $uv$ space.

Figure 4.5.2: $uvw$ track derived from the simulation and projection in the $uv$-plane.

The track in $uvw$ space are curves and the projection in the $uv$ plane are arcs. Let us focus on the track's projection in this plane. To get observation-independent knowledge of the track we can try to combine the three equations of $u$, $v$ and $w$, the aim being to eliminate $H$ from the equation. We end up with an equation linking $u$, $v$, $X$ and $Y$ (the full derivation can be found in $\S$ A.3 ➞):

$$\boxed{u^2 + \left[ \frac{v -\frac{Z}{\lambda} \cos \delta}{\sin \delta} \right]^2 = \left[ \frac{X}{\lambda} \right]^2 + \left[ \frac{Y}{\lambda} \right]^2}$$

One can note that in this particular case, the $uv$ track takes on the form of an ellipse.

TLG:GM: Check if the italic words are in the glossary.

This ellipse is centered at $(0,\frac{Z}{\lambda} \cos \delta)$ in the ($u$,$v$) plane.

The major axis is $a=\frac{\sqrt{X^2 + Y^2}}{\lambda}$.

The minor axis (along the axis $v$) will be a function of $Z$, $\delta$ and $a$. We can check this by plotting the theoretical ellipse over the observed portion of the track. (You can fall back to the duration of the observation to see that the track is mapping this ellipse exactly).

Figure 4.5.3: The blue (resp. the red) curve is the $uv$ track of the baseline $\mathbf{b}_{12}$ (resp. $\mathbf{b}_{21}$). As $I_\nu$ is real, the real part of the visibility $\mathcal{V}$ is even and the imaginary part is odd making $\mathcal{V}(-u,-v)=\mathcal{V}^*$. It implies that one baseline automatically provides a measurement of a visibility and its complex conjugate at ($-u$,$-v$).

4.5.1.2 Special cases

4.5.1.2.1 The Polar interferometer

Let settle one baseline at the North pole. The local zenith corresponds to the North Celestial Pole (NCP) at $\delta=90^\circ$. As seen from the NCP, the baseline will rotate and the projected baseline will correspond to the physical baseline. This configuration is the only case where this happens.

If $\mathbf{b}$ rotates, we can guess that the $uv$ tracks will be perfect circles. Let's check:

Let's compute the $uv$ tracks of an observation of the NCP ($\delta=90^\circ$):

Let's compute the uv tracks when observing a source at $\delta=30^\circ$:

Figure 4.5.4: $uv$ track for a baseline at the pole observing at $\delta=90^\circ$ (NCP) and at $\delta=30^\circ$ with the same color conventions as the previous figure.

When observing a source at declination $\delta$, we still have an elliptical shape but centered at (0,0). In the case of a polar interferometer, the full $uv$ track can be covered in 12 hours only due to the symmetry of the baseline.

4.5.1.2.2 The Equatorial interferometer

Let's consider the other extreme scenario: this time, we position the interferometer at the equator. The local zenith is crossed by the Celestial Equator at $\delta=0^\circ$. As seen from the celestial equator, the baseline will not rotate and the projected baseline will no longer correspond to the physical baseline. This configuration is the only case where this happens.

If $\mathbf{b}$ is not rotating, we can intuitively guess that the $uv$ tracks will be straight lines.

Figure 4.5.5: $uv$ track for a baseline at the equator observing at $\delta=0^\circ$ and at $\delta=10^\circ$, with the same color conventions as the previous figure.

An equatorial interferometer observing its zenith will see radio sources crossing the sky on straight, linear paths. Therefore, they will produce straight $uv$ coordinates.

4.5.1.1.3 The East-West array

The East-West array is the special case of an interferometer with physical baselines aligned with the East-West direction in the ground-based frame of reference. They have the convenient property of giving a $uv$ coverage which lies entirely on a plane.

If the baseline is aligned with the East-West direction, then the Elevation $\mathcal{E}$ of the baseline is zero and the Azimuth $\mathcal{A}$ is $\frac{\pi}{2}$. Eq. 4.5.1 ⤵ then simplifies considerably:

The only non-zero component of the baseline will be its $Y$-component.

\begin{equation} \frac{1}{\lambda} \begin{bmatrix} X\\ Y\\ Z \end{bmatrix} = |\mathbf{b_\lambda}| \begin{bmatrix} \cos L_a \sin 0 - \sin L_a \cos 0 \cos \frac{\pi}{2}\nonumber\\ \cos 0 \sin \frac{\pi}{2} \nonumber\\ \sin L_a \sin 0 + \cos L_a \cos 0 \cos \frac{\pi}{2}\\ \end{bmatrix} = \begin{bmatrix} 0\\ |\mathbf{b_\lambda}|\\ 0 \\ \end{bmatrix} \end{equation}

If we observe a source at declination $\delta_0$ with varying Hour Angle, $H$, we obtain:

\begin{equation} \begin{pmatrix} u\\ v\\ w\\ \end{pmatrix} = \begin{pmatrix} \sin H & \cos H & 0\\ -\sin \delta_0 \cos H & \sin\delta_0\sin H & \cos\delta_0\\ \cos \delta_0 \cos H & -\cos\delta_0\sin H & \sin\delta_0\\ \end{pmatrix} \begin{pmatrix} 0\\ |\mathbf{b_\lambda}| \\ 0 \end{pmatrix} \end{equation}\begin{equation} \begin{pmatrix} u\\ v\\ w\\ \end{pmatrix} = \begin{pmatrix} |\mathbf{b_\lambda}| \cos H \\ |\mathbf{b_\lambda}| \sin\delta_0 \sin H\\ -|\mathbf{b_\lambda}|\cos\delta_0\sin H\\ \end{pmatrix} \end{equation}

when $H = 6^\text{h}$ (West)

\begin{equation} \begin{pmatrix} u\\ v\\ w\\ \end{pmatrix} = \begin{pmatrix} 0 \\ |\mathbf{b_\lambda}|\sin\delta_0\\ |\mathbf{b_\lambda}|\cos\delta_0\\ \end{pmatrix} \end{equation}

when $H = 0^\text{h}$ (South) \begin{equation} \begin{pmatrix} u\ v\ w\

\end{pmatrix}

\begin{pmatrix} |\mathbf{b_\lambda}| \\ 0\\ 0\\ \end{pmatrix}

\end{equation}

when $H = -6^\text{h}$ (East)

\begin{equation} \begin{pmatrix} u\\ v\\ w\\ \end{pmatrix} = \begin{pmatrix} 0 \\ -|\mathbf{b_\lambda}|\sin\delta_0\\ -|\mathbf{b_\lambda}|\cos\delta_0 \end{pmatrix} \end{equation}

In this case, one can notice that we always have a relationship between $u$, $v$ and $|\mathbf{b_\lambda}|$:

$$ u^2+\left( \frac{v}{\sin\delta_0}\right) ^2=|\mathbf{b_\lambda}|^2$$

4.5.1.3 Sampling the visibility plane with $uv$-tracks

4.5.1.3.1 Simulating a baseline

When we have an EW baseline, some equations simplify.

Firstly, $XYZ = [0~d~0]^T$, where $d$ is the baseline length measured in wavelengths.

Secondly, we have the following relationships: $u = d\cos(H)$, $v = d\sin(H)\sin(\delta)$,

where $H$ is the hour angle of the field center and $\delta$ its declination.

In this section, we will plot the $uv$-coverage of an EW-baseline whose field center is at two different declinations.

TLG:AC: Add the following figures. This is specifically for an EW array. They will add some more insight.

4.5.1.3.2 Simulating the sky

Let us populate our sky with three sources, with positions given in RA ($\alpha$) and DEC ($\delta$):

• Source 1: (5h 32m 0.4s,60$^{\circ}$-17' 57'') - 1 Jy
• Source 2: (5h 36m 12.8s,-61$^{\circ}$ 12' 6.9'') - 0.5 Jy
• Source 3: (5h 40m 45.5s,-61$^{\circ}$ 56' 34'') - 0.2 Jy

We place the field center at $(\alpha_0,\delta_0) = $ (5h 30m,60$^{\circ}$).

We then convert the ($\alpha$,$\delta$) to $l,m$: TLG:AC:Point to Chapter 3.

• $l = \cos \delta \sin \Delta \alpha$
• $m = \sin \delta\cos\delta_0 -\cos \delta\sin\delta_0\cos\Delta \alpha$
• $\Delta \alpha = \alpha - \alpha_0$

The source and phase centre coordinates are now given in degrees.

Figure 4.5.6: Distribution of the simulated sky in the $l$,$m$ plane.

4.5.1.3.3 Simulating an observation

We will now create a fully-filled $uv$-plane, and sample it using the EW-baseline track we created in the first section. We will be ignoring the $w$-term for the sake of simplicity.

We create the dimensions of our visibility plane.

We create our fully-filled visibility plane. With a "perfect" interferometer, we could sample the entire $uv$-plane. Since we only have a finite amount of antennas, this is never possible in practice. Recall that our sky brightness $I(l,m)$ is related to our visibilites $V(u,v)$ via the Fourier transform. For a bunch of point sources we can therefore write:

$$V(u,v)=\mathcal{F}\{I(l,m)\} = \mathcal{F}\{\sum_k A_k \delta(l-l_k,m-m_k)\} = \sum_k A_k e^{-2\pi i (ul_i+vm_i)}$$

Let's compute the total visibilities for our simulated sky.

Below we sample our visibility plane on the $uv$-track derived in the first section, i.e. $V(u_t,v_t)$.

TLG:RC: The graphs below intersect. Axis labels inside other graphs.

Figure 4.5.7: Real and imaginary parts of the visibility function. The black curve is the portion of the $uv$ track crossing the visibility.

We now plot the sampled visibilites as a function of time-slots, i.e $V(u_t(t_s),v_t(t_s))$.

Figure 4.5.8: Real and imaginary parts of the visibility sampled by the black curve in Fig. 4.5.7, plotted as a function of time.

Figure 4.5.9: Amplitude and Phase of the visibility function. The black curve is the portion of the $uv$ track crossing the visibility.

Figure 4.5.10: Amplitude and Phase of the visibility sampled by the black curve in Fig. 4.5.7, plotted as a function of time.

4.5.1.3.4 "Real-life" visibility

In the following figure, we present a collection of visibility measurements taken with different baselines, as a function of time. These measurements come from a real LOFAR dataset observing Cygnus A (Fig. 4.4.11 ⤵), a powerful radiosource. Each color corresponds to a different baseline measurement, and consequently, a different sampling of the same visibility function along different uv-track.

Figure 4.5.11: Cygnus A at 21 cm.

Figure 4.5.12: Visibility amplitude as a function of time.

Fig. 4.5.12 ⤵ shows a plot of the amplitudes of all the visibility samples from our observation of Cygnus A. The large number of antennas makes its interpretation difficult. Even the inspection of single visibility's amplitude (i.e. a single $uv$ track) is hard to interpret due to the source's intrinsic complexity. Let us see what happens if we plot the same information as a function of the $uv$-distance, $r_{uv}$.

Figure 4.5.13: Visibility amplitude as a function of $r_{uv}$.

Fig. 4.5.13 ⤵ display the same information as Fig. 4.5.12 ⤵ this time as a function of $r_{uv}$. It should be quite clear that, as in $\S$ 4.4 ➞, we are stacking the radial plots of the visibility function. The interpretation of these radial plots provides us with information about the size of the source. For Fig. 4.5.13 ⤵ in particular, when the amplitude of the visibility goes to zero, one characteristic size of the source has been resolved.

From these plots, it is clear that the more baselines we have, the better the sampling of the visibility function. In the next section, we discuss how astronomers improve their $uv$ coverage.

Important things to remember

• Each individual baseline samples the visibility function along a single $uv$ track.
• The $uv$ tracks are ellipses whose parameters depends on the latitude and declination of observation.
• The polar (resp. equatorial) interferometer gives circular (linear) $uv$ tracks.
• Accumulating samples over time enhances the sampling of the visibility function, thus improving our knowledge of the source.

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• Next: 4.5.2 UV Coverage: Improving Your Coverage

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