Import standard modules:
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import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from IPython.display import HTML
HTML('../style/course.css') #apply general CSS
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Import section specific modules:
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from IPython.display import HTML
HTML('../style/code_toggle.html')
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In the last section, we briefly summarised the mathematical tools we will need in this section. This section is intended to explain the physics underpinning interferometry. Before we can dig into what actually interests us - i.e. physics - we will therefore have to describe interferometers more explicitly.
Before we can get into the details of the relationship between the visibilities measured by each baseline and the sky brightness astronomers are interested in, we will need to define some key concepts. We will begin with the baseline: is the separation vector between two antenna elements in an interferometric array. It is a fundamental concept in interferometry: a baseline directly measures a point in Fourier space.
An interferometric array consists of several baselines (formed by every pair of antennas in the array). The baselines are thus determined by the array's configuration. In this chapter, we will introduce projected baselines, and derive expressions for the baseline vector in various coordinate systems.
TLG:GM: Check if the italic words are in the glossary.
Let us start by defining the notation and vector definitions we will use throughout this course. They are designed to take us from the local and instantaneous observer frame of reference (local $x$,$y$, $z$ on Earth) to the sky coordinates in the equatorial frame ($H$, $\delta$) through a series of coordinate changes:
TLG:GM: Check if the italic words are in the glossary.
The physical baseline is a geometric construct based on the separation between two antenna elements in 3-D space, whereas the projected baseline is the mapping of this 3-D physical baseline onto a 2-D plane (determined by the direction of the observation). While the physical baseline is a constant in the terrestrial reference frame (the acronym ITRF - In Terrestrial Reference Frame - is commonly used), the projected baseline changes as the Earth rotates (in the frame of reference of a source fixed in the sky - a.ka. the "sky frame of reference").
A position can be described in 3-D with a vector. Let us define an origin $O$, an orthogonal basis $\mathcal{B}$ and a local Cartesian reference frame attached to the ground $\mathcal{R}$ ($O$, $\mathcal{B}$($\mathbf{\hat{e}_x}, \mathbf{\hat{e}_y}, \mathbf{\hat{e}_z}$)) (see Fig. 4.2.1 ⤵).
Let us now consider two antennas in this frame of reference. They are defined by their position in $\mathcal{B}$:
$$\textbf{a}_1=\vec{OA}_1=x_1 \mathbf{\hat{e}_x} + y_1 \mathbf{\hat{e}_y} + z_1 \mathbf{\hat{e}_z}$$$$\textbf{a}_2=\vec{OA}_2=x_2\mathbf{\hat{e}_x} + y_2 \mathbf{\hat{e}_y} + z_2 \mathbf{\hat{e}_z}$$In other words, all the information on the position of our antennas is given by ($x_1$,$y_1$,$z_1$) and ($x_2$,$y_2$,$z_2$), which are defined in the basis $\mathcal{B}$.
Let us assume that the distance between the antennas is small enough to consider the curve of the Earth between them to be flat. The Earth then becomes a plane, on which the basis vectors can be translated without any rotation. The local physical baseline in Cartesian coordinates is defined as the difference vector
$$\mathbf{b} = \mathbf{a}_2 - \mathbf{a}_1 = \vec{A_1A_2} $$It is important to note that the physical baseline is a vector: it depends only on the relative positions of two antenna elements, and is independent of the origin of the reference frame.
Figure 4.2.1: Vectors $a_1$ and $a_2$ in a local 3-dimensional Cartesian coordinate system.
Unfortunately, in our formulation thus far, the directions of $\mathbf{\hat{e}_x}, \mathbf{\hat{e}_y}, \mathbf{\hat{e}_z}$ are arbitrary. We can do better: let us attach our coordinate basis to an "absolute" point of reference on Earth. To do this, we use a terrestrial reference frame: a new Cartesian basis defined by the cardinal points. Thus:
The baseline vector is still expressed in a local reference frame, chosen to coincide with the geographical North and East (see Fig. 4.2.2 ⤵). TLG:GM: Check if the italic words are in the glossary. Remember italic words can only be glossary definitions.
As seen in $\S$ 3.4 ➞, an observer located somewhere on Earth can define a direction in the sky in terms of local azimuth $\mathcal{A}$ and elevation $\mathcal{E}$. A baseline can similarly be expressed in these coordinates.
Let us set Antenna 1 as the origin of the reference frame on the ground. The North, East and Up axes are defined as per $\S$ 4.2.1.2 ⤵.
The azimuth is the angle contained in the plane of the local ground measured clock-wise from North to East. The elevation is the angle measured from the horizon to the local zenith. TLG:GM: Check if the italic words are in the glossary. Remember italic words can only be glossary definitions.
Figure 4.2.2: Relation between the horizontal frame ($\mathcal{A}$, $\mathcal{E}$) and the ($E$, $N$, $U$) Cartesian frame.
The baseline vector is expressed in the basis $\mathcal{B}'$:
\begin{equation} \mathbf{b}_{\text{ENU}} = \lvert \mathbf{b} \rvert \begin{bmatrix} \sin \mathcal{A} \cos \mathcal{E}\\ \cos \mathcal{A} \cos \mathcal{E}\\ \sin \mathcal{E} \end{bmatrix} \end{equation}The baseline is fully described in the $ENU$ system using the azimuth $\mathcal{A}$ and elevation $\mathcal{E}$. As shown in $\S$ 3.1 ➞, for an observer located at latitude $L_a$, the extension of the direction of the baseline defines a position on the sky. This position can be associated with Equatorial coordinates ($H$, $\delta$), where $H$ is the hour angle and $\delta$ the declination.
To generalize the baseline further (and to ease subsequent derivations), we need to define a set of reference frames which will map the baseline onto sky coordinates on the celestial sphere. To do so, we define an intermediate frame of reference $XYZ$ with basis ($\mathbf{\hat{e}_X}$,$\mathbf{\hat{e}_Y}$,$\mathbf{\hat{e}_Z}$). We attach this frame of reference to the Earth. We can now position these axes with respect to the Equatorial coordinates $(H; \delta)$. Let us define the axes of the $XYZ$ frame as:
Fig. 4.2.3 ⤵ shows that the plane associated with the array elements (red) can be related to the plane of the celestial sphere (blue) via a coordinate transformation.
We can convert ($\mathcal{A}$, $\mathcal{E}$, $L_a$) into $(H, \delta)$ in this new frame:
\begin{equation} \begin{bmatrix} X\\ Y\\ Z \end{bmatrix} = \begin{bmatrix} \lvert \mathbf{b} \rvert \cos \delta \cos H\\ -\lvert \mathbf{b} \rvert \cos \delta \sin H\\ \lvert \mathbf{b} \rvert \sin \delta \end{bmatrix} = \lvert \mathbf{b} \rvert \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 3.1: Conversion from baseline vector $\mathcal{E},\mathcal{A}$ to $XYZ$ . $\mathbf{b}$: amplitude of baseline vector. $H$: Hour angle, $\delta$: Declination, $L_a$: latitude of the array.
Now that we have defined a baseline in a terrestrial XYZ frame we are ready for the final transformation to the celestial uvw reference frame. Let ($H_0$, $\delta_0$) be the point on the celestial sphere in the direction of a source $\mathbf{s_0}$. We now define a new set of axes:
The ($u$,$v$) axes form a 2-D plane perpendicular to $\mathbf{s_0}$.
The transformation from the ($X$,$Y$,$Z$) frame to the ($u$, $v$, $w$) frame requires two succesive rotations applied to the equatorial coordinates of the baseline:
Figure 4.2.4: Relation of the (X,Y,Z) frame to the ($u$,$v$,$w$) frame. $Z$: local zenith. ($X$,$Y$) = Celestial plane $\perp$ to the NCP.
where \begin{equation}
\end{equation}
and \begin{equation}
\end{equation}
This results in the following transformation matrix: \begin{equation} \begin{pmatrix} u\ v\ w
\begin{pmatrix} X\\ Y\\ Z \end{pmatrix} \end{equation}
This section was dedicated to writing a mathematical expression the baseline vector in astronomy-friendly reference frames. This will later allow us to quickly link the physical length of a baseline between two receivers to a quantity measured on the celestial sphere. In $\S$ 4.3 ➞, we will focus on a simple one dimensional 2-element interferometer. We will address the correlation of two signals collected from a remote point source, and construct the corresponding visibility function which will be described in greater detail in $\S$ 4.4 ➞.