Useful Antenna Equations¶
This section includes a number of equations that are useful in radio astronomy in general.
Cane Noise Brightness¶
As for any other experiment, it is important to understand your background. In the case of radio antennas at the South Pole, we expect that radio frequency interference (RFI) will be small. Thus the main source of background will be radio emission from the sky (extra-terrestial sources, not the atmosphere itself). The Cane Model describes the spectral brightness (also known as specific intensity), \(B\), of the sky, on average, for a given frequency, \(f\).
The brightness is the differential power, \(P\), per solid angle, \(\Omega\), per area, \(A\), per frequency.
\(B = \frac{dP}{(\cos \theta\,dA)\,df\,d\Omega}\)
The \(\cos\theta\) has been included here to account for the decreasing projected size with increasing angle with the normal vector. This is a particularly useful quantity because it does not depend on the distance to the source, i.e. the brightness of the sun is a constant quantity if you measure it on Mercury or on Earth.
This model splits the brightness into the sum of a galactic, \(B_\text{Gal}(f)\), and extra-galactic, \(B_\text{Ex-Gal}(f)\) term.
\(B_\text{Gal}(f) = 2.48^{-20} \times \left(\frac{f}{\text{MHz}}\right)^{-0.52} \frac{1 - \exp\left( -\tau(f) \right)}{\tau(f)}\)
\(B_\text{Ex-Gal}(f) = 1.06^{-20} \times \left(\frac{f}{\text{MHz}}\right)^{-0.80}\exp\left( -\tau(f) \right)\)
\(\tau(f) = 5\times\left(\frac{f}{\text{MHz}}\right)^{-2.1}\)
\(B_\text{Cane}(f) = B_\text{Gal}(f) + B_\text{Ex-Gal}(f)\)
Brightness Temperature¶
The brightness of an object can be converted to an equivalent temperature using the Rayleigh-Jeans Law, known as a brightness temperature. The Rayleigh-Jeans Law is simply the Taylor expansion of black-body radiation for low frequencies, \(hf \ll k_b T\),
\(B = \frac{2hf^3}{c^2} \frac{1}{\exp{\frac{hf}{k_b T}} - 1} \simeq \frac{2f^2 k_b T}{c^2}\)
The brightness temperature, \(T_B(f)\) can be found by rearranging this equation. In the case of the case of the Cane Model, the equivalent temperature is given by
\(T_{\text{Cane}}(f) = \frac{1}{2 k_\text{b}} B_\text{Cane}(f) \left(\frac{c}{f}\right)^2\).
This quantity is one of note because it is common for the absolute calibration of antennas to be compared to a load of a specific temperature.
The noise one expects to measure in an antenna at the South Pole should be a combination then of the sky noise, and other sources of noise, which can be characterized by their own brightness temperature.
\(T_\text{Total}(f) = T_{\text{Cane}}(f) + \sum T_{\text{Other}, i}(f)\)
Using the Rayleigh-Jeans Law, we can convert this back to a brightness.
\(B_\text{Total}(f) = 2 k_b \left(\frac{f}{c}\right)^2 T_\text{Total}(f)\)
The Power in an Antenna¶
To find the power in an antenna, we begin with the spectral brightness to get the power per hertz, \(\mathcal{P}\),
\(d\mathcal{P} = B (\cos \theta\,dA)\,d\Omega\)
\(\mathcal{P} = \int d\Omega \int B \cos \theta\,dA\)
The notion of an area of an antenna is typically described by its effective area, \(A_\text{eff}\). This is the area an ideal antenna (which collects all of the incident electromagnetic radiation) would have to equal the amount of power is collected by the real antenna. Note that in the above equation, \(\theta\) refers to the angle between the normal of the infinitesimal area, \(dA\), and the zenith. However, to generalize this equation to any coordinate system (and not one that is defined by the antenna geometry), we replace \(\cos\theta A \longrightarrow A_\text{eff}\),
\(\mathcal{P}(f) = \int B_\text{Total}(f, \theta, \phi) A_\text{eff}(f, \theta, \phi)\,d\Omega\)
If one is considering only isotropic noise, such as for the Cane noise, then the brightness can be pulled out of this integral. Further, the effective area can be replaced by its equivalent gain. This gain describes the relative ability of an antenna to receive/transmit power with respect to that of an ideal, isotropic antenna,
\(A_\text{eff}(f, \theta, \phi) = \left(\frac{c}{f}\right)^2 \frac{G(f, \theta, \phi)}{4 \pi}\).
For the Cane model, this gives us the integral,
\(\mathcal{P}(f) = \left(\frac{c}{f}\right)^2 \left(\frac{f}{c}\right)^2 \frac{2 k_b T_\text{Total}(f)}{4 \pi} \times \int d\phi \int d\theta\ G(f, \theta, \phi) \sin(\theta)\).
In the radcube code, this integral is approximated by a summation,
\(\mathcal{P}(f) \simeq \frac{2 k_b T_\text{Total}(f)}{4 \pi} \times \frac{2 \pi}{N_\phi} \frac{\pi}{2N_\theta} \sum\limits_i^{N_\theta} \sum\limits_j^{N_\phi} G(f, \theta_i, \phi_j) \sin(\theta_i)\),
where \(N_\phi\) and \(N_\theta\) are the number of integration bins. The power in one frequency band, \(\Delta f\), is, \(P(f) = \mathcal{P}\,\Delta f\), and the power accepted from a single polarization is,
\(P_\text{pol} (f) = \frac{P(f)}{2} = k_b\,\Delta f\, T_\text{Total}(f) \times \frac{G_\text{int}(f)}{4 \pi}\).
Calculation of the Voltage in an Antenna¶
In this section \(\Theta\) and \(\Phi\) give the direction of an electric field’s wave vector in spherical coordinates and \(\theta\) and \(\phi\) will refer to the components of electric field. These are related, but it will be easy to confuse them here so I leave them with different names.
The voltage (\(V\)) in an antenna is the dot product of the electric field (\(E\)) and the vector effective length (\(\mathcal{L}\)):
\(V(f,\Theta,\Phi) = \vec{E}(f) \cdot \vec{\mathcal{L}}(f,\Theta,\Phi)\)
The vector effective length can be decomposed into \(\hat{\theta}\) and \(\hat{\phi}\):
\(\vec{\mathcal{L}}(f,\Theta,\Phi) = \mathcal{L}_\theta\ \hat{\theta} + \mathcal{L}_\phi\ \hat{\phi}\)
where each component, \(j \in \{\theta,\phi\}\), of the VEL is a complex quantity
\(\mathcal{L}_j = \sum\limits_j |\mathcal{L}_j |\ e^{-i\Psi_{j}}\)
Where each component, of \(\mathcal{L}\) follows:
\(|\mathcal{L}_j(f,\Theta,\Phi)|^2 = \frac{\lambda^2\ G_j(f,\Theta,\Phi)}{4\pi} \frac{Z_\text{ant}}{Z_0}\)
We are given the gain, \(G\), in power:
\(G_{j}(f,\Theta,\Phi) = \frac{P_j}{P_{0,j}} = g_{j}\ e^{i\psi_j}\)
Finally we add a phase delay to the signal
\(\mathcal{L}_j(f,\Theta,\Phi) = \sqrt{ \frac{\lambda^2}{4\pi} \frac{Z_\text{ant}}{Z_0} g_j} \times e^{i\psi_j}\)
Likewise, one can get the electric field from the voltage in the two arms, \(V_1\) and \(V_2\), by simply inverting equation voltage/E-field relationship:
\(E_{\hat \theta} = \frac{V_1 \mathcal{L}_{2, \hat\phi} - V_2 \mathcal{L}_{1, \hat\phi}} {\mathcal{L}_{1, \hat\theta} \mathcal{L}_{2, \hat\phi} - \mathcal{L}_{1, \hat\phi} \mathcal{L}_{2, \hat\theta}},\qquad \qquad E_{\hat \phi} = \frac{V_2 - E_{\hat \theta} \mathcal{L}_{2, \hat\theta}} {\mathcal{L}_{2, \hat\phi}}\)
