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• Outline
• Glossary
• 1. Radio Science using Interferometric Arrays
  • Previous: 1.1 Remarks on Basic Astrophysics
  • Next: 1.3 Radiation Transport

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1.2.1 Electromagnetic Waves

As astronomers, we are interested in the light from celestial objects: we are interested in the electromagnetic waves these objects emit. Light is tricky because it behaves both as a wave (EM wave) and a particle (photon); this behaviour is best explored in a particle physics course. When it comes to radio astronomy, the light we observe is sufficiently low-energy that it is best treated as a wave rather than a particle; the wave-particle dichotomy will thus not be covered here, and we will treat light purely as a transverse wave in the EM field.

1.2.1.A Derivation of electromagnetic waves

The wavelike nature of light is a trivial development from Maxwell's laws of electrodynamics. Mawxwell's equations are as follows:

(1)

$ \nabla \cdot E = \frac{\rho}{\epsilon_0} $ </large>

(2)

$ \nabla \cdot B = 0 $ </large>

(3)

$ \nabla \times E = - \frac{\partial B}{\partial t}$ </large>

(4)

$ \nabla \times B = \frac{J}{\epsilon_0 c^2} + \frac{1}{c^2} \frac{\partial E}{\partial t}$ </large>

where

$ J = \sigma E $, $\sigma$ is the electrical conductivity, and $\epsilon_0$ is a constant known as vacuum permissivity. $c_0^{-2}=\mu_0 \epsilon_0$, where $\mu_0$ is a constant known as vacuum permeability.

These equations can be simplified by considering a region with no charges ($\rho=0$) and no current ($J=0$). Let us do so, and take the curl (i.e. matrix product with $\nabla$) of equations (3) and (4):

(5)

$\nabla \times (\nabla \times E) = \nabla \times (-\frac{\partial}{\partial t} B) = -\frac{\partial}{\partial t} (\nabla \times B) $ </large>

(6)

$\nabla \times (\nabla \times B) = \nabla \times (\mu_0 \epsilon_0 \frac{\partial}{\partial t} E) = \mu_0 \epsilon_0 \frac{\partial}{\partial t} (\nabla \times E)$ </large>

Note that we have put the $\nabla$ operator within the $\frac{\partial}{\partial t}$, since the two operators commute. Let us now perform the time-honoured trick of physics professors everywhere, and pluck a convenient vector identity from the ether to turn a thorny problem into an elegant formula:

(7)

$\nabla \times (\nabla \times X) = \nabla (\nabla \cdot X) - \nabla^2 X) $ </large>

This allows us to write our last two lines as follows:

(8)

$\nabla(\nabla \cdot E) - \nabla^2 E = -\frac{\partial}{\partial t} (\nabla \times B)$ </large>

(9)

$\nabla(\nabla \cdot B) - \nabla^2 B = \mu_0 \epsilon_0 \frac{\partial}{\partial t} (\nabla \times E) $ </large>

From Eqs (1) and (2), we know that $\nabla \cdot E = \nabla \cdot B = 0$. Our expressions are thus

(10)

$ -\nabla^2 E = -\frac{\partial}{\partial t} (\nabla \times B)$ </large>

(11)

$ -\nabla^2 B = \mu_0 \epsilon_0 \frac{\partial}{\partial t} (\nabla \times E) $ </large>

We are nearly done! Now, all that remains is to substitute equations (4) and (3) into equations (10) and (11) respectively:

(10)

$ -\nabla^2 E = -\frac{\partial}{\partial t} (\frac{1}{c^2} \frac{\partial E}{\partial t})$ </large>

(11)

$ -\nabla^2 B = \mu_0 \epsilon_0 \frac{\partial}{\partial t} (- \frac{\partial B}{\partial t}) $ </large>

which gives us (as $c_0^{-2}=\mu_0 \epsilon_0$) a set of two second-order differential equations:

(12)

$\frac{1}{c^2_0} \frac{\partial^2 E}{\partial t^2} - \nabla^2 E = 0$ </large>

(13)

$\frac{1}{c^2_0} \frac{\partial^2 B}{\partial t^2} - \nabla^2 B = 0$ </large>

One particular solution to this form of differential equation should be well-known to you:

(14)

$ E = const.\sin(c_0t + r + const.)$ </large>

1.2.1.B Complex Form of EM Wave

The form given above is annoying to work with as a physicist, since argument of the sine here has units of $r$! Assigning physical meaning to the constants of integration therefore begins with returning to a dimensionless value within the sine. Our particular solution thus becomes:

(15)

$ E(r,t) = E_0 \cos(\omega t - k \cdot r + \phi_0 ) $ </large>

where $\omega$ is the angular frequency of our wave, $k=\frac{\omega}{c}$ is the wave vector, and $\phi_0$ is the phase angle. This solution can be checked by inspection. Note that the magnetic field (B-field) has a similar solution, with $B_0$ substituted for $E_0$.

The last physical concern we have with this solution is that it varies in time and space: since we want to encode conservation of energy into our equation, we would prefer a form with

(16)

$|E(r,t)| = const.$ </large>

We thus tend to write our wave equation as a complex wave, of the following form:

(15)

$ E(r,t) = E_0 e^{i(\omega t - k \cdot r + \phi_0 )} $ </large>

This elegantly encodes the fact that the actual E-field oscillation associated with a given photon is varies in time and space, but that its energy is finite and constant in time and space.

We have thus shown that the very nature of Maxwell's equations mean that electromagnetic waves must exist, a result reached by Maxwell himself. He also noted that the speed he predicted - based on measuring permissivity and permeability - was in perfect agreement with the value of the speed of light, known through an independent measurement by astronomer Ole Roemer! This was strong indication - later vindicted in an experiment by Heinrich Hertz - that radio waves (and therefore light) were oscillations of an electric field.

1.2.1.C Polarisation information

The final step to fully represent the signal our antennas monitor is to encode polarisation information into our EM wave. Here, we will only introduce the mathematical formalism with which this is usually done, in order to complete our discussion on the mathematical description of EM waves.

Polarisation describes the situation where the frame of reference of your EM-wave (e.g. $E_x,E_y$) is rotated with respect to the frame of reference of your E-field ($E_{0x},E_{0y}$). This can occur when light travels through interstellar magnetic fields, such as ones found in ionised plasma.

Polarisation information is encoded in our final wavefunction using Jones calculus. This is not quite the most general framework in which to treat polarisation, but it will suffice for our needs (for keen readers, Mueller calculus is an extension of Jones calculus which allows users to treat polarisation more fully).

The idea is basically to characterise the polarisation as a phase shifts ($\phi_0$ in Eq (15) ) from the frame of reference of your E-field. In other words, the complex amplitude of E becomes:

$\begin{pmatrix} E_x(t) \\ E_y(t) \\ E_z(t) \end{pmatrix} = \begin{pmatrix} E_{0x}e^{kz-\omega t +\phi_x} \\ E_{0y}e^{kz-\omega t +\phi_y} \\ 0 \end{pmatrix} = \begin{pmatrix} E_{0x}e^{i\phi_x} \\ E_{0y}e^{i\phi_y} \\ 0 \end{pmatrix} e^{i(kz-\omega t)} $

This framework will be extended 7.1. Jones notation.

Having shown - hopefully convincingly! - that radio waves are electromagnetic waves, we will now consider how they are analyzed.

1.2.2.Electromagnetic Radiation and Astronomical Quantities

Astronomical observations are all about measuring the radiation from astronomical sources to infer the physical properties of these sources. So we'll first discuss how measurement of radiation from astronomical sources is quantified.

The total energy per unit time emitted by a source is its luminosity, typically denoted as $L$. This has units of Watts per second - or ergs per second if you prefer CGS units. This is the energy added over all the frequencies at which the source emits radiation (no astronomical source emits radiation at a single frequency). This is also known as the 'bolometric luminosity'. More often, however, the luminosity of a source at a particular frequency is given. This is the 'spectral luminosity' of the source, denoted as $L_{\nu}$ ($\nu$ will denote frequency, throughout the book). The suffix is to remind us that : $$L = \int L_{\nu}\, d\nu \qquad .$$

The spectral luminosity gets an extra $Hz^{-1}$ (Hertz) unit, since it is power (energy per unit time) at a given frequency.

Typically, astronomers cannot measure the luminosity of the source (since only a part of the total energy is intercepted by any telescope), but infer it under the assumption of a certain geometry of the radiation field (often assumed to be isotropic).

The amount of power which we receive at the telescope depends on the collecting surface of a telescope. The quantity independent of the measurement is a different one, namely the (spectral) power flux, or the so-called flux density (sometimes flux is used synonymously) $S_\nu$, which, for an isotropic radiation field of a point source, can be written as :

$$ S_{\nu} = \frac{L_{\nu}}{4\pi \, D^{2}} \qquad ,$$

where $D$ is the distance to the source. The inverse is trivially always true:

$$ \begin{align} L_{\nu} \,&=\, \int S_{\nu}(R, \theta, \phi)\,d\Omega\\ &=\, \int S_{\nu}(R, \theta, \phi) \sin{\theta} \,d\phi d\theta \qquad ,\\ \end{align} $$

where R is a constant distance and $S_\nu(R,\, \theta,\, \phi)$ denotes the flux density generated by the source at the position described by $R$, $\theta$, and $\phi$.

The flux density has units of ${\rm W}\,{\rm m}^{-2} \,{\rm Hz}^{-1}$. Radio astronomers prefer to measure the flux density in units of Jansky, denoted by the symbol 'Jy', with $$1 \, {\rm Jy}\, =\, 10^{-26}\, {\rm W}\, {\rm m}^{-2}\, {\rm Hz}^{-1} $$ ! This is due to radio signals from extragalactic sources being rather weak. For example, Cygnus A, the closest extragalactic radio source has a flux of $1590$ Jy at $1.4$ GHz (see Photometric Data for Cygnus A ⤴).

The flux density measures the spectral flux of a single source without giving any information about the direction from which the radiation is emitted and about the source structure. Obviously, though, a measurement contains that information.

What is hence measured is the average of another quantity over a limited solid angle (the instrumental function, ideally, of course the quantity is measured directly). The (specific) intensity, or brightness specifies the flux density per solid angle from a certain direction of the sky. It is defined via the infinitesimal power $dP$ permeating an infinitesimal surface $dA$ from the direction of the solid angle element $d\Omega$ over the infinitesimal frequency range $d\nu$

$$ \begin{align} dP \,&=\, I_\nu(\theta,\,\phi) \,dA_{\rm eff}\,d\nu\,d\Omega\\ &=\, I_\nu(\theta,\,\phi)\,\cos{\theta}\,dA\,d\nu\,d\Omega\qquad {\rm ,}\\ \end{align} $$

where $\theta$ and $\phi$ are polar coordinates, $\theta$ is also the angle between the solid angle element and the normal of the surface element, $A_{\rm eff}$ the surface crosssection in the direction of the observed solid angle element.

Specific intensity has a remarkable property - it does not change with distance (as long as the radiation is neither emitted nor absorbed on it path).

The usual way to show this is to consider the power flux through through two infinitesimal surface elements $dA_1$ and $dA_2$, the normals of which include the angles $\theta_1$ and $\theta_2$ with the connecting line.

For a distance r, the two surface elements appear under solid angle elements $d\Omega_1$ and $d\Omega_2$

$$ d\Omega_1 = \frac{cos{\theta_2}\,dA_2}{r^{2}}\\ d\Omega_2 = \frac{cos{\theta_1}\,dA_1}{r^{2}} $$

The power flux through both surfaces is equal when the intensity is integrated over the apparent solid angle under which the mutual surface elements appear. If we insert the definition of the intensity, we get

$$ \begin{align} dP\,&=\,I_\nu^1\,\cos{\theta_1}\,dA_1\,d\Omega_1\,d\nu\\ &=\,I_\nu^1\,r^2\,d\Omega_2\,d\Omega_1\,d\nu\\ &=\,I_\nu^2\,r^2\,d\Omega_2\,d\Omega_1\,d\nu\\ &=\,I_\nu^2\,\cos{\theta_2}\,dA_2\,d\Omega_2\,d\nu\qquad ,\\ \end{align} $$

and hence $I_\nu^1\,=\,I_\nu^2$. In other words, again, if intensity is not absorbed or generated, it is independent of the distance from the source. What happens in the case of emission or absorption is briefly covered in the next chapter.

Specific intensity is also known as the brightness, spectral radiance, spectral intensity etc. The units of spectral intensity are $ {\rm W} {\rm m}^{-2} {\rm Hz}^{-1} {\rm sr}^{-1}$ or more usually in radio astronomy, ${\rm Jy} \,{\rm sr}^{-1}$ and it is denoted generally as $I_{\nu}$. It is not uncommon that the intensity is normalised to the effective solid angle of an instrumental funtction, the "beam". In that case, the unit of intensity becomes ${\rm Jy}\, {\rm beam}^{-1}$, where "beam" be substituted by the effective solid angle of the observing function.

Specific intensity is the quantity which is mapped in radio interferometric images of radio sources. How exactly that is done is dicussed in Chapter 5).

In the next section, we'll discuss how electromagnetic radiation is generated, especially in astrophysical scenarios.

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