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 Image

1.2. Electromagnetic Radiation and Astronomical Quantities

Astronomical observations are all about measuring the radiation from astronomical sources to get an idea about 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 frequnecies at which the source emitts radiation (no astronomical source emitts radiation at a single frequency). This is also known as the 'bolometric luminosity'. More often, however, the luminosity of a source at a frequncy is given. This is the 'spectral luminosity' of the source, denoted as $L_{\nu}$ ($\nu$ will denote frequency, throught 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 $d\nu$

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Image(filename='figures/solid_angle_1.png', width=500)
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$$ \begin{align} dP \,&=\, I_\nu(\theta,\,\phi) \,dA_{\rm eff}\,d\nu\,d\Omega\\ &=\, I_\nu(\theta,\,\phi)\,\cos{\theta}\,dA\,d\nu\,d\Omega\qquad {\rm ,}\\ \end{align} $$

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.

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Image(filename='figures/solid_angle_2.png', width=500)
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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.

Future Additions:
  • add basics of light as a complex wave
  • interactive: change the phase and wavelength/frequency of light