Angular Broadening

Figure 16.4: Angular broadening.

As discussed in the previous sections, the small scale fluctuations of electron density in the ionosphere lead to an excess phase for a radio wave passing through it. This excess phase is given by

$$\begin{eqnarray*} \phi(x) &=& {2\pi \over \lambda} \int \Delta \mu dz, \\ \phi(x) &=& { C \lambda} \int \Delta n(x,z) dz, \end{eqnarray*}$$

where $\Delta \mu$ is the change in refractive index due to the electron density fluctuation, $C$ is a constant and $\Delta n(x,z)$ is the fluctuation in electron density at the point (x,z) and the integral is over the entire path traversed by the ray (see Figure 16.4).

If we assume that $\phi(x)$ is a zero mean Gaussian random process, with auto-correlation function given by $\phi_0^2 \rho(r)$, where $\rho(r) = e^{-r^2/2a_\phi^2}$, then from the relation above for $\phi(x)$ we can determine that $\phi_0^2 \propto \lambda^2 \Delta n^2 L$, where L is the total path length through the ionosphere^16.2. Let us assume that a plane wavefront from an extremely distant point source is incident on the top of such an ionosphere. In the absence of the ionosphere the wave reaching the surface of the earth would also be a plane wave. For a plane wave the correlation function of the electric field (i.e. the visibility) is given by $\bigl<E_i(x){E\_i}^*(x+r)\bigr> = {E_i}^2$, i.e. a constant independent of $r$. On passage through the ionosphere however, different parts of the wave front acquire different phases, and hence the emergent wavefront is not plane. If $E(x)$ is the electric f ield at some point on the emergent wave, then we have $E(x) = E_i e^{-i\phi(x)}$. Since $E_i$ is just a constant, the correlation function of the emergent electric field is

$$\bigl<E(x)E^*(x+r)\bigl> = E_i^2 \bigl<e^{-i(\phi(x) - \phi(x+r))}\bigr>.$$

From our assumptions about the statistics of $\phi(x)$ this can be evaluated to give

$$\bigl<E(x)E^*(x+r)\bigr> = E_i^2 e^{-2\phi_0^2(1-\rho(r))}.$$ (16.5.6)

If $\phi_0^2$ is very large, then the exponent is falls rapidly to zero as $(1-\rho(r))$ increases (or equivalently when $r$ increases). It is therefore adequate to evaluate it for small values of $r$, for which $\rho(r)$ can be Taylor expanded to give $\rho(r) \simeq 1 - 1/2 r^2/a_\phi^2$. and we get

$$\bigl<E(x)E^*(x+r)\bigr> = E_i^2 e^{-\phi_0^2{r^2 \over a_\phi^2}}.$$

The emergent electric field hence has a finite coherence length (while the coherence length of the incident plane wave was infinite). From the van Cittert-Zernike theorem this is equivalent to saying that the original unresolved point source has got blurred out to a source of finite size. This blurring out of point sources is called ``angular broadening'' or ``scatter broadening''. If we define $a = a_\phi/\phi_0$ then the visibilities have a Gaussian distribution given by $e^{-i r^2/a^2}$, meaning that the characteristic angular size $\theta_{scat}$ of the scatter broadened source is $\sim \lambda/a \propto \lambda^2 \sqrt{\Delta n^2 L}$. $\theta_{scat}$ is called the ``scattering angle''.

On the other hand if $\phi_0^2$ is small then the exponent in eqn 16.5.6 can be Taylor expanded to give

$$\begin{eqnarray*} \bigl<E(x)E^*(x+r)\bigr> &=& E_i^2 \bigl[1 - 2\phi_0^2(1-\rho(... ...1 - 2\phi_0^2) + 2\phi_0^2 e^{-r^2\over 2 a_\phi^2} \bigr]. \\ \end{eqnarray*}$$

This corresponds to the visibilities from an unresolved core (of flux density $E_i^2~(1-2\phi_0^2)$) surrounded by a weak halo.

Footnotes

... ionosphere^16.2

This follows from the equation for $\phi(x)$ if you also assume that $<\Delta n(x,z)\Delta n(x,z^{'})> \ =\ \Delta n^2 \delta(z,z^{'}).$