Scintillation Notes

I gave myself a crash course on interstellar scintillation (ISS) today. (Sounds fancier than “twinkling”.) For posterity, here are some quick notes of the key results I found.

First of all, scintillation is twinkling. For my work, it’s in the radio and caused by plasma in the ISM. The Narayan paper (below) was a very helpful reference for booting up.

The basic model is an infinitely distant point source with incoming plane waves. There’s a zero- thickness phase screen at distance D defined by its phase change as a function of position phi(x,y). We’re observing at wavelength lambda. The formal expression for the effect of the scattering screen is the Fresnel-Kirchoff integral (cf Narayan eq 2.1):

\psi(X,Y) = \frac{e^{-i\pi/2}}{2\pi r_F^2}\int\int exp\left[i \phi(x,y) + i \frac{(x-X)^2 + (y - Y)^2}{2r_F^2}\right]dx dy

Here r_F is the Fresnel length, which can easily be interpreted as an angle at the distance of the screen:

r_F = \sqrt{\lambda D / 2 \pi}

\theta_F = \sqrt{\lambda / 2 \pi D} \ll 1 \textrm{ for this to work}

The Fresnel length is the transverse displacement on the scattering screen that causes a change in the arrival phase of the incoming wave due to path length differences. There’s a right triangle with adjacent of length D and opposite of length r_F, and you’re thinking about the phase change of the hypotenuse versus the adjacent. Assuming a gentle (or absent) phase screen, the dominant contribution to the integral comes from within r_F, where the phase isn’t changing much. As you get farther out, phase variations become rapid and signals tend to interfere and cancel.

For ISS and other natural cases, the phase screen has some length scale r_d on which it affects phases by ~1 radian. The usual assumption is that the phase structure function is Kolmogorov-distributed

D_\phi(x,y) = \langle[\phi(x'+x,y'+y) - \phi(x,y)]^2\rangle_{x',y'} = (r / r_d)^{5/3}

but the precise distribution doesn’t seem to matter so long as D_phi increases with r and there’s some characteristic scale r_d.

There are two main scintillation cases: weak and strong.

If r_d > r_F, we have weak scintillation. As mentioned above, most of the contribution to the received flux comes from within r_F, and the phase screen is nearly constant over that scale. The fluctuations will mostly be on this spatial scale, and so if the scattering screen is moving transversely at velocity v, the characteristic timescale will be r_F / v. A point source will broaden to the Fresnel angular scale,

\theta_s \approx \theta_F \approx \sqrt{\frac{\lambda}{D}}

If r_d < r_F, we have strong scintillation. The incoming phase is stirred up even within the Fresnel patch, so r_F basically is irrelevant. The “size” of a scatterer is the phase coherency length, so the typical angular scale is

\theta_s = \frac{\lambda}{r_d}

This is what point sources get scatter-broadened into, and it’s bigger than theta_F. (Smaller scatterer -> more diffraction.)

Strong scintillation has contributions from two components. First of all, there’s the variation on those r_d scales — this is “diffractive strong scintillation”. A source of angular size

\theta_d > \frac{r_d}{D}

has larger angular extent than the phase coherency scale and so crazy things start happening (cf Narayan). Note that this is not the same as theta_s: it is much smaller. The fluctuation timescale for a pointlike source here with transverse motion is r_d / v. Narayan says this effect is narrowband.

There’s also “refractive strong scintillation”. This is because a point source has that angular size theta_s, which backprojects to

r_r = \theta_s D = r_F^2 / r_d \gg r_d

and so is sensitive to variations on that lengthscale in the scattering screen. There’s a consequent timescale of r_r / v. Narayan says this effect is broadband.