ISS Simulations

Using code developed locally we can simulate the scintillation process. This document summarizes our simulations and presents examples.

• Examples

• Brief Summary of Code

Examples

These images show examples of different regimes of scintillation. Shown is one realization of the two-dimensional spatial distribution of intensity, in essence these images are the diffraction patterns from the medium.

A common measure of the strength of scintillation is the scintillation index, $m$_I = s_I/<I>, where $s$_I is the rms intensity and is the mean intensity. Three scintillation regimes can be identified

Weak scintillation

$m$_I is less than unity and increases with distance as $D**(11/6)$ or decreases with frequency as $f**(-17/6)$. For typical radio astronomical observations with near 1 GHz, only sources nearer than 50 pc (163 lt yrs) will display weak scintillation.

Transition regime

$m$_I exceeds unity, then begins to decrease toward unity. At 1 GHz, sources nearer than about 0.5 kpc (1,630 lt yrs) will display transition regime scintillation.

Strong scintillation

$m$_I approches unity asymptotically, i.e. the fluctuations in the intensity equal the average intensity. Essentially the entire Galaxy, except for a region approximately 0.5 kpc (1,630 lt yrs) in size near the Sun, is in the strong scintillation regime.

The images were produced assuming propagation through a random medium whose scattering strength is characteristic of the local interstellar medium [$C$_n² = 10**(-3.5) kpc m**(-20/3)]. We cite the frequencies and propagation distances assumed in producing these simulations; other combinations of observing frequency and distance can yield a similar diffraction pattern. All of these images are formed from a thin-screen approximation.

This very weak scintillation image has $m$_I = 0.15. It was produced assuming an observational frequency of 2 GHz and a propagation distance of 0.02 pc (0.065 lt yrs). [Full image is 261 kB.]

This weak scintillation image has $m$_I = 0.4. It was produced assuming an observational frequency of 1 GHz and a propagation distance of 10 pc (32 lt yrs). [Full image is 254 kB.]

This transition regime image has $m$_I = 1.2. It was produced assuming an observational frequency of 1 GHz and a propagation distance of 50 pc (163 lt yrs). Note the localized regions of enhanced intensity and the large regions of depressed intensity. [Full image is 173 kB.]

This transition regime image has $m$_I = 1.5. The observational frequency and propagation distance are the same as in the previous image. However, the random medium is assumed to have additional scattering structures on a scale of about 10**(11) cm. These contribute to the additional fringing seen. [Full image is 128 kB.]

This strong scintillation image has $m$_I = 1.0. It was not produced from our simulations, but rather by using the theoretical expression for the intensity distribution in asymptotically strong scintillation. It serves to illustrate how such a diffraction pattern might appear, were such simulations possible. [Full image is 208 kB.]

Brief Summary of Process

Our simulations are similar to those described in Cordes, Pidwerbetsky, & Lovelace (1986; see also Martin & Flatté 1988,1990; Martin 1994; Flatté, Bracher, & Wang 1994). The geometry is as follows: A source at $(0,\; 0,\; -z$_src) emits monochromatic radiation. The wave field propagates from the source to an observer in the plane $($r, z_obs), where $$r = (x, y) is the transverse coordinate, through a random medium extending from $z\; =\; 0$ to $z\; =\; z$_obs.

The random medium is approximated as a series of thin, phase-changing screens. Each screen is assumed to alter only the phase of the wave incident upon it and the space between screens is vacuum. We have considered two different approximations for the random medium. In the first, the entire medium is approximated by one screen. This characterization of random media has proven quite useful in understanding properties of scintillation (e.g. Cohen et al. 1967; Salpeter 1967; Lovelace 1970; Lee 1976) and is termed the "thin screen" approximation. In the second, the medium is approximated by 20 screens (Martin 1992, private communication). This "extended medium" approximation is obviously more realistic, although less discussed in the literature and more computationally intensive.

Propagation from screen to screen or from screen to observer is accomplished with the Fresnel-Kirchoff diffraction integral (Born & Wolf 1980). We treat an incident, scalar wave field. The effect of the phase screen is $exp[i\; f($r)] where $f($r) is the phase perturbation imposed by the screen. The propagation from source to medium, within the medium, and from the medium to the observer is treated by a free-space propagator which includes the effects of diffraction; it is given by $P($r,z) = exp[i pi (r/R)²]/R², where $R\; =\; sqrt\{cz/\; nu\}$ is the Fresnel radius.

Both the construction of phase screens and propagation are performed in the transform domain. A phase screen is constructed by multiplying a complex random number with Gaussian statistics at each point in the wavenumber domain by an envelope function; the envelope function is chosen to force the ensemble phase spectrum to have the form required by the assumed density spectrum. Propagation was accomplished by multiplying the discrete transform of the incident wave by the discrete approximation to the continuous transform of the propagator and transforming back.

For small to intermediate scattering strengths, a fully three-dimensional simulation could be performed, i.e. one dimension of propagation and two-dimensional screens. Due to memory constraints, larger values of scattering could be obtained only at the expense of using one-dimensional screens, i.e. $f($r) -> f(x).

• Born, M. & Wolf, E. 1980, Principles of Optics, 6^th ed. (Oxford: Pergammon Press)

• Cohen, M. H., Gundermann, E. J., Hardebeck, H. E., & Sharpe, L. W. 1967, ApJ, 147, 449

• Cordes, J. M., Pidwerbetsky, A., & Lovelace, R. V. E. 1986, ApJ, 310, 737

• Flatté, S. M., Bracher, C. & Wang, G.-Y. 1994, J. Opt. Soc. Am. A, 11, 2080

• Lee, L. C. 1976, ApJ, 218, 468

• Lovelace, R. V. E. 1970, Ph.D. thesis, Cornell University

• Martin, J. 1993, in Wave Propagation in Random Media (Scintillation), eds. V. I. Tatarskii, A. Ishimaru, & V. U. Zavorotny (Bellingham, Washington: SPIE) p. 463

• Martin, J. M. & Flatté, S. M. 1990, J. Opt. Soc. Am. A, 7, 838

• Martin, J. M. & Flatté, S. M. 1988, Applied Optics, 27, 2111

• Salpeter, E. E. 1967, ApJ, 147, 433

T. Joseph W. Lazio <lazio@rsd.nrl.navy.mil>
Last modified: Fri Jul 16 10:23:43 1999