Two scintillation régimes are distinguished by a scattering strength parameter, uLF so , which compares the coherence scale, so, to the Fresnel scale, LF (Rickett 1990; 2001). Strong scattering (u> 1)
Strong scattering induces significant phase fluctuations (> 1 rad) in the radio wave front on scales smaller than the Fresnel scale (so< LF), and is associated with relatively distant ISM scattering screens, and low observing frequencies. There are two strong ISS phenomena:
diffractive interstellar scintillation (DISS) andrefractive interstellar scintillation(RISS). DISS is the diffraction pattern that results from interference between components of the scattered wave spectrum. The spatial scale of the phase fluctuations induced in the radio wave emerging from the scattering screen is effectively the coherence scale, sdso, and the spatial intensity pattern produced at the observer has peaks that occur on the same scale as the phase fluctuations (Rickett, 1990). This is a narrow band effect, with the intensity peaks corresponding to interference fringes with a characteristic frequency band (Walker, 1998). In contrast to this, RISS is a broad band effect. It occurs when wave front fluctuations cause focusing and defocusing of the radio signal. The scattering discis the region of radio wave above an observer that influences the signal intensity, and its radius sets the RISS spatial scale, sr, which is thus larger than the DISS spatial scale, sd(Rickett, 1990).
The strong scattering spatial pattern for a point source thus has peaks at scales of sd, due to DISS, with peaks grouped into clumps at scales of sr, due to RISS. These scales are related to the Fresnel scale through the relation sdsr= (LF)2(e.g. Rickett, 1990).
Weak scattering (u< 1)
As the observing frequency increases, the DISS and RISS effects steadily converge, and become equal at the transition frequency, discussed in the next section. Higher frequencies are associated with the weak scattering régime, which leads to weak interstellar scintillation
(WISS),a broad band effect which for sources off the Galactic plane is associated with scattering screens less than ~500 pc from Earth, and observing frequencies above ~3 GHz (Rickett, 2001), at least for lines of sight off the Galactic plane.
In WISS, the phase coherence scale is larger than the Fresnel scale (so> LF), since the ISM induces only small phase changes (< 1 rad) in the wave front on scales comparable to LF. Physically, large scale electron density fluctuations focus and defocus radio waves to produce a spatial intensity pattern observed on Earth (e.g. Beckert et al., 2002). However, although so> LF, the focusing and defocusing action sets LF as the size of a coherent patch of wave front, so swLF(Narayan, 1992).
Scintillation Angular Scales and Cut-Offs
Table 2.2 summarises the characteristic spatial and angular scales for WISS, DISS, and RISS (e.g. Rickett, 2004).
Spatial scale Angular scale
DISS sd 1
kscatt
d sd R 1
Rkscatt
RISS sr Rscatt R
ksd
r sr R scattWISS s L R k
F
w / w F sw R 1 Rk
Table 2.2 Characteristic interstellar scintillation scales.
In Table 2.2, scatt is the scattering angle, kis the wavenumber, and sr sd LF R k
2
. The scintillation scales vary with scattering screen distance, R, and with observing frequency,
kc 2 . The form of the variation depends on the ISM free electron density fluctuation spectrum, which is responsible for inducing phase fluctuations in radio waves that translate into the scintillation phenomenon.
The electron density fluctuation spectrum is usually assumed to follow the Kolmogorov power law that describes a turbulence (i.e. eddy) spectrum across the inertial range of scales that lies between the outer scale at which energy is input to the ISM, and an inner energy dissipation scale (Kolmogorov, 1941; Pope, 2000). Armstrong et al. (1995) show that a Kolmogorov power law is consistent with local ISM observations over a wavenumber range of 10-13to 10-8m-1, although at a fine level of detail the local ISM is quite patchy.
A key assumption is that density is a passive tracer for turbulence, so the electron density fluctuation spectrum is similar to the ISM turbulence spectrum, whereby the inertial range between the inner and outer scales is characterised by large eddies breaking into smaller ones. This assumption is good for modelling ocean turbulence (Dr Trevor McDougall, CSIRO, pers. comm., 2003), but it is not an obvious assumption since turbulence can occur in an incompressible fluid in the absence of density variations. Since much ISM turbulence is generated by violent processes such as supernovae (Elmegreen & Scalo, 2004), it might be expected that ISM plasma compressibility should be assumed when modelling turbulence, and departures from the Kolmogorov spectrum are known to exist along some lines of sight from the Earth (Cordes & Lazio, 2006b). However, it is generally believed that Alfvén turbulence in incompressible magnetohydrodynamics is able to explain the ISM turbulence spectrum and aspects of interstellar scintillation such as the anisotropic nature of the scattering structures (Norman & Ferrara, 1996; Goldreich & Sridhar, 1995; 1997; Spangler, 1999; Chandran & Backer, 2002; Luo & Melrose, 2006).
Cordes & Lazio (2006b) concluded that a Kolmogorov ISM free electron density spectrum appears to be a reasonable assumption overall, and it is a good starting point for predicting how angular scattering parameters vary with screen distance, R, and observing frequency, . If the Kolmogorov spectrum is combined with the cold plasma dispersion relation, then (Narayan, 1992): 6 . 0 2 . 1
R
s
o
Substituting this into the expressions set out in Table 2.2 gives the scaling relationships, summarised in Table 2.3. 6 . 1 2 . 1 R d 2.2 0.6 R r 0.5 0.5 R w or d 1.2 R1.6 6 . 0 2 . 2 R r 0.5 0.5 R w
Table 2.3. Variation of angular scattering sizes with screen distance and observing wavelength or frequency, for a Kolmogorov ISM electron density spectrum.
The scaling relationships in Table 2.3 define angular cut-off sizes for the various modes of scintillation. Figure 2.12 shows the angular cut-off sizes varying with observing frequency,
, with the relationships linearised by the log-log scale. The gradients of the d, r, andw
cut-off lines are 1.2, -2.2, and -0.5 respectively.
Figure 2.12 Typical ISS angular cut-offs, after Rickett( 2004). Example: If WISS is present in emissions observed at 10 GHz, for a source with a transition frequency of ~ 3 GHz, the source size must be less than ~ 0.05 mas.
Time scale Source
Diameter Strong Weak Months 10 mas Days 1 mas Hours 0.1 mas Minutes 0.01 mas
Seconds Ceduna 0.001 mas
1 3 10 30 Frequency (GHz) 0.3
w
-0.5d
1.2r
-2.2Figure 2.12 supports the saying that pulsars scintillate, but AGN may not,counterpart to the adage stars twinkle, planets don’t. Pulsars are effectively point emission sources, and ISS in all forms has been observed in pulsar radio signals (e.g. Gupta, 2001). However, AGN emission regions have extended angular structure which restricts the ISS effects that an ISM scattering screen can generate from an AGN’s radio emissions.
The restrictions occur because a source of angular size blurs the scintillation pattern at distance Rover a spatial scale R. This quenches ISS when is large enough that ISS patterns due to emissions from different parts of the source overlap and smear each other out (e.g. Rickett 1990; 2001; Narayan, 1992). Thus WISS is quenched when is comparable to the characteristic angular scale, w, and similarly for DISS and RISS (Rickett, 2004).
Figure 2.12 also shows that AGNs are associated with WISS and RISS. DISS requires sources of smaller angular size than WISS, since sd< LF, and AGN angular sizes almost always blur out DISS. But sr> sdso the cut-off for RISS is less stringent than for DISS. Thus a source of angular size d< <rquenches DISS effects at the small scale on which
strong scattering induces diffractive phase perturbations, but the source is still small enough to exhibit the RISS effects associated with the larger scattering disc size (Rickett, 1990).