We discuss the possibility of the Mach cone formation involving the modified dust-acoustic waves in a dusty magnetoplasma composed of electrons and positively charged dust grains. It is shown that the modified dust-acoustic waves, whose wavelength is of the order of (or shorter than) the electron gyroradius, are one of the viable candidates for the formation of Mach cones in Saturn's dusty rings.
 The structure and dynamics of Saturn's rings continue to present surprises and challenges to celestial mechanics. The first surprise came in October 1980 when Voyager 1 sped past Saturn and sent back lots of information and images of Saturn's rings [Smith et al., 1981; Hill and Mendis, 1981; Morfill et al., 1983]. The CASSINI spacecraft, which will go into the orbit around Saturn on 1 July 2004, promises to yield even more detailed information on the plasma and optical properties of Saturn's dusty rings. Since a direct probing of Saturn's dense rings is not practically possible due to the danger of collisions, one must resort to remote sensing to investigate the physical conditions within such dense rings. One can do this by the stellar occultation measurements, or through the observations of absorption of high-energy particles or radio-waves or of light scattering by dust particles at different angles. It has been proposed that the characteristics (viz. opening angle) of Mach cones in dusty plasmas of Saturn's rings might play an interesting role as potential diagnosis method [Havnes et al., 1995, 2001; Brattli et al., 2002] and can be used for deducing information regarding the physical state of the ambient dusty plasma of Saturn's rings, since Mach cones can be directly viewed from outside the plasma system. On this point of view, Mach cones by dust-associated waves (viz. dust-acoustic and dust-magnetoacoustic waves) in unmagnetized [Havnes et al., 1995] and magnetized [Mamun et al., 2003; Shukla and Mamun, 2003] dusty plasmas containing negatively charged dust particles, ions, and electrons have been investigated.
 The dust grains in space are charged positively mainly by photoemission in the presence of a flux of UV photons with energy larger than the work function of the grains, but lower than the ionization potential of the background gas [Rosenberg et al., 1999]. The amount of photo-electrons, i.e., positive charges on a dust grain depends on i) the wavelength of the incident photons, ii) the surface area of the dust grain, and iii) the properties of the dust grain material. We note that various metals typically have photo-electric work function Wf < 5 eV, such as Ag (Wf = 4.46 eV), Cu (Wf = 4.45 eV), Al (Wf = 4.2 eV, Ca (Wf = 3.2 eV), and Cs (Wf = 1.8 eV). There are also a number of low work function materials, e.g., carbides (binary compounds of carbon and more electro-positive metals) with work functions Wf ≃ 2.18–3.50 eV, borides (binary compounds of boron and more electro-positive metals) with work functions Wf = 2.45–2.92 eV, metal oxides with work functions ranging from Wf = 1 eV (Cs) to Wf = 4 eV (Zirconium). The space dust grains are composed of low-work function materials, and can therefore be easily positively charged by photoemission in the presence of a flux of UV photons [Rosenberg et al., 1999]. It has been shown by Rosenberg et al. [Rosenberg et al., 1999] that as a result of only the photoemission process a dust grain of few micron size can acquire a positive charge of the order of 102–105 proton charges.
 It is important to mention here some features of an electron-dust (ED) plasma [Shukla, 2000; Khrapak and Morfill, 2001], which significantly differ from those of an electron-ion-dust (EID) plasma. The characteristic length of an ED plasma without an external magnetic field is the electron Debye radius λDe, contrary to the effective Debye radius λD = λDeλDi/(λDe2 + λDi2)1/2 in an EID plasma, where λDi is the ion Debye radius. The scalelength in a magnetized ED plasma is λDe or the electron thermal gyroradius ρe = VTe/ωce, contrary to the ion thermal gyroradius ρi = VTi/ωci or the ion-acoustic gyroradius ρs = Cs/ωci in a magnetized EID plasma, where VTe (VTi) is the electron (ion) thermal speed, Cs is the ion-acoustic speed, and ωce (ωci) is the electron (ion) gyrofrequency. On the other hand, the dust-acoustic wave (DAW) frequency in unmagnetized ED and EID plasmas are ∼ kλDeωpd and ∼ kλDωpd, respectively, where k is the wavenumber and ωpd is the dust plasma frequency. In a magnetized EID plasma, the frequency spectra have a broad range [Shukla and Mamun, 2002, 2003]. In this Letter, we investigate the formation of Mach cones in a magnetized ED plasma associated with the modified DAWs which involve magnetized hot electrons and unmagnetized cold positively charged dust grains. Thus, spatial and temporal scales for the Mach cone formation in an ED magnetoplasma are significantly different from those in an EID magnetoplasma.
2. Modified Dust-Acoustic Waves
 Let us consider the propagation of low-frequency (ω ≪ ωce, ωce = eB0/mec, B0 is the magnitude of the external magnetic field B0, me is the electron mass, e is the magnitude of the electron charge, c is the speed of light in vacuum, and is the unit vector along the z-direction) electrostatic waves in an ED plasma composed of electrons and positively charged dust particles [Shukla, 2000; Khrapak and Morfill, 2001; Fortov et al., 2003]. Thus, at equilibrium we have ne0 ≃ Zdnd0, where ne0 (nd0) is the equilibrium electron (dust) number density and Zd is the number of proton charges residing onto the dust grain surface.
 When the wavelength of the electrostatic wave is shorter than or comparable to the electron thermal gyroradius ρe, one must employ a kinetic theory for calculating the electron density perturbation ne1. Thus, using a kinetic theory [Brambilla, 1998] in the presence of such a low-frequency (ω ≪ ωce) electrostatic wave potential ϕ, the electron density perturbation ne1 is given by ne1 = (k2/4π e) χeϕ, where the electron susceptibility χe for two-dimensional electron motion in a plane perpendicular to is
where Γ0,1 = I0,1 exp(−be), I0(I1) is the modified Bessel function of zero (first) order, and be = k2ρe2.
 We now suppose that the perturbation wave frequency is much larger than the dust gyrofrequency, so that positively charged dust grains are unmagnetized. Accordingly, one obtains the dust number density perturbation nd1 = (k2/4πZde)χdϕ, where the dust susceptibility is
where VTd = (Td/md)1/2 is the dust thermal speed.
 Using equations (1) and (2) in 1 + χe + χd = 0, the dispersion relation involving the perturbation wave phase speed Vp = ω/k can be expressed in the form
where CD = λDeωpd = (ZdTe/md)1/2 is the dust acoustic speed [Shukla, 2000] and α = 1 + k2λDe2. In deriving equation (3), we assumed that ω2 ≫ 3 k2VTd2. Equation (3) is the general dispersion relation for the modified DAWs [Rao et al., 1990; Shukla, 2000] in a magnetized ED plasma including finite electron gyroradius effect. This is valid for arbitrary (short or long) wavelength modified DAWs. In the short wavelength limit (viz. be ≫ 1), we have a Boltzmann electron density perturbation ne1 = ne0eϕ/Te, which dictates that unmagnetized electrons in the DAW potential follow a straightline orbit across . In such a situation, equation (3) reduces to Vp = CD/. We note that a Boltzmann electron response also appears for be ≪ 1 and ω ≪ kzVTe, ωcekz/k⊥ when kz ≠ 0, where kz is the magnetic field aligned wavenumber.
3. Dust Dynamics
 We consider a positively charged dust particle of mass md and charge Zde moving in a field which includes Keplerian gravity, corotating planetary magnetic field (taken to be aligned centered dipole) with concomitant induced electric field [Mendis et al., 1982; Howard et al., 1999]. We examine the motion of a single dust particle and neglect the radiation pressure, plasma drag, planetary oblateness, charge fluctuations, and collective effects. The dynamics of such a positively charged dust grain is governed by the combined gravitational, magnetic, and electric forces. The orbital angular velocity ωd of the positively charged dust particle can therefore be expressed as [Mendis et al., 1982; Howard et al., 1999]
where r is the dust particle position normalized by the planet radius Rp, ωcd = ZdeB0/mdc is the dust gyrofrequency evaluated at a point on the planetary equator, B0 is the magnetic field strength on the planetary equator, Ωk = (GMp/Rp3)1/2 is the Kepler frequency evaluated at a point on the planetary equator, Mp is the planet mass, G is the universal gravitational constant, and Ωp is the angular rotation of the planet. We note that in deriving equation (4) the planetary magnetic field B is assumed to be dipolar, i.e., B = B0/r3 is used in the definition of ωcd for r > 0.
 The + (−) sign in equation (4) represents the prograde (retrograde) motion of the dust particle. We are interested here in the prograde motion of the dust particle. We can consider equation (4) in both magnetic and Keplerian regimes [Howard et al., 1999]. Equation (4) implies that in a magnetic (Keplerian) regime the dust grain radius rd must be smaller (larger) than a critical value rd0, where rd0 = 1.563(ΩpZdB0/Ωk2ρm)1/3 nm. Here Ωp and Ωp are in units of rad/s, B0 is in units of G, and ρm (dust material mass density) is in units of gm/cm3. It is found that rd0 ≃ 42 nm for Saturn, where Ωp = 1.691 × 10−4 rad/s, Ωk = 4.16 × 10−4 rad/s, B0 = 0.2 G, Zd = 100, and ρm ≃ 1 gm/cm3}. Therefore, the dust boulders, whose gyrofrequency is obviously negligible compared to their Kepler frequency, move at the Keplerian velocity. Hence, a dust boulder and a small dust particle will move at difference speeds. The difference in speeds is
When Vd is larger than the modified DAW phase speed Vp, i.e., Vd/Vp > 1, the effect of the dust boulder will be equivalent to that of a body moving through the medium with a super-dust-acoustic speed [Havnes et al., 1995].
 To approximate ωd in equation (4) for small (micron, even submicron sized) dust particles we take ωcd ≪ Ωk, Ωp, which is well satisfied for the dusty plasma parameters (Zd = 100, B0 = 0.2 G, rd = 0.5 μm, i.e., ωcd ≃ 6 × 10−7 rad/sec) in Saturn. Hence, equation (4) can be approximated as ωd = r−3/2Ωk + ωcdr−3(1 − r3/2Ωp/Ωk)/2, which can be substituted into equation (5) to obtain
It is obvious from equation (6) that Vd = 0 for r = r0 = (Ωk/Ωp)2/3, and for a particular planet ∣Vd∣ > 0 for both r < r0 and r > r0, where r0 is known as the synchronous distance. It is found that r0 = 1.822 for Saturn.
4. Formation of Mach Cones
 The Mach cones can be formed by any perturbation (e.g., modified DAWs in our case) if the perturbing object (viz. a dust boulder in our case) speed Vd is larger than the modified DAW phase speed Vp, i.e., Vd/Vp > 1. If this condition is satisfied, the Mach cone opening angle is θ = sin−1(Vp/Vd), where Vd is given by equation (5) or (6) and Vp = ω/k is defined by equation (3). It is obvious that Mach cones can only be formed by those dust particles which are either inside (r < r0) or outside (r > r0) the synchronous distance r0.
 To analyze the possibility for the formation of Mach cones associated with the modified DAWs defined by equation (3), we have numerically analyzed the relative speed Vd defined by equation (5), and the DAW phase speed Vp defined by equation (3) for typical ED plasma parameters of Saturn [Mendis et al., 1982; Howard et al., 1999; Shukla and Mamun, 2002; Verheest, 2000]: Te = 100 eV, nd0 = 10 cm−3}, Zd = 100, B0 = 0.2 G, Mp = 5.688 × 1026 kg, Rp = 60300 km, and for dust particles inside (r < r0) the synchronous distance r0. The numerical results are displayed in Figure 1. The upper plot of Figure 1 shows that for rd = 1 μm, the maximum wavelength (λm) of the modified DAWs, for which the Mach cones are formed, are ∼6.81 m, ∼3.94 m, and ∼1.92 m for r = 1.5, r = 1.6, and r = 1.7, respectively. The lower plot of Figure 1 indicates that for r = 1.6, the maximum wavelength are ∼10.7 m, ∼3.94 m, and ∼2.08 m for rd = 0.5 μm, rd = 1 μm, and rd = 1.5 μm, respectively.
 We have also analyzed the Mach cone formation at locations outside (r > r0) the synchronous distance r0. Our analysis reveals that for rd = 1 μm, the maximum wavelength (λm) of the modified DAWs, for which the Mach cones are formed, are ∼1 m, ∼2.14 m, and ∼3 m for r = 1.9, r = 2.0, and r = 2.1, respectively. We have also found that for r = 2, the maximum wavelength are ∼6 m, ∼2.14 m, and ∼1.15 m for rd = 0.5 μm, rd = 1 μm, and rd = 1.5 μm, respectively.
Figure 1 also shows how the Mach cone opening angle θ = sin−1(Vp/Vd) varies with kρe, r, and rd. We have estimated the Mach cone opening angle θ associated with the modified DAWs of wavelength ∼7.5 m (corresponding to kρe ≃ 1, since ρe ≃ 1.193 m for Te = 100 eV and B0 = 0.2 G) in both cases of r = 1.5 < r0 and r = 2.5 > r0 for rd = 0.5 μm. This is found to be ∼25° and ∼29° in the cases of r = 1.5 < r0 and r = 2.5 > r0, respectively.
 We have discussed the possibility of the Mach cone formation involving the modified dust-acoustic waves, with a perpendicular wavelength of the order of (or shorter than) ρe, in a magnetized ED plasma. We have numerically estimated the maximum wavelength (λm) of the modified DAWs for which Mach cones are formed in Saturn's rings. We have found that λm increases as r decreases (increases) in the cases of r < r0 (r > r0), and λm increases as rd decreases in both the cases of r < r0 and r > r0. We have estimated the Mach cone opening angle θ associated with the modified DAWs of wavelength ∼7.5 m (corresponding to kρe ≃ 1) in both the cases of r = 1.5 < r0 and r = 2.5 > r0 for rd = 0.5 μm and dusty plasma parameters corresponding to Saturn's rings. This is found to be ∼25° and ∼29° in the cases of r = 1.5 < r0 and r = 2.5 > r0, respectively.
 We have shown that in magnetized dusty plasmas of Saturn's rings, Mach cones are formed due to resonance interactions between the modified DAWs and the relative motion of the small dust particles to that of a dust boulder. The relative motion is caused by the balance between the gravitational force and the Lorentz force associated with the planetary magnetic field. The speed of small dust particle relative to that of the dust boulder depends on i) the sign of the dust, ii) the size of the dust rd, iii) the location of the dust r, and iv) whether the dust is inside (r < r0) or outside (r > r0) the synchronous distance r0. Our present investigation has clearly shown how the relative speed Vd of positively charged dust particles (participating in the formation of Mach cones) varies with the size as well as with the position of the dust particle in both inside (r < r0) and outside (r > r0) the synchronous distance.
 We expect that the NASA/ESA space probe CASSINI can make direct observations of Mach cones involving the modified dust acoustic waves, which we have reported in this Letter. The opening angles of the corresponding Mach cones, which we have estimated herein, can be used for obtaining the dust mass density, dust charge, dust composition, and the optical depth of Saturn's dense rings.