Temporal evolution of MHD shocks in the interstellar medium

We undertake calculations of the time-dependent structure of shock waves propagating in dark and diffuse interstellar clouds. The results of the time-dependent model are compared with those obtained by means of an independent steady-state code and found to agree well at sufficiently late times. Discontinuities in the flow of the neutral fluid are handled by introducing a pseudo-viscosity. Special procedures are adopted to correct for the associated widening of the discontinuity, in order not to distort the role of inelastic collision processes. We find that, in dark clouds, C shocks will tend to predominate, but are unlikely to have attained steady state in the cloud lifetime. On the other hand, in diffuse clouds, steady state may be reached but the discontinuity in the flow of the neutral fluid remains. We find no evidence for the existence of C* shocks, in which the neutral fluid undergoes a continuous transition from supersonic to subsonic flow (in the reference frame of the shock wave). Attention is drawn to the possible importance of these results for the interpretation of H₂ rovibrational line intensities.     

Time-dependent simulations of steady C-type shocks

Using a time-dependent multifluid, magnetohydrodynamic code, we calculated the structure of steady perpendicular and oblique C-type shocks in dusty plasmas. We included relevant processes to describe mass transfer between the different fluids, radiative cooling by emission lines and grain charging, and studied the effect of single- and multiple-sized grains on the shock structure. Our models are the first of oblique fast-mode molecular shocks in which such a rigorous treatment of the dust grain dynamics has been combined with a self-consistent calculation of the thermal and ionization structures including appropriate microphysics. At low densities, the grains do not play any significant rôle in the shock dynamics. At high densities, the ionization fraction is sufficiently low that dust grains are important charge and current carriers and, thus, determine the shock structure. We find that the magnetic field in the shock front has a significant rotation out of the initial upstream plane. This is most pronounced for single-sized grains and small angles of the shock normal with the magnetic field. Our results are similar to previous studies of steady C-type shocks showing that our method is efficient, rigorous and robust. Unlike the method employed in the previous most detailed treatment of dust in steady oblique fast-mode shocks, ours allow a reliable calculation even when chemical or other conditions deviate from local statistical equilibrium. We are also able to model transient phenomena.     

Revealing the “fingerprints” of the magnetic precursor of C-shocks

We present the first C-shock and radiative transfer model that calculates the evolution of the line profiles of neutral and ion species like SiO, H¹³CO⁺ and HN¹³C for different flow times along the propagation of the shock through the unperturbed gas. We find that the line profiles of SiO characteristic of the magnetic precursor stage have very narrow linewidths and are centered at velocities close to the ambient cloud velocity, as observed toward the young shocks in the L1448-mm outflow. Consistently with previous works, our model also reproduces the broad SiO emission detected in the high velocity gas in this outflow, for the downstream postshock gas in the shock. This implies that the different velocity components observed in L1448-mm are due to the coexistence of different shocks at different evolutionary stages.     

Resolved shocks in clumpy media

We study the structure of shocks in clumpy media, using a multifluid formalism. As expected, shocks broaden as they weaken: for sufficiently weak shocks, no viscous subshock appears in the structure. This has significant implications for the survival of dense clouds in regions overrun by shocks in a wide range of astrophysical circumstances, from planetary nebulae to the nuclei of starburst galaxies.     

Three-dimensional simulations of molecular cloud fragmentation regulated by magnetic fields and ambipolar diffusion

We employ the first fully three-dimensional simulation to study the role of magnetic fields and ion–neutral friction in regulating gravitationally driven fragmentation of molecular clouds. The cores in an initially subcritical cloud develop gradually over an ambipolar diffusion time while the cores in an initially supercritical cloud develop in a dynamical time. The infalling speeds on to cores are subsonic in the case of an initially subcritical cloud, while an extended (≳0.1 pc) region of supersonic infall exists in the case of an initially supercritical cloud. These results are consistent with previous two-dimensional simulations. We also found that a snapshot of the relation between density (ρ) and the strength of the magnetic field ( B ) at different spatial points of the cloud coincides with the evolutionary track of an individual core. When the density becomes large, both the relations tend to   B ∝ρ^0.5  .     

The contributions of J-type shocks to the H2 emission from molecular outflow sources

We present the results of modelling of the H₂ emission from molecular outflow sources, induced by shock waves propagating in the gas. We emphasize the importance of proper allowance for departures from equilibrium owing to the finite flow velocity of the hot, compressed gas, with special reference to the excitation, dissociation and reformation of H₂. The salient features of our computer code are described. The code is applied to interpreting the spectra of the outflow sources Cepheus A West and HH43. Particular attention is paid to determining the cooling times in shocks whose speeds are sufficient for collisional dissociation of H₂ to take place; the possible observational consequences of the subsequent reformation of H₂ are also examined. Because molecular outflow sources are intrinsically young objects, J-type shocks may be present in conjunction with magnetic precursors, which have a C-type structure. We note that very different physical and dynamical conditions are implied by models of C- and J-type shocks which may appear to fit the same H₂ excitation diagram.     

Simulations of the wardle instability of c-type shock waves

C-type shocks in the partially ionized ISM are modelled by numerical simulations. Under certain conditions the shocks are subject to the Wardle instability, which initially makes the shock front rippled, then in the non-linear stage can produce density variations in both the ion and neutral fluids. A systematic search in the numerically accessible parameter space is done to determine the wave vector k_max and the growth rates _max of the fastest growing modes. The neutral Alfvén number, and the angle _sbetween the shock normal and the upstream magnetic field determine the strength and obliqueness of the shock, as well as the dimensionless parameters of the fastest growing mode. The results confirm and extend Wardle's linear analysis.The non-linear evolution shows saturation of the instability and the formation of high density regions that detach from the shock front with the downstream flow. Numerical difficulties are partially solved by an implicit treatment of the ion-neutral friction terms, but strong shocks still can not be modelled efficiently. A fully implicit method for the ions and the magnetic field is used to model C-type shocks with low fractional ionization and high ion Alfvén speed.     

A simple model for H2 line profiles in bow shocks

We present a model for empirically reproducing line profiles of molecular hydrogen emission in bow shocks. The model takes into account bow velocity, dissociation limit, a cooling function, viewing angle, bow shape and a limited form of extinction. Our results show that both geometrical factors and shock physics can significantly affect the profile morphology. In a companion paper we will apply this model to Fabry–Perot observations of bow shocks in the Orion BN–KL outflow.     

Continuum-driven shocks in active galactic nuclei

In this paper, we extend the study of instabilities in flows driven by the radiation pressure of an ionizing continuum to flows that are not plane parallel. It is well known that the plane-parallel instability leads eventually to the formation of continuum-driven shocks backed by a sonic transition. If these structures are thin, we find that they are unstable to a corrugation mode, and evolve to form sharp-peaked triangular profiles. Once this has occurred, the thin-shock approximation is no longer valid.
We study the further development of the shocks by numerical hydrodynamic simulations. The flow tends to break up into numerous discrete bow-shaped components. The speed of these components through the upstream material is almost constant. As a result, the maximal velocity of radiatively driven shocks through the upstream gas may be determined by instabilities rather than by other physical effects. Interactions between gas in the wings of neighbouring bowshocks can, however, form subsequent generations of bowshocks that are faster and more acute than their predecessors.
One likely location where continuum-driven shocks may occur is in the broad-line regions of active nuclei. We discuss the application of our results to such flows.     

Wisps and knots in the central Crab nebula

The fast-spinning Crab pulsar (∼30 turn s⁻¹), which powers the massive expansion and synchrotron emission of the entire Crab nebula, is surrounded by quasi-stationary features such as fibrous arc-like wisps and bright polar knots in the radial range of 2×10¹⁶≲ r ≲2×10¹⁷ cm, as revealed by high-resolution (∼0.1 arcsec) images from the Wide Field and Planetary Camera 2 (WFPC2) on board the Hubble Space Telescope ( HST ). The spin-down energy flux (∼5×10³⁸ erg s⁻¹) from the pulsar to the luminous outer nebula, which occupies the radial range 0.1≲ r ≲2 pc, is generally believed to be transported by a magnetized relativistic outflow of an electron–positron e^± pair plasma. It is then puzzling that mysterious structures like wisps and knots, although intrinsically dynamic in synchrotron emission, remain quasi-stationary on time-scales of a few days to a week in the relativistic pulsar wind. Here we demonstrate that, as a result of slightly inhomogeneous wind streams emanating from the rotating pulsar, fast magnetohydrodynamic (MHD) shock waves are expected to appear in the pulsar wind at relevant radial distances in the forms of wisps and knots. While forward fast MHD shocks move outward with a speed close to the speed of light c , reverse fast MHD shocks may appear quasi-stationary in space under appropriate conditions. In addition, Alfvénic fluctuations in the shocked magnetized pulsar wind can effectively scatter synchrotron beams from gyrating relativistic electrons and positrons.     

Mean-field dynamo in partially ionized plasmas – I

There are several astrophysical situations where one needs to study the dynamics of magnetic flux in partially ionized turbulent plasmas. In a partially ionized plasma, the magnetic induction is subjected to the ambipolar diffusion and the Hall effect in addition to the usual resistive dissipation. In this paper, we initiate the study of the kinematic dynamo in a partially ionized turbulent plasma. The Hall effect arises from the treatment of the electrons and the ions as two separate fluids and the ambipolar diffusion due to the inclusion of neutrals as the third fluid. It is shown that these non-ideal effects modify the so-called α effect and the turbulent diffusion coefficient β in a rather substantial way. The Hall effect may enhance or quench the dynamo action altogether. The ambipolar diffusion brings in an α which depends on the mean magnetic field. The new correlations embodying the coupling of the charged fluids and the neutral fluid appear in a decisive manner. The turbulence is necessarily magnetohydrodynamic with new spatial and time-scales. The nature of the new correlations is demonstrated by taking the Alfvénic turbulence as an example.     

Energy, angular momentum and wave action associated with density waves in a rotating magnetized gas disc

Both fast and slow magnetohydrodynamic (MHD) density waves propagating in a thin rotating magnetized gas disc are investigated. In the tight-winding or WKBJ regime, the radial variation of MHD density-wave amplitude during wave propagation is governed by the conservation of wave action surface density which travels at a relevant radial group speed C _g. The wave energy surface density and the wave angular momentum surface density are related to by = and = m respectively, where is the angular frequency in an inertial frame of reference and the integer m , proportional to the azimuthal wavenumber, corresponds to the number of spiral arms. Consequently, both wave energy and angular momentum are conserved for spiral MHD density waves. For both fast and slow MHD density waves, net wave energy and angular momentum are carried outward or inward for trailing or leading spirals, respectively. The wave angular momentum flux contains separate contributions from gravity torque, advective transport and magnetic torque. While the gravity torque plays an important role, the latter two can be of comparable magnitudes to the former. Similar to the role of gravity torque, the part of MHD wave angular momentum flux by magnetic torque (in the case of either fast or slow MHD density waves) propagates outward or inward for trailing or leading spirals, respectively. From the perspective of global energetics in a magnetized gas sheet in rotation, trailing spiral structures of MHD density waves are preferred over leading ones. With proper qualifications, the generation and maintenance as well as transport properties of MHD density waves in magnetized spiral galaxies are discussed.     

Near-infrared imaging of H2 in molecular outflows from young stars

In an attempt to identify the molecular shocks associated with the entrainment of ambient gas by collimated stellar winds from young stars, we have imaged a number of known molecular outflows in H₂ v=1-0 S(1) and wide-band K. In each flow, the observed H₂ features are closely associated with peaks in the CO outflow maps. We therefore suggest that the H₂ results from shocks associated with the acceleration or entrainment of ambient, molecular gas. This molecular material may be accelerated either in a bow shock at the head of the flow, or along the length of the flow through a turbulent mixing layer.     

Studies of accretion flows around rotating black holes – II. Standing shocks in the pseudo-Kerr geometry

Standing, propagating or oscillating shock waves are common in accretion and winds around compact objects. We study the topology of all possible solutions using the pseudo-Kerr geometry. We present the parameter space spanned by the specific energy and angular momentum and compare it with that obtained from the full general relativity to show that the potential can work satisfactorily in fluid dynamics also, provided the polytropic index is suitably modified. We then divide the parameter space depending on the nature of the solution topology. We specifically study the nature of the standing Rankine–Hugoniot shocks. We also show that as the Kerr parameter is increased, the shock location generally moves closer to the black hole. In future, these solutions can be used as guidelines to test numerical simulations around compact objects.     

H2 emission from shocks in molecular outflows: the significance of departures from a stationary state

We have computed the time dependence of the H₂ rovibrational emission spectrum from molecular outflows. This emission arises in shock waves generated by the impact of jets, associated with low-mass star formation, on molecular gas. The shocks are unlikely to have attained a state of equilibrium, and so their structure will exhibit both C- and J-type characteristics. The rotational excitation diagram is found to provide a measure of the age of the shock; in the case of the outflow observed in Cepheus A West by the ISO satellite, the shock age is found to be approximately 1.5×10³ yr. Emission by other species, such as NH₃ and SiO, is also considered, as are the intensities of the fine-structure transitions of atoms and ions.     

Instabilities in two-fluid magnetized media with inter-component drift

We analyse the stability of a magnetized medium consisting of a neutral fluid and a fluid of charged particles, coupled to each other through a drag force and exposed to differential body forces (for example, as the result of radiation forces on one phase). We consider a uniform equilibrium and simple model input physics, but do not arbitrarily restrict the relative orientations of the magnetic field, slip velocity and wavevector of the disturbance. We find several instabilities and classify these in terms of wave resonances. We briefly apply our results to the structure of SiO maser regions appearing in the winds from late-type stars.     

Hydrodynamic simulations of molecular outflows driven by slow-precessing protostellar jets

We present hydrodynamic simulations of molecular outflows driven by jets with a long period of precession, motivated by observations of arc-like features and S-symmetry in outflows associated with young stars. We simulate images of not only H₂ vibrational and CO rotational emission lines, but also of atomic emission. The density cross-section displays a jaw-like cavity, independent of precession rate. In molecular hydrogen, however, we find ordered chains of bow shocks and meandering streamers which contrast with the chaotic structure produced by jets in rapid precession. A feature particularly dominant in atomic emission is a stagnant point in the flow that remains near the inlet and alters shape and brightness as the jet skims by. Under the present conditions, slow jet precession yields a relatively high fraction of mass accelerated to high speeds, as also attested to in simulated CO line profiles. Many outflow structures, characterized by HH 222 (continuous ribbon), HH 240 (asymmetric chains of bow shocks) and RNO 43N (protruding cavities), are probably related to the slow-precession model.     

Excitation and kinematics in H2 bow shocks: near-infrared observations of HH 99 and VLA 1623A (HH 313)

We present a comprehensive near-infrared study of two molecular bow shocks in two protostellar outflows, HH 99 in R Coronae Australis and VLA 1623A (HH 313) in Rho Ophiuchi. New, high-resolution, narrow-band images reveal the well-defined bow shock morphologies of both sources. These are compared with two-dimensional MHD modelling of molecular bows from which we infer flow inclination angles, shock speeds and the magnetic field in the pre-shock gas in each system. With combined echelle spectroscopy and low-resolution K -band spectra we further examine the kinematics and excitation of each source. Bow shock models are used to interpret excitation (CDR) diagrams and estimate the extinction and, in the case of VLA 1623, the ortho–para ratio associated with the observed H₂ population. For the first time, morphology, excitation and kinematics are fitted with a single bow shock model.
Specifically, we find that HH 99 is best fitted by a C-type bow shock model (although a J-type cap is probably responsible for the [Fe  ii ] emission). The bow is flowing away from the observer (at an angle to the line of sight of ∼45°) at a speed of roughly 100 km s⁻¹. VLA 1623A is interpreted in terms of a C-type bow moving towards the observer (at an angle to the line of sight of ∼75°) at a speed of ∼80 km s⁻¹. The magnetic field associated with HH 99 is thought to be orientated parallel to the flow axis; in VLA 1623A the field is probably oblique to the flow axis, since this source is clearly asymmetric in our H₂ images.     

The influence of the Mach number on the stability of radiative shocks

We study the stability properties of hydrodynamic shocks with finite Mach numbers. The linear analysis supplements previous analyses which took the strong shock limit. We derive the linearized equations for a general specific heat ratio as well as temperature and density power-law cooling functions, corresponding to a range of conditions relevant to interstellar atomic and molecular cooling processes. Boundary conditions corresponding to a return to the upstream temperature  ( R = 1)  and to a cold wall  ( R = 0)  are investigated. We find that for Mach number   M > 5  , the strong shock overstability limits are not significantly modified. For   M < 3  , however, shocks are considerably more stable for most cases. In general, as the shock weakens, the critical values of the temperature power-law index (below which shocks are overstable) are reduced for the overtones more than for the fundamental, which signifies a change in basic behaviour. In the   R = 0  scenario, however, we find that the overstability regime and growth rate of the fundamental mode are increased when cooling is under local thermodynamic equilibrium. We provide a possible explanation for the results in terms of a stabilizing influence provided downstream but a destabilizing effect associated with the shock front. We conclude that the regime of overstability for interstellar atomic shocks is well represented by the strong shock limit unless the upstream gas is hot. Although molecular shocks can be overstable to overtones, the magnetic field provides a significant stabilizing influence.     

MHD study of temporal and spatial evolution of simulated interplanetary shocks in the ecliptic plane within 1 AU

We utilize a 21/2-D MHD time-dependent model to perform a parametric study of interplanetary shock propagation to 1 AU. The input conditions are represented by the following variables:(1) initial shock velocity, (2) duration of the driving pulse, and (3) width of the pulse at the near-Sun position (18 solar radii). The total net energy added to the solar wind was calculated for each pulse. The forward shock's travel time to, and the peak dynamic pressure at, 1 AU as a function of location along the shock front have been studied over a range of total input pulse energies from 10²⁹ to 10³² ergs. For input pulses with modest angular width and temporal duration, we find that the propagation of the resulting interplanetary fast forward shock waves depends primarily upon the net input energy. The dependence of the transit time upon energy is a power law with a -1/3 index which corresponds to the classical, piston driven case. Reverse shocks are also formed behind all but the lowest energy shocks. Their properties, although also a function of input energy, depend upon the specific values of the input pulse shock velocity, width and duration. We also briefly discuss the propagation of the shocks out to 1 AU, and the conditions for which the interplanetary shocks depart from being symmetric about the input pulse central meridian due to magnetic and dynamic effects.