Pergamon Vistas in Asmnovy Vol. 4 I. No. I. pp. 79--X5. I997

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RELATIVISTIC SIMULATIONS OF SUPERLUMINAL SOURCES

J. L. GbMEZ, ‘.?.* J. M. MARTi, 3 A. l? MARSCHER 2 and J. M. IBhEZ 3

’ Department of Physics and Astronomy, University of Manchester, Manchester Ml3 9PL. U.K.

’ Department of Astronomy, Boston University, Boston, MA 02215, U.S.A. 3 Departamento de Astronomia y Astrofisica, Universidad de Valencia, Valencia. Spain

Abstract- We present numerical simulations of the radio emission from hydrodynamical relativistic jets. The quiescent-state jet emission consists of quasi-periodic knots of high emission, associated with internal rec- olhmation shocks. Superluminal components can be reproduced by in- troducing a square-wave perturbation in the injection velocity of the jet. Strong interactions of the resulting moving shock and the standing recol- limations result in a “drag” and increase in emission of the latter. @ 1997 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Very Long Baseline Interferometric (VLBI) observations have revealed the existence of jets with relativistic bulk velocities to be a common phenomenon in Active Galactic Nuclei (AGN) (e.g. Blandford, 1990) and accreting binary star systems, as recently observed in the galactic sources GRS1915+105 (Mirabel and Rodriguez, 1994) and GRO 51655-40 (Tingay et al.. 1995), in what seems to be a reduced version of the same phenomenon (Sams et al.,

1996). Sudden bursts of emission at multiple wavelengths are commonly observed in these sources, in what is thought to be an indication of the ejection of new VLBI components. This behaviour has been successfully explained in terms of travelling shock waves, approximated by square waves (Marscher and Gear, 1985; Hughes et al., 1991; Gomez et al., 1994). However. in order to obtain a description of the generation, evolution, and influence of these shocks in the radio emission of jets, a more realistic description of the flow hydrodynamics is needed. This has become possible only recently thanks to the development of time- dependent hydrodynamical simulations of relativistic jets (Marti et al., 1994, 1995; Duncan and Hughes. 1994), and its implementation in the calculation of radio emission (Gbmez et

*Present address: Instituto de Astrofisica de Andalucia. CSIC. Granada. Spain

79

80 .I L. Ghez et al.

Fig. I. Logarithm of the rest-mass density, logarithm of the specific internal energy, and bulk Lorentz factor of the quiescent jet model.

al., 1995, 1996; Hughes et al., 1996; Momisarov and Falle, 1996). This allows us to make predictions, test the current paradigm, and improve our knowledge of the physics relevant to relativistic jets and their environments.

2. THE MODEL

The hydrodynamic code solves the relativistic conservation equations of rest-mass, mo- mentum, and total energy for a perfect fluid. The flow is assumed to be axisymmetric, allowing a two-dimensional representation of the conservation equations in cylindrical co- ordinates. The code is based on an approximate Riemann solver which relies on the spec- tral decomposition of the Jacobian matrices of the multidimensional system of relativistic equations. The spatial accuracy of the code is set by a conservative monotonic piecewise- parabolic interpolation of pressure, proper rest-mass density, and flow velocity components within the numerical cells. For further details of this code we refer the reader to Marti et

al. (1996). where an exhaustive testing of the code can also be found. To obtain the synchrotron emission from the jet whose relativistic hydrodynamics is cal-

culated as above, we distribute the internal energy among the relativistic electrons follow- ing the usual power law N(E) dE = N,E-PdE, with Em,,, I E I Emax. Radiative energy losses can be neglected at radio frequencies, and therefore the ratio between the maximum and minimum energy can be considered constant all along the jet, and the power law is then fully determined (Gomez et al., 1995). We assume that the magnetic energy density remains a fixed fraction of the particle pressure, and that the magnetic field direction is predominantly randomly oriented.

The absorption and emission coefficients for the synchrotron radiation are computed in the fluid’s frame, taking into account relativistic aberration and Doppler transformation of the frequency of observation. To account for time delay effects, the coefficients are calculated using a retarded time that accounts for the delays within the jet. Therefore, time evolution information of the parameters that determine the emission is needed for every cell in the computational grid. The coefficients are then Lorentz transformed into the observer’s frame, where the Stokes parameters that determine the emission are calculated by integrating the transfer equations for the synchrotron radiation through all different lines of sight. We refer the reader to Gomez et al. (1993, 1994, 1995) for a detailed discussion of the emission calculations.

Relativistic Simulations of Superluminal Sources

9.3x10-=

4.7x10-2

Fig. 2. Total intensity maps corresponding to Fig. 1 model. The right map shows the same jet convolved with a Gaussian beam.

3. THE QUIESCENT-STATE JET STRUCTURE

Much of the physics of jets and their environments can be deduced from the quiescent- state jet structure. To avoid the formation of the bow shock ahead of the jet, we use as initial condition a jet of constant radius, velocity, and thermodynamic properties, propagating through a given atmosphere. To allow radial expansion of the jet, a decreasing pressure gradient in the atmosphere along the axial direction is introduced. The code is then run until a stationary solution is found. Figure 1 shows the distributions of rest-mass density, specific internal energy, and Lorentz factor for such a stationary relativistic jet. Due to the initial overpressure of the jet, over-expansions and over-contractions caused by inertial overshooting past equilibrium of the jet lead to the formation of standing oblique shocks. Note that because of the negative external pressure gradient, these would occur even if the pressure were matched at the injection point.

Figure 2 shows the synchrotron radiation for the jet model shown in Fig. I when viewed at 10” with respect to the jet axis, and optically thin frequency. The right map in Fig. 2 shows the appearance of the same jet when convolved with a Gaussian beam. The unconvolved total intensity map in Fig. 2 (left map) consists of a regular pattern of knots of high emission. These knots are associated with internal oblique shocks (see Fig. l), where the specific internal energy and rest-mass density are increased. The intensity of the knots decreases along the jet due to the expansion resulting from the gradient in external pressure. The maps in Fig. 2 reflect the different resolutions we may obtain with centimeter-wave, ground- based VLBI (convolved map) and space or millimeter-wave VLBI (unconvolved map). If these stationary components are to be detected (in sources where the jet is close to pressure equilibrium with the external medium), we might require very high linear resolutions that are possible with space or millimeter-wave VLBI observations (especially of nearby sources). Recently. indications of the possible detection of these standing shocks have been reported by Junor and Biretta (1995) in the jet of 3C 274 (M87). Another strong evidence of these standing knots in the emission have been observed in the galactic superluminal source GRO 51655-40 (Tingay et al., 1995) close to what is thought to be core of the jet,

4. OUTBURSTS IN RADIO EMISSION

In order to study the generation and evolution of superluminal components, Fig. 3 shows five subsequent epochs in the evolution of the pressure and bulk Lorentz factor for the jet model of Fig. 1 in which the injection Lorentz factor has been temporarily increased from the initial value off - 3 to f - 9.

Figure 3 shows a complex, time-variable structure in the perturbation, consisting of mul- tiple shocks and rarefactions. The time evolution of the dynamics in the jet is largely deter- mined by the interactions of the perturbation with the standing recollimation shocks and

82 J L. Gcimez et al.

-1.13 -1.60 -2.08 Ii

Fig. 3. Evolution of the logarithm of pressure (top five panels), and bulk Lorentz factor

(bottom five panels) when an increase in the injection velocity is introduced in the jet.

rarefactions. When the perturbation passes through a standing shock the latter is “dragged” downstream owing to the increased Mach number. The interaction with a rarefaction is similar, with the additional feature that a reverse shock is formed to separate the perturba- tion from the lower pressure, higher velocity fluid in the rarefaction. These reverse shocks propagate relatively slowly down the jet. Eventually the jet recovers its initial steady state,

as long as the jet maintains a constant level of energy and velocity at the input. Figure 4 shows the total intensity maps corresponding to six subsequent epochs after the

ejection of the perturbation for the jet model shown in Fig. 3. The maps have been obtained for a viewing angle of 10” and optically thin frequency of observation, and have been convolved with a circular Gaussian beam. The total intensity maps are dominated by the

presence of a well-defined component associated with the perturbation. The enhancement of the emission is due to an increase in the Doppler boosting and internal energy density in the perturbation. Part of this flux is also coming from the secondary features. especially the

reverse shocks, all blended by the convolving beam. The peak in the intensity maps moves at an apparent velocity of - 7c.

Time delay effects introduce a significant variation in the emission maps compared with

previous calculations neglecting these effects (Gomez et al., 1996). Due to time delays, the main emission feature appears much longer as viewed in the observer’s frame, resulting in components with higher emission and an elongated structure for the perturbation consid- ered here. (A weaker perturbation would allow the “core” emission near the input point to be more prominent. while higher resolution would allow the complex structure of the mul- tiple shocks to be revealed.) Observations of circular or transversely elongated components must therefore represent very short-lived variations in the jet inlet.

Relativistic Simulations qf Superluminal Sources 83

Fig. 4. Simulated total intensity images corresponding to six epochs (from left to right) in

the evolution of the perturbation. The input point of the jet is at the top of each panel.

5. COMPONENTS EVOLUTION IN SYMMETRIC SOURCES: GRS 1915+105

The observation of the jet and counter-jet in relativistic sources may provide significant information on the physical properties of both jets. Of special interest is the case where the ejection of plasma seems to occur symmetrically in both jets. The differences in the observed properties of both components have to reside exclusively (under the assumption of complete symmetry) on relativistic effects.

To determine the expected ratio of the flux for the approaching and receding components R. consider two truncated cones at the source’s frame of length 1 and minor diameter d moving at velocities rfis at a viewing angle of 8, through jets of half-opening angle 47. Taking into account time delays, the ratio of longitudinal sizes for the approaching and receding components is given by

sine+ (d/l+tang,)l~s-cosOl 1 +/Jscos8 RI, =

sm 0 + (d/l + tan q)(fis + cos 8) 1 - /3s cos 8 (1)

and for optically thin emission

R = R,J2+& = Re I

where o( is the emission spectral index, 6, is the ratio of Doppler factors between both components, and fir is the velocity of the fluid inside the components in the source’s frame.

Note that equation (2) differs from the results presented by Bodo and Ghisellini (1995) only in the consideration of the opening angle of the jet, which is necessary in order to allow adiabatic expansion of the components as they move along the jets. Contrary to the assumption made in Bodo and Ghisellini (1995) of dominant synchrotron losses, adiabatic losses should represent the major cooling mechanism.

84 J L. Gdmez et al.

We need to determine how the length 1 of the component changes as it moves along the jet. We assume, as suggested by polarization data (Rodriguez et al.. 1995) that the components are associated with travelling shock waves. (The case of clouds of plasma being responsible for the components will be considered in the future within our numerical model presented previously.) Two types of shock waves can be considered, forward shocks moving faster that the upstream flow, and reverse shocks that decelerates the plasma after it passes through the shock front. In any case, the length of the shocked plasma is I = Ifir - ps I ct. c is the light speed, and I is time. Therefore the ratio of fluxes for the approaching and receding components is finally determined by

R= sine+ I+&) ( tan(PlBs-cosQl 1 +/?scosB

sine+ (1 + fi > tanq,(&+cos@ 1 -BscosQ (3)

We can use equation (3) to obtain an estimation of the velocity of the plasma in the components ejected by the March flare in GRS 1915+ 105 using 8 = 70”. Bs = 0.92. o( = -0.8, Q, = 3”, and R = 8 as given by Mirabel and Rodriguez (1994). For a reverse shock (not solution is found for a forward shock) we obtain a preliminary value of

0.96 I BF 5 0.98 (4)

in contrast to the value of 0.92 obtained by Mirabel and Rodriguez (1994), and of 0.73 by Bodo and Ghisellini (1995). Due to this velocity of the plasma fluid, any emission lines will be shifted by bv, with 6 given by

l Approaching: 0.3 I 6, I 0.417 l Receding: 0.149 I 6, I 0.2 11

6. SUMMARY AND CONCLUSIONS

Besides revealing the existence of standing shocks and the internal structure of movmg components, our simulations show that a single perturbation can create several distinct emission features as it propagates downstream. Some indications of this effect are suggested by the C4 component of 3C 345 (see Lobanov and Zensus. 1996). In addition. the interaction of moving components with standing shocks may result in a temporary “drag” of the latter. An indication of this behavior has been detected in component Kl of 0735+178 by Gabuzda et al. (1994). Observations of standing components possibly associated with the recollimations shocks have been made in the jets of M87 (Junor and Biretta, 1995) and GRO J1655-40 (Tingay et al., 1995), and more sources are expected to show these as the observational resolution increases.

Time-dependent hydrodynamical simulations of relativistic jets provide a new, more re- alistic description of the physics involved in superluminal sources. We are in the process of exploring a greater range of parameter space so that the rich morphological detail re- vealed by the results of our simulations can be used to interpret the wealth of information provided by high-frequency and space VLBI imaging and multiwaveband monitoring of

variations in blazars.

Relativistic Simulations of Superfuminal Sources 85

ACKNOWLEDGEMENTS

This research is supported in part by NASA grant NAG%2508 and the Spanish DGI- CYT (refs. PB94-0973, PB94-1275). JLG gratefully acknowledges a postdoctoral Fulbright fellowship from the Spanish Ministry of Education and Fulbright Commission, as well as the receipt of a PPARC associateship by the University of Manchester.

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