Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mrao.cam.ac.uk/~krause/krause04.apss.pdf Дата изменения: Mon Feb 14 22:19:56 2005 Дата индексирования: Mon Oct 1 22:19:18 2012 Кодировка:

Large scale simulations of the jet-IGM interaction
Martin G. H. Krause
Landessternwarte KЁ onigstuhl, 69117 Heidelberg, Germany December 15, 2003 Abstract. In a parameter study extending to jet densities of 10-5 times the ambient one, I have recently shown that light large scale jets start their lives in a spherical bow shock phase. This allows an easy description of the sideways bow shock propagation in that phase. Here, I present new, bipolar, simulations of very light jets in 2.5D and 3D, reaching the observationally relevant scale of > 200 jet radii. Deviations from the early bow shock propagation law are expected because of various effects. The net effect is, however, shown to remain small. I calculate the X-ray appearance of the shocked cluster gas and compare it to Cygnus A and 3C 317. Rings, bright spots and enhancements inside the radio cocoon may be explained. Keywords: extragalactic jets, jet-IGM interaction, hydrodynamics, simulations, very light jets

1. Intro duction X-ray studies of galaxy cluster centers containing a radio jet have shown that the jets have a considerable impact on the cluster gas (Carilli et al., 1994; Blanton et al., 2001; Smith et al., 2002), forming rings, apparent spirals, and aligned features. Such systems have b een claimed to b e asso ciated with very light jets (Clarke et al., 1997; Rosen et al., 1999; Krause and Camenzind, 2003). Parameter studies of very light jets have b een carried out recently (Carvalho and O'Dea, 2002a; Carvalho and O'Dea, 2002b; Saxton et al., 2002; Krause, 2003; Zanni et al., 2003). There, it has b een shown that very light jets first form spherically symmetric b ow sho cks (Krause, 2003). In that phase, they follow the expansion law (derived for a strong b ow sho ck, which is applicable here):
r 0 t t

M(r )r dr = 2

dt
0 0

E (t )dt ,

(1)

where M(r ) is an arbitrary spherically symmetric ambient gas mass distribution and E (t) is the energy injection law. Here, I show 3D and 2.5D large scale simulations, up dating the X-ray app earance of the sho cked IGM, and exploring the accuracy of the spherical expansion law for the b ow sho ck at late times.
c 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1. Logarithm of number density (slice) for the 3D run at 2.04 Myr.

2. 3D Simulation 2.1. Simulation Setup A cylindrical grid was used for the jet simulation (compare Krause and Camenzind (2002)). The size of the computational domain was: Z [-69 kp c, 69 kp c], R [0, 57 kp c] and [0, 2 ]. 2042, 805, and 57 grid p oints were used in the Z,R and directions, resp ectively. The full extend was not reached b ecause of long computation times and breakdown of the sup ercomputer. With a jet radius of r j = 0.55 kp c, this gives a resolution of 8 p oints p er b eam radius (ppb). The grid was initialised with an isothermal King cluster atmosphere: e (r ) = -3 /2 r2 , where r = R2 + Z 2 denotes the spherical radius, e,0 1 + a2 e,0 = 1.2 в 10-25 g/cm3 is the characteristic density, = 0.75 and a = 35 kp c is the core radius. In order to break the bip olar and axial symmetry, this density profile was mo dified by random p erturbations. The jet is injected in the middle of the grid in the region Z [-0.55, 0.55 kp c], R [0, 0.55 kp c], and [0, 2 ]. This region has the constant values: jet = 6.68 в 10-28 g/cm3 , vZ = ±0.4c, c b eing the sp eed of light. The kinetic jet luminosity is L kin = 1.04 в 1046 erg/s for b oth jets together. The pressure was set in order to match the external pressure at that p osition. This gives a slightly varying density contrast across the grid of = jet /ext 7 в 10-3 and an internal Mach numb er M = 10. The temp erature in the external medium is set to 3 в 10 7 K. The co oling time in the sho cked cluster gas is approximately 100 Myr. The jet is exp ected to propagate through the whole volume in 10 Myr. So, co oling by bremsstrahlung marginally influences the state of the gas. This was taken into account. In order to keep the system in hydrostatic equilibrium, gravity by an assumed dark matter distribution had to b e applied. The non-relativistic co de NIRVANA (Ziegler and Yorke, 1997) was used for the computation.

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Figure 2. Bow shock shapes for the 3D run, compared to ellipse and parabola.

2.2. Results The final snapshot at t = 2.04 Myr is shown in Fig. 1. The co co on is nicely placed around the jet b eam. Shear instabilities show up prominently. They still grow towards the center, and develop into long fingers at the innermost p ositions. The pressure shows a regular spacing of sho ck compressed and rarefaction zones in the b eam. High pressure regions are small and show up only at the end of the b eams, where the Mach disk is lo cated. The central region, with a diameter of roughly 10 kp c, is now dynamically calm. No large Mach numb ers are observed there, and the pressure is approximately constant. 2.2.1. The shape of the bow shock Figure 2 shows the b ow sho ck shap e for the final snapshot in detail. It has an axisymmetric part in the middle, where it can b e well represented by an ellipse. The elliptical shap e ends at |Z | 10 where two cigar like extensions join the b ow sho ck. They are 3D in nature, and can b e represented, on average, by a parab ola of rank three. 2.2.2. Emission maps The emission due to bremsstrahlung was integrated for different viewing angles (see Fig. 3). The general X-ray emission prop erties of sho cked ambient gas have b een discussed by Clarke et al. (1997), which has b een up dated recently by Zanni et al. (2003). The idea is that the gas is pushed aside by the jet co co on. Dep ending on its compression, it may form X-ray deficits at the lo cation of the co co on, and bright shells at the edges. The critical parameter is the relative shell thickness, , defined as the width of the sho cked ambient gas region divided by the lo cal b ow sho ck radius. X-ray deficits could b e observed for > 38%, for sources at inclination i = 90 . Here, is comparatively low. Hence, at high i the X-ray surface brightness never falls b elow that of the undisturb ed King atmosphere, but the deficit is pronounced for low i (compare Fig. 3).

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0.8 0.7 0.6 erg/s/cm2 erg/s/cm2 -8 -6 -4 -2 0 X [kpc] 0.6 0.55 0.5 0.45 erg/s/cm2 0.4 0.35 0.3 0.25 0.2 0.15 0.1 -8 -6 -4 -2 0 X [kpc] 0.45 0.4 0.35 0.3 0.25 0.2 0.15 -8 -6 -4 -2 0 X [kpc] 0.45 0.4 0.35 0.3 0.25 0.2 0.15 -8 -6 -4 -2 0 X [kpc] 0.45 0.4 0.35 0.3 0.25 0.2 0.15 -8 -6 -4 -2 0 2 X [kpc] 4 6 8 0.45 0.4 0.35 erg/s/cm2 erg/s/cm2 0.3 0.25 0.2 0.15 0.1 0.05 -20 -10 0 Z [kpc] 10 20 2 4 6 8 0.45 0.4 0.35 erg/s/cm2 erg/s/cm2 0.3 0.25 0.2 0.15 0.1 0.05 -25 -20 -15 -10 -5 0 Z [kpc] 5 10 15 20 25 2 4 6 8 0.4 0.35 0.3 0.25 0.2 0.15 0.1 -15 2 4 6 8 erg/s/cm2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 -10 -5 0 Z [kpc] 5 10 2 4 6 8 0.5 0.4 0.3 0.2 0.1 0 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -8 -6 -4 -2 0 Z [kpc] 2 4 6 8

erg/s/cm2

erg/s/cm2

-10

-5

0 Z [kpc]

5

10

15

Figure 3. Bremsstrahlung emission maps for the 3D run at t From top to bottom, the viewing angle is 0 , 10 , 30 , 60 , and column shows the emission map, the middle one a vertical, and a horizontal slice through the center. The undisturbed emission

= 2.04 Myr. 90 . The left the right one is indicated.

The two phases of the b ow sho ck, cigar and elliptical (see sect. 2.2.1), show up prominently in the emission maps. They form circular and elliptical rings, dep ending on the viewing angle. Where the rings partially overlap, they are brighter, pro ducing the impression of ring segments (e.g. Fig. 3, 10 ). The structures can also intersect on the line of sight, pro ducing bright sp ots (Fig. 3, 30 ). The p ole on figures show at least two rings: one from the cigar phase and one from the inner elliptical part.

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3. 2.5D simulation 3.1. Simulation Setup In order to study the jet evolution on larger scale, an axysymmetric (2.5D) simulation was p erformed with initial conditions similar to the 3D simulation in the previous section. The 20 ppb simulation was run for 20 Myrs of simulation time. During that time the jet reached an extent of 110 kp c which corresp onds to 220 jet radii. The King atmosphere parameters are: e,0 = mp /cm-3 , a = 10 kp c, = 0.75, and T = 3 в 107 K. The jet is injected with a density of jet = 10-4 в e,0 , a sound sp eed of 0.2c, and an internal Mach numb er of M = 3. 3.2. Results Logarithmic density and integrated X-ray emission after 20 Myr are presented in Fig. 4. Krause (2003) could only reach an evolution to up to 5 Myr. In this early phase, the b ow sho ck is spherical, its radius following an expansion law given by (1). At later times the co co on transforms via a conical phase towards a cylindrical one (Fig. 4). 3.2.1. Pressure evolution The average jet pressure is the driving force of the inner elliptically shap ed part of the b ow sho ck. The pressure in the jet system monotonically decreases with radius. Close to the axis, the pressure is higher b ecause of sho cks in the b eam region. In the sho cked ambient gas region, a new equilibrium of gravity and pressure app ears. The smallest pressure values are lo cated at the b ow sho ck, roughly 20% b elow the average. It has b een found that the sideways expansion of the b ow sho ck follows the blastwave's equation of motion (Krause, 2003), in the spherical phase. The accuracy of this law will b e checked in the following also for the larger 2.5D simulation. In this case, the b ow sho ck has propagated more than three core radii in the sideways direction. In the spherical approximation, the average pressure inside of the b ow sho ck is given by (neglecting energy stored in the b eam): pj = ( - 1)(Lt - Mv 2 /2)/Vj , (2)

where Vj is the jet volume (everything inside the b ow sho ck). The p ower L includes all sources of energy, i.e. the flux of kinetic and internal energy through the jet channel, the flux of internal energy entering through the surface of the b ow sho ck, and the energy lost by work against the gravitational field. Then, (2) can b e evaluated, where v

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Figure 4. Final snapshots of the 2.5D simulation. Top: Line-of-sight integrated X-ray emission due to bremsstrahlung. Bottom: Logarithm of the number density.

and t are given by (1): v= Lt2 , t= Mr 3 L
r 0 1/3

M(r )r dr

.

(3)

Figure 5 shows this analytical estimate together with the data from the simulation. Here, r was related to time via measurement from the simulation. The agreement is quite go o d, in general. The analytical formula follows the slop e of the simulation data, but underestimates it by up to 20%. 3.2.2. Sideways motion of the inner bow shock part The b ow sho ck propagation was lo cally fitted by a function of typ e a + b tc . The resulting parameters for the different regions are shown in Table I. Usually, a = 0 (fixed), since only the late evolution of the jet is studied. Only for the time span up to 5 Myr a fit with a = 0 has b een included b ecause here it is p ossible that effects from the initial conditions still dominate the propagation. For comparison, also fits to the detailed spherical approximation are given, computed by application of (3). According to that, the exp onent for the first five million

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Table I. Fit parameters for the bow shock position in the 2.5D simulation. The star denotes a fit with fixed a = 0. Two stars denote fits to the spherical approximation with fixed a = 0. time range [Myr] [0 : 5] [0 : 5] [0 : 5] a 2.00 0 0 b 2.67 4.66 3.22 c time range [Myr] a b c



0.76 0.35 [15 : 20] 0.67 [15 : 20]



0 0

2.99 2.36

0.78 0.81

100

0 - 5 Myr fit 15 - 20 Myr fit 15 - 20 Myr fit (spherical approximation) spherical approximation measured radial bow shock position

10-9 dyn/cm2, %

10

r [kpc] 0-5 Myr fit 15-20 Myr fit relative deviation average pressure (spherical approximation) average pressure (simulation) 1 time [Myr] 10

10

1

1 t [Myr]

10

Figure 5. Left: Average jet pressure over simulation time. The stars show the values measured in the simulation, crosses mark the pressure according to a spherical approximation, plusses show the relative difference of the former in percent. Corresponding symbols are connected with solid lines. The lines show fits to the pressure measured from the simulation. The fits are: 32.84 t-0.69 (0-5 Myr, solid line), and 227.95 t-1.66 (15-20 Myr, dashed line). The best fit for the spherical approximation in the range 15-20 Myr is: 126.05 t-1.50 (not shown). Right: Bow shock radius at Z=0 versus time (squares) with fits and compared to spherical approximation, including all power sources (see text, plus-signs). The three fits are: 4.66 t0.53 (0-5 Myr), 2.99 t0.78 (15-20 Myr), 2.36 t0.81 (15-20 Myr, spherical approximation).

years should b e 0.67. Using the pure p ower law, an exp onent of 0.35 is achieved in the simulation data. Allowing for the radial offset gives a b est fit exp onent of 0.76. Since the exp onent of 0.35 is much b elow any exp ectation, it follows that the initial conditions are still imp ortant in that phase, and the fit with offset is more appropriate. The concurrence of the curves increases with time. For the last five million years, the exp onent for the spherical approximation (0.81) exceeds the one for the p ower law fit of the simulation data (0.78). From the increasing asp ect ratio, an exp onent lower than the one of the spherical approximation was exp ected. The simulation shows that the effect is small.

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4. Discussion The bip olar simulations in connection with detailed studies of the early lives of very light jets reveal two parts of the b ow sho ck: an inner elliptical part, and an outer cigar shap ed one. These parts also app ear in the X-ray emission. When viewed from an appropriate angle, they can app ear as partial rings and bright sp ots. I suggest that the two X-ray rings in 3C 317 (Blanton et al., 2001) are caused by this effect. The X-ray app earance of the 2.5D simulation repro duces some imp ortant details of Cygnus A's X-ray emission (Smith et al., 2002): the elliptical deformation in the sho cked ambient gas region, the bright X-ray filaments inside the co co on, and the fork like structure around the co co on. The b ow sho ck can b e lo cated at the interface, where the elliptical X-ray contours meet the spherical ones from the unaffected cluster gas. This go o d agreement also supp orts the idea of a very light, relativistic, and magnetised jet in Cygnus A.

Acknowledgements This work was supp orted by the Deutsche Forschungsgemeinschaft (SFB 439). I thank the HЁ hstleistungsrechenzentrum Stuttgart for sup eroc computing time.

References
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