Jet bending and velocity profiles
aa r X i v : . [ a s t r o - ph . H E ] D ec JET BENDING AND VELOCITY PROFILES
I. Liodakis , V. S. Geroyannis , V. G. Karageorgopoulos Department of Physics, University of Crete, Greece , Department of Physics, University of Patras, Greece [email protected], [email protected], [email protected] 10, 2018 Abstract
In this study, we consider the problem of geometry of curved jets. Wework with a simplified two-dimensional kinematical model assuming thatthe jet velocity is varying. We prescribe the velocity variation and studytwo cases concerning parsec and kilo-parsec scale jets by applying somesimplified numerical simulations. For parsec scale jets, we use some gen-eral expressions and derive the shape of jets for certain numerical cases.For kilo-parsec scale jets, we study the cases of 4C53.37 and 1159+583of Abell 2220 and Abell 1446 galaxy clusters, respectively, in an effort totest our model and compare our numerical results with observational data.
Keywords: galaxies: jets (bending, parsec scale, kilo-parsec scale, veloc-ity variation); individual objects (4C53.37, 1159+583, Abell 2220, Abell1446)
Jets are an interesting issue in astrophysics. We know how they are producedand propagate through space, but there are still some questions regarding theirinteraction with their environment. In theory, galactic jets seem to be streamsof plasma directed perpendicular to an accretion disk surrounding a supermas-sive black hole. Observations reveal curved jets. Such jet bending is due tothe interaction of the jet with a nebula, a gas cloud, a density anisotropy, orany other cause that can be expressed as a pressure gradient while the jet isjourneying through the galactic center or the intergalactic medium.The bending of a jet depends on its velocity and on the external pressuregradient ([1], Secs. I and III). Assuming that (i) the pressure gradient is distance-dependent ([2], Fig. 5.4), P = P y /y , (1)1here P and y are model parameters, and that (ii) this behaviour is the samefor any jet, then the difference in jet bending is due to the velocity profile.Even though small scale jets are usually considered to be accelerating tospeeds near the speed of light, velocities for parsec and kilo-parsec scales are oforder ∼ km s − (cf. [3], Table 1); and they are generally considered constant.For galactic jets owing to AGN, which can reach lengths of up to a few mega-parsecs, it is probably of lower significance to consider time-varying velocities.Still, given the scales involved, we cannot assume that they are constant, espe-cially when their interaction with the pressure gradient is taken into account.Accordingly, we have to examine two scenarios: either an accelerating jet, untilit becomes relativistic or destroyed after interacting with an external pressure,or a decelerating one. For our jet model, we adopt a two-dimensional kinematical model ([2], Eq. 5.17;we use here the definitions and symbols of this investigation), d fdx " (cid:18) dfdx (cid:19) − / = 1 ρu ∂P∂R | ⊥ , (2)where ([2], Eq. 5.18) ∂P∂R | ⊥ = sin( a ) ∂P∂x − cos( a ) ∂P∂y , (3)with ([2], Eq. 5.19) tan( a ) = dfdx . (4)Assuming that the pressure gradient is a function of the y -coordinate, we obtainthe dimensionless form ([2], Eq. 5.22) x = x + Z y h L e − P ( f ) /ρ u − i − / df, (5)where x is the integration constant and L a parameter associated with theangle the jet meets the pressure gradient ([2], Eq. 5.23),tan( i ) = ( L − − / . (6)There is a critical value L crit ([2], remarks following Eq. (5.27) and Fig. 5.5),such that, when below that value, the jet stops propagating forward and returnstowards its source. 2 The Computations
DGAUS8 , which integrates real functions in one vari-able along finite intervals on the basis of an adaptive 8-point Legendre-Gaussalgorithm, and which is appropriate for high accuracy integrations.
After some preliminary tests of the code, we examine four different cases forthe velocity profile. Two cases have to do with an accelerating jet (linear andnonlinear), and further two with a decelerating one (linear and nonlinear). Forthe nonlinear case we adopt a simplified exponential behaviour, u = u exp( ± y ) . (7)We examine several velocity profiles and scales for all cases and show theirgraphs in Fig. 1. Regarding the form we expect to have for this model, it isclear that the jets must be decelerating linearly, a conclusion also derived in[4] (Sec. 3.3). With that result in mind, we proceed in examining the cases forparsec and kilo-parsec scale jets. In the case of parsec scale jets, we assume that the bending is caused by anexternal pressure gradient interacting with the jet while moving through thegalactic center. We examine sixteen different sets of parameters. In detail, weintroduce four “velocity coefficients”, λ = 0 . λ = 0 . λ = 0 . λ = 1 .
0, inorder for the velocity to vary from one to two orders of magnitude ([4], Sec. 4) u i = u − λ i y (8)In addition, for each velocity coefficient λ i , we use four values of the parameter L , L = 2 . L = 4, L = 6 . L = 10 . i = π/ i = π/ i = π/
8, and i = π/
10, respectively. Numerical experimentswith several angles show that these particular values lead to distinguishablegraphs (since close angles yield almost identical curves).3umerical results for λ = 0 . λ = 0 . λ = 0 . λ = 1 . L = 2 . L = 4, Fig. 4 the case L = 6 . L = 10 . X and Y valuesare given in pc.We can see how the velocity affects the shape of the jet, and how the inter-action angle affects the final length of the jet. Moreover , in all cases computed,all values X corresponding to Y = 100 pc are of order(s) below the accuracyof the computations, so considered to be zero. All X values corresponding to Y = 200 pc are of order(s) 10 − —10 − . These values correspond to a diversionof approximately 10 —10 km; even thought it is very small compared to thescale of the jets, it cannot be neglected. We can conclude that the bending ofthe jet begins at a distance between 100 pc and 200 pc from the source.Knowing the speed of the jet, and reducing observational data to scale, wecan draw information and conclusion about the velocity variation and interac-tion angle. We consider the well-known jets of the structures 4C53.37 and 1159+583 in thegalaxy clusters Abell 2220 and Abell 1446, respectively. The bending of thejets is now due to the relative movement of the galaxy through the “interclustermedium” (ICM). Since, in this case, we have a much larger scale (hundredsof kilo-parsecs), we adopt a nonrelativistic hydrodynamic expression for thevelocity given by the Euler equation ∂u∂t + ( u ∇ ) u = − ∇ Pρ + g. (9)Following a similar analysis as before, the expression for the velocity becomes([6], Eq. (7); we use here the definitions and symbols of this investigation) u = " R k T n rρ t r c (1 + r /r c ) / + 2 R U m p n ρ t r t (1 + r /r c ) / / , (10)Burns and Balonek ([6], Eq. (9)) have managed to reduce Eq. (10) to the form u j = U gal (cid:18) n ICM n j Rh (cid:19) / , (11)where u j and n j are the velocity and density of the jet, U gal the velocity ofthe host galaxy, n ICM the mean intercluster density, R the radius of curvature,and h the distance from the source at which the interaction with the pressuregradient begins (scale height). For the case of ICM with high density, h is equalto the radius of the galactic center; for the case of ICM with low density, on theother hand, h coincides with the length of the jet.4able 1: Calculated values with velocity coefficients λ = 0 . λ = 0 . X and Y are given in pc. X L X L X L X L X L X L X L X L Y10 − × − × − × − × − × − × − × − n = n ICM /n j for each jet. Results for 4C53.37 and 1159+583are shown in Table 3 and in Figs. 6 and 7, respectively.The observational values for n in [6] are n = 0 . n = 12for 1159+583. Figs. 6 and 7 show how the shape of the jet does change withdifferent intercluster densities. For the case of 4C53.37 there is a formidablevariation in the shape of the curves between density ratios, whereas for the caseof 1159+583 the differences are small. Comparing our results with observationaldata given by [7] and [8] for 4C53.37 and 1159+583, respectively, and takinginto account only the main jet, even though the jet in 4C53.37 seems to havelost its shape, if we imagine the path it would have followed, both jets seem tobe in agreement with our computations. Approximating a two dimensional kinematical model by some simplified simu-lations, we have studied the velocity profiles of large scale jets by assuming thattheir speed is not constant. We have seen that, when the velocity of the jetdecreases linearly, the angle of interaction affects the length of the jet; and thebending begins at distances over 100 pc from the source. Regarding kilo-parsecscale jets, we can say that, morphologically, our simplified numerical simulationsgive results resembling relevant observations.5able 2: Calculated values with velocity coefficients λ = 0 . λ = 1 for allinteraction constants. Details as in Table 1. X L X L X L X L X L X L X L X L Y2.8 × − × − × − × − × − × − × − × − n = 0 . n = 0 . n = 1. For 1159+583: n = 5, n = 9, n = 12. Details asin Table 1. X n X n X n X n X n X n Y0.408 1.3 × − × − − − eferences [1] R. Fiedler and R. N. Henriksen, The Astrophysical Journal , , 554-559(1984).[2] V. Icke, “From nucleus to hotspot: nine powers of ten”. In “Beams andJets in Astrophysics”, P. A. Hughes (ed.), Cambridge University Press , , 232-277 (1991).[3] G. Pelletier and H. Sol, Monthly Notices of the Royal Astronomical Society , , 635-646 (1992).[4] D. S. De Young, The Astrophysical Journal , , 69-81 (1991).[5] A. P. Marscher, “Relativistic Jets in Active Galactic Nuclei”, AmericanInstitute of Physics Conference Series , P. A. Hughes & J. N. Bregman(eds.), Vol. 856, pp. 1-22 (2006).[6] J. O. Burns and T. J. Balonek,
The Astrophysical Journal , , 546-556(1982).[7] J. O. Burns and F. N. Owen, The Astronomical Journal , , 204-214(1980).[8] J. O. Burns, F. N. Owen, and L. Rudnick, The Astronomical Journal , , 1683-1693 (1979). 7 .2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Arbitrary Units0.20.40.60.81.0 A r b i t r a r y U n i t s A r b i t r a r y U n i t s A r b i t r a r y U n i t s A r b i t r a r y U n i t s Figure 1: Velocity profiles corresponding to four cases. Top left: accelerat-ing exponentially. Top right: accelerating linearly. Bottom left: deceleratingexponentially. Bottom right: decelerating linearly. y ( p a r s e c ) λ =0 . λ =0 . λ =0 . λ =1 Figure 2: Jet paths with interaction constant L = 2 .
718 for all velocity coeffi-cients λ i . 8
50 100 150 200 250 300x (parsec)2004006008001000 y ( p a r s e c ) λ =0 . λ =0 . λ =0 . λ =1 Figure 3: Jet paths with interaction constant L = 4 for all velocity coefficients λ i . y ( p a r s e c ) λ =0 . λ =0 . λ =0 . λ =1 Figure 4: Jet paths with interaction constant L = 6 . λ i . 9
20 40 60 80 100 120 140 160 180x (parsec)2004006008001000 y ( p a r s e c ) λ =0 . λ =0 . λ =0 . λ =1 Figure 5: Jet paths with interaction constant L = 10 .
472 for all velocity coeffi-cients λ i . y ( k il o - p a r s e c ) y ( k il o - p a r s e c ) y ( k il o - p a r s e c ) y ( k il o - p a r s e c ) n=0.1n=5n=1 Figure 6: Jet paths for three density ratios. Top left: n = 1. Top right: n = 0 . n = 0 .
1. Bottom right: all n values.10
50 100 150 200 250x (kilo-parsec)150200250300350400450500550600 y ( k il o - p a r s e c ) y ( k il o - p a r s e c ) y ( k il o - p a r s e c ) y ( k il o - p a r s e c ) n=5n=9n=12 Figure 7: Jet paths for three density ratios. Top left: n = 5. Top right: n = 9.Bottom left: n = 12. Bottom right: all nn