Mon.Not.R.Astron.Soc.000,1–11(0000) Printed1stFebruary2008 (MNLATEXstylefilev2.2) Formation of internal shock waves in bent jets S. Mendoza1 & M.S. Longair2 1 Institutode Astronom´ıa, Universidad Nacional Aut´onoma de M´exico, AP 70-264, DistritoFederal 04510, M´exico 2 Cavendish Laboratory, Madingley Rd., Cambridge CB3 OHE, U.K. 2 0 1stFebruary2008 0 2 n ABSTRACT a Wediscussthecircumstancesunderwhichthebendingofajetcangenerateaninternal J shock wave.The analysis is carriedout for relativistic and non–relativisticastrophys- 0 ical jets. The calculations are done by the method of characteristics for the case of 1 steady simple waves.This generalisesthe non–relativistictreatmentfirst usedby Icke (1991).We show that it is possible to obtain an upper limit to the bending angle of a 2 jetinordernottocreateashockwaveattheendofthecurvature.Thislimitingangle v 5 hasa value of∼75◦ for non–relativisticjets with a polytropicindex κ=4/3,∼135◦ 2 for non–relativistic jets with κ=5/3 and ∼50◦ for relativistic jets with κ=5/3.We 1 alsodiscuss under which circumstancesjets willform internalshock wavesfor smaller 9 deflection angles. 0 1 Key words: hydrodynamics – relativity – galaxies: active – galaxies:jets. 0 / h p - 1 INTRODUCTION analysis presented in this article generalises that made by o Icke (1991) by introducing relativistic effects into the flow. r t In a previous article (Mendoza & Longair 2001), we dis- This generalisation has a drastic effect on the results. Rel- as cussedamechanismbywhichgalacticandextragalacticjets, ativistic jets cannot bend as much as non–relativistic jets. : can change their original straight trajectory if they pass As we will see below this occurs because, when relativis- v through a stratified cold and high density region. For the tic effects are taken into account, the characteristic lines in i X case of galactic jets this could be a cloud in the vicinity of the flow are beamed in the direction of the flow velocity. anH–Hobject.Forextragalacticjets,thisregioncouldbea Thebeamingincreasesasthevelocityoflightisapproached r a nearbygalaxy,theinterstellarmediumofthehostgalaxyor by the flow. In other words, the chances for an intersection theintraclustermediumitself.Physically,whatisimportant between characteristic lines of uniform plane–parallel flow isthatthemediuminteractingwith theexpandingjet from (before the bending) and a curved flow increase because of thesource has a non–uniform density. thisbeaming. As mentioned by Icke (1991) and Mendoza & Longair In order to analyse in detail the bending of relativis- (2001),whenajetbends,itisindirectcontactwiththesur- tic jets we discuss the following points in each section. We roundingsandoneshouldexpectthatentrainmentfromthe write down the equations of relativistic hydrodynamics in surrounding gas might cause a severe disruption of the jet section 2 and their non–relativistic counterparts. This sets itself. Assumingthat thisentrainmentis not important, for the scene for discussing the propagation of disturbances in examplebyanefficientcooling, what itsleft isahighMach flows which leads naturally to the definition of the rela- numberflowinsideacollimatedflowthatbends.Whenasu- tivistic Mach number and characteristics. This generalises personic flow bends, the characteristics emanating at each thetraditionalapproachtogasdynamicsandcharacteristic pointof theflowtendtointersect (Landau&Lifshitz 1987; surfacessuchasthatdiscussedfornon–relativistichydrody- Courant & Friedrichs 1976). Since every hydrodynamical namics by Landau & Lifshitz (1987). The relativistic Mach quantityhasaconstant valueon agivencharacteristic line, numberwasfirstintroducedbyChiu(1973) andsomeofits this intersection causes the different physical quantities in propertiesarewelldescribedbyK¨onigl(1980).Themostim- theflow,suchasthedensityorpressure,tobemultivalued. portantresulttobeprovedinthissectionisthebeamingof This situation cannot occur in nature and a shock wave is characteristiclinesinthedirectionofmotion oftheflowfor formed. relativisticflows.Section4analysestherelativisticandnon– Theformationofshockwavesinsideajetarepotentially relativistic cases for a flow depending on one angular vari- dangerous. These shock waves could give rise to subsonic able, knownas Prandtl–Meyer flow.This leads naturally to flowinthejet andcollimation mightnolongerbeachieved. the generalisation of a typeof flow called rarefaction waves Thepresentarticlediscussesthecircumstancesunderwhich innon–relativistichydrodynamics(Landau&Lifshitz1987) aninternalshockwaveshouldbeexpectedinabentjet.The whichinturnisessentialfortheunderstandingofflowsthat 2 S. Mendoza & M.S. Longair movethrough a certain angle. In section 5 we use themain effectsarenottakenintoaccount.Inthecaseofanultrarel- resultsof thePrandtl–Meyerflowto describesteady simple ativisticphotongasithasavalueof4/3.Itisnotdifficultto waves.Thesewavesform naturallyforsteadyplaneparallel showthatforapolytropicgas,thespeedofsoundaandthe flow at infinity when the jet turns through an angle with a enthalpy per unit mass, the specific enthalpy, w of are re- certaincurvedprofile.Itisthenpossibletoapplytheneces- latedtoeachotherbythefollowingformula(Stanyuokovich saryphysicalingredientstothesteadysimplewavesformed. 1960): We then calculate an upper limit to the deflection of a jet in order to avoid the formation of a shock at the end of its c2 1 a2 curvature.Thislimitdoesnotmeanthatajetwhichcurves w =1− κ 1c2. (5) through a small deflection angle is safe from generating an − The quantities in the relativistic case are defined with internal shock wave. So, we also calculate a lower limit, for respect to the proper system of reference of the fluid, which a shock could form at the onset of thecurvature.Fi- whereas in classical mechanics these quantities are referred nally,insection7wediscusstheastrophysicalconsequences to the laboratory frame. In the relativistic case the ther- of the results described above. modynamic quantities, such as the internal energy density e, the entropy density σ and the enthalpy density ω are all defined with respect to the proper volume of the fluid. In 2 BASIC EQUATIONS non–relativistic fluiddynamics, thesequantitiesare defined in units of the mass of the fluid element they refer to. For Inordertounderstandtheformationofinternalshockwaves instance, the specific internal energy ǫ, the specific entropy inabentjet itisnecessarytounderstandsomeofthebasic sandthespecificenthalpywareallmeasuredperunitmass properties of thegas dynamicsof relativistic flows. inthelaboratoryframe.Whentakingthelimitinwhichthe Theequationsofmotionforanidealrelativisticfloware speed of light c tendsto infinity we must also bear in mind describedbythe4–dimensionalEuler’sequation(Landau& that the internal energy density e includes the rest energy Lifshitz 1987): density nmc2, where m is the rest mass of the particular fluid element under consideration. Therefore the following ωuk∂ui = ∂p u uk ∂p , (1) non–relativistic limits shouldbetakenin passingfrom rela- ∂xk ∂xi − i ∂xk tivistic to non–relativistic fluid dynamics: in which Latin indices take the values 0, 1, 2, 3. The vec- tor xk = (ct,r), where r represents the three dimensional mn ρ 1 v2/c2 ρ 1 v2/2c2 , radius vector, c the speed of light and t the time coordi- −c−→−∞→ − ≈ − nate. The Galilean metric gik for flat space time is given e npmc2+ρǫ ρc2 (cid:0) 1ρv2+ρǫ(cid:1), by g =1, g =g =g = 1 and g =0 when i= k. −c−→−∞→ ≈ − 2 00 11 22 33 ik − 6 The pressure is represented by p, ω =e+p is the enthalpy ω p mc2+m ǫ+ m c2+w , (6) per unit proper volume and e is the internal energy den- n −c−→−∞→ ρ ≈ (cid:18) (cid:19) sity. The four–velocity uk = dxk/ds where the relativistic (cid:0) (cid:1) where ρ is the mass density of the fluid in the laboratory interval ds is given by ds2 = g dxkdxl. The values of the kl frame. different thermodynamic quantities are measured in their local proper frame. Thespacecomponentsofeq.(1)givetherelativisticEu- 3 CHARACTERISTICS AND MACH NUMBER ler equation: The properties of subsonic and supersonic flow, are com- γω ∂v v ∂p pletely different in nature. To begin with, let us see how +v gradv = gradp , (2) c2 ∂t · − − c2 ∂t perturbations with small amplitudes are propagated along (cid:26) (cid:27) theflowforbothsubsonicandsupersonicflows.Forsimplic- where v is the three dimensional flow velocity and γ is the ity in the following discussion we consider two dimensional Lorentz factor for a flow with this velocity. flow only. The relations obtained below are easily obtained For an ideal flow, in the absence of sources and sinks, for thegeneral case of three dimensions. the relativistic continuity equation is given by (Landau & Ifagasinsteadymotion receivesasmallperturbation, Lifshitz 1987): this propagates through the gas with the velocity of sound relative to the flow itself. In another system of reference, ∂nk the laboratory frame, in which the velocity of the flow is v =0, (3) along the x axis, the perturbation travels with an observed ∂xk velocity u whose x and y components are given by: where the particle flux 4–vector nk=nuk and the scalar n is theproper numberdensity of particles in thefluid. acosθ+v A polytropic gas obeys therelation: ux = 1+avcosθ/c2, (7) γ−1asinθ p nκ, (4) uy = 1+avcosθ/c2, (8) ∝ wherethepolytropic index κisaconstantandhasthevalue according to the rule for the addition of velocities in spe- 5/3 for an adiabatic monoatomic gas in which relativistic cial relativity (Landau & Lifshitz 1994). The polar angle θ Formation of internal shock waves in bent jets 3 u On the other hand, since tanα = uy/ux, it follows from eqs.(7)-(8) and eq.(10) that: u aeˆ′ aeˆ′ r r α v a2 θ θ tanθ= 1 , 0 v 0 v −ar − v2 and so eq.(10) gives a relation between the angle α, the velocity of theflow v and its sound speed a: tanα=γ−1 a/v (11) (a) (b) 1 (a/v)2 − Figure1. Regionofinfluenceofsmallamplitudeperturbations. This variation of the angqle α with the velocity of the Aperturbationofsmallamplitudeisproducedintheflowatsome flow is plotted in Fig. 2 for the case in which the gas is point 0. This is carried by the flow which has velocity v. In the assumedtohavearelativisticequationofstate,thatis,when case of subsonic flow, as shown in panel (a), the perturbation is p=e/3.Theimportant featuretonotefrom theplot isthat abletopropagatetothewholespace.Whentheflowissupersonic the aperture angle of the cone of influence is reduced when the perturbation is propagated only downstream inside a cone with aperture angle 2α. The speed of sound a and the angle θ thevelocity of theflow approaches that of light. aremeasured inaframeof reference inwhichthe flow isat rest From eq.(11) it follows that, as the velocity of the flow –the proper frame of the flow. The vector u is the relativistic approaches that of light, the angle α vanishes. In other additionofvectorsvandaeˆ′r,whereeˆ′r isaunitradialvectorin words, as the velocity reaches its maximum possible value, theproperframeoftheflow. the perturbation is communicated to a very narrow region along thevelocity of theflow. In studies of supersonic motion in fluid mechanics it andthevelocityofsoundaarebothmeasuredintheproper is very useful to introduce the dimensionless quantity M frame of the fluid. Since a small disturbance in the flow definedas: moves with the velocity of sound in all directions, the pa- rameter θ can have values 06 θ 6 2π. This is illustrated 1 γ a pictorially in Fig. 1. sinα= a , (12) M ≡ γ v Let us consider first the case in which the flow is sub- sonic, as illustrated in case (a) of Fig. 1. Since by defini- accordingtoeq.(11).Thequantityγ =1/ 1 (a/c)2isthe a tion v < a and c2 > av, it follows from eqs.(7)-(8) that Lorentz factor calculated with the velocity of−sound a. The ux(θ=π)<0 and uy(θ=π)=0. In other words, the region number M has the property that M 1 pas v a and M influenced by the perturbation contains the velocity vector as v c. It also follows that M>1→if and o→nly if v>a.→ v.Thismeansthattheperturbationoriginatingat0isable ∞ Th→e surface bounding the region reached by a distur- to betransmitted to all partsof theflow. bance starting from the origin 0 is called a characteristic When the velocity of the flow is supersonic, the situa- surface (Landau& Lifshitz1987). Inthegeneralcase ofar- tion is quite different, as shown in case (b) of Fig. 1. For bitrary steady flow, the characteristic surface is no longer this case, ux(θ=π)>0 and uy(θ=π)=0. In other words, a cone. However, exactly as it was shown above, the char- thevelocityvectorv isnotfullycontainedinsidetheregion acteristic surface cuts the streamlines at any point at the ofinfluenceproducedbytheperturbation.Thisimpliesthat angle α. onlyaboundedregionofspacewillbeinfluencedbytheper- Let us briefly discuss the non–relativistic limit of the turbation originating at position 0. For the case of steady differentphysicalcircumstancespresentedabove.Todothis, flow, this region is evidently a cone. Thus, a disturbance we use the relations in eq.(6) with c and, as usual for arising at any point in supersonic flow is propagated only thenon–relativisticcase,werepresent→th∞espeedofsoundby downstream inside a cone of aperture angle 2α. By defini- c. tion, the angle α is such that it is the angle subtended by ThedimensionlessnumberM satisfiesthefollowing re- the unit radius vector eˆr with the velocity vector v at the lation: point in which the azimuthal unit vector eˆ is orthogonal α with the tangent vector d(aeˆ′)/dθ to the boundary of the r 1 c region influenced by the perturbation. The unit vector eˆ′ =sinα (13) r M −c−→−∞→ v is the unit radial vector in the proper frame of the flow. In and is called in non–relativistic hydrodynamics the Mach other words, the angle α obeys the following mathematical number.1 relation: d(aeˆ′) 1 TherelativisticgeneralisationoftheMachnumberaspresented dθr ·eˆα =0. (9) in eq.(12) was first calculated by Chiu (1973), who reduced the problemofsteadyrelativisticgasdynamicstoanequivalentnon– Substitution of eqs.(7)-(8) into eq.(9) gives: relativisticflow.Fromeq.(12)itfollowsthatthisnumberisinfact adefinitionoftheproper Mach number sinceitisdefinedasthe ratio of the space component of the relativistic four–velocity of 1 av tanα= γ + . (10) theflowγvtothesamecomponentoftherelativisticfour–velocity − (cid:26)tanθ c2sinθ(cid:27) ofsoundγaa(Ko¨nigl1980). 4 S. Mendoza & M.S. Longair dv v = r, (14) φ dφ d(v nγ) v nγ+ φ =0, (15) r dφ ωγ/n=const, (16) where v and v are the components of the velocity in the r φ 2α radial and azimuthal directions respectively. Eq.(16) is the v relativistic Bernoulli equation (Landau & Lifshitz 1987) for thisproblem. Using thedefinition of thespeed of sound, ∂p a=c (17) ∂e (cid:18) (cid:19)σ in eq.(15) and eq.(16) it is found that: dv a2 c2 d v + φ +v 1 ln(ω/n)=0. (18) r dφ φ − c2 a2dφ (cid:18) (cid:19) Ontheotherhand,differentiationofγ−2withrespecttothe Figure 2. Region of influence of a perturbation for different azimuthal angle φ and using eq.(16), gives: values of the velocity of a relativistic gas with a sound speed a=c/√3.Fromlefttorighttheclosedloopscorrespondtovalues tloihfgehthtpecro.vpTeelhorcesitypysetrvetmuorfbo0af.t0rio,enfe0r.i2esn,acs0es.4uo,mf..eth.de,t0ofl.8oowrini.gIiunnnaitttheseaotcfattshheeeoosfprisegueipdneoor-ff vφ(cid:18)vr+ ddvφφ(cid:19)+c2 1− vr2c+2vφ2!ddφln(ω/n)=0. (19) sonicflow,theregionofinfluenceoccursonlydownstream inside Multiplication of eq.(18) by v and substracting this from φ a cone with aperture angle 2α. This cone surrounds the corre- eq.(19) gives: sponding loop and is tangent to it. When the flow is subsonic, the perturbation is transmitted to all the flow. As a particular example, the cone and the velocity vector have been drawn for v2 =a2 1 vr2 . (20) thecaseinwhichthevelocityisv=0.8c. φ − c2 (cid:18) (cid:19) Bernoulli’sequation,eq.(16),togetherwiththevalueof thespecificenthalpyforapolytropicgasgivenineq.(5),can The results obtained concerning the relativistic and berewritten non–relativistic Mach number M can be rewritten in the following way:thedimensionless MachnumberM increases withoutlimitsasthevelocityoftheflowtakesitsmaximum 1 vr2+vφ2 1 1 a2 2 = 1 1 a20 2, possible value. This maximum value is the speed of light − c2 !(cid:18) − κ−1c2(cid:19) (cid:18) − κ−1c2(cid:19) in the relativistic case and infinity in the non–relativistic (21) one. The Mach number tends to zero as the velocity of the in which it has been assumed that at some definite point, flow vanishes,and tendsto unityas thevelocity of theflow theflowvelocityvanishesandthespeedofsoundhasavalue tendstothevelocityofsound.TheMach numberisgreater a there.Itisalwayspossibletomakethevelocityzeroata 0 than one for supersonic flow and less than unity when the certainpointbyasuitablechoiceofthesystemofreference. velocity of the flow is subsonic in both, the relativistic and Eqs.(20)-(21) can besolved in terms of v and v : r φ non–relativistic cases. v2/c2 =1 F2(a), (22) r − v2 =a2F2(a), (23) φ 4 PRANDTL–MEYER FLOW where Let us describe briefly the exact solution of the equations 1 a2 2 a2 −1 1 a2 −2 ofhydrodynamicsforplanesteadyflowwhichdependsonly F2(a)= 1 0 1 1 . − κ 1c2 − c2 − κ 1c2 on the angular variable φ only. This problem was first in- (cid:18) − (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19) (24) vestigated by Prandtl and Meyer in 1908 (Landau & Lif- shitz1987;Courant&Friedrichs1976)forthecaseinwhich Because v dv = c2F(a)F′(a)da, eq.(14) gives the re- r r relativistic effects were nottaken intoaccount.The fullrel- quired solution (Kolosnitsyn & Stanyukovich1984): ativistic solution to the problem is due to Kolosnitsyn & Stanyukovich(1984). F′(a)da Forthiscase,Euler’sequation,eq.(1)andthecontinuity φ+φ = c . (25) 0 equation, eq.(3) can be written: ± Z a 1−F2(a) p Formation of internal shock waves in bent jets 5 This equation gives the speed of sound as a function of the v azimuthalangle.Fromeqs.(22)-(23)itfollowsthattheradial andazimuthalvelocitiescanbeobtainedasafunctionofthe sameangleφ.Asaresult,alltheremaininghydrodynamical vφeˆφ α vreˆr variables can be found. The sign in eq.(25) can be chosen to be negative by measuring theangle φ in the appropriate direction and we will dothat in what follows. Let us consider now the case of an ultrarelativistic gas and integrate eq.(25) byparts, toobtain: c da φ χ φ+φ = arccosF(a)+c arccosF(a). (26) 0 a a2 Z 0 For the case of an ultrarelativistic gas, the speed of sound a is given by (Stanyuokovich 1960) a=√κ−1 c . In other Figure3. Relationbetweenthevelocityvectorv=vreˆr+vφeˆφ words,thisvelocityisconstantandsotheintegralineq.(26) andtheangleχ,asafunctionoftheazimuthal angle φ.χisthe is a Lebesgue integral. Since this integral is taken over a anglethatthevelocityvector makeswithcertainfixedaxiswith boundedandmeasurablefunctionoverasetofmeasurezero, originO. its valueis zero. Using eqs.(22)-(23) and eq.(26) the desired solution is c2dp/a2 combined with the first law of thermodynamics, it obtained (Kolosnitsyn & Stanyukovich1984; K¨onigl 1980): follows that dn/dφ<0. Also, using eqs.(22)-(23) it follows that dv/dφ da/dφand necessarily dv/dφ>0. ∝− v =csin √κ 1(φ+φ ) , (27) Ontheotherhand,theangleχthatthevelocityvector r 0 − vφ =√κ−1 c(cid:8)cos √κ−1(φ+(cid:9)φ0) , (28) mazaimkeustwhaitlhansogmleeφfixbeyd: axisisrelatedtothevelocityandthe for an ultrarelativistic equati(cid:8)on of state of the(cid:9)gas. For the non–relativistic case, in which c , eq.(25) →∞ χ=φ+arctan(v /v ) (33) gives for a polytropic gas with polytropic index κ: φ r asitshowninFig.3.Thus,sincetheφcomponentofEuler’s κ+1 dζ equation, eq.(1) implies that: φ+φ = , 0 − κ 1 1 ζ2 r − Z − ∂v γ2v ω ∂(ω/n) where p vr+ ∂φφ nφ +c2 ∂n =0, (cid:18) (cid:19) a κ+1 1 a2 2 −1/2 it follows that dχ/dφ= v2γ2ω/c2 −1dp/dφ. ζ 1 1 0 , − ≡ crκ−1( −(cid:18) − κ−1c2(cid:19) ) In other words, we ha(cid:0)ve proved(cid:1)that for the flow with whichweareconcerned,thefollowing inequalitiesaresatis- andso,therequiredsolutionis(Kolosnitsyn&Stanyukovich fied: 1984): dp/dφ<0, dn/dφ<0, dv/dφ>0, dχ/dφ>0. (34) κ+1 c φ+φ = arccos , (29) 0 rκ−1 (cid:18)c∗(cid:19) A flow with these properties is described as a rarefaction where the speed of sound a has been rewritten as c to be wave in non–relativistic fluid dynamics (Landau & Lifshitz consistent in the non–relativistic case. The critical velocity 1987) and we will use this name in what follows. of sound c∗ is given by (Landau & Lifshitz 1987): Another, very important property of this rarefaction wave is that the lines at constant φ intersect the stream- lines at the Mach angle, that is, they are characteristics. 2 c2∗ = κ+1c20. (30) Indeed, from Fig. 3, it follows that the angle α between the line φ = const and the velocity vector v is given by The value for the velocities can thus be calculated from sinα = v /v. Using eqs.(22)-(24) it follows that this rela- φ eq.(14) and eq.(23) with F(a)=1: tion can be written as eq.(12). Because all quantitiesin the problemarefunctionsofasinglevariable,theangleφ,itfol- κ+1 κ 1 lows that every hydrodynamical quantity is constant along vr = κ 1c∗sin κ−+1(φ−φ0), (31) thecharacteristics. r − r κ 1 vφ =c=c∗cos κ−+1(φ−φ0). (32) r 5 STEADY SIMPLE WAVES Some important inequalities must be satisfied for the flow under consideration. First of all, eq.(25) together with Let usnowconsider thetwo dimensional problem of steady eq.(5) and the first law of thermodynamics, d(ω/n) = plane parallel flow which then turns through an angle as Td(σ/n)+(1/n)dp (Landau & Lifshitz 1987), imply that it flows round a curved profile. A particular case of this dp/dφ < 0. Using this inequality and the fact that de = problem,whentheflowturnsthroughan angleisdescribed 6 S. Mendoza & M.S. Longair by Landau & Lifshitz (1987). For this particular situation L thePrandtl-Meyerflowisobviouslythesolutionandso,the A’ y A hydrodynamical quantities depend on a single variable, the H angle φ measured from a defined axis at the onset of the curvature. Because of this, all quantities can be expressed as functions of each other. Since this case is a particular solution to the general problem, it is natural to seek the K solutions of the equations of motion in which the quanti- φ1 ties p, n, v , v can be expressed as a function of each x y φ∗ P other.Evidentlythisimposesarestrictiononthesolutionof α1 θ φ∗−φ theequationsof motion since for twodimensional flow, any x O quantity depends on two coordinates, x and y, and so any chosenhydrodynamicalvariablecanbewrittenasafunction Figure 4. Supersonic uniform flow at the left of the diagram of any other two. bends around a curved profile OH. The Mach angle α1 is the Because of the fact that the flow is uniform at infinity, anglemadebythecharacteristics andthestreamlinesbeforethe where all quantities are constant, particularly the entropy, onsetofthecurvature.Thecharacteristicsmakeanangleφ1with andbecausetheflowissteady,theentropyisconstantalong the“continuation”oftherarefactionwaveformedattheonsetof a streamline. Thus, if there are no shock waves in the flow, thecurvatureandtheangleφismeasuredfromtheline0A’.The the entropy remains constant along the whole trajectory of curvature causes the characteristic lines to intersect eventually theflow and in what follows we will use this result. andthisoccursatpointKinthediagram,givingrisetoashock Inthiscase,Euler’sequation,eq.(2),andthecontinuity waverepresentedasthesegmentKL. equation, eq.(3), are respectively: at constant pressure: (∂y/∂x) . Since every hydrodynamic ∂v ∂v c2 ∂p p v x +v x = , quantity is assumed to be a function of the pressure, then x ∂x y ∂y −γω∂x in the previous set of equations it necessarily follows that v ∂vy +v ∂vy = c2 ∂p, ∂y/∂x is a function which depends only on the pressure, x ∂x y ∂y −γω∂y that is (∂y/∂x) =f (p).Therefore: p 1 ∂ ∂ (γv n)+ (γv n)=0. ∂x x ∂y y y=xf (p)+f (p). (35) 1 2 Rewriting these equations as Jacobians2 we obtain: Nofurthercalculationsareneededifweusethesolution forthecaseinwhichararefactionwaveisformedwhenflow ∂(v ,y) ∂(v ,x) c2 ∂(p,y) v x v x = , turns through an angle (Landau & Lifshitz 1987). This so- x ∂(x,y) − y ∂(x,y) −γω∂(x,y) lutionisgivenbytheresultsofsection4.Aswasmentioned ∂(v ,y) ∂(v ,x) c2 ∂(p,x) in that section, all hydrodynamical quantities are constant v y v y =+ , x ∂(x,y) − y ∂(x,y) γω∂(x,y) along the characteristic lines φ=const. The particular so- ∂(γv n,y) ∂(γv n,x) lution of the flow past an angle obviously corresponds to x y =0. thecase in which f (p)=0in eq.(35). The function f (p) is ∂(x,y) − ∂(x,y) 2 1 determined from theequations obtained in section 4. We now take the coordinate x and the pressure p as in- For a given constant value of the pressure p, eq.(35), dependent variables. To make this transformation we have gives a set of straight lines in the x–y plane. These lines tomultiplythepreviousset ofequationsby∂(x,y)/∂(x,p). intersectthestreamlinesattheMachangle.Thisoccursbe- Thismultiplication leavestheequationsthesame,butwith cause the lines y=xf (p) for the particular solution of the 1 the substitution ∂(x,y) ∂(x,p). Expanding this last re- flow through an angle have this property. In other words, → lation and because all quantities are now functions of the one family of characteristic surfaces correspond to a set of pressure p but not of x,it follows that: straight lines along which all quantities remain constant. However,forthegeneralcase,theselinesarenolongercon- ∂y dv c2 ∂y current. v v x = , y− x∂x dp γω∂x The properties of the flow described above are anal- (cid:18) (cid:19) ogous to the non–relativistic equivalent known as simple ∂y dv c2 ∂y v v y = , waves (Landau & Lifshitz 1987). In what follows we will y− x∂x dp −γω∂x (cid:18) (cid:19) usethis name to refer to such a flow. v v ∂y d(γn) +γn dvy ∂ydvx =0. Letusnowconstructthesolutionforasimplewaveonce y− x∂x dp dp − ∂x dp the curved profile is fixed. Consider the profile as shown in (cid:18) (cid:19) (cid:26) (cid:27) Fig.4.Planeparallel steadyflowstreamsinfromtheleftof Here we have taken ∂y/∂x to mean the derivative point O and flows around the curved profile. Since we as- sumethattheflow issupersonic, theeffect of thecurvature 2 TheJacobian∂(a,b)/∂(x,y)isdefinedas: startingatOiscommunicatedtotheflowonlydownstream of the characteristic OA generated at point O. The char- ∂(a,b) ∂a/∂x ∂a/∂y ∂(x,y) =det ∂b/∂x ∂b/∂y . acteristics to the left of OA, region 1, are all parallel and (cid:20) (cid:21) intersect thex axis at theMach angle α given by eq.(12): 1 Formation of internal shock waves in bent jets 7 1 (v1/c)2 a v2 =c2 1−(2−κ)cos2√κ−1φ , (43) sinα1 = q1− (a/c)2 v1, (36) θ=φ∗−φ−α(cid:8), (cid:9) (44) − =φ∗ φ arctan √κ 1cot√κ 1φ , q − − − − where the velocity v1 is the velocity of the flow to the left p=p0(2 κ)−κ/2(κ−(cid:8)1)cos−κ/(κ−1)√κ 1(cid:9)φ. (45) of thecharacteristic OA.In eqs.(25)-(32) theangle φ of the − − characteristics is measured with respect to some straight Since the angle φ∗ φ is the angle between the char- − line in the x–y plane. As a result, we can choose for those acteristics andthex axis, it follows that theline describing equations the constant of integration φ 0. This means thecharacteristics is: 0 ≡ that the line from which the angle φ is measured has been choseninaratherspecialway.Inordertofindthelinewhich y=xtan(φ∗ φ)+G(φ). (46) is the characteristic for φ = 0, let us proceed as follows. − When φ = 0 and the gas is ultrarelativistic, eqs.(27)-(28) Thefunction G(φ)isobtainedfrom thefollowing argu- show that the velocity v = a and for the non–relativistic mentsforagivenprofileofthecurvature(Landau&Lifshitz case,itfollowsfromeqs.(31)-(32)thatthevelocitytakesthe 1987). If the equation describing the shape of the profile is value v=c. In both cases this means that the line φ=0 given by the points X and Y where Y =Y(X), thevelocity corresponds to the point at which the flow has reached the of thegas is tangential to this surface, and so: value of the local velocity of sound. This, however, is not possible since we are assuming that the flow is supersonic dY tanθ= . (47) everywhere.Nevertheless,iftherarefactionwaveisassumed dX to extend formally into the region to the left of OA, we Now,theequationofthelinethroughthepoint(X,Y)which canusetheserelationsandthenthecharacteristiclinemust makes an angle φ∗ φ with thex axis is: correspond to a valueof φ given by: − y Y =(x X)tan(φ∗ φ). (48) κ+1 c − − − φ = arccos 1 , (37) 1 rκ−1 (cid:18)c∗(cid:19) Eq.(48) is thesame as eq.(46) if we set: for a non–relativistic gas according to eq.(29), and G(φ)=Y Xtan(φ∗ φ). (49) − − c 1 (v /c)2 If we start from a given profile Y = Y(X) then, using 1 φ = arccos − , (38) 1 a p1 (a/c)2 eq.(47) we can find the parametric set of equations: X = − X(θ) and Y =Y(θ).Substitutionofθ=θ(φ)fromeq.(41)or for the ultrarelativistic case apccording to eq.(21), eq.(24) eq.(44) depending on whether the gas is non–relativistic or and eq.(26). The angle between the characteristics and ultrarelativistic, we find X=X(φ) and Y =Y(φ). Substitu- the x axis is then given by: φ∗ φ, where φ∗ = α1 + tion of this in eq.(49) gives therequired function G(φ). − φ , and the angle α is the Mach angle in region 1. The If the shape of the surface around which the gas flows 1 1 x and yvelocitycomponentsintermsoftheazimuthalangle is convex, the angle θ that the velocity vector makes with θ are given by: thexaxisdecreasesdownstream.Theangleφ φ∗ between − the characteristics leaving the surface and the x axis also decreases monotonically. In other words, characteristics for vx =vcosθ, vy =vsinθ, (39) this kind of flow do not intersect resulting in a continuous and rarefied flow. and the values for the magnitude of the velocity, the angle On the other hand, if the shape of the surface is con- θ and thepressure are given by: cave as shown in Fig. 4, the angle θ increases monotoni- cally and so does the angle the characteristics make with the x axis. This means that there must exist a region in 2 κ 1 v2 =c2∗ 1+ sin2 − φ , (40) the flow in which characteristics intersect. The value of the ( κ−1 rκ+1 ) hydrodynamical quantities is constant for every character- θ=φ∗ φ α, istic line. This constant however changes for different non– − − parallel characteristics. In other words, at the point of in- κ 1 κ 1 (41) =φ∗ φ arctan − cot − φ , tersectiondifferenthydrodynamicalquantities,forexample, − − (rκ+1 rκ+1 ) the pressure, are multivalued. This situation cannot occur and results in the formation of a shock wave. This shock p=p∗cos2κ/(κ−1) κ−1φ, (42) wave cannot be calculated from the above considerations, κ+1 r sincetheywerebasedontheassumptionthattheflowhadno for a non–relativistic gas according to eqs.(29)-(32) and us- discontinuitiesatallbecausetheentropywasassumedtobe ing the fact that the Poisson adiabatic for a polytropic gas constant.However,thepointatwhichtheshockwavestarts, meansthat:pc−2κ/(κ−1)=const.Inthecaseofan ultrarela- thatispoint Kin Fig.4, canbecalculated from thefollow- tivisticgas,eqs.(26)-(28)togetherwithBernoulli’sequation ing considerations. We can work out the inclination of the and the fact that the enthalpydensity ω=κp/(κ 1) give: characteristics φ as a function of the coordinates x and y. − 8 S. Mendoza & M.S. Longair Thisfunctionφ(x,y)becomesmultivaluedwhenthesecoor- where dinatesexceedcertainfixedvalues,sayx and y .Atafixed 0 0 x=x0 thecurvegiving thevalue of φ as afunction of y be- (κ 1)/(κ+1) if thegas is non–relativistic, µ − ocorm(∂esy/m∂uφl)tiv=al0u.edIt. Tisheavtidise,ntthtehadteraitvatthiveep(o∂inφt/∂yy=)xy=t∞he, ≡(p√κ−1 for an ultrarelativistic gas. x 0 (52) curveφ(y)mustlieinbothsidesoftheverticaltangent,else thefunctionφ(y)wouldalreadybemultivalued.Thismeans As was mentioned above, if the jet is sufficiently nar- thatthepoint(x ,y )cannotbeamaximum,oraminimum 0 0 row, it appears that it can safely avoid the formation of an of the function φ(y) but it has to be an inflection point. In internal shock. However, differentiation of eq.(51) with re- otherwords,thecoordinatesofpointKinFig.4canbecal- specttotheanglethevelocityvectormakeswiththexaxis, culatedfrom thesetofequations(Landau&Lifshitz1987): that is the deflection angle θ, implies that: (cid:18)∂∂φy(cid:19)x =0, (cid:18)∂∂φ2y2(cid:19)x =0. (50) ddαθ = 21(cid:18)Γ−1+ MΓ2+−11(cid:19), (53) When the profile is concave, the streamlines that pass with above thepoint O in Fig. 4 pass through a shock wave and the simple wave no longer exist. Streamlines that pass be- κ if thegas is non–relativistic, low this point seem to be safe from destruction. However, Γ (54) theperturbingeffectfromtheshockwaveKLinfluencesthis ≡(κ/(2 κ) for an ultrarelativistic gas. − regionalso,andsoitisnotpossibletodescribetheflowthere as asimple wave.Nevertheless,since theflow issupersonic, The Mach number M is given by eq.(13) and eq.(12) re- the perturbing effect of the shock wave is only communi- spectively. As the Mach number M 1, then the deriva- → cated downstream. This means that the region to the left tive dα/dθ . This means that the rate of change of the →∞ ofthecharacteristicPK,whichcorrespondstotheotherset Machanglewithrespecttothedeflectionanglegrowswith- of characteristics emanating from point P, does not notice out limit as the Mach number decreases and reaches unity. thepresenceoftheshockwave.Inotherwords,thesolution Onabend,theMach numberdecreases and care isneeded, mentionedabove,inwhichasimplewaveisformedarounda orelsethecharacteristicswillintersectattheendofthecur- concaveprofileisonlyvalid totheleft ofthesegmentPKL. vature.Thereis only onespecial shape for which thiseffect is bypassed and this occurs when the increase of θ matches exactlywith theincrease of α(Courant &Friedrichs1976), butofcourse,thisisquiteauniquecase.Itappearshowever, thatwhateverthethicknessofthejetitcannotbebentmore than the point at which dα/dθ exceeds the rate of change 6 CURVED JETS of θ with respect to the bending angle θ. In other words, dα/dθ 6 dθ/dθ =1 (Icke 1991). From this last inequality Let usnow usetheresultsobtained in sections 4and 5 and and eq.(53) a value of theMach numbercan beobtained: apply them to the case of jets that are curved due to any mechanism, for example the interaction of the jet with a cloudaswasdiscussedbyusinapreviouspaper(Mendoza& 2 M = . (55) Longair2001),orduetotherampressureoftheintergalactic ⋆ √3 Γ gas as in the core of radio trail sources. − The greatest danger occurs when the jet forms inter- If theMach numberin the jet decreases in such a way that nal shock waves. This is because, after a shock, the normal thevalue M⋆ is reached, then a terminal shock is produced velocity component of the flow to the surface of the shock andthejetstructureislikelytobedisrupted.Itisimportant becomes subsonic and the jet flares outward. Nevertheless, tonotethatthisterminalshockisweaksinceM&1andso, aswehaveseen insection5,theshockthatformswhengas itmightnotbetoodisruptive.Nevertheless,thismonotonic flows around a curved profile (such as a bent jet due to ex- decrease of the Mach number makes the jet flare outwards, ternalpressuregradients)doesnotstartfromtheboundary even if theterminal shock is weak. of thejet. It actually forms at an intermediate point to the Let us now calculate an upper limit for the maximum flow.Inotherwords,itispossiblethat,ifajetdoesnotbend deflection angle for which jets do not produce terminal toomuchtheintersectionofthecharacteristiclinesactually shocks.Inordertodoso,werewriteeq.(51)inthefollowing occurs outside the jet and the flow can curve without the way: production of internal shocks. Aswehaveseeninsection5theMachangleoftheflow, 1 1 relativisticandnon-relativistic,doesnotremainconstantin θ=arcsin + arctan µ M2 1 φ∗ (56) − M µ − − the bend (see for example eq.(41) and eq.(44)). The Mach n p o numbermonotonically decreases as thebend proceeds. To eliminate the constant φ∗ from all our relations, we can Eq.(41) and eq.(44) imply that: compare the angle θ evaluated at the minimum possible value of the Mach angle M =M with θ evaluated at its ⋆ maximum value M = . In other words, the angle θ max tanα=−µcotµ(α+θ−φ∗). (51) definedas: ∞ Formation of internal shock waves in bent jets 9 y A dθ α D R d + α dθ α x O Figure 5. Sketch of a curved jet of radius D that develops a shockatthebeginningofthecurvature.Thecurveisassumedto be acirclewith radiusR, sothat this approximation is validon asufficiently small regionabout the onset of the curvature. The MachangleofthejetisαattheleftofthecharacteristicOAthat emanates fromthepointwherethebendingstarts. θ θ(M=M ) θ(M= ) max ⋆ ≡ − ∞ ◦ 74.21 for thenon–relativistic case, = (57) ◦ (47.94 for an ultrarelativistic gas. are upper limits to the deflection angle. Jets which bend morethanthislimitingvalueθmaxdevelopaterminalshock. Figure 6. Plot of the maximum ratio D/R as a function of the This upper limit however, does not mean that the jet difference θ θ⋆ where θ is the deflection angle and θ⋆ is the − isimmunefromdevelopinganinternalshockifitisbentby maximumbendingangleajetcanhaveinordernottoproducea a smaller angle. Indeed, let us suppose that the jet bends terminalshock.Theplotreferstothepointsforwhichashockat and that the curvature it follows is a segment of a circle as thebeginningofthecurvature(whichwasassumedtobeacircle) hasdeveloped.Jetswithparameterswhichliebelowthecurvein it isshown in Fig. 5.Thecircle can beconsidered tobethe anycasedonotdevelopanyinternalshocksatallforthispartic- circleofcurvatureofthejet’strajectoryformedattheonset ularcirculartrajectory.Theplotatthetopwascalculated using of thebending.According tothefigure,theequation of the theresultsinwhichthegas isnon–relativisticanditspolytropic characteristic OA that emanates from the point O, where index is 5/3. The plot at the bottom was made by considering thecurvaturestarts is: the gas to be ultrarelativistic and relativistic effects in the bulk motionoftheflowweretakenintoaccount. Forthissecondplot, thepolytropicindexwasassumedtobe4/3.Thenumbersineach y=xtanα (58) plotcorrespondtothevaluesoftheMachnumberintheflow. Oncetheflow hascurveddθ degrees, thecharacteristic at this point is given by: between thedeflection angle θ andthemaximumdeflection angle θ θ(M ), as is shown in Fig. 6. ⋆ ⋆ y=(x Rdθ)tan(α+dα+dφ) Jets≡forwhichtheratioD/Rliesbelowthecurvedonot − xtanα+x(dα+dθ)/cos2α Rdθtanα, develop any shocks at all. For example, consider a jet with ≈ − a given Mach number for which its ratio D/R is given. As where R is the radius of curvature of the circular trajec- the width of the jet increases (or the radius of curvatureof tory. The intersection of this characteristic and that given the profile decreases), it comes a point in which a shock at by eq.(58) occurs when they coordinate has a value: theonsetofthecurvatureisproduced.Inthesameway,jets with a fixed ratio D/R for a given Mach numberwhich are Rsin2α initiallystable,sothattheyliebelowthecurve,candevelop D= . 1+dα/dθ a shock at the beginning of thecurvatureby increasing the bendingangle of thecurve. Substitution of eq.(53) gives (Icke1991): D = 2 M2 1 M−4. (59) R Γ+1 − 7 DISCUSSION (cid:0) (cid:1) Using eq.(56) and eq.(59) it is possible to make a plot The relativistic Mach angle is smaller for a given value of inwhichtwozonesseparatethecasesforjetswhichdevelop thevelocityoftheflowthanitsnon–relativisticcounterpart shocks at the onset of the curvature, and the ones that do aswasprovedinsection3.Thisfactisextremelyimportant not. Indeed,we can plot theratio of thewidth of thejet D whenanalysingthepossibilityoftheintersectionofdifferent toradiusofcurvatureRasafunctionofthedifferenceθ θ characteristics in a bent jet. This intersection is what gives ⋆ − 10 S. Mendoza & M.S. Longair risetothecreationofshockwaves.Forarelativisticflow,the bulkmotionisnon–relativistic,theshockisgeneratedwhen ◦ characteristics,whichmakeanangleequaltotheMachangle the bending angle is more than 135 . Jets with a poly- to the streamlines, are always beamed in the direction of tropic index of 5/3 that move no∼n–relativistically generate theflow.Thus,whenajetstartstobend,thepossibilitiesof ashockifthebendingangleexceedsthevalue 50◦.These ∼ intersection between some characteristic line in the curved anglesareonlyupperlimitsandthepreciseconditionsunder jetandtheonesbeforeitcurves,becomemoreprobablethan which a shock is produced have to be calculated individu- for their non–relativistic counterpart. ally. However using the diagrams presented in Fig. 6 and This differenceresultsin asevereoverestimation ofthe that presented by(Icke1991) it is possible tosee if a shock maximumbendingangleθ whenanon–relativistictreat- is produced at theonset of thecurvature. max mentismadetotheproblem.Forexample,Icke(1991)used All radio sources in which bendings of radio jets have the non–relativistic analysis in the discussion of the gener- been observed appear to satisfy the upper limits discussed ation of internal shocks due to bending of jets. Using the above. So, it seems that jets are perhaps unstable if an in- non–relativisticequationsdescribedabove,butwithapoly- ternal shock is generated in a curvature. However, various tropicindex κ=4/3, thevaluefor themaximumdeflection observationsandtheoreticalworkingalacticandextragalac- angleisθ =134.16◦.Thisismuchgreaterthanthevalue ticsources(seeforexampleCantoetal.1989; Falle&Raga max of θ =74.21◦ obtained with a full relativistic treatment. 1993,1995;Komissarov&Falle1998,andreferenceswithin) max TheanalysismadebyIcke(1991)isimportantforjetsin showthatinternalshockswithinajetareagoodmechanism whichthemicroscopicmotionoftheflowinsidethejetisrel- that,undercertain circumstances, can collimate thejets. ativistic,butthebulkmotionoftheflowisnon–relativistic. The question of whether an internal shock will lead to Radiotrailsources(Begelmanetal.1984)showconsid- disruptionofthejetisunknownandstillamatterofdebate. erable bending of their jets with deflection angles of about Weaim togivetotheproblemananswerinafuturepaper. ◦ 90 in many cases. Since the bending is produced by the proper motion of the host galaxy with respect to the in- tergalactic medium, the deflection angle cannot be greater than 90◦. The results presented in eq.(57) show that jets 8 ACKNOWLEDGEMENTS which have a relativistic equation of state and a bulk rela- SM would like to thank Paul Alexanderfor providing ideas tivistic motion of thegas within its jet, cannot bedeflected ◦ totheproblemdiscussedinthispaperwhilegivingaseminar morethan 50 .Sincethedeflectionsofradiotrailsources ∼ at theCavendish Laboratory in Cambridge. Healso thanks are greater than this value, this result would imply that support granted by the Cavendish Laboratory and the Di- most radio trail sources should generate shocks at the end recci´on General de Asuntos del Personal Acad´emico at the of their curvature. However, observations (see for example Universidad Nacional Aut´onomade M´exico. Eileketal.1984;O’Dea1985;deYoung1991,andreferences within) show that the velocity of the material of the jets . 0.2–0.3c. Therefore, the bulk motion of the flow is non– relativistic, despitethefactthatthegasinsidethejethasa References relativistic equation of state.Aswesaw before, thisimplies that the valueof the maximum angle is θ =134.16◦. Begelman M., Blandford R., Rees M., 1984, Rev. Mod. max In other words, these type of jets develop a terminal Phys.,56, 255 shock if their jets bend more than 135◦. This seems to Best P.N.,Longair M.S., Rottgering H.J.A., 1997, MN- ∼ be the reason why radio trail sources are able to bend so RAS,286, 785 muchwithoutresultinginaninternalshockwavethatcould Canto J., Raga A. C., Binette L., 1989, Revista Mexicana potentially cause disruption of its structure. deAstronomia y Astrofisica, 17, 65 Inapreviouspaper(Mendoza&Longair2001),wedis- Chiu H.H., 1973, Physics of Fluids, 16, 825 cussedthepossibilityofabentjetintheradiogalaxy3C34 Courant R., Friedrichs K., 1976, Supersonic Flow and using the observations of Best et al. (1997). According to Shock Waves. Applied mathematical sciences; vol.21, In- these observations, the radio source lies more or less in the terscience Publishers plane of the sky, and so, if the western radio jet is curved, de YoungD. S., 1991, ApJ, 371, 69 this must be of the order of 10◦. The value θ 10◦ is well Eilek J. A., Burns J. O., O’Dea C. P., Owen F. N., 1984, belowtheupperlimitof 50◦ calculatedineq∼.(57),sothat ApJ,278, 37 ∼ no terminal shock would be produced by the deflection of Falle S.A. E. G., Raga A.C., 1993, MNRAS,261, 573 thejet.Fromthelower plotof Fig.6andbecausetheangle Falle S.A. E. G., Raga A.C., 1995, MNRAS,272, 785 θ 1 for a high relativistic flow, it follows that if the tra- IckeV.,1991,inHugesP.A.,ed.,BeamsandJetsinAstro- ⋆ ≪ jectory of the jet in 3C 34 is circular, then in order not to physics. From nucleus to hotspot. Cambridge University produceaninternalshockattheonsetofthecurvature,the Press, pp 232–277 ratio D/R has tobe less than 0.08. KolosnitsynN.I.,StanyukovichK.P.,1984,PMMJournal ∼ In the analysis made above, we have calculated how of Applied Mathematics and Mechanics, 48, 96 shocks can begenerated inside abent jet. These shocks are Komissarov S. S., Falle S. A. E. G., 1998, MNRAS, 297, specialinthesensethattheydonotreachthesurfacebound- 1087 ary of the jet. Instead they are generated away from the K¨onigl A.,1980, Physics of Fluids, 23, 1083 walls of the jet. For jets with an ultrarelativistic equation Landau L., Lifshitz E., 1987, Fluid Mechanics, 2nd edn. of state that possess a relativistic bulk motion, a shock is Vol.6 of Course of Theoretical Physics, Pergamon ◦ internally generated if they bend more than 50 . If their Landau L., Lifshitz E., 1994, The Classical Theory of ∼