The physical structure of radio galaxies explored with three-dimensional simulations

7.7 MB·

by  Justin Donohoe

MNRAS000,1–17(2014) Preprint1February2016 CompiledusingMNRASLATEXstylefilev3.0
The physical structure of radio galaxies explored with
three-dimensional simulations
Justin Donohoe 1? & Michael D. Smith 1†
1CentreforAstrophysics&PlanetaryScience,TheUniversityofKent,Canterbury,KentCT27NH,U.K.
Accepted.....Received.....;inoriginalform.....
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1
0 ABSTRACT
2
n We present a large systematic study of hydrodynamic simulations of supersonic adiabatic
a jets in three dimensions to provide a definitive set of results on exploring jet density, Mach
J numberandprecessionangleasvariables.Werestricttheset-uptonon-relativisticpressure-

9 equilibrium flows into a homogeneous environment. We first focus on the distribution and
2 evolution of physical parameters associated with radio galaxies. We find that the jet density
has limited influence on the structure for a given jet Mach number. The speed of advance

] variesbyasmallfactorforjetdensitiesbetween0.1and0.0001oftheambientdensitywhile
A
thecocoonandcavityevolutionchangefromnarrowpressurebalancedtowideover-pressure
G astheratiofalls.Wealsofindthatthefractionofenergytransferredtotheambientmedium
. increaseswithdecreasingjet-ambientdensityratio,reaching≈80%.Thisenergyispredom-
h
inantly in thermal energy with almost all the remainder in ambient kinetic form. The total
p
energy remaining in the lobe is typically under 5%. We find that radio galaxies with wide
-
o transversecocoonscanbegeneratedthroughslowprecessionatlowMachnumbers.Weex-
r plore a slow precession model in which the jet direction changes very slowly relative to the
t
s jetflowdynamicaltime.Thisrevealstwoseparatedbowshockspropagatingintotheambient
a medium,oneassociatedwiththeentirelobeexpansionandtheotherwiththeimmediateim-
[
pactzone.Thelobesgeneratedaregenerallyconsistentwithobservations,displayingstraight

1 jetsbutasymmetriclobes.
v
2 Keywords: hydrodynamics–radiogalaxies
5
0
8
0
1 INTRODUCTION justified the use of two-dimensional numerical simulations (e.g.
.
1 Carvalho&O’Dea2002)althoughitisunlikelythatsuchsimula-
0 Radiogalaxiesaresomeofthelargest(Willisetal.1974;Malarecki tionscanaccuratelyreproducetheconsequencesofturbulenceand
6 etal.2013)andmostpowerfulstructures(O’Deaetal.2009)that
fluid instabilities (Carvalho & O’Dea 2002) even in this context
1 weobserveintheuniverse.Theirpowerisderivedfromprocesses
(Krause & Camenzind 2001). Moreover, the first such adiabatic
: in the vicinity of supermassive black holes which lie at the heart
v hydrodynamic study demonstrated that the shape of the cocoon
of active galactic nuclei (Rees et al. 1982; Rawlings & Saunders
i and hot spots continuously change structure (Smith et al. 1985)
X 1991). Their power is channelled out through jets containing an
asthepressureoftheexpelledgasfeedsbackontotheapproach-
r unknownmixtureofhigh-energyplasma,relativisticparticlesand
ing jet (Norman et al. 1982). Nevertheless, Hardcastle & Krause
a magneticfield,asdeducedfromradiohotspotsandlobeswherea
(2013)demonstratedthattwo-dimensionalcalculationscanprovide
smallfractionoftheenergyisconvertedintosynchrotronradiation
theframeworkinwhichtodiscusstheoverallenergyandpressure
(cf.Smith2012).Theyarealsocosmologicallyimportantasevolv-
distributions within a non-uniform environment corresponding to
ingbeaconsathighredshift(Malareckietal.2013),asregulators
propagationthroughagalacticmedium.Theassociatedhighspa-
ofgalaxygrowthandwithimplicationsforprocessesinvolvedwith
tialresolutionthatcanbeachievedintwodimensionsalsopermits
blackholephysics(Smith2012).
thesimulationstoberuntogreaterjetlengths.
There are many types of radio galaxies and much effort has
goneintotheclassificationsystemandassociateddynamicalpro-
cesses. Classical examples display approximate jet-axial mirror Asystematicstudyinthreedimensionsathighresolutionmay
symmetry (in addition to twin-lobe mirror symmetry) which has helpverifyorcontradictthetrendsfoundintwo-dimensionalsim-
ulations.Weexpectsymmetrytobebrokenbyjetprecession(Ek-
ersetal.1978)orreorientation(Hodges-Kluck&Reynolds2012),
? E-mail:[e-mail], interstellarpressuregradients(Smith&Norman1981),intraclus-
† E-mail:[e-mail] tergasmotions(Begelmanetal.1979),inter-clustergasdynamics
(cid:13)c 2014TheAuthors
2 J.Donohoe&M.D.Smith
(Lokenetal.1995)orsimplythroughinstabilityattheimpactin- fiedambientmediumwithinfiveruns.TheyfoundthatlowMach
terface. numberjetsaresomewhatmoreeasilydisrupted.
However, with only a few three dimensional studies per- Thirdly,thesmoothre-alignment,wobblingorprecessingof
formed,manypropertieshaveyettobesystematicallyinvestigated. thejetdirectionhasnotbeengreatlyexploredinthecontextofgi-
Thefirstattemptsweredirectedatjetbending(Balsara&Norman antradiogalaxies.Whilecrucialtothestructuresformedbyheavy
1992;Lokenetal.1995).O’Neilletal.(2005)foundthatenergyis (ballistic) jets, the short jet dynamical time in comparison to the
transferredefficientlytotheambientmediumwithapproximately lobe dynamical time for light jets implies that precession-related
halfofthejetenergybeingconvertedtothermalenergyintheam- phenomena may be more difficult to identify (Gong et al. 2011).
bientmedium.Mendygraletal.(2012)analysedthree-dimensional Thus,ratherthangeneratingahelicalstructure,weshouldobserve
magnetohydrodynamical simulations of intermittent jets in a tur- curvedorarcuatelobesasifapaintbrushhassweptacrossacan-
bulentintraclustermedium.Thisweatherdistortsthejetsandlobes vas (Ekers et al. 1978) or a distinct X-shaped structure (Cheung
intomorphologiessimilartowide-angletailradiosourceswhilethe 2007). A combination of re-alignment and intracluster wind may
intermittencyleadstofeaturessimilartothoseindouble-doublera- benecessary(Hodges-Kluck&Reynolds2011).
diogalaxies.Finally,Hardcastle&Krause(2014)alsoconsidera Welimitthepresentstudytothatofanon-relativisticadiabatic
clusterenvironment,takingalowMachnumberjet.Theydemon- flowofauniformsupersonicjet.Thejetisinjectedintoauniform
stratethepolarisationpropertiesandshowamajordifferencebe- stationaryambientmedium.Itbeginsperfectlycollimatedfroma
tweensynchrotronandinverse-Comptonemissionimages. nozzlewithasimulatedcircularcross-section.Wedonotinclude
Inthispaper,weareinterestedinthedynamicalstateofthe pulsations,shearoranorbitingnozzle.Inaddition,magneticand
injected jets and the corresponding physical structures generated gravitationalforcesarealsoignored.
ratherthantheinfluenceoftheambientmedium.Inasubsequent
paper, we investigate the observational predictions including the
associatedX-raycavityandtheroleoftheangleoforientationto
thelineofsight. 2 METHOD
Inthisprogramme,theultimategoalistoprovideasetofra-
dioandX-rayimagesofradiogalaxiesatvariousorientationsand 2.1 TheCodes
distancesfromtheobserver.Howradiogalaxiesappearathighred-
Twoefficientcodesforcomputationalfluiddynamics,suitablefor
shiftandclosetothelineofsightwillprovideameansofinterpreta-
simulationsurveys,aretestedandemployed.ZEUS-3Disagrid-
tionofdatafromthenextgenerationoftelescopes.Canwededuce
basedsecond-orderEulerianfinitedifferencecode(Stone&Nor-
the causes of observed features, and constrain the mass, momen-
man 1992) that uses Van Leer advection and consistent transport
tumandenergybudgets?Toachievethis,wefirstrequirethede-
ofthemagneticfield.WithvonNeumannandRichtmyerartificial
pendenceontheintrinsicflowparameterstobeestablishedbefore
viscosityandanupwindedscheme,itisidealforproblemsinvolv-
proceedinginafollow-upworktopresentthemorphologies.
ingsupersonicflowandisversatile,robustandwell-tested(Clarke
Fundamental injection parameters which have yet to be sys-
2010).Althoughhigherordercodesarepotentiallymoreaccurate,
tematically studied within three dimensional simulations include
thehighspeedofthealgorithmsmeansthatlargeproblemscanbe
thedensity,precessionangleandtheMachnumber.Thefirstissue
solvedathighresolution.
weconfronthereisthedependenceonthejet-ambientdensityratio,
Weemployversion3.5ofZEUS-3D(dzeus35)whichisfreely
η.Thedensityratioisthoughttobecriticaltotheradiogalaxymor-
available for use by the scientific community and can be down-
phologywithlowerdensityratiosgeneratingwiderlobesinaself-
loadedfromtheInstituteofComputationalAstrophysics(ICA)at
similaranalysiswhichassumesthelobesmaintainahighpressure St.Mary’sUniversity,NovaScotia,Canada1.
incomparisontotheambientmedium.However,itisnotsupported
PLUTO is similarly grid-based but incorporates modern
bythetwodimensionalsimulationsofKrause(2003).Inaddition,
Godunov-type shock-capturing schemes (Mignone et al. 2007).
thereflectionboundaryconditionalongtheplanecontainingthejet
After comparing the results of numerous options, we chose a
nozzle is crucial since the cocoon material is trapped on the grid
fast linear interpolation Hancock time-stepping (denoted HLLC)
andawidecavitybuildsup;whereaswithanoutflowcondition,the scheme2.
cocoonremainsquitecylindricalanduniformintime.Hardcastle
Forthesimulationsthatinvolveonlyadiabatichydrodynamics
&Krause(2013)discusstherelevanceoftheself-similarsolutions
(HD),sixpropertiesarerecordedtofile:thedensity,ρ,pressure,p,
(Begelman & Cioffi 1989), as first raised by Scheuer (1974), in
threevelocitycomponents,v ,v andv ,andamass-weightedjet
x y z
whichthejet-suppliedcocoonmaintainsahighpressureandcon-
tracer,χ.
cludes that it can be dismissed on observational grounds. This is
because,afterabriefinitialstage,thecocoonmaterialhastimeto
expandandreadilyreachesapproximatepressurebalancewiththe
ambientmedium. 2.2 Scaling
Secondly,thejetMachnumberisacrucialparameterasfirst
Inordertoadequatelycoverwideprecessingradiosources,webase
realisedbyNormanetal.(1983)andstudiedbyBicknell(1985). the simulations on a D3 = 30x30x30 unit volume where the jet
ThelatterworkexploredthepossibilitythattheMachnumberde-
radius,r ,issettooneunit.Theambientmediumistakentobe
jet
termines whether a radio source is of type FR-I or FR-II as de-
uniform with a sound speed, c , of one unit. This sets the time
amb
finedbyFanaroff&Riley(1974).IfhotspotsarepresentinFR-Is,
scale,r /c ,alsotooneunit.Givenanambientdensityofone
jet amb
theyoccurclosetothehostgalaxy.whilstFR-IIshaveprominent
hotspotsthatoccurfurtherawayfromthehostgalaxyFR-Iradio
galaxiestendtohavevisiblejetswhilstFR-IIjetstendtobefaint
idatalldetectable.Holdingthejetdensityconstant,O’Neilletal. 1 http://www.ica.smu.ca/zeus3d/
(2005)lookedattheinfluenceofjetMachnumberandthestrati- 2 http://plutocode.ph.unito.it/
MNRAS000,1–17(2014)
Thephysicalstructureofradiogalaxiesexploredwiththree-dimensionalsimulations 3
Table1.Theinitialconditions:bothnon-dimensionalparametersandtheirexamplescaledinterpretationstakingajetwithadensityof1%oftheambient
density.Theparameternp,amb,isthehydrogennuclei(freeproton)densityintheambientmedium.
- unit Compact Giant
- value Source Source
D 30 75.0kpc 750kpc
rjet 1 2.5kpc 25kpc
Mjet 6.0 6.0 6.0
ρamb 1 2.3410−26 gcm−3 9.3710−28 gcm−3
camb 1 6.72107 cms−1 8.23107 cms−1
np,amb n/a 1.010−2 cm−3 4.010−4 cm−3
Tamb n/a 2.0107 K 3.0107 K
uamb 0.9 9.5310−11 ergcm−3 5.7210−12 ergcm−3
pamb 0.6 6.3510−11dynecm−2 3.8110−12dynecm−2
pjet/pamb 1.00 1.00 1.00
ρjet/ρamb 0.01 0.01 0.01
vjet 60.0 4.03109cms−1 4.94109cms−1
M˙jet 1.88 2.80M(cid:12)yr−1 13.7M(cid:12)yr−1
Pram 113 7.121035dyne 4.271036dyne
Ljet 3,562 1.511045ergs−1 1.111046ergs−1
to=rjet/camb 1 3.64Myr 29.7Myr
tprecession=2π/ω 4 14.64Myr 118.8Myr
tlobe-dynamic=D/U 5 18.18Myr 148.4Myr
tjet-dynamic=D/vjet 0.5 1.82Myr 14.8Myr
unitand Thevelocitycomponentsarethen
c =rγ.pamb, (1) vy =vjet.sin(θ).sin(ωl.t)
amb ρ v =v .sin(θ).cos(ω.t) (4)
amb z jet l
p
yields a pressure pamb = 0.6 and internal energy per unit volume vx = vjet2−(vy2+vz2)
uamb=0.9forthespecificheatratioofγ=5/3since Theprecessionisaddedatthenozzleofthejetasitentersthe
inflowboundary.AsseeninEquation(4),theangleofprecessionis
p =(γ−1)u . (2)
amb amb θ.Tocoverawiderangeofscenarios,precessionanglesof0.25◦,
1◦,5◦,10◦and20◦areused.Thesmallestprecessionof0.25◦isap-
We assume adiabatic media so that all quantities can be scaled.
plied to break up any numerical fluctuation when we simulate a
We may thus consider whether our simulations represent both a
straight jet. By default precession of 1◦, 10◦and 20◦are used for
quite compact radio galaxy and a giant source. For the example
alldensities;theprecessionof0.25◦and5◦areforamorein-depth
parametersdetailedinTable1,thescalesizeanddynamicaltime
lookfortheparticularcaseofthedensityratioof0.1.Therateat
scalesare75kpcand18Myr(Compact)and750kpcand148Myr
whichthejetsprecessissettoadefaultofonceperfourtimeunits.
(Giant)
The ambient medium parameters are specified through the
numberdensityandtemperature,bothconstrainedfromX-raydata. 2.3 Mass,momentumandenergy
Wespecifythehydrogennucleidensityhereandaddon10%ofhe-
liumnuclei,assumingbothspeciestobefullyionised. Themassfluxinjectedintothesystemis
Inthiswork,wedumpthedataevery0.1units,butthisisal- M˙ =ρ ·v ·A, (5)
jet jet x
teredaccordingtotherateofpropagationofthejetacrossthegrid.
Thistranslatesto1dumpfileroughlyevery0.36Myrs(Compact where ρjet is the input jet density, vx is the jet velocity normal
Source)and3.0Myr(GiantSource). to the boundary, which is obtained from Equation (3), and A =
Amajoraimistoinvestigatetheinfluenceofjetprecessionon (1−µ)πrj2et isthejetarea.Here,µrepresentsasmalladjustment
themorphologyoftheradiogalaxy.Theprecessionisaddedinto sincethenumericalnozzleprofileisanapproximationtoacircle.
thesystembysplittingthevelocityupintocomponents.Toachieve Therewillbealinearincreasewithtimeforallthesimulationswith
this,thefulljetspeedistakentobe reflective outflow boundaries because the mass influx is constant
throughoutthepresentsetofsimulations.Withanoutflowbound-
r r
p ρ arycondition,thecocoonbackflowremovesmassfromthegrid,
v =M ·c =M · jet · amb ·c , (3)
jet jet jet jet p ρ amb asdiscussedbelow.
amb jet
In order to test and calibrate the results, we introduce the
wherep ,ρ andM arethepressure,densityandMachnum- SteadyPropagationModel.Themodelassumesthatthejetploughs
jet jet jet
berofthejet,respectively.ThejetspeediscalculatedusingMach into the ambient medium, advancing a high-pressure hot spot at
numbersof2,4,6,8,12,24and48. a constant speed. To understand how the jet and lobe propagate
MNRAS000,1–17(2014)
4 J.Donohoe&M.D.Smith
Table2.Breakdownofsimulationnamesandmainparametersthatareusedinthispaper.”z”inthefilenamesignifiesareflectiveinflowboundary
CodeUsed FileName DirectoryName Resolution DensityRatio Mach Precession π (rad)
180
Both zaa xyz1 75x75x75 0.1 6 1
Both ba xyz1 150x150x150 0.1 6 1
Both bb xyz01 150x150x150 0.01 6 1
Both bc xyz001 150x150x150 0.001 6 1
Both bd 10xyz1 150x150x150 0.1 6 10
Both bg 20xyz1 150x150x150 0.1 6 20
Both bm xyz0001 150x150x150 0.001 6 1
Both bn 10xyz0001 150x150x150 0.1 6 10
Both bo 20xyz0001 150x150x150 0.1 6 20
Both zbm Rxyz0001 150x150x150 0.001 6 1
Both zbn 10Rxyz0001 150x150x150 0.1 6 10
Both zbo 20Rxyz0001 150x150x150 0.1 6 20
Both zba Rxyz1 150x150x150 0.1 6 1
Both zca Rxyz1 225x225x225 0.1 6 1
Both zda Rxyz1 300x300x300 0.1 6 1
PLUTOCODE ea xyz1 150x150x150 0.1 2 1
PLUTOCODE ed 10xyz1 150x150x150 0.1 2 10
PLUTOCODE ee 20xyz1 150x150x150 0.1 2 20
PLUTOCODE fa xyz1 150x150x150 0.1 4 1
PLUTOCODE fd 10xyz1 150x150x150 0.1 4 10
PLUTOCODE fe 20xyz1 150x150x150 0.1 4 20
PLUTOCODE ga xyz1 150x150x150 0.1 8 1
PLUTOCODE ha xyz1 150x150x150 0.1 12 1
PLUTOCODE ia xyz1 150x150x150 0.1 24 1
PLUTOCODE id 10xyz1 150x150x150 0.1 24 10
PLUTOCODE ie 20xyz1 150x150x150 0.1 24 20
PLUTOCODE ja xyz1 150x150x150 0.1 48 1
throughtheambientmedium,wealsoassumeherethatthejetden- Acocooncanusuallybeeasilyidentifiedwitharadiogalaxy
sity is low and the Mach number is high. We then write the jet simulation. This is filled with the jet material which has been
momentumflowratealongtheaxisintermsoftheramforce shocked at the hot spot and spills out into a lobe or cocoon im-
mediately surrounding the jet. The cocoon is thus distinct from
P˙ =ρ ·v2·A. (6)
jet x theshockedambientmaterial,asoriginallydefined(Normanetal.
Thiscanbewrittenintheform 1982),butdoesnotincludetheshockedambient(astakenbyCioffi
&Blondin(1992)).
v2
P˙ =ρjet· vj2xet ·M2·c2jet·A. (7) rj2et)DIfwthheecreocRoco(nt)isiscythlienrdoroict-aml,ethaen-vsoqluuamreeatvheernaggerocwoscoaosnπr(aRdc2iu−s.
Inthismodel,wetakep =p sothat Wealsoassumethatthejetflowisbrakedatthehotspotbyastrong
jet amb
shockwithacompressionratioof(γ+1)/(γ−1)=4andahot-
P˙ =ρ ·c2 · vx2 ·M2 ·A. (8) spotpressureof2γMj2etpjet/(γ +1).Thisisfollowedbyanadia-
amb amb v2 jet baticpressurefallbacktotheambient/jetpressure.Theassociated
jet
densitydecreasedirectlyyieldsthecocoondensityas
Thatmeansthatthemomentumflowrateisroughlyaconstantfor
afixedMachnumber,independentofthejetdensity.Remarkably, (cid:20) (cid:21)1/γ
γ+1 γ+1
thisimpliesthatthejetcrossingtimeofalllow-precessionsimu- ρc = γ−1 2γM2 ρjet, (11)
lationsofagivensizeandMachnumberisaconstant.Thespeed jet
scale,Ufortheadvanceoftheradiogalaxyisgivenbythemomen- or,forγ =5/3,
tumbalanceformula,
(cid:16)4(cid:17)3/5
U2 = ρρjet ·vx2 =Mj2et·c2amb, (9) ρc =4 5 Mj−et6/5ρjet. (12)
amb
Underareflectionboundarycondition,weequatetheinjected
whichassumesthatthejetremainscollimatedandpropagateswith
masstothecocoonmass,whichthenyieldsthecocoonvolumeand
adragcoefficientofunity.Thisyieldsaradiogalaxycrossingtime
manipulationthenyieldstheaveragecocoonradiusthrough
of
D/U =D/(Mjet·camb). (10) R =r (cid:20)1+ 53/5M6/5(ρamb)1/2(cid:21)1/2. (13)
c jet 48/5 jet ρ
jet
Therefore, we expect the jet crossing time to be approximately 5
unitsfortheD=30gridandM =6jet. Itcanthusbeseenthatevenwithjet-ambientdensityratiosofbe-
MNRAS000,1–17(2014)
Thephysicalstructureofradiogalaxiesexploredwiththree-dimensionalsimulations 5
tween 0.1 and 0.0001, a Mach 6 jet would generate an averaged dumpswiththereflectionboundaryconditionapplied.Thetracer
cylindricallobeofradiusofbetween3.0and15.7r . providesanaverageforzoneswheremixingoccurswiththeambi-
jet
Theassumedpressureequilibriumwiththeambientmedium, entmediumwhileambientmasslossfromthegridatlatetimesis
however, cannot be established if the cocoon is too wide, occu- discounted.Jetmaterialdoesnotleavethegridsincethereflection
pyingtheregionwhichtheshockedambientmediumwouldhave boundary condition is applied across the entire inner plane upon
expandedinto.Insteadtheshockedambientmediumdoesnotre- whichthejetnozzleisthensuperimposed.
expand and so applies a surface pressure on the cocoon. Equa- Figure1showsthatthereisacleardiscrepancywiththetheo-
tion13impliesthatthiswilloccuratverylowjetdensitiesorex- reticalmassinflowof5.96perunittimeassumingaperfectcircular
tremelyhighMachnumbers. nozzleofradiusr asshownbythesolidlines.Thecauseofthe
jet
Theenergypumpedontothegridisconvertedordivertedinto discrepancyismadeclearfromthedependenceontheresolution.
severalcomponents.Theaddedenergyiscontainedintheambient Astheresolutionisincreased,themassinflowrateconvergesto-
medium, cocoon or jet. For each, we have contributions to both wards a value within a few per cent of the theoretical value for
thethermalandkinetic(turbulent)energy.Toobeyconservationof the PLUTO code (right panel). However, this value is lower for
energy,weneedtoaccountforanyenergylostthroughthebound- theZEUSruns(leftpanel):theZEUS-3Drunsconvergeunderthe
aries,althoughthisiszeroprovidednodisturbanceshavereached theoretical line whereas the PLUTO code is converging onto the
theouterboundariesandwehaveimposedareflectioninnerbound- theoreticalline.Inbothcases,wecanseetheinjectedmassstarts
arycondition.Thetotalpoweraddedtothegridis: toconvergeataresolutionof1503.
Eachcodesetsupthecircularjetnozzleonthesquarefaces
(cid:18) (cid:19) oftheentryzonesbyapplyingadifferentapproximationscheme.
1 1 p
L= 2vj2et+ γ−1ρjet ·vx·ρjet·A, (14) Asmoothingprofileisusedtofixthespeedandmassmixinginthe
jet interfacezones.Theresolutionthusinfluencesthemassinflowac-
which,ontakingv =v forsimplicity,canbewritten cordingtothenumberofzonesacrossthejetdiameterN/Dwhere
x jet
N is the grid resolution. Fig. 1 shows that the mass discrepancy
(cid:18) (cid:19) (cid:18) (cid:19)1/2
L= 1 + 1 1 1 · ρamb ·M3·ρ ·c3 ·A. (15) is approximately ∝ (D/N)2. This result highlights the fact that
2 γγ−1Mj2et ρjet jet amb amb jet simulations are extremely sensitive to the frayed edges of the
jetsurfacewhichmayintroduceinstabilitiesandsmall-scaleturbu-
Again,notethelowdependenceonthedensityratioandthestrong
lence on larger scales much further downstream. For this reason,
dependenceonMachnumber.InSection5,wewilldeterminehow
wedonotexpectanytwocodes,letaloneanytworesolutionswith
thisenergyisredistributed.
the same code, to generate exactly the same physical structures.
RadiomapsandX-rayimageswillbederivedfromthesesim-
However, some of this sensitivity is clearly eliminated at higher
ulationsinafollowingwork.Here,weutilisepseudo-synchrotron
resolutions.
radioemissionbytakingtheemissionperunitzonevolumeas
Inthelowresolutionruns,thelackofdetailandtheaveraging
Eradio ∝χ·p2, (16) oftheeddycurrentsthatoccurattheinterfaceofthejetandambient
materialareclear.Asonewaytoquantifythis,westudythemaxi-
wherethetracerχissetsothattheambienthasavalueof0and
mumintensityofthe“hotspot”onthepseudo-synchrotronimages,
100% jet is set to a value of 1, thus only selecting zones where
andthedistanceofthathotspotfromthesourceasafunctionof
thereismaterialoriginatingfromjetinjectionbutnotaccounting
the resolution. This is a surface brightness and, hence, would be
forshockacceleration.Bysummingtheemissivitythroughaspe-
expectedtoincreasegraduallyastheresolutionincreases.Thisis
cificdirectionweobtainemissionmapsandcanthusfindtheloca-
displayedinFigure2forZEUS-3DandPLUTO.Thereareconsid-
tionofthemaximumintensitywhichweusebelowasaquantitative
erablevariationsasexpectedalthoughthesevariationsdecreaseas
measureofthenumericalconvergence.
thejetradiusbecomesbetterresolved.Thereisanindicationthat
Alistofsimulationsandtheirnamingconventionsisprovided
PLUTOstartstoconvergeatresolutiongreaterthan2253.
inTable2.Therearetwomainnamesforeachjetsimulation.The
This is further supported when the advance of the ambient
first is the file name that ZEUS-3D uses which is carried over to
shock front is plotted, as shown in Fig. 3. The advance speed is
thePLUTOcode.Thesecondiscomprisedofthreeparts[preces-
stablebetweentheresolutionrunsafteraninitialset-upperioddur-
sion][coordinatesystem][densityratio].Also,ifthereisareflective
ingwhichthespeedoftheabruptentrancedependsonthegridzone
boundaryaddedtothesimulationthenan“R”isaddedtothecorre-
sizeasthecodesmoothsoverthesteepgradientsintroduced.The
spondingcoordinate.Forexample,20Rxyz01refersto20◦preces-
advancespeedsettlestoaconstantvalueof∼4.7and4.3forthe
sioninCartesiancoordinatewithareflectioninthestartingplaneof
ZEUS-3DandPLUTOcodes,respectively.Thisislowerthanthe
thejetwithadensityratioof0.01.Notethatthereflectioncondition
valueofU =6fortheSteadyPropagationModel.However,aswill
isonlyappliedtotheboundarywherethejetisinitialized.
befoundbelow,advancespeedsinexcessofU arefoundforlower
jet densities and, in contrast, can also be higher for the PLUTO
code.Insummary,withspecificreservations,the1503resolutionis
3 RESOLUTION&CONVERGENCE
thepreferredchoiceforafullnumericalsurvey.
Resolutions studied span the range from 753 to 3003. Our
workhorseisthe1503simulation.Thispermitsustocoverasmuch
groundaspossibleandfollowupwithhigherresolutionsonanyin-
teresting findings. The wide range allows us to test the influence 4 PARAMETERSTUDY
oftheresolutiononthemorphologyofthejet.Allthesimulation
4.1 Density
whichareillustratedhavetheconditionsoutlinedinTable2.
The accumulated mass injected as a function of time is dis- The dependence on the jet-ambient density ratio is illustrated in
played in Fig. 1. It is calculated from the grid density and tracer thedensityslicesdisplayedinFigs. 4&5andthecorresponding
MNRAS000,1–17(2014)
6 J.Donohoe&M.D.Smith
Figure1.Thetotalmassonthegridasafunctionoftimeforthereflectionboundaryconditionontheinflowboundary,generatedfortheindicatedfour
resolutions.Thedensityratioisρjet/ρamb =0.1andtheprecessionangleis1◦.TheleftpaneldisplaysZEUS-3DandtherightshowsthePLUTOcode.The
solidblacklinecorrespondstothetheoreticalvaluethatshouldbeinjectedintothesystemthroughaperfectcircularcross-section.
Figure2.Themaximumintensityofthehotspotandthedistanceofthathotspottothesource.LeftgraphshowsZEUS-3D;rightshowsPLUTOresults.
Thesetestswereperformedwithareflectioninflowboundary.
Figure3.Aresolutionstudyofthelocationoftheadvancingshockintotheambientmediumforthecasewithjet-ambientdensityratioof0.1andoutflow
boundarycondition.Theresolutionsshownincreasefrom753(solid,black)to3003(dot-dash,green)withtheinitialconditionsoutllnedinTable2.Theleft
panelisZEUS-3DcodeandtherightpanelisthePLUTOcode.
MNRAS000,1–17(2014)
Thephysicalstructureofradiogalaxiesexploredwiththree-dimensionalsimulations 7
(a) (b) (c)
Figure4.Densitydependence.Densityvolumetricslicesthroughthemid-planefromZEUS-3Dwithequivalentconditionsapartfromthejet-ambientdensity
ratioof0.1(a),0.01(b)and0.001(c).AllhaveajetMachnumberof6,asmallprecessionangleof1◦andperiodof4simulationunitsinordertobreakthe
symmetry.ConditionsareoutlinedinTable2.Thelimitsofthecolourscalearesettothespecificsimulationminimumandmaximumvalues.
(a) (b) (c) (d)
Figure5.Densitydependence.DensityvolumetricslicesfromthePLUTOcodewithequivalentconditionsapartfromthejet-ambientdensityratioof0.1(a),
0.01(b),0.001(c)and0.0001(d).ConditionsareasstatedinFig.4.
velocityslicesofFigs. 6&7.Thesecorrespondtostraightcolli- oncomparisonofslice(b)betweenFigs. 4&5.Thereisatime
matedjetswithasuperimposedsmall-anglelong-periodprecession differenceof0.8unitsor2.9MyrsfortheCompactSourcedespite
tobreakthesymmetry. thesameinitialcondition.ItcanbeseenfromthePLUTOfigures
Thedensityslicesshowthatthereisamodestincreaseinthe thatthejetsthemselvesaremorestablecomparedtotheZEUScode
volumeofthecocoonoccupiedbyjetmaterialasthedensityratio simulations.Thismeansthatthemomentumisefficientlypushing
decreasesbelow0.01.Thebowshockintheambientmediumalso theheadofthejetintonewambientmaterialratherthandissipating
becomesprogressivelyblunterasthejetdensityfalls. theenergytocreateawideningplume.
Thenarrowcocoonassociatedwithdensityratiosabove0.001 Theadvancespeedincreasesbyasignificantfactorastheden-
ensuresthatthejetpropagateswithlittledissipationacrossthejet- sityratioisloweredfrom0.1.Butthistrendisreversedforthevery
cocoon vortex sheet. In contrast, at lower densities, the cocoon lowest density where the asymmetric structure leads to a spread-
broadening is associated with the generation of large asymmetric ingandpartialdisruptionofthejetwhichslowsdowntheadvance.
vorticeswhichfinallydominateforthedensityratioof0.0001.This Incomparison,fromtheSteadyPropagationModelwewouldan-
broadcocoonexpansionforverylightjets,typicallytoalmostthe ticipateaconstantspeedofadvance.Thedifferenceisassociated
samesizeasthebowshock,wasfirstfoundinaxisymmetricsim- withpressurefeedbackontothejetwhichsqueezesthejetasseen
ulationsofKrause(2003)forMachnumbersaboveapproximately from the velocity structure in panels (b) and (c) of Figs. 6 & 7.
three.Thewidecocoonsupportsahighpressureandstrongpres- The higher advance speed also tends to streamline the cocoon as
surevariationswhichfeedbackontothejet.Thejetaxialvelocity material flowing into the cocoon does not have to expand so far
becomes limited to a narrower and convergent-divergent channel laterally.Afurtherconsequenceisthatthecocoon-jetdensityratio
(seeFigs. 6and7)asthecocoonpressuredominates. generallyincreasesasthejetdensitydecreases(seetheassociated
colourbars).
Thedifferencesbetweenthetwocodesarequitemodestwith
thePLUTOsimulationssomewhatmoreaerodynamicinshapeand We thus confirm here that a wide range in jet density has a
speedacrossthegridforaspecificdensityratio.Thisisconsistent relativelysmallaffectonthepropagationspeedofthelobes.This
withthebettersmoothingprofileacrossthenozzleinterfaceofthe is because the injected momentum flow rates are the same. This
PLUTO code. The difference between the two codes is apparent isduetothejetdensitybeinginverselyproportionaltothesound
MNRAS000,1–17(2014)
8 J.Donohoe&M.D.Smith
(a) (b) (c)
Figure6.VelocityvolumetricslicesfromZEUS-3Dwithequivalentconditionsbarfromthedensityratioof0.1(a),0.01(b)and0.001(c).Otherconditions
areasstatedinFig.4.
(a) (b) (c) (d)
Figure7.VelocityvolumetricslicesfromPLUTOCODEwithequivalentconditionsbarfromthedensityratioof0.1(a),0.01(b),0.001(c)and0.0001(d).
OtherconditionsareasstatedinFig.4.
speedsquaredofthejet.Sodecreasingthedensityratioincreases thejetisindependentofthejetdensityintheSteadyPropagation
thevelocityofthejetsincetheMachnumberisheldataconstant Model(Eq.7).However,themomentumflowratethroughthejet
value,asseenfromEquation3.Thisisalsoevidentfromtheveloc- isproportionaltotheMachnumbersquared.Therefore,theMach
ityslicesdisplayedinFigs. 6and7.Theresultisapparentwhen numbershouldbetheparameterwhichcontrolsthejetstrengthand
lookingatthefourdensityratiosimulationsofPLUTO(Fig.5).It advancespeed.ThisisindeedthecaseasshowninFig.10forthe
showsthatthetimeittakesforthedensityratioof0.1,0.01,0.001 densityratioof0.1.
and 0.0001 correspond to 18.1, 11.6, 11.9 and 15.2 Myr, respec- At Mach numbers below 4, the cocoon is stripped from the
tively when interpreted as compact radio sources. Thus the mo- jet through Kelvin-Helmholtz instabilities. This confirms the 2D
mentumofthejetisthekeyfactorinthepropagationdistanceof resultsfirstdiscussedbyNormanetal.(1982)andtheworkdone
theheadofthejet. byRossietal.(2008)whereinlowerMachnumberswillresultin
Thepressuredistributionsdisplaymorevariety,asdisplayed FR-ItypesduetothedecelerationofjetmaterialclosertotheAGN
in Figs. 8 and 9. While similar ambient bow shocks are present source.
for the jet-ambient density ratio of 0.1 (left panels), the ambient
AthighMachnumbers,theentirestructurereacheshighas-
bow is wider in the ZEUS runs although the maximum pressure
pect ratios. At Mach numbers in excess of 8, the feedback effect
reachedislower.Hence,theresultsaresensitivetotheprecisecon-
from the cocoon becomes increasingly evident. However, the dy-
ditionswiththePLUTOcodegeneratingslightlymorecollimation
namical time for the ambient medium is just t = 1 (3.64 Myr
o
and less jet-cocoon turbulence at the lower densities. Finally, the
forthecompactsystem).ThisimpliesthehighMachnumberflows
high pressure created at the lowest density (right panel) perturbs
penetrate all the way through the ambient medium before sound
theunder-pressuredjet,leadingtothestrongpressurevariationsall
signalscancrossadistanceequaltothejetradius.Inthiscase,the
alongthejet,typicalofaconvergent-divergentnozzle.
cocoon has insufficient time to expand and the pressure is high.
Consequently, there is high pressure feedback from the cocoon
whichsqueezesonthejetbeforetheterminatinghotspotasthese
4.2 ParameterStudy:Machnumber
jetsareclearlystillintheinitialblastphase.
For a fixed Mach number and pressure balance between the in- In summary, the Mach number has a profound influence on
jectedjetandambientmedium,themomentumflowratethrough the morphology of a radio galaxy. High Mach numbers generate
MNRAS000,1–17(2014)
Thephysicalstructureofradiogalaxiesexploredwiththree-dimensionalsimulations 9
(a) (b) (c)
Figure8.PressurevolumetricslicesfromZEUS-3Dwiththeconditionsthesameapartfromthedensityratio.Slice(a)showsadensityratioof0.1,(b)isa
ratioof0.01and(c)isaratioof0.001.OtherconditionsareasstatedinFig.4.
(a) (b) (c) (d)
Figure9.PressurevolumetricslicesfromPLUTOwiththeconditionsthesameapartfromthedensityratio.Slice(a)showsadensityratioof0.1,(b)isaratio
of0.01and(c)isaratioof0.001and(d)isaratioof0.0001.OtherconditionsareasstatedinFig.4.
highlyaerodynamicandsymmetricsourcesaswellasnarrowX-ray wheretheprecisenumericalnozzleshape,turbulenceandinstabil-
cavities.LowMachnumbersgenerateasymmetricturbulentlobes. itydominateoverdynamicalconditions.
Alowrateofprecessionincomparisontothedynamicaltime
generates structure similar to the standard straight jet simulation
(e.g.rightpanelsofFig.13).Ahighprecessionraterelativetothe
4.3 ParameterStudy:Precession
dynamicaltimeistheleadingcauseofthechangeofmorphology
ofthejets.Thestandardperiodofprecessionof4simulationunits
Withtheprecession-inducedjetwobbling,asdefinedbyEquations
impliesthatthereisonlyjustoveronecompleteturnofthejetover
3&4,newstructureoccurs.Themostobviousisthegreateramount
thetypicalpropagationtimeofthestructurewhilethejetgastakes
ofmixingthatoccursascomparedtoarelativestraightjet.Thisis
onlyD/v =1.7unitsforthejet-ambientdensityratioof0.1and
achieved through both of the defining parameters, the first being jet
Machnumberof6.Hence,whilethecocoonisdistorted,thejets
theangleatwhichthejetprecessesandthesecondistheratethe
donotdisplaysignaturesofacorkscreworhelicalstructure.
jetprecesses.
Theoverallstructureofthelobe/cocoonchangesastheangle Asthejetprecesses,newpartsarecomingintocontactwith
at which the jet precesses increases, as shown in Figs. 11 & 12. older ejected parts of the jet as it impacts the ambient material,
Theanglespreadsthemomentumandsolimitsthedistancethejet asseenfromthemultiplebowshocksinFigures11&12.Thisis
propagates with time. It also causes the cocoon of the jet to ex- causing a distorted but continuous bow shock to trace where the
pandandincreasesthetotalamountofmixingthatcanoccur.The jetmaterialiscomingintocontactwithdensermaterial.Asthejet
twofiguresestablishthatthecodeemployedhasminimalinfluence thenreturnstoapreviouslyexcavatedcavity,ithastoexpendless
onthestructuregeneratedoncealargeprecessionangleisapplied. energyasthecavityisalsoexpandingtowardstheambientmedium.
Theinitialstateisonlyimportantfortheonedegreeprecessioncase This then results in the cocoon being made of stacked layers of
MNRAS000,1–17(2014)
10 J.Donohoe&M.D.Smith
(a) (b) (c)
(d) (e) (f)
(g)
Figure10.Machnumberdependence.DensityvolumetricslicesfromthePLUTOcodewithequivalentconditionsbarfromMachnumbersof2(a),4(b),6
(c),8(d),12(e),24(f)and48(g).Thesamecolourbarappliestoallslices.OtherconditionsareasstatedinFig.4.
jet material rather than being just the result of Kelvin-Helmholtz thus expect such slow precession to lead to interesting radio and
instabilities. X-raystructures.
Theseeffectsarehighlightedwhenstudyingtheprecessionat
Astheprecessionangleincreasesthereisalsoanincreasing high Mach numbers. Figure 13 takes a sample of Mach number
cone-shapedindentedregionatthesymmetricheadofthejet,the fromFig.(10)andshowshowprecessionconfinestheheadofthe
pointatwhichthereis”symmetry”aroundthecentreofprecession. jetclosertothesourceofthematerial.TheMach24jet,column
Thisregiondiminishesovertimedependingontheratethejetpre- ’c’onFigure13,beingoneoftheupperendMachnumbers,has
cesses. This region plays an interesting role with precessing jets. its propagation length roughly halved when precessed from 1◦to
As the jet precesses we are left with a void that traces the wake. 20◦.Ontheotherhand,thelowMachnumbercasesdisplayedin
Thislowpressureregionisthenfilledinbysurroundingmaterial. column‘a’,displayatrendofincreasingbreak-upandplumeflow
Thismaterialcancomefromtheconeregioneffectivelydetaching astheprecessionangleincreases.
partsofthecocoonfromthemainstructure.
Remarkably,twobowshockscanbedistinguishedintheam-
bientmedium.Thefirstisthestandardadvancingbowshockwhich
5 ENERGY
envelopstheentireoutflow.Thesecondfollowingbowshockisas-
sociatedwiththepresentpointofimpactofthejet.Thisinnerbow Thedistributionsoftheenergyinajet-drivensystemhasbeenstud-
isnarrowerbutseenprominentlyinthepressurepanelofFig.14 ied by many authors in order to provide evidence on how radio
andisconfirmedaspureambientmaterialviathetracerpanel.We galaxiesandtheambientmediaevolve.Oneconcernisthedynam-
MNRAS000,1–17(2014)