Irradiation of an Accretion Disc by a Jet: General Properties and Implications for Spin Measurements of Black Holes

0.71 MB·English

by  Thomas Dauser

Mon.Not.R.Astron.Soc.000,1–15(2012) Printed31January2013 (MNLATEXstylefilev2.2)
Irradiation of an Accretion Disc by a Jet: General Properties and
Implications for Spin Measurements of Black Holes
T. Dauser1⋆, J. Garcia2, J. Wilms1, M. Bo¨ck1,3, L. W. Brenneman4, M. Falanga5,
3
6 2
1 K. Fukumura , and C. S. Reynolds
0 1Dr.KarlRemeis-ObservatoryandErlangenCentreforAstroparticlePhysics,Sternwartstr.7,96049Bamberg,Germany
2 2DepartmentofAstronomyandMarylandAstronomyCenterforTheoryandComputation,UniversityofMaryland,CollegePark,MD20742,USA
n 3Max-Planck-Institutfu¨rRadioastronomie,AufdemHu¨gel69,53121Bonn,Germany
a 4Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138,USA
J 5InternationalSpaceScienceInstitute,Hallerstrasse6,3012,Bern,Switzerland
6AstrophysicsScienceDivision,NASAGoddardSpaceFlightCenter,Code663,Greenbelt,MD20771,USA
0
3

] 31January2013
E
H
. ABSTRACT
h
X-rayirradiationoftheaccretiondiscleadstostrongreflectionfeatures,whicharethenbroad-
p
enedanddistortedbyrelativisticeffects.We presentadetailed,generalrelativisticapproach
-
o tomodelthisirradiationfordifferentgeometriesoftheprimaryX-raysource.Thesegeome-
r triesincludethestandardpointsourceontherotationalaxisaswellasmorejet-likesources,
t
s whichareradiallyelongatedandaccelerating.IncorporatingthiscodeintheRELLINEmodel
a forrelativisticlineemission,thelineshapeforanyconfigurationcanbepredicted.Westudy
[
how different irradiation geometries affect the determination of the spin of the black hole.

2 Broademissionlinesareproducedonlyforcompactirradiatingsourcessituatedclosetothe
v blackhole.Thisistheonlycasewheretheblackholespincanbeunambiguouslydetermined.
2 Inallothercasesthelineshapeisnarrower,whichcouldeitherbeexplainedbyalowspinor
2 anelongatedsource.We concludethatforthosecasesandindependentofthequalityofthe
9 data,nouniquesolutionforthespinexistsandthereforeonlyalowerlimitofthespinvalue
4 canbegiven.
.
1
Keywords: Accretion,AccretionDiscs,blackholephysics,Galaxies:Nuclei,galaxies:ac-
0
tive,Lines:Profiles
3
1
:
v
i
X
1 INTRODUCTION tor,andourviewingdirectiononthesystem,parametrisedthrough
r anappropriateinclinationangle, i.Additionallyaprimarysource
a DuetothevicinityoftheX-rayemittingregioninActiveGalactic
of radiation has to exist in order to produce the observed reflec-
Nuclei(AGN)andX-raybinariestothecentralcompactobject,it
tionand also theunderlying continuum. Initialassumptions were
is expected that the observed X-ray spectrum will show signs of
that theprimarysource consists of ahot corona around the inner
relativisticeffects(Fabianetal.1989).Sucheffectswerefirstseen
regionsofthedisc,asComptonizationofsoftdiscphotonsinsuch
in the skew symmetric shape of the fluorescent Fe Kα line from
a corona naturally produces a power law spectrum which fitsthe
these objects (Tanakaetal. 1995; Reynolds&Nowak 2003, and
observations (Haardt 1993; Doveetal.1997).Under theassump-
referencestherein).Theserelativisticlinesarepresent inasignifi-
tionthattheintensityofthehardradiationscatteredbackontothe
cantfractionofAGNspectra(Guainazzi,Bianchi&Dovcˇiak2006;
discbythecoronaisproportionaltothelocaldiscemissivity,the
Nandraetal. 2007; Longinottietal. 2008; Patricketal. 2011) irradiationoftheaccretiondiscwouldbeI(r) r−3fortheouter
and Galactic black hole binaries (Miller 2007; Duroetal. 2011; ∝
parts,andgraduallyflattentowardstheinneredgeofthediscfora
Fabianetal.2012a).Recent workhasbeen applyingamoreself-
standardShakura&Sunyaev(1973)disc.
consistent approach by including models for the relativistic dis-
tortion of the full reflection spectrum during the data analysis With the advent of high signal to noise data from satellites
(see,e.g.,Zoghbietal.2010;Duroetal.2011;Fabianetal.2012b; such as XMM-Newton, however, measurements showed a dis-
Dauseretal.2012). agreement with the Fe Kα line profiles predicted by this coro-
Therelativisticdistortionofthereflectionspectrum isdeter- nal geometry. For many sources, the data favoured disc emis-
minedbythespinoftheblackhole,a,thegeometryofthereflec- sivities that are much steeper in the inner parts of the accretion
disc (see, e.g., Wilmsetal. 2001; Milleretal. 2002; Fabianetal.
2002; Fabianetal. 2004, 2012a; Brenneman&Reynolds 2006;
⋆ E-mail:[email protected] Pontietal. 2010; Brennemanetal. 2011; Galloetal. 2011;
(cid:13)c 2012RAS
2 T. Dauseret al.
Dauseretal.2012).Variabilitystudiesofthebroadironlinespose 2 THEORY
additional problems for standard corona models. In such studies
2.1 Introduction
the time variability of the continuum flux, i.e., the primary hard
X-rayradiation,iscomparedtothefluxinthelines,whicharepro- We calculate the shape of the relativistic line in the lamp-post
ducedbythereflectedradiation.Thisallowstoprobetheconnec- geometry by following the radiation emitted from the primary
tionbetweentheprimaryandthereflectedradiation.Inacoronal source on the axis of symmetry of the accreting system to the
geometry, oneexpects apositivecorrelationbetweenthestrength accretion disc and from there to the observer. Due to the deep
oftherelativisticallydistortedreflectionspectrumandtheprimary gravitational potential close to the black hole, the photon trajec-
continuum (Martocchia&Matt 1996).Thisisincontrast towhat tories are bent and their energies red-shifted. Moreover the rela-
isobserved:MeasurementsofMCG 6-30-15(Fabian&Vaughan tivistic movement of the accretion disc alters the energy flux in-
−
2003; Miniuttietal. 2003) revealed large variations of the direct cident on the disc and the shape of the observed line through
radiation,whilethereflectedcomponentremainedconstant. the relativistic Doppler effect. Using techniques introduced by
Cunningham (1975), the flux seen from a certain element of the
accretion disc under a specific inclination can be predicted (e.g.,
Speith,Riffert&Ruder 1995) and summed up to the complete
As shown by Martocchia,Matt&Karas (2002), spectrumofthesource.Thisproblemhasbeenextensivelystudied
Fabian&Vaughan (2003), Miniuttietal. (2003), and inthepastandthereareseveralmodelsavailabletopredicttherel-
Vaughan&Fabian (2004) for the case of MCG 6-30-15, in ativisticsmearing given a certain emissivity of the accretion disc
−
a geometry in which the illuminating continuum is assumed (e.g., Fabianetal. 1989; Laor 1991; Dovcˇiak,Karas&Yaqoob
to be emitted from a source on the rotational axis at height h 2004;Brenneman&Reynolds 2006; Dauseretal. 2010).Inthese
above the black hole, strong light bending yields properties of modelstheintensityemittedfromtheaccretiondisc(theso-called
the reflected radiation that are consistent with the observations. “emissivity”) is parametrised as a power law r−ǫ with index ǫ,
Figure1illustratesthis“lamppost”geometry(Matt,Perola&Piro wherer isthedistancetotheblackhole.Thestandardbehaviour
1991; Martocchia&Matt 1996). In general, data and predicted isǫ = 3,whichisproportionaltotheenergyreleaseinastandard
line shapes show very good agreement (see Wilkins&Fabian Shakura&Sunyaev(1973)disc.
2011;Duroetal.2011;Dauseretal.2012).Thelamppost model Thisemissivitycanalsobecalculateddirectlyfromanirradi-
also explains the observed connection between the luminosity atingsource, theso-called “primarysource”. Inthefollowingwe
andthereflectionstrength:Foraprimarysourceveryclosetothe willusea sourcesituated on therotational axisof theblack hole
blackhole,mostofthephotonsarefocusedontheaccretiondisc, forthispurpose.Byapplyingthesameray-tracingtechniquesused
producing a strong reflection component. Therefore less photons totracephotonsfromthedisctotheobserver,wecanalsocalculate
are left over to contribute to the continuum component, which is theproperirradiationoftheaccretiondiscbytheprimarysource.
directly emitted towards the observer (Miniutti&Fabian 2004). Theradial dependency of thisirradiation isequal to thereflected
Foran increasing height of thehardX-raysource thiseffect gets radiation,i.e.,theemissivity,whichwaspreviouslymodelledbya
weaker and thus more photons can escape, which strengthens powerlaw. Inthispaperweconcentrate onthe irradiation of the
the continuum radiation and, depending on the flux state of the accretiondisc.Inordertobeabletocompareourresultstoobser-
X-raysource,weakensthereflectedflux(Miniutti&Fabian2004; vationaldata,wealsoneedtocalculatetheraytracingfromtheac-
Miniutti2006). cretiondisctotheobserver,whichisdonewiththeRELLINE-code
(Dauseretal.2010).
Basedontheearlierworkonthelamppostgeometrypresented
2.2 PhotonTrajectoriesintheLampPostGeometry
above, Fukumura&Kazanas (2007) provided a more detailed
treatmentoftheemissivityforarbitraryspinandanisotropicemis- Inthefollowingwewillconcentrateonasimplifiedgeometry by
sionoftheprimarysource.Off-axissourceswerefirstinvestigated assumingapoint-like,photonemittingprimarysourceataheight
by Ruszkowski (2000). Using similar methods Wilkins&Fabian habovetherotationalaxisoftheblackhole.Thissourceirradiates
(2012) presented araytracing method working on Graphics Pro- athin,butopticallythickaccretiondisc(Fig.1).Relativisticpho-
cessingUnits(GPUs),whichcancalculateirradiationprofilesfor tontrajectoriesinthelamppostgeometrywerefirstinvestigatedby
almost arbitrary geometries of the primary sources, now also in- Matt,Perola&Piro(1991)andusedbyMartocchia&Matt(1996)
cludingsourcesextendedalongandperpendiculartotherotational inordertoexplaintheverylargeequivalent widthoftheironKα
axis.Inthispaperwepresentacompletemethodtoderiveirradi- line in some AGN. A more detailed discussion of effects in this
ationprofilesinthelamppostgeometry(Sect.2),including radi- geometrywaspresentedbyMartocchia,Karas&Matt(2000),in-
allyextendedandacceleratingsources(Sect.3).Usingthisformal- cludingadiscussionoftheinfluenceoftheblackhole’sspinonthe
ism, we introduce an implementation of the lamp post geometry overallspectra.
as afittingmodel forrelativisticreflection, which can beapplied As a good physical explanation of this hard X-ray source
tomorerealisticexpectationsoftheoreticaljetmodels(Sect.4.1). on the rotation axis is the base of a jet (Markoff&Nowak
In Sect. 4.2 we analyse the shape of the reflection features pre- 2004), we call this geometry also the “jet base geometry”.
dicted by the different types of irradiating sources. In particular This interpretation is highly supported by the earlier work of
weconcentrateontheimplicationsforspinmeasurements,thedif- Ghisellini,Haardt&Matt(2004),whereitisshownthatallAGN
ferent assumptions for the geometry of the primary source have are capable of forming jets. This is achieved by inventing the
(Sect.4.3).Theseresultsarequantifiedbysimulatingsuch obser- concept of “aborted” jets for radio-quite quasars and Seyferts,
vationsforcurrentinstruments(Sect.4.4).Finally,wesummarise which are produced when the velocity of the outflowing mate-
themainresultsofthepaperinSect.5. rialissmallerthantheescapespeed.Thereforesuchajetextends
(cid:13)c 2012RAS,MNRAS000,1–15
Irradiationofan AccretionDiscbyaJet 3
havetorotateatthespeedoflight.1 Notethatallequationsinthis
paperaregiveninunitsofG M c 1.Thedifferentsignsin
≡ ≡ ≡
equation(1)areforincreasing(uppersign)anddecreasing (lower
sign)valuesofrandθ.Heretheconservedquantitiesarethetotal
(cid:14) energy,E,theangularmomentumparalleltotherotationalaxisof
theblack hole, λ, and theCarter(1968) constant, q. Thelatteris
givenby
h
h2 2h+a2
q=sinδ − . (5)
r h2+a2
(cid:0)i
Assuming the source to be located on the symmetry axis of the
system(θ = 0)simplifiesthecalculationsignificantly,asp does
θ
onlytakearealvalueifλ=0,andthereforeequation(1)simplifies
to
p =E( 1, √V /∆, E q, 0) . (6)
µ r
r − ± ± | |
Thenthetrajectorycanfinallybecalculatednumericallyfromthe
Figure1.Aschematicdrawingofthecomponentsinthelamppostgeom- integralequation
etry.Theprimarysourceofphotons(blue)issituatedabovetherotational
axisoftheblackholeandisemittingphotons(red),whichhittheaccretion r dr′ = θ dθ′ . (7)
disc. Zh ±√Vr′ Z0 ±√Vθ′
(following Chandrasekhar 1983; Speith,Riffert&Ruder 1995).
only asmalldistance fromtheblack hole, producing only aneg- Notethat thesign changes at turning points of thephoton trajec-
ligible amount of radio flux while at the same time it strongly tory.Inthiscasetheleftpartofequation(7)hastobesplitintotwo
irradiates the inner accretion disc in X-rays, which produces the integrals, each going from and to the turning point, respectively.
observed, highly relativistic reflection (Ghisellini,Haardt&Matt Notethatnoturningpointsintheθdirectionhavetobetakeninto
2004). This interpretation is encouraged by works showing that account,asphotonswhichinitiallyflytowardsthedisc,willnotex-
direct and reprocessed emission from such a jet base is equally hibitaturningpointbeforecrossingtheequatorialplaneandthus
capable in describing the observed X-ray spectrum as a corona beforehittingtheaccretiondisc.
above the accretion disc and also yields a self-consistent expla-
nation of the full radio through X-ray spectrum of many com-
2.3 IlluminationoftheAccretionDisc
pactsources(Markoff,Nowak&Wilms2005;Maitraetal.2009).
In addition a direct connection between the X-rays and the ra- In order to calculate the incident intensity on the accretion disc,
diocanexplainthecorrelationbetweenobservedradioandX-ray wefirsthavetoconsiderthegeometriceffectsintrinsictothelamp
flaresof Microquasars such asGX339 4 (Corbeletal.2000)or postsetup.Withoutanyrelativisticeffectstheintensityimpinging
−
CygX-1(Wilmsetal.2007).Wenotethatasimilarkindofconnec- ontheaccretiondiscforanisotropicprimaryemitterisgivenby
tionisalsoindicatedinsomemeasurements ofAGNlike3C120
cosδ h
(Marscheretal.2002)or3C111(Tombesietal.2012).Addition- Ii(r,h)∝ r2+hi2 = (r2+h2)23 . (8)
ally,evidence of adirectinfluence of thejeton theblackholein
Microquasarsisgrowingrecently.Namely,Narayan&McClintock Thismeansthatalreadyforflatspacetimetheirradiatedintensity
(2012)observeadirectcorrelationbetweenthejetpowerandthe stronglydependsontheradius.
spinvaluefromanalysingasmallsampleofsources. Duetothestronggravitythephotontrajectorieswillbesignif-
In the following we will briefly summarise the most im- icantlybent,i.e.,inoursetupthephotonswillbe“focused”ontothe
portant equations required for deriving the photon trajectories innerregionsoftheaccretiondisc(Fig.1),modifyingtheradialin-
from a source on the rotational axis of the black hole. As we tensityprofile.Notethatthisfocusingdependsonboth,theheight,
are dealing with potentially rapidly rotating black holes, we h,andtheinitialdirectionofthephoton,parametrisedbytheangle
choose the Kerr (1963) metric in Boyer&Lindquist (1967) co- betweenthesystem’saxisofsymmetryandtheinitialdirectionof
ordinates to describe the photon trajectories. Following, e.g., thephoton,δ.
Bardeen,Press&Teukolsky (1972), the general photon momen- Using the equations of the previous section we developed a
tumisgivenby ray-tracing code using similar techniques as those presented in
Dauseretal.(2010).Withthiscodeweareabletocalculatephoton
p = E, p = E√V /∆, p = E√V , p =Eλ, (1)
t − r ± r θ ± θ φ trajectoriesfromthepointofemission(h,δ)attheprimarysource
with totheaccretiondisc,yieldingthelocation(r,δi)wherethisspecific
photonhitsthedisc.Astheprimarysourceislocatedontherota-
∆ = r2 2r+a2 (2) tionalaxisoftheblackhole,thetrajectoryofthephotonisuniquely
−
Vr = (r2+a2)2 ∆(q2+a2) (3) determinedbytheq-parameter(equation5).Theincidentpointis
−
λ thencalculatedbysolvingtheintegralequation(7)forrinthecase
V = q2 cos2θ a2 (4)
θ − hsin2θ − i ofθ=π/2.
foracertaindistancerandspinaoftheblackhole.Thespinisde-
finedinsuchawaythatitsabsolutevaluerangesfrom06 a 61, 1 Thorne(1974)showedthattherealisticupperlimitofthespinismore
| |
whereat themaximal value, amax = 1,theevent horizonwould likelyamax=0.998.
(cid:13)c 2012RAS,MNRAS000,1–15
4 T. Dauseret al.
Knowing wheretheisotropically emittedphotons hittheac- Thisis in line with the results obtained by Fukumura&Kazanas
cretiondisc,wecanderivethephotonfluxincidentonitssurface. (2007)2.
Asthephotonsaredesignedtobeemittedatequallyspacedangles Inordertounderstandtheinfluenceofthedifferentrelativis-
δ,thedistance ∆r between thesepointsisrelatedtotheincident tic parameters on the incident intensity, Fig. 2 shows the single
intensity.Photonsemittedin[δ,δ+∆δ]aredistributedonaring components of equation (18). A similardiscussion of these com-
ontheaccretiondiscwithanareaofA(r,∆r).Theproperareaof ponentsisalsogivenbyWilkins&Fabian(2012).Weassumethat
sucharingatradiusrwiththickness∆risgivenby theprimarysourceisanisotropicemitter.Alleffectsarestrongest
forsmallradiiandwillthereforebemostimportantforhigh spin,
r4+a2r2+2a2r where the accretion disc extends to very low radii. First, length
A(r,∆r)=2πr ∆r (9)
·r r2 2r+a2 contractionreduces theareaof theringasseenfromtheprimary
−
in the observer’s frame of rest (Wilkins&Fabian 2012). In or- source. In the rest frame of the accretion disc, this “contraction”
der to calculate the irradiation in the rest frame of the accre- implies an effectively larger area and therefore the incident flux
tion disc, we have to take into account its rotation at relativis- decreases with increasing vφ proportional to the inverse Lorentz
tic speed. The area of the ring will therefore be contracted. Us- factor1/γ(φ)(Fig.2a).Whencomparedtoflatspacetime,thearea
ing the Keplerian velocity profile deduced from the Kerr metric ofdiscclosetotheblackholeisadditionallyenhancedintheKerr
(Bardeen,Press&Teukolsky1972),thedisc’sLorentzfactoris metric(Fig.2b).Interestingly,thiseffectisalmostindependentof
the spin of the black hole3. However, compared to the effect in-
γ(φ) = √r2−2r+a2(r3/2+a) (10) ducedbytheenergyshift(equation14),thechangeinareaisonly
r1/4 r√r+2a 3√r√r3+a2r+2a2 aminoreffect.Dependingonthepowerlawindex, Γ,theenergy
p − shiftofthephotonshittingthediscisthestrongestfactorinfluenc-
(Bardeen,Press&Teukolsky 1972, see also Wilkins&Fabian
ingthereflectionspectrum.Forasourceontherotationalaxisofthe
2011, 2012). Taking into account that the photons are emitted at
blackholethechangeinirradiatedfluxcanbeaslargeasafactorof
equally spaced angles, we finally find that for isotropic emission
100(Fig.2c),dependingstronglyonthesteepnessoftheprimary
thegeometriccontributiontotheincidentintensityhastobe
spectrum. This amplification factor depends on the height of the
sinδ emitting source (it becomes larger for increasing height) and de-
Igeo = . (11)
i A(r,∆r)γ(φ) creasesforlargerradii.Especiallyforlow hlightbendingfocuses
photonstowardsthediscandadditionallyenhancestheirradiation
Becauseoftherelativemotionoftheemitterandtheaccretiondisc,
oftheinnerregions.ThedashedlinesinFig.2dshowhow 1/∆r
aswellasbecauseofgeneralrelativisticeffects,theirradiatedspec-
decreases in flat space just due to geometrical reasons following
trumwillbeshiftedinenergy(Fukumura&Kazanas2007).Using
equation(8).Thefullyrelativistictreatment(solidlines)revealsa
theinitialfour-momentumattheprimarysource
focusing of the photons towards theblack hole. But compared to
uµ =(ut,0,0,0) (12) the“effectivelyenhanced”irradiationoftheinnerpartsduetothe
h h
energyshift(Fig.2c),therelativisticfocusingisonlyaminoreffect.
andthecorrespondingfour-momentumontheaccretiondisc
Insummary,thepowerlawindex,Γ,hasthestrongestinfluenceon
uµ =ut(1,0,0,Ω) (13) theirradiationprofileatsmallradii,whiletheheightoftheemitting
d d
sourcemostlyaffectstheouterpartsofthedisc.
togetherwiththephoton’smomentum(equation6),theenergyshift Finally, Fig. 2e combines all effects and shows the incident
is fluxintherestframeoftheaccretiondisc.Ingeneralthisplotcon-
glp= EEei = ppµνuuµdνh = √r(cid:0)r√rr2−+3ar(cid:1)+√h22a√−r2√hh+2a+2a2 (14) afioturmtleosrwopuahrretoisg.vheFtroasrltlraounnngidnleycrrseitraarnsaiddniignagtheoetifhgtehhteinolnafemtrhpeppasorotssutrbgcueeotmmaloemrteroysa:tnSdnoomutrcotherees
The components of pthe four-velocities are calculated photonshittheouterpartsoftheaccretiondiscandanincreasingre-
from the normalising condition u uµ = 1 (see, e.g., gionofmoreconstantirradiationatroughlyh/2iscreated.Inorder
µ
Bardeen,Press&Teukolsky1972). − tocheckoursimulationforconsistency,athoroughcheckagainst
Asthenumberofphotonsisconservedwecanwrite thecalculationsofFukumura&Kazanas(2007)wasdoneandwe
couldvalidatetheresultfromFig.2eathighprecision.The same
Ne(ph)∆te∆Ee=const.=Ni(ph)∆ti∆Ei , (15) istrueforthestationarypointsourcesolutionofWilkins&Fabian
(2012).
whereN(ph)(N(ph))istheemitted(incident)photonflux.Assum-
e i
ingapowerlawshapeoftheemittedradiation
Nph =E−Γ , (16)
e e 2.4 EmissivityProfilesintheLampPostGeometry
thephotonfluxontheaccretiondiscisgivenby
Since for a simple accretion disc the local disc emissivity is
Nph(r,a)=E−Γ g (r,a)Γ , (17) roughly r−3, in the description of observations it is common
i i · lp toparam∝etrisethediscemissivityprofilethrough
as∆E /∆E =1/g and∆t /∆t =g .Duetotherelativistic
e i lp e i lp
energyshift,theincidentphotonfluxnowalsodependsonwhere F(r,h) r−ǫ (19)
∝
thephotonhitstheaccretiondisc(r)andwhichspintheblackhole
has.Usingthisresult,wecanfinallycalculatetheincidentfluxon
theaccretiondisc
2 NotethatFukumura&Kazanas(2007)usethespectralindexα,whereas
F(r,h)=Igeo gΓ = sinδglΓp . (18) weusethephotonindexΓ.BothquantitiesarerelatedbyΓ=α+1.
i i · lp A(r,∆r)γ(φ) 3 deviationsarelessthan0.2%
(cid:13)c 2012RAS,MNRAS000,1–15
Irradiationofan AccretionDiscbyaJet 5
spqtuvwxyzxyy{|}x~(cid:127)(cid:128)~(cid:129)(cid:130)(cid:131)(cid:132)(cid:133) %p&’a(fo)co))*+,o/;</>?@AB
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(cid:138)(cid:134) (cid:0)=(cid:17)0:99 m
(cid:137) E
(cid:135)(cid:136) (cid:2)(cid:3)+(cid:4)(cid:5)(cid:4)(cid:4)
(cid:29)
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(cid:7).(cid:18)
(cid:30)
(c)
(cid:25)(cid:26)(cid:26)
(cid:19)(cid:20)(cid:24)
(cid:21)(cid:22)(cid:23) (cid:31)
(cid:144) (cid:145) (cid:27)(cid:28) (cid:19)(cid:20)3 (cid:26) (cid:24)(cid:25) (cid:22)(cid:23)(cid:23) (cid:20)(cid:21)(cid:21)(cid:21)
(cid:143)
" ["#]
Figure3.Theemissivityindexǫasdefinedinequation(19)oftheradiation
(cid:29)
irradiatingtheaccretiondiscfromaprimarysourceatdifferentheightsh.
&’ (cid:14)d(cid:15)
(cid:2)(cid:3)+(cid:4)(cid:5)(cid:6)(cid:6)
whereǫiscalledtheemissivityindex.Notethatinthisrepresenta-
(cid:29)
tiontheinformationofthenormalisationoftheemissivity profile
!"#$%
(cid:149) islost.Butasusuallytheluminosityoftheirradiatingsourceisnot
(cid:148) (cid:7).(cid:18) h(cid:30)(cid:31) rg
(cid:147) known,thisisnotveryimportantforreflectionstudies.Ourcalcu-
(cid:146)
lations easily allow to determine the radius dependent emissivity
(cid:11)(cid:12)(cid:11)*
indexandthusphraseourresultsinalanguagethatisdirectlycom-
46 parablewithobservations.
,/
Figure3showstheemissivityprofilefordifferentheightsof
(cid:25)(cid:26)(cid:26) (e)
theprimarysource.Regardlessofthespecificheight,threedifferent
7;< UVWXYZ[
radial zones are visible in all profiles. Firstly, for large radii the
NOPQRST
EFMJIJKL indexconvergesalwaystowardsitsvalueinflatspace(ǫ = 3,see
1 EFGHIJKL equation8).Theclosertheemittingsourceistotheblackhole,the
(cid:159)(cid:160) >?@AABACD fasterǫconvergestowardsthisvalue.
(cid:156)(cid:157)(cid:158)(cid:155) byaTsthroenzgonsetecelpoesneirntghaonf2thregefmroimssitvhietyblparcokfihleolteowisacrdhsartahceteirnisneedr
(cid:154) (cid:11)(cid:12)(cid:11)*
edge of the accretion disc. Except for an extremely low height,
thissteepeningisalmostindependent oftheheightoftheprimary
source4.Hence,alargeemissivityatlowradii,whichisusuallyin-
4i
,/ terpretedas“strongfocusingofalowheightemitter”,isnotdirectly
relatedtotheheight oftheprimarysource. Instead, thesteepness
‘ _ f] ^’ \] (cid:25)(cid:26)(cid:26) almost solely depends on the relativistic boosting of the primary
(cid:150) (cid:151)(cid:150)(cid:152)(cid:153) photonsandespeciallyonthesteepnessΓoftheprimaryspectrum
Figure2.Relativistic factorswhichinfluencetheincidentfluxontheac- (seeFig.2c).
cretion disccompared totheemittedintensity attheprimarysource(see As has been mentioned in the introduction (Sect. 1), many
equation 18).Ifnotstated inthefigureexplicitly weusea = 0.99and sources are observed to have very steep emissivity indices, i.e.,
assumethattheprimarysourceisanisotropicemitter.Theverticaldashed
valuesofǫ=5-10arenormal.Similartotheemissivityprofilein
linesindicatethelocationoftheinnermoststablecircularorbit(ISCO)for
Fig. 3, we can also derive the maximal possible emissivity index
certain valuesofspin.(a)Theinversebeamingfactor(red),whichdeter-
foracertainvalueofspinandsteepnessoftheinputspectrum.For
minestheinfluenceoflengthcontraction(seeequation10)ontheincident
astandardlamppostsourceataheightofatleast3r ,thisinfor-
flux.(b)Theimpact ofthe properarea. (c)Geometric intensity distribu- g
tion on the accretion disc for the relativistic (solid) and Newtonian case
(dashed).(d)Energyshift,whichthephotonexperiences whentravelling
fromtheprimarysourcetotheaccretiondisc,takentothepowerofΓ(blue). 4 Clearly, the absolute flux foracertain luminosity is highest foralow
(e)Combinedirradiatingfluxontheaccretiondiscforaprimarysourceat emitter(seeFig.2e),butweusuallydonotmeasuretheabsoluteintensity,
differentheightsbutequalluminosity. butonlytheemissivityindex.
(cid:13)c 2012RAS,MNRAS000,1–15
6 T. Dauseret al.
by Duroetal. (2011) to find a unique and consistent solution to
(cid:0)(cid:1)(cid:2)(cid:3)998 describethereflectionspectrumofCygX-1.
(cid:5)(cid:6) a=0:(cid:4)(cid:4)0
(cid:0)(cid:1)(cid:2)(cid:3)9(cid:2)(cid:2)
a=0:500 2.5 Theincidentangle
(cid:5)(cid:7)
The incident angle δ of the irradiated radiation is important for
i
modelling the reflected spectrum, as it determines the typical in-
(cid:5)(cid:8) teraction depth of the reflected photon and therefore strongly
influences the limb-darkening of the reflected radiation (e.g.,
)
((cid:15) Svobodaetal.2009).Constructinganormalvectoronthedisc,δi
(cid:14) (cid:9)
(cid:13) isgivenby
m
µ
(p ) n(θ)
6 p⊥ d µ d q
cosδi = |p| = (pd)ν(cid:16)(ud)ν(cid:17) (cid:12)(cid:12)(cid:12)θ=π/2= rutd(r,a) . (20)
(cid:10) (cid:12)
Figures5aandbshowδ fordiffe(cid:12)rentheightsoftheprimarysource
i
andassuminganisotropicprimaryemitter.Formostdiscradiithe
(cid:11) photonshitthediscatashallowangle,exceptforasmallfraction
of disc. The location and width of the steeper in-falling photons
dependsontheheightoftheprimarysource.
1 2 3 4
The effect of the incidence angle of the illumination in the
PhotonIndex(cid:12)
reflected spectrum from an accretion disc has been discussed in
Figure4.Themaximumpossibleemissivityindexattheinnerregionsof
thedisc(r<10rg)foracertainphotonindexΓ.Theheightofthesource Garc´ıa&Kallman(2010).Inthecalculationofreflectionmodels,
waschosentobeh=10rg.However,notethatweshowedinFig.3thatfor theboundaryatthesurfaceofthediscisdefinedbyspecifyingthe
h>3rgthesteepeningattheinneredgeofthediscisalmostindependent intensity of the radiation field that illuminates the atmosphere at
oftheheight. aparticularangle.Usingtheirequation(19) and(37),this canbe
expressedas
2n ξ
mation isplotted in Fig. 4. In this case, no large emissivities are I = , (21)
inc 4π cosδ
expectediftheblackholeisnotamaximalrotator,i.e.,roughlyfor (cid:16) (cid:17) i
a < 0.9. If theinput spectrum ishard (Γ < 2.5), theemissivity where n is the gas density (usually held fixed), and ξ is the ion-
indexisflatterthanthestandardindexof ǫ = 3.Evenforhighly isation parameter that characterises a particular reflection model
rotatingblack holestheactual emissivityat theinneredge of the (Tarter,Tucker&Salpeter 1969). Consequently, for a given ioni-
accretion disc is only steep if the incident spectrum is very soft sation parameter, varying the incidence angle varies the intensity
(Γ>2.5). oftheradiationincidentatthesurface.Thishasinterestingeffects
IfwewanttoapplytheinformationofFig.4toacertainobser- ontheionisationbalancecalculations.Ifthephotonsreachthedisc
vation,wehavetotakeintoaccountthattheemissivityisgenerally at a normal angle (δi = 0), the intensity has itsminimum value,
parametrisedinformof abroken powerlaw.Indetailthismeans but the radiation can penetrate into deeper regions of the atmo-
thatbelowacertain“breakradius”(r )onesingleemissivityin- sphere producing more heating. On the contrary, for grazing in-
br
dexisusedtodescribethesteepemissivityandabovethatitisusu- cidence(δi = 90)Iinc increases,resultinginahotteratmosphere
allyfixedtothecanonicalr−3behaviour.Generally,breakradiiare nearthesurface;buttheradiationfieldthermalizesatsmallerop-
foundtobeintherangeof3r (e.g.,in1H0707 495, Dauseretal. ticaldepths,whichyieldslowertemperatureinthedeeperregions
g
2012) up to 6r (e.g., in NGC3783, Brenn−emanetal. 2011).5 ofthedisc.Evidently,thesechangesintheionisationstructurewill
g
Hence,theemissivityindexwemeasureinobservations accounts alsoaffect thereflected spectrum(see Fig.5c). Thenarrow com-
fortheaveragesteepnessintherangeofr tor andwillthere- ponent of theemissionlinesareexpected tobeemittedrelatively
in br
forebelowerthanthemaximalemissivity.Wenotethattheemis- near the surface, where photons can easily escape without being
sivityindicesfoundinmanyobservationsareclosetoorabovethe absorbedorscattered.Ontheotherhand,thebroadcomponentof
maximalallowedemissivityindexasdefinedinFig.4.Forexample theemissionlines,andinparticulartheonesfromhighZelements
ǫ 5forΓ=1.8inNGC3783(Brennemanetal.2011),ǫ 6for such as iron, are produced at larger optical depths (τ 1), and
Γ≈=2inMCG 6-30-13(Brenneman&Reynolds2006),ǫ≈>6.8 thereforearemorelikelytobeaffectedbychangesinthe∼ionisation
forΓ = 1.37in−CygnusX-1(Fabianetal.2012a),orǫ 10for structureoftheslab.
Γ = 3.3 in1H0707 495 (Dauseretal.2012). Asthee≈missivity However, Garc´ıa&Kallman (2010) showed that in a gen-
index obtained from−thesemeasurements isaveraged overthe in- eral sense, reflected spectra resulting from models with large in-
nermost few r these emissivities can therefore not be properly cidence angles tend to resemble models with higher illumina-
g
explainedsolelybythelamppostgeometry.Despitetheseissues, tion. This means that the changes introduced by the incidence
however,insomecasesitispossibletouseFig.4todecideifamea- angle can be mimicked by correcting the ionisation parame-
suredvaluefortheemissivityindexcanreasonablybeexplainedin ter to account for the difference introduced in the illumination.
thelamppostgeometry.Thismethodhasbeensuccessfullyapplied The current analysis shows that below 7rg the incidence angle
can vary as much as 25 80 degrees, equivalent to a change
−
in the ionisation parameter by more than a factor of 5. Fig-
5 Notethatthevalueofthebreakradiusishighlycorrelatedwiththeemis- ure 5c shows the reflected spectra for these two incidence an-
sivityindexwhentryingtoconstrainbothbyobservation. gles predicted by the XILLVER code (Garc´ıa&Kallman 2010;
(cid:13)c 2012RAS,MNRAS000,1–15
Irradiationofan AccretionDiscbyaJet 7
30(cid:14) 4(cid:0)(cid:1) 60(cid:14) 75(cid:14)
LMj 9"(cid:1) (cid:15)b(cid:16) FGG (c) -i.>?<
(cid:6)(cid:4) (a) #(cid:0)(cid:1) -i./;<
18
1(cid:7) $%& D(cid:2)
mo
(cid:8)(cid:9) mu
QRTS (cid:8)(cid:10) UV4(cid:0)(cid:1) ‘fld
P _ H
1(cid:4) ’%&
(cid:11)
(cid:12) *(cid:0)(cid:1) (cid:20)(cid:27)(cid:21)(cid:28)(cid:22)(cid:29)(cid:23)(cid:30)(cid:24)(cid:31)(cid:25) (cid:26)! IJK
(cid:13) +, h=(cid:17)(cid:18):(cid:19)rg
(cid:5) 1(cid:4) 1(cid:3) 2(cid:2) 1(cid:4) 2(cid:2) ABC E @ (cid:5) D(cid:2) (cid:6)(cid:4) (cid:3)(cid:4)
N [NO] N [NO] WneXYyZke\^
Figure5.(a)2dimageshowingtheincidentangleδ ofphotonsontheaccretiondisc.Thespinoftheblackholeisa=0.998.(b)incidentangleforthethree
i
lines(solid,dashed,dotted)markedinsubfigure(a).(c)Samplereflectionspectrafordifferentincidentanglesδ =25◦(red)and 80◦(blue)calculatedwith
i
theXILLVERcode(Garc´ıa&Kallman2010;Garc´ıa,Kallman&Mushotzky2011).
Garc´ıa,Kallman&Mushotzky2011)usingthesameionisationpa- where e(ν) are the tetrad basis vectors for ν = t,r,θ,φ (see
µ
rameter.Theeffectoftheincidenceangleisevident. Bardeen,Press&Teukolsky1972).
Inordertocalculatethetrajectoryof aphoton emittedat an
angle δ′ from themoving source, wetransform fromthe moving
frametothestationary,locallynon-rotatingframeatthesamelo-
3 ANEXTENDEDRAY-TRACINGCODE cation.Thismeansthatthephoton isemittedatanangleδ inthe
stationarysystemaccordingto
Sofarweassumedthattheemittingprimarysourceisatrestwith
respecttotheblackhole.Iftheprimaryemitteristhebaseofthe cosδ′ β
cosδ= − , (24)
jet, however, then it is far more likely that the primary source is 1 βcosδ′
moving. Typical speeds at the jet base can already be relativistic −
depending on the velocity of the source. Following, e.g., Krolik
(McKinney 2006). Furthermore, if the irradiating source is a jet,
(1999), itcan beeasilyshown that thisapproach impliesthat the
then we need to relax our assumption of a point-like emitter and
intensityobservedinthestationaryframewillbealteredbyafactor
includetheradialextentofthejet.Asweshowinthissectionby
of
takingintoaccountbothoftheseextensions,thelineshapeissig-
nificantlyaffected. 2 , (25)
D
where isthespecialrelativisticDopplerfactor,whichisdefined
D
inourcaseas
3.1 AMovingJetBase
1
Firstinvestigationsofamovingsourceirradiatingtheaccretiondisc = , (26)
D γ(1+βcosδ)
anditsimplicationfortheFeKαweredonebyReynolds&Fabian
(1997).Beloborodov(1999)investigatedthecouplingbetweenthe with γ = 1/ 1 β2 being the Lorentz factor of the moving
movingprimarysource andthereflectedradiation. Usinggeneral source.Usingtphetr−ansformedintensity,wecannowcanapplythe
relativistic ray tracing techniques, Fukumura&Kazanas (2007) stationarycalculationsfromSect.2.3tothenewemissionangleδ
andWilkins&Fabian(2012)calculatedtheilluminationprofileon toobtaintheirradiationoftheaccretiondiskbyamovingsource.
theaccretiondisc. Additionally,wehavetocalculatetheproperenergyshiftbe-
Wefirstassumethattheemittingprimarypointsourceismov- tweentheaccretiondiscandthemovingsource,withthe4-velocity
ingataconstantvelocityβ =v/c.Themostprominenteffectofa givenbyequation(22).Similarlytoequation(14)thisenergyshift
movingjetbasecomparedtoajetbaseatrestistheDopplerboost- isgivenby
ingofradiationinthedirectionofthemovingblob,i.e.,awayfrom
theaccretion disc. Thisboosting alsomeans that theenergy shift g = (pd)µuµd = glp(β=0) . (27)
ofthephotonbetweenprimarysourceandaccretiondiscchanges lp (ph)νuνh γ 1 √(h2+a2)2−∆(q2+a2)β
withvelocity.The4-velocityisthengivenby (cid:16) ∓ h2+a2 (cid:17)
uµh =uth 1,ddrt,0,0 (22) lNimotiet,tghlapt=for lagrlgpe(βhe=igh0ts).gUlpsisnigmtphleifireessutoltsitwsespaelcreiaaldyreolabttiaviinsteidc
(cid:16) (cid:17) D
forastationaryprimarysource(seeSect.2.3),wecanfinallywrite
Thevelocityβ asseenbyanobserveratthesamelocationinthe
downthetotalfluxtheaccretiondiscseesfromamovingsource:
locallynon-rotatingframe(LNRF)isconnectedtodr/dtthrough
2F(r,h,β=0)
β= ee(µ(νrt))uuνhµh = r2+∆a2 · ddrt , (23) Fi(r,h,β)= hγ(cid:16)1∓D√(h2i+ah22)2+−a∆2(q2+a2)β(cid:17)iΓ . (28)
(cid:13)c 2012RAS,MNRAS000,1–15
8 T. Dauseret al.
spinKLfoMcoMMeNOoPdQPRSTUV {|}~(cid:127)(cid:128)(cid:129)(cid:130)(cid:131)(cid:132)(cid:130)(cid:131)(cid:131)(cid:133)(cid:134)(cid:135)(cid:130)(cid:136)(cid:137)(cid:138)(cid:136)(cid:139)(cid:140)(cid:141)(cid:142)(cid:143)
+,99 * &’ EFGG * &’
(a) /b6
(cid:23)(cid:24)(cid:24)(cid:24) (cid:10)(cid:11)3(cid:12)(cid:13)
@
(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)
(cid:0)(cid:1)25(cid:2)(cid:3)
(cid:25)(cid:26)(cid:26) h=100rg
(cid:19)
(cid:27)(cid:28) z
qw A
k
j
x (cid:29) ‘
Jlu ^v^ty B
(cid:30).(cid:31) ^__
m
\ >
! "
-% C
#$
-4
#$ D
(cid:20) (cid:19) (cid:22)(cid:17) (cid:18)(cid:17) (cid:16)(cid:17) (cid:21)(cid:15)(cid:15) (cid:14)(cid:15)(cid:15) > < (cid:27)(cid:28) ;(cid:28) :(cid:28) ?88 788
H [HI] W XWYZ
Figure6.(a)Irradiatingfluxand(b)EmissivityprofilessimilartoFig.3,whichshowtheimpactofamovingjetbasewithvelocitiesv=0c(solid),v=0.5c
(dotted),andv=0.9c(dashed).
Figure6showsthedependencyoftheemissivityindexonthe The influence of changing the location of the jet base while
velocity of the jet base. In general the irradiating flux decreases fixingthetopheightisdepictedinFig.7a,b.Interestinglytheemis-
significantlywithincreasingspeed ofthejetbase, asthephotons sivity profile is not very sensitive to the location of the jet base.
areboostedawayfromtheblackhole(Fig.6a). On the other hand, fixing the jet base at a low value (3r ) and
g
Comparingtheshapeoftheemissivityprofilesforamoving thenincreasingtheradialextentofthejet(Fig.7c)stronglyalters
jetbase(Fig.6b),theintermediateregionoftheaccretiondisc(3– theemissivityprofile.Themorethetopisawayfromthebase,the
100r )experiences anincreaseof irradiationwithincreasingve- largerthedeviationsbecomecomparedtotheprofileofthejetbase
g
locity of thejet base. On the other hand the emissivity profileof (red, dashed line) and the more the irradiation resembles the one
theveryinnerregionsoftheaccretiondiscdoesnotdependonthe fromtheupperpartofthejet.Comparingtheextendedprofilesto
movementofthejet.Forhighspin(a>0.9)theaccretiondiscex- theonesforapoint-likeprimarysource(dashedlines)revealsthat
tendsdowntotheseverysmallradiiandduetothesteepemissivity theextendedemissioncreatesanirradiationpatternthatcouldhave
most of thereflected radiation comes fromthere. If the accretion similarlybeenproducedbyapointsourceatanintermediateheight
disconlyextendsdownto6r ,asisthecasefora=0,theirradi- in between h and h , too. This implies that if we measure
g base top
ationoftheinnermostregionscandifferalmostuptoafactorof2 anemissivityprofilesimilartooneforalowsourceheight(Fig.7,
inemissivityindexdependingonthevelocityofthejetbase. dashedredline),thejetcannotbeextendedorthereisnosignifi-
cantamountofradiationfromtheupperpartsirradiatingthedisc.
Onewaytoexplainthelackofphotonsisthatthejetdoesnothave
auniformvelocity,butthejetbaseisatrestandfromthereonthe
3.2 IrradiationbyanElongatedJet
particlesareveryefficientlyaccelerated(see McKinney2006)such
It is straightforward to extend the previous discussion of a thattheradiationisbeamedawayfromtheaccretiondisc.
moving jet base to the the case of an extended jet (see also
Wilkins&Fabian2012).Wesimplydescribetheincidentradiation
for an elongated jet by many emittingpoints at different heights,
3.3 JetwithConstantAcceleration
weightedbythedistancebetweenthesepoints.Emissivityprofiles
fortheextendedjetareshowninFig.7.Ingeneral,theshapeofthe Havinganalysedtheeffectofamovingprimarysourceandthepro-
emissivityprofileinthecaseofanextendedsourcedoesnotdiffer fileofanextendedjet,weareabletocombinetheseeffectstoform
significantlyfromthatofapoint-likesource.Similartothemoving amorerealisticapproach.Itislikelythattheactualbaseofthejet
jet base(seeSect. 3.1),the irradiationof theinnerregions of the isstationaryorhasatleastavelocitynormaltothediscplanethat
accretion disc (r < 2r ) only differs in normalisation but not in ismuchlessthanthespeedoflight.Abovethejetbasetheparti-
g
shape(Fig.7).However,theregionsofthediscthatarealittlebit clesareefficientlyaccelerated(McKinney2006)tohigherenergies
furtheroutwards(>3r )areaffectedatamuchgreaterfractionby andintheendtoveryfastvelocitiesseen,e.g.,inVeryLongBase-
g
extendingtheemissionregion.Butdespitetheselargedifferences, lineInterferometry(VLBI)measurements(see,e.g., Cohenetal.
theoverallshapeoftheemissivityprofiledoesnotchange,i.e.,that 2007).Inthefollowingwewillassumethesimplestcasebyusing
thegeneralpropertiesanalysedinSect.2.4arestillvalidinthecase aconstantacceleration oftheparticles.Inthiscasethevelocity
A
ofelongatedjets. evolvesas(see,e.g.,TorresDelCastillo&Pe´rezSa´nchez2006)
(cid:13)c 2012RAS,MNRAS000,1–15
Irradiationofan AccretionDiscbyaJet 9
LMinNOfPQRPQQSTUPVdWVXYZ\^ LMinNOfPQRPQQSTUPVdWVXYZ\^ LMinNOfPQRPQQSTUPVdWVXYZ\^
(cid:18)(cid:19)99 (cid:30) (cid:28)(cid:29) (cid:18)(cid:19)99 (cid:30) (cid:28)(cid:29) (cid:18)(cid:19)99 (cid:30) (cid:28)(cid:29)
(cid:13)(cid:14) (a) (cid:31) ! DcE
(cid:1)(cid:7)(cid:8)(cid:9)(cid:10) (cid:2)3(cid:5)(cid:6) (cid:1)(cid:7)(cid:8)(cid:9)(cid:10) (cid:2)3(cid:5)(cid:6) (cid:1)(cid:7)(cid:8)(cid:9)(cid:10) (cid:2)3(cid:5)(cid:6)
hb(cid:0)se =10rg "#$%& ’67,/ > HHHHHH >
(cid:15) (cid:151)hb(cid:151)(cid:0)se(cid:151)=(cid:151)2(cid:151)5(cid:151)rg "8#8$%&8’8*8+8,/ ? mq (cid:1)(cid:1)ttoopp (cid:2)(cid:2)F(cid:3)G(cid:4)(cid:5)(cid:5)(cid:6)(cid:6) ? mq
(cid:1)top (cid:2)(cid:3)(cid:4)(cid:4)(cid:5)(cid:6) (cid:1)top (cid:2)(cid:3)(cid:4)(cid:4)(cid:5)(cid:6) vw (cid:1)top (cid:2)(cid:3)(cid:4)(cid:4)(cid:5)(cid:6) vw
(cid:16).(cid:17) w w
@ vy @ vy
Klux (cid:18)(cid:19)(cid:18)(cid:20) A vz{|} A vz{|}
~ ~
(cid:127) (cid:127)
-(cid:23) B (cid:128) B (cid:128)
(cid:21)(cid:22) (cid:129) (cid:129)
-4 (cid:15) (cid:15)
(cid:21)(cid:22)
(cid:26)(cid:27) C C
(cid:24)(cid:25)
(cid:15) (cid:13)(cid:14) (cid:11)(cid:12)(cid:12) < :; (cid:11)(cid:12)(cid:12) < :; (cid:11)(cid:12)(cid:12)
I [IJ] _ ‘_jk _ ‘_jk
Figure7.EmissivityprofilessimilartoFig.3,whichshowtheimpactofanelongatedjetcomparedtoapointsource.Forcomparisonthedashedlinesshow
theemissivityofapoint-likeemittingsourcewithsamecoloursasinFig.3,i.e.,red,blue,green,andorangefor3rg,10rg,25rg,and100rg,respectively.
Forvaryingthebaseoftheemittingregiontheirradiatingflux(a)andtheemissivityprofile(b)isshown.In(c)thetopheightofthejetisaltered.
β(t)= At , (29) Thecalculationsinthelamppostgeometryareusedtodeter-
√c2+ 2t2 minetheproperirradiationprofileandreplacetheartificialbroken
A
wherethetimetisgivenby powerlawemissivityintheRELLINEmodel.Besidesthestandard
point source inthelamp post geometry, wealsoincluded theex-
x2 x tended geometries presented in Sect. 3, i.e., elongated and mov-
t= +2 (30)
rc2 ingprimarysources. Astheinformationof theraytracingis tab-
A ulated,the RELLINE LPmodel isevaluated veryquicklyandthus
forx = h h .Thereforetheaccelerationtoreachaspecific
base wellsuitedfordatamodelling.
−
velocityβatheighthisgivenby
γ 1
= − , (31)
A h h 4.2 Influenceofthelamppostparametersontheshapeofthe
base
− reflectionfeatures
whereγistheLorentzfactor.Figure8adisplaysthevelocity(equa-
tion29)insidethejetforconstantacceleration. WiththeRELLINE LPmodelitispossibletocalculatethepredicted
Theirradiationoftheaccretiondiscinthissetupisshownin lineshapesofbroademissionlinesforthedifferentparametersde-
Fig.8b.Theeffectoftheacceleratedmovementshowsupatlarger terminingthesetupinthelamppostgeometry.Figure9showsthat
radiibysteepeningtheemissivitycomparedtothestationaryjetand thelineshapeisverysensitiveforcertainparametercombinations
the for all heights the profile gets more similar to a point source andalmostindependentinothercases.
at the jet base. Thisresult confirms the general picture that for a The line shape is highly sensitive to a change in height of
large acceleration the accretion disc sees only the lowest part, as the primary source (Fig. 9a and b). Especially when assuming a
mostoftheupperpartofthejetisstronglybeamedawayfromthe rapidly rotating black hole, the line shape dramatically changes
disc.Thismeansthatifwemeasurealocalised,lowheightofthe from a really broad and redshifted line to a narrow and double-
emittingsource,itcouldalsobethebaseofastronglyaccelerating peakedstructurewhenincreasingtheheightofthesource(Fig.9a).
jet.Additionally,Fig.8crevealsthatthestrongertheacceleration, Thesamebehaviourcanbeobservedforanegativelyrotatingblack
themoretheemissivityprofilesresemblesthecanonical r−3 case hole(Fig.9b),butheredifferencesarenotaslargeasintheprevi-
forallbuttheinnermostradii(r>2rg). ouscase.Broadlinesseenfromaconfiguration ofalow primary
source and a highly rotating black hole are also sensitive to the
photonindexΓoftheincidentspectrum(Fig.9c).Inthiscasethe
lineshape gets broader fora softer incident spectrum. When fix-
4 DISCUSSION ingtheheightoftheirradiatingsource(Fig.9d–f),foralowsource
heighttheshapeisstillsensitivetothespin(Fig.9d).Butalready
4.1 RELLINE LP—Anewrelativisticlinemodel
foramediumheightof25r thelineshapesvirtuallycoincidefor
g
UsingtheapproachmentionedinSect.2.3,thelamppostgeometry allpossiblevaluesofblackholespin.Inthiscase,evenforarapidly
wasincorporatedintheRELLINEmodel(Dauseretal.2010).This rotatingblackhole,thelineshapedoesnotdependonthesteepness
model was designed to be used withcommon data analysis tools ofincidentspectrum(Fig.9f).
suchasXSPEC(Arnaud1996)orISIS(Houck&Denicola2000)for UsingtheRELLINE LP-code,wearealsoabletocompareline
modelling relativisticreflection. It caneitherpredict asingleline shapes for moving (Sect. 3.1), elongated (Sect. 3.2), and acceler-
shapeoritcanbeusedasaconvolutionmodelsmearingacomplete ating(Sect.3.3)primarysources.Achangeinvelocityonlyalters
ionised spectrum (such as the REFLIONX model Ross&Fabian thelineshape ifthesource isat low height andtheblack hole is
2007). The new RELLINE LP model can be downloaded from rapidly rotating (Fig. 9g). Changing either of these to negatively
http://www.sternwarte.uni-erlangen.de/research/rerloltaitinneg/(.Fig.9h) oralargersource height (Fig.9i),different ve-
(cid:13)c 2012RAS,MNRAS000,1–15
10 T. Dauseret al.
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H*i+,/7:;v*J*/B7<*x[>?] L MLNO L MLNO
Figure8.(a)Velocityofthejetassumingaconstantaccelerationandthejetbaseatrest.Thespinoftheblackholeisa=0.998,andthepowerlawindex
Γ=2.Theaccelerationisparametrisedbyspecifyingthevelocityv100thejethasat100rgabovethejetbase.Linesareplottedforv100 =0.9c(dashed),
v100 =0.99c(dotted),andv100 =0.999c(dashed-dotted).(b)and(c)Emissivityprofilesforanextendedjetwiththeconstantaccelerationshownin(a).
Parametersarethesameasina).Thesolidlinedisplaystheemissivityprofileforastationaryjet.Despitethedifferentheights,eachjetisassumedtohavean
equalluminosity.
locitiesoftheirradiatingsourceonlyhaveamarginaleffectonthe Insummary,Fig.9confirmsthattherelativisticreflectionfea-
line shape. Similarly, measuring the radial extent of the primary tureissensitivetoverydifferentparametersinthelamppostgeom-
sourceisalsonotpossibleforallparametercombinations. Firstly, etry.However,assoonastheprimarysourceisnotveryclosetothe
therestrictionsofmeasuring aandΓforlargerheights,asseenin blackholeoriselongatedintheradialdirection,thedependencyon
Fig.9eandf,alsoapplyhere.Thereforewefixthesetwovaluesat parameterssuchasthespinoftheblackholeortheincidentspec-
a = 0.99andΓ = 2.0.Settingthebaseofthesourceat3r and trumisnotlarge.
g
alteringitsheight (Fig.9j) doesindeed result ingreat changes in
thelineshape.Thesechangesareverysimilartoachangeinheight
ofapoint-likeprimarysource(seeFig.9a).Hence,anelongatedjet
4.3 ImplicationsforMeasuringtheSpinofaBlackHole
producesareflectionfeaturesimilartoapoint-likesource withan
effectiveheight.Asimilarbehaviourcanbeobservedwhenchang- If we are interested in measuring the spin of the black hole by
ing the base of the primary source and leave the upper boundary analysing the relativistic reflection, Fig. 9 and the discussion in
constantat100rg.However,asthelargeupperpartnowdominates theprevioussectionhelpustodecideunderwhichconditionswe
theirradiationoftheaccretiondisc,theprofileisnotverysensitive arecapable in doing so. Generally, these conditions can be sepa-
to the location of the base of the source. Finally, an accelerating rated into two classes: Either the primary source is compact and
jet(Fig.9l)onlyinfluencesthelineshapeifalreadyrelativelyhigh veryclosetotheblackhole,ortheirradiatingemissioncomesfrom
velocitiesareobtainedataheightof100rg (v100 > 0.99c),asin largerheightsoranelongatedstructure.Ashasbeenshowninthe
theupperpartofthejetmoreandmorephotonsarebeamedaway previoussection,thelattercase,alargeheightoftheprimarysource
fromtheaccretiondiscduetothehighlyrelativisticmovementof and an elongated structure, produces avery similarirradiation of
the emitting medium. The line profiles for a strong acceleration theaccretion disc and therefore weuse followingpicture forthis
thusresembleverycloselytheonesforalowertopoftheemitting discussion:Thebaseoftheprimarysourceisalwayslocatedat3r
g
source(Fig.9j),whichmeansthatinrealitywewillnotbeableto andsimplythetopheight oftheelongatedstructureisallowedto
measureiftheemittingsourceisacceleratingwithoutknowingthe change.Forsimplicitywecallita“compact jet”ifthetopheight
fullgeometryofsuchaprimarysource. ofthesourceiscloseto3r and“extendedjet”forlargervaluesof
g
In some sources with a low mass accretion rate, the disc thetopheight.
mightbetruncatedfurtherawayfromtheblackholethantheISCO Asanexample,Fig.10showshowalineprofilefordifferent
(see Esin,McClintock&Narayan 1997 and, e.g., the observa- spinwouldlooklikeforacompactandextendedjet.Inthecaseof
tionsbyMarkowitz&Reeves 2009;Svoboda,Guainazzi&Karas acompact jetthesensitivityonthespinishigh(seeFig.9d)and
2010). Generally, relativistic emission lines from such truncated thereforewecanclearlyseethedifferenceofareallybroadlinefor
discsareevennarrowerthanfornonornegativelyspinningblack highspin(Fig.10a)comparedtoamuchnarrowerlineforlowspin
holes.However,suchlinescanalsobeexplainedbytheirradiation (Fig. 10b). If the irradiating source is elongated, the line profiles
fromanlargelyelevated (Chiangetal.2012)orelongated source forhighandlowspinareverysimilarandespeciallyreallynarrow
(Fig.9k). (Fig. 10c,d). At first this underlines what was already discussed
Besidesthespin,theinclinationofthesystemhasalsoastrong aboveinthecontextofFig.9:Ifwehaveanelongatedsourcewe
effectonthelineshape(see,e.g., Dauseretal.2010).Astheincli- wewillnotbeabletoconstrainthespinatall.
nationismainlydeterminedbydefiningthemaximalextentofthe ButthediagraminFig.10revealsamuchlargerproblemwe
lineattheblueside(Fig.9),thesteepdropisalwaysatthesame havetodealwith.NamelythreeofthefourlinesintheFigurewill
locationforafixedinclinationandcanthereforebedeterminedal- bedetected asnarrow linesand when fittedwithany model sim-
mostindependentlyofthegeometry. ulating broad lines, will result in very low spin values. But this
(cid:13)c 2012RAS,MNRAS000,1–15