Variability of X-ray binaries from an oscillating hot corona

0.56 MB·English

by  C. Cabanac

Mon.Not.R.Astron.Soc.000,1–12(2009) Printed13January2010 (MNLATEXstylefilev2.2)
Variability of X-ray binaries from an oscillating hot corona.
C. Cabanac1,2,3⋆, G.Henri3, P.- O Petrucci3, J. Malzac1, J. Ferreira3 and T. M. Belloni4
0 1Centre d’E´tude Spatiale des Rayonnements, CNRS-UPS, 9 Avenue du Colonel Roche, 31028 Toulouse Cedex 4, France
1 2School of Physics and Astronomy, Universityof Southampton, Southampton SO17 1BJ, UK
0 3Laboratoire d’Astrophysique de Grenoble–Universit´e Joseph-Fourier/CNRS UMR 5571 –BP 53, F-38041 Grenoble, France
2 4INAF-Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate (LC), Italy

n
a
J Accepted 2010January12.Received2010January12;inoriginalform2009September 09

3
1
ABSTRACT

] The spectral and timing properties of an oscillating hot thermal corona are investi-
E
gated. This oscillation is assumed to be due to a magneto-acoustic wave propagating
H withinthecoronaandtriggeredbyanexternal,nonspecified,excitation.Acylindrical
. geometry is adopted and, neglecting the rotation, the wave equation is solved in for
h
differentboundaryconditions.TheresultingX-rayluminosity,throughthermalcomp-
p
tonization of embedded soft photons, is then computed, first analytically, assuming
-
o linear dependence between the local pressure disturbance and the radiative modu-
r lation. These calculations are also compared to Monte-Carlo simulations. The main
t
s results of this study are: (1) the corona plays the role of a low band-pass medium,
a
its response to a white noise excitation being a flat top noise Power Spectral Den-
[
sity (PSD) at low frequencies and a red noise at high frequency, (2) resonant peaks

1 are present in the PSD. Their powers depend on the boundary conditions chosen
v and, more specifically, onthe impedance adaptationwith the externalmedium at the
6 corona inner boundary. (3) The flat top noise level and break as well as the resonant
1 peak frequencies are inversely proportional to the external radius r. (4) Computed
j
1
rmsandf-spectraexibitanoverallincreaseofthevariabilitywithenergy.Comparison
2
with observed variability features, especially in the hard intermediate states of X-ray
.
1 binaries are discussed.
0
Key words: Accretion, accretion discs – X-rays: binaries.
0
1
:
v
i
X
1 INTRODUCTION abilityishighandthePSDharbouraBand LimitedNoises
r (BLNs) shape extending up to a break frequency ν . Large
a X-ray binaries (XRBs) exhibit large variability on various b
peaks,thesocalled Quasi-PeriodicOscillations (QPO),can
timescales. While their spectral states and their accretion
also be observed. In the soft state the Poissonian noise is
ratesaretypicallychangingfromweekstodays(seee.g.the
usually dominating on all the frequency range and no or
differentcanonicalstatesobservedin blackholebinaries, as
weak QPOs are detected. Different types of QPOs (called
definedbyMcClintock & Remillard2003,Homan & Belloni
A, B and C) can be identified depending on the value of
2005,Belloni2009andreferencestherein),theirlightcurves
theirfrequency,theirstrength andeventheirtimelags (see
exhibit drastic changes from hours to millisecond. Several
againvan der Klis2004orCasella et al.2004andreferences
toolsarenowusedinordertoanalysethesetimingfeatures
therein). However, if QPOs are remarkable features, the
(see e.g. van derKlis 2004 for a review) such as e.g. cross-
major partof thevariability isusually aperiodic. Asfor en-
correlation between different energy band (which allows to
ergyspectra,variabilityevolvesduringtime:ingeneral, the
infer the so-called time-lags), but the most commonly used
overall variability decreases as the spectrum softens, with
is still the Fourier analysis via the computation of Power
frequenciesincreasing,untilinthesoftstatewheretheBLN
Spectral Distribution (hereafter PSD).
reachesalowlevel(seee.g.Belloni et al.2005;Belloni2009).
Despite its known limitations (e.g. signal phase lost in
the analysis), any attempt to model the physics of XRBs
hastotakeintoaccountthevariousfeaturesobservedinthe Several models intending to interpret variability
PSDanditsevolutionwhenthesourcetransitsfromastate features focuses on the QPO phenomena. For the high
toanother.Intheso-called hardstate,thelevelof thevari- frequency QPOs (ν > 100 Hz), lense-thirring precession
(seee.g.Stella & Vietri1998)orbeatingfrequencybetween
particularorbits(Lamb & Miller2003)havebeenproposed.
⋆ E-mail:[email protected](CC) For the low frequency QPOs (0.01 < ν < 100 Hz), fewer
2 C. Cabanac, G. Henri, P.-O. Petrucci, J. Malzac, J. Ferreira, T. M. Belloni
z
δP
0
δP0 (cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)z(cid:0)(cid:1)=(cid:0)(cid:1)0(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)C(cid:0)(cid:1)O(cid:0)(cid:1)(cid:0)(cid:1)R(cid:0)(cid:1)O(cid:0)(cid:1)N(cid:0)(cid:1)A(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)r(cid:0)(cid:1)h δP0
δP
0
0 r r
i j
Figure 1.Sketchofthemodelgeometry.Thecoronaisassumedtoharbourcylindricalsymmetry.Itextends fromitsinnerradiusri to
itsouterradiusrj anditsheightishc.Non-specifiedwhitenoiseexcitations areassumedtooccuratrj thentriggering(magneto-)sonic
wavesinthecorona.TheresultingX-rayluminosityisthencomputedthroughthermalcomptonization ofembeddedsoftphotons.
models are available (see e.g. Tagger & Pellat 1999 or ence between the radial and vertical epicyclic frequencies
Titarchuk & Shaposhnikov 2005, but also Stella & Vietri where half the forcing frequencies.
1998),but theyusually try to explain theobserved correla- One of the very first attempt to evaluate the effect of
tion between frequencies and/or other observables, without propagation in a comptonising region on the timing be-
taking into account the whole emitting process. However, haviour was examined by Miyamoto et al. (1988). They
it is worth noting that it is in the highest energy bands showed that the tight period dependence of the observed
that the X-ray flux is observed as being highly variable. In time lags in Cyg X-1 could not be accounted for by the
contrast in most of theavailable models, thesource of vari- inverseComptonscatteringsprocessonly.Inanotherframe-
ability lies in the geometrically thin accretion disc, which work,Z˙ycki& Sobolewska(2005)testedtheirmodelofvari-
emit mainly at lower energies. Note that some observations ability (Z˙ycki 2003) where the variable emission responsi-
also suggest directly that the disc is less variable than the blefor thenoise component is attributed tomultiple active
corona (seee.g.Churazov et al.2001,Rodriguez et al. 2003 regions/perturbations moving radially towards the central
howeversee also Wilkinson & Uttley 2009). In consequence black hole. The QPOs are obtained by modulating either
a proper model for variability in BHB has to deal with the thereflection amplitude, the heating rate, the covering fac-
radiative transfer between the disc and thecorona. tor of the reprocessor or the column density value. They
predictfor eachof thesecases thecorrespondingpower and
f spectra (for a definition of f-spectra, see Revnivtsev et al.
A few models have already been proposed in the liter- 1999), and the time lag energy dependencies. In a subse-
ature.Forinstance, afullmodelling oftheBLN component quentpaper,Sobolewska & Z˙ycki(2006)testedtheirmodel
has been attempted by Misra (2000) in the framework of on real data and concluded that within this framework the
a “transition disc model”: following Nowak et al. (1999)’s QPO spectra in the hard state are always softer than nor-
idea, an acoustic wave is propagating within the accretion malaveragespectraandcolddiscoscillationsmightthenbe
disc.Inthisframework,theauthorconsidersonlythepropa- responsible forthelow frequencyQPO.Ontheotherhand,
gationtooccurinonedirectiontowardsacentralsink.This when the energey spectra gets softer, the QPO spectra are
modelisthenusedtoexplainqualitativelythegeneralshape harder and the low frequency QPO might then originate
of the power spectra and lag energy dependencies observed from thehot plasma.
in Cyg X-1. Ina morerecent study,Schnittman et al. (2006) tryto
Psaltis & Norman (2000) tried to model the filter effect of model the oscillation of a torus in Kerr metric, via three
anarrowannulusinageometrically thindisc.However,the radiative processes: a thin emission line then a thick one,
natureof thisannulusisnot specified anditstypicalexten- andfinallyanopticallythickthermalemissionprocess.They
sion is δr/r < 10−2. They also neglect the contribution of manage to reproduce some of the the properties of C-type
theradialpressureforces. Inthisframework anddepending QPOs, especially the observed increase of the amplitude
onthemodeofoscillation chosen,theexternalperturbation with theinclination of thesystem.
is shown to be modulated in amplitude according to the Inthispaperwepresentanewapproachthatdealswith
exciting frequency. The square of the response in pressure theradiativetransferintoanoscillatingcorona.Theseoscil-
exhibits Lorentzians which could account for the observed lations are assumed to be due to a magneto-acoustic wave
behaviourinBHorNSbinaries.Inasimilarspirit,Lee et al. propagating within the corona, modulating the efficiency
(2004) studiednumerically theresponse to radial perturba- of the comptonisation process on embedded soft photons.
tions in an accretion torus. They showed that resonances These basic ingredients give a promising framework to re-
could occur and become larger when the frequency differ-
Variability of X-ray binaries from an oscillating hot corona. 3
produce the main timing features of the X-ray binary in 2.2 Wave equation
hardandhard-intermediatestatessuchastheBandLimited
In cylindrical 1D geometry, in absence of local damping or
Noise continuum and C-type LFQPO. The assumptions of
excitation,andneglectingtherotation,thebasicwaveprop-
themodels and wave equation solutions are detailed in sec-
agation equation can be written as:
tion 2. An analytical study in the linear approximation is
discussedinsection3andcomparedwithMonte-Carlosim- 1 ∂ ∂p 1 ∂2p
r − =0,. (2)
ulations in section 4. We finally discuss the main results of r∂r ∂r c2 ∂t2
(cid:18) (cid:19) s
thistoymodelandthecomparisontoobservationinsection
p(r,t) is the perturbation in pressure given by, in complex
5.
notation and for a given frequency ν =ω/2π:
p(r,t)=p (r)expı(−ωt). (3)
r
ω
Introducing the new variable x= r and putting Eq. 3 in
c
s
Eq.2, we get:
2 THE MODEL
d2p dp
2.1 Basic assumptions x2 dx2r +xdxr +x2pr =0 (4)
The general structure of the model is sketched in Fig. 1. A The general solution of those equations are linear combina-
hot optically thin cylindrical medium is assumed, hereafter tion of Hankel’s function of the zeroth order H1 and H2
0 0
called the “corona”. It is limited radially by its inner and (Abramowitz & Stegun 1964):
outer radii r and r respectively. This corona has thus a
i j p(x,t)= αH1(x)+βH2(x) e−ıωt, (5)
ring shape of typical height h . This geometry agrees with 0 0
c
theobservationsthatsuggesttheX-raycoronatobeclosely αandβbeingdeterm(cid:2)inedbythebounda(cid:3)ryconditions.Note
linkedtothebaseofthejetinXRB(e.g.Markoff et al.2001; that the above solution is similar to the one obtain by
Fenderet al. 2004; Markoff et al. 2005). Then the inner ra- Nowak et al. (1999), or Titarchuk & Shaposhnikov (2005)
dius ri can be identified as thelast stable orbit close to the despitethefact thatthoseauthorsused onlyoneinstead of
central compact object. On the other hand rj can be com- a linear combination of both Hankel’s function as solution.
pared to the transition radius between an outer standard
accretion disc and the inner hot corona, a geometry com-
2.3 Boundary conditions: total reflection at the
monly invoked in XRB(e.g. Esin et al. 1997, Ferreira et al.
internal radius
2006,Done et al. 2007).
Inthispaper,nohypothesisisdoneonthephysicalori- Attheexternalradiusofthecoronar,weassumeaconstant
j
gin of these radii. For simplicity, the corona is assumed to excitation p 1 (white noise hypothesis):
0
have,atrest,aconstanttemperatureT anddensityn and
consequentlyaconstantpressureP0.W0ethenconsider0that p(xj,t)= αH01(xj)+βH02(xj) e−ıωt =p0e−ıωt. (6)
pressure instabilities at the external radius r generate a
j where xj = x(cid:0)(r = rj). The clos(cid:1)ure relationship will be
sound wavewithin this thermalised plasma. Theorigin and
given by the behaviour of the wave at the internal radius
the nature of those instabilities are not discussed in this
r. Wewill, as a primary assumption, consider that there is
i
paper since we focus only on the radiative response of the
notransmission ofthewaveandhencetotal reflectionin r.
i
corona. We assume these instabilities to have a white noise
Ifr isequalorclosetotheLastStableOrbit,onewouldex-
i
spectrum(i.e.sameamplitudeforallexcitationfrequencies)
pectindeedthedensityofthecoronatodropquicklyinside
which corresponds to a Dirac perturbation in the temporal
thisradius and hencethepressure as well. It thusgives:
domain,i.e,welimitourstudytothecoronatransfertfunc-
tion . For seek of simplicity we also restrict our calculation p(xi,t)= αH01(xi)+βH02(xi) e−ıωt =0. (7)
to the 1D case i.e. the wave will only propagate radially at
where xi = x(r =(cid:0)ri).The system of E(cid:1)qs. 6 and 7 can be
thesound velocity:
solved in order to obtain α and β:
cs =skmTp0 ≃3.1×108(cid:18)100T0keV(cid:19)1/2cm.s−1. (1) α = p0H02(xi)H01(x−HjH)02−1((xHxi)0)1(xi)H02(xj) (8)
β = p 0 i (9)
In the last expression T is in keV. It is important to note 0H2(x )H1(x )−H1(x )H2(x )
0 0 i 0 j 0 i 0 j
that there are simplifications which make this paper only a
firststep.Onceadditionalcomplicationssuchasincorporat-
ing the effects of rotation are introduced, they might lead 3 ANALYTICAL SOLUTIONS IN THE LINEAR
tosomerevisionofthescenariopresentedhere.Thiswillbe AND ZERO-PHASE APPROXIMATIONS
addressed in a forthcoming paper. Note however that the
In this section, we infer the shape of the expected PSD in
nature of the corona and hence its exact rotation profile is
the simple case where the radiative response of the corona
still mainly unknown.
dependslinearly on thelocal perturbation. This appears to
Finallywesupposeablackbodyseedphotonfieldoftemper-
atureT emittedisotropicallyatthecoronamidplane(i.e.
seed
inz =0seeFig.1).Theseseedphotonswillbecomptonized 1 Note the difference between p0, the pressureperturbation im-
in thecorona then producing a variable X-ray emission. posedinr=rj andP0 thecoronapressureatrest
4 C. Cabanac, G. Henri, P.-O. Petrucci, J. Malzac, J. Ferreira, T. M. Belloni
10+0 10+0 10+0
ν−2 10−1
2P (%rms)ν111000−−−642 ν2*P (%rms)νc1111100000−−−−−65432 2P (%rms)ν111000−−−642
10−8 ν/10−7 10−8
10−10 10−8 10−10
10−9
0.001 0.01 0.1 1.0 10.0 100.0 1000. 0.001 0.01 0.1 1.0 10.0 100.0 1000. 0.001 0.01 0.1 1.0 10.0 100.0 1000.
ν/νj ν/νj ν (Hz)
Figure 2. Left: Power spectrum plotted in reduced frequency xj =ν/νj (with νj =2πcs/rj) for a singlezone corona with ξ =10−2,
Clin = 1 and δP0/P0 = 0.1. Center: same spectrum plotted in νPν. Right: PSD plotted in true frequency and its evolution when
rm(%jo=rvmin6sg8)02r(j@Rignνw==ar10d0)s9solcingmlhy,.tlryHededr:eecr,rjeMa=se=2s0a01n0dRMgνb⊙=i,n3crri×ea=1s0e2s8wRcmhge,n=btlhu3ee×:erx1jt0e=6rnca6ml8rRaadngidu=skoT1f00t8h=ecm1c0o,0rgornkeeaeVnd:e(crhrjeen=acsee1s4,cstRh=ogu=g3h.12t×h×e11e00ff87eccctmmi.s.s−t(iN1n)yo.thBeeltrahec.ak)t:.
give analytical results in good agreement with our Monte with L =πη h (r2−r2)=πη h x2c2s(1−ξ2), and:
Carlo simulations detailed in thenext section. 0 0 c j i 0 c jω2
The corona being optically thin, the luminosity dL0 ξ≡ ri = xi. (15)
emitted locally at rest in a ring of radius r, width dr and r x
j j
height h is proportional to its emissivity per unit volume
c Note that relationship 14 is only valid if the emissivity per
η :
0 unit volume η of the corona, at rest, is uniform, which
0
dL (r)=η 2πrdrh . (10) is a direct consequence of our assumptions of constant
0 0 c
temperatureand density T and n .
0 0
Inpresenceofthesonicwave,theluminositydL(r,t)ofthis
ringvariesintimeandwewillmaketheassumptionthatits
relativevariation (dL−dL )/dL isalinearfunction ofthe
0 0
relative pressure perturbation p(r,t)/P : 3.1 Power Spectra obtained with large vertical
0
wavelength and total reflection in r
dL(r,t)−dL (r) p(r,t) i
0 =C . (11)
dL0(r) lin P0 UsingEq.5,theexpressionoftheperturbationinluminosity
C is the constant of proportionality which is in complete L∼ becomes
lin
generality a function of the perturbation frequency. How- 2L C e−iωt xj
ever, if the delays implied by the multiple diffusions of the L∼ = 0P linx2(1−ξ2) αH01(x)+βH02(x) xdx.
photoninsidethecoronaarelow compared totheperiod of 0 j Zξxj
(cid:0) (cid:1) (16)
the wave, little phase delay is expected between the pres-
TheHankel’sfunctionspresentinαandβ(seeEqs.8and9)
sure wave and the luminosity. Hence C will be a real
lin are easily integrated (see Abramowitz & Stegun 1964) and
number independent of ω. This is what we call the “zero-
hencetheprevious equation leads to:
phaseapproximation”.Itisusuallyverifiedinopticallythin
ep.lga.smMaalwzahcer&eJpohuortdoanisnt2r0a0v0el).aTfheewnhacrobuegfohreesetsimcaaptiengof(tsheee L∼ = 2L0CPlinp0x2(e1−−iωtξ2)Mxj,ξ (17)
0 j
time spent by the photons inside the corona compared to
with,
theperiod 2π/ω of the wavegives:
H2 (H1 −ξH1 )−H1 (H2 −ξH2 )
2ωπhcc = 21πhrc ccsx (12) Mxj,ξ =xj 0,i 1,j H02,iH10,1i,j−H001,,iiH021,j,j 1,i . (18)
It is generally much smaller than 1 in the cases we are H2 and H1 are the Hankel’s function of first order and the
1 1
interested in (i.e. corona aspect ratio h /r < 1 and corona
c index i or j corresponds to the point where the Hankel’s
temperature of a few tens to hundred of keV) unless x
function is evaluated i.e. x or x respectively. It is also
i j
becomes of theorder of 100 or 1000.
easy to demonstrate that the function M is real in case of
total reflection (both numerator and denominator are pure
Combining Eqs 10 and 11, we obtain:
imaginaries and hencethe ratio is real).
p(r,t)
dL(r,t)=η0(cid:18)1+Clin P0 (cid:19)2πrdrhc. (13) The PSD Pν ≡|L∼|2/L20 can then be directly deduced
from Eq. 17:
Previous equation integrated on the whole volume of the
corona gives: p M(x ,ξ) 2
L=L0+η02πhPc0Clin Zrirjp(r,t)rdr, (14) It is direcPtlνy≡pr|oLp∼o|r2t/ioLn20a=l, d(cid:12)(cid:12)(cid:12)(cid:12)2uCeltinoPo00urx2jli(n1e−ajrξa2s)s(cid:12)(cid:12)(cid:12)(cid:12)um.ption(,19to)
the input perturbation through the term 2C p /P but it
≡L∼ lin 0 0
| {z }
Variability of X-ray binaries from an oscillating hot corona. 5
0.20 3.2 Wave transmission in r
i
The oscillations present above the break frequency in the
PSD (see Fig. 2) are due to the infinite resonances at the
0.15 eigenfrequenciesofthecoronaduetotheassumptionoftotal
ξ)| reflection in ri and no wave damping. This total reflection
,
0 hypothesiscanberelaxedbyassumingthatpartofthewave

>
− 0.10 istransmitted in ri inamedium of differentsoundvelocity.
x j Animpedanceadaptation thenoccursbetweenthetwome-
P( dia, depending on the value of the acoustic impedance of
| the system Z =c /c , i.e. the ratio of the corona sound
0.05 s,2 s,1
speed c to the sound speed c below r . The general
s,1 s,2 i
form of the solution in the corona (medium 1) is similar to
theoneobtainedbefore(Eq.5)andhencecanbewrittenin
0.00 reduced unitsx=ωr/c as:
s,1
0.0050.01 0.02 0.05 0.1 0.2 0.5 1.0
p (x,t)= αH1(x)+βH2(x) e−ıωt, (21)
1 0 0
ξ
whereas,inthemediu(cid:2)m2,thetransmitted(cid:3) waveisonlypro-
gressive and hencehas thefollowing form:
Figure 3.Dependency ofthe reduced PSD power atlowest fre-
x
quencies (xj → 0) PxjC(lxinjPp→00 0) (see Eq. 19) in function of the Inordertoconstrpa2i(nxα,t,)β=anζHd0ζ1,(cid:16)wZe(cid:17)neee−dıωnt.owthreediffe(r2en2)t
ratioξ=ri/rj. equations.Thefirstonescomesfromthepressurecontinuity
in r and r :
i j
x
αH1(x )+βH2(x ) = ζH1 i , and (23)
0 i 0 i 0 Z
is also modulated by the intrinsic response of the corona
through the function M(xj,ξ)/(x2j(1−ξ2)). αH01(xj)+βH02(xj) = p0. (cid:16) (cid:17) (24)
Inthezero-phaseapproximation(seeabove)Clinisreal The mass conservations in ri gives thethird relation:
and we can find asymptotic expressions of the PSD for low
(ρ S v ) =(ρ S v ) , (25)
and high frequencies. For low frequencies, the function M 1 1 1 ri 2 2 2 ri
can beapproximated to (see appendix A): with S1 = 2πrih1 (respectively S2 = 2πrih2) being the
vertical surface in r in medium 1 (respectively medium
i
M(x ,ξ) ∼ x2 1 − ξ2−1 . (20) 2) and v1 (respectively v2) the corresponding flow velocity.
j xj→0 j(cid:18)2 4ln(ξ)(cid:19) The link between vm and the pressure perturbations pm in
each medium is obtained by applying the Euler equations
Hence, the PSD P (which is inversely proportional to x2) (m={1,2}):
ν j
tendstoaconstant.Athigh frequencies, theHankel’sfunc-
∂v
tionstendtocosinefunctionswhoseamplitudesarepropor- ρm ∂tm =−∇~pm. (26)
tional tox−1/2 and thusP ∝x−2.
j ν j Hence,by using Eqs. 21 and 22, we get:
Examples of PSD given by Eq. 19 are plotted in Fig.
ω
2 in function of the reduced frequency xj = ν/νj, with −iωρ1v1 = c αH11(x)+βH12(x) and (27)
ν = 2πc /r . As expected a flat-top noise component at s,1
j s j
ω (cid:2) x (cid:3)
low frequencies and a red noise at high frequencies are −iωρ v = ζH1 . (28)
present. The break frequency scales like νb∼2.5νj ∝cs/rj 2 2 Zcs,1 h 1(cid:16)Z(cid:17)i
when ξ → 0 and νb ∼πνj/(1−ξ) when ξ → 1 (see some Combining these two equations with Eq.25 give then:
examplesplottedinrightpanelofFig.2).Todeterminethis S S x
value of the break frequency, we fitted the analytical PSD c 1 αH11(xi)+βH12(xi) = Zc2 ζH11 Zi . (29)
s,1 s,1
obtainedwithazerocenteredLorentzian,followingthedefi- We need(cid:2) then to constrain t(cid:3)he coronahheigh(cid:16)t in(cid:17)iboth side
nitionofBelloni et al.(2002).ThePSDalsoexhibitsapeak
ofr .Wethereforeassumethatthecorona isinhydrostatic
around the break frequency and oscillations above it as ex- i
equilibrium and in Keplerian motion. Consequently, h/r =
pected from Hankel’s function. The ratio between the fre-
c /v (r) in each medium. As a result, at the internal
quencies of the first peak and the second one is predicted s Kepl
radius r , S /c = 2πr h /c = 2πr2/v (r ). This
tobeclose to2.Notetherefore thatthefrequenciesdepend i m s,m i m s,m i Kepl i
latter value is independent from the value of m and hence
onthevalueoftheouterradiusoftheinnerregion.Wealso
we obtain that S /c = S /c = S /(Zc ). The mass
noticedthatinourmodeltheQPOpeaksaroundthebreak 1 s,1 2 s,2 2 s,1
conservation (Eq.29) therefore reduces to:
frequency.
x
Moreover,thePSDplateau,atlowfrequencies,slightly αH1(x )+βH2(x )=ζH1 i , (30)
1 i 1 i 1 Z
decreaseswhenξincreases.Thisisduetothebehaviorofthe
(cid:16) (cid:17)
modulationfunctionM(x ,ξ)/(x2(1−ξ2))atlowfrequency The resolution of the previous system gives then the full
j j
(i.e. when x tends to 0). It is plotted versus ξ for x = 0 solution in pressure within thecorona:
j j
in Fig.3and showsadecrease from about 25% for ξ=0to AH1(x)+BH2(x)
p (x,t)=p 0 0 e−ıωt, (31)
about 5% for ξ=1. 1 0AH1(x )+BH2(x )
0 j 0 j
6 C. Cabanac, G. Henri, P.-O. Petrucci, J. Malzac, J. Ferreira, T. M. Belloni
10+0 10+0
10−2 10−2
2 2
s)10−4 s)10−4
m m
%r %r
P (ν10−6 P (ν10−6
10−8 10−8
10−10 10−10
10−6 10−4 10−2 10+0 10+2 10+4 10+6 10−6 10−4 10−2 10+0 10+2 10+4 10+6
ν/νj ν/νj
Figure 4.Left:power spectrum obtained forthe coronawhen thesound velocities inmedia1and2arethesame(Z =cs,2/cs,1 =1).
Right:PSDevolutionwhenchangingthevalueoftheacousticimpedance. Thickgreycurve:Z =1.2,black:Z=10.
with, thecorona mid plane(z =0).The temperatureand optical
depthofthecoronaatrestarefixedtoT andτ respectively
ξx ξx 0 0
A = H12(ξxj)H01 Zj −H02(ξxj)H11 Zj and,(32) implying a uniform pressure P0. The wave propagation be-
(cid:18) (cid:19) (cid:18) (cid:19) ingsupposedadiabatic,theperturbationinopticaldepthδτ
B = H1 ξxj H1(ξx )−H1 ξxj H1(ξx ). (33) and temperature δT are given by:
1 Z 0 j 0 Z 1 j
(cid:18) (cid:19) (cid:18) (cid:19) δT γ−1 p
= , and (38)
In thelinear approximation, we can obtain theouting PSD T γ P
0 0
by following the same steps as in section 3.1. The power δτ 1 p
= . (39)
spectrum from the main corona is therefore very similar to τ γP
0 0
Eq. 19:
Then we impose sine perturbations in r = r with dif-
j
P = 2C p0 M1(xj,ξ) 2, (34) ferent frequencies but same relative amplitudes ǫ ≡ p0/P0
xj,1 linP x2(1−ξ2) (whitenoise).Weassumetotal reflection at theinternalra-
(cid:12) 0 j (cid:12)
(cid:12) (cid:12) diusand the pressure (and hencetemperature and density)
with (cid:12)(cid:12) (cid:12)(cid:12) profile that the photon encounters while travelling within
C+D thecoronaisthengivenbyEqs.5,8and9.Forthosesimu-
M (x ,ξ) = x , (35)
1 j jAH1(x )+BH2(x ) lations,boththelinearandzero-phaseapproximationstud-
0 j 0 j
C = A H1(x )−ξH1(ξx ) and (36) ied in the previous section were released. Hence the pho-
1 j 1 j ton will encounter during its travel, and scatterings after
D = B(cid:2)H12(xj)−ξH12(ξxj)(cid:3), (37) scatterings, either some part of the corona where the per-
turbation in pressure is positive, or negative. The number
A and B being giv(cid:2)en by Eqs. 32 and 33(cid:3).
of positive and negative zones in the corona is linked to
SomeexamplesofPSDareplottedinFig.4fordifferent
the wavelength and hence the exciting frequency. The full
valuesoftheacousticimpedanceZ =c /c .Asexpected
s,2 s,1
Klein-Nishina cross section is taken into account. The tem-
theresonancesarecancelledwhenthereisimpedanceadap-
poral evolution of the corona during the photon motion is
tation (i.e., Z = 1) between both media (see left panel of
also fully taken into account but appears to be negligible
Fig. 4). The sound wave is in that case totally transmitted
compared to the wave frequencies used, in agreement with
in r . Then thestrength of theresonances are tunedby the
i
ourzero-phaseapproximationadoptedinSect.3.Wefinally
impedance value as soon as it is different from unity and
fit the corona X-ray emission with a sine function in order
the higher the difference between both sound velocity, the
stronger the resonances. Two PSD examples with Z = 1.2 to obtain its amplitude L∼ and potential phase delay. We
repeat this procedure for different frequencies in order to
and Z =10 are displayed on theright panel of Fig. 4).
build a power spectrum.
4.2 Results
4 MONTE-CARLO SIMULATIONS
Some examples of power spectra are plotted in Fig. 5
4.1 A linear comptonisation code
for different values of the external radius r and different
j
In order to check the validity domain of the linear approx- energy band indicated on the different figures. The other
imation that is used in the previous section, we performed parameters are r = 3×106 cm, kT = 100 keV, τ = 1.4
i 0 0
Monte-Carlo simulations of the radiative response of the andkT =0.25keV.Theamplitudeofthemodulationin
seed
corona.Thecomptonisationcodeusedislinear,i.e,nofeed- pressure is set to an arbitrary value ǫ=0.15.
back of the computed high energy flux on the state of the
corona is taken into account. The code uses the weighted For such low value of ǫ, the PSD behaviour appears in
Monte-Carlo technique (see e.g. Pozdniakov et al. 1983). A good agreement with the linear hypothesis and zero-phase
monothermalblackbodydistributionoftemperatureT is approximation studied in the previous section. Note that,
seed
assumed fortheseed photonswhich arerandomlydrawnin thanks to the Monte-Carlo simulations, we are also able to
Variability of X-ray binaries from an oscillating hot corona. 7
10+0 10+0 10+0
a b c
10−1 10−1 10−1
2moy)]avg10−2 2moy)]avg10−2 2moy)]avg10−2
s/ s/ s/
m m m
D [(r10−3 D [(r10−3 D [(r10−3
PSν r = 3x10 8 cm PSν r = 1.5x10 8 cm PSν r = 3x10 7 cm
j j j
10−4 10−4 10−4
10−5 10−5 10−5
0.01 0.1 1.0 10.0 100.0 0.01 0.1 1.0 10.0 100.0 0.01 0.1 1.0 10.0 100.0
ν(Hz) ν(Hz) ν(Hz)
Fréquence en Hz Fréquence en Hz Fréquence en Hz
10+0 10+0 10+0
d e f
10−1 10−1 10−1
2moy)]avg10−2 2moy)]avg10−2 2moy)]avg10−2
s/ s/ s/
m m m
D [(r10−3 D [(r10−3 D [(r10−3
PSν PSν PSν
0.1−1 keV 5−25 keV 45−100 keV
10−4 10−4 10−4
10−5 10−5 10−5
0.01 0.1 1.0 10.0 100.0 0.01 0.1 1.0 10.0 100.0 0.001 0.01 0.1 1.0 10.0 100.0
ν(Hz) ν(Hz) ν(Hz)
Fréquence en Hz Fréquence en Hz Fréquence en Hz
Figure5. Evolutionofthepowerspectrumobtainedwitheitherrj ortheenergyband.1σerrorsarisingfromthefitontheamplitude
ofthewaveinfluxarealsoplotted.Seetextfortheinputparametersvalue.Upperpanel, fromlefttoright:PSDobtainedfor25-45keV
band. TheQPOfrequency movesupwardfrom0.33to0.67and3.3Hzwhentheexternal radiusofthecorona rj moves inward.Lower
panel:EvolutionofthePSDwiththeenergyband.Thevalueoftheexternalradiusisrj =3×108 cm.Notetheincreaseoftheoverall
variabilitywith the energy. In Fig. 5 f The dashed curves are Lorentzians usually employed to fit the different components in PSDs of
XRB.
L NL L NL
30 2.
25 10.0 %) 1.0
0
C(E) vs C(E)linquad 112050 C(E)| vs |C(E)|linquad 1.0 χε=0.2 C /Cux linéaire à plus de 8max lin quadth0000.0...5125
5 | 0.1 (flmax0.02
ε
0 0.01
0.1 1.0 10.0 100.0 0.1 1.0 10.0 100.0 0.1 1.0 10.0 100.0
EEnneerrggiey e(kne kVe)V EnEenrgeireg ey n( kkeeVV) ÉneErngeierg eyn ( kkeeVV)
Figure6. Left:Clin(circle)vsCquad(cross)valuewithlinearcoordinatesontheyaxis(seetextfortheparameterofthecoronainthe
steadystate).Underabout3keV(the“pivot”)thevalueofClin isnegativeThisisadirectconsequence ofphotonnumberconservation
in the compton scattering (see text). Center: same as right plot, but in absolute value and y axis in log coordinates, to compare the
magnitude.Above10keV,Clin ishigherthan1,emphasisingtheeffectofmodulationinthefluxbytheperturbationinpressure.Right:
Plotofǫmax spectrum(seetextforexplanation).Here,χth=0.2.Ife.g.ǫ=0.15asplotted(dashedhorizontalline),twolinearandnon
linearzones canbeobserved.
8 C. Cabanac, G. Henri, P.-O. Petrucci, J. Malzac, J. Ferreira, T. M. Belloni
study the evolution of the power spectra with energy. In- To further investigate the domains of non linearities,
terestingly, the overall shape of the power spectra can also let’s assume an arbitrary threshold χ of the ratio of the
th
be mimicked by using sets of Lorentzians especially at high quadratic to the linear terms in Eq. 40, i.e. Cquadǫ, above
energy as shown in Fig. 5 f. Clin
which we estimate that the linearity hypothesis is no more
valid. This, in turn, determines different area in the ǫ/χ
th
4.3 When does the linear approximation become vs energy plane (see Fig. 6 ) that agrees or disagrees with
invalid? the linear approximation. For example, assuming a thresh-
oldχ of20%meansthatweestimatethatlinearityiswell
th
Asshownintheprevioussection,forlowvaluesofthemod- C ǫ
satisfiedassoonas quad <0.2.Forthesetofparameters
ulation amplitude ǫ, the Compton emission of the corona C
lin
agreesrelativelywellwiththelinearapproximation.Weex- used in Sect. 4.2 and for which we obtain Fig. 6 this con-
pect however some deviation from linearity for larger mod- strains the perturbation relative amplitude ǫ to be smaller
ulation amplitudes, deviation that should also depends on than ǫmax ≃ 50% =2.5χth in the 0.1−3 keV energy band
the energy. This aspect is investigated here by increasing ǫ and even much smaller above 10 keV. For high values of ǫ,
in the simulations and then adjusting the corona spectral say e.g. ǫ=0.7, thewhole energy domain is non-linear. For
emission atdifferentenergyE withaquadraticpolynomial, lower values, say e.g. ǫ = 0.15 as plotted by the horizontal
i.e.: linein right Fig. 6, twolinear and two nonlinear zones can
benoted.
L(ǫ,t)−L δL(ǫ,t)
0 ≡
L L
0 (cid:12)E 0 (cid:12)E
(cid:12)(cid:12)(cid:12) = Clin(E)(cid:12)(cid:12)(cid:12)ǫ+Cquad(E)ǫ2 e−iwt, (40) 4.4 Energy dependence of the variability
Hence, the energy-de(cid:2)pendent parameters C(cid:3) vs C Finally, the strength of the variability and its evolution
lin quad
with energy is investigated. For that purpose, we com-
allows to “quantify” the linear and quadratic behaviour of
theperturbation2. puted and plotted on Fig. 7 the so called “RMS spectra”
and “f-spectra” . Those “variability spectra” were com-
Starting from a reference spectrum corresponding to
puted for three spectral ranges: at the lowest frequencies,
kT =0.75 keV, kT =75 keV and τ =1.2, we simulate
9 dsieffederent spectra wit0h values of ǫ ranging from 0 to 60% wherethePDSisflat(10−3−10−1 Hz),neartheresonance
(10−1−0.7 Hz) and after (0.7−1.5 Hz).
and fit the data with the polynomial given by Eq. 40 for
As expected from the behaviour of C , the presence
different energy bins. The corresponding values of C and lin
lin
C are plotted in Fig. 6 in function of the energy. They of the pivot lead to a drastic decrease of therms at around
quad
2 keV for all the frequencies studied, which is a strong pre-
appear strongly energy dependent. Noticeably they both
diction of the model, even if the position of this pivot in
cancelandchangesignatamediumenergyofabout2keV,
signature of a “pivot” in the variable spectral emission of energydependsonseveralinputparametersuchastheseed
photontemperature.Afterit,thevariabilityincreaseswith-
thecorona.
outanycut-offathighenergy.Thisduetothefactthatthe
This pivot is also clearly visible when comparing light
curves at low and high energies such as those plotted in wave in the corona also modulates the temperature of the
high energy electrons and hence the position of the cut-off
Fig. 7. Whereas the high energy lightcurve respond to the
in the high energy spectrum. The higher the modulation,
excitation coherently, the low energy one (under the pivot)
the larger the cut-off. An increase of the variability of the
has a phase lag of π. This pivot is due to the compton up-
0.7−1.5 Hz component is also predicted above∼100 keV.
scattering of low energy photons which naturally produces
F-spectra were also computed, following definition of
a decrease of the number of soft photons and, simultane-
Revnivtsevet al. (1999), and adopting the same three fre-
ously, an increase of the number of the high energy ones.
quency bands as above. Such spectra were then fitted be-
Consequentlythereisaπ phaselagbetweenthetwoenergy
tween 2 and 150 keV by a powerlaw. Their photon indices
domains.ThequadratictermC cancelsasecondtimeat
quad
aredecreasing with increasing frequenciesas theyareequal
higherenergy(∼90keVinthesimulationplottedinFig.6).
to1.61±0.02forthe10−3−10−1Hzband,1.53±0.01forthe
10−1−0.7Hzbandand1.48±0.05forthe0.7−1.5Hzband.
By examining the trend at high energy in center panel
Incontrast,thetimeaveragedspectrumhasahigherphoton
of Fig. 6, it is worth noting that the value of C is higher
lin
index (Γ ∼2.1). The ratio of theobtained f-spectra to a
than 1 above 10 keV. This means that any fluctuation in avg
powerlawofindex1.53isplottedinFig.7,andalsotheratio
pressure will produce an amplified radiative response of
of the time averaged spectrum to a powerlaw of index 2.1.
the corona at high energy. This is due to the pivoting of
Thetimeaveragedandthef-spectraexhibitaquitesensible
the spectrum, which acts as a ”lever” arm: the higher the
deviation from the simple power-law model. This discrep-
energy bin, with respect to the pivot energy, the larger the
ancy with a power-law is however more emphasised in the
luminosity variation.
f-spectra than in thetime averaged spectra.
2 Note that the parameter Clin of Eq. 40 is not exactly the 5 DISCUSSION
sameastheoneusedinSect.3.Italsoincludesallthegeometry-
dependent partof the corona response (i.e.the term 2Mxj,ξ The timing response of an oscillating corona in cylindrical
x2(1 ξ2) geometry is investigated, the corona acting as a filter when
j −
presentinEq.17) responding to a perturbation in pressure generated at
Variability of X-ray binaries from an oscillating hot corona. 9
1100 10+0
5.
n) 1000 10−1 2.
bi
minosity L (cts/ 789000000 2P (%rms) ν1100−−32 S(E,f)/PLij 0100....1025
Lu 600 10−4 0.05
500 0.02
10−5 0.01
0 500 1000 1500 2000 0.1 1.0 10.0 100.0 0.1 1.0 10.0 100.0
Time (s) Energy (keV) Energy (keV)
Figure 7. Left: Light curves obtained for the lowest frequencies probed (ν =10−3 Hz), fortwo energy bands, and their best fit by a
sinefunction(dashedcurve).Upperplot:0.25 1keV.Lowerplot:2 3keV.Notethephaselagofπbetweenthehighandlowenergies.
Center:“RMSspectra”fordifferentfrequency−bands.N:10−3 10−−1 Hz, :10−1 0.7Hz.(cid:3):0.7 1.5Hz.Right:3lowercurves:Ratio
− • − −
ofthe f-spectratoapower lawmodel withphoton indexα=1.54, fordifferentfrequencies (samesymbolcode as center panel). Upper
curve(⋆):ratioofthetimeaveragedspectrumtoapowerlawofindexΓavg =2.1.Timeaveragedand0.7 1.5Hzspectrawererescaled
−
forclarity.
its external radius. In the case of total reflection at the a fraction of the observed anti-correlation. Indeed, the BH
internal radius, infinite resonances may occur. They may anti-correlationtranslatesthefactthatinνP ,thedifferent
ν
be damped in the case of acoustic impedance adaptation observedLorentziansarepeakingatroughlythesamelevel,
between the corona and the external medium. We have or equivalently that the rms integrated variability remains
shown that the filtering effect of the corona leads to power roughlyconstant.Anextraconditionontheexcitingprocess
spectrawithabroadbandshapewellfittedbyaLorentzian is therefore required, with a rms level that must decrease
centered in ν = 0 (see Fig. 5). Moreover, the resonances when the size of the corona decrease as well. The nec-
produce a major peak at the PSD break and smaller peaks essaryconditionwouldhencebethatδP ∝rα,withα>0.5.
j
at higher frequency. The characteristic frequencies of the
PSD break and peaks scale with ν = 2πc /r . For a 10
j s j
As shown in Fig. 7 we expect a π phase lag between
solar mass black hole, an outer corona radius r = 200 R
j g
theverylowenergyandthehighenergylightcurves,which
and a corona temperature kT = 100 keV (i.e. a sonic
0
velocity c = 3.1×108 cm.s−1), this gives ν = 0.17 Hz. is a strong prediction of the model. This is a direct conse-
s j
quenceof theassumed geometry, especially of the fact that
These power spectra are thus relatively similar to the one
the source of seed soft photons is assumed at the corona
observed in the HIMS (Hard Intermediate State) of X-ray
midplane. This source is necessarily on the line of sight of
transients, whose shape are well fitted by 3-4 Lorentzians
the observer. Then due to the photon number conservation
peaking at different frequencies from a tenth of hertz (for
during the Compton scattering process, the disappearance
the break and the low frequency QPO, hereafter LFQPO)
of the soft photons is directly compensated by an increase
to hundreds of hertz (for the high frequency QPO, see e.g.
of the hard ones. Moreover it depends on the pivot energy
Nowak 2000 or Pottschmidt et al. 2003).
whichcruciallydependsontheinputparameters(especially
Interestingly our low frequency peak is always of the order
kT andkT ).Inageometrywherethesoft seedphotons
ofthefrequencybreakν (bothscalingwithν ).Thisisalso c soft
b j
areproducedoutsidethecorona,thisconstraintcanbeeas-
inagreementwiththeobservedcorrelationsbetweenν and
b
ily relaxed. This would be the case of a soft photon field
the low frequency QPO ν (Wijnands & van der Klis
LFQPO
produced by an outer accretion disc. Such geometry (inner
1999; Belloni et al. 2002; Klein-Wolt & van derKlis 2008).
hot corona surrounding by an outer accretion disc) is in-
We note however that ν is usually 5 times higher
LFQPO
deedgenerally believedtoqualitativelywell reproducedthe
than ν , especially in Wijnands & van derKlis (1999) rela-
b
inner region of the accretion flow around compact objects
tionship.Wenotehoweverthatintheequivalentcorrelation
(i.e.Esin et al.1997;Doneet al.2007).Notethatwithsuch
plotted on Fig. 11 of Belloni et al. (2002), thelower branch
geometry, a decrease of the outer corona radius r (which
shows a correlation between the break frequency ν and j
b
correspondstotheinneraccretiondiscone)wouldimplyan
the typical frequency of the “hump” (ν ), with a ratio
h
increase of the seed photons temperature and flux i.e. an
close to one. Finally, the ratio between the first two peak
increaseof thecorona coolings. Consequentlyasofteningof
frequencies is close to 2 in our model, as it is observed for
the X-ray spectrum should happen. Since a decrease of r
several typeC QPO. j
impliesalso,inourmodel,anincreaseoftheresonancepeak
frequencies, a correlation between the X-ray photon index
Our model also predicts a decreasing PSD power and andtheQPOfrequenciesisexpected.Suchcorrelationisin-
an increasing resonance peak frequencies when decreasing deedobservedindifferentXRB(Titarchuk & Shaposhnikov
the outer corona radius r . This is here again consistent 2005).
j
with the general trend observed in BHB where the PSD Thepositionofthepivotcanalsobedrasticallychanged
powerdecreaseswhentheobservedfrequenciesincrease(BH and moved towards lower energies if the seed photons typi-
anti-correlation, Belloni & Hasinger 1990). However in our calenergyisfarlowerthanconsideredhere.Forexample,as
model, this effect is low and could hence only account for suggestede.g.inMalzac & Belmont(2009),thesoft photon
10 C. Cabanac, G. Henri, P.-O. Petrucci, J. Malzac, J. Ferreira, T. M. Belloni
contribution could originate from the synchrotron emission explain the main features observed in the PSD of XRB,
in optical and UVfrom thehot plasma. significant improvements are needed to include different,
We also demonstrated that C increases after the and potentially important, physical effects.
lin
pivot whereas C is decreasing again around 20 keV. For instance we have shown that the effect of trans-
quad
The general increase of δL/L with the energy after the mitting the wave in an inner medium results in softening
pivot (visible e.g. in the rms-spectra plotted in Fig. 7) the resonances and the gaps in the power spectra. But
couldberesponsibleinpartfortheobservedincreaseofthe the kinematics of the corona could also play an important
variabilitywithenergyfortheQPOandthecontinuum(see role. Assuming the corona located in the inner region of an
e.g. for XTE J1550-564 Cui et al. 1999). Alternatively, the accretion flow, some (differential) rotation is expected and
observed decrease of the QPO harmonic strength in Fig. we might expect the emitted frequency by each part of the
5 of Cui et al. (1999) could also be well explained by the corona to be Doppler shifted, smoothing the PSD in the
drop in the value of C after 30 keV. We can also note samewayemissionlinescanbesmoothedinenergyspectra.
quad
thesimilarity between theRMSspectra plottedin Fig. 5of In the close vicinity of the central engine, the general
Cui et al. (1999), when the source is in its HIMS, and the relativistic effects should also significantly contribute to
one we obtained with our model. The increase of the QPO blurred the PSD. The use of a non uniform sound velocity
and continuum variability between 2 and 20 keV is then a profile (due to e.g. non uniform corona temperature)
direct consequenceof thecomptonisation process. along the radial or vertical directions is also expected to
Wealsoobserveadecreaseofthepowerlawslopeinthe have some impact in this respect. But more importantly,
f-spectrawhenthefrequenciesgethigher.Thisisconsistent including all these effects will result in a very different
with the results obtained for Cyg X-1 in Revnivtsevet al. dispersion equation followed by the acoustic waves and
(1999). It is worth noting that those authors explain the thus to a significantly different behavior of the radiative
presence of “wiggles” in the f-spectra as a consequence response of thecorona.
of reflexion features on an optically thick material. Our We also used a single sound wave, but other acoustic
Monte-Carlo simulations shows howeverthatthosefeatures wavescouldbepresent.Forexample,inthecaseofmagnetic
can begenerated by thecomptonisation process only. plasma Alfv´en and both slow and fast magnetosonic waves,
The non-linear domain is then investigated as a func- with velocities vA, c− and c+, should be used. The former
tion of the energy. We demonstrated that if a reasonable do not generate density or pressure perturbation. They
arbitrary value of the non-linear to linear ratio is chosen, are not expected to change the comptonisation efficiency,
it is expected that the non-linear effects of the radiative contrarytothemagnetosonic ones.Inthecaseofplasmain
response will occur only at veryhigh and medium energies. equipartition, c2s = vA2, hence c− = 0.54cs and c+ = 1.31cs
and we do not expect significant changes compared to the
The choice of a cylindrical geometry has also an sound wave case studied here3. However, for plasma far
impact on the degree of loss of the input signal due to from equipartition, c− ∼ cs 6= c+ ∼ vA. Therefore if both
scatterings. The following arguments are similar to those wave propagates in the medium, each component would be
discussed in previous studies, such as in e.g Miller (1995). responsible for its own band limited noise and resonances
Indeed, unless one global oscillation takes place in the in the power spectra. Multiple perturbating waves may
corona (which is the case at very low exciting frequencies), then explain the multiple Lorentzian components generally
aphotonmayencounteralternatively positiveandnegative needed to fit the PSD of XRB in hard states (e.g. Nowak
perturbationsinpressureifittravelsintheradialdirection. 2000; Pottschmidt et al. 2003). Note however that in the
It would consequently kill the effect of the input signal HIMS, only two Lorentzians are necessary. If we identify
on the output lightcurve. We even expect this effect to be in our framework the low frequency Lorentzian (Lb, see
emphasised when looking at high energies, as, regardless Belloni et al. (2002) for the labels of the different PSD
of the optical depth of the medium chosen, a high energy components) with the propagation of a slow magnetosonic
photon encountered a large number of scatterings. On the wave and the lower upperfrequency (Ll) with the fast one,
contrary,theoscillationwillnotbesmearedoutifthetravel it would require a very high magnetisation parameter for
of the photon remains local. theplasma.Indeedtheobservationsgivesroughlyνl ∼50νb
As we chose a cylindrical geometry, the optical depth (see e.g. tables 2 and 3 in Belloni et al. 2002), and since
τ ∼ 1 is relative to the vertical direction and hence this c− ∼cs and c+ ∼vA, it would imply that vA2/c2s ∼2500.
impliesthatintheazimuthalandradialdirectionsτ ≫1.
r,θ
As a consequence, this is forcing the locality, relative to We also did not investigate the implied time-lags gen-
the radial wavelength of the pertubation, which in turn erated by the comptonisation process. It is however known
preserves the signal, whatever the output photon energy. that such lags can be reproduced by pivoting in the spec-
As one goes towards a more spherical geometry, where τ tra (see e.g. Poutanen & Fabian 1999 or K¨ording & Falcke
now becomes a measure of the radial optical depth, τ in 2004),andourmodelspredictsthepresenceofsuchapivot.
various directions becomes much less anisotropic. Looking Morefundamentally,asweprobethetimingresponseof
at higher energies means looking at a larger range of thecorona,itcanbeusedasacomplementtoothervariabil-
sampled locations, and hence smearing of the imparted ity models as the shape of the input excitation is supposed
variability signal for sufficiently small wavelengths of the here to be a white noise. The RMS-flux scaling found in
perturbation. bothAGNand XRB(seee.g. Uttley & McHardy2001)has
While this simple model already provides very in-
teresting timing behavior and appears very promising to 3 TheabovecalculationsweredonewithvA,poloidal=1/√2vA.