Observational Signatures of Tilted Black Hole Accretion Disks from Simulations

0.83 MB·English

by  Jason Dexter

Draftversion January21,2011
PreprinttypesetusingLATEXstyleemulateapjv.8/13/10
OBSERVATIONAL SIGNATURES OF TILTED BLACK HOLE ACCRETION DISKS FROM SIMULATIONS
Jason Dexter
DepartmentofPhysics,UniversityofWashington, Seattle, WA98195-1560, USA
P. Chris Fragile
DepartmentofPhysics&Astronomy,CollegeofCharleston,Charleston,SC29424, USA
Draft version January 21, 2011
1
1 ABSTRACT
0
Geometrically thick accretion flows may be present in black hole X-ray binaries observed in the
2
low/hard state and in low-luminosity active galactic nuclei. Unlike in geometrically thin disks, the

n angular momentum axis in these sources is not expected to align with the black hole spin axis.
a We compute images from three-dimensionalgeneralrelativistic magnetohydrodynamicsimulations of
J
misaligned (tilted) accretion flows using relativistic radiative transfer, and compare the estimated

9 locations of the radiation edge with expectations from their aligned (untilted) counterparts. The
1 radiation edge in the tilted simulations is independent of black hole spin for a tilt of 15◦, in stark

contrast to the results for untilted simulations, which agree with the monotonic dependence on spin
]
E expected from thin accretion disk theory. Synthetic emission line profiles from the tilted simulations
depend strongly on the observer’s azimuth, and exhibit unique features such as broad “blue wings.”
H
Coupledwithprecession,theazimuthalvariationcouldgeneratetimefluctuationsinobservedemission
. lines,whichwouldbeaclear“signature”ofatiltedaccretionflow. Finally,weevaluatethepossibility
h
p thattheobservedlow-andhigh-frequencyquasi-periodicoscillations(QPOs)fromblackholebinaries
- could be produced by misaligned accretion flows. Although low-frequency QPOs from precessing,
o tilted disks remains a viable option, we find little evidence for significant power in our light curves in
r the frequency range of high-frequency QPOs.
t
s Subject headings: accretion, accretion disks — black hole physics — radiative transfer — relativity
a
[

1 1. INTRODUCTION This method has been used to look for high frequency
v In standard thin disk accretion theory quasi-periodic oscillations (HFQPOs) in simulated data
3 (Schnittman et al. 2006) and to create radiative models
(Shakura & Sunyaev 1973; Novikov & Thorne 1973),
8 ofSagittariusA*(Noble et al.2007;Mo´scibrodzka et al.
the angular momentum axis of the accretion flow is
7 2009; Dexter et al. 2009, 2010).
assumed to be aligned with the black hole spin axis.
3 All of this work assumed alignment between the an-
Bardeen & Petterson (1975) found that even if the
. gular momentum axis of the accretion flow and the
1 initial angular momentum axis of the accretion flow
black hole spin axis. Fragile et al. (2007, 2009); Fragile
0 is misaligned from the black hole spin axis, the inner
1 part of the disk will still align on the viscous timescale. (2009) were the first to do GRMHD simulations of disks
1 However, this so-called “viscous” regime only operates with a tilt between these two axes. These new simula-
: when H/R . α, where H/R is the scale height of the tions yielded a number of unexpected features. First,
v
the main body of the disk remained tilted with re-
i accretion disk, and α is the parameterized viscosity
X (Papaloizou& Lin 1995). This is applicable in active spect to the symmetry plane of the black hole; thus
there was no indication of a Bardeen-Petterson effect
r galactic nuclei (AGN) and the high/soft or thermal
a state of black hole X-ray binaries. On the other hand, in the disk at large. The torque of the black hole
instead principally caused a global precession of the
advection-dominated accretion flows (ADAFs) are
main disk body (Fragile & Anninos 2005; Fragile et al.
expected in the low/hard state of black hole X-ray
2007). The time-steady structure of the disk was also
binaries (Narayan& Yi 1995; Esin et al. 1997) and in
warped, with latitude-dependent radial epicyclic motion
low-luminosityAGN.ADAFs areunabletocoolthrough
driven by pressure gradients attributable to the warp
efficient radiation, and are geometrically thick. It is
(Fragile & Blaes 2008). The tilted disks also truncated
likely that the accretionflow in many of these sources is
at a larger radius than expected for an untilted disk.
misaligned, or “tilted.”
In fact, based on dynamical measures, the inner edge of
Contemporary general relativistic MHD simulations
these tilted disks was found to be independent of black
(GRMHD, De Villiers & Hawley 2003; Gammie et al.
holespin(Fragile2009),insharpcontrasttotheexpecta-
2003) currently provide the most physically realistic de-
tion that accretion flows truncate at the marginally sta-
scription of the inner portion of accretion flows around
ble orbit ofthe black hole. Finally, Henisey et al.(2009)
spinning black holes. Radiation can be calculated
foundevidencefortrappedinertialwavesinasimulation
from these simulations in post-processing by assuming
with a black spin a = 0.9, producing excess power at a
thatitisdynamicallyandthermodynamicallynegligible.
frequency 118(M/10M⊙)−1 Hz.
In this work we use relativistic ray tracing to produce
[email protected]
2 Dexter & Fragile
tion,andinjected entropyatshocks. Sucha formulation
does not conserve energy, and produces a more slender,
cooler torus than conservative formulations which cap-
ture the heat from numerical reconnection of magnetic
fields (Fragile & Meier 2009). The scale height spanned
the range H/R 0.05 0.1 in these simulations, with
∼ −
larger scale heights for higher spin simulations.
2.2. Ray Tracing
Relativistic radiative transfer is computed from simu-
lation data via ray tracing. Starting from an observer’s
camera,raysaretracedbackwardsintimeassumingthey
Figure 1. Sample images of the thermal emissionmodel for the are null geodesics (geometric optics approximation), us-
90h (left) and 915h (right) simulations at 60◦ inclination. The ing the public code described in Dexter & Agol (2009).
observedphotonenergyisE0=10keVfora10M⊙ blackhole,and Inthe regionwhereraysintersectthe accretionflow, the
eachpanelis54M across. Thecolorscaleislinear,increasingfrom
radiative transfer equation is solved along the geodesic
bluetoredtoyellowtowhite.
(Broderick 2006) in the form given in Fuerst & Wu
(2004), which then represents a pixel of the image. This
procedure is repeated for many rays to produce an im-
Table 1
SimulationParameters age,andatmanytimestepsofthesimulationtoproduce
time-dependent images (movies). Light curves are com-
Simulation a/M Tilt Grid puted by integrating over the individual images. Sam-
Angle ple images of two simulations are given in Figure 1.
0Ha 0 ... Spherical-polar Dopplerbeamingcausesasymmetryintheintensityfrom
315Hb 0.3 15◦ Spherical-polar approaching (left) and receding (right) fluid. Photons
50Ha 0.5 0◦ Cubed-sphere emitted from the far side of the accretion flow are de-
515Ha 0.5 15◦ Spherical-polar
715Hb 0.7 15◦ Spherical-polar flected toward the observer, causing it to appear above
90Hc 0.9 0◦ Spherical-polar the black hole. The thick, central ring is due to gravita-
915Hc 0.9 15◦ Spherical-polar tionallensingfrommaterialpassingundertheblackhole,
a while the underresolved circular ring is caused by pho-
Fragileetal.(2009)
b tons that orbit the black hole one or more times before
Fragile(2009)
c Fragileetal.(2007) escaping. These ring features are in excellent agreement
with the predictions made by Viergutz (1993).
imagesandlightcurvesofsomeofthesenumericallysim-
To calculate fluid properties at each point on a ray,
ulatedtiltedanduntiltedblack-holeaccretiondisks. Our
the spacetime coordinates of the geodesic are trans-
goalin this paper is to discuss observable differences be-
formed from Boyer-Lindquist to the Kerr-Schild coordi-
tweenthetwotypesofaccretionflows,andtoidentifyob-
nates used in the simulation. Since the accretion flow is
servationalsignaturesoftiltedblackholeaccretiondisks.
dynamic, light travel time delays along the geodesic are
2. METHODS taken into account. Data from the sixteen nearest zone
centers(eightonthesimulationgridovertwotimesteps)
2.1. Simulation Data
are interpolated to each point on the geodesic. Between
ThesimulationsusedherearefromFragile et al.(2007, levels of resolution near the poles on the spherical-polar
2009);Fragile(2009). TheparametersaregiveninTable grid,datafromthehigherresolutionlayerareaveragedto
1. All of the simulations used the Cosmos++ GRMHD create synthetic lower resolution points, which are then
code (Anninos et al. 2005), with an effective resolution interpolated. Very little emission originates in the un-
of 1283 for the spherical-polar grid (except near the derresolvedregions of the simulation.
poleswherethegridwaspurposefullyunderresolved)and The simulations provide mass density, pressure, veloc-
128 64 64 6 for the cubed-sphere grid. The simula- ity and magnetic field in code units. These are con-
× × ×
tions wereinitializedwith ananalyticallysolvable,time- vertedintocgsunitsfollowingtheproceduredescribedin
steady, axisymmetric gas torus (De Villiers & Hawley Schnittman et al. (2006) and Dexter et al. (2009). The
2003), threaded with a weak, purely poloidal magnetic length- and time-scales are set by the black hole mass,
field that follows the isodensity contours and has a min- taken to be 10M⊙ throughout.
imum P /P = 10 initially. The magnetorotational We consider two emission models. The thin line emis-
gas mag
instability (MRI) arose naturally from the initial condi- sivity from Schnittman et al. (2006) is a toy model that
tions, and the disk quickly became fully turbulent. The traces the mass density in the accretion flow. The ther-
simulations were all evolved for 8000M, or 40 orbits mal emission model from Schnittman et al. (2006) uses
∼ ∼
at r = 10M in units with G = c = 1. Only data from free-freeemissionandabsorptioncoefficients,andisused
the final 2/3 of the simulation are used in this analysis, as a model for the high/soft state. Although we do not
once the disks arefully turbulent asmeasuredby apeak expect tilted disks to accurately represent the high/soft
in the accretion rate and in the mass inside of r =10M. state, this model may be appropriate for sources radiat-
This is chosen to utilize as much of the simulation data ing at an appreciable fraction of Eddington, where the
aspossible,andnoneofourresultsdependonwhichtime infalltimeisshorterthantheradiativediffusiontimeand
interval in the simulation is used. the accretion flow becomes geometrically “slim.” When
These simulations all evolvedan internalenergy equa- taking the temperature from the ideal gas law rather
Signatures of Tilted Accretion Disks 3
Figure 2. Comparison of relative intensities for all simulations Figure 4. Comparison of relative intensities for all simulations
using the thin line emissivity. The flux from grids of images over usingthethermalemissivityatE0=1keV.Thefluxfromgridsof
observer time, inclination and azimuth for each simulation have imagesover observertime,inclinationandazimuthforeachsimu-
beenaveraged tocreatethesecurves. lationhavebeenaveragedtocreatethesecurves.
Figure 3. Radiationedgeasafunctionofspinforuntilted(open) Figure 5. Radiation edge as a function of spin for untilted
andtilted(solid)simulationsforthethinlineemissivity. Theerror (open) and tilted(solid)simulations forthe thermal emissivityat
barsshow theone standarddeviation timevariabilityinthe radi- E0 =1keV. The error bars show the one standard deviation time
ation edge, averaged over other parameters. The solid line is the variability in the radiation edge, averaged over other parameters.
marginallystableorbit. Thesolidlineisthemarginallystableorbit.
than the radiation-dominated equation of state used in
that the disk has a sharp cutoff at the innermost sta-
Schnittman et al. (2006), this model may be qualita-
ble circularorbit,whichdepends onspin (Bardeen et al.
tively appropriate for modeling the low/hard state in
1972). Fragile(2009)usedfour differentdynamicalmea-
X-ray binaries or low-luminosity AGN.
suresfromKrolik & Hawley(2002)tocomparetheinner
In 3.1, we consider emission from inside of r = 15M,
§ edges of simulated tilted and untilted accretion disks.
while in 3.2 and 3.3 fluid inside of r = 25M is used for
§ Here we use the ray traced models to locate the “radi-
the ray tracing. For all results here, we take the tem-
ation edge,” the radius inside of which the contribution
perature from the ideal gas law rather than assuming
to the total flux is negligible.
a radiation-dominated equation of state. All of our re-
For each emission model, images are calculated for all
sultsarequalitativelyidenticalwhenusingtheradiation-
simulations over a grid of observer inclination, observer
dominated equation of state to calculate the tempera-
time, and observer azimuth (for the tilted simulations).
ture.
We then compute images cutting out fluid inside of suc-
3. RESULTS cessive values of the radius rin. The radiation edge is
functionally defined as the radius where the ratio of in-
3.1. Radiation Edge
tensities, F(r )/F(0), drops below an arbitraryfraction
in
Inferringtheinneredgeofaccretionflowsisimportant f, chosen so that the untilted radiation edge agrees as
for attempts to measure spin from broad iron lines (e.g. well as possible with r , the marginally stable orbit.
ms
Wilms et al.2001)orcontinuumfitting(e.g.Shafee et al. Figure 2 shows a plot of F(r )/F(0) as a function of
in
2006; Davis et al. 2006). Such measurements assume r averagedoverobservertime, azimuthandinclination
in
4 Dexter & Fragile
forthethinlineemissivity. Fromthesecurvesweextract spond to assuming the emitted line flux is proportional
values of the radiation edge, r . Results are shown in to the incident flux from an irradiating source on the
edge
Figure 3, where the error bars are computed from the spin axis and to local dissipation of heat, respectively
standard deviation of r as a function of time, aver- (e.g., Fragile et al. 2005). This simple form allows us to
edge
aged over the other parameters. This result agrees well focus on general features to be expected from emission
withthedynamicalmeasuresfromFragile(2009). While lines from tilted black hole accretion disks.
the radiation edge moves in towards the black hole with Figure6showssamplelineprofilesforaninclinationof
increasing spin for untilted simulations, there is no such i=60◦ forfourobserverazimuthsfromthe915hsimula-
trend in the tilted simulations. Instead, the radiation tion. Only a single observer azimuth from the 90h sim-
edge appears to be independent of spin. ulation is shown, since the time-averaged emission line
Figures 4 and 5 show the same plots for the thermal is independent of observer azimuth for untilted simula-
emission model with observed photon energy E = 1 tions. Inallcases,the lines consistofastrongpeak near
0
keV. The conclusions are identical with this emission the rest energy of the line (g E /E = 1), a smaller
0 em
≡
model. The untilted simulations have radiation edges peakatlowerenergyanda“redwing,”whoseextentand
which agree quite well with r , while the tilted simula- strengthdepends onthe amountofemissionarisingvery
ms
tionsshownocorrelationbetweenspinandr . Again, close to the black hole (small g). The location of the
edge
theseresultsareconsistentwithFragile(2009),although “blue” peak (large g) depends on the maximum veloc-
we find no trend of increasing radiation edge with spin, ity along the line of sight in the accretion flow. For an
aswasfoundforacoupleofthedynamicalmeasuresused untilted disk, this corresponds directly to the observer’s
in Fragile (2009). Plots from other observed photon en- inclination angle, since all fluid velocities are essentially
ergies are not shown; although the relative flux falls off in the equatorial plane.
much more quickly with increasing r at higher photon For the tilted model shown in Figure 6, the loca-
in
energies, the results for the radiation edge remain com- tion and strength of the blue peak changes significantly
pletely unchanged. with observer azimuth. When the angular momentum
axis of the accretion flow is in the plane of the sky
3.2. Emission Line Profiles
( π/2.φ . π/4,dependingonthesimulationtime),
0
− −
SpectrafromAGNandX-raybinariestypicallyinclude its fluid velocities are maximally aligned with the ob-
strongemissionandabsorptionfeatures. Astheobserved server’s line of sight, leading to the largest blueshifts.
lineshapesaresensitivetoboththevelocityofthe emit- This is the same condition as an untilted disk being
ting/absorbing fluid, and also to the local gravitational viewed edge-on. For other orientations, the blue tail
redshift,theycanprovideinformationaboutthedynam- canextend to significantly higher photon energies in the
ics of the accretion flow (Fabian et al. 1989; Laor 1991). tilted simulations because the largest effective inclina-
UntiltedaccretionflowshavenearlyKeplerianvelocity tion is approximately i = i+β. When the accretion
eff
distributions outside the marginally stable orbit, where flow is not edge-on, there will exist orientations where
thevelocitiessmoothlytransitiontoplunging. Simulated i >i,andthe blue peakforatilted simulationwilloc-
eff
tiltedaccretiondisks,ontheotherhand,showthreema- cur at higher energy than possible for untilted accretion
jor differences. The Keplerian velocity structure is now flows. The red wing, on the other hand, remains largely
tilted. unchanged with observer azimuth, since it is caused by
Secondly,thewarpedstructureofthetilteddisksleads gravitationalredshiftsratherthanDopplerboosts. Since
to epicyclic motions with velocity magnitudes compara- the radiation edge for the 915h simulation was found to
bletothelocalgeodesicorbitalvelocity(Fragile & Blaes occuratsignificantly largerradiusthan thatof90h,it is
2008). Finally, the larger radiation edge values of the expected that the red wing should extend further in the
tilted disks identified in 3.1 means that the transition 90h simulation. The effect is subtle, but identifiable in
§
toplungingorbitsoccursatlargerradiusthaninuntilted Figure 6.
disks. To quantify these trends, for all simulations we com-
These effects indicate that we should expecta number putetheextentofthelineprofile,aswellasthestrengths
of differences in line profiles from tilted accretion flows andlocationsoftheirredandbluepeaks. Mostclearare
(Fragile et al. 2005). The maximum blueshift should be the results for the line extents, shown for i = 60◦ in
largerfor tilted accretiondisks, exceptfor edge-onview- Figures 7 and 8. As expected, the red wing extends to
ing. For i < 90◦ β, where i is the observer’s incli- lowerphotonenergiesathigherspinsforuntiltedsimula-
−
nation angle and β is the initial tilt angle, both rela- tions,whilethereisnosimilartrendforthetiltedmodels.
tive to the black hole spin axis, the tilted accretion flow Alsoasexpected,thebluewingextendstosystematically
should mimic an untilted one with a larger inclination. higher photon energies in the tilted simulations because
In contrast, the red wing should be less pronounced in of the difference between i and i noted above and the
eff
the tilted disks due to their larger truncation radii. On epicyclic motion in the tilted simulations.
the redshifted side, tilted disks behave similar to lower Perhaps the most striking feature of the line profiles
spin, untilted disks. is the variation with observer azimuth seen in all tilted
Producingadetailedreflectionspectrumwouldrequire simulations. These changes in line shape between differ-
a significantnumber of assumptionsto model the metal- ent observer azimuths are typically larger than the full
licity, ionization levels, and incident X-ray flux through- rangeofchangesseenbetweendifferentspinsforuntilted
out the accretion flow. For simplicity, we instead use simulations. Thissuggeststhatthemostpowerfulmeans
toy model emissivities of the form j ρr−s, where ρ of recognizing a tilted accretion disk may be to measure
∝
is the fluid mass density, j is the photon-energy inte- changes in an emission line profile over time as the disk
grated emissivity and s = 2,3. The two values corre-
Signatures of Tilted Accretion Disks 5
Figure 6. Emissionline profiles for simulations with a=0.9. The emissivityis j ∝ρr−3 and the observer inclination is 60◦ inall cases.
Thedottedlinesshowthe1σ range,takenfromthetimevariability.
precesses.
3.3. Variability
X-ray timing of black hole binaries has allowed
the characterization of power spectra and the
detection of transient QPOs (for a review, see
Remillard & McClintock 2006). High-frequency QPOs
are seen in the steep power law state (SPL), while
low-frequency QPOs have been observed in both the
hard state and the SPL. The geometry of the accretion
flow in both these states is uncertain, and there is
no reason to assume complete alignment between the
accretion flow angular momentum and black hole spin
axes in these states. Given the time-dependent nature
of the ray tracing, we can analyze the variability of the
simulated accretion flows for the simplistic emission
models used here to analyze the shape of their power
spectra and to look for possible QPOs.
Figure 7. Minimum line energy vs. spin for all simulations.
The tilted (untilted) simulations are denoted by solid (open) cir- Thebesttimesamplingofthesimulationsisin90hand
cles, and four observer azimuths are plotted for the tilted simu- 915h,whichareusedhereat8observerazimuths,3incli-
lations. The open diamonds are from a thin disk in the equa- nationsand3observedphotonenergiesusingthethermal
torial plane with an emissivity j ∝ r−3 similar to that used in
emissionmodel. Eachlightcurvecapturesroughly6(20)
Schnittman&Bertschinger (2004); Dexter &Agol (2009). The
minimum lineenergy is defined as the lowest energy contained in orbits at r = 25M (10M), corresponding to a total ob-
thesetofintensitiescomprising99%ofthetotallineintensity. The server time, ∆t = 0.23M s, where M is the black
obs 10 10
1σ errorsaretakenfromthetimevariability. hole mass in units of 10M⊙. This duration is about 1/8
of the total precession period for the torus in the 915h
simulation.
Figure 9 shows sample light curves and power spec-
tra from the thermal emission model at 10keV for an
observer inclination of i = 60◦. The secular trend is re-
moved by subtracting the linear best fit from the light
curve before computing the power spectrum.
Allpowerspectraarewellfitbybrokenpowerlawmod-
els of the form:
P(ν)=Aν−γ1 ν ν
b
≤
=Aν γ2−γ1ν−γ2 ν >ν , (1)
b b
where γ , γ are power law indices and the break fre-
1 2
quency, ν , lies near 100Hz M−1 in both simulations.
b 10
The tilted disk power spectra tend to flatten out at the
highest sampled frequencies, 1000Hz M−1. Figure 10
∼ 10
showsmedian power spectra for the three different incli-
nation angles from each simulation. The error bars are
estimated from the standarddeviation in logPower over
Figure 8. AsinFigure7,butforthemaximumlineenergy. observer azimuths and photon energies. At higher incli-
nations,thepeaksinthepoweraround100Hzgrow,espe-
6 Dexter & Fragile
Figure 9. Sample lightcurveand linear fit(left), lightcurve withlinearfit subtracted (middle) and power spectrum (right). Theunits
arescaledtoa10solarmassblackhole.
Figure 10. Medianpowerspectra fori=30◦,60◦,90◦ fromthe90hand915h simulations. Theerrorsareestimated fromthestandard
deviations of the set of power spectra at observed photon energies of 1, 3, 10 keV at four observer azimuths. All power spectra are well
describedbythebrokenpowerlawmodel,withbreakfrequencies around100Hz.
cially for the tilted simulations. This would be expected overall plots are shifted according to the mean ratio be-
from a source of excess power in the inner radii, where tween power spectra.
the larger Doppler shifts at higher inclination would en- At almost all frequencies, these ratios are within
hance the signal. 2σ, and are unlikely to be observed as significant fea-
±
To quantitatively compare the power spectra between tures. However,thereareafewnoteworthyfeaturesnear
the two simulations, the ratio between median power 100M−1Hz. These are particularly interesting given the
10
spectra in untilted and tilted simulations is plotted for finding by Henisey et al. (2009) that the tilted simula-
eachinclination inFigure 11. The values arenormalized tion 915h contains excess power due to trapped inertial
to the combined uncertainties at each frequency. The waves at 118M−1Hz.
10
Signatures of Tilted Accretion Disks 7
Table 2
BrokenPowerLawFitParameters
90h 915h
30◦ 60◦ 90◦ 30◦ 60◦ 90◦
γ1 1.6±0.6 0.7±0.3 0.4±0.7 1.2±0.3 0.9±0.3 0.7±0.9
γ2 3.3±0.6 3.8±0.3 4.0±0.7 3.4±0.8 3.2±1.0 3.7±0.8
νb 80±20 90±5 100±20 90±10 90±20 100±30
Figure 11. Differenceinthelogarithmsofmedian90hand915h
power spectra, normalized to their combined standard deviations
for i = 30◦, 60◦, 90◦. The median 90h power spectra are shifted
toaccount fortheirlowermeanpower.
Figure 13. Fractional difference in shell-averaged angular mo-
mentumbetweentiltedanduntiltedsimulationswithsimilarspins.
Theuntiltedsimulationsarenearlygeodesic, whilethetiltedsim-
ulationsareincreasinglysub-geodesicwithdecreasingradius.
quencies in the fit can favor models with break frequen-
cies 500 M−1Hz, steep initial slopes and shallow post-
∼ 10
break slopes. This occurs due to the denser sampling
of the PSD at high frequencies. Simply ignoring the
highest frequencies gives better results than a variety
of more complicated weighting schemes. The features
near 100M−1Hz from Figure 12 never show up at more
10
than99%significance. In general,while the feature near
Figure 12. Samplepowerspectrum(solid),bestfitbrokenpower 50M−1Hz in the tilted simulations is more convincing
law model (dot-dashed) and upper and lower 99% (dotted) and than1a0nything from the untilted simulations, it does not
99.9%(dashed)significancecontours.
appear at high enough significance at enough observer
To assess the significance of possible features in the frequencies and azimuths to be identified as a QPO.
PSDs, the power spectrum is fit with a broken power Finally, fitting the sets of power spectra provides a
law model. The parameters from the best fit are used generalidea forthe rangeofbestfitvalues ofthe broken
tosimulatemanyrandomlightcurveswiththe samepa- power law parameters. The median parameters found
rameters,andwhichcontainnosignificantfeatures. The from the tilted and untilted simulation are listed in Ta-
significance is determined by comparing the values for ble 2, where the quoted uncertainties are the standard
the power at each frequency for each model power spec- deviations from light curves with different observer az-
trum with the distribution of randomones. An example imuthsandfrequencies. Breakfrequencieshavetheunits
is shown in Figure 12, where a single power spectrum M−1Hz. The break in slope becomes more pronounced
10
from the 915h simulation is shown, as well as the best at higher inclination as the initial slope becomes shal-
fit broken power law model and upper and lower 99.9% lower while the post-break slope becomes steeper. The
confidenceintervalsfromsimulatingrandomlightcurves. post-breakslopeisslightlyshallowerinthetiltedsimula-
No obvious QPO features show up in this analysis. tions, while the initial slope is more strongly dependent
In several of the 915h light curves, the feature near on inclination in the untilted case.
50M−1Hz shows up as 99.9% significant. It appears at
high1s0ignificanceinmoreofthelightcurvesathighincli- 4. PHYSICALCAUSEOFDISKTRUNCATION
nations. Inthe90hsimulations,almostallsignificantfea- The observable signatures of tilted disks discussed so
tures are found at very high frequencies 1000M−1Hz. far are, for the most part, due to two main differences
∼ 10
These are spurious, caused by slight errors in the fit to between tilted and untilted disks: tilted disks precess,
the post-break slope incurred by ignoring all frequen- andthey are truncatedoutside r . Fragile et al.(2007)
ms
cies larger than 800M−1Hz. Including the highest fre- alreadydiscussedwhy the simulatedaccretionflows pre-
10
8 Dexter & Fragile
mentumprofilesofthetiltedsimulationswiththestand-
ing shocks can be found from looking at the time-
dependence of the shell-averaged angular momentum.
While the untilted simulation remains nearly geodesic,
the tilted simulations are continuously transporting an-
gular momentum outward from r 10M for the first
∼
5000M before reaching a steady state, as would be
∼
expected from a dynamical mechanism. Finally, verti-
callyintegratedcontourplotssuchasFigure15showthat
the angular momentum in the tilted simulations is non-
axisymmetrically distributed. The regions of depleted
angular momentum correspond to the standing shocks,
which appear as regions of excess entropy in the bottom
panels of Figure 15.
Following Fragile & Blaes (2008), we postulate that
the standing shocks are caused by deviations from cir-
cular orbits near the black hole. Figure 16 shows the
shell-averagedeccentricities of the orbits in each simula-
Figure 14. Shell-averaged entropy distributions for all simu-
lations. Excess entropy inside r ∼ 10M is generated by non- tion, estimated at one scale-height in the disk using
axisymmetricstandingshocksinthetiltedsimulations.
r ∂(βsinγ)
cess. It is our interest to better understand the physical e= , (3)
−6M ∂r
cause for the large truncation radius.
The first thing to note is that rapidly rotating black where β is the tilt and γ is the precession of each or-
holes provide more centrifugal support to an accretion bital shell.1 All quantities are calculated from fitting
disk than slowly rotatingblack holes. Therefore,the an- the shell-averaged disk tilt and twist (Eqs. 32 and 41
gular momentum extraction mechanism at play in the of Fragile et al. 2007) with power laws, and using the
tilted disks must be more effective at higher spin. This resulting expressions in Equation (3). The increase in
is confirmed in Figure 13, where we plot the difference eccentricity toward smaller radii leads to a crowding of
in density-weighted, shell-averaged specific angular mo- orbits near their apocenters (Ivanov & Illarionov 1997),
mentumfortiltedanduntiltedsimulationsofcomparable whichleadstotheformationofthestandingshocks. The
spin. The angular momentum is defined as ℓ= u /u , eccentricityislargerforhigherblackholespin,exceptin-
φ t
where uµ is the fluid four-velocity and the shell-−average side the plunging regionwhere the fits become poor and
of a quantity x is given by, the eccentricity is ill-defined. Equation (3) may indicate
how these results depend on the initial tilt of the simu-
1 lations. If we assume that the strongestdependence of e
x = dΩ√ gx, (2)
h i AZ Z − on tilt is through β and that ∂β/∂r and ∂γ/∂r remain
unchanged for different tilts, then equation (3) suggests
where Ω is the coordinate solid angle, g is the metric that the eccentricity of the orbits should vary roughly
determinantandA= dΩ√ g. Thedensity-weighted linearly with the initial tilt, at least for small angles.
−
shell-averageofxisdeRfinRedas ρx / ρ . Theangularmo- This prediction is tentatively confirmed by a simulation
h i h i
mentum profiles for the untilted simulations are nearly we have done that started with an initial tilt of 10◦.
geodesicoutsideofr 5M.Insideofr 10M,the tilted
simulations become i∼ncreasingly sub-ge∼odesic, with the 5. DISCUSSION
higherspincasesdeviatingmorethanthelowerspinones.
Tilted accretion flows will inevitably be present in a
The same trend holds when comparing the tilted simu-
significant fraction of black hole sources with L/L .
edd
lations to the analytic result for the angular momentum
0.05andpossiblyL/L &0.3(thickorslimdisks). Us-
edd
profileofmaterialongeodesicorbitsinanequatorialdisk
ingrelativisticraytracingandasetofsimpleemissivities,
inclined 15◦ to the black hole spin axis.
we have compared the radiationedge, emission line pro-
Fragile & Blaes (2008) suggested that the non-
files and power spectra of simulated black hole accretion
axisymmetric standing shocks that occur in the inner flowswithatiltof15◦ totheiruntiltedcounterparts. We
radii above and below the midplane of the disk may
findthe radiationedgeis independentofblackholespin,
enhance the outward transport of angular momentum,
while the untilted simulations agreed with the expected
causing fluid to plunge from outside the marginally sta-
qualitativetrendofdecreasinginnerradiuswithincreas-
ble orbit. To connect the enhanced angular momentum
ingspin. Theseresultsfortheradiationedgeconfirmthe
lossof the tilted disks withthe standing shocks,we next
workof Fragile (2009), who used dynamical measures to
lookataplotofthe density-weighted,shell-averageden-
locate the inner edge.
tropy profiles in Figure 14. Since these simulations con-
Due to the independence of inner edge on spin, the
serve entropy except across shocks, the excess inside of
red wing of tilted accretion flow emission line profiles is
r 7M in the tilted simulations signifies the presence of
also fairly independent of spin. This introduces a possi-
∼
extrashocks. Thesteepnessoftheentropygradientgives
blecomplicationforattempts tomeasureblackholespin
somemeasureofthe strengthofthese shocks. Again, we
see that the effect is greatestin the simulations with the 1 ThisdefinitionofediffersfromIvanov&Illarionov(1997)by
fastest spinning black holes. aphasefactorofπ/2inγ. Fragile&Blaes(2008)usedtheformula
Further evidence linking the sub-geodesic angular mo- fromIvanov &Illarionov(1997)withoutmodification.
Signatures of Tilted Accretion Disks 9
be quite common, and this effect is unlikely to signifi-
cantlydependonaccuratereflectionspectrummodeling.
Although the simulations can only be run for a short
time compared to the precession time scale, precession
isa possiblesourceoflowfrequencyquasi-periodicoscil-
lations when the accretion flow is optically thin due to
the modulation of Doppler shifts as the velocities in the
accretionflowalignandmisalignwiththeobserver’sline
of sight (see Ingram et al. 2009, for more discussion of
QPOs from precessing tilted disks).
Finally, we have studied power spectra for our
simple models. We find broken power law spec-
tra with break frequencies around 100M−1Hz and
10
power law indices in the range 0-2 (3-4) pre-
(post-) break for both tilted and untilted simula-
tions. Previous studies (Armitage & Reynolds 2003;
Noble & Krolik 2009) found single power laws with in-
dex 2. Armitage & Reynolds (2003) found that power
∼
spectrafromindividualannuliarewelldescribedbybro-
ken power laws where the break frequency is close to
the local orbital frequency – the averaging of many an-
Figure 15. Verticallyintegratedcontourplotsofspecificangular
nuli with an emissivity that falls with radius smooths
momentum (top) and entropy (bottom) for snapshots of the 90h
(left) and 915h (right) simulations. The color scale is linear, in- the power spectrum into a single power law. We see
creasingfrombluetoredtoyellowtowhite. Thenon-axisymmetric thesamebehaviorinoursimulations;thebreakfrequen-
shocks in915h correspondtoregions withdeficit (excess) angular cies from power spectra of individual radial shells agree
momentum(entropy).
withthelocalorbitalfrequencyforbothsimulations90h
and 915h. A break frequency 100M−1Hz then implies
10
a radius of r 16M. Our broken power law spectra are
≈
therefore likely due to the fact that our emissivity peaks
relatively near the outer radius used for the ray tracing,
r = 25M. A larger radial domain would likely shift the
break to smaller frequencies.
Observed break frequencies in the low/hard state are
typically ν 0.1 1Hz, which may be caused by the
b
∼ −
transition from a thin disk to a thicker, ADAF flow
(Esin et al. 1997). That would imply a transition radius
r 200 1000M2/3M. Our results for pre- and post-
t ≃ − 10
break slopes from both tilted and untilted simulations
agreewith those foundin CygnusX-1 (Revnivtsev et al.
2000) for an inclination i = 30◦. In GRO J1655-40
(Remillard et al.1999) our pre-breakslopes agreefor all
inclinations. However, the PSD for that source is well
described by a single power law.
There is no clear evidence in our work for high fre-
Figure 16. Shell-averagedorbitaleccentricitiesforalltiltedsim- quency QPOs due to the trapped inertial waves identi-
ulations,estimatedatonescale-heightinthedisk. Theincreasing
fied by Henisey et al. (2009), although there are more
eccentricity of orbits toward smaller radii leads to a crowding of
orbits at their apocenters, which, in turn, can generate standing features in power spectra from the 915h simulation at
shocks. higher significance than in 90h. Even when computing
PSDs for sets of spherical shells from the simulations,
from sources which may be geometrically thick. In gen-
there are no clear features in the tilted power spectra
eral,measurementsofsmallspin(largeinnerradius)may
thatarenot alsopresentin the untilted case. It is possi-
be unreliable unless the disk is known to be untilted. A
ble that this result could depend on the chosen emissiv-
reliable estimate of a large black hole spin (small inner
ity. Alternatively, the excess power in trapped inertial
radius),incontrast,couldruleoutthepresenceofatilted
waves could be insufficient to rise above the red noise
disk.
continuum.
The blue wing can be much broader for tilted ac-
Theindependence ofthe innerradiusofthetiltedsim-
cretion flows, and the tilted-disk line profiles depend
ulations on black hole spin is attributable to the extra
strongly on the observer azimuth as well as inclination.
angularmomentumtransportprovidedbythe asymmet-
Since a tilted disk is expected to precess (Fragile et al.
ricstandingshocks. Theseshocksareonlypresentinthe
2007), highly variable emission line profiles could signify
tilted simulations. Their strength scales with black hole
the presence of a tilted accretion flow, as pointed out
spin, which is a necessary condition for countering the
by Hartnoll & Blackman (2000) for warped thin disks.
greater centrifugal support at higher spins. The stand-
Since many LLAGN and X-ray binaries in the low/hard
ingshocks,inturn,appeartobeattributabletoepicyclic
stateshouldbetilted,time-variableemissionlinesshould
10 Dexter & Fragile
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