Impact of inclination on quasi-periodic oscillations from spiral structures
AAstronomy & Astrophysics manuscript no. Spiral © ESO 2018November 7, 2018
Impact of inclination on quasi-periodic oscillationsfrom spiral structures
P. Varniere and F. H. Vincent , AstroParticule & Cosmologie (APC), UMR 7164, Universit´e Paris Diderot, 10 rue Alice Domon et Leonie Duquet,75205 Paris Cedex 13, France. [email protected] Observatoire de Paris / LESIA, 5, place Jules Janssen, 92195 Meudon Cedex, France. Nicolaus Copernicus Astronomical Center, ul. Bartycka 18, PL-00-716 Warszawa, Poland.Received / Accepted
ABSTRACT
Context.
Quasi-periodic oscillations (QPOs) are a common feature of the power spectrums of X-ray binaries. Currentlyit is not possible to unambiguously differentiate the large number of proposed models to explain these phenomenathrough existing observations.
Aims.
We investigate the observable predictions of a simple model that generates flux modulation: a spiral instabilityrotating in a thin accretion disk. This model is motivated by the accretion ejection instability (AEI) model for low-frequency QPOs (LFQPOs). We are particularly interested in the inclination dependence of the observables that areassociated with this model.
Methods.
We develop a simple analytical model of an accretion disk, which features a spiral instability. The disk isassumed to emit blackbody radiation, which is ray-traced to a distant observer. We compute pulse profiles and powerspectra as observed from infinity.
Results.
We show that the amplitude of the modulation associated with the spiral rotation is a strong function ofinclination and frequency. The pulse profile is quasi-sinusoidal only at low inclination (face-on source). As a consequence,a higher-inclination geometry leads to a stronger and more diverse harmonic signature in the power spectrum.
Conclusions.
We present how the amplitude depends on the inclination when the flux modulation comes from a spiralin the disk. We also include new observables that could potentially differentiate between models, such as the pulseprofile and the harmonic content of the power spectra of high-inclination sources that exhibit LFQPOs. These mightbe important observables to explore with existing and new instruments.
Key words.
X-rays: binaries, accretion disks
1. Introduction
Quasi-periodic oscillations (QPOs) are a common featureof the power spectrum of black-hole X-ray binaries (seee.g. the reviews by Remillard & McClintock 2006; Doneet al. 2007, and references therein). Early on, QPOswere observed outside of the X-ray band, for example inthe optical (Motch et al. 1983) and recently in the in-frared (Kalamkar et al. 2015). They are characterized bya Lorentzian component in the power spectrum of thesource, distinguished from broader noisy peaks by a con-dition on the coherence parameter ( Q = ν LF / FWHM (cid:38)
Q > ∼ . −
30] Hz. LFQPOs are seen when thepower-law component of the spectrum is strong (such as thehard, hard/intermediate states). This suggests that theseoscillations are not only linked to the thermal disk but alsoto a non-thermal component which could be either a hotinner flow (Done et al. 2007) or some kind of a hot coronaabove the disk. Different types of LFQPOs have been intro-duced based mainly on their characteristics. In this articlewe are focusing on the most frequent one, namely the type-C LFQPO, which has the interesting property of having a strongly varying frequency ν LF that seems to be corre-lated with the disk flux. This may be a hint that LFQPOshave an origin in the disk while also being reprocessed in acorona or hot inner flow.While we have a lot of observations, no consensus onthe origin of the modulation has yet been achieved. Thereare several classes of models, depending on what causes themodulation. Among all the families of models, one origi-nated from Stella & Vietri (1998) which advocated thatthe Lense-Thirring precession of particles around a rotat-ing black hole could produce LFQPOs. This family wasfurther extended by Schnittman et al. (2006) and more re-cently Ingram et al. (2009) and Nixon & Salvesen (2014).Another family is based on different kinds of instabilities oc-curing in either the disk or the corona. Among those Tagger& Pellat (1999) looked at a spiral instability driven bymagnetic stresses (the accretion-ejection instability, AEI).Chakrabarti & Manickam (2000) propose that QPOs aredue to radiation emitted by oscillating shocks in the diskwhile Titarchuk & Fiorito (2004) consider a transitionlayer that comptonizes the outer disk. In yet another ap-proach, Cabanac et al. (2010) consider a magneto-acousticwave propagating in the corona while O’Neill et al. (2011)present MHD simulations of blackhole accretion flow ex-hibiting a dynamo cycle with a characteristic time that co- a r X i v : . [ a s t r o - ph . H E ] A p r . Varniere and F. H. Vincent: Observables to differentiate QPO models incides with LFQPOs. While several models have links withemission outside the X-ray band, for example in the caseof the AEI (Varni`ere & Tagger 2002), the most developedmodel at the moment is the one by Veledina et al. (2013,2015), which explains this multi-wavelength phenomena byelaborating on the precessing hot flow model of Ingramet al. (2009). It is worth noting that their curve of rmsas a function of the inclination of the system is quite dif-ferent from the one presented here, and observation mightbe able to distinguish between them.Despite the amount of ongoing modeling, up to nowQPO models have focused mainly on explaining the fre-quency observed in the power spectrum (PDS). While thisis essential, it is only the first observable we had accesstoo and all of the models fulfilled this requirement sincethey were created to answer it. It is therefore interesting tolook for other observables that might help to differentiatebetween models. Some of those we already have access tothrough observation, such as the amplitude, but also somewe do not have yet complete access to, such as the pulseprofile.In this article we are interested in revisiting theAEI model and expand on a previous Newtonian ap-proach (Varni`ere & Blackman 2005) to include a widerrange of frequencies and general-relativistic effects. In par-ticular, we want to see how the observables are affectedby the general relativistic effects in higher-inclination sys-tems. Already with this simple model, we aim at studyingthe evolution of the amplitude of the spiral modulation as afunction of the QPO frequency and to do this for high- andlow-inclination systems. We are also interested in how thisaffects the shape of the pulse profile and hence the harmoniccontent of its power spectrum. Section 3 presents our simplemodel of a disk featuring a spiral instability. Section 4 showsthe evolution of the amplitude with inclination and QPOfrequency. Section 5 analyzes the pulse profile for variousinclinations and discusses the corresponding power spectra.
2. The accretion-ejection instability as a model forLFQPO
In a nutshell, the accretion-ejection instability is a globalspiral instability which occurs in the inner region of a fullymagnetized (close to equipartition) accretion disk (Tagger& Pellat 1999). The energy is extracted from the disk(hence allowing accretion) through the spiral wave andthen stored in a Rossby vortex, which is located at thecorotation radius of the spiral (the point where the spiraland accretion disk rotate at the same velocity), which istypically a few times the inner radius of the disk. In thepresence of a low-density corona, the Rossby vortex willtwist the footpoint of the magnetic field line. This causesan Alfven wave to be emitted toward the corona, thereforelinking accretion and ejection (Varni`ere & Tagger 2002)which, in turn, has the unique consequence of linking whatis happening in the thermal disk with what is happeningat higher energy in the corona. This way the modulationin the disk will be transmitted to the corona.Here we will summarize salient points and observationaltests of the AEI as the origin of the LFQPO. We focus onthe X-ray data which represent the bulk of the observations: - the rotation frequency of the spiral is the orbital fre-quency at the corotation radius, r c , which is predictedto be a few tenths of the orbital frequency at the inneredge of the disk. This frequency is consistent with theLFQPO frequency (Tagger & Pellat 1999). - as the position of the inner edge of the disk evolves dur-ing the outburst, the rotation frequency of the spiralalso changes, which can be directly compared with ob-servation (Varni`ere et al. 2002). - by including general relativity through the existence of alast stable orbit and orbital velocity profile, the AEI ex-plains the observed turnover in the correlation betweenthe color radius (= inner disk radius, as determined bythe spectral fits) and the LFQPO frequency (Rodriguezet al. 2002; Mikles et al. 2009). - in the Newtonian approximation, the AEI is able to cre-ate a thermal flux modulation (Varni`ere & Blackman2005) in the range of the observed one. - once the AEI is established, 2D MHD simulations showa nearly steady rotation pattern, and is thus able toaccount for persistent LFQPOs - the AEI transfers energy and angular momentum towardthe corona by Alfven waves, thus providing a supplyof Poynting flux that may produce the compact jet of-ten observed in the low-hard state (Varni`ere & Tagger2002).Because of these characteristics, the model based on theassimilation of the AEI with the origin of the LFQPO hassince been expanded, first as a scenario for the β class ofGRS 1915+105 (Tagger et al. 2004), then as a way of clas-sifying blackhole states (Varniere et al. 2011) and, morerecently, a possible explanation for the different types ofLFQPO (Varniere et al. 2012).
3. Amplitude of the modulation for a spiral model
Rather than looking at the formation of the spiral insta-bility in a disk, we decided to focus on its consequenceonthe emission. Consequently, rather than taking full MHDsimulations of the AEI in the different conditions we wishto explore, we decided to create a simple, analytical modelthat mimics the temperature profile of the AEI to test thedifferent parameters more cleanly. Indeed, in a full fluidsimulation, changing one parameter in the initial conditioncan have repercussions on several observable parametersand, therefore, it is harder to study the different effectsseparately. We consider a geometrically thin accretion disk,introduce a spiral feature in this disk at a higher temper-ature, and consider the emission of blackbody radiation.Also, through the AEI, energy and angular momentum issent toward the corona (Varni`ere & Tagger 2002), but theexact amount depends on many parameters that are noteasily constrained by observation. Here we take a simplerapproach; rather than adding multiple unconstrained pa-rameters, we decide to stick to a very simple, analytic modelof the thermal emission in the disk for which we have moreconstraints. This means that we are able to predict thelight curve of LFQPOs in the lower energy part of the spec-trum (1keV), leaving a more in-depth study of the LFQPOrms-spectrum as a function of energy for a more complex,simulation-based, model.We have therefore taken a simple, analytic model of adisk that features a spiral structure, which is able to repro-
2. Varniere and F. H. Vincent: Observables to differentiate QPO models duce the main aspects of the AEI model as seen in simu-lations (see for example Varniere et al. 2012), and see howthe flux is thus modulated.
We consider a geometrically thin disk surrounding aSchwarzschild blackhole of mass M . The disk extends froma varying inner radius r in to a fixed outer radius r out =500 M . To keep a simple structure, we choose to have thetemperature profile T ( r ) ∝ r − η . In agreement with thethin disk blackbody model, we took η = 0 .
75 as the equi-librium temperature profile. This profile is then fixed bychoosing the temperature at the innermost stable circularorbit (ISCO), labeled T ISCO .This equilibrium disk is assumed to give rise to a spi-ral instability. We model this phenomena in a very simpleway, just describing the hotter spiral structure rotating inthe equilibrium disk, in agreement with numerical simula-tions of the AEI (see for example Varniere et al. 2012). Thetemperature of the disk with this added spiral feature reads T ( t, r, ϕ ) = T ( r ) (1) × (cid:34) γ (cid:16) r c r (cid:17) β exp (cid:32) − (cid:18) r − r s ( t, ϕ ) δ r c (cid:19) (cid:33)(cid:35) , where the perturbation term between brackets describes thespiral pattern. The parameter γ encodes the temperaturecontrast between the spiral and the surrounding disk. Thequantity r c is the corotation radius of the spiral. The spiraltemperature is thus following a power-law decrease whenmoving away from r = r c , with an exponent β . This ensuresthat the spiral will fade away into the disk after a few turns.In the latter Gaussian term, r s is a shape function encodingthe spiral feature. It ensures that the spiral’s width is afactor δ times the corotation radius. The shape functionreads r s ( t, ϕ ) = r c exp ( α ( ϕ − Ω( r c ) t )) , (2)where α is the spiral opening angle and Ω( r c ) is theKeplerian frequency at r c . Ultimately it is the rotation fre-quency of the spiral and the frequency at which the flux ismodulated. The spiral parameters are chosen to (1) be in agreementwith numerical simulations of the AEI and (2) produce rea-sonable values of amplitude. The present study is not de-voted to the detail of the dependency of observables on themodel parameters (see Varni`ere & Blackman 2005, for a pa-rameter study in the Newtonian approximation). Instead,we fix the parameters that represent the spiral and thenstudy, in a frozen framework, how the inclination impactsthe amplification, and this for a sample of modulation fre-quencies representative of the type-C LFQPO.In this respect, the temperature at the ISCO is takento be T ISCO = 10 K, thus emitting blackbody radiationmainly around 1 keV and the disk is geometrically thinwith an aspect ratio of
H/r = 0 .
01. The power-law expo-nent encoding the temperature decrease away from r = r c is set to β = 0 .
5. This ensures that the spiral will be negli-gible within a few turns. The opening angle is chosen to be α = 0 . , in agreement with numerical simulations (Varniereet al. 2011, 2012). Then we decided to use and freeze the pa-rameters of the spiral so that a QPO of about 10 Hz had anamplitude of about 10% at an inclination of 70 ◦ . This gives γ = 2 . δ = 0 . , which are conservative compared tothe maximum reached in numerical simulations (Varniereet al. 2012). These same parameters gave an amplitude ofabout 3% at an inclination of 20 ◦ .In the rest of the paper, only the inclination at which thesystem is observed and the position of the inner edge of thedisk will be varied. This produces a large set of QPO fre-quencies. Indeed, as shown by Tagger & Pellat (1999), thecorotation radius of the spiral is a few times the inner radiusof the disk. Here we chose to keep r c /r in fixed at a typicalvalue of 2, so that, as the inner radius varies, so the corota-tion radius. We consider an inner edge of the disk varyingin the range [1 . r ISCO , r ISCO ] which creates modulationfrequencies in the range [1 Hz ,
42 Hz] for M = 10 M (cid:12) . Wedo not compute very-low-frequency QPOs (below 1 Hz) be-cause, in our model, they demand both a very large disk(the corotation radius for this kind of frequency is far fromthe last stable orbit) and a very fine grid resolution, whichtranslates into overly long computing time. We also con-sider one simulation at 42 Hz, thus above the maximumobserved LFQPO frequency, to capture the tendency of therms amplitude towards the highest frequency LFQPOs. The whole disk is assumed to simply emit as a blackbody atthe temperature T ( t, r, ϕ ). So the specific intensity emittedat some position in the disk is I em ν = B ν ( ν em , T ) , (3)where the superscript em refers to the emitter’s frame, i.e. aframe corotating (at the local Keplerian frequency) with thedisk. This emitted intensity is then transformed to the dis-tant observer’s frame using the constancy along geodesicsof I ν /ν . Thus I obs ν = g I em ν , (4)where g = ν obs /ν em is the redshift factor. This redshiftfactor is, in particular, responsible for the so-called beamingeffect, which makes the observed specific intensity strongerwhen the emitter travels towards the observer and fainterin the opposite case. In the following we use hν obs = 1 keV.To compute maps of specific intensity I obs ν , weuse the open-source general relativistic ray-tracing codeGYOTO (Vincent et al. 2011) into which we added theparametrized disk profile that was defined in the previoussection. Null geodesics are integrated in the Schwarzschildmetric, backwards in time from a distant observer at someinclination with respect to the disk. Inclination is equal tothe angle between the observer’s line of sight and the nor-mal to the black hole’s equatorial plane. From such maps ofspecific intensity, the light curve (flux as a function of time)is derived by summing all pixels weighted by the elementof solid angle, which is subtended by each pixel.Figure 1 shows, in logscale, a spiral with a corotationradius located at r c = 10 r ISCO . It shows the spiral armfading into the background disk following the power-lawdecrease of the amplitude as the spiral expands. Before itreaches the end of the disk, the spiral has fully faded. Most
3. Varniere and F. H. Vincent: Observables to differentiate QPO models
Fig. 1.
Logscale view of the inner region of an accretiondisk, having a hotter m = 1 spiral wave with a corotationradius of r c = 10 r ISCO , seen at an inclination of i = 70 ◦ .of the emission actually comes from the first two turns ofthe spiral, close to r = r c , and we only consider the fullspiral for consistency.
4. Impact of frequency and inclination on theamplitude
We can now take advantage of our analytical model andtest how the amplitude of the modulation that comes fromthe rotating spiral behaves with respect to the positionof the corotation radius, namely the frequency and theinclination of the system. While it seems easy to obtainobservational data in the first case (amplitude versus ν QP O ), the second case (amplitude versus inclinationof the system) seems less attainable. Here we need toclarify that, even this first case is not as easily compared toobservation as it seems. Indeed, we are trying to look at theimpact of one parameter at a time, therefore no temporalevolution is taken into account. We are not ”monitoring”the evolution of the instability as the inner edge of the diskmoves but, instead, looking at the same spiral in differentlocations, meaning all the other parameters are frozenwhile we take snapshots with different inner edges of thedisk.To be able to compare the different lightcurves on thesame plot we renormalize them with the time being t/T where T is the period of each modulation, and the flux being f /f mean , where f mean is the mean flux of each particularcurve. This allows for an easier comparison of the shape ofeach lightcurve independantly of its total flux and period.For all models, we define the amplitude asamp = f max − f min f max , (5)where f min and f max are the extremal values of flux. In the case of a spiral, or any non-axisymetrical structureorbiting in the disk, one would get a modulation as a com-bination of two effects. First, as the spiral orbits around the disk, its observed intensity is modulated by the beam-ing effect: this will be stronger on the approaching sideand fainter on the receding side. In addition, the projectedemitting area on the observer’s sky varies with time as thespiral rotates, which also creates a flux modulation. Fromthis, it appears that the most important parameter of thespiral, as far as amplitude is concerned, is its temperaturecontrast with the disk (encoded in the γ parameter). Thehigher the contrast with the disk, the greater the amplitudeof the modulation.In Figure 2 we show the renormalized lightcurves forthree positions of the inner edge (and consequently of thespiral’s corotation radius) for a disk seen under an inclina-tion of 20 ◦ (close to face-on). Several facts already appearon this plot. While all of the spirals have the same ampli-tude, γ , the amplitude increases as the corotation radius isfurther away in the disk. rc = 4 incli 20rc = 10 incli 20rc = 20 incli 20 fl u x / m ean Fig. 2.
Renormalized light curves obtained for the samespiral with three values of the inner edge of the disk andassociated corotation radii. Here the system is viewed atan inclination i = 20 ◦ and the black dots represent r c =4 r ISCO , blue triangles r c = 10 r ISCO and red squares r c =20 r ISCO .This can easily be understood as follows. Given the tem-perature profile in Eq. 1, it is straightforward to obtain thatthe ratio between the maximum temperature along the spi-ral (i.e. at the corotation radius r = r c ) and the tempera-ture at the inner edge is T spi /T in = (1 + γ ) / γ = 2 . B ν ( T spi ) /B ν ( T in ) as a functionof the value of the inner radius. This ratio is a stronglyincreasing function of the inner radius. As a consequence,the spiral dominates more and more the inner regions ofthe disk as the inner radius increases. Thus, at similar spi-ral parameters, the amplitude increases with a receding disk .It is important to note that this is valid for similar spiralparameters and that, for widely varying spiral parameters,the amplitude behavior with respect to the inner edge ofthe disk could be reversed. On the same note, if we relaxthe constraint on the corotation r c = 2 r in , we can see forthe same r c and spiral parameters, the modulated part ofthe flux is constant while, changing the position of the in-ner edge of the disk changes the unmodulated part of theflux, hence impacting the total rms amplitude of the mod-ulation. In the AEI framework, this would be the equivalent of say-ing that the disk physical paramaters, such as the density andmagnetic field, are changing.4. Varniere and F. H. Vincent: Observables to differentiate QPO models
Another interesting point is that, at low inclination, themodulation closely resembles a sine function. It takes aFourier decomposition to see the presence of small ampli-tude harmonics and is often limited to the first few har-monics (more about this in Section 5). Those also becomestronger similarly to the amplitude when the inner edge ofthe disk is further away from the ISCO.
Following the observation from Fig.2. that the same spiralhas a stronger amplitude as it is further away in the disk, wecan plot the evolution of the amplitude of the modulationcreated by the same-parameter spirals as a function of thecouple ( r in , r c = 2 r in ). With the parameters we fixed, weget an approximate 3% amplitude at a frequency of 10 Hzfor the low inclination ( i = 20 ◦ ) case. a m p li t ude sun )/M Hz1 10 Fig. 3.
Evolution of the amplitude of the modulation asfunction of the QPO frequency, meaning Ω( r c = 2 r in ), foran inclination i = 20 ◦ .As we can see in Fig. 3, we follow a wide range of fre-quencies and see a steady increase in the amplitude as thefrequency gets lower. This is in part linked to the fact thatwe are keeping the inner egde of the disk and the corota-tion radius in a locked ratio of r c = 2 r in . Indeed, as thefrequency increases it means that the inner edge of the diskgets closer to the last stable orbit, hence gets hotter andemits more in the observation band. As a consequence themodulation amplitude over the total emission is lower.Here we need to be cautious, since this plot is donefor the same spiral parameters but looking at different fre-quencies, meaning that it does not represent an evolutionof the spiral responsible for the modulation. In the case ofa temporal evolution, such as the one would get in a fullMHD simulation and/or a full outburst, the spiral wouldalso change (e.g. its amplitude) hence it would influence theprecise shape of this plot. Nevertheless, it shows a strongpropensity of the amplitude to decrease with frequency. One parameter that can change a lot, both the beaming ef-fect and the projected emitting surface (and thus, the mod-ulation) is the inclination at which the system is observed.Using observation alone this is not something that caneasily be studied. However, Motta et al. (2015) show oneof the first statistical studies demonstrating that higher-inclination systems tend to have a higher amplitude forthe QPO than is the case for lower-inclination systems. It is therefore interesting to determine how the amplitude inan almost edge-on system (inclination 70 ◦ ) and an almostface-on system (inclination 20 ◦ ) would compare for exactlythe same spiral parameters . For this reason we focus on theratio between amplitudes at different inclinations and nottheir actual value.Figure 4 shows the evolution of the ratio of the ampli-tude at inclination 70 ◦ over the amplitude at inclination 20 ◦ as a function of the position of the structure in the disk,namely r c , which is related to the frequency through themass of the central object. Indeed, as in observation, onecannot observe the same system at different inclinations,it seems easier to produce a plot that could be scaled tothe different masses of the objects. While high inclination a m p / a m p c /r ISCO
Fig. 4.
Evolution of the ratio of the amplitude at inclina-tion 70 ◦ over the amplitude at inclination 20 ◦ as a functionof the position of the structure given in units of the ISCOradius.system always have a higher amplitude than low inclina-tion system for the same physical parameters, we see thanthe ratio is dependant on the frequency of the modulation,linked to the position of the co-rotation radius. We see thatthe ratio decreases from ∼ . r c , which trans-lates to a higher frequency, to ∼ . r c , meaninga lower frequency. Indeed, this is related to the fact thatthe flux modulation is, in particular, due to the beamingfactor. The flux will be boosted when the hottest parts ofthe spiral are on the approaching side of the disk, while theflux will be deboosted when these hottest parts are on thereceding side. This boosting is a strong function of inclina-tion: it will be bigger for close to edge-on views because theemitter will then be traveling almost exactly towards theobserver. Furthermore, as we get far from the blackhole,the effect of the beaming will decrease, and will thus beless important for lower QPO frequencies (higher r c ).Figure 4 is impossible to compare directly with the re-sults of Motta et al. (2015) since these authors average sev-eral sources, hence different blackhole masses, which in turnmean the same QPO frequencies do not correspond to thesame distance in the disk, and to different inclinations anddifferent times of their outburst evolution. On the contrary,here we look at the amplitude dependence on inclinationfor exactly the same system at different QPO frequencies,which is clearly not what happens in an outburst, since thesource evolves dynamically. Nevertheless, it is already in-teresting to see that the same system, under the same con-ditions, does indeed have a different amplitude, dependingon the inclination of the system, and that this differenceevolves, depending on the position of the spiral in the disk.
5. Varniere and F. H. Vincent: Observables to differentiate QPO models
5. Case of the pulse profile in high-inclinationsystems
While the previous section focused on the amplitude of theQPO, there is more to QPOs than just a frequency and anamplitude. Indeed, here we have access to the exact pulseprofile. This allows us to study the more direct impact ofinclination on the lightcurve.
In Fig. 5, we show one period of three pulse profiles forthree different locations of the spiral, as seen from an incli-nation of 70 ◦ . We have chosen the same position/frequencyas in Fig. 2 so that it is easier to compare both cases. Themain difference with the 20 ◦ inclination case is that, athigh inclination, the detailed shape of the pulse is clearlynot sinusoidal. This implies that any Fourier decomposi-tion will contain more than one frequency, even when theinitial signal in our model only has one. This effect comesfrom Doppler boosting and the varying apparent area ofthe spiral on the observer’s sky, and it gets stronger withinclination. We explore how this should be visible on thePDS in the next section . rc = 4 incli 70rc = 10 incli 70rc = 20 incli 70 fl u x / m ean Fig. 5.
Same as Fig. 2 for an inclination of 70 ◦ .However, it is worth mentioning another possible way ofdetecting this change in the shape of the pulse profile, evenwithout having access to its details. Indeed, in Fig. 5 wesee that for high-inclination systems, the lightcurves arenot symmetrical with respect to the mean values. Whilethis is only a few % and hardly detectable all the time, wemay find some high-inclination systems that have a strongQPO around a few Hz in a steady enough observation inwhich we can try to assess the time spent above and belowthe mean value. Perhaps this would be worth investigatingin, for example, the Plateau state of GRS 1915+105? Whilethe resolution needed to look at it on a pulse timescale isbeyond the capacity of past and present satellites, it wouldbe an interesting new observable to differentiate betweenmodels for future missions. As seen in the previous section, the shape of the pulse pro-file departs more and more from a sinusoid as the inclina-tion of the system increases from almost face-on to almostedge-on. When looking at it in the Fourier space, this de-parture translates into more harmonics that might be moreeasily detectable than a difference in the pulse profile. To compute the power density spectrum, we extendedour lightcurve to 20 orbits and added a 1 /f noise. Also, aswe are interested in comparing the extent of the harmonicstructure with respect to inclination, we re-normalized theamplitude of the fundamental peak in each case to 1. Thisallows us to compare only the relative strength of the dif-ferent peaks.In the case of a QPO frequency of about 2 . r ISCO ), we already have a firstharmonic present even at the inclination of 20 ◦ , while thepulse profile looks sinusoidal (see Fig.2.). In the case of aclose to edge-on view, we get a much richer and strongerharmonic content. Indeed, we have not one, as in the closeto face-on view, but several additional and stronger peaks,while the simulation we ran had one mode present (one-arm spiral). This behavior of richer harmonic content forhigh-inclination systems does not seem to depend on thefrequency of the modulation, though it would be easier todetect for higher amplitude fundamentals, hence smallerfrequencies.However, care must be taken when interpreting this re-sult since instabilities that give rise to spiral waves in a disktend to have more than one mode and therefore the originalsignal could be more complex than the case with only onemode that is presented here. The different modes are in aclose to harmonic relationship and could therefore lead toharmonic peaks in the PDS. Nevertheless, these instabili-ties also tend to have such a degree of competition betweenmodes (more as a transition from one to another) and itshould not be difficult to detect this.
6. Conclusion
In this paper we looked at a simplified version of the spi-ral formed by accretion ejection instability and studied thelightcurves that are observable from this type of system.Our main goal was to study how the amplitude of the ob-served lightcurve depends not only on the frequency of theQPO, but also on its inclination, with respect to the ob-server.Our model predicts that, for a similar spiral struc-ture, at a given frequency, the amplitude is strongly de-pendent on inclination and increases at high inclination.Using parameters in agreement with both numerical simu-lations and observations, our simple model is able to pro-duce amplitudes that differ by an amount of ∼ . − . ◦ ) with respect to low-inclination(20 ◦ ) sources. Thus it is in agreement with the recent studyof Motta et al. (2015), which shows higher amplitudes fromhigh-inclination sources, although it is not possible to di-rectly compare the actual amplitude ratio predicted by oursimulations with observations, since we do not know if theobserved sources were in a state that is similar to our model.Another interesting point raised by our ability to pro-duce detailed lightcurves is how the pulse profile of theseQPO changes with frequency and inclination. It becomesless and less sinusoidal as the source inclination increases.This translates into a richer harmonic content in the powerspectrum. While there is no statistically significant proofthat high-inclination sources have a significantly higher har-monic content than low inclination sources, it is coherentwith the data gathered in Motta et al. (2015) (private com-munication). This would be worth exploring in more detail,especially if other QPO models have distinct predictions.
6. Varniere and F. H. Vincent: Observables to differentiate QPO models
Acknowledgements.
PV acknowledges financial support from theUnivEarthS Labex program at Sorbonne Paris Cit´e (ANR-10-LABX- 0023 and ANR-11-IDEX-0005-02). FHV acknowledges fi-nancial support by the Polish NCN grant 2013/09/B/ST9/00060.Computing was partly done using the Division Informatique del’Observatoire (DIO) HPC facilities from Observatoire de Paris (http://dio.obspm.fr/Calcul/) and at the FACe (Francois Arago Centre) inParis.
References