MSc Astronomy and Astrophysics
Track Astronomy and Astrophysics
M
ASTER
T
HESIS
A spectral-timing comparison of accreting
black holes and neutron stars
by
Sarah Brands
6056334
August 2017
54 ECTS
September 2015 - August 2017
Supervisor:
Dr. Phil Uttley
Examiners:
Dr. Phil Uttley
Dr. Anna Watts
Anton Pannekoek
Institute
The accreting compact object within a low mass X-ray binary can be either a neutron star or a black hole. Variability studies suggest that if the magnetic field of an accreting neutron star is weak, the accretion flow around it behaves very similar to the flow around a black hole. The flow can be examined further by measuring time lags between variations seen in different X-ray energy bands. The length of the lags may be influenced by the surface of the compact object, which is solid for a neutron star and presumably an event horizon for a black hole. We aim to observe a time lag surface signature by comparing the lags of seventeen neutron star and three black hole sources using archival RXTE data. We find that typically neutron stars show no lags, whereas the lags of black holes are long. Our findings support a recently proposed model which attributes the lags to propagation delays and to pivoting of the power-law component observed in the energy spectrum of these sources. The power-law emission is associated with an optically thin corona, which is assumed to be present in both neutron star and black hole systems. In the case of a neutron star, additional emission coming from the solid surface changes the pivoting behaviour of the power-law in such a way that the time lags are expected to be zero. We find further support for this model in the covariance spectra of two sources, which show that the variable components in both black hole and neutron star spectra have the shape of a power-law. Our findings provide indirect evidence for the existence of event horizons.
Once trapped inside a black hole there is no way to escape. Even light, which makes that we can see the things around us, will get imprisoned when it ends up inside a black hole. Because of this it is impossible to see black holes, no matter how advanced our telescopes are, or ever will be. This is a pity, because black holes are very interesting, for example to check whether we have a good understanding of the theory of gravity. Luckily for us, there are often stars in the vicinity of black holes. Sometimes a star is so close, that the outermost layers of this hot ball of gas are pulled towards the black hole. When this happens the gas gets even hotter than it normally is, causing it to emit in X-rays. The combination of the star and the black hole, together with the gas that is flowing from one to another, is called an X-ray binary.
An X-ray binary can also consist of a normal star and a neutron star, instead of a normal star and a black hole. In that case X-rays can emerge in the exact same manner. Despite its name, a neutron star is more similar to a black hole than it is to a normal star. This is because the particles that make up a neutron star are packed extremely close together. Neutron stars weigh about as much as the Sun – which is about half a million times as heavy as the Earth – but their cross section is not much bigger than that of Amsterdam. So much material in such a small space makes that the gravity around neutron stars is extremely strong. Just like in the vicinity of black holes.
Figure 1. X-ray binary. When a normal star is in the vicinity of a black hole
or neutron star, gravity pulls gas off the surface of the star, which will start flowing towards the surface of the black hole or neutron star. This gas will become extremely hot and will start emitting X-rays.
living creatures this radiation is harmful. However, astronomers who want to study X-ray binaries have to be creative to bypass this problem: they have to use space telescopes to do their job. These telescopes are orbiting around the Earth, so they do not bother about the atmosphere.
When we look at X-ray binaries with space telescopes we cannot see them as clearly as in Figure 1. Actually, we do not see much more than a tiny dot in the sky. Luckily, we can still learn a lot from these tiny dots. For example, we can distinguish different types of X-ray emission: cold gas emits a different type of X-rays than hot gas does. Because of this, we know that the gas around neutron stars and black holes can have different temperatures. Another thing that we can measure is the brightness of the emission: how bright is an X-ray binary at a certain moment, and how does this change in time? It turns out that the brightness of X-ray binaries changes constantly. This is a hint that sometimes a lot of gas is flowing from the normal star to the black hole or neutron star, but at other times only little.
Figure 2. The Rossi X-ray Timing Explorer space telescope.
For my Master’s thesis I investigated whether there is a difference be-tween the emission coming from an X-ray binary with a black hole, and that coming from one with a neutron star. In order to do this, I examined how the brightness of several X-ray binaries changes in time, using the Rossi X-ray Timing Explorer space telescope (see Figure 2). I was especially interested in comparing the different types of X-ray emission.
The difference between neutron stars and black holes was very clear. An explanation for this is that a neutron star can emit X-rays, but a black hole cannot. Once the hot gas has reached the black hole it will be trapped inside and we can no longer see it. On the other hand, we can see a neutron star: if hot gas will reach a neutron star, its surface will become super hot and will start emitting X-rays, just like the gas itself. So, even though we only see X-ray binaries as tiny dots in the sky, they taught us that there is a difference between neutron stars and black holes: in the case of a neutron star the surface radiates, whereas in the case of a black hole the radiation is absorbed. It falls right through the black hole boundary – nothing will stop it. This is exactly what the theory of gravity predicts: something that is caught inside a black hole will be stuck in there forever.
Eenmaal in een zwart gat is er geen enkele mogelijkheid meer tot ont-snappen. Voor niets en niemand niet. Ook licht, wat ervoor zorgt dat wij dingen kunnen zien, raakt opgesloten als het in een zwart gat belandt. Hierdoor is het – hoe geavanceerd onze telescopen ook zijn, of ooit zullen worden – onmogelijk om zwarte gaten te zien. Onhandig, want zwarte gaten zijn heel interessant, bijvoorbeeld om te controleren of we wel goed begrijpen hoe zwaartekracht werkt.
Een geluk bij een ongeluk is dat er in de buurt van zo’n zwart gat vaak sterren te vinden zijn. Soms is er een ster zelfs zo dichtbij, dat de buitenste lagen ervan – een ster is immers een bol van heet gas – naar het zwarte gat worden toegetrokken. Wanneer dit gebeurt wordt het gas nog veel heter dan normaal, waardoor het Röntgenstraling gaat uitzenden. De combinatie van de ster en het zwarte gat, samen met het gas dat van de een naar de ander stroomt, noemen we een Röntgendubbelster.
Een Röntgendubbelster kan ook bestaan uit een gewone ster en een neutronenster, in plaats van een gewone ster en een zwart gat. Er kan dan op precies dezelfde manier Röntgenstraling ontstaan. Neutronen-sterren lijken eigenlijk meer op zwarte gaten dan op gewone Neutronen-sterren. De deeltjes waarvan ze gemaakt zijn, zitten namelijk extreem dicht op elkaar gepakt. Neutronensterren wegen ongeveer evenveel als de zon – die ongeveer een half miljoen keer zo zwaar is als de aarde – maar zijn qua doorsnede niet veel groter dan Amsterdam. Doordat er zo veel materiaal in zo’n kleine ruimte zit, is de zwaartekracht rondom zo’n neutronenster heel erg sterk. Net als in de buurt van een zwart gat.
Figuur 1. Röntgendubbelster. Wanneer een gewone ster dicht bij een zwart
gat of neutronenster staat, valt door de zwartekracht gas van de gewone ster op het zwarte gat of de neutronenster. Dit gas wordt extreem heet en gaat Röntgenstraling uitzenden.
goed ook, want te veel van deze straling is gevaarlijk voor onze gezond-heid. Voor sterrenkundigen is het onpraktisch, maar ruimtetelescopen bieden een oplossing: deze bekijken het heelal vanuit een satelliet in een baan om de aarde en hebben geen last van de atmosfeer.
Met ruimtetelescopen is de Röntgenstraling van vele dubbelsterren gemeten. Zoveel detail als in Figuur 1 kunnen we in het echt niet zien: we zien eigenlijk niet meer dan een klein stipje aan de hemel op de plek waar de dubbelster staat. Toch kunnen we hier veel van leren. We kunnen bijvoorbeeld verschillende typen Röntgenstraling onderscheiden: koud gas geeft een ander soort Röntgenstraling af dan heet gas. Zo weten we dat niet al het gas rondom neutronensterren en zwarte gaten dezelfde temperatuur heeft. Iets anders dat we kunnen meten is de helderheid van de straling: hoeveel straling zien we op een bepaald moment, en hoe verandert dit in de loop van de tijd? Het blijkt dat de helderheid van dubbelsterren erg wisselend is. Dit is een aanwijzing dat er soms veel, en dan weer weinig gas van de gewone ster naar het zwarte gat of de neutronenster stroomt.
Figuur 2. De Rossi X-ray Timing Explorer ruimtetelescoop.
Voor mijn afstudeeronderzoek heb ik aan de hand van waarnemingen met de Rossi X-ray Timing Explorer ruimtetelescoop (zie Figuur 2) ge-keken of er een verschil is tussen de Röntgenstraling afkomstig van dubbelsterren met een zwart gat, en dubbelsterren met een neutronen-ster. Ik deed dit door te kijken naar veranderingen van de helderheid van de straling in de loop van de tijd. Het was vooral interessant veran-deringen van de verschillende soorten Röntgenstraling te vergelijken.
Ik vond een duidelijk verschil tussen neutronensterren en zwarte gaten. Het verschil kan begrepen worden door te bedenken dat een neutronenster zelf kan stralen, maar een zwart gat niet. Eenmaal bij het zwarte gat valt het hete gas namelijk naar binnen, en kunnen we er niets meer van zien. De neutronenster kunnen we wel zien: het oppervlak is verschrikkelijk heet door al het gloeiende gas dat erop terecht komt en gaat, net als het gas, ook stralen in Röntgen.
Het oppervlak van de hemellichamen speelt dus een cruciale rol voor de straling die we zien: bij een neutronenster straalt het oppervlak, waar in het geval van het zwarte gat de straling er dwars doorheen valt – om nooit meer terug te keren. Dit is wat de zwaartekrachttheorie al
voorspelde: iets dat in een zwart gat belandt, komt er niet meer uit. viii
1 Introduction 1
2 Theory and background 5
2.1 Accretion onto compact objects. . . 5
2.2 Accretion disks in low mass X-ray binaries . . . 6
2.2.1 The alpha-disk model . . . 6
2.2.2 Disk emission . . . 7
2.2.3 The viscous time scale . . . 7
2.3 Low mass X-ray binaries in outburst . . . 8
2.3.1 Energy spectra . . . 8
2.3.2 Disk geometry and the origin of the power-law . . 12
2.4 X-ray spectral-timing . . . 14
2.4.1 Short time scale variability . . . 15
2.4.2 Lag observations . . . 16
2.4.3 Origin of the lags . . . 18
2.5 A propagative flow model for hard lags. . . 19
2.5.1 The fluctuating-accretion model of Lyubarskii . . . 19
2.5.2 Fluctuating-accretion models in practice . . . 21
2.5.3 Hard lags from a compact corona . . . 23
2.5.4 Effect of the neutron star surface . . . 28
3 Methods 31 3.1 Fourier analysis techniques . . . 31
3.1.1 Discrete Fourier transform . . . 31
3.1.2 Power spectral density . . . 33
3.1.3 Cross spectrum, coherence and lags . . . 35
3.1.4 Energy dependent timing spectra . . . 38
3.2 State classification with power colours . . . 39
3.2.1 Power colours . . . 40
3.2.2 Black holes and neutron stars in the PCC . . . 40
3.2.3 Binning . . . 41
3.2.4 Hardness . . . 46
3.3 Data selection . . . 47
3.3.1 Rossi X-ray Timing Explorer . . . 47
3.3.2 Targets of lag analysis . . . 48
3.3.3 Data selection for the covariance spectrum . . . 50
3.3.4 Data selection for the lag-energy spectrum . . . 51
3.4 Lag calculation . . . 51
3.4.1 Data reduction with chromos . . . 51 ix
3.5 Energy dependent timing spectra . . . 55
3.5.1 Further adaptions to chromos. . . 55
3.5.2 Adaptions of spectime . . . 56
3.5.3 Calculation and response matrix . . . 57
3.5.4 Fitting the spectrum . . . 58
4 Lag and power spectral comparison 59 4.1 Timing spectra of individual sources . . . 59
4.2 A comparison of Aquila-X1 and GX-339-4 . . . 64
4.3 Averaging per compact object type . . . 66
4.4 Soft low frequency lags . . . 75
5 Energy dependent timing spectra 79 5.1 Covariance spectra . . . 80
5.2 Lag-energy spectra . . . 83
6 Discussion 87 6.1 A spectral-timing surface signature . . . 87
6.2 Soft low frequency lags . . . 89
6.3 Limitations and weaknesses . . . 90
6.3.1 Neutron star magnetic fields . . . 90
6.3.2 Central object mass. . . 91
6.3.3 Error calculation with low number statistics . . . . 91
6.3.4 Potential bias due to count rate selection . . . 92
6.3.5 Scatter in the PCC of state bin 1. . . 92
6.4 Suggestions for further research . . . 93
6.4.1 Energy dependent timing spectra, soft lags . . . 93
6.4.2 Tracing flow parameters throughout an outburst. . 93
6.4.3 Lag spectra as a differentiating diagnostic. . . 93
6.4.4 Extending the lag comparison to the SIMS and SS . 94
7 Conclusion 95
Bibliography 97
Appendices 101
A Power spectra and lags per state 103
B Fast Fourier Transform 125
1.1 Low mass X-ray binary . . . 1
2.1 Idealised accretion disk . . . 6
2.2 Outbursts of Aquila-X1 . . . 9
2.3 Effect of the boundary layer on the energy spectrum . . . 10
2.4 Spectral states of Cyg X-1 and 4U1705-44 . . . 10
2.5 Energy colour-colour diagram of 4U 1608-52. . . 11
2.6 Hardness intensity diagram of GX-339-4 . . . 12
2.7 Disk geometry in the hard and the soft state . . . 13
2.8 Geometry of the source of the hard X-ray photons . . . 14
2.9 Short time scale variability of Cyg X-1 . . . 15
2.10 Power spectra of Cyg X-1 . . . 16
2.11 Hard lags in Cyg X-1 . . . 17
2.12 The main components of a BH LMXB energy spectrum. . . . 18
2.13 Propagation of mass fluctuations through the disk. . . 20
2.14 Propagative flow model for short time scale variability . . . . 21
2.15 Lag-energy spectra of GX-339-4. . . 23
2.16 Cooling mechanisms in the corona . . . 25
2.17 Lags from a compact corona: black holes vs. neutron stars . . 30
3.1 Fourier transform . . . 32
3.2 Types of noise . . . 34
3.3 Coherence visualed with vectors. . . 37
3.4 Determining power colours from the power spectrum . . . . 40
3.5 Schematic power colour-colour diagrams (PCCs) . . . 41
3.6 Power colour-colour diagram and examples of power spectra 42 3.7 Neutron star and black hole observations in the PCC . . . 43
3.8 Comparison of state binning methods . . . 44
3.9 State binning using a polynomial fit . . . 45
3.10 Hardness hue diagram . . . 47
4.1 All observations of our sample in the PCC . . . 59
4.2 Power spectra and phase lags of GX-339-4 . . . 60
4.3 Location and associated states of the bins in the PCC . . . 61
4.4 Power spectra and phase lags of Aquila-X1 . . . 63
4.5 Lag difference of GX-339-4 and Aquila-X1 . . . 64
4.6 Power spectra and phase lags of GX-339-4 and Aquila-X1 . . 65
4.7 Powerspectra and phase lags of black holes and neutron stars 67 4.8 Average power spectra and phase lags of black holes . . . 68
4.9 Average power spectra and phase lags of neutron stars. . . . 69
4.10 Average lag difference of black holes and neutron stars . . . 70 xi
4.13 Comparison of high and low inclination black hole lags . . . 74
4.14 Spectra of bin 1 for neutron star sources . . . 76
4.15 Spectra of bin 1 for black hole sources. . . 78
5.1 Unfolded covariance spectra of Aquila-X1 . . . 81
5.2 Unfolded covariance spectra of GX-339-4. . . 82
5.3 Lag-energy spectra of IGR J17480-2446 and 4U 1705-44. . . . 84
5.4 Lag and power spectra of IGR J17480-2446 and 4U 1705-44 . 84 6.1 Examples of spectra with different values forminp . . . 92
List of Tables
3.1 Targets of the lag analysis. . . 493.2 Targets and observations of the covariance analysis . . . 49
3.3 Targets and observations of the lag-energy spectrum . . . 50
3.4 PCU selection for the covariance spectrum . . . 55
3.5 Binned channels for the covariance spectrum . . . 57
4.1 χ2test results of Aquila-X1 compared to GX-339-4 . . . 66
4.2 χ2 redvalues of individual sources compared to NS average . . 71
4.3 χ2 redvalues of individual sources compared to NS average . . 73
4.4 χ2test results of NS averages compared to BH averages . . . 77
cs Speed of sound
CXY,n Cross spectrum of light curves x(t)and y(t)at fn ¯CXY(νj) Cross spectrum at binned frequency νj
Cv(νj) Covariance at binned frequency νj
D(R) Dissipation rate: rate of energy that is radiated per unit
plane surface area
E Energy
Es Energy of seed photons fn Fourier frequency fNyq Nyquist frequency
F Flux
G Gravitational constant
h Hue
H Vertical scale height
Hhs Hardness
I Number of frequencies in binned Fourier spectrum
J Angular momentum
K Number of frequencies in a bin (binned Fourier analysis) k Element counter (Fourier analysis)
Lacc Total accretion luminosity Ldisk Total disk luminosity
Ls Cooling luminosity of power-law region Lh Heating luminosity of power-law region Mc Mass of the accreting body/central object
M Number of light curve segments in a bin (binned Fourier analysis)
˙
M Mass accretion rate
M Solar mass
m Segment counter (binned Fourier analysis) n Frequency counter (binned Fourier analysis) n2 Bias term needed for coherence correction N Total amount of elements in data series N(E) Photon flux density at energy E
p Statistical p-value P Pressure within the disk
Pn Periodogram of DFT at frequency νj
Pnoise Poisson noise level in the power spectrum
¯P(νj) Power spectrum at binned frequency νj R Radius: distance from the central object R∗ Radius of the accreting body
Rg Schwarzschild radius tvisc Viscous time
Ts Surface temperature of the disk Tobs Total duration of a time series
v Velocity
vff Free-fall velocity vK Keplerian velocity
vR Radial ‘drift’ velocity or infall velocity
hxi Average count rate of light curve x(t)
Xn Value of the DFT at fnof light curve x(t) Yn Value of the DFT at fnof light curve y(t)
α Dimensionless measure for viscosity α∗ Power-law index
β Constant in expression for power-law index, 1/6 for XRBs ∆Cv(νj) Error on the covariance at νj
∆t Temporal resolution of a time series
∆τνj) Error on the time lag between two curves at νj ∆φ(νj) Error on the phase lag between two curves at νj
γ2 Coherence
Γ Power-law index
Γ0 Constant in expression for power-law index, 7/3 for XRBs
ν Viscosity
νdof Degrees of freedom νj Binned frequency
ρ Density
σY2(νj) Noise-subtracted absolute rms-squared of time series y(t)
τ(νj) Time lag between two curves at νj τd Optical depth of the disk
φn Phase of the cross spectrum at fn
φ(νj) Phase lag between two curves at νj χ2 Chi-squared value (statistical test)
χ2red Reduced chi-squared value (statistical test) ΩK Keplerain angular velocity
AGN Active galactic nucleus AMP Accreting millisecond pulsar AP Accreting pulsar
ASM All-Sky Monitor
BH Black hole
BRB Broad reference band CCD Colour-colour diagram CCF Cross correlation function DFT Discrete Fourier transform FFT Fast Fourier transform
HEXTE High Energy X-ray Timing Experiment HID Hardness-intensity diagram
HIMS Hard intermediate state
HS Hard state
IC Individual channel LMXB Low mass X-ray binary
NS Neutron star
PC Power colour
PCA Proportional counter array PCC Power colour-colour diagram PCU Proportional counter unit PSD Power spectral density QPO Quasi Periodic Osciallation RXTE Rossi X-ray Timing Explorer
SS Soft state
SIMS Soft intermediate state
XRB X-ray binary
1
Introduction
In a low mass X-ray binary (LMXB) a compact object is accreting matter from a low mass companion star (Van Paradijs & Van der Klis,2001). Around the compact object the accreting material forms a geometrically thin, optically thick disk (see Figure1.1). During periods of high mass accretion rate, referred to as outbursts, a lot of gravitational energy is released in the disk as heat and then radiated, mainly in the form of X-rays (Shakura & Sunyaev,1973). LMXBs are unique laboratories that can give us insight into the effects of extremely strong gravitational fields, where event horizons are expected. Furthermore, by studying LMXBs we can strengthen our understanding of the physics of accretion. This is important for many astrophysical processes, ranging from star and planet formation to accretion onto super massive black holes.
The accreting compact object within a LMXB can be either a neutron star (NS) or a black hole (BH). The compactness of these objects is comparable, therefore the accretion flow around NS and BHs can be expected to be similar, assuming the NS magnetic field strength is low (van der Klis, 1994b). Whereas the gravitational potential of the two objects is almost identical, there is also an important difference between the two: while a BH is assumed to have an event horizon, a NS has a solid surface.
The differences and similarities of these two types of source are reflected in their energy spectra. Typically both spectra contain a black-body component, which is associated with the optically thick disk (e.g.
Figure 1.1: Artist impression of a low mass X-ray binary. Figure adapted
from Rob Hynes.
Mitsuda et al.,1984), as well as a power-law component (see e.g. Remil-lard & McClintock,2006;Lin et al.,2007). The latter is associated with non-thermal processes occuring in the regions closest to the compact object, such as (inverse-)Compton scattering. In the case of NS LMXBs, an additional blackbody-like component of the hot boundary layer is present in the spectrum (e.g.White et al.,1988). The exact properties of the power-law emitting region are still under debate, but observations seem to suggest that it should be centrally located and very compact (e.g. Uttley et al.,2011). Possibilities include a hot, optically thin, geometri-cally thick corona (see e.g.Sunyaev & Truemper,1979), jets (Markoff et al.,2005) or a geometrically thick hot inner flow (Shapiro et al.,1976; Zdziarski et al.,1998).
In the course of an outburst LMXBs exhibit different spectral states. In the so called hard state the powerlaw is dominating, whereas the blackbody radiation dominates the spectrum in the soft state. These spectral changes are associated with changes in the geometry of the inner accretion flow (e.g.Done et al., 2007). The different states are also observed when the variability of LMXBs is considered, which is typically done by analysing the shape of the power spectrum. The fact that BH and NS systems show very similar short time scale variability stresses the similarity of their accretion processes (van der Klis,2005).
Spectral-timing measurements, in which spectral and variability information is combined, vastly improved our knowledge of the inner accretion flow as well as that of the variability mechanism of LMXBs. The combination of spectral and variability information can for example be used to measure the time lags between two energy bands (as was done for the first time for BHs and NSs byPriedhorsky et al. 1979and Hasinger & van der Klis 1989, respectively). Furthermore, it is possible to very precisely measure the variable part of the spectrum by calculating the covariance spectrum (seeWilkinson & Uttley,2009). Using these methods,Wilkinson & Uttley(2009) found that the disk is intrinsically variable, an important finding concerning the origin of the observed variability.
Numerous lag measurements have been carried out. A notable observation was that of hard (i.e. hard band lagging the soft band), long lags in BH systems. Lags as long as thousands of gravitational radii of light-travel time are measured (e.g.Uttley et al.,2011). Such lags cannot be be explained by light-travel time due to Compton scattering in a corona, since the corona should then be very large, which is energetically unfeasible (Poutanen,2001). Building on the work ofLyubarskii(1997), Kotov et al.(2001) interpret the hard long delay as a propagation delay, that arises due to the the viscous propagation of fluctuations through the disk. Uttley & Malzac (in prep.) further extend this model: they attribute changes in intensity in the different energy bands not only to overall luminosity changes, but also to pivoting of the power-law, which slope is assumed to be determined by the ratio of cooling and heating luminosities (Beloborodov,1999;Uttley & Malzac,in prep.). In this lag model the corona is presumed to be compact, an assumption that is
supported by microlensing studies of AGN (Dai et al.,2010;Chartas et al.,2012), in which the accretion flows are assumed to behave similar to those in LMXBs.
In this thesis we for the first time carry out a spectral-timing compar-ison between NS and BH LMXBs. We do this by analysing the lag spectra of the two types of source in a range of variability states. Where we expect to find hard long lags for BHs, for NSs we expect an approximate zero lag. This difference is expected because of the solid surface of NSs, as opposed to the event horizon that is assumed to be the boundary of a BH. The photons coming from the NS surface provide an extra source of cooling photons, which is expected to alter the pivoting behaviour, leading to a zero lag.
A comparison of NS and BH lag spectra could provide us with indirect evidence for the presumed event horizon around black holes, since the key difference between the NS and BH lags is expected to arise from the surface properties of the compact object. Since no direct evidence for the existence of event horizons is expected to be found in LMXBs, each piece of indirect evidence is valuable. Furthermore, the comparison of NS and BH lag spectra will help us to test and constrain the hard lag model ofUttley & Malzac(in prep.), which would provide further support for propagating fluctuation models, and strengthen the evidence for a compact corona.
A difference in lag behaviour between NS and BH could serve as an additional diagnostic to differentiate between NS and BH LMXBs. Several of these diagnostics already exist. For example a very massive (>3M ) central object indicates the object is a BH, whereas if we observe
bursts (Grindlay et al.,1976) or pulsations (Giacconi et al., 1971) the central object is likely to be a NS. However, since we cannot distinguish the two types of source directly, having an additional diagnostic based on the lag spectrum would be valuable.
Outline
The content of this thesis is as follows. Chapter2contains an overview of the theory and background of LMXBs. Especially Section2.5is of importance, since it gives background and a description of the hard lag model ofUttley & Malzac(in prep.). Chapter3describes our methods, and includes an overview of relevant Fourier techniques as well as information on the diagnostic of power colours of Heil et al. (2015), which allowed us to compare NS and BH sources in the same spectral state. Chapter 4 contains the results of the lag and power spectral comparison, whereas Chapter5contains covariance spectra that could provide support for the interpretation of the lag spectra. In Chapter6
the results as well as suggestions for further research are discussed. We end with our conclusions in Chapter7.
2
Theory and background
2.1 Accretion onto compact objects
In the process of accretion, matter falls into the gravitational potential of an object, during which its gravitational energy is converted into heat. The amount of energy that is released scales with the compactness of the accreting body, Mc/R∗, where Mcand R∗ are the mass and the radius
of the compact object, respectively (Frank et al.,2002). Due to their high compactness, black holes (BHs) as well as neutron stars (NSs) are very efficient in releasing energy through accretion: they typically convert 10% of the rest mass energy of the accreting matter to radiation (Gilfanov & Merloni,2014).
An X-ray binary (XRB) can form when one of the stars of a double star system forms a compact object (BH or NS) at the end of its life. The BH or NS may accrete matter from its companion star, a process in which a large amount of energy is released, largely but not exclusively in the form of X-rays (Shakura & Sunyaev,1973). Based on the mass of the companion star XRBs can be divided into two categories: high mass X-ray binaries (HMXBs), having a companion of mass& 10M , and
low mass X-ray binaries (LMXBs), having a companion of mass.1M
(Psaltis,2006). In the case of a HMXB the companion star loses mass due to a strong stellar wind, which is partly trapped by the gravitational field of the compact object and consequently accreted. In a LMXB on the other hand, mass transfer occurs through Roche lobe overflow. The intermediate case (1-10M ) is rare because generally such stars do not
have a strong enough wind to feed the compact object significantly, nor do they fill their Roche lobes (Van Paradijs & Van der Klis,2001).
Accretion onto compact objects not only occurs on a stellar scale in XRBs, but also on galactic scale in the center of galaxies. Observa-tions strongly suggest that the accretion processes in these so called active galactic nuclei (AGN) are very similar to those governing LMXBs, even though the sources can differ a factor∼ 108in mass (McHardy
et al.,2006). Source properties such as luminosity, system size and time scales on which changes in the system occur, scale with the mass of the accreting object (Done & Gierli ´nski,2005).
Figure 2.1: Schematic representation of an accretion disk around a compact
object. Gas will orbit the object in almost Keplerian orbits and in that process slowly spiral inwards until it reaches the compact
object. Figure credits:Frank et al.(2002).
2.2 Accretion disks in low mass X-ray binaries
The systems that are the subject of this research are all LMXBs. The X-ray radiation we receive from these systems is mainly released in their optically thick accretion disks. Furthermore an optically thin corona and/or a jet might be present in the vicinity of the compact object. In the case of a NS system, an additional source of radiation is present: the NS surface. In this section a very brief overview of the physics of accretion disks in LMXBs is presented. More details and derivations can be found in for exampleFrank et al.(2002).2.2.1 The alpha-disk model
In LMXBs the companion star can be either a main sequence star, a giant star or a white dwarf (Van Paradijs & Van der Klis,2001). A tight orbit or a large size of the companion allow the companion star to fill its Roche lobe, the area in which the matter is gravitationally bound to the star. Once the Roche lobe is filled, the matter will flow from the surface of the companion star towards the compact object (see e.g. Kippenhahn & Weigert,1990). Initially the matter follows an elliptical trajectory around the compact object, but the orbit is precessing and will eventually intersect itself, after which the orbits of the matter circularise (Frank et al.,2002).
As the matter spirals inwards (see Figure2.1), angular momentum has to be transported outwards. This is ensured by viscous torques that emerge due to the differential rotation of the disk. The magnetorota-tional instability (MRI,Balbus & Hawley,1991,1998) and the resulting turbulence seem to play a crucial role in the viscous processes within accretion disks. However, the exact details of these processes, and thus the magnitude of the viscosity, are stil under debate. The alpha-disk model (Shakura & Sunyaev,1973) allows one to mathematically describe the disk while bypassing all the physical details of the viscosity. This is done by capturing the magnitude of the viscosity in a dimensionless parameter, α. The viscosity ν can then be expressed as:
ν= αHcs, (2.1)
where α.1, H is the disk scale height, and csis the sound speed. Using this parameterisation the disk can be described with a closed set of
equations (see e.g.Shakura & Sunyaev,1973;Frank et al.,2002, p. 90). Disk characteristics such as density and temperature can be expressed as functions of four system properties: accretion rate ˙M, mass of the central object Mc, distance from the central object R and the viscosity parameter α.
2.2.2 Disk emission
The energy released in the process of accretion, Lacc, is given by: Lacc= η2GMc
˙ M
R∗ (2.2)
(Frank et al.,2002). Here η is the efficiency, G the gravitational constant, Mcthe mass of the central object, ˙M the mass accretion rate and R∗ the
surface (or event horizon) of the central compact object. Efficiency is defined as the fraction of the rest mass energy of the accreting matter that is released in the process of accretion. A typical value for the efficiency for NSs as well as for BHs is η∼0.1.
We now have a look at the equation for the dissipation rate as function of radius D(R): D(R) = 3GMcM˙ 8πR3 1− R∗ R , (2.3)
for a thin disk in a steady state (seeFrank et al.,2002, for a derivation). This equation shows that the dissipation rate is strongly dependent on the radius and that most of the energy is released very close to the central object. It is this region of the disk where the temperature is the highest and where most of the X-ray emission we observe comes from. Furthermore, integrating this equation gives us the total disk luminosity Ldisk. This turns out to be half of the total luminosity:
Ldisk = 12Lacc. (2.4) This implies that the other half of the energy is radiated when the matter reaches the surface of the compact object. This so called boundary layer emission will only be released when the compact object is a NS: in the case of a BH, the matter just passes through the event horizon and the energy is lost to the BH.
2.2.3 The viscous time scale
The radial ‘drift’ velocity (or shortly radial velocity), the effective velocity at which the radial displacement of matter occurs, is given by:
vR= −23Rν 1− R∗ R −1 . (2.5)
(Frank et al.,2002). Note that the radial velocity is strongly dependent on the magnitude of the viscosity. Assuming R as the typical length
scale that has to be traversed in the disk, we can now estimate the time it takes for the matter to radially drift inwards. This time scale, called the radial drift time scale or the viscous time scale, can be expressed as:
tvisc∼ |vR R| ∼
R2
ν . (2.6)
(Frank et al.,2002). Alternatively, it can be expressed in terms of relative disk thickness H/R and the Keplerian angular velocityΩK:
tvisc∼ " α H R 2 ΩK #−1 (2.7) (Lyubarskii,1997). The viscous time scale is the typical time scale on which changes in the surface density propagate through the disk. This will be of importance in Section2.5.1.
2.3 Low mass X-ray binaries in outburst
Based on their long term variability (weeks to years), LMXBs can be categorised into persistent and transient sources. The persistent sources continuously have a high luminosity, while the transient sources have an increased luminosity occasionally. In periods of increased luminosity these transients are said to be ‘in outburst’ and in the low luminosity periods they are referred to as being ‘in quiescence’. During an outburst, which typically lasts weeks up to years, the luminosity of the LMXB increases by several orders of magnitude (Klein Wolt,2004). The NS LMXB Aquila-X1 for example shows this behavior: Figure2.2shows a light curve covering roughly six years in which five outbursts occurred.
No consensus has yet been reached on the origin of the state evolu-tion.Meyer-Hofmeister et al.(2005) suggest that the balance between Compton cooling and heating is crucial for the state transition. Alterna-tively,Petrucci et al.(2008) explain how the presence of a non-chaotic magnetic field, the magnitude of which is closely related to the accretion rate, could drive the changes. A third explanation is given bySmith et al. (2002), who propose a model in which the disk consists of two flows that exist in separate vertical regions: a thin disk in which fluctuations in mass accretion rate propagate slowly, and a hot halo on top of the thin disk, through which fluctuations propagate much faster.
2.3.1 Energy spectra
The energy spectra of LMXBs generally consist of a thermal and a non-thermal component (see e.g.Remillard & McClintock,2006;Lin et al., 2007). The thermal part peaks around. 2 keV and is therefore also
called the soft component1. The shape resembles a multi-colour
black-body spectrum2the origin of which is generally attributed to the
accre-tion disk (Mitsuda et al.,1984). In the case of NS systems, the boundary
1In the soft state (SS, see below) the thermal peak lies around 2 keV, whereas in the hard state (HS, see below) the peak lies at lower energies.
Figure 2.2: Outbursts of Aquila-X1 over a period of approximately six years
(1996-2001), revealed by monitoring with ASM/RXTE. Figure
adapted fromŠimon(2002).
layer of its surface is often considered as an extra blackbody component (Mitsuda et al.,1984;White et al.,1988). This is illustrated in Figure2.3.
The shape of the hard component, which generally reaches up to much higher energies, can be approximated by a power-law, N(E) ≈
E−Γ, that can have a break or a cutoff (Remillard & McClintock,2006;
Klein Wolt,2004). This power-law emission is often associated with inverse-Compton scattering in an optically thin region of the system (see Section2.3.2). The presence or absence of a thermal cutoff in this part of the spectrum could provide information about the nature of the electron distribution (i.e. thermal or non-thermal).
A third component that can be present in the spectrum is a reflective component. This component arises when disk material is irradiated by X-rays and backscatters a fraction of this radiation. The reflected spectrum consists of a continuum topped with lines (Fabian & Ross, 2010). The most prominent feature of the reflective component is the iron line which peaks at 6.4 keV (Uttley et al.,2014).
During an outburst, the ratio between the hard and soft components of the spectrum changes, behaviour that is observed for BH as well as NS systems (Tanaka & Shibazaki,1996). Spectra of Cyg X-1 (BH) and 4U1705-44 (NS) in two different spectral states are shown in Figure2.4. Apart from the hard state (HS) and the soft state (SS), we can distinguish a hard intermediate state (HIMS) and a soft intermediate state (SIMS)3.
The change of the spectrum as the system evolves through the differ-ent states of the outburst can be illustrated by two types of diagrams: a colour-colour diagram (CCD, introduced into X-ray astronomy by Ostriker,1977) and a hardness-intensity diagram (HID, introduced in this context byMiyamoto et al. 1995, popularised byHoman et al. 2001;
3There are several systems for classifying/naming the different states. In this work we will be consistently using the HS/SS/HIMS/SIMS classification because it is universally applicable to NS as well as BHs which is important for our analysis. Furthermore, it is also the one adopted byHeil et al.(2015), who developed a method for determining the state that proves to be essential for our analysis.
Figure 2.3: A typical spectrum of a NS LMXB in the soft state (solid black
line). The boundary layer of the NS adds an extra component (red dotted line) to the thermal part of the energy spectrum, which is originating from the accretion disk (blue dashed line). Figure credits:Done et al.(2007).
Figure 2.4: Energy spectra of BH system Cyg X-1 (left) and NS system
4U1705-44 (right), both in the soft (red) and hard (blue) state. We see that the blackbody component peaks in the lower energy range of the spectrum and dominates in the soft state, wheras the power-law component is strong in the higher energy range of the spectrum, and dominates the hard state. Also note the reflective feature present in all spectra at 6.4 keV (iron line). Figure adapted from
Figure 2.5: CCD of a Z-source (left) and an atoll-source (right). The clustering
of observations relates to the different states that the systems can be in. Three branches can be identified in the CCD of the Z-source: the horizontal branch (HB), the normal branch (NB) and the flaring branch (FB). The atoll-sources typically evolve from the island state (IS), through the lower banana state (UB) to the
upper banana state (LB). Figure credits:Schnerr et al.(2003).
Belloni et al. 2005).
In a CCD the hardness and softness of the spectrum – calculated from ratios of count rates in different energy bands4– are plotted against
each other. The CCD is commonly used to distinguish different types of NS systems and their states. Typically they trace either a z-shaped or a u-shaped pattern in the CCD, which classifies them as either Z-sources or atoll-Z-sources, respectively (Hasinger & van der Klis,1989). Initially,Hasinger & van der Klis(1989) suggested the strength of the NS magnetic field to be the key difference between these systems, being weak in the case of atolls and strong in the case of Z-sources. However, recent work on XTE J1701-462, which transitions between Z and atoll behaviour, suggests that the differences in behaviour may be linked to differences in mass accretion rate (Homan et al.,2010). Figure2.5
shows the difference in appearance of an atoll and a Z-source in the CCD. Other classes of NS systems are the accreting pulsar (AP) and accreting millisecond pulsar (AMP) systems. For this work we will use observations of atoll-sources and AMPs rather than those of Z-sources and APs, because their timing properties turn out to be similar to those of BHs in the HS and HIMS (Done & Gierli ´nski 2003, see Sections2.4.1
and3.3.2).
Whereas CCDs are suitable for tracing NS states, the states of BH systems are better shown in a HID, in which the hardness of a source is plotted against its total intensity. During an outburst a BH system traces a q-shaped path counter-clockwise, passing through the states as following: HS→ HIMS→ SIMS→ SS →SIMS →HIMS (Miyamoto
et al.,1995;Fender et al.,2004). An example is shown in Figure2.6.
4There is no convention as of how to calculate these ratios. One could use two, three or four energy bands, and plot for example A against B (in the case of two bands), C/A against C/B (in the case of three bands) or A/B against C/D (in the case of four bands). The boundaries of the energy ranges can also vary.
Figure 2.6: Hardness-Intensity diagram (HID) of GX-339-4. The system enters
the outburst in the hard state (HS). It will thereafter follow a q-shaped path counter-clockwise, tracing the hard intermediate state (HIMS), the soft intermediate state (SIMS), the soft state (SS), the SIMS again, to end with the HIMS. The HID is especially useful to trace the outbursts of BH systems. Figure adapted from
Hiemstra et al.(2011).
2.3.2 Disk geometry and the origin of the power-law
The different components of the energy spectrum of LMXBs are assumed to originate from different physical components in the system. The ther-mal (multi-colour blackbody) component of the spectrum generally is associated with the geometrically thin, optically thick disk as described inShakura & Sunyaev(1973), while the non-thermal component of the spectrum is associated with an optically thin ‘power-law region’. The exact origin and geometry and physical location of the power-law region are still a subject of debate. A simple two component model5is that of
the combination of a thin disk and a hot, optically thin ‘corona’, which are alternately dominating in the SS and the HS (Done et al.,2007).
According to this model, in the SS the disk reaches all the way to the innermost stable orbit. The disk emits as a multi-colour blackbody and thus is the likely origin of the thermal component that is so prominent in the SS spectrum. Additionally, a relatively weak power-law component is observed in the SS spectrum (typically having a slope ofΓ ∼ 2.3−
2.5,Malzac & Belmont 2009). An optically thin, hot cloud of gas that is located in the innermost regions of the system, referred to as the corona, is often assumed to be responsible for the power-law emission.
5Detailed models of the corona often also invoke two components within the corona itself: one component being the proton population, the other that of the electrons, the two having different temperatures (e.g.Beloborodov,2001). This is not the two component model we are referring to.
Figure 2.7: Schematic illustration of the central corona model as described
in the text. In the soft state (SS, top figure) the disk dominates the spectrum and is assumed to reach to close to the compact object. The corona is small and is located very centrally above and below the disk. In the hard state (HS, bottom figure) the corona is dominating the spectrum. The disk is assumed to be truncated and the corona is extending over the large region within the disk inner edge and the central object.
The corona is heated by the gravitational energy that is released in the process of accretion. The energy is channeled from the disk to the corona through magnetic field lines (Done et al.,2007), heating the ions within the medium. The electrons in turn, are heated through Coloumb collisions with those hot ions. Cooling of the corona occurs through soft seed photons originating from the disk, that are gaining energy by inverse-Compton scattering, a process in which energy is transferred from electrons to photons. It is these photons that are assumed make up the non-thermal power-law component of the emission (see e.g.Sunyaev & Truemper,1979).
In the hard state the disk emission is less prominent in the spectrum. The disk may be truncated, meaning that it ends farther away from the compact object (tens to a few thousand gravitational radii,Malzac & Belmont, 2009). In the HS, the corona is assumed to be more promi-nently present than it is in the SS. This results in power-law emission dominating the spectrum in the hard state (in this case typically having a slope ofΓ∼1.5−1.9,Malzac & Belmont 2009). The disk geometry of
the central corona model in the two states is illustrated in Figure2.7. Instead of a central corona, other components that would explain the power-law emission are proposed.Shapiro et al.(1976) propose a hot inner flow. In this model the inner parts of the disk are assumed to evaporate into a hot, geometrically thicker flow (see alsoZdziarski et al., 1998). Alternatively, under very specific conditions a corona on top of the disk (either a homogeneous slab or a patchy corona), could reproduce the observed power-law spectrum. Another model assumes the base of a jet to be emitting the power-law (Markoff et al.,2005): properties from magnetised coronas are very similar to the plasma properties of the base of jets, hence the two could be related. Where the seed photons
Figure 2.8: Four proposed geometries of the source of the hard power-law
photons in LMXBs, as described in the text. Timing experiments suggest that the power-law region is located centrally, leaving the central corona, hot inner flow and jet model as the most plausible
models (Gilfanov,2010;Cassatella et al.,2012).
for the inverse-Compton scattering in a corona are coming from the disk, in the case of a jet they are, additionally, coming from synchroton photons originating in the jet itself.Markoff et al.(2005) show that their jet models fit the hard component of the energy spectrum as well as Compton corona models.
An overview of the different models for the power-law emitting component is presented in Figure2.8. Although no consensus is reached on the exact shape and mechanism of the power-law emitting region, timing experiments strongly suggest that it is located centrally, leaving the central corona, hot inner flow and jet model as the most plausible models (Cassatella et al., 2012). This is discussed in more detail in Section2.5.3.
2.4 X-ray spectral-timing
The emission of LMXBs varies not only on the long time scales associated with the periods of quiescence and outburst, but also on much shorter time scales. This short term variability can be monitored in the periods of outburst, when luminosity is high enough to allow for obtaining signal in observations with a high temporal resolution. The properties of the short time scale variations of a source (or shortly ‘timing prop-erties’) are, additional to the energy spectrum, a probe for the physical characteristics of LMXBs. The discipline of ‘X-ray timing’ concentrates on the analysis of these timing properties. When the analysis involves the comparison of timing properties of different energy bands, we speak
Figure 2.9: Short time scale variability of Cyg X-1. Shown is a fragment
(totalling 20 seconds of observation) of a light curve of Cyg X-1 in the HS. We see variations on a range of time scales. Figure credits:
Churazov et al.(2003), labels enlarged.
of ‘X-ray spectral-timing’, or shortly ‘spectral-timing’.
This section starts with a brief introduction to the observed short time scale variability, after which the focus will be on one of the key aspects of spectral-timing: the time lags that are observed between the light curves in different energy bands. We note that this section is dedicated to the qualitative aspects and physical interpretations of X-ray timing and spectral-timing and that technical details can be found in Section3.1of the Chapter onMethods.
2.4.1 Short time scale variability
An example of a light curve showing short time scale variations of the BH LMXB Cyg X-1 is shown in Figure2.9. We see that the course of the light curve looks very chaotic and no clear patterns can be distinguished. We cannot predict what will be the next value of the light curve, however, we can describe the underlying probability distribution: the underlying process is a so called stochastic process. To describe the probability distribution of the signal, the short time scale variability is studied in the frequency domain rather than the time domain. One can then plot the power spectrum of a light curve, which shows the contribution of each of the frequencies to the total power of the signal (see e.g.Leahy et al.,1983). The power spectra of LMXBs change during their outburst cycle, just as the energy spectra do. A power spectrum of Cyg X-1 both in the soft and the hard state is presented in Figure2.10. Frequencies considered in X-ray timing of LMXBs are typically in the range of 10−3−103Hz.
In the power spectrum two components can be identified. First of all broadband noise can be present, possibly spanning decades of Fourier frequencies and typically being flat or following a (broken) power-law. This is visible in the soft (blue) power spectrum of Cyg X-1 in Figure2.10. Additionally, a power spectrum can contain peaked features, as can be seen in the hard (red) power spectrum. Whereas
Figure 2.10: Power spectra of Cyg X-1 in the soft (blue) and hard (red) state.
Note that the power density is multiplied by the Fourier frequency (this ensures that in the log-log plot the area under the graph represents the power). We see that the slope of the power in several frequency ranges is constant, e.g. in the soft spectrum
we see that for frequencies up to≈10 Hz the spectrum is flat (or
∝ f−1if we look at power density rather than power density×f ).
Figure adapted fromGilfanov et al.(2000).
the features in the power spectrum of of Cyg X-1 are relatively broad, sharper peaks are observed in the power spectra of many other LMXBs. These sharper peaks are referred to as Quasi Periodic Osciallations (QPOs)6. A wide variety of QPOs exists. They can be classified according
to their frequency, amplitude, their peakedness, among other features (see e.g.Casella et al.,2005). Whereas low frequency QPOs are seen in both NS and BH systems, kHz QPOs are typical for NS sources (Klein Wolt,2004).
While the power spectra of NS and BH sources can show large differences, similarities are also observed, especially between the NS atoll-sources (see Section2.3.1) and BH sources, implying similarities in the accretion flow (van der Klis,1994a). This is not that surprising, given the fact that the gravitational potential of a NS is not expected to differ much from that of a BH.Done & Gierli ´nski(2003) find that atoll-sources especially show much similarity with BH sources in the hard state7.
2.4.2 Lag observations
A powerful tool in X-ray timing is to compare the variability observed in different X-ray energy bands to map time lags between the two.
6The sharpness (or peakedness) of a peak can be quantified with the q-value, which is the ratio of peak-frequency and the FWHM. For the peaks observed in Cyg X-1 typically q.1, whereas for a peak to be classified as a QPO, it is required that q>3.
7Done & Gierli ´nski(2003) use different names for the outburst states than we do here, and refer to the hard state as the ‘low bright state’.
Figure 2.11: Time lag versus frequency of the soft band (2−4 keV) versus
the hard band (8−13 keV) in Cyg X-1 in the hard state. For all
frequencies, the hard band is lagging the soft band. We see that lags generally decrease with increasing frequency, however three
step-like features are clearly present in the data. Nowak(2000)
show that these steps correspond to Lorentzian peaks present in the power spectrum of this observation (see the red plot in
Figure2.10). Figure credits:Uttley et al.(2014).
Priedhorsky et al.(1979) were the first to observe lags in a LMXB. The asymmetric cross correlation functions (CCFs) of several soft and hard bands of the persistent source Cyg X-1 indicated a lag between the bands8. NS lags were first observed in Cyg X-2 byHasinger(1987). The
CCF was asymmetric and additionally showed a sinusoidal pattern, which was ascribed to the QPOs showing up in the power spectrum of the source.
Numerous lag observations of LMXBs have since been carried out. The majority of the observed lags are hard, meaning that the hard band is lagging the soft band (Kotov et al., 2001; Poutanen, 2001). More specifically, the lags are found to depend approximately logarithmically on the energy separation between the two bands (Miyamoto & Kitamoto, 1989). On the other hand, soft lags are observed too, for example in the soft state of GX-339-4 (Miyamoto et al.,1991).
The analysis of lags with the cross-spectrum of the two light curves9
rather than a CCF, as first done byvan der Klis et al.(1987), allowed for the measurement of the dependence of lag on frequency. The lags generally increase for decreasing frequency, consistent with an approxi-mate constant phase lag (Miyamoto et al.,1988). Deviations from this power-law slope for the time lag are also frequently observed, as can for example be seen in Figure2.11.
8The CCF measures the displacement or lag of two time series relative to each other. When different frequencies are present in a time series, the CCF will be the sum of a series of peaks, each corresponding to one of these frequencies. The peaks of the short frequencies, having short lags, dominate the CCF, causing the peak to be very close to zero. The lower frequencies have longer lags and their peaks are wide but further from zero: this causes the assymetry in the CCF.
Figure 2.12: The main components of the energy spectrum of a BH LMXB:
blackbody radiation of the optically thick disk (blue), Comp-tonised emission from the hot optically thin power-law region (region shown in pink, emission in red) and emission due to reflec-tion (green). Colours of the plot (left) and the schematic geometry
(right) are in correspondence. Figure credits:Gilfanov(2010).
2.4.3 Origin of the lags
The origin of the (different types of) lags is still a topic of discussion. The characteristics of the lags, such as their magnitude and frequency and energy dependence, are related to the geometry of the system and to the radiative processes in the disk and the power-law emitting region.
For identifying the cause of energy dependent lags, the energy spectrum is a practical starting point. In Figure2.12a spectrum and a possible geometry of the disk/power-law region are depicted. In the figure we see three components. First of all, there is the thermal disk radiation (blue). This radiation is reaching the observer directly from the disk and will have a blackbody spectrum consisting mostly of soft X-ray photons. Part of this radiation is assumed to be up scattered by inverse-Compton interactions towards higher energies, this is the Componized component in the spectrum, which often has a power-law shape. Some fraction of this high energy emission will leave the system, whereas another fraction will be intercepted by the disk10. This radiation
can then be reflected towards the observer. We refer to reflection if the radiation is either backscattered fluorescent emission or radiation that is reprocessed by the disk (Uttley et al.,2014).
In a system with these components lags between the soft and the hard band naturally arise. Soft lags could arise from reflection because of the light-travel time from the power-law region to the disk or because of the time it takes to thermally reprocess the energy (thermal lag). The majority of the observed lags, however, are hard (Kotov et al.,2001). The origin of the hard lags was initially assumed to be related to the time it
10The precise fractions will depend on the shape of the optically thin Comptonising region (location, density distribution) as well as that of the disk (degree of truncation, being either flat or flared).
takes for soft photons to gain energy in the process of inverse-Compton scattering (Poutanen,2001). Variations are first taking place in the soft part of the spectrum, after which a fraction of these photons gains energy through inverse-Compton scattering. The time it takes to scatter (light-travel time through the power-law region) could then be assumed to be the cause of the hard lag. The more scatterings a photon undergoes, the higher its final energy, but also the longer it takes for the photon to escape. This model naturally explains the observed logarithmic energy dependence of the lags. However, the size of the Comptonising cloud that is needed to produce this lags is 103-105R
g, which is energetically unrealistic (see e.g.Poutanen,2001, and references therein). Therefore, Comptonisation cannot explain the observed long hard lags, despite playing a role in the formation of the power-law emission.
An important aspect of interpreting the observed lags is the physical origin of the stochastic variations. Initially, the variability was described by shot noise models, in which the observed fluctuations in luminosity covering a wide range of time scales were associated with uncorrelated shots of energy (Terrell,1972). The shots of energy were typically associ-ated with magnetic flares in the corona (Uttley & McHardy,2001). In simple shot noise models the time scale over which these shots decay is constant, but more complicated models also exist (Poutanen,2001). The shot noise model of flares in a corona matched the view that the variation was originating in the power-law emitting region, since the soft spectrum was observed to be less variable than the hard spectrum (Miyamoto et al.,1991;Poutanen,2001). However, the rms-flux relation, as discovered byUttley & McHardy(2001), shows that the variability amplitude of LMXBs scales linearly with flux and thus that the power spectra of these systems are non-stationary on all time scales. This is in contradiction with shot noise models, which predict that on short time scales the power spectrum should not vary (Uttley & McHardy,2001).
2.5 A propagative flow model for hard lags
An alternative model for the origin of the stochastic noise, in agreement with the rms-flux relation, can be found in fluctuating-accretion models, in which the luminosity changes are attributed to variations in the accretion rate (Uttley & McHardy, 2001). In this case the variations are not completely independent, such as in the additive shot noise models, but multiplicative instead: they propagate through the flow until they reach the inner parts of the system, where most of the emission is produced (Lyubarskii,1997). In these propagation flow models the observed long hard lags in BH LMXBs are naturally produced, without the need for an extremely large corona.2.5.1 The fluctuating-accretion model of Lyubarskii
In fluctuating-accretion models the observed fluctuations in the radia-tion intensity are attributed to variaradia-tions in the mass accreradia-tion rate in
Figure 2.13: Propagation of a mass fluctuation through the disk. The evolution
of the surface densityΣ as a function of radius at four different
moments is depicted, assuming a mass fluctuation (having a
δ-function profile) being ‘injected’ at t=0 at R=30Rg. The matter
will spread through the disc on the viscous time scale, which is the reason why fluctuations on shorter time scales are damped: such fluctuations will be indistinguishable, since they will be washed out by each other.
the inner regions of the disk11. Fluctuations in mass accretion rate and
dissipation rate are closely related: the source of the dissipated energy is the accretion of matter. Variations in the magnitude of the viscosity, which is encapsulated in the parameter α, are thought to cause the dissi-pation rate fluctuations, which magnitude is assumed to be independent of the mass accretion rate. Such variations could be attributed to the random nature of the turbulent process producing the viscosity.
If we look at a typical LMXB power spectrum (Figure2.10), we see that the slope of the power spectrum is similar in a range of frequencies. This suggests that the fluctuations of different frequencies are generated by the same underlying physical mechanism (Revnivtsev, 2008). A propagative flow model, such as the one proposed byLyubarskii(1997), can explain how fluctuations governing such a wide range of frequencies could have a common physical origin. In the model it is assumed that fluctuations in ˙M arise at a wide range of radii ri, each radius being associated with a certain typical frequency≈ fi. The typical frequency
is assumed to be of the order of the viscous frequency, being the inverse of the viscous time scale12, fluctuations on shorter time scales would be
damped (Churazov et al.,2001). This is illustrated in Figure2.13, where we see the evolution of surface density as a function of time and radius.
While the matter travels inwards, the mass accretion rate variations emerging at different radii in the disk propagate inwards, and multiply. Finally, when the accreting matter reaches the innermost regions of
11Remember from Equation (2.3) that the dissipation rate scales inversely with radius cubed (approximately), which ensures that the radiation coming from the inner radii is dominating the total luminosity.
Figure 2.14: Simplified, idealised illustration of a propagative flow model for
variability. Fluctuations in the mass accretion rate ˙M originate
at different radii (green) and have a typical frequency associated with them. While propagating through the disk the variations multiply, until they reach the innermost regions of the disk, where the X-ray radiation is coming from (red). The luminosity will thus reflect the fluctuations coming from several parts of the disk, even though the radiation is mainly generated very close to the compact object.
the disk and the amount of dissipated energy is high – resulting in X-rays being radiated – the accretion rate will be equal to the product of the accretion rates of all radii in the regions outside that radius. Mathematically, the accretion rate at the inner regions of the disk ˙Min can be described as:
˙ Min = M˙out iout
∏
i=iin (1+ ˙mi(t)) (2.8)(Lyubarskii, 1997; Uttley et al., 2005). Here ˙Mout is the ‘intial’ mass accretion rate at the outer edge of the disk, and ˙mi(t)represents the fractional mass accretion rate variations at time t corresponding13 to
each radius i. This is illustrated in Figure2.14.
If we assume the variations to arise through viscous instabilities that can be translated into a varying α parameter, we find that if the ampli-tudes of the variations in α are equal for all ri then the Fourier spectrum follows a power-law: f−1. Other dependencies of the variations in α on radius could explain different power-law indices (Lyubarskii,1997).
2.5.2 Fluctuating-accretion models in practice
The feasibility of the fluctuating-accretion model ofLyubarskii(1997) was tested byKotov et al.(2001), who extended the model analytically and compared it to observations of BH LMXBs. Their model is mainly based on two assumptions. First of all, the perturbations originating at the different radii are assumed to be independent of each other, and their time scales are assumed to be a function of radius. Secondly, the
13Note that t indicates the time at the moment that the fluctuation emerges, which is different for each radius.
spectrum is assumed to be hard near the compact object, whereas it is soft at larger distances from the center. The emission of the hard and the soft components, H(r)and S(r), respectively, is related through
the hardness function h(r). This function defines the ratio between the
hard and the soft band, and is a decreasing function of radius (virtually representing the power-law slope at a certain radius). In the model two prescriptions were adopted: h(r)being either a step or smooth
function. The emission profile of each of the bands is then found by scaling it to the total emission at a given radius e(r), which is assumed
to vary inversely with radius squared, resulting in S(r) = 1/r2 and
H(r) = h(r)/r2. The frequency of fluctuations at the radius r0 scales
with the radius at which they originate in the same fashion as the inverse of the viscous time does: f(r0)∝ r−03/2. The propagation of fluctuations
through the disk is described by a Green function, which describes the response of the surface density of the disk on a mass injection. The particular Green function used byKotov et al.(2001) was the one of Lynden-Bell & Pringle(1974)14, see Figure2.13.
In the model, the dependence of the lags on frequency can be adapted by changing the hardness function as well as the time scales of the perturbation and propagation. Tuning these parameters, the model ofKotov et al.(2001) roughly reproduces the observed lag-frequency spectrum. However, their model has limitations: the dependence of lags on the emissivity profile is not examined and neither are flow parameters such as α or the relative scale height taken into account.
Arévalo & Uttley(2006) extended the model ofLyubarskii(1997) andKotov et al.(2001) and performed a closer investigation of the model by means of a numerical simulation. In this model the propagation ve-locity as well as the frequency of the fluctuations are coupled to the disk parameters α and H/R. The key parameters of the model are i) the emissivity profiles of the hard and the soft band15, ii) disc structure
pa-rameters, encapsulated in the value of α(H/R)2, being either constant or
varying with radius, iii) power spectrum of the input signals and iv) the damping coefficient, which specifies the distance after which fluctuation of a certain frequency is damped, this way replacing the Green function as was used byKotov et al.(2001). Arévalo & Uttley(2006) find that the amplitude of the lags depends strongly on the difference between emissivity indices of the energy bands, but only weakly on the damp-ing and structure parameters. A comparison to Cyg X-1 observations yields good fits to the lag and power spectra simultaneously16, showing
14This Green function is, for a fluctuation at radius r
0, defined as: G(r, r0, t) ∝ r−1/4 t exp −r1/2l0 +r1/2l 4t Il r1/4lr1/4l 0 2t
, where Ilis the Bessel function of imaginary argu-ment (also referred to as ‘modified Bessel function of the first kind’) and l = 1/3
(appropriate assuming H/r is constant).
15The emissivity profiles are parameterized through e(r) =r−γ(1−p
Rmin/r)where
γ=3 for the soft band and γ>3 for the hard band. Comparing this description of the emissivity to the format ofKotov et al.(2001), this is equivalent of a hardness function h(r) =rγsoft−γhard.
