Iron line reverberation mapping of black hole binaries: A quantitative study of accretion disks using frequency resolved power spectra

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University of Amsterdam

Iron line reverberation mapping of

black hole binaries

A quantitative study of accretion disks using frequency resolved

power spectra

by

Karel Temmink

Report Bachelor Project Physics and Astronomy, size 15 EC Conducted between 01 - 05 – 2015 and 30 - 06 – 2015

University of Amsterdam Faculty of Science The Netherlands Student number: 10432965 Supervised by: A.R.Ingram, Ph.D.

prof. M.B.M. van der Klis, Ph.D. Second assessor: A.L.Watts, Ph.D.

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Abstract

An F e-Kα line is regularly observed in BHBs as a result of reflection of x-rays off of inner

regions of the Keplerian accretion disk. Variations in flux from the inner regions lead to variations in the iron line, smeared out by some small light-crossing lag; Very fast variations in the continuum flux will be distorted largely by differences in path length for different reflected photons, in a way that depends on the source geometry. Therefore, studying the relative amplitude of reflection in the spectrum as a function of Fourier frequency can place constraints on the accretion disk geometry.

Here, we examine frequency-resolved spectra for three BHs; Cygnus X-1, GRS 1915+105, and GX 339-4. Using iron line reverberation, we confirmed earlier research from Revnivtsev et al. (1999) showing suppressed iron line variability at high ν for Cygnus X-1, consistent with a disk truncated at a radius larger than the innermost stable circular orbit (Rin∼ 100RG).

We also studied two observations of GRS 1915+105 with strong QPOs. We found that sup-pression of high ν iron line variability is consistent with the truncation radius being larger when the QPO frequency is smaller, as predicted by the truncated disk model. We also saw suppression of iron line variability at low frequencies, which might be explained by spectral pivoting, as well as the fact that merely two additive models were used. We also observed small feature at the first overtone of the second observation. This feature is statistically not interesting, and we conclude that the QPO in GRS 1915 is most likely of non-geometric origin.

Examining GX 339-4, we found a large feature at νQP O, meaning that this QPO has a different

mechanism than the broad band noise. The line variability is consistent with the truncated disk geometry.

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Popular scientific summary

Een R¨ontgendubbelster bestaat uit een compact object (in dit verslag een zwart gat) en een begeleidende ster. Het komt regelmatig voor dat de zwaartekracht ten gevolge van het zwarte gat sterk genoeg is om materiaal van de begeleidende ster weg te trekken.

Figuur 1: Een zwart gat en begeleidende ster. Materiaal stroomt van de begeleider naar de accretor en vormt een accretieschijf.

Materiaal van de begeleidende ster valt dan niet rechtstreeks het zwarte gat in, maar vormt een zogenoemde ‘accretieschijf’ rondom het zwarte gat (in een dergelijke si-tuatie ook wel ‘accretor’ genoemd), ten ge-volge van beweging van de twee hemellicha-men (zie figuur 1). Materie in een accretie-schijf volgt nagenoeg cirkelvormige banen, en spiraalt langzaam het zwarte gat in. Derge-lijk materiaal wordt door de zwaartekracht versneld, en wordt heter. Het binnenste ge-deelte van de accretieschijf wordt z´o heet, dat het r¨ontgenstraling uit kan zenden. Uit onderzoek blijkt dat deze straling vaak niet

constant is, maar in een Quasi Periodieke Oscillatie (QPO) oscilleert rondom een gemiddelde waarde.

Een deel van deze straling reist direct naar de waarnemer toe. Een ander deel reflecteert van de accretieschijf af naar de waarnemer toe, dit wordt waargenomen als een spectrale piek bij de energie van ijzer (6-7 keV).

Als de flux van de binnenste accretieregio fluctueert in de tijd, zal ook de flux van de ijzerpiek fluctueren. “Reverberation” is een techniek waar men fluctuaties in de gereflecteerde emissie (de ijzerpiek) vergelijkt met fluctuaties in de directe emissie van de binnenste regio, waaruit men de geometrie van de accretieschijf kan bepalen. In dit project werd deze techniek ge-bruikt om te bepalen of de QPO een geometrische oorzaak heeft, of dat slechts de intensiteit van de R¨ontgenbron variabel is in de tijd.

In deze scriptie zijn de accretieschijven van drie zwarte gaten bestudeerd; Cygnus X-1, GRS 1915+105, en GX 339-4. We hebben eerder onderzoek aan Cygnus X-1 door Revnivstev et al

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(1999) bevestigd. Dit onderzoek vond onderdrukte ijzer-lijn variabiliteit op grotere tijdscha-len, en toonde hiermee aan dat er op grote tijdschalen minder reflectie optreedt.

Dit is consistent met een theoretische berekening van de ijzerlijnvariabiliteit voor een simpel ‘platte schijf’-model (zie figuur 2). Daar Cygnus X-1 geen QPO vertoont, werd verder geen onderzoek verricht aan dit zwarte gat.

Figuur 2: Het theoretische model van een ac-cretieschijf; een platte, dunne schijf met een binnen- en buitenstraal.

Tevens zijn twee observaties van GRS 1915+105 beschouwd waarin duidelijke QPO signalen in aanwezig waren. We hebben ont-dekt dat de mate waarin reflectie onderdrukt wordt consistent is met een kleinere binnen-straal als de QPO-frequentie lager is, zo-als het schijfmodel voorspelt. Verder namen we onvoorspelde onderdrukking van de ijzer-lijnflux waar bij kortere tijdschalen, hetgeen vermoedelijk het gevolg is van al te versim-pelde aannamen aangaande het schijfmodel en de manier waarop variabiliteit optreedt. Op een kleine afwijking na volgt de ijzer-lijnvariabiliteit de theoretische voorspelling, waaruit we concluderen dat de QPO naar alle waarschijnlijkheid geen geometrisch verschijnsel is.

Een studie van data van GX 339-4 leidde tot de waarneming van significante afwijkingen van de theoretische berekening rond de QPO frequentie. Dit betekent dat de QPO hier een geometrische oorsprong heeft.

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CONTENTS CONTENTS

1 Introduction

This thesis is centred about the study of Black hole binary system properties. Black hole binary systems (hereafter Black Hole Binaries (BHB)) consist of a black hole (BH), and a companion. The companion, which is usually a star, but could also be a brown dwarf, a planet, or an asteroid, and the BH are so close that their gravitational interaction causes them to orbit about a common centre of mass (see figure 3).

Figure 3: Artist’s rendition of a black hole binary system

Black holes, by their very definition, cannot be observed directly; the strongest evidence for black holes comes from BHBs in which a visible star can be shown to be orbiting a massive but unseen companion. A BH’s strong gravitational interaction with its surroundings can cause matter to flow from the companion towards the black hole. Matter streaming into black holes will form an accretion disk due to relative individual motion of the celestial bodies in the system. Most BHBs are X-ray sources, because the accreting matter will be ionized and greatly accelerated, producing X-rays. The X-rays are produced mainly in the inner region of the accretion disk, making the inner region the most interesting region of the accretion disk.

Unfortunately, this inner region is too small to be imaged using conventional optical tech-niques and technology. Nonetheless, extensive studies of several properties of BHBs and their radiative emission has led to a new manner of investigating this otherwise inaccessible accretion region.

The signal we observe from a BHB is a light curve; the amount of photons detected per second over the course of the observation.

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1 INTRODUCTION

Figure 4: Light curve from GRS 1915+105 Displayed in figure 4, a light curve typically

appears fairly random and distorted, riddled with noise. In 1984, Van der Klis discovered that an x-ray emitting neutron star exhib-ited a certain quasi periodicity in its emis-sion intensity, called a Quasi Periodic Oscil-lation (QPO). Since then, a variety of BHs have been discovered to have the same sort of QPO in the signals they emit. Presently, the precise mechanism causing the QPOs is not widely agreed upon, and no definitive de-scription of their origin currently exists.

Though different in the details of their properties, accretion disks do share several spectral qualities. The spectra we observe generally look like the figure on the right. Roughly, the spectra all consist of three components, depicted in figure 5:

(a) A typical BHB spectrum. The line col-ours denote different spectral components (Ingram 2012);

black - the total observed spectrum

blue- comptonised original emission

green- reflection spectrum

red- black body radiation

(b) BH accretion disk schematic.

The arrows, coloured according to the spec-tral lines in a), all point towards the observer.

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1 INTRODUCTION

The primary source, approximated as being a point source located above the disk axis, emits photons, which experience Compton scattering as they move through the highly ionised ma-terial close to the BH and collide with electrons. This leads to a classic broad “Compton-ised” hump in the spectrum, denoted by the blue line.

Some of the primary radiation will interact with and reflect off of the accretion disk. The material in the disk is in some state of ionisation. This results in a typical reflection spectrum (green line), with several peaks and dips, corresponding to the materialistic make-up and ionisation state of the accreting matter. The most interesting part of this component is the so called “Iron line” at 6-7 keV.

Lastly, there exists a region of the disk, hot enough to emit its own X-rays. This appears as a black body spectrum corresponding to the disk temperature (red line).

The aim of this thesis is to map the inner accretion region and to infer qualitative properties of the QPOs. We will try to accomplish this by using a technique called iron line reverbera-tion mapping.

When the brightness of the inner region fluctuates, the flux of the iron line, which we as-sume to originate from reflection only, will also fluctuate in response. Reverberation is the technique of comparing the fluctuations we see in the reflected emission (the iron line) to the fluctuations we see in the direct emission from the inner region, in order to map the geometry of the accretion disk. We will be using this technique to try and determine if the geometry itself is changing over the course time, or if the geometry stays the same and it is only the brightness that changes with time. In doing so, we hope to reveal the origins of the QPOs.

In this thesis, three black holes and their accretion disks will be studied, to wit; Cygnus X-1, GRS1915+105, and GX 339-4. Cygnus X-1 has been studied using this technique before, by Revnivtsev et al. (1999). We will try to scientifically verify their conclusions. This black hole is devoid of a QPO signal, meaning that using the reverberation technique, we will be able to use this system as a means of comparison for the other BHs we examined. GRS1915+105 and GX 334-9 are both well-known for their QPO signals, and will therefore be studied and compared to Cygnus X-1.

We expect to find a difference in QPO origin between all black holes, as BHs tend to go through different states of accretion.

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2 THEORY

2 Theory

2.1 Fourier analysis

In modern x-ray spectral analysis, one can not do without Fourier Analysis. Using Fourier techniques, it is possible to analyse data in a different way:

For a given frequency ν, one can define the values (A,B) or (C,φ), such that the function A cos(ωt) + B sin(ωt) = C cos(ωt − φ), where ω = 2πν, best fits the original data x(t). Let xk be a time series of N numbers, of duration T . In this thesis, xk will always represent

the total amount of detected photons in bin k. then the Fourier decomposition of x(t) is given by repeating the above for a large enough amount of frequencies ω (van der Klis 1988):

x(t) = 1 N X j aj cos(ωjt − φj) (1) = 1 N X j Aj cos(ωjt) + Bj sin(ωjt) (2)

Now, let xk equal the data point at time t = tk, then the components Aj and Bj can, trivially,

be calculated as follows: Aj = X k xk cos(ωjtk) (3) Bj = X k xk sin(ωjtk) (4)

This is called the Fourier Transform (FT) of the function x(t). The above equations can be represented by a single expression, using complex numbers:

aj =

X

k

xkeiωjtk (5)

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2.2 Convolution 2 THEORY

The signals used in this thesis are equitemporally spaced, hence tk = kT_N and ωj = 2π_Tj.

Substituting these expressions for ωjtk in equation (5) leads to the following, most general,

expression (Ingram (2012), van der Klis (1988)):

aj =

X

k

xke2πijk/N (6)

The change in time, and as such the temporal resolution, is δt = _NT and the change in frequency, or the frequency resolution, is δν =_T1.

Since νj = _{N δt}j , the lowest non-zero frequency we can study is simply _T1, this is the same as

the frequency the original data was ’sampled’ at. The highest frequency, also known as the Nyquist frequency, is _2δt1 . It is noteworthy to mention that, at zero frequency, equation (6) simply denotes the total amount of detected photons, across all bins: P

kxk= Nγ.

For completeness, we shall provide the FT for a continuous function, though we will not make use of it in this thesis:

ˆ x(ν) = F (x(t)) = Z ∞ −∞ x(t)e−2πitνdt (7) 2.2 Convolution

The convolution of two functions x and h, written as x ∗ h, is defined as follows:

x(t) ∗ h(t) ≡ Z ∞ −∞ f (τ )g(t − τ )dτ (8) = Z ∞ −∞ f (t − τ )g(τ )dτ (9)

For functions that are defined only on certain domains, the integration limits can be altered. As we are dealing with discrete functions only in this thesis, we need the discrete form of the convolution: xn∗ hn≡ ∞ X m=−∞ xmhn−m (10) = ∞ X m=−∞ xn−mhm (11)

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2.3 Impulse response 2 THEORY

An useful theorem, called the Convolution Theorem, states that the FT of the convolution of two functions g(t) and h(t) is equal to the multiplication of the individual FTs of the functions:

F {g ∗ h} = F {g} · F {h} (12)

2.3 Impulse response

An impulse response function, or simply impulse response, of a system is the systems output, when presented with a brief input signal, called an impulse. More generally, and mathem-atically, for a system with input x(τ ) = δ(τ ), impulse response function h(τ ) and output y(t):

y(t) = Z ∞

−∞

δ(τ )h(t − τ )dτ = h(t) (13)

or, if the system is discrete:

y(t) =

∞

X

m=−∞

δm,nhn−m= hn (14)

This way, we can determine a systems reaction (output) to an input that is not an impulse: The time behaviour of a linear time-invariant system, with input x(t) and output y(t), is given by the convolution ( eqs (8) and (10)) of the input with the impulse response function of the system h(t):

y(t) = x(t) ∗ h(t) (15)

In this thesis, we shall work with the so called transfer function, which is the impulse response for a reflecting body. It is common practise to denote this function as a capital letter ’t’; T (t).

2.4 Power Spectra

The discrete power spectrum is defined as follows:

P(νj) = |aj|2 (16)

Where one can use an arbitrary normalisation. In this thesis, however, we adopt the normal-isation of Ingram (2012), which we shall now introduce.

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2.4 Power Spectra 2 THEORY

Parseval’s Theorem states that:

N −1 X k=0 |x_k|2 = 1 N N/2−1 X j=−N/2 |a_j|2 (17)

The Variance of a discrete series xk of N numbers, V ar(xk), is defined as follows:

V ar(xk) = N −1 X k=0 (xk− ¯x)2 (18) = N −1 X k=0 x2_k− 1 N( N −1 X k=0 xk)2 (19)

Applying Parseval’s Theorem, this becomes:

V ar(xk) = 1 N N/2−1 X j=−N/2 |a_j|2− 1 Na 2 0 (20) = 1 N N/2−1 X j=−N/2 j6=0 |a_j|2 (21)

Using the fact that rms2 = _N1V ar, and using from section 2.1 that the frequency step is equal to δν = 1/T , we can now normalise the power spectrum as follows:

P(ν_j) = 2T µ2_N2|aj| 2 _(22a) P(νj) = 2T N2|aj| 2 _(22b) Where j = 0, ...,N₂.

These normalisations, which differ only by a factor of µ2 _{(µ denotes the mean of the count}

rate), allow one to obtain the fractional rms2 (equation (22a)), or the usual rms2 (equation (22b)) over a certain frequency range ν1< ν < ν2, simply by integrating the power spectrum

over that frequency range.

It is for this reason that we shall plot power spectra multiplied by frequency against frequency, instead of the power spectra themselves against bin number; it allows the viewer to instantan-eously understand the contribution of every frequency interval to the total (fractional) rms2. It is noteworthy to mention that a time signal that is periodic with some Fourier frequency νp, such as the discrete function xk= A sin(2πνptk), will show up in the power spectrum as a

peak at that frequency. This way, a signal containing a quasi-periodic oscillation (QPO) will lead to a power spectrum consisting of at least one peak, provided that the QPO is sufficiently strong, or more, at the harmonics of the QPO frequency νQP O (see figure 6).

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(a) Original signal (b) Distorted signal

(c) Power spectrum

Figure 6: Recovery of original periodicity using Fourier analysis. The original signal is a sine wave with a frequency of 64 Hz and is distorted by a collection of randomly generated numbers. A clear spike can be seen in the power spectrum, allowing us to reconstruct the original periodicity

2.4.1 Error calculation

As P(νj) is a discrete function, there is a Poisson counting error on the data bins. According

to van der Klis (1988), this error contributes a virtually constant power to the spectrum. This so-called white noise is given by:

Pnoise =

2(µ + B)

µ2 (23)

for the normalisation of (22a), where B is the background. In the interest of examining variability without instrumental effects, one needs to consider P(νj)−Pnoise. P(νj), calculated

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The dispersion on the bins in P(νj) is roughly described by a χ2 distribution with two degrees

of freedom; this means that for every error in the power bins: ∆P(νj) ∼ P(νj), which results

in 100% error in the bins. This is not very practical, and better estimates for P(νj) are

required (Ingram (2012), van der Klis (1988)): Ensemble averaging

As the name suggests, this technique requires averaging the white noise subtracted P(νj) function over S segments:

P(νj) = 1 S S X i=1 Pi(νj) (24) with error: dP(νj) = 1 S v u u t S X i=1 (Pi(νj) − P(νj))2 (25)

With enough smoothing, i.e. enough segments, this error approaches the standard counting error:

dP(νj) ' P(νj)/

√

S (26)

(Logarithmic) Re-binning

The second technique involves re-binning the data, such that the amount of data points in the jth frequency bin is:

Mj ≤ cj (27)

where c is an arbitrary constant, and c > 1. In practise, it is common to choose 1 < c < 2. This means that cj does not, in general, equal an integer number.

In this case, Mj is chosen to equal ceil(cj), ceil(x) is equal to the nearest integer higher

than x. The width of the jth frequency bin is now simply equal to ∆fj = Mj· df . The

power in this bin, and its error are then: P(ν_j) = Pxmax xminP(νx) M_j (28a) ∆P(νj) = q Pxmax xmin dP 2 (νx) Mj (28b) Where xmin and xmax are the first and last original bin number within the new bin.

This error calculation assumes that ensemble averaging has been done on the data first. If this is not the case, the error is given by the standard deviation around the mean.

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2.5 Reverberation mapping 2 THEORY

2.5 Reverberation mapping

Reverberation mapping is the technique of mapping a certain geometry by comparing vari-ations in reflected radiation to the original continuum. This technique heavily relies on the concept of path length difference:

Figure 7: Physical representation of the truncated disk model. The figure depicts light travelling to the observer directly, and via reflection. It is clear that the two paths are not, in general, equal. This causes smearing of the original signal, described by the geometry’s transfer function

When presented with a primary continuum consisting of a δ-function flash, any geometry will reflect the signal off of different parts towards the observer. For most geometries, this will lead to a smearing of the original signal over time; the signal, which originated from one point, now travels towards the observer along different paths. The exact manner in which the original signal becomes distorted by this phenomenon is highly geometry dependent. The so-called response of a geometry to such a δ-function flash can be calculated theoretically, and is known as the transfer function or response function of the geometry.

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(fig 7). The accretion region is approximated as an infinitely thin, flat disk, truncated at in-ner radius Rin and extending outwards to the outer radius Rout. In practise, Rout is only

used to constrain the largest path length difference, and, consequently, the largest time delay observed, and has little effect on the actual distortion of the original signal. As it is presently unknown what causes the observed x-ray radiation, we approximate the x-ray source as being a point source, located at height h on the disk axis, which is tilted at inclination angle i.

Assuming that the disk processes the original signal without delay and without altering the intensity or energy, the transfer function can be calculated by evaluating the following integral (Poutanen 2002): T (t) = 2h π Z pM pm dp p2_{p(p − p}₋_)(p +− p) , tmin < t < tmax (29) Where: p2 = r2+ h2 p±= q ± s q = t − h cos(i) cos2_(i) , s = sin(i)pt(t − 2h cos(i)) cos2_(i) pm = max( q R2_in+ h2_{, p}₋₎ pM = min( q R2_out+ h2_{, p} +)

tmin = max(2h cos(i),

q R2_in+ h2_{+ h cos(i) − R} insin(i)) tmax = q R2

out+ h2+ h cos(i) + Routsin(i)

As this integral does not have an exact solution for non-zero Rin, we have evaluated it

numerically, for different values of the truncation radius. The results are displayed in figure 8.

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(a) Transfer function

(b) FT of the transfer function

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For any primary signal F0(t), the reflected radiation is given by the convolution of the original

signal with the geometry’s transfer function:

R(t) = F0(t) ∗ T (t) (30)

Taking the FT of this equation using the convolution theorem leads to:

ˆ

R(ν) = ˆF0(ν) × ˆT (ν) (31)

This means that the iron line equivalent width, which is equivalent to the ratio of rms amplitudes of the reflected variations to the primary continuum variations in a given Fourier frequency range, when determined from a Fourier frequency resolved spectrum (Gilfanov et al. 2000), is, as such, equivalent to the absolute value of the FT of the transfer function:

EW (ν) ∝ | ˆR(ν)| | ˆF0(ν)|

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3 CYGNUS X-1

3 Cygnus X-1

3.1 Introduction

Cygnus X-1 (Cyg X-1) is a well-known source of cosmic X-ray radiation, located within the Cygnus constellation. Cyg X-1 is one of the strongest X-ray sources known to man, and was the first X-ray source widely accepted to be a black hole, hence the suffix “X-1”. The binary system is, as such, amongst the most studied celestial systems, especially within its category. It is precisely this reason that Cyg X-1 was chosen as research subject in this thesis. The study of this section is targeted at repeating research done by Revnivtsev et al. (1999) and Gilfanov et al. (2000). This repetition provides an opportunity for both scientific verification of previously obtained results and education, that future black hole studies may be carried out with more ease.

Another reason for studying this system is the fact that the signal coming from Cyg X-1 contains no measurable QPO. This means that this section is perfect to serve mainly, but not solely, as a “warming-up” for future studies, as the data has a low probability of pos-ing unforeseen complications. This section will mostly follow the methods outlined in the aforementioned articles, using the exact same data sets. We expect to be able to replicate the results from Revnivtsev et al. (1999) and conclude much the same things. Results from Gilfanov et al. (2000) will be used in later sections. The main results from these two articles will also be provided.

3.2 Data analysis and processing (spectral analysis)

In this part of this thesis, publicly available data from the Rossi X-ray Timing Explorer (RXTE) was used. A total of four different observations were implemented, from the P10238 set. The observations were performed between 26-03-1996 and 31-03-1996. Each observa-tion consists of 64 energy channels, each contained within one light curve file. The temporal resolution in every observation was δt = ₆₄1 s, and the observations span the energy range ∼ 0.1 − 83 keV .

A previous study of the same data found that the PCA background is negligible for the time averaged spectra of Cyg X-1 in the energy band of interest, 3–13keV . The PCA background contribution to the frequency resolved spectra at the frequencies 0.03–30Hz is even less im-portant (Revnivtsev et al. 1999). Hence, no use will be made of files containing background curves.

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3.2 Data analysis and processing (spectral analysis) 3 CYGNUS X-1

In order to generate frequency resolved spectra, the RXTE light curves were processed ac-cording to the theory outlined in sections 2.4 and 2.4.1: The signal of the first observation was extended with the other observations, and averaged over 1953 segments of length 1024 bins, for a total observational time of ∼ 31ks. This was done for each of the 64 energy bands.

Thereafter, the power spectrum was calculated for each band. The 64 power spectra were individually, but equitably re-binned, using a binning constant c = 1.5. Each frequency bin was integrated over its frequency range, to calculate the total rms2 in said bin. From this, the rms was trivially calculated.

As the frequencies of the bins were the same for every power spectrum, an rms(E) plot could be constructed, for each νj. Using the normalisation mentioned above, the rms is in units

of count rate, meaning it could be fit using Xspec. The rms(E) data points were fitted as intensity versus energy spectra to the same suitable model. This resulted in different values for the equivalent width (EW) of the Gaussian model component, which were plotted against frequency. The EW was then compared to the theoretical transfer function calculation.

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3.3 Results 3 CYGNUS X-1

3.3 Results

The power spectra, plotted in units of power × f requency, are shown in figure 9. This fig-ure contains merely four of the 64 calculated spectra, for the reader’s convenience, so that a quantitative view of the processed data is possible.

Firstly, one can clearly see that the power spectra are different for different energy bands. It is also noteworthy to mention that, as energy increases, the noise/signal ratio significantly increases, especially in the higher energy regions, which become almost entirely smeared out. Secondly, the spectra are almost completely devoid of any significant peaks or dips.

(a) Energy band 0.144802 − 1.6381 keV (b) Energy band 4.33994 − 4.64127 keV

(c) Energy band 19.0663 − 20.0053 keV (d) Energy band 80.8647 − 83.1019 keV

Figure 9: Power density spectra in different energy bands

In order to investigate the frequency dependence of spectral elements shown in figure 9, the spectral data was logarithmically re-binned into ten frequency bins, ranging from 0.19 to 20.5 Hz, using said binning constant c = 1.5.

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For each frequency bin, we have calculated the energy dependent absolute rms by integrating each bin over its frequency range. The results are displayed in figure 10:

Figure 10: Absolute variability in different frequency bands

In their present form, the data in figure 10 are not very illuminating; the data are largely distorted by the telescope’s response at different energies. We have calculated the ratio of these frequency resolved spectra to a simplistic power law model with a photon index of 1.8. The resulting ratio spectra are shown in figure 11 below.

It is clear that the spectral shape changes with frequency; The reflective characteristics, the broad peak at 5 − 7keV and the smeared edge at energies above that, as mentioned by Re-vnivtsev et al. (1999), decrease in apparent strength of their amplitudes as Fourier frequency increases. These features are completely absent above a frequency of ν ∼ 20Hz.

In order to, quantitatively, be able to discover more about the accretion disk surrounding Cygnus X-1 and the radiation it emits, we have fitted the frequency resolved spectra for each frequency bin within the same energy band as Revnivtsev et al. (1999), to wit 3.0 − 13.0keV , to a simplistic model consisting of three components; phabs ∗ (ewgaus ∗ powerlaw), as de-scribed in section 7.

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Figure 11: Ratio of spectra to models (re-scaled for clarity).

Three types of variability are displayed; slow (black), middle (red), and fast (green). It is immediately clear that the slow variability does not display an iron line at all, and that reflective features decrease with increasing Fourier frequency.

In this model, which was fitted with different parameters for every data set, the centroid energy of the Gaussian line was kept fixed at 6.4keV , across all sets. The parameter σ was allowed to vary, but kept tied across the different spectra, as was nH. Other parameters were

allowed to go free in the fit.

The resulting shape of the theoretical model for every frequency bin can be seen in figure 12. This figure contains the spectra themselves as well.

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Figure 13: Equivalent width versus Fourier Frequency

The aforementioned fitting process resulted in different values for the fitting parameters per spectrum. Figure 13 shows the Fourier frequency dependence of the EW of the Gaussian line used to model the iron line in the spectra.

It is immediately clear that the EW is not a constant across all examined frequencies; the EW remains constant up to ν ∼ 1Hz, and starts decreasing at higher frequencies. At a frequency of ν ∼ 10Hz, the EW is reduced to half of its maximum value. No distinct peaks or dips are present in this figure. Please note that the shape and amplitude of the EW (ν) points always depends on the fit (reduced χ2, fitting parameters, models used). Also, several energy bins had to be excluded from fitting, as the power within could not be constrained to be above the Poisson noise, rendering them useless. The general trend in the figure should, however, not change too much.

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3.4 Discussion 3 CYGNUS X-1

3.4 Discussion

The deteriorating quality of the data as energy increases is due to the relatively small amount of counts registered, meaning that the signal/noise ratio decreases.

Figure 11 clearly shows that the simplistic model of an unabsorbed, single slope power law does, in fact, deviate too much from the observed spectra. This is a common occurrence and is generally ascribed to reflection from an an optically thick, neutral medium (Revnivtsev et al. 1999). Such a medium possesses reflective features that have been extensively studied and are well understood (Basko et al. 1974): a thin iron line at 6.4keV (EW ∼ 100eV ), an absorption edge at E ∼ 7.1keV and a reflected continuum (’reflection hump’ ) at higher energies (∼ 20 − 30keV ).

It is clear, from literature and figures 11 and 12 that X-ray binaries’ spectra are more com-plicated than this. The line energy, kept fixed at 6.4keV , is, in reality, often not a constant across all frequencies. The expected sharp absorption edge is present in the spectra, however broadly smeared out.

These differences also indicate something else: a complicated ionisation state of iron and relative motion of parts of the reflecting media, which both vary with frequency. Letting the line energy and σ vary without bounds in the more complicated model containing a power law, absorption and a Gaussian line lead to unacceptable values. The use of this last model is justified only for quantitative analysis, physically, it turns out that the model is virtally useless.

Figure 11 does, however, illustrate a very important thing: the reflection features are sup-pressed as the characteristic time scale decreases.

The last model allowed us to plot the EW as a function of the examined frequencies EW (ν). From this, several features of the accretion disk can be estimated.

Examining figure 13, it is clear that this figure also displays the aforementioned change in reflective feature presence. This was expected, as the EW is a measure of the ratio of the amplitudes of variations of the reflected component to the primary emission. The EW steadiness at frequencies ν . 1Hz implies that the amplitude of variations of the reflected component has the same frequency dependence as that of the radiation from the source.

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To understand this effect, we re-examine the simple truncated disk model; an accretion disk centered at an X-ray primary, with an inner and outer radius (Rin and Rout, respectively).

These radii are to be interpreted as marking the part of the disk capable of producing an iron fluorescence line. In practice, Rout has no impact on results, except for changing the

maximum value of time lag.

The most trivial explanation of the suppression of reflection lies in the difference of path length that light has to travel before reflecting and reaching the observer. The reflecting media have a finite size, causing light from the outer regions of the disk to reach the observer in a different time range than light reflected off of the inner regions. This model assumes that the X-ray source, depicted in figure 14, contains within the unreflected spectrum itself all the characteristic variability features (including the QPO signal), smeared out by differences in light travel time in the EW (ν) plot.

Figure 14: Physical representation of the disk model

Because of difference in local velocity due to higher temperatures and loss of potential energy, the inner most regions of the disk will vary more than the outer regions. As the inner regions are hotter than the outer regions, they will emit the most hard radiation and will therefore be more present in the short time scale spectra. The flux on the inner parts consists mainly of direct source photons and will therefore not show a clear reflection spectrum.

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The outer parts of the disk, on the contrary, experience a flux consisting mainly of photons that have undergone Compton-scattering as they traveled through the disk to get to the outer part. This means that the outer parts will show spectra containing more distinct reflection features than the inner part.

The cause of the local viscous differences between inner and outer parts of the disk may be a smaller solid angle of the reflecting medium in the higher frequency range or a different geometry entirely.

Independent of which explanation is correct, the approach used in this thesis, frequency resolved spectroscopy, provides a way to study spectra of X-ray binaries as a function of the radius of the accretion disk (Ingram 2012).

To see if this truncated disk model is valid, we imposed upon our measured EW (ν) calculated ˆ

T (ν) for different truncation radii in figure 15.

Figure 15: A comparison of the measured EW (ν) and the theoretically calculated ˆT (ν), assuming h = 10RG and i = 50°

It is clear that the model is valid and may be used to describe the general accretion disk geometry. Like Gilfanov et al. (2000), we find that a truncation radius of order r ∼ 100RG

best fits the data. Research by the same group whose results we replicated here on different spectral states of Cygnus X-1 has concluded that the truncation radius Rin decreases as the

spectrum softens (Gilfanov et al. 2000). Several observations have given rise to the suspicion that the QPO frequency increases as the spectrum softens. This is an interesting premise that we will test for the next black hole.

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3.5 Conclusion 3 CYGNUS X-1

3.5 Conclusion

Using techniques outlined in the previous sections, we were able to study the frequency-resolved spectra from Cygnus X-1.

Cygnus X-1 does not show a distinct QPO peak in any spectrum from the observations used. This research confirms the results found by Revnivtsev et al. (1999), to wit an understandable EW trend.

The underlying process of the EW (ν) plot is still unknown, but two options currently exist, both explained by a simple disk model; the EW shape may be caused by differences in light travel time from various parts of the accretion disk, meaning that all of the variability ori-ginates from the X-ray source itself. An other possibility is that large time scale variations appear in geometrically different regions than the high frequency variations. Presumably, the large time scale variations originate from the outer regions, and the small time scale variations originate from the inner regions. This might be caused by a different solid angle, as ‘seen’ by reflection photons.

Furthermore, frequency-resolved spectroscopy seems to provide a way to probe the reflection features as a function of radius from the black hole center.

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4 GRS1915+105

4.1 Introduction

The results of the study of frequency resolved spectra from Cygnus X-1 agree well with the results of Revnivtsev et al. (1999). This naturally gives rise to the question whether all known black holes and their reflective properties can be understood this way or not.

GRS1915+105 forms an interesting object for a such spectral study, as its spectra contain a more distinct QPO peak than the spectra from Cygnus X-1 (section 4.3). One might expect a strong QPO signal to cause extraordinary variation in the EW (ν) plot at the cor-responding frequency νQP O, which would reveal interesting characteristics of the accretion

disk surrounding the black hole. Alternatively, no notable difference would lead to much the same conclusions as in the last section, meaning that this may be a common accretion disk property.

4.2 Data analysis and processing

To study GRS1915+105, data from two observations were used. Both observations were car-ried out by the aforementioned RXTE, equipped with a PCA.

The first observation was performed on 09/02/1997, possesses observational ID 2040201-15-00, and shall, hereafter, be referred to as Observation 1. The second observation was performed on 06/03/2002 , has observational ID 60701-01-28-00, and shall henceforth be known as Obervation 2. These observations contain one light curve for each energy band, for a total of 16 light curves per observation. Both observations have temporal resolution δt = ₁₂₈1 s and span the energy range ∼ 1.5 − 15 keV .

As the timing resolution of the Cygnus X-1 observation (δt = 1.5625 · 10−2s) is twice the size of the resolution of the GRS1915+105 observations (δt = 7.8125 · 10−3s), the length of the averaged segments needed to be increased twofold, in order to be able to probe down to the same frequencies (from section 2.4: νlowest= _{N δt}1 ).

This lead to the data being averaged over 634 segments, for a total observational time of ∼ 10ks for observation 1, and 603 segments and a total observational time of ∼ 9.6ks for observation 2.

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4.2 Data analysis and processing 4 GRS1915+105

The data were processed in the same manner as before;

• Calculating the power spectrum for every energy band, for each observation, averaged over all segments.

• Logarithmically re-binning the power spectra and integrating the power bins over their respective frequency ranges across all energy bands. This resulted in an rms(E) plot for every frequency bin.

• Fitting said spectra in Xspec to a suitable model. • Plotting EW (ν)

Our research on Cygnus X-1 in the previous section provided us with a number of things to test for other BHs. First of all, we need to see if the EW (ν) does indeed still has the predicted declining shape. As GRS 1915+105 does have a QPO, it is interesting to see what, if any, difference this makes for the EW (ν) trend. Lastly, we need to see if the inferred relation that Rin decreases as νQP O increases holds. From the results, we will be able to infer the QPO

origin.

No ratio plot was made for this Black Hole, as figure 11 served only an illustrative point, which does not need to be proven again. The rms(E) plots have been excluded from this thesis as well, for the shape of an rms(E) plot is entirely instrument dependent and does not in the slightest provide insight into the data. As the processing is identical to the Cygnus X-1 procedure in all other aspects, the results shall be addressed only briefly.

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4.3 Results 4 GRS1915+105

4.3 Results

As before, only a selection of all spectra is presented to the reader in this section, in figure 16. This time, each sub-figure contains two spectra; one from each observation, plotted yet again in units of power × f requency. As the energy channels of the individual observations are different, we plotted spectra corresponding to comparable average bin energy.

(a) Energy ∼ 1.6 keV (b) Energy ∼ 5 − 6 keV

(c) Energy ∼ 8 − 9 keV (d) Energy ∼ 12 − 15 keV

Figure 16: Power density spectra at comparable average energies

At first glance, the data from GRS1915+105 seem to be of higher quality than the data from Cygnus X-1 (figure 9). As frequency increases, so does the noise/signal ratio, still. As energy increases, however, the data retain their quality and shape.

A total of four peaks can be observed in the spectra, two per observation. These peaks, at the QPO frequencies and their second harmonics, have the following frequencies:

• Observation 1: νQP O ∼ 2.25Hz and νharmonic∼ 4.5Hz

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The data were fitted to a slightly different model, phabs ∗ smedge ∗ (ewgaus ∗ powerlaw). Good fits were achieved for both observations:

For observation 1, χ2 was 122.52/168, while keeping the line energy of the Gaussian line fixed at 6.4 keV . Again, σ was allowed to vary, but kept tied across the different spectra, as were nH, Ec and W . α and β were fixed at −2.67 and 0.3, respectively. Any other parameters

were allowed to go free in the fit.

For observation 2, χ2 was 88.6/112, the line energy was also kept fixed at 6.4 keV . Other parameters were treated as presented above. As the data were of unacceptably bad quality in the higher frequency bands, some rms-spectra had to be excluded from fitting, resulting in fewer data points in the EW (ν) plot.

From these fits, the EW was calculated across all frequency bins, for each observation. The results are displayed below, in figure 17:

(a) Observation 1 (b) Observation 2

Figure 17: EW as function of Fourier frequency, for two observations

Observation 1: The general trend of figure 17a moderately resembles a Gaussian function. The EW remains constant up to ν ∼ 0.6Hz and starts increasing after. At ν ∼ 1.5Hz the EW reaches a state of well-nigh constancy, until ν ∼ 15Hz Subsequently, the EW declines again in magnitude. at frequencies ν ∼ (1 , 25)Hz, the EW is at half its maximum value.

Observation 2: This figure’s trend, too, bears resemblance to a Gaussian function, albeit smeared and broadened, as well as less pronounced. The EW increases from the largest time scale, rising until ν ∼ 3Hz and falling forthwith.

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It is impossible to determine when the EW reaches half its maximum value from figure 17b.

Both EW (ν) plots show some deviations from the general trend, none of which are statistically interesting, due to relatively large errors. For better comparative study of both Black Holes, we plotted all EW, individually re-normalised, in a single figure below:

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4.4 Discussion 4 GRS1915+105

4.4 Discussion

The observed νQP O and νharmonic values agree well with the values found by Ingram and Van

der Klis (2015), confirming that these are the right frequencies.

Both GRS1915+105 EW (ν) trends are completely different from the EW (ν) trend found for Cygnus X-1. Examining figure 18, it is clear that, at higher frequencies, both observations from the second Black Hole resume the same declining trend as the first Black Hole.

At lower frequencies, however, the trends could not be more dissimilar. Despite the fact that one could view observation 2 as a mere oddity, due to the near constancy of the EW at large time scales, observation 1 clearly displays utterly different behaviour.

The fact that the EW declines at low frequencies indicates that reflection is suppressed, not only at high frequencies, as expected, but at low frequencies as well. Hence the simple disk model, introduced in section 3.4 will not work for this Black Hole. The EW trend of ob-servation 1 especially suggests a complicated non-stationary geometry of the accretion disk, implying that the QPOs have a geometric origin. Presently, no known models exist to explain the observed EW behaviour. The most likely explanation to the unexpected low-frequency reflection suppression is found in the spectral make-up; the flux we observe consists of two components; the continuum, and the reflected flux. f (t) = c(t) + r(t). Taking the square of the absolute value of both sides (as, indeed, we did to calculate our power spectra) leads to cross-terms: | ˆf (ν)|2_{= |ˆ}_c(ν)|2_+|ˆ_r(ν)|2_+2<(c∗_{(ν)r(ν)). In this thesis, a simple two-component}

model was used to fit the data with; one for the continuum, and one for the reflection. At high frequencies, when r → 0, there is no problem. At low frequencies, however, r 9 0, and we fit using the wrong set of components. It is precisely at these frequencies that we observe unexpected behaviour.

Another explanation may be found in a phenomenon called “spectral pivoting”. We assumed that spectral variations would occur equally for each energy. In reality, the hardest spectrum is not necessarily the brightest in the region of interest (6 − 7keV ), meaning that we observe more or less reflection than we think there is, as a harder continuum leads to more reflective features; more photons present with a high enough energy to ionise the accreting matter. Recent articles (Motta et al. (2015), Heil et al. (2015)) have studied signals with QPOs from a variety of objects and concluded that the QPO strength increases with inclination angle i, which affirms the theory that the QPOs have a geometric origin.

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4.4 Discussion 4 GRS1915+105

For completeness, as well as improved statistical validity, we explicitly investigated the power spectral bins at and about the QPO and first harmonic peak frequencies, without logarithmic re-binning of the data. Whilst examining this new data, we neglected EW (ν) at lower fre-quencies, as it remains mostly unexplained and is likely to be erroneous.

Ignoring the EW (ν) at lower frequencies leads to the conclusion that the QPO is most likely caused by variation in flux from the X-ray source; any deviation in the EW (ν) trend would mean a deviation in the transfer function of the geometry, which, in turn, would indicate a difference in geometry at that time scale. Figure 19 shows which bins were selected for investigating the EW at better frequency resolution.

The numbers of the selected bins were kept constant across all energy channels, and in the familiar way the EW was calculated from a spectral fit.

Figure 19: Bin selection about the QPO and first harmonic peaks

We furthermore imposed this last data on top of the already presented figure 18. This resulted in the data points displayed in figure 20.

Is is clear that, for observation 1, this last attempt at clarification was almost to no avail; the uncertainty in EW magnitude is still too high to definitively conclude anything different from the already discussed.

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de-4.4 Discussion 4 GRS1915+105

Figure 20: Comparison of relative EW from different Black Holes

viation from the general trend about νharmonic. This hints at the possibility that the QPO

originates (partially) from geometric effects, instead of solely originating from variability in the X-ray source. A similar conclusion was reached in the paper from Ingram and Van der Klis (2015), which shows hints of differences in the iron line shape during the QPO cycle.

A general explanation for both observations would be a non-constant accretion rate, which would result in a non-constant geometry. Further research on the matter is required, as are better data quality and more advanced processing techniques.

Lastly, we are able to confirm that Rin ↓ as νQP O ↑. Remembering the Rin-dependence

of ˆT (ν) from figure 8b, we see that observation 1 (νQP O ∼ 2.25Hz) is clearly represented

best by a model with a smaller truncation radius than observation 2 (νQP O ∼ 0.5Hz); even

though the data points at low frequencies remain unexplained, it is clear that the EW (ν) starts declining at higher frequencies in observation 1, indicating a smaller truncation radius. We did not explicitly compare the theoretical transfer function calculation to the measured EW (ν), because of the unexplained behaviour at low frequencies.

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4.5 Conclusion 4 GRS1915+105

4.5 Conclusion

Using the same techniques as in sections 2.4 and 3.2, we have studied the frequency-resolved spectra of BHB GRS1915+105 for two different observations.

The EW , obtained from frequency-resolved spectroscopy, performed on data from GRS1915+105, possesses a completely different – statistically valid – trend from Cyg X-1. Both observations for this black hole differ substantially from one another, indicating that the black hole state changes considerably over the course of five years.

Though different, both Observation 1 and Observation 2 have one thing in common; they seem to portray a suppression of reflective features, at frequencies lower than ∼ 1.5Hz and ∼ 3Hz, respectively. This is a deviation from the expected trend we observed in Cyg X-1 (sections 3.3 and 3.4). The explanations offered in section 3.4 are only partially valid for this system. The larger time scale part of the observations remains unexplained. As no discernible difference was discovered at the QPO frequencies, these results strongly hint at a geometric origin of the QPO. This result is backed by recent papers, that found that the QPO strength changes with inclination angle (Motta et al. (2015), Heil et al. (2015)).

Further inspection of the power around the QPO and harmonic peaks provided new results, which did not alter aforementioned conclusions. We did, however, find another hint that the QPO may not be originating from the X-ray source itself, in Observation 2 only. This nicely connects to a recent paper, that found hints of variations of the iron line shape during a QPO cycle (Ingram and Van der Klis 2015). The fact that the QPO may have a different origin in both observations affirms the fact that the black hole state changes substantially over a relatively short period of time. This may be due to a change in in-flow of accreting material.

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5 GX339-4

5.1 Introduction

The last BH to have its properties examined in this thesis is GX 339-4. Chosen for its exceptionally strong QPO signal, we expect this BH to lead to results that differ from the BHs we already studied.

5.2 Data analysis and processing

To study GX339-4, data from an RXTE observation were used.

The observation used was performed from 11/05/2002 to 12/05/2002, and possesses observa-tional ID 70109-04-01-01.

The timing resolution of this is of the same size as the resolution of the GX1915+105 obser-vations (δt = 7.8125 · 10−3s = ₁₂₈1 s). This means that the length of the averaged segments needed to be increased twofold again, in order to be able to probe down to the same frequen-cies. The data were averaged over 1036 segments, for a total observational time of ∼ 16ks. The data were processed in much the same manner as before, which will not be listed here.

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5.3 Results 5 GX339-4

5.3 Results

Figure 21 below contains four of the calculated power spectra:

(a) Energy ∼ 1.8 keV (b) Energy ∼ 6 keV

(c) Energy ∼ 9.5 keV (d) Energy ∼ 14.4 keV

Figure 21: Power density spectra at comparable average energies

The data seems to be of moderate quality, becoming slightly distorted at higher energies. The graphs exhibit a strong QPO signal at a frequency νQP O ∼ 0.45Hz.

The data were fitted to the following model: phabs ∗ smedge ∗ (ewgaus ∗ nthComp). A reasonable fit was achieved, with χ2 = 155.13/128. The line centroid energy was kept fixed at 6.4keV . The parameter σ was allowed to vary, but tied across different frequencies, as were nH, Ecand W .

The values of α, β, inp-type, redshift, and kTe were fixed at −2.67, 0.3, 0, 0, and 100,

respectively. Such a high value for kTe was chosen, as the data only extend out to about

Emax∼ 15 keV , and do not contain a high energy cutoff. Any other parameters were allowed

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The resulting EW is plotted in figure 22 below.

Figure 22: Equivalent width versus Fourier Frequency

It is difficult to discern a clear trend in figure 22, it is largely obscured by error bars of considerable magnitude. The EW does have a generally declining trend. Large deviations from the trend can be seen at νQP O. The EW reaches half of its maximum value at a

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The model we used allows us to plot a new parameter against Fourier frequency; the photon temperature in the disk:

Figure 23: Disk temperature versus Fourier Frequency

The temperature has an almost constant value, except for at νQP O, where large deviations

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5.4 Discussion 5 GX339-4

5.4 Discussion

It is immediately clear from figures 22 and 23 that GX 339-4 is in yet another state of ac-cretion than the other black holes observed. The large magnitude of the errors on the EW , ∆EW , is most likely the result of the model component ewgaus. Similar ∆EW problems were experienced by Ingram and Heil.

On the subject of the EW itself; the EW (ν) trend is strikingly similar to the one found for Cyg X-1; declining and without noticeable deviations. That is, of course, excluding the points at νQP O. Here, the data points deviate significantly from the general trend, indicating

something interesting may be occurring at this frequency. As the EW is essentially a measure for the absolute value of the transfer function, a significant deviation at a certain frequency is not expected. One would generally expect the transfer function to be relatively insensitive to the frequency of the original signal. The fact that we observed precisely such deviations could be a strong indication that the QPO signal from GX 339-4 arises from a completely different mechanism than the broad band noise surrounding it in frequency space. This is affirmed by the trend in figure 23, that clearly shows that the photon temperature is significantly higher at νQP O.

The correlation between Rin and νQP O cannot be tested at this point; the errors on the data

obscure the precise trend of the EW (ν) too much. For this reason, we have also not compared the data to the theoretical calulation.

Presently no viable explanation is known for the deviations. The results found in this thesis do not necessarily mean all black holes are completely different regarding their QPOs. It might be the case that all black holes and their accretion disks change over time in a similar manner. The black holes studied in this thesis would merely be at a different point along a common trajectory.

5.5 Conclusion

Having studied the frequency-resolved spectra from GX 339-4, we conclude the following: GX 339-4 generally displays expected iron line changes with frequency, except around νQP O,

where strange iron line shape modulations take place. This is a very strong indication that the QPO originates from a different physical mechanism than the primary continuum, which is not yet understood. Although we do presently not understand the actual mechanism, we did discover that it is geometric in nature.

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6 CONCLUSION

6 Conclusion

We examined the frequency-resolved spectra of three BHs, Cygnus X-1, GRS 1915+105, and GX 339-4, using iron line reverberation mapping. We used variations in the F e-line to infer the geometries of these BHs and compared them to a theoretical transfer function calculation for a simple truncated disk model. We found that Cygnus X-1 is well described by this model, consistent with a truncation radius Rin∼ 100RG. Our results confirmed earlier research done

by Revnivtsev et al. (1999).

GRS 1915+105 is not well represented by the truncated disk model; we studied two observa-tions, and found suppression of reflective features at low frequencies in both observations. At higher frequencies, both observations behaved as expected from a theoretical transfer function calculation. This suppression can be explained by any one or a combination of wrongfully made approximations; Spectral pivoting and neglected cross terms in fitting. Of course there always exists the possibility that the geometry differs from a truncated disk. Presently, no geometry is known to be able to reproduce the EW (ν) trends found for both observations.

We also found that the geometry of GX 339-4 fits well with the truncated disk model, though the results are hard to interpret due to large errors on the data.

Cygnus X-1 is the only BH we studied that does not have a QPO. GRS 1915+105 shows a QPO in both observations used, at νQP O ∼ 2.25Hz and νQP O ∼ 0.5Hz for observations 1 and

2, respectively. Ignoring the lower frequency suppression, we found little to no statistically valid deviation from the general trend, save for a small feature at the second harmonic peak in observation 2. This strongly implies that, presently, the QPO observed in this BH is of non-geometric origin, as any geometrically different situation at the QPO time scale would show as a deviation in the EW (ν) at the corresponding frequency. The small overtone feature may be regarded as a hint of an accretion phase change from or to a state where the QPO is caused by a geometric phenomenon. A geometric origin is consistent with several recent papers (Motta et al. (2015), Heil et al. (2015)). We were furthermore able to confirm that, generally, the truncation radius is smaller when the νQP O is larger.

Even though data from GX 33-9 are largely obscured by large errors, we found clear deviations from the general trend at νQP O. This leads to the conclusion that the QPO in this BH is

definitely of geometric origin. The actual mechanisms behind QPOs in BHs are presently unknown.

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7 APPENDIX - XSPEC MODELS

7 Appendix - Xspec models

In this thesis, spectra were fitted to a combination of a variety of model components, below is a short description of each one:

Absorption (phabs)

An absorption function using the photo-electric cross-section (without Thomson scat-tering) for a column of Hydrogen gas. As a formula, the absorption function takes the following shape:

f (E) = e−nHσ(E) ₍₃₃₎

This model takes only one parameter, nH, which represents the equivalent column of

Hydrogen through which the emitted light travels whilst being partially absorbed by the gas before reaching the observer. The units of nH are of 1022 atoms cm–2. The

component uses standard abundances to calculate absorption.

Power law (powerlaw, or po)

A simple photon power law function, describing the most trivial form of an emission spectrum. The formula for this has the following form:

g(E) = KE−Γ (34)

This component takes two parameters: Γ, the so called photon index, which is a dimen-sionless number, and K, a normalisation factor, which is the amount of photons keV–1 cm–2 s–1 at E = 1keV .

Gaussian line (gaus)

A Gaussian line, often used for fitting spectral peaks, as we shall in this thesis: g(E) = N 1

σ√2πe

−(E − El)2/2σ2 ₍₃₅₎

This component takes three parameters: El, the centroid energy of the line, σ, the line’s

width, and N , a normalisation factor, defined as being the total amount of photons cm−2 s−1 enclosed by the line.

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7 APPENDIX - XSPEC MODELS

This thesis will mostly feature a similar model, ewgaus. This model, created by Ingram and Heil, differs from gaus in the fact that it is a multiplicative model, and, instead of N , takes in the equivalent width of the line as parameter. The model is defined as multiplicative, in order to calculate the equivalent width, which requires a reference continuum.

Smeared edge (smedge)

A model for simulating a smeared reflection edge, as from the Ebisawa PhD thesis:

f (E) =    1 if E < Ec e−β(E/Ec)α _{· (1 − e}(Ec−E)/W₎ _{if E ≥ E} c

This component takes four parameters: Ec, the threshold energy in keV , β, the

max-imum absorption factor at threshold energy, α, an index for the photo-electric cross-section (virtually always -2.67), and W , the smearing width in keV .

Thermally comptonized continuum (nthComp)

This is a thermally comptonized continuum model, from Zdziarski, Johnson and Mag-dziarz (1996, MNRAS, 283, 193), as extended by Zycki, Done and Smith (1999, MNRAS 309, 561). It takes the following parameters: Γ, the aforementioned photon index, kTe,

the electron temperature, kTbb, the seed photon temperature, inp − type, which equals

0 or 1 for blackbody or disk-blackbody seed photons, respectively, redshif t, N , a nor-malisation factor.

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8 BIBLIOGRAPHY

8 Bibliography

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L. M. Heil, P. Uttley, and M. Klein-Wolt. Inclination-dependent spectral and timing prop-erties in transient black hole X-ray binaries. MNRAS, 448:3348–3353, April 2015. doi: 10.1093/mnras/stv240.

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A. R. Ingram. A physical model for the variability properties of X-ray binaries. PhD thesis, 2012.

S. E. Motta, P. Casella, M. Henze, T. Mu˜noz-Darias, A. Sanna, R. Fender, and T. Belloni. Geometrical constraints on the origin of timing signals from black holes. MNRAS, 447: 2059–2072, February 2015. doi: 10.1093/mnras/stu2579.

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M. Revnivtsev, M. Gilfanov, and E.Churazov. Frequency resolved spectroscopy of cyg x-1: fast variability of the f ekα line. Astronomy and Astrophysics, 1999. doi: arXiv:astro-ph/

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Iron line reverberation mapping of black hole binaries: A quantitative study of accretion disks using frequency resolved power spectra

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