1. Quasi Periodic Oscillations in Black Hole Binaries
Jeroen Meringa – 6277349
Essay Bachelor project Physics and Astronomy, extending 12 EC,
executed between 13-05-2013 and 19-07-2013.
API Institute
Faculty of Science (FNWI)
Handed in on 19-07-2013
Supervisor:
L. Heil
Second judge:
M. van der Klis

2. Abstract
In this essay, an analysis is presented of the short timescale variations in the
properties of the strong quasi periodic oscillation observed in the objects MAXI
J1659-152, GX 339-4 and H1743-322. A linear rms-flux relation is commonly seen in
the broad band noise of black hole binary systems, but when the rms-flux
dependence in a QPO was first analysed with XTE J1550-564 (L.M. Heil, S.
Vaughan, P. Uttley 2010) different correlations as well as no correlation at all had
been observed.1
In response, three more objects are analysed in this report on
timescales shorter than about 3000 seconds. We found similar behaviour in H1743-
322, while GX 339-4 showed a very different relation.
Samenvatting
Zwarte gaten zijn objecten zie zó zwaar zijn, dat hun zwaartekracht sterk genoeg is
dat zelfs licht niet meer kan ontsnapen. Soms draait er een ster om een zwart gat
heen, dat we kunnen zien omdat de ster wel licht uitzend, en om een onzichtbaar
punt heen draait. Soms valt er materiaal van de buitenste atmosfeer van de ster naar
het zwarte gat, omdat deze zo’n grote aantrekkingskracht heeft. Dit materiaal
(gaswolken) vormt dan een schijf om het zwarte gat heen voordat het naar binnen
valt. Bij het gat in de buurt beweegt het materiaal zó heftig, dat het heel heet wordt
en straling uitzend. Met een satteliet om de aarde meten we deze straling. Er zit veel
informatie in deze straling, zoals hoe snel de wolk van materiaal rond het zwarte gat
draait.
In dit onderzoek is gekeken naar dit signaal, en er is gevonden dat er iets is, een
onbekend object, dat telkens met ongeveer dezelfde snelheid (frequentie) rond het
zwarte gat draait. Als er meer licht (straling) van de schijf om het gat heen af komt,
dan zien we ook dat ons onbekende object sneller om het zwarte gat heen draait. Bij
eerdere onderzoeken is al eens gekeken naar de straling van de schijven om zwarte
gaten, en iedere keer is er een verband gevonden tussen de frequentie (hoe snel het
materiaal in de schijf om het zwarte gat heen draait) en hoe sterk het licht is dat
uitgezonden wordt. Maar in ons onbekende object is dat verband niet altijd hetzelfde.
Volgens sommige modellen kan het object een klompje met extra materiaal zijn om
het zwarte gat heen. Dit verklaart waarom er meer licht af komt, en waarom het
sneller draait als het dichter bij het midden komt, maar niet alle verbanden worden
hiermee verklaart. Meer onderzoek is dus nog nodig.

3. Index
Page 4 1 Introduction and Background
Page 4 1.1 Motivation and Introduction
Page 4 1.2 Black Hole Binary Systems
Page 5 1.3 X-ray Detection
Page 6 1.4 Light Curve and Power Spectrum
Page 7 2 Theory
Page 7 2.1 Quasi Periodic Oscillations
Page 9 2.2 RMS-Flux Relation
Page 10 2.3 Black Hole States
Page 11 2.4 Fast Fourier Transform
Page 12 2.5 Lense–Thirring precession model
Page 13 3 Observation Method and Equipment
Page 13 3.1 Rossi X-ray Timing Explorer Satellite
Page 15 3.2 The Proportional Counter Array
Page 15 3.3 Experiment Data System
Page 16 3.4 Data manipulation
Page 17 3.5 Data Analyses
Page 18 4 Results
Page 18 4.1 RMS-Flux Relation
Page 19 4.2 Central Frequency-Flux Relation
Page 21 4.3 Mean Frequency – RMS-Flux Gradient Relation
Page 22 4.4 Intercept from Linear fit of the RMS-Flux relation
Page 24 5 Discussion
Page 24 5.1 The RMS-Flux Relation Analyses
Page 25 5.2 The Central Frequency-Flux Relation
Page 25 6 Conclusion
Page 28 Attachments

4. 4
1 Introduction and Background
1.1 Motivation and Introduction
The RMS-flux relation in broad-band noise is a known and studied phenomenon.
When this rms-flux dependence was first analysed in a QPO (XTE J1550-564, L.M.
Heil, S. Vaughan, P. Uttley 2010) different correlations as well as no correlation at all
had been observed.1
This research had been conducted to see if this relation always
exists, or whether it differs per object. For that, three more objects had been
analysed (MAXI J1659-152, and H1743-322).
1.2 Black Hole Binary Systems
Black holes (BH) are difficult to detect, as they are so dense that their escape
velocity is greater than light, meaning that no information can escape from it.
Locating such object that can not be observed directly can be difficult. Indirect
methods must be used, such as looking at the effects black holes have on their
surroundings, particularly when they are part of a binary system. A binary system is
a system of two astronomical objects such as stars, brown dwarfs, black holes but
also galaxies or asteroids. If they get close enough, their gravitational interaction
causes them to orbit around a common centre of mass. In this case, we are
interested in binary systems consisting of a black hole and a star. These kind of
systems can be detected whenever a visible star can be seen orbiting a massive, but
unseen companion. This companion does not have to be a black hole per se; it might
as well be a dwarf or neutron star.
When a star with an initial mass of at least about ten times our Sun’s mass is near
the end of its lifetime and can no longer maintain nuclear fusion in its core, gravity
will cause the outer shell to collapse in upon the core. This quick collapse triggers a
supernova, and the outer material of the star is expelled outwards. What remains,
depends on the mass of the remaining core. One possibility is a neutron star. Stars
with masses less than about two to three solar masses (the Tolman-Oppenheimer-
Volkoff limit 2,3
) should remain neutron stars, and slowly radiate away their energy. If
the stellar core remaining after the supernova is heavier than that, it will collapse into
a singularity known as a black hole.
If, when a star is seen in orbit around an invisible companion, the mass of that binary
companion is bigger than three solar masses, it must therefore be a black hole.
Using Kepler’s laws of motion, the mass of the unseen companion can be calculated
from the orbital motion of the visible star. This way, Black Hole Binary (BHB)
systems can be found. The black hole Cygnus X-14
is a good example of the search
for black holes in this way. This is also the object in which the RMS-flux relation was
initially observed 20, 21, 22, 23
(see section 2.2).

5. 5
1.3 X-ray Detection
When a star system is in orbit along our line of sight, spectroscopy can be used to
detect it. The information about a spectroscopic binary system can be found in the
periodic Doppler shifting of the lines in its spectrum.30
When the visible star moves
towards us, the spectral lines are blue shifted, whereas when it’s moving away, the
lines are red shifted. From the way in how the spectral lines shift, one can get the
orbital speed, and the period can be calculated from the frequency. This information
can be used to determine the mass of the invisible companion, and identify it as a
potential black hole candidate.
Another way of detecting black holes is by looking at the X-rays that are generated
around it. These X-rays are analysed in this essay. Regular stars are not bright at all
in the X-ray spectrum, which is highly energetic radiation with wavelengths much
shorter than visible light. Since normal stars hardly emit these rays, some other
process must be causing it. If a black hole is part of a binary system, its huge
gravitational field will draw away material from the outer atmospheres of the
companion star. Due to the orbital angular momentum of the pair, the gas swirling
inwards towards the black hole will form a flat disc, called an accretion disc. The
material in the disc is accelerated to very high velocities and compressed. Friction
and turbulence will cause the material to heat up to temperatures close to 10 million
Kelvin,31
high enough to emit X-rays and gamma-rays.
Figure 1: schematic view of a black
hole binary system where gas is
flowing inwards to the black hole due
to its gravity. A disc forms around the
black hole, and turbulence will heat up
the material enough for it to emit X-
rays.
Ross Mays 2010
This scenario is not limited by black holes, as it can very well happen with neutron
stars as well, which was suspected to be the case most of the time. However,
detection methods have improved over time, and right now it is very well possible to
distinguish between black holes and neutron stars regarding X-ray emission.

6. 6
1.4 Light Curve and Power Spectrum
A satellite can record the X-ray emission by counting the incoming photons and
recording their energy and time of detection. From this data, we can see how the flux
varies over time, and the spectrum of energies.
Light Curve
Figure 2: example data
taken with the
proportional counter
array from the satellite
(see section 3.2). This
light curve shows the
number of counts
(detected photons) per
second at the indicated
time.
When a black hole accretes (material is flowing from the companion towards the BH,
forming an accretion disc) it shows variations over a wide range of different
timescales. Regarding the light curve (figure 2) this means that we see many
different variations interacting with each other. By performing a Fast Fourier
Transform (FFT; see section 2.4), power spectra can be created (see section 3.4).
These spectra tell us how well a certain sine frequency is represented in the Fast
Fourier Transform of the light curve.
Power Spectrum
Figure 3: example power
spectrum, data taken from
H1743-322. The x and y axis
are on a logarithmic scale, and
represent respectively the
power (in counts per second)
and the frequencies (in Hz).

7. 7
2 Theory
2.1 Quasi Periodic Oscillations
If at the source, there would be a strictly periodic effect, it would (in the ideal case,
after taking enough observations) show in the power spectrum as a Dirac-Delta peak
at the corresponding frequency.
Figure 4: illustrated example of
how a strictly periodic effect
would show in a power spectrum.
However, in the case of Quasi Periodic Oscillations (QPOs) there is a much broader
peak, around a central frequency. This is why the effect is called “quasi” periodic,
rather than periodic.
Figure 5: illustration of a quasi-
periodic effect. There is a
distribution around a central
frequency rather than a Dirac
peak in the power spectrum.

8. 8
As stated, accreting black holes show variations over a range of timescales. Their
power spectra includes not just QPOs and their (sub) harmonics, but also broadband
noise and band-limited noise.1
The exact reason for the observed QPOs is not yet
known, though there are models describing them (see section 2.5). QPOs are not
only observed strictly in BHBs, but in neutron stars as well. QPOs are commonly
classified in three types; type A, B and C.6, 7, 8, 9, 10
Type A and B are seen in the soft
intermediate states (see section 2.3) while type C is observed in hard states. Here,
mainly type C QPOs will be discussed, as those are the observed ones. The
frequency of these QPOs are known to be correlated with the spectral properties of
the source, disc fluxes and power law, disc inner radius, photon index and disc
temperature.11, 12
The details of the physical origin is not yet clear, although there are
models (such as the Lense-Thirring model, see section 2.5) describing them.

9. 9
2.2 RMS-Flux Relation
The different timescale variations seen in Figure 2 are the result of underlying
processes and effects taking place in the accretion disc of the source. In the outer
edges of the source, long timescale effects occur. While the outer material flow falls
inwards, more volatile interactions will cause shorter timescale effects on top of the
already existing long timescale ones. This is visible in the light curve in the form of
variations with short periods stacked on top of ones with longer periods. The waves
interact multiplicatively, so that when the mean flux is higher, the amplitude of the
short timescale variations increases.
In general, the RMS-flux relation is a simple and stable relation, connecting the RMS
amplitude of variations to the mean flux with a positive linear correlation that can be
seen over a wide range of timescales, as long as the power spectra stay similar
enough.20
This relation is characteristic to short-term variability with an (almost)
stationary power-spectral shape. Changes in the longer-term RMS might be caused
by changes in the source, evolving through different spectral states over time.
However, these are different to the RMS-flux relation which seems to be a
fundamental feature of the variability process itself.
We also see that the RMS-flux relation flattens. In terms of the light curve, this
means that the fluctuation stops getting bigger as the flux increases. A conclusion
from the original paper was that the QPO just gets to a strength where it can no
longer increase with increasing flux. A possible explanation for the difference seen in
the objects could have to do with the difference in inclination. GX 339-4 is a low-
inclination system, whereas H1743-322 has a very high inclination. Why this would
affect the relation between the central frequency and the flux in the observed
fashion, is unclear. No clear trend is seen in MAXI J1659-152. The signal-to-noise
ratio was very low, meaning that the observations had too much noise to see clear
trends. In high inclination sources there seem to be stronger QPOs than in the lower
inclination ones, so perhaps the angle at which the QPO are seen is what causes the
differences. It could be that in lower inclination sources the QPO does not get strong
enough to observe this effect, or maybe if the Lense-Thirring model is correct we
generally see a bigger occultation at high inclinations, but eventually the precessing
region gets so small that we see the more emission in the dips that we do when the
region is larger. So the amplitude of the oscillation stays over all the same even
when the emission in the peaks is higher.
For the analysis, the (sub) harmonics of the QPOs had not been analysed. They
generally show the same relations as the main peak, and should not influence the
result. They do have been fitted, as that will affect the RMS values, though the
relations should remain the same regardless.
An explanation for the RMS-flux relation found in the broad-band noise of accreting
systems is the “propagating fluctuation” model.20, 22, 23, 24
This model describes that
long timescale fluctuations in the accretion rate come from processes at large radii in
the accretion disc. This could be due to random viscosity fluctuations. While
propagating inwards, shorter timescale variations coming occurring at smaller radii 25,
26, 27
interact with the already existing longer timescale ones, until the mixed effects
become visible at the innermost regions, where the X-rays are produced. This would
explain why the X-ray emission varies over such wide timescales, and a linear RMS-
flux relation because of the multiplicative interactions of variations on all timescales.
It is not clear what this would imply for the QPO though, or how it would fit into this
model.

10. 10
2.3 Black Hole States
The observation of correlated and periodic timing and spectral behaviour led to the
introduction of the canonical states for black hole systems. The definition of those
states have changed a lot over the years,7,15,16,17
as first they were determined
mainly by luminosity, but later on spectral hardness dominated. The states used here
are based on the relative location in the Hardness-Intensity Diagram (HID) and the
properties of the power spectra. Systems go through different states and behave
differently over time. Low-mass X-ray binaries with low-mass stellar companions are
transient and go through outbursts. The spectral evolution through these outbursts
can be described by the HID. The classification based on the HID is done by looking
at the energies of the emitted X-rays, thus the energy carried by each individual
photon emitted from the BHB source. If these photons carry a lot of energy they are
indicated as hard photons. Likewise, low-energy photons are soft. This separation is
arbitrary, however usually energies ranging from 4-6 keV are considered soft, and
from 9-20 keV are hard. A state is then identified as a Hard State (HS) if the ratio
high-to-low energy photons is high, meaning that if there are relatively many hard
photons emitted at some point in time, it is in the hard state. Visa versa, a flux with
relatively many low-energy photons is in the Soft State (SS). The hard state is further
characterized with an X-ray spectrum dominated by a hard power-law and strong,
band-limited noise over a wide range of timescales in the power spectrum.18
For the
soft state, there’s a soft, disk blackbody dominated X-ray spectrum and little rapid
variability. There are also intermediate states (IMS), where both the power law as
well as disk blackbody contribute significantly to the power spectrum. In these states,
the most complex variability characteristics are shown, which includes most of the
QPOs. All three states happen over a wide range of luminosity, but the very lowest
level of luminosity occurs mostly when
the system is in quiescence.
Figure 6: a hardness-intensity diagram. It
is followed chronologically in an anti-
clockwise direction along the line, starting
at the top right in the hard state. Over
time, it evolves through intermediate
states to the soft state. At times indicated
by the jet line, jets are observed from the
black hole. This track is an example of
those followed by black hole transients in
outburst, but any persistent source can
follow parts of the track.18
The grey area is
the area of the intermediate states. This
can be split in the hard intermediate state
and the soft intermediate state.
Adapted from M. Klein-Wolt and M. van der Klis, 2007.18

11. 11
For this research, data from when the system was in a hard state had been
analysed. This occurs mostly during an outburst, when the systems were easier to
detect. In the lower hard states, the QPO mean frequency was too low for this
analysis. Often, the noise level is too high in some observations in order to be able to
clearly distinguish the QPO amongst the noise. However, the QPO is the strongest in
the high hard state, which is when it’s observed best, so focussing mainly on those
regions will not compromise the results. The QPO frequency is too low in the lower
hard state, and the QPO disappears in the low state and most of the intermediate
states. So the focus is on the hard intermediate state.
2.4 Fast Fourier Transform
In order to create a power spectrum from the light curve, a Fast Fourier Transform
must be performed. The Fourier Transform (FT) is used to convert time-varying
signals to the frequency domain. A FFT is an algorithm for calculating the Discrete
Fourier Transform (DFT) and its inverse. The FFT is used to convert time to
frequency (and the other way around) much faster than other transformations. The
FFT computes the DFT with the exact same result as when the DTF would be
evaluated from its definition directly, but the FFT performs this calculation much
quicker. The DFT is defined as:
In order to evaluate this, O(N2
) operations are required, as there are N outputs Xk,
and each of them require a sum of N terms. An FFT is any method to compute the
same result, only with O(N log(N)) operations.
The fundamental idea of the FFT is to split the Fourier Transform into smaller parts.19
For example, if there would be 2N
data points, the FT is split into 2N
/ 2 chunks, each
containing only two numbers. Proceed to FT every pair and put it back together
again. This results in much less operations, as stated before. For our computation,
this means fewer round off errors in the final result (as there were fewer steps
involved), so numerical accuracy improves by using the FFT. Further, computing
time gets greatly reduced as well.
By performing this FFT on the light curve, power spectra can be created. These
spectra tell us how well a certain sine frequency is represented in the Fast Fourier
Transform of the light curve. These power spectra have then been analysed for
QPOs.
For this research, the FFT algorithm from the IDL programming language had been
used. This is based on one in Numerical Recipes (Press et. al. 1998). For the IDL
program used, see the attachments.

12. 12
2.5 Lense–Thirring precession model
The positive flux-frequency relation observed finds a possible explanation in the
Lense–Thirring precession model. This is a general-relativistic model concerning
frame-dragging where the hot, optically thin inner flow in the region closest to the
black hole is not aligned with the accretion disc and precesses on its own. The spin
axis of the black hole is misaligned with the disc allowing this to happen.
Figure 25: schematic diagram of the considered geometry. The inner flow is indicated with
grey, and the blue vectors are the angular momentum. The inner flow precesses about the
black hole, indicated with a black angular momentum vector. The outer disc (orange/red)
remains aligned with the binary partner.
Image by 29
Adam Ingram, Chris Done and P. Chris Fragile, 2009.
The QPO can be explained using this model, by stating that there would be a spot
with cluttered material, a region of higher density in the accretion disk. More material
means more things that can send out the X-rays, thus resulting in a higher flux. As
indicated, the inner region precesses. When it faces the satellite (with the higher
density region) a higher flux will be detected. If due to the precession the geometry
faces away from the observation point (in this case, the satellite) the flux diminishes.
This would cause the oscillation in flux and result in a QPO.
In order to better understand the flux-frequency relation, the inner regions need to be
considered. Due to conservation of angular momentum, material in the disc spins
faster when it’s closer to the black hole and slower when it is further away. In the
outer regions of the disc the mass concentration contributes relatively little to the
total mass in that outer orbit, as there is much more other mass in that ring of the
flow. But as the concentration moves inwards, there is fractionally more mass at
these radii, so the inner hot flow precesses faster. Note that this will result in the
entire inner body rotating faster. The extra mass contributes more and more relative
to the total mass the closer it moves to the black hole’s most inner stable orbits. The
extra mass will result in a higher flux, because there is just more material to send
photons towards us, and because the region precesses faster, the QPO frequency
goes up as well. Recall that the inner region isn’t aligned with the rest of the disc, but
rather at an angle receding to and from us, causing the oscillation in flux. This would
explain the Frequency-flux relation: relatively more mass at a closer orbit, thus
higher flux with increasing frequency. This model doesn’t tell us anything about the
power though.

13. 13
Figure 26: illustration of the
inner regions around the black
hole. The material concentration
contributes relatively more to
the total mass in the inner
orbits, making for a higher flux
and causing the whole inner
flow to precess at higher
velocity, which in turn results in
a higher frequency observed.
3 Observation Method and Equipment
3.1 Rossi X-ray Timing Explorer Satellite
For this research, the measurements and observations of the Rossi X-ray Timing
Explorer (RXTE) have been used. The RXTE was a mission sponsored by the Office
Space Science and Applications and managed by NASA’s Goddard Space Flight
Center.5
It was launched in 1996 on a Delta II rocket. Its primary objective was to
study temporal and broad-band spectral phenomena with regard to galactic and
stellar systems having compact objects. This includes white dwarfs, black holes and
neutron stars. Its instruments span an energy range of 2-200 keV, and can study
timescales from microseconds to years. The design also facilitates multifrequency
observations.
Figure 7: animated picture of the Rossi X-
ray timing Explorer. It orbits the Earth at
an altitude of 580 km, with an orbital
period of 90 minutes and an inclination of
23 degrees.13
Image by Nasa.gov

14. 14
The satellite is equipped with three major instruments. First there’s the Proportional
Counter Array (PCA). This instrument is made from five large proportional counters
with anticoincidence features. Electronic anticoincidence is a method to lower
unwanted background events. An event that we want to study (in this case, a high-
energy interaction due to a gamma ray) occurs, and is then detected by the
electronic detector. This creates an electronic pulse, but the desired events get
mixed up with a large number of other events created by background processes, and
the detector can not distinguish between relevant events and ones created by
background. The anticoincidence feature is an arrangement of other photon
detectors to intercept the unwanted background events, producing simultaneous
pulses that can be used with fast electronics to reject the unwanted background. A
mechanical hexagonal collimator (a device that narrows the incoming beam of
photons to become more aligned in a specific direction) provides 1 degree (FWHM)
collimation (adjusted to the line of sight of the optical device). This means that
sources as faint as 1/1000 of the Crab nebula could be detected within just a few
seconds.
The second instrument is a High-Energy X-ray Timing Experiment (HEXTE). This
features a large area and low background with a 1 degree field of vision co-aligned
with the PCA field of vision. Eight phoswich detectors (detectors developed to detect
very low-intensity X-rays) are arranged in two clusters, each of which rocks on and
off the source. This, together with automatic gain control for each of the eight
detectors together, makes for a well determined background. This means that it can
take spectral measurements of a faint source at 100 keV in about 24 hours.
Last, the RXTE is equipped with an All Sky Monitor (ASM). The ASM alerts RXTE to
flares and changes of state in X-ray sources. It has three rotating Scanning Shadow
Cameras (SSC) that scan about 80% of the sky in 1.5 hours. The cameras measure
intensities of about 75 known celestial sources per day and can measure the position
of a new source with about 3` precision.

15. 15
3.2 The Proportional Counter Array
In this experiment, the data from the PCA has been used. It consists of five large
detectors, spanning a total net area of 6250 cm2
. Every detector is a bigger (about
50%) version of the HEAO-1 A2 HED (an earlier X-ray experiment) sealed detector.
These are filled with xenon gas and are capable of getting low background due to
efficient anti-coincidence schemes. These include side and rear chambers and
propane top layers. The three signal detection layers have xenon of 3,6 cm thick at a
pressure of 1 bar. As a quench gas (a gas to ensure that each pulse discharge
terminates), methane has been used. There are aluminized Mylar windows of 25 μm
on the front window, and a window separating the xenon/methane chambers. The
propane layer can also be used as a signal layer in the energy range of 1-3 keV.
Figure 8: the proportional counter array. It
measures photons with energies between 2
and 60 keV, and consists of five X-ray
detectors filled with gas. The background noise
is kept low due to anti-coincidence chambers
on four sides of the detection chamber. The
energy is calibrated with an on-board
radioactive source.
Figure by H. Bradt, M. Halverson et al.5
3.3 Experiment Data System
The PCA X-ray data is pre-analysed by the Experimental Data System (EDS) before
it sends down the data to the ground, as the telemetry bandwidth is limited.
Telemetry is the highly automated communications process by which measurements
are made and other data collected at remote or inaccessible points and transmitted
to receiving equipment for monitoring.14
The EDS bins data and analyses it
according to criteria that can be changed for each observation. It is also in control of
the ASM rotation, and processes the ASM data. Further, it is equipped with six Event
Analysers (EA) that can independently analyse the entire PCA data stream at the
same time.

16. 16
3.4 Data manipulation
After obtaining several light spectra for each observation, power spectra were
created. This is done by manipulating the spectra data taken by the satellite. The
light curves have been segmented into two-second pieces each.
Figure 9: the light curves are segmented
into pieces of two seconds.
Each of those segments has then undergone a FFT. This means that each segment
of the light curve got decomposed into sine waves and their respective strengths and
frequencies.
Figure 10: the FFT of each two second
segment is taken.
The mean counts per second value was taken for each segment, and then the
segments got binned together according to flux. If the satellite observations were
long enough to create at least four power spectra in the final output, then those
observations have been analysed.

17. 17
3.5 Data Analyses
After creating the power spectra as described in section 3.4, they have been
analysed for QPOs. In order to measure the strength of the QPO peaks, the spectra
have been fitted using a broken power law to fit the continuum, and Lorentzians to fit
the QPO peak and (sub) harmonics. A Lorentzian distribution was chosen over a
Poisson one due to the former better fitting the wings of the peak, and not because
the physical interpretation of the QPO peak would call for a Lorentzian rather.
Figure 11: the example data of H1743-322.
The continuum is fitted using a broken power
law (red), and the QPO and harmonic have
been fitted with Lorentzians (green and blue
respectively).
The strength of the QPO is then indicated by the area under the peak. To calculate
the integrated area, a custom program named ana_ascii had been used. The extra
area from the runaway to the sides of the QPO peak is neglectable, as the axes are
scaled logarithmically and the extra contribution off the sides of the X-axis is not
influencing the result. Further, for the relations that have been focussed on (Root
Mean Square (RMS)-Flux relation and how the central frequency shifts over time),
the absolute and exact strength of the peak is not as important as how trends
change over time.
The RMS of the integrated area is taken next (for the program, see Attachments)
and then plotted against the flux. This way, there is a clear view of the relation (if
any) between the flux and the strength of the QPOs.

18. 18
4 Results
4.1 RMS-Flux Relation
The relation between the RMS and the flux had been analysed for the objects MAXI
J1659-152, GX 339-4 and H1743-322 to see how they would compare to the first
time this relation had been observed in QPOs, in object XTE J1550-564.1
The
gradient of the RMS-flux relation for each observation flattened with increasing
frequency as well in XTE J1550-564.
Figure 12: RMS-Flux plot from the observations of GX 339-4.
Figure 13: RMS-Flux plot from the observations of MAXI J1659-152.

19. 19
Figure 14: RMS-Flux plot from the observations of H1743-322.
4.2 Central Frequency-Flux Relation
The relation between the central frequency of the QPO and the flux has also been
analysed.
Figure 15: central frequencies of the QPOs plotted against the flux for each observation of
object GX 339-4.

20. 20
Figure 16: central frequencies of the QPOs plotted against the flux for each observation of
object MAXI J1659-152.
Figure 17: central frequencies of the QPOs plotted against the flux for each observation of
object H1743-322.

21. 21
4.3 Mean Frequency – RMS-Flux Gradient Relation
The gradient of the RMS-flux relation had been plotted to compare it with the earlier
studied1
XTE J1550-564. The gradient was calculated with a custom IDL program
(see Attachments).
Figure 18: the gradient of the RMS-flux relation plotted against the mean QPO frequency of
the relevant observation for GX 339-4.
Figure 19: the gradient of the RMS-flux relation plotted against the mean QPO frequency of
the relevant observation for MAXI J1659-152.

22. 22
Figure 20: the gradient of the RMS-flux relation plotted against the mean QPO frequency of
the relevant observation for H1743-322.
4.4 Intercept from Linear fit of the RMS-Flux relation
The intercept of the linear fit to the RMS-flux relation had been analysed in order to
see whether it is consistent with zero in all cases or not, within error. This could tell
whether the relation is directly proportional or just shows that there is a relation. The
intercept had been calculated using a custom IDL program (see Attachments).
Figure 21: plot of the intercept values with error of GX 339-4. For this object, 9 linear plots
had been analysed, as indicated by the x-axis.

23. 23
Figure 22: plot of the intercept values with error of MAXI J1659-152. For this object, 8 linear
plots had been analysed, as indicated by the x-axis.
Figure 23: plot of the intercept values with error of H1743-322. For this object, 11 linear plots
had been analysed, as indicated by the x-axis.

24. 24
5 Discussion
5.1 The RMS-Flux Relation Analyses
In GX 339-4 we see that as the flux increases, so does the RMS. This relation
seems to be positively linear, as expected. However, in the case of H1743-322, there
are some negative relations too, as well as relations within a single observation that
don’t seem to be linear at all. This doesn’t seem to be the result of a problem with
the data nor due to a poor signal to noise ratio.
As for the mean frequency – RMS-flux gradient relation, we can see with H1743-322
that the gradient flattens (gets closer to zero) as the frequency increases. This is
consistent with what had been observed in a previous study (see 1
L.M. Heil, S.
Vaughan, P. Uttley – 2010) with object XTE J1550-564.
The QPO frequency at which the gradient of the RMS-flux relation becomes zero is
at about the same frequency for both objects as well, around 5,5 Hz. This could
mean that there is something fundamental occurring at a certain distance from the
black hole.
Figure 24: comparison of the
gradient-frequency plot of
XTE J1550-564 (by 1
L.M.
Heil, S. Vaughan, P. Uttley –
2010) and H1743-322. Not
only is the trend similar, but
so are the gradient values.
Also, at around 5,5 Hz the
gradients become zero in
both objects. This is a
remarkable similarity.
Top plot by 1 L.M. Heil, S. Vaughan, P.
Uttley – 2010.

25. 25
However, despite H1743-332 showing remarkably similar characteristics, GX 339-4
shows no obvious relation with the frequency, within errors. Apart from inclination,
there should be no real differences between the object. Maxi J1659-152 again shows
too much noise to support any sensible conclusions.
5.2 The Central Frequency-Flux Relation
In the case of H1743-322, it is very clear in some observations that as the flux
increases, so does the central frequency of the QPO. There are models such as the
Lense–Thirring precession model 28, 29
describing this phenomenon (see section
2.5). However, with object GX 339-4, the relation seems the opposite way around.
There is a negative correlation, as when the flux goes up, the central frequencies of
the QPOs decrease. Again, no trend is seen in MAXI J1659-152. The signal-to-noise
ratio was just too low.
The intercept of the linear fit to the RMS-flux relation had been analysed for
consistency with zero. This could tell whether the relation is directly proportional or
just shows that there is a relation. Nearly all points are zero within three sigma. This
would indicate that the frequency-flux relation is directly proportional.
6 Conclusion
There doesn’t always seem to be a linear RMS-flux relation in the QPO. This relation
is obeyed only in the case of H1743-322. This object is also the one showing a
positive frequency-flux correlation, and the gradients of the RMS-flux relation behave
similar (both qualitatively and quantitatively) to what was observed earlier1
in XTE
J1550-564. The flattening in the RMS-flux relation may be because the QPO just
gets to a strength where it can no longer increase with increasing flux, though this
has not been observed with all objects. The frequency-flux correlation might be
explained with the Lense–Thirring precession model. However, GX 339-4 shows
almost opposite correlations. MAXI J1659-152 had too much noise to say anything
conclusive. This is new behaviour, and further research is needed. A possible
explanation for the differences might have to do with the inclinations with which the
objects are observed. H1743-322 is a high-inclination source, while GX 339-4 is low-
inclination. Further research could include looking at objects with high inclinations
and check if they all exhibit the same behaviour, and ditto for low-inclination objects.
The power spectra can also be looked at next, to see whether they differ per
observation.
Acknowledgements
With sincere thanks to supervisor L. Heil and secondary supervisor M. van der Klis
for the guidance and constant supervision as well as for providing necessary
information regarding the project, and also for the support in completing the project.

26. 26
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27. 27
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28. 28
Attachments
IDL Program

29. 29

30. 30