Bachelor Thesis
GRS 1915+105: rms-flux relation of the
quasi-periodic oscillations and broadband
noise
Author:
Monica van Santbrink 10202889
Supervisor: Lucy Heil Second Supervisor: Michiel van der Klis
University of Amsterdam Faculty of Science Anton Pannekoek Institute
Report Bachelor Project Physics and Astronomy, size 15 EC, conducted between 31-03-2014 and 18-07-2014
Abstract
The rms-flux relation for the QPO and the broadband noise of GRS 1915+105 have been analysed.
The rms-flux relation for the QPO has been compared to that of XTE J1550-564. The results hint that their rms-flux relations could behave in a similar manner. The rms-flux relation for GRS 1915+105 displays one constraint positive gradient below 3 Hz. At frequencies around 3 Hz the gradient becomes clearly flat and then possibly negative, but the high frequencies cannot be seen in the used observations. The turn over seems to be at a lower frequency than for XTE J1550-564. What is more, the rms has been corrected for a frequency dependent filter that might exist, this revealed that the rms-flux relations of the type C QPO in GRS 1915+105 are not linear. By using Kendalls tau the results hint that the behaviour still might be similar. The rms-flux relations for the broadband noise were all positive linear, this agrees with results from other black hole binaries. Moreover, the broadband noise and QPO differ in terms of their rms-flux relation, this hint that they might have a different formation mechanism and could possibly be explained by Lense-Thirring precession.
Samenvatting
In het heelal komen objecten voor die zeer compact zijn, dat wil zeggen dat er veel massa in een relatief klein gebied zit. In het meest extreme geval kunnen deze objecten zo zwaar zijn dat zelfs licht er niet uit kan ontsnappen, dan wordt er gesproken van een zwart gat.
Sommige zwarte gaten komen voor in dubbelstersystemen, dat wil zeggen dat een zwart gat en een begeleidende ster, simpelweg de begeleider genoemd, om elkaar heen draaien. Hierbij kan massa worden overgedragen van de begeleider naar het compacte object. Bij dit proces wordt een schijf, de zogeheten accretieschijf, gevormd. Bovendien wordt er r¨ontgenstraling uitgezon-den, deze straling komt van het binnenste van de accretieschijf. In dit geval spreekt men dan van een r¨ontgendubbelster.
Figure 1: Een illustratie van een r¨ontgendubbelstersysteen waarbij massa wordt overgedragen en er een accretieschijf wordt gevormd (NASA)
Een merkwaardige r¨ontgendubbelster is GRS 1915+105, in dit systeem is de accretieschijf veel groter dan in andere gevallen. Daarnaast is de verandering van de r¨ontengenstraling in de tijd niet zoals die wordt waargenomen bij andere systemen.
Zowel bij het analyseren van de straling van GRS 1915+105 als andere r¨ontgendubbelsterren wordt op bepaalde momenten een quasi-periodieke oscillatie (QPO) geobserveerd, wat zichtbaar is als een piek in het spectrum. Daarnaast wordt een brede component waargenomen.
Hoe de QPO’s precies ontstaan, is niet duidelijk. Er zijn grofweg twee modellen waarbij een van de modellen een zelfde oorzaak voor de QPO en de brede component voorspelt, terwijl de andere component een andere oorzaak aanwijst.
In dit onderzoek wordt de uitgezonden straling van GRS 1915+105 onderzocht. Hierbij wordt naar twee aspecten gekeken. Enerzijds wordt de QPO vergeleken met die van een andere r¨ontgendubbelster, XTE J1550-564. Anderzijds wordt het gedrag van de QPO vergeleken met die van de brede component om een uitspraak te kunnen doen over het gedrag van beide com-ponenten, wat informatie zou kunnen opleveren over het ontstaan van QPO’s.
Uit de geanalyseerde data komt naar voren dat de QPO’s waargenomen in GRS 1915+105 hetzelfde gedrag lijken te vertonen als in XTE J1550-564. Bovendien is in beide gevallen gecor-rigeerd voor een filter die eventueel effect zou kunnen hebben, na deze correctie suggereren de data nog steeds dezelfde conclusie.
Het andere aspect van het onderzoek, het vergelijken van de QPO en de brede component, laat duidelijk zien dat deze twee van elkaar verschillen. Dit kan er op duiden dat ze een andere fysische oorzaak hebben.
Abstract i
Contents iii
1 Introduction 1
1.1 Black hole binaries . . . 1
1.1.1 Accretion . . . 1
1.1.2 Behaviour in X-ray emission . . . 2
1.2 GRS 1915+105 . . . 4
1.3 Power spectra . . . 4
1.4 Quasi-perodic oscillations . . . 5
1.4.1 Origin . . . 6
1.5 The rms-flux relation . . . 7
1.5.1 Rms-flux relation for the QPO . . . 8
1.5.1.1 Frequency dependent filter . . . 9
1.6 Motivation . . . 9
2 Data Analysis 11 2.1 Rossi X-ray Timing Explorer . . . 11
2.2 Data selection . . . 11
2.3 Power spectra . . . 12
2.4 Fitting the power spectra . . . 12
3 Results 15 3.1 Rms-flux relation of the QPO . . . 15
3.2 Rms-flux relation for the broadband noise . . . 18
4 Discussion 22 4.1 Comparison between GRS 1915+105 and XTE J1550-564 . . . 22
4.1.1 Filter . . . 23
4.2 Comparison between broadband noise and QPO . . . 29
5 Conclusion 30
References 31
Appendix 33
Introduction
1.1
Black hole binaries
When the X-ray binary Cygnus X-1 was studied in both the X-ray and the optical band, the observations hinted that this was the first detected black hole (Bolton 1972). More of these X-ray binary systems are now known, these systems contain a compact object with a mass too heavy to be a neutron star (M > 3M ) along with a secondary stellar component and are
labeled as black-hole binaries (BHBs).
1.1.1 Accretion
In BHBs the primary component, the black hole, is accreting mass off the secondary star by Roche-lobe overflow. To explain this accretion mechanism it is useful to describe the system in a coordinate system that is rotating together with the objects in it. In this frame next to gravity there is an additional centrifugal force. As a result, the potential φ consists of three terms as can be seen in equation 1.1, where the first two terms represent the gravity terms and the third the centrifugal force.
φ = − GM1 r − r1 − GM2 r − r2 − 1 2(Ω × r) 2 (1.1)
From this formula it is clear that the potential φ depends on the position vector in the center of mass frame r and the two position vectors r1 and r2, the respective masses of both objects
in the binary and the period P, since Ω = 2πP . The force on a gas particle is simply given by F = m∇φ. The particle will not rotate in the system if F = 0 or in other words if ∇φ = 0, the latter corresponds with the maxima/minima and the saddle points of the potential. These points are labeled Lagrangian points and can be seen in the schematic Figure 1.1 (L1, .., L5)
Figure 1.1: Schematic illustration of the five Lagrangian points (Figure on the courtesy of J. Vink)
For the accretion mechanism Roche-lobe overflow, L1 is the most important point. As the
companion star expands due to stellar evolution it can fill its Roche-lobe up to L1. Since there
is no force at this point, mass can easily flow from the companion star towards the compact object. This is called Roche-lobe overflow.
The matter that is transferred from the secondary star to the black hole, will have angular momentum since the objects are rotating around each other. As a result the incoming mass cannot go directly into the black hole, instead it will rotate around the compact object and an accretion disk will be formed. This disc is geometrically thin and optically thick (Pottschmidt. et al 2003)
1.1.2 Behaviour in X-ray emission
X-ray emitting BHBs which have a secondary star with low mass, for example spectral type K or M, are called low mass X-ray binaries (LMXBs). They go into outbursts which typically last on the order of months (Remillard & McClintock 2006). During such an outburst the accretion flow changes and as a result the X-ray emission changes. The variation in the flux of this radiation on short time scales, on the order of seconds, is called the variability.
The energy spectral evolution of these objects can be displayed in a hardness-intensity diagram (HID). This diagram has on its horizontal axis the hardness which is the ratio of hard energies
over soft energies, where hard energy corresponds to high energy, and on its vertical axis the total count rate.
Figure 1.2: Hardness intensity diagram (Belloni, Motta & T. Mu˜noz-Darias 2011)
The BHBs follow counterclockwise a q-shaped path in the HID. This behaviour is explained in terms of spectral states and the transitions between them, where the definition of the states is based on spectral and timing properties. These spectral states are linked to different compo-nents in the accretion flow, which consists of a hot, optically thin corona and a cooler optically thick disc.
Generally at the beginning of an outburst, the BHB starts in the Low-Hard State (LHS), this is the vertical branch on the right side of the HID (Belloni, Motta & Mu˜noz-Darias 2011). As implied by its name, the energy spectrum is hard and its shape is a power law. This hard spectrum is due to thermal Comptonisation, which takes places in the optically thin corona. After the hard state, the BHB moves in the diagram to the Hard-Intermediate State (HIMS), the upper horizontal branch. In this state the energy is softer than in the previous hard state. At low hardness a thermal accretion disc component starts contributing in its spectrum. As the hardness decreases to a certain value, the object enters the Soft-Intermediate State (SIMS). This state is similar to the HIMS when comparing their energy spectra, however the timing properties are different. From here the BHB evolves to the last state, namely the High-Soft State (HSS), this is the left vertical branch in the HID. The energy that is emitted during this state is soft and black-body like, coming from the optically thick disc. From this last state the
object moves in the HID on the lower horizontal branch back to its starting point, the LHS, and will start again tracing the q-shaped path.
1.2
GRS 1915+105
In 1992 the black hole binary system GRS 1915+105 was discovered as it went into outburst with the X-ray all-sky monitor WATCH on board GRANAT, an initially Soviet (later Russian) space observatory. The binary system contains a spinning black hole, with mass 14.0 ± 4.4 M
and a K-M III giant star with mass 0.8 ± 0.5 M (Harlaftis & Greiner 2004). In this system
mass-exchange takes place, mainly by Roche-lobe overflow. The object is located at a distance of ∼11 kpc (Yan et al. 2013). It was immediately clear that the source was very variable in the X-ray band (Belloni 2002). However as well as X-ray emission, the system produces radio jets and emits radiation in the infrared.
When it was discovered it was labeled as a transient black hole binary, however the source has been in outburst since its discovery and has extraordinarily large variability. As stated in Sec-tion 1.1.2, the duraSec-tion of an outburst of a typical black hole binary is on the order of months. The fact that the outburst of GRS 1915+105 is significantly longer than the other BHBs could be explained by its large binary separation (Deegan, Combet & Wynn 2009). The result of this large separation is that a large accretion disc can be formed around the black hole, therefore a enormous amount of mass is available to preserve the outburst.
Moreover, not only is the duration of the outburst unusual, the energy spectral evolution is peculiar too. The source does not follow the characteristic q-shaped path in the HID and its evolution cannot be defined using the canonical states. However its behaviour can be classified in twelve classes, which can be brought back to three basic states labeled A, B and C (Reig, Belloni & van der Klis 2003).
1.3
Power spectra
Much of the information that is detected from an accreting black hole binary is contained in the rapid variations in the X-ray intensity, the variability. For the time series measured from these objects it is typical to use a power-density spectrum (PSD) as an analysis tool. This spectrum describes the average variance per unit frequency. Such a spectrum is obtained by splitting the measured light curve of the object in N time segments and then performing a fast Fourier transform on each segment. By squaring the amplitude of the latter and taking the average over the N segments, one gets the final PSD. The integrated area of the spectrum is equal to
the variance, by taking the square root of this, one gets the root mean square (rms).
In Figure 1.3 eighteen power spectra of six different BHB are displayed. The power spectra are from different spectral types, numbers 1-7 belong to the hard state, followed by the intermediate states and numbers 17 and 18 belong to the soft state.
Figure 1.3: Power spectra in different spectral states, starting in the hard state through the intermediate states and ending in the soft state (Heil, Uttley & Klein-Wolt 2014)
It can be seen from Figure 1.3 that in the hard state there is substantial variability over the whole frequency range, whereas for the soft states the variability is clearly less abundant. The important state for this research is the HIMS, this state is in Figure 1.3 labeled by the numbers eight up to twelve. In these power spectra the most unmistakable feature is a narrow peak, which is known as quasi-periodic oscillation (QPO). Next to this characteristic component, the power spectra consists of a broad component which is referred to as the broadband noise.
1.4
Quasi-perodic oscillations
QPOs are, as their name suggests, oscillations within a time series who’s frequencies change very slightly within an observation. Generally the QPOs are classified in two groups, the ones having a frequency ν < 30 Hz are called low frequency QPOs and the ones having frequency ν > 40 Hz are referred to as high frequency QPOs (Remillard & McClintock 2006). The low-frequency QPOs are regularly observed with higher peaks which are harmonically related to the fundamental peak.
What is more, the low-frequency QPOs (LFQPOs) are classified in three groups labeled A, B and C. Type A and B are associated with the transition from the HIMS to the SIMS, whereas type C is associated with the two states HIMS and LHS (Soleri, Belloni & Casella 2008). Despite the fact that different black hole binaries significantly differ in their masses, the frequencies at which the type C QPOs appear, are consistent.
1.4.1 Origin
The origin of the QPOs is poorly understood. In general there are two models that suggest a possible formation mechanism. In the first model a mechanism in the accretion flow could cause the QPO. In this model a mode becomes excited for some reason and this results in an oscillating signal being emitted from the accretion flow. This suggests that the broad component, the broadband noise, and the QPO have the same underlying formation mechanism.
The other model uses Lense-Thirring precession, a gravitomagnetic frame-dragging effect, to explain the QPO. In this case the origin is not a result of some action in the accretion flow, but the accretion flow itself. This is illustrated with Figure 1.4.
Figure 1.4: A suggested model to explain the origin of QPO is Lense-Thirring precession (Ingram, Done & P.C. Fragile 2009)
Here the accretion disc is represented by the red/orange features, the corona is the grey torus and the black hole is represented by a black dot. The two vectors display the angular momentum, the blue one of the accretion flow and the black one of the black hole. This model describes that the orbits in the corona precess around the black hole and consequently the whole flow rotates creating the QPO. The orbits precess here as a solid body due to the black hole torque which is misaligned with the disc, as is illustrated in Figure 1.4 (Ingram, Done & Fragile 2009). Unlike the previous model, this suggests that the broadband noise and the QPO have different formation mechanisms.
1.5
The rms-flux relation
Uttley & McHardy (2001) discovered that Cygnus X-1 displays a positive linear relation between the amplitude of the broadband noise and the flux, this was called the rms-flux relation. This relation is displayed in Figure 1.5, where the flux is plotted on the horizontal axis and the rms on the vertical axis.
Figure 1.5: The positive linear rms-flux relation for the broadband noise in Cygnus X-1 (Uttley & McHardy 2001)
After this the light curves of nine more BHBs were studied to look for this correlation and in these objects the linear relation was also found consistently within a large sample of observations (Heil, Vaughan & Uttley 2012). The linear relation between these two physical quantities can be explained by the propagation fluctuation model. This model describes the accretion rate in the accretion disc by stating that the long time-scale fluctuations appear at large radii and propagate inwards towards smaller radii as material loses angular momentum within the disc. As the material moves inwards short time-scale fluctuations are superimposed on longer timescale ones, including the fluctuations of the innermost part of the disc where the X-rays are produced. This predicts a positive linear relation between the rms and flux if the fractional amplitude of the variations in the accretion rate is not dependent on the actual accretion rate (Uttley & McHardy 2001). This is illustrated in Figure 1.6.
Figure 1.6: Propagation fluctuation model: long-time scale fluctuations at large radii move towards the blackhole and shorter-time scales fluctuations are superimposed on them
In this figure the blue feature represents the accretion disc and the half black circle represents the black hole. The variability on long time scales from the outer radii propagates inwards towards the black hole. Since the superimposing of the shorter time scales variability on the long-time scales can be viewed as a multiplication in the final emission light curve, the peaks display considerably more variability than the dips.
1.5.1 Rms-flux relation for the QPO
In 2011 it was found by Heil et al. (2011) that the variability of the QPO (rms) in XTE J1550-564 displays a flux dependent relation. This relation was not only confirmed to be linear, but also frequency dependent. In the range ∼1-5 Hz the gradient of the function was positive linear, at around 5 Hz the relation was flat and at higher frequencies up to 10 Hz, the gradient became negative. This is illustrated in Figure 1.7, where the flux is plotted on the horizontal axis and the rms on the vertical axis.
Figure 1.7: The rms-flux relation for the QPO of XTE J1550-564 (Heil, Vaughan & Uttley 2011)
1.5.1.1 Frequency dependent filter
On long time scales, it has been found that a frequency dependent filter, which reduces the rms of the power spectra at high frequencies, may exist: the fractional rms, which is the absolute rms normalised by the flux, of all components in the power spectra rise until ∼3 Hz, as the frequency continues to rise the fractional rms falls off steeply (Pottschmidt et al. 2003). In Figure 1.8 the factional rms of the broadband noise components have been plotted against the frequency for Cygnus X-1.
Figure 1.8: Fractional rms of the broadband noise steeply falls off at frequencies higher than ∼3 Hz (Pottschmidt et al. 2003)
Therefore Heil et al. (2011) investigated as to whether the change of the gradient sign in XTE J1550-564 is the effect of the increasing QPO frequencies with flux. This resulted in the conclusion that the rms-flux relation in the QPO is still frequency dependent but the gradient of this relation tends to zero at higher frequencies, instead of becoming negative.
1.6
Motivation
In this project two aspects of the variability of GRS 1915+105 will be studied.
Firstly, the behaviour of the rms-flux relation of the QPO will be compared to the rms-flux relation of the black hole binary XTE J1550-564. In order to do this I present in Section 3.1
the rms-flux relation for the QPO. This relation will be fitted and the frequency dependence will be evaluated.
Secondly, since the origin of the quasi-periodic oscillations is not known, it is interesting to compare the the rms-flux relations of the QPO to the broadband noise. From this it can be concluded whether the behaviour of both components are similar or not and from this there might be an indication about the formation mechanism of the QPO. In order study this, I present in Section 3.1 the rms-flux relations for the QPO and the frequency dependence of this relation. In Section 3.2 the rms-flux relation for the broadband noise and its frequency dependence are presented together with the results of the QPO in order to be able to compare them.
Data Analysis
2.1
Rossi X-ray Timing Explorer
The observations used were taken by the Proportional Counter Array on board Rossi X-ray Timing Explorer (RXTE). RXTE was launched at the end of 1995 from the Kennedy Space Center of NASA. It was designed to observe the variability of X-ray emission for sources with moderate spectral resolution on different time scales, from microseconds to months. Initially RXTE was meant to observe for two years, however it turned out to be more than 16 years. The Proportional Counter Array (PCA) consists of five proportional counters. A proportional counter is basically a box made out of a material that admits X-rays, it is filled with gas and contains a high voltage wire. If an incoming X-ray has sufficient energy, it will, by means of the photoelectric effect, eject an electron from the atoms in the gas. This is called the primary electron. If the energy was large compared to the amount that was needed to eject the electron, the left over energy is used by the electron to ionize other gas atoms. As a result a collection of secondary electrons is formed. The population of both generations of electrons move towards the high voltage wire, the anode, and accelerate due to the electric field. During this process the energy increases and if their energies reach a certain value, they will ionize more gas atoms. This results in a rapidly increasing electric field, which is measured by the high voltage wire (Jahoda et al. 1996).
2.2
Data selection
A sample consisting of 22 observations was selected for the initial analysis, the ObsIDs of which are listed in the Appendix. The energy range of the data was 2-13 keV and it was studied over the frequency range 1-10 Hz. All observations had to be long enough to contain sufficient segments of light curve to measure the rms-flux relation. As will be discussed in Section 2.3,
the light curves were split up in to segments of 3 seconds, this limited the usage of PSDs having frequencies below 0.3 Hz.
The observations used in Heil et al. to study the QPO of XTE J1550-564 were taken in the hard intermediate State. Since the motivation is partly to compare this object to GRS 1915+105, it was ensured that the observations were in the χ state, which is the only similar states between the two objects.
2.3
Power spectra
As discussed in Section 1.3, a common tool for doing analysis on X-ray binaries is a power spectrum (PSD). In order to convert the data to a power spectrum, the measured light curves were split up in to segments of 3 seconds. Then on each segment a Fast Fourier transform was performed. One gets the PSD for each segment by squaring the previous result. To get the total PSD, the average of the power spectra of the segments was taken, where the averaging was done according to flux and it was ensured that each flux bin contained 300 PSDs. The amount of 300 PSDs per flux bin was needed in order make the scatter approximately Gaussian so they could be fit with a chi-squared fitting mechanism.
The power spectra for each observation had an absolute normalisation, meaning that the in-tegrated area of the PSD is equal to the variance. The measured light curves have however a stochastic nature, meaning that the power values are not the true values, but scatter around them according to a χ2 distribution with two degrees of freedom. This effect can be reduced by binning the data (Uttley et al. 2014).
2.4
Fitting the power spectra
The power spectra were fitted with Lorentzian components, each spectrum required a broad feature for the broadband noise and at least one strong narrow feature for the fundamental QPO. One of the fitted power spectra is displayed in Figure 2.1.
Figure 2.1: Power spectrum which contains a broadband noise component, QPO and harmonic
Some of the spectra needed an additional narrow feature, as can be seen in Figure 2.1. This ex-tra component is the harmonic. The narrowness of these components is defined by the Q-value, which is the peak frequency divided by the full width at half-maximum of the Lorentzian. The QPO was defined being the feature having the highest rms and a large Q-value. The Poisson noise differs physically from the broadband noise, since the broadband noise is a real signal whereas the Poisson noise is not, the latter is a noise level. Poisson noise is a form of uncer-tainty due to the discrete particle nature of radiation. Poisson noise leads to flattening of the PSD at high frequencies, in order to correct for this noise level one can subtract the constant Poisson level from the observed PSD (Uttley et al. 2014).
Within each observation the same amount of features was needed to produce a good fit, the latter meaning that most of the fits have a reduced chi-squared of 1.0 and 1.1, some of them were slightly lower and others slightly higher. The values lower than 1 show that the data are slightly over-fitted, whereas the values bigger than 1 indicate that the fit has not fully cap-tured the data. However it was ensured that the QPO was fitted properly, the imperfections of the fit were on the broadband noise, these fits were not used as will be explained in Section 3.2
The low frequency PSDs were often difficult to fit, this is due to the fact that the light curves were split up into segments of 3 seconds. As a result, some of the QPOs ended up being binned into two flux bins instead of one and therefore the peak was poorly constrained and could not be fit nicely with a Lorentzian.
Results
3.1
Rms-flux relation of the QPO
For each fit and each component, the Lorentzian properties, namely the integrated power, the Q-value and the peak frequency, were recorded. The integrated power of the QPO was converted to the root mean square (rms) in absolute units by taking the square root. The errors of the variance were divided by two times the rms in order to calculate the errors of the rms: σerror = σ
2 error
2σ . Here σ represents the rms and σ2 the variance.
The rms was used to evaluate the rms-flux flux relation for each observation. This relation was then fitted with a linear function, which was chosen to avoid over fitting. As a check the p-value was calculated, this result supported the linear function. The p-value is a probability that shows how extreme the observed result is, assuming the null hypothesis is true. A small p-value signifies that the data is not consistent with the null hypothesis. In this case the null hypothesis is the statement that each rms-flux relation can be fitted with a linear relation. The p-values all being bigger than 0.1 supported this null hypothesis.
In the Figure 3.1 the rms-flux relation for the QPO is presented. The horizontal axis is labeled as the flux, and the vertical axis as the rms. The units are in cts/s/PCU, this means counts per second per detector.
Figure 3.1: Rms-flux relation for the QPO
As can be seen the rms-flux relation is not easily described, it behaves in a complex manner. To investigate its frequency dependence, the gradients of the linear fits are plotted in Figure 3.2. The horizontal axis represents the frequency and the vertical axis the gradient. The gradients are plotted at the mean frequency of the QPO of an observation.
Figure 3.2: The gradients of the linear fitted rms-flux relation for the QPO
In Figure 3.2 it appears that the rms-flux relation for the QPO may be frequency dependent. There is one point with constrained gradient which is positive. At around ∼3 Hz the gradients are mostly flat within 2 sigma errors and as the frequency rises the sign of the gradients is constrained to be negative.
However, there are two points at ∼4 Hz that have a constrained positive gradient, these points belong to the observations 90105-07-01-00 and 90105-07-02-00. They will be discussed in Chap-ter 4.
In Figure 3.3 are next to the gradients of the rms-flux relation for the QPO, the gradients of the linear relation for the harmonic displayed. The gradients of the harmonic have not been plotted at the frequency of the harmonic itself, but at the frequency of the QPO and are shown in red.
Figure 3.3: The gradients of the linear fitted rms-flux relation for the QPO (black) and harmonic (red) plotted against the frequency of the QPO
Since not all observations needed a harmonic to fit the power spectra as discussed in Section 2.4, there are less harmonic points. The errors on the harmonic are bigger than for the QPO, because the rms of this component is harder to measure. It can be seen that the gradient of the harmonic appears to behave in a similar fashion as the QPO.
3.2
Rms-flux relation for the broadband noise
As stated in Chapter 2 the fits to the broadband noise were not used since they did not always capture the shape of the noise. In order to study the rms-flux relation for the broadband noise, first the rms was calculated. This has been done by reading out the power values of the frequency bins up to 30 Hz where each bin size was 1/3 Hz. Thus a full spectrum was created, including the QPO component. Therefore the QPO was modeled for each observation and subtracted off each spectrum. This is illustrated in Figure 3.4. Here the red line represents the full spectrum, the modeled QPO is displayed in green and the remaining spectrum, the broadband noise component, is represented by the blue line.
Figure 3.4: Full power spectrum (red) from which a modeled QPO (green) was subtracted and the result is the broadband noise (blue)
The remaining power values were summed in the range 0 - 30 Hz and multiplied by the bin size 1/3. This value was then corrected for the Poisson noise in order to get the variance. By taking the square root of the variance, the rms was found. To calculate the error on the rms, formula 3.1 was used. In this formula ∆f represents the binsize which equals 1/3, M is the 300 power spectra used to reduce the stochastic effects as discussed in the section 2 and yj is the power
value. σerror = ∆f 2σ√M s X j < yj2> (3.1)
In Figure 3.5 the rms-flux relation of both the broadband noise, in blue, and the QPO, in black, have been plotted, to be able to compare them. The vertical axis represents the rms and the horizontal axis the flux.
Figure 3.5: The rms-flux relations for the QPO (black) and broadband noise (blue)
From Figure 3.5 it is clear that the QPO and the broadband noise behave differently in terms of their rms-flux relations. The broadband noise displays a clear linear positive relation, whereas the QPO behaves in a more complex way as discussed earlier in this chapter. The amplitude of both features differ, this is simply the result from the fact that the broadband noise is a larger component than the QPO.
These rms-flux relations for the broadband noise displayed indisputably a positive linear rela-tionship. Therefore these this relation was chosen to fit the correlation. To be able to compare the frequency dependence of the rms-flux of the broadband noise to the QPO, the gradient of the linear fit was plotted against the same frequency, namely the frequency of the primary QPO. This result can be seen in Figure 3.6, where the broadband noise components have been plotted at the same frequencies as for the QPO.
Figure 3.6: The gradients of the linear fits of the rms-flux relations for the QPO (black) and broadband noise (blue)
The frequency dependence of the rms-flux relations is different. Where the gradient of the QPO follows the decreasing pattern as discussed earlier in Section 3.1, the gradients of the broadband noise are consistently positive.
Discussion
This research was meant to explore the rms-flux relation of the quasi-periodic oscillations and the broadband noise for GRS 1915+105. One could question why GRS 1915+105, which is well known to be peculiar, has been chosen to study the rms-flux relations for the QPO and broadband noise instead of a typical transient black hole. The reason this object has been chosen is firstly due to the fact that other, typical black hole binaries, are only for a short period of time in the hard intermediate state: the state that was needed in this research. GRS 1915+105 has enough observations in this corresponding state.
Secondly it was not known what the behaviour of GRS 1915+105 was in terms of its rms-flux relations for both the QPO and broadband noise. Therefore it was interesting to see whether this peculiar object has similar behaviour compared to a typical black hole binary.
4.1
Comparison between GRS 1915+105 and XTE J1550-564
The analysis of the studied observations of GRS 1915+105 suggests that the rms-flux relations for the quasi-periodic oscillations which are observed in the Hard Intermediate State can be fitted with a linear function. The gradients of these relations are not uniform, they seem to depend on the frequency of the QPO. There is one well constraint positive gradient at a fre-quency lower than ∼3 Hz. Around 3 Hz most of the relations become flat, meaning having a gradient of 0. At higher frequencies the gradients of the rms-flux relations are constrained to be negative. There are two anomalous observations at around 4 Hz, clearly having a positive gradient. These points belong to observations which are observed in two consecutive days in 2005, it could be that for some reason something anomalous happened around this time. In some of the observations a harmonic was needed to fit the power spectra properly. The fre-quency dependence of the rms-flux relation of the harmonic appears to show similar behaviour
to the QPO.
When comparing these results to XTE J1550-564, one can see that they might exhibit similar behaviour in the frequency range 0-10 Hz. The rms-flux relation for the QPO in XTE J1550-564 is dependent of the frequency of the QPO, as the the frequency rises, the gradient decreases. This might be happening in GRS 1915+105 in a similar manner. However the frequency where the gradient changes sign is not the same for both objects, for GRS 1915+105 it seems to be around 3 Hz whereas for XTE J1550-564 it is at a higher frequency namely 5 Hz.
However, it cannot be confirmed that these objects behave in the same manner in terms of their frequency dependent rms-flux relation. Since the limited amount of observations, it can only be confirmed that these observations suggest that GRS 1915+105 acts in the same way. As been discussed in Section 2, the observed light curves of the black hole binary were separated in segments of 3 seconds, this limited most of the low QPO frequency observations in such way that they could not be used. For further research it is interesting to analyse more power spectra with low QPO frequency, so the similar behaviour between GRS 1915+105 and XTE J1550-564 can be confirmed. In order to do this, the light curve should be split up in longer time segments.
As has been mentioned in Section 1.2 the mass of the black hole in the system GRS 1915+105 is examined to be 14 + 4.4 M (Harlaftis & Greiner 2004), whereas the black hole in XTE
J1550-564 is well constraint at 10 M (Orosz et al. 2011). There could be some mass dependence, but
due to the large error on mass of the compact object of GRS 1915+105 it is hard to evaluate. The possible mass dependence might be an interesting point for further research.
The dates at which the observations of GRS 1915+105 are taken, span a large period of time ∼10 years. The observations that have been used for the published results of XTE J1550-564 however, are from a single outburst in 1998. This significantly large difference in time, could be an interesting point of further research. One could check whether the behaviour of the systems are more similar if the observations are from a comparable length of time.
Moreover, the period of time around the two anomalous observations as discussed before, having constraint positive gradients at around 4 Hz, could be another interesting point of further research. The gradients of the linear rms-flux relations can be compared to other observations of this time.
4.1.1 Filter
As discussed in Section 1.5.1.1 the frequency dependence of the QPO’s, in XTE J1550-564 may be influenced by a filter. To find out whether this might affect the results of GRS 1915+105, it should first be examined if there is frequency shifting within the observations. In order to
explore this frequency shifting, the frequency of the QPO was plotted against the flux. This relation is displayed in the Figure 4.1.
Figure 4.1: Frequency dependence of the flux
As can be seen from Figure 4.1 there clearly is frequency shifting within the observations. Therefore it is worth checking the results one would get without the filter. In order to do this the fractional rms, which is the rms divided by the mean flux, was plotted against the mean frequency, this relation was fitted with a power law. The value of the power was estimated by converting the fractional rms and the mean flux to log scale and then fit the linear relation between these quantities. Figure 4.2 displays the power law fit of the fractional rms and the mean frequency on non-log scale.
Figure 4.2: The fractional rms plotted against the mean frequency (yellow) and this relation has been fit with a power law (green)
The value of the acquired power-law index was -0.98. Subsequently the rms was divided by the frequency to the power -0.98 or in formula form: σb = νσαa, where σa is the rms without the
correction, ν the frequency, α the index -0.98 and σb the corrected rms value. The error on
the corrected rms was calculated assuming the estimate for the power-law index was adequate, therefore it was calculated in the same way the rms was corrected.
In Figure 4.3 the rms-flux relation of both the uncorrected, in black, and corrected, in pink, for the frequency dependent filter are displayed. As for the previous plots, on the horizontal axis the flux is plotted and on the vertical axis the rms.
Figure 4.3: The rms-flux relations for the QPO, uncorrected (black) and corrected for the frequency filter (pink)
Since the corrected rms was calculated by dividing the uncorrected by the frequency to the power -0.98, this is the expected increase in amplitude. The flux relation for the frequency filter corrected rms, displays a more complicated relation than the uncorrected. To show this more clearly, Figure 4.4 displays the corrected rms-flux relation for one observation.
Figure 4.4: The rms-flux relation for one observation where the rms has been corrected for the frequency dependent filter
As can be seen in Figure 4.4 the relation is not linear, it appears to contain a curvature. In Figure 4.3 the rms-flux relations of the uncorrected and the, as said before, more complex filter-corrected are displayed. The relations reveal a substructure, with the rms-flux relation appearing to turn over during some observations.
This analysis of the filter-corrected rms-flux relations of GRS 1915+105 yield that these corre-lations are not suitable for linear fitting, instead a fit with a polynomial would be a suggestion for further research. Since a polynomial is completely defined by its coefficients, it might be an idea to plot the range of these coefficients against the frequency of the QPO.
Since the rms-flux relations are not suitable to be fit with a linear relation, the statistic corre-lation coefficient Kendall’s tau has been calculated in order to compare the non-corrected and corrected rms-flux relations. Kendall’s tau is a rank correlation that describes the agreement between two rankings, if the agreement is perfect, meaning if in both rankings the (n+1)thvalue is higher than the nth the tau-coefficient equals 1, this is the highest tau-value. One gets the
the (n + 1)th value is lower than the nth value, in this case the correlation coefficient equals -1. If the tau-value equals zero, it means that the two rankings are independent. The values found for Kendall’s tau have been plotted in Figure 4.5, on the horizontal axis is the frequency plotted and on the vertical axis the τ -value. Like for the gradient-plot, this has been done for the uncorrected rms-flux relations, in black, and the corrected ones, in pink. Some observations yielded the same value for Kendall’s tau, these points are half black, half pink.
Figure 4.5: Kendalls tau for the non-corrected rms-flux relations (black) and the filter cor-rected (pink)
In Figure 4.5 one can see that at low frequencies the filter frequency corrected rms-flux relations strongly positively correlate. As the frequency goes up, both the uncorrected as the corrected show a change of sign in Kendall’s tau, however the uncorrected relations become more strong negatively correlated than the corrected ones. This is similar to the results of XTE J1550-564 as discussed in Section 1.5.1.1. However, the τ -value still provides a negative outcome at higher frequencies for GRS 1915+105, whereas the gradient of the linear fits of the other object become zero at higher frequencies.
4.2
Comparison between broadband noise and QPO
The broadband noise component of the power spectra of GRS 1915+105 has been studied and from the results it is clear that there is a linear relation between its rms and the flux. As dis-cussed in Section 1.5, this has been confirmed in other BHBs too and therefore GRS1915+105 is similar to typical BHBs in terms of the rms-flux behaviour of the broadband noise. The linear relation between these two physical quantities can be explained by the propagation fluctuation model as discussed in Section 1.5.
When comparing the behaviour of the rms-flux relations of the QPO and the broadband noise it is clear that they differ. Both could be fitted with a linear relation, but where the rms of the QPO gets suppressed at higher frequencies and shows a change of the sign of the gradient dependent on frequency, the broadband noise does not. The gradient of the latter is exclusively positive.
In some formation mechanism models the QPO is part of the total variability, in others it is not an inherent process, it is not known why they might produce different rms-flux relations for the QPO and the broadband noise. If the QPO has a different formation mechanism than the broadband noise it could be easier to explain the difference in their rms-flux behaviour.
The difference in the rms-flux relations for the QPO and broadband noise as presented in Figure 3.5 yields an indication that the underlying formation mechanisms of the broadband noise and the QPO are different. This could be explained by the Lense-Thirring precession, where the QPO is formed as a result of precessing orbits around the black hole in the corona as explained in Section 1.4.1, instead of being part of the rest of the variability in the accretion flow.
Conclusion
In this research the rms-flux relations of the quasi-periodic oscillations and the broadband noise of the peculiar black hole binary GRS 1915+105 have been studied.
The rms-flux relations of the QPO have been compared to that of XTE J1550-564 which is a typical X-ray binary. The rms-flux relation of GRS 1915+105 could be fitted with a linear func-tion and the results hinted that the gradients of these linear relafunc-tions were frequency dependent, in a similar way compared to XTE J1550-564. In both systems the gradients are positive at low frequencies, as the frequency rises the value of the gradient decreases, first becoming zero and then negative. However the frequency at which the gradient changes sign is different for the two black hole binaries.
Furthermore the effects of a frequency filter have been studied, the results suggest a similar behaviour compared to XTE J1550-564. However, the filter corrected rms-flux relations were not as suitable as the other relations to fit with a linear function.
In addition I have also compared the rms-flux relation of the QPO to that of the broadband noise. It can be concluded that the rms-flux relations of both features are different, the QPO has the frequency dependence as described above, whereas the broadband noise displays only a positive linear relation in its rms-flux. The latter agrees with rms-flux relations in the broadband noise for other black-hole binaries. The difference between the behaviour of the QPO and the broadband noise hints that they might have a different physical origin.
Acknowledgments
I would like to thank my supervisor Lucy Heil for her kind support during this bachelor project and her suggestions for this thesis. I would also like to thank the API for letting me use their facilities.
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Appendix
ObsID Date Average QPO frequency (Hz) 1 10258-01-05-00 20/08/1996 6.2 2 10408-01-24-00 16/07/1996 2.4 3 10408-01-30-00 18/08/1996 5.1 4 10408-01-31-00 25/08/1996 3.9 5 10408-01-33-00 07/09/1996 5.5 6 10408-01-45-00 29/10/1996 3.3 7 20187-02-05-00 06/10/1996 4.3 8 20402-01-08-00 24/12/1996 3.9 9 30182-01-02-00 09/07/1998 3.5 10 30703-01-33-00 15/09/1998 3.3 11 40117-01-01-00 24/04/2000 3.5 12 40703-01-02-00 08/01/1999 3.6 13 40703-01-05-00 12/02/1999 4.2 14 50125-01-03-00 15/07/2000 3.3 15 50703-01-56-00 26/04/2001 3.0 16 50703-01-63-00 13/06/2001 4.0 17 60405-01-03-00 05/08/2001 2.8 18 80127-05-06-00 14/05/2004 4.7 19 90105-07-01-00 12/04/2005 4.0 20 90105-07-02-00 13/04/2005 3.9 21 90701-01-18-00 18/07/2004 3.5 22 92702-01-30-00 14/10/2006 4.1
