black holes: the effects of the
impulse response on the power
spectrum
Stian Huseby
Anton Pannekoek Institute for Astronomy
University of Amsterdam
The Netherlands
September 2014
X-ray reverberation mapping of black holes: the
effects of the impulse response on the power
spectrum
Stian Huseby
Abstract
Any observed light from an accreting black hole will be a combination of direct and reflected light (from the accretion disc). By applying Fourier techniques to this observed combined light curve and taking the amount of reflection per energy band into consideration this can be used to find the actual time delay or lag between the two light curves. This time lag can be used to reveal the structure around the black hole from the broad line region (optical observations) to regions closer than one rg(X-ray observations). This effect called reverberation also affects the amplitude of
variability, providing extra info by measuring the power spectrum, which shows the amplitude as a function of Fourier frequency or equivalently, time-scale. The goal of this paper is to see if this power spectrum can reveal similar patterns as seen in the time lags over the last few years. This is done by simulating a realistic light curve and changing the properties of the surroundings of the black hole. The height of the source seem to change the location of the first oscillation. A higher source would move the location down to lower frequencies. The spin seemed to change the amplitude of all of the oscillations and the inclination seemed to change the amplitude in a similar way as the spin. A higher inclination reduced the amplitude of the first oscillation but increased the amplitude of the other oscillations. A PSD of NGC 4051 was also looked at, but show no sign of the patterns seen in the simulations. This could be mostly due to the low average count rate of 0.165 around the iron Kα line and 5.57 in the soft band (0.3-1 keV). Simulations show that a count rate of about 300 would be necessary to reduce the noise to a level that would show the patterns seen when no noise is present, meaning that these patterns will most likely not be found in the near future.
Licht vanuit de omgeving van een zwart gat kan door observatoria gezien worden en uit eerder onderzoek blijkt dat dit licht uit twee componenten bestaat: `e`en direct vanuit de lichtbron en `e`en gereflecteerd vanaf de schijf die om het zwart gat heen zit. Deze twee componenten zijn te onderscheiden door naar verschillende energiebanden van het licht te kijken. Door wiskundige technieken toe te passen kan een tijdsver-schil gemeten worden tussen de twee componenten. Dit kan vervolgens gebruikt worden om de structuur om het zwarte gat heen in kaart te brengen. In dit onder-zoek wordt er gekeken naar het effect van dit fenomeen op het vermogensspectrum (lichtspectrum gevonden door het toepassen van wiskundige technieken), door een simulatie te bouwen waarin de structuur om het zwart gat als variabelen genomen worden. Daarna wordt er gekeken of hier patronen zichtbaar zijn die vergelijkbaar zijn met de patronen die eerder in tijdsvertragings metingen werden gezien. Het bli-jkt dat de hoogte van de lichtbron, de draai van het zwarte gat en de inclinatie van de schijf allen invloed hebben op het vermogensspectrum. Er is ook gekeken naar het vermogensspectrum van een echte observatie om te kijken of de patronen zoals gezien in de simulaties hier ook zichtbaar waren. Dit bleek niet het geval te zijn, wat waarschijnlijk door de lage gemeten waarden kwam. Simulaties lieten zien dat de gemeten waarde rond de 300 moest zijn, maar deze was slechts 5.57 in de beste energie band. Dit betekent dat de in de simulatie gevonden patronen waarschijnlijk pas in de verre toekomst te zien zullen zijn.
Declaration
The work in this thesis is based on research carried out at the Anton Pannekoek Institute for astronomy, the Netherlands. No part of this thesis has been submitted elsewhere for any other degree or qualification and it is all my own work unless referenced to the contrary in the text.
First of all I would like to thank my advisor Dr. Phil Uttley, for his ability to inspire me to get enthusiastic about every single part of the project. His patience, motivation, enthusiasm, and immense knowledge has been invaluable throughout this project. I would also like to thank Dr. Sera Markoff for agreeing to proofread the paper.
I would also like to thank Prof. Chris Reynolds for providing the general relativis-tic impulse response code used in this project, and PhD student at the University of Amsterdam, Catia de Jesus Silva for providing the light curves from NGC 4051.
Contents
Abstract ii
Popular Dutch Summary iii
Declaration iv Acknowledgements v 1 Introduction 1 2 Analysis 7 2.1 Fourier Transform . . . 7 2.2 Cross spectrum . . . 8
2.3 Power Spectral Density . . . 9
2.4 Convolution . . . 11
3 Modeling 14 3.1 Generating The Power law Noise . . . 14
3.2 Dilution . . . 17
3.3 Simple ”Top Hat” Model . . . 19
3.4 Phase Wrapping . . . 20
3.5 General Relativistic Model . . . 20
3.5.1 The variables of the simulation . . . 23
3.6 NGC 4051 . . . 24
4 Results 26 4.1 Effects On The PSD . . . 26
4.1.1 Height Of The Source . . . 26
4.1.2 Spin Of The Black Hole . . . 27
4.1.3 Inclination Of The Disc . . . 29
4.2 NGC 4051 . . . 30
5 Summary 34 5.1 Short Summary . . . 34
List of Figures
1.1 Schematic diagram showing the corona above the accretion disc, or-biting a black hole. The reverberation signal is the time lag that is caused by the light-travel time difference between the direct power law signal and the reflection spectrum. The black holes gravitational field bends the light and this also adds to the time delay seen in the measured signal. Source: Uttley et al. [2014] . . . 2 1.2 The reflection spectrum from an ionized disc. Showing the three
characteristic parts; Soft excess, the broad iron line and the Compton hump. Source: Uttley et al. . . 3 1.3 The lag as a function of Fourier frequency between 4-5 keV and 6-7
keV energy bands for the black hole MCG5-23-16. Positive lag indi-cates hard lag where the harder band, 6-7 keV(iron peak) is lagging the softer band, 4-5 keV(dominated by continuum). Source: Zoghbi et al. [2013] . . . 5
2.1 Example time lag plot. Taken from a GR(General relativistic) situ-ation with a dilution of 1, a source height of 10 rg, an inclination of
30 degrees and a spin of 0.1. . . 9 2.2 Example PSD with a power law index of -2, made from a random
walk light curve. . . 11
2.3 Top left: Standard light curve with an average count rate of 56. Top Right: The general relativistic impulse response for the NGC 4051(spin:0.23, inclination:31 degrees and height: 6.3 rg) represented
to a time of 100 seconds. Bottom: Convolution of the light curve and the impulse response. The count rate has changed significantly and the light curve also appears more noisy. . . 13
3.1 Example of unsmoothed PSD with a power-law break. This PSD includes the negative frequencies from the Nyquist frequency at 0.5 and up. . . 15 3.2 Example PSD of a GRIR case with a count rate of 100 with Poisson
noise. It can be seen that with the Poison noise the PSD flattens out at a certain value that equals approximately 2/(count rate) (0.02 in this case). Inclination is 30 degrees, spin is 0.998 and source height is 4 rg. . . 17
3.3 Time lags with different values of dilution. Impulse response used is a top hat with a centroid of 1000 and a width of 500 seconds. Red represents R = 1.5, green represents R = 1 and blue represents R = 0.5. . . 18 3.4 Left: Diagram showing the reprocessing by a spherical shell of
ra-dius R. The path length difference between the direct emission (dot-ted line) and the reprocessed emission (dashed line) is R(1 + cos), and hence the time delay for a given position on the sphere is τ = (1+cos)R/c. An isodelay surface is shown in blue. Right: The cor-responding impulse response is a simple top hat function extending from the minimum delay (τ = 0) to the maximum delay at θ = 180, which is τ = 2R/c. Source: Uttley et al. [2014] . . . 19 3.5 Graphic representation of the lamp post model. Light travels from
the source (orange) to the observer (satellite), directly and via the ac-cretion disc. h is the height of the source, θ is the inclination and rinis
the innermost stable circular orbit(ISCO). Source: Emmanoulopoulos et al. [2014] . . . 21
List of Figures x
3.6 Ray traced image of the reflected light from a disc around a black hole. Darker color means that the reflected light is further red shifted. The light from behind the black hole is warped due to the gravitational effects. Source: Uttley et al. [2014] . . . 22 3.7 Left: 2-D plot of the complete impulse response simulated for the
situation; spin = 0.1, height = 4 rg and inclination = 30 degrees.
The numbers on the left is a representation of the redshift of the light, a lower number represents a higher energy and vice versa. 0 is approximately 10 keV an 800 closing in on 0. The energy of the iron peak is located around 400. Right: the combined 1-D impulse response created from line 350 to 650 from the 2-D plot on the left. . 23
4.1 The green line represents a source height of 10 rg and the blue line
a height of 4 rg. The difference between the heights can be seen
as a difference in the frequency where the first oscillation occurs. The oscillations start at lower frequencies with a higher source. The other variables of this figure are a spin of 0.1 and an inclination of 18 degrees. Figure created using a power law index of -1 on frequencies lower than 10−3 and an index of -2 on frequencies higher than 10−3. There was no Poisson noise used in this simulation. . . 27 4.2 The green line represents a maximally spinning black hole, and the
blue line represents a low spinning black hole. The difference between the two can be seen as the difference in depth of the first dip. The other variables of this figure are a height of 4 rg and an inclination of
18 degrees. Figure created using a power law index of -1 on frequencies lower than 10−3 and an index of -2 on frequencies higher than 10−3. There was no Poisson noise used in this simulation. . . 29
4.3 The green line represents an inclination of 60 degrees and the blue line represents an inclination of 30 degrees. The difference can be seen in the amplitude of the oscillations, the first being smaller for the higher inclination. The other oscillations are on the other hand are larger for the higher inclination. The other variables of this figure are a height of 4 rg and a spin of 0.998. Figure created using a power
law index of -1 on frequencies lower than 10−3 and an index of -2 on frequencies higher than 10−3. There was no Poisson noise used in this simulation. . . 30 4.4 Left: A simulated PSD with a count rate of 300. The red line
rep-resents the flat part of the curve that the PSD should hit at approx-imately 6 × 10−2, but the small bump might reveal the effect of the impulse response. Right: A simulated PSD with a count rate of 1114. The red line represents the flat part of the curve that the PSD should hit at approximately 6 × 10−2. The bump here clearly reveals the ef-fect of the impulse response showing the first peak of the interference pattern. Figures created using a power law index of -1 on frequen-cies lower than 10−3 and an index of -2 on frequencies higher than 10−3. Both plots Based on the parameters of NGC 4051(spin:0.23, inclination: 31 and height of source: 6.3 rg) including Poisson noise. . 31
4.5 Left: The PSD of NGC 4051 from the energies between 6 keV and 10 keV. The average count rate of this dataset is 0.165. No effect of the impulse response can be seen in this figure Right: The PSD of NGC 4051 from the energies between 1.5 keV and 4 keV. The average count rate of this dataset is 1.02. No effect of the impulse response can be seen in this figure . . . 32 4.6 The PSD of NGC 4051 from the energies between 0.3 keV en 1 keV.
The average count rate of this dataset is 5.57. No effect of the impulse response can be seen in this figure. . . 33
Chapter 1
Introduction
It is well known that most, if not all galaxies contain a central supermassive black hole (Kormendy and Richstone [1995]). Most of these black holes are, like the one in the center of our own Milky Way, quiet. This means that it can only be detected by the effect of its gravity on the surroundings, like the movement of the local stars (see Gillessen et al. [2008] for more in depth explanation on this subject). Gas that fall into such a black hole inevitably has some angular momentum and thus forms an accretion disk. Viscosity causes the disk gas to spiral slowly inward, heating up (with temperatures up to 109 K, Cao [2008]) and radiating away its gravitational potential energy, until it reaches the last stable orbit and falls in (Sparke and Gallagher [2011]). When this happens the black hole ”activates” and the resulting phenomenon is called an active galactic nucleus (AGN). A corona may then form above the black hole due to the magnetic fields generated by the accretion disc. This corona consists of very hot plasma that start to radiate as a power law continuum (Matteo [1999]).
The light from this radiation can be observed by telescopes and satellite obser-vatories. But the light that is observed comes not only from the source itself, there is also reflection from the accretion disc and other gas particles around the black hole. This reflected light is also influenced by the general relativistic laws that are in effect close to the black hole. Light will tend to bend around an object that has a high density like a black hole creating an even longer path from the source to the disc and then to us. This is effect is called gravitational lensing and can be used to measure the mass of i.e. clusters of galaxies (Hoekstra et al. [2013]). This effect
also redshifts the light. The light is even further red shifted and broadened by the Doppler effect due to the spinning of the disc around the black hole(see example in figure 3.6).
Figure 1.1: Schematic diagram showing the corona above the accretion disc, orbiting a black hole. The reverberation signal is the time lag that is caused by the light-travel time difference between the direct power law signal and the reflection spectrum. The black holes gravitational field bends the light and this also adds to the time delay seen in the measured signal. Source: Uttley et al. [2014]
The observed light will reflect better at some energies than others as can be seen in figure 1.2. The reason for these different reflections are mainly two competing effects, Compton scattering where light gets reflected by the gas (one reflection model is presented by Cassatella et al. [2012]), and absorption where the photons get absorbed by the atoms in the gas. At some energy ranges the atoms are very likely to absorb the photons as i.e. between about 1-4 keV. Then above 6-7 keV the light is more likely to be reflected (Uttley et al. [2014]). Relatively cold matter in the near vicinity of an astrophysical black hole will inevitably find itself irradiated by a spectrum of hard X-rays. The result can be a spectrum of fluorescent emission lines, the most prominent being the Kα line of iron at an energy of 6.40 - 6.97 keV(Reynolds and Nowak [2003]). The energy range of this peak is of high interest since the amount of reflected light is fairly high there. The fact that the X-rays from the corona also irradiate the near vicinity of the black hole means that when these lines are observed, the innermost regions of the accretion disc is revealed, and can thus be mapped. There is also another area of the energy band that might be
Chapter 1. Introduction 3
of interest, namely the soft excess. This area is usually pure continuum light, but it was discovered by Fabian et al. [2009] that this was composed of relativistically broadened photo ionized reflection features, namely the iron L emission line, and possibly the oxygen line, as well. The individual lines in the soft excess are in general hard to detect as there are many emission lines at these low energies and there might also be contributions from the disc black-body emission as well (Uttley et al. [2014]).
Figure 1.2: The reflection spectrum from an ionized disc. Showing the three characteristic parts; Soft excess, the broad iron line and the Compton hump. Source: Uttley et al.
The concept of emission-line reverberation mapping was initially put forward by Blandford and McKee in 1982. But the predicted effects first started to reveal themselves in the years prior to 1988, when a large number of observations indicated that the broad optical emission lines in Seyfert galaxies varied in response to the continuum on surprisingly short timescales. Then in 1988-89 the first successful re-verberation campaign was carried out. This was done by combining UV spectra with optical ground based spectra (Peterson and Horne [2004]). This program achieved a resolution of a few days in several continuum and emission-line time series. From this project several important results arose. One important result was that over all August 12, 2015
measured wavelengths the variations of the continuum seemed to be in phase. Any lags seen between bands were no longer then a few days. Another important result was that the highest ionization lines responded more rapidly and lower ionization lines more slowly to the variations. During the time from the late 80’s until now, it has become clear that the H-β emission lag is a dynamic quantity that varies with time. This meant that there should be much more nuclear gas at large scale (1000’s of rg where rg = GM/c2 is the gravitational radii) then was initially thought.
In 2004, Peterson et al. completed a re-analysis of 117 independent reverberation mapping data sets on 37 AGN’s. They measured the emission line lags, line widths and black hole masses for all but two of these sources, thus improving the mass database significantly. This was however still done using optical observations, thus probing the broad line region (BLR) of the accreting black hole. This gave a bet-ter understanding of this region but left the regions close to the black hole still to be unknown. Fabian et al. found an X-ray reverberation lag of approximately 30 seconds, something comparable to the light crossing time for the innermost radii around a supermassive black hole. This indicates that the emission lines came from an area within one gravitational radius. By combining these lags a radius of the disc can be estimated, and adding the line width, the mass of the black hole can be found by using Kepplarian gas dynamics(Peterson [2013]). Zoghbi et al. [2013] found time lags in the spectrum of the black hole MCG5-23-16 between the bands 4-5 keV and 6-7 keV as can be seen in figure 1.3. It was clear that the 6-7 keV lagged the 4-5 keV band at frequencies below 10−4 Hz. This indicated the discovery of reverberation signals from this black hole(see figure 1.3).
Chapter 1. Introduction 5
Figure 1.3: The lag as a function of Fourier frequency between 4-5 keV and 6-7 keV energy bands for the black hole MCG5-23-16. Positive lag indicates hard lag where the harder band, 6-7 keV(iron peak) is lagging the softer band, 4-5 keV(dominated by continuum). Source: Zoghbi et al. [2013]
De Marco et al. [2013] found that the negative lag (seen just above 10−4 Hz in figure 1.3) shifts to lower frequencies as the mass of the black hole increases. This revealed a connection between black hole mass and the time lags found in the soft X-ray band. In a reverberation scenario this is to be expected, given that the gravitational radius light crossing time scales linearly with mass of the central object(tc = rg/c). Agis-Gonzalez et al. [2014] used the time lags to estimate the
spin and inclination of the Seyfert 1.5 galaxy ESO 362-G18 and determined a spin August 12, 2015
of ≥ 0.92 with at a 99.99 percent confidence level and an inclination of 53 degrees with a confidence level of 90 percent. This and other results show that the X-ray time lags can be used to map the inner region of the black hole including spin, inclination and source height. Besides the lags, reverberation also affects the am-plitude of variability, as will be demonstrated in this paper, providing extra info by measuring the power spectrum, which shows the amplitude as a function of Fourier frequency or equivalently, time-scale. This power spectrum can easily be obtained, and if any structure in this spectrum that resembles the structures seen in time lags can be found, this would yield a second way to map the small scale area around the black hole, and could increase accuracy when the two methods are compared. First we take a look at the available mathematical tools that will be used during this paper like the Fourier transform and how this can be used to get the time lags and power spectrum. We then take a look at the simulations and explain how a random light curve can be simulated and discuss the impulse response models used. Then the results of the simulations are revealed and at the end discussed. We also look into an observation of the active galaxy NGC 4051 to see if any results from the simulations can be seen in the data.
Chapter 2
Analysis
This chapter takes into consideration the different methods used to analyze the observed light curves. The methods discussed below contain the advantages and uses of the Fourier transforms of the observed or simulated light curves.
2.1
Fourier Transform
The Fourier transform maps a function depending on time x(t) onto a function depending on frequency X(ν). This is done by performing the integration:
X(ν) = Z
x(t) e−iνtdt (2.1.1) This can be used when dealing with waves for frequency analysis, analysis of crystal structures or to solve differential equations. There is also an inverse Fourier transform that will transform a function back to its original state. To perform this inverse Fourier transform simply use the function below:
x(t) = Z
X(ν) eiνtdν (2.1.2) The Fourier transform is a complex number that can be written as follows in polar form:
X(ν) = α(ν) eiφ(ν) (2.1.3)
In this function
α(ν) = pX∗(ν)X(ν) (2.1.4)
represents the amplitude of the function, where the asterisk denotes the complex conjugate of the function. Also
φ(ν) = arctan Im(X(ν)) Re(X(ν))
(2.1.5)
is the complex phase of the spectrum. Here Im and Re are the imaginary and real parts of the function X(ν) respectively. After this mapping onto the frequency domain, the curve of the Fourier transform lets you see patterns which would not be visible by just looking at the light curve. Frequently occurring oscillations will show up as a peak at the frequency of the oscillation.
2.2
Cross spectrum
If we want to look at the difference between two light curves, we could measure something called a cross spectrum. If we have two light curves x(t) and y(t), we take the Fourier transform of them to get two frequency dependent functions X(ν) and Y(ν). We can now combine these into a cross spectrum C(ν) by taking the complex conjugate of one of these and multiply this complex conjugate with the normal Fourier transform of the other:
C(ν) = X(ν) Y∗(ν) (2.2.6) From the equation we can see that this can be used to find the frequency de-pendent phase difference between the two light curves. If we look at the polar representation
2.3. Power Spectral Density 9
where µ(ν) equals [φ(ν) − ψ(ν)] and is the phase difference between the two light curves. This phase difference between the light curves can be used to find the actual time lag at the different frequencies. This is done by performing the equation:
τ (νi) =
µ(νi)
2 π νi
(2.2.8) at each frequency i. An example of this time lag can be seen in figure 2.1. The plot was made using a general relativistic impulse response function in the simulation software created in this project. We will look deeper into these models in section 3.
Figure 2.1: Example time lag plot. Taken from a GR(General relativistic) situation with a dilution of 1, a source height of 10 rg, an inclination of 30 degrees and a spin of 0.1.
2.3
Power Spectral Density
The power spectral density (PSD) is created simply by taking a cross spectrum of a light curve with itself, leaving the real valued modulus squared of the light curve: August 12, 2015
P (ν) = X(ν) X∗(ν) (2.3.9) This is called the periodogram. To estimate the PSD from this periodogram, it has to be normalized to have the same units as the PSD, thus in fractional variance per Hz(Uttley et al. [2014]). This normalization is called the rms-squared normal-ization and looks like this:
Pn(ν) =
2 ∆t
< x >2 N P (ν) (2.3.10)
Here ∆t is the width of a single time bin, < x > is the average value of the light curve before it is Fourier transformed and N is the total number of bins. The PSD is thus a simple curve to construct once a light curve is acquired. This makes it a very interesting curve to look for features associated with the ones that have been found in the time lags by i.e. Cackett et al. [2013]. The PSD is usually a power law structure with a functional form of ν1, breaking to ν12 at high frequencies. An
example PSD is shown in figure 2.2. The spot on the curve where this break between
1 ν and
1
ν2 is located can be used to get an indication of the mass of the black hole. A
break at a frequency of about 10−3 Hz indicates a black hole mass of approximately 106M (McHardy et al. [2006]). This mass will be used in the simulations discussed
2.4. Convolution 11
Figure 2.2: Example PSD with a power law index of -2, made from a random walk light curve.
2.4
Convolution
The convolution product is an operation on two functions that creates a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated. The convolution product can be performed by taking the Fourier transform (eq 2.1.1) of two functions and multiplying these. Lets say we have a normal light curve x(t) and an impulse response i(t). The convolution product would then look like:
(x ∗ i)(t) = Z
x(ν) i(t − ν) dm = X(ν)I(ν) (2.4.11)
where the * indicates a convolution product. When these functions are convolved, the end result is a light curve that has been affected by the impulse response. The impulse response represents the reaction of the disc to the light, this means that August 12, 2015
when the light curve is convolved with the impulse response, the result is a light curve that has been affected by the disc, a reprocessed or reflected light curve. This light curve can be simulated and it is thus possible to test different scenarios simply by convolving several different impulse responses with the light curve, and looking at the predicted outcomes. One example of the convolution is shown in figure 2.3. The effect that the impulse response has on the light curve is that the count rate changes and the light curve itself looks noisier. It is important to note that for all but the sharpest impulse responses, the effect of the convolution is not only to delay the underlying time-series, but also to smear it out on time-scales comparable to or shorter than the width of the impulse response(Uttley et al. [2014]). When convolving two parameters of the impulse response (general relativistic or top hat), points that become important are the central point or centroid and width of the impulse response. The centroid determines the maximum time delay in a time lag plot, and the width determines where the phase wrapping (where the time lag goes from positive to negative) seen starts (see the example time lag plot in figure 2.1). Phase wrapping will be further discussed in section 3.4.
2.4. Convolution 13
Figure 2.3: Top left: Standard light curve with an average count rate of 56. Top Right: The general relativistic impulse response for the NGC 4051(spin:0.23, inclination:31 degrees and height: 6.3 rg) represented to a time of 100 seconds. Bottom: Convolution of the light curve and the impulse
response. The count rate has changed significantly and the light curve also appears more noisy.
Modeling
This chapter takes in to consideration how the simulated model was built up, from building the first power law light curves up to the impulse response models and how they are applied to create a complete simulation of the observed light curve from an AGN.
3.1
Generating The Power law Noise
The first step to simulate these effects is to simulate a power law noise distributed random light curve. This can be done in several ways, but we chose to use the method used in ”On creating power law noise” by Timmer and Konig [1995]. The light curve of course needs to be the same every time it is run so that there are no statistical errors on the light curve itself, and the results, if any, will not be influenced by differences between the simulated light curves. The algorithm to simulate a power law light curve contains four steps.
First, the power of the power law spectrum must be chosen as (1ν)α, with α as
the chosen power.
The second step is to decide what frequencies to use in the distribution up to the Nyquist frequency (defined as νN = 1/(2∆t)), as ν and insert this in the function
given above. Now random Gaussian distributed numbers have to be chosen and multiplied withpP (ν)/2 where P(ν) is the underlying power law or broken power law function. This has to be done twice, once for the real part and once for the
3.1. Generating The Power law Noise 15
imaginary part of the complex number that is the power law distribution.
The third step is to take the complex conjugate of this power law distribution. This complex conjugate then has to be reversed (last number goes first, and first last and so on) and added to the end of the first distribution. It should look like figure 3.1 when it is done correctly. This complex conjugate represents the negative frequencies of the spectrum. These are needed to create the light curve via the inverse Fourier transform, but will be removed at the end of the simulation because negative frequencies are of no physical interest.
Figure 3.1: Example of unsmoothed PSD with a power-law break. This PSD includes the negative frequencies from the Nyquist frequency at 0.5 and up.
The imaginary part of the Fourier transform at the Nyquist frequency is set to zero to make the resulting light curve real. The fourth step is to get the time dependent light curve. All that needs to be done is to take the inverse Fourier transform (eq 2.1.2) of the power-law function.
Since we were working on a kind of broken power law distribution, two different powers have to be chosen and combined. The two distributions will be:
S1(ν) = N0 ν0 ν α (3.1.1) S2(ν) = N0 ν0 νbr α νbr ν β (3.1.2)
Where S1 is the distribution before the break and S2 the one after the break. This
will give a light curve that has a mean of zero. This means that half of the values will be below zero. This has to be corrected for, and the way to do this is to take the numbers from the light curve and use an exponent on them like efi with f
i as the flux
of the light curve. This ensures that the new count rate is always positive and since the old curve was centered on 0, it gives a light curve that has an average count rate that is around 1, giving the opportunity to give the light curve any average count rate wanted simply by multiplying it with the wanted count rate. To add further realism to the light curves, one long simulation was made of 10 times the actual wanted length of the ”observation”, before cutting this into segments of the wanted length. The segments were treated separately when convolving with the impulse response. After this, a PSD was created from each segment of the light curve, and these PSD segments were then combined to one averaged PSD. Now there is only one more thing that has to be added to have a complete realistic light curve, the Poisson noise. This noise flattens out the PSD at a certain level depending on the re-normalized count rate chosen. The way this was done was simply by taking the count rate in each bin and then picking a Poisson distributed random number around this count, and used that number instead of the original number in the light curve before the PSD was calculated. This gave a flattening out of the PSD as expected, see figure 3.2 for an example. The PSD was also binned up into increasingly larger bins to compensate for the logarithmic scale and reduce the scatter inherent to the stochastic process. If this is not done, the curve seen will be too noisy and no patterns would be recognized.
3.2. Dilution 17
Figure 3.2: Example PSD of a GRIR case with a count rate of 100 with Poisson noise. It can be seen that with the Poison noise the PSD flattens out at a certain value that equals approximately 2/(count rate) (0.02 in this case). Inclination is 30 degrees, spin is 0.998 and source height is 4 rg.
3.2
Dilution
Adding dilution is an important factor in simulating these light curves. Dilution is the effect of reflected light in the observed direct continuum. This is of course a natural effect when actually observing these light curves. What is does is that it lowers the amplitude of the time lags but it can also change the shape of the lag spectrum (Uttley et al. [2014]). If we use the definition given in the review paper by Uttley et al., that the relative amplitude of the reflected flux compared to the direct flux is R. Then R = 1 means that the flux of the reflected light and direct light are the same and have a 50/50 distribution. If R = 0, this indicates that there is no reflected light in the light curve, and if R = ∞ there is no direct light. This means that using the reflection distribution from figure 1.2, the different energy bands can
be simulated by just changing R. An R = 0.2 was used to simulate the low energy range around 3 keV, and R = 1 to simulate the iron Kα peak. An example of different degrees of dilution can be seen in figure 3.3, here R = 0.5, R = 1 and R = 1.5 is shown for the top hat function that will be looked into in the next paragraph. This specific top hat function was made with a centroid of 1000 seconds and a width of 500 seconds. The green line represents R = 1, and the 50/50 distribution can be seen by the max lag being at 500 seconds, half of the predicted lag from the centroid.
Figure 3.3: Time lags with different values of dilution. Impulse response used is a top hat with a centroid of 1000 and a width of 500 seconds. Red represents R = 1.5, green represents R = 1 and blue represents R = 0.5.
One thing to note about these lags is that the first zero-crossing point does not change no matter what the dilution is, but the lag at low frequencies change by a factor of R / (1+R). In this thesis we mainly use the time lags to be able to compare the simulations with ones done by others. Doing this it can be seen if there are any errors in the simulation, and corrections can be made if the time lags dont agree
3.3. Simple ”Top Hat” Model 19
with other simulations.
3.3
Simple ”Top Hat” Model
To start off a simple impulse response model was used. This simple model consists of a reflecting spherical shell around the black hole (see figure 3.4).
Figure 3.4: Left: Diagram showing the reprocessing by a spherical shell of radius R. The path length difference between the direct emission (dotted line) and the reprocessed emission (dashed line) is R(1 + cos), and hence the time delay for a given position on the sphere is τ = (1+cos)R/c. An isodelay surface is shown in blue. Right: The corresponding impulse response is a simple top hat function extending from the minimum delay (τ = 0) to the maximum delay at θ = 180, which is τ = 2R/c. Source: Uttley et al. [2014]
At a given time τ after a flare the reflection from a spherical shell will look like:
τ = (1 + cosθ)r
c (3.3.3)
with r as the radius of the shell and θ is the angle seen from the observers point of view(Uttley et al. [2014]). This spherical shell is represented by a simple delta function that is called the top hat function, assuming an equal emission at all places from the shell. In this function the centroid of the top hat represents the distance r/c, thus the time delay in the path length of the radius. The first bend of the lag smooths out, the wider the top hat function is. The reason for this lies in the Fourier transform of the top hat function. This Fourier transform is a sinc(ν∆τ /2) August 12, 2015
function with ∆τ as the width of the top hat function. This of course is not a realistic approach since AGN’s usually don’t have a spherical shell but an accretion disc around it that causes the reflection. This model was thus just used the test the simulations at a simple level before the realistic general relativistic model was introduced.
3.4
Phase Wrapping
Consider a simple delta function impulse response like the top hat. The lag (see figure 3.3) is constant until the lag time corresponds to a half-wave shift in phase (a shift of π). At this point, the waveform could have been shifted either backwards or forwards by half a wave, and since the phase of the wave is defined to be in the range −π to π, the phase wraps around and the measured lag becomes negative. This is a type of anti-aliasing effect similar to the wagon-wheel effect seen when wheels rotate in movies. A delta function at t = 0 has a unity Fourier transform over all frequencies. Adding a time-shift and applying the shift-theorem of Fourier transforms multiplies this value by eiωτ0 where ω = 2πν and τ is the time-shift. In
the complex plane, as frequency increases the corresponding vector rotates around the origin at a constant rate with frequency, causing successive crossings of the π/−π boundary (Uttley et al. [2014]). This effect is called phase wrapping. Physically this can be seen as an interference pattern between the two light curves (reflected and original) where the waves cancel each other when the lag drops to zero and builds each other at the other peaks after the first oscillation. The reason that the two light curves dont show any interference before this point is because of light travel times. The light needs to travel from the source down to the disc and get reflected back to be able to interfere with the other light curve causing a delayed effect.
3.5
General Relativistic Model
It is now time to look at a model with a light source located above the black hole with a accreting disc around it. This is known as the lamp post model, and the
3.5. General Relativistic Model 21
setup looks like figure 3.5. In this setup the general relativistic effects also start to have an impact on the path lengths close to the black hole since the space itself is curved in this area. The impulse response simulation for this situation was created by Chris Reynolds and contains 3 stages of simulation to get an impulse response. This model is based on ray tracing. This means that it follows the path of individual photons to reconstruct images of the system as seen by a distant observer.
Figure 3.5: Graphic representation of the lamp post model. Light travels from the source (orange) to the observer (satellite), directly and via the accretion disc. h is the height of the source, θ is the inclination and rin is the innermost stable circular orbit(ISCO). Source: Emmanoulopoulos et al.
[2014]
The first stage involves the setup of the situation we want to simulate. This involves giving the simulation information about the height of the source, the in-clination of the disc seen from the observer and the spin of the black hole. Other factors that do not play a significant role were kept at a standard value. The vari-ables that were changed will be discussed in section 3.5.1. The ones that were kept at a standard value were among others the dimensions of the image plane and the maximum radius to calculate the point source illumination. Keeping these at a set value gave the opportunity to focus on the factors that most likely would have an impact on the PSD based on earlier findings on the time lag by i.e. Cackett et al. August 12, 2015
[2013].
The second stage of the simulation involved tracing the rays of the light from the source above the black hole. This was done by releasing a flash with 4 × 106 photons
in random directions and tracing every single one to map out the reaction of the disc to this flash. This was done in a ”small” area (out to 100 rg), where general
relativity may have an effect. As can be seen in the image of such a ray tracing event(figure 3.6), the light bends significantly close to the black hole and even the parts of the disc behind the black hole are visible from the front.
Figure 3.6: Ray traced image of the reflected light from a disc around a black hole. Darker color means that the reflected light is further red shifted. The light from behind the black hole is warped due to the gravitational effects. Source: Uttley et al. [2014]
The third stage is extending this small scale impulse response out to a greater distance to make a Newtonian approximation of the impulse response at large dis-tances from the black hole. This distance was also kept at a standard value, large enough that any further extension was unnecessary.
This then created a 2-d impulse response as in figure 3.7. This 2-d image con-tains information on how the disc responds to the different energies of the photons. Following Uttley et al. [2014] we assume that the shape of the impulse response is energy-independent (as expected for a broad reflection continuum), but since the dilution is energy-dependent, the different bands are simply represented by changing the dilution according to the strength of the reflected emission. Since the significant area of the impulse response lies between line 350 to 650 (seen on the left in
fig-3.5. General Relativistic Model 23
ure 3.7), I chose to combine this range into one impulse response. This can then be used for all energy ranges by simply changing the dilution of the light curve. These numbers are a representation of the redshifts of the light, with the iron line (6.4 keV) centered at approximately 400. Higher energies have lower numbers and lower energies have higher numbers. The resulting impulse response looked like the right figure in figure 3.7. This impulse response was different for each situation discussed below in section 3.5.1.
Figure 3.7: Left: 2-D plot of the complete impulse response simulated for the situation; spin = 0.1, height = 4 rg and inclination = 30 degrees. The numbers on the left is a representation of the
redshift of the light, a lower number represents a higher energy and vice versa. 0 is approximately 10 keV an 800 closing in on 0. The energy of the iron peak is located around 400. Right: the combined 1-D impulse response created from line 350 to 650 from the 2-D plot on the left.
3.5.1
The variables of the simulation
Here all the variables that would be changed in the simulations run are introduced. Later the findings will be compared with the ones of Cackett et al. [2013] The main variables that were tested were the height of the source, the spin of the black hole and the inclination of the disc. The other variables will not be discussed here since they will not be changed as mentioned above.
The height of the source is expected by Emmanoulopoulos et al. [2014] to be the best constraint to look for when looking at the time lags. The effect on the time lags from this variable is where on the frequency spectrum the time lags start to August 12, 2015
oscillate. Emmanoulopoulos et al. find that the source location is usually close to the black hole ( 4rg). This indicates that we should test this height to see if this
would be visible on the PSD. A higher source height of 10rg is also tested.
Reynolds et al. [1999] predict a bump in the energy-sliced impulse response (i.e. the spectrum at a given time delay) when dealing with a high spin black hole. Emmanoulopoulos et al. predict a small effect of the spin on the time lags that will be difficult to fit. However their data do suggest that there might be two groups of black holes, the rapid spinning ones (>0.75 spin) and slowly spinning ones (<0.5 spin). This indicates that this also could be visible on the PSD and two extremes of these situations are tested by using a spin of 0.1 and one of 0.998.
We also test different inclinations of the disc, this could have a significant effect because of the physical differences that the inclination brings with it. When the inclination increases, the path length of the light on the ”back” side of the disc increases and the path length of the one on the ”front” side of the disc decreases seen for the observer. This could change the way the oscillations are seen on the PSD. The inclinations tested are ∼ 18, 30, 60 and ∼ 72 degrees.
3.6
NGC 4051
To test the simulations we also take a look at the PSD of a real light curve taken from the AGN NGC 4051. NGC 4051 is a narrow line Seyfert 1 galaxy with high amplitude X-ray variability. This variability makes this a good target to see if any patterns that are found in the simulations can be found in the data. Alston et al. [2013] studied the time delays that can be seen in NGC 4051. Using a combination of two top-hat models as one impulse response, the maximum time lag in the hard and soft band were consistent with each other at a value of approximately 1600s, placing the outer region of the reprocessor at 160 rg, assuming a mass of 2 ×106M . Using
a top-hat plus power law model the maximum time lag found was 600s, placing the reprocessor at 60 rg. This means that the distance used in the simulations of
100 rg lie approximately in the middle of these and will be used for the simulation
3.6. NGC 4051 25
regions of many AGN’s including NGC 4051. They found a mass of approximately 1, 9 × 106M
, a minimum spin of 0.23, an inclination of 31 degrees and a source
height of approximately 6,3 rg. These values were tested to simulate NGC 4051,
along with the other standard values mentioned above.
Results
This chapter reviews the results gained from the simulations and the light curves from NGC 4051. The effects on the PSD from the different situations is looked into. The top hat results are not physically realistic for a relativistic disc and will not be further discussed. By testing the different values of R, it was found that at the low energies (R=0.2) the effects of the impulse response are very small and will therefore not be mentioned further. The rest of the results are thus based on an R of 1, corresponding to the reflection seen in the iron K-α band around 6.4 keV.
4.1
Effects On The PSD
Here we take a look at the effects of the GRIR (general relativistic impulse response) on the PSD, and compare these with effects seen in the time lags simulated by Cackett et al. we expect to see some similar patterns when changing the three different parameters of height, spin and inclination.
4.1.1
Height Of The Source
The height of the source is expected to have an effect on the location of the first oscillation on the PSD based on the findings by Cackett et al. The increase in height of the source causes the frequency of the point where the PSD starts to oscillate to move to a lower frequency. This is because of the increased light travel time from the source to the disc. This creates a greater delay before the interference between
4.1. Effects On The PSD 27
the two light sources (original source and the reflection from the disc) may start and the oscillation in the spectrum starts at different times when the height of the source is changed. This effect is visible in the simulations run, and to be able to see this in a real observation, a high source height will be preferred to have the oscillations start before the spectrum gets too noisy and the PSD flattens out.
Figure 4.1: The green line represents a source height of 10 rgand the blue line a height of 4 rg. The
difference between the heights can be seen as a difference in the frequency where the first oscillation occurs. The oscillations start at lower frequencies with a higher source. The other variables of this figure are a spin of 0.1 and an inclination of 18 degrees. Figure created using a power law index of -1 on frequencies lower than 10−3 and an index of -2 on frequencies higher than 10−3. There was no Poisson noise used in this simulation.
4.1.2
Spin Of The Black Hole
The spin of the black hole influences the innermost stable circular orbit(ISCO) and is thus expected to have an effect on how the PSD behaves at high frequencies. It was found by Cackett et al. that the spin has a small effect on the lags. They found August 12, 2015
that it gave a larger peak response for a maximally spinning black hole. This is because a maximally spinning black hole has a disc that extends closer to the black hole (small ISCO) and thus has a larger surface area. This extra surface area is also in the area that is highly affected by general relativistic effects, thus having a more significant Shapiro delay, causing the effect to be slightly later than by a non-spinning black hole. They also found that the spin had very little effect on the frequency where phase wrapping started. For the PSD similar effects as the ones found by Cackett et al., were found in the simulations. The spin had no effect on where phase wrapping occurred, but it did have an effect on the amplitude of the oscillations. This effect is small as can be seen in figure 4.2 but it should be enough to tell the difference between a very low and a very high spin black hole by using this method if the depth of the first dip is considered.
4.1. Effects On The PSD 29
Figure 4.2: The green line represents a maximally spinning black hole, and the blue line represents a low spinning black hole. The difference between the two can be seen as the difference in depth of the first dip. The other variables of this figure are a height of 4 rgand an inclination of 18 degrees.
Figure created using a power law index of -1 on frequencies lower than 10−3 and an index of -2 on
frequencies higher than 10−3. There was no Poisson noise used in this simulation.
4.1.3
Inclination Of The Disc
Cacket et. al found that for a higher inclination the maximum lags are reduced by a small amount. The PSD shows a very different effect. Figure 4.3 shows that a higher inclination seem to smoothen out the first oscillation, and deepens the other oscillations. This effect might disturb a measurement of the height of the source, since with a high inclination disc the point where phase wrapping first occurs might not be as visible as the second one, and one might mistake the two indicating a much lower height of the source than it actually is. This means that it is a great advantage to find the inclination from a different type of measurement to get an idea of which oscillation that would be the most significant. If the height is found August 12, 2015
by i.e. the time lags, the inclination of the disc may be found by looking at the PSD as long as the errors are small enough. It is also worth mentioning that from looking at the figures of spin and inclination that the effects are similar. A high inclination disc shows similar effects as a low spin black hole, and a low inclination show similar effect as a high spin black hole.
Figure 4.3: The green line represents an inclination of 60 degrees and the blue line represents an inclination of 30 degrees. The difference can be seen in the amplitude of the oscillations, the first being smaller for the higher inclination. The other oscillations are on the other hand are larger for the higher inclination. The other variables of this figure are a height of 4 rg and a spin of 0.998.
Figure created using a power law index of -1 on frequencies lower than 10−3 and an index of -2 on frequencies higher than 10−3. There was no Poisson noise used in this simulation.
4.2
NGC 4051
When looking at the PSD of NGC 4051 it is clear that the effect is not visible(fig 4.5). The average count rate observed of 0.165 around the iron line is simply too low, meaning that the effects that are expected to occur around a frequency of
4.2. NGC 4051 31
∼ 2 × 10−2 are occurring after the PSD has flattened out completely, showing no
effect. There is simply too much noise. For the energy range between 1.5 keV and 4 keV the reflected amount is comparable to the R factor of 0.2, and hence no patterns are expected. The simulations show that the count rate that would be needed to reduce the noise to a level that could reveal the patterns seen would need to be around 300 as can be seen in figure 4.4. Also included in the figure is a simulation with an average count rate of 1114(200 times the observed soft excess value of 5.57). Here the first bump is very clear, while with a count rate of 300 it can be seen that something might be going on in the right frequency range. This was confirmed by releasing the random seed in the light curve generator to eliminate the possibility that the small bump was caused by a random effect in that specific seed.
Figure 4.4: Left: A simulated PSD with a count rate of 300. The red line represents the flat part of the curve that the PSD should hit at approximately 6 × 10−2, but the small bump might reveal the effect of the impulse response. Right: A simulated PSD with a count rate of 1114. The red line represents the flat part of the curve that the PSD should hit at approximately 6 × 10−2. The bump here clearly reveals the effect of the impulse response showing the first peak of the interference pattern. Figures created using a power law index of -1 on frequencies lower than 10−3and an index of -2 on frequencies higher than 10−3. Both plots Based on the parameters of NGC 4051(spin:0.23, inclination: 31 and height of source: 6.3 rg) including Poisson noise.
Compared to the average of 0.165 for the iron line the needed value of 300 would mean an increase of a factor of about 2 × 103. This is such a significantly high
number that the patterns will most likely not be found in the iron band in the near future.
Figure 4.5: Left: The PSD of NGC 4051 from the energies between 6 keV and 10 keV. The average count rate of this dataset is 0.165. No effect of the impulse response can be seen in this figure Right: The PSD of NGC 4051 from the energies between 1.5 keV and 4 keV. The average count rate of this dataset is 1.02. No effect of the impulse response can be seen in this figure
When looking at the PSD around the energies of the soft excess (0.3 - 1 keV), the count rates are much higher with an average of 5.57. This results in a lower Poisson noise level as can be seen in figure 4.6. This means that for this energy range the increase in count rate needed would be much lower. However, the needed count rate of 300 is still a factor of 60 away from the observed count rate of 5.57.
4.2. NGC 4051 33
Figure 4.6: The PSD of NGC 4051 from the energies between 0.3 keV en 1 keV. The average count rate of this dataset is 5.57. No effect of the impulse response can be seen in this figure.
Summary
5.1
Short Summary
In this paper we have looked at how the PSD can be used to map the inner regions of an AGN. We started out by testing a simple top hat model to simulate a spherical shell around the black hole. We then looked into several GR situations and how they could affect the PSD of an observed light curve. The spin seemed to make the amplitude of the oscillations vary in size. This may be because the higher spin black hole has a much lower ISCO and the area of the disc that is affected is the area that is mostly influenced by the extreme gravity of the black hole and thus increases the path length of the light here. The height of the source determines where on the frequency range the oscillations begin. This could come from the longer path length from the source to the disc and hence changing the time it takes before the interference patterns can occur, moving the oscillations along the frequency spectrum. The inclination of the disc had an effect on the amplitude of the oscillations causing the first oscillation to be less significant, and the other ones to be more significant with a higher inclination. This is also expected to have to do with the difference in path length on both ”sides” of the black hole. The light on the ”back” side would interfere with the source at a later time than the light on the ”front” side, hence changing the amplitude of the oscillations. Lastly we looked at a light curve taken of the AGN NGC 4051, a Seyfert 1 galaxy. The PSD of this light curve unfortunately shows no signs of the patterns simulated during this paper.
5.2. Concluding remarks 35
This was because the part of the PSD where the patterns were expected to be, was completely covered by the Poisson noise level. This meant that higher count rates need to be obtained. Simulations show that this count rate needs to be at around 300 before the noise level is low enough for these patterns to occur.
5.2
Concluding remarks
These simulations have shown that the PSD of an AGN could show patterns that could reveal the structure of the inner areas around a black hole. The main problem with the light curves today are the low count rates. This gives a high amount of noise that makes any patterns at high frequencies invisible. Higher count rates would lower this noise and might reveal the wanted patterns, and give a new way to map these highly interesting regions close to the black hole. Simulations show that the count rates needed would be too high to be achievable in the near future. However future missions such as the ATHENA, that will increase the collecting area by more than an order of magnitude compared to the XMM-Newton (Uttley et al. [2014]) should give much higher count rates, but even this might still be too low to see the wanted patterns in the PSD. This shows however that new observatories could eventually reach a point where these patterns may be detected in the soft excess part of the energy spectrum. The iron band may unfortunately still be far away given the low count rates found today.
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