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University of Amsterdam
MSc Astronomy & Astrophysics
Astronomy and Astrophysics
Master Thesis
Lense-Thirring Precession Around Neutron Stars With
Known Spin
by
Marieke van Doesburgh
5743222
December 2014
54 ECTS
2013-2014
Examiner:
Supervisor:
Prof. dr. M.B.M. van der Klis
Dr. R.A.W. Wijnands
Astronomical Institute Anton Pannekoek
i
Abstract
Quasi periodic oscillations (QPOs) between 300 and 1200 Hz in the X-ray emission from
low mass X-ray binaries have been linked to Keplerian orbital motion at the inner edge
of accretion disks. Lense-Thirring precession is precession of the line of nodes of inclined
orbits with respect to the equatorial plane of a rotating object due to the general relativistic effect of frame dragging. The Lense-Thirring model of Stella and Vietri (1998)
explains QPOs observed in neutron star low mass X-ray binaries at frequencies of a few
tens of Hz by the nodal precession of the orbits at the inner disk edge at a precession
frequency, νLT , identical to the Lense-Thirring precession of a test particle orbit. A
quadratic relation between νLT and the Keplerian orbital frequency, and a linear dependence on spin frequency are predicted.
In early work (van Straaten et al., 2003) this quadratic relation was confirmed to remarkable precision in three objects of uncertain spin. Since the initial work, many neutron
star spin frequencies have been measured in X-ray sources that show QPOs at both low
and high frequency.
Using archival data from the Rossi X-ray Timing Explorer, we compare the LenseThirring prediction to the properties of quasi periodic oscillations measured in a sample
of 19 low mass X-ray binaries with known neutron star spin frequencies that are known
to show QPOs in their X-ray emission.
We find that in the range predicted for the precession frequency, we can distinguish two
different oscillations. In previous works, these two oscillations have often been confused.
The Lense-Thirring precession model is inconsistent with the observed frequencies, as
the required specific moment of inertia of the neutron star exceeds values predicted for
realistic equations of state. Also, we find correlations characterized by power laws with
indices that differ significantly from the prediction of 2.0. We find no evidence that the
neutron star spin frequency affects the QPO frequencies.
Acknowledgements
First and foremost, I would like to offer my sincere gratitude to my supervisors Michiel
van der Klis and Diego Altamirano. Michiel, thank you for your guidance throughout
this project, sharing and encouraging my enthousiasm, and for giving thorough and
instructive feedback. Diego, thank you for all the ”Jajaja’s” you replied to my e-mails,
your friendship and your support, even though you just started a new job in England.
I am grateful to many more people that contributed to this work. In particular to Emilie
Majoor, for your culinary support and unrivalled friendship. To my parents, for your
eternal advice that the bow cannot be ready to fire at all times. To Claartje, Iris and
Jan, for sharing your bikes, phones, food and laughter. To my fellow students, for the
stimulating and fruitful conversations. And to my friends, for never doubting the wild
horoscopes I predicted for you over the years.
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Nederlandse Samenvatting
De eenvoudigste theorie die overeenkomt met waarnemingen van de beweging van hemellichamen is de algemene relativiteitstheorie als opgesteld door Albert Einstein. De extreem vervormde ruimtetijd rond compacte objecten als neutronensterren en zwarte
gaten vormt het ultieme laboratorium om deze theorie te toetsen aan de werkelijkheid.
Een manier om dat te doen is door de Röntgenstraling te bestuderen die wordt uitgestraald als materie op een compact object valt.
In dit project concentreren we ons op dubbelstersystemen bestaande uit een neutronenster en een donorster, die een hoofdreeksster, een witte of bruine dwerg of een rode reus
kan zijn. Als materie van de donorster voldoende dichtbij de neutronenster komt dat
zijn zwaartekrachtspotentiaal de overhand krijgt, ontstaat er een materiestroom naar de
neutronenster. In Figuur 1.1 is een impressie te zien.
De donorster en de neutronenster draaien om een gemeenschappelijk zwaartepunt dat
dichtbij de neutronenster ligt, en door het behoud van impulsmoment kan de aangetrokken
materie niet direct op de neutronenster vallen.
Een accretieschijf, een differentieel
roterende schijf van heet gas, wordt gevormd waarin de materie impulsmoment naar
buiten transporteert. Zo kan het materiaal de neutronenster steeds dichter naderen, tot
het uiteindelijk op het oppervlak valt. Voor een neutronenster van ∼10 km doorsnede
wordt ∼90% van de gravitiationele potentiele energie van accreterende materie vrijgemaakt binnen ∼100 km van het oppervlak. Het binnenste van de accretieschijf wordt
extreem warm (> 107 K) en gaat stralen in het Röntgengebied van het elektromagnetisch
spectrum.
De materiestroom zal niet vloeiend zijn, maar variaties vertonen, door bijvoorbeeld
turbulentie. Deze variaties zijn zichtbaar in de emissie die we ontvangen en bevatten
informatie over de banen die beschreven worden rond de neutronenster in het binnenste
deel van de accretieschijf. Omlooptijden in dit gebied zijn in de orde van milliseconden,
en waargenomen kHz variabiliteit wordt dan ook in verband gebracht met Keplerbanen
in het binnenste van accretieschijven.
Ook kunnen we door het bestuderen van deze banen de massa en straal van de neutronenster achterhalen, om zo een idee te krijgen van hoe materie zich gedraagt onder extreme
omstandigheden. Het verband tussen druk, temperatuur en dichtheid met elkaar in verband staan heet de toestandsvergelijking van de materie. In deze exotische uithoek van
de parameterruimte is deze niet bekend.
Een voorspelling van de algemene relativiteitstheorie is de vervorming van de ruimtetijd
rond een compact object dat om zijn as draait, het frame-dragging effect. Een compact
object dat harder om zijn as draait, en dus een hogere spinfrequentie heeft, zal een
sterkere frame-dragging teweeg brengen. De vervorming veroorzaakt Lense-Thirring
iv
precessie; het precederen van banen van testdeeltjes die niet in het equatorvlak liggen.
Quasi-periodieke verschijnselen die te zien zijn in de emissie van Röntgen-dubbelstersystemen zijn in verband gebracht met deze precessie-voorspelling voor testdeeltjes door
Stella and Vietri (1998). De precessiefrequentie van tientallen Hz verhoudt zich in het
model kwadratisch met de omloopfrequentie van Keplerbanen (∼kHz) in de accretieschijf, en is bij gegeven ster-massa lineair afhankelijk van de spinfrequentie van de
neutronenster.
In 2003 werd een vrijwel identiek kwadratisch verband gemeten in drie dubbelstersystemen door van Straaten et al. (2003). Met een waarde van 2.01±0.02 lag de macht
verrassend dichtbij de voorspelling van 2. De drie bronnen hebben alle een inmiddels
gemeten, verschillende spinfrequentie. Het samenvallen van het gemeten verband in deze
drie bronnen valt moeilijk te rijmen met het spin-afhankelijke precessiefenomeen dat ten
grondslag ligt aan het model. Wij hebben in dit project bovengenoemde relatie opnieuw
onderzocht in een sterk uitgebreide dataset, ontleend aan het archief van de Rossi X-ray
Timing Explorer, van Röntgendubbelstersystemen met neutronensterren met bekende
spinfrequenties die de eerdergenoemde quasi-periodieke oscillaties vertonen. We komen
tot de conclusie dat het verband zoals voorspeld door Stella and Vietri (1998) niet in
overeenstemming is met de data. We zien dat de oorspronkelijke conclusie van van
Straaten et al. (2003) geen stand houdt doordat er twee verschillende frequenties zijn
die in aanmerking komen de precessiefrequentie te vertegenwoordigen. We vinden in
de bronnen machtswetten met machten die significant verschillen van 2. Een duidelijk
effect van de verschillende spinfrequenties van de neutronensterren is niet te ontdekken.
De voorspelling die we toetsen is door Stella and Vietri (1998) rechtstreeks ontleend aan
die van een testdeeltje in de lege ruimte. We hebben in werkelijkheid, zoals genoemd,
te maken met een turbulente schijf van heet gas, waar hydrodynamische, magnetische
en stralingseffecten niet verwaarloosd kunnen worden. De toevoeging van deze effecten
zouden de voorspelde precessie dermate kunnen beı̈nvloeden dat overeenstemming wordt
bereikt met onze bevindingen. Het is echter ook mogelijk dat er een ander effect dan
precessie ten grondslag ligt aan de lage frequentie QPOs.
Contents
Abstract
i
Acknowledgements
ii
Dutch Summary
ii
Contents
v
List of Figures
viii
1 Introduction
1.1 Introduction . . . . . . . . . . . . . . . .
1.1.1 Accretion . . . . . . . . . . . . .
1.1.1.1 Source Types . . . . . .
1.1.1.2 Frequency Correlations
1.1.2 Lense-Thirring precession . . . .
1.1.3 Motivation and Outline . . . . .
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2 Method
2.1 Instrumentation: The Rossi X-ray Timing Explorer . . . . . . .
2.1.1 PCA: Technical Specifications and Instrument Response
2.1.1.1 Proportional Counters . . . . . . . . . . . . . .
2.1.2 Data System . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Instrument History and Response Calibration . . . . . .
2.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Data Filtering . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Timing Analysis . . . . . . . . . . . . . . . . . . . . . .
2.2.3.1 Fourier Transforms . . . . . . . . . . . . . . .
2.2.3.2 Power Spectral Analysis . . . . . . . . . . . . .
2.2.4 Timing Behaviour and Accretion State . . . . . . . . . .
2.2.5 Fitting Routines . . . . . . . . . . . . . . . . . . . . . .
2.2.5.1 Spearman’s Rank Correlation . . . . . . . . . .
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3 Data selection & Results
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3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
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Contents
3.2
3.3
3.4
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Bursters . . . . . . . . . . . . . . . . . . . .
3.2.1 4U 1728-34 . . . . . . . . . . . . . .
3.2.1.1 Comparing to van Straaten
3.2.2 4U 1636-53 . . . . . . . . . . . . . .
3.2.3 4U 1608-52 . . . . . . . . . . . . . .
3.2.3.1 Comparing to van Straaten
3.2.4 4U 0614+09 . . . . . . . . . . . . . .
3.2.4.1 The Flaring State . . . . .
3.2.4.2 Comparing to van Straaten
3.2.5 4U 1702-43 . . . . . . . . . . . . . .
3.2.5.1 The Flaring State . . . . .
3.2.6 KS 1731-260 . . . . . . . . . . . . .
3.2.7 SAXJ1750.8-2900 . . . . . . . . . . .
3.2.8 Aquila X-1 . . . . . . . . . . . . . .
3.2.9 Bursters with poor statistics . . . .
Pulsars . . . . . . . . . . . . . . . . . . . . .
3.3.1 SAXJ1808.4-3658 . . . . . . . . . . .
3.3.1.1 The Flaring State . . . . .
3.3.2 HETEJ1900.1-2455 . . . . . . . . . .
3.3.3 IGRJ17480-2446 . . . . . . . . . . .
3.3.4 XTEJ1807-294 . . . . . . . . . . . .
3.3.5 IGRJ17511-3057 . . . . . . . . . . .
3.3.6 SAXJ1748.9-2021 . . . . . . . . . . .
Summary of results . . . . . . . . . . . . . .
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4 Interpretation
4.1 The Lense-Thirring model . . . . . . . . . . .
4.2 Fitting the Data . . . . . . . . . . . . . . . .
4.2.1 Hypothesis A . . . . . . . . . . . . .
4.2.2 Hypothesis B . . . . . . . . . . . . . .
4.3 Comparing to van Straaten et al. (2003): 4U
4U 1728-34 combined . . . . . . . . . . . . . .
4.4 Other Possibilities and Future Work . . . . .
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5 Conclusion
A Color-Color Diagrams
A.1 4U1636-53 . . . . . .
A.2 4U1608-52 . . . . . .
A.3 4U0614+09 . . . . .
A.4 4U1702-43 . . . . . .
A.5 KS1731-260 . . . . .
A.6 SAXJ1750.8-2900 . .
A.7 Aql X-1 . . . . . . .
A.8 SAXJ1808.4-3658 . .
A.9 HETEJ1900.1-2455 .
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Contents
vii
A.10 IGRJ17480-2446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A.11 XTEJ1807-294 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.12 IGRJ17511-3057 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
B Digital Appendix
135
Bibliography
136
List of Figures
1.1
1.2
2.1
2.2
2.3
2.4
2.5
Artist impression of a low mass X-ray binary by M. van der Sluys using
BinSim, a program developed by Rob Hynes. . . . . . . . . . . . . . . . .
From van Straaten et al. (2003). Correlation of Lh with Lu for three
LMXBs, a power law index of 2.01±0.02 was found, in striking correspondence with the Lense-Thirring precession model prediction. . . . . . .
Schematic drawing of the cross-section of a PCU. RXTE’s Proportional
Counter Array contained five of these proportional counter units. ‘1-3L’
and‘1-3R’ indicate the 6 signal chains of wire grids in the gas chamber.
V1 an V2 were the chained anodes forming the xenon/methane anticoincidence layer and the propane anti-coincidence layer (see main text).
‘ALPHA’ indicates the radioactive americium 241 calibration anode signal
chain. The collimator was situated on top of the PCU (Jahoda et al., 2006).
The energy spectrum of a 5 ks observation of burster 4U 0614+09, starting
on MJD50197 (ObsID:10095-01-02-00). To calculate the hard color we
divide the count rate in D by the count rate in C (= 0.63) , and for the
soft color we divide the count rate in B by the count rate in A (= 0.96).
This observation shows up in region D of the color-color diagram, see
Figure 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Color-color diagram for 495 observations of 4U0614+09.‘HLF’ stands for
high luminosity flaring. Errors on hard and soft color are smaller than
the symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A power spectrum of the LMXB 4U 1728-43 fitted with multiple Lorentzians,
plotted in two representations. On the left we plot power vs. frequency,
in this representation the maximum of a Lorentzian is at the centroid frequency ν. On the right we multiply the power with frequency and plot it
vs. frequency, enhancing features at high frequency. The maximum of the
Lorentzian in this representation is at νmax . ”B” refers to region B in the
color-color diagram, See Figure 3.1. The power spectra of observations
from region B were averaged to obtain the power spectrum depicted here.
From Altamirano et al. (2008a). Left: Color color diagram for 4U 163653, with the names of atoll source states indicated in pink. Centers of
regions are indicated with the letters A to N. Right: Identification of
features in the average power spectrum of observations from regions B
and J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
3
7
11
14
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19
22
List of Figures
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Color-color diagram for 423 observations of 4U1728-34. Each dot represents the energy spectral shape of one observation of a few ks. The
position in this diagram correlates with the shape of the source power
spectrum, as can be seen in Figure 3.2. Error bars are smaller than the
symbols (typically <0.5%). . . . . . . . . . . . . . . . . . . . . . . . . . .
Representative fitted power spectra for different regions of the color-color
diagram (see 3.1). Observations from regions Ai , B, C, D, Ei and Eii
were used for further analysis, as the hump feature or LF QPO, and kHz
QPOs appear here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Centroid frequencies (ν) plotted against the centroid frequency of the upper kHz QPO (νu ) of 4U 1728-34. We plotted these values for all other
sources in our sample in lighter colors as a reference. All components
exceed a 3σ confidence level except for the 2.0σ detection in a selection of
observations (from region Ei in the color-color diagram) of a Lorentzian
with ν = 65 Hz that follows the same relation with νu as Lh . The simultaneously present Lorentzian with ν = 35 Hz, detected at 3σ, follows the
same relation as LLF . When νu exceeds 800 Hz, Lh is not detected. . . . .
Multi-Lorentzian fit to 1290 averaged power spectra (see Table 3.2) from
5 subsequent observations starting at MJD 51133.3. of 4U 1728-34. We
find a Lorentzian characterized by ν = 65 Hz at a 2σ confidence level.
We identify this to be Lh , and νLF = 35 Hz. This power spectrum is the
result of averaged power spectra from observations that are close both in
time and color, a careful subselection of the rough averaged power spectra
we show in Figure 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fractional rms amplitudes and Q-factors of Lh and LLF fitted in power
spectra of 4U1728-34 plotted against their centroid frequency. We plot
fractional rms amplitudes and Q values of all other sources in our sample
(including pulsars) in lighter colors as a reference. All components exceed
a 3σ confidence level except for the 2.0σ detection of a Lorentzian with ν
= 65 Hz. Lh is harder to distinguish from LLF in power spectra characterized by an upper kHz QPO at >802 Hz, or when its centroid frequency
>40 Hz. See the main text for a discussion. . . . . . . . . . . . . . . . . .
Measurements of Lh from the multi-Lorentzian timing study by van Straaten
et al. (2002) (only of 4U 1728-34), converted from νmax to ν, plotted with
our measurements of νLF and νh of 4U 1728-34. The best fit from van
Straaten et al. (2003) to a combination of data from 4U 0614+09, 4U
1728-34 and 4U 1608-52 with a power law index of 2.01±0.02 is drawn,
as well as our best fit power law indices (only to data of 4U 1728-34)
(αh =2.67, αLF =2.76). The frequency of the 2σ Lorentzian was not included in this fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Left: Fit with a 5-Lorentzian model to the power spectrum of an observation of 4U1728-34 (ObsID: 10073-01-07-000) as in Di Salvo et al.
(2001), van Straaten et al. (2002), (χ2 /dof =425/327). Right: Fit with 6
Lorentzians (χ2 /dof =343/324). . . . . . . . . . . . . . . . . . . . . . . . .
Power spectrum reproduced from van Straaten et al. (2002) of 4U 1728-34.
In their ”interval 7” Lorentzians are fitted with ν=26.7 Hz and νu =706
Hz (with νmax converted to centroid frequency). . . . . . . . . . . . . . .
ix
29
30
32
33
34
36
37
37
List of Figures
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
Left: Fit with a 4-Lorentzian model to the power spectrum of observation
40033-06-02-04 of 4U 1728-34 as in van Straaten et al. (2003) (see Figure
3.8, χ2 /dof =446/329). Right: Fit with 5 Lorentzians (χ2 /dof =363/326).
In analogy to Figure 3.3. νh and νLF plotted vs. νu for 4U 1636-53.
When νu exceeds 610 Hz, Lh cannot be identified unambiguously. . . . . .
In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of 4U 1636-53 plotted against their centroid frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In analogy to Figure 3.3, for 4U 1608-52. All components exceed a 3σ
confidence level except for the 2.2σ detection in interval B of a Lorentzian
with ν=12 Hz. It follows the same relation with νu as νh . The simultaneously present Lorentzian with ν=3.9 Hz, detected at 3σ, follows the
same relation with νu as νLF . When νu exceeds 520 Hz, Lh is not detected.
In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of 4U 1608-52 plotted against their centroid frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In analogy to Figure 3.6, for 4U 1608-52. Measurements of Lh from the
multi-Lorentzian timing study by van Straaten et al. (2003) (only of 4U
1608-52), converted from νmax to ν, plotted with our measurements of νLF
and νh . The best fit to combined data of 4U 0614+09, 4U 1728-34 and
4U 1608-52 with a power law index of 2.01±0.02 is drawn, as well as our
best fit power law indices (αh = 2.43 ± 0.16 and αLF = 2.40 ± 0.24). For
these fits only data from 4U 1608-52 was used. Our attempt to recreate
the frequencies measured by van Straaten et al. (2003) is plotted in pink.
The gold colored centroid frequency belongs to the Lorentzian drawn with
a solid line in Figure 3.15 (left). It was omitted from the assessment of a
possible correlation with νu by van Straaten et al. (2003). See main text
for a discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power spectra from van Straaten et al. (2003) of 4U 1608-52. Left: LLF is
detected simultaneously with Lh in region ’B’ of their color-color diagram.
Right: in region ’C’ of their color-color diagram Lorentzians are fitted
with νmax,h =19.8 and νmax,u =474 Hz, or converted to centroid frequency
νh =8.5 Hz and νu =400 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . .
Left: Fit with a 3-Lorentzian model to the power spectrum of observations 10094-01-[09-01][10-000/00/010/01/020] of 4U 1608-52 as in van
Straaten et al. (2003)(Figure 3.15, χ2 /dof =438/333). Right: Fit with 4
Lorentzians (χ2 /dof =424/330). . . . . . . . . . . . . . . . . . . . . . . . .
In analogy to Figure 3.3, for 4U 0614+09. When νu exceeds 605 Hz, Lh
is not detected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of 4U 0614+09 plotted against their
centroid frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multi-Lorentzian fit to 592 averaged power spectra from 4U 0614+09
starting on MJD 51788.6. We fit a Lorentzian characterized by ν = 4.46
Hz with 3σ confidence level in the integral power. We identify this to be
LLF , Lh has ν = 7.77 Hz (13.8σ). . . . . . . . . . . . . . . . . . . . . . . .
x
38
40
41
44
45
46
48
48
51
52
53
List of Figures
xi
3.20 Multi-Lorentzian fit to 1016 averaged power spectra from 4U 0614+09
starting on MJD 52948.6. We fit a Lorentzian characterized by ν =15.61±
0.34 Hz with a 3.8σ confidence level in the integral power. We identify
this to be LLF , Lh has ν =26.59 ± 1.6 Hz (3.5σ). . . . . . . . . . . . . . . 53
3.21 Multi-Lorentzian fit to 1611 averaged power spectra from all observations
of 4U 0614+09 found to be in the flaring state (χ2 /dof =961/872). In
Figure 3.22 the residuals of this fit are presented. . . . . . . . . . . . . . 55
3.22 Residuals of a multi-Lorentzian model fit to 1611 averaged power spectra
from the flaring state seen in 4U 0614+09, see Figure 3.21. . . . . . . . . 55
3.23 Multi-Lorentzian fit to 508 averaged power spectra from the flaring state
of 4U 0614+09 starting on MJD 51173.9. We fit Lorentzians characterized
by ν = 20.52±0.96 Hz (10σ), ν = 40.53±0.0.62 Hz (8σ), ν = 107.4±2.4 Hz
(6σ), and ν = 1304±25 Hz (5.04σ). For a discussion on the identification
of these features see Section 3.2.4.1. . . . . . . . . . . . . . . . . . . . . . 56
3.24 In analogy to Figure 3.6, for 4U 0614+09. Measurements of Lh from the
multi-Lorentzian timing study by van Straaten et al. (2002), converted
from νmax to ν, plotted with our measurements of νLF and νh . Points
plotted in this figure are all measured in 4U 0614+09. The best fit from
van Straaten et al. (2003) to 4U 0614+09, 4U 1728-34 and 4U 1608-52
with a power law index of 2.01±0.02 is drawn, as well as the best fit
power law indices to our data of 4U 0614+09 with αh = 2.65±0.14 and
αLF =2.54±0.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.25 Left: Fit with a 4-Lorentzian model to the power spectrum of observations
50031-01-01-01/02/03/04/06/07 of 4U 0614+09 as in van Straaten et al.
(2002), χ2 /dof =484/329). Right: Fit with 5 Lorentzians (χ2 /dof =456/326). 58
3.26 Power spectrum from van Straaten et al. (2002) of 4U 0614+09. In ”interval 4” Lorentzians are fitted with ν=22.6 Hz and νu =623.8 Hz (with
νmax converted to centroid frequency). Note that the lowest frequency
measured here is ∼0.008 Hz, the lowest frequency probed in our analysis
is 0.0625 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.27 Left: Fit with a 4-Lorentzian model to the power spectrum of observation 90422-01-01-01 as in van Straaten et al. (2003) (see Figure 3.26,
χ2 /dof =348/330). Right: Fit with 5 Lorentzians (χ2 /dof =327/327). . . . 59
3.28 In analogy to Figure 3.3, for 4U 1702-43. When νu exceeds 550 Hz, Lh is
not detected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.29 In analogy to Figure 3.5; fractional rms levels and Q-factors of Lh and
LLF fitted in power spectra of 4U 1702-43 plotted against their centroid
frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.30 Multi-Lorentzian fit to 1193 averaged power spectra of 4U 1702-43 starting on MJD 53209.4. We fit a Lorentzian characterized by ν =7.87 ± 0.28
Hz (3.3σ). We identify this to be LLF , Lh has ν =13.77 ± 0.65 Hz (10.1σ). 63
3.31 Multi-Lorentzian fit to 4468 averaged power spectra from the flaring state
of 4U 1702-43 (see Figure A.7-H) starting on MJD 53022.0. We are able
to identify Lb2 , Lb , LLF , LhHz , L` and Lu , from left to right, excluding
the low frequency noise at ∼0.2 Hz. See Table 3.6 forν values and the
main text for a discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
List of Figures
3.32 Multi-Lorentzian fit to 4068 averaged power spectra from KS 1731-260
starting on MJD 51815.7. We identify Lb , LhHz and Lu , at low frequency
the identification of features is not straightforward. See Table 3.7 for
measured frequencies, and the main text for a discussion. . . . . . . . . .
3.33 In analogy to Figure 3.3, for KS 1731-260. When νu exceeds 490 Hz, Lh
is not detected. At low frequency the identification of fitted Lorentzians
is not straightforward. Harmonics of LLF are included in this figure.
Because of the scarcity of obtained points and unclear identification, we
opt not to fit a power law to these data. See main text for a discussion. .
3.34 In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of KS 1731-260 plotted against their
centroid frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.35 Multi-Lorentzian fit to 2424 averaged power spectra of 4U 1608-52. We
identify Lb , LLF ,LLF2 , LhHz and Lu , see Table 3.4 forν values, and the
main text for a discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.36 In analogy to Figure 3.3. Due to the scarcity of obtained points we opted
not to fit a power law to these data. See main text for a discussion. . . .
3.37 In analogy to Figure 3.5; rms and Q values of Lh and LLF fitted in power
spectra of SAXJ1750.8-2900 plotted against their centroid frequency. . . .
3.38 In analogy to Figure 3.3, for Aql X-1. When νu exceeds 500 Hz, Lh is not
detected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.39 In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of AqlX-1 plotted against their centroid
frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.40 Multi-Lorentzian fit to 4068 averaged power spectra starting on MJD
52092.6. We identify Lb , LLF , Lh and Lu . See Table 3.9 for measured
frequencies, and the main text for a discussion. . . . . . . . . . . . . . . .
3.41 Multi-Lorentzian fit to 2758 averaged power spectra of AqlX-1. We identify Lb , LLF ,Lh , LhHz and Lu , see Table 3.9 for measured ferquencies, and
the main text for a discussion. . . . . . . . . . . . . . . . . . . . . . . . . .
3.42 In analogy to Figure 3.3 for SAXJ1808.4-3658. When νu exceeds 700 Hz,
Lh is not detected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.43 In analogy to Figure 3.5, for SAXJ1808.4-3658; fractional rms amplitudes
and Q-factors of Lh and LLF fitted in power spectra of SAXJ1808.4-3658
plotted against their centroid frequency. . . . . . . . . . . . . . . . . . . .
3.44 Power spectrum from van Straaten et al. (2005) of SAXJ1808.4-3658 in
which Gaussians are fitted to the 1-10 Hz features (dashed lines, indicated
as ’G1 ’), and Lorentzians are fitted with ν ∼10 and 40 Hz. νu is not detected.
3.45 In analogy to Figure 3.3, for HETEJ1900.1-2455. When νu exceeds 500
Hz, Lh is not detected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.46 In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of HETEJ1900.1-2455 plotted against
their centroid frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.47 In analogy to Figure 3.3 for IGRJ17480-2446. We identify the Lorentzian
we fit at ν =44.4 Hz as LLF , based on the correlation of νLF with νu in
other sources in our sample, plotted in lighter colors in this figure. . . . .
3.48 In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of IGRJ17480-2446 plotted against their
centroid frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii
65
65
67
67
68
69
72
72
74
74
77
78
78
81
81
84
84
List of Figures
3.49 In analogy to Figure 3.3, for XTEJ1807-294. We identify the Lorentzian
we fit in this source at ν =17 Hz as Lh , based on fractional rms amplitude
and Q-factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.50 In analogy to Figure 3.5; fractional rms amplitude and Q-factors of Lh
and LLF fitted in power spectra of XTEJ1807-294 plotted against their
centroid frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.51 In analogy to Figure 3.3, for IGRJ17511-3057. We interpret the Lorentzian
at ν = 2.3 Hz as Lh , based on rms levels and Q-factor (see Figure 3.52).
3.52 In analogy to Figure 3.5; fractional rms amplitude and Q-factor of the
Lorentzian fitted in power spectra of IGRJ17511-3057 plotted against its
centroid frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.53 νLF and νh from subselection A plotted against νu . Open symbols indicate frequencies measured in pulsars. Filled symbols indicate frequencies
measured in bursters. All components exceed a 3σ confidence level. We
plot frequencies not belonging to subselection A in lighter colors. . . . .
3.54 fractional rms amplitudes and Q-factors of Lh and LLF from subselection
A plotted against their centroid frequencies. All Lorentzians exceed a 3σ
confidence level. Open circles indicate indicate rms and Q of Lorentzians
fitted in pulsars. Filled circles indicate rms and Q of Lorentzians fitted in
bursters. All components exceed a 3σ confidence level. We plot rms and
Q of Lorentzians not belonging to subselection A in lighter colors. . . .
4.1
4.2
4.3
4.4
4.5
xiii
. 87
. 87
90
. 90
. 94
. 95
νLF and νh scaled by neutron star spin vs. νu . Triangles are measurements
of νLF , circles of νh . Filled symbols refer to bursters, open symbols to
pulsars. We also plot the range in which I45 m−1 of the neutron star would
be acceptable in the Lense-Thirring interpretation of Stella and Vietri
(1998). The black line marks I45 m−1 = 2 for an observed frequency (νLF ,
νh ) equal to νLT , or I45 m−1 = 1 for νLF , νh equal to twice νLT . . . . . . . 99
A zoom in on Figure 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Left: values for κα=2 plotted against spin frequency of the neutron stars.
Right: ∆κ ≡ κα=2,h - κα=2,LF plotted against spin frequency of the neutron star. Filled symbols refer to bursters, open symbols to pulsars. . . . 103
Best fit power laws to νh and νLF vs. νu of all sources in our sample
in one plot. To unclutter the plot we scale as indicated in the legend.
Triangles are measurements of νLF , circles of νh . Open symbols refer to
pulsars, filled symbols to bursters. . . . . . . . . . . . . . . . . . . . . . . 105
Measurements of Lh from the multi-Lorentzian timing study by van Straaten
et al. (2002), converted from νmax to ν, plotted together with our measurements of νLF and νh . The best fit from van Straaten et al. (2003) to
combined data of 4U 0614+09, 4U 1728-34 and 4U 1608-52 with a power
law index of 2.01±0.02 is drawn, as well as our best fit power laws to the
combined data of the three sources. . . . . . . . . . . . . . . . . . . . . . . 108
List of Figures
A.1 Top: Color-Color diagram for 1555 observations of 4U1636-53. We opted
to limit our analysis of the regions in this diagram to the ones where LLF ,
Lh and Lu were indicated to be present by Altamirano et al. (2008a).
Bottom: Representative fitted power spectra for different regions of the
color-color diagram. Observations from regions A, B, C, D and E were
used for further analysis, as the hump feature, and the LF and kHz QPOs
appear here (Altamirano et al., 2008a). Observations from the boundary
between region E and F were also included. . . . . . . . . . . . . . . . .
A.2 Color-Color diagram for 495 observations of 4U1608-52. . . . . . . . . .
A.3 Representative fitted power spectra for different regions of the color-color
diagram (see Figure A.2). Regions A, B, C and D were used for further
analysis, as the hump, LF and kHz QPOs only appear here (also identified
by van Straaten et al. (2003)). . . . . . . . . . . . . . . . . . . . . . . . .
A.4 Color-color diagram for 495 observations of 4U0614+09.‘HLF’ stands for
high luminosity flaring. . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.5 Representative fitted power spectra for different regions of the Color-Color
diagram (see Figure A.4). Regions A, B, and Cii were used for further
analysis, as the hump, LF and kHz QPOs only appear here.‘HLF’ stands
for high luminosity flaring. . . . . . . . . . . . . . . . . . . . . . . . . . .
A.6 Color-color diagram for 495 observations of 4U1702-43. Observations
from A - Cii were used for further analysis, as the hump feature, LF
QPO and kHz QPO appear here. . . . . . . . . . . . . . . . . . . . . .
A.7 Representative fitted power spectra for different regions of the Color-color
diagram. Regions A - Cii were used for further analysis, as the hump
Lorentzian, LF QPO and kHz QPO appear here. . . . . . . . . . . . . .
A.8 Top: color-color diagram for 86 observations of KS1731-260. Bottom:
Representative fitted power spectra for different regions of the ColorColor diagram. Regions A - C were used for further analysis, as the
hump feature, and LF and kHz QPOs appear here. . . . . . . . . . . . .
A.9 Color-Color diagram for 131 observations of SAXJ1750.8-2900. . . . . .
A.10 Representative fitted power spectra for different regions of the Color-Color
diagram (see Figure A.9). Regions A - C were used for further analysis,
as the hump feature, and LF and kHz QPOs appear here. . . . . . . . .
A.11 Color-color diagram for 566 observations of AqlX-1. . . . . . . . . . . .
A.12 Representative fitted power spectra for different regions of the color-color
diagram. Regions B and C were used for further analysis, as the hump
feature, LF QPO and kHz QPO appear here. . . . . . . . . . . . . . . .
A.13 Color-color diagram for 493 observations of SAXJ1808.4-3658. . . . . .
A.14 Representative fitted power spectra for different regions of the color-color
diagram. Regions F,G and H were used for further analysis, as the hump
feature, LF QPO and kHz QPO appear here. . . . . . . . . . . . . . . .
A.15 Top: Color-color diagram for 354 observations of HETEJ1900. Bottom:
Representative fitted power spectra for different regions of the color-color
diagram. RegionsB and C were used for further analysis, as the hump
feature, LF QPO and kHz QPO appear here. . . . . . . . . . . . . . . .
A.16 Top: Color-color diagram for 151 observations of IGRJ17480-2446. Bottom: Representative fitted power spectra for different regions of the colorcolor diagram. Region B was used for further analysis, as the LF QPO
and kHz QPO appear here. . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
. 113
. 114
. 115
. 116
. 117
. 118
. 119
. 121
. 122
. 123
. 124
. 125
. 126
. 127
. 129
. 131
List of Figures
xv
A.17 Top: Color-color diagram for 111 observations of XTEJ1807-294 Bottom:
Representative fitted power spectra for different regions of the color-color
diagram. Region B was used for further analysis, as the LF QPO and
kHz QPO appear here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.18 Top: Color-color diagram for 281 observations of IGRJ17511-3057. Bottom: Representative fitted power spectra for different regions of the colorcolor diagram. Regions B and C were used for further analysis, as the
hump feature, LF QPO and kHz QPO appear here. . . . . . . . . . . . . . 134
Chapter 1
Introduction
1.1
Introduction
The detection of X-rays from outside the Solar system in 1962 by Giacconi et al. (1962),
marked the beginning of the research field of X-ray astronomy. The brightest point
sources in the 2-10 keV range of the electromagnetic spectrum are X-ray binaries, that
were extensively studied by numerous satellite missions over the years1 .
X-ray binaries are the rare evolutionary end phase of a stellar binary comprising a compact object and a companion star. They are small in number because the mass of the
binary partners, the orbital separation and the evolutionary phase of the system have
to be just right for such a system to evolve. Formation scenarios also include the tidal
capture of a passing stellar companion by the compact object. The compact object
is either a neutron star or black hole that accretes matter from its stellar companion
resulting in the emission of X-rays via the release of gravitational potential energy.
X-ray binaries are classified depending on the mass of the stellar companion. Accretion in high mass X-ray binaries (HMXBs) can occur via the capture of stellar material
that is either flung off the equatorial region of rapidly rotating Be stars or that comes
from the radiatively driven stellar wind of massive stars (O, B and blue supergiant stars
of 10 M or more).
The typical accretion mechanism found in low mass X-ray binaries (LMXBs) is Roche
lobe overflow. The low mass stellar companion of the compact object, typically a Solar
type star, white or brown dwarf, or a red giant, evolves and expands. Matter flows over
the inner Lagrange point of the system into the gravitational potential of the compact
object, as depicted in Figure 1.1.
1
To name a few: UHURU, ARIEL-V, GINGA, EXOSAT, ROSAT and RXTE.
1
Chapter 1. Introduction
2
In this project we focus on neutron star LMXBs. When we use the abbreviation LMXB,
we always mean a binary comprising a neutron star and a main sequence star, white or
brown dwarf, or a red giant.
Angular momentum conservation prevents the accreting material from falling onto the
compact object directly. The accreting matter forms an accretion disk. The disk emits
radiation in many wavelengths, the shortest of which predominently come from the inner
regions. Assuming the Shakura and Sunyaev (1973) description of an axially symmetric
stationary accretion disk we can approximate a maximum temperature of 107 K to occur
at ∼1 13 of the inner disk radius of 12 km for a 1.4 M neutron star2 . The emission will
mostly be in X-rays at these high temperatures. The emission is time-variable as inhomogeneities occur in the accretion flow due to turbulence and the magnetorotational
instability (Balbus and Hawley, 1991).
General relativity describes the motion of particles in gravitational fields. It can therefore
be tested in extreme conditions by studying the X-ray variability detected from X-ray
binaries, as spacetime is strongly curved in the vicinity of compact objects. Neutron
stars can contain 1.4 to 2.2 times the mass of the Sun, compressed in a ∼10 km radius.
The equation of state (EOS) of matter, the relation between pressure, temperature and
density, is unknown under these extreme conditions. Different radii are predicted for a
certain neutron star mass with different EOS. The analysis of orbital motion constraining mass and radius of the star can therefore help to constrain the equation of state.
In this thesis we test a prediction of general relativity, Lense-Thirring precession. This
frame dragging effect arises in misaligned orbits of test particles with respect to the spin
axis of a central compact object. In the framework of this theory we observe modulation
of the emission due to Keplerian orbital motion from a ring of accreting matter, and due
to the precession of the orbit itself around the spin axis of the compact object.
1.1.1
Accretion
The amount of energy that can be released by accreting a mass m onto an object with
mass M and radius R∗ is the gravitational potential energy, for which the Newtonian
expression is E =
GMm
R∗ .
This way of extracting energy is very efficient; nuclear fusion
would only render 5% of the energy extracted via accretion. As mentioned, we focus
on low mass X-ray binaries in this thesis, comprising an accreting neutron star and a
2
We use T 4 = (1 −
p
ri /r)
3GM Ṁ
8πσr 3
for the effective temperature, with r the radius in the disk,
Ṁ the mass acretion rate approximated at 1018 gram/second, M the mass of the compact object, G
the gravitational constant, ri the inner disk radius, and σ the Stefan-Boltzmann constant (Frank et al.,
2002).
Chapter 1. Introduction
3
donor that is either a main sequence star like our Sun, a white or brown dwarf or a
red giant. When the system evolves, the donor fills its Roche lobe, and matter passes
through the inner Lagrange point into the gravitational potential of the neutron star.
The binary partners orbit a common center of mass that lies closely to (or within) the
neutron star. Due to the angular momentum of the accreting matter, it cannot fall onto
the neutron star in a straight trajectory and starts to orbit it. As stated in Frank et al.
(2002), from the perspective of the neutron star, it is as though matter is squirted at it
from a nozzle rotating in the binary plane. The initial formation of an accretion disk is
attributed to the elliptical orbit of the matter that precesses due to the presence of the
stellar companion. The stream of matter will intersect itself, and lose energy via shocks.
Consequently, the matter will orbit the primary at the lowest energy possible for a given
angular momentum; in a circular orbit.
Figure 1.1: Artist impression of a low mass X-ray binary by M. van der Sluys using
BinSim, a program developed by Rob Hynes.
Energy can dissipate from the system through radiation as a result of heating by collisions in the gas, convection, and magnetic processes. Viscosity that is linked to the
magnetic field and turbulence in the disk enables angular momentum transport outward.
Matter in the disk can approach the compact object as a result of angular momentum
loss (Shakura and Sunyaev, 1973).
When the neutron star has a strong magnetic field, the accretion disk flow can be
Chapter 1. Introduction
4
disrupted. At the radius where the magnetic pressure exceeds the pressure from the infalling gas, the Alfvén radius, matter will follow the magnetic field lines to the magnetic
poles of the neutron star. When the magnetic poles do not coincide with the spin axis of
the neutron star, we observe the source as an X-ray pulsar, as we detect emission from
localized rotating hot spots on the stellar surface (Truemper et al., 1978). The pulse
frequency reflects the spin frequency of the neutron star.
In addition to persistent X-ray emission, LMXBs can show sudden increases in their
X-ray luminosity, so called X-ray bursts. Type 1 X-ray bursts are thought to arise from
sudden unstable nuclear burning of accreted material on the neutron star surface. They
are characterized by a sharp rise and gradual decay of the luminosity, and last from
seconds to minutes. The time between bursts is typically hours to days.
Type 2 X-ray bursts last from seconds to tens of minutes and are repetitive on shorter
timescales of seconds to hours. They are attributed to accretion disk instabilities.
Strong magnetic fields can improve heat transport and stabilize nuclear burning on the
neutron star surface. Therefore, if we detect Type 1 X-ray bursts, this constrains the
magnetic field strength of the neutron star to be <1010 G (Joss and Li, 1980). Surface
phenomena such as bursts and pulsations indicate that the compact object cannot be a
black hole, which is surfaceless.
The spin frequency of the neutron star can be inferred from burst oscillations, as the
nuclear burning starts at a specific location on the surface. For an instance the neutron
star appears pulsar-like with a localized hot spot. Detections of burst oscillations in an
X-ray pulsar confirm that they occur at the neutron star spin frequency (Chakrabarty
et al., 2003).
In the field of X-ray astronomy a source can be described as ’being in outburst’. This
terminology indicates that the source is observable in X-rays; it can show all sorts of
short term variability. A source can suddenly return to quiescence, in which case little
or no emission is detected. This type of longterm variability (up to years) is thought to
be caused by episodic accretion, attributed to instabilities in the accretion disk. Sources
that are inactive most of the time are called transients; persistent sources show the
opposite behaviour.
1.1.1.1
Source Types
We analyze LMBXs using spectral and temporal information. The energy spectra tell
us something about the accretion state of the source. Studying variability of the source
over time, or timing analysis, in the frequency domain yields information on periodic and
aperiodic phenomena in the system. RXTE (the Rossi Timing X-ray Explorer) detected
the predicted (Shvartsman, 1971) ubiquitous millisecond variability from X-ray sources.
Chapter 1. Introduction
5
If this variability is caused by orbital motion and related phenomena in the accretion
disk, this means hot gas is orbiting the primary with velocities of ∼0.5c, which is in
agreement with general relativistic predictions of orbital motion close to neutron stars.
Changes in the strength and number of components that make up the variability in the
frequency domain (see for instance Figure 2.5) are related to changes of the accretion
state of the system, sampled by the X-ray spectral shape. We use the coupling of the
two in this project. A subdivision in classes based on the spectral and timing behavior
of X-ray sources exists. The sources can be divided in Z and atoll sources, based on
energy spectral behaviour and luminosity. The changes in energy spectrum are traced
by X-ray colors, which is the ratio of counts in two X-ray bands. We calculate the soft
color at low (2-6 keV) and hard color at at high energy (6-16 keV). When we plot soft
vs. hard color, sources move through this color-color diagram over time and trace out a
pattern, see Figures 2.2 and 2.3. Originally these patterns were thought to be characteristic for different sourcetypes. Z-sources trace out a ”Z” shaped pattern, and atolls
trace out an atoll (named after partly submerged ring-shaped coral reefs) shaped pattern
(Hasinger and van der Klis, 1989). However, sources have been discovered that show
both Z-type behaviour at high luminosity, and atoll type behaviour at low luminosity
(Homan et al., 2010). Z-sources are all very luminous (∼1038 erg/s), atolls span a range
of luminosities (∼1036−38 erg/s). The main difference between the two source types are
the timing characteristics in the frequency domain when the source is in a particular
accretion state. Both sources show quasi periodic oscillations (QPOs) and band limited
noise in their power spectra. Z-sources show characteristic power spectra in the three
arms of the Z-shape in the color-color diagram (the Horizontal Branch, Normal Branch
and Flaring Branch). Atoll sources show characteristic power spectra in five different
regions of the color-color diagram, for an example see Figure 2.5, and for an overview
see van der Klis (2006).
Generally, the position in the color-color diagram predicts timing characteristics accurately (Kuulkers et al., 1994, Mendez et al., 1997). To increase signal to noise, we use
this correlation to select power spectra that show similar characteristics, by selecting in
color.
1.1.1.2
Frequency Correlations
When we Fourier transform the time-variable signal from LMXBs to the frequency domain, we can obtain power spectra (power vs. frequency) with characteristic features
(more details can be found in Section 2.2.3.2). A delta peak in the power spectrum at
a certain frequency represents a persistent periodic function in the time domain. The
Chapter 1. Introduction
6
pulse frequency of a pulsar will for instance show up as such a peak in the power spectrum. The humps we see in the power spectra of LMXBs cover more than one frequency,
and are therefore called quasi-periodic oscillations, or QPOs see Figure 2.4. These could
be caused by a number of phenomena in the time domain that do not persist on long
timescales, one of which is a damped harmonic oscillator3 . The natural phenomena that
cause the variability in the emission of an LMXB are unknown, but we can imagine
for instance that we are observing a hot blob of accreting matter orbiting the compact
object, disintegrating over time. Different source types (Z, atoll, containing a black hole
or a neutron star, with a high magnetic field or without) show remarkably similar power
spectral features that correlate with each other in the same way. Accretion disks are
a common trait between these different source types, and it is there that theoreticians
look for the origin of the variability.
Well known correlations between power spectral features are the WK-relation at low
frequency (Wijnands and van der Klis, 1999), and the PBK-relation at high and low
frequency (Psaltis et al., 1999a), named after their discoverors. Identification of features
in different sources is often done by using these correlations as a template. At high
frequency, kHz QPOs are detected. In the Lower Left Banana state (see Figure 2.5),
twin kHz QPOs are observed. The frequency difference between these two peaks has
been linked to the spin frequency of the neutron star, in the context of a beat frequency
model (van der Klis, 2006).
1.1.2
Lense-Thirring precession
Lense-Thirring precession is a predicted general relativistic frame dragging effect, that
arises due to the angular momentum of a rotating central object affecting test particles
in orbits tilted with respect to its equatorial plane. The relativistic precession model of
Stella and Vietri (1998) predicts a QPO with a centroid frequency νLT depending on
the Keplerian orbital frequency νK in the same way as the Lense-Thirring precession of
a test particle:
νLT =
2 ν
8π 2 IνK
s
2
= 4.4 × 10−8 I45 m−1 νK
νs ,
c2 M
(1.1)
where M and I are the mass and moment of inertia, νs is the spin frequency of the
neutron star, m is the mass in units of solar mass M , and I45 is the moment of inertia
in units of 1045 g cm2 . Blobs of hot gas in a tilted orbit with respect to the spin axis
of the neutron star are predicted to be detectable at the Keplerian orbital frequency
at the inner disk edge (kHz) and the precession frequency (∼30-50 Hz), giving rise to
3
Others are listed in van der Klis (2006).
Chapter 1. Introduction
7
the QPOs in the power spectra at these particular frequencies. van Straaten et al.
(2003) searched his sample of RXTE data of three sources for the correlation proposed
in Equation 1.1. They found a striking correlation of a Lorentzian in the ∼30-50 Hz
range, Lhump with the upper kHz QPO, Lu (see Figure 2.5). They fit a power law with
an index of 2.01±0.02, see Figure 1.2. As noted by van Straaten et al. (2003), this result
is remarkable due to the different spin frequencies then inferred from Type 1 X-ray
bursts for two of these sources. The power law index is in striking correspondence to
the Lense-Thirring model prediction. However, one would have expected different spin
frequencies to yield different values for I45 /m × νs , which was not seen. As the relation
between spin frequency and burst oscillation frequency had not fully been established
at the time, no strong conclusions were drawn.
Figure 1.2: From van Straaten et al. (2003). Correlation of Lh with Lu for three
LMXBs, a power law index of 2.01±0.02 was found, in striking correspondence with
the Lense-Thirring precession model prediction.
1.1.3
Motivation and Outline
Motivated by this, we set out to check the findings of van Straaten et al. (2003) with
data on other sources and extended data sets on their three sources. As the LenseThirring precession frequency depends on the spin frequency of the neutron star, and
the Keplerian orbital frequency is identified with the upper kHz QPO we focus on all
sources with known spin frequencies that are known to show kHz QPOs in their power
Chapter 1. Introduction
8
spectra. In Chapter 2 we discuss the methods used to obtain our result. We discuss
instruments used and data reduction techniques. We offer a brief explanation of Fourier
techniques. In Chapter 3 we present our results for each source. For some sources
we present coincidental finds, like the detection of a very high frequency QPO in 4U
0614+09. We also attempt to link power spectral features at ∼100 Hz seen in flaring
states of 4U 1702-53, 4U 0614+09 and SAXJ1808.4-3658 to features at low (<40 Hz)
frequency seen in non-flaring states. We compare our findings to the findings of van
Straaten et al. (2003) for each of the three sources reported therein. In Chapter 4, we
offer an interpretation in the framework of the Lense Thirring precession model, and
explore possibilities to reduce the discrepancy between the predictions and what we
observe. We conclude in Chapter 5.
Chapter 2
Method
In this chapter we discuss the instruments and techniques used to acquire and analyze
data. We introduce the telescope that observed the X-ray sources studied in this project,
and elaborate on energy spectral and timing analyses.
2.1
Instrumentation: The Rossi X-ray Timing Explorer
The data for this project were acquired from the Rossi X-ray Timing Explorer (RXTE)
public data archive, found at NASA’s High Energy Astrophysics Science Archive Research Center (HEASARC)1 . The Rossi X-ray Timing Explorer was2 a satellite in a circular low-Earth orbit that was used to observe time-variable astronomical X-ray sources
from 1995 to 2012. It featured a time resolution of several microseconds in combination
with moderate energy resolution (∆E/E) of ∼18% at 6 keV and 15% at 60 keV in the 2
to 250 keV band. The satellite carried three instruments, the Proportional Counter Ar-
ray (PCA), the All Sky Monitor (ASM) and the High Energy X-ray Timing Experiment
(HEXTE). The All Sky Monitor monitored variations in source intensities covering more
than 70% of the Sky in each 90 minute orbit, and enabled rapid response to transient
phenomena. HEXTE and PCA were pointed instruments with complementary energy
ranges. HEXTE covered the 15-250 keV band and PCA covered the 2-60 keV part. The
instruments had time resolutions of 8 µs and 1 µs respectively and their field of view
was ∼1◦ (FWHM) (Jahoda et al., 1996, Zhang et al., 1993). As only data from the PCA
were used in this thesis, we now focus on this instrument and discuss it in further detail.
1
http://heasarc.gsfc.nasa.gov/
RXTE is expected to re-enter Earth’s atmosphere anywhere between now and 2023. Accurately
predicting the re-entry date is not straightfoward due to the unpredictable Solar wind-induced expansion
of Earth’s atmosphere.
2
9
Chapter 2. Method
2.1.1
2.1.1.1
10
PCA: Technical Specifications and Instrument Response
Proportional Counters
The PCA consisted of five identical proportional counter units (PCU’s) with a geometrical collection area of ∼1600 cm2 each. A proportional counter is a device that in
response to an incident photon produces an electrical pulse output that is proportional
to the energy of the photon (Fraser, 1989). It comprises a sealed volume filled with a
mixture of inert gas and quench gas. When an energetic photon enters the volume it
excites the inert gas producing ‘primary’ ion pairs consisting of an electron and a positively charged atom. A positively charged anode wire attracts the electrons, while the
positive ions drift towards a negatively charged kathode. While passing through the gas,
the charged particles leave a trail of ‘secondary’ ion pairs. The number of ion pairs produced is proportional to the energy of the incident photon, and therefore proportional
to the measured charge. The scaling of the amount of primary electrons to the total
amount (primary + secondary) of electrons, is called the gas gain, and can be tuned by
adjusting the voltage across the detector. In the recovery time after a detection, the
proportional counter cannot detect any events for a short time.
The following account is based on the work of Zhang et al. (1993). The proportional
counters that made up RXTE’s PCA were gas-filled (methane and xenon) chambers,
each containing a stack of four wire grids. Xenon acted as inert gas and was ionised
by incoming photons. Methane gas acted as the quench gas. The chamber was divided
into detector cells of ∼1.3 cm × 1.3 cm × 1 m, so that the detection probability did
not vary greatly throughout the detector. The cells had an anode wire in their center
and shared ’walls’ of 16 kathode wires, see Figure 2.1. The 26 cells at the sides and
bottom of the volume acted as an anti-coincidence layer. When an event (or X-ray) was
detected in this layer and elsewhere in the counter simultaneously, it was registered as
a background event. An additional detector layer between two thin polyester windows
filled with propane gas sat on top of the main gas chamber. The polyester windows were
transparent to X-rays but prevented the propane gas from escaping. The layer served as
a front anti-coincidence layer containing a fifth wire grid. In addition, the propane gas
absorbed incident electrons reducing background electron flux (from the Solar wind) in
the top layer of the counter.
A radioactive americium 241 source was mounted on the back plate of the PCU in a small
2-cell alpha particle counter. The X-rays accompanying alpha decay of the americium
source were detected, as well as the alpha particles. Simultaneous detection of both an
alpha particle and an X-ray enabled continuous energy calibration. A collimator was
situated on top of the anti-coincidence propane layer limiting the field of view to ∼1◦
Chapter 2. Method
11
(FWHM). The collimator filtered incoming radiation, off-axis X-rays would not reach
the detector.
Figure 2.1: Schematic drawing of the cross-section of a PCU. RXTE’s Proportional
Counter Array contained five of these proportional counter units. ‘1-3L’ and‘1-3R’
indicate the 6 signal chains of wire grids in the gas chamber. V1 an V2 were the
chained anodes forming the xenon/methane anti-coincidence layer and the propane
anti-coincidence layer (see main text). ‘ALPHA’ indicates the radioactive americium
241 calibration anode signal chain. The collimator was situated on top of the PCU
(Jahoda et al., 2006).
2.1.2
Data System
Each PCU had 9 output channels; 6 from the anode chains in the center of the counter,
two from the anti-coincidence layers, and one from the americium alpha counter, see
Figure 2.1. An analog to digital converter converted pulse height information of an
event to a 256 channel (8 bit) digital pulse height. This conversion caused most of the
deadtime of the detector (the time the detector was inactive after a detection). The gas
recovered within this time-window (Jahoda et al., 2006).
In addition, information from the anti-coincidence layers was passed to the Experiment
Data System (EDS). Information of every detected event in all 5 PCU’s was sent to each
of the 6 Event Analyzers (EA) in the EDS. The EDS added information on the PCU
number and timed the data to a microsecond. The EA’s were programmed in different configuration modes, and produced data that were binned in time and energy in 6
specific ways. Two of the EA’s were configured in standard modes, in order to provide
every observation with uniformly processed data products.
An ”observation” typically contains a few hours of recorded count rate of a source,
in which the data have a homogeneous format. A new observation was started when
the configuration mode of the EA’s changed, they were rebooted, or when a different
source was observed. Each observation was given a name (ObsID) in a specific format;
NNNNN-TT-VV-SSX, where each letter represents a number. NNNNN refers to the
proposal, TT is the target number assigned by the guest observer facility. VV is the
Chapter 2. Method
12
so-called viewing number, corresponding to different observations of the same target.
SS is a sequence number that indicates different pointings of the same recording of a
source. X is used to indicate abnormalities in instrumentation (for instance when the
telescope was slewing to the target during an observation).
Standard 1 data has 0.125 s time resolution, Standard 2 data contain pulse heights in
129 energy channels with a 16 s time resolution(Jahoda et al., 2006, Zhang et al., 1993).
We use the Standard 2 data for the color (spectral) analysis of the sources in our sample.
The other 4 EA configuration modes were programmed according to user-preference. We
use modes that have a time resolution of at least 125 µs to analyse the timing behaviour
of the sources in our sample.
2.1.3
Instrument History and Response Calibration
RXTE’s PCA had a number of technical issues (for example due to ageing of the detector
and meteorite hits) that divide its observing history in 5 ‘epochs’. Epochs differ in
detector gain (see section 2.1.1.1). Consequently, the relation between pulse height and
photon energy varied between epochs, and the amount of observations where all PCU’s
were fully functioning dropped over time (Jahoda et al., 2006). Corrections for these
‘major’ response changes were provided by the RXTE team. In addition, within epochs
gradual changes in response occurred, for which we correct as described in section 2.2.2.
2.2
Data Analysis
2.2.1
Data Filtering
Not all data in the RXTE archive are suitable for further analysis. The field of view
might have been obscured by Earth for part of an observation, the satellite could have
just passed through the South Atlantic Anomaly3 or the instrument might have been
slewing to a target. For this project, we use standard RXTE-recommended data filtering
to create Good Time Intervals (GTI’s). In the GTI, data satisfy the following criteria:
1. The pointing elevation is above 10 degrees. A bright Earth has been known to
affect data where the elevation was around 5 degrees. To be on the safe side, the
3
The South Atlantic Anomaly is a near-Earth region where there is a dip in Earth’s magnetic field.
Satellites are subject to strong radiation in the SAA that can damage sensitive instruments which are
therefore switched off. Charged particles from the Solar wind can interact with spacecraft materials to
produce isotopes that decay radioactively over time and increase the background. Data taken within 30
minutes after SAA passage are therefore discarded.
Chapter 2. Method
13
RXTE-team recommends filtering out data with elevation below 10 degrees. Here,
zero elevation is defined as pointing at the horizon, negative is pointing below the
horizon (i.e. at Earth) and positive is pointing above the horizon (i.e. at the sky).
2. The pointing offset is less than 0.02 degrees.
For data with a countrate below 40 counts per second we use additional criteria:
3. The time since SAA-passage is more than 30 minutes.
4. The electron contamination rate in the propane layer of the detector (see Section
2.1.1.1) is less than 10% of the event rate in the xenon/methane volume. A high
electron flux causes high background, limiting the quality of the data.
5. The number of coincident events in the top layer of the xenon/methane volume in
PCU 0 in epoch 5, and PCU 1 after 2007, is less than a factor of the number of
coincidences in all deeper layers. With this criterion we filter out ’good’ data from
PCU 0 and PCU 1 taken after the loss of the propane layer due to a meteorite hit.
2.2.2
Spectral Analysis
Analyzing the spectral variations of X-ray sources provides an insight into their accretion
states. In this project we quantify the broad band X-ray spectral shape using photometry to define X-ray colors. This method enables us to track source states over short
timescales. Another approach to defining spectra is fitting a model. This, however,
requires co-adding spectra to improve statistics, sacrificing temporal information. We
define an X-ray color as the photon count ratio between two energy bands. In effect,
color is a measure of spectral slope. We calculated two X-ray colors per spectrum, selecting four energy bands from the total spectral energy range. We define the ‘soft’ color
as the ratio between counts in the energy band from 3.5 to 6.0 keV (B) and 2.0 to 3.5
keV (A), and the ‘hard’ color as the ratio between the 9.7 to 16.0 keV (D) and 6.0 to
9.7 keV (C) bands. The energy spectrum of a 5 ks observation of burster 4U 0614+09 is
depicted in Figure 2.2, this observation has a hard color (D/C) of 0.63, and a soft color
(B/A) of 0.96. This observation will show up as a dot in region D of the color-color
diagram, see Figure 2.3.
We extracted a lightcurve and spectrum for each observation using color analysis,
a script developed by the X-ray Timing Group at the Anton Pannekoek Institute. It
utilizes NASA’s FTOOLS package. FTOOLS is a software package with utility programs
designed by NASA to examine data files in the Flexible Image Transport System (FITS)
14
Intensity × (counts/s/keV)2 Hz−1
Chapter 2. Method
Energy (keV)
Figure 2.2: The energy spectrum of a 5 ks observation of burster 4U 0614+09, starting
on MJD50197 (ObsID:10095-01-02-00). To calculate the hard color we divide the count
rate in D by the count rate in C (= 0.63) , and for the soft color we divide the count
rate in B by the count rate in A (= 0.96). This observation shows up in region D of
the color-color diagram, see Figure 2.3.
format. Much used examples are f dump, which is used to convert headers and data of
a FITS table extension to ASCII format; f plot, which plots columns from a FITS file
using a QDP/PLT plot package; f lcol, which lists FITS table column information; and
saextrct, which creates a lightcurve and/or spectrum from RXTE data, using FITS or
ASCII good time intervals (see section 2.2.1).
When we run color analysis on the raw Standard 2 (see 2.1.2) data in the good time
interval, the script first subtracts a time-variable background. It applies pcabackest,
a tool that uses the good time interval and a model4 to create a background file in
the Standard 2 format. color analysis now uses the FTOOL saextrct to extract a
lightcurve and spectrum both from the raw Standard 2 and the Standard 2 background
file, and subtracts the latter from the former. When type I X-ray bursts occur in the
light curve, they are removed5 . color analysis calculates X-ray colors using specified
energy bands (4 bands from 2.0 to 16.0 keV). It performs a linear interpolation between
photon count rates in the detector channels to obtain the count rates in these exact
4
[elv.gt.10.and.offset.lt.0.02.and.num pcu on.ne.0], the most current background model for faint (<
40 counts/s/PCU in the full energy band) and bright observations created by the PCA instrument team
was used. For every observation of a source the average count rate per PCU is determined, and the
appropriate background model is applied.
5
When the local count rate is more than twice the mean of an observation, the program traces the
lightcurve back in time until the difference in count rate between two points is less that 4σ, marking the
start of the burst. The end of the burst is taken to be the point in time where the difference in count
rate between two points is less than 0.5σ. A type I X-ray burst has a rapid rise and a slower decline,
which motivates our choice of σ-values.
Chapter 2. Method
15
1.1
A
E
B
F
Ci
G
Cii
HLF
D
1
Hard Color (Crab)
0.9
0.8
0.7
0.6
0.5
0.4
0.9
0.95
1
1.05
1.1
Soft Color (Crab)
Figure 2.3: Color-color diagram for 495 observations of 4U0614+09.‘HLF’ stands for
high luminosity flaring. Errors on hard and soft color are smaller than the symbols.
energy ranges, based on the channel-to-energy calibration provided by the RXTE team.
We now have the count rates in each energy band per PCU, per 16 seconds. The total
count rate, or intensity, over the energy band from 2.0-16.0 keV is also calculated.
The gain and effective area of the PCUs were slightly different and changed over time
(see Section 2.1.3). Different observations were done with different selections of PCUs
on and off. To correct for this we selected RXTE observations of Crab taken close in
time to, and in the same gain epoch as, our observations and calculated X-ray colors
and intensity for this source in the same way. The spectrum of Crab is supposed to be
constant in the energy range we use to calculate X-ray colors, so when variations occur
in Crab’s X-ray color, they are due to changes in detector response (Hasinger and van
der Klis, 1989, Kuulkers et al., 1994). The count rates of Crab per PCU per 16 seconds
are averaged for each day, and we obtain Crab’s X-ray colors and intensity per PCU per
day. We divide the colors and intensity obtained from our data by the Crab colors and
intensity in the same gain epoch for the corresponding PCU. We average the colors and
intensity over all active PCUs. Summarizing, we now have hard and soft X-ray colors
and intensity corrected for changes in detector response, background, and type I X-ray
bursts, for every 16 seconds, averaged over all active PCUs.
Using these data, we create a color-color diagram in units of Crab (see Figure 2.3 and
Appendix A) by plotting the average hard vs. soft color per observation. These diagrams
are used in this project to assess the accretion state of a source.
Chapter 2. Method
2.2.3
16
Timing Analysis
We use Fourier analysis to assess the variability in the X-ray emission from LMXBs. We
transform the signal from the time to the frequency domain, decomposing it into sine
waves. The subtleties of and reasoning behind using this technique are discussed in van
der Klis (1989). We give a short introduction and overview of techniques used to obtain
our results.
2.2.3.1
Fourier Transforms
In the 18th century, J.B. Fourier realized that any continuous signal could be described
by a sum of sinusoids. He formulated a mathematical transform, mapping functions
from the time domain to the frequency domain and vice versa:
∞
Z
f (t) =
A(ν)e2πiνt dν
(2.1)
f (t)e−2πiνt dt
(2.2)
−∞
Z
∞
A(ν) =
−∞
We can regard the Fourier transform A(ν), amplitude as a function of frequency, as
a different representation of the time domain function or signal f (t)6 . The Fourier
transform allows us to navigate between coordinate systems, depending on how we want
to view our signal7 .
Ultimately, we want to investigate what frequencies make up the X-ray signal from
LMXBs, and link them to frequencies that are predicted by physical models of natural
phenomena. We take the Fourier transform of the signal in the time domain, and view
it in the frequency domain. Because the sampling of our data is not continuous, and
they do not stretch to infinity, we cannot take the transform as given in equations
2.1 and 2.2. We need to use the Discrete Fourier Transform, or DFT instead. The
DFT takes discrete sampling as an input, in our case the number of photons (Nph )
detected in a number of time intervals (NT )8 , and gives the amplitudes of N = Nph sine
waves. To obtain the power in a frequency interval, we apply Parseval’s theorem9 and
square the absolute value of the amplitude we obtain from the DFT. We now have a
6
For a crash course introduction to Fourier Transforms we refer the reader to the Brian Douglas’
YouTube channel: https://www.youtube.com/watch?v=1JnayXHhjlg
7
A complex number describes both phase and amplitude, in the complex plane the length of the line
from the number to the origin is the amplitude, and the angle of this line with the real axis is the phase.
The function A(ν) in equations 2.1 and 2.2 are complex valued. If f (t) is real, the imaginary terms
rendering phase information cancel out when integrating over frequency, as in that case the imaginary
part of A(ν) is an odd function (van der Klis, 1989).
8
If a time interval
R ∞of 16 s contains
R ∞gaps, it is discarded.
9
Total power = −∞ |f (t)|2 dt = −∞ |A(ν)|2 dν, in the Leahy normalization: P (ν) = N2ph |A(ν)|2
Chapter 2. Method
17
power spectrum P (ν). Because of the discrete sampling, the power spectra we obtain
have a number of characteristics. The highest frequency in the power spectrum, the
Nyquist frequency, is half of the sampling rate (and the inverse of the time resolution).
The lowest frequency is the inverse of the length of the time interval (T). To reduce
calculation time, the FFT-algorithm (Fast Fourier Transform) was developed by Cooley
and Tukey (1965). Creatasum, a script developed by the X-ray Timing Group at the
Anton Pannekoek Institute, uses a similar algorithm (radix8) to calculate power spectra.
The calculated power is Leahy normalized (Leahy et al., 1983), in this normalization the
Poisson counting noise level is expected to be approximately 2 (see van der Klis (1989)
for details). In this project, we use data products with a time resolution of 122 µs or
higher10 , take all energy channels into account
11 ,
and choose the time interval T to
be 16 seconds. That gives us a Nyquist frequency of 4096 Hz, and makes the lowest
frequency (and frequency resolution) in the power spectra 0.0625 Hz.
A typical observation of 3 hours, will render 675 power spectra which we average to
increase signal to noise.
2.2.3.2
Power Spectral Analysis
Before calculating the power spectra, we do not perform any background or deadtime
(instrumental effect, see section 2.1.1.1) corrections. We only correct for them after averaging the Leahy normalized power spectra. To do so, we subtract a predicted counting
noise spectrum incorporating dead-time effects, based on the work of Zhang et al. (1995).
The method we use was developed by Klein-Wolt et al. (2004). In order to compare between different power spectra, we renormalize them such that the square root of the
integrated power in the spectrum
equals the fractional root mean square (rms) of the
qR
P (ν)dν = rms, van der Klis (1989)).
variability in the signal (
The result of all the corrections and calculations described is an averaged power spectrum for every observation (see Section 2.1.2) we obtain from the RXTE archive. We
show a typical power spectrum in Figure 2.4, for the source 4U 1728-34, in two different representations. We see characteristic features, humps that peak at a particular
frequency. These features move through the power spectra differently over time, depending on the accretion state (as diagnosed by the shape of the energy spectrum) of
the source. As the peak frequency of a hump increases, its width typically decreases.
10
In section2.1.2 we mentioned the 4 different user modes, or time binnings, that are output to the
RXTE data system. We use Event, Single Bit and Good Xenon modes, all with time resolution of 122
µs or higher.
11
We can opt to include a particular energy range, as source flux and background flux compare
differently depending on the energy of the radiation. High frequency features are typically stronger at
high energy, and low frequency features at low energy, as we are interested in both the low and high
frequency domain, we use all energy channels.
Chapter 2. Method
18
On the right in Figure 2.4 we multiply the renormalized power ((RMS/Mean)2 Hz−1 )
with frequency. By doing this, we enhance features at high frequency.
A delta peak in the power spectrum at a certain frequency represents a persistent periodic function in the time domain. The humps we see in the power spectra of LMXBs
could be caused by a number of phenomena in the time domain that do not persist
on long timescales, one of which is a damped harmonic oscillator12 . An exponentially
damped sinusoid in the time domain transforms to a Lorentzian in the frequency domain,
a function that fits the humps in power spectra well:
P (ν) ∝
(ν − ν0
)2
1
+ ( F W2HM )2
(2.3)
The natural phenomena that cause the variability in the emission of an LMXB are
unknown, but we can imagine for instance, as mentioned in Chapter 1, that we are
observing a hot blob of accreting matter orbiting the compact object, disintegrating
over time.
A model consisting of a combination of several Lorentzians can describe the power
spectra of LMXBs, and enables us to track the different features over time. The quality
factor Q is a measure of the ’sharpness’ (Q=ν0 /F W HM , the higher Q, the sharper)
of the Lorentzian, FWHM is the full width at half maximum, and ν0 is the centroid
frequency of the Lorentzian, where P (ν) reaches its maximum. The centroid frequency
is the frequency at which the Lorentzian peaks in the power vs. frequency representation,
as shown on the left in Figure 2.4.
Very broad Lorentzians are characterized by their FWHM more than their centroid
frequency,
q to deal with this Belloni et al. (2002) introduced the characteristic frequency,
νmax = ν02 + ( F W2HM )2 . This frequency expresses the highest frequency covered by
the Lorentzian. On the right in Figure 2.4 the peak of a Lorentzian is at frequency νmax .
A Lorentzian with a high Q (small FWHM) has a centroid and characteristic frequencies
that are almost equal. In this project, we fit power spectra using centroid frequency,
R
FWHM and integral power ( P (ν)dν, integrating the Lorentzian from −∞ to ∞) to
characterize the Lorentzians. We indicate the centroid frequency of a specific Lorentzian
as νx with x being an identification label. For instance, the upper kHz QPO fitted with
Lorentzian Lu has centroid frequency νu . In case we use νmax , we will specifically mention
this. When we use the symbol ν we always mean centroid frequency, as defined above.
Because of the normalization
of the power spectra, we can recover fractional rms (%) of
qR
the Lorentzian (100
P (ν)dν = rms (%)). When Lorentzians have a low Q, or a large
FWHM, we overestimate the integral power because of our integration boundaries. We
compared our findings to the case where we integrate from 0 to ∞. We find that for
12
Others are listed in van der Klis (2006)
Chapter 2. Method
19
Q = 0.6, we overestimate the rms by 10%. For Q = 0.9, we overestimate by 8%. The
overestimation is negligible when Q = 2. We give the calculated rms-levels, keeping in
mind that we overestimate them by ∼10% for low values of Q.
In the power spectra of pulsars, a strong delta peak occurs at the spin frequency of the
neutron star. Before we fit these power spectra, we always take out the pulsar spike by
removing the frequency bins in which the excess power is concentrated, before rebinning
(RMS/Mean)2 Hz−1
Frequency × (RMS/Mean)2 Hz−1
in frequency.
Frequency (Hz)
Frequency (Hz)
Figure 2.4: A power spectrum of the LMXB 4U 1728-43 fitted with multiple
Lorentzians, plotted in two representations. On the left we plot power vs. frequency,
in this representation the maximum of a Lorentzian is at the centroid frequency ν. On
the right we multiply the power with frequency and plot it vs. frequency, enhancing
features at high frequency. The maximum of the Lorentzian in this representation is
at νmax . ”B” refers to region B in the color-color diagram, See Figure 3.1. The power
spectra of observations from region B were averaged to obtain the power spectrum
depicted here.
Chapter 2. Method
2.2.4
20
Timing Behaviour and Accretion State
We analyze both energy and timing behaviour of the sources in our sample, because
the two are correlated. When a source is in a particular accretion state, it shows a
characteristic energy and power spectrum (Kuulkers et al., 1994, van der Klis, 1989).
In order to study timing behaviour at both low and high frequency in detail, we need to
increase the signal to noise ratio. To realize this, we average power spectra of different
observations. Averaging power spectra that do not have similar characteristics, means
risking loss of information. We prevent this by using the correlation between X-ray
color (or accretion state) and timing characteristics as explained below. A source shows
similar timing features in the averaged power spectra of two different observations when
they have similar X-ray colors(Kuulkers et al., 1994, van der Klis, 1989).
For most sources, the shape traced out over time in the color color diagram enables us
to sort the observations either in hard color or soft color. We first determine regions
in which we can sort the observations in a particular way (the hard or soft color is
roughly constant). We put the observations in order of increasing color (hard or soft)13 .
Other methods to determine regions in the color-color diagram include parametrizing
the shape traced out in the color-color diagram with a spline S and selecting boxes along
the spline, often used successfully for Z-sources (for example Altamirano et al. (2008a),
Di Salvo et al. (2001)). This method however, is not always successful for atoll sources,
that trace out a pattern in the color-color diagram on longer timescales than Z-sources
(for instance 4U1608-52 van Straaten et al. (2003)). The majority of sources in our
sample are atolls, and to standardize our method we opted not to use a spline.
Sources that do not trace out a pattern in the color-color diagram suitable for sorting by
color, typically do not have a lot of observations. In these cases we sort the observations
in time, which reflect changes in color more accurately (see for instance the color color
diagram of KS 1731-260 in Appendix A, where a region is a set of observations close in
time). Subsequently, we create a catalogue of the power spectra of all observations in
the sorting order. We roughly determine regions in the color-color diagram in which the
power spectra of observations have similar characteristics. When we have an indication
of the boundary of regions from previous research with similar data reduction methods,
we use it. For instance for 4U 1636-53, we used the analysis of Altamirano et al. (2008a),
and concentrated on the observations with a hard color > 0.85. Below this hard color,
the features we are focusing on in this project were not detected.
13
We assessed if the timing characteristics in the power spectra of observations of a source at a
particular color changed over time. We compared a power spectral fit to observations of 4U 1636-53
from the beginning of the RXTE mission to power spectral fits to observations at the end of the mission.
The characteristics of the fits changed within errors.
Chapter 2. Method
21
After a preliminary assessment we can decide to further subdivide the regions into subregions in some of which the power spectra show additional features that do not occur
in the rest of the region. We indicate the regions with latin capitals and the subregions
with Roman numeral subscripts: A, Ai , Aii etc. See Appendix A for all color-color
diagrams and the definition of all regions and subregions.
After determining regions in the color-color diagram14 , we average the power spectra
of all observations within a (sub)region, and fit the resulting power spectra with the
multi-Lorentzian model described in the previous section. These power spectra give us
an indication of the timing behaviour of the source accross the color-color diagram, and
allow us to find the regions where the features we are interested in, occur. We identify
the features depending on their location in the spectra using the identification scheme
and naming convention from Altamirano et al. (2008a) and van Straaten et al. (2002),
as Lb , Lb2 , LLF , LLF2 , Lh , LhHz , L` or Lu . This relies on the similarities in timing
behaviour between different sources (i.e. van Straaten et al. (2002)). In Figure 2.5 we
show an example of the application of the identification scheme. The features we want
to analyze in more detail at low frequency (ν < 80 Hz) named Lh and LLF gradually
become less prominent in power spectra as hard color decreases.
Note that this is the sole purpose of the procedure op to this point. No use is made of
the fitted parameters of the ’region’ power spectra in the final analysis. We only want
an indication of the timing behaviour of the source accross the color-color diagram to
guide the further, more precise, analysis described below.
After selecting the regions that have average power spectra containing the features we are
interested in (those with ν < 80 Hz, specifically Lh and LLF ), we create new catalogues
containing the power spectra of individual observations within these regions, again sorted
in color or time. Not to miss any detections of Lh and LLF described above, we do the
same for adjacent regions.
We fit power spectra of the single observations where Lh or LLF , or both, are present,
with the multi-Lorentzian model. So, we limit the risk of broadening narrow features by
only averaging power spectra from a single observation, i.e. close in time. We always use
the full observation, we never split them into shorter time-intervals. To increase signal to
noise, we average power spectra from more than one observation under certain stringent
conditions. We always use all observations that were obtained within a continuous time
interval shorter than 200 ks, with hard and soft colors that change by less than 10%,
and that have similar power spectra. For sources that are highly variable, this means
fewer power spectra can be averaged.
14
We do not aim for perfection here, we only want an indication of the timing behaviour of the source
accross the color color diagram.
Chapter 2. Method
22
Figure 2.5: From Altamirano et al. (2008a). Left: Color color diagram for 4U 1636-53,
with the names of atoll source states indicated in pink. Centers of regions are indicated
with the letters A to N. Right: Identification of features in the average power spectrum
of observations from regions B and J.
As a result, we have measurements of the FWHM, ν, and fractional rms level of fitted
Lorentzians (and Q=ν/FWHM). The identification of features at low frequency (ν <
80 Hz) ( i.e. answering the question, is a feature we see in one power spectrum the
same feature we see in another?) is done based on these parameters. We plot νh and
νLF against the centroid frequency of the upper kHz QPO (νu ), and apply the method
of Motta et al. (2011) of plotting fractional rms and Q-factor against ν to discriminate
between features. We further discuss the identification of features in each source in
Chapter 4.
Ultimately we obtain a set of measurements of the centroid frequency of each of three
Lorentzians through different accretion states of 14 sources, see Table 3.15. We proceed
to asses if these features correlate with one another as predicted by the Lense-Thirring
precession model, by fitting power laws to these frequency measurements of individual
sources, and to the frequencies of all sources together; for details see Chapter 4.
2.2.5
Fitting Routines
We fit power laws to the (νLF , νu ) and (νh , νu ) pairs:
Chapter 2. Method
23
νx = κx × νuαx ,
(2.4)
where we substitute LF and h for x. We only include frequencies of Lorentzians in our
fits detected at >3σ. We plot one measurement against another and consequently obtain
error bars in two dimensions. These error bars are not strictly symmetric. Asymmetric
error bars can have a statistical or systematic origin. If we assume that in our case the
asymmetry has a systematic origin, then an explanation could be that our results are
non linearly dependent on - for instance - a calibration constant (Barlow, 2004). We
regard the difference in errors to be small (negative errors are on average 5% larger
than positive errors), and we symmetrize them by taking a mean between the positive
and negative errors. We can opt to take a geometric, arithmetic or quadratic mean,
however none of these methods is physically well-motivated. As a compromise we take
the arithmetic mean, which renders neither the highest nor the lowest value.
We use a least squares fitting procedure based on the numerical recipe for straight-line
data fitexy, with errors in both coordinates (Press et al., 1992). We convert our frequencies to logspace by taking the natural logarithm. In logspace, the index of a power
law is the slope of a linear function. We obtain best fit values for the index and normalization of the power law (α and κ), and χ2 /dof .
To assess if a model with more parameters (in our case a power law with index α
and normalization constant κ), fits our data significantly better than a model with less
parameters (in our case when we fix the index of the power law) we compare between
χ2 /dof obtained in both cases by using an F-test for additional parameters:
F =
(χ21 − χ22 )dof2
,
(dof1 − dof2 )χ22
(2.5)
where χ21 , dof1 are obtained in the case where we have less parameters or more degrees
of freedom, and χ22 , dof2 are obtained in the case where we have more parameters or less
degrees of freedom. The F-test compares the relative decrease of the sum of squares to
the relative decrease in degrees of freedom, when adding parameters. When F=1, the
relative decrease in sum of squares equals the relative decrease in degrees of freedom,
and we can conclude that the model with an additional parameter does not describe
the data any better. We compare the F-value to the F-distribution, computed with
Numerical Recipe routine betai (Press et al., 1992). The probability (P) is returned
that we obtain our F-value under the assumption that an additional parameter will not
improve the fit. We accept that an addition of parameters improves our fit significantly
when P<0.01.
Chapter 2. Method
2.2.5.1
24
Spearman’s Rank Correlation
We use Spearman’s rank correlation to test if two parameters correlate with each other
at a basic level, as in Altamirano et al. (2010). If one parameter seems to increase
monotonically with the other, Spearman’s rank correlation assesses how well a monotonic function describes the data. The test returns a value ρ that can be tested for its
significance (we use the double tail T-distribution) for a certain number of degrees of
freedom:
ρ=1−
6Σd2i
,
n(n2 − 1)
(2.6)
where n is the size of the sample, and di is the difference in rank number of a value
in both sets. If the lowest value in the first set corresponds to the lowest value in the
second set, the rank numbers are equal and di = 0.
Chapter 3
Data selection & Results
3.1
Results
In this chapter we discuss the results obtained by applying the methods described in
the previous chapter to a sample of 19 X-ray sources containing neutron stars with
known spin frequencies that show kHz QPOs in their power spectra. We divide the
sources into two groups: pulsars (6) and bursters (13), see Table 3.1. When a source
shows millisecond pulsations in its X-ray emission sufficiently persistent to measure its
orbit we classify it as a pulsar, even though in some cases these sources also show X-ray
bursts and burst oscillations. The neutron star spin frequencies of the remaining sources
are known from their burst oscillations1 , so they all show X-ray bursts. It was shown
by van Straaten et al. (2005) that the timing behaviour of at least some millisecond
pulsars is different from other atoll and low luminosity LMXBs. The frequency-frequency
correlations appear to be shifted. The strong magnetic field of pulsars has been suggested
as the cause of this shift (van Straaten et al., 2005). The inner edge of the accretion disk
cannot get as close to the neutron star as in bursters. If this were the cause however,
we would expect to see the effect in all pulsars, which is not the case.
We use 4U 1728-34 as an example of the presentation of our results. We reiterate
once more that all frequencies reported in this thesis are Lorentzian centroid frequencies
and that the subscript ”0” is not used to label frequencies as such. We discuss the
identification of features at low (ν < 80 Hz) frequency in the timing behaviour of each
source. For the color-color diagrams and the fits to the average power spectra from
observations in the regions of the color-color diagram, we refer the reader to Appendix
A.
1
AqlX-1 and 4U 1636-53 were found to show millisecond pulsations during one single brief episode
each by Casella et al. (2008) and Strohmayer and Markwardt (2002) respectively. However, we classify
these sources as burst oscillators because their timing behaviour is very similar to 4U 1728-34, 4U
0614+09 and 4U 1608-52 Altamirano et al. (2008a)
25
Chapter 3. Data selection & Results
Source
4U 1728-34
4U 1636-53
4U 1608-52
4U 0614+09
4U1702-43
KS 1731-260
SAXJ1750.8-2900
Aql X-1
EXO 0748-676
IGRJ17191-2821
XTEJ1739-285
A 1744-361
4U 1915-05
SAXJ1808.4-3658
HETEJ1900.1-2455
IGRJ17511-3057
XTEJ1807-294
IGRJ17480-2446
SAXJ1748.9-2021
26
Classification Spin Freq. Orbital Period
(Hz)
(days)
XB
∼364
0.007
XB
∼581.9
0.158
XB
∼619
0.537
XB
∼414.7
0.034
XB
∼330
XB
∼524
XB
∼601
XB
∼550.3
0.789
XB
∼552.5
0.190
XB
∼292.8
XB
∼1122
XB
∼529
0.036
XB
∼270.3
0.035
XB, XP
401
0.084
XB, XP
377.3
0.058
XB, XP
244.8
0.145
XP
190.6
0.028
XB, XP
11
0.886
XP
205.9
0.039
Table 3.1: Summarizing table of all sources included in our sample. Spin frequency
refers to the neutron star spin frequency, either inferred from burst oscillations or
pulsations. ’XB’ and ’XP’ indicate that the source is an X-ray pulsar and/or an X-ray
burster (Ritter and Kolb, 2003).
3.2
3.2.1
Bursters
4U 1728-34
4U 1728-34 is an atoll source (Hasinger and van der Klis, 1989) and an X-ray burster
(Hoffman et al., 1976). The burst oscillation frequency of the neutron star varies between
360.5-364.2 Hz (Strohmayer et al., 1997). No optical counterpart has been identified and
the orbital period is unknown. The distance to the source is estimated at 4.4-4.8 kpc
using photospheric radius expansion bursts (Galloway et al., 2003).
In timing analysis of 298 ks of data from 1996 focusing on features at low (8-50 Hz)
frequency by Ford and van der Klis (1998) a strong correlation between these features
and the upper kHz QPO is reported. These authors use the same representation of
Lorentzians (ν, FWHM) as we do in our study, and find a power law with index 2.11±0.06
for the relation of νLF with νu , with νLF ranging between 9.0-41.5 Hz. In this paper,
a single Lorentzian is fit at low frequency. The authors mention an indication of an
additional feature in the power spectra at low frequency.
A different multi-Lorentzian timing study of 4U 1728-34 was conducted by van Straaten
et al. (2002) in the νmax -Q representation, using a dataset of 455 ks from 1996 and 1997.
A power law was fitted to a possible relation between features at low frequency and the
upper kHz QPO from combined measurements of these features in van Straaten et al.
(2003) (νmax was converted toν ) in 4U 1728-34, 4U 0614+09 and 4U 1608-52. An index
Chapter 3. Data selection & Results
27
of 2.01 ± 0.02 was found.
We retrieved all data for this source from the RXTE archive, which amounted to 423
observations comprising 1.7 Ms of usable data, between 1996 and 2011. We create a
color-color diagram using all observations, and assess the timing characteristics of the
source along its pattern in this color-color diagram. We find that the hump feature, the
low frequency QPO and upper kHz QPO (Lh , LLF and Lu ) are present in regions Ai ,
Aii , B, C, D, Ei , and Eii of the color color diagram (see Figure 3.1).
We now look for power spectra of single observations within these and adjacent regions,
in which Lh and LLF are present, and fit them with the multi-Lorentzian model. The
details of these fits are presented in Table 3.2. We indicate the region of the color-color
diagram where the observations occur, the starting date in MJD, centroid frequency
(ν) of all fitted Lorentzians, fractional rms amplitudes and Q-factors for Lh and LLF 2 ,
and the observation identification number (ObsID). 30042-01-[01-01/02/03][02-03/04]
for instance indicates that 5 observations were used, namely with the IDs: 30042-0101-01, 30042-01-01-02, 30042-01-01-03, 30043-01-02-03 and 30042-01-02-04. As noted in
Section 2.2.4, when we average power spectra from multiple observations for one fit, we
always use all observations that were obtained within a continuous time interval, and
with hard and soft colors that change by less than 10%.
The identification of features at low frequency (ν < 80 Hz) is done based on these parameters. We plot measurements of νh and νLF against the centroid frequency of the
upper kHz QPO (νu ), and additionally use the method applied in Motta et al. (2011)
of plotting fractional rms amplitudes and Q-factor against ν to discriminate between
features, see Figures 3.3 and 3.5. In power spectra characterized by νu < 700 Hz, we
fit two significant Lorentzians between 5-41 Hz. We name the Lorentzian with low centroid frequency LLF , and the one with high centroid frequency Lh , in accordance to the
naming convention used in Altamirano et al. (2008a) and van Straaten et al. (2002). In
this range of νu , νLF < 30 Hz and νh < 50 Hz and QLF >Qh . As a function of their
own centroid frequency, the two Lorentzians have clearly different Q and fractional rms
amplitude, see Figure 3.5, so at frequencies < 30 Hz we can also distinguish between
the two based on these parameters.
When νu exceeds 800 Hz (see Table 3.2), we can fit only a single Lorentzian around 40
Hz, and based on the centroid frequency we are able to identify it as LLF , even though
identification in Figures 3.3 and 3.5 is not secure, as Q-factors and fractional rms amplitudes of LLF and Lh converge for higher centroid frequencies.
2
The Q-factors and fractional rms amplitudes of the other fitted Lorentzians can be found in the
digital appendix (see B) (see B).
Chapter 3. Data selection & Results
28
A fit to a power spectrum in which the upper kHz QPO is at νu =802 Hz, where we
detect Lorentzians at ν =35 Hz (3σ) and tentatively ν=65 Hz (2σ) (see Figure 3.4 and
Table 3.2) offers an additional clue for the identification of the single Lorentzian we fit
when νu exceeds 800 Hz. When plotting all detected centroid frequencies against the
upper kHz QPO, we see that the 65 Hz Lorentzian follows the Lh relation, and that
the simultaneously present ( 3σ) Lorentzian with ν=35 Hz follows the LLF relation, see
Figure 3.3, confirming our identification of the low frequency component when νu > 800
Hz as LLF . The Q of this component is high, similar to that of LLF at lower νu .
It is possible, however, that the single Lorentzian when νu > 800 Hz is a blend of Lh
and LLF . We do not detect both Lorentzians in this frequency domain of νu . For these
reasons, we opted not to include this feature in our assessment of the power law indices
of νh or νLF vs. νu .
In the other sources in our sample, we use this result to limit our analysis to the frequency
domain in which both LLF and Lh are present, where we have two diagnostics for
identification of a feature.
Fitting a power law as defined in Equation 3.1 using νLF and νh , yields: αh = 2.67±0.09
( χ2 /dof =11.4/9) and αLF = 2.76±0.07 ( χ2 /dof =3.7/8).
νx = κx × νuαx
(3.1)
Finally, we note that, when both Lorentzians are present in the same power spectrum,
the ratio of νLF :νh in this source on average is 0.595±0.055.
Chapter 3. Data selection & Results
29
1.1
Ai
G
Aii
H
1.05
B
I
C
J
D
K
Ei
L
Eii
M
F
1
Hard Color (Crab)
0.95
0.9
0.85
0.8
0.75
0.7
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Soft Color (Crab)
Figure 3.1: Color-color diagram for 423 observations of 4U1728-34. Each dot represents the energy spectral shape of one observation of a few ks. The position in this
diagram correlates with the shape of the source power spectrum, as can be seen in
Figure 3.2. Error bars are smaller than the symbols (typically <0.5%).
30
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Figure 3.2: Representative fitted power spectra for different regions of the color-color
diagram (see 3.1). Observations from regions Ai , B, C, D, Ei and Eii were used for
further analysis, as the hump feature or LF QPO, and kHz QPOs appear here.
453/326
457/326
375/326
387/327
343/324
395/326
360/326
309/327
330/326
363/326
377/326
296/323
379/323
330/326
329/326
363/323
χ2 /dof
Lb
ν (Hz)
0.54±0.05
0.63±0.05
0.65±0.12
0.33±0.05
2.34±0.20
1.05±0.09
0.80±0.15
0
1.21±0.40
1.45±0.52
1.87±0.81
15.65±0.54
18.72±0.36
18.17±0.87
22.21±0.72
21.32±0.32
Lb2
ν (Hz)
0
2.49±0.29
2.33±0.54
3.79±0.43
6.4±1.4
2.64±0.70
LLF
ν (Hz)
5.59±0.08
6.18±0.13
8.08±0.13
9.45±0.16
11.04±0.09
14.60±0.13
20.01±0.51
22.7±0.3
27.2±0.3
26.1±0.4
35.4±0.3
41.6±0.4
44.06±0.89
46.8±0.3
45.5±0.7
LLF
rms (%)
4.06±0.40
2.40±0.29
3.50±0.39
6.00±0.72
3.74±0.26
4.89±0.34
5.29±1.01
6.45±0.45
5.38±0.41
4.79±0.81
5.64±0.28
4.35±0.26
3.74±0.56
4.90±0.98
3.46±0.31
LLF
LLF2
Lh
Lh
Lh
LhHz
L`
Lu
Q
ν (Hz)
ν (Hz)
rms (%)
Q
ν (Hz)
ν (Hz) ν (Hz)
2.97±0.47 8.81±0.11 10.48±0.52 14.67±0.63 0.71±0.06
0
380±12
4.44±0.99
9.54±0.22 16.70±0.31 0.59±0.03 39.5±30.0
412±13
4.00±0.88
13.43±0.36 14.42±0.42 0.79±0.06 62.8±38.1
467±12
6.25±0.12 16.05±0.45 0.66±0.04
143±7
324±13
1.74±0.31
16.30±0.60 14.64±0.53 0.75±0.06
129±7
479±8
4.29±0.60
17.71±0.35 14.07±0.29 0.78±0.04
140±5
501±7
2.83±0.34
24.1±0.6 11.67±0.37 0.93±0.06
142±6
556±4
3.16±1.17
35.5±3.0 8.69±1.83 1.33±0.63 177±28
625±10
2.24±0.34
38.8±1.6 6.59±0.65 1.69±0.33 126±14
654±5
2.96±0.41
43.4±1.4 5.66±0.58 1.87±0.38 128±10
698±6
3.98±1.31
40.7±4.5 7.86±1.71 1.15±0.46 134±11
689±6
2.42±0.20
65.1±1.5∗ 2.29±0.72 4.20±2.22 62.3±16.3
802±1
3.20±0.39
82.7±6.4 518±22 847±1
4.52±1.27
65.5±25.3
869±3
3.16±1.17
103.4±21.6
873±3
3.67±0.69
92.6±4.5 567±14 901±2
Table 3.2: Parameter values of the multi-Lorentzian fits to power spectra of 4U 1728-34. All Lorentzians exceed a 3σ confidence level, the errors use
∆χ2 = 1.0. The ∗ starred fit value was detected at a 2.0σ level. ”0” means the feature is present in the spectrum, but is (fixed to be) zero-centered.
MJD
ObsID
Number
Region
Start
of PS. of CCdiagram
50143.1
10073-01-10-01
781
Ai
51086.3
30042-03-01-00
594
Ai
51127.7
30042-03-10-00
255
Ai
51198.1
40033-06-01-00
589
Aii
50136.9
10073-01-07-000
1056
B
51128.9
30042-03-13-00
622
B
51201.9
40033-06-02-01
1008
B
54142.7
92023-03-45-10
114
B
52336.2
70028-01-01-070
1092
C
51205.9
40033-06-02-04
398
C
51727.5
50023-01-30-00
191
D
51133.3 30042-03-[14-00/01/02][15-00] 1290
Ei
51209.7
40033-06-03-020
940
Ei
53828.6
92023-03-16-00
139
Ei
51669.5
50023-01-14-00
158
Ei
50128.7
10073-01-01-00
658
Ei
Chapter 3. Data selection & Results
31
Chapter 3. Data selection & Results
32
100
hump
LF
hump 2σ
ν (Hz)
40
10
5
1
100
200
500
1000
νu (Hz)
Figure 3.3: Centroid frequencies (ν) plotted against the centroid frequency of the
upper kHz QPO (νu ) of 4U 1728-34. We plotted these values for all other sources in
our sample in lighter colors as a reference. All components exceed a 3σ confidence level
except for the 2.0σ detection in a selection of observations (from region Ei in the colorcolor diagram) of a Lorentzian with ν = 65 Hz that follows the same relation with νu
as Lh . The simultaneously present Lorentzian with ν = 35 Hz, detected at 3σ, follows
the same relation as LLF . When νu exceeds 800 Hz, Lh is not detected.
33
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Figure 3.4: Multi-Lorentzian fit to 1290 averaged power spectra (see Table 3.2) from 5
subsequent observations starting at MJD 51133.3. of 4U 1728-34. We find a Lorentzian
characterized by ν = 65 Hz at a 2σ confidence level. We identify this to be Lh , and νLF
= 35 Hz. This power spectrum is the result of averaged power spectra from observations
that are close both in time and color, a careful subselection of the rough averaged power
spectra we show in Figure 3.2.
Chapter 3. Data selection & Results
34
hump
25
hump
LF
LF
hump 2σ
hump 2σ
10
15
Q
rms (%)
20
10
1
5
0
10
20
30
40
ν (Hz)
50
60
70
80
0
10
20
30
40
50
60
70
80
ν (Hz)
Figure 3.5: Fractional rms amplitudes and Q-factors of Lh and LLF fitted in power
spectra of 4U1728-34 plotted against their centroid frequency. We plot fractional rms
amplitudes and Q values of all other sources in our sample (including pulsars) in lighter
colors as a reference. All components exceed a 3σ confidence level except for the 2.0σ
detection of a Lorentzian with ν = 65 Hz. Lh is harder to distinguish from LLF in
power spectra characterized by an upper kHz QPO at >802 Hz, or when its centroid
frequency >40 Hz. See the main text for a discussion.
Chapter 3. Data selection & Results
3.2.1.1
35
Comparing to van Straaten et al. (2002)
In the analysis of the timing behaviour of 4U 1728-34 by van Straaten et al. (2002)
the partly overlapping features we identify separately as LLF and Lh , are fitted and
identified with one Lorentzian. We convert the νmax values reported in van Straaten
et al. (2002), originally measured by Di Salvo et al. (2001), to centroid frequency ν and
plot them together with our measurements of νh and νLF , see Figure 3.6. We see that
when νu ≈500, van Straaten et al. (2002) report frequencies intermediate to ours. The
data analysis in Di Salvo et al. (2001) is slightly different from ours. To calculate the
hard and soft colors for every 16 s of data, energy bands 2.0-3.5, 3.5-6.4, 6.4-9.7 and
9.7-16.0 keV were used. We calculate colors for each observation, and use the bands
2.0-3.5, 3.5-6.0, 6.0-9.7 and 9.7-16.0 keV. Di Salvo et al. (2001) select data segments
with similar colors, and produce 19 power spectra for 455 ks of data. On average, they
add more data to produce one power spectrum and have larger error bars on the colors
than we do. We fit averaged power spectra of single observations, so that there is less
risk that secular changes in the relation between colors and QPO frequencies affect the
average power spectra (Prins and van der Klis, 1997, van Straaten et al., 2003). On the
other hand, we ignore any spectral changes within a single observation.
We expect these differences in data analysis to have only a small effect on the power
spectrum of a single observation, and analyze an observation that was part of the dataset
analyzed by Di Salvo et al. (2001); 10073-01-07-000. We fit the power spectrum of this
observation with 5 Lorentzians as in Di Salvo et al. (2001), and compare the result to
our best fit with 6 Lorentzians, see Figure 3.7. We do an F-test (see Section 2.2.5) to see
if the inclusion of a 6th Lorentzian is a significant improvement and find F(ν1 , ν2 )=26
and P=0.53×10−14 ; we conclude that it is. When we use 5 Lorentzians, we fit LLF and
Lh with one Lorentzian, with centroid frequency 11.1 Hz. This centroid frequency is
lower than νh (16.3 Hz), and higher than νLF (9.5 Hz), νu =497 Hz. This explains the
frequencies reported in Di Salvo et al. (2001), van Straaten et al. (2002) intermediate to
ours; the Lorentzian we identify as LLF is not fitted in those analyses and consequently
their ’hump’ feature fits a blend of LLF and Lh . The importance of properly accounting
for blends in power spectral fits was previously emphasized by Klein-Wolt and van der
Klis (2008).
We detect frequencies identified as νh ranging from ∼20-45 Hz with νu ≈500-700 Hz that
are not reported by van Straaten et al. (2002); see Figure 3.6. When we fit the power
spectrum of observation 40033-06-02-04 with 4 Lorentzians as in van Straaten et al.
(2002) instead of 5, we fit a single broad Lorentzian with ν=28.35±0.65 Hz instead of
two Lorentzians with ν=27.2 Hz and ν=43.4 Hz (χ2 /dof =446/329), see Figure 3.9. This
is close to the frequency reported in van Straaten et al. (2002) (ν=26.7 Hz) for ”interval
7”, characterized by νu =700 Hz (see Figure 3.8) and to νLF we detect in our fit with
Chapter 3. Data selection & Results
36
5 Lorentzians. The inclusion of a 5th Lorentzian fits the data significantly better, with
F(ν1 , ν2 )=24.8 and P=0.17×10−13 . So, in this case van Straaten et al. (2002)’s value
is not intermediate to our two frequencies, but instead close to the lower one, which
dominates due to its higher Q.
In light of this comparative analysis, we conclude that LLF becomes stronger compared
to Lh with increasing νu . The single Lorentzian fitted in van Straaten et al. (2002) to
the two partly overlapping Lorentzians we identify as LLF and Lh , has an intermediate
centroid frequency to νLF and νh at low νu , and has a similar centroid frequency to LLF
at higher νu (see Figure 3.6).
100
hump
LF
Van Straaten 2002
hump 2σ
ν (Hz)
40
10
5
200
500
1000
νu (Hz)
Figure 3.6: Measurements of Lh from the multi-Lorentzian timing study by van
Straaten et al. (2002) (only of 4U 1728-34), converted from νmax to ν, plotted with
our measurements of νLF and νh of 4U 1728-34. The best fit from van Straaten et al.
(2003) to a combination of data from 4U 0614+09, 4U 1728-34 and 4U 1608-52 with a
power law index of 2.01±0.02 is drawn, as well as our best fit power law indices (only
to data of 4U 1728-34) (αh =2.67, αLF =2.76). The frequency of the 2σ Lorentzian was
not included in this fit.
37
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Figure 3.7: Left: Fit with a 5-Lorentzian model to the power spectrum of an observation of 4U1728-34 (ObsID: 10073-01-07-000) as in Di Salvo et al. (2001), van Straaten
et al. (2002), (χ2 /dof =425/327). Right: Fit with 6 Lorentzians (χ2 /dof =343/324).
Figure 3.8: Power spectrum reproduced from van Straaten et al. (2002) of 4U 172834. In their ”interval 7” Lorentzians are fitted with ν=26.7 Hz and νu =706 Hz (with
νmax converted to centroid frequency).
38
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Figure 3.9: Left: Fit with a 4-Lorentzian model to the power spectrum of observation 40033-06-02-04 of 4U 1728-34 as in van Straaten et al. (2003) (see Figure 3.8,
χ2 /dof =446/329). Right: Fit with 5 Lorentzians (χ2 /dof =363/326).
Chapter 3. Data selection & Results
3.2.2
39
4U 1636-53
Classified as an atoll source based on spectral and timing analysis of observations with
EXOSAT by Hasinger and van der Klis (1989), 4U 1636-53 is the most frequently observed source in our sample. It is a Galactic source at a distance of 5.5 kpc (van Paradijs
and White, 1995) and has an orbital period of ∼3.8 hours (van Paradijs et al., 1990).
It has a secondary of 0.36 M (assuming it is a main sequence star). The neutron
star spin frequency is 578-582 Hz as inferred from burst oscillations (Giles et al., 2002),
most probably 581.9 Hz as seen during a brief interval of pulsations during a superburst
(Strohmayer and Markwardt, 2002).
Altamirano et al. (2008a) conducted a multi-Lorentzian timing study (in the νmax , Q
representation) using a dataset from between 2001 and 2004. Lh , LLF and Lu were
identified in this study. We retrieved all data for this source from the RXTE archive,
which amounted to 1555 observations comprising 4.4 Ms of usable data, from between
1996 and 2011. Our analysis of the timing behaviour of 4U 1636-53 was performed in the
same way as that of 4U 1728-34 described in the previous section. But for 4U 1636-53,
relying on the timing analysis performed by Altamirano et al. (2008a), we only analyze
averaged fitted power spectra for regions of the color-color diagram in which Lh and LLF
were indicated to be present by the authors, see Appendix A.
The details of the fits to single observations are summarized in Table 3.3. Fitting the
power law from equation 3.1 yields αh = 2.53±0.17 and αLF = 2.44±0.15.
Chapter 3. Data selection & Results
40
100
hump
LF
ν (Hz)
40
10
5
1
100
200
500
1000
νu (Hz)
Figure 3.10: In analogy to Figure 3.3. νh and νLF plotted vs. νu for 4U 1636-53.
When νu exceeds 610 Hz, Lh cannot be identified unambiguously.
Chapter 3. Data selection & Results
41
hump
hump
LF
LF
25
10
15
Q
rms (%)
20
10
1
5
0
10
20
30
40
ν (Hz)
50
60
70
80
0
10
20
30
40
50
60
70
80
ν (Hz)
Figure 3.11: In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of 4U 1636-53 plotted against their centroid frequency.
0
-
Lb
Lb2
ν (Hz)
ν (Hz)
0.69±0.04
0.94±0.19
2.41±0.16
0
1.01±0.50 5.95±0.12
353/330
0
293/324 2.61±0.16
374/327
0
361/327
0
359/326
0
364/324
376/329
270/324
373/323
χ2 /dof
1.31±0.20 179.75±20.68
0.70±0.07
105±18
0.89±0.52
0.80±0.20 225.2±15.9
3.10±1.51
251±15
-
LLF
LLF
Lh
Lh
Lh
LhHz
Ll
rms (%)
Q
ν (Hz)
rms (%)
Q
ν (Hz)
ν (Hz)
5.39±0.43 2.75±0.38 4.47±0.12 14.06±0.75 0.71±0.08
0
3.87±0.77 3.64±1.41 10.85±0.90 15.64±0.91 0.55±0.10
9.35±1.99 0.98±0.36 13.60±1.16 14.03±1.07 0.71±0.13 162.86±9.27
4.44±1.59
11.39±1.00 15.61±0.97 0.50±0.08 142.85±15.59
-
23.57±0.73 7.62±0.61
6.58±0.13 6.37±0.80 1.86±0.38 14.48±0.60 14.81±0.59
8.98±0.14 3.20±0.49 4.95±1.57 20.85±0.57 10.94±0.80
21.34±2.25 8.04±1.61
14.72±0.19 2.49±0.49 8.42±4.41 34.05±1.09 3.56±1.00
LLF
ν (Hz)
1.93±0.02
5.03±0.11
5.53±0.54
3.08±0.59
502±8
440±13
450±14
556±5.1
602±5
Lu
ν (Hz)
289±33
360±32
425±17
418±20
Table 3.3: In analogy to Table 3.2: parameter values of the multi-Lorentzian fits to power spectra of 4U 1636-53. Distinguishing between Lb and
Lb2 is not straightforward in some cases due to poor statistics.
MJD
ObsID
Number
Region
Start
of PS. of CCdiagram
53816.2
91152-05-01-00
1628
A
55336.1
93082-06-04-00
902
A
53028.7
80425-01-04-02
278
B
55084.1 [94087-01-32-10][94437-01-02-000/00] 3274
B
[94437-01-01-00/000/010/010]
52295.8
60032-05-10-000/00
1661
C
52545.9
60032-05-21-000/00
1255
C
53612.1
70043-01-01-000/00
1885
C
52295.8 60032-05-[10-000/00][11-01/00/02]
3170
D
52299.1
60032-05-11-02
907
E
Chapter 3. Data selection & Results
42
Chapter 3. Data selection & Results
3.2.3
43
4U 1608-52
4U 1608-52 is a LMXB that shows outbursts on timescales of 100 days to years (Lochner
and Roussel-Dupre, 1994). It was first discovered in the Norma constellation by Tananbaum et al. (1976). No optical counterpart was found. Based on its X-ray spectral
variability Hasinger and van der Klis (1989) classified 4U 1608-52 as an atoll source.
The spin frequency of the neutron star is ∼619 Hz as inferred from burst oscillations. In
van Straaten et al. (2003) source states were studied using all RXTE data from March
1996 to May 2000. LLF was not detected. As described above in Section 3.2.1, a power
law was fitted to combined data in that study (νmax was converted to ν ) of 4U 1728-34,
4U 0614+09 and 4U 1608-52 and an index of 2.01 ± 0.02 was found.
We retrieved 1072 observations comprising 2.1 Ms of usable data from the RXTE archive.
We detect Lh and Lu in regions A, B and C of the color-color diagram (see Appendix
A), and focus on observations from these and adjacent regions. The details of the multiLorentzian fits to these data can be found in Table 3.4. We fit LLF or Lh and Lu
simultaneously in power spectra where νu is between ∼200 and 500 Hz. For rms levels
and Q see Figure 3.13.
A fit to νh and νLF vs. νu yields: αh = 2.43 ± 0.16 (χ2 /dof = 6.8/2) and αLF =
2.40 ± 0.24 (χ2 /dof =3.7/2).
3.2.3.1
Comparing to van Straaten et al. (2003)
(In analogy to Section 3.2.1.1) In the above-mentioned analysis of the timing behaviour
of 4U 1608-52 by van Straaten et al. (2003), a narrow low frequency Lorentzian and Lh
were simultaneously detected in the averaged power spectrum of region ’B’ of their colorcolor diagram, see Figure 3.15 (left). They combined data of 4U 1728-34, 4U 0614+09
and 4U 1608-52 to assess the frequency correlation with the upper kHz QPO. LLF was
not detected in the other sources, while the timing behaviour overall showed very similar characteristics. The narrow low frequency Lorentzian was therefore omitted from
the analysis of the frequency correlation, and identified as LLF . We convert the νmax
reported in van Straaten et al. (2003), to centroid frequency ν and plot them together
with our measurements of νh and νLF , see Figure 3.14. We see that when νu ≈400
Hz, van Straaten et al. (2003) measure a Lorentzian with a centroid frequency of ≈8.5
Hz (νmax =19.8 Hz) that we do not. The data reduction in van Straaten et al. (2003)
is almost identical to ours. The authors calculated colors per 16 second time interval,
where we calculate them per observation.
Chapter 3. Data selection & Results
44
100
hump
LF
hump 2.2σ
ν (Hz)
40
10
5
1
100
200
500
1000
νu (Hz)
Figure 3.12: In analogy to Figure 3.3, for 4U 1608-52. All components exceed a 3σ
confidence level except for the 2.2σ detection in interval B of a Lorentzian with ν=12
Hz. It follows the same relation with νu as νh . The simultaneously present Lorentzian
with ν=3.9 Hz, detected at 3σ, follows the same relation with νu as νLF . When νu
exceeds 520 Hz, Lh is not detected.
We attempt to reproduce the power spectrum in which the ∼8.5 Hz Lorentzian is fitted in van Straaten et al. (2003), using observations from around MJD 50440 (in the
decay of the 1996 outburst of the source). We use observations 10094-01-[09-01][10000/00/010/01/020] starting on MJD 50444. To increase signal to noise, we need to
average power spectra from observations with hard and soft color that differ by more
than 20%. Therefore we do not use these results in our analysis elsewhere and for our
interpretation.
The resulting power spectrum is similar to the one reported in van Straaten et al. (2003),
see the right panel (C) of Figure 3.15 and Figure 3.16, although we fit Lu at a somewhat lower frequency (ν=380 Hz instead of 400 Hz). We fitted the power spectrum of
this observation with 3 Lorentzians, as in van Straaten et al. (2003), and compare to
a fit with 4 Lorentzians, see Figure 3.16. We do an F-test for additional terms (see
Section 2.2.5) to see if the inclusion of the 4th Lorentzian is a significant improvement.
We find F=3.75 and P=0.011; we conclude that it is close to significant. When we
use 3 Lorentzians, we fit the partly overlapping LLF and Lh with one Lorentzian, with
ν=15.6±1.9 Hz (νmax =19.4±1.8 Hz). The centroid frequency of this Lorentzian is lower
Chapter 3. Data selection & Results
45
hump
25
hump
LF
LF
hump 2.2σ
hump 2.2σ
10
15
Q
rms (%)
20
10
1
5
0
10
20
30
40
ν (Hz)
50
60
70
80
0
10
20
30
40
50
60
70
80
ν (Hz)
Figure 3.13: In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of 4U 1608-52 plotted against their centroid frequency.
than νh (20.7±2.4 Hz) in the fit with 4 Lorentzians, and higher than νLF (11.8±0.35 Hz).
νu is 340±34 Hz (νmax =368±27 Hz). In Figure 3.14 we plot these frequencies in pink.
We see that the lowest and highest frequencies, thought to be νLF and νh respectively,
do not coincide with the correlations traced out by other frequencies measured. We attribute this discrepancy to the averaging of dissimilar power spectra indicated by hard
and soft colors that differ by more than 10%. We also note that we do not reproduce
the ∼8.5 Hz frequency as reported in van Straaten et al. (2003). We do not know the
reason for this discrepancy. In section 3.2.1.1 we present similar findings for 4U 1728-34.
Chapter 3. Data selection & Results
46
100
hump
LF
Van Straaten 2003
comparison
LF detected in Van Straaten 2003
ν (Hz)
40
10
5
1
100
200
500
900
νu (Hz)
Figure 3.14: In analogy to Figure 3.6, for 4U 1608-52. Measurements of Lh from
the multi-Lorentzian timing study by van Straaten et al. (2003) (only of 4U 1608-52),
converted from νmax to ν, plotted with our measurements of νLF and νh . The best fit
to combined data of 4U 0614+09, 4U 1728-34 and 4U 1608-52 with a power law index
of 2.01±0.02 is drawn, as well as our best fit power law indices (αh = 2.43 ± 0.16 and
αLF = 2.40 ± 0.24). For these fits only data from 4U 1608-52 was used. Our attempt
to recreate the frequencies measured by van Straaten et al. (2003) is plotted in pink.
The gold colored centroid frequency belongs to the Lorentzian drawn with a solid line
in Figure 3.15 (left). It was omitted from the assessment of a possible correlation with
νu by van Straaten et al. (2003). See main text for a discussion.
509/468
523/324
299/324
254/326
362/329
χ2 /dof
Lb
Lb2
ν (Hz) ν (Hz)
0.19 ±0.07
0.32±0.07
2.02±0.11
4.65±0.43
1.04 ±0.77
-
LLF
ν (Hz)
2.13±0.04
2.38±0.02
3.91±0.04
10.02±0.13
-
LLF
rms(%)
1.98±0.39
2.80±0.26
6.71±0.61
6.28±0.92
-
LLF
LLF2
Lh
Q
ν (Hz)
ν (Hz)
5.11±2.11 4.31±0.072 5.58±0.34
5.97±1.25 4.93±0.07
5.17±0.34
5.07±0.93 8.20±0.18∗ 12.01±0.95∗∗
4.54±1.26
23.50±1.62
28.53±0.61
Lh
rms (%)
11.82±1.12
13.24±1.30
8.06±2.67
9.90±1.04
3.79±0.55
∗
Lu
ν (Hz)
230±15
279±13
317±22
482±24
519±7
values were detected at a
Lh
LhHz
Ll
Q
ν (Hz)
ν (Hz)
0.65±0.12 19.01±5.71
0.52±0.10
0
1.48±0.66
0
1.08±0.25
4.4±1.5 194.45±24.66
-
Table 3.4: In analogy to Table 3.2: fit parameters of the multi-Lorentzian fits to power spectra of 4U 1608-52. Starred
2.8σ confidence level, ∗∗ at 2.2σ.
MJD
ObsID
Number
Region
Start
of PS. of CCdiagram
52236.3 60052-03-02-00/01/02/03
2424
A
50925.5
30062-02-02-000/00
1195
A
54270.5
92401-01-09-00
50
B
54392.8
93408-01-17-00
54
D
52233.3 60052-[02-06/07-00][03-01-00]
462
D
Chapter 3. Data selection & Results
47
Chapter 3. Data selection & Results
48
Frequency × (RMS/Mean)2 Hz−1
Figure 3.15: Power spectra from van Straaten et al. (2003) of 4U 1608-52. Left: LLF
is detected simultaneously with Lh in region ’B’ of their color-color diagram. Right:
in region ’C’ of their color-color diagram Lorentzians are fitted with νmax,h =19.8 and
νmax,u =474 Hz, or converted to centroid frequency νh =8.5 Hz and νu =400 Hz.
Frequency (Hz)
Figure 3.16: Left: Fit with a 3-Lorentzian model to the power spectrum of observations 10094-01-[09-01][10-000/00/010/01/020] of 4U 1608-52 as in van Straaten
et al. (2003)(Figure 3.15, χ2 /dof =438/333).
Right: Fit with 4 Lorentzians
(χ2 /dof =424/330).
Chapter 3. Data selection & Results
3.2.4
49
4U 0614+09
4U 0614+09 is an atoll source (Mendez et al., 1997) and an X-ray burster Swank et al.
(1978). The spin frequency of the neutron star, inferred from burst oscillations, is ∼414.7
Hz (Strohmayer et al., 2008). 4U 0614+09 is thought to be an ultracompact binary com-
prising a white dwarf or non-degenerate Hydrogen-deficient stellar secondary, for reasons
summarized in Hakala et al. (2011) and Kuulkers et al. (2010). It is not understood how
the source can show Type I X-ray bursts (due to unstable burning of Hydrogen or Helium). The distance to the source is 3.2 kpc Kuulkers et al. (2010).
van Straaten et al. (2000) used 265 ks of data recorded between 1996 and 1999 to study
correlations between timing and spectral properties in 4U0614+09. Another timing
study of 4U 0614+09 was done later by van Straaten et al. (2002) in the νmax , Q representation, using an additional dataset from September 2000 and May 2001. As noted
above, a power law was fitted to combined data in this latter study (νmax was converted
to ν) of 4U 1728-34, 4U 0614+09 and 4U 1608-52 and an index of 2.01 ± 0.02 was found.
We retrieved all data for this source from the RXTE archive, which amounted to 494
observations comprising 2.75 Ms of usable data, between 1996 and 2011.
LLF or Lh and Lu occur simultaneously in regions A, B and Cii of the color-color
diagram. The details of the fits to (a selection of) single observations from these regions
are summarized in Table 3.5.
In the literature (i.e. van Straaten et al. (2002)) no distinction is made between LLF
and Lh in timing analyses of 4U0614+09, as power spectra where the two features
are simultaneously present are rare. We fit two Lorentzians at low frequency (<10 Hz,
confidence exceeds a 3σ level) when νu ≈324 Hz, as can be seen in Figure 3.19. Moreover,
we fit two significant features in the decaHz domain when νu is between 500-900 Hz, see
Figure 3.20. A comparison to the timing behaviour accross the color-color diagram of
4U1728-34 (3.2.1), supports our identification of these features as LLF and Lh .
Power law fits to νh and νLF vs. νu yield: αh =2.65±0.14 (χ2 /dof =17/6) and αLF =2.54±0.14
(χ2 /dof =0.9/3), see Figure 3.17. The detections from the flaring state were not included
in these fits.
3.2.4.1
The Flaring State
In the lower banana (flaring) state of 4U0614+09 (see Figure A.4) we see power spectra
that are characterized by a strong broad component around 10 Hz and a QPO around
Chapter 3. Data selection & Results
50
100 Hz, see Figure 3.21 for the average power spectrum of all observations showing these
timing characteristics. We investigate if any of the features present can be identified as
Lh or LLF .
We make a subselection, and fit 508 averaged power spectra of 3 observations in this
region of the color-color diagram with start MJD ranging from 51173.9-51175.9 (ObsIDs 40030-01-04/05/06-00) (see Figure 3.23). We fit a kHz QPO with ν=1303 Hz (at
a 5.0σ confidence level)3 , and Lorentzians with ν =20.5, 40.5 Hz, and 107.4 Hz. The
characteristics of power spectra in the flaring state emerge suddenly and at high frequency, which complicates comparison of features accross the color-color diagram (see
Figure 3.18 for Q-factors and fractional rms amplitudes in the flaring state depicted in
green). However, we are able to identify the feature around 100 Hz as LLF based on
the correlation of νLF with νu (see Figure 3.17). The upper kHz QPO is at such high
frequency that we rely on the extrapolation of the correlation we find at low frequency.
The hHz Lorentzian has a centroid frequency that usually lies ∼100 Hz. A comparison
to the flaring state in 4U 1702-43, however, where we suspect we see both LhHz and
LLF in a power spectrum that is very similar to what we see in 4U 0614+09 (see Figure
3.31), supports our identification of the component at ∼100 Hz in 4U 0614+09 as LLF .
If we regard the Lorentzian characterized by ν=107.4 Hz as LLF , as we do in Figure
3.23, that would alter the parameters of the power law we fit to κLF =1.03±0.4×10−5
and αLF =2.27±0.054 (χ2 /dof =23.1/5).
Bult and van der Klis (2014) fitted power spectra of the LMXB SAXJ1808.4-3658 in
the flaring state using a power law with exponential cut off (a Schechter function),
accounting for the steep slope at low frequency of the broad component. We set out
to compare their findings with the features we see in 4U 0614+09. The residuals of
our fit to 1611 averaged power spectra (see Figure 3.21) indicate that the slope at low
frequency is marginally steeper than can be fitted with a Lorentzian (see Figure 3.22).
The fractional rms amplitude between 0-50 Hz is ∼10%, while in the flaring state of
SAXJ1808.4-3658, it is ∼26% and decays gradually after outburst. Bult and van der
Klis (2014) discuss several mechanisms explaining these timing features and argue that
the ‘dead-disk’ accretion instability proposed by D’Angelo and Spruit (2010) matches
their findings best. Whether the characteristic timing features seen in the flaring state
of pulsars (Bult and van der Klis, 2014) and LMXBs that are not pulsars (this work)
are caused by the same mechanism remains to be determined.
3
The prior detection of Lu in this source with ν >1300 Hz by van Straaten et al. (2000) was described
as a statistical artefact by Boutelier et al. (2009). We note that we do not fit the 5.0σ Lu in a single
observation, but only in an addition of three. In a single observation with start MJD 51175.9 (ObsID:
40030-01-06-00) we fit Lu with ν=1369±50 at a 3.8σ level. When selecting in energy, only including
2-20 keV, averaging power spectra from 3 observations [40030-01-04/05/06-00]; we fit a 1275 Hz QPO
with a FWHM of 280±70 at 5.8σ.
Chapter 3. Data selection & Results
51
hump
LF
LF 2.5σ
flaring state
ν (Hz)
100
10
1
100
200
500
1000
1500
νu (Hz)
Figure 3.17: In analogy to Figure 3.3, for 4U 0614+09. When νu exceeds 605 Hz, Lh
is not detected.
3.2.4.2
Comparing to van Straaten et al. (2002)
(In analogy to Sections 3.2.1.1 and 3.2.3.1) In the above-mentioned analysis of the timing
behaviour of 4U 0614+09 by van Straaten et al. (2002) the partly overlapping features
we identify separately as LLF and Lh , are identified as, and fitted with, one Lorentzian.
We convert the νmax reported in van Straaten et al. (2002) of 4U 0614+09, to centroid
frequency ν and plot them together with our measurements of νh and νLF , see Figure
3.24. We see that when νu ≈350 Hz, van Straaten et al. (2002) report a ν≈6.5 Hz,
which is intermediate to our measured frequencies. The data analysis in van Straaten
et al. (2002) of 4U 0614+09 is similar to ours. Where we calculate X-ray colors for every
observation, they split the data in time intervals of 2500 seconds and calculate colors for
every time interval. The shape traced out in the color-color diagram is parametrized by
a spline Sa in van Straaten et al. (2002), along which data with similar X-ray colors are
selected. We analyze an averaged power spectrum of a selection of observations that was
also used in van Straaten et al. (2002); 50031-01-01-01/02/03/04/06/07 starting on MJD
51789.1 (September 2000), see Table 3.5. We fit the power spectrum with 4 Lorentzians
(χ2 /dof =484/329), see Figure 3.25. We find, by doing an F-test for additional terms
(see Section 2.2.5), that the inclusion of a fifth Lorentzian is a significant improvement
Chapter 3. Data selection & Results
52
hump
25
hump
LF
LF
LF 2.5σ
LF 2.5σ
flaring state
flaring state
10
15
Q
rms (%)
20
10
1
5
0
20
40
60
ν (Hz)
80
100
120
0
20
40
60
80
100
120
ν (Hz)
Figure 3.18: In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of 4U 0614+09 plotted against their centroid frequency.
of the fit, with F(ν1 ,ν2 )=6.673 and P=0.0002. The centroid frequency of the Lorentzian
that fits the partly overlapping LLF and Lh in the fit with 4 Lorentzians (ν=6.9±0.14
Hz), is lower than νh (8.1±0.3 Hz) in the fit with 5 Lorentzians, and higher than νLF
(5.4±0.2 Hz). The Lorentzian we identify as LLF is not fitted in van Straaten et al.
(2002), which explains the intermediate frequency they report.
Furthermore, van Straaten et al. (2002) do not report our measured frequencies at ∼40
Hz with νu ranging between 500-610 Hz. As in Section 3.2.1.1, we refit a power spectrum characterized by νu within this frequency range, with 4 instead of 5 Lorentzians
(ObsID: 90422-01-01-01,χ2 /dof =348/330, see Figures 3.27 and 3.26). In the fit with 4
Lorentzians, we fit a single Lorentzian with ν=19.4±3.5 Hz. Close to this frequency,
we fit two Lorentzians in the fit with 5 Lorentzians characterized by νLF =21.42±0.55
and νh =37.24±1.82, νu =602±7 Hz. The inclusion of a 5th Lorentzian is a significant improvement to the fit with 4 Lorentzians, quantified by F(ν1 , ν2 )=6.99 and P=0.14×10−3 .
This result supports our conclusion in Section 3.2.1.1; LLF becomes stronger with increasing νu . When fitting a single Lorentzian to the partly overlapping Lorentzians we
identify as Lh and LLF , its centroid frequency is intermediate to the centroid frequencies
we report at low (100-400 Hz) νu . As LLF becomes stronger, at high (>400 Hz) νu the
centroid frequency of the single Lorentzian will lie close to νLF . This can be seen in
Figure 3.24.
53
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Frequency × (RMS/Mean)2 Hz−1
Figure 3.19: Multi-Lorentzian fit to 592 averaged power spectra from 4U 0614+09
starting on MJD 51788.6. We fit a Lorentzian characterized by ν = 4.46 Hz with 3σ
confidence level in the integral power. We identify this to be LLF , Lh has ν = 7.77 Hz
(13.8σ).
Frequency (Hz)
Figure 3.20: Multi-Lorentzian fit to 1016 averaged power spectra from 4U 0614+09
starting on MJD 52948.6. We fit a Lorentzian characterized by ν =15.61 ± 0.34 Hz
with a 3.8σ confidence level in the integral power. We identify this to be LLF , Lh has
ν =26.59 ± 1.6 Hz (3.5σ).
456/326
393/329
363/329
374/326
349/329
324/327
318/327
359/327
327/327
377/326
365/323
382/327
χ2 /dof
Lb
Lb2
ν (Hz)
ν (Hz)
0.16±0.04
0.12±0.010
0.14±0.021
0.24±0.085
0.62±0.24
0
0
0
0
13.63±0.80 1.79±0.43
19.7±1.80 0.014±0.0020
40.5±0.6
20.5±0.9
LLF
ν (Hz)
5.45±0.15*
4.46±0.11
13.47±0.64
15.61±0.34
18.84±0.61
21.42±0.55
36.92±0.52
45.28±1.67∗
107.4±2.4
LLF
rms (%)
4.8±0.1
3.1±0.6
10.6±1.8
8.06±1.18
7.14±1.82
6.08±0.98
5.83±0.06
2.7±0.95
13.8±1.2
LLF
LLF2
Q
ν (Hz)
1.74±0.50
5.39±2.30
1.26±0.39
2.12±0.55
2.47±1.27
3.09±0.98
3.72±0.82
5.46±4.79
4.14±1.07
-
Lh
Lh
Lh
LhHz
Ll
Lu
ν (Hz)
rms (%)
Qν (Hz)
ν (Hz)
ν (Hz)
8.07±0.33 20.2±0.6 0.59±9.04
66±11
346±9
1.27±0.03 25.0±1.0 0.41±0.02
4.8±1.37
140±8
1.33±0.14 24.4±1.0 0.38±0.05 8.89±2.48
175±31
7.77±0.27 20.00±0.72 0.67±0.06 65.3±20.4
324±12
12.23±1.03 20.7±1.2 0.49±0.08 119.25±8.43
441±13
26.93±1.64 9.81±1.75 1.61±0.55 112.1±11.1
491±19
26.6±1.6
9.3±1.5 1.55±0.45 95.26±27.97
535±9
36.81±2.08 8.5±1.5 1.97±0.79 118.60±13.09
564±8
37.24±1.82 6.2±1.1 2.34±0.93 96.41±21.8
602±7
70.54±10.93
746±4
93.67±5.73 574.30±5.0 855±4
1304±25
Table 3.5: In analogy to Table 3.2.fit parameters of the multi-Lorentzian fits to power spectra of 4U 0614+09. The starred (∗ ) value was at a 2.5σ
confidence level.
ObsID
Number
Region
of PS. of CCdiagram
51789.1 50031-01-01-01/02/03/04/06/07 4184
A
52054.0
50031-01-04-[00-18][120]
5761
A
52057.4
50031-01-04-13
809
A
51788.6
50031-01-01-02
592
A
53281.7
90422-01-01-00
738
B
54350.7
93404-01-05-00/02
955
B
52948.6
80037-01-01-02
1016
B
50195.8
10095-01-01-00
580
Cii
53282.7
90422-01-01-01
566
Cii
50951.0
30054-01-01-02
516
Cii
50197.5
10095-01-02-00
1923
Cii
51173.9
40030-01-04/05/06-00
508
HLF
MJD
Chapter 3. Data selection & Results
54
55
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Residual (σ)
Figure 3.21: Multi-Lorentzian fit to 1611 averaged power spectra from all observations
of 4U 0614+09 found to be in the flaring state (χ2 /dof =961/872). In Figure 3.22 the
residuals of this fit are presented.
Frequency (Hz)
Figure 3.22: Residuals of a multi-Lorentzian model fit to 1611 averaged power spectra
from the flaring state seen in 4U 0614+09, see Figure 3.21.
56
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Figure 3.23: Multi-Lorentzian fit to 508 averaged power spectra from the flaring
state of 4U 0614+09 starting on MJD 51173.9. We fit Lorentzians characterized by ν
= 20.52±0.96 Hz (10σ), ν = 40.53±0.0.62 Hz (8σ), ν = 107.4±2.4 Hz (6σ), and ν =
1304±25 Hz (5.04σ). For a discussion on the identification of these features see Section
3.2.4.1.
Chapter 3. Data selection & Results
57
100
hump
LF
Van Straaten 2002
ν (Hz)
40
10
5
1
100
200
500
900
νu (Hz)
Figure 3.24: In analogy to Figure 3.6, for 4U 0614+09. Measurements of Lh from
the multi-Lorentzian timing study by van Straaten et al. (2002), converted from νmax
to ν, plotted with our measurements of νLF and νh . Points plotted in this figure are all
measured in 4U 0614+09. The best fit from van Straaten et al. (2003) to 4U 0614+09,
4U 1728-34 and 4U 1608-52 with a power law index of 2.01±0.02 is drawn, as well as
the best fit power law indices to our data of 4U 0614+09 with αh = 2.65±0.14 and
αLF =2.54±0.14
.
58
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Figure 3.25: Left: Fit with a 4-Lorentzian model to the power spectrum of observations 50031-01-01-01/02/03/04/06/07 of 4U 0614+09 as in van Straaten et al. (2002),
χ2 /dof =484/329). Right: Fit with 5 Lorentzians (χ2 /dof =456/326).
Figure 3.26: Power spectrum from van Straaten et al. (2002) of 4U 0614+09. In ”interval 4” Lorentzians are fitted with ν=22.6 Hz and νu =623.8 Hz (with νmax converted
to centroid frequency). Note that the lowest frequency measured here is ∼0.008 Hz,
the lowest frequency probed in our analysis is 0.0625 Hz.
59
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Figure 3.27: Left: Fit with a 4-Lorentzian model to the power spectrum of observation
90422-01-01-01 as in van Straaten et al. (2003) (see Figure 3.26, χ2 /dof =348/330).
Right: Fit with 5 Lorentzians (χ2 /dof =327/327).
Chapter 3. Data selection & Results
3.2.5
60
4U 1702-43
4U 1702-43 is an atoll source (Oosterbroek et al., 1991) and an X-ray burster (Makishima
et al., 1982). It has no known optical counterpart. The neutron star spin frequency
inferred from these bursts is 330 Hz (Markwardt et al., 1999).
In an analysis of 101 ks of data from July 1997 Markwardt et al. (1999) discuss a possible
correlation between a ∼35 Hz feature and the kHz QPOs in the context of a Lense-
Thirring precession model. They approximate the Kepler frequency at the inner disk
edge to equal the centroid frequency of the lower kHz QPO (ν` ) plus the spin frequency
of the neutron star, and identify νh as νLT (see Equation 4.1). They fit a power law
index of 2.3 ± 0.1 when plotting the component of ∼35 Hz against the inferred Kepler
frequency.
We retrieved all data for this source from the RXTE archive, which amounted to 225
observations comprising 1.22 Ms of usable data, between 1996 and 2011. We present
the color-color diagram (see Figure A.6), details of the fits to (a selection of) single
observations are summarized in Table 3.6.
A fit to νh and νLF vs. νu yields: αh = 2.81±0.22 (χ2 /dof =5.9/8), and αLF =2.12±0.23
(χ2 /dof =2.8/3), see Figure 3.28.
3.2.5.1
The Flaring State
In region H of the color-color diagram (Figure A.7-H, Appendix A) , we analyzed 4468
averaged power spectra from 11 subsequent observations. In analogy to Section 3.2.4.1,
we investigate if any of the power spectral features can be identified as LLF or Lh . We
fit Lorentzians with ν=10 Hz (2.3σ), 32 Hz (2.8σ), 85 Hz, 221 Hz, 789 Hz (L` ) and 1223
Hz (Lu ) (χ2 /dof =274/320, see Figure 3.31). We are able to identify the Lorentzian
with ν=85 Hz as most likely LLF , based on the centroid frequency, which follows the
extrapolation of the correlation of νLF with νu at low frequency (see Figure 3.28). We
would then identify the Lorentzian with ν=221.1 Hz as LhHz or Lh . If we include this
possible detection of LLF in our analysis, the parameters of the fit toνLF vs. νu changes
to κLF =1.62±1.2 ×10−5 and αLF = 2.18 ± 0.07 ( χ2 /dof =4.56/6).
Chapter 3. Data selection & Results
61
hump
LF
LF 2.2σ
flaring state
ν (Hz)
100
10
1
100
200
500
1000
1400
νu (Hz)
Figure 3.28: In analogy to Figure 3.3, for 4U 1702-43. When νu exceeds 550 Hz, Lh
is not detected.
hump
hump
LF
25
LF
LF 2.2σ
LF 2.2σ
flaring state
flaring state
10
15
Q
rms (%)
20
10
1
5
0
20
40
60
ν (Hz)
80
100
0
20
40
60
80
100
ν (Hz)
Figure 3.29: In analogy to Figure 3.5; fractional rms levels and Q-factors of Lh and
LLF fitted in power spectra of 4U 1702-43 plotted against their centroid frequency.
80033-01-17-01/02/03
50030-01-10-00/01/02/03
50030-01-04-00
80033-01-17-00/000
[50029-27-01-00/000/01]
[50030-01-03-00/01/02]
80033-01-01-00/01/02/08
80033-01-21-000
80033-01-[19-01/02]
80033-01-18-00
80033-01-20-02
50025-01-01-00
80033-01-[02-00/000][03-00/01]
[01-03/030/04/05/06/07/08]
53209.0
51961.5
51854.8
53209.4
51769.6
548
1190
509
785
907
682
4468
B
Ci
Cii
Cii
Cii
Cii
H
Number
Region
of PS. of CCdiagram
1951
A
902
A
892
A
1193
A
2538
B
Lb
ν (Hz)
0.33±0.12
2.24±0.11
0.17±0.07
0.51±0.13
0.19±0.10
Lb2
ν (Hz)
-
505/471 0.37±0.07
361/326 0.54±0.27
355/326 0.44±0.30
342/326 0.53±0.36
329/326 0.097±0.578
352/329
0
∗∗∗
273/320 31.74±3.47
10.19±1.37∗∗
374/329
407/330
362/326
532/326
369/326
χ2 /dof
6.77±0.07
12.95±0.21
13.24±0.36
12.80±0.53
15.16±0.51
22.97±1.41
84.95±2.19
1.55±0.32
5.93±0.63
6.58±0.93
6.56±1.29
7.06±1.41
9.59±1.92
5.05±0.51
14.67±8.37
3.46±0.69
1.99±0.46
1.83±0.56
1.93±0.69
1.08±0.35
2.90±0.74
-
LLF
LLF
LLF
LLF2
ν (Hz)
rms (%)
Q
ν (Hz)
7.87±0.28 3.91±0.75 2.89±0.94
6.78±0.23∗ 2.57±0.90 3.99±2.51
9.85±0.32
22.66±0.87
23.76±1.22
22.72±1.85
27.21±1.39
-
Lh
ν (Hz)
10.73±0.47
5.87±0.24
5.56±0.22
13.77±0.65
10.53±0.52
17.04±0.43
10.19±0.96
11.16±0.95
11.16±1.66
9.16±1.06
-
Lh
rms (%)
17.63±0.52
17.55±0.61
16.88±1.06
15.07±0.74
16.16±0.62
∗
118.83±9.33
108.31±13.78
125.45±12.87
117.06±21.79
144.92±12.73
88.5±43.5
221.1±20.8 790±21
LhHz
Ll
ν (Hz)
ν (Hz)
130.72±7.54
0
126.89±11.77
119.01±11.95
126.87±10.39
-
386±17
507±14
512±12
527±11
539±11
639±8
1223±30
Lu
ν (Hz)
407±18
284±22
312±24
388±30
401±18
values were detected at
0.57±0.04
1.04±0.30
1.08±0.20
1.05±0.31
1.60±0.40
-
Lh
Q
0.51±0.04
0.51±0.05
0.57±0.07
0.75±0.10
0.61±0.06
Table 3.6: In analogy to Table 3.2: Fit parameters of the multi-Lorentzian fits to power spectra of 4U 1702-43. Starred
a 2.2σ confidence level, ∗∗ at 2.3σ, ∗∗∗ at 2.8σ.
52957.3
53311.4
53211.0
53210.6
53212.7
51781.2
53022.0
ObsID
MJD
Chapter 3. Data selection & Results
62
63
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Frequency × (RMS/Mean)2 Hz−1
Figure 3.30: Multi-Lorentzian fit to 1193 averaged power spectra of 4U 1702-43
starting on MJD 53209.4. We fit a Lorentzian characterized by ν =7.87 ± 0.28 Hz
(3.3σ). We identify this to be LLF , Lh has ν =13.77 ± 0.65 Hz (10.1σ).
Frequency (Hz)
Figure 3.31: Multi-Lorentzian fit to 4468 averaged power spectra from the flaring
state of 4U 1702-43 (see Figure A.7-H) starting on MJD 53022.0. We are able to
identify Lb2 , Lb , LLF , LhHz , L` and Lu , from left to right, excluding the low frequency
noise at ∼0.2 Hz. See Table 3.6 forν values and the main text for a discussion.
Chapter 3. Data selection & Results
3.2.6
64
KS 1731-260
KS 1731-260 is an X-ray burster (Syunyaev et al., 1990). The neutron star spin frequency
as inferred from burst oscillations is ∼524 Hz (Smith et al., 1996), and an upper limit on
the distance to the source is 7.8 kpc (Muno et al., 2000). The optical counterpart was
observed by Wijnands et al. (2001). The source turned to quiescence in February 2001,
and became active again in 2013 (Kuulkers et al., 2013). Wijnands and van der Klis
(1997) studied the timing behaviour of KS 1731-260 and discovered two simultaneous
kHz QPOs.
We retrieved all data for this source from the RXTE archive, which amounted to 86
observations comprising 464 ks of usable data, between July 1996 and August 2007. See
Figure A.8 for the color-color diagram. The details of the fits to averaged power spectra from observations that were close both in time and color are summarized in Table 3.7.
In region A of the color-color diagram the identification of features at low frequency
is not straightforward. We fit LLF and LLF2 , in addition to Lb , Lh , LhHz , Lu and an
unidentified Lorentzian, see Figure 3.32. It is unclear which of the components at low
frequency is comparable to what we call LLF in other sources. In Altamirano et al.
(2008b) similar harmonics of the LF QPO are seen in the timing analysis of the LMXB
1E 1724-3045. They identify the LF QPO with the highest characteristic frequency as
LLF . We also see similar features at low frequency in the timing analysis of 4U 1608-52
(see Figure 3.35) and in the horizontal branch of the color-color diagram of Z-sources
(Jonker et al., 2002), which supports the comparison between LLF and the horizontal
branch oscillation (HBO). We suspect that when we do not fit the harmonic of the
LF QPO, or the unidentified feature, as we do in Figure 3.32, the fitted Lorentzian is
broadened and the centroid frequency turns out lower than the centroid frequency of Lh
as fitted in combination to the low frequency QPOs (see Figure 3.32). The Lorentzian
we interpret to be Lu in region A likely fits both LhHz and Lu , as it turns out low.
We opted not to fit a power law to these data, because we are only confident of the
identification of features in power spectra with νu ≈500 Hz.
65
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Figure 3.32: Multi-Lorentzian fit to 4068 averaged power spectra from KS 1731260 starting on MJD 51815.7. We identify Lb , LhHz and Lu , at low frequency the
identification of features is not straightforward. See Table 3.7 for measured frequencies,
and the main text for a discussion.
100
hump
LF
ν (Hz)
40
10
5
1
100
200
500
1000
νu (Hz)
Figure 3.33: In analogy to Figure 3.3, for KS 1731-260. When νu exceeds 490 Hz,
Lh is not detected. At low frequency the identification of fitted Lorentzians is not
straightforward. Harmonics of LLF are included in this figure. Because of the scarcity
of obtained points and unclear identification, we opt not to fit a power law to these
data. See main text for a discussion.
50031-02-02-00/01/02/03/04/
05/06/07/08
50031-02-02-080
50031-02-01-00
30061-01-02-00/01/02/03/04/05
30061-01-02-06
51815.7
926
484
2513
906
A
A
C
C
370/323 0.19±0.02
338/326 0.32±0.04
326/250
0
402/329 0.28±0.39
-
LLF2
ν (Hz)
1.66±0.01
0.79±0.02 5.08±0.70 2.21±0.48 1.71±0.03∗∗
1.53±0.04 3.51±0.58 3.20±1.01
9.37±0.37 3.07±0.34 2.15±0.89
-
Number
Region
χ2 /dof
Lb
Lb2
LLF
LLF
LLF
of PS. of CCdiagram
ν (Hz) ν (Hz)
ν (Hz)
rms (%)
Q
4068
A
324/320 0.18±0.01
0.81±0.013 6.24±0.62 1.44±0.19
-
1.87±0.12
3.45±0.10
16.75±0.43
15.65±0.79
14.63±0.85
14.08±1.12
12.02±0.62
13.41±0.82
LhHz
ν (Hz)
8.59±1.20
∗
-
181±33
203±22
484±7
459±15
Ll
Lu
ν (Hz)
162±12
values were detected
0.62±0.12 2.09±3.10
0.76±0.10 3.41±6.37
0.82±0.09 147.02±21.83
0.73±0.09 191.34±35.55
L?
Lh
Lh
Lh
ν (Hz)
ν (Hz )
rms (%)
Q
2.23±0.02∗ 2.07±0.41 13.63±1.14 0.40±0.12
Table 3.7: In analogy to Table 3.2. Fit parameters of the multi-Lorentzian fits to power spectra of KS 1731-260. Starred
at a 2.9σ confidence level, ∗∗ at 2.5σ.
51817.0
51762.6
51088.1
51092.0
ObsID
MJD
Chapter 3. Data selection & Results
66
Chapter 3. Data selection & Results
67
hump
hump
LF
LF
25
10
Q
15
10
1
5
0
10
20
30
40
50
60
70
80
0
10
20
ν (Hz)
30
40
50
60
70
80
ν (Hz)
Figure 3.34: In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of KS 1731-260 plotted against their centroid frequency.
Frequency × (RMS/Mean)2 Hz−1
rms (%)
20
Frequency (Hz)
Figure 3.35: Multi-Lorentzian fit to 2424 averaged power spectra of 4U 1608-52. We
identify Lb , LLF ,LLF2 , LhHz and Lu , see Table 3.4 forν values, and the main text for a
discussion.
Chapter 3. Data selection & Results
3.2.7
68
SAXJ1750.8-2900
SAXJ1750.8-2900 is an X-ray burster (Natalucci et al., 1999) with a neutron star spin
frequency of ∼601 Hz as inferred from burst oscillations (Kaaret et al., 2002). An upper
limit on the distance to the source is ∼6.8 kpc (Galloway et al., 2008). Kaaret et al.
(2002) first reported QPOs in the 500-1020 Hz range.
We retrieved all data of this source from the RXTE archive, amounting to 129 observations comprising 350 ks of usable data. In regions A, B and D of the color-color
diagram we detect LLF or Lh and Lu simultaneously. We present multi-Lorentzian fits
to observations from these and adjacent regions in Table 3.8. The upper kHz QPO was
detected > 3σ in power spectra from observations in region D only. The signal to noise
ratio is too low in most single observations of this source to detect it. In the power
spectrum of the observation starting on MJD 52008.4 we only fit Lh . We suspect we
underestimate the centroid frequency of this Lorentzian because we do not fit LLF due
to poor statistics. We do discern LLF by eye, and it is significantly present in a similar
power spectrum we fit of an observation starting three days earlier (MJD 52005.5). We
do not fit a power law to these data, due to a lack of measurements.
100
hump
LF
ν0 (Hz)
40
10
5
1
100
200
500
1000
ν0u (Hz)
Figure 3.36: In analogy to Figure 3.3. Due to the scarcity of obtained points we
opted not to fit a power law to these data. See main text for a discussion.
Chapter 3. Data selection & Results
69
hump
hump
LF
LF
25
10
15
Q
rms (%)
20
10
1
5
0
10
20
30
40
ν (Hz)
50
60
70
80
0
10
20
30
40
50
60
70
80
ν (Hz)
Figure 3.37: In analogy to Figure 3.5; rms and Q values of Lh and LLF fitted in
power spectra of SAXJ1750.8-2900 plotted against their centroid frequency.
Table 3.8: In analogy to Table 3.2.Fit parameters of the multi-Lorentzian fits to power spectra of SAXJ1750.8-2900. Starred ∗ values were detected
at a 2.8σ confidence level, ∗∗ at 2.2σ.
MJD
ObsID
Number
Region
χ2 /dof
Lb
Lb2
LLF
LLF
LLF
LLF2
Lh
Lh
Lh
LhHz
Ll
Lu
Start
of PS. of CCdiagram
ν (Hz) ν (Hz)
ν (Hz)
rms(%)
Q
ν (Hz)
ν (Hz)
rms (%)
Q
ν (Hz)
ν (Hz) ν (Hz)
52005.5 60035-01-02-00
168
D
229/250
0
17.52±0.28 2.95±0.81 10.31±7.85
28.36±4.24 10.68±1.76 0.81±0.33 246.7±23.72
579±23
52008.4 60035-01-02-01
189
D
251/250 2.34±0.43
22.25±3.02 11.69±1.26 0.61±0.19 189.8±17.5
649±14
Chapter 3. Data selection & Results
70
Chapter 3. Data selection & Results
3.2.8
71
Aquila X-1
Aquila X-1 (Aql X-1) is an X-ray burster (Priedhorsky and Terrell, 1984). The source
shows distinct similarities in timing and spectral behaviour with atoll sources, like 4U
1608-52 (Reig et al., 2000). Casella et al. (2008) found a short time interval (≈150 s) with
550.3 Hz pulsations in an observation from October 1998. They propose several possible
causes for the pulsation, such as a magnetic field channeling the accretion flow to the
magnetic poles as in AMXPs, or a temporary asymmetry on the neutron star surface.
Reig et al. (2000) investigated timing properties and spectral states of Aquila X-1, using
all data between 1997 and 2002 from the RXTE archive, but do not investigate the
correlations between the centroid frequencies of fitted Lorentzians.
We retrieved all data for this source from the RXTE archive, amounting to 583 observations comprising 1.9 Ms of usable data. In regions Ai , Aii , B and C of the color-color
diagram, we detect Lh or LLF and Lu simultaneously. We present multi-Lorentzian fits
to (a selection of) single observations from these and adjacent regions in Table 3.9. In
a power spectrum from an observation starting on MJD 52092.6 (Februari 2001) we fit
a 115 Hz QPO (3.7σ, Q=19.7±19), that was not reported in any previous study. We
identify it as LhHz , see Figure 3.40.
In a fit to the combined power spectra of observations 91028-01-[05-22]-00 (see Figure
3.41), we measure two similar centroid frequencies; νLF =3.38±0.05 and νh =3.46±0.09.
We are unsure of the reason for this, a harmonic of LLF could be present in this power
spectrum, similar to what is seen in 4U 1608-34 and KS 1731-260 (see Figures 3.35 and
3.32), that we do not detect due to poor statistics. Consequently, Lh could be broadened
and a lower centroid frequency is measured. We omit this measured νh from our assessment of the index of the power law fitted to a possible correlation, and from subselection
A, see Section 3.4.
A fit to νh vs. νu yields: αh =3.14±0.29 (χ2 /dof =3.5/3).
Chapter 3. Data selection & Results
72
100
hump
LF
LF 2.2σ
40
ν (Hz)
10
5
1
100
200
500
1000
νu (Hz)
Figure 3.38: In analogy to Figure 3.3, for Aql X-1. When νu exceeds 500 Hz, Lh is
not detected.
hump
25
hump
LF
LF
LF 2.2σ
LF 2.2σ
10
15
Q
rms (%)
20
10
1
5
0
10
20
30
40
ν (Hz)
50
60
70
80
0
10
20
30
40
50
60
70
80
ν (Hz)
Figure 3.39: In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of AqlX-1 plotted against their centroid frequency.
457/328
381/329
367/329
307/327
316/330
0
0
0
0
0
Lb
ν (Hz)
419/329
0
χ2 /dof
LLF
rms(%)
-
LLF
Q
-
17.01±0.36
15.96±1.09
14.31±0.82
11.09±1.36
13.54±0.88
LhHz
ν (Hz)
7.55±5.00
∗
-
265±12
299±52
409±30
436±26
488±20
Ll
Lu
ν (Hz) ν (Hz)
272±19
values were detected at a
0.46±0.03
0
0.72±0.08 12.25±15.27
0.72±0.09
0
1.20±0.38 115.6±0.8
0.71±0.12 150.22±15.47
Lh
Lh
Lh
ν (Hz)
rms (%)
Q
3.53±0.11 16.93±0.71 0.55±0.05
3.38±0.05 1.79±0.26 7.11±2.17 3.49±0.09
5.89±0.16
15.08±0.74
12.98±0.39∗ 3.43±0.99 6.14±4.05 21.66±1.42
19.95±1.40
LLF
ν (Hz)
-
Table 3.9: In analogy to Table 3.2: Fit parameters of the multi-Lorentzian fits to power spectra of Aql X-1. Starred
2.2σ confidence level.
MJD
ObsID
Number
Region
Start
of PS. of CCdiagram
53150.9 90017-01-[04/05-00][06-00/01] 1319
B
[07-00/01][08-00/01/01]
53469.7
91028-01-[05-22]-00
2758
B
53715.1
91414-01-08-00/08
356
B
51317.9
40047-01-01-01
129
C
52092.6
60054-02-02-02
172
C
53164.4
90017-01-09-02
141
C
Chapter 3. Data selection & Results
73
74
Frequency × (RMS/Mean)2 Hz−1
Chapter 3. Data selection & Results
Frequency (Hz)
Frequency × (RMS/Mean)2 Hz−1
Figure 3.40: Multi-Lorentzian fit to 4068 averaged power spectra starting on MJD
52092.6. We identify Lb , LLF , Lh and Lu . See Table 3.9 for measured frequencies, and
the main text for a discussion.
Frequency (Hz)
Figure 3.41: Multi-Lorentzian fit to 2758 averaged power spectra of AqlX-1. We
identify Lb , LLF ,Lh , LhHz and Lu , see Table 3.9 for measured ferquencies, and the
main text for a discussion.
Chapter 3. Data selection & Results
3.2.9
75
Bursters with poor statistics
For the bursters EXO 0748-676, IGRJ17191-2821, XTEJ1739-285, A 1744-361 and 4U
1915-05 statistics were too poor to conduct our analysis. We created color-color diagrams
and reduced all data for these bursters from the RXTE archive. The results can be
found in the digital appendix (see B). A different analysis of these sources might reveal
additional information to the methods applied in this project. We suggest splitting
observations in shorter time intervals, and creating colors and power spectra for each
interval prior to averaging power spectra with similar (to within 10%) colors, as for
instance in Di Salvo et al. (2001). Variations over time and secular changes in the
relation between colors and QPO frequencies are not taken into account in this suggested
approach. The method applied in this project renders a more pure result in that sense.
Here we discuss our findings for each source briefly. 4U 1915-05 has 56 observations in the
RXTE archive comprising 0.4 Ms of usable data. A positive identification of Lorentzians
at low frequency is not straightforward due to poor statistics. We encountered similar
issues in XTEJ1739-285 (9 observations, 24 ks of data), A 1744-31 (53 observations, 150
ks of data) and IGRJ17191-2821 (19 observations, 70 ks of data). In EXO0748-676 (749
observations, 2000 ks of data) we could not detect the upper kHz QPO in (selections
of) single observations above 3σ.
3.3
Pulsars
Pulsars have known spin frequencies due to the presence of coherent X-ray pulsations,
presumably due to their strong non-aligned magnetic field. Accreting matter follows
magnetic field lines and accretes onto a region of the neutron star surface around the
magnetic poles. Because the analysis of these sources was performed in a similar manner
as the sources in the previous sections, we keep using 4U 1728-34 as an example of our
methods.
We take out the pulsar spike in the power spectra of pulsars by removing the frequency
bins in which the spike is concentrated, before rebinning in frequency. We refer the reader
to Appendix A for the color-color diagrams and representative fitted power spectra for
the regions in the color-color diagram.
3.3.1
SAXJ1808.4-3658
SAXJ1808.4-3658 is a LMXB discovered with the BeppoSAX satellite in 1996 (in ’t Zand
et al., 1998). It was the first X-ray source found to show millisecond pulsations (401 Hz)
and type 1 X-ray bursts (Wijnands and van der Klis, 1998a). The pulsations proved
Chapter 3. Data selection & Results
76
the presence of rapidly spinning neutron stars in LMXBs, predicted by the discovery of
radio pulsars in the early 1980’s (Alpar et al., 1982). It also linked oscillations in type
1 (thermonuclear ) X-ray bursts to the spin frequency of the neutron star. The orbital
period of the binary is 120.8 minutes (Wijnands and van der Klis, 1998a).
The timing behaviour of the source has been extensively studied over the years, and
similarities with atoll sources such as 4U 1728-34, 4U 0614+09 and 4U 1705-44 were
established (Wijnands and van der Klis, 1998b).
Relations between timing features in millisecond pulsars were analyzed by van Straaten
et al. (2005). Data from the 1998 and 2002 outbursts of SAXJ1808.4-3658 were included
in this study. The authors find that the correlations of timing features with νu are shifted
with respect to other sources in their sample. This shift is best described by a shift of the
upper and lower kHz QPO frequency by a factor 1.5. The magnetic field of millisecond
pulsars is offered as an explanation for this shift. If νu reflects the Keplerian frequency
at the inner edge of the accretion disk, the magnetic field prevents this inner edge from
coming as close to the neutron star as it does in non-pulsating X-ray sources.
van Straaten et al. (2005) suggest the identification of a ∼40 Hz Lorentzian in the flaring
state of SAXJ1808.4-3658 as Lh , see Figure 3.44. The upper kHz QPO is not detected
in this power spectrum, and an identification based on the correlation of Lh with Lu is
therefore not possible. They fit a power law to the correlations of Lh and LLF with Lu
of combined data of XTEJ1814-338, XTEJ1751-305, XTEJ0929-314 and SAXJ1808.43658, in the νmax representation. Before fitting, all correlations are shifted to coincide.
They find power law index αLF =2.92±0.63 and αh =2.43±0.03.
We retrieved all data from the RXTE archive, amounting to 493 observations comprising
1.6 Ms of usable data. In all power spectra, we ’weed’ out the pulsar spike at 401 Hz
before fitting it with a multi-Lorentzian model. We find Lh or LLF and Lu simultaneously
in regions B, G, H and I of the color-color diagram, see Figure A.14. The fits to (selections
of) single observations from these, and adjacent regions of the color-color diagram are
presented in Table 3.10. We see a shift in correlations of νLF and νh vs. νu with respect
to the other sources in our sample, similar to the shift reported in van Straaten et al.
(2005), see Figure 3.42 In Figure 3.43, we plot fractional rms amplitudes and Q-factors
of detected Lorentzians. We see that at low centroid frequency, these values coincide
with what we find in bursters (plotted in lighter colors). When νh and νLF exceed ∼30
Hz, rms and Q values deviate.
We find αh =2.13±0.04 (χ2 /dof =20/5), and αLF =2.11±0.03 (χ2 /dof =1.6/2) when
fitting power laws to νh and νLF vs. νu .
Chapter 3. Data selection & Results
77
100
hump
LF
flaring state
ν (Hz)
40
10
5
1
100
200
500
1000
νu (Hz)
Figure 3.42: In analogy to Figure 3.3 for SAXJ1808.4-3658. When νu exceeds 700
Hz, Lh is not detected.
3.3.1.1
The Flaring State
In the above-mentioned analysis of SAXJ1808.4-3658 by van Straaten et al. (2005), the
authors suggest the identification of a feature in a power spectrum in the flaring state
as Lh , see Figure 3.44. Lu is not detected, therefore the identification of the Lorentzian
based on the correlation of Lh with Lu is not possible.
We find these power spectra in region D of the color-color diagram, see Figure A.14D. We set out to compare the centroid frequency of the Lorentzian we fit at ∼20 Hz
with the correlation we find of Lh and LLF with Lu . In single observations we do not
detect Lu > 3σ. We use the power spectrum of all averaged observations in region D
of the color-color diagram. νu =431 Hz. If we assume we weeded out the pulsar spike
efficiently, we find that the 21±3 Hz Lorentzian we fit, falls in between the correlation
of νh with νu and the correlation of νLF and νu (see Figure 3.42). We obtain a high
35±3% fractional rms amplitude, and low Q-factor of 0.33±0.08 for this Lorentzian. We
average power spectra of many observations, which has a broadening effect on power
spectral features. This explains the low Q-factors and high fractional rms amplitudes
obtained, and does not help in discriminating between LLF and Lh . We do not include
these findings in the assesment of the correlation of Lh and LLF with Lu .
Chapter 3. Data selection & Results
78
40
hump
hump
LF
LF
flaring state
flaring state
35
10
30
Q
rms (%)
25
20
15
1
10
5
0
10
20
30
40
ν (Hz)
50
60
70
80
0
10
20
30
40
50
60
70
80
ν (Hz)
Figure 3.43: In analogy to Figure 3.5, for SAXJ1808.4-3658; fractional rms amplitudes
and Q-factors of Lh and LLF fitted in power spectra of SAXJ1808.4-3658 plotted against
their centroid frequency.
Figure 3.44: Power spectrum from van Straaten et al. (2005) of SAXJ1808.4-3658
in which Gaussians are fitted to the 1-10 Hz features (dashed lines, indicated as ’G1 ’),
and Lorentzians are fitted with ν ∼10 and 40 Hz. νu is not detected.
16.38±0.20
34.16±0.26
21.00±0.39
18.68±0.40
16.64±0.26
16.36±0.51
14.73±0.41
16.47±0.81
13.42±0.86
14.49±0.53
0.84±0.05
1.34±0.07
0.79±0.07
1.45±0.19
1.05±0.07
38.25±20.34
62.82±33.19
58.69±33.28
0
16.40±17.68
-
338±7
493±7
382±13
354±16
328±4
-
419/329
380/326
376/329
362/326
409/322
1.31±0.05
1.64±0.32
1.54±0.09
1.53±0.09
10.89±0.61 3.87±1.15 2.77±1.53
0
2.27±0.09 10.52±0.19 2.93±0.63 3.61±1.31
Lu
ν (Hz)
680±4
664±4
337±7
Lb
Lb2
LLF
LLF
LLF
LLF2
Lh
Lh
Lh
LhHz
Ll
ν (Hz)
ν (Hz)
ν (Hz)
rms(%)
Q
ν (Hz)
ν (Hz)
rms (%)
Q
ν (Hz)
ν (Hz)
514/326 7.29±0.47 2.80±0.02 48.29±0.47 14.52±0.57 1.42±0.12
141.81±67.18
359/323 13.91±0.41 3.72±0.49 45.48±0.46 7.14±0.36 2.36±0.23
73.40±1.22 7.12±0.50 2.23±0.31 190.47±23.87
381/329 1.33±0.06
18.30±0.20 16.56±0.52 0.95±0.06 60.93±18.10
-
χ2 /dof
Table 3.10: In analogy to Table 3.2. Fit parameters of the multi-Lorentzian fits to power spectra of SAXJ1808.4-3658.
MJD
ObsID
Number
Region
Start
of PS. of CCdiagram
55872.8
96027-01-01-01/02/03/04/05
2474
F
52562.0 70080-[01-01-00/000/01/03/04][03-04-00] 2497
G
52570.3
70080-01-[03-000/00/04/05]
3344
G
[02-12/13/14/15/16/17/18/22]
52564.5 70080-01-[01-02][02-01/02/03/05/06]
3344
G
53530.6
91056-01-[02-08][03-00/01/02/03]
2526
G
53523.5
91056-01-02-00/01/03/04
2154
G
54731.8
93027-01-01-[00/01/02/03/08/080]
2000
G
52566.1
70080-01-02-00/000/04
2505
H
Chapter 3. Data selection & Results
79
Chapter 3. Data selection & Results
3.3.2
80
HETEJ1900.1-2455
HETEJ1900.1-2455 is a LMXB that shows intermittent 377.3 Hz X-ray pulsations (Morgan et al., 2005) with an orbital period of 83.3 minutes (Kaaret et al., 2006). It was
first detected with HETE-2 (the High Energy Transient Explorer 2) (Vanderspek et al.,
2005). Assuming a 1.4 M helium burning neutron star, the distance to the source is
estimated to be 5 kpc (Kawai and Suzuki, 2005). HETEJ1900.1-2455 has been reported
to be an atoll source based on its energy spectral behaviour (Papitto et al., 2013).
We retrieved all data from the RXTE archive, amounting to 354 observations comprising
1 Ms of usable data. We find Lh and Lu simultaneously in regions B, C and D of the
color-color diagram. Multi-Lorentzian fits to (a selection of) single observations from
these and adjacent regions are presented in Table 3.11.
Although we cannot identify detected features based on a possible correlation with Lu ,
Q and rms levels support the identification of the Lorentzian with centroid frequencies
∼10-15 Hz as Lh . We see that comparing to other sources in our sample in this regard
can be a diagnostic in identifying a power spectral feature, for pulsars and bursters alike.
A power law we fit to the frequencies of Lorentzians detected in this source has very
poor statistics; formally, κh =0.52±0.02 and αh =0.53±0.61 (χ2 /dof=2.4/1), we therefore
omit it from our interpretation, Section 4.2.
Chapter 3. Data selection & Results
81
100
hump
40
ν (Hz)
10
5
1
100
200
500
1000
νu (Hz)
Figure 3.45: In analogy to Figure 3.3, for HETEJ1900.1-2455. When νu exceeds 500
Hz, Lh is not detected.
hump
hump
25
10
15
Q
rms (%)
20
10
1
5
0
10
20
30
40
ν (Hz)
50
60
70
80
0
10
20
30
40
50
60
70
80
ν (Hz)
Figure 3.46: In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of Lh
and LLF fitted in power spectra of HETEJ1900.1-2455 plotted against their centroid
frequency.
Lb
Lb2
LLF
LLF LLF LLF2
Lh
Lh
Lh
LhHz
Ll
ν (Hz) ν (Hz) ν (Hz) rms(%) Q ν (Hz)
ν (Hz)
rms (%)
Q
ν (Hz)
ν (Hz)
318/329 0.21±0.34
14.18±1.37 16.26±1.64 0.65±0.15 124.00±16.73
335/330
0
10.67±2.04 19.84±1.79 0.40±0.12 130.06±10.82
333/329 0.40±0.38
11.12±1.45 19.00±1.69 0.54±0.13 112.15±7.86
-
χ2 /dof
Table 3.11: In analogy to Table 3.2. Fit parameters of the multi-Lorentzian fits to power spectra of HETEJ1900.1-2455.
MJD
ObsID
Number
Region
Start
of PS. of CCdiagram
53548.4 91015-01-04-03/04/06 1011
B
53608.9 91057-01-04-00/02
420
B
53842.7
92049-01-11-00
199
B
Lu
ν (Hz)
450±14
490±31
340±47
Chapter 3. Data selection & Results
82
Chapter 3. Data selection & Results
3.3.3
83
IGRJ17480-2446
IGRJ17480-2446 is an accreting X-ray pulsar in a binary with a 21.3 hr period (Papitto
et al., 2010), pulsating at 11 Hz (Strohmayer and Markwardt, 2010). It was discovered
in the globular cluster Terzan 5 and shows X-ray spectral and timing behaviour characteristic of both atoll and Z-sources (Altamirano et al., 2010). Its 11 Hz spin frequency is
atypical; other accreting low magnetic field (∼108 Gauss) neutron stars with measured
spins have frequencies 100-700 Hz. This might indicate that the magnetic field is of
intermediate strength (Papitto et al., 2010).
Altamirano et al. (2012) found a QPO ∼35-50 Hz and kHz QPOs in power spectra of
Terzan 5, in a data set of 48 RXTE observations. They argue that the power spectral
characteristics resemble those typical of Z-sources, and identify the ∼35-50 Hz QPO
with the horizontal branch oscillation (HBO). The Lense-Thirring precession model is
tested, using the ∼35-50 Hz QPO as νLT , see Equation 4.1. They conclude that if the
kHz QPO is identified as νK , νLT should be <0.82 Hz, which is much lower than the
frequencies observed. Classical Lense-Thirring precession is therefore not the mechanism
responsible for this QPO.
We retrieved all data from the RXTE archive, amounting to 151 observations comprising
500 ks of usable data. We identify LLF , Lh and Lu in regions B and C of the color-color
diagram, see Figure A.16.
Statistics were too poor to detect Lorentzians significantly in many (selections of) single
observations, we find only one, see Table 3.12. We identify the 44 Hz QPO (which is the
∼35-50 Hz QPO found in the analysis by Altamirano et al. (2012)) as LLF , as it falls on
the correlations of νLF with νu found in other sources in our sample4 , see Figure 3.47.
Fractional rms aplitude and Q-factor are consistent with this identification, but provide
no firm confirmation, see Figure 3.48. The measured Lorentzian is therefore omitted
from our interpretation, in Section 4.2.
4
HBO seen in Z-sources follow the same frequency-frequency (WK and PBK) relations as LLF and
Lh in atolls van der Klis (2006).
Chapter 3. Data selection & Results
84
100
LF
40
ν (Hz)
10
5
1
100
200
500
1000
νu (Hz)
Figure 3.47: In analogy to Figure 3.3 for IGRJ17480-2446. We identify the Lorentzian
we fit at ν =44.4 Hz as LLF , based on the correlation of νLF with νu in other sources
in our sample, plotted in lighter colors in this figure.
LF
LF
25
10
15
Q
rms (%)
20
10
1
5
0
10
20
30
40
ν (Hz)
50
60
70
80
0
10
20
30
40
50
60
70
80
ν (Hz)
Figure 3.48: In analogy to Figure 3.5; fractional rms amplitudes and Q-factors of
Lh and LLF fitted in power spectra of IGRJ17480-2446 plotted against their centroid
frequency.
Table 3.12: In analogy to Table 3.2. Fit parameters of the multi-Lorentzian fits to power spectra of IGRJ17480-2446.
MJD
ObsID
Number
Region
χ2 /dof
Lb
Lb2
LLF
LLF
LLF
LLF2
Lh
Lh
Lh
LhHz
Ll
Lu
Start
of PS. of CCdiagram
ν (Hz) ν (Hz)
ν (Hz)
rms(%)
Q
ν (Hz) ν (Hz) rms (%) 3.57±1.10 ν (Hz) ν (Hz) ν (Hz)
55490.5 95437-01-09-00
408
B
415/321 4.35±1.37
44.41±0.73 2.83±0.34 3.57±1.10
0
851±5
Chapter 3. Data selection & Results
85
Chapter 3. Data selection & Results
3.3.4
86
XTEJ1807-294
XTE J1807-294 is an AMXP discovered with RXTE in 2003, with a neutron star spin
frequency of 190.6 Hz(Markwardt et al., 2003a). It has a short orbital period of 40
minutes (Markwardt et al., 2003b). No X-ray bursts were reported for this source.
A multi-Lorentzian timing study (in νmax , Q representation) by Zhang et al. (2006)
revealed a similar shift in the frequency-frequency correlations as previously seen in
SAXJ1808.4-3658 (van Straaten et al., 2005), see Section 3.3.1. Zhang et al. (2006)
confirm that the shift exists in the upper kHz QPO frequency and find Lh with νmax
ranging from 17 to 39 Hz.
We retrieved all data from the RXTE archive, amounting to 116 observations comprising
500 ks of usable data. We find Lh and Lu to be simultaneously present in power spectra
from regions C and D of the color-color diagram (see Figure A.17) . We present our multiLorentzian fits to (a selection of) single observations from all color-color diagram regions
in Table 3.13, following the method explained in Section 2.2.4. The identication of Lh is
based on a comparison to centroid frequency ν, Q-factor and fractional rms amplitudes
of other sources, see Figure 3.50. Only for the Lorentzian with ν≈17 Hz, identification
as Lh is supported by rms amplitude and Q-factor (see Figure 3.50). We are convinced
that we measure the same Lorentzian in all three power spectra of single observations,
due to the clear correlation traced out when plotting their centroid frequency vs. νu , and
we therefore identify it as Lh regardless of the unclear identification of two Lorentzians
based on Q-factor and rms amplitude. We find a similar range of centroid frequencies
for Lh as reported in Zhang et al. (2006), with νh between 16 and 39 Hz. The similarity
of νmax and centroid frequency is due to the high Q-factor of detected Lorentzians in
this source.
We fit a power law to νh vs. νu . We find αh =1.77±0.17 (χ2 /dof =0.3/1). As explained
in Section 4.2, we do not use this result in the analysis of a possible correlation followed
by all sources (see Section 4.2.2), due to the observed shift in the frequency-frequency
correlation found in XTEJ1807-294 (among other pulsars) with respect to bursters. We
do however assess the power law index found for this source in Section 4.2.1.
Chapter 3. Data selection & Results
87
100
hump
40
ν (Hz)
10
5
1
100
200
500
1000
νu (Hz)
Figure 3.49: In analogy to Figure 3.3, for XTEJ1807-294. We identify the Lorentzian
we fit in this source at ν =17 Hz as Lh , based on fractional rms amplitude and Q-factor.
40
hump
hump
35
10
30
Q
rms (%)
25
20
15
1
10
5
0
10
20
30
40
ν (Hz)
50
60
70
80
0
10
20
30
40
50
60
70
80
ν (Hz)
Figure 3.50: In analogy to Figure 3.5; fractional rms amplitude and Q-factors of
Lh and LLF fitted in power spectra of XTEJ1807-294 plotted against their centroid
frequency.
Lb
Lb2
LLF
LLF LLF
Lh
Lh
Lh
LhHz
Ll
ν (Hz) ν (Hz) ν (Hz) rms(%) Q
ν (Hz)
rms (%)
Q
ν (Hz) ν (Hz)
398/329 1.19±0.88
- 16.96±0.99 17.44±1.62 0.67±0.13
0
326/331
0
39.4±2.2 11.73±1.31 1.83±0.90
0
301/331
0
26.4±0.9 11.31±1.72 1.89±1.11
0
-
χ2 /dof
Table 3.13: In analogy to Table 3.2. Fit parameters of the multi-Lorentzian fits to power spectra of XTEJ1807-294.
MJD
ObsID
Number
Region
Start
of PS. of CCdiagram
52702 80145-01-[02-00/01/02/03][03-02/03] 1642
B
52711
80145-01-02-05
744
B
52697
70134-09-02-00
216
B
Lu
ν (Hz)
362±4
585±4
458±3
Chapter 3. Data selection & Results
88
Chapter 3. Data selection & Results
3.3.5
89
IGRJ17511-3057
IGRJ17511-3957 is an AMXP Markwardt et al. (2009) with a neutron star spin frequency
of 244.8 Hz. The source shows type 1 X-ray bursts. The burst oscillations are within 1
Hz of the spin frequency of the neutron star (Altamirano et al., 2010), offering evidence
that burst oscillation frequencies reflect the spin frequencies of the neutron stars in
burst sources. A conservative upper limit on the distance is 7.1 kpc (Altamirano et al.,
2010), and the orbital period is ∼207.5 min. Kalamkar et al. (2011) investigated the
timing behaviour of IGRJ17511-3957 with a multi-Lorentzian model in the νmax , Q
representation. They used a dataset comprising 500 ks of data or 71 observations taken
in 2009, and fit a Lorentzian in the 5-14 Hz range they call Lh . In their analysis they
produce 9 power spectra for 500 ks of data, adding together more data than we do for our
final result. They do not fit a power law to the correlation of the maximum frequency
of Lh with Lu .
We retrieved all data from the RXTE archive, amounting to 290 observations or 1.3 Ms
of usable data. We find Lh and/or LLF to be present with Lu in regions B and C of the
color-color diagram, see Figure A.18. In Table 3.14 we present multi-Lorentzian fits to
(selections of) single observations from this region and adjacent regions. Due to poor
statistics we only fit one single observation from region B in the color-color diagram
where fitted Lorentzians exceed a 3 σ confidence level (the signal to noise in most observations is low). The Lorentzian with ν=2.3 Hz roughly falls on the correlation we find
in other sources of νh with νu (see Figure 3.51), and corresponds to the Lorentzian fitted
in Kalamkar et al. (2011) when converting their measurements to centroid frequency,
but by itself this result is unconvincing. In Figure 3.52 we see that we can identify
the Lorentzian as Lh when comparing fractional rms amplitude and Q-factor to other
sources. Due to a lack of fitted Lorentzians, we do not fit a power law to these data,
and we omit this source from our analysis of possible frequency-frequency correlations
in Section 4.2.
Chapter 3. Data selection & Results
90
100
hump
ν0 (Hz)
40
10
5
1
100
200
500
1000
ν0u (Hz)
Figure 3.51: In analogy to Figure 3.3, for IGRJ17511-3057. We interpret the
Lorentzian at ν = 2.3 Hz as Lh , based on rms levels and Q-factor (see Figure 3.52).
hump
hump
25
10
15
Q
rms (%)
20
10
1
5
0
10
20
30
40
ν0 (Hz)
50
60
70
80
0
10
20
30
40
50
60
70
80
ν0 (Hz)
Figure 3.52: In analogy to Figure 3.5; fractional rms amplitude and Q-factor of
the Lorentzian fitted in power spectra of IGRJ17511-3057 plotted against its centroid
frequency.
Table 3.14: In analogy to Table 3.2. Fit parameters of the multi-Lorentzian fits to power spectra of IGRJ17511-3057.
MJD
ObsID
Number
Region
χ2 /dof
Lb
Lb2
LLF
LLF LLF LLF2
Lh
Lh
Lh
LhHz
Ll
Lu
Start
of PS. of CCdiagram
ν (Hz) ν (Hz) ν (Hz) rms(%) Q ν (Hz) ν (Hz)
rms (%)
Q
ν (Hz) ν (Hz) ν (Hz)
52373.0 70131-01-[05-00/000/01/02/03][06-00/01] 3345
B
324/330 0.16±0.02
2.33±0.06 14.71±0.40 0.64±0.03
0
194±15
Chapter 3. Data selection & Results
91
Chapter 3. Data selection & Results
3.3.6
92
SAXJ1748.9-2021
We do not include results from our analysis of SAXJ1748.9-2021 in this report as statistics are poor. We do not detect LLF and Lh in any region of the color-color diagram.
We retrieved 174 observations from the RXTE archive, comprising 500 ks of usable
data. The color-color diagram can be found in the digital appendix (see B). For analysis
suggestions see Section 3.2.9.
3.4
Summary of results
In the previous sections we presented all fitted Lorentzians in power spectra of (a careful
selection of) single observations for each source in our sample. Distinguishing between
LLF and Lh is important when assessing a possible correlation of their centroid frequencies with νu . To identify a Lorentzian as LLF or Lh we use two characteristics.
Firstly, we compare the fitted power spectrum in a single observation to the timing
behaviour of the source in the region of the color-color diagram where the observation
occurs. Relying on the similar timing behaviour of different sources, we apply the identification scheme discussed in section 2.2.4, and use the condition that always νh > νLF .
We find that when we plot νh against νu a correlation is traced out that is approximately
parallel to, but at higher frequency than the correlation of νLF with νu , see for instance
Figure 3.3.
Secondly, we use the relation of the Q-factor and fractional rms amplitudes of LLF and
Lh with their own centroid frequency. At low (40 Hz), we see a clear distinction between
Lh and LLF in this regard: LLF has higher Q-factors and lower fractional rms amplitudes. Distinguishing between the two Lorentzians based on Q-factor and fractional rms
amplitude becomes harder with increasing centroid frequency, see e.g. Figure 3.54. In
practice we find that Lorentzians with centroid frequencies >40 Hz detected in power
spectra with νu >700 Hz in 4U 1728-34 and 4U 0614+09, and with νu >600 Hz in 4U
1702-43 are difficult to identify based on these criteria.
In section 3.2.1 on 4U 1728-34, we present a power spectrum (see Figure 3.4) with
νu =802 Hz in which we detect a 2σ Lorentzian with a centroid frequency of 65.1 Hz following the same relation with νu as νh . In the same spectrum we detect a 3σ Lorentzian
with a centroid frequency of 35.4 Hz that follows the same relation with νu as νLF .
This possible detection of Lh supports the identification of the Lorentzians with much
lower centroid frequencies of ∼40-50 Hz in power spectra where we only detect one
Lorentzian between ∼25-80 Hz (νu ≈800-900 Hz) as LLF . Additionally, they follow the
same relation with νu as νLF .
Chapter 3. Data selection & Results
93
Some pulsars are known to behave differently from bursters with regard to frequencyfrequency correlations (for instance the correlations of SAXJ1808.4-3658 and XTEJ1807294 appear to be shifted compared to those of bursters(van Straaten et al., 2005, Zhang
et al., 2006). We come to the same conclusion based on our data, see Figure 3.53. However, we see strong similarities between the two goups when comparing the behaviour
of Q-factor and fractional rms amplitudes with regard to centroid frequency of LLF and
Lh , see Figure 3.54. The identification of Lorentzians we fit in power spectra of pulsars
HETEJ1900.1-2455, IGRJ17480-2446, XTEJ1807-294 and IGRJ17511-3057, is therefore
based on the behaviour of the Q-factor; see for instance Figure 3.52.
With the purpose of selecting the best measured and most unambiguous set of frequencies
possible, we define ”subselection A” as the Lorentzians that were unambiguously identified based on Q-factor and fractional rms amplitude. This selection therefore excludes
Lorentzians detected in power spectra with νu >700 Hz in 4U1728-34 and 4U0614+09,
and with νu >600 Hz in 4U1702-43. We also exclude from subselection A Lorentzians
detected in power spectra with νu < 250 Hz in KS1731-260, where we fit harmonics of
LLF that complicate identification, and Lorentzian detected in power spectra of pulsar
IGRJ17480-2446, where identification is ambiguous based on Q-factor and fractional rms
amplitude.
In Table 3.15 we present an overview of all fitted centroid frequencies of LLF , Lh and
Lu for each source. The best fit power law indices to νLF vs. νu and νh vs. νu (with
normalization and χ2 /dof) can be found in Section 4.2.2. For these fits we only use
Lorentzians from subselection A that also exceed a 3σ confidence level.
Chapter 3. Data selection & Results
94
100
hump
ν (Hz)
LF
10
1
100
200
500
1000
νu (Hz)
Figure 3.53: νLF and νh from subselection A plotted against νu . Open symbols
indicate frequencies measured in pulsars. Filled symbols indicate frequencies measured
in bursters. All components exceed a 3σ confidence level. We plot frequencies not
belonging to subselection A in lighter colors.
Chapter 3. Data selection & Results
95
40
hump
LF
35
30
rms (%)
25
20
15
10
5
0
10
20
30
40
50
60
70
80
70
80
ν (Hz)
hump
LF
Q
10
1
0
10
20
30
40
50
60
ν (Hz)
Figure 3.54: fractional rms amplitudes and Q-factors of Lh and LLF from subselection
A plotted against their centroid frequencies. All Lorentzians exceed a 3σ confidence
level. Open circles indicate indicate rms and Q of Lorentzians fitted in pulsars. Filled
circles indicate rms and Q of Lorentzians fitted in bursters. All components exceed a
3σ confidence level. We plot rms and Q of Lorentzians not belonging to subselection A
in lighter colors.
Chapter 3. Data selection & Results
Source
Spin Freq. Subselection
(Hz)
360.5-364.2
A
A
A
A
A
A
A
A
A
A
A
A
LLF
Lh
ν (Hz)
ν (Hz)
4U 1728-34
5.59±0.08
10.48±0.52
6.18±0.13
9.54±0.22
8.08±0.13
13.43±0.36
6.25±0.12
9.45±0.16
16.30±0.60
11.04±0.09 17.71±0.35
14.60±0.13
24.1±0.6
20.01±0.51
35.5±3.0
22.7±0.3
38.8±1.6
27.2±0.3
43.4±1.4
26.1±0.4
40.7±4.5
35.4±0.3
65.1±1.5∗
41.6±0.49
44.06±0.89
46.8±0.3
45.5±0.7
4U 1636-53
578-582
A
1.93±0.02
4.47±0.12
A
5.03±0.11
10.85±0.90
A
5.53±0.54
13.60±1.16
A
5.95±0.12
11.39±1.00
A
23.57±0.73
A
6.58±0.13
14.48±0.60
A
8.98±0.14
20.85±0.57
A
21.34±2.25
A
14.72±0.19 34.05±1.09
4U 1608-52
619
A
2.13±0.04
5.58±0.34
A
2.38±0.02
5.17±0.34
A
3.91±0.04 12.01±0.95∗∗
A
10.02±0.13 23.50±1.62
A
28.53±0.61
4U 0614+09
414.7
A
1.27±0.03
A
1.33±0.14
A
5.45±0.15*
8.07±0.33
A
4.46±0.11
7.77±0.27
A
12.23±1.03
A
13.47±0.64 26.93±1.64
A
15.61±0.34
26.59±1.6
A
18.84±0.61 36.81±2.08
A
21.42±0.55 37.24±1.82
36.92±0.52
45.28±1.67∗
FS
107.4±2.4
4U 1702-43
326.5-330.5
A
10.73±0.47
A
5.87±0.24
A
5.56±0.22
A
7.87±0.28
13.77±0.65
∗
A
6.78±0.23 ∗ 10.53±0.52
A
6.77±0.07
9.85±0.32
A
12.95±0.21 22.66±0.87
A
13.24±0.36 23.76±1.22
A
12.80±0.53 22.72±1.85
A
15.16±0.51 27.21±1.39
22.97±1.41
FS
84.95±2.19
KS 1731-260
522.5-525
0.81±0.01
2.07±0.41
0.79±0.02
1.87±0.12
1.53±0.04
3.45±0.10
A
9.37±0.37
16.75±0.43
A
15.65±0.79
SAXJ1750.8-2900 599.5-601
17.52±0.28 28.36±4.24
22.25±3.02
Aql X-1
550.3
A
3.53±0.11
A (only νLF ) 3.38±0.05
3.49±0.09
A
5.89±0.16
A
15.08±0.74
∗∗
A
12.98±0.39
21.66±1.42
A
19.95±1.40
SAXJ1808.4-3658
401.0
A
48.29±0.47
A
45.48±0.46 73.40±1.22
A
18.30±0.20
A
16.38±0.20
A
34.16±0.26
A
21.00±0.39
A
10.89±0.61 18.68±0.40
A
10.52±0.19 16.64±0.26
HETEJ1900.1-2455
377.3
14.18±1.37
10.67±2.04
11.12±1.45
IGRJ17480-2446
11.0
44.41±0.73
XTEJ1807-294
190.6
A
17.44±1.62
A
39.4±2.2
A
26.4±0.9
IGRJ17511-3057
244.8
A
2.33±0.06
96
Lu
ν (Hz)
380± 12
412±13
467±12
324±13
479±8
501±7
556±4
625±10
654±5
698±6
689±6
802±1
847±1
869±3
873±3
901±2
289±33
360±32
425±17
418±20
502±8
440±13
450±14
556±5
602±5
230±15
279±13
317±22
482±24
519±7
140±8
175±31
346±9
324±12
441±13
491±19
535±9
564±8
602±7
746±4
855±4
1304±25
407±18
284±22
312±24
388±30
401±18
386±17
507±14
512±12
527±11
539±11
639±8
1223±30
162.2±11.5
180.8±32.7
202.8±21.8
484.4±6.6
459.4±14.6
579±23
649±14
272±19
265±12
299±52
409±30
436±26
488±20
680±4
664±4
337±7
338±7
493±7
382±13
354±16
328±4
450±14
490±31
340±47
851±5
362±4
585±4
458±3
194±15
Table 3.15: Summary table of all fitted centroid frequencies of LLF , Lh and Lu for
each source. Spin frequencies were taken from articles cited in previous sections. All
Lorentzians exceed a 3σ confidence level, except where noted otherwise. The errors use
∆χ2 = 1.0. The ∗ starred fit value in 4U 1728-34 was detected at a 2.0σ confidence
level, those in 4U 0614+09 were detected at a 2.5 σ confidence level, and the ∗∗ fit
values in 4U 1608-52, 4U 1702-43 and Aql X-1 at 2.2σ. ”A” indicates subselection A,
see Section 3.4 for a discussion. ”FS” indicates detections in flaring states. No attempt
was made to fit power spectra outside subselection A for sources other than 4U 1728-34,
4U 0614+09, 4U1702-32 and IGRJ17480-2446.
Chapter 4
Interpretation
In this chapter we present the interpretation of our results. The principal issue we assess
is how well the Lense-Thirring precession model as proposed by Stella and Vietri (1998)
for QPO frequencies describes our data. We also discuss what other conclusions we can
derive from power law fits to our data.
4.1
The Lense-Thirring model
As noted in Chapter 1, the relativistic precession model of Stella and Vietri (1998) predicts a QPO with a centroid frequency νLT depending on the Keplerian orbital frequency
νK in the same way as the Lense-Thirring precession of a test particle:
νLT =
2 ν
8π 2 IνK
s
2
= 4.4 × 10−8 I45 m−1 νK
νs ,
c2 M
(4.1)
where M and I are the neutron star mass and moment of inertia, νs is the spin frequency
of the neutron star, m is the mass in units of solar mass M , and I45 is the moment of
inertia in units of 1045 g cm2 . When assuming that in our analysis either νh or νLF is
representative of νLT and that νu represents νK , we can scale the correlations we find
by the spin frequency νs , and write:
νLF,h
= 4.4 × 10−8 I45 m−1 νu2 .
νs
(4.2)
Models of rotating neutron stars predict values of I45 m−1 between 0.5 and 2, for any
acceptable equation of state and mass (Cook et al., 1994, Friedman et al., 1986, Stella
and Vietri, 1998). We can plot a range of I45 m−1 within which the values of νLF or νh
97
Chapter 4. Interpretation
98
plotted vs. νu should fall for a given spin frequency of the neutron star. It was proposed
that the frequency we observe could be twice the precession frequency (Vietri and Stella,
1998), and we account for this by letting I45 m−1 vary between 0.5 and 4. When scaling
by spin, all frequencies of either LLF or Lh , or both, for all sources in our sample should
then fall within this range of I45 m−1 , see Figure 4.1 and 4.2.
We use all measurements of νLF and νh in Table 3.15 in this scaling analysis, except
those that were obtained from fits to power spectra in the flaring states of 4U0614+09
and 4U 1702-43, and measurements that did not exceed a 3 σ confidence level.
We see in Figures 4.1 and 4.2 that our measurements of both νLF and νh in pulsars
SAXJ1808.4-3658, IGRJ17480-2446, XTEJ1807-294 and IGRJ17511-3057 fall above the
range of acceptable I45 m−1 , with IGRJ17480-2446 (in Terzan 5) an extreme outlier due
to its spin frequency of 11 Hz (upper right corner of Figure 4.1), cf. Altamirano et al.
(2012). For these sources we can conclude, based on the scaling analysis, that LenseThirring precession as proposed by Stella and Vietri (1998) is not the cause of the power
spectral features detected. Only if we see more than twice the precession frequency,
we could reconcile the model prediction with our data. We note that all these sources
are pulsars, and might therefore have stronger magnetic fields than the bursters in our
sample.
For bursters 4U 1728-34, 4U 1702-43 and 4U 0614+09 some or all νh are also out of
range, but all νLF are within the range of acceptable I45 m−1 . Finally, for bursters 4U
1636-53, SAXJ1750.8-2900, 4U1608-53 and KS1731-260, and pulsars HETEJ1900.1-2455
and Aql X-1 all measurements of both νLF and νh fall within the range.
The majority of measured centroid frequencies within the range, fall above the black
line marking I45 m−1 = 2, indicating that νLF or νh can only be identified with the
Lense-Thirring precession frequency νLT if we see twice the precession frequency.
Chapter 4. Interpretation
99
I45 /m from 0.5 to 4
4U1728-34
4U1636-53
4U1608-53
4U0614+09
4U1702-43
1
KS1731-260
SAXJ1750.8-2900
Aql X-1
SAXJ1808.4-3658
HETEJ1900.1-2455
IGRJ17480-2446
XTEJ1807-294
IGRJ17511-3057
ν
νspin
0.1
0.05
0.02
0.01
0.001
100
300
500
1000
νu (Hz)
Figure 4.1: νLF and νh scaled by neutron star spin vs. νu . Triangles are measurements
of νLF , circles of νh . Filled symbols refer to bursters, open symbols to pulsars. We also
plot the range in which I45 m−1 of the neutron star would be acceptable in the LenseThirring interpretation of Stella and Vietri (1998). The black line marks I45 m−1 = 2
for an observed frequency (νLF , νh ) equal to νLT , or I45 m−1 = 1 for νLF , νh equal to
twice νLT .
Chapter 4. Interpretation
100
I45 /m from 0.5 to 4
4U1728-34
4U1636-53
4U1608-53
4U0614+09
4U1702-43
KS1731-260
SAXJ1750.8-2900
0.1
Aql X-1
SAXJ1808.4-3658
HETEJ1900.1-2455
ν
νspin
XTEJ1807-294
0.05
0.02
300
500
νu (Hz)
Figure 4.2: A zoom in on Figure 4.1.
Chapter 4. Interpretation
4.2
101
Fitting the Data
While the scaling analysis in Section 4.1 provides an indication of whether or not the
measurements in a given source can be in accordance with the Lense-Thirring interpretation, the full predicted frequency-frequency correlation is not tested there. In this
section we test the νh,LF ∝ νu2 νs prediction directly, and also consider an alternative
interpretation. For this analysis we use subselection A (see Section 3.4). In particular,
we test two hypotheses in this section, regarding the power law fits to correlations of νh
and νLF with νu :
A Our data are in agreement with the Lense-Thirring prediction. The power law
index α is consistent with 2 in each source and the spin frequency affects the
normalization of the power law.
B All sources fall on the same power law relation, regardless of differences in neutron
star spin frequency.
By fitting the data with power laws varying in degrees of freedom we answer the following
detailed questions;
A1 In each source, does a power law fit?
A2 In each source, does our best fit power law fit the data better than when we fix α
to 2?
A3 In each source, when we fix α to 2, does the normalization (κx ) of the power law
fall within the acceptable range of I45 m−1 ?
A4 Among all sources, are the normalizations we obtain when we fix α to 2 correlated
to νs ?
B1 Can the relation between νh or νLF and νu be described by a power law, when
combining the data of all sources in our sample?
B2 Do individual source power law fits fit the data significantly better than the power
law fit to combined data?
Chapter 4. Interpretation
4.2.1
102
Hypothesis A
We perform power law fits of the form:
νx = κx × νuαx
(4.3)
to assess a possible relation of νh and νLF with νu of each source, where x stands for
LF or h. The details (χ2 /dof ,κx ) of these fits can be found in Table 4.1. We can fit the
data of each source with a power law (A1 ). While reduced χ2 values tend to be >1, in
view of the often low number of degrees of freedom, only one of the best power law fits
is rejected at better than 3σ confidence (for νh in SAXJ1808.4-3658). We note that the
pulsars XTEJ1807-294 and SAXJ1808.4-3658 show a lower power law index than the
bursters, see Figure 4.4.
We now fix α to 2, and compare these fits to the ones with α free, using an F-test for
one additional term (see Section 2.2.5). We find that for νh in bursters 4U1728-34 and
4U1702-43 the α = 2 power law fits are significantly worse (P<0.01). For νLF , this is
only the case for burster 4U1728-34 (A2 ). So, we can reject Hypothesis A for these cases.
The remaining sources are: 4U 1636-53, 4U 1608-52, 4U 0614+09, XTEJ1807-294, AqlX1 and SAXJ1808.4-3658 for νh vs. νu , and 4U 1702-43, 4U 1636-53, 4U 1608-52, 4U
0614+09 and SAXJ1808.4-3658 for νLF vs. νu .
For the remaining sources, we assess if we can reject Hypothesis A based on unrealistic
values of I45 m−1 . In the Lense-Thirring prediction, κ = 4.4 × 10−8 I45 m−1 νs . Using
the κ values obtained when fitting a power law with α = 2 and the spin frequency of
the neutron star, we obtain the values for I45 m−1 presented in Table 4.1. For νh , all
sources have I45 m−1 of more than 2 which is only acceptable if we see the precession
frequency twice and if it is <4. For νLF , 3 of the 5 bursters have I45 m−1 of more than 2.
Frequencies measured in pulsars SAXJ1808.4-3658 and XTEJ1807-294 all have I45 m−1
of more than 4, we can reject Hypothesis A for these sources. The same holds for νh
measured in burster 4U 0614+09 (A3 ).
Of the remaining sources, we are left with 4U 1636-53, 4U 1608-52 and Aql X-1 for νh
vs. νu , and 4U 1702-43, 4U 1636-53, 4U 1608-52, 4U 0614+09 for νLF vs. νu . We cannot
reject Hypothesis A in these cases.
From these two tests, for bursters 4U 1728-34, 4U 1702-43 and 4U 0614+09 (only for
νh in the last two sources), and pulsars SAXJ1808.4-3658 and XTEJ1807-294 we can
state that the Lense-Thirring prediction as proposed by Stella and Vietri (1998) does
not agree with our data.
Chapter 4. Interpretation
103
When we plot the normalization obtained by a power law fit with α fixed to 2 (κα=2 ) vs.
the spin frequency of the neutron star (νs ) we do not see any systematic increment or
decrement. In the Lense-Thirring model κα=2 reflects the expression 4.4×10−8 I45 m−1 νs ,
and we therefore expect a linear dependence on spin. However, mass differences between
neutron stars are expected to disturb this relation through I45 m−1 variation. We note
that when we plot the difference between κα=2,h and κα=2,LF (∆κα=2 ), there seems to
be a systematic increment of this quantity with spin frequency for the bursters in our
sample (see Figure 4.3) (A4 ). However, the Spearman rank correlation coefficient for
∆κα=2 vs. νs is ρ=0.9, corresponding to P=0.4 (see Section 2.2.5.1); so the correlation
between these two parameters is not significant.
200
100
hump
bursters
LF
pulsars
80
∆κα=2(×10−6)
κα=2(×10−6)
150
100
60
40
50
20
0
0
200
300
400
500
νspin (Hz)
600
700
250
300
350
400
450
500
550
600
650
700
νspin (Hz)
Figure 4.3: Left: values for κα=2 plotted against spin frequency of the neutron stars.
Right: ∆κ ≡ κα=2,h - κα=2,LF plotted against spin frequency of the neutron star. Filled
symbols refer to bursters, open symbols to pulsars.
4.2.2
Hypothesis B
As mentioned in section 3.4, pulsars have different frequency-frequency correlations from
bursters van Straaten et al. (2005). For testing hypothesis B we therefore do not combine
all data, but focus on the bursters in our sample, using the data in subselection A (see
Section 3.4). We fit power laws to νh vs. νu and νLF vs. νu of all bursters together. We
find αh = 2.45±0.05, κh = 4.7 ± 0.6 × 10−6 ( χ2 /dof = 230/49), αLF = 2.70±0.05, and
κLF = 5.6 ± 0.5 × 10−7 (174/31). We can reject the hypothesis that all sources follow
Chapter 4. Interpretation
104
the same power law correlation based on these large χ2 , P<0.0001 in both cases (B1 ).
Of course these bad fits could be due to individual sources fitting well to power laws
that are different, or to power laws being bad fits even in individual sources. To check
this, we fix the slope and normalization constant of a power law to these best fit values
for the combination of all bursters, and compare the resulting χ2 to those from our fits
to individual sources. The χ2 /dofκ,αf ixed values are given in Table 4.1. We perform
an F-test for two additional free parameters (see Section 2.2.5) with χ2 /dofκ,αf ixed and
χ2 /dofκ,αf ree , and list F and P values. We see that for νh we cannot reject B2 , as all
sources have P >0.01 that random scatter led to the better fit with both parameters
free. In the case of νLF we can reject B2 , as sources 4U1636-53 and 4U0614+09 have
P<0.01 that random scatter led to the better fit with both parameters free.
That all sources have low probability for these hypotheses (B1 ,B2 ) may indicate that
Hypothesis B does not apply to any of them. We note that we might be underestimating
the errors of the measured centroid frequencies due to the exclusion of systematic errors
when determining the error on the fit parameters (Ford and van der Klis, 1998, van
Straaten et al., 2003).
Chapter 4. Interpretation
Figure 4.4: Best fit power laws to νh and νLF vs. νu of all sources in our sample
in one plot. To unclutter the plot we scale as indicated in the legend. Triangles are
measurements of νLF , circles of νh . Open symbols refer to pulsars, filled symbols to
bursters.
105
Source
4U 1728-34
4U 1636-53
4U 1608-52
4U 0614+09
4U 1702-43
Aql X-1
SAXJ1808.4-3658
XTEJ1807-294
4U 1728-34
4U 1636-53
4U 1608-52
4U 0614+09
4U 1702-43
SAXJ1808.4-3658
νs
362.4
580
619
414.7
328.5
550.3
401
190.6
362.4
580
619
414.7
328.5
401
NL
11
9
4
8
10
5
7
3
11
7
4
5
4
4
κ (×10−6 )
1.10±0.8
2.80±2.0
7.1±5.5
1.7±0.8
0.55±0.4
0.08±0.01
67.1±10
511±334
0.38±0.2
2.3±0.6
3.6±2.5
1.9±1.0
23.5±23.0
56.23±14
α
2.67±0.09
2.53±0.17
2.43±0.16
2.65±0.14
2.81± 0.22
3.14± 0.29
2.13±0.04
1.77±0.17
2.76±0.07
2.44±0.15
2.40±0.24
2.54±0.14
2.12± 0.60
2.09±0.04
χ2 /dof κα=2 (×10−6 ) I45 m−1 χ2 /dofα=2
11.4/9
76 ±9
4.8
92.9/10
18.9/7
89±2
3.5
31.6/8
6.8/2
102±3
3.7
15.6/3
17.0/6
95±3
5.2
48.0/7
5.9/8
81±3
5.6
29.2/9
3.5/3
68±1
2.8
21.5/4
20.0/5
155±2
8.8
29.4/6
0.3/1
125±3
14.9
2.36/2
3.7/8
51±1
3.2
216.3/9
9.8/5
39±1
1.8
22.8/6
3.7/2
37±2
1.4
7.3/3
0.9/3
56±1
3.1
17.3/4
2.8/3
50±2
3.5
2.82/4
7.15/3
103±1
5.8
7.2/3
F
64.34
4.73
2.59
10.91
31.34
15.42
2.33
6.87
466.09
6.65
1.95
52.09
10.03
7.05
P
χ2 /dofκ,αf ixed
F
P
0.22 ×10−4
29.7/11
7.26 0.013
0.066
38.4/9
3.61 0.08
0.25
74.8/4
10.00 0.09
0.016
60.5/8
7.67 0.02
0.51×10−3
14.8/10
6.03 0.03
0.29×10−1
0.19
0.23
0.22×10−7
4.6/10
1.08 0.38
0.049
74.3/7
16.45 0.006
0.29
9.4/4
1.54 0.39
0.55
68.2/5
112.16 0.002
0.87
14.9/5
6.48 0.08
0.12
-
Table 4.1: Results of F-tests comparing χ2 /dof obtained in three cases for each source. 1; both α and κ are free. 2; κ is free and α=2. And 3; α
and κ are both fixed to the values fitted to the best fit to data of all sources combined. NL are the number of detected Lorentzians (Lh or LLF ).
LF
hump or LF
h
Chapter 4. Interpretation
106
Chapter 4. Interpretation
4.3
107
Comparing to van Straaten et al. (2003): 4U 1608-52,
4U 0614+09 and 4U 1728-34 combined
van Straaten et al. (2003) took three sources into consideration; 4U 1608-52, 4U 0614+09
and 4U 1728-34. The same power law relation provided a satisfactory fit to all three
sources together, and a power law index of 2.01±0.02 was measured. When plotting the
measurements of van Straaten et al. (2003) together with our own, see Figure 4.5, we
see that our discrimination between LLF and Lh yields a different result for the power
law slope fitted to the combined data of all three sources. We find αh = 2.41 ± 0.05
(χ2 /dof = 227.7/21, P<0.05) and αLF = 2.61 ± 0.05 (χ2 /dof = 63.1/17, P<0.0005). In
Sections 3.2.1.1 (4U 1728-34), 3.2.3.1 (4U 1608-52) and 3.2.4.2 (4U 0614+09), we discuss
the differences between our analysis and the analysis of van Straaten et al. (2003) in
more detail for each source. The large reduced χ2 values indicate that power laws do
not fit the combined data of these three sources, even when separating LLF and Lh .
We fix the power law index in these three sources to αh = 2.41 and αLF = 2.61 and
refit the data of each source. We do an F-test for one additional free parameter with
the obtained χ2 /dof with α fixed, and the χ2 /dof with α free. The only value of P<0.01
is P=0.0074, for 4U 1728-34 where we fixed αLF = 2.61. For νh we cannot reject the
hypothesis that these three sources correlate similarly with νu . We can however state,
that for νLF these three sources do not all show the same correlation with νu .
4.4
Other Possibilities and Future Work
The Lense-Thirring model links the orbital motion of individual test particles in empty
space to QPOs that are detected in the emission of accretion disk. In an accretion
disk, where many interacting particles are present, hydrodynamics will cause deviations
from model predictions. We find values of I45 m−1 for most sources that are larger than
predicted by realistic equations of state when assuming that we see the fundamental of
the precession frequency, represented by νLF or νh . We could, as suggested in section
4.2.1, be seeing twice the precession frequency due to a twofold symmetry existing in the
accretion disk. This would however only reconcile the model with observations for some
and not all sources in our sample. In the extreme case of IGRJ17480-2446 we would
have to see the fundamental frequency an implausible 140 times. XTEJ1807-294, offers
another unrealistic case with 14 times the fundamental frequency.
A latitude-dependent radiation field1 , as proposed by Miller (1999), could produce a
precession frequency significantly greater than the Lense-Thirring precession frequency
1
This can occur when accretion does not take place on the entire surface of the neutron star but for
instance only onto the equatorial region.
Chapter 4. Interpretation
108
100
hump
LF
Van Straaten 2002
ν0 (Hz)
40
10
5
1
100
200
500
1000
ν0u (Hz)
Figure 4.5: Measurements of Lh from the multi-Lorentzian timing study by van
Straaten et al. (2002), converted from νmax to ν, plotted together with our measurements of νLF and νh . The best fit from van Straaten et al. (2003) to combined data
of 4U 0614+09, 4U 1728-34 and 4U 1608-52 with a power law index of 2.01±0.02 is
drawn, as well as our best fit power laws to the combined data of the three sources.
through radiation forces, even in low luminosity sources. If radiation forces dominate
the orbital and precession frequencies of accreting gas, the correlation of the precession
frequency with the upper kHz QPO frequency would be partly attributable to radiation
forces and not just to frame dragging effects (Psaltis et al., 1999b).
In Psaltis et al. (1999b) a power law index of on average ∼1.8, is found when fitting the
centroid frequency of the horizontal branch oscillation νHBO vs. νu in five Z-sources.
Unrealistically high values of I45 m−1 are found. While our pulsars, the least luminous
sources in our sample show an index of (αLF,h =1.77-2.13), the atoll sources in our sample,
which are intermediate in luminosity to the pulsars in our sample and the Z sources in
(Psaltis et al., 1999b), show an index that is significantly greater than 1.8.
This discrepancy might possibly be explained by a mix up of harmonics of the HBO,
similar to the mix up of LLF and Lh we described above. It could also mean that the
Chapter 4. Interpretation
109
HBO in Z-sources cannot be compared to LLF or Lh in atolls.
We do not find the linear dependence of the precession frequency on the spin that is
predicted by the Lense-Thirring model. A possible explanation is offered when regarding
the hot inner flow of the disk to precess as a solid body (a torus) between an inner and
outer radius (Ingram and Done, 2010). In this scenario, the sound crossing timescale
through the disk is shorter than the precession timescale, and the material in the disk
is coupled via pressure waves. As a result of the misalignment of the flow with the
spin-axis of the neutron star, the inner radius of the precessing torus is not determined
by the innermost stable orbit, but depends on spin. The dependence of the precession
frequency on the spin is counteracted, as the inner radius is larger for faster spinning
compact objects. The correlation of the precession frequency with the Keplerian orbital
frequency is expected to change, but how exactly is not clear. The frequency detected
in the extreme outlier IGRJ17480-2446 (in Terzan 5) with a spin of 11 Hz cannot be
reconciled with this model, and would form an exception (Altamirano et al., 2012).
We do not take classical and magnetic precession into account in this work. They could
however affect or fully determine the precession frequency. Magnetic fields can deform
an accretion disk (Shirakawa and Lai, 2002), resulting in precession that is independent
of spin frequency. For neutron stars with suggested strong magnetic fields such as
pulsars, the retrograde magnetic precession could dominate over the prograde LenseThirring precession. Classical (Newtonian) precession is due to the quadrupole moment
of a deformed oblate neutron star, and is retrograde. Under the assumption that the
deformation is due to stellar rotation only, the corrections are of the order of the LenseThirring precession frequency (Morsink and Stella, 1999). The combination of the three
precession mechanisms could account for the different behaviour of pulsars and bursters
in our sample.
Chapter 5
Conclusion
In Chapter 3 we have presented our measurements and described the behaviour of frequencies in the power spectra of a sample of 19 sources. We find two different power
spectral features in the frequency domain predicted for the Lense-Thirring precession
frequency by Stella and Vietri (1998). We place the findings of van Straaten et al.
(2003) in the context provided by the expansion of the dataset in Sections 3.2.1.1 (4U
1728-34), 3.2.3.1 (4U 1608-52) and 3.2.4.2 (4U 0614+09). We see that in each of the
three sources presented in van Straaten et al. (2003), for which a correlation was found
with a power law index of 2.01, the two features we find are mixed up. We find higher
power law indices when discriminating between them. In Chapter 4 we have offered an
interpretation of our findings in the framework of the Lense-Thirring model. We find
that the required specific moment of inertia is too high for realistic equations of state.
We find power laws with indices that describe the data significantly better than the
Lense-Thirring prediction of index 2 in five sources, of which three are pulsars. The
power law indices are larger in non pulsating bursters than in pulsars included in our
sample. We do not find the predicted dependency of the precession frequency on the
spin frequency of the neutron star. In Section 4.4 we briefly discuss how the inclusion
of radiation forces, and magnetic and classical precession might lead to models more
compatible with our findings.
110
Appendix A
Color-Color Diagrams
This Appendix contains the color-color diagrams and representative fitted power spectra
for the different color regions for each source. Each point in the diagram represents the
energy spectral shape of one observation of a few ks. The position in the diagram traces
the appearance of the source power spectrum, as can be seen in Figure A.1. Error
bars are smaller than the symbols (typically <0.5%). These results were used to map
the timing behaviour of the source, and assess which regions of the color-color diagram
contain power spectra showing Lh and Lu . None of the fitted Lorentzians shown in this
appendix were used to determine the power law index of correlations between νLF , νh
and νu .
111
Appendix A. Color-Color diagrams
A.1
4U1636-53
112
Appendix A. Color-Color diagrams
113
1.2
A
D
B
E
C
Not used
1.1
Hard Color (Crab)
1
0.9
0.8
0.7
0.6
0.5
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
Frequency × (RMS/Mean)2 Hz−1
Soft Color (Crab)
Frequency (Hz)
Figure A.1: Top: Color-Color diagram for 1555 observations of 4U1636-53. We opted
to limit our analysis of the regions in this diagram to the ones where LLF , Lh and Lu
were indicated to be present by Altamirano et al. (2008a).
Bottom: Representative fitted power spectra for different regions of the color-color
diagram. Observations from regions A, B, C, D and E were used for further analysis, as
the hump feature, and the LF and kHz QPOs appear here (Altamirano et al., 2008a).
Observations from the boundary between region E and F were also included.
Appendix A. Color-Color diagrams
4U1608-52
1.2
1.1
A
F
B
G
C
H
D
I
E
1
0.9
Hard Color (Crab)
A.2
114
0.8
0.7
0.6
0.5
0.4
0.7
0.8
0.9
1
1.1
1.2
1.3
Soft Color (Crab)
Figure A.2: Color-Color diagram for 495 observations of 4U1608-52.
115
Frequency × (RMS/Mean)2 Hz−1
Appendix A. Color-Color diagrams
Frequency (Hz)
Figure A.3: Representative fitted power spectra for different regions of the color-color
diagram (see Figure A.2). Regions A, B, C and D were used for further analysis, as
the hump, LF and kHz QPOs only appear here (also identified by van Straaten et al.
(2003)).
Appendix A. Color-Color diagrams
A.3
116
4U0614+09
1.1
A
E
B
F
Ci
G
Cii
HLF
D
1
Hard Color (Crab)
0.9
0.8
0.7
0.6
0.5
0.4
0.9
0.95
1
1.05
1.1
Soft Color (Crab)
Figure A.4: Color-color diagram for 495 observations of 4U0614+09.‘HLF’ stands for
high luminosity flaring.
117
Frequency × (RMS/Mean)2 Hz−1
Appendix A. Color-Color diagrams
Frequency (Hz)
Figure A.5: Representative fitted power spectra for different regions of the ColorColor diagram (see Figure A.4). Regions A, B, and Cii were used for further analysis,
as the hump, LF and kHz QPOs only appear here.‘HLF’ stands for high luminosity
flaring.
Appendix A. Color-Color diagrams
A.4
118
4U1702-43
A
1.05
E
B
F
Ci
G
Cii
H
D
I
1
0.95
Hard Color (Crab)
0.9
0.85
0.8
0.75
0.7
0.65
0.6
1.15
1.2
1.25
1.3
1.35
1.4
Soft Color (Crab)
Figure A.6: Color-color diagram for 495 observations of 4U1702-43. Observations
from A - Cii were used for further analysis, as the hump feature, LF QPO and kHz
QPO appear here.
119
Frequency × (RMS/Mean)2 Hz−1
Appendix A. Color-Color diagrams
Frequency (Hz)
Figure A.7: Representative fitted power spectra for different regions of the Colorcolor diagram. Regions A - Cii were used for further analysis, as the hump Lorentzian,
LF QPO and kHz QPO appear here.
Appendix A. Color-Color diagrams
A.5
KS1731-260
120
Appendix A. Color-Color diagrams
1.2
121
A
D
B
E
C
F
1.1
Hard Color (Crab)
1
0.9
0.8
0.7
0.6
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
Frequency × (RMS/Mean)2 Hz−1
Soft Color (Crab)
Frequency (Hz)
Figure A.8: Top: color-color diagram for 86 observations of KS1731-260.
Bottom: Representative fitted power spectra for different regions of the Color-Color
diagram. Regions A - C were used for further analysis, as the hump feature, and LF
and kHz QPOs appear here.
Appendix A. Color-Color diagrams
SAXJ1750.8-2900
1.05
A
F
B
G
C
D
E
1
0.95
0.9
Hard Color (Crab)
A.6
122
0.85
0.8
0.75
0.7
0.65
0.6
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Soft Color (Crab)
Figure A.9: Color-Color diagram for 131 observations of SAXJ1750.8-2900.
123
Frequency × (RMS/Mean)2 Hz−1
Appendix A. Color-Color diagrams
Frequency (Hz)
Figure A.10: Representative fitted power spectra for different regions of the ColorColor diagram (see Figure A.9). Regions A - C were used for further analysis, as the
hump feature, and LF and kHz QPOs appear here.
Appendix A. Color-Color diagrams
Aql X-1
1.2
1.1
1
0.9
Hard Color (Crab)
A.7
124
0.8
0.7
0.6
Ai
Aii
B
C
0.5
D
E
F
0.4
0.8
0.9
1
1.1
1.2
1.3
1.4
Soft Color (Crab)
Figure A.11: Color-color diagram for 566 observations of AqlX-1.
125
Frequency × (RMS/Mean)2 Hz−1
Appendix A. Color-Color diagrams
Frequency (Hz)
Figure A.12: Representative fitted power spectra for different regions of the colorcolor diagram. Regions B and C were used for further analysis, as the hump feature,
LF QPO and kHz QPO appear here.
Appendix A. Color-Color diagrams
SAXJ1808.4-3658
1.15
1.1
A
F
B
G
C
H
D
I
E
1.05
Hard Color (Crab)
A.8
126
1
0.95
0.9
0.85
0.8
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Soft Color (Crab)
Figure A.13: Color-color diagram for 493 observations of SAXJ1808.4-3658.
127
Frequency × (RMS/Mean)2 Hz−1
Appendix A. Color-Color diagrams
Frequency (Hz)
Figure A.14: Representative fitted power spectra for different regions of the colorcolor diagram. Regions F,G and H were used for further analysis, as the hump feature,
LF QPO and kHz QPO appear here.
Appendix A. Color-Color diagrams
A.9
HETEJ1900.1-2455
128
Appendix A. Color-Color diagrams
129
1.1
1
Hard Color (Crab)
0.9
0.8
0.7
0.6
A
B
C
D
0.5
E
0.4
0.9
0.95
1
1.05
1.1
1.15
Frequency × (RMS/Mean)2 Hz−1
Soft Color (Crab)
Frequency (Hz)
Figure A.15: Top: Color-color diagram for 354 observations of HETEJ1900.
Bottom: Representative fitted power spectra for different regions of the color-color
diagram. RegionsB and C were used for further analysis, as the hump feature, LF
QPO and kHz QPO appear here.
Appendix A. Color-Color diagrams
A.10
IGRJ17480-2446
130
Appendix A. Color-Color diagrams
131
1.4
A
B
C
1.3
D
E
1.2
Hard Color (Crab)
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.8
1
1.2
1.4
1.6
1.8
Frequency × (RMS/Mean)2 Hz−1
Soft Color (Crab)
Frequency (Hz)
Figure A.16: Top: Color-color diagram for 151 observations of IGRJ17480-2446.
Bottom: Representative fitted power spectra for different regions of the color-color
diagram. Region B was used for further analysis, as the LF QPO and kHz QPO appear
here.
Appendix A. Color-Color diagrams
A.11
132
XTEJ1807-294
1.15
A
B
C
1.1
Hard Color (Crab)
1.05
1
0.95
0.9
0.85
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
Frequency × (RMS/Mean)2 Hz−1
Soft Color (Crab)
Frequency (Hz)
Figure A.17: Top: Color-color diagram for 111 observations of XTEJ1807-294
Bottom: Representative fitted power spectra for different regions of the color-color
diagram. Region B was used for further analysis, as the LF QPO and kHz QPO appear
here.
.
Appendix A. Color-Color diagrams
A.12
IGRJ17511-3057
133
Appendix A. Color-Color diagrams
134
1.2
A
B
C
D
E
1.1
Hard Color (Crab)
1
0.9
0.8
0.7
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Frequency × (RMS/Mean)2 Hz−1
Soft Color (Crab)
Frequency (Hz)
Figure A.18: Top: Color-color diagram for 281 observations of IGRJ17511-3057.
Bottom: Representative fitted power spectra for different regions of the color-color
diagram. Regions B and C were used for further analysis, as the hump feature, LF
QPO and kHz QPO appear here.
Appendix B
Digital Appendix
The digital appendix can be found at [email protected]:/scratch/marieke/
digital_appendix. To login and for more information, contact Marieke van Doesburgh
at [email protected] or Michiel van der Klis at [email protected].
135
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