Lense-Thirring Precession Around Neutron Stars With Known Spin

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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

Supervisor:

Prof. dr. M.B.M. van der Klis

Examiner: Dr. R.A.W. Wijnands

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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

rela-tivistic 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

depen-dence on spin frequency are predicted.

In early work (van Straaten et al.,2003) this quadratic relation was confirmed to

remark-able 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 Lense-Thirring 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.

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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 hemel-lichamen is de algemene relativiteitstheorie als opgesteld door Albert Einstein. De ex-treem 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 R¨ontgenstraling te bestuderen die wordt

uit-gestraald als materie op een compact object valt.

In dit project concentreren we ons op dubbelstersystemen bestaande uit een neutronen-ster en een donorneutronen-ster, die een hoofdreeksneutronen-ster, 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 Figuur1.1is 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}

vrijge-maakt binnen _{∼100 km van het oppervlak. Het binnenste van de accretieschijf wordt}

extreem warm (> 107 K) en gaat stralen in het R¨ontgengebied 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 neutronen-ster 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 ver-band 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

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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 R¨

ontgen-dubbelster-systemen 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}

accreti-eschijf, 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

dubbelstersys-temen 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 R¨ontgendubbelstersystemen 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 doorStella 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 be¨ınvloeden 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.

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List of Figures

1.1 Artist impression of a low mass X-ray binary by M. van der Sluys using BinSim, a program developed by Rob Hynes. . . 3 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} corre-spondence with the Lense-Thirring precession model prediction. . . 7 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-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). 11 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. . . 14 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. . . 15 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 fre-quency ν. On the right we multiply the power with frefre-quency 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. 19 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. . . 22

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List of Figures ix

3.1 Color-color diagram for 423 observations of 4U1728-34. Each dot rep-resents 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%). . . 29 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. . . 30 3.3 Centroid frequencies (ν) plotted against the centroid frequency of the

up-per 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

simul-taneously present Lorentzian with ν = 35 Hz, detected at 3σ, follows the same relation as LLF. When νu exceeds 800 Hz, Lh is not detected.. . . . 32

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. . . 33 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

charac-terized by an upper kHz QPO at >802 Hz, or when its centroid frequency >40 Hz. See the main text for a discussion. . . 34 3.6 Measurements of Lhfrom the multi-Lorentzian timing study by van Straaten

et al. (2002) (only of 4U 1728-34), converted from νmaxto 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

in-cluded in this fit. . . 36

3.7 Left: Fit with a 5-Lorentzian model to the power spectrum of an

ob-servation 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). . . 37 3.8 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

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List of Figures x

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). 38 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. . . 40

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

cen-troid frequency. . . 41 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

simulta-neously 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. 44

and LLF fitted in power spectra of 4U 1608-52 plotted against their

cen-troid frequency. . . 45 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 νmaxto 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. . . 46 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. . . 48

3.16 Left: Fit with a 3-Lorentzian model to the power spectrum of obser-vations 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). . . 48 3.17 In analogy to Figure 3.3, for 4U 0614+09. When νu exceeds 605 Hz, Lh

is not detected. . . 51 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. . . 52 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). . . 53

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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

”in-terval 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

obser-vation 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

start-ing 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

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List of Figures xii

3.32 Multi-Lorentzian fit to 4068 averaged power spectra from KS 1731-260 starting on MJD 51815.7. We identify Lb, LhHzand 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. . . 65 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. . 65 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. . . 67 3.35 Multi-Lorentzian fit to 2424 averaged power spectra of 4U 1608-52. We

identify L_b, LLF,LLF2, LhHz and Lu, see Table 3.4 forν values, and the

main text for a discussion. . . 67 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. . . . 68 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. . . . 69 3.38 In analogy to Figure 3.3, for Aql X-1. When νu exceeds 500 Hz, Lh is not

detected. . . 72 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. . . 72 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. . . 74 3.41 Multi-Lorentzian fit to 2758 averaged power spectra of AqlX-1. We

iden-tify Lb, LLF,Lh, LhHzand Lu, see Table 3.9 for measured ferquencies, and

the main text for a discussion.. . . 74 3.42 In analogy to Figure 3.3 for SAXJ1808.4-3658. When νu exceeds 700 Hz,

Lh is not detected. . . 77

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. . . 78 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. νuis not detected. 78

3.45 In analogy to Figure 3.3, for HETEJ1900.1-2455. When νu exceeds 500

Hz, Lh is not detected. . . 81

and LLF fitted in power spectra of HETEJ1900.1-2455 plotted against

their centroid frequency. . . 81 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. . . 84 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

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List of Figures xiii

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. . . 87 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. . . 87 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). 90

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. . . 90 3.53 νLF and νh from subselection A plotted against νu. Open symbols

indi-cate frequencies measured in pulsars. Filled symbols indiindi-cate frequencies measured in bursters. All components exceed a 3σ confidence level. We plot frequencies not belonging to subselection A in lighter colors. . . 94 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. . . 95 4.1 νLF and νhscaled 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 I45m−1 of the neutron star would

be acceptable in the Lense-Thirring interpretation of Stella and Vietri (1998). The black line marks I45m−1= 2 for an observed frequency (νLF,

νh) equal to νLT, or I45m−1 = 1 for νLF, νh equal to twice νLT. . . 99

4.2 A zoom in on Figure 4.1. . . 100 4.3 Left: values for κα=2 plotted against spin frequency of the neutron stars.

Right: ∆κ_{≡ κ}α=2,h- κα=2,LF plotted against spin frequency of the

neu-tron star. Filled symbols refer to bursters, open symbols to pulsars. . . . 103 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 4.5 Measurements of Lhfrom the multi-Lorentzian timing study by van Straaten

et al. (2002), converted from νmax to ν, plotted together with our

mea-surements 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

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List of Figures xiv

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. . . 113 A.2 Color-Color diagram for 495 observations of 4U1608-52. . . 114 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)).. . . 115 A.4 Color-color diagram for 495 observations of 4U0614+09.‘HLF’ stands for

high luminosity flaring. . . 116 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.. . . 117

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. . . 118 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. . . 119 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. . . 121 A.9 Color-Color diagram for 131 observations of SAXJ1750.8-2900. . . 122 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. . . 123 A.11 Color-color diagram for 566 observations of AqlX-1. . . 124 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. . . 125 A.13 Color-color diagram for 493 observations of SAXJ1808.4-3658. . . 126 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. . . 127 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. . . 129 A.16 Top: Color-color diagram for 151 observations of IGRJ17480-2446.

Bot-tom: 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. . . 131

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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.

Bot-tom: 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.. . . 134

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Chapter 1

Introduction

1.1 Introduction

The detection of X-rays from outside the Solar system in 1962 byGiacconi 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 com-pact 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. Accre-tion 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 Figure1.1.

1_{To name a few: UHURU, ARIEL-V, GINGA, EXOSAT, ROSAT and RXTE.}

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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 theShakura and Sunyaev(1973) description of an axially symmetric

stationary accretion disk we can approximate a maximum temperature of 107K to occur

at_∼11₃ 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 in-homogeneities 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 constrain-ing 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 −}_pr i/r) 3GM ˙M 8πσr3

for the effective temperature, with r the radius in the disk, ˙

M the mass acretion rate approximated at 1018 gram/second, M the mass of the compact object, G the gravitational constant, rithe inner disk radius, and σ the Stefan-Boltzmann constant (Frank et al.,

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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 inFrank 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 col-lisions 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).

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disrupted. At the radius where the magnetic pressure exceeds the pressure from the

in-falling gas, the Alfv´en 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

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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

charac-teristic 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 (_∼1038erg/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

seevan der Klis (2006).

Generally, the position in the color-color diagram predicts timing characteristics

accu-rately (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 do-main, 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

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pulse frequency of a pulsar will for instance show up as such a peak in the power spec-trum. 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 Figure2.4. 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 = 8π2Iν_K2νs c2_M = 4.4× 10 −8_I 45m−1νK2νs, (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}

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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 byvan 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 Lense-Thirring 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

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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 Chapter4, 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

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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

Re-search Center (HEASARC)1. The Rossi X-ray Timing Explorer was2 a satellite in a

cir-cular 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-Ar-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/

2

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.

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Chapter 2. Method 10

2.1.1 PCA: Technical Specifications and Instrument Response

2.1.1.1 Proportional Counters

The PCA consisted of five identical proportional counter units (PCU’s) with a

geomet-rical collection area of _{∼1600 cm}2 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 pos-itively charged atom. A pospos-itively 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 pro-duced 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

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(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 differ-ent 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

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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 section2.1.1.1). 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 section2.2.2.

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.

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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 photom-etry 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, se-lecting 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 Figure2.3.

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)

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Chapter 2. Method 14 In tensit y × (coun ts/s/k e V) 2 Hz − 1 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 Figure2.3.

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 section2.2.1).

When we run color analysis on the raw Standard 2 (see2.1.2) 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.

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Chapter 2. Method 15 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.9 0.95 1 1.05 1.1 H ar d C ol or (C ra b)

Soft Color(Crab)

A B Ci Cii D E F G HLF

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

AppendixA) by plotting the average hard vs. soft color per observation. These diagrams

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2.2.3 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 invan

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:

f (t) = Z ∞ −∞ A(ν)e2πiνtdν (2.1) A(ν) = Z ∞ −∞ f (t)e−2πiνtdt (2.2)

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 2.1and 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 of 16 s contains gaps, it is discarded.

9 Total power =R∞ −∞|f (t)| 2 dt =R∞ −∞|A(ν)| 2

dν, in the Leahy normalization: P (ν) = 2 Nph|A(ν)|

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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 byCooley

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 section2.1.1.1) corrections. We only correct for them after

av-eraging the Leahy normalized power spectra. To do so, we subtract a predicted counting

noise spectrum incorporating dead-time effects, based on the work ofZhang et al.(1995).

The method we use was developed byKlein-Wolt et al.(2004). In order to compare

be-tween 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 variability in the signal (

q

R P (ν)dν = rms,van der Klis (1989)).

The result of all the corrections and calculations described is an averaged power

spec-trum 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

dif-ferent representations. We see characteristic features, humps that peak at a particular frequency. These features move through the power spectra differently over time, de-pending 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.

Lense-Thirring Precession Around Neutron Stars With Known Spin

University of Amsterdam

MSc Astronomy

& Astrophysics

Astronomy and Astrophysics

Master Thesis

Lense-Thirring Precession Around Neutron Stars With

Known Spin

Abstract

Acknowledgements

Nederlandse Samenvatting

Contents

List of Figures

Chapter 1

Introduction

1.1

Introduction

Chapter 2

Method

2.1

Instrumentation: The Rossi X-ray Timing Explorer

2.2

Data Analysis