High-Speed Astrophysics: Chasing Neutron-Star Oscillations

Ricky Nilsson

Lund Observatory Lund University 2005-EXA05

Degree project of 20 credit points (for a degree of Master) March 2005

Lund Observatory Box 43 SE-221 00 Lund Sweden

High-Speed Astrophysics: Chasing Neutron-Star Oscillations

Ricky Nilsson

Lund Observatory Lund University, Sweden 2005 Abstract The potential of high time-resolution optical observation methods in the search for high-speed astrophysical phenomena, specifically neutron-star (NS) oscillations, has been studied. Related evaluation of a digital hardware-correlator instrument in conjunction with single-photon-counting avalanche photo-diode (APD) detectors has been performed, both in laboratory experiments and in actual astronomical observations with the 1.3-m telescope at Skinakas Observatory on Crete. Subsequent preliminary analysis of the observation data was carried out, and additional simulations done to estimate the observability of faint emission from NSs using future Extremely Large Telescopes (ELTs) and its possible signatures of pressure-driven (p-mode) oscillations. It was demonstrated that the “correlation method” and use of high time-resolution hardware correlators offer several advantages compared to traditional photometric measurements, e.g. real-time analysis possibilities and instant data reduction, of which the latter is crucial if we are to handle the extremely high data rates expected during high time- resolution measurements with ELTs. Furthermore, this method appears to be favourable in the search for NS p-mode oscillations, which have periods on the order of 0.1 ms. Although the amplitude of any oscillation signature is difficult to predict, further simulations could tell us which precision in the determination of oscillation frequencies is needed in order to infer NS parameters (like mass, radius, and equation of state) with a certain accuracy.

Populärvetenskaplig sammanfattning I och kring väldigt kompakta astronomiska objekt, såsom neutronstjärnor, vita dvärgar och svarta hål, existerar oerhört extrema miljöer med stark gravitation och kraftiga magnetfält. För att studera de mycket snabba förlopp som kan äga rum i dessa omgivningar behöver man instrument som kan registrera snabba variationer i ljuset från dessa. Detta innbär att man måste ha ytterst bra tids- upplösning, i många fall ner mot några få tusen- eller miljondelar av en sekund. I detta examensarbete har jag speciellt studerat hur vibrationer i neutronstjärnor (med förväntade svängningsperioder på ca 100 miljondelar av en sekund) skulle kunna upptäckas med en typ av instrument som kallas korrelator. Detta instrument jämför i realtid den ljusintensitet som registreras av en eller ﬂera detektorer med hög tidsupplösning på olika tidsskalor, för att på så sätt hitta variationer i ljuset från ett observerat objekt. Om man skulle kunna upptäcka vibrationer i neutronstjärnor och mäta dess svängningsperioder, så innebär det att man kan få information om dessa speciella stjärnors inre beskaﬀenhet, på samma sätt som seismologi har avslöjat hur Jordens inre är uppbyggt. Utöver programmering av ett styr- och datainsamlingsprogram till korrelatorn och lab- oratorietester utförda här i Lund, samt astronomiska observationer vid ett mindre teleskop på Kreta (tillsammans med en forskargrupp från München), har även datorsimuleringar gjorts för att uppskatta de möjligheter som kan komma att erbjudas av framtidens stora, markbaserade, optiska teleskop. Dessa kan samla in mångfaldigt mer ljus inom en viss tid och ger på så sätt större chanser till upptäckt av små variationer i det svaga ljuset från svängande neutronstjärnor och andra snabba fenomen kring kompakta objekt. De slutsatser som kan dras av arbetet är att den instrumentering som vi använt för observationer med hög tidsupplösning för detektering av snabba astrofysikaliska förlopp verkar mycket lovande även på existerande teleskop och att jakten på neutronstjärnors svängningar troligen kan bära frukt med framtidens extremt stora teleskop.

i Contents

1 Introduction 3 1.1 High-SpeedAstrophysics...... 3 1.2 ObjectiveofThesis...... 5

2 Astrophysics of Neutron Stars 6 2.1 Supernovae - the Birth of Neutron Stars ...... 6 2.2 TheStructureofNeutronStars ...... 7 2.2.1 EquationofState...... 7 2.2.2 InternalComposition...... 9 2.3 Emission Mechanisms and Observations of Neutron Stars ...... 11 2.3.1 Pulsars ...... 11 2.3.2 Other Observations of Neutron Stars ...... 14 2.3.3 CoolingandThermalEmission ...... 15 2.3.4 General Relativistic Light Deﬂection ...... 16 2.4 Neutron-StarOscillations ...... 17 2.4.1 Stability and Non-Radial Oscillations in General ...... 17 2.4.2 P-modeOscillations ...... 23 2.4.3 Excitation...... 25 2.4.4 Damping ...... 26 2.4.5 Manifestations in Observables ...... 27 2.5 Probing the Interior of Neutron Stars ...... 28

3 Instrumentation 31 3.1 Detectors ...... 31 3.2 Correlators ...... 32 3.3 QVANTOS/OPTIMA ...... 33 3.3.1 QVANTOSinstrumentation...... 33 3.3.2 OPTIMAInstrumentation...... 38 3.3.3 Laboratory Testing QVANTOS/OPTIMA ...... 39

4 Observations 41 4.1 PreparingtheObservations ...... 41 4.1.1 CrabPulsarSimulation ...... 41 4.2 SkinakasObservatory ...... 44

1 4.3 Measurements...... 45 4.3.1 ExperimentalSetup ...... 45 4.3.2 ObservedObjects...... 47 4.3.3 SourcesofError ...... 47 4.4 Analysis...... 48 4.4.1 ReductionofRawData ...... 48 4.4.2 OpticalLight-Curves...... 49 4.4.3 CorrelationFunctions ...... 51 4.5 ResultsandDiscussion...... 53

5 Simulated Observations with ELTs 55 5.1 Observability of Faint Objects Using ELTs ...... 55 5.1.1 OWLExposureTimeCalculator ...... 56 5.2 Observability of P-mode Oscillating Neutron Stars ...... 56 5.2.1 Simulated Correlation Functions and Power Spectra ...... 56 5.3 ResultsandDiscussion...... 58

6 Summary and Conclusions 62

A Manual and User Guide for the Digital Correlator Flex01-05D and Control-Program PhoCorr 69 A.1 Introduction...... 69 A.2 Basiccorrelatorspeciﬁcations ...... 69 A.2.1 Mode A: Single auto/cross correlation ...... 69 A.2.2 Mode B: Quadcorrelation ...... 70 A.2.3 Mode C: Dual auto/cross correlation ...... 70 A.2.4 Photonhistoryrecorder ...... 70 A.2.5 Calculating the correlation functions ...... 70 A.3 LabVIEW...... 71 A.4 Flexx01 ...... 71 A.4.1 USBInitialize...... 72 A.4.2 USBStart...... 72 A.4.3 USBUpdate ...... 72 A.4.4 USBStop...... 73 A.4.5 USBFree ...... 73 A.5 PhoCorr...... 73 A.5.1 Interpretingtheblockdiagram ...... 73 A.5.2 HowtousePhoCorr ...... 74 A.6 Appendix ...... 77

B MATLAB Scripts 82

2 Chapter 1

Introduction

Observational astrophysics is constantly probing deeper and deeper into the study of the nature of our marvellous Universe and the objects and phenomena within it. Continuous development of new technology allows these advances to be made by constructing new and more effective instruments to explore the “parameter domains” of astrophysics (see figure 1.1). Effectively all current observations in astronomy, disregarding cosmic and gravitational radiation, neutrino studies and occasional in situ measurements or sample return missions in our solar system, can be categorized into the areas of imaging, spectroscopy and photometry, combined in various ways, analyzing the information we receive in the electromagnetic radiation from space. By pushing the limits of these areas, i.e. increasing spatial resolution and field of view, increasing the spectral resolution and expanding the observable wavelength regions, and increasing the temporal resolution, knowledge is gained in regions previously inaccessible to observations.

1.1 High-Speed Astrophysics

When exploring the time domain of astrophysics and reaching temporal resolution down to milli-, micro- and nanosecond timescales, a number of novel phenomena might be encoun- tered, generally in or near compact objects. Variability on these timescales can originate from e.g. instabilities in accretion disks around or flows onto white dwarfs, neutron stars (NSs) and black holes, optical emission from millisecond pulsars, “photon bubbles” in exceptionally luminous stars, radial oscillations in white dwarfs or perhaps non-radial oscillations in NSs. An additional benefit of high time-resolution is the prospect of revealing small-scale structures e.g. spot-like features of NS surface magnetic fields (Geppert et al. 2003). Giant pulses with temporal widths smaller than 15 ns observed in pulsars (Soglas- nov et al. 2004) would correspond to a structure size of about 5 m, if they originate from the surface of the NS, and microstructure revealed in several radio observations of pulsars (Kostyuk et al. 2003) are almost certainly present in other wavelength regions. The key to high time-resolution photometry lies in large photon quantity. As the time-bins get smaller more photons are needed (in any given time interval) in order to obtain useful statistics. Consequently, telescope- and detector-systems that manage to collect and register many photons efficiently are required if one hopes to catch the most

3 Figure 1.1: The “parameter domains” of astrophysics. (Figure courtesy of D. Dravins) rapid variability in astronomical objects. Moreover, the advantage of using the optical wavelength range, compared to e.g. X-rays, is twofold. First of all, existing large ground based optical telescopes have a light collecting power greatly exceeding projected X-ray satellites or corresponding instruments in other wavelength regions, differing by several orders of magnitude in count rates (see e.g. comparison by Dravins 1994). Secondly, it seems like some fundamentally new properties of light could be reachable by pushing time- resolution into the domains of quantum optics (see again figure 1.1). With nanosecond resolution one could use 2:nd or higher order correlation functions to measure statistical multi-photon properties of astrophysical objects, like bunching of photons in thermal light (inferred classically by Hanbury Brown & Twiss 1956), and possibly acquire thermody- namic information about the emission process or redistribution of light from a plasma. The above discussion of the prospects of high time-resolution astrophysics is, as mentioned, principally dependent on the construction of high efficiency telescope- and detector- systems, and such instruments are evidently on their way, considering the ongoing projects of several Extremely Large Telescopes (ELTs), the most prominent one perhaps being ESO’s 100-meter Overwhelmingly Large Telescope (OWL). In addition, optical photon- counting avalanche diode detectors with nanosecond resolution and high quantum efficiency extending into the infrared, are being developed to supplement today’s generation of CCD-detectors in optical astronomy.

4 1.2 Objective of Thesis

The purpose of this thesis is to investigate the potential of high time-resolution optical observations to explore high-speed astrophysical phenomena, focusing on NS oscillations, and also which methods and instruments that should be required for detection of such. In addition, the thesis contains an account of the initial testing of a high time-resolution photon-correlator, intended to be part of a future mobile observation instrument, QVAN- TOS (Quantum-Optical Spectrometer) Mark II, and the programming of its control- and data acquisition-program. We had the opportunity to test the correlator both in actual astronomical observations at Skinakas Observatory in Crete together with the OPTIMA- group from Max Planck Institute for Extraterrestrial Physics (MPE) in Garching near Munich and their OPTIMA-detector instrument, and prior to that in the optics laboratory at Lund Observatory. Subsequent preliminary analyses of data from the observations at Mt. Skinakas are also presented. To comprehend the implications of NS oscillation detection on the research of NS structure and nuclear physics, some introductory theory on these subjects will be given, concentrating on p-modes in the oscillation part. My hope is that this thesis will serve as a presentation of one possible application oﬀered by the promising future of high time- resolution astrophysics.

5 Chapter 2

Astrophysics of Neutron Stars

In this chapter, basic theory of NSs, focusing on NS oscillations, will be presented according to current scientific models, together with brief explanations (in footer) of related terms and concepts. This is to serve as a background to the subject of observability of NS oscillations with high time-resolution optical instruments and its possible application to NS seismology, i.e. the probing of NS interiors. The logical starting point, the formation of a NS, is portrayed quite schematically, partly because it is not essential to the main topic of the thesis and also because a detailed description of the supernova process is yet to be established. Next I have tried to give a more thorough description of NS structure and composition, followed by an account of theories for NS emission mechanisms. The final and major part deal with stellar oscillations, specifically in NSs, possible manifestation of these in observables, and lastly the potential to draw conclusions on the stellar interior by studying stellar oscillation modes.

2.1 Supernovae - the Birth of Neutron Stars

1 As a high-mass (≥8 M⊙ ) star approaches the end of its life-cycle, it has an onion-like structure with layers of diﬀerent composition separated by nuclear burning shells. The core of the giant star will consist mainly of iron. Since iron is the most tightly bound nucleus, all nuclear sources of energy will be exhausted and there will be no thermal pressure to support the core. After 100 million years of holding sway against gravity, the core of the star will collapse in a fraction of a second and the released gravitational binding energy will (with a non-negligible contribution from neutrinos) expel the outer layers in an immense Type II2 supernova explosion. The contraction of the core halts when the density of the core becomes high enough for free protons and electrons to combine and form neutrons together with escaping neutrinos, creating a supporting pressure from a degenerate neutron gas as the lowest neutron energy levels are getting ﬁlled and the densely packed neutrons

1This is the notation for one solar mass and equals approximately 1.9891 · 1030 kg. 2Supernovae are classiﬁed into Types I and II, and these types can be further subclassiﬁed, e.g. Ia (which are believed to be accreting white dwarfs that explode when they exceed the critical Chandrasekhar mass limit in binary systems). Type II supernovae show traces of hydrogen in their spectrum, while Type I do not.

6 are forced into higher energy levels. The super dense neutron-rich remnant is called a neutron star. However, if the mass of the remaining core is larger than the so called Oppenheimer-Volkov mass, it will go on contracting to a point where not even light can escape from it, and a black hole is formed. As a result of the conservation of angular momentum and magnetic ﬂux from the supernova progenitor, a newborn NS will spin rapidly, typically 50 times per second, and possess an immense surface (dipole-)magnetic ﬁeld, on the order of 108 Tesla (T). In addition, the temperatures (considering the small radiative surface area) will be equally extreme, perhaps up to 106 Kelvin (K). Asymmetries in the supernova explosion could also produce a violent “birth kick”, sending the NS soaring through space with velocities of 100-1,000 km s−1 (Lyne & Lorimer 1994).

2.2 The Structure of Neutron Stars

Already in 1932, the same year as Chadwick discovered the neutron, Landau proposed the formation of “one giant atomic nucleus” in stars exceeding the critical Chandrasekhar mass3, and was followed by Baade & Zwicky who in 1934 suggested a connection of supernovae to the birth of NSs. NSs offer a possibility to explore “extreme physics”, which is not available in our laboratories. Thus, astronomical observations of NSs can answer questions in the fields of dense matter physics, nuclear physics, particle physics and astrophysics, which can not be answered elsewhere. However, it is the interplay between observations and theoretical progress that drives our scientific knowledge forward. A developed theoretical model can later be constrained or rejected on basis of observations.

2.2.1 Equation of State Perhaps the most important piece of the puzzle in theoretically calculating properties of a NS is the determination of an equation of state (EOS) for dense matter, i.e. the relation between pressure P and density ρ. This functional relation determines the range of masses possible, as well as the mass-radius relationship of the star. In addition to this, the EOS is important when establishing the internal structure of the NS. Physicists often speak of “hard” and “soft” EOS. The former is when the pressure rises steeply (with increasing density) at normal nuclear density, while the latter corresponds to a smaller EOS derivative at nuclear density. The ﬁrst calculations of a simpliﬁed model of a NS was performed by Oppenheimer & Volkov (1939), who described the star as a sphere of cold, non-interacting neutrons, i.e. as a pure Fermi gas of neutrons. Solving the EOS for such a sphere within the framework of general relativity yielded a few gravitational equilibrium states for a range of masses, the maximum mass being ∼0.7 M⊙. Depending on the rest mass density of NS matter, the neutrons can be either non-relativistic (ρ ≪ 6 · 1015 g cm−3) or extremely relativistic

3 This is the maximum possible mass of a degenerate star, which is 1.44M⊙ for a white dwarf. Degenerate stars having masses greater than this limit collapse to a NS or black hole.

7 (ρ ≫ 6 · 1015 g cm−3) and are then described by a simple polytropic4 EOS, P = Kργ with γ = 5/3 or 4/3, respectively (Oppenheimer & Volkov 1939). However, these results and suggested EOS fail to explain observed NS masses. Different EOSs can be coupled together to give a more realistic description of the star structure. By solving the differential equations of general relativistic hydrostatic equilibrium and mass distribution for a spherical object, the so called Tolman-Oppenheimer- Volkov equations: dP G(m(r) + 4πr3P/c2)(ρ + P/c2) = − , (2.1) dr r(r − 2Gm(r)/c2) and dm(r) = 4πρr2, (2.2) dr where P is the pressure, G is the gravitational constant, ρ is the mass-energy density, and m(r) is the gravitational mass enclosed within a radius r, one can obtain total masses and radii for specific EOSs. One set of such mass-radius relations calculated for realistic EOSs by Lattimer & Prakash (2001) is shown in figure 2.1.

Figure 2.1: Diagram of mass-radius relations calculated for diﬀerent EOS (including two models for strange quark matter - SQM). (Lattimer & Prakash 2001)

Today, general consensus is that the maximum (Oppenheimer-Volkov) mass is somewhere between 1.6 and 2.4 M⊙, the typical NS mass is about 1.5 M⊙, which is also in agreement with observations from binary pulsar systems, and corresponding radii 8 km

8 states of lepton-rich NSs, giving a limit of about 1M⊙ (Haensel et al. 2002). Obtaining re- liable measurements of radius is much more difficult, but a simple estimate yields a radius of approximately 12km (Heiselberg & Pandharipande 2002). The rapid rotation of the NS causes an enormous centrifugal force, which has to be counteracted by gravity to keep the star from being ripped apart. Thus, one can by hand-waving conclude that the density must be very high and, for a given mass, the radius quite small (see e.g. Glendenning 1997). The large variation in predictions of NS radii is a consequence of the uncertainties 14 −3 in the EOS near nuclear matter equilibrium density ρ0 > 10 g cm . NS matter is distinctly different from ordinary nuclear matter, whose properties are quite well known. The distinction lies in the fraction of protons, which is nearly 50% in nuclear matter, while only being a few percent in NS matter. This lack of symmetry gives a yet undetermined contribution to the energy of the matter, called the symmetry energy function Sν(n) (where n is the number density of nucleons), making the pressure at ρ0 hard to establish. Even though the microscopic properties are uncertain, constraints on the EOS by inferring radius, can be made using macroscopic properties like the maximum rotation rate of NSs (see e.g. Friedman et al. 1986, Haensel et al. 1995), observations of X-ray burst oscillations from X-ray binaries (Strohmayer 2004), gravitationally redshifted spectral lines (originating from the NS atmosphere) (Cottam et al. 2002) or thermal emission. The observational key is then obviously to make precise measurements of mass and radii, to find an accurate mass-radius relationship and consequently a determination of the EOS. As will be described later, this approach to EOS inference is also offered by observation of NS oscillations.

2.2.2 Internal Composition NS models predict the interior as being built up of ﬁve diﬀerent regions of varying density and composition (e.g. Lattimer & Prakash 2004):

• Inner core

• Outer core

• Crust

• Envelope

• Atmosphere

The inner and outer cores contain the bulk of the mass, about 99%. Very little is known about the physical properties of matter in the inner core, where the density can reach values far above nuclear density at 1014 g cm−3. Existence of various exotic particles at these supra-nuclear densities has been proposed, for example strangeness-bearing

9 Figure 2.2: A carved out neutron star reveals its interior structure. (http://www.astroscu.unam.mx/neutrones/NS-Picture/NStar/NStar.html) baryons5 (Glendenning 1997), condensed pions6 or kaons7 (Kaplan & Nelson 1986) or even deconfined quarks8 (Glendenning 1992a), which would be immersed in a sea of mainly su- perfluid9 neutrons but also superfluid (and hence superconducting) protons. Alcock & Olinto (1988) and others have even suggested the existence of pure “strange quark matter” (SQM) stars, composed entirely of quark matter with up, down and strange quarks, which at sufficiently high density could be the definitive ground state of matter. Such

5Baryon is the name for all subatomic particles consisting of three quarks, the most familiar of course being protons and neutrons. 6Are unstable mesons (made up of quark-antiquark pairs) which are bosons as opposed to baryons which are fermions. 7These are another type of mesons, made up of different quark-antiquark pairs. 8Quarks are the basic building blocks of matter (together with the leptons) and they come in six “flavours”. Two of these flavours, the up and down quarks, exist in ordinary atomic nuclei. When nuclear matter is compressed to sufficiently high density the nucleons “merge” and turn into a liquid of uniform quark matter. 9At temperatures near absolute zero, some liquids, e.g. helium, turn into a superfluid i.e. a fluid with zero viscosity and high heat conductivity.

10 stars could also be self-bound, i.e. not requiring gravity to be held together. Also the outer core is thought to consist of nucleons in a superfluid state mixed up with “normal” electrons and muons10. This fairly homogeneous liquid would have a density between 2·1014 and 8·1014g cm−3 (Carroll & Ostlie 1996) and extend out to the core→crust transition phase (depicted in the upper part of figure 2.2). In this transition phase, by some called the nuclear pasta(!), a quantity of the superfluid nucleonic matter will transform into various geometrical structures of nuclear lattices. Going to lower densities, increasingly more matter will be able to reside in nuclei, finally reaching a density below the so called neutron drip density 4 · 1011 g/cm3 where neutrons are prevented from leaking out of nuclei. The solid crust, consisting of a Coulomb lattice of heavy neutron-rich nuclei and a relativistic degenerate electron gas, has a thickness on the order of 1 km, perhaps with “mountain ranges” extending a few centimeters above the surface of the NS. The chemical composition of the surface (or envelope) and atmosphere is quite uncertain. Since the gravitation gradient at the surface is enormous, elements will be separated according to their atomic weight, thus heavy elements will sink and light elements will reside in the atmosphere. Another factor which significantly affects the composition of the outer layers of a NS is possible accretion of hydrogen (H) or helium (He) and other light elements from the interstellar medium or a companion star, and to what extent this matter has been burnt or ejected from the surface (Pavlov et al. 1996). This also determines the existence of a surface fluid “ocean” layer. Dense atmospheres composed of fully ionized H and He have been proposed, as well as heavy element (Fe) atmospheres. The main properties of the surface and atmosphere, i.e. composition and magnetic field strength, strongly influence the photon emission spectrum, as will be discussed in the next section.

2.3 Emission Mechanisms and Observations of Neutron Stars

2.3.1 Pulsars Most NSs are observed as pulsars, first discovered as a source of regularly pulsed radio emission by Jocelyn Bell and Anthony Hewish in 1967. Thomas Gold (1968) was the first to explain the origin of the observed radio pulses as coming from rapidly rotating NSs, although his initial theory of the pulsar emission mechanism was soon revised. Most current models support the theory that the emission originates from charged particles accelerating in the intense magnetic field near the magnetic poles of the NS. Only a narrow beam of radiation is emitted since the relativistic beaming of so called synchrotron radiation11 is directed along the particle motion, roughly parallel to the magnetic axis (see figure 2.3). The dipole magnetic field geometry, where the magnetic field axis does not coincide with the rotation axis, produces a lighthouse-effect where the radiation cone occasionally sweeps by an observer who happens to be positioned in the path of the radiation cone (see e.g. van Leeuwen 2004 for a review of pulsar emission mechanisms).

10Muons are leptons just like electrons, although more massive. 11When a charged particle is accelerated, e.g. along a curved magnetic ﬁeld line, it will emit electromagnetic radiation. If the speed is close to that of light, the radiation will be emitted in a narrow cone in the forward direction. Synchrotron radiation is very intense, highly polarized, and covers a wide range of the electromagnetic spectrum.

11 Figure 2.3: Schematic picture of pulsar geometry. (http://www.physics.hku.hk/∼nature/CD/regular e/lectures/images/chap16/pulsar.jpg)

However, this emission mechanism alone does not account for the rate of the observed spin-down among most pulsars. As the pulsar radiates it loses rotational energy Erot, which leads to an increase in rotation period P with time. The period and the time derivative of the period P˙ , are important NS parameters and have been measured for 3 quite a few pulsars. Estimating the loss of rotational energy – E˙ rot ∝ P˙ /P – from observations of these parameters, yields a value that would require an emission rate orders of magnitude larger than that from synchrotron radiation. A possible explanation would be emission of magnetic dipole radiation generated by the rapid changes in the magnetic field as the tilted NS dipole magnetic field spins around the rotation axis (Prialnik 2000). This radiation would then have a frequency equal to the spin frequency. It should also be mentioned that the pulsar light-curves typically displayed in basic textbooks (e.g. the one of the Crab pulsar in figure 2.5), are often average pulse profiles, created by folding the measured intensity over several rotation periods. Individual pulses can look quite differently, and regularly display e.g. amplitude modulation and phase drift of sub-pulses in consecutive periods. The origin of these have been studied by e.g. Deshpande & Rankin (1999), and identified as a system of drifting sub-beams circulating around the magnetic axis of the star. Furthermore, the number of components in the average pulse profile varies between one and five, (being two in the Crab pulsar profile), which tells us that the geometry of the beaming is, without doubt, rather complex (Gil 1993). Sudden spin-ups, so called glitches, observed in the otherwise monotonically increasing period of e.g. the Vela and Crab pulsars, are thought to be a result of sharp changes in the moment of inertia of the crust, due to “starquakes”. The cracking of the crust may be a consequence of growing shear stresses as the oblate shape of the NS decreases when it spins

12 Figure 2.4: Overview of the possible evolutionary paths of a pulsar, showing how a neutron star can be observed in X-ray and radio emission. (http://www.astrodomi.com.ar/universo/estrellas/pulsar/pulsar.jpg) down (see Link et al. 1998). Alternatively, it comes from the frictional coupling between the crust and the superfluid (Anderson & Itoh 1975). Study of the gradual recovery of the spin-down rate as the superfluid and the rest of the star returns to rotational equilibrium, could then also impose constraints on the nuclear EOS (Link et al. 1993a). In addition to the discussed radio pulsars, a new type of pulsar was discovered by the first specialized X-ray satellite Uhuru in the beginning of the 1970s. These X-ray pulsars had been predicted prior to discovery by several Russian astrophysicists and were modelled as accreting NSs in binary systems with normal stars (Shvartsman 1970). Several features distinguish X-ray pulsars from radio pulsars. The transfer of matter from the normal companion star via the inner Lagrangian point12 and subsequent fall in toward the NS, also transfers a significant amount of angular momentum to the pulsar, leading to a spin-up of the NS with time. Moreover, the emission spectrum of X-ray pulsars exhibit signatures of thermal radiation from high temperature (108K) plasma, possibly from the accretion hot spots, in contrast to the clearly non-thermal origin of the radio pulsar emission (although weak thermal emission components have been found also in radio pulsars, see e.g. Shearer & Golden 2002). Besides the short pulsations of X-rays, variations associated with the orbital period, due to eclipses and accretion rate modulations as a result of separation

12In a gravitational two-body system there exists 5 equilibrium points, so called Lagrangian points, of which the inner is the meeting point of the equipotential surfaces (Roche lobes) of the two bodies.

13 Figure 2.5: Pulse profile for Crab pulsar. (Perryman et al. 1999) changes of the binary components, have been observed. Pulsar emission is not confined solely to the radio and X-ray wavelength regions but can be observed also in other parts of the electromagnetic spectrum. To this point, five optical pulsars have been confirmed, including the Crab (the brightest optical pulsar), Vela and Geminga (Shearer et al. 1998). The origin of the pulsed component of optical radiation from young and middle-aged pulsars may be somewhat different from that of the radio emission, although still thought to come from relativistic electrons radiating by synchrotron processes; it could be generated in the outer regions of the magnetosphere (see e.g. Shearer & Golden 2002 for a review of emission zone geometry). Furthermore, Geminga was, subsequent to its detection as an X-ray pulsar, also discovered as a bright γ-ray source (Bignami & Caraveo 1992), pulsating with a period of 0.237 seconds.

2.3.2 Other Observations of Neutron Stars Some objects of another class of X-ray sources, X-ray bursters, with quasi-periodic bursts of X-ray emission, have also been associated with NSs in binary systems. In this case thermonuclear bursts from accreted matter on the NS surface ﬁt the observations, and are seen against a roughly constant background of X-ray ﬂux. These object constitute a sub-class of the X-ray binaries which are NSs (or in a few cases black holes) in binary systems with a low-mass (making a low-mass X-ray binary, LMXB) or high-mass (making

14 a high-mass X-ray binary, HMXB) companion star. Other exotic objects that have been linked to NSs are the magnetars. These rare (only ten have been found so far) and relatively newly discovered objects, are thought to be furiously spinning NSs with magnetic ﬁeld strengths around 1011 T, (three orders of magnitude higher than in ordinary pulsars), which beam out high-energy X-rays or γ-rays. Recent papers by Eikenberry et al. (2004) and Figer et al. (2005) have suggested that magnetars originate from extremely massive stars which have evaded the destiny of black holes by great mass loss prior to the supernova explosion, keeping their high angular momentum. Thus, the rapid rotation of the star would power a dynamo that generates superstrong magnetic ﬁelds. Magnetars comprise both the soft γ-ray repeaters (SGRs), observed as γ-ray bursts with oscillatory decays, and the anomalous X-ray pulsars (AXPs), which are probably not members of binary systems but still exhibit pulsed X-ray emission.

2.3.3 Cooling and Thermal Emission Following the NS creation, since it has no remaining source of energy production, the star will unconditionally start to cool off. Initially it will have temperatures in excess of 1011 K, cooling to surface effective temperatures of 105 K in 105 − 106 years. During this period the interior of the star will cool rapidly by energy loss from neutrino emission, either through the efficient direct Urca process13 or the less efficient modified Urca process14. Which process that occurs depends on the proton fraction of matter in the NS interior, which in turn depends on the previously mentioned symmetry energy function Sν(n) at interior densities (Pethick 1992). Since the density dependence of Sν is currently uncertain, as discussed earlier, the dominant process is unknown, as well as issues concerning the existence of hyperons, pions, kaons and quarks in NS interiors (Lattimer & Prakash 2004). The surface temperature is about two orders of magnitude lower than the core temperature and follows the cooling of the interior, causing the relatively young (≤ 106 yr) NS to emit surface thermal radiation mostly in the extreme ultraviolet (EUV) or X-ray regions. As the bulk of the pulsar emission from radio to X-ray wavelengths is non-thermal, generated in the NS’s magnetosphere, it does not give any direct information concerning internal structure, composition and evolution. To derive such information, from global parameters like mass, radius and temperature, one must instead try to utilize the thermal emission, either in the “unpulsed” component of pulsar emission or from “non-pulsating”, preferentially old isolated NSs. While young hot NSs might give a higher thermal flux, they have the disadvantage of being surrounded by hot glowing gas (called the pulsar wind nebula) from the supernova remnant, which seriously contaminates (if not to say even dominates over) light from the actual NS. However, although old isolated NSs are claimed to have been detected (Walter et al. 1996), a detailed optical study of such faint sources would certainly require ELTs. Expected magnitudes in the optical/UV range are approximately 26-28 mag. at a distance of roughly 100 pc (Pavlov et al. 1996). So how does one determine mass and radius of a NS from observations of its thermal emission? Well, by assuming black body radiation, one can calculate the so called radiation radius R∞ (Lattimer & Prakash 2004) by starting with the definition of effective

13 − + Direct Urca: n → p + e +ν ¯e, p → n + e + νe 14 − + Modiﬁed Urca: n + (n, p) → p + (n, p) + e +ν ¯e, p + (n, p) → n + (n, p) + e + νe

15 temperature: 2 L∞ 4 R∞ F∞ = = σ T , (2.3) 4πd2 B eff,∞ d −23 −1 where σB = 1.3807 · 10 JK is the Boltzmann constant. If the ﬂux or luminosity observed on Earth (F∞ and L∞) can be established together with estimates of the distance d (e.g. by parallax measurement) and eﬀective temperature Teff,∞ (from Wien’s displacement law), then R∞ can be roughly determined. Obviously we need more information, and this we get from gravity, which plays a decisive role by gravitationally redshifting the light that leaves the NS surface. The gravitational redshift can be expressed as

− 2GM 1/2 z = 1 − − 1, (2.4) Rc2 where G is the gravitational constant, and M and R the mass and radius of the NS, respectively. Consequently, if the redshift of identified spectral lines has been determined (and thereby enabled corrections to the flux and effective temperature), both the mass and radius can be calculated. Unfortunately, NSs are not black bodies, thus the spectral distribution of thermal emission from their surfaces deviates considerably from a Planck function. First of all, the temperature is expected to be non-uniformly spread over the surface of the NS because of the anisotropic magnetic field (Pavlov et al. 1994), which affects the surface-integrated spectrum. Secondly, the thin (0.1-10cm) but dense (0.1-100gcm-3) atmosphere that covers the surface, as discussed earlier, substantially redistributes the flux. Interpretation of the observed thermal emission therefore requires atmospheric models, which can determine how the atmosphere shapes the emergent spectrum. Numerous such models have been constructed (see e.g. Rajagopal & Romani 1996, for low magnetic field NSs; Pavlov et al. 1995, for medium-field NSs; and Ho & Lai 2001, for high-field NSs) and their spectra have been fitted to observed data, predominantly in the X-ray wavelength region. Since identification of spectral lines in many cases is impossible due to low-resolution spectra, often only the radiation radius can be calculated. Generally H and He atmospheres with low magnetic fields yield spectra with an overall shape which deviates greatly from black body spectra, especially in the high frequency part, while Fe atmosphere spectra lie closer to a black body profile, except for photo-ionization edges and dense absorption lines (Pavlov et al. 1996). A black-body fit of the former atmosphere would then overestimate the effective temperature, and as a result underestimate the radiation radius. A problem with atmosphere models for strongly magnetized NSs is that there exists no detailed theory and understanding of the properties of elements heavier than H in strong magnetic fields.

2.3.4 General Relativistic Light Deflection As a last note on the subject of how NSs are observed, I feel it appropriate to mention the effects of general relativity on the photon trajectories. A NS distorts the fabric of space- time into a deep gravity well by its enormous density, which apart from the mentioned relativistic redshift, also causes light to bend when emitted at an angle relative to the surface normal. The amount of deflection is determined by the ratio of the NS’s radius to

16 its Schwarzschild radius15. Figure 2.6 graphically illustrates how light is deﬂected due to relativistic “ray bending”. This property certainly inﬂuences how NS are observed, e.g. the appearance of pulsar light-curves.

Figure 2.6: Deﬂection of light from the surface of a neutron star to a distant observer. (Nollert et al. 1988)

2.4 Neutron-Star Oscillations

Everything around us can vibrate, from the chair you are sitting on to the stars in space. They can all act like musical instruments, oscillating with characteristic eigenfrequencies, so called normal modes. NSs should be no exception, although the existence of NS oscillations has not yet been confirmed. Stellar pulsations were well known already in the first half of the 20th century through the study of pulsating variables such as Cepheids and RR Lyrae stars, where self-driven periodic volume changes cause brightness variations, and the research field was further spurred by the discovery of the solar five-minute oscillation by Leighton, Noyes and Simon (1962). The latter type of stellar oscillation is a non-radial oscillation where the spherical shape of the star is not preserved, as opposed to the radial oscillation of pulsating variables in which the star expands and contracts around its equilibrium figure, while maintaining its spherical shape. In fact the radial oscillation may be regarded as a special case of the general non-radial oscillations, with the so called spherical harmonic index, l = 0, as will be described later.

2.4.1 Stability and Non-Radial Oscillations in General If a dynamically stable system is momentarily perturbed from its equilibrium conﬁguration it will start to oscillate, often with several superimposed normal modes. Depending on

15The Schwarzschild radius is the gravitational radius of a static and spherically symmetric mass and is 2GM given by RS = c2 , where G is the gravitational constant, M is the mass, and c is the speed of light. If the radius of a star is smaller than its Schwarzschild radius, neither particles nor light can escape from it, thus it is called a black hole.

17 whether the system is also vibrationally stable or unstable (overstable) the oscillation will either damp out, assuming that no continuous periodic perturbation is applied, or continue to grow. The reason of the oscillation is of course that when material is displaced in a dynamically stable system, like a star, a restoring force can “push back” the material and cause a wave motion. In a NS several different restoring forces act on a displaced mass element. Besides the pressure gradients and gravity, causing the acoustic p-mode and gravitational g- and f-mode in ordinary stars like our Sun, elastic forces in the crust, magnetic fields, centrifugal and Coriolis forces exist in rotating NSs, resulting in a wide variety of possible oscillation modes. Depending on the internal composition of structural layers in the NS, several sub-classes of modes can be classified, so it is hard to offer a comprehensive presentation of all possible modes. The most commonly discussed oscillation modes are presented in table 2.1. The p-, f-, g- (core and surface), s- (shear/transverse) and i- (interfacial) modes, can be brought together in the general category of spheroidal modes, while the torsional oscillations (r- and t-modes), which are normal modes of elastic waves in the solid crust, can be categorized as toroidal modes. If the NS was completely superfluid, non-rotating and non-magnetic, the toroidal modes would all be degenerate at zero frequency (McDermott et al. 1988). It is important to notice that the oscillation properties of a newly born hot so called proto-NS are very different from its cold NS descendant because of the thermal and chemical evolution which changes the composition. For example, g-modes due to composition gradients in the inner core of cold NSs have typical periods above 5 ms, while corresponding modes in hot proto-NSs have shorter periods owing to larger composition gradients (Ferrari et al. 2003). Furthermore, relativistic treatment of NS oscillations introduces an additional family of modes, e.g. gravitational wave w-modes. NSs are clearly relativistic objects, but basic features of the oscillation modes can nonetheless be established by a Newtonian treatment. The following description of non-radial oscillation modes in NSs will mainly build on results from an extensive Newtonian pulsation analysis by McDermott et al. (1988) who used a NS three-component model with a fluid core, solid crust, and thin surface fluid “ocean” together with a relatively soft EOS, and their comparisons with two pure fluid models with Newtonian and relativistic pulsation treatment, respectively. For a fully general relativistic computation for a wider range of NS models, the reader is referred to e.g. Lindblom & Detweiler (1983) or Finn (1990). McDermott et al. express the deformation of a solid body by Lagrangian displacements u(x,t) of infinitesimal mass elements from their equilibrium positions, and define the strain tensor uik as 1 ∂u ∂u u = i + k = u , (2.5) ik 2 ∂x ∂x ki k i and then continue with the equations governing the motion of a mass element:

• The continuity equation ∂ρ + ∇· (ρ~v) = 0, (2.6) ∂t

18 Oscillation modes Description Predicted period [ms] p-modes Resonant acoustic (sound) waves 0.1 trapped in the star. Pressure is the restoring force. f-modes These so called fundamental 0.1 - 0.8 modes are surface gravity waves (not to be confused with gravitational waves), similar to rip- ples in a pond, were gravity acts through buoyancy as the restoring force. They can also be regarded as p- or g-modes without radial nodes i.e. with n = 0. g-modes Internal gravity waves. Restor- 10 - 400 ing force is gravity. gc-modes Long-periodic gravity waves ap- > 10, 000 pearing in the fluid core of neutron stars with solid crust. r-modes Torsional modes occurring in ro- Same order as rotation period tating stars. They are retrograde of star. waves with oscillation amplitude along the surface where the Cori- olis force acts as a restoring force. sbr-modes These transverse shear waves are < 2 largely confined to the crust and depend on the crust thickness. i-modes Interfacial modes at the bound- Strongly dependant on local aries of different layers were mass density and temperature at elements move in elliptical or- the interfaces. bits causing a travelling wave along the surface, penetrating to a depth of one wavelength. t-modes Torsional modes confined to the < 20 crust with material motions tan- gential to the surface. ssf -modes Modes confined to the superfluid Shorter than in ordinary fluid interior of the star. modes.

Table 2.1: Presentation of the most commonly discussed classes of NS oscillation modes.

• The momentum equation

∂~v 1 + (~v · ∇)~v = ∇· ~σ − ∇Φ, (2.7) ∂t ρ

19 • Poisson’s equation ∇2Φ = 4πGρ, (2.8) where ρ is the bulk density, v is the velocity of the mass element, ~σ is the strain tensor, Φ is the gravitational potential, and G is the gravitational constant. Pulsation equations are derived by applying oscillatory time dependent perturbations to these equations and the equations describing the elastic solid (McDermott et al. 1988 and references therein), and we arrive at the linear, adiabatic wave equation for non-radial oscillations of an elastic sphere (see McDermott 1985 for details):

Γ p 1 ν2~u = −∇ 1 0 ∇· ~u − ∇ ~u · ∇p ρ ρ 0 0 0 Γ p 1 2 −~e A 1 0 ∇· ~u + ∇Φ′ + ∇ µ∇· ~u − (∇µ · ∇) ~u r ρ ρ 3 0 0 "

−∇ (~u · ∇µ) + (~u · ∇) ∇µ − µ ∇2~u + ∇ (∇· ~u) . (2.9) # Here p0 and ρ0 are the equilibrium pressure and density, respectively, ν is the pulsation frequency, Γ1 is the so called adiabatic index, and A is defined by 1 dρ 1 dp A ≡ 0 − 0 . (2.10) ρ0 dr Γ1p0 dr Without going into any details of the monstrous eigenvalue equation 2.9, I will just state that separation of the angular dependence of the adiabatic wave equation can be done in two ways, yielding the radial equations for spheroidal and toroidal oscillation modes. The solutions to the eigenvalue equations of the spheroidal separation are proportional to the m well-known spherical harmonics Yl (θ, φ) , which will be described in more detail in the section dealing with p-mode oscillations. Basic features of non-radial oscillations can be obtained even if the perturbations of the gravitational field (caused by stellar pulsations) are ignored (Cowling 1941). Although this so called “Cowling approximation” may not hold for an accurate description of modes formed deep within the star, e.g. gc-modes, it is often used since it greatly simplifies the calculations. The solution to the global three-component pulsation problem is given by the solution to the equations in the fluid ocean and core, and solid crust coupled together at the interfaces by suitable boundary conditions. These are given by the conditions that both the radial displacement and the tractions (the friction between a body and the surface on which it moves) are continuous at the interfaces. Since the p-modes are basically normal modes equivalent to acoustic waves, their period Πp depends on the speed of sound in the stellar matter, and can be estimated as the travel time of an acoustic wave across the star. This of course depends on the mean density and temperature of the NS, but a simple approximation with the longitudinal wave speed 10 -1 6 10 -1 cl ≈ 10 cm s gives Πp ≈ R∗/cl ≈ (10 cm)/(10 cm s )≈ 0.1 ms. The local analysis calculation by McDermott et al. (1988) estimate that the periods of low-order p-modes

20 Figure 2.7: Periods of several normal modes, calculated for a model with M = 0.5 M⊙, R∗ = 10.1 km and 7 Tc = 1.03 · 10 K (McDermott et al. 1988) are given by:

3 1/2 14 −3 1/2 0.5 ms R6 0.5 ms 5 × 10 g cm Πp ≈ ≈ , (2.11) kR∗ M∗/M⊙ kR∗ ρ¯ 6 where R6 = R∗/10 cm and k is the wavenumber. Estimating that the transverse wave 8 -1 speed is ct ≈ 10 cm s , the corresponding relation for s-modes is: 6 ms ∆r Π ≈ , (2.12) s k∆r 1 km where ∆r is the crust thickness. In a local approximation the mass motions in the p-modes are primarily radial, while mass motions in the shear modes are mainly transverse. The periods of the core g-modes are roughly:

1/6 c kR∗ −1 −1/2 M∗ Πg ≈ 35 s T7 R6 , (2.13) 1/2 M⊙ [l (l + 1)] 7 where T7 = T/10 K. Figure 2.7 shows a summary of the period calculations by McDermott et al. (1988) for several normal mode oscillations in three diﬀerent NS models, and ﬁgure 2.8 displays

21 Figure 2.8: Overview of the l = 2 oscillation spectrum for two different models. (McDermott et al. 1988) an overview of the l = 2 oscillation spectrum of both the pure fluid and three-component model. Comparison with period values derived by Strohmayer et al. (1991) for the same model, but with alternative crust properties shows no significant deviation in the f-, p- and surface g-modes, while increasing by some 10-30% in i-, s- and t-modes. An additional Newtonian investigation by Lee (1995) for a NS with a superfluid core and normal fluid envelope displays slightly longer periods for both the f- and p-modes. Besides the hy- drodynamic approach, a nuclear elastodynamic calculation of non-radial oscillations for a homogenous model has been done by Bastrukov et al. (1996), yielding, in general, shorter periods by almost a factor of two compared to a hydrodynamical model. To summarize: NSs can sustain a wide variety of different classes of modes. Each class has (theoretically infinitely) many normal modes, characterized by a set of quantum numbers. Many thousands of these can be present simultaneously and occupy a narrow frequency range, which makes distinction between specific modes quite difficult. In addition, the actual frequencies of the modes principally depend on the EOS in NS matter (see figure 2.9 from Kokkotas et al. 2001).

22 Figure 2.9: Diagram showing how the mode frequencies depend on the mass of the NS, for 12 diﬀerent equations of state. (Kokkotas et al. 2001)

2.4.2 P-mode Oscillations In this section, let us concentrate on the p-mode oscillations, which are acoustic oscillations with pressure as the restoring force. A simplified model of a NS as a dynamically and vibrationally stable, spherically symmetric, non-rotating and non-magnetic star, yields possible normal modes which, as briefly mentioned before, are described by the eigenfunc- tions that are proportional to the familiar spherical harmonics, which have a real part defined as:

m 2l + 1 (l − m)! m Yl (θ, φ) ≡ Pl (cos θ) sin(mφ), (2.14) s 4π (l + m)! where l (l = 0, 1, 2,...) is the spherical harmonic index or angular quantum number, which determines the eigenfrequency , and m (m = 0, ±1, ±2,..., ±l) is the azimuthal quantum number. θ ∈ [0,π] and φ ∈ [0, 2π] are the spherical polar coordinates, i.e. the colatitudinal angle from the “north-pole” through which the z-axis is deﬁned and the longitudinal angle m from the positive x-axis, respectively. Pl are the associated Legendre functions (see Sparr & Sparr 1999). If rotation or magnetic ﬁeld is introduced the (2l+1)-fold degeneracy in m is split, equivalent to “Zeeman splitting” in atomic physics. Additionally, the number of nodes in the radial displacement from the centre to the surface of the star contributes with

23 the radial quantum number n, which is equal to 0 for the fundamental mode, followed by 1 for the ﬁrst overtone mode and 2 for the second overtone mode, etc. The radial modes

Real part of the Spherical Harmonic: Y12 15

0.8

0.6

0.4

0.2

3 0 x −0.2

−0.4

−0.6

−0.8

−1 1

0.5 1 0.5 0 0 −0.5 −0.5 −1 −1 x 2 x 1

Figure 2.10: Visualization of a neutron-star p-mode oscillation. (The figure was created with the MATLAB package SPHEREDEMO written by Matthias Conrad 2001, with minor modifications of the code done by the author) are special cases of l = 0, as mentioned earlier, while the spherical harmonics l = 1 and 2 are labelled dipole and quadrupole oscillations, respectively. The displacements of mass elements during a normal mode oscillation describe a wave pattern on the surface of the star, as well as internally. The spherical harmonic index l specifies how many longitudinal nodal lines exist, while the azimuthal quantum number m

24 indicates how many times these nodal lines cross the equator. Since the radial quantum number n only affects the internal pattern, it will not influence the surface appearance. A 12 NS oscillating with a p15-mode could resemble something depicted in figure 2.10, although the amplitude in this plot has been highly exaggerated. In fact, the amplitudes of NS oscillations are extremely difficult to estimate and partially depend on the strength of the excitation mechanism and its correlation to the mode eigenfunction. This also makes it hard to predict the strength of any possible oscillation signature in observations, regardless of which the observables are. However, typical time-averaged pulsation energies normalized to relative amplitude at the surface, have been approximated by McDermott et al. (1988).

2.4.3 Excitation The energy channelled into pulsations must of course be acquired from somewhere. A supernova core collapse is one obvious candidate, since it releases an extreme amount of energy. By comparing the energy released in a supernova explosion, some 1053 ergs16, with the energy of the pulsations calculated by McDermott et al., also displayed in figure 2.7, we conclude that only a fraction of that energy would be needed to excite a majority of the modes, including p-modes, to significant amplitudes. A newborn proto-NS can be expected to oscillate quite violently in several normal modes, even though these oscillations might be damped out within a short amount of time, as I will come to in the next section. Another driving mechanism for non-radial oscillations can be the starquakes associated with observed pulsar glitches. The released energy in a glitch process, either from crust cracking or sudden transfer of angular momentum, has been estimated to 1038 − 1046 ergs (see Ruderman 1991; Van Riper et al. 1991; or Mock & Joss 1998). This would be enough to initiate considerable surface g-mode oscillation, but probably no large amplitude p- mode oscillation. But then again, it is still not known how large the amplitude of any perturbation has to be in order to produce a visible effect in NS observables. A third likely excitation mechanism is accretion of matter onto the NS in an X-ray binary. Here energy can be supplied from the actual accretion process e.g. in the flow onto accretion hot spots, as well as from the energy liberation during an X-ray burst when accreted matter undergoes explosive nuclear burning in the surface layers of the NS (Strohmayer 1993). According to McDermott et al. (1988), the latter would release approximately 1039 ergs, which would be sufficient for onset of at least surface g-mode oscillations or easily driven r-modes (discussed by Lee & Strohmayer 1995). Yet another possible excitation source have been proposed in NS binary systems, where the two NSs continuously spiral inwards due to emission of gravitational radiation, and eventually merge. In late stages of the merger process, when the stars are close together, the tidal interactions are strong and the orbital period will sweep past the oscillation period time-scales for several normal modes, perhaps briefly exciting them when in resonance (Reisenegger & Goldreich 1994; Kokkotas & Schäfer 1995). Also the coalescence could form an oscillating NS, but presumably it produces a black hole (Baumgarte et al. 1996a, b). Lastly, I should mention the possibility of a sudden phase transition (e.g. formation of

16The unit of energy in the CGS system. 1 erg = 10−7 Joule (J)

25 pion or kaon condensate, or strange star) and following mini-collapse leading to a release of maybe as much as 1051 ergs. Part of the release of gravitational binding energy could be channelled into oscillations of the NS (Kokkotas et al. 2001).

2.4.4 Damping The kinetic energy of matter undergoing oscillations will gradually dissipate through various processes, and the oscillations will cease unless they are continuously excited. Several damping mechanisms exist, and their eﬀectiveness varies depending on matter parameters (EOS) and oscillation mode. Damping times τ, i.e. the time required for the amplitude of oscillations to decay to 1/e of its original value, for three of the mechanisms presented below, are tabulated in ﬁgure 2.11,

Figure 2.11: Calculated damping times for the three diﬀerent NS models presented in the ﬁgure below. (McDermott et al. 1988)

Figure 2.12: Data for the models used in calculation of damping times by McDermott et al. above.

Gravitational radiation damping is one mechanism which is currently extensively discussed. Non-radial quadrupole or higher oscillation modes of compact objects like NSs

26 are expected to emit gravitational waves in the kilohertz frequency range. Although gravitational radiation should damp the oscillation, r-modes in rotating NSs could actually be amplified by it, causing the star to become unstable (Chandrasekhar 1970; Friedman & Schutz 1978). Energy losses through other mechanisms seem to suppress this so called CFS-instability, thus it threatens to destabilize only extremely rapidly spinning stars. The gravitational damping times for low-order p-modes are a few tenths of milliseconds, while they can be significantly longer for other classes of modes and generally shorter for higher overtones (the latter applying to essentially all damping mechanisms). Neutrino emission is another possible source of damping. Departure from chemical equilibrium due to the density changes during pulsations will alter the relative concentra- tions of neutrons, protons and electrons, causing pulsed neutrino emission (see e.g. Osaki & Hansen 1973 for the case of white-dwarf oscillations). However, this damping process is not regarded as being very efficient, except for young hot NSs with higher overall neutrino emission and then specifically for core g-modes. Damping times for this mechanism solely are consequently very long, as can be seen in figure 2.11. Damping from electromagnetic radiation generated by an oscillating magnetic field is an additional contributor, discussed by e.g. McDermott et al. (1984a). The field, which is effectively frozen into the matter on the pulsation time-scale, will produce electromagnetic radiation with a frequency corresponding approximately to the normal modes. The dissipation of energy due to this mechanism depends on the strength of the magnetic field, and 8 yields, assuming no other simultaneous contributing process and B0 = 10 T, damping times from a few days up to a couple of years for low-order p-modes. Internal friction in the NS core and crust would probably also cause energy dissipation in the oscillations. Even NS star matter in a superfluid state would have viscous contribution from electron-electron scattering, and closer to the crust proton-electron scattering and scattering from neutrons would alter the viscosity. In 1987 Cutler & Lindblom actually demonstrated that NS matter in a superfluid state would be more viscous than an ordinary neutron fluid, due to the various scattering processes. Damping times for this mechanism are hard to establish but probably are on the order of years (McDermott et al. 1988). Consideration should also be taken to non-adiabatic effects in matter during oscillations. This could convert kinetic energy from the pulsations into thermal energy, causing temperature variations in the interior and on the surface of the NS, and probably luminosity fluctuations in the thermal emission. Again, damping times are hard to estimate, and hardly any studies of thermal damping have been made, other than crude approximations for surface g-modes (McDermott et al. 1988).

2.4.5 Manifestations in Observables So far, no detections of NS oscillations have been made, but as described in the above sections, there are quite a few observables which could be explored for signatures of the non-radial oscillation, i.e. emission of gravitational radiation, electromagnetic radiation, and neutrinos. However, not all of these are plausible candidates. Billions of neutrinos pass through the Earth every second, most of these are solar neutrinos, while some originate from supernovae and background relic radiation, and some are produced in the atmosphere

27 by collisions with high-energy cosmic γ-rays. The likelihood of detecting a short ﬂuctuation in the neutrino emission from a pinpoint location on the sky, with today’s large, low- eﬃciency, low-resolution neutrino detectors, are naturally small. However, the other two observables seem very promising. Ongoing collaborating projects like LIGO, VIRGO, GEO and TAMA with recently constructed laser-interferometer detectors, are hoping to actually detect gravitational waves from astrophysical sources. Thus far, these have only been inferred e.g. from the exact prediction of the orbital decay in the binary pulsar PSR 1913+16 (Hulse & Taylor 1974). Gravitational radiation signals are expected to be very weak, even from powerful events like supernovae, and black hole- or NS-mergers, so the space-time “echo” from vibrating NSs should be very weak indeed, with an amplitude of approximately (Kokkotas et al. 2001):

1/2 1/2 −20 Egw 10 kpc 1 kHz 1 ms A ≈ 2.4 × 10 − , (2.15) 10 6M⊙c2 r f τ where Egw is the energy released in the normal mode, r is the distance between the source and detector, f is the frequency of the mode, and τ is the damping time. Kokkotas et al. (2001) conclude that the next generation of gravitational wave detectors should have a good chance of detecting oscillating NSs if their sensitivity was increased in the 5-10 kHz range. When it comes to observing oscillation features in the electromagnetic radiation, we should perhaps concentrate on the isolated NSs. The assumed luminosity variations originating from temperature ﬂuctuations on the surface of the star, or modulations in the radio pulsar emission due to distortion of the magnetosphere (Timokhin 2004), during oscillations, are quite promising aspirants. Although burst oscillations in LMXBs have been associated with surface r-modes (see e.g. Lee 2003), there are probably several other processes active in X-ray binaries which could interfere on the time-scales of interest. To narrow it down further, the optical wavelength region shows potential due to high count-rates, especially with future ELTs, and should be the superior observation window for high time-resolution measurements, as discussed in the introduction. With ELTs we can also access the faint thermal emission from the large fraction of cooling NSs not observed as pulsars. According to Pavlov et al. (1996) our Galaxy should be populated by 108 − 109 NSs, and so far only about 10−6 − 10−5 of these have been detected (mostly as pulsars). High resolution optical spectroscopy of isolated NSs could also provide us with information about gravitational redshift, in addition to data about the chemical composition of NS atmospheres (at least in old NSs where the magnetic ﬁeld could have decayed enough to allow use of non-magnetic atmosphere models).

2.5 Probing the Interior of Neutron Stars

Let us assume that we have detected an oscillation mode in the optical signal from a NS. Which are the relevant parameters that we want to measure if we are to reveal the supranuclear EOS, and with what precision can they be determined? Kokkotas et al. (2001) have presented a “ﬁngerprint analysis” using the frequencies and damping times

28 of the f- and p-modes, for the case of gravitational radiation, but which is also partially applicable to optical investigations. The accuracy of frequency and damping time measurements, in a signal of a certain amplitude, principally depend on the instrumentation used in the observations. This could be explored quantitatively for different instruments, and will be treated more in depth in a later chapter, although even then I will only perform simulations with a qualitative approach. For now, let us just say that with high time-resolution and high signal-to-noise ratio (SNR), we could undoubtedly infer very accurate parameters. As was described in the section about the EOS, constraints are imposed by accurately deducing mass and radii of NSs. In theory, the frequency and damping time for any mode could yield values for both of these. However, Andersson & Kokkotas (1998) numerically demonstrated that the empirical relations, both between the stellar parameters and the frequencies, and between the stellar parameters and damping times, for many modes (including p-modes) are EOS dependent, thus precluding them from being used alone to determine mass and radius. To accomplish this they instead employ a procedure using the fundamental f-mode together with the first p-mode. In short, the frequencies ff and fp of the f- and p-mode, respectively, are utilized to identify the most likely EOS, and then the detected fp is used to infer the compactness of the star. To be slightly more specific, we can study the empirical relation for the f-mode frequency:

f M¯ 1/2 f ≈ (0.78 ± 0.01) + (1.63 ± 0.01) , (2.16) 1kHz R¯3 where M¯ ≡ M/1.4M⊙ and R¯ ≡ R/10 km, and the two diagrams in figure 2.13. From equation 2.16, which shows that the frequency scales with the mean density of the NS, one can estimate the mean density if one has measured ff . As a result, if fp is also known, the most likely EOS can be picked out from the left diagram of figure 2.13. From the right diagram, which shows how the frequency of the first p-mode depends on the stellar compactness M/¯ R¯, one can then estimate the compactness of the star. The mean density and the compactness together yield the mass and radius of the NS.

29 Figure 2.13: Dependence of the frequency of the ﬁrst p-mode on the square root of the mean density of a NS (to the left), and on the compactness (to the right), for diﬀerent EOS. (Kokkotas et al. 2001)

30 Chapter 3

Instrumentation

In this chapter, I will review the main photometric instrumentation used during the course of this thesis work. The central component was a digital hardware correlator, which was initially tested together with equipment in the optics laboratory at Lund Observatory, and then also, prior to the observations, evaluated jointly with the OPTIMA research group at MPE as part of their OPTIMA detector system.

3.1 Detectors

The leading detectors in optical astronomy today are Charge Coupled Devices (CCDs), and they definitely are beneficial as imaging detectors, owing to their high quantum efficiency and ability to obtain wide field pictures. However, they suffer from one major drawback if they are to be used in high time-resolution photometry, namely long read-out times. As the CCD technology is still in rapid progress the development of high-speed CCDs will probably change this to a certain extent, however, faster read-out also contributes with more noise, which puts a fundamental limit to the application of CCDs in very high time resolved measurements. On the other hand, detectors which are currently widely used for single-photon counting, e.g. photomultipliers, have lower quantum efficiency, particularly when extending toward the infrared. Fortunately, there exist solid state detectors which combine single-photon counting on sub-millisecond timescales with high quantum efficiency. These silicon avalanche photo- diodes (APDs) operate in a Geiger counter mode, which means that an absorbed photon initiates an avalanche of photoelectrons driven by the bias voltage. To shorten the dead time of the detector the avalanche can be actively quenched, reducing the dead time to 50ns, as opposed to nearly 1µs for passive quenching. The recovery time, of course, varies between different models, as well as the wavelength sensitivity, which depends on the employed absorbing material. As an example of range and sensitivity of a single-photon counting module (type SPCM-AQR-15-FC) from PerkinElmer, see the quantum efficiency diagram in figure 3.1. The typical active diode area is small, with a diameter of about 200 µm, and peak photon detection efficiency in excess of 70% at 650-700 nm (slightly varying over the active area) for some models (see e.g. www.perkinelmer.com). Maximum count rates for actively quenched detectors are some 10 MHz, above which saturation may

31 Figure 3.1: Quantum Eﬃciency diagram for APD module (type SPCM-AQR-15-FC) from PerkinElmer (Kanbach et al. 2003) occur, and the timing of individual photons can be done with a resolution of 20 ps. APD detector properties relevant for high-speed astronomical photometry have been studied by Dravins et al. (2000).

3.2 Correlators

As a consequence of very high time-resolution, the data rates are expected to become extremely high, possibly several megabytes per second (Dravins 1994). To record a full resolution light-curve would require data processing and storage capabilities out of the ordinary. Another possible and more applicable way to handle the analysis of these vast amounts of data is to carry out some real-time data reduction to statistical functions, e.g. 2:nd order correlation functions. The strength of a correlation signal would also increase with the second power of the intensity. Such a measurement is described by (Dravins 2000): (2) G [~r1,t1; ~r2,t2]=, (3.1) where <> denotes time average and I = is the measured intensity in a spatial location and point in time given by ~r and t, respectively. For r1 6= r2 and t1 = t2 equation 3.1 gives the quantity measured by an intensity interferometer (Hanbury Brown & Twiss 1958), while the corresponding measurement for r1 = r2 and t1 6= t2 is performed by a correlation spectrometer. The latter is obviously the instrument which we want to utilize in our high time-resolution measurements. However, before I describe the workings of the instrument in more detail, let us review the mathematics of correlation functions.

32 (2) 1 To calculate the auto-correlation Gac (τ) of a real function f(t) we determine the integral +∞ (2) Gac (τ)= f(t) · f(t − τ)dt. (3.2) Z−∞ In words, this means that the auto-correlation is given by the integral as the function f(t) is swept over an analogous function. Any change in the function on a speciﬁc time-scale will appear as a variation in the auto-correlation at the corresponding delay time τ. The auto-correlation is apparently independent of the order, but this is not the case for the (2) cross-correlation Gcc (τ) between two functions f(t) and g(t), which is given by

+∞ +∞ (2) Gcc (τ)= f(t − τ) · g(t)dt = f(t) · g(t + τ)dt. (3.3) Z−∞ Z−∞ The useful differences between the auto- and cross-correlation functions will become apparent when discussing the experimental setup. Often the correlation functions are normalized by dividing with the central value, thus total correlation equals one. The principal function of a correlator instrument is to make an estimate of the “true” correlation by approximating the integral by a summation, which means sampling the counts from the detector in some time interval T , shifting the time bins by the delay time, multiplying overlapping bins, and adding up. This yields a raw correlation value at the specific delay time, which has to be normalized, e.g. by dividing with the number of samples. Since the time series is finite the decreasing number of non-zero terms for longer delay times also has to be compensated for in the normalization. In principal all this can be done by a software correlator, however, such a system would become very slow since it would again imply storing and processing a large amount of data, which brings us back to our initial problem. With a hardware correlator simultaneous calculations of the products of the last sample count and the number of counts of the last say L samples, can be performed within a short sample time by using L electronic multipliers. Alternatively, a clipped correlation technique can be used to obtain even higher speeds (Saleh 1978).

3.3 QVANTOS/OPTIMA

The collaboration between QVANTOS and OPTIMA emerged as a result of previous contacts between thesis supervisor Dainis Dravins and MPE-group leader Dr. Gottfried Kanbach, discussing instrument compatibility and mutual beneﬁts of joint observations. In addition to the mentioned group leader, the group from MPE consisted of Fritz Schrey and graduate student Alexander Stefanescu, while I and fellow diploma worker Helena Uthas participated as part of the QVANTOS-group from Lund Observatory.

3.3.1 QVANTOS instrumentation In the optics laboratory at Lund Observatory two diﬀerent silicon APD detectors, one older passively quenched SPCM-100-PQ unit and one newer actively quenched SPCM-161-AQ

1If the function is complex, the delayed function will be complex conjugated.

33 unit, manufactured by PerkinElmer, were used in the process of familiarizing ourselves with the hardware correlator. The former has a stated peak quantum eﬃciency, QE, around 43% at λ = 643 nm and active diode diameter d, of 200 µm, while corresponding properties for the latter are QE ≈ 70% at λ = 630 nm and d = 180 µm. Each detected photon triggers a standard TTL pulse (of 2.5V minimum) as output from BNC connectors at the rear of the module. These pulses are transferred via BNC cables to either in-channel A or B on our correlator, which calculates high time-resolution correlation functions and lower resolution intensity traces, and ultimately transmits the data through USB cables to the host computer where it is stored. After this summary of the detection–storage chain, let us take a look at the speciﬁc characteristics and functions of the QVANTOS correlator.

Flex01-05D Digital Correlator & LabVIEW Instrument Control The correlator Flex01-05D was purchased in 2002 from the company Correlator.com and is a small (some 10x10x3 cm) multi-function instrument, with delay-channel resolutions down to 5 ns, and delay time range from 5 ns to 23 or 183 min, depending on mode. One photon history recorder mode and three diﬀerent correlator modes are available, and a comprehensive description of these is given in the manual and user-guide by Nilsson & Uthas (2004) included as appendix A. Moreover, this appendix contains a discussion of the method and normalization schemes used when computing the correlation functions, as well as an account of the LabVIEW control and data acquisition program PhoCorr written by me and H. Uthas. Nevertheless, I will brieﬂy explain vital features of the correlator

Figure 3.2: Correlator unit Flex01-05D to the right, and data acquisition computer with control-program PhoCorr running.

34 and software in the paragraphs below. PhoCorr was written to improve the control interface and enhance the real-time graphical display quality and data acquisition, compared to the more basic original program. The programming was not trivial, partly because of inadequate correlator documentation, and partly because neither I nor H. Uthas had any prior experience of the LabVIEW programming environment. Hence, some time was initially spent to examine (through trial-and-error) this “little black box” we had been handed. To verify our progression, both the original program and our program were run with known identical signals fed into the correlator input channels, and the stored data were compared. PhoCorr has been continuously ﬁne-tuned during the course of the thesis work, and the main functions associated with the control of the correlator have been satisfactorily implemented. The user can choose to run the photon correlator in:

• Mode A: Single auto/cross correlation - Calculates the normalized auto (AxA) or cross (AxB) correlation function of the incoming signals (A and B) for delay times from 5 ns to 23 min in 1088 channels. This mode gives the highest resolution correlation function. PhoCorr displays correlation and lower resolution intensity series (0.04194304 s) in real-time, while storing to disc.

• Mode B: Quad correlation - Calculates the normalized auto (AxA and BxB) and cross (AxB and BxA) correlation functions for delay times from 5 ns to 23 min in 288 channels. PhoCorr displays correlations and intensity series in real-time, while storing to disc.

• Mode C: Dual auto/cross correlation - Calculates the normalized auto (AxA and BxB) or cross (AxB and BxA) correlation functions for delay times 5 ns to 183 min in 608 channels. PhoCorr displays correlations and intensity series in real-time, while storing to disc.

So called multiple tau computation is employed to calculate the correlation functions in all of the above modes. This means that sample and delay times are monotonically increased in geometrical progression in consecutive channel segments giving a wider temporal range of the correlation function for a single measurement, compared to “linear” correlators. (Again the reader is referred to appendix A for more details.)

Laboratory Testing and Simulations The experimental setup in the optics laboratory at Lund Observatory during the initial tests consisted of the two mentioned APD detectors (henceforth labelled DA and DB), mounted on an optical bench, moderately covered to avoid over-exposure of light, but still enabling detection of stray light from an incandescent lamp. BNC cables were coupled from the outputs of the two detectors to input A and B, respectively, on the correlator unit. As a result of the 50 Hz alternating voltage supply leading to 100 Hz modulated light from the incandescent light source, an oscillation signature at 10 ms was clearly distinguishable in the cross-correlation between the DA and DB signals (see ﬁgure 3.3). The advantage of a cross-correlation measurement between two detectors monitoring the

35 same light source is that noise is greatly suppressed due to the non-correlation of short time-scale random ﬂuctuations in two diﬀerent detectors. In an auto-correlation the noise would instead be enhanced with the second power of the noise intensity. Additional

Figure 3.3: A screenshot of the PhoCorr user interface, showing the 100 Hz modulation of light from an incandescent lamp. experiments were performed by using a pulse generator and a 128-bit word-generator, both to transmit a signal of “artificial photon pulses” directly to the correlator, and to modulate light from a light emitting diode (LED). Pulses generated by the pulse generator were selected by assigning three hexadecimal digits, representing the delay in microseconds between consecutive pulses, to the word-generator. The TTL pulse shape and amplitude was monitored and adjusted by a connected oscilloscope to ensure a safe signal. In our first experiment with the LED we flashed double pulses with a specific period (similar to a pulsar). Just a short (30 s) autocorrelation measurement of the modulated light from the LED still displayed a clear sign of the 55 ns pulse width, the 1 µs two- pulse delay, and the 25 µs period modulation (see figure 3.4). Unfortunately the word- generator failed before we had any chance to simulate emission from real astronomical objects with the LED, nevertheless, the initial tests distinctly demonstrated the potential of the hardware correlator for detection of short time-scale optical variability.

36 Figure 3.4: Auto-correlation of ﬂashing LED. By zooming in we can see the time between the second pulse and the ﬁrst pulse (of the next phase) at a delay time of approximately 24 µs. This of course corresponds to the period minus the time between the two pulses, i.e. (25 − 1) µs.

Computer simulations of correlation functions and power spectra Simulations of correlation functions and power spectra from observations of astronomical objects constitute an important part of the thesis, both in the preparations before the observing campaign on Mt. Skinakas and in the theoretical work concerning neutron-star oscillations. The goal of the simulations is, in the former case, to make some kind of prediction on what we can expect to see in the measurement of the correlation signal from speciﬁc objects during the observation using our correlator equipment. This is needed in order to make some kind of immediate interpretation of what we are seeing. In the latter case, i.e. in the theoretical part, the simulations (described in chapter 5) are supposed to give a hint on how non-radial oscillation of neutron stars can be detected in future ELTs using the “correlation method”. As well as making computer simulations of realistic emission signals from astrophysical sources, it can be quite instructive to make simple simulations of correlation functions and power spectra of e.g. harmonic functions superposed on some constant background level and noise. Already from the discussion of correlation functions earlier, you might suspect that there is a close relation between the autocorrelation and the power spectrum of a function f(t). The power spectrum is given by:

P (ν)= |F (ν)|2 , (3.4) where F (ν) is the Fourier transform function of f(t), i.e. the function which describes the frequency content of the original function f(t), and ν is the frequency. The Fourier transform F {f} of a complex, one-dimensional function f(t) is deﬁned by (see e.g. Spanne 2000): ∞ F {f} = F (ν)= f(t)e−i2πνtdt. (3.5) Z−∞

37 It can be shown2 that the power spectrum is actually the Fourier transform of the auto- correlation, i.e.

∞ (2) −i2πντ P (ν) = Gac (τ)e dτ. (3.6) Z−∞ For a sampled signal the integral above is replaced by a discrete summation, and the highest frequency which can be represented is given by half the sampling frequency, called the Nyquist-frequency. Frequencies above the Nyquist-frequency are folded to lower frequencies. Another peculiarity which arises when calculating Fourier transforms of signals measured for a finite time period (not only in the discrete case), is that the abrupt transition at “the edges” of a signal introduce sidebands in the frequency spectrum. The reason for this is that the signal shape is equivalent to a multiplication of an infinite function with a square window function. A Fourier transform of the square window yields the well- known sinc-function which introduces artifacts in the signal transform. One can overcome this problem to a certain extent by multiplying with a smoother function, e.g. Hanning or Hamming windows. Since direct computation of the Fourier coefficients (DFT) is quite time consuming, an effective algorithm called Fast Fourier Transform (FFT) has been implemented to perform the calculation in many computer programs. The FFT uses certain symmetries in the definition of the discrete Fourier transform to speed-up the calculation (see e.g. Bennewitz 2003). Simple simulations of correlation functions and power spectra were performed both in SigmaPlot and MATLAB, using the FFT algorithms, just to allow us to get acquainted with their properties, and to serve as a basis for more realistic simulations of emission from astrophysical objects.

3.3.2 OPTIMA Instrumentation Optical Pulsar Timing Analyser (OPTIMA) is a photon counting, high-speed photometer, constructed by the γ-ray group of the Max Planck Institute for Extraterrestrial Physics (MPE). It is a stand-alone instrument, which can be used in the visitor focus of telescopes at various observatories. The main purpose of the instrument is, as the name implies, high time-resolution (10−6 s) study of the optical emission from pulsars, however, OPTIMA has been successfully employed to study other interesting objects with confirmed or supposed rapid optical variability, e.g. cataclysmic variables and black hole candidates, often in conjunction with X- or γ-ray observations. Figure 3.5 illustrates schematically the hardware layout of OPTIMA, which consist of a CCD pointing control unit for target acquisition and pointing confirmation, an assembly of eight fibre fed high-speed APD detectors, a GPS timing receiver, and separate PC control units. A hexagonal bundle of seven optical fibres is centred at the astronomical target

in the focal plane to minimize atmospheric background ﬂux contribution to the signal,

Ò Ó h i Ê Ê ∗ 2 (2) ∞ ∞ i2πντ

F Gac = −∞ f(t) −∞ e f(t − τ)dτ dt = [t = −y]

h i

¢ £

Ê Ê Ê ∗ ∞ ∞ −i2πνy ∞ i2πνt ∗ = −∞ f(t) −∞ e f(t + y)dy dt = −∞ f(t) F (ν)e dt = F ∗(ν)F (ν) = |F (ν)|2.

38 and the additional eighth ﬁbre monitors the adjacent sky brightness. The APD detectors, which are more or less identical to the newer one used by us in the optics laboratory at Lund Observatory, and the acquisition computer can record arrival times of individual photons with an accuracy of almost 4µs, owing to the precision of the GPS timing system.

3.3.3 Laboratory Testing QVANTOS/OPTIMA During one week in June 2004 supervisor Dainis Dravins, Helena Uthas, and myself, visited MPE in Garching, Germany, to verify the compatibility of our QVANTOS equipment with the OPTIMA detector system, and discuss possible arrangements for a joint observation campaign. Such collaboration would complement OPTIMA with a higher time-resolution instrument (2-3 orders of magnitude) and real-time analysis display, while offering an evaluation of the correlator hardware and control-software intended for QVANTOS Mark II. In addition we could explore the limitations and possibilities of the OPTIMA/QVANTOS system at a fairly large telescope. The practical details of the tests at MPE dealt with how the signals from the detectors could be shared between the correlator and the OPTIMA data acquisition system. Fritz Schrey and Alexander Stefanescu had mounted OPTIMA in an MPE laboratory with an enclosed, fixed light source which illuminated the fibre bundle through a tube, so we merely had to connect the signal from the detectors to our portable correlation system. The OPTIMA system seemed to be fully compatible with our Flex01-05D correlator, however some precautions had to be taken when splitting the signal, so that the signal amplitude was not damped to such an extent that pulses were ignored by the correlator. Additional test were performed by sending signals from multiple detectors to the same correlator input channel, however in that case the signals of course had to be damped enough to ensure a safe signal in the event of overlapping pulses, but still enough to trigger a count for every TTL-pulse. These test demonstrated that coupling two detectors to the same correlator input seemed possible, while any additional detectors would require insufficient pulse-voltage amplitude. Following the successful laboratory tests, further discussions were held with project leader Gottfried Kanbach and the rest of the OPTIMA group at MPE, on the issue of joint observations and astrophysical advantages thereof. Since OPTIMA was going to be transported to Crete and mounted on the 1.3-meter telescope at Skinakas Observatory in mid-October, it was decided that we (the QVANTOS group) should bring our equipment and join the OPTIMA team there. Although the MPE group originally intended to perform observations of the optical afterglow from γ-ray bursts (GRBs), detected with NASA:s SWIFT satellite, during this October campaign, this came to naught because of delayed launch of the satellite. Consequently, more time was open for study of other fascinating objects.

39 Figure 3.5: Schematic illustration of the hardware layout of the OPTIMA detector system. (Straubmeier et al. 2001)

40 Chapter 4

Observations

In this chapter an account of the observations at Skinakas Observatory is presented along with a subsequent preliminary analysis of the acquired data. The preparations included selecting suitable objects for observation, as well as performing computer simulated observations of some of these. Of course any observation program ultimately had to be established in consultation with the MPE participants.

4.1 Preparing the Observations

The selection of appropriate observing objects was mainly done by H. Uthas, while I focused on constructing a MATLAB script for computer simulations. By using existing calendars (like the one Helmut Steinle’s OPTIMA site1) to check which objects are in principle visible from the observatory in late October and if they are bright enough, a list of objects could be compiled together with a summary of relevant information from articles about them. Related ﬁnding-charts were also put together by H. Uthas.

4.1.1 Crab Pulsar Simulation An example of one, from Skinakas, visible object with a well established optical light curve is the Crab pulsar. It is by far the brightest of the five confirmed optical pulsars (Crab, PSR 0540-69, Vela, Geminga, and PSR 0656+14) found to date, with an apparent B magnitude peak corresponding to a flux of 90, 000 photons/sec on a VLT sized (8.2-m) telescope (Shearer & Golden 2002). By performing a correlation simulation of an idealized Crab pulsar light curve, we can predict the general appearance of a measured correlation function, thus enabling one to immediately notice during observations whether there is any distinct deviation from the expected shape. Additional simulations with added artificial rapid variability to the light curve e.g. sharp spikes, were also carried out. Light-curve data from numerous measurements of the Crab pulsar were compared to estimate the pulse profile, which was then constructed in Simulink (an extension to MATLAB). With Sim- Calc.m auto-correlations and power spectra could be calculated and plotted. A summary of the results is presented in figures 4.1 and 4.2, and the MATLAB scripts can be found

1http://mpe.mpg.de/∼hcs/OPTIMA/

41 in appendix B. The simulations correspond to a measurement with 1 µs resolution, where the signature of the 15 µs spikes in the light-curve (figure 4.2) is seen as a “bump” in the correlation function at delay times of a few microseconds, compared to figure 4.1. This sign of microsecond variability would be visible even if the spikes were drifting, although they in this simulation were fixed in phase through consecutive periods. Transferring this scenario into nanosecond time-scales brings us to where we are today, with hardware correlators reaching nanosecond resolution and pushing toward the sub-nanosecond time domains. After testing the Flex01-05D correlator at MPE, Dainis Dravins contacted

Lightcurve 3000

2000

1000 Relative intensity 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Time t [s]

> 5 2 x 10 Auto−correlation of lightcurve 2

1.5

0.5

0 −6 −5 −4 −3 −2 −1 0 10 10 10 10 10 10 10 Relative auto−correlation

5 10 Power [Hz

0 10 −2 −1 0 1 2 3 4 5 10 10 10 10 10 10 10 10 Frequency [Hz]

Figure 4.1: Schematic Crab pulsar light-curve (in the topmost graph) together with the simulated auto- correlation and power spectrum. the manufacturer Correlator.com to discuss the purchase of a supplementary correlator Flex03LQ-01 of similar type, but with notably higher time-resolution (1.56 ns). Making use of both correlators during the observations would give us four in-channels instead of two and permit e.g. simultaneous cross-correlation measurements between several spectral

42 Lightcurve 8000

6000

4000

2000 Relative intensity 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Time t [s]

> 5 2 x 10 Auto−correlation of lightcurve 2

1.5

0.5

0 −6 −5 −4 −3 −2 −1 0 10 10 10 10 10 10 10 Relative auto−correlation

5 10 Power [Hz

0 10 −2 −1 0 1 2 3 4 5 10 10 10 10 10 10 10 10 Frequency [Hz]

Figure 4.2: Crab pulsar light-curve with 15 µs wide spikes; one near the main-pulse maximum and one at the trailing edge of the main-pulse. colour bands (from a simple modification with a prism in the OPTIMA set-up). Probably this new hardware correlator could be run with our PhoCorr control program, requiring only minor adjustments of the software. However, the manufacturer could not guarantee delivery of this unit before the departure to Crete, so we worked only on polishing up the user interface and implementing more effective data storage procedures for the control program of the first correlator. Luckily the new correlator arrived the week before departure, and was tested with the enclosed original software on the lap-top computer that was going to be used during the observations. Unfortunately the hardware caused some trou- ble, rendering both correlators to be unrecognized by the computer, just two days before leaving for observations! After several hours on the phone with the manufacturer, poking around in Windows system registry, D. Dravins managed get the lap-top to cooperate with the old correlator again.

43 4.2 Skinakas Observatory

Skinakas Observatory is beautifully situated on a plateau in the Ida mountains in central Crete, some 1750 m above sea-level, overlooking the city of Heraklion to the north-east. The Observatory was planned and built by the University of Crete, the Foundation for Research and Technology - Hellas (FORTH), and MPE in the mid 80s, mostly for the purpose of education and research of extended sky objects, thus, the completion of the first building with a 0.3 m flat-field telescope was nicely timed with the arrival of comet Halley in the Spring of 1986. Almost ten years later a 1.3 m Ritchey-Chrétien telescope was installed in a new dome, and it has ever since been successfully used to study galaxies, star clusters and gaseous nebulae.

Figure 4.3: Skinakas Observatory with the guesthouse in the foreground, visitor building and 1.3-m telescope dome to the right and left, respectively, in the background.

The seeing inﬂicted by atmospheric turbulence has been found to be relatively small, approximately 0.6-0.7 arcseconds (Boumis et al. 2001), implying good conditions for high (spatial) resolution observations.

44 Figure 4.4: A bunch of cheerful astronomers posing next to the 1.3-m telescope at Skinakas. From left to right: Dainis Dravins, Helena Uthas, Ricky Nilsson (Lund); Fritz Schrey, Alexander Stefanescu, and Gottfried Kanbach (MPE).

4.3 Measurements

After the slight mishap two days prior to departure, we were all pleased to discover that the hardware correlator was recognized by the lap-top computer, and the ﬁrst real astrophysical signal appeared on the screen in the evening of October 13, 2004. The observations spanned over a period of nine nights in total, and generally the ﬁrst measurements started at 7 p.m. and the last ended at 7 a.m. The duration of individual target measurements varied between 5 min and 4 hours depending on, among other things, weather conditions and periods of eclipsing objects.

4.3.1 Experimental Setup OPTIMA was firmly mounted in the focus of the 1.3-m telescope, and the signals from the detectors were sent to the control room, where the data acquisition and detector control units were located. Since 98% of the received flux from the observed object enters the central fibre when the hexagonal bundle is accurately positioned (Straubmeier et al. 2001), the TTL signal from its connected detector was the primary target of our measurements. This signal was split through an electronic analog splitter and sent to OPTIMA’s data acquisition computer and input channel A of our digital hardware correlator Flex01-05D, respectively. Input channel B was for the most part connected to that detector which received light from the sky background. The telescope positioning and guiding system

45 Target Object type description Cyg X-1 High mass X-ray binary (HMXB). Stellar-mass black RA: 19 58 21.6 hole candidate in binary system with a blue super- Dec: +35 12 05.7 giant. Accretion from normal star onto black hole causes X-ray emission. Crab Pulsar (PSR 0531+21 ) Young, rapidly spinning NS embedded in the Crab RA: 05 34 31.9 Nebula. Dec: +22 00 52.0 HU Aqr Cataclysmic variable of eclipsing polar type. This RA: 21 07 58.2 is a small binary system with a white dwarf and a Dec: −05 17 39.4 normal (red dwarf) star. The normal star overfills its Roche lobe, matter falls toward the white dwarf and is funnelled onto hot spots by the intense magnetic field of the white dwarf. New CV in Pisces Cataclysmic variable. RA: 00 25 11.2 Dec: +12 17 11 V404 Cyg Low mass X-ray binary (LMXB). Quiescent, accret- RA: 20 24 03.7 ing stellar-mass black hole candidate in binary sys- Dec: +33 52 03.2 tem with low mass (M < 2 M⊙) main sequence star. X Persei HMXB. Accreting NS in binary system with high RA: 03 55 23.0 mass star. Dec: +31 02 45.0 V2275 Cyg Cataclysmic variable of nova type. Accreting white RA: 21 03 01.9 dwarf in binary system with low mass normal star. Dec: +48 45 52.8 Exhibits unpredictable outbursts when accumulated hydrogen-rich matter burns explosively and ejects a large part of the outer envelope. SDSS 0155+00 Cataclysmic variable of eclipsing polar type. RA: 01 55 43.4 Dec: +00 28 07.2 V1974 Cyg Cataclysmic variable of nova type. RA: 20 30 31.6 Dec: +52 37 50.8 LW Cam Cataclysmic variable of polar (AM Her) type. RA: 07 04 09.9 Dec: +62 03 27.9 S5 0716+71 Active Galactic Nucleus (AGN) of BL Lac type. Ob- RA: 07 21 53.3 ject exhibiting strong featureless continuum emis- Dec: +71 20 36 sion from radio to X-ray frequencies, and significant brightness variations over days or weeks.

Table 4.1: Observed targets during the observation campaign at Skinakas Observatory. was operated entirely by the MPE group and the attending night-assistant.

46 4.3.2 Observed Objects Since eleven diﬀerent astrophysical targets were observed during the campaign, a comprehensive description of all of these would be quite time consuming, and moreover lie outside the scope of this thesis. What is presented is a table stating the name of the target, along with a short description of this type of object and some additional speciﬁc information (table 4.1). Two of the targets; the Crab pulsar and eclipsing polar SDSS 0155+00, will be dealt with somewhat more thoroughly, one being a NS and the other showing some interesting features in its correlation.

4.3.3 Sources of Error What does one actually see in a measured correlation function obtained with the 1.3-m telescope at Skinakas? To interpret the signal from the correlator is not trivial. Many features may originate not from fluctuations in the observed astronomical source, but rather from noise sources, such as vibrations in the telescope optics or background light variation. All such possible causes have to be excluded prior to any attempt of an astrophysical explanation. This work can be simplified if one during the observations also studies objects that are not expected to have any significant variations in their emitted light e.g. “normal” stars. When it comes to variations on 100 ms − 100 ns time scales, contribution from atmospheric scintillation becomes an important factor, especially for small telescopes. Ex- tensive correlation measurements (with QVANTOS Mark I) of atmospheric scintillation have been performed by Dravins et al. (1997a, 1997b, 1998), showing a general decrease of power in the power spectra for larger telescope apertures, due to spatial averaging of turbulence components over a larger area. Although scintillation effects may be significant, and ought to be calibrated for, an obvious “bump” in the correlation function on the relevant time scale should also be visible in the correlation of the calibration star, if of terrestrial origin. Thus, measurements of a calibration star lying close to the object of interest were made prior to any target observation. Clouds drifting past the observed sky region produced brightness variations on the order of seconds, however these effects could easily be distinguished by comparing the intensity series of the target (channel A) with the one of the sky background (channel B). If the perturbation was confirmed in both channels the integration was terminated and only the correlation function accumulated up to then was later analyzed. In addition to the variability from atmospheric effects, mechanical vibrations in the telescope structure can produce rapid changes in light intensity. Even though the telescope is firmly mounted, winds hitting the cross-section exposed through the dome aperture can dump a considerable amount of energy into the telescope, causing it to move from its equilibrium position. As a result the telescope will start to vibrate at the combined resonant frequency of its moving parts, and possibly at higher overtone frequencies. Telescope vibrations can also be excited by the motor drive. Depending on the amplitude of the vibration some varying amount of light will spill outside the central fibre, and generate an intensity signal amplitude which fluctuates with a frequency corresponding to the telescope resonance.

47 Several measurements during windy nights at Skinakas Observatory displayed a 7 Hz signature in the correlation function, most likely originating from wind induced telescope vibrations. Figure 4.5 shows two such correlations; one for a 2000 s measurement of V404 Cyg and one for a 600 s exposure of its calibration star, obtained October 17.

Figure 4.5: Telescope vibrations produce these oscillation features at τ ≈ 0.147 s, corresponding to approximately 6.8 Hz, in the measured auto-correlation function of both V 404 Cygni and its calibration star.

4.4 Analysis

In this section, preliminary analysis of data from the observations will be presented, mainly concentrating on the Crab pulsar and SDSS 0155+00. This does by no means imply that the other objects are uninteresting, but rather that they did not exhibit any correlation signature which attracted our immediate attention.

4.4.1 Reduction of Raw Data More than 45 hours of effective target observations gave us a great amount of raw data to work with. All measured correlation functions and intensity values were stored once every second to separate plain-text files on the acquisition computer. Due to some built-in properties of the DLL-file (which acts as a communication link between the correlator and the LabVIEW software, see appendix A), the intensity time series are saved with a varying number of appended zeros (and additional voids) each second. These were removed by using a MATLAB script (see appendix B, IntExtr.m) to read the relevant lines of data and save as a new reduced TXT-file. If clouds or other unanticipated non-astrophysical signatures were visible in the intensity series, we could, after plotting the light-curves in MATLAB or SigmaPlot, determine which of the accumulated correlation functions saved prior to abortion to pick out for further analysis.

48 4.4.2 Optical Light-Curves Among the obtained light-curves, the ones from eclipsing objects are interesting in that they provide information about system geometry and certain dimension parameters. How- ever, such estimations are somewhat outside the topic of high-speed astrophysics and will be left for later research. Quite some time was nevertheless spent on analyzing the intensity series from the eclipsing polar system SDSS 0155+00, for reasons that will become apparent.

SDSS 0155+00 Unlike ordinary cataclysmic variables (CVs), a polar (or AM Herculis) system has an accretion and rotation geometry completely shaped by the extraordinary magnetic ﬁeld strength of the primary white dwarf. Characteristic for these systems are strong linear

Figure 4.6: Magnetic cataclysmic variable with accretion onto the magnetic poles of the white dwarf (also called polar or AM Herculis object). (http://www.star.le.ac.uk/∼julo/research.html#CV) and circular polarization of emitted light, varying smoothly over the orbital period. For recent reviews of SDSS 0155+00, see Wiehahn et al. (2004) or Schmidt et al. (2004). The rotation of the white dwarf is synchronized to the orbital period due to magnetic interaction with the secondary main sequence star. Furthermore, the accretion flow from the secondary star does not form an ordinary accretion disc around the white dwarf, but is instead funnelled by the intense fields onto one (or both) of the magnetic poles (figure 4.6). A light-curve for SDSS 0155+00 obtained during measurements at Skinakas Observa- tory October 22 (figure 4.7) shows three eclipses and an orbital period of approximately 87 min. When studying the real-time display of the correlation function during this observation, we could distinguish a small but noticeable bump near a delay time of 3 s. This indication of a 3-s modulation seemed to appear in a particular region of the orbital phase, viz. the eclipse ingress, and was later “washed out”. Certainly, the oscillation feature called for further investigations, although not being even a sub-second variability. Since our intensity series is sampled at almost 24 Hz, it is definitely adequate for confirm- ing any 300 mHz oscillation using e.g. Fourier analysis or auto-correlation calculations.

49 Such oﬀ-line examination of our SDSS 0155+00 data was performed upon returning to Lund Observatory.

Light−curve of SDSS 0155+00; Skinakas 041022 [23:01] 3500

3000

2500

2000

1500

Intensity [counts/second] 1000

500

0 0 5000 10000 15000 Time [s]

Figure 4.7: Three distinct drop-oﬀs in intensity are seen when the white dwarf and accretion stream disappears behind the normal (red dwarf) companion star, in our line-of-sight. Each eclipse lasts for approximately 6.5min and the total orbital period is roughly 87min, making it the shortest-period eclipsing polar known.

The initial search for quasi-periodic oscillations (QPOs) in the extracted light-curve proceeded as follows:

1. The intensity was corrected for background light contribution. Since the moon was descending there was a general decrease in the signal prior to the correction.

2. The intensity vector was divided into segments of 2048 elements, corresponding to roughly 100 s.

3. Each segment was analyzed by calculating and plotting the power spectrum, which gave us 171 spectra to study.

Of course, this may not have been the optimal separation, thus another segmentation was employed to analyze more in detail the ingress, egress, and actual eclipse. By studying segments of 2N (where N is a positive integer) elements spanning the three ingress, egress, and eclipse phases individually, we obtained nine diﬀerent power spectra, none of which showed any signiﬁcant peaks near 0.33 Hz. We also calculated auto-correlations of our intensity data, both in segments spanning phases of the light-curve in which the 3-s oscillation feature had appeared in the measured

50 auto-correlation, as well as in 10-min segments overlapping 5 min throughout the entire acquired intensity series. In the latter we should observe any gradual development of a “bump” by visually scanning consecutive segments. After testing diﬀerent smoothing procedures in MATLAB we could distinguish a weak indication of a signal, however the most promising signal was found in a power spectrum (depicted in ﬁgure 4.8) of a 10-min segment spanning minutes 5-15 of the measured light-curve from October 22.

Power spectrum for 5−min−delayed 10−min−segm nr. 1 of the SDSS 0155+00 light−curve; Skinakas 041022 [23:01]

15 Power spectral density

0 0.1 0.2 0.3 0.4 0.5 0.6 Frequency [Hz]

Figure 4.8: Power spectrum of a 10-min segment from the eclipsing polar SDSS 0155+00 showing a quite prominent peak at 0.312 Hz.

One might suppose that a power spectrum of the SDSS0155+00 signal calculated from the measured correlation function would be useful for the investigation of alleged QPOs, however normalization issues have so far precluded this approach.

4.4.3 Correlation Functions The most informative auto- and cross-correlation functions were obtained when observing the Crab pulsar. Several acquisitions were made monitoring alternative fibre signals, and not the customary target/background signals, e.g. combinations of central (target) fibre and adjacent fibres in the hexagonal bundle, and also colour cross-correlations.

The Crab Pulsar In 1054 Chinese astronomers (and other stargazers) witnessed the birth of the NS we today call the Crab pulsar, and still the nebulous remains from the supernova explosion, some 2 kpc away, shine quite brightly. The well determined rotation period P of the pulsar is approximately 33 ms, continuously increasing by P˙ = 4 · 10−13 s/s. As mentioned earlier it is the brightest of all observed optical pulsars, and it has a characteristic two-pulse proﬁle. Owing to the strong ﬂux in the optical, all of Stokes parameters describing the polarization states of electromagnetic radiation can be measured throughout the rotation

51 period of the pulsar. However, no polarization measurements were conducted during the observation campaign, since the motorized rotating polarization ﬁlter in the OPTIMA setup was unavailable.

Auto−correlation of Crab pulsar light−curve 1.15

Pre−observation simulation of Crab pulsar auto−correlation Measured Crab pulsar auto−correlation; Skinakas 041023 [05:29]

1.1 > 2

1.05 Relative auto−correlation

0.95 −6 −5 −4 −3 −2 −1 0 10 10 10 10 10 10 10 Delay time τ [s]

Figure 4.9: Comparison a between simulated and a measured (3,000-s integration) auto-correlation of the Crab pulsar light-curve. Note that the scaling diﬀers for delay times above milliseconds, since the hardware correlator uses a “biased” normalization (i.e. doubled sampling time in consecutive segments yields lower “amplitude” of the correlation ﬂuctuations for longer delay times), while the simulation was scaled by dividing the raw correlation with Ntot − Nlag, where Ntot is the total number of data points in the light-curve and Nlag is the number of sample-interval steps for the relevant delay time.

The measured auto-correlation function of the Crab pulsar is plotted in ﬁgure 4.9, together with a computer simulated auto-correlation which was done prior to the observations. The simulated function has been rescaled to approximately correspond with the normalization of the measured function at shorter delay times. One can clearly notice the rise at the shortest time-scales, which are caused by correlated afterpulsing in the detector. Of course, some of the features hiding in this region could be astrophysical signals, however it is hard to say before having carefully compared them to calibration sources. To elucidate what we are really seeing in the correlation proﬁle, let us recall the shape of a folded Crab pulsar light-curve and compare it to the auto-correlation. At delay time τ = 0 the two correlated light-curves overlap completely, thus we see the steep decline in

52 the main- and inter-pulse fall-off as a variation (decrease) in the correlation when moving toward millisecond lag. Any smaller amplitude variation on these times-scales would show up as a fluctuation superposed on the first “central” bump or slope. The next peak in the correlation function (when moving toward longer delay times) represents the time elapsed between the main- and inter-pulse maxima, which appears at τ = 14 ms, and is shortly followed by a peak at the time corresponding to the inter-pulse→main-pulse separation. Evidently the next peak in the correlation, at τ = 33 ms, is the period of the Crab pulsar. Subsequent peaks correspond to the “higher harmonics” of the periodic signal. In multi-channel optical fibre systems there is a possibility of undesirable crosstalk between individual fibres. Such effects were studied by measuring the cross-correlation between the central target fibre (in-channel A) and an adjacent fibre (in-channel B) monitoring a field (in the nebula) close to the Crab pulsar. If the cross-correlation function revealed any sign of correlated variability between the two signals, this could indicate crosstalk interference (or some mysterious sky brightness modulation). However, also effects from light spilling outside the central fibre, due to telescope- or instrument-vibration, and into the neighbouring fibre, would inflict common fluctuations in the received light. During the observations one of the outer fibres in the hexagonal bundle seemed to exhibit the discussed effect, both in the Crab measurement and in a similar cross-correlation measurement of a calibration star in the vicinity of the Crab. Low-resolution spectral cross-correlation measurements were performed by inserting a prism spectrograph with four output channels into the OPTIMA setup. Light from the target fibre is dispersed through the prism and enters four aligned output fibres whose diameters determine the spectral bandwidth, allowing photometry in four colours. Since the dominant part of the optical radiation from the Crab pulsar is thought to originate from the same pulsar emission process we do not expect to see any significant difference between the colour cross-correlation functions and the auto-correlation of the full optical signal. However, the fact that we are correlating signals from two different detectors greatly suppresses the contributions from correlated afterpulsing at nano- and microsecond time-scales. Unfortunately, as a result of the low overall efficiency of the spectrometer (5- 10% according to Kanbach et al.), much longer integration times are required to get a strong correlation signal, thus our short (600 s) measurements of the Crab pulsar did not fully exploit this advantage of cross-correlations. However, in the cross-correlation measurements of bright sources, like Cygnus X-1, this effect was more apparent.

4.5 Results and Discussion

Overall, the joint QVANTOS/OPTIMA observations at Skinakas Observatory during two weeks in October 2004 can be considered successful, although we did not unravel any pro- found astrophysical mysteries. Of course this was neither expected nor our ambition from measurements with a 1.3-m telescope, considering the fact that extreme time-resolution requires high photon ﬂux. Our aim was instead to evaluate our high time-resolution equipment on a moderately large telescope in order to explore the potential of a QVAN- TOS/OPTIMA setup (or similar instrument) when extrapolating to larger optical telescopes. Apart from this general aspiration, we also wanted to test the developed control-

53 and data-acquisition program PhoCorr, in real observations in order to identify any over- looked weakness in the data storage chain or real-time display. An important part of the campaign was to investigate sources of error that can influence high time-resolution measurements. The issue of telescope vibrations demonstrate the need of very stable and well understood instrument structure if we expect to reduce its effects. Since everything vibrates (on some level) the concern is to minimize the amplitude of vibrations. Future ELTs will have large cross-sections, offering a large area for the wind to deposit its energy onto and as a consequence excite oscillations. Furthermore these planned large telescopes have intricate designs with complex light paths via multiple segmented mirrors, correctors, active- and adaptive-optics etc., which all can affect the temporal distribution of light. Detector noise and atmospheric scintillation are other important non-astrophysical causes for rapid variability in measured signals, which have to be considered. The immediate identification of telescope oscillation features also shed light upon the significance of on-line data processing. To discover that your calculated power spectrum is cluttered by instrumental resonance frequencies upon returning home after observation, may neither be especially amusing nor particularly time effective. Thus, the hardware correlator together with the real-time display proved very useful for immediate preliminary interpretation of acquired data. It was truly fascinating to see the characteristic auto-correlation of the Crab pulsar emerge already within the first second of a measurement, without need for accurate timing and off-line light-curve folding techniques. Moreover, it is important to note that any temporary non-periodic variation, e.g. a drifting hot spot, will not be detected in a folding procedure, whereas it can be revealed in a correlation measurement. Although it is hard to distinguish real astrophysical signatures from noise on short time-scales in the measured correlation functions, due to low SNR, we can nonetheless conclude that features do exist (see again figure 4.9), and that future large telescopes might provide us with more decisive information. One can also contemplate the potential of an observation of the weak (mV = 25.4) Geminga pulsar with a 100-m OWL telescope, compared with the Crab pulsar measurements performed with a 1.3-m telescope. This will be examined further in the next chapter of this thesis. When it comes to SDSS 0155+00 we actually did not find any hard evidence for a 3-s oscillation, neither in the calculated auto-correlation of the low-resolution intensity series nor in the computed power spectra of the intensity. Existence of high frequency QPOs are not rare in CVs (or other accreting binaries for that matter), however they are more common in the Nova (33-333 mHz) and intermediate polar type of CVs which have an accretion disc, and then often with lower frequencies (14-30 mHz) in the systems with stronger magnetic field white dwarfs (see Mauche 2004). Certainly a more detailed investigation of our observation data have to be conducted for SDSS 0155+00, as well as for all other observed objects, when more time is available.

54 Chapter 5

Simulated Observations with ELTs

There are many things one has to consider when constructing the initial “input” signal to the emulated correlator. First of all, the intrinsic flux from the astronomical light source at different wavelengths, i.e. the spectral energy distribution, and the distance to the object determines how many photons will eventually enter the Earth atmosphere (neglecting interstellar absorption). The amount of light entering the telescope aperture of course depends on the size of the aperture (considering also the obscuring optics) and is a sum of the atmospherically distorted light from the object and the overall light from the sky. By isolating the target with a size-adapted optical fibre in the focal plane of the telescope, like in OPTIMA, background contribution can be minimized. Further, wavelength dependent losses are introduced by the telescope optics guiding the light to the detectors. Finally the wavelength sensitivity of the detector decides how many of the photons hitting it will be detected and hence the strength of the signal which is to be correlated. Apart from this, different sorts of noise (astrophysical and instrumental) obviously affect the measured signal (see e.g. Dravins et al. 2000 for relevant information concerning APDs). For the initial simulations of light-curves from oscillating NSs it is not realistic to aim for a quantitative treatment since it would be almost impossible to predict absolute light fluxes considering the diverse contributions from different physical processes in the immediate vicinity of the neutron star. However a reasonable estimation of the light flux should be done and a qualitative approach using relative variations expressed in percentages would be suitable and also more realistic within the frame of the thesis. For the simulations of results for different telescope aperture sizes one could use the OWL Exposure time calculator1 to estimate fluxes from real objects and then add light from e.g. fictitious surface temperature alterations due to oscillations, which will inflict rapid light variations.

5.1 Observability of Faint Objects Using ELTs

Before we can go hunting for NS oscillations we have to locate the NS with our telescopes. It seems reasonable that one must be able to establish a signal from the NS in an imaging

1http://www.eso.org/observing/etc/doc/owl/helpowl.html

55 CCD-detector, within a sensible integration time, if any correlation measurements are to be carried out. Thus, the observability of weak light-emitting objects (like isolated NSs) with diﬀerent telescopes is essential.

5.1.1 OWL Exposure Time Calculator A useful tool in our investigation of observability of faint objects was the mentioned OWL exposure time calculator (OWL ETC), which provides estimations of the capability of telescopes with different aperture sizes and system configurations. The OWL ETC was employed to perform simulations of SNR dependence on CCD integration time for seven telescopes, including the Skinakas 1.3-m telescope, the Nordic Optical Telescope (NOT) 2.6 m, the Very Large Telescope (VLT) 8.2 m, and four ELTs with 30-, 50-, 75-, and 100-m main mirrors, respectively. Geminga was taken as an example of a weak optical emitter with a V-magnitude of mV = 25.4 and was compared to the Crab pulsar having mV = 16.8. The system configuration parameters were set to default values (check the site for more detailed information), but with zero detector read-out noise in order to make an evaluation not dependent on the number of read-outs. The central obscuration (largely equivalent to the secondary mirror size) was set to approximately 1/3 of the main mirror size in each simulation, and the sky brightness magnitude was 21.8 per square arcsecond with a variability of ±0.1 mag. OWL ETC was used to calculate the SNR over an area of diameter corresponding to an adaptive-optics corrected image, for varying integration times, and the data could be accessed in an ASCII table and saved for further analysis. Presented in the figures below are the results of the investigated SNR simulations.

5.2 Observability of P-mode Oscillating Neutron Stars

For observation of a NS comparable to Geminga in optical brightness on a 100-m telescope, the total number of photoelectrons (e−) per second produced over the PSF area in a CCD- detector (by V-band photons) is approximately 400, and the contribution from the sky background is some 0.2 ± 0.02 e−/second (computed with OWL ETC). This can be taken as coarse estimate of the minimum count-rate in a measurement with an APD-detector, although the actual value should be higher as a consequence of a somewhat better QE. Additionally, observation over the detector’s whole efficient wavelength region outside the V-filter will result in even higher count-rates. By assuming an average flux of some 4000 counts per second (and correspondingly higher sky brightness contribution), in a correlation measurement of a NS light-curve, we can make simulations to establish what requirements are needed in order to detect variability of a certain amplitude, against the background noise.

5.2.1 Simulated Correlation Functions and Power Spectra An example of the above mentioned simulations is presented below. Here a simple sine- wave with an amplitude of 10 (corresponding to 0.25% of the mean ﬂux from the object), frequency of 10 kHz (matching a typical p-mode frequency), and oﬀset of 4000, is taken

56 Figure 5.1: OWL ETC simulations of Crab pulsar observability. to represent the light-curve of the oscillating NS. Superposed on this is random Gaussian noise with a variance of 0.4, representing the sky background variability.

57 Figure 5.2: OWL ETC simulations of Geminga observability.

5.3 Results and Discussion

Comparing the simulated SNR of measurements of the Crab pulsar performed at Skinakas Observatory with the correlation functions we actually observed, can serve as an indication of what the simulated SNR for a Geminga observation on an ELT would really allow us to see in a correlation measurement. When making that evaluation we also have to remember that the calculated SNR belongs to the intensity, thus the gain in a correlation signal will

58 Figure 5.3: Comparison between Geminga and Crab pulsar observability from OWL ETC simulations. be higher by a power of two. Measuring over a wider wavelength region than just the V-band would increase the SNR further. Evidently, the construction of a 100-m OWL telescope would greatly facilitate high time-resolution optical study of faint objects. From the simpliﬁed initial simulations of p-mode oscillating neutron stars one can conclude that it would in principle be possible to identify oscillation features in correlation measurements of an isolated NS with a brightness comparable to Geminga on a 100-m

59 Simulated lightcurve of 10−kHz oscillating Neutron Star 4020

4010

4000

3990 Relative intensity 3980 0 0.005 0.01 0.015 0.02 0.025 0.03 Time t [s]

> 7 2 x 10 Auto−correlation of lightcurve 1.6001

1.6

1.5999

1.5998 −6 −5 −4 −3 −2 −1 0 10 10 10 10 10 10 10 Relative auto−correlation

4 10 Power [Hz

2 10 −2 0 2 4 10 10 10 10 Frequency [Hz]

Figure 5.4: Undamped oscillation signature in simulated NS light-curve with superposed sky background noise. telescope and an amplitude modulation of a few tenths of a percent, if we neglect instrumental noise and assume a low variability of the sky brightness. Having established a stable oscillation signal, the accuracy of the oscillation frequency determination would then principally depend on the time-resolution of our instrument in the speciﬁc delay time range. With a multiple-tau correlator unit this would actually mean lower uncertainty

60 for higher frequencies. For a more likely damped oscillation the frequency uncertainty is of course determined by the number of observed periods. Undoubtedly, these simulations should be elaborated on further for more realistic light-curves, possibly proceeding from a NS emission model, taking into account the eﬀects of general relativity and rapid rotation, and integrating over the stellar disc. From such simulations one could more correctly determine the observability of NS oscillations with ELTs, and from instrumental consid- erations estimate the accuracy of the frequency determinations, thus give a hint on how precisely stellar parameters and EOS can be derived from observations thereof.

61 Chapter 6

Summary and Conclusions

In this thesis I have assessed the prospects of high time-resolution optical observations and their application to the study of high-speed astrophysical events, and in particular neutron-star (NS) oscillations. I started off with a thorough description of what knowledge we hitherto have gained about NSs, which questions still remain unanswered, and how we could go about solving these problems. Although quite a few methods (e.g. spectral analysis of surface thermal emission) have been proposed to determine global NS parameters like mass and radius, most of them have limited precision due to lack of applicable theory and shortage of observations. As a consequence, the equation-of-state (EOS) at supranuclear densities remains uncertain, along with the internal structure and composition of the star. Perhaps a more promising method is offered by NS seismology, i.e. the probing of the NS interior by studying the frequencies of oscillations excited via events which channel extreme amounts of energy into resonance vibrations of the dense star. This implies leaving the spectroscopic techniques and instead exploring the time domains in which these frequencies reside by pushing the resolution into nano-, micro-, and millisecond time-scales. Most NS models suggest periods of non-radial oscillations in the sub-millisecond range, with typical p-modes having periods around 0.1 ms. Although possible manifestation of these oscillations in observables still remain to be comprehensively examined, signatures are supposed to be found in the electromagnetic radiation or possibly gravitational radiation emanating from the NS. Since the optical wavelength window seems most favourable, allowing high count-rates due to effective avalanche photo-diode (APD) detectors and future extremely large telescopes (ELTs), this might be the region in which we want to focus or search. While waiting for ELTs to emerge and open up new fields for exploration, e.g. astrophysical applications of quantum optics, we should obviously evaluate the potential of high time-resolution instrumentation on existing telescopes and observe interesting objects within our current brightness limit. During this thesis work, one such instrument, namely the hardware correlator Flex01-05D, played a central role. By not only offering extreme time-resolution (5 ns), but also presenting an effective solution to the problem of handling large data rates, this type of instrument promises to be vital for the future of high-speed astrophysics. Furthermore, the constructed LabVIEW control- and data

62 acquisition-program PhoCorr demonstrated the advantages of real-time graphical displays during measurements. In spite of the only moderately large 1.3-m telescope at Skinakas, the joint QVAN- TOS/OPTIMA observation campaign turned out to reveal several important issues concerning high-time resolution measurements, e.g. implications of telescope vibrations. In addition to this, the obtained correlation function of the Crab pulsar could later be used as a reference point in simulations of SNR at telescopes of diﬀerent sizes, to estimate the strength of a correlation signal from a faint object (with mV ≈ 25 mag.) in ELTs. We can conclude that a sub-microsecond study (being the practical limit in the performed observations of the Crab pulsar) of Geminga would require at least a 75-m telescope. While the observability depends largely on telescope size and integration time, the accuracy of any oscillation frequency determination principally depends on the number of periods observed in the correlation measurements, thus the damping times of the oscillations establish the limits of precision of inferred astrophysical information. However, further investigations involving detailed inspection of NS thermal emission and simulation of NS light-curves will be needed in order to more accurately predict the amplitude of an oscillation signal. We conclude that the potential of high-time resolution optical observations in our pur- suit of NS oscillations and other high-speed astrophysical phenomena seem very promising indeed, and that the implications of time domain exploration might be a deeper understanding of the physics surrounding compact objects. In addition, the future might hold answers to questions about multi-photon properties of light emission in astrophysical sources, when entering the realm of quantum optics.

63 Acknowledgements

I would like to thank Prof. Dainis Dravins for being an inspiring supervisor, an astronomy visionary, and a skilled mountain-road driver. His insightful thoughts and suggestions have been invaluable during the work of this thesis. Also at Lund Observatory, research engineer Bo Nilsson is thanked for advice and help with electronics in the optics laboratory, and at Lund Institute of Technology, examiner Ragnar Bengtsson is thanked for managing administrative issues. Big thanks of course also to Dr. Gottfried Kanbach and the rest of the MPE-group for great and successful teamwork, and for letting us take part in the observation campaign at Skinakas Observatory. Special credit goes to Fritz Schrey for cooking delicious meals, and to Alexander Stefanescu for his comments on simulated auto-correlations, and his swiftness and precision in target acquisition and telescope pointing. Naturally I would like to express my gratitude to Helena Uthas for being an excellent co-worker and cheerful office roommate. Last, but certainly not least, I am utterly grateful for the support and love of my wonderful fiancéJennie Berg who followed me home in the blistering cold when I had worked late nights at the office, as well as my family who throughout this work have been considerate and encouraging.

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68 Appendix A

Manual and User Guide for the Digital Correlator Flex01-05D and Control-Program PhoCorr

By: Ricky Nilsson & Helena Uthas

A.1 Introduction

This short manual is intended to give a basic introduction to the correlator model Flex01- 05D and the software we have developed in the LabVIEW programming environment. The correlator was purchased in 2002 from the company Correlator.com for use in the optics-lab at Lund Observatory.

A.2 Basic correlator speciﬁcations

The correlator has two connectors (in-channel A and B) for transferring standard TTL pulses from the system detectors and it can be operated in 3 correlator-modes and one photon history recorder mode.

In all the correlation modes described below an intensity series is also transferred to the host computer via USB. The number of photon pulses within the interval 0.04194304 seconds is registered and divided by the time interval to give the intensity in units of Hz.

A.2.1 Mode A: Single auto/cross correlation Calculates the auto (AxA) or cross (AxB) correlation function and transfers the normalized values for delay times from 5 ns to 23 min in 1088 channels, via a USB cable to the host computer. The ﬁrst 64 channels contain the values for delay times 5 ns (being the minimum sample time) to 64 · 5 ns, the next 32 channels contain values for delay times 64 · 5 +10 ns up to 64 · 5 + 32 · 10 ns etc, with sample time doubling every 32 channels. Thus, the delay time for the last channel of every 32-channel segment is given by

69 5(64 + 32(2 + 22 + 23 + ... + 2n)) where n is the number of the 32-channel segment. In this mode there are 32 segments, i.e. maximum delay time is 23 min. In addition to the correlation function the intensity series for channel A and B are also transferred to the host computer.

A.2.2 Mode B: Quad correlation Calculates the auto (AxA and BxB) and cross (AxB and BxA) correlation functions and transfers the normalized values for delay times from 5 ns to 23 min in 288 channels for each function. The ﬁrst 16 channels contain the values for delay times 5 ns (minimum sample time) to 16 · 5 ns, the next 8 channels contain values for delay times 16 · 5 + 10 up to 16 · 5 + 8 · 10 ns etc, with sample time doubling every 8 channels. Delay time for last channel of every 8-channel segment (total of 34 segments) is 5(16+8(2+22 +23 +...+2n)) .

A.2.3 Mode C: Dual auto/cross correlation Calculates the auto (AxA and BxB) or cross (AxB and BxA) correlation functions and transfers the normalized values for delay times from 5 ns to 183 min in 608 channels for each function. The ﬁrst 32 channels contain the values for delay times 5 ns (minimum sample time) to 32 · 5 ns, the next 16 channels contain values for delay times 32 · 5 + 10 ns up to 32·5+16·10 ns etc, with sample time doubling every 16 channels. Delay time for last channel of every 16-channel segment (total of 36 segments) is 5(32+16(2+22+23+...+2n)).

A.2.4 Photon history recorder Used as a photon history recorder the correlator transfers the time diﬀerences between successive photon events to the host computer hard drive. You can select to record the photon stream from either channel A or both channel A and B.

A.2.5 Calculating the correlation functions The correlator uses the multiple tau method to calculate high resolution correlation functions, which means that sample and delay times are varied (in this case doubled) in consecutive channel segments giving a wider temporal range of the correlation function for a single measurement. Thus the correlator uses so called MT-64, MT-32, and MT-16 schemes in mode A, B and C respectively, as described above. The number of photon counts within the sample time represents the intensity within the specific bins. By multiplying intensities of two bins differing in time by an amount equal to the delay time in question and adding the products of all the pairs of multiplied bins, a raw value of the correlation function at this delay time is calculated. The values are then normalized according to two different schemes; symmetric normalization for late delay channels and asymmetric normalization for early delay channels. Symmetric normalization: The value of the correlation function for a given sample time τ and delay time T is

70 È N · i=1 Ai Bi+T /τ

N È Corr(T )= È N N (A.1) i=1 Ai i=1 Bi+T /τ N · N

where { Ai,i = 1 to N + T/τ } and { Bi,i = 1 to N + T/τ } are the intensity series and N is the number of elements. The summation in the numerator represents the raw correlation function value. In the denominator the average intensity for A is multiplied with the average delayed intensity for B. This is not the case in the asymmetric scheme. Asymmetric normalization: Here the value of correlation function is given by

È N · i=1 Ai Bi+T /τ

N È Corr(T )= È N N (A.2) i=1 Ai i=1 Bi N · N

The result is the same as in the symmetric scheme when T/τ <

A.3 LabVIEW

LabVIEW is a graphical programming language produced by National Instruments. Two different windows are used; The front panel is the user interface and it is used for data displaying and for controlling different functions that are programmed on the block diagram. The block diagram is where these functions and objects are linked together with wires. Different tools as function palette and tool palette are used to produce the different functions. The possibilities of making programs for special applications in LabVIEW are many, but LabVIEW is mainly used for controlling an external instrument. For further information; www.labview.com

A.4 Flexx01

A couple of sample programs in VB (Visual Basic), C++ and LabVIEW were provided for programming the correlator, and for our interests, the sample program Flexx01 was a good start for programming in LabView. The following describes what this sample program consisted of and its basic functions. The program communicates with the correlator by calling specific library functions in the flex01-05d.dll-file. The functions are listed below with some explanations of what they do and which output parameters that are required.

All functions, USB Initialize, Start, Update, Stop and Free are organized in a Stacked Sequence Structure and they will execute in a speciﬁc order. The USB Update is placed in a For Loop that will execute N times before it continues with the stacked sequence

71 structure. A function Wait is placed inside the For Loop and it will wait a given number of milliseconds before the For Loop runs again.

A.4.1 USB Initialize mode A (Single correlation) = 65 mode B (Quad correlation) = 66 mode C (Dual correlation) = 67

This function is called by a Call Library function node on the block diagram and it is initialized by the input values above depending on in which mode you want to run the correlator. The input signal to the Call library function is of type numeric and data type unsigned 8-bit integer.

A.4.2 USB Start Auto = 1 and Cross = 0 (only for single and dual mode)

A Boolean (false or true) from the auto/cross button is converted to 0 or 1 and wired to the input parameter of USB Start. The value sent to this library function node will decide if the speciﬁed mode calculates the auto or cross correlations.

A.4.3 USB Update To get correlation functions and intensities you need to call the function USB Update. There are some parameters of diﬀerent data type corresponding to the diﬀerent outputs of the function; elapsed time, trace count, intensity trace A and B and the correlation functions.

• Elapsed Time starts counting the time as soon as USB Start is called. It is of numeric type and the data type, 4-byte single.

• Trace Count counts the number of new trace values. The number depends on the speciﬁed number of milliseconds in the function Wait. It is of numeric type and data type unsigned 16-bit integer.

• Correlation function 1-4, intensity trace A and trace B are of array type and data type 4-byte single. The arrays which are going to contain the correlation functions, trace A and trace B are created and initialized with zeros and sent as input parameters to USB Update. The corresponding output parameters are pointers to the diﬀerent arrays and these are sent to an indicator display. The dimension size of the array for the diﬀerent parameters are;

– Correlation 1: 1088 elements

72 – Correlation 2: 608 elements

– Correlation 3 and 4: 288 elements each

– Trace A and B must have array sizes at least equal to the trace count which depends on the time set in the Wait function in the For Loop.

In single mode correlation 2, 3 and 4 are ignored and in dual mode correlation 3 and 4 are ignored.

A.4.4 USB Stop This function will stop the signal.

A.4.5 USB Free Is called to clean up before exit.

A.5 PhoCorr

Based on the sample program Flexx01 which was designed only to run in single mode and without any graphical representation of the measured correlation functions and intensity series, we have developed a more flexible LabVIEW program which can run the correlator in all the specified modes and display correlation functions and intensity series in graphs. The program also saves the obtained data in a txt-file for further analysis e.g. calculation of power spectra. For the Photon History Recorder mode a separate program, called Photon, is used since the DLL-file (flex01-05d.dll) written for use with LabVIEW lacks this function.

A.5.1 Interpreting the block diagram For possible future modiﬁcation of the program and easier interpretation of the block diagram a short description of the latter is presented in the following section. The basic structure is as mentioned earlier, a Stacked Sequence, and the functions in the diﬀerent sequences will therefore be presented separately. For further information and detailed explanations of how to program in LabVIEW the user is referred to www.labview.com or other online LabVIEW tutorials.

Sequence 0: The correlator is initialized with the corresponding value of the chosen mode. This value is also passed on to the next sequence.

Sequence 1: Contains the main part of the program code enclosed by a While Loop which runs until elapsed time reaches the speciﬁed run time. The wait function is set to 1000 ms. During this time the correlator gathers intensity trace values and calculates correlation functions which are transferred to the computer when USB Update is called in the next iteration. Since it takes some time to run the subdiagram the actual run time might

73 exceed the specified with a few tenths of a second as the final iteration finishes. Initially the Library function USB Start is called (outside the loop) and it starts the correlator in dual or cross mode. The 8 output parameters from USB Update are as mentioned; elapsed time, trace count, correlation function 1 (corr1), corr2, corr3, corr4, intensity trace A and intensity trace B, in that order. Data from these parameters are wired to different subVIs which are subprograms (VI-files just like the main VI PhoCorr.vi, Figure A.3) saved in the same folder as the main VI. Each subVI is shown as a connector pane with a set of input and output terminals that correspond to the controls and indicators of that VI. The intensity traces A and B are sent to intensityAsub.vi and intensityBsub.vi respectively. These subVIs (Figure A.1) also include a function that saves the intensity trace values in files named intensityA.txt and intensityB.txt. Feedback nodes are employed to transfer new trace values from one iteration to the next. The new trace values are appended to another array which grows in size with the added elements each iteration. Array indices are translated to elapsed time in the graph axis settings in the mode subVI that receives the intensity data. Singlesub.vi, Dualsub.vi and Quadsub.vi are placed inside separate Case Structures and are thus executed and launched as separate front panels only if the selected mode number corresponds to 65 (in case of Single), 66 (in case of Quad) or 67 (in case of Dual). In the block diagram of Singlesub.vi (Figure A.2) the arrays of intensity values for channel A and B are wired to waveform graphs which are displayed in the front panel. The array containing correlation function values are sent to yet another subVI, singlecor- rfunc1sub.vi, in which the channel index is replaced with the corresponding delay time values as x-values and sent to a XY-graph. In this subVI the values of the correlation function are also saved to a file named singlecorrfunc1.txt. Besides the correlation function and intensities, the elapsed time and number of new trace values are wired to numeric indicators and displayed on the front panel of the subVI. The subVIs for Dual and Quad mode are built up in a similar way, but obviously with a different number of correlation functions and related delay times.

Sequence 2: The correlator is stopped.

Sequence 3: The correlator is “cleaned up” and the latest channel values are removed.

A.5.2 How to use PhoCorr When starting the program PhoCorr a front panel containing a short list of how to run the program is shown. Start by choosing mode from the list (single, quad or dual) and then choose auto or cross correlation. Specify run time and then press the white arrow (run button) in the subVI menu bar. A new window opens that show correlation functions and intensity traces. In the left frame, elapsed time, number of new trace values and amplitudes for intensity traces and correlation functions are shown. Change the plot range by writing new values directly into the graph or use the Graph Palette, on top of the graphs, to activate the zoom function. Make sure you have disabled this function by pressing the zoom button again when ﬁnished resizing. Pressing the white hand on the

74 Graph Palette makes it possible to move the graph. On top of the correlation functions is a Cursor Legend. Specific graph positions can be marked and the exact x and y values can be acquired. Choose Bring to center or Go to cursor in the cursor menu to make a cursor. The first brings the cursor to center position of the graph and the second moves the graph to the cursor. The cursor can be moved when the cross button is chosen in Graph Palette or by clicking up/down/left/right in the directional button (a white and blue square). Cursor style as colour and linewidth can be chosen in the menu. Lock the cursor to a specific position by choosing Lock to plot in menu. To get multiple cursors change to edit mode in the subVI menu Operate >> Change to Edit mode and choose Show Tools Palette from Window. A toolbox will appear. Use the arrow to enlarge the cursor window, and more Cursor Legends will appear depending on window size. Change to run mode again when finished, Operate >> Change to Run mode. To choose visible items such as Plot Legend, x-scrollbar or autoscaling, right click on Graph palette and a menu will appear. Interrupt the running process by pressing the red button situated on the subVI menu bar. Remember to rename the saved files after each execution to keep them from being overwritten when the program is run again. Alternatively change the save settings in the Write LVM File function on the specific subVI block diagram by clicking Operate >> Change to Run mode and then double click on the function icon.

75 Bibliography

[1] Klaus Sch¨atzel, Single Photon Correlation Techniques, Dynamic Light Scattering: The method and some applications, edited by Wyn Brown, Clarendon Press, Oxford, P 76, 1993.

[2] Klaus Sch¨atzel et., Noise in Multiple-Tau Photon Correlation Data, SPIE Vol. 1430, P109, Photon Correlation Spectroscopy: Multicomponent Systems, 1991.

[3] Klaus Sch¨atzel, New Concept in Correlator Design, Inst. Phys. Conf. Ser. No. 77, P175, 1985.

[4] Klaus Sch¨atzel et., Photon Correlation Measurements at Large Lag Times, Journal of Modern Optics, Vol. 35, No. 4, P711, 1988.

[5] Getting started with your new Flex01-05D USB based Digital correlator, Correla- tor.com, 2004

76 A.6 Appendix

The ﬁgures referred to in the text are presented on the following pages.

Figure 1 : Block diagram of intensity A subVI

Figure 2 : Block diagram of single correlation subVI

Figure 3 : Main block diagram

Figure 4 : Front panel of single cross correlation run

77 Figure A.1: Block diagram of intensity A subVI

78 Figure A.2: Block diagram of single correlation subVI

79 Figure A.3: Main block diagram

80 Figure A.4: Front panel of single cross correlation run

81 Appendix B

MATLAB Scripts

82 83 84 85 86 87 88 89