University of BielefeldFaculty of Physics

BACHELOR THESIS

Scintillation studies of PSR B0809+74

Hauke Jung

November 12, 2014

Supervisor and first referee: JProf. Dr. Joris VerbiestSecond referee: Dr. Stefan Os lowski

Contents

1 Introduction & Theory 21.1 Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 PSR B0809+74 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Pulsar Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 LOFAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Interstellar scintillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Description & Techniques 112.1 PSRCHIVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 RFI zapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Standard templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Times of Arrival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 TEMPO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Dynamic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 Secondary spectrum and scintillation arcs . . . . . . . . . . . . . . . . . 16

3 Analysis & Results 173.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 DM variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Scintillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Conclusion & Outlook 23

1

Chapter 1

Introduction & Theory

In the last ten years, a massive resurgence in low-frequency radio astronomy has come about.With telescopes like LOFAR (the LOw Frequency ARray), the LWA (Long Wavelength Array)and the MWA (Murchison Widefield Array) coming into operation, low-frequency radio astron-omy research is enabled at far higher sensitivity than ever before. One key area of interest,is that of pulsar astronomy and in particular the study of pulsar emission at low frequencies.This is particularly beneficial for studies of the ionised interstellar gas, since effects from thisionised medium on the radio pulses of pulsars, are most powerful at the lowest frequencies. Inthis thesis we investigate how LOFAR data can be used to investigate pulsar scintillation inthe bright and nearby pulsar PSR B0809+74 (J0814+7429). In the following, we will brieflyintroduce pulsars in general, their origin (Section 1.1 and PSR B0809+74 in specific (Section1.2), explain the basics of the LOFAR telescope (Section 1.4) and finally describe how interstel-lar scintillation affects pulsar timing (Section 1.3) and pulsar radiation at LOFAR frequencies(Section 1.5).Software dedicated to pulsar analysis made research remarkably easier. Two important stan-dard software packages were used in this thesis and will be introduced in sections 2.1 and 2.2along with a description on data processing as well as dynamic and secondary spectra (Section2.3).Chapter 3 will provide the results and analysis of the work and chapter 4 gives a short summaryand an outlook on possible future work with this pulsar.

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1.1 Supernovae

A massive (more than 8M�1) star has nearly finished its life when it doesn’t have enough

hydrogen left to keep up the hydrogen fusion process. The helium atoms resulting from thefusion process then will fuse and this process is continued until the resulting fusion product isiron. No energy can be gained by fusing iron atoms. Therefore the fusion process stops andgravity makes the core collapse.It will continue to collapse until the electron degeneracy pressure doesn’t allow further collapseto happen. The falling outer layers get shocked by the sudden halt and cause a rebound whichis observable to us as a nebula.During the explosion the brightness is rises up to a billion times compared to the regular bright-ness of the star, the outer shells of the star are blown away and the only remaining part is thecore.

For core masses beyond the Chandrasekhar mass (≈ 1.39M�) electron degeneracy pressure isinsufficient to sustain the core and it will further collapse until electron degeneracy pressure islarge enough to prevent a further collapse. In this case a neutron star is formed.For masses above the Tolman–Oppenheimer–Volkoff limit (ill-defined: 1.5 - 3M�) even electrondegeneracy pressure provides insufficient support the core and it will further collapses into ablack hole.These values hold for cores before they go supernova. At the end neutron stars will have massesbetween 1 and 2M�. The difference is the

1.2 Pulsars

In this section we will give a short historical note on the discovery of pulsars and an introductionto pulsars and their properties.

1.2.1 Discovery

In 1967, Jocelyn Bell discovered a highly periodic signal while searching the sky for radiosources. She and her doctoral advisor Anthony Hewish first thought this could be artificial,sent out by other civilizations. Therefore they first named this radio source LGM1 (LittleGreen Men 1). They published their results in the Nature magazine (”Observation of a RapidlyPulsating Radio Source”, Hewish et al., 1968).Pulsars are radio sources and due to their strong magnetic field, they continuously emit radiowaves from the magnetic poles. As the pulsar also rotates around its rotation axis which differsfrom the magnetic axis, we see an effect that is best described as a lighthouse in the sense thatthe continuous emission is seen as a series of pulses.”Pulsar” is the short form for pulsating source of radio emission. The star itself doesn’t pulsate,but the signal we see is pulsating.Pulsars were found to be fast rotating neutron stars. These have been predicted by Baade andZwicky (1934) but were not expected to be observable.The rapid rotation of pulsars is explained by the huge but slowly rotating progenitor star.Conservation of angular momentum leads to a rapidly rotating pulsar if the size decreases.

1solar masses, the mass of the sun

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1.2.2 Basic properties

Figure 1.1: P − P diagram (Source: Lorimer andKramer, 2005)

In Figure 1.1 we can see the pulseperiod derivative vs. the pulse periodderivative.The first thing to notice is thatthere are two groups of points inthe figure. The second thing isthe range of the y-axis which goesfrom approximately 10−22 to 10−10.This means, a typical pulse pe-riod derivative P for the groupat the lower left is 10−20 and forthe other group 10−15. These twomain groups correspond to ”nor-mal” (slow) pulsars at the upperright and fast spinning millisecondpulsars (MSPs) at the lower left.Normal pulsars have a spin pe-riod (the time they need to doa complete spin around its rota-tional axis) of several seconds downto 30 ms. If pulsars have a com-panion, they can accrete matterfrom it. This increases their ro-tational momentum and thereforethey accelerate their spin. Pul-sars which have gone through thisscenario are called MSPs and havepulse periods down to approximately1 ms.

The life of a newborn pulsar begins at the upper left of the diagram and will move downwardsaccording to the diagonal lines denoting the age. Normal pulsars have a typical age of 10 millionyears. The emission of magnetic dipole radiation causes the pulsar to lose energy. Consequentlythe pulsar slows down (pulse period is getting longer), represented by a shift to the right in thediagram. When the spin period gets too long, the emission process becomes inefficient, so thatwe can no longer detect the pulsar and it is declared radio-quiet. In the diagram it falls intothe gray ”graveyard” area.When a companion recycles a pulsar it moves from the graveyard to the lower left group ofMSPs. In the process of recycling its spin increases and its magnetic field gets weaker, causingless dipole radiation (i.e. lower P - the period gets more constant).Since their discovery, continuous observations across a wide range of observing frequencies haveuncovered a vast amount of information about pulsars. Consequently, we now know that theirmasses cover a much wider range than expected from the work by Baade and Zwicky (1934)from less than 1.17M� (Janssen et al., 2008) at the low end, to more than 2.01M� (Antoniadiset al., 2013) at the high end. Based on these values and models for equations of state forextremely dense matter (Lattimer and Prakash, 2004), the radii of these objects are predictedto be of the order of 10 km. Finally, while still primarily studied at radio frequencies, pulsarshave now been detected across the entire electromagnetic spectrum

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1.2.3 PSR B0809+74

This section gives a short overview of the basic properties of this special pulsar. Specifically,we will explain why this pulsar is a good choice for scintillation studies.

PSR B0809+74 is a nearby (≈ 430pc) and very bright2 pulsar. Therefore it is easy to detect.It is a ”normal” pulsar with a rotational period2 of 1.29 s and moves with a transverse velocity2

of 102.67 km/s with respect to the solar system.With a declination of +74° it is circumpolar in Europe. That means that a continuous 24-hourobservation is possible. Most pulsars are in the plane of the galaxy while PSR B0809+74 isnot. The pulsar is nearby, but outside of the so-called ”local bubble” which is a cavity inthe interstellar medium (ISM) where the density of dust and electrons is very low. Our SolarSystem resides in the local bubble.If we now observe this pulsar we could expect to see scintillation right from the edge of thelocal bubble. Most pulsars have multiple peaks in their profile and they differ from pulse topulse. In contrast the peaks of PSR B0809+74 have a regular pattern and are constant inwidth and height (van Leeuwen et al., 2003). In Rickett et al. (2000) the authors also deal withthe scintillation of PSR B0809+74 exclusively. They try to find out where the scattering of thewaves that causes the scintillation happens.

Nulling & drifting

Figure 1.2: An example for nulling and driftingin our observation.

PSR B0809+74 has features like nullingand drifting which are interesting to in-vestigate and are briefly explained be-low.

Nulling is a sudden lack of a signal for oneor more pulse periods. An instrumental fail-ure is not the cause of this. The pulsar stopsemitting and after a short time it starts again.In the case of PSR B0809+74 the nulling frac-tion (percentage of nulls) is 1.4% (van Leeuwenet al., 2003).Drifting is a periodic change in arrivaltime of the pulse. For a characteris-tic time (called P3) the pulse is comingearlier or later each pulse. After thatthe pulse comes again at the time whereit started drifting before. This is re-peated indefinitely. Many normal pulsarsare nulling but the nulling fraction is dif-ferent for each pulsar. Some are notnulling at all whereas some only ”turnon” for a short time (i.e. they aremostly nulling). Pulsars that fall intothe last category are called rotating radiotransients (RRATS, see McLaughlin et al.,2006)

2see ATNF Pulsar Catalogue at http://www.atnf.csiro.au/

5

An example of both nulling and drifting is shown in Figure 1.2. The null can be seen approxi-mately at the edge of the lower third of the figure. The drifting is the slope of the pulse fromthe bottom right to the top left, repeated multiple times.

Carousel model

The question about the cause for drifting pulses leads to the carousel model. It currently is themost favored model to explain the drifting phenomenon.In Figure 1.3 we can see an illustration of the emission geometry that is proposed for thecarousel model.It assumes that we don’t have one compact beam which is emitted from one single place butinstead there are multiple subbeams emitted from different regions. Theses are depicted ascircles in the figure. Adding to that the emitting regions aren’t static but moving in a circlelike a carousel does.Each drifting band (length of P3 as depicted in the figure) is from the same subbeam but comesat a slightly different phase every pulse period because the carousel has shifted slightly.Rankin and Rosen (2014) recently confirmed the carousel model is good in explaining the drift-ing behavior for PSR B0809+74. Unfortunately it is a phenomenological model which doesn’tgive any physical insight.

Figure 1.3: Illustration of the carousel mechanism. Source: van Leeuwen et al. (2003)

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1.3 Pulsar Timing

Pulsar timing is the process obtaining a pulsar’s accurate measurements of its period and otherintrinsic properties through repeated observations.A model for a pulsar includes multiple properties that affect the arrival times of its pulses onearth. The first property is derived by initially pointing the telescope at the pulsar. From therewe already get the right ascension and declination. By watching the pulses for some time wecan get the periodicity and from their frequency dependence the dispersion measure (which willbe discussed in section 3.2.1).

If we have such a ”timing” model we can calculate what the model predicts for when the pulsesshould arrive and compare the prediction to the actual observations. Figure 1.4 shows an il-lustration of how this could possibly look like, if the pulse period is wrong. The actual pulses(in red) come later than expected. The difference between predicted and actual pulse is calleda timing residual.The linearly increasing residuals shown in the illustration describe a wrong period in the model.Every wrongly described parameter in the model results in its own characteristic timing signa-ture.

1.4 LOFAR

Figure 1.4: Illustration of how a deviation from the modellooks in case the pulse period is incorrect in the timing model.

LOFAR is the low frequencyarray. As the name already sug-gests, it is an array of low fre-quency antennas. The projectconsists of multiple stations inEurope. The majority of thestations are in the Netherlands.It consist of several hundredsof immovable antennas detect-ing the radio waves at the verylowest frequencies detectable from the earth right before the ”atmospheric cut-off” as seen inFigure 1.5.

Stations have low-band antennas (LBA) which operate from 10-80 MHz (30-3.7 m wavelength)and high-band antennas (HBA) operating from 110-240 MHz (2.7-1.2 m). This means that theLOFAR array is placed right of the telescope depicted in Figure 1.5.LOFAR can be used to observe with all stations simultaneously, with a subset of all stations orjust with one single station. The data used in this thesis were taken with the station locatedin Effelsberg, Germany.

All antennas are static, therefore they wouldn’t be able to track objects in the common waywhere telescopes are moved. Instead LOFAR is a digital telescope. This means, if the wavearrives at the station coming from an angle not perpendicular to the surface, the wave arrivesslightly later at the far end of the antenna field.Using this difference, the incoming wave can be combined from all antennas in phase, whicheffectively ”points” the telescope to the source. For pulsar observations this is how LOFARis used and so it essentially does not matter where the telescopes are placed. The resolutionof LOFAR is dependent on the longest baseline between LOFAR antennas. This is the reasonsome LOFAR stations are located very far from the rest of the LOFAR network (as far as

7

central France, southern Sweden and soon Ireland and Poland). Further details on the LOFARsystem are given by van Haarlem et al. (2013).

Gamma rays, X-rays and ultraviolet

light blocked by the upper atmosphere

(best observed from space).

Visible light

observable

from Earth,

with some

atmospheric

distortion.

Most of the

infrared spectrum

absorbed by

atmospheric

gasses (best

observed

from space).

Radio waves observable

from Earth.

Long-wavelength

radio waves

blocked.

0 %

50 %

100 %

0.1 nm 1 nm 10 nm 100 nm 1 µm 10 µm 100 µm 1 mm 1 cm 10 cm 1 m 10 m 100 m 1 km

Wavelength

Atm

ospheric

opacity

Figure 1.5: ”Atmospheric electromagnetic opacity” by NASA (original); vector-ized by User:Mysid; Source: http://commons.wikimedia.org/wiki/File:Atmospheric_

electromagnetic_opacity.svg

Figure 1.6: Low band (left, ©W. Reich, MPIfR Bonn) and high band (right, ©ASTRONDwingeloo ) LOFAR dipole antenna.

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1.5 Interstellar scintillation

Interstellar scintillation is an effect that causes variations in the measured intensity of electro-magnetic waves (e.g. radio waves coming from pulsars) at the telescope. It is similar to thetwinkling of the stars which is observable in the night sky. While the scintillation of (visible)stars is caused by air turbulence in the atmosphere, the interstellar scintillation of radio sourcesis caused by the interstellar medium between Earth and the radio source.From basic physics we know the concept of diffraction & interference: a point source emits awave and the wave is sent out isotropically (uniformly in all directions). If the wave then hitsan object it will be diffracted. The diffracted wave can then interfere with a wave coming fromthe point source or other density inhomogeneities. If we then measure the intensity at a lineperpendicular to the direction in which the wave traveled we can see an interference pattern.This concept can be applied to interstellar scintillation. Pulsars are very small and thereforecan be treated as point sources. The radio waves coming from their magnetic poles traversespace and at some point they will interact with the free electrons of the ionized interstellarmedium. The waves will be diffracted and then can interfere with each other at the observeron Earth. Since the Earth moves through space we can observe the scintillation pattern.

Because the ISM is responsible for the interference pattern, we can study the ISM due to thisdiffraction. As the line of sight (the line connecting the pulsar and the observer at Earth)sweeps through the inhomogeneous and turbulent ISM, the interference pattern changes.

The inhomogeneous ISM is often approximated by a thin-screen model. The large scale of theISM can theoretically be condensed into a thin wall. Then only this thin screen is responsiblefor the interference. This model (contrary to what one might believe) is able to explain thebasic effects of the observed scintillation.Depending on the scale of the distance between two interfering waves we can distinguish twodifferent types of scintillation: diffractive (DISS) and refractive (RISS) interstellar scintillation.DISS takes place if closely spaced waves scattered from the thin screen interfere with each otherand create an interference pattern with small maxima (also called ”scintles”) and minima atthe observer (also seen in Figure 1.7).RISS corresponds to larger scales in the scattering screen. This causes long term intensitymodulations.

Figure 1.7: Illustration of the thin screen model and scintillation (Source: Lorimer and Kramer,2005, after Cordes, 2002).

9

The free electrons in the ISM also delay the waves in a strongly frequency-dependent way. Thisphysical principle is called dispersion and says, that the group velocity of a wave is dependent ofthe wavelength (and therefore the frequency2). See section 3.2.1 for a more detailed descriptionof this particular phenomenon.

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Chapter 2

Description & Techniques

In this chapter we discuss the processing techniques and software that were used for the dataanalysis. In particular, we describe the PSRCHIVE package (Section 2.1) which was used fordata optimization (2.1.1) and generation of pulse-arrival-times (2.1.2 and 2.1.3); the TEMPO2package (Section 2.2) which was used to derive the derived pulse-arrival-times; and MatLab(Section 2.3.2) which was used for investigation of the scintillation properties of the pulsar.

2.1 PSRCHIVE

PSRCHIVE1 is an open-source package with the sole purpose of providing tools and algorithmsfor pulsar data analysis. It consist of several command-line programs as well as graphicalinterfaces. The parts of the package that were relevant to our work will be briefly explained inthe following sections.An introduction to PSRCHIVE can be found in van Straten et al. (2012).

2.1.1 RFI zapping

As LOFAR is designed to observe at low frequencies, there is a high chance that it also detectsradio broadcasts, military radar and other types of artificial, man-made radio waves which areemitted near the telescope.These signals are orders of magnitude higher in intensity than the astronomical signals andtherefore they can outshine the pulsars. Depending on the signal we can remove these disturbingsignals by deleting or zero-weighting different chunks of data.If there is a continuous stream of radio-frequency interference (RFI) at a well-defined frequency,this can be easily dealt with by ”zapping” (i.e. removing) the relevant frequency channel. Anexample of this kind of RFI is shown in Figure 2.1.Another possibility is an impulsive signal consisting of small bursts which may be radar. Othersources can be satellite or point-to-point communication. All of this RFI needs to be removed aswell as possible. This is done by zapping scripts and manually with the ”pazi” program whichis an interactive program that displays either frequency vs. pulse phase or time vs. pulse phase.The data files are stored with information about frequency, time (pulse phase), polarizationand intensity. To be able to plot this information in a 2D graph the data needs to be scrunched(i.e. averaged over the dimension that is not plotted).A screenshot of pazi is shown in Figure 2.1. Here we see the frequency vs. pulse phase view andtwo types of RFI. One is seen at 121.5 MHz and slightly below. Here we see, that one completefrequency channel is distorted. 121.5 MHz is the international aircraft emergency frequency.

1http://psrchive.sourceforge.net/

11

Apparently there was an emergency call during a short time of this observation. We can easilydelete that particular frequency channel and keep the rest of the data. As noted in the lastparagraph in section 1.5, radio waves are dispersed. Waves with higher frequency arrive earlierthan those with lower frequency. PSRCHIVE can correct for these inter-channel delays andalign the pulses (and at will disperse the pulses again). If we now have a short burst of RFIthis would result in a straight vertical line. But as we correct for dispersion by shifting thefrequency channels, the former straight line becomes a parabolic line. The sloped lines in thefigure are this kind of short, impulsive RFI. Sometimes there are multiple lines and it’s notfeasible to get rid of all of them. In those cases we only deleted the RFI with highest intensity.

Figure 2.1: Example of two different kinds of RFI, displayed in pazi.

2.1.2 Standard templates

After the RFI is finally zapped we can create a so-called ”standard template”. It is an analyticmodel of the average pulse shape. The tool ”paas” from PSRCHIVE can create such a model byinteractively fitting multiple von Mises functions to the pulse shape. Using standard templatesis a quite new approach. The method commonly used before was to use the best observation asa template to time against. By doing this the best observation needs to be excluded from thedataset. To exclude the best observation is not favorable. The standard template containing acomposition of multiple von Mises functions is beneficial here.In Figure 2.2 we see a stacked, aligned and color-coded view of all four templates with the114 MHz-template in black, 137 MHz in red, 161 MHz in green and 185 MHz in blue. It isvisible that the pulse shape changes slightly with frequency. This may be due to interstellarscattering or it can be intrinsic to the pulsar.

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Figure 2.2: Templates aligned and colored to see differences.

2.1.3 Times of Arrival

Now that we have the RFI-zapped data and the analytic model we can do the pulsar timingdescribed in section 1.3. First we use the standard template to determine the ToAs (Times ofArrival). These are the times at which the different pulses arrive at the LOFAR-station. Nor-mally, multiple pulses get averaged before an ”averaged” ToA is derived, but in case of PSRB0809+74 individual pulses were possible to use because it is very bright. The command ”pat”from PSRCHIVE creates a ”tim”-file containing the ToAs. Now, if we measure at two differenttimes, the radio wave will reach us at two different points in time. Considering that the Earthmoves around the Sun, the waves reach us at a different point in time than if the earth wouldstand still. This would affect the ToAs. A conversion to a barycentric arrival time (which is thetime at which the wave would arrive at the barycenter2 of the Solar System), is needed. This iscalled the Roemer delay correction and is the biggest correction needed to be made. This is animportant step when analyzing pulsar data. Other steps include corrections for offsets betweenthe telescope clock and the international time standard, corrections for the rotation of the Earth(and the Doppler shift caused by this) and corrections for space-time distortions induced by themajor planets (mostly Jupiter). The data is prepared now and can be analyzed with TEMPO2.

2center of mass

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2.2 TEMPO2

TEMPO2 3 is a software package designated for pulsar timing. It was mainly developed byGeorge Hobbs & Russell Edwards and significantly enhanced by Joris Verbiest and variouspulsar astronomers subsequently. It is based on the TEMPO software package and is written inC/C++. Designed to be expandable through plugins it is now used as the global standard. Ifno plugins are loaded it will output values of variables characteristic to pulsar (e.g. rotationalfrequency and derivative, dispersion measure, eccentricity) as well as statistical values of thefit algorithms. These values are derived from the input ToAs and timing model. Output valuescan be altered and expanded by the user to include more information. A very important pluginis called ”plk” and provides the possibility to plot the ToAs with uncertainties as a functionof different variables (e.g. frequency, time, position angle) as well as fitting the timing modelparameters to the residuals. An example image of what the plk plugin looks like is shown inFigure 2.3. The plottable variables for the x- and y-axes are at the left side and the timingmodel variables that can be fitted are at the top.

Figure 2.3: Example (color inverted) image of the TEMPO2 plugin ”plk”.

3http://www.atnf.csiro.au/research/pulsar/tempo2/, Hobbs, Edwards, and Manchester (2006)

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2.3 Spectral Analysis

In this section we will describe what the dynamic and secondary spectra are and show howthey were investigated.

2.3.1 Dynamic Spectrum

The dynamic spectrum is a frequency vs. time plot in which you can see the frequency spectrumfor each subintegration4. That means, in this plot we see how intense the received radio wavesare at a given frequency and time. If we have scintillation or other changes in the ISM weare able to see these in the dynamic spectrum in the form of intensity fluctuations. In thedynamic spectrum we are able to see so-called ’scintles’ if scintillation is present. Scintles areareas of higher intensity. The goal of this thesis is to measure the size of these scintles and toinvestigate how they change with frequency. Naively one could suggest to just measure the sizee.g. by counting the pixels of the image. But there is a more sophisticated quantitative methoddescribed by Cordes et al. (1986) and Rickett (1990).At first, a 2D autocorrelation of the dynamic spectrum is generated. Then we take a slice atthe middle frequency and middle time-step. We now have two functions and we will take thehalf-width at half-maximum for the time-slice and the half-width at 1

efor the frequency-slice.

The value for the time-slice corresponds to the decorrelation bandwidth (size of the scintles inthe frequency dimension) and the value for the frequency slice corresponds to the decorrelationtime or scattering broadening time (size of the scintles in the time dimension).

Autocorrelation function

The autocorrelation function is a function which shows a measure of the correlation of a functionor signal with itself. It is created by shifting the signal by a value τ and then integrating overthe product of the original signal and its shifted version. The value τ is called the time lag.Because the dynamic spectrum is a matrix of intensities and not a one-dimensional functionof intensities, we need to do a 2D-autocorrelation. In practice this can be detained using theConvolution theorem and the Wiener-Khinchin-Theorem, resulting in:

ACF = abs {(fftshift(ifft2 [(fft2(M) · conj(fft2(M))]))} /(n ·m) (2.1)

where M is the matrix of intensity values, n and m are this dimensions of the matrix, ifft2 isthe two-dimensional inverse fourier transformation, conj(fft2(M)) is the conjugate of the fouriertransformation and fftshift shifts the center lags to the middle of the matrix.

We then want to fit the theoretical predicted functions to our derived ACFs. The functions are(following Coles et al. (2010) and Lorimer and Kramer (2005)):

ρ(τ) = exp

(−(

τ

τ0

)53

)(2.2)

ρ(ν) = exp

(− ln 2(

ν

ν0

)

)(2.3)

where τ0 and ν0 are the decorrelation time and bandwidth.

4A subint essentially means averaging of multiple pulse. While this is needed for many pulsars to be able toactually see the pulse, it was not necessary for this thesis because PSR B0809+74 is so bright that we can seeindividual pulses.

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2.3.2 Secondary spectrum and scintillation arcs

A secondary spectrum is the power spectrum of the dynamic spectrum. When investigatingthe secondary spectrum Stinebring et al. (2001) found that sometimes they contain parabolicfeatures. These ”scintillation arcs” correspond to different scattering screens along the line ofsight.An example of scintillation arcs are shown in Figure 2.4. It shows a dynamic spectrum atthe top and the corresponding secondary spectrum at the bottom. Scintillation arcs can beseen in the secondary spectrum as two symmetric features drifting away from zero conjugatefrequency. We indicated them with the red line. They change with time as the line of sighttraverses the ISM. Depending on how inhomogeneous the ISM is and how fast the pulsar movesthey can change on timescales from minutes to hours. It should be noted that the intensitycolor gradient is on a logarithmic scale. This already is a hint that scintillation arcs are noteasy to detect.

Figure 2.4: Example dynamic (left) and secondary spectrum (right). Original figures takenfrom Stinebring et al. (2001).

Matlab

For the calculation of the autocorrelation functions and the secondary spectrum a matlabscript originally written by Bill Coles for different types of data, was slightly adapted to ourdata format to compute the secondary spectrum.

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Chapter 3

Analysis & Results

3.1 Data

For this thesis we had a continuous 24-hour set of data files from September 2013. It wasobserved at the LOFAR station in Effelsberg with the higher frequency band. Which meanswe observed at the spectrum of 110-240 MHz.The data was already pre-processed. This means, the data we got included files each having500 subints split into four different frequency bands. They were centered approximately at 114;137; 161; and 185 MHz. The first thing we then did was deleting as much RFI as possible likedescribed in section 2.1.1. Then we concatenated the whole observation into a single file foreach four frequency bands and created four standard templates. For further progression, thefour standard templates needed to be aligned so that we have one standard template for thewhole observation. This can be done with the tool ”paas” from PSRCHIVE.

3.2 Timing

We used the process described in section 2.1.2, 2.1.3 and 2.2 to derive ToAs which were usedto get a dispersion measure and align the pulses for the whole observation.

3.2.1 DM variations

Waves that are scattered by the ISM arrive later at lower frequency. The time delay can beinferred through

∆t = D · DMf 2

where the Dispersion Measure, DM , is defined as

DM =

∫ l

0

ne · dl

and the Dispersion constant, D, equals

D =e2

2πmec.

The DM value has multiple uses. If we have a DM value from observing a pulsar and calculatingthe time delay between two frequencies, then assuming we have a model of the free electrondistribution for the galaxy we can calculate ne and furthermore we can calculate the distancel to the pulsar.

17

If we have a DM value and measured the distance e.g. by the parallax method we can updateour model for the free electron distribution.We measured the DM based on the time delay across our observing bandwidth at 15-minute

5.7

5.72

5.74

5.76

5.78

5.8

5.82

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

DM

[pc c

m]

Time [MJD-56548.4]

DataData without the lower half 114MHz band

Figure 3.1: DM variations across the 24h data.

intervals, shown in Figure 3.1. The intent for this figure was to see if the DM value changesover time or is constant. Another reason was to check if the DM value is correct.The original value is derived from the ATNF catalogue and we saw that it was incorrect. Theeffect is that pulses were folded too early at lower frequencies which is also visible in Figure 2.1where the pulses drift to the left side at lower frequencies.Also obvious are some significant changes in the DM value on short time scales between MJDs56548.5 and 56548.6, as well as near MJD 56548.8 and MJD 56549.1. These correspond torapid changes in the ISM.Because the sensitivity is lower at the frequency band edges, the RFI plays a bigger role. TheRFI in the 114 MHz band affected the DM values extraordinarily (DM value is proportionalto f−2 - so lower frequencies affect the value more than higher frequencies) and therefore weexcluded the lower half of the 114 MHz band. It is visible that the uncertainties are also reducedby that step.

18

3.3 Scintillation

MJD−56548.4507890790

Fre

quency [M

Hz]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

110

120

130

140

150

160

170

180

190

10

20

30

40

50

60

70

80

Figure 3.2: Dynamic spectrum with ∼10 s subintegrations

On of the most interesting images is Figure 3.2. It shows the dynamic spectrum of PSR B0809+74for the whole 24-h observation.The x-axis is time and the y-axis is frequency with low frequencies at the top and high frequen-cies at the bottom. The color gradient is proportional to the intensity measured at the LOFARstation.Quite apparent are the small bright spots that are visible almost everywhere. These are theintensity maxima (or scintles) from the interstellar scintillation described in section 1.5.If we look how the plot changes from left to right we see bright scintles on the left, then fainterscintles and again bright scintles at later times. This structure is caused by the refractive in-terstellar scintillation (RISS). The period of this scintillation appears to be approximately oneday, but that is not necessarily so. This change could also be some instrumental effect, but weverified that it is not caused by the elevation-dependent sensitivity of the antennas.At the top and bottom we can see the intensity vanishing. The cause again is the low sensitivityat the frequency band edges. There is almost no information until approximately 110 MHz andfrom approximately 190 MHz on.Also there is a gap in the data at the lower right. We don’t have data for that frequency bandand amount of time because of failure of the recording computers.

To improve the data and make the following steps easier we first took the average of the wholedynamic spectrum and subtracted that value from the whole spectrum.

19

Autocorrelation functions

Next we divided the dynamic spectrum into four separate parts (each consisting of full timeresolution and a quarter of the frequency bandwidth) and calculated an autocorrelation functionfor each of the subset spectra. After that we did the same, but divided the spectrum into 16parts (quarter time, quarter frequency bandwidth)Furthermore we peak-normalized the obtained ACFs to be in the range from 0 to 1. This wayfitting was easier because there is one fitting parameter less. Hence we can see a time andalso frequency dependence of the scintillation timescale and frequency scintillation bandwidth.First we fitted the functions 2.2 and 2.3 to all of the ACFs. This was done by writing a gnuplotscript for each ACF and setting individual starting conditions for the fitting routine.A selection of the ACFs are shown in Figure 3.3. The top four figures are frequency slices,therefore they have time as a variable (a subint was ∼10 s). It is visible that the 114 MHzband (subfigures 1 and 5) carries very little small information and does not have scintles at all.Therefore the slices from the 114 MHz band only show the sharp peak (delta function - whichis a general characteristic of ACFs).When viewing the slices of the 137 and 161 MHz bands (subfigures 2, 3, 6, and 7) the typicalexponential decay becomes apparent. In the 185 MHz band (subfigures 4 and 8) we also seean exponential in the middle which has a triangle shape at the bottom. This is caused by themissing data in the bottom right of the dynamic spectrum but does not effect the other valuesfor the scintillation parameters.All ACFs were fitted and the values for the scintillation timescale and frequency bandwidthwere plotted in Figure 3.4 and 3.5 as a function of observing frequency.We see a clear trend in both figures. The theoretically expected frequency-dependence is ν1.2

for the scintillation timescale and ν4 for the frequency bandwidth (see Lorimer and Kramer,2005).There are a few outliers which are far away from the fitted curve. They can possibly beexplained by RFI, refractive scintillation and changes over short time scales.Typical values of the scintillation parameters finally are 18.1 min as the median scintillationtimescale and 2.55 MHz as the median frequency bandwidth.

20

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500

Inte

nsity

Subint #

F Slice from the ACF of the 1st time and 1st frequency intervalexp(-t/t0)

(5/3)*L

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500

Inte

nsity

Subint #

F Slice from the ACF of the 3rd time and 2nd frequency intervalexp(-t/t0)

5/3)*L

0

0.2

0.4

0.6

0.8

1

0 500 1000 1500 2000 2500

Inte

nsity

Subint #

F Slice from the ACF of the 2nd time and 3rd frequency intervalexp(-t/t0)

5/3)*L

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Inte

nsity

Subint #

F Slice from the ACF of the 4th frequency intervalexp(-t/t0)

5/3)*L

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

Inte

nsity

Frequency [MHz]

T Slice from the ACF of the 1st time and 1st frequency intervalexp(-f/f0))*C

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

Inte

nsity

Frequency [MHz]

T Slice from the ACF of the 3rd time and 2nd frequency intervalexp(-f/f0))*C

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

Inte

nsity

Frequency [MHz]

T Slice from the ACF of the 3rd time and 3rd frequency intervalexp(-f/f0))*C

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

Inte

nsity

Frequency [MHz]

T Slice from the ACF of the 4th time and 4th frequency intervalexp(-f/f0))*C

Figure 3.3: A selection of all autocorrelation functions showing different behavior. Top fourare slices through the ACF at the middle frequency (f-slices), bottom four are slices throughthe ACF at middle time/subint (t-slices).

21

Scintillation timescale

∆t

[min

]

10

12

14

16

18

20

22

24

26

10

12

14

16

18

20

22

24

26

ν[MHz]

110 120 130 140 150 160 170 180 190

110 120 130 140 150 160 170 180 190

quarter lengthfull lengthν1.2 fitting curve

Figure 3.4: Time bandwidth ∆t with ∼ 60 s subints and band edges removed.

Frequency bandwidth

∆f

[MHz]

1

2

3

4

5

6

1

2

3

4

5

6

ν[MHz]

130 140 150 160 170 180 190

130 140 150 160 170 180 190

quarter length

full length

NichtlinearFit1

NichtlinearFit1

quarter length

full length

ν4 fitting curve

Figure 3.5: Frequency bandwidth ∆f with ∼60 s subints and band edges removed.

22

Chapter 4

Conclusion & Outlook

PSR B0809+74 was the choice to investigate because it is nearby, bright and it is a scintillatingpulsar. As this pulsar is thought to be just outside of the local bubble it is expected to seescintillation coming from this place.

The DM value was almost constant throughout the observation with just slight changes. It wascompared with the value in the ATNF catalogue and a difference was apparent.One reason why we are able to see short time variations in the DM value is that LOFAR isobserving at a low frequency. With the LWA and the MWA coming into operation we couldget even more sensitive to short time variations.

Having a dynamic spectrum with very visible scintillation effects we were hoping to see scin-tillation arcs coming from the edge of the local bubble. But even with varying sensitivity andresolution we could not see any.Knowing that scintillation arcs are not easy to detect because they come and go frequently, it isrecommendable to keep looking. It is also possible to further increase sensitivity by decreasingthe subint length. 10 s were chosen because of hardware limitations. The LOFAR data for PSRB0809+74 enable us to use single pulses (∼1.29 s) although with this short subint length onewould need to create dynamic spectra for shorter timescales, or increase computing power. Butin essence it would be interesting to keep looking and as far as possible automate this analysis.Also repeating this process on different pulsars could be valuable. Another aspect of continuingto observe is to look for changes in the scintillation parameters. As the pulsar moves, our lineof sight traverses the structure of the ISM in between. If the line of sight changes we also seedifferent structures of the ISM because of the scintillation. It would be interesting to see onwhich scales the scintillation parameters change over time, if at all.

23

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25

Acknowledgements

I would like to thank Joris Verbiest for offering the opportunity to write this thesis, for havinga solution for almost any problem that came up and for his support throughout the last phaseof writing.Secondly I would like to thank Stefan Os lowski for always being helpful when dealing withproblems in PSRCHIVE!

26

Declaration

I hereby declare and confirm that this thesis is entirely the result of my own original work.

Where other sources or information have been used, they have been indicated as such and

properly acknowledged. I further declare that this or a similar work has not been submitted

for credit elsewhere.

Hauke Jung

27