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LOFAR Studies of the Ionised Interstellar Medium at the Smallest Scales Master’s Thesis Julian Donner March 14, 2017 University of Bielefeld Faculty of Physics Referee: Prof. Joris Verbiest Second Referee: Dr Caterina Tiburzi

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LOFAR Studiesof the Ionised Interstellar Medium

at the Smallest Scales

Master’s Thesis

Julian Donner

March 14, 2017

University of Bielefeld

Faculty of Physics

Referee: Prof. Joris Verbiest

Second Referee: Dr Caterina Tiburzi

Contents1 Introduction 1

1.1 Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Millisecond Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Pulsar Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Pulsar Timing Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Interstellar Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Faraday Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 DM Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.4 Structure Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.5 Chromatic DMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.6 Extreme Scattering Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Aims and Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Hardware and Observations 122.1 The LOw Frequency ARray (LOFAR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Observations Used in this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 DM Comparison Between Different GLOW Stations . . . . . . . . . . . . . . . . . . . . 16

3 Intra-day DM Variability 18

4 Chromatic DMs 21

5 Extreme Scattering Event in the Direction of PSR J2219+4754 235.1 Timing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 DM variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2.1 Impact of Scattering on the DMs . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2.2 Profile Shape Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2.3 Structure Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2.4 Extreme Scattering Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.3 RM variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.4 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.5 Comparison to Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Long-Term DM Variability 346.1 Pulsars with Constant DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 Pulsars with Variable DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7 Discussion and Conclusions 377.1 Consequences for High-Precision Pulsar Timing Experiments . . . . . . . . . . . . . . . . 377.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Acknowledgements 39

References 43

A Appendix 44A.1 Calculation of the Structure Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Abstract

In this thesis, I present measurements of the interstellar dispersion of pulsed emission from neutronstars. With the German LOFAR telescopes, which operate at very low frequencies (10 - 250 MHz),the dispersion can be measured very precisely, as the delay it induces scales with the inverse squareof the observing frequency. Also, these LOFAR telescopes are frequently available, allowing a highsampling rate and thus a very precise time evolution of the Dispersion Measure (DM), down to variationson daily time scales. I find short-term DM variations on the order of a few 10−3 cm−3 pc, which inducedispersive delays sufficiently high to significantly influence higher-frequency pulsar timing data. Thisis problematic, because it is often the case that pulsar timing experiments use frequency-unresolved datawith one observation per month and thus are not able to account for a variable dispersive delay accurately.I use pulsars with a known high DM precision to be able to quantify how often short-term variations atthis level of significance occur and show that DM excesses are rather common and can even be found forpulsars currently used by Pulsar Timing Arrays.

1 IntroductionThis thesis is about turbulence in the interstellar medium (ISM), which can be analysed with pulsar obser-vations. The basic introduction below, on pulsars, pulsar timing, and the ISM, is based on the descriptionof Lorimer and Kramer (2005).

Measurement uncertainties are always given in brackets after the actual measurement value, in units ofthe last digit. They indicate the estimated standard deviation of the measurement, i.e. the 68% confidenceinterval. Equations containing electromagnetic quantities are in cgs units.

1.1 Pulsars

0

0.25

0.5

0.75

1

0.45 0.5 0.55

flux (

arb

itra

ry u

nits)

rotational phase

Figure 1: Zoom-in on the pulse profile ofPSR J0332+5434 observed at 150 MHz with abandwidth of 95 MHz. To get a high signal-to-noise ratio (S/N), subsequent pulses have beenadded over 5.8 hrs. The units in x are chosen suchthat one full rotation of the pulsar corresponds toone unit in x.

Radio pulsars where first discovered by JocelynBell (Hewish et al., 1968) in 1967. These objectsemit electromagnetic radiation in regular pulses asseen from Earth. The current understanding of thisphenomenon is that pulsars are neutron stars – theremnants of massive stars that underwent a super-nova. During such an event, the outer layers of thestar get ejected and the core collapses. If the grav-itational pressure is large enough, it can overcomethe electron degeneracy pressure and push the elec-trons into the nuclei, converting most of the matterinto neutrons. The radius of a neutron star is con-strained to lie between roughly 10 km and 20 kmand their mass is larger than the Sun’s. That meansthat neutron stars are so dense that they are only bya factor of a few larger than their Schwarzschildradius, which is the radius below which an ob-ject turns into a black hole. Due to the collapseand the conservation of angular momentum, pul-sars rotate very rapidly. The fastest known pulsarJ1748–2446ad rotates 716 times per second. Theslowest currently known pulsar J1841–0456 has arotational period of 11.8 s, which is a difference of four orders of magnitude in rotational period. Thesepulsars were detected by Hessels et al. (2006) and Vasisht and Gotthelf (1997), respectively.

Pulsars have a very strong magnetic field. At their magnetic poles, they emit two beams of radio wavesdue to the interaction of their magnetic field with charged particles. The emission is very strong at radiofrequencies, which is why these objects are mostly studied with radio telescopes. As the rotation andmagnetic axes are generally not aligned, the beam is pointed in different directions over time, similar to alighthouse. If this beam is by chance directed towards Earth once per rotation, we see short and regularpulses of radiation. The shape of these pulses (in an intensity vs rotational phase plot, see Fig. 1) variesfrom pulse to pulse, but the average shape of hundreds of pulses is usually very constant in time. It is

1

1e-22

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

0.001 0.01 0.1 1 10 100

dP

/dt

(ss

-1)

P (s)

singlebinary

Figure 2: P-P diagram of all 2536 pulsars known to date. It is striking that there are two ma-jor populations of pulsars. Also, most short-period pulsars are in a binary system. Data takenfrom the ATNF Pulsar Catalogue (psrcat) web-page (http://www.atnf.csiro.au/people/pulsar/psrcat/) by Manchester et al. (2005).

characteristic for the pulsar and can differ a lot between different pulsars. It varies from simple, thin peaksto wide and complex multi-peaked shapes. If the angle between the magnetic and rotational axes is close to90 degrees, we receive two pulses of radiation per rotation. If the two axes are nearly aligned, the receivedradiation is no longer pulsed as it is detected throughout the whole rotation. In addition to the very stablepulse shape, the rotational period of pulsars is very stable as well, so pulsars are like galactic clocks and wecan precisely predict when the next pulse arrives at Earth. Slight deviations from these predictions allow usto study the interstellar medium, the pulsar’s motion, or even test general relativity.

As pulsars emit radiation, they have to lose energy, which leads to a spindown. To get an idea of pulsarpopulations, the pulse period derivative P is often plotted against the pulse period P. The so-called P-Pdiagram (Fig. 2) shows that there are two major groups of pulsars: the slowly-rotating normal pulsars inthe centre of the plot with pulse periods of the order of seconds and the fast-rotating millisecond pulsars(MSPs) in the bottom left corner with pulse periods down to milliseconds. The usual evolution of a pulsarin the diagram is the following: the pulsar starts its life in the top left of the main population and, due tothe spindown, moves down to the bottom right until it becomes so faint that it is no more visible as a pulsar.MSPs do not fit into this scenario. In the P-P diagram it is striking that they are usually found in binarysystems which is related to their stellar evolution history, as explained below.

1.1.1 Millisecond Pulsars

When two stars orbit each other in a binary system, the more massive star undergoes a supernova first, asit burns its nuclear fuel faster. The explosion can rip the binary apart, but if it does not, the remnant of themore massive star continues to orbit its companion. Depending on its initial mass, it can become either awhite dwarf (low mass), a neutron star, or a black hole (high mass). In the second scenario (neutron star) theremnant may be seen as a pulsar from Earth. When the companion comes to the end of its life, it becomesa red giant. In this state, the radius of the star has grown so much that its outer layers are only looselybound by gravity, hence allowing the neutron star to accrete parts of those outer layers when it comes close

2

enough. The accreted matter forms a disk around the neutron star and heats up, such that the binary is thenseen in X-rays until the disk dissipates. In this state, the system is also known as a low-mass X-ray binary(see Karttunen et al., 2007). During this process, orbital angular momentum is transferred to the pulsarwhich consequently spins up, which is why MSPs are also called recycled pulsars. At the same time, thepulsar’s magnetic field is weakened, which leads to a lower energy loss from magnetic dipole radiation. Asthe companion has a lower initial mass than the pulsar, its supernova explosion should not rip the binaryapart either, so it will remain the companion of the pulsar as either a white dwarf or a neutron star. It cannotend up as a black hole, as its initial mass would have to be larger than the pulsar progenitor’s initial mass,in which case it would also have exploded first.

During the accretion, the pulsar’s rotation becomes increasingly stable and the spindown becomes verysmall (see Fig. 2), which makes MSPs very useful tools for high-precision timing measurements, eventhough they are on average much fainter than normal pulsars. The high rotational frequency also impliesshorter pulses and thus allows a more precise measurement of the pulses’ arrival times.

Some MSPs (around a third, see Tauris, 2014) are not in binary systems. One idea is that the companionhas been completely dissipated by the pulsar wind, a stream of highly energetic particles, but this is still anopen question. There are binary systems with extremely-low-mass companions, the so-called black widowsand redbacks, which could be the progenitors of isolated MSPs. Addressing this question is one goal of thefuture SKA (Square Kilometre Array) telescope (see Tauris et al., 2015).

There are also ‘mildly’ recycled pulsars, the double neutron stars (DNSs). These are created when thetwo initial stars have a similar mass and thus explode within a relatively short timespan. In this scenario,the pulsar can only accrete a relatively small amount of matter until its companion explodes and thus onlyspins up a limited amount.

Around 13% of all known pulsars are MSPs (when only considering pulsars with rotational periodsbelow 30 ms, see Fig. 2). As they have a slow spindown, they are usually long-lived and thus on averagerather old, which is also a reason why they are expected to have a larger scale height than the normal pulsars(i.e. are on average further away from the disk).

1.1.2 Pulsar Timing

As mentioned earlier, the regular pulses we receive from pulsars are like clocks that can be used to studyvarious physical properties. For this purpose, the accurate arrival times of the pulses are analysed, whichis called ’pulsar timing’. In order to get measurements in pulsar timing, a mathematical model is fitted tothe times of arrival (ToAs) of the pulses. This so-called ‘timing model’ contains all parameters needed toprecisely model the pulsar’s rotation, position, and movement, but also accounts for interstellar effects likedispersion. To analyse and optimise the parameters used in the timing model, we study the timing residuals,which are the difference between the model-predicted and actual ToA. This way, even small deviationsbetween the model and the observations can become visible.

In astronomy, dates are usually given in the form of the Modified Julian Date (MJD), which counts thedays that passed from a given point in time, 17 November 1858 00:00 UTC. This format will be used inthis thesis, as it is easier to use than Gregorian dates. The most important parameters in the timing modelare the following:

Pulsar Spin. The pulsar’s spin frequency is the most basic parameter. When this parameter is incor-rect, the timing residuals show a linear trend, as subsequent pulses arrive increasingly earlier or later thanexpected. Due to the pulsar’s spindown, a rotational frequency derivative is also needed. Higher-orderspin derivatives can be needed, the number of which depends on the pulsar’s rotational stability and on thetiming precision.

Position. Due to the movement of the Earth around the Sun, the distance to the pulsar changes with aone-year period. The phase and amplitude of this effect in the timing is determined by the pulsar’s rightascension and declination. For pulsars with a high proper motion or a long observing timespan, the propermotion in both right ascension and declination also need to be added to the timing model.

Dispersion. Dispersion delays pulses at different frequencies differently. To avoid frequency-dependenttiming residuals this effect has to be modelled, so the dispersion measure (see Section 1.2.1) is alwaysincluded in the timing model.

Other parameters, like for example the motion of the pulsar in its binary system or even relativisticcorrections in tight binaries, are not of high relevance for this thesis and are thus not discussed in detail. Anoverview can be found in Edwards et al. (2006).

3

1.1.3 Pulsar Timing Arrays

Pulsars are not only useful tools by themselves. The timing residuals of many pulsars can be used to forma so-called Pulsar Timing Array (PTA). With a sufficiently large sample of pulsars in all directions onthe sky, one can try to detect gravitational waves. These waves deform the space differently in differentdirections and thus a passing wave would delay some pulsar signals while others would arrive earlier. Sothere should be a correlation or anti-correlation of the timing residuals of the different pulsars dependingon their relative positions on the sky. As MSPs can be timed precisely over tens of years (Verbiest et al.,2009), PTAs should be sensitive to low-frequency gravitational waves, which are also expected to be themost powerful gravitational waves (see Jenet et al., 2005, and references therein). Thus, PTA observationsof gravitational waves are complementary to ground-based experiments such as LIGO and VIRGO, whichcan only observe high-frequency waves (roughly around 100 Hz, see, e.g. Abramovici et al., 1992).

To be able to detect the weak signal of a gravitational wave, very high timing precision and stabilityis needed, and thus only MSPs are of interest for PTAs. As described by Jenet et al. (2005), timing 20to 40 MSPs at a precision of 100 ns over 5 years should be sufficient to detect gravitational waves. Sincethen, predictions have changed. Sesana et al. (2016) showed that the amplitude of the gravitational wavebackground has been overestimated, which increases the challenge of PTA efforts. As a precision of 100 nsis hard to reach (usually rather 1 µs, see Verbiest et al., 2016), the observing timespan must be increased.The pulsars are usually observed once per month at each telescope. All effects that have an impact onthe ToAs have to be measured precisely, such that they can be subtracted to reveal the signals of interest.Time-dependent effects have to be monitored constantly. Some of these effects have an interstellar origin,as described below (see also Lentati et al., 2016).

1.2 Interstellar EffectsThe interstellar space, which is the space between the stars of a galaxy, is not completely void. It is filledwith gas at various densities and temperatures, which is spread out through the entire galactic disk with ahigher concentration in the spiral arms and consists mostly of hydrogen and helium. The overall densityof the gas is extremely low, more than 20 orders of magnitude lower than in our planetary atmosphere.Still, the interstellar gas makes up 10 - 15% of the mass of our Galaxy (see Ferriere, 2001). Our currentknowledge of the medium is mostly based on our own Galaxy. As described by Ferriere (2001), the ISMconsists of four main phases:

Molecular Gas. The coldest phase of the ISM (down to 10 K) allows the formation of dense cloudsof molecules (e.g. CO) and dust. These clouds only make up 1 - 2% of the Galactic volume and can beobserved through spectral line emission and absorption. Also, they are confined very close to the disk, witha FWHM (full width at half maximum) on the order of 100 pc.

Neutral Atomic Gas. This phase is spread out much more evenly through the Galaxy. Its temperature ison the order of 104 K and it consists mostly of neutral hydrogen atoms. It is easily observed at a wavelengthof 21 cm, the hyperfine structure transition of hydrogen, which is why many radio telescopes operate at thiswavelength. The FWHM is about twice as high as for the molecular gas, but it is still rather confined to thegalactic disk.

Warm Ionised Gas. In the vicinity of massive stars, atoms can become ionised by the high-energy(UV) photons those stars emit. These so-called Hii regions (Hii = ionised hydrogen) can be seen at opticaland UV frequencies due to spectral line emission (recombination of ions and electrons) and absorption(ionisation of atoms). As massive stars are usually formed in spiral arms and do not live long, the scaleheight (the height at which the density becomes 1/e of the peak density) is very low, on the order of 70 pc.There is also a diffuse component of the warm ionised gas outside of Hii regions. It is spread evenly throughthe Galaxy with a scale height of ∼ 900 pc in the Solar environment. Its origin is probably the cooled-downhot gas (see below).

Hot Ionised Gas. This phase consists of ionised gas at temperatures up to 106 K. It is very tenuousand its scale height is hard to measure (due to the lack of background sources) but is estimated to be on theorder of a few kpc. The origin of this hot gas is most likely supernovae, which release a lot of energy andthus can heat the gas up and push it out of the Galactic disk.

In the following, the interactions of electromagnetic waves with the ionised parts of the ISM are de-scribed. The basic concepts of dispersion and Faraday rotation are explained in Section 1.2.1 and Sec-

4

tion 1.2.2, respectively, after which turbulences in the medium and possible frequency-dependent effectsfrom those are discussed in Section 1.2.3 to Section 1.2.6.

1.2.1 Dispersion

The warmer, ionised parts of the medium affect the group velocity of electromagnetic radiation. As de-scribed by Lorimer and Kramer (2005), this effect can be quantified by looking at the frequency-dependentrefractive index µ of the medium:

µ =

√1 −

(fp

f

)2

, fp =

√e2ne

πme' 8.5 kHz

( ne

cm−3

)1/2. (1)

Here me and e are the mass and charge of the electron, respectively, ne is the electron density, f is theobserved frequency and fp is the plasma frequency of the medium, which is the highest frequency at whichno electromagnetic radiation can pass through. This is due to the relation of the the refractive index µ andgroup velocity, vg = cµ, which becomes 0 at f = fp.

As I will show shortly, a typical electron density in the ISM is 0.03 cm−3, which corresponds to aplasma frequency of 1.5 kHz (see Eq. 1). When comparing this value to the observing frequencies of radiotelescopes, which are typically higher than 100 MHz, it becomes clear that µ (and thus also 1/µ) is veryclose to 1 and thus can be approximated using a first-order Taylor expansion around µ = 1 ( fp/ f = 0):

= 1/√

1 −(

fp

f

)2

' 1 +f 2p

2 f 2 . (2)

The relative error of this approximation on the final result is on the order of 10−10 for observing frequenciesof 150 MHz, which are used in this thesis. This systematic error is much lower than the measurementprecision, so this approximation is valid.

The extra time τ it takes the radiation to travel a certain path of length d in comparison to undispersedpropagation is:

τ =

∫ d

0

dlvg−

dc

=

∫ d

0

dlcµ−

dc

=1c

∫ d

0

1 +f 2p

2 f 2

dl −dc

=e2

2πmec

∫ d0 nedl

f 2 ≡ DDMf 2 . (3)

In this formula, the dispersive time delay is only dependent on the observing frequency and the DispersionMeasure

DM =

∫ d

0nedl , (4)

which is the electron density integrated over the line of sight. D ' 4.149 · 103 MHz2pc−1cm3s is thedispersion constant. With this formula, one can calculate the average electron density along the line of sightby dividing the DM by the distance to the pulsar. Using data from psrcat for all pulsars with currentlyknown DM and distance, one can see that typical electron densities in our Galaxy are between 10−2 cm−3

and 10−1 cm−3, with a median value of 0.03 cm−3

From Equation 3, we can derive the difference in the pulse arrival time between two frequencies:

∆τ = τ( f2) − τ( f1) = D · DM 1

f 22

−1f 21

. (5)

Figure 3 shows the dispersion of the pulse profile of PSR J1136+1551. As expected from Equation 3,the pulse is more strongly dispersed at lower frequencies. Although this pulsar has a rather low DM ofonly 4.85 cm−3 pc and a rather long pulse period of about 1.2 s (see Table 1), the difference in the dispersivedelay between the two ends of the observed bandwidth is on the order of one pulse period. In pulsar timingexperiments, the data are usually integrated in frequency to increase the S/N, so the DM has to be calculatedat first to shift the frequency channels correctly before adding them.

As pulsars emit pulses of radiation (as seen from Earth), the travel times of electromagnetic radiationcan be compared at different frequencies and thus the dispersion can be measured. The result can be usedto directly relate the average electron density along the line of sight and the distance to the pulsar. With

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Figure 3: Dispersed pulse profile of PSR J1136+1551. The logarithmic greyscale shows the inten-sity of the pulsed emission at the given frequency and pulse phase. For this plot, pulses have beenadded up in time for one day to get a very high S/N. The gap at around 180 MHz is due to removalof RFI-affected frequency channels (see Section 2.3).

the use of independent distance estimates (e.g. from parallax measurements or associations with supernovaremnants), one can model the free electron density in the Galaxy, which has been done for example byCordes and Lazio (2002) and Yao et al. (2017). With the help of these models, an estimate for the distanceto a pulsar can be calculated when the DM is known, which is usually very easy to measure. Local inho-mogeneities are a big problem for these models, and thus the distances they provide have to be taken as arather rough estimate. Still, for many pulsars, this is the only currently available distance estimate.

Pulsars have, in comparison to normal stars, a very high velocity. Lyne and Lorimer (1994) attributethis phenomenon to small asymmetries in the supernovae the pulsars were born in, which can lead to asubstantial kick velocity. Some pulsars are even so fast that they exceed the escape velocity of the gravi-tational potential of the Galaxy. This high velocity induces significant motion of the line of sight, whichthus samples different parts of the ISM at different times. As a consequence, turbulence in the ISM can bemeasured by monitoring the time evolution of the DM.

1.2.2 Faraday Rotation

Another effect of the ISM on electromagnetic radiation is Faraday Rotation. This effect occurs when thereis, in addition to free electrons, a magnetic field component parallel to the propagation direction of theradiation. The result is the rotation of the polarisation plane due to the different group velocities of left- andright-hand circularly polarised radiation. Below, I give a description of the most important formulae. Formore details and derivations, see Lorimer and Kramer (2005).

The rotation of the polarisation angle as a function of the wavelength is:

∆Ψ = λ2RM , RM =e3

2πm2ec4

∫ d

0neB||dl . (6)

The Faraday rotation only depends on the observed wavelength λ and the Rotation Measure (RM), whichis the integrated magnetic field along the line of sight, multiplied by the electron density. As the Faraday

6

rotation is frequency dependent, any observed radiation gets depolarised when integrated in frequency.Similar to the DM correction for dispersion, the RM is used to correct for Faraday rotation, when one isinterested in the polarisation properties of a pulsar.

By combining the measurements of DM and RM, one can furthermore calculate the average magneticfield strength along the line of sight, assuming a constant electron density, with the following formula:

⟨B||

⟩=

∫ d0 neB||dl∫ d

0 nedl= 1.23 µG

(RM

rad m−2

) (DM

cm−3 pc

)−1

. (7)

This formula gives a measurement of the magnetic field strength along the line of sight, but has two majorproblems: firstly, the result can be strongly affected by dense regions of ionised particles and thus maynot reflect the overall magnetic field. Secondly, a reversal of the magnetic field leads to a sign flip in B||,which reduces the absolute value of this measurement, leading to an underestimation of the average absolutestrength. To solve this problem, multiple sources in similar directions at different distances are needed. Theconsequence of these two problems is that the results of the measurements have to be taken with care andshould rather be taken as a first estimate.

The pulsar emission mechanism is not entirely understood, but is related to the interaction of the mag-netic field with charged particles and is observed to be polarised. Assuming the emitted polarisation angleto be equal at all frequencies, the RM can be used to get an idea of the Galactic magnetic field along theline of sight.

1

2

3

4

56400 56600 56800 57000 57200 57400 57600

RM

ion (

rad m

-2)

MJD

Figure 4: Ionospheric corrections in the direc-tion of PSR J2219+4754 for the observations dis-cussed in Chapter 5, using the ionospheric elec-tron content map code.

The very outer parts of Earth’s atmosphere areionised by highly energetic particles and radiationfrom the Sun. The so-called ionosphere thus con-tains charged particles and free electrons. In com-bination with Earth’s magnetic field, incoming ra-diation from space undergoes Faraday rotation inthe ionosphere, which affects the measured RMsignificantly. The effect is strongly dependent onthe Solar activity and thus has to be measured fre-quently as the pulsar is always visible at differenttimes and positions on the sky. Figure 4 showsthe ionospheric RM contribution in the direction ofPSR J2219+4754 for the observations discussed inChapter 5. The annual structure is primarily due tothe different times of day at which the pulsar canbe observed with LOFAR (see Section 2.1) duringthe year. As a reference, typical pulsar RM valuesare tens to hundreds of rad m−2, but their variationsdue to inhomogeneities in the ISM are only on theorder of 1 rad m−2 or less. This is the reason why the ionospheric corrections, which are of the same orderor even bigger, have to be measured precisely.

The measurements of the atmospheric electron content are provided by different sources. In this thesis,the maps from the Center for Orbit Determination in Europe (CODE), the International GNSS Service(IGSG), and the Royal Observatory of Belgium (ROB) are used. Sotomayor-Beltran et al. (2013) describethe open source python program ionFR, which calculates the ionospheric RM contribution for a giventelescope position, time, source position, and electron density map. They model the ionosphere as a thinspherical shell surrounding Earth.

1.2.3 DM Variations

The time-dependence of the DM has already been observed by various groups (see references below). Theusual DM precision of an observation is not given in most of these papers but can be estimated from theaverage DM they give and the number of observations, or (in some cases) from their DM time series.

The largest sample has been studied by Hobbs et al. (2004), who observed 374 pulsars usually overmore than 10 years with the Lovell Radio Telescope at the Jordell Bank Observatory. Their typical DMprecision for one observation is about 10−1 cm−3 pc. Petroff et al. (2013) analysed DM variability of 168

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19.612

19.614

19.616

19.618

19.62

19.622

-250 -200 -150 -100 -50 0 50 100 150 200 250

DM

MJD - 56558

DM variations for PSR J1509+5531

Data

fit

Figure 5: DM variations in the direction ofPSR J1509+5531. In addition to an overall in-creasing trend, there is a large scatter in the mea-surement points which significantly excesses theDM measurement precision. Figure taken fromDonner (2014).

19.6177

19.6178

19.6179

19.618

19.6181

19.6182

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

DM

MJD - 56899

Full-day observation for PSR J1509+5531

Datafit

Figure 6: DM variations for the same pulsar asin Fig. 5, but for a day-long observation startedon 20 August 2014. There is a clear trend forthe DM derivative with a value of dDM/dt =

0.18(5) cm−3 pc/yr and the measurement uncer-tainties do appear to accurately quantify Gaus-sian noise. Figure taken from Donner (2014).

pulsars, usually over 5 years. Their dataset is complementary to Hobbs et al. (2004) in the sense that theyuse the Parkes Radio Telescope, which observes in the Southern Hemisphere. Their typical DM precisionis on the order of 1 cm−3 pc. A higher precision (usually better than 10−3 cm−3 pc) was achieved by Keithet al. (2013), who observed 20 MSPs for about 5 years. Coles et al. (2015) found Extreme Scattering Events(see Section 1.2.6) in the DM time series of four MSPs, which they monitored for a few years with a DMprecision typically better than 10−3 cm−3 pc.

All of these groups detected gradients in the DM for various pulsars. Hobbs et al. (2004) have derivedthe following empirical relation between the DM and its derivative:∣∣∣∣∣dDM

dt

∣∣∣∣∣ = 0.0002√

DM, (DM in cm−3 pc, t in yrs). (8)

This relation however fits the data with large scatter, so individual measurements can strongly differ fromit. Still, it can be used to estimate long-term DM variability.

In my previous thesis work (Donner, 2014), highly precise DM variations for a large sample of pulsarsover a time span of about one year were analysed based on low-frequency data. Many of the pulsars showedgradual trends in DM, but some of the more precisely measured DM time series showed a scatter of themeasurement points, which was much larger than expected from the measurement uncertainties (see Fig. 5).This can be either explained by uncertainty underestimation or by significant small-scale turbulence. Froma long observation started on 20 August 2014 (see Fig. 6), it became evident that there are indeed significantvariations over periods as short as a day, at least towards some of these pulsars.

Correcting for DM variations

The effects of DM variability on PTA experiments have often been underestimated in the past. In thefollowing, I will show how to quantify these effects and how they would affect the required timing precision.From Equation 3 one can equate the additional time delay t from a DM offset ∆DM:

t = D · ∆DM1f 2 . (9)

At the typical observing frequency of 1.4 GHz and with a DM offset of 10−3 cm−3 pc, this extra delay wouldbe 2.1 µs. To correct for this extra time delay, the observed frequency band has to be split into at least twoparts and the DM can be calculated from the two resulting ToAs:

∆t = t2 − t1 = D · ∆DM 1

f 22

−1f 21

. (10)

8

To relate the delay ∆t across the band to the additional time delay t, we can do the following calculation, byusing f = ( f1 + f2)/2:

t∆t

=

4( f1+ f2)2

1f 22− 1

f 21

=

4( f1+ f2)2

f 21 − f 2

2f 21 f 2

2

=

4 f 21 f 2

2( f1+ f2)2

( f1 − f2)( f1 + f2)=

4 f 21 f 2

2

( f1 + f2)3( f1 − f2)=: γ . (11)

So now we can calculate t = ∆t · γ, γ being only dependent of the observational setup. Its value can belowered by increasing the bandwidth while keeping the centre frequency the same, as doing so reduces thevalue of f1 f2, keeps the value of ( f1 + f2) constant, and increases ( f1 − f2).

Assuming equal ToA uncertainties σt1 = σt2 for both bands, which are worse than the total ToA uncer-tainty σToA of the full band ToA by a factor of

√2, we get the following uncertainty for the DM time delay

correction:σ∆t =

√2σt1 = 2σToA, σt = σ∆t · γ = 2γσToA =: γσToA. (12)

With a common observational setup of 1.4 GHz centre frequency and 250 MHz bandwidth (which resultsin a frequency difference of the upper and lower half band of 125 MHz), we get γ = 11, so the correctionof the DM values would be 11 times worse than the ToA precision and thus worsen it by this factor. As aconsequence, the initial timing precision would have to be a factor of 11 better. One way to improve thisfactor is to use a higher bandwidth. With a total bandwidth of 1 GHz in the example above, γ would become5. To avoid the worsening of the ToA precision, the DM can be averaged across multiple observations toincrease the precision of its measurement. However, this only works if the DM variations are slow enoughwith respect to the observation cadence. Alternatively, lower-frequency DM measurements could be takensimultaneously to correct for the DM in the higher-frequency data.

1.2.4 Structure Functions

Figure 7: Structure function of the interstellarelectron density at a wide range of spatial scales.Image taken from Armstrong et al. (1995).

To quantify interstellar turbulence, it is useful tostudy its structure function, which is defined as (seeYou et al., 2007):

DDM(τ) =⟨[DM(t + τ) − DM(t)]2

⟩. (13)

The angle brackets indicate the calculation of themean value. DDM(τ) is a measure of how much theDM varies on average after a given time interval τ.

The numerical calculations and the weightingof the DM values are discussed in detail in Ap-pendix A.1.

According to Armstrong et al. (1995), the struc-ture function is known to follow a Kolmogorov tur-bulence spectrum, i.e.

DDM(τ) ∝ τ5/3. (14)

In their paper, they combine structure functionsof the interstellar electron density from differenttypes of measurements and conclude that it fits theKolmogorov spectrum rather well across the entirespectrum that they computed (see Fig. 7).

You et al. (2007) analysed the structure func-tion for various pulsars. Their results based on DMvariations span time scales of months to years, butthey also used scintillation measurements, whichprobe time scales that are shorter than an hour. Thegap between those two types of observations couldbe filled with frequent high-precision DM measurements, as already presented in my Bachelor’s thesis(Donner, 2014) down to time scales of one week. In this thesis, I will further expand on those results.

9

1.2.5 Chromatic DMs

The DM does not only vary with time, but also with frequency. As described by Cordes et al. (2016), thereason behind this effect is scattering. When electromagnetic radiation hits a turbulent medium, it getsdeflected in many directions. The angle of this effect is frequency-dependent in a way that low-frequencywaves are deflected more strongly. When we observe a pulsar, we do not only receive radio waves directlyemitted towards Earth, but also waves that were emitted in a slightly different direction, which are thendeflected towards Earth. As the scattering is stronger at lower frequencies, more radiation is received fromrather large angles than at high frequencies, so effectively a larger part of the sky is sampled at once. Whena very small structure passes through the line of sight, it will affect the low frequencies for a longer timethan it affects the high frequencies, but also not as strongly, as the lower-density surroundings of the smallstructure also affect the low-frequency DM measurement. This leads to a smoothing of the DM time seriesas very small structures only fill a small part of the sampled area and thus do not completely dominatethe measurement when passing through the line of sight. Also, the low-frequency DM can be affected bystructures that come close to the line of sight and do not move directly through it, so it can in some casesshow structure that higher frequencies do not show.

In this model, a chromatic DM can only occur when ISM is turbulent in the direction of the pulsar, andthus requires a time-dependent DM to be observable.

However, this effect has not been observed yet. Hassall et al. (2012) tried to detect chromaticity acrossa very large band from 40 MHz to 8 GHz in observations of four bright pulsars, but concluded that theycannot find any evidence of chromaticity.

1.2.6 Extreme Scattering Events

Figure 8: Flux variations for the quasar0954+658 at 2695 MHz and 8085 MHz. Therather symmetric structure at the centre is iden-tified as an extreme scattering event. The figureis taken from Fiedler et al. (1987).

Extreme Scattering Events (ESEs) were first dis-covered by Fiedler et al. (1987). They ob-served variations in the flux density of the quasar0954+658 (see Fig. 8) and found a short period (afew months long) of strong variations they explainwith some sort of lens moving through the line ofsight. ESEs are a rare phenomenon in their dataset.They only span 395 days in about 160 years of totalobserving time. With pulsar observations, not onlycan we observe flux density variations, but alsovariations in electron density which show up di-rectly in the dispersion of the pulse (see, e.g. Coleset al., 2015). As pulsars are point sources (in con-trast to AGN) the effect of scattering can also beanalysed by calculating the diffractive time scaleand bandwidth (see Coles et al., 2015). Strong scat-tering also leads to a widening of the pulse pro-file, especially at low frequencies (see Bhat et al.,2004). Observing an ESE in several of these ob-servables at once can tell us more about them. An-other advantage of pulsars is that they have a (high)proper motion, as they are part of our Galaxy and move rather rapidly due to asymmetries in the supernovathey were born in. This allows us to sample the ISM faster and also makes it possible to see clouds thatare stationary or only moving slowly with respect to Earth, as the pulsar can pass behind them. All thesereasons make pulsars much more useful to observe ESEs than extragalactic sources. Still, only a handfulof pulsar observations of ESEs have been published (Cognard et al., 1993; Keith et al., 2013; Coles et al.,2015).

Maitia et al. (2003) were the first (and currently only ones) to observe an ESE on a time scale of a fewyears. The event started in 1997 and was observed at 1.41 GHz and 1.28 GHz. Like Fiedler et al. (1987),they only observed flux density variations, but their event is three years long. Since long events like this onehave not been observed by other groups, it is evident that such long events are very rare or hard to observe.

10

A good example of ESEs detected simultaneously in different of the parameters mentioned above isprovided by Coles et al. (2015), who looked at the DM, the diffractive time scale and the scintillationbandwidth. Every one of their events however only consists of a few observations.

1.3 Aims and Structure of this ThesisThe primary aim of this thesis is to analyse the short-term DM variations discussed in Section 1.2.1, tocheck how frequently they occur, and to quantify their effect on pulsar timing experiments.

In Chapter 2, I present the basic concepts of the telescopes and data used, the observational methods,and the basic data analysis.

Chapters 3 to 6 show the results for DM variations on different time scales, starting with observationswith a length of a few days and ending with overall trends as long as three years. The intra-day variabilitiesanalysed in Chapter 3 are a new field of research and probe exactly the missing time scales in the structurefunction of a few hours to a few days. A three-year-long ESE in the direction of PSR J2219+4754 isanalysed in DM, pulse width, and pulse shape in Chapter 5. The observations are made with the LOwFrequency ARray (LOFAR) single stations of the German LOng Wavelength consortium (GLOW), whichprovide extremely precise DM measurements as they observe at low frequencies and the dispersion scaleswith f −2 (see Section 1.2.1). Also, the pulse shape is much more strongly affected by scattering at thesefrequencies, as the pulse broadening scales with f −4.4 for a turbulent medium that follows a Kolmogorovspectrum (see Bhat et al., 2004). The usually weekly observations yield a continuous and reliable dataset,which most other ESE detections do not have, Fiedler et al. (1987) being an exception with their highsampling rate and long campaign length.

Chapter 7 summarises the results and discusses the consequences of this work for future research.

11

Figure 9: Positions of the LOFAR stations in Europe. Image copyright: Astron, theNetherlands Institute for Radio Astronomy. Downloaded from https://www.astron.nl/lofar-telescope/lofar-telescope.

2 Hardware and ObservationsThe observations used in this thesis were taken with the German modules (usually called ’stations’) ofthe LOw Frequency ARray (LOFAR), operated by partners of the German LOng Wavelength consortium(GLOW). They are placed in Effelsberg (DE601), Unterweilenbach (DE602), Tautenburg (DE603), Pots-dam (DE604), Julich (DE605), and Norderstedt (DE609). In this chapter, the basic principles of the LOFARtelescopes are explained in Section 2.1 and detailed information on the observations is given in Section 2.2.In Section 2.3, the procedure of the basic data analysis for this thesis is described. As this thesis focuses onDM measurements, the consistency of these measurements between the different stations is checked with acouple of simultaneous observations described in Section 2.4.

2.1 The LOw Frequency ARray (LOFAR)As described by van Haarlem et al. (2013), LOFAR consists of 40 stations in the Netherlands and 12 inter-national stations (see Fig. 9), including the recently (2014) built Bielefeld-Hamburg station in Norderstedtand the three Polish stations which are operational since 2016. The inner part of the core (also known as‘Superterp’) in the Netherlands consists of 6 stations.

LOFAR works as an interferometer. Every station consists of multiple dipole antennas, which areplaced at fixed positions. To be able to point the telescope at different positions in the sky, the signalsfrom the antennas get different phase offsets, which simulates a longer path length and thus only signalsfrom a specific direction interfere constructively. It is also possible to use multiple stations together as alarge interferometer to get very high resolution, but this is difficult in practice due to variable ionosphericpropagation delays.

The stations observe with two different types of antennas (see Fig. 10). The low-band antennas (LBAs)cover the frequency range from 10 MHz to 90 MHz. The high-band antennas (HBAs) work at frequenciesbetween 110 MHz and 240 MHz and are used in this thesis. The international stations consist of 96 LBAsand 96 HBAs, whereas the Dutch stations only contain half the number of antennas. Additionally, the HBAsof the inner core are divided into two parts.

12

Figure 10: GLOW station DE601 in Effelsberg operated by the Max-Planck-Institut fur Radioas-tronomie in Bonn. The LBAs are in the foreground, the black boxes in the background con-tain the HBAs. Image downloaded from https://www.glowconsortium.de/index.php/en/lofar-about/stations-featured.

2.2 Observations Used in this ThesisThe observations with the six GLOW stations used in this thesis were taken between 25 February 2013 and25 November 2016. Early observations until 20 August 2013 have a total bandwidth of 47.7 MHz and acentre frequency of 138.8 MHz. Some of these observations have one or more quarters of the bandwidthmissing due to instrument failure or instabilities in the observing system. To improve the measurementprecision of the DM, the bandwidth was doubled to 95.3 MHz, centred at 149.9 MHz. On 30 January2015, the bandwidth was reduced to 71.5 MHz (centred at 153.8 MHz), because the outer parts of the band(where the instrumental sensitivity is very low) were removed in order to reduce the data rate for all stationsbut DE601. The duration of an observation is usually between 15 minutes and 3 hours. The pulsars weremainly observed with weekly cadence over a period of about three years, primarily with the LOFAR stationsin Effelsberg, Julich, and Tautenburg.

The long-term DM time series I computed were used to be able to see overall DM variations or abruptchanges in DM, that indicate small-scale variations which are the main subject of this thesis. The mostpromising pulsar candidates to show short-term DM variability were additionally observed intensivelythroughout several weekends between 27 November 2015 and 22 August 2016, mainly with the LOFARstations in Potsdam and Norderstedt. Each of the selected pulsars was observed around 15 to 25 times forthree hours during each weekend, which equals a usually continuous observation of the pulsar for up toabout 3 days. This way, multiple short-term series of high-precision DM measurements are obtained forthose pulsars.

In my Bachelor’s thesis (Donner, 2014), I searched for DM variations towards 80 pulsars using GLOWobservations. From this sample, I selected the pulsars with the highest DM measurement precision tomaximise the chances of detecting short-term variations. In addition, pulsars with clearly visible trends (likePSR J2219+4754) or with strongly scattered DM measurements (like PSR J1509+5531) were selected.

Table 1 shows basic information on the observed pulsars taken from psrcat. The sample includes avariety of pulsars with different properties: PSRs J1300+1240 and J2145–0750 are MSPs, PSR J2145–0750

13

Table 1: Basic information on the pulsars observed in this thesis, taken from psrcat. The tableshows the pulsars’ names, the pulse periods P, the proper motions µ and the DM. σDM is themedian DM measurement uncertainty from Donner (2014).

Name Name P µ DM σDM(J2000) (B1950) (s) (mas/yr) (cm−3 pc) (cm−3 pc)

J0332+5434 B0329+54 0.71 19.5(4) 26.76 1.9e-04J0700+6418 B0655+64 0.20 10(8) 8.78 1.6e-04J0814+7429 B0809+74 1.29 50.1(4) 5.73 9.5e-04J0953+0755 B0950+08 0.25 29.53(8) 2.97 2.0e-04J1136+1551 B1133+16 1.19 375.5(4) 4.85 1.5e-04J1300+1240 B1257+12 0.0062 96.15(7) 10.17 1.1e-04J1509+5531 B1508+55 0.74 96.68(7) 19.62 1.1e-04J1921+2153 B1919+21 1.34 36(6) 12.44 3.0e-04J2018+2839 B2016+28 0.56 6.7(4) 14.20 2.1e-04J2145–0750 - 0.016 13.1(3) 9.00 1.5e-04J2219+4754 B2217+47 0.54 32(7) 43.50 0.6e-04

being frequently observed by PTAs (see, e.g. Verbiest et al., 2016). Two of the selected pulsars, namelyPSRs J0700+6418 and J2145–0750, are in binary systems. PSR J1300+1240 is a pulsar with three knownplanets, which are the first extrasolar planets ever discovered. The other pulsars are mostly very bright slowpulsars (e.g. PSR J0332+5434, which is the brightest pulsar in the northern sky). Due to the high S/N of theobservations, they offer very precise DM measurements, which are as accurate as those taken for preciseMSPs.

2.3 Data ProcessingThe data analysis has been carried out using the psrchive (Hotan et al., 2004) and tempo2 (Hobbs et al.,2006) software packages. Plots and simple fits are created with gnuplot and the psrchive program psrplot.

Each observation is collected in datacubes, commonly called ’archive’, where the flux density of thereceived radio waves is stored as a function of the pulsar’s rotational phase, time, frequency and polarisationparameter. In the used data, the entire rotational phase is divided in 1024 equally spaced steps called ’bins’.Besides this, the data are organized such that a series of subsequent pulses for a total of 10 seconds ofintegration is coherently added and averaged to give a ’sub-integration’. The entire frequency band isdivided in 195 kHz-wide subparts called ’channels’. Additionally, the polarisation information per bin,sub-integration and frequency channel is stored in 4 parameters.

As a first step, the data were cleaned from radio frequency interference (RFI). Different man-made sig-nals can be detected by the telescopes and are typically much stronger than the source to observe. Theexcision of these signals has been carried out by using the script coastguard by Lazarus et al. (2016). Fig-ure 11 shows an example of how removing the RFI can reveal a pulsar signal. Even though the observationis rather bright and the pulsar is detected in all sub-integrations and frequency channels (after removingRFI), the RFI completely dominates the integrated flux as it is much brighter than the pulsar signal, whichbecomes clear when looking at the bottom two images of the figure.

The data were then time-averaged. To avoid adding slightly shifted sub-integrations, an updated timingmodel was first applied to the data. To get more homogeneous data, the data with a bandwidth of 95.3 MHzhave been cut down to match the 71.5 MHz bandwidth data. This way, the subsequent investigation ofthe DM and pulse-shape variations is more reliable, for example because any frequency-dependent profileevolution or chromatic DMs affect all post-August 2013 data equally. Still, different telescopes have dif-ferent RFI bands, which are always removed for this specific telescope, so the dataset is never completelyhomogeneous, as there are also observation-specific RFI channels that are removed.

To be able to calculate the ToAs of the different observations, a method is needed to measure the exactposition of the pulse in the data file, i.e. the time that passed from the start of the observation until the pulseis detected. A very basic method is to measure the centre of flux in the observation, i.e. by calculating thepulse phase at which the integrated flux density beforehand and afterwards is the same. The resulting time

14

Figure 11: Lowest third of the band for an observation of PSR J2145−0750 from 19 August 2016 intime (top), frequency (centre), and flux (bottom) plotted against pulse phase before (left) and after(right) RFI removal. The greyscale indicates the flux in the given sub-integration or frequencychannel in the first two plots, darker areas implying a stronger signal.

15

is then added to the MJD of the observation to get the exact pulse arrival time. A more accurate approach isto use a standard template pulse profile with frequency resolution. These standard templates were createdby adding a few subsequent observations with a high S/N and contain the full frequency information ofthe observations. Therefore, any frequency dependence of the pulse shape is contained in the template anddoes not bias the resulting ToAs if, for example, the pulse is wider at some frequencies. This way, intrinsicvariations of the pulse shape across the band can be modelled by the template and do not corrupt the results.

For a few very faint pulsars (like PSR J2145–0750), a full frequency-resolved template is not possibleas the noise is too high to be able to detect the pulsar in the individual frequency channels. In this case, thefull resolution template is split into three bands which are then integrated in frequency, i.e. three frequency-unresolved templates cover the frequency ranges from 118 MHz to 142 MHz, 142 MHz to 166 MHz, and166 MHz to 190 MHz. To further reduce the problem of noise in the template, analytic templates are oftenused. The psrchive program paas fits a number of von Mises functions (which look similar to Gaussians,but are periodic) to a given observation. Those analytic templates are then aligned by cross-correlationwith the program pas. This method has also been applied as a first fast analysis to get an overview of thepotential results. For the interesting candidates, frequency-resolved templates were used to improve theToA precision, especially when the pulse profile varied with frequency.

The ToAs of the pulses for all channels were then calculated by cross-correlating the pulse profiles ofeach channel with the corresponding frequency channel of the standard template using the psrchive programpat.

Around 30 outlier ToAs (out of a total of 366 ToAs) were typically removed per observation. Theseoutliers occur when the S/N in a frequency channel is so low that the cross-correlation finds a noise peak,or when the channel has been removed during the process of RFI excision.

All DM measurements were taken by fitting Equation 5 to the ToAs of the different frequency channelsof an observation with tempo2. The program automatically multiplies the resulting uncertainty by the squareroot of the reduced χ2 of the fit to account for unmodelled structure in the residuals. This can occur due to afrequency-dependent pulse shape, which can be either intrinsic to the pulsar or caused by effects of the ISMlike scattering. A chromatic DM (see Section 1.2.5) can also lead to frequency-dependent residuals afterfitting for DM. Since all of these effects can vary with time, they may not be absorbed into the standardtemplate.

2.4 DM Comparison Between Different GLOW StationsMeasurements of the DM are the main focus of this thesis, therefore the consistency of the results fordifferent stations should be checked. For this purpose, simultaneous observations were started on 10 May2016 with all of the GLOW stations except Norderstedt. Four pulsars were observed two or three times,with each observation lasting between one and two hours.

The DM was calculated for every observation as described in Section 2.3. The calculations were carriedout for every pulsar and telescope separately. The results are shown in Figure 12. It is clear that the DMmeasurements are consistent between the different stations for all pulsars except PSR J0332+5434. For thispulsar, the Julich station has a significantly lower DM value than all other stations, which provide consistentresults. Looking at the long-term observations of this pulsar (see Fig. 13), it becomes evident that there isa drop in DM for this telescope between 25 July and 2 August 2015. Before this drop, the data betweendifferent telescopes are consistent, but afterwards they are not, so there seems to be some problem withthe observations of this pulsar with the Julich station from that date on. The sudden drop also coincideswith the time when the observing mode for this pulsar was changed in Julich in order to provide higherfrequency resolution, so this change probably caused the offset somehow. As the technical investigation ofthese problems lies outside the scope of this thesis, the Julich data are ignored from that point on for thispulsar.

16

0.0e+00

5.0e-04

1.0e-03

1.5e-03

2.0e-03

0 0.02 0.04 0.06

∆D

M (

cm

-3pc)

MJD - 57519.45

PSR J0332+5434

DE601DE602DE603DE604DE605

0.0e+00

2.0e-04

4.0e-04

6.0e-04

0 0.05 0.1 0.15 0.2

∆D

M (

cm

-3pc)

MJD - 57518.69

PSR J1136+1551

DE601DE602DE603DE604DE605

0.0e+00

2.0e-04

4.0e-04

6.0e-04

0 0.05 0.1 0.15 0.2

∆D

M (

cm

-3pc)

MJD - 57518.94

PSR J1509+5531

DE601DE602DE603DE604DE605

0.0e+00

1.0e-04

2.0e-04

3.0e-04

0 0.05 0.1 0.15 0.2

∆D

M (

cm

-3pc)

MJD - 57519.19

PSR J2219+4754

DE601DE602DE603DE604DE605

Figure 12: DM measurements for four different pulsars compared between five different GLOWstations. The stations give consistent measurements, except for the Julich station (DE605) whenobserving PSR J0332+5434. All DM values are in units of cm−3 pc.

-2

-1

0

1

2

56400 56600 56800 57000 57200 57400

2013.3 2013.8 2014.4 2014.9 2015.5 2016.0

∆D

M (

10

-3cm

-3p

c)

MJD

year

DE601DE603DE605DE609

Figure 13: DM variations in the direction of PSR J0332+5434. The different point types indicate differenttelescopes. It is striking that the Julich data show a drop around MJD 57230, after which it is inconsistentwith the other station used at that time, namely Norderstedt.

17

-2

-1

0

1

2

57355

∆D

M (

10

-3cm

-3p

c)

57389 57397 57502 57524 57530 57537 57543

-2

-1

0

1

2

57558

∆D

M (

10

-3cm

-3p

c)

57572 57578 57594 57600 57607 57614 57621

Figure 14: DM variations for PSR J1509+5531 on daily time scales. The x-axis is not continuousand is divided into 16 long observations, each lasting 2 to 3 days, centred on the given MJD. The(subtracted) baseline DM value is 19.619 cm−3 pc for all sub-plots. The five observations whichshow a significant trend when fitting a straight line to the data have that fit overplotted. Some ofthe early DE609 observations (e.g. on MJD 57397) are strongly affected by RFI such that manyfrequency channels had to be removed, which is why the uncertainties of their DM values are veryhigh.

3 Intra-day DM VariabilityIn this chapter, DM variations on time scales of a few days or less are analysed in continuous observationsof PSRs J1509+5531 and J0332+5434, i.e. those pulsars have been observed several times for two to threedays. The occurrence of these variations and their impact on pulsar timing experiments is quantified.

Figure 14 shows the intra-day DM variations for PSR J1509+5531 in a sample of 16 continuous obser-vations. A linear model was fit to each long observation through a least-square fitting routine in gnuplot.If the slope of the linear fit is larger than three times its uncertainty, the fit is plotted as well in the figure.For five of the 16 observations, such a significant trend is detected, four of which have a slope at leastfive times larger than the uncertainty. Figure 32 shows that the average value of the DM over the entiredataset (about three years long) does not evolve strongly on the large time scales. However, the DM mea-surements derived from individual observations are largely scattered. Indeed, besides the strong intra-dayvariability, Figure 14 also shows that significant offsets between these observations occur, indicating thatnon-negligible DM gradients took place during the (usually) five days of separation between the observingsessions.

The pulsar is located at a Galactic latitude of Gb = 52.29o at a distance of 2.10 kpc (see Chatterjee et al.,2009), which is far outside the Galactic plane. As most of the Galactic gas is rather close to the Galacticplane (see Section 1.2), the gas causing the strong DM variations is probably rather nearby, inside or atthe edge of the local bubble, the low-density region of the ISM in which the Solar system is located. Intheir paper, Lallement et al. (2003) map the edges of the local bubble by measuring Nai (neutral sodium)absorption lines in spectra of nearby stars with known distances. They present three perpendicular cross-sections of their map, one of which can be used to see the shape of the local bubble in the direction ofPSR J1509+5531, as it is located at a galactic longitude of about 90o (Gl = 91.33o). Figure 15 containstheir graphic with the addition of the line of sight direction for PSR J1509+5531. It is striking that the localbubble is rather open in the direction of the pulsar, although there is a small overdensity of neutral gas inthe line of sight, at the edges of which turbulence in the ionised medium is likely to exist: when there isan overdensity and thus turbulence in the neutral gas, there is probably also turbulence in the ionised gas,as the two interact with each other at their interface. Another support for the idea of nearby small cloudscausing the DM variations is the rather weak large-scale variability in the DM time series, which should bemuch stronger in the case of standard Kolmogorov turbulence in the Galactic halo.

18

+

LOS towardsPSR J1509+5531

Figure 15: Cross section through the local bubble with the addition of the line of sight (LOS) to-wards PSR J1509+5531. The Galactic centre is located at the position of the viewer. The horizontalaxis lies in the Galactic plane and the y-axis goes through the Galactic poles. The units on the axesare pc. The Solar System is at the origin. The dashed and continuous contours indicate the regionswhere the equivalent width of the Nai spectral line is lower than 20mÅ and 50mÅ, respectively,indicating more absorption and thus more neutral gas outside these regions. The ’+’ on the lineof sight indicates the position of the turbulence as estimated by Wucknitz et al. (2017). The dotsindicate the stars towards which the Nai measurements were made. Figure taken from Lallementet al. (2003).

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Figure 16: DM variations for PSR J0332+5434 on daily time scales. The x-axis is not continuousand is divided into 16 long observations, each lasting 2 to 3 days, centred on the given MJD. The(subtracted) baseline DM value is 26.7633 cm−3 pc for all sub-plots.

Using the proper motion µ = 97 mas/yr from Table 1, one can calculate the transverse size x of thestructures that cause the short-term variability on a weekly scale (t = 7 days). The distance to these structuresis estimated to be d = 124 pc, which is the value Wucknitz et al. (2017) obtained by observing this pulsarwith the VLBI (Very Long Baseline Interferometry) technique using the GLOW telescopes. This distanceis consistent with the overdensity of neutral gas in the local bubble along the line of sight seen in Figure 15.This results in:

x = d · µ · t = 0.2 AU. (15)

Structures of this scale have already been detected by Brisken et al. (2010), who analysed the scintilla-tion of PSR B0834+06. In their paper, they conclude that the scintillation is due to scattering by inhomo-geneous filaments with dimensions of 0.05 AU by 16 AU. However, the filaments they observed are muchfurther away, outside the local bubble and likely have a different origin, as the structures discussed in thisthesis seem related to the edge of the local bubble.

Figure 16 shows the intra-day DM variations for PSR J0332+5434. In the whole dataset of 16 con-tinuous observations there are none that show a significant (> 3σ) trend, although there is some scatter insome of the observations. Also overall the DM values are very constant with a WRMS (Weighted RootMean Square) value of 3 · 10−4 cm−3 pc. Outliers can happen due to RFI that is difficult to remove (e.g. onMJD 57439) or short observation durations down to only 15 min (e.g. on MJD 57494). One thing to noteis that this pulsar’s proper motion is smaller than the one of PSR J1509+5531 by a factor of about 5 (seeTable 1), so it is less likely for this pulsar to show extremely fast DM variability.

The DM variations on the order of 10−3 cm−3 pc discussed in this chapter lead to a dispersive delay ofabout 2 µs at 1.4 GHz (see Equation 3). This is more than the precision that PTAs aim for (see Section 1.1.3)and thus has to be dealt with. Due to the very short time scale of the DM variability, each (monthly taken)PTA observation would be affected by a different DM offset. As shown in Section 1.2.3, correcting for DMvariations with the high frequency data with single observations severely decreases the precision and thussimultaneous (more precise) low-frequency observations would be needed to correct for this type of DMvariability precisely. More details on these corrections are given in Section 7.1.

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Figure 17: DM variations in the direction of PSR J2219+4754 for the upper and lower half of theband. Note that until MJD 56524, the observing bandwidth was only 47 MHz (see Section 2.2),which leads to a worse DM precision and less sensitivity to chromatic DM differences.

4 Chromatic DMsAs described in Section 1.2.5, the DM can be frequency dependent due to the larger volume of spacesampled at lower frequencies compared to higher frequencies. To investigate this effect, the observationswere split into two bands with equal bandwidth, namely 118-154 MHz and 154-190 MHz. The DM wascalculated as discussed in Section 2.3 for the two bands separately for every observation. By definition,the lower- and upper-frequency DMs are equal for observations that are temporally close to the standardtemplate, so a direct comparison of these two is not possible as the ‘real’ DM value is correlated with theprofile’s pulse shape evolution in frequency. However, the differences in the DM variations do containinformation on the chromaticity of the DM.

Figure 17 shows the chromatic DM variations of PSR J2219+4754. The full description of the analysisfor this pulsar is described in detail in Chapter 5. A very good example of the lower frequency showing asmoothed-out version of the higher-frequency DM time series (see Section 1.2.5) is the central peak nearMJD 57000. This peak is much narrower at high frequencies, as the overdensity also affects the lowerfrequencies even when it has moved out of the line of sight of the higher frequencies. What does not fit thisexplanation is the fact that the peak value is the same for the two bands, while the lower-frequency peakshould have a lower DM value in the chromatic DM framework. This indicates that the low-frequency DMvalues are shifted upwards due to the choice of the template on MJD 57162, which in turn implies thereshould be some level of chromaticity left at MJD 57162. Also, it is striking that the steep drop after thiscentral peak is steeper at low frequencies, which is in contrast to the idea of a smoothed-out version of thehigher-frequency DM presented by Cordes et al. (2016). Between MJDs 57300 and 57500, the two bandsshow an opposing trend for over 200 days. Both of these problems could be explained by an overdensity(or hole) coming close to the line of sight, which has not come close enough to be detected in the high-frequency data. Further observations of this pulsar could further clarify the mechanisms of chromaticity,but probably the model described in Section 1.2.5 is too simplistic to provide a self-consistent explanation.

Figure 18 illustrates the frequency dependence of the timing residuals of two observations after fittingfor DM. It becomes clear that for the observation on MJD 57162, where no chromaticity is detected, theresiduals are completely flat (as expected since the template profile is taken from an observation close to

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Figure 18: Frequency-dependent timing residuals for PSR J2219+4754 after fitting for DM onMJD 57162 (left) and MJD 57494 (right). The continuous lines are smoothing splines to illustratethe overall trend.

this date, implying that any remaining chromaticity at that date has been interpreted as frequency evolutionof the pulse profile), whereas for the observation on MJD 57494, there is a non-quadratic residual structurein the residuals due to the chromaticity of the DM. These two observations are representative of the entiredataset and are chosen because they illustrate the two extreme cases well.

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Figure 19: DM variations in the direction ofPSR J1136+1551 for the upper and lower half ofthe band.

With the exception of PSRs J2219+4754 andJ1509+5531, I did not detect chromaticity in anyof the pulsars in my sample (see Fig. 19 for anexample). This is probably caused by a lowersensitivity: the DM precision is lower than forPSR J2219+4754, while the DM variations are lesspronounced, which makes a clear detection impos-sible. The only exception is PSR J1509+5531,which shows a strong scatter of very precise mea-surement points (see Fig. 32). Most low-band ob-servations of this pulsar have a higher DM (withrespect to the template observation) than the high-frequency observations. Also the DM gradients ofthe two bands sometimes differ for this pulsar.

This could also explain why Hassall et al.(2012) did not find any evidence for chromaticDMs, as all of their pulsars (PSRs J0332+5434,J0814+7429, J1136+1551, and J1921+2153) wereobserved in this thesis as well and their result of nodetection of chromaticity for those pulsars is confirmed. Note, however, that their observations were takenin late 2009, which pre-dates any of the data contained in this thesis. Since none of these pulsars show DMvariability at the beginning of our dataset (see Fig. 32), it might just be the case that they observed pulsarswithout DM variability, which are therefore less likely to show chromaticity by definition, as turbulencein the medium is required for a chromatic DM (see Section 1.2.5). Also, it is difficult (perhaps even im-possible) to disentangle chromatic DMs and frequency-evolution of the profile with a single observation,whereas the change of the chromaticity with time can be measured reliably, as shown above.

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Figure 20: Timing residuals for the full timespan of PSR J2219+4754, observed with the GLOWstations, plotted against MJD (left) and frequency (right). Each point indicates the average residualof the ToAs of one observation (usually ∼330 ToAs averaged, left plot) or the average residualof all ToAs within a frequency interval of 1 MHz (usually ∼600 ToAs averaged, right plot). Thereduced χ2 of the fit (based on the non-averaged ToAs not shown here) is 8.7.

5 Extreme Scattering Event in the Direction of PSR J2219+4754As already seen in Chapter 4, the DM measurements for PSR J2219+4754 vary significantly over theobserved timespan and show very clear trends. In this chapter, this phenomenon is analysed in more detailand a possible explanation is given.

5.1 Timing ResultsThe ToAs of the J2219+4754 dataset have been calculated as described in Section 2.3 with a frequency-resolved standard template created from six observations taken on MJDs 57161 and 57162 (each lastingtwo to three hours), when the pulsar was observed continuously with the GLOW station in Effelsberg.Figure 20 shows the timing residuals for the entire observed timespan. The fitted parameters are the pulsarposition, the rotational frequency and its first two derivatives, and offsets between the different telescopesused (DE601, DE603, and DE605). Every observation has been corrected for its DM as calculated inSection 5.2 to avoid the DM variations being partly absorbed by other parameters, which corrupts theirvalues. From the figure it becomes clear that there are several unmodelled structures in the residuals.Figure 21 shows the variations in DM towards J2219+4754, and it is striking that the unmodelled structurein the timing residuals is less prominent during the times of little DM variations from MJD 57100 to 57600.This structure may partly be explained by variations of the pulse shape, as discussed in Section 5.2, whereI demonstrate that realistic pulse-shape variations can lead to ToA differences of up to 100 µs. Anotherelement to be considered is the chromaticity of the DM, as the low frequencies have a stronger effect onthe DM measurements and thus the average DM values used to correct the ToAs are biased by the lowfrequencies, which can be seen in the smoothing function of the right-hand plot of Figure 18, where thehigh-frequency residuals are worse than the low-frequency ones. Another possible reason for the structurein the timing residuals is the so-called timing noise, which is caused by intrinsic variations of the rotationalperiod of the pulsar, and usually shows up on longer time-scales. The reduced χ2 of the timing is 8.7, whichconfirms the sub-optimal fit. Improving the model would be very difficult due to the effects just discussed,and as the timing model does not negatively impact our DM measurements, a further improvement of thetiming model lies outside the scope of this thesis.

Measuring the proper motion with a dataset as short as this one is very difficult, as its effect on the ToAsis very small. A more reliable value can be obtained from much longer observations of the pulsar, whichhas been done by Michilli et al. (2017). Their values are:

µα = −12.2(5)masyr

, µδ = −19.5(5)masyr

. (16)

This gives a total proper motion value of µ = 23.0(5) mas/yr.

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Figure 21: DM variations in the direction ofPSR J2219+4754. The vertical line indicates thetime at which the dataset became homogeneousin frequency. The two arrows indicate the obser-vations used in Fig. 22. The (subtracted) baselineDM value is 43.48205 cm−3 pc.

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5.2 DM variationsThe DM was calculated for every observation as described in Section 2.3. The median reduced χ2 of thesefits was 2.62, which indicates that there is some weak frequency-dependent structure in the residuals ofmost observations, which cannot be removed by fitting for DM. The reason for this structure can be thechromaticity of the DM (see Section 4), but it can also be an intrinsic variation of the pulse shape (seeMichilli et al., 2017) that can behave differently at different frequencies. This implies that the uncertain-ties are slightly underestimated, but it should not influence the results significantly, as the deviation fromχ2/nfree = 1 is only small.

Figure 21 shows the results of the initial analysis. There is a very clear three-peaked feature visible,overlaid by a generally decreasing trend. In the following paragraphs, I will discuss possible origins of thisstructure in order to assess its physical meaning. Also, I will quantify possible corruptions of the obtainedDM values.

5.2.1 Impact of Scattering on the DMs

Taking only one standard template from a few observations leads to some problems. When observing aturbulent medium, the pulse profile can get widened by scattering, as the scattered photons travel alongslightly longer paths and thus arrive later than the unscattered ones. The widening is one-sided, so a cross-correlation between a scattered profile and the unscattered standard template leads to a delay in the arrivaltimes. Furthermore, this delay is frequency dependent and thus corrupts the DM. Another problem can bean intrinsic variation of the pulse shape (see Michilli et al., 2017), as such variations affect the measuredToAs, and if frequency-dependent, also influence the DM measurement. Figure 22 shows two pulse profilesof PSR J2219+4754 from two different MJDs. Besides the disappearing scattering tail, there is a secondpeak appearing in the later data, indicating intrinsic pulse-shape variations. This ”postcursor” componentis known to appear and disappear over time (see Suleymanova and Shitov, 1994). It should not corrupt thetiming significantly, given its width, low amplitude, and lack of sharp features.

The wider tail however does affect the DM measurements. Figure 23 shows the width of the pulse(which is mainly affected by the scattering tail) as a function of MJD for the different frequency bands. Themost striking feature in this plot is the gradual decrease in pulse width until the time of the end of the secondDM peak (around MJD 57100). In addition to this general trend, there is an increase in pulse width visiblein all three frequency bands during that second DM peak, as well as an increase in the width of the lowestband during the third DM peak, which is also only visible at low frequencies in the frequency-resolvedDM time series (see Fig. 17). So while there is evidence for a correlation between pulse width and DM, itis still unclear why there is so much excess pulse width in early data. Nevertheless, an interstellar origin(scattering) of the pulse width variations seems very likely due to the correlations with DM just discussed.

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Figure 23: Pulse width variations at the 10% level of the peak value in different frequency bands.The vertical line indicates the point in time after which all observations have identical centrefrequency and bandwidth. The centre frequencies of the three channels from that point on are130 MHz, 154 MHz, and 178 MHz, with a bandwidth of 24 MHz. The grey crosses with error barsshow the DM variations from Fig. 21 as a reference.

One could argue that the pulse width is affected by the DM variations, as integrating the pulse shapeover the full band with a slightly wrong DM would misalign the frequency channels and thus lead to anincrease in pulse width. To test this, the pulse width of an observation has been calculated for two differentDM values with a difference of 3 · 10−3 cm−3 pc, which is the amplitude of the second peak of the DM timeseries. This difference in DM only leads to an increase in width of 10−5 pulse periods, which is completelynegligible as it is below the phase resolution of data. So the pulse-shape variations have a different origin,which can be scattering or intrinsic to the pulsar, the former being likely due to the correlations in time withthe DM variations, as described earlier.

To quantify the corruption of the DM measurements due to pulse-shape variations, the observation withthe worst goodness-of-fit in the timing (from MJD 56570), which has a high DM value and a wide (scat-tered) profile, has been compared to an observation that fits the standard template very well (MJD 57162).The profile of the scattered observation was aligned in phase with the unscattered observation by align-ing the peaks and the leading edge. Then both the unscattered profile and the shifted scattered one weretimed against the standard profile. The scattered profile leads to an extra delay of 105.6 µs at 130.0 MHz,79.2 µs at 153.8 MHz, and 56.6 µs at 177.6 MHz. The spectral index of these values is -1.96, so it behavesexactly like additional dispersion in this case. Fitting for the DM in both cases leads to a difference of∆DM = 3.5 · 10−4 cm−3 pc. In the following analysis, this is assumed to be the maximum extra measureddispersion that the scattering can cause.

5.2.2 Profile Shape Evolution

To be able to quantify how badly the scattering affects the timing, the profile shape difference between theobservations and the standard template are shown in Figure 24. For this analysis, all frequency channelswere averaged and the residuals are calculated after aligning and normalising the peaks. The plot showsthat the pulse shape differs from the standard profile for early observations only. Additionally, it becomesobvious that the pulse shape varies exclusively after the peak, which supports the idea of a scattering tail

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Figure 24: Pulse-shape variations for PSR J2219+4754. The plot shows the difference of the pulseshape to the standard template. The vertical line indicates the position of the peak of the pulse.For a clearer view, only every third observation is plotted. The selected MJD values on the rightindicate the first observation, the first two peaks, the template observations, the third peak, and thelast observation, respectively.

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Figure 26: DM variations for PSR J2219+4754.The maximum estimated corruption due to pulse-shape variations is indicated by the dashed errorbars (which appear below the points since scatter-ing can only increase the DM).

discussed in Section 5.2.1.The maximum and minimum residuals from Figure 24 of all observations are shown in Figure 25. It

becomes clear that only early observations are affected by pulse-shape variations, and that these variationsare mainly an excess of power. As a good approximation, the scattering decreases linearly until MJD 57000,at which point it basically disappears and only noise is left, as the minimum and maximum residuals of allobservations after this date are similar to the ones for observations that are temporally very close to thestandard profile on MJD 57162. The scattering of the points can be explained by the different S/N in thedifferent observations. In the following, the corruption of the DM values due to scattering is estimated todecrease linearly until MJD 57000 as well.

Figure 26 shows the measured DM variations from Figure 21, with the uncertainties adjusted to visualisethe maximum expected effect of the scattering for each observation. As scattering can only increase the DM,this correction appears as dashed error bars below the (unmodified) points. The scattering does affect thetiming precision and corrupts the DM, but from the plot it becomes evident that this corruption does notchange the basic outcome of the analysis. When subtracting the estimated DM corruption due to scattering,the amplitude of the first peak is reduced by an almost negligible amount. More difficult to explain is thefact that the scattering decreases linearly (see Fig. 25) during a double-peaked DM time evolution. Still,DM and profile variations do appear related as the scattering disappears at the end of the large initial DMexcess. Earlier data would have been useful to see how the DM and profile variations started, but are notavailable.

Overall, this analysis confirms the study of the pulse width from Section 5.2.1, but with a lower precisionand without frequency-resolution, which makes it impossible to see the details like the small excess in pulsewidth in the lowest band during the third peak of the DM time series. However, it shows very clearly thatthe increase in pulse width is one-sided, i.e. a scattering tail and thus ISM-related.

5.2.3 Structure Function

The structure function of the DM variations is shown in Figure 27. It is striking that the structure functionfollows a Kolmogorov spectrum very well up to timespans of ∼200 days. For longer timespans the spectrumflattens between 200 and 300-day-long intervals, which can be explained by the size of the major features(peaks and dips) in the DM variations, which is about 200-300 days (see Fig. 21). For the longest timespans,the overall slope of the structure function is consistent with a Kolmogorov spectrum again although with adifferent amplitude. One should note that the longest timespans are not very well sampled by the dataset(because they are only defined by the early and late observations), and thus the structure function couldchange a lot for these timespans when adding more samples. So overall it is valid to claim that the structureis consistent with Kolmogorov, as the later data points in the structure function are not that reliable. The

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Figure 27: Structure function of the DM variations towards PSR J2219+4754. The spatial scaleis estimated for turbulence half way to the pulsar. The straight line is a fit for the amplitude of aKolmogorov turbulence spectrum with a fixed spectral index of 5/3. The fit has been carried outfor timespans τ up to 200 days. The tick marks on the axes are for factors of 2,5, and 8.

very shortest time intervals are dominated by the noise level of the DM measurements and thus are flat, soit is expected that the Kolmogorov spectrum does not fit those points well.

The spatial scale estimate in the plot is calculated based on the assumption that the turbulence is locatedhalf-way to the pulsar. More details on modelling efforts and the pulsar distance are given in Section 5.4.

As described in Appendix A.1, calculating a structure function without weighting the DM measurementsincreases the value of the structure function especially at short time intervals. This effect is clearly visiblein Fig. 27 and it also becomes clear that this effect makes the spectrum flatter and non-Kolmogorov. Thisshows that weighting of the DM values is essential when calculating the structure function for a datasetwith DM measurements at varying levels of precision, which is usually the case. Even if the observationalsetup is fixed, the pulsar’s brightness can vary a lot due to scattering, RFI, or its elevation in the sky, whichaffects our sensitivity.

5.2.4 Extreme Scattering Event

The three-peaked structure in the DM time series from Figure 21 can be interpreted as an ESE, where threeclouds move through the line of sight. The strong scattering especially in the early data also supports thisinterpretation, but it also shows that the observed structure is probably not the start of the DM variability.The shape of the DM time series (peaks and dips) is reminiscent of what Coles et al. (2015) found forPSR J1939+2134, but with a larger amplitude. In their higher-frequency dataset, they also see correlationsof the DM with scintillation parameters.

The observed DM variations can be interpreted as excess power in the structure function for time scalesup to approximately 200 days, or perhaps the ESE can be considered as part of the Kolmogorov turbulenceitself. Further observations could clarify this by making the larger scales of the structure function morereliable or by observing the end (if there is to be one) of the strong DM variations.

Generally, ESEs should be monitored over longer timespans, as they might not be as isolated as ex-pected. The extremely high DM precision of the data presented in this thesis allows investigations withmuch more detail and could be very useful when catching the end of an ESE to see how it behaves.

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Figure 28: Polarisation of the pulse profile of PSR J2219+4754 for an observation taken on 29November 2015 at 130 MHz with 24 MHz bandwidth before (left) and after (right) correcting forRM. The lower frame in each plot shows the flux density in arbitrary units for the total intensityand the linearly and circularly polarised fraction. The upper frame shows the evolution of thepolarisation angle as a function of pulse phase, the y-axis having tick marks at steps of 20o. Itsprecision is much higher after the RM corrections.

5.3 RM variationsWhen calculating RM values, the first step is always to calibrate the polarisation of the data, as the sensi-tivity of the two orthogonal polarisation directions depends on the angle at which the radiation reaches theantennae and thus the position of the pulsar in the sky (see Noutsos et al., 2015).

Once calibrated the data, one can compute the RM values and apply them to the observations. Figure 28shows the polarisation profile of a frequency-integrated observation plotted as a function of pulse phasebefore and after RM correction. Before the correction, the linear polarisation fraction is basically zero,whereas it is significant after the correction. This is due to the frequency dependence of the Faraday rotation,which depolarises the linearly polarised signal when integrating in frequency (see Section 1.2.2). Applyingpolarisation angle corrections from an RM measurement to the data before integrating in frequency thereforeincreases the linear polarisation fraction.

The psrchive program rmfit is used to calculate the RM. It takes a range of RM values and for each ofthose values, corrects the data file for the Faraday Rotation and calculates the linear polarisation fraction.The RM that results in the highest polarisation fraction is then considered as the measured RM. For thismethod to work, an initial estimate is needed around which RM values are tested. It is taken from psrcat(−35.93 rad m−2), but could also be acquired from an rmfit call with a large RM range, like [−1000 rad m−2:1000 rad m−2]. One problem with this program is that the calculated uncertainties are usually highly under-estimated and often even exactly zero.

As Earth’s ionosphere has an impact on the RM which is not negligible (see Section 1.2.2), this impactis to be subtracted to get the final results. For this purpose, three different ionospheric electron content mapsfor the ionospheric corrections are used (namely code, rob, and igsg). This way, a more reliable view of theRM time series is achieved as it is known that there are some complications with those maps, but they arethe best currently available. The values of code and igsg are always consistent, with the igsg uncertaintybeing a bit smaller. The rob however always gives a smaller value of the ionospheric electron content thanthe other two, with a much smaller uncertainty.

Figure 29 shows the RM variations over the observed timespan. For a better visualisation of the trends, anatural smoothing spline has been calculated with gnuplot, weighting the data points by the inverse of theirvariance. In the third plot, all data points are weighted equally because the uncertainties of the ionosphericcorrections seem underestimated, as they are sometimes even exactly zero, which sets the weight of thosepoints to infinity.

Around 10 outlier RM measurements were removed, half of which were completely wrong and withlarge uncertainties. This can happen when the degree of polarisation in the observation is very low and thecorrect RM cannot be found. Other outliers could be caused by observing the pulsar far from the zenith,

29

where the polarisation calibration and ionospheric correction become more difficult and unreliable. Thisusually happens when the pulsar is observed for a longer timespan, like on MJD 57162, the date on whichthe template observation was recorded.

Overall, the RM is rather constant. The uncertainties look overestimated for the first two ionosphericmaps, as the scatter of the measurements (with the exception of a few outliers) is much less than the sizeof the error bars. Still, there is some weaker structure visible consistently in all three plots, mainly thetemporary drop in RM (which is an increase in absolute numbers as the RM is negative) shortly beforeMJD 57000, which coincides with the second main peak in DM. The first peak in DM also shows thisbehaviour, but not as clearly due to the lower measurement precision. This indicates that the overdensity inthe ISM might be magnetically confined. Another striking feature is the increase in RM at the end of thedataset.

Through rmfit and the ionospheric correction, I computed an average RM value of −35.8 rad m−2. Whencombined with an average DM value of 43.5 cm−3 pc, and by using Equation 7, one obtains the averagemagnetic field strength along the line of sight, which is < B|| >= −1.0 µG. As the second peak in the DMtime series shortly before MJD 57000 coincides with the dip in RM (see Fig. 29), the magnetic field strengthinside this overdensity can be calculated. The extra DM of this overdensity with respect to the following100 days of constant DM as baseline is ∆DM = 3 · 10−3 cm−3 pc, while the RM offset is about ∆RM =

−0.2 rad m−2. From Equation 7 follows an average magnetic field strength of < B|| >= −80 µG inside theoverdensity, which is much higher than the average value along the entire line of sight to the pulsar. Notethat this measurement is rather uncertain because of the large uncertainties of the RM measurements, but itis still significant.

5.4 ModellingThe three peaks in the DM time series (see Fig. 21) can be modelled as three clouds moving through the lineof sight. In the following, a simplistic analysis of their properties is carried out and the results are comparedto similar analyses from literature.

Due to the high cadence and precision in DM, finding a model that fits the data well is very complicatedgiven the amount of detail in the data. The simplest model is to just assume spherical, homogeneous blobsof ionised matter that do not move with respect to Earth. One problem is that the actual DM baseline withoutthe overdensities is unknown as the start of the event has not been observed and it is still ongoing. For arough picture of the individual blobs, the second peak is modelled considering the successive 100 days ofconstant DM as baseline, which is also the baseline DM for all plots and contains the template observations.The time from the maximum to the end of the peak is 150 days, so the width of that blob is estimated tobe 300 days. Using a proper motion of 23.0 mas/yr (see Michilli et al., 2017), the angular size can becalculated as θ = 19 mas. To get the physical size, the distance to the cloud is needed. This distance canonly be estimated, but it has to be lower than the distance to the pulsar, which is 2.2 kpc in the NE2001model from Cordes and Lazio (2002) (see https://www.nrl.navy.mil/rsd/RORF/ne2001/). Theymodel the Galactic electron density and thus can calculate a distance estimate for a pulsar from its positionand DM. Recently, Yao et al. (2017) published a new Galactic electron density model (see http://119.78.162.254/dmodel/), which gives a consistent distance estimate of 2.4 kpc. Combining the angular sizeof the peak and the distance (from the NE2001 model) to the pulsar, the maximum size of the cloud (if itwas directly in front of the pulsar) is 42 AU.

Due to the assumption of a spherical object, the maximum pathlength through the medium is equal to itssize, and from this pathlength and the maximum DM offset (with respect to the baseline), one can calculatethe average excess electron density in the medium. From the upper limit on the cloud size follows a lowerlimit on the average extra electron density, which is 15 cm−3. Figure 30 shows the estimated cloud size andits electron density depending on its distance to Earth. Some literature values of similar estimates are addedfor comparison: the first ESE observed by Fiedler et al. (1987, without distance estimate), an ESE observedin flux density and timing residuals modelled as two clouds by Cognard et al. (1993), the three-year-longESE observed in flux density only by Maitia et al. (2003, without uncertainties), and the models of fourESEs observed by Coles et al. (2015, without distance estimates).

When comparing the size and density estimates to typical values from literature, this particular ESEcould be anywhere in our line of sight without falling out of the sample, as the densities and sizes inliterature span the whole range of possible values. All calculations for this simple model are only veryrough estimates, as it is difficult to disentangle the different components of the DM time series and to find

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Figure 30: Estimated cloud size and electron density for the second component of the ESE de-pending on its distance to Earth, represented as solid lines in the plot. Horizontal lines representvalues from literature without distance estimate. The lowest cloud density estimate by Coles et al.(2015) actually represents two different ESEs with electron densities of 3.7 cm−3 and 3.75 cm−3 notdistinguishable in the plot. The cloud size estimates of Cognard et al. (1993) for the two cloudsforming the ESE are also very similar (0.050 AU and 0.094 AU) and thus indistinguishable in theplot.

the right baseline. Probably, there are multiple clouds of various sizes, which can overlap. The steepest DMdecrease of the second peak, which starts shortly after MJD 57000, can be interpreted as the edge of onecloud and lasts about 75 days. Estimating the cloud to be located half-way to the pulsar, this edge is about5 AU thick. This is roughly on the order of what Brisken et al. (2010) found for elongated filaments, so itcould be the case that the filaments they see are actually the edges of ionised clouds.

5.5 Comparison to LiteratureAhuja et al. (2005) observed this pulsar from 2001 to 2002 with the Giant Metrewave Radio Telescope(GMRT) at the two following frequency bands: 325 - 341 MHz and 601 - 626 MHz. Due to the higherfrequencies, they have less DM precision, but they still see significant DM variations during that period.Figure 31 shows their dataset and an overview in combination with the dataset presented in this thesis.Their data show a gradual increase of about 2 · 10−2 cm−3 pc over one year, followed by a turnover point.Compared to the steepest parts of the ESE described in Section 5.2, which have slopes on the order of7 · 10−3cm−3 pc/yr, the gradient found by Ahuja et al. (2005) is about three times as steep and lasts aboutthree times as long. Also, their average DM is about 3.5 · 10−2 cm−3 pc higher. This offset is too large tobe explained by profile shape evolution, but may be a consequence of gradients in the turbulent ISM, as itmatches the trend dDM/dt = 0.0002

√DM proposed by Hobbs et al. (2004) quite well. This leads to the

conclusion that the observed structure could be a part of a much bigger inhomogeneity and more variationscan be expected in future observations.

5.6 ConclusionsI have presented strong DM variations in low-frequency data of PSR J2219+4754, which I ascribe to agroup of interstellar clouds typically referred to as an ESE. The ESE in this thesis is one of the longest ob-

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served to date. It is also the first one observed in electron density and scattering at the same time althoughthe quantitative analysis of the scattering is beyond the scope of this thesis, given the complexity of the in-duced pulse-shape variations. The frequent observations with LOFAR allow for detailed and highly precisemonitoring of the DM evolution. Coles et al. (2015) for example only had about five measurements acrossany ESE, whereas in this paper there are over 150 observations during the entire event, and about 40 duringthe modelled second peak. The high measurement precision complicates any modelling efforts, which arealready complex due to the large number of unknown parameters like the distance or proper motion of theclouds and the DM baseline. The simple spherical model discussed in this chapter does not tell much aboutthe ESE, but does provide some limits on its electron density and size. Further observations of ESEs willhelp to improve the constrains on the size as the distance does not dominate the measurement when a largesample of observations is given, assuming a homogeneous distribution of ESEs in the Galaxy.

Still, these results can be used to show how events like this can influence pulsar timing and createunmodelled structures in ToAs, even at much higher observing frequencies than the ones at which LOFARoperates. This structure can be partly absorbed by other parameters in the timing model like the positionor proper motion, which can therefore be badly corrupted. Although the timing model used in this thesisis rather poor (high reduced χ2, see Section 5.1), the impact of DM variations on other parameters canbe shown with the example of the pulsar’s position: with the DM corrections, the fit of the timing modelgives a significantly different position (82σ difference in right ascension, 63σ in declination), compared toa fit without DM corrections. Also, the reduced χ2 goes down from 12 to 8.7, so correcting for the DMvariability improves the timing solution significantly.

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6 Long-Term DM VariabilityFor all of the 11 observed pulsars (see Table 1), long-term DM variability over more than 2 years has beenanalysed. In this chapter, the results for all of these pulsars except PSR J2219+4754 (see Chapter 5) arepresented and divided into pulsars with constant and variable DM. The DM time series for all of thesepulsars is shown in Figure 32.

6.1 Pulsars with Constant DMA couple of the analysed pulsars show very stable DM values for more than 2 years. The most constraininglimit on the DM variations is measured for PSR J0700+6418, which is stable with a WRMS of only 2 ·10−4 cm−3 pc. The DM values for PSRs J0814+7429 and J0953+0755 are stable on with a WRMS of1 · 10−3 cm−3 pc and 7 · 10−4 cm−3 pc, respectively, which is much smaller than the variations observed forPSR J2219+4754 (see Chapter 5). There is some indication of intra-day variability in the latter two pulsars,as the median DM uncertainties for those are 4 · 10−4 cm−3 pc and 2 · 10−4 cm−3 pc, respectively, which is afactor of a few smaller than the WRMS. As both of these pulsars are very nearby (433 pc and 262 pc, seeBrisken et al., 2002), it is likely that this small-scale structure is caused by turbulence at the edge of thelocal bubble, similarly to PSR J1509+5531 (see Chapter 3).

Pulsars with such constant DM values (especially PSR J0700+6418) are very useful for high-precisionpulsar timing experiments, as the ToAs are not influenced by any DM variations and pulse-shape variationsinduced by scattering.

6.2 Pulsars with Variable DMMost of the pulsars observed in this thesis show DM variability in various forms.

PSR J0332+5434 shows a rather stable DM time series, but there is a 200 days long small gradientbetween MJDs 57000 and 57200, after which not many measurement points are available, but the DMseems to remain rather constant. Also it becomes clear from the analysis in Chapter 3, that the DM is veryconstant at that time.

PSRs J1136+1551 and J1921+2153 show sudden increases on the 10−3 cm−3 pc scale after more thana year without variations. The scale of the features is similar to that of the ESE in the direction ofPSR J2219+4754 discussed in Chapter 5, although the DM excess is smaller by a factor of 2 to 3. Fur-ther observations are needed to see how long these possible events last and how they develop.

PSR J1300+1240 shows a generally decreasing trend with a dip at the end. The overall variations areon the order of 5 · 10−4 cm−3 pc and non-trivial to model as part of a timing model. This is different forPSR J2018+2839, where a clear linear trend is visible for observations after MJD 56600, which can be welldescribed by a DM derivative in the timing model.

PSR J1509+5531 is a very special pulsar and it shows the strongest DM variability of all pulsars con-tained in this study. The short-term variations discussed in Chapter 3 are on the order of 10−3 cm−3 pcover one week, but there is also large-scale structure visible (the DM rises, drops, and rises again). Oneshould note that this is a pulsar with a rather high proper motion (see Table 1), which makes short-term DMvariations more likely to happen as the pulsar samples the ISM rather fast.

PSR J2145–0750 is a MSP, which is also used by PTAs (see, e.g. Verbiest et al., 2016). It is very faintand thus the data have been partially integrated in frequency to be able to measure the ToAs for four differentparts of the band, with which the DM has then been calculated. Calculating the DM over such few points cancause underestimation of error bars when the data by chance fit the function very well, as tempo2 multipliesthe resulting uncertainty by the square root of the fit’s reduced χ2. This explains the few extremely precisepoints, the uncertainty of which thus should not be believed. For PTA experiments, timing stability iscrucial. Shortly after MJD 57400, there is a temporary increase in DM of 8 · 10−4 cm−3 pc (obtained fromfitting a Gaussian), which could be problematic for the timing when the ToAs are not corrected for thisexcess in DM. This is an important finding, as this extra dispersion is very difficult or even impossibleto accurately measure with the PTA data, as they are typically taken at much higher frequencies than ourobservations. However, this excess is not related to turbulences in the ISM, but rather to Solar activity.The DM peaks exactly at the time when the Solar angle (the angle between the pulsar and the Sun in thesky) goes down to only 6o. The pulsar’s radiation passes very close to the Sun and thus gets dispersed bythe charged particles in the Solar wind. tempo2 attempts to correct for the impact of the Solar wind on

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the ToAs, but the model is not accurate enough as the Solar activity and thus also the Solar wind varieswith time and is highly turbulent and unpredictable. The effect already starts to become visible at a Solarangle between 20o and 40o. Usually, only observations with a Solar angle below 5o (so none of the onesfrom this thesis) are considered to be affected by unmodelled Solar wind effects (see latest IPTA releaseby Verbiest et al., 2016). Still, the observed DM excess of 8 · 10−4 cm−3 pc corresponds to an extra timingdelay of 1.7 µs at 1.4 GHz, which is not negligible. Deller et al. (2016) have shown that proper motion andparallax measurements for two pulsars including PSR J2145–0750 are inconsistent with model-independentmeasures from VLBI and ascribe this to the Solar wind. For the other times when the pulsar comes thisclose to the Sun during the observed period, no clear DM excess is visible, beyond that already correctedfor by the Solar-wind model implemented in tempo2.

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Table 2: Fraction of data that is affected by peak-to-peak dispersive delays below or above 1 µs,rescaled to an observing frequency of 1.4 GHz. The last row shows in how much data any of thethree types of variability occurred.

fraction of dataaffected by > 1 µs <= 1 µs

ESEs 18% 82%slow variations 37% 63%rapid variations 27% 73%

total 55% 45%

7 Discussion and ConclusionsIn this thesis, I have detected DM variations at all of the observed time scales ranging from a couple of daysto a couple of years. For the following statistics, the DM variability of all of the 11 observed pulsars is usedto quantify how often different types of DM variability can be expected to occur. Note that these statisticsare somewhat affected by the limited number of lines of sight that were investigated, but also note that thesample is randomly spread through the Galaxy and so may be representative of the pulsar population asa whole. At the shortest observed scales (2-3 days), 16% of the observations show significant variationsin DM. This number however is obtained from only two pulsars, both of which were expected to showshort-term variations, and thus is probably biased.

Table 2 shows how much data is affected by ESEs, slow variability (i.e. slow enough to be modelled withone observation per month), and fast variability. The values are obtained by identifying the different typesof variability through visual inspection in Figures 21 and 32. A deeper, statistical analysis could reveal evenmore variability as it may be possible to find structures in the parts that look rather noisy to the eye. Anyvariability, which leads to a peak-to-peak dispersive delay of more than 1 µs at 1.4 GHz observing frequency,is considered in the following statistics: 37% of the observations are affected by slow DM variability, 27%of the observed timespans show significant short-term variability, and 18% of the observations were takenduring (possible) ESEs. Only 45% of the observations are not significantly affected by any of the threetypes of DM variability. This supports the findings of Verbiest et al. (2016), who also found DM variabilityto be highly common even though they only have sensitivity to slow variations.

Overall, some kind of long-term variability has been observed in 8 out of the 11 pulsars, which wereselected on their DM measurement precision. This clearly shows that DM variations are very common.

I detected a very long and intense potential ESE in the direction of PSR J2219+4754, which is present inthe entire dataset and thus probably has not ended yet. I analysed it in detail and many different parameters,like the pulse width, which is affected by scattering. Also, I have detected weak variations in the RM duringthe ESE, which allows the first estimate of the magnetic field strength of an ESE. The main difficulty inthis analysis is Earth’s ionosphere, which has a time-dependent impact on the RM measurements that ismeasured with rather poor precision. Two other pulsars, namely PSRs J1136+1551 and J1921+2153 showthe beginning of possible ESEs at the end of the analysed dataset.

The DM variations in the direction to PSR J2219+4754 are consistent with a Kolmogorov turbulencespectrum, at least up to time scales up to the length of the large features in the DM time series. This showsthat a Kolmogorov spectrum can describe interstellar turbulence very well in some cases.

In addition to the DM variations at various scales, I have detected clear DM chromaticity for the veryfirst time. The effect was only theoretically predicted before and it turns out to be more complex to explainthan expected, which is partly also caused by the unprecedented DM measurement precision and cadenceachievable with the GLOW telescopes. I have found that the Solar wind models used for pulsar timing areinsufficient even at Solar angles beyond 5o, and thus the extra DM from the Solar wind can corrupt theToAs.

7.1 Consequences for High-Precision Pulsar Timing ExperimentsVariations in the DM that cannot be accurately and precisely measured and modelled are a problem for pul-sar timing, as they add a time-dependent extra delay to the ToAs. During an ESE, such variations happen

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and badly corrupt astrophysically relevant parameters or can significantly reduce sensitivity to astrophys-ically interesting signals (see, e.g. Keith et al., 2013) if their impact on the data is not removed by, forexample, calculating the DM for every epoch and correcting for it. This correction can only be done withsimultaneous multi-frequency or low-frequency data, which are not always available (see, e.g. the latestIPTA release, Verbiest et al., 2016). In the ESE discussed in this thesis, the maximum difference in theextra time-delay across the whole dataset (from a DM difference of 6 · 10−3 cm−3 pc) at the commonly used21-cm wavelength would be 13 µs, which is definitely a problem for high-precision pulsar timing as theprecision required to reach PTA goals is far below this (see Section 1.1.3). The ToA difference across a250 MHz bandwidth centred at 1.4 GHz due to the extra dispersion is 2 µs, so the ToA precision has tobe substantially better to correctly measure and correct the impact the ESE has on the data. As shown inSection 1.2.3, the precision of this correction would however be an order of magnitude worse than the ToAprecision and averaging the DM values to increase precision is typically not a valid solution because of theusually low sampling rate and possible short time scales of the variations. Thus, lower-frequency data areneeded to correct for the DM variations in high-frequency data. However, this correction could suffer fromDM chromaticity, if the effect is not quantified correctly. As chromaticity is strongest at low frequenciesand LOFAR has a large fractional bandwidth, it should be detectable in the LOFAR data if it would haveany impact on the corrections of the higher-frequency data. Also, the structure function indicates a steepspectrum of the DM variations with more power at large spatial frequencies (see Donner, 2014, for exam-ple), which is expected to be similar for gravitational waves, whereas the chromaticity primarily plays arole at high spatial frequencies. So the chromaticity affects the data on scales that are not of interest forgravitational wave detection efforts, and thus should not be a problem.

7.2 Future ResearchThe observations of PSR J2219+4754 should be continued as the ESE towards this pulsar is still ongoing.With more data and a more sensitive analysis, the relation between DM variability, chromaticity, scattering,and pulse-shape variability could be clarified. Additionally to these parameters, secondary spectra can belooked at, like Coles et al. (2015) did it in their paper on ESEs. This pulsar is also very useful to studychromatic DMs, as it is the first discovery of this effect and it is clearly visible. To be able to correct for DMvariations in high-frequency data with low-frequency data, the understanding of chromaticity is crucial andthis should thus be analysed in more detail. Other precisely monitored pulsars with DM variations couldalso help to understand chromaticity.

For PSRs J1136+1551 and J1921+2153, the start of possible ESEs has been detected and thus theobservations of those pulsars should be continued. Although the ESEs are (until now) less pronounced thanin the PSR J2219+4754 data, they are likely to further improve our understanding of ESEs.

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AcknowledgementsThis thesis used data from the six GLOW stations: DE601, the LOFAR station of the Max-Planck-Institutfur Radioastronomie at the Effelsberg observatory; DE602, the LOFAR station of the Max-Planck-Institutfur Astrophysik in Unterweilenbach; DE603, the LOFAR station of the Thuringer Landessternwarte inTautenburg; DE604, the LOFAR station of the Leibniz-Institut fur Astrophysik in Bornim; DE605, theLOFAR station of the Ruhr-Universitat Bochum and the Forschungszentrum Julich in Julich; and DE609,the LOFAR station of the Universitat Hamburg and the Universitat Bielefeld in Norderstedt. I thank thestaff and station owners for the technical support and for making this campaign possible. Most observationswere taken by Dr Stefan Osłowski and Dr Caterina Tiburzi. I like to thank Prof. Dr Joris Verbiest and DrCaterina Tiburzi for supervising me.

Lorenz Haase, Philip Bergjann, and Max Kreie created a good atmosphere in the office, discussingphysical and non-physical topics.

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A Appendix

A.1 Calculation of the Structure FunctionAs discussed in Section 1.2.4 (Equation 13), the structure function is defined as:

DDM(τ) =⟨[DM(t + τ) − DM(t)]2

⟩. (17)

As the DM measurements are not continuous, τ has been quantised. To get equally distributed points on alogarithmic scale, the τ values are separated by a constant factor β. In Equation 17, two measurements attimes t and t′ are considered to be separated by a time span of τ, if t′ ∈ [t + τ/

√β : t + τ ·

√β].

Weighted Mean

As the precision of the DM measurements in a given dataset can vary a lot depending on the data quality, itis useful to use a weighted mean in Equation 17. Without weighting, imprecise points - which are usuallymore scattered than precise ones - show larger variations and thus corrupt the structure function, especiallyfor short time intervals τ, where the value of the structure function is rather small. The choice of the weightscan have a big impact on the outcome. A very common method of weighting a set of values xi is to weightthe xi by their inverse variance, i.e.

〈x〉 =

∑ni=1 xi · wi∑n

i=1 wi=

∑ni=1 xi ·

1σ2

i∑ni=1

1σ2

i

. (18)

In this case, this weighting leads to some problems, as the variance of (∆DM)2 is proportional to the valueof ∆DM itself, i.e. σ2

(∆DM)2 = |2 ∆DMσ∆DM|2 (assuming Gaussian error propagation). This leads to an

overweighting of small DM differences in the structure function, i.e. coincidental measurements that have∆DM = 0 will be weighted infinitely strongly regardless of their uncertainty. The same effect occurs whencalculating the variance of (∆DM)2 with Monte Carlo simulation, so the assumption of Gaussian errorpropagation is not the cause for this unwanted effect. To avoid this overweighting of small values, thevariance of ∆DM is used for weighting instead of the variance of (∆DM)2, so the weights are calculated asfollows:

wi =1

σ2DM(t+τ) + σ2

DM(t)

. (19)

Structure Function Uncertainty

The uncertainty of DDM(τ) is calculated using Monte Carlo simulations: the initial DM values are variedby adding Gaussianly distributed random numbers with a standard deviation equal to the measurement un-certainty. These new measurements are then inserted into Equation 17, using a weighted mean as discussedabove. After repeating the whole process 10000 times, the actual value and uncertainty of DDM(τ) arecalculated as the mean and standard deviation of the large sample of DDM(τ) results.

Final DeclarationI confirm that I have written this thesis on my own and that I documented all sources and materials used.

——————————————————————–

(Julian Donner)

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