Moding Statistics for PSR B0329+54 at LOFAR Frequencies · 1 Introduction to pulsar astronomy The most rapidly rotating pulsar (PSR J1748-2446ad) known today has a rotational period

Moding Statistics for PSR B0329+54 atLOFAR Frequencies

Max Kreie

November 7, 2016

Bachelor thesis / Bachelor of science: PhysicsFaculty of Physics

Bielefeld University

Supervisor: Jun.-Prof. Dr. Joris P.W. VerbiestSecond Supervisor: Dr. Caterina Tiburzi

Abstract

Pulsars are rapidly rotating neutron stars. Like a lighthouse, their emission beams sweep throughspace, causing them to be observed as pulsating source. Because of their long-term stability, thestudy of these pulses allows a vast array of scientific investigations. Even though, their pulseshapes are highly variable on short time-scales. In this thesis, we investigate a particular formof pulse-shape change, called “moding”.

Moding describes the phenomenon that the pulse shape sometimes switches between a fewtypical shapes. This phenomenon is the main topic of this thesis. Previous work for PSRB0329+54 was done by Chen et al. (2011) at observing frequencies above 1 GHz. Analysisat LOFAR frequencies (∼ 154 MHz) gives insight into moding at low frequencies.Principal component analysis leads us to RFI -identification and -cleaning. Applying PCA ontothe cleaned data allows us to investigate different pulsar modes. Removing (or identifying) thefailed observations allows us to apply statistical tools.We find PSR B0329+54 to spend for ∼ 82.45% of the observed time in the normal mode and∼ 17.55% of the observed time in the abnormal mode and get typical mode lengths for the nor-mal mode of (18.7± 1.8) minutes and for the abnormal mode (7.7± 0.7) minutes though weshows that these durations are likely biased and shorter than the true values.

Contents

1 Introduction to pulsar astronomy 11.1 Supernovae and the birth of pulsars . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Pulse emission and the lighthouse effect . . . . . . . . . . . . . . . . . . . . . 41.3 Changes of the pulse profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Moding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3.2 Nulling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Interstellar propagation effects . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 PSR B0329+54 (PSR J0332+5434) . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Introduction to principal component analysis (PCA) 112.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Practical Example of PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Observation and Data Analysis 143.1 PSRCHIVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 The Low Frequency Array (LOFAR) . . . . . . . . . . . . . . . . . . . . . . . 153.3 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.1 RFI cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.2 Frequency averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3.3 Pulse summing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Mode and RFI identification using psrpca . . . . . . . . . . . . . . . . . . . . 18

4 Identify modes and RFI with PCA 194.1 Definition of a mode change . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Definition of the mode length . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Removal of failed observations . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Moding Statistics 245.1 Moding fraction and rough estimate of mode length . . . . . . . . . . . . . . . 245.2 Correction for false-positives . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Offset in Mode length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Discussion and conclusions 28

7 Appendix 307.1 Single pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 Mode changing examples for PSR B0329+54 . . . . . . . . . . . . . . . . . . 317.3 Different principal components . . . . . . . . . . . . . . . . . . . . . . . . . . 337.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

References 38

Statutory declaration 41

1 Introduction to pulsar astronomy


This chapter is (unless mentioned otherwise) based on Lorimer and Kramer (2005).

A radio pulsar (pulsating source of radio emission; short: PSR) is a rapidly rotating neutronstar, a remnant of a core collapse supernova (SN) at the end of a massive star’s life.

Neutron stars were first assumed to not be observable when they were predicted by Baade andZwicky (1934). But later, when Hewish et al. (1968) observed the first radio pulsar, this was dis-proven. Until this day, the emission of beamed radio waves emitted by these highly magnetizedneutron stars is not understood in detail but astronomers make good use of it (see e.g. Stairs,2006).A pulsar is detectable as a highly periodic signal at all electromagnetic frequencies. This prop-erty allows studies of not only neutron star physics, but also of general relativity, alternativegravitational models and theories, magnetic fields, the interstellar medium (see § 1.4), celestialmechanics, planetary physics and cosmology. Because the pulsar emits electromagnetic wavesalong its magnetic axis only and the magnetic axis is not aligned with the rotational axis, theso-called lighthouse effect leads to a periodic signal detected on earth (see § 1.2).

Figure 1: A single-pulse time-series from PSR B0301+19 from the Arecibo radio telescope. Thefour boxes on the bottom show individual pulses. Taken from Lorimer and Kramer(2005).

As shown in the upper half of Fig. 1, the detected signal of the pulsar is strongly periodic,but (as shown in the lower half) the pulse profile1 differs stochastically for each individual pulse(see § 1.3). For the best of our knowledge this is true for all pulsars.

1The pulse profile is the observed flux from the pulsar as a function of the phase of the pulsars rotation or time.

1


The most rapidly rotating pulsar (PSR J1748-2446ad) known today has a rotational period of1.39 ms and the slowest (PSR J1841-0456) a period of 11.79 s (PSRCAT-v1.54, 2015). In total,there are currently more than 2500 pulsars known (PSRCAT-v1.54, 2015).

The observed brightness of the known pulsars covers a huge range; from easily detectablesingle pulses to faint detections with the largest and most sensitive radio telescopes on earth. Asthe distances to them spread over more than one order of magnitude, that is not really surprising.

1.1 Supernovae and the birth of pulsars

Pulsars are rapidly rotating neutron stars; remnants of supernovae. Low- and intermediate-massstars evolve differently than more massive stars. Stars with a mass above approx M ∼ 8M�explode in a supernova type Ib, Ic or II at the end of their lives: After different burning stages,the gravity causes a collapse of the star because the radiation pressure is not able to balance thegravity any more (Langer, 2016).In several respects, the core is in a strange state at that time; because of neutrino cooling (inthe late stages of massive stellar evolution), the degree of electron degeneracy in the core ofthe star becomes considerable. This causes a high temperature and a high density of about(ρ ≥ 109g/cm3

), so the energy of the electrons becomes relativistic (Langer, 2016).

With increasing density ρ ∼ 109 gcm3 and temperature T ∼ 109 K inside a massive star’s core

while it collapses, the inverse β-decay (electron capture process) e + p → νe + n occurs moreoften and the increased neutron richness with decreasing electron pressure causes a further core-collapse (ibid.). The Chandrasekhar mass2 of the core

(MCh = 5.83µ−2

e ≈ 1.26M�)

decreasesbecause of the increasing average mass per free electron (µe) and exceeds the core mass.At that time, this process is called supernova and the lighter elements get blown out. The neu-trinos ejected by the inverse β-decay can be detected at the earth. This is the first indicator of asupernova, because neutrinos propagate with a velocity near the speed of light.

2The Chandrasekhar mass is the maximum mass supportable by electron-degeneracy pressure (which in practicemeans it’s the maximum mass of white dwarfs).

2


Figure 2: The initial mass - remnant mass relation as a highly uncertain sketch. Typically as-sumed remnants depend on the initial mass as follows: Mi ≤ 8M� : white dwarf.8M� ≤ Mi ≤ 60M� : neutron star. Mi ≥ 60M�: black hole. Taken from Binneyand Merrifield (1998).

1e-22

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-03 1e-02 1e-01 1e+00 1e+01 1e+02

P1

in s

/s

P0 in s

Figure 3: The P -P -diagram. P0 is the pulse period and P1 its time derivative P . The phe-nomenon of millisecond pulsars (MSPs; millisecond pulsars (MSPs) are pulsars witha spin-period of a few milliseconds) is mainly discovered in binary systems (the clusteron the left corresponds to MSPs). Red crosses indicate solitary pulsars. Green squaresidentify pulsars in binary systems. Data taken from PSRCAT-v1.54 (2015).

As shown in Fig. 2, a star with an initial mass of Mi ≈ 8 − 60M� can form a neutron star

3


with a remnant mass of about Mr ≈ 1.4M�. Some time ago (Stairs, 2006, e.g.) observationsseemed to support this narrow mass range for neutron stars, but more recent work (Demorest etal., 2010 and Antoniadis et al., 2013) demonstrate some higher-mass pulsars. Demorest et al.(2010) for example present the mass for PSR J1614-2230 to be (1.97± 0.04)M�.The remnant core of a star with a mass above the Chandrasekhar limit of 1.4M� that explodedin a supernova can become a neutron star, if it is not significantly higher than a certain limitabove which it can become a black hole.

Neutron stars are (as far as we know) one of the most dense forms of matter in our Uni-verse (Demorest et al., 2010). After the core-collapse, the density is about ρ ∼ 104 g

cm3 in thecrust and ρ ∼ 4 · 1014 g

cm3 in the center (Hanslmeier, 2014), so the radius of the neutron star isabout R ∼ 10− 12 km.

Because the angular momentum of the star is largely conserved, the core of the supernovaremnant (the neutron star) rotates with a high frequency.

1.2 Pulse emission and the lighthouse effect

The pulsed emission we detect on earth is generally thought to be continuously emitted by thepulsar (however, see sec 1.3.2).

Figure 4: The emission regions of a neutron starwith the magnetic axis not aligned to therotational axis. Taken from Lorimer andKramer (2005).

The so called “lighthouse effect” (seeFig. 4) causes that a pulse is only visiblein the short time during which the “beam”of the pulsar crosses our line of sight.Because of this effect, the time betweendetected pulses always gives the rotationperiod of the pulsar in this very simplepicture.Conservation of angular momentum dur-ing the core-collapse causes the rapid ro-tation of the neutron star.

The emission of radiation along themagnetic axes of the pulsar causes a lossof rotational energy of the pulsar. Thisconsequently causes a slight spin-downof the pulsar. See Fig. 3 for the P − Pdiagram where P is the time derivative ofthe pulse period P .In a few cases, pulsars show emission allthe time. This suggests the magnetic axisto be basically aligned to the rotationalaxis and our line of sight.

Some pulsars show so-called interpulses, separated by about 180◦ in pulse phase. They can beexplained to originate from the opposite magnetic pole of the neutron star. In this case bothmagnetic poles pass our line of sight (which implies they are ∼ 90◦ offset from the rotation axis

4


= “orthogonal rotators”).

1.3 Changes of the pulse profile

As the detected radiation emitted by pulsars is very weak and the shape of single pulses variesfrom pulse to pulse, it is necessary to integrate pulses (sum a few or more pulses together) toget a constant pulse shape. These integrated pulses can be seen as a “fingerprint” for a pulsar3.Sometimes they are very complex in shape and sometimes very simple.

According to Dicke (1946), for Tsys the root mean square fluctuation is:

∆Tsys =Tsys√npt∆f

(1)

This is the radiometer equation with the observing bandwidth ∆f , the integration time tint,np = 1 for observations with single-polarisation and np = 2 for orthogonal polarisations.

The S/N (signal-to-noise ratio) taken from Lorimer and Kramer (2005) of an observed pulsewith S/N =

Tpeak∆Tint

is

S/N =√nptint∆f

(Tpeak

Tsys

) √W (P −W )

P(2)

with the integration time tint, the observing bandwidth ∆f , the noise fluctuations ∆Tint, pulseperiod P and pulse width W .

1.3.1 Moding

Some pulsars show a strange phenomenon called “moding” whereby the profile shape switchfrom the so called “normal mode” to the “abnormal mode” on a short timescale (Zhang et al.,1996). The pulse profiles for this two different shapes are well defined (Lyne et al., 2010). Thisphenomenon was first discribed a few years after the discovery of pulsars by Backer (1970) forPSR B1237+25. For some pulsars, a predominant feature of mode switching is the change ofrelative intensity between the different components of the pulse profile (Chen et al., 2011). Thephenomenon of moding is the main topic of this thesis.

Kramer et al. (2006) and Lyne et al. (2010) showed that changes of the pulsar spin-down P(ν in their case) are linked to changes in the pulse shape and they are caused by changes in themagnetospheric current flow. The timescale for mode changing is on the order of a few secondsor a few pulses.The increase of the breaking torque observed in some pulsars is caused by an enhanced stream of(charged) particles what increases the amount of emitted electromagnetic radiation and thereforecauses changes in the pulsars pulse shape (Lyne et al., 2010).

3Because they are usually reproducible (i.e. are identical between observations).

5


1.3.2 Nulling

The nulling phenomenon was first described by Backer (1970) and there defined as a decrease ofpulse energy by more than a factor of ten. They suggested that not only the pulse energy dropssuddenly at a null, but there is not pulsed radiation in the null pulses (ibid.).Nulling may be an extreme case of mode changing (moding) and only differ in the magnitudeof change in the pulsars magnetospheric current flow (Lyne et al., 2010). There are a lot of openquestions because nulling is not well understood. According to Young et al. (2015), nulling mayoccur due to four different scenarios:

1. “deep nulls”: A pulsar undergoes cessation completely.

2. While transiting from a brighter to a weaker emission state.

3. The radio beam moves out of our line of sight.

4. Not enough electron-positron pairs in the acceleration zone.

Rotating Radio Transients (RRATs) are pulsars which pulses can only be detected sporad-ically. As nulling is not very well understood, RRATs may be (an extreme) case of nullingpulsars with a high nulling-fraction4 (Burke-Spolaor, 2012).There are more than 70 RRATs known (ibid.), what indicates that if all sporadic emission ofthem comes from nulling, the Galactic population of steady-emitting pulsars formed in core-collapse super novae was highly underestimated (ibid.).Burke-Spolaor (2012) describe three types of nulling:

1. “Standard” nulling with NF from a few percent to ∼ 95% and timescales of activity fromseconds to minutes.

2. High-fraction nulling with NF upwards of 99%. The nulling timescale is larger than theemitting timescale.

3. Intermittent pulsars with nulling and emission timescales of weeks, months or years.

According to Wang et al. (2007), a high nulling fraction (NF > 65%) gives a higher tendencyfor a pulsar to be older than 5 · 106 years. Burke-Spolaor (2012) propose this to be an indicatorthat extreme nulling may somehow be an evolutionary state for a pulsar.

4High nulling fraction (NF) means: The pulsar spends a majority of the time without being detectable at earth.

6


1.4 Interstellar propagation effects

(a) Dispersed pulse. The line is not continuous becausethe intra-channel dispersion has been corrected.

(b) Dedispersed pulse.

Figure 5: Integrated pulse with and without dedispersion. RFI-cleaned (see § 3.3.1).

Traversing the interstellar medium (short: ISM) consisting of cold, ionised plasma, cosmic rays,atoms and dust, radio signals are affected by dispersion: higher radio frequencies arrive earlierat the earth than lower radio frequencies (see Fig. 5a).This effect can be used for studying the ISM (as explained below).

The refractive index for electromagnetic radiation is

µ =

√1−

(fpf

)2

(3)

with the wave frequency f and the plasma frequency

fp =

√e2neπme

' 8.5kHz( ne

cm−3

) 12. (4)

me, ne and e are the electron mass, number density and charge. In the ionized, warm ISM theelectron number density is typically of order ne ∼ 0.03 cm−3. With the refractive index (eq. 3)µ < 1, the group velocity is less than the speed of light c.The frequency-dependent time delay with respect to a signal of infinite frequency travelling thelength d to Earth is

t =

d∫0

dlvg

− d

c. (5)

7


So we find

t = D × DMf2

(6)

with the dispersion measure

DM =

d∫0

nedl (7)

expressed in cm−3 pc and the dispersion constant

D ≡ e2

2πmec= (4.148808± 0.000003)× 103 MHz2 pc−1 cm3s. (8)

We find the delay between two frequencies f1 and f2, both in MHz:

∆t ' 4.15× 106 ms×(f−2

1 − f−22

)× DM. (9)

Applying this equation to the pulse in Fig. 5a with a known dispersion measure (DM), thededispersed pulse looks like the pulse in Fig. 5b.Based on this correction, it is possible to distinguish between radio-frequency interference(short: RFI) and the radio pulsar’s signal: Since RFI is produced on Earth it is not affectedby the ISM and not dispersed.For more details on the RFI-cleaning procedure used in this thesis, see § 3.3.1.

1.5 PSR B0329+54 (PSR J0332+5434)

PSR B0329+54 is a radiopulsar with a period of P ≈ 0.714s and an age of about 5.53Myr. Itis one of the brightest pulsars discovered yet5 and the brightest one in the northern hemisphere,seen in the constellation of Camelopardalis.

5Discovered by Hewish et al. (1968).

8


property value uncertaintyPSR J-name J0332+5434 −PSR B-name B0329+54 −RAJ [hh:mm:ss.s] 03:32:59.368 5.0 · 10−3

DECJ [+dd:mm:ss] +54:34:43.57 7.0 · 10−2

PMRA [mas/yr] 17.0 0.3PMDEC [mas/yr] −9.5 0.4DM [cm−3 pc] 26.7641 1.0 · 10−4

RM [rad m−2] −63.7 0.4F [Hz] 1.399541538720 6.0 · 10−12

F [s−2] −4.011970 · 10−15 1.4 · 10−20

F [s−3] 5.3 · 10−28 1.5 · 10−28

P [s] 0.714519699726 4.0 · 10−12

P [1] 2.048265 · 10−15 8.0 · 10−21

age [yr] 5.53 · 106 −distance [kpc] 1.00 −

Table 1: Properties of PSR B0329+54 from PSRCAT-v1.54 (2015).

The template profile for PSR B0329+54 with a high S/N of 8909.4 at a center frequency of153.6133 MHz and with a bandwidth of 23.4375 MHz is shown in Fig. 6. It was averaged over276438.69 seconds of observations (76.79 hours; 386887 pulses) between MJD 57160.759085and MJD 57164.418865 (mid May 2015) and is used as the template6 for the further analysis.Pulses affected by RFI are not included (see § 3.3.1).As described for example by Chen et al. (2011) and Liu et al. (2006), PSR B0329+54 is knownto show “moding”. They observed the pulsar at frequencies of about 1540 MHz, so observingwith LOFAR (see § 3.2) would deal with this phenomenon at much lower frequencies.

6We need a template profile for PCA to work. See § 2 for more information.

9


Figure 6: The template profile for PSR B0329+54. 276438.69 seconds (76.79 hours; 386887pulses) of observation with DE605 at a center frequency of 153.6133 MHz and with abandwidth of 23.4375 MHz in arbitrary units. See § 3.2 for further details (e.g. whatis DE605?).

The template profile for PSR B0329+54 (Fig. 6) shows mainly three peaks, the second peakis about 23 times higher than the first peak and about 12 times higher than the third peak. Onthe leading edge, the main peak (the second) is not as sharp as on the trading edge; a bump isvisible at the leading edge of the main peak.

As described in a lot of publications, PSR B0329+54 is known to show different modes (Chenet al., 2011).

10

2 Introduction to principal component analysis (PCA)

Figure 7: The normal (integration time ∼ 24 hr; left) and the abnormal (integration time ∼ 2 hr;right) modes for PSR B0329+54 at ∼ 153 MHz.

In Fig. 7 the two modes for PSR B0329+54 are shown. The integration length for the normalmode is 85338.88 s (∼ 24 hr; 120196 pulses) and 6849.99 s (∼ 2 hr, 9594 pulses) for theabnormal mode.The normal mode shows three peaks, whereas in the abnormal mode the third peak is missing(or very much smaller).


2.1 Theoretical background

Principal component analysis (short: PCA) is an analysis method that is used in a wide rangeof fields of modern science. As a (simple) method, the main aim is to reveal a hidden structurein a sometimes huge and very complex dataset (Shlens, 2014), it gives the opportunity to findpatterns in a high dimensional dataset (Smith, 2002).

The goal of this section is to describe how PCA works and not why it works. Therefore, I willpresent the main steps of PCA according to Smith (2002), as applied to my work:

Step 1: Data

The used data are 1024 dimensional as every pulse (or here: pulse-integration) consists of 1024phase-bins.The data at first has to be adjusted so, that it best matches the template using the best-fittingphase shift (Osłowski et al., 2011).The data are the pulsed intensity as a function of spin phase (in 1024 phase bins), for all availableobservations.

11


For any given observation, the 1024 intensity values are considered as a vector, a single datapoint in 1024 dimensions for the principal component analysis.

Step 2: Subtraction of the mean

The mean for every bin has to be subtracted from every bin for each observation (Osłowski et al.,2011).Therefore, we have to define the pulse7 pi =

(pi1, ..., p

i1024

)with its 1024 components (=bins).

We define the mean of bin j (j = 1 to 1024) for all the N pulses pi (with i = 1 to N ) as

pj = 1N

N∑i=1

pij . This gives N mean-subtracted pulses for N given pulses, marked with a prime:

pij ′ = pij − pj ∀ 1 ≤ j ≤ 1024 (10)

The result is pi′, the mean-subtracted pulse number i.

Step 3: Covariance matrix

The covariance matrix is a 1024× 1024 matrix and is calculated/defined as follows:

Ckl =

N∑j=1

pjk′ pjl ′

N − 1(11)

Step 4: Eigenvectors and eigenvalues of the covariance matrix

For the square matrix Ckl, which has full rank in case of N ≥ 1024 (Osłowski et al., 2011), wecan calculate the eigenvalues λj as described by Fischer (2010):

Ckl · x = λj · x (12)

The eigenvectors vj are in this case given as:

(Ckl − λj1) · vj = 0 with |v| = 1 (13)

The eigenvectors vj now have to be sorted by decreasing eigenvalue λj to determine the mostsignificant variations in the first eigenvalues/eigenvectors (Osłowski et al., 2011).

7We note down as follows: lower indices describe the bin for every observation. To distinguish observations, thehigher indices are used. Vectors (in 1024 dimensions) are notated bold.

12


Step 5: Projecting the profiles onto the eigenvectors

We now can project each template-difference profile onto the orthonormal basis formed by theeigenvectors. These projections are quantified by the projection coefficients αij for the i-thprofile onto the j-th eigenvector, defined as (Osłowski et al., 2011):

αij =(pi′)Tλj (14)

One can think of these coefficients to be the residual of pulse i in the orthonormal basis spannedby the eigenvectors λi. They are called “principal components” (PC).The i-th principal component, PCi, corresponds to vi which belongs to the i-th largest eigen-value.

2.2 Practical Example of PCA

This example was reproduced based on (Smith, 2002).

To start, we take a 2-dimensional dataset:

i p1i p1

i ′ p2i p2

i ′1 2.6 0.77 2.35 0.662 0.4 −1.43 0.61 −1.083 2.3 0.47 2.82 1.134 2 0.17 2.3 0.615 3.2 1.37 2.9 1.216 2.2 0.37 2.8 1.117 1.9 0.07 1.5 −0.198 1.1 −0.73 1 −0.699 1.5 −0.33 −0.32 −2.010 1.1 −0.73 0.9 −0.79

Table 2: The used data for this example(p1i and p2

i

)and the data with the mean subtracted(

p1i ′ and p2

i ′).

13

3 Observation and Data Analysis

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5

(a) The data for the example.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-1.5 -1 -0.5 0 0.5 1 1.5

(b) The normalized data and the eigenvectors overlayedas dashed lines.

Figure 8: Plots of the data. Reproduced based on Smith (2002).

Calculating the covariance matrix as described above gives us a 2× 2-matrix:

C =

(0.67566 0.720680.72068 1.23594

)(15)

Calculation of the eigenvalues gives:

λ1 = 0.182586 and λ2 = 1.72901 (16)

And the eigenvectors are:

v1 =

(−0.825320.564666

)and v2 =

(−0.564666

0.82532

)(17)

The eigenvectors v have to be interchanged so that they are sorted with decreasing eigenvalueλj as described above.Last step is project the data pi onto the new basis.


The analysis consisted of two main parts; first the determination of the eigenvectors with PCAusing data recorded with DE605 (see § 3.2) in Julich next to Forschungszentrum Julich and sec-ond PCA applied on a large set of data recorded using different “Low Frequency Array” (short:LOFAR) stations in Germany.The observation frequency, bandwidth and telescopes used for all observations in our analysisare given in Table 6.

14


The analysed data at first consisted of 122 frequency channels in the given bandwidth andabout 1024 bins per pulse in the time domain. For the further analysis, the data had to be“scrunched” in frequency, so that there is only one frequency channel consisting of the sum ofall the 122 channels. More details are given in § 3.3.2.

For the first step of analysis, the determination of the eigenvectors for the principal componentanalysis (PCA), there is a dataset consisting of about 4 days of quasi-continuous observationsfor PSR B0329+54 at a frequency of ∼ 153 MHz and a bandwidth of ∼ 23 MHz given as singlepulse observations. It consists of about 0.4 million single pulses. This dataset is called “datasetI” in the subsequent discussion.The later analysis of mode switching and RFI is based on the data described in Table. 6. Thisdataset is called “dataset II” further on. Dataset I is a subset of dataset II.

3.1 PSRCHIVE

Most of the analysis was made with the so called PSRCHIVE package, an open-source softwarepackage developed for pulsar astronomy in C++. It contains many tools and algorithms forpulsar timing, scintillation studies, polarimetric calibration, single-pulse work, etc. (PSRchive,2013).

3.2 The Low Frequency Array (LOFAR)

The Low Frequency Array (LOFAR) is a radio interferometer constructed in the northern Nether-lands and across Europe, covering the largely unexplored frequency range from 10− 240 MHz(wavelengths of about 1.2 − 30 m). It core is spread around near the village of Exloo, a fewmore stations in the Netherlands, six more stations throughout Germany and one each in France,Sweden, Poland and the UK (van Haarlem et al., 2013). LOFAR consists of two main frequencyranges, represented by two different antenna designs consisting of 96 dipoles at each interna-tional station8:

LBA The “Low Band Array” (LBA) elements have a frequency range of 10−80 MHz. They aremade of one dipole antenna to detect polarisation. The wires have an angle of 45 degreeto the ground. See Fig. 24.

HBA The “High Band Antenna” (HBA) have a frequency range of 110−240 MHz. A single tileconsists of a square, 4x4 array of dipoles with dual polarisation and build-in amplifiers.Each 16-element tile is 5x5 meters in size and made of an expanded polystyrene structure.To protect the hardware from weather, it is covered by two overlapping polypropylene foillayers (van Haarlem et al., 2013). See Fig. 9. A part of this band was used for this thesis.

8Half that number for Dutch stations.

15


(a) LOFAR HBA single antenna element. (b) Zoom onto LOFAR HBA element.

Figure 9: A LOFAR HBA element in Tautenburg near Jena at Thuringer Landessternwarte(DE603). The weather protection has been removed to expose the hardware.

LOFAR is one of the first radio telescopes to feature automated processing to deliver fullycalibrated products to its users. This and the new capabilities, techniques and modus operandi ofa large interferometric array make it an important pathfinder for the Square Kilometre Array (vanHaarlem et al., 2013).In this thesis only single international stations are used to observe PSR B0329+54. A list of thestations used is given in Table 3.

station name identifier location additional informationEffelsberg DE601 next to the 100m radiotelescope –Tautenburg DE603 Tautenburg near Jena –

Julich DE605 near “Forschungszentrum Julich” while analysing the used data itwas obvious, that it is affectedby RFI more by day than bynight. Maybe “Forschungszen-trum Julich” is a problem.

Norderstedt DE609 Norderstedt near Hamburg regular observations com-menced in 2016 but some olderdata are used in this thesis;PCA-analysed observationsfrom this station look strange.

Table 3: The LOFAR stations used in this thesis.

16


3.3 Preprocessing

The given LOFAR data have to be prepared for the further analysis and the principal componentanalysis, as described in detail in this section. Special care was taken for the identification ofRFI in a test data set (dataset I), which allowed automatic RFI identification in the rest of thedata (see Section 4).

3.3.1 RFI cleaning

Radio-frequency interference (RFI) is a big problem at the low frequencies used as some emer-gency pagers work at ∼ 165 MHz and some other man-made signals were detectable there.Because the international LOFAR stations are not shielded very much against RFI and theyare typically not build in so-called “radio-quiet zones”9, for some observations (especially withshort integration lengths, faint pulsars or single pulses) the pulsar signal is not visible as RFIwas much brighter.The main problem of RFI for the further analysis is that man-made RFI can of course havea structure and thereby can be seen as internal structure of the pulsar if it is not identified orspecially marked.

At first, RFI in every single-pulse observation was removed by the clean.py script of theCoastguard package.To find an automatic way to identify RFI affected pulses in dataset II, every single pulse indataset I was checked by eye and marked, if it was still affected by RFI after running the scriptlike in Fig. 10a (if just a few frequency channels out of 122 channels were affected, this channelswere deleted).

(a) A pulse with RFI. (b) Most RFI was deleted.

Figure 10: RFI has to be deleted to allow optimal analysis of the data.

After this process one can be sure the unmarked pulses are not affected by RFI. Because ofthat, it is possible to look at RFI–affected pulses again in the later analysis (see § 4).

9Except the Swedish station in Onsala.

17


3.3.2 Frequency averaging

After cleaning RFI, for every pulse the 122 frequency channels have to be summed together tohave just one channel, so that the principal component analysis can work without having toomany dimensions to handle (for more information see § 2). This process was done using thepam program from the PSRCHIVE package.

3.3.3 Pulse summing

Every single pulse was then summed up to 80 pulse integrations using psradd and pam fromthe PSRCHIVE package (see § 3.1). This integration length of about 56s is long enough forpulse-to-pulse variations to be averaged enough that they only cause variations on levels wellbelow the level at which the moding is active. Coincidentally, the integration length is similar tothat used by Chen et al. (2011).

3.4 Mode and RFI identification using psrpca

Because PCA (as described in § 2) is able to reveal hidden structures in a complex dataset, wewant to use PCA to identify modes and delete RFI in a huge dataset.Using psrpca of PSRCHIVE, the goal is to apply PCA to the 80-pulse integrations to see dif-ferent clusters or islands in the first principal components (PCs) which correspond to differentmodes and to see scattering in higher-order PCs to be able to delete RFI-affected pulses auto-matically.

The first step will be to apply PCA to the RFI-cleaned dataset I (so the pulses in this datasetwhich are not marked as affected by RFI) and thereby looking for clusters in the first principalcomponents (where we assume the most information about mode changing to be).

Using the eigenvectors calculated with the RFI-less dataset I, we can project the RFI-affected80-pulse integrations onto the calculated basis and see whether they behave differently.

The next step is to use these eigenvectors again and to project dataset II onto this new basisand look at the PC where RFI-affected points behave differently than clean ones. In this PC itshould be possible to identify RFI without looking at all the data by eye. Deleting the pointsthat are obviously RFI-affected should help to see clusters or islands in a lower-order principalcomponent: looking at the principal component where we saw different modes allows us toidentify modes and do statistics on them.

18

4 Identify modes and RFI with PCA


Figure 11: Principal components 2 (top), 1023 (middle) and 1024 (bottom) as function of time(80pulse integrations). Left: Zoomed in to see the “cleaned” data (red). Right:Zoomed out to see all the points (blue points are RFI-affected).

As we apply PCA to cleaned dataset I and to the RFI-affected pulses of dataset I (using theeigenvectors of the first calculation), we can look for modes in the first few PCs and wether RFIbehaves differently in the last few PCs.In the second principal component (for a discussion why it is not the first PC see § 6) we seeclearly the points scattered around zero but with a few clusters near y = 0.4 (see Fig. 11). If welook at the pulse profiles corresponding to these two clusters (one cluster around y = 0 and onearound y = 0.4), it is obvious that these two clusters correspond to different modes. We definethe points around zero to show the normal mode (because y = 0 is predominant) and the pointsaround y = 0.4 to show the abnormal mode (see Fig. 7; this figure was produced by integratinga lot of pulses of the normal mode or the abnormal mode together).

19


Studying higher-order principal components (ex. PC1023), the RFI-affected points (the blueones) are fount to be a lot more spread out (Fig. 11). We interpret pulses too far away from zeroto be RFI-affected10 and because we don’t want them to play a role in the further analysis, wedelete them above or below a certain threshold

(i.e. |y| > 3 · 10−7

).

Looking at PC 1024 (see Fig. 11) some periodic structure becomes clear: At night (∼ integerMJD), the points are a lot more scattered; by day (∼ half-integer MJD) they are a lot closerto zero. A simple explanation is that the circumpolar pulsar B0329+54 is much closer to thehorizon by night, which decreases the effective collecting area and thus the S/N.

The next step is to apply principal component analysis to dataset II (all available data) usingthe eigenvectors (i.e. the basis) calculated in the first step. Note that only the projection of thepulses in dataset II is used for the further analysis.This projection leads to Fig. 12 where as expected the two modes become visible in PC2. Be-cause of the large scatter in PC1022 we assume PC1022 to give us the possibility to identifyRFI-affected pulses. The reason to use PC1022 and not PC1023 is that the structure is basi-cally the same but PC1022 shows a lot more scatter from MJD 57390 (early 2016) on. UsingPC1022 to delete RFI-affected pulses deletes a lot more recent observations than using PC1023and thereby it makes the analysis a lot more robust (assuming that the data from MJD 57360 onare not useful for the analysis as they don’t show a lot of structure in PC2 and are scattered inevery inspected PC by many more orders of magnitude than older data are).

4.1 Definition of a mode change

The previously mentioned threshold in PC2 of about y = 0.2 is now used to identify the modes.At this stage, every point with y > 0.2 is expected to be a pulse-integration showing the abnor-mal mode and every point y < 0.2 to show the normal mode.If we now want to deduce information about the modes (e.g. mode length), we have to analysethis principal component. Therefore, we define a mode change at pulse pk as follows (for thenotation see § 2):

αk2 > 0.2 ∧ αl2 ≤ 0.2 with l = k ± 1 (18)

But be careful! Considering false positives11 would lead to a mistaken analysis. To read howwe avoid this, see § 4.2.

4.2 Definition of the mode length

To first order, for every mode change as defined above at pm and mode change at pn withm < n,we define three values for the mode length:

10This assumption is valid because we marked RFI affected pulses blue in this Plot.11In Fig. 12 we see that the points are scattered around 0 and around 0.2 in PC2. Defining a mode change as in eq. 18

would give a false alarm if one point lies nearly below/above the threshold and the next point is back above then.

20


• mode length in pulse integrationsThe mode is l = n−m pulse integrations long. This leads to a mode length of l · tint withthe integration length of tint = 57.12 s.

• mode length in days ∆TThe mode is ∆T = MJD (pn)−MJD (pm) days1213 long.

• mode length in days TThe mode isT =

(MJD((p+1)m)−MJD(pn)

2 + MJD (pn))−(

MJD(pm)−MJD((p+1)n)2 + MJD ((p+ 1)n)

)days long.

Secondly (to avoid false positives), we have to ignore a mode length of l = 1 (respectivelyl = 2) so we change our assumption from the very beginning of this subsection to m + 1 < n(or m+ 2 < n)14.

4.3 Removal of failed observations

First step

Because we have to consider the case that the observation may end before the mode ends(Fig. 13), we have to give up some data. Plotting the mode length in days T against the modelength in pulse integrations (as defined in § 4.2), one can define a threshold for the observedmode to be recorded accurately or the observation to be failed (Fig. 14).

12MJD() means the modified Julian date of the pulse between the brackets.13As we calculate the mode length in days, we can easily calculate the mode length in hours or minutes. Every time

we talk about mode length in days/hours/minutes in this thesis, we use this definition.14Note: If there was a mode length of l = 1 and we define m + 1 < n, the modechange would not count; so the

outlier would be counted to belong to the other mode.

21


Figure 12: Principal components 2 (top), 1022 (middle) and 1023 (bottom) as a function of timefor 80pulse integrations. Left: Zoomed in to see the modes or RFI. The thresholdsfor the further analysis are: PC1022: Every point y with |y| > 0.001 is deleted.PC2: Every point above 0.2 shows the abnormal mode; below 0.2 shows the normalmode. At MJD 57160 (here x = 0) is the time where the 4 days quasi-continuousobservation was recorded (dataset I).

22


-0.6

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Tabnormal Tnormal

PC

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(a) Overview.

0.1

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0.7

-337.07 -337.06 -337.05 -337.04 -337.03 -337.02 -337.01

∆Tabnormal

PC

2

MJD - 57160

(b) Zoomin onto the left cluster in (a). With thethreshold at y = 0.2, the system would detect a modechange. Changing the definition of a mode changeto ignore a mode with a length of l = 1 and l = 2would lead the system not to detect a mode change inthis case.

Figure 13: Clearly visible two different observations whereby in both cases obviously not thewhole mode was recorded: Observation stopped before we see a mode change. In(b) we see that our analysis would detect the abnormal mode to be recorded betweenMJD 57160− 337.064 and MJD 57160− 337.024. This gives: Mode length in pulseintegrations l = 70 = l · tint = 1.1 hr. T = 6.1 days. ∆T = 0.04 days.

0.0001

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Data abn. modeData norm. mode


(b) linear scale

Figure 14: Mode length in pulse integration on the x-axis and in days on the y-axis (definitionin § 4.2). The dashed line is the expectation: Every pulse within the mode length wasrecorded. The green line is the threshold; data above the solid line are sorted out inthis step (later marked in green). For an explanation why a few points lie below thedashed line see text. The observed abnormal mode from Fig. 13 would lie at x = 70pulses, y = 6 days. Because of the zoom in in b, not all points are shown.

Maybe the five points below the dashed line in Fig. 14 correspond to a mode that was recordedsimultaneously at different LOFAR-stations and thereby the mode length in pulse integrations l

23

5 Moding Statistics

multiplied by the integration length t = 0.0006 days is longer than the mode length in days T 15:T < l · t. The solid line marks this calculation in the case of t = 0.0012 days (so if every secondpulse was recorded).

Second step

Secondly, it is possible that the recorded mode length ∆T highly exceeds the mode length inpulse integrations l multiplied by the integrations length tint. Therefore, we plot l · tint against∆T and delete all data above a certain threshold (delete point, if l · tint · 2 < ∆T ).

0.0001

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modenomode


(a) logscale

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norm

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al

modelength in pulse integrations converted in days

modenomode


(b) linear scale

Figure 15: Plotting l·tint against ∆T leads us to replete a few more points for the further analysis.In b, a lot of points are not shown because of the zoom in onto the most denselypopulated area.

5 Moding Statistics

Using PCA, we are now able to identify modes automatically. Using the analysed data, we canapply statistical tools onto it.After calculating the mode length and the fraction of time in the different modes, we compareour results with literature. Interesting is especially the mode length for both, the normal andthe abnormal mode. Once we identified modes, we can calculate this value and calculate thefraction of time it stays in the normal/abnormal mode.

5.1 Moding fraction and rough estimate of mode length

We want to see how often the abnormal mode occurs. Therefore we can compare the number ofobservations identified as normal mode with the number of observations showing the abnormalmode: overall we find that 82.45% of the observed pulse integrations show the normal mode,while 17.55% of the observed pulse integrations show the abnormal mode.

15Our analysis is still valid because we see in Tab. 6 this not to happen very often.

24

5 Moding Statistics

1

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Modelength for the normal mode in hours

Chen et al. 2011

99%

95%

modelengthmean

(a) Histogram for the normal mode.

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Modelength for the abnormal mode in hours

Chen et al. 2011 99%95%

modelengthmean

(b) Histogram for the abnormal mode.

Figure 16: Histogram of the mode length in hours. Every mode change was counted and falsepositives were in this case not avoided. Modes were only counted if the observationsduring the mode sampled at least half the mode length. The identification of modesis described in § 4. The 95th and the 99th percentiles are indicated.

Table 4 gives a typical mean mode length for the normal mode of (19.3 ± 5.5) minutes andfor the abnormal mode of about (7.2± 2.7) minutes.

The histogram in Fig. 16 shows that a majority of modes (a few hundred) do have a modelength of a few 80-pulse integrations for both the normal and the abnormal mode. This corre-sponds to the points ≤ 0.1 hours.For the normal mode, there are still a few outliers: One mode with a length of about 5.2 hours,one with 4.3 hours and four modes with a length between 3 and 4 hours.For the abnormal mode, 2 modes have a length of more than one hour and about 16 have a lengthbetween half an hour and one hour.

In total this gives at 153 MHz a typical time scale for the normal mode of a few minutes toabout half an hour and of a few minutes for the abnormal mode. To avoid looking at modes witha length of about one integration length (which can be due to noise), we next ignore a modelength of one or two integrations (see § 5.2).

5.2 Correction for false-positives

Modes for which the start and/or end were missed could be considered as lower limits onthe mode length. By considering those lower limits, insights could be gained on the short-observation bias and how it affects our results. Fig. 17 shows a histogram for the mode length.As one can see in Fig. 17, most of the detected normal modes still last on time scales of less than

17The 95th percentile means the value in the corresponding histogram, where 95% of the pulses lie below thisthreshold.

17Like the 95th percentile but with 99%.

25

5 Moding Statistics

value normal normal mode abnormal abnormal modemode (corr. mode change) mode (corr. mode change)

fraction of 78.2% 84.1% 21.8% 15.9%observed time

95th percentile16 0.709 hr 0.958 hr 0.208 hr 0.444 hr≈ 42.5 min ≈ 57.5 min ≈ 12.5 min ≈ 26.6 min

99th percentile17 1.986 hr 1.986 hr 0.764 hr 0.792 hr≈ 119.2 min ≈ 119.2 min ≈ 45.8 min ≈ 47.5 min

mean mode length (0.185 ± 0.014) hr (0.321 ± 0.092) hr (0.050 ± 0.004) hr (0.120 ± 0.028) hr≈ (11.1 ± 0.8) min ≈ (19.3 ± 5.5) min ≈ (3.0 ± 0.2) min ≈ (7.2 ± 2.7) min

mean mode length 12.895 ± 1.034 19.630 ± 1.867 4.380 ± 0.282 8.037 ± 0.755in 80-pulse int.

number of 875 363 949 269modes

Table 4: Moding Statistics. “Corr. mode change” stands for the definition of the mode changegiven in § 4.1 with a mode change detected if three points lie above/below the threshold(y = 0.2). The uncertainty for the mean length of the mode is calculated with the root-mean-squared. As we do not consider a mode change if the “mode” was shorter than 3integrations, the uncertainties in columns 3 and 5 are greater than the others.

one hour, but there are a few outliers with lengths of more than one hour.For the abnormal mode one can see a similar picture on a shorter time scale and with only a few(3 in black and 6 in green) modes longer than one hour. Most of the modes in this case still lastshorter than 10 minutes.

26

5 Moding Statistics

1

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Modelength for the normal mode in hours

Chen et al. 2011

99%95%

95% 99%

modelength + deletedmodelength

mean for the black

(a) Histogram for the normal mode.

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Chen et al. 2011

99%95%

95% 99%

modelength + deletedmodelength

mean for the black

(b) Histogram for the abnormal mode.

Figure 17: Histogram of the mode-length in hours. In this histogram modes are only counted, ifthey are longer than 80-pulse-integrations and if the observation time during the modeis at least half as long as the time the mode lasts. Modes below the threshold (if themode is shorter and lies above the line in Fig. 14) are marked in green (if T > l·tint·2).The values plotted in green are effectively lower limits for the mode length. Moreabout the identification of modes in § 4. The mode length proposed by Chen et al.(2011), the 95th and the 99th percentiles are marked. The green percentiles (90%and 95%) are calculated for both, the regular data in black and the partialy observedmodes.

5.3 Offset in Mode length

The 95th and 99th percentile in black (Fig. 17) do lie far more left (at shorter mode lengths) thanthe green ones (mode lengths sorted out according to Fig. 14). Because we assume the greenvalues (not fully recorded modes) to represent a lower limit on the mode length, this distributionindicates that our presented mode length is biased by too short observation lengths.

We want to look how long our observations are and compare this values to Chen et al. (2011).In Tab. 5 one can see our total usable observation length to be much shorter than observationsused by Chen et al. (2011), as in our case data from 2016 were mainly deleted because they wererecorded with DE609 and looked different in the analysed principal components. This fact leadsto much shorter observation lengths (a larger fraction of the total observation time was spent insingle observations with a duration shorter than 2 hours) because the longest observations wererecorded in 2016 (see Tab. 6).

The fact that the black percentiles are smaller than the green percentiles, indicates that themode lengths in black have been underestimated. This implies that our results are biased by thelimited length of the observations. This is interesting because (Chen et al., 2011) give muchlonger indicative time scales, which suggests that they are not (as) sensitive to short-observationbiases.

27

6 Discussion and conclusions

length < 2 hr 2 hr ≤ length < 24 hr 24 hr < length total obs. lengthChen et al. (2011) 7.4% 23.7% 69.1% 521.0 hr

this thesis 27.1% 19.8% 53.1% 611.0 hrthis thesis without 42.9% 24.9% 32.3% 311.9 hrdata from 201618

(DE609)

Table 5: Overview of the observation length used by Chen et al. (2011) and in this thesis. Onecan see the data used by them to consist of longer continuous observations.


The aim of our reseach was to find out if pulsar moding is a broadband phenomenon or not. Pre-vious studies have been hampered by small-number statistics and are consequently contradictory.As it is still in debate weather mode changing occurs simultaneously at different frequencies(e.g. Gajjar et al. (2014) who showed that nulling and related events do occur simultaneously atdifferent frequencies versus Bhat et al. (2007) who showed that nulling does not occur simulta-neous at all frequencies), it is still unclear whether the time scale of moding published by Chenet al. (2011) holds for 150 MHz.

Our values for the overall fraction of the observing time (after correcting for false-positives)for PSR B0329+54 are: ∼ 15.9% in the normal mode and ∼ 84.1% in the abnormal mode.These values are nearly similar to the values given by Chen et al. (2011): 15.1% and 84.9%.This indicates that moding may indeed be broad-band. Having a closer look at e.g. Fig. 11(PC2) and Fig. 19 & 20 one can propose that mode changing for PSR B0329+54 does not oc-cur abruptly but takes time scales about a minute (because of the gradually rising), wherebythe time resolution of our 80 pulse integrations cannot reveal structures on time scales of lessthan a few minutes. Mode changing on time scales of less than our used integration time wouldyield gradually rising values in PC219, too. Based on this measure, the statistics at high and lowfrequencies seem identical, i.e. moding seems to be broad-band.

There are more (relevant) statistics to look at. Specifically one can compare the mode length:We see clearly that our analysis gives smaller mode lengths for the normal and abnormal modethan found by Chen et al. (2011). Thereby we have to consider that our observation lengths areoften shorter than the mode length published by them (see Tab. 5 and Fig. 18): more than∼ 80%of the observations were shorter than 2 hours (this is about 43% of the total observing time; seeTab. 5)20. Excluding data from the further analysis as described in § 4.1 and § 4.3 decreases the

18As discussed in Tab. 3, PCA-analysed data recorded in 2016 look very strange (and thereby was deleted consider-ing PC1022 as described in § 4).

19A gradually rising of the third peak (where the difference between the mode lies) on time scales of seconds wouldbe averaged by integrating pulses to 80-pulse integrations in a way that it would lead to a gradually rising in PC2.

20If we ignore the data recorded in 2016 which was deleted anyway, it is about 43% of the observing time.

28


number of mode changes, caused by the short observation length but still highly underestimatesthe mode length because our observations were too short and we identified a bias in our data.Further research based on longer continuous observations (more observations with a durationlonger than a few hours) would solve this problem as many complete modes and mode changeswould be recorded.We have to consider the order of our analysis again: at first (after deleting RFI affected data) weanalysed the data with respect to the mode length. Secondly (as described in § 4.1), we did notallow a mode change with a length less than three integrations. After that, we excluded modesthat were not adequately sampled, i.e. that only had observations during less than half the modelength. Maybe interchanging point two and three would affect the further analysis in a verycomplex way.

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Observation length in hours

Figure 18: Histogram of the observation length of the data used in this thesis.

We have to note that Chen et al. (2011) observed PSR B0329+54 at ∼ 1540 MHz and ouranalysis is based on LOFAR data at ∼ 150 MHz.

Finally, we have performed a proof-of-concept investigation of using PCA for mode-analysison LOFAR data but a definite answer on the consistency of mode lengths at LOFAR and higherfrequencies could be obtained providing longer, uninterrupted observations. DE609 data alreadyprovides a significant number of those but our present analysis was not able to deal with these,due to practical yet minor issues.

29

7 Appendix

7 Appendix

7.1 Single pulses

As PSR B0329+54 is a very bright pulsar, it is clearly visible in single pulses. The followingcontinuous scene (Fig. 19 and 20) was recorded on 19th of May 2015 in the very early afternoon(∼ 2 pm) and shows 30% of the pulse phase (the on-pulse region).We see clearly that the pulse shape varies a lot between those single pulses: The first pulse showsfour different components and the number of pulses decreases then from pulse to pulse and thepulsar is not visible then for four or five single pulses and switches back on (with differentcomponents).

Figure 19: Single pulses.

30

7 Appendix

Figure 20: Single pulses for the observed PSR B0329+54. The time between the single pulses is0.714s = P ; the time between two pulses.

7.2 Mode changing examples for PSR B0329+54

Here we show a few practical examples of what modes look like in our data. Specifically, inFig. 21a, a mode change is clearly visible and we can roughly estimate the mode length: ∼ 0.5hours for the abnormal mode and ∼ 1.5 hours for the normal mode.

31

7 Appendix

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PC

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(a) The pulsar is in the normal mode, the abnormal mode occurs and it goes back to thenormal mode after ∼ 0.02 days. About 0.06 days later, the abnormal mode occurs again.

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(b) One mode change recorded.

Figure 21: A few fully recorded mode changes. Clearly visible in principal component 2 is thegradually rising and some periodic behavior in the normal mode.

32

7 Appendix

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Figure 22: This is what we normally have in our observations: The normal mode is detectedonce and the abnormal mode twice. But because we do not have any mode changesactively detected, we can only determine a lower limit on the mode length.

7.3 Different principal components

In Fig. 23 we clearly see the two main principal components used in this thesis: PC2 to identifymodes and PC 1022 to identify RFI.

33

7 Appendix

-80

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PC1022

(b) Zoom-in. Clearly visible are the two modes: Nor-mal mode at y ≈ 0 and abnormal mode at y ≈ 0.4.

Figure 23: PC1022 against PC2 for all the data. One point corresponds to one 80-pulse inte-gration. Thereby, PC1022 gives an estimate of the RFI and PC2 an estimate for themodes. Note the order of magnitude for the zoom-in (by a factor of 104 in x and 102

in y).

34

7 Appendix

7.4 Observations

obs. Date MJD obs. length telescope freq. bandwidth obs. Date MJD obs. length telescope freq. bandwidthyyyy-mm-dd hr MHz MHz yyyy-mm-dd hr MHz MHz2012-12-04 56265.66 0.08 DE601 125.78 11.91 2014-02-11 56699.77 0.47 DE603 161.82 23.832012-12-04 56265.66 0.08 DE601 148.83 11.91 2014-02-18 56706.72 0.47 DE603 161.82 23.832012-12-04 56265.97 0.71 DE601 148.83 11.91 2014-02-24 56712.7 0.47 DE603 161.82 23.832012-12-10 56271.63 0.03 DE601 148.83 11.91 2014-03-07 56723.68 0.47 DE605 161.82 23.832012-12-10 56271.94 0.08 DE601 160.74 11.91 2014-03-09 56725.7 0.47 DE603 161.82 23.832013-01-14 56306.63 0.16 DE601 172.66 11.91 2014-03-14 56730.73 0.47 DE605 161.82 23.832013-01-28 56320.72 0.33 DE601 184.57 11.91 2014-03-16 56732.73 0.47 DE605 161.82 23.832013-01-28 56320.74 0.25 DE601 148.83 11.91 2014-03-23 56739.67 0.47 DE605 161.82 23.832013-01-28 56320.75 1.33 DE601 148.83 11.91 2014-03-30 56746.62 0.47 DE605 161.82 23.832013-02-04 56327.74 0.08 DE601 148.83 11.91 2014-04-06 56753.64 0.47 DE605 161.82 23.832013-02-11 56334.43 0.19 DE601 148.83 11.91 2014-04-11 56758.43 0.07 DE603 161.82 23.832013-02-11 56334.82 0.25 DE601 172.66 11.91 2014-04-11 56758.49 0.07 DE601 161.82 23.832013-02-11 56334.83 0.08 DE601 148.83 11.91 2014-04-11 56758.54 0.07 DE603 161.82 23.832013-02-11 56334.84 0.25 DE601 156.64 11.91 2014-04-11 56758.59 0.47 DE605 161.82 23.832013-02-26 56349.54 0.08 DE601 156.64 11.91 2014-04-13 56760.55 0.47 DE603 161.82 23.832013-03-05 56356.52 0.08 DE601 156.64 11.91 2014-04-18 56765.58 0.03 DE605 161.82 23.832013-03-12 56363.47 0.07 DE601 156.64 11.91 2014-04-21 56768.66 0.47 DE603 161.82 23.832013-03-18 56369.52 0.05 DE601 156.64 11.91 2014-04-25 56772.39 0.08 DE605 161.82 23.832013-03-25 56376.64 0.12 DE601 156.64 11.91 2014-04-25 56772.39 0.08 DE603 161.82 23.832013-03-26 56377.66 0.07 DE601 156.64 11.91 2014-04-25 56772.54 0.47 DE605 161.82 23.832013-03-28 56379.7 0.05 DE601 156.64 11.91 2014-05-04 56781.55 0.47 DE603 161.82 23.832013-03-29 56380.56 0.17 DE601 156.64 11.91 2014-05-09 56786.51 0.11 DE605 161.82 23.832013-04-09 56391.45 0.02 DE601 120.9 11.91 2014-05-11 56788.53 0.47 DE605 161.82 23.832013-04-23 56405.49 0.05 DE601 156.64 11.91 2014-05-18 56795.41 0.47 DE603 161.82 23.832013-04-23 56405.56 0.05 DE601 156.64 11.91 2014-05-19 56796.29 0.47 DE603 161.82 23.832013-04-23 56405.65 0.3 DE601 156.64 11.91 2014-05-23 56800.47 0.47 DE605 161.82 23.832013-04-30 56412.77 0.12 DE601 156.64 11.91 2014-05-25 56802.5 0.47 DE603 161.82 23.832013-04-30 56412.97 0.05 DE601 156.64 11.91 2014-05-30 56807.47 0.47 DE605 161.82 23.832013-05-03 56415.7 0.02 DE601 156.64 11.91 2014-06-01 56809.45 0.97 DE605 161.82 23.832013-05-03 56415.79 0.17 DE601 156.64 11.91 2014-06-14 56822.94 0.97 DE605 161.82 23.832013-05-04 56416.86 0.01 DE601 156.64 11.91 2014-06-21 56829.37 0.97 DE603 161.82 23.832013-05-07 56419.92 0.02 DE601 156.64 11.91 2014-06-21 56829.38 0.28 DE603 161.82 23.832013-05-13 56425.44 0.03 DE601 156.64 11.91 2014-06-28 56836.39 0.47 DE605 161.82 23.832013-05-13 56425.75 0.07 DE601 156.64 11.91 2014-06-30 56838.35 0.82 DE603 161.82 23.832013-05-14 56426.59 0.07 DE601 156.64 11.91 2014-07-05 56843.35 0.47 DE605 161.82 23.832013-05-22 56434.63 0.07 DE601 156.64 11.91 2014-07-06 56844.35 0.97 DE605 161.82 23.832013-06-17 56460.54 0.08 DE601 156.64 11.91 2014-07-13 56851.27 1.48 DE605 161.82 23.832013-06-26 56469.66 0.08 DE601 156.64 11.91 2014-07-20 56858.3 1.48 DE605 161.82 23.832013-07-01 56474.44 0.03 DE601 156.64 11.91 2014-07-27 56865.27 1.15 DE605 161.82 23.832013-07-01 56474.81 0.5 DE601 156.64 11.91 2014-08-03 56872.25 1.58 DE605 161.82 23.832013-07-02 56475.58 0.02 DE601 156.64 11.91 2014-08-09 56878.22 1.67 DE605 161.82 23.832013-07-09 56482.45 0.05 DE601 156.64 11.91 2014-08-25 56894.22 1.28 DE605 161.82 23.832013-07-17 56490.38 0.03 DE601 156.64 11.91 2014-08-30 56899.13 1.48 DE605 161.82 23.832013-07-23 56496.55 0.03 DE601 156.64 11.91 2014-09-08 56908.14 1.1 DE605 161.82 23.832013-07-25 56498.38 0.03 DE601 156.64 11.91 2014-09-13 56913.09 1.3 DE605 161.82 23.832013-07-26 56499.89 0.08 DE601 156.64 11.91 2014-09-21 56921.05 1.58 DE605 161.82 23.832013-07-27 56500.29 0.5 DE601 156.64 11.91 2014-09-29 56929.06 1.15 DE605 161.82 23.832013-07-28 56501.29 0.5 DE601 156.64 11.91 2014-10-05 56936 1.1 DE605 161.82 23.832013-08-07 56511.23 1 DE601 156.64 11.91 2014-10-12 56942.99 1.1 DE605 161.82 23.832013-08-12 56516.41 0.05 DE601 156.64 11.91 2014-10-20 56950.05 1.68 DE605 161.82 23.832013-08-16 56520.36 0.03 DE601 156.64 11.91 2014-10-26 56956.91 1.65 DE601 161.82 23.832013-08-16 56520.42 0.03 DE601 156.64 11.91 2014-10-31 56961.8 3.86 DE605 161.82 23.832013-08-19 56523.52 0.07 DE601 185.64 23.83 2014-11-03 56964.99 1.1 DE601 161.82 23.832013-08-19 56523.56 0.08 DE601 161.82 23.83 2014-11-07 56968.69 5.93 DE605 161.82 23.832013-08-26 56530.42 0.05 DE601 161.82 23.83 2014-11-09 56970.96 0.08 DE605 161.82 23.832013-09-04 56539.18 0.22 DE601 71.68 11.91 2014-11-16 56977.94 1.32 DE605 161.82 23.832013-09-05 56540.14 0.5 DE601 161.82 23.83 2014-11-21 56982.93 1.1 DE605 161.82 23.832013-09-05 56540.17 0.5 DE601 161.82 23.83 2014-11-29 56990.96 1.1 DE605 161.82 23.832013-09-06 56541.42 0.03 DE601 137.99 23.83 2014-12-05 56996.88 1.13 DE605 161.82 23.832013-09-15 56550.7 1.5 DE601 161.82 23.83 2014-12-12 57003.85 1.45 DE605 161.82 23.832013-09-18 56553.07 1.97 DE601 161.82 23.83 2014-12-21 57012.87 1.47 DE605 161.82 23.832013-09-30 56565.43 0.03 DE601 161.82 23.83 2014-12-22 57013.85 1.58 DE605 161.82 23.832013-10-04 56569.1 0.97 DE601 161.82 23.83 2014-12-25 57016.83 1.97 DE605 161.82 23.832013-10-08 56573.44 0.02 DE601 161.82 23.83 2015-01-03 57025.83 1.1 DE605 161.82 23.832013-10-18 56583.03 0.38 DE601 161.82 23.83 2015-01-05 57027.77 1.1 DE605 161.82 23.832013-11-11 56607.99 0.22 DE601 161.82 23.83 2015-01-18 57040.77 1.1 DE605 161.82 23.832013-11-18 56614.97 0.22 DE601 161.82 23.83 2015-01-24 57046.76 1.1 DE605 161.82 23.832013-12-04 56630.68 0.05 DE601 161.82 23.83 2015-01-30 57052.73 1.25 DE605 153.81 23.832013-12-04 56630.86 0.05 DE601 161.82 23.83 2015-02-06 57059.71 1.23 DE605 153.81 23.832013-12-11 56637.43 0.1 DE601 161.82 23.83 2015-02-14 57067.69 0.05 DE605 153.81 23.832013-12-11 56637.91 0.22 DE601 161.82 23.83 2015-02-22 57075.73 1.1 DE605 153.81 23.832013-12-25 56651.83 0.47 DE601 161.82 23.83 2015-03-01 57082.68 1.1 DE605 153.81 23.832013-12-26 56652.75 2 DE601 161.82 23.83 2015-03-06 57087.64 1.12 DE605 153.81 23.832013-12-31 56657.66 0.08 DE603 161.82 23.83 2015-03-13 57094.56 1.35 DE605 153.81 23.83

35

7 Appendix

obs. date MJD obs. length telescope freq. bandwidth obs. date MJD obs. length telescope freq. bandwidthyyyy-mm-dd hr MHz MHz yyyy-mm-dd hr MHz MHz2015-03-21 57102.56 1.78 DE605 153.81 23.83 2015-12-12 57368.93 1.77 DE605 153.81 23.832015-03-27 57108.57 1.2 DE605 153.81 23.83 2015-12-18 57374.91 1.77 DE609 153.81 23.832015-04-10 57122.55 2.07 DE605 153.81 23.83 2015-12-19 57375.01 53.16 DE609 153.81 23.832015-04-18 57130.57 1.77 DE605 153.81 23.83 2015-12-24 57380.43 47.47 DE605 153.81 23.832015-04-24 57136.51 1.98 DE605 153.81 23.83 2015-12-26 57382.85 2.55 DE609 153.81 23.832015-05-09 57151.51 1.77 DE605 153.81 23.83 2015-12-30 57386.39 47.47 DE605 153.81 23.832015-05-16 57158.5 1.77 DE605 153.81 23.83 2016-01-02 57389.81 2.37 DE609 153.81 23.832015-05-18 57160.44 22.44 DE605 153.81 23.83 2016-01-08 57395.47 0.97 DE609 153.81 23.832015-05-19 57161.44 0.45 DE605 153.81 23.83 2016-01-08 57395.56 0.97 DE605 153.81 23.832015-05-19 57161.56 0.73 DE605 153.81 23.83 2016-01-08 57395.78 2.05 DE609 153.81 23.832015-05-19 57161.64 0.01 DE605 153.81 23.83 2016-01-09 57396.44 0.97 DE609 153.81 23.832015-05-20 57162.56 12.82 DE605 153.81 23.83 2016-01-09 57396.61 0.97 DE609 153.81 23.832015-05-21 57163.19 0.04 DE605 153.81 23.83 2016-01-09 57396.78 0.97 DE609 153.81 23.832015-05-21 57163.31 4.25 DE602 153.81 23.83 2016-01-09 57396.95 0.97 DE609 153.81 23.832015-05-22 57164.45 2.97 DE605 153.81 23.83 2016-01-10 57397.11 0.97 DE609 153.81 23.832015-05-23 57165.43 1.25 DE605 153.81 23.83 2016-01-10 57397.28 0.97 DE609 153.81 23.832015-05-30 57172.44 1.77 DE605 153.81 23.83 2016-01-10 57397.57 0.97 DE609 153.81 23.832015-06-20 57193.41 1.77 DE601 161.82 23.83 2016-01-10 57397.73 0.97 DE609 153.81 23.832015-06-27 57200.31 2.8 DE605 153.81 23.83 2016-01-10 57397.9 0.97 DE609 153.81 23.832015-06-27 57200.39 1.77 DE601 161.82 23.83 2016-01-11 57398.07 0.97 DE605 153.81 23.832015-06-28 57201.31 2.97 DE601 161.82 23.83 2016-01-16 57403.78 1.95 DE605 153.81 23.832015-06-29 57202.32 1.41 DE605 153.81 23.83 2016-01-24 57411.81 1.77 DE605 153.81 23.832015-07-04 57207.37 1.77 DE605 153.81 23.83 2016-01-29 57416.74 1.87 DE609 153.81 23.832015-07-12 57215.32 1.77 DE605 153.81 23.83 2016-02-02 57420.66 0.97 DE609 153.81 23.832015-07-25 57228.26 1.77 DE605 153.81 23.83 2016-02-02 57420.83 0.97 DE609 153.81 23.832015-08-02 57236.25 1.77 DE605 153.81 23.83 2016-02-03 57421 0.97 DE609 153.81 23.832015-08-08 57242.22 1.77 DE605 153.81 23.83 2016-02-03 57421.17 0.97 DE605 153.81 23.832015-08-24 57258.2 2.08 DE605 153.81 23.83 2016-02-07 57425.78 1.93 DE609 153.81 23.832015-08-31 57265.18 2.08 DE605 153.81 23.83 2016-02-12 57430.63 0.97 DE609 153.81 23.832015-09-06 57271.18 1.77 DE605 153.81 23.83 2016-02-12 57430.8 0.97 DE609 153.81 23.832015-09-14 57279.13 1.87 DE605 153.81 23.83 2016-02-12 57430.97 0.97 DE609 153.81 23.832015-09-16 57281.24 1.77 DE605 153.81 23.83 2016-02-13 57431.13 0.97 DE605 153.81 23.832015-09-26 57291.09 1.88 DE605 153.81 23.83 2016-02-14 57432.78 1.77 DE609 153.81 23.832015-10-04 57299.07 1.98 DE609 129.98 23.83 2016-02-20 57438.52 4.56 DE609 153.81 23.832015-10-04 57299.07 1.98 DE605 153.81 23.83 2016-02-20 57438.77 4.57 DE609 153.81 23.832015-10-18 57313.09 1.88 DE609 129.98 23.83 2016-02-21 57439.02 4.54 DE609 153.81 23.832015-10-18 57313.09 1.88 DE609 153.81 23.83 2016-02-21 57439.27 4.68 DE609 153.81 23.832015-10-24 57319.02 2.13 DE605 153.81 23.83 2016-02-21 57439.52 4.56 DE605 153.81 23.832015-10-24 57319.03 2.13 DE605 153.81 23.83 2016-02-21 57439.7 1.77 DE609 153.81 23.832015-10-30 57326 1.87 DE605 153.81 23.83 2016-02-21 57439.77 4.71 DE609 153.81 23.832015-11-06 57332.97 2.22 DE609 153.81 23.83 2016-02-22 57440.02 5.93 DE605 153.81 23.832015-11-07 57333.01 2.22 DE605 153.81 23.83 2016-02-26 57444.66 2.02 DE609 153.81 23.832015-11-15 57341.96 1.77 DE605 153.81 23.83 2016-03-04 57451.72 57.19 DE609 153.81 23.832015-11-21 57347.95 1.77 DE605 153.81 23.83 2016-03-07 57454.2 2.97 DE609 153.81 23.832015-11-28 57354.93 1.77 DE605 153.81 23.83 2016-03-18 57465.74 1.17 DE609 153.81 23.832015-12-05 57361.92 1.77 DE601 161.82 23.83 2016-03-18 57465.91 56.37 DE609 153.81 23.832015-12-10 57366.88 1.97 DE601 161.82 23.83 2016-04-01 57479.39 62.3 DE609 153.81 23.83

Table 6: Observations used in this thesis. The sum of the observation length is 611.02 hr.

36

7 Appendix

Figure 24: A single LOFAR LBA-antenna (foreground) and other antennas from the same field(DE603) in the background. Height of the pole in the middle ≈ 1.5m.

37

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Statutory declaration

I declare that this Bachelor Thesis is completely written by myself. All contents, that originatefrom others or literal quotation are marked. Furthermore this Thesis was not part of any otherpublication nor used to achieve any other academic degree.

Bielefeld, November 7, 2016 Signum

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Documents

Moding Statistics for PSR B0329+54 at LOFAR Frequencies · 1 Introduction to pulsar astronomy The most rapidly rotating pulsar (PSR J1748-2446ad) known today has a rotational period