Monte Carlo simulating the magnetar central enigne model for further constraining using LOFAR

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Bachelor Thesis

Bachelor of Physics and Astronomy (Joint Degree)

Nathan Peters student number: 11272341

VU, University Amsterdam Anton Pannekoek Instituut, Department of Physics and Astronomy

Faculteit der Natuurwetenschappen, Wiskunde en Informatica and faculteit der B´etawetenschappen Daily supervisor: Kelly Gourdji

Supervisor: Dr. Antonia Rowlinson Examinator: Pof. Dr. Ralph Weijers

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Abstract

Using a Monte Carlo simulation, the magnetar central engine model is sampled in order to find and visualise the most constrain-ing parameters of the model, and to point out useful constraints for further observations using LOFAR. The incoming flux density has been compared with an achieved lower flux density limit of 1.7 mJy. The model has also been adapted for the cosmological redshift effect. The visualisations show that the efficiency is a constraining parameter. More observations have to be done to be able to rule out the magnetar central engine model. For an efficiency of 10−4 this would have to be 7.3 two-hour observations. Also, the spin period is more constraining than the magnetic field strength and the cosmo-logical redshift effect is not significant in comparison with the effect of other parameters.

Neutronsterren zijn compacte sterren met een massa rond 1.4 keer die van de zon, en hebben een sterk magnetisch veld. Neut-ronsterren met een bijzonder sterk magnetisch veld, rond 1015 keer zo sterk als die van de aarde, worden magnetars genoemd. Het kan voorkomen dat twee neutronsterren samensmelten. Wat hierna ge-vormd wordt, is nog onbekend. Het magnetar model voorspelt dat een magnetar gevormd wordt uit de twee samengesmolten neutron-sterren. Er wordt verwacht dat deze magnetars ook radiostraling vanuit hun polen uitzenden. Omdat de magnetar snel draait, met een periode tussen de 0.55 ms en 8300 ms, worden de radiogolven als regelmatige pulsen ontvangen. Het magnetar central engine model is een model dat de meetbare flux van een magnetar voorspelt, aan de hand van de fysische parameters van de magnetar. In dit on-derzoek is het model een miljoen keer gesimuleerd met willekeurige parameterwaardes binnen een bepaald bereik. Met de resultaten hier-van kunnen voorspellingen gedaan worden over toekomstige radio-observaties met de LOFAR radiotelescoop.

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1 Introduction

Gamma ray bursts (GRBs) are bright bursts in the gamma-ray spectrum emerging from distant galaxies. (NASA, 2010) A specific kind of GRBs, short GRBs (SGRBs), are bursts lasting typically less than two seconds. These SGRBs have been associated with compact binary mergers. Binary mergers are neutron star mergers or a black hole and neutron star merger. (Chu et al., 2016) During these mergers, it is thought that the collapse of the two structures into one is an origin of SGRBs. (Nakar, 2007).

The nature of the merger remnant remains unknown. In (Rowlinson et al., 2013), there is a suggestion that the merger of two neutron stars can form a millisecond magnetar, emitting electromagnetic waves. Depending on the masses of the progenitors, the remnant will either remain a stable magnetar or will collapse into a black hole after losing some of its rotational energy trough electromagnetic and gravitational waves. The mass and the rotational period at which this would happen, depends on the magnetars equation of state. If the magnetar remains stable, the pulses will gradually decrease in luminosity as it spins down. Figure 1 illustrates this hypothesis.

Figure 1: This cartoon is taken from (Rowlinson et al., 2014) and edited. It shows the possible courses of a neutron star merger or a neutron star and black hole merger. The magnetar starts emitting radio waves directly after the merger.

There are multiple indications that a magnetar is the remnant of a binary merger, as shown in (Rowlinson et al., 2013). However, decisive evidence still remains to be found. Rowlinson and Anderson (2019) expect newly formed magnetars to emit copious amounts of radio waves. Searching for these waves using radio telescopes could place stringent constraints on the magnetar theory. The magnetar central engine model in (Rowlinson and Anderson, 2019) predicts the flux density of the emitted radio waves

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from the magnetar. This project aims to visualise the different parameters of this model, to point out useful constraints and to help give direction to future observations. This is done using a Monte Carlo sampling strategy where a sample population of magnetars is created.

2 Theoretical background and method

2.1 Magnetars and pulsars

Magnetars are a subcategory of neutron stars - dense star remnants with a typical mass of 1.4 M and a radius of 10 Km - distinguishing them-selves by having a magnetic field typically 103 to 105 stronger than regular magnetars. (Kaspi and Beloborodov, 2017) After about 10.000 years, their magnetic field decays. Pulsars are another subcategory of neutron stars emitting electromagnetic radiation at the poles. Because neutron stars ro-tate, this radiation is detected like pulses as the beam passes the earth. The magnetar theory predicts a neutron star that is a magnetar and a pulsar.

2.2 Magnetar central engine model

Rowlinson and Anderson (2019) predict the flux density of the radio emis-sion, in (Jy), to be:

Fv ∼= 8 × 107v_obs−1rD_lum−2 B152 R 6 6P

−4

−3J y (1)

In equation 1, v_obs−1 is observing frequency in MHz. r is the efficiency at which the magnetar converts its rotational energy into radiation. This depends on its emmission mechanism, which is not known, and remains uncertain. Dlum is the luminosity distance in Gpc. B15 is the field strength

B

1015 in Gauss. R6 is the radius of the magnetar ₁₀R6 in centimeters. P−3 is

the spin period in ms.

2.2.1 cosmological redshift

The magnetar model does not incorporate the cosmological redshift effect, that would effect the flux density. As the radiation travels trough space, it gets lengthened as space expands. This causes the peak of the spectrum to redshift. At large luminosity distances this has a notable impact. The

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magnetar model 1 does not incorporate this effect. Therefor it has been adapted to: Fv ∼= 8 × 107v−1_obsrD_lum−2 B152 R 6 6P −4 −3 × ( 1 1 + z) −α_{J y} ₍₂₎

Here, z is the redshift and α is the spectral index. This constant is currently not well known. For this model, the redshift z is found using Astropy’s z at value function. (Astropy Collaboration et al., 2018) The modeled universe used is a flat, cold dark matter universe with a cos-mological constant. The used Hubble constant H0 = 69.6 Km/s/Mpc, the temperature of the cosmic background radiation Tcmb0 = 2.725 K and

Ω0 = 0.286. This model will be referred to as the cosmological magnetar central engine model. As the redshift only goes up to 1, it is expected that its effect only significantly affects the radio magnetar remnants sampled at highest redshift.

2.3 Sampling strategy and parameters

The sampling of the population of magnetars has been done using a Monte Carlo sampling method using Python 2.7. (van Rossum, 1995) The code can be found in the appendix. This method samples random values in between a given range for some variable. A population of samples is created by sampling multiple variables. In this paper, every visualisation has been created by sampling the basic parameters for the spin period and magnetic field strength. Then per plot other parameters like the luminosity distance, the efficiency and the spectral index have been sampled. These parameters are discussed in the following subsections.

2.3.1 spin period, magnetic field strength and luminosity dis-tance

Every simulation is based on the sampling of the spin period and the mag-netic field strength. The magmag-netic field strength range used, is from 1014_G to 1017 _{G. This is a plausible range coming from (Rowlinson et al., 2014).} The magnetic field strength and spin period remain uncertain because it is directly related to the equation of state of the magnetar, which is cur-rently not well known. The used range of spin periods is from 0.55 ms to 8300 ms. (Lattimer, 2004), (Rowlinson et al., 2014). The luminosity

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distance is also sampled, ranging from 40 Mpc to 6701.1 Mpc or a red-shift range from 9.2 × 10−3 to 1. This range comes from the distribution of known SGRBs.(Berger, 2014) The cosmological calculator of Wright (2006) is used to calculate the luminosity distance. The spin period, the magnetic field strength and the luminosity distance are sampled in logarithmic scale. 2.3.2 Flux density limits, observation configurations and

ob-serving frequencies vobs

For a better understanding of the possibilities of the different LOFAR ob-serving configurations, a flux density limit is included in the different visu-alisations. In the visualisations, the limit of 1.7 mJy at 144 MHz is used, coming from (Rowlinson et al., 2020), where the LOFAR Full dutch array was used. However, this limit highly depends on several factors such as the observing setup, the duration of observation, the sky condition and the location of the source relative to LOFAR’s field of view. Also, a tool is created that incorporates different theoretical flux density limits of the dif-ferent LOFAR observing configurations at difdif-ferent observation frequencies vobs coming from (ASTRON, 2021). This could also give more insight on LOFAR’s capabilities.

2.3.3 spin down efficiency r

In the magnetar central engine model, r dictates efficiency of the radio magnetar remnant converting its rotational energy into radiation. This ef-ficiency is typically thought of to be around ∼ 10−4.(Rowlinson and Ander-son, 2019) However, these efficiencies can vary and are poorly constrained. In this paper, the efficiency in the the central engine model has also been varied between 10−10 ≤ r ≤ 10−2 as is done in (Rowlinson and Anderson, 2019) and found in (Szary et al., 2014).

2.3.4 Spectral index α

In the cosmological magnetar central engine model 2, the spectral index α plays a crucial role in the weight of the redshift adaptation of the magnetar central engine model. Because this index is not well known, this is also varied. In these simulations, the spectral index is varied between −1 and −3.0. (James J. Condon, 2018)

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3 Results

3.1 Magnetar model

The Monte Carlo simulations of magnetar model have resulted in the fol-lowing visualisations. First, figure 2, figure 3 and figure 4 show different plotted parameters made using the original magnetar model. Figure 5 has been made using the adapted cosmological magnetar central engine model.

Figure 2: This figure shows the flux densities in mJy at different luminosity distances in megaparsec of the million sampled magnetars. The blue marks indicate the sampled radio magnetar remnant population. The red line indicates the flux density limit of the LOFAR full Dutch array at 1.7 mJy at 144 MHz. Of this sampled population, 42.6(1)% of the radio magnetar remnants would be detectable. A r of 10−4 has been assumed. The minimum spin period used 0.55 ms. The maximum spin period used is 8300 ms. The minimum magnetic field strength used is 1014 G and the maximum magnetic field strength of the sampled population is 1017 G.

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In figure 2 The possible measured flux densities in mJy is plotted against the luminosity distance in megaparsec. Also, the red line indicates the flux density limit of LOFAR. In total, the simulations predict 42.6(1) % of ex-isting radio magnetar remnants in the given luminosity distance limits to be detectable with the LOFAR full Dutch array. It is clearly visible that at all luminosity distances in the used range, radio magnetar remnants would be detectable with the LOFAR full dutch array with a flux density limit of 1.7 mJy.

Figure 3: This figure shows the magnetic field strength plotted against the spin period of a sampled population of magnetars. The colour gradient indicaties the flux density. The red line indicates the flux density limit of the LOFAR full Dutch array at 1.7 mJy at 144 MHz. The sample is taken at a luminosity distance of 3370.55 Mpc. For this population, a r of 10−4 has been assumed. The minimum spin period used 0.55 ms. The maximum spin period used is 8300 ms. The minimum magnetic field strength used is 1014 G. The maximum magnetic field strength of the sampled population is 1017 G.

Figure 3 is a colour bar plot showing the possible different magnetic field strengths and spin periods of the sampled magnetar population at a specific distance of 3370.55 Mpc. The colour gradient indicates the

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associ-ated flux density of the magnetar. The red line indicates the minimum flux density limit. This figure shows the relation between the spin period and the magnetic field strength. This figure clearly shows that the spin period is much more constraining on the detectability of the radio magnetar rem-nant than the magnetic field strength. At a spin period of 2 × 102 _{ms at} the above-mentioned luminosity distance, no radio magnetar remnant is detectable, no matter its magnetic field strength.

Figure 4: This figure shows the effect of different possible efficiencies r on the possible flux densities at different luminosity distances. The blue marks indicate the sampled radio magnetar remnant population. The red line indicates the Flux density limit of the LOFAR full Dutch array at 1.7 mJy at 144 MHz. The effi-ciencies (eff) are shown in the figure per subplot. The detectable percentages of the sampled magnetars is also shown per plot. For this simulation the minimum spin period used was 0.55 ms. The maximum spin period used is 8300 ms. The minimum magnetic field strength used is 1014 G. The maximum field strength of the sampled population is 1017 G.

Figure 4 gives a visualisation of possible flux densities at different lu-minosity distances with the spin down efficiency r varied. This shows a range of possible detectable portions of the radio magnetar remnants and the effect of the uncertain spin down efficiency on the detectable radio spectrum. At an efficiency of 10−10, only 9.3(1) % of the radio magnetar remnants would be detectable, compared to 42.7(1) % at the expected ef-ficiency of 10−4. It is clear that the efficiency is a constraining parameter

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in the model, giving a difference of 33.4(2) %. Notable is that even at an efficiency of 10−10 some of the radio magnetar remnants at the highest redshift would be detectable.

3.2 Cosmological magnetar central engine model

While the above-mentioned plots had no incorporation of cosmological ef-fects, the next visualisation does incorporate cosmological effects. In figure 5, the flux density is plotted against the luminosity distance of the sampled population of magnetars with varying spectral indices α. At a spectral in-dex of −1, the detectability of the radio magnetar remnants only differs with 0.5(1) % from the magnetar central engine model without the redshift correction. At the lowest spectral index of −3, the difference is 1.5(2) %.

Figure 5: This figure shows the effect of different spectral indices on the detect ability of the sampled population of 106 magnetars. The blue marks indicate the sampled radio magnetar remnant population. The red line indicates the Flux density limit of the LOFAR full Dutch array at 1.7 mJy at 144 MHz. For this simulation the minimum spin period used was 0.55 ms. The maximum spin period used is 8300 ms. The minimum magnetic field strength used is 1014 _G. The maximum magnetic field strength of the sampled population is 1017 G.

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4 Discussion

A sample population of magnetars using the magnetar central engine model 1 applied in (Rowlinson and Anderson, 2019) and the adapted magnetar central engine model 2 has been created using a Monte Carlo sampling method. Different visualisations of these models give insight on constrain-ing the magnetar model. Figure 2 shows that, with an assumed r of 10−4, it would be possible to find 42.6(1) % radio magnetar remnants at any luminosity distance in the expected luminosity distance range of 40 Mpc to a redshift of 1, at certain combination of spin period and magnetic field strength, given a lower flux density limit of 1.7 mJy achieved in (Rowlin-son et al., 2020). Figure 3 shows the spin-magnetic field strength relation and the clearly stronger constraining impact of the spin period on the de-tectability of the radio magnetar remnant. Especially for radio magnetar remnants at a distance close to a redshift of 1, only those with the smallest spin period would be detectable. Observing the pulses in other electromag-netic wave bands could also help constrain other variables in the magnetar central engine model.

By varying the efficiencies in figure 4, different scenarios are presen-ted for detectable magnetars. The efficiency has a considerable impact on the amount of detectable radio magnetar remnants. At a difference of 10−6, the difference in the detectable percentage is 34.4(2) %. But even at 10−10, radio magnetar remnants could be detectable at all luminosity dis-tances in the given range with a small enough spin period and high enough magnetic field strength. Through observations, this parameter can also be constrained. In (Rowlinson et al., 2020) for example, they constrain the efficiency to be r ≤ 6 × 10−8, as no fast radio burst was detected. This places a possible limit on the maximum efficiency of the magnetar.

In figure 5, the possible cosmological redshift effect is shown. In compar-ison with other factors such as the efficiency, the spin period, the magnetic field and the luminosity distance, the cosmological redshift has a small im-pact on the flux density. This can be easily explained by looking at the used range of luminosity distances. At the maximum used redshift of z = 1, the cosmological term in the cosmological magnetar central engine model 2

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scales the original flux density with 8 with a spectral index α = −3. Even with a spectral index α = −5, higher than reported in (Jun Han, 2017), the flux density at a redshift of 1 only scales with a factor of 32. It is important to note that the redshift has little effect on this sample population because luminosity distance is sampled logarithmically. This places most of the population at a relatively small redshift on the used scale. Therefore, the redshift effect is is relatively small on most samples.

With the detectable percentages coming from these simulations, pre-dictions can be made for future observations. If the efficiency is 10−4, and the observations attain a lower flux density limit of 1.7 mJy, resulting in 42.6(1) % detectable radio magnetar remnants, 8.3 two-hour observations like in (Rowlinson et al., 2020) would have to conducted to achieve a 99 % confidence interval. However, with an efficiency of r ≤ 6 × 10−8, at least 17.3 two-hour observation would have to be conducted to achieve a 99 % confidence interval. Of course, even with the model being correct, other factors could severely disturb expected results. First of all, the emission could be hindered in its path. The emission could be unable to escape the magnetars opaque surface. Also, the beaming angle of the radio emission could be less aligned than assumed. However, this seems unlikely as (Rowl-inson et al., 2014) predicted. It could also be possible that no magnetar was created in the first place. Finally, the model in (Rowlinson and Anderson, 2019) could be incorrect.

5 Conclusion

The magnetar central engine model in (Rowlinson and Anderson, 2019) has been modeled using a Monte Carlo sampling method creating a sample population of 106 radio magnetar remnants. This has also been done with the cosmological magnetar central engine model. The visualisations give an overview of different parameters that could be constrained with fur-ther observations using LOFAR. By using a flux density limit obtained in (Rowlinson et al., 2020), percentages of detectable radio magnetar rem-nants can be found. At an efficiency of 10−4, 8.3 two-hour observations would be necessary to obtain a 99 % confidence interval. However, if the efficiency is lower, as found in (Rowlinson et al., 2020), at least 17.3

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two-hour observations would have to be conducted to achieve a 99 % confidence interval. Furthermore, the spin period has a stronger constraining impact on the flux density than the magnetic field strength. Finally, the redshift effect has a relatively small impact on the flux density in comparison with other parameters like the efficiency and the spin period. More observations have to be conducted to place more stringent constraints on the magnetar model.

6 Acknowledgements

I would like to thank ms. Kelly Gourdji for the daily supervision of my work, as well as dr. Rowlinson for being my supervisor and Professor dr. Ralph Weijers for being my exterminator. I would also like to thank the Anton Pannekoek institute Transients Group for their input and support.

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A

code

This code is one of the programs made to simulate the radio magnetar remnant population by using a Monte Carlo sampling method. This code also varies the efficiencies of the magnetar populations resulting in figure 4.

# s a m p e l i n g a r a d i o magneter remnent and v a r i e n g i t s e f f i c i e n c y

import random a s rd

import m a t p l o t l i b . p y p l o t a s p l t import numpy a s np

import i n q u i r e r

# o v e r r u l i n g f r e q c u e n c y and o b s e r v a t i o n q u e s t i o n s t o u s e 1 . 7 mJy l i m i t a t 144 MHz coming from R o w l in s o n 2020

q u e s t i o n s = [ i n q u i r e r . L i s t ( ’ o v e r r u l e ’ , message=” O v e r r u l e o t h e r q u e s t i o n s and u s e 144 MHz w i t h 1 . 7 mJy l i m i t ? ” , c h o i c e s =[ ’ Yes ’ , ’ No ’ ] , ) , ] a n s w e r s = i n q u i r e r . prompt ( q u e s t i o n s )

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#t r a n s l a t i o n o f answer i n b i n a r y c o d e o v e r r u l e d = { ’ Yes ’ : i n t ( 1 ) , ’ No ’ : i n t ( 0 ) } o v e r r u l e = o v e r r u l e d [ a n s w e r s [ ” o v e r r u l e ” ] ] # a c t i n g on p r e v i o u s q u e s t i o n i f o v e r r u l e < 1 : q u e s t i o n s = [ i n q u i r e r . L i s t ( ’ o b s e r v a t o r y ’ ,

message=”Which o b s e r v a t o r y do you want t o u s e ? ” , c h o i c e s =[ ’ S u p e r t e r p ’ , ’NL Core ’ , ’ F u l l NL ’ , ’ F u l l EU ’ ] , ) , ] a n s w e r s = i n q u i r e r . prompt ( q u e s t i o n s ) print a n s w e r s [ ” o b s e r v a t o r y ” ] # t r a n s l a t i o n o f o b s e r v a t o r i e s o b s d = { ’ S u p e r t e r p ’ : i n t ( 0 ) , ’NL Core ’ : i n t ( 1 ) , ’ F u l l NL ’ : i n t ( 2 ) , ’ F u l l EU ’ : i n t ( 3 ) } obs = o b s d [ a n s w e r s [ ” o b s e r v a t o r y ” ] ] q u e s t i o n s = [ i n q u i r e r . L i s t ( ’ f r e q u e n c y ’ , message=”And a t what f r e q u e n c y ? ” , c h o i c e s =[ ’ 30 Mhz ’ , ’ 45 Mhz ’ , ’ 60 Mhz ’ , ’ 75 Mhz ’ , ’ 120 Mhz ’ , ’ 150 Mhz ’ , ’ 180 Mhz ’ , ’ 200 Mhz ’ , ’ 210 Mhz ’ , ’ 240 Mhz ’ ] , ) , ] a n s w e r s = i n q u i r e r . prompt ( q u e s t i o n s ) print a n s w e r s [ ” f r e q u e n c y ” ] # t r a n s l a t i o n o f f r e q u e n c i e s f r e q d = { ’ 30 Mhz ’ : i n t ( 0 ) , ’ 45 Mhz ’ : i n t ( 1 ) , ’ 60 Mhz ’ : i n t ( 2 ) , ’ 75 Mhz ’ : i n t ( 3 ) , ’ 120 Mhz ’ : i n t ( 4 ) , ’ 150 Mhz ’ : i n t ( 5 ) , ’ 180 Mhz ’ : i n t ( 6 ) , ’ 200 Mhz ’ : i n t ( 7 ) , ’ 210 Mhz ’ : i n t ( 8 ) , ’ 240 Mhz ’ : i n t ( 9 ) } f r e q = f r e q d [ a n s w e r s [ ” f r e q u e n c y ” ] ] # a r r a y o f a l l s e n s i t i v i t i e s p e r o b s e r v a t o r y s e n s i t i v i t i e s = np . a r r a y ( [ [ 3 6 . 0 , 2 9 . 0 , 2 5 . 0 , 4 4 . 0 , 1 . 5 ,

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1 . 3 , 1 . 5 , 2 . 5 , 2 . 5 , 5 . 6 ] , #s u p e r t e r p [ 9 . 0 , 7 . 4 , 6 . 2 , 1 0 . 8 , 0 . 3 8 , 0 . 3 1 , 0 . 3 8 , 0 . 6 2 , 0 . 6 2 , 1 . 4 ] , #NL c o r e [ 5 . 7 , 4 . 7 , 3 . 9 , 6 . 8 , 0 . 3 , 0 . 3 , 0 . 3 , 0 . 4 8 , 0 . 4 8 , 1 . 1 ] , # F U l l NL [ 3 . 8 , 3 . 1 , 2 . 6 , 4 . 5 , 0 . 2 , 0 . 1 6 , 0 . 2 0 , 0 . 1 6 , 0 . 2 , 0 . 3 2 , 0 . 3 2 , 0 . 7 3 ] ] ) # f u l l eu # l i s t o f o b s e r v i n g f r e q u e n c i e s i n Mhz L V obs = [ 3 0 . , 4 5 . , 6 0 . , 7 5 . , 1 2 0 . , 1 5 0 . , 1 8 0 . , 2 0 0 . , 2 1 0 . , 2 4 0 . ] V obs = L V obs [ f r e q ] #i n p u t o f c h o i s e f o r f r e q u e n c y i s p u t a s c o n s t a n t f l u x d e n s i t y l i m i t = s e n s i t i v i t i e s [ obs ] [ f r e q ] # f i n d i n g f l u x d e n s i t y l i m i t e l s e : V obs = 144 #MHz f l u x d e n s i t y l i m i t = 1 . 7 #mJy # f u n c t i o n c a l c u l a t i n g f l u x

def t e s t 7 (M, R, min B , max B , min spinP , max spinP , V obs , e p s i l o n r , D lum min , D lum max , f l u x d e n s i t y l i m i t ) :

# t o t a l t r i e s f o r E and P N = 1000 # l i s t o f v a r i a b l e s # p a r t 1 L P = [ ] L B = [ ] # p a r t 2 L D lum = [ ] L Flux = [ ] L f d l = [ ] P e r c e n t a g e c o u n t e r = 0 . f o r n in range (N) :

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# random B and P g e n e r a t e d

RN1 = rd . random ( ) ∗ (max B − min B ) + min B

RN2 = rd . random ( ) ∗ ( max spinP − min spinP ) + min spinP RN3 = rd . random ( ) ∗ ( D lum max − D lum min ) + D lum min

Random B = 1 0∗∗(RN1) ∗10∗∗−15 # B in 10ˆ15

Random P = 1 0∗∗(RN2) # P in

Random D = 1 0∗∗(RN3)

Flux = 8∗10∗∗(7) ∗ V obs ∗∗( −1) ∗ e p s i l o n r ∗ Random D ∗∗( −2) ∗ Random B ∗∗2 ∗ Random P∗∗( −4) ∗ 10∗∗3 # R i s a l r e a d y 10 ∗∗6 ( in mJy)

# Added t o l i s t s L B . append ( Random B ) L P . append ( Random P )

L Flux . append ( Flux ) L D lum . append ( Random D )

L f d l . append ( f l u x d e n s i t y l i m i t ) # c o u n t i n g amount o f f l u x i s h i g h e r t h a n t h e f l u x d e n s i t y l i m i t i f Flux >= f l u x d e n s i t y l i m i t : P e r c e n t a g e c o u n t e r = P e r c e n t a g e c o u n t e r +1 # c a l c u l a t i n g p e r c e n t a g e v i s i b l e p e r c a n t a g e = P e r c e n t a g e c o u n t e r / (N) ∗100

return L P , L B , L Flux , L D lum , L f d l , v i s i b l e p e r c a n t a g e

# LIST OF CONSTANTS M = 2 . 1∗ 1 . 9 9 ∗ 1 0 ∗ ∗ ( 3 0 ) R = 10∗∗6 # in cm min B = 14 # i n l o g G max B = 17 # i n l o g g m i n s p i n P = np . l o g 1 0 ( 0 . 5 5 ) # i n l o g ms max spinP = np . l o g 1 0 ( 8 3 0 0 ) # i n l o g ms e p s i l o n r = 10∗∗−4 D lum min = np . l o g 1 0 ( 0 . 0 4 0 0 0 ) # i n l o g Gpc D lum max = np . l o g 1 0 ( 6 . 7 0 1 1 ) # i n l o g Gpc

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# l i s t o f d i f f e r e n t p o s s i b l e e f f i c i e n c i e s ( e p s i l o n R) L e f f i c i e n c i e s = [ 1 0∗∗ −10 , 10∗∗−7, 10∗∗−4, 10∗∗ −2]

d a t a = t e s t 7 (M, R, min B , max B , min spinP , max spinP , V obs , e p s i l o n r , D lum min , D lum max , f l u x d e n s i t y l i m i t )

L P , L B , L Flux , L D lum , L f d l , v i s i b l e p e r c a n t a g e = d a t a

L o b s e r v a t o r i e s = [ ’ S u p e r t e r p ’ , ’NL Core ’ , ’ F u l l NL ’ , ’ F u l l EU ’ ]

def p l o t t i n g ( L e f f i c i e n c i e s ) : Colums = len ( L e f f i c i e n c i e s )

f i g , a x s = p l t . s u b p l o t s ( 1 , Colums , s h a r e y=True , s h a r e x=True ) f i g . s u p t i t l e ( ’ P o s s i b l e Flux d e n s i t i e s a t d i f f e r e n t l u m i n o s c i t y d i s t a n c e s w i t h v a r y i n g e f f i c i e n c i e s . With a f l u x d e n s i t y l i m i t o f %.2 f mJy a t %.1 f Mhz . ’ % ( f l u x d e n s i t y l i m i t , V obs ) , f o n t s i z e = 1 7 ) f o r i in range ( Colums ) :

d a t a = t e s t 7 (M, R, min B , max B , min spinP , max spinP , V obs , L e f f i c i e n c i e s [ i ] , D lum min , D lum max , f l u x d e n s i t y l i m i t ) L P , L B , L Flux , L D lum , L f d l , v i s i b l e p e r c a n t a g e = d a t a x = L D lum y = L Flux # C r e a t e f o u r p o l a r a x e s and a c c e s s them t h r o u g h t h e r e t u r n e d a r r a y a x s [ i ] . p l o t ( x , y , ’ bo ’ , m a r k e r s i z e = 0 . 0 1 ) a x s [ i ] . p l o t ( x , L f d l , ’ r ’ ) # c r e a t i n g t h e f l u x l i m i t l i n e p l t . y s c a l e ( ’ l o g ’ ) p l t . x s c a l e ( ’ l o g ’ ) a x s [ i ] . s e t t i t l e ( ’ e f f = %.2E , %.1 f%% d e t e c t a b l e ’ % ( L e f f i c i e n c i e s [ i ] , v i s i b l e p e r c a n t a g e ) ) # Adding l a b e l s t o t h e a x e s

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f i g . a d d s u b p l o t ( 1 1 1 , frameon=F a l s e ) # h i d e t i c k and t i c k l a b e l o f t h e b i g a x i s p l t . t i c k p a r a m s ( l a b e l c o l o r= ’ none ’ , t o p=F a l s e , bottom=F a l s e , l e f t =F a l s e , r i g h t=F a l s e ) p l t . x l a b e l ( ’ L u m i n o s i t y d i s t a n c e (Mpc) ’ , f o n t s i z e = 1 7 ) p l t . y l a b e l ( ” Flux d e n s i t y F i n (mJy) ” , f o n t s i z e = 1 7 ) p l t . show ( ) p l o t t i n g ( L e f f i c i e n c i e s )

B

link

This link gives access to all codes created during this project:

Monte Carlo simulating the magnetar central enigne model for further constraining using LOFAR

Contents

1

Introduction

2

Theoretical background and method

2.1

Magnetars and pulsars

2.2

Magnetar central engine model

2.3

Sampling strategy and parameters

3

Results

3.1

Magnetar model

3.2

Cosmological magnetar central engine model

4

Discussion

5

Conclusion

6

Acknowledgements

References

A

code

B

link