Summer Research Fellowship Programme-2012

Project Title- To estimate the magnetic field

strength of X-ray Binary Systems(LMXB)

BySambit Kumar Panda

Phys267

July 5, 2014

Contents

1 ABSTRACT 2

2 Introdution to Neutron Stars and pulsars and the source Swift-J1749 2

3 Some properties of Accreting Neutron stars 3

4 RXTE Observation data of Swift J1749.4-2807 and its analysis 5

5 Description of the models 8

6 The Disk-Magnetosphere interaction in the acretion poweredmillisecond pulsar SWIFT J1794.4-2807 106.1 Limits on the magnetic dipole moment . . . . . . . . . . . . . . . 11

7 Limits on magnetic fields 127.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1

1 ABSTRACT

In this project I present the study and data analysis report of a Low MassX-Ray Binary(LMXB) which is also the first eclipsing accretion powered mil-lisecond pulsar discovered, named as Swift J1749.4-2807. This source is situatedin the Saggitarius constellation at an Right Ascension(RA) of 267.368 degs andDeclination(Dec) of -28.121 degs. The distance of the source is estimated tobe around 6.7 (±1.3) kpc. The neutron star present in this binary sytem ro-tates at a frequency of ∼ 518Hz and has a mass between 0.8 M to 2.2 M,where M is the mass of Sun. The orbital period of the sytem is 8.8 hr. Ithas a projected semi-major axis of ∼ 1.90 lt-s and an inclination between therange ∼ 74.4 - 77.3. The final value of the magnetic field was found to be1.438 × 107G < µ < 3.136 × 108G. For the study of this source I have usedthe data from the Rossi X-Ray Timing Explorer(RXTE ) telescope. I have re-duced the data by creating filter files and good time intervals, that selects thebest potentially useable data, and then produced light curves and spectrumsfor different observations. To estimate the magnetic field of the LMXB, I usedthe disk-magnetosphere interaction model which gives us the maximum andminimum limit of the magnetic field of the neutron star. I hereby give thefull description of my work on this source with the results that I got from thespectral fitting of the data using the chi-squared statistics.

2 Introdution to Neutron Stars and pulsars andthe source Swift-J1749

Neutron Stars- It is a type of stellar remnant which can result from thegravitational collapse of a massive star. According to the Chandrasekhar Limit,if the mass of a star is less than 1.4 M(mass of the Sun) then it would end upas a white dwarf, if the mass is between 1.4 to 3 solar masses then it would endup as a neutron star. The most exotic end product is a Black hole which is thestellar remnant of a star which has a mass greater than 3 M. The Neutronstars are one of the most compact and dense objects in the Universe. A typicalneutron star has a mass between 1 and 2 solar masses and a radius of about12 km. Neutron stars have overall densities of 3.7 × 1017 to 5.9 × 1017 kg/m3

(2.6 × 1014 to 4.1 × 1014 times the density of the Sun), which compares withapproximate density of the atomic nucleus of 3.1× 1017 kg/m3.

Pulsars- These are rapidly rotating strongly magnetized neutron stars whoseenergy emissions are powered by the loss of rotational kinetic energy. They typ-ically have a mass and radius of 1.4M(3.1× 1030 kg) and 10 km, respectively.The pulsed appearance of their emissions is due to the fact that the magneticdipole is inlcined with respect to the star’s rotation axis. Beacause of the inter-stellar plasma, the pulse arrives later at a lower radio frequency. The “dispersionmeasure” computed from the frequency dependence of pulse arrival times con-tains information about the integrated density of the free electrons over theline of sight to the pulsar, and is often used to estimate the distance to thepulsar. On continued observations, we can notice that the pulsars are slowingdown. The spin-down rates, along with the spin periods of pulsars, constitutethe most important body of data on which our current ideas of pulsar evolutionis based. The measured spin down rate P of the pulsar in combination with its

2

pulse period P allows one to estimate its surface dipole magnetic field strengthBs

Bs =

(3c3I

8π2R6

) 12

PP ' 3.2× 1019(PP )12 G (1)

where the numerical coefficient results from using the standard values forthe neutron star radius R' 106 cm and moment of inertia I' 1045 gcm2, withP expressed in seconds (P is dimensionless). The assumptiom behind this ex-pression is that the loss of rotational energy from the pulsar equals the amountof magnetic dipole radiation emitted by a neutron star in vacuum with its spinaxis perpendicular to the magnetic axis.

A millisecond pulsar is defined as a pulsar whose period is less than 10ms. They have much shorter pulse periods and much weaker surface magneticfields than the bulk of the radio pulsars. They derive this fast rotation fromthe accretion process, so they are also called Accreting Millisecond X-ray Pul-sars(AMXP). The first AMXP was discovered in 1988 (SAX J1808.4-3658) andsince then 13 AMXPs have been found and studied in detail. Most AMXPsshow near sinusoidal profiles during most of their outburts. Our source, SwiftJ1749.4-2807, which happens to be an AMXP, was detected in a bright ther-monuclear Type I X-ray outburst in 2006, June 2 by the Swift Burst AlertTelescope(BAT). It was also reported by NASA’s RXTE satellite, whose firstobservtion was performed on April 14 which lasted for about 16 ks. The pulsarand the companion star in this case are separated by 1.22 million miles. Thesekinds of binary systems are called X-ray Binaries.

X-ray Binaries consist of a neutron star and a normal star. When thenormal star exceeds its Roche Lobe, matter from this star starts flowing intothe neutron star due to strong gravitational force and forms a disk around theneutron star, known as the accretion disk. In this process matter is funelledonto the magnetic poles which heats up the matter and as a result producesX-rays. These X-ray binaries are further categorised as Low mass and HighMass. Our source Swift J1749.4-2807 is a Low Mass X-ray Binary(LMXB).

3 Some properties of Accreting Neutron stars

For a given accretion rate M the spin-up of the pulsar will end when it reachesits so-called “equilibrium” spin period Peq given by

Peq = 2.4ms.B967M

−57

(M

MEdd

)−37

R6167 (2)

where B9, M and R6 are the surface dipole magnetic field strength of the neutronstar in units of 109 G, its mass in solar masses, and its radius in units of 106 cm,respectively. MEdd is the maximum possible “Eddington-Limit” accretion rateat which the accretion luminosity Lacc equals the Eddington Luminosity LEdd.The latter is defined as the luminosity at which the radiation pressure force onionized hydrogen plasma near the star balances the gravitational accelerationforce exerted by the star on this plasma:

GM

r2=

Lσel4πr2mpc

(3)

3

where r is the distance to the star, mp is the proton mass and σel is the Thomp-son scattering cross section of the electron(dominant opacity source in an ionizedhydrogen plasma). Equation (3) yields

LEdd =4πGMmpc

σel= 104.5

(M

M

)L (4)

where M and L denote the mass and the Luminosity of the sun. TheEddington limit sets an upper limit to the accretion luminosity Lacc of an com-paqt object, since for Lacc > Ledd, further accretion of matter will be inhibitedby the radiation pressure. So, the maximum accretion rate MEdd possible for aneutron star of mass M and radius R is given by

Lacc =GMM

R= 104.5

(L

R6

)(M

M

)(M

1.5× 10−8Myr−1

)(5)

Combination of eqs(4) and (5) yields

MEdd = R6.1.5× 10−8Myr−1 (6)

for a “standard” neutron star with R6=1, MEdd=1.5× 10−8Myr−1.The derivation of expression (2) for the equilibrium spin period Peq follows

from a very simple derivation given below. The matter flowing in through theaccretion disk gets coupled to the field lines of the neutron star at the so-calledAlfven Radius RA. RA is defined by the condition that the energy density ofthe magnetic field equals the kinetic energy density of the inflowing plasma, i.e.:

B2(r)

8π=

1

2ρ(r)υ2(r) (7)

where υ(r) is the Keplerian velocity (or-which is similar-the escape velocity) ata distance r from the neutron star.

υ(r) =

(2GM

r

) 12

. (8)

Converting the acceleration rate M into gas density ρ(r) by the continuity equa-tion M=4πr2ρ(r) × υ(r), and assuming the neutron star magnetic field to bedipolar, we get

B(r) = Bs

(Rsr

)3

(9)

where Bs and Rs are the surface dipole magnetic field strength and the neutronstar radius, respectively. Using this in eq (7) we obtain the expression for AlfvenRadius as

RA =

(Bs

2Rs6

M√

2GM

) 27

=

(µ4

2GMM2

) 17

(10)

For r < RA the magnetic field dominates the motion of the matter, whichis therefore forced to corotate with the neutron star. For r > RA, the disk

4

matter will not move freely in Keplerian orbits. The disk matter enters themagnetosphere at r = RA along the “open” field lines above the magneticpoles. Clearly, it can only enter the magnetosphere if the rotational angularvelocity of this star is not larger than te Keplerian angular velocity at r = RA.The equilibrium spin period Peq is defined as the spin period at which these twoangular velocities are exactly equal:

2π

Peq= Ωk(r = RA) =

(GM

R3

) 12

. (11)

For P > Peq, accretion is possible and the angular momentum of the accretedmatter will cause the neutron star to spin up. For P < Peq, matter piling up nearthe Alfven Radius will exert braking torques on the magnetosphere, causing theneutron star to spin down. As a result one expects the rotation of the accretingneutron star to finally always settle near P ' Peq.

4 RXTE Observation data of Swift J1749.4-2807 and its analysis

The data from the 2010 observation was downloaded from the RXTE website.A sofware called HEASOFT, by NASA, was used to reduce and analyse thedata. The data contained different proposal directories e.g P95085, P92016etc. For this project, P95085 proposal was chosen which contained the 2010observations. The proposal directories had different Observation Ids e.g 95085-09-01-00, 95085-09-02-00 etc. which contained the obseravtions at differentpoints of time. Each Observation Id was separately worked on and reducedusing heasoft by selecting the science data from pca directory which was recievedfrom an array of five proportional counter units(pcu) on board RXTE, knownas Proportional Counter Unit Array(PCA). It has a total collecting area of 6500square cm. Events detected by the PCA are processed on board before insertioninto the telemetry system. There are only two basic formats for science data:science array and science event. All RXTE data files are in Flexible ImageTransport System (FITS) format and have the following parts:

1. Primary Header contains information about the mission, the instrument,the observation and the initial processing. As with all FITS headers, this infor-mation is in the form of keywords with assigned values.

2. Primary Image Array is blank.3. First Extension Header contains keywords which provide a complete and

detailed description of the contents of the first extension. For convenience, italso contains some of the same information as the primary header.

4. First Extension contains the scientific data. In the case of science arrayfiles, the first extension is called XTE SA. In the case of science event files, itis called XTE SE.

5. Second Extension Header contains keywords which provide a completeand detailed description of the contents of the second extension. For conve-nience, it also contains some of the same information as the primary header.

6. Second Extension lists the standard good time intervals(gti), i.e. the startand stop times of all the potentially useable data in the file. It is called GTI.

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7. Third Extension Header contains keywords which provide a complete anddetailed description of the contents of the third extension. For convenience, italso contains some of the same information as the primary header.

8. Third Extension lists the good time intervals, i.e. the start and stop timesof all the data in the file, regardless of whether the data are actually useable.Like the second extension, it is called GTI.

Science Array Files : Basic Structure- The science array format is usedfor data binned at regular intervals by the spacecraft electronics. Examplesare the PCA Standard 2 configuration, present in the pca directory , whichcontains 129-channel spectra accumulated every 16 seconds, and HEXTE multi-scalar Bin mode, which contains light curves in 1-8 spectral bands. The dataoccupy the XTE SA extension in the form of regularly accumulated arrays orhistograms.

The two GTI extensions- RXTE science array files have two additional ex-tensions containing good time intervals (GTI). Both are called GTI.

The first GTI extension, the second extension in the file, contains the ANDedGTI corresponding to the times when:

1. Telemetred data are present2. The satellite is pointing at the nominal source position, as derived from

the spacecraft attitude3. The nominal source position is not occulted by the Earth, as predicted

by mission operations4. The satellite is outside the South Atlantic Anomaly, as predicted by

mission operationsThe second GTI extension, the third extension in the file, contains the GTI

corresponding to when telemetred data are present, i.e. just the first of the fourcriteria in the first GTI extension.

Filter Files-RXTE filter files contain the values of various housekeeping dataand derived quantities with which good data can be selected

All these files are found in the standard product directory (stdprod) whichare finished products. But for this project, we reduced the data ourselves bycreating our own good time intervals(gti) and filter (.xfl) files by using the datafrom the pca directory. We used the pcu2 data and the Standard2 files for thebest data selection.

The filter files were created by using the xtefilt command while running theheasoft software and the gti files were created by using the maketime com-mand. We used time bins of 16 secs and an offset less than 0.02 with el-evation greater than 10 while creating the gti files. Light curves and spec-trum were then extracted by using the saextrct command which producedlightcurves as .lc files and spectrums as .pha files. As we used only the pcu2 data,only the layers of pcu2 were included while extracting the lightcurves, namelyX1LSpecPcu2, X1RSpecPcu2, X2LSpecPcu2, X2RSpecPCU2, X3LSpecPcu2and X3RSpecPcu2. The lightcurves were then examined for selecting the ob-servation ids with maximum and minimum counts. It was found that obsid95085-09-02-05 had the minimum counts and 95085-09-01-00 had the maximumcounts. The lightcurves of these Observation Ids are shown in Fig 2 and 3.

Apparently these observations include the background noise/flux as well.Hence to get true photon counts and flux from the source, these backgroundreadings need to be subtracted. So, for that we created background files first,

6

by using the Pcabackest package in the heasoft. Pcabackest is a tool that pro-duces synthetic background data by matching the background conditions of theobservation with those in various model files. These model files contain ac-tual background observations sorted by quantities such as the position of thespacecraft with respect to the South Atlantic Anomaly(SAA). The output ofpcabackest is in the form of data files very similar in structure to Standard2data files. They can be put through saextrct to produce background light curvesand spectra. Since the average counts in all observations were less than 40, weused the pca bkgd cmfaintl7 eMv20051128.mdl to produce the background filealong with pca saa history which contains the information regarding the SAA.The background counts in case of 95085-09-01-00 and 95085-09-02-05 are shownin Fig 4 and 5.

After extracting the lightcurve and spectrum from the background file us-ing saextrct, we produced the net lightcurve by using lcmath which takes thelightcurve(with background) as the first parameter and the background lightcurveas the second parameter. It is clearly evident that the counts decreased afterthe subtraction of the background.

Interestingly we also found the eclipse in three observation ids- 95085-09-02-02, 95085-09-02-04, and 95085-09-02-11. These observations were used in theestimation of background counts/flux by selecting the portion between eclipseingress and egress from the lightcurve and the spectrum. For example, in theobs id 95085-09-02-02, the region between 0-500 s was chosen to estimate thebackground flux. The lightcurve of this obs id is shown in Fig 8.

To find out the flux from the source, and the bolometric luminosity, thespectrums were needed to be analysed by fitting them with physical models.For this purpose, we used the package called Xspec. Xspec is a command-driven, interactive, X-ray spectral fitting program designed to be completelydetector independent so that it can be used for any spectrometer. It takes thespectrums(.pha files) of the source and the background along with the responsematrix as its input parameters. PCA response matrices are notionally dividedin two parts: the Ancillary Response File (ARF), which accounts for the detec-tor windows and collimator response; and the Redistribution Matrix Function(RMF), which accounts for the redistribution of photon energy amongst detec-tor channels by the detecting medium. The product of the two is known as theresponse (RSP) and is used by xspec for spectral analysis. The rsp files wereproduced by running a Perl script called pcarsp. We included all the layers ofthe pcu2 namely LR1,LR2 and LR3 while running pcarsp.

After feeding the spectrum and the background into the xspec, a modelspectrum is applied to the spectrum which has different parameters. More thanone models are also apllied when required. The model parameters are thenvaried to find the parameter values that give the most desirable “fit” statistics.These set of parameters are called the best-fit parameters. The commonly usedfit statistic is the χ2 statistics given by

χ2 = Σ(C(I)− Cp(I))

2

σ(I)2(12)

where σ(I) is (generally unknown) error for channel I, C(I) is the actualcounts and Cp(I) is the model count. σ(I) is usually taken as

√Cp(I), assuming

Poissonian statistics. The χ2 statistic provides a well-known-goodness-of-fit

7

criterion for a given number of degrees of freedom (ν, which is calculated as thenumber of channels minus the number of model parameters) and for a given

confidence level. If the reduced χ2(χ2

ν ) is approximately equal to one, then themodel is considered to be a good fit. If its value exceeds 1 by a large amountthen it is a poor fit, or else if it is less than 1, then the errors are said to beoverestimated.

Using this principle, we have applied different models, such as wabs, pow,gauss, and diskbb, to the spectrums to get a good fit .

5 Description of the models

1. wabs is a photelectric-absorption model defined by

M(E) = exp (−nHσ(E))

where (a)σ(E) is the photo-electric cross-section(NOT including Thomsonscattering).

(b)par1 = nH represents the equivalent hydrogen column(in units of 1022

atoms cm−2)

2. powerlaw(pow) is a simple photon power law model given by

A(E) = KE−α

where (a)par1 = α is the photon index of power law(dimensionless)

(b)norm = K is the photons keV−1cm−2s−1

3. gaussian(gauss) is a simple gaussian/iron line profile. If the width is ≤ 0,then it is treated as a delta function. It is defined as

A(E) = K1

σ√

2πexp

(− (E − El)2

2σ2

)where (a)par1 = El is the line energy in keV

(b)par2 = σ is the line width in keV

(c)Norm=K in total photons cm−2s−1 in the line.

4. disk blackbody(diskbb) is accretion disk model which considers the spec-trum from it to be comprised of multiple-black body components.

(a)par1 is the temperature at inner radius (in keV) (b)par2 is the normwhich is given by

norm =

(Rin

km

)(D

10kpc

)2

cos θ

where Rin is “an apparent” inner disk radius, D the distance to the source,and θ the angle of the disk(θ = 0 is face on).

8

Model par Model comp Component Parameter Unit Value1 1 wabs nH 1022 3.87946 ± 0.5661892 2 powerlaw PhoIndex - 1.95829 ± 3.62742E-023 2 powerlaw norm - 0.151583 ± 1.33415E-024 3 gaussian LineE keV 6.53636 ± 3.16927E-025 3 gaussian sigma kev 0.278828 ± 6.74574E-026 3 gaussian norm - 2.3719E-03 ± 1.61738E-04

Table 1: Model fit Parameters without cflux

Model par Model comp Component Parameter Unit Value1 1 wabs nH 1022 3.87754 ± 0.02 2 cflux Emin keV 0.1 frozen3 2 cflux Emax keV 100 frozen4 2 cflux lg10flux cgs -9.20234 ± 0.05 3 powerlaw PhoIndex - 1.95820 ± 0.06 3 powerlaw norm - 100 frozen7 4 gaussian LineE keV 6.53654 ± 0.08 4 gaussian Sigma keV 0.279507 ± 0.09 4 gaussian norm - 1.47696 ± 0.0

Table 2: Model fit parameters with cflux

While fitting the curves, the model combination used was: wabs*(pow+gauss).This gave a fairly good fit to the spectrum, a reduced chi square little greaterthan 1, in most of the cases. But in some cases, where reduced chi square wasnear 2 or more, we had to include diskbb in the model to make the fit better.The reason of choosing both gauss and pow and not just a single powerlaw wasa small and thin ‘bump’ in the spectrum near 6.5 keV energy bin, which canbe clearly interpreted as an iron(Fe) line, which was not being fit by a simplepowerlaw model. The spectrum of obsid 95085-09-01-00 is shown in Figs 9 and10. The Fig 9 shows the data itself without any models fitted to it. Fig 10 showsthe spectrum(in logarithmic scales) fit with the models wabs*(pow+gauss).

The reduced chi square of this observation is 1.005, with 43 degrees of free-dom. A systematic error of 0.01 was introduced into the fit paramters. The fitparameters are shown in Table 1.

We used flux command with energy range between 0.1 to 100 keV to getthe total flux from the source which can be converted into the bolometricluminosity. To find the errors and the unabsorbed flux at the same time,a new convolution model named cflux was used with the above models as :wabs*cflux*(pow+gauss). Using cflux*wabs*(pow+gauss) would have given usthe absorbed flux instead. The parameter values are shown in Table 2 and thespectrum of the fit with cflux is shown in Fig 11.

The reduced chi square in this case was 1.005 with 43 degrees of freedom.The errors in the model fit and in the estimation of flux are given in Table 3.

To verify our results we checked the flux from the model fit with cflux andwithout cflux separately. As expected, both the values were exactly the same,thus verifying our models.

Note- There is one small residue in the chi square fit at around 7 keV(refFig.10), which was also found in other observation ids. We think that it may

9

Parameter Confidence Range (2.706)1 2.94218 4.79936 (-0.934837,0.922337)4 -9.21685 -9.18778 (-0.0145095,0.0145605)5 1.89895 2.01649 (-0.0592135,0.0583245)7 6.48641 6.58774 (-0.0500856,0.0512512)8 0.122925 0.405292 (-0.156696,0.125671)9 1.16895 1.87788 (-0.308281,0.400651)

Table 3: Errors in the estimation of the flux

need one more ‘gaussian’ to fit in, but since the flux contribution of this smallbump is very low, we have neglected it in our analysis. However, more detailedanalysis can be done to test the physical significance of this small ‘bump’.

6 The Disk-Magnetosphere interaction in the acre-tion powered millisecond pulsar SWIFT J1794.4-2807

The accretion powered pulsars, with a rotating neutron star accreting matterfrom a binary companion, have magnetic fields strong enough to disrupt theaccretion flow above the stellar surface. When threaded by the stellar magneticfield, the accreting gas is brought into corotation with the star and is channeledalong field lines to the polar caps, releasing its potential and kinetic energymostly in X-rays. The rotation of these hot spots through our line of sightproduces X-ray pulses at the spin frequency of the neutron star.

The presence of coherent pulsations from a weakly magnetic neutron starover a wide range of accretion rates places strong constraints on models ofthe disk-magnetosphere interaction which in turn can be used to get good con-straints on the magnetic field of the neutron star. Hence we used this to estimatethe maximum and minimum magnetic field of the neutron star in Swift J1749.4-2807. We assumed that most of the accreting gas around Swift J1749.4-2807 isconfined in a geometrically thin accretion disk before interacting with the pulsarmagnetic field. We also neglected the effect of wind mass loss from the inneraccretion disk and of radiation drag forces, as well as all general relativisticeffects.

The radius r0 at which magnetic stresses remove the angular momentum ofthe disk flow and disrupt it can be estimated by balancing the magnetic andmaterial stresses,

(BPBφ

4π

)4πr0

2∆r0 = MΩr02 (13)

where BP and Bφ are the poloidal and torroidal components of the magnetic

field, ∆r0 is the radial width of the interaction region, M is the mass transferrate through the inner disk, and Ω(r) is the angular velocity of the gas at radiusr. Assuming the poloidal magnetic field is dipolar with magnetic moment µ andthat the accretion flow is Keplerian, we obtain

10

r0 = γB27

(µ4

GMM2

) 17

(14)

where r0 is the accretion disk cutoff radius, with

γB =

(BφBP

)(∆r0r

)(15)

The expression of r0 is related to the Alfven radius as

r0 = RA ×

(γB

27

2−17

)(16)

However, if the disk-magnetosphere interaction takes place in a region ofthe accretion disk where all physical quantities have a powerlaw dependence onthe radius and eqn (13) describes angular momentum balance in the interactionregion, then the Keplerian Orbital frequency at r0 can be written as

νK0 =

(γB

−37

2π

)(GM)

57µ

−67 M

37 s−1 (17)

The Luminosity of the neutron star is given by

L =GMM

Rergs/s (18)

In normalized form, the expression for the maximum and minimum lumi-nosity can be written as

Lmax = 1.8688× 1036(

M

1.4M

)(Mmax

1016

)(Rs

10Km

)−1

ergs/sec (19)

Lmin = 1.8688× 1036(

M

1.4M

)(Mmin

1016

)(Rs

10Km

)−1

ergs/sec (20)

6.1 Limits on the magnetic dipole moment

For a rotating star to appear as an accretion-powered pulsar, the stellar mag-netic field must be strong enough to disrupt the Keplerian disk flow above thestellar surface. Therefore, the stellar magnetic field must be strong enough todisrupt the disk flow at radii a little larger than the neutron star radius R atthe maximum mass accretion rate Mmax for which coherent pulsations weredetected. This leads to lower limit on the magnetic dipole moment,

µmin = 1.169× 1025 × (γB)−12

(Mmax

1016g/s

) 12(

M

1.4M

) 14(

Rs10Km

) 74

Gcm3 (21)

11

At the same time, the stellar magnetic field must be weak enough thataccretion is not centrifugally inhibited at the minimum mass accretion rate Mmin

for which coherent pulsations were detected. Therefore, the orbital frequencyνk0 must be larger than the spin frequency of the neutron star νs. This leadsto an upper limit on the magnetic dipole moment,

µmax = 4.25× 1026 × (γB)−12 ×

( νs100

)−76

(M

1.4M

) 56

(M

1016g/s

) 12

Gcm3 (22)

7 Limits on magnetic fields

The maximum and minimum flux obtained from 95085-09-01-00 and 95085-09-02-05 obs ids are 9.1262×10−2ergs/cm2/s and 4.7396×10−10ergs/cm2/s respec-tively . The background flux was estimated to be 3.8595 × 10−10ergs/cm2/s.So, the net background subtracted flux values are:

Max flux= 5.2667× 10−10ergs/cm2/sMin flux= 0.8801× 10−10ergs/cm2/sThus the maximum and minimum luminosities are 2.8274× 1036ergs/s and

0.4725 × 1036ergs/s. Thus putting these values in eqns (19) and (20) we getthe maximum and minimum accretion rates as Mmax= 1.513× 1016g/s (2.39×10−10Myr

−1) and Mmin= 0.2528× 1016g/s (4.0083× 10−11Myr−1).

Now putting the value of Mmax in the expression for µmin,given by

µmin = 1.169× 1025 × (γB)−12

(Mmax

1016g/s

) 12(

M

1.4M

) 14(

Rs10km

) 74

Gcm3

the lower limit on the magnetic diplole moment was calculated as µmin =1.438× 1025Gcm3 and Bmin = 1.438× 107G.

Similarly, putting the value of Mmin in the expression

µmax = 4.25× 1026 × (γB)−12 ×

( νs100

)−76

(M

1.4M

) 56

(M

1016g/s

) 12

Gcm3

the upper limit on the magnetic dipole moment was found to be µmax = 3.136×1026Gcm3 and Bmax = 3.136×108G. Hence we get decent, but wide range of themagnetic field of the neutron star in Swift J1749.4-2807. So, we need more dataand precise measurements for better results. To check the consistency of ourresults with the theoretical assumptions, we estimated the inner and outer radiusof the accretion disk around the neutron star using our results.

Using the expression

r0 = γB27

(µ4

GMM2min

) 17

(23)

the outer radius of the accretion disk was found to be Rout=r0 = 26.8km whichfairly agrees with the theoretical assumption that it should be equal to Coro-tation Radius of neutron star as the Keplerian frequency(νk0) of the matter is

12

equal to the spin frequency(νs) at this radius. Similarly, putting Mmax in eqn(27) we got the inner radius of the accretion disk as Rin = r0 = 10km, whichfollows our arguement that the accretion disk is cut-off just before the surfaceof the neutron star.

7.1 Conclusion

The final values of Bmax and Bmin are 3.136×108G and 1.438×107G respectively.Taking into account all the above arguements regarding the Corotation radiusand mass accretion rates, we can conclude that our estimates of the magneticfield and the accretion disk are fairly good and can be improved with more dataand correct observations.

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