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Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December
© 2017. The Astronomical Society of the Pacific. All rights reserved. Printed in the U.S.A.
On the Geometry of the X-Ray Emission from Pulsars.
I. Model Formulation and Tests
Rigel Cappallo1,2, Silas G. T. Laycock1,2, and Dimitris M. Christodoulou1,3
Lowell Center for Space Science and Technology, 600 Suffolk Street, Lowell MA, 01854, USA
Department of Physics and Applied Physics, University of Massachusetts Lowell, Lowell MA, 01854, USA;,
Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell MA, 01854, USA;
Received 2017 April 10; accepted 2017 September 20; published 2017 October 24
X-ray pulsars are complex magnetized astronomical objects in which many different attributes shape the pulse
profiles of the emitted radiation. For each pulsar, the orientation of the spin axis relative to our viewing angle, the
inclination of the magnetic dipole axis relative to the spin axis, and the geometries of the emission regions all play
key roles in producing its unique pulse profile. In this paper, we describe in detail a new geometric computer model
for X-ray emitting pulsars and the tests that we carried out in order to ensure its proper operation. This model
allows for simultaneous tuning of multiple parameters for each pulsar and, by fitting observed profiles, it has the
potential to determine the underlying geometries of many pulsars whose pulse profiles have been cataloged and
made public in modern X-ray databases.
Key words: accretion, accretion disks – methods: numerical – pulsars: general – stars: magnetic field – stars:
neutron – X-rays: binaries
Online material: color figures
1. Introduction
This phenomenology makes XRPs a rich environment for the
study of high-energy astrophysics.
The pulse profiles of XRPs vary greatly between objects in
the class and they are both energy and geometry dependent
(Basko & Sunyaev 1976; Nagel 1981; Pavlov et al. 1994;
Ferrigno et al. 2011). They display a variety of morphologies
when viewed in different energy bands; and they often exhibit
contrasting patterns due to variations of the emission regions
near the surfaces of the NSs or projection effects (Karino 2007;
Klus 2015). This phenomenology suggests strongly that
various physical processes become more/less important at
distinct energy thresholds (Hong et al. 2017).
Many factors modifying pulse profiles have been investigated with a variety of empirical models built by many
researchers in the past (Wang & Welter 1981; Mészáros &
Nagel 1985a, 1985b; Mészáros & Riffert 1988; Riffert &
Meszáros 1988; Parmar et al. 1989; Riffert et al. 1993; Kraus
et al. 1995; Leahy & Li 1995; Beloborodov 2002, to name a
few key investigations), but these models limit their parameter
spaces in different ways by introducing various assumptions
about the emitted beams, or relativistic effects, or the geometry
of the emitting regions and their magnetic fields. In hopes of
further understanding the structures of the emission regions and
the various physical processes that dictate differing beam
patterns at various energies in XRPs, we have developed a
computer code that simulates a variety of orientations and
emission geometries through the use of multiple free
Pulsars are magnetized neutron stars (NSs) that rotate at
various spin periods (PS ∼ 10−3–104 s) while emitting beams
of radiation from regions believed to be coincident with their
magnetic poles (e.g., Mereghetti 2001; Frank et al. 2002; Lyne
& Graham-Smith 2012). If these beams happen to cross our
line of sight (LOS) and the NS spin and magnetic axes are not
aligned, then our telescopes detect periodic pulsations from
these objects (Lyne & Graham-Smith 2012). Pulsars have been
observed emitting in all areas of the electromagnetic spectrum,
from radio waves to gamma-rays (Thompson 2000). These
types of NSs all have differing mechanisms causing their
particular emission energies (Lewin et al. 1995; Mereghetti &
Stella 1995; DeDeo et al. 2001; Eggleton 2001; Mereghetti
2001; Cordes et al. 2004; Becker & Wolff 2005; Kylafis
et al. 2012).
The focus of this paper is on modeling the pulse profiles of
NSs that emit strongly in the X-ray regime, known as X-ray
pulsars (XRPs), and especially on the subset of Be/X-ray
pulsars (Coe et al. 2010; Reig 2011; Klus et al. 2014; Coe &
Kirk 2015; Klus 2015; Haberl & Sturm 2016). These objects
are found in binary systems where matter accretion from their
stellar companions results in highly energetic phenomena.
Their spin periods vary in time t, exhibiting both spin-up
(dPS/dt < 0) and spin-down (dPS/dt > 0) states, as well as
transitions between these states (Christodoulou et al. 2017a).
Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December Cappallo, Laycock, & Christodoulou
parameters to be determined by direct fits to observed pulse
profiles. The profiles created from this model will subsequently
be fitted to existing large catalogs of observed pulse profiles so
that the underlying physical mechanisms responsible for the
X-ray emission patterns from each source may become clearer.
Similar models to our own have been developed previously
by Wang & Welter (1981), Parmar et al. (1989), and
Beloborodov (2002); yet, these models include various
additional constraints that limit the number of free parameters
and their applicability to the entire class of XRPs. For example,
in Beloborodov (2002) the emission regions were assumed to
be antipodal and isotropic sources of radiation, and in Parmar
et al. (1989) the magnetic-dipole axis did not pass through the
center of the NS, resulting in some new complex features in the
modeled profile. Our model is capable of reproducing results
from these models by fixing the same geometrical parameters
that these authors held constant. But the new model also takes
the next step by providing the ability to fine-tune a larger
number of free parameters. As such, it will enable us to fit large
databases of widely varying pulse shapes and it will ultimately
give us a better understanding of the orientations and
underlying geometries of the emission regions in many pulsars.
In what follows, we describe our theoretical framework for
modeling XRP emission (Section 2), the geometry and the
computer code of our model (Section 3), and some preliminary
tests and comparisons with previous landmark results from the
literature (Sections 4 and 5). We conclude in Section 6 with a
discussion of the applicability of our new model and future
applications. In paper II of this series, we intend to present our
detailed model fits to the pulse profiles of the Be/X-ray
Magellanic pulsars for which we have created a new library of
all previous X-ray observations in the past 15 years (Yang
et al. 2017).
Figure 1. Orientations of pencil and fan beams in an accreting NS. AC
represents the accretion columns and z is the rotation axis. The magnetic dipole
axis is along the ACs.
(A color version of this figure is available in the online journal.)
Christodoulou et al. 2016, 2017b), exotic physics may possibly
occur in this realm (Bachetti et al. 2014; Klus et al. 2014;
Tendulkar et al. 2014). A better understanding of how these
emission regions are structured and behave will eventually lead
to a clearer grasp of the physics in very high-energy regimes.
2.1. Hot Spot Geometry
It is believed that there are two underlying, fundamental
components to the HS geometry. At lower luminosities
(LX < 1037 ergs−1), and for certain magnetic field strengths
and orientations, there exists a “pencil-beam” component
which consists of radiation directed along the local magnetic
magnetic field lines (Zavlin et al. 1995). At higher luminosities,
a “fan-beam” component begins to emerge which exhibits
maximum radiation perpendicular to the magnetic dipole axis.
This orthogonal component explains the double-peaked
structures that are found only in profiles from high luminosity
(LX > 1037 ergs−1) sources (Ferrigno et al. 2011; Klus
et al. 2014; Klus 2015). These double peaks appear when the
local energy flux exceeds the Eddington flux (defined as the
first-order moment of the radiation field) by a factor of ∼100,
forming a shock front above the NS surface with photons
“leaking out” of the sides of the column of slowly falling
plasma contained below the front (see Becker 1998, and our
Figure 1).
2. Theoretical Framework
X-ray binary (XRB) pulsars consist of a NS and a stellar
companion supplying material (by wind outflow or Roche lobe
overflow, or when the NS is immersed in circumstellar
material) onto an accretion disk that forms, in many cases,
around the NS. The plasma in the inner accretion disk
redistributes its angular momentum and some of the matter
with reduced angular momentum descends toward the surface
of the NS following magnetic-field lines. As this matter is
finally decelerated on impact, it releases large amounts of
gravitational potential energy in the form of high-energy
radiation. For this reason, the electromagnetic emission regions
of pulsars are thought to coincide with the positions of their
magnetic poles. The geometry of such emission regions, known
as hot spots (HSs), is believed to be complicated. Due to the
high energies of the particles and the large magnitudes of
the NS magnetic fields (∼1012–13 G; Stella et al. 1986;
Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December Cappallo, Laycock, & Christodoulou
Figure 2. Geometry in the light-bending approximation of Beloborodov
(A color version of this figure is available in the online journal.)
The alignment of the dipolar magnetic field lines in the HSs
is also an issue that must be considered. The field lines could be
normal to the NS surface or, alternatively, they could be
aligned with the magnetic dipole axis, which may not be
passing through the NS center (e.g., Burnard et al. 1988;
Parmar et al. 1989). Only for an antipodal arrangement does the
dipole axis pass through the NS center and then the above two
field line geometries coincide. This simplified arrangement has
been used in many models, including Beloborodov (2002) and
Wang & Welter (1981), but not in Parmar et al. (1989) who
produced very different profiles resulting from such an offcentered magnetic field (among other considerations as well).
Some results from these models have been reproduced with our
new code and they are discussed in Sections 3.4 and 5 below.
Figure 3. Far side of a NS (outer circle). The shaded regions represent the
unobserved sections of the surface and correspond to different ratios of rg/RNS.
The circle colored in dark blue and black is for a ratio of 1/3 and the inner
black circle represents the canonical NS with a ratio of 0.4135.
(A color version of this figure is available in the online journal.)
normal vector to the point of emission; and α is the angle
between the initial direction of photon emission and the same
normal vector (Figure 2). Equation (1) is used to determine α
for a choice of ψ and the ratio rg/RNS.
For the canonical NS, rg=4.135 km and the term in the first
parentheses in Equation (1) is reduced to a constant, viz.
2.2. Gravitational Considerations
The gravitational field of a typical NS is quite strong near its
surface (g ∼ 1012 m s−2 for the canonical NS with mass 1.4
Me and radius 10 km) and the exterior space deviates markedly
from Euclidean space. For this reason, gravitational lightbending effects onto the emitted radiation must be taken into
account. For a relatively slowly rotating NS, frame dragging
can be neglected and then the Schwarzschild metric is a
sufficient approximation for the surrounding space (Pechenick
et al. 1983). Although the general relativistic equations
governing electromagnetic wave propagation in the Schwarzschild metric cannot be solved analytically, a simple yet
powerful approximation for light bending has been developed
by Beloborodov (2002). This approximation is represented by
the equation
rg ⎞
1 - cos a = ⎜1 ⎟ (1 - cos y) ,
1 - cos a = 0.5865 (1 - cos y).
(2 )
All of the profiles created by our model in this paper use
Equation (2), but the model generally retains the ratio rg/RNS
as a free parameter. This ratio is effectively related to the
gravitational redshift zg of the emitted radiation, that is
rg ⎞- 2
z g = ⎜1 ⎟ - 1.
(3 )
For comparison purposes, zg=0.3058 for the canonical NS
with rg/RNS=0.4135.
Equation (2) reproduces the results of the results of
Pechenick et al. (1983) to within an accuracy of „3% for
α„90° and the error decreases to „1% for α„75°
(Beloborodov 2002). For our model, errors of 3% or less are
acceptable when fitting pulse profiles since larger errors are
inherent in the observations. But we retain four significant
figures in the above equations to avoid introducing systematic
errors of order of 1%.
(1 )
where rg = 2GM c 2 is the Schwarzschild radius for a body of
mass M, gravitational constant G, and light speed c; RNS is the
radius of the NS; ψ is the angle between the LOS and the
Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December Cappallo, Laycock, & Christodoulou
Figure 5. Screen shot of the visual representation of our model. This specific
geometry uses asymmetric HSs and produces the profiles shown in Figures 16
and 17 below. The two HS vectors are green and the rotation axis is red.
(A color version of this figure is available in the online journal.)
3. Model and Computer Code
The computer code, named Polestar,4 is written in Python
2.7 and VPython 3.0 and simulates radiation emission from a
single NS with an arbitrary number of HSs radiating a
combination of pencil and fan beams, along with the capability
of incorporating more complex beam structures. The inclusion
of multiple HSs facilitates the implementation of extended
regions of emission on the surface of the NS rather than just
two point sources (Ho 2007). The model NS rotates at a given
angular velocity w , the magnetic field B can be chosen to be be
misaligned in a number of different ways, and the resulting
pulse profile of the radiation that reaches the observer is then
computed as a function of phase. Finally, VPython 3.0 is used
to create an animated, three-dimensional, visual representation
of the geometry and the orientation of the source relative to our
LOS, such as that shown in Figure 5.
The mathematical framework that forms the basis of the
Polestar code is described below.
Figure 4. Pulse profile generated by two antipodal isotropic HSs (a) without
and (b) with gravitational light bending included. The differences are described
in the text.
As a consequence of light bending, more than half of the
surface of a NS is observed at any time implying that there may
be points in the pulse profile where both poles are visible. The
terminator that separates the region of the surface that cannot
be observed is a circle on the far side of the NS (Figure 3)
whose radius depends on RNS and the NS mass
(Beloborodov 2002).
The inclusion of gravitational light bending in the model is
necessary because the effect alters pulse profiles in a dramatic
fashion. Figure 4 shows an example of two antipodal
isotropically emitting HSs with and without light bending.
Whenever both poles are visible, the observed flux remains
constant (panel (b)) producing the plateaus that fill the deep
troughs which dominate in the nonrelativistic case (panel (a)).
It should be noted however that this behavior emerges only for
antipodal HS orientations.
3.1. Mathematical Framework
Conceptually, the model is vector-based and consists of a
line-of-sight unit vector L, an arbitrary number of HS normal
unit vectors mi (i = 1, 2, ...), and a geometric beam (intensity)
pattern for each HS (see Figure 6). The rest frame is taken to be
the NS frame, so L is rotated in incremental steps Δξ through
one full revolution about the rotation axis w of the NS
(Figure 6). This axis is defined as the z-axis in this system. At
each Δξ step, the intensities Ii of the HSs are propagated in
space obeying Equations (1) or (2), and then they are used to
populate the resulting pulse profile.
An interactive web interface for the Polestar code is publicly available at the
Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December Cappallo, Laycock, & Christodoulou
Table 1
Input and Output Parameters in the Polestar Code
Input Parameters
Figure 6. Vectors of the model. The z axis is the NS rotation axis. Two
additional angles are not marked for clarity: angle ψ in Equations (1) and(2) is
the angle between the two vectors, and the inclination angle i0 is the colatitude
of f.
(A color version of this figure is available in the online journal.)
å Fi
i=1, 2, ...
Free parameter (Figure 6)
Free parameter (Figure 6)
Figure 2
Equations (1) and(3)
Equations (1) and(2)
i=1, 2, ...
Equations (11) and(12)
Equations (11) and(12)
Flux from each HS to observer
Sum of fluxes from all HSs to
Phase of the profile
i=1, 2, ...
For each Δξ step
Modulo 2π
L (x + Dx ) = R (Dx ) L (x ).
(4 )
(7 )
Then a flux value F is calculated by summing the two
contributions from the HSs along the new LOS vector:
F = (L · m1) I1(a1) + (L · m 2) I2 (a2) ,
(8 )
where I1 and I2 are the beam intensities of the two HSs. The
intensities are free parameters of the model and they may also
be functions of the angles αi between the LOS vector and each
HS vector mi (Nagel 1981). The dot products must be positive.
If any of them is negative, then the corresponding HS is not
visible and the term is reset to zero.
The above process is iterated for a total of 2π/Δξ steps in
order to produce a complete pulse profile, and then F is
normalized by the maximum computed flux value. A detailed
description of the steps in the code is given in the following
section and a complete listing of the input and the output
parameters of Polestar is shown in Table 1.
(5 )
The rotation matrix R (Dx ) is used to rotate L through an angle
Δξ, where
⎛ cos (Dx ) -sin (Dx ) 0 ⎞
R (Dx ) = ⎜⎜ sin (Dx ) cos (Dx ) 0 ⎟⎟.
⎝ 0
Modulo 2π
Free parameter (Figure 6)
Colatitude of f, Equation (13)
In the typical case with two HSs and only pencil-beam
emission, each iteration by Δξ involves the following sequence
of steps:
First, L(ξ) is rotated about the z-axis by an angle Δξ to
L (x + Dx ):
and the HS vectors as
⎛ cos qi cos gi ⎞
mi = ⎜ sin qi cos gi ⎟.
⎝ sin gi ⎠
This axis is fixed
Initially set on xz-plane
Initially set to zero (Figure 6)
Output Parameters
The orientation of the LOS vector L is specified by the
latitude f and the azimuthal angle x = å Dx (modulo 2π); this
vector always begins on xz-plane (ξ = 0) of the rest frame. This
is done for simplicity and manifests itself as a definition of
phase which can, in turn, be shifted to any other time once the
profile is complete. Furthermore, the orientation of each HS
vector mi is specified by a latitude γi and an azimuthal angle θi.
Based on Figure 6, the LOS vector can be expressed as
⎛ cos x cos f ⎞
L = ⎜ sin x cos f ⎟ ,
⎝ sin f ⎠
Rotation axis of NS
LOS vector, ∣L∣ = 1
Azimuth of L and phase of the
Advance of ξ to new phase
Latitude of LOS vector
Inclination of LOS to NS spin
axis z
HS vectors, ∣ mi ∣ = 1
Azimuth of mi vector
Latitude of mi vector
Angle between LOS and mi
Compactness parameter
Angle between initial photon
path and mi
HS intensities
Pencil-beam coefficient of Ii
Fan-beam coefficient of Ii
(6 )
The size of the increment Δξ is an input parameter and it
allows for the model to have higher resolution than the
observed pulse profiles.
Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December Cappallo, Laycock, & Christodoulou
3.2. Computer Code
The input parameters to Polestar (see Table 1) are held fixed
if a specific geometry is desired, or they are varied if the code is
in the process of fitting an observed profile. There is also an
option to fix any number of the input parameters if the user
wishes to explore the influence of the remaining parameters to
the overall profile. On initialization, the code calculates the
initial angles between the HS unit vectors mi and the LOS unit
vector L. After initialization, Polestar runs an iterative loop
over angle ξ (modulo 2π) in which the calculation of the pulse
profile takes place.
The iterative loop carries out the following sequence of
(1) It updates the angles ξ and ψ (Table 1).
(2) It calculates angle α (Equation (1) or Equation (2)).
(3) It calculates the dot and cross products between L and mi ,
that is L · m i = cos ai for each pencil beam and
L ´ m i = sin ai for each fan beam, respectively. If any
of these values is negative, then the corresponding beam
is not visible to the observer and the product is reset
to zero.
(4) The pencil and fan intensity components from each HS
are calculated and added together in order to obtain the
total contribution from each HS (see Equation (11)
(5) The total flux at this phase is finally determined by
summing the intensities of all the HSs.
Figure 7. Emission pattern from a single HS with flux proportional to either a
cosine (simple pencil in red) or a sine (simple fan in blue) function. Here,
I1=1 in Equations (9) and(10). The dashed line represents the magnetic axis
through the two HSs.
(A color version of this figure is available in the online journal.)
of fan to pencil fluxes shown in Figure 7, viz.
F1,total (a1) = (Cp cos a1 + Cf sin a1) I1(a1) ,
where Cp and Cf are tuning coefficients that obey the relation
Cp + Cf = 1.
In the next step, L is rotated through an angle Δξ and the
computation runs again for the new phase.
The simplest and most widely used approximation is that of
emission from a single point on the NS surface (e.g.,
Beloborodov 2002; Karino 2007). This mimics a single pencil
beam and can be achieved by taking the inner product between
the LOS vector L and the HS vector m1; then the flux is
(9 )
where ∣ L ∣ = ∣ m1 ∣ = 1, angle α1 is determined from
Equation (1) orEquation (2) for a given LOS, and cos a1 must
be positive (otherwise it is reset to zero).
On the other hand, the geometry of the fan-beam component
can be represented by a sine function (Nagel 1981; Wang &
Welter 1981; Karino 2007), in which case we can write its flux as
F1,fan (a1) = I1(a1) sin a1,
More complex beam architectures can also be accommodated through manipulation of the beam-generating function
within the code. This function effectively controls the
intensities Ii of any number of HSs and allows for more
complicated radiation patterns than the simple sin ai and cos ai
geometric dependencies of the fluxes discussed above (see also
Section 3.4). For example, narrowing of the emission peaks can
be accomplished by using higher powers of these sine and
cosine functions in Ii(αi), and the higher the power the
narrower the peak. This is illustrated in Figure 8, where the
solid lines show fluxes F1 = cos10 a1 (Cp = 1) and F1 = sin10 a1
(Cf = 1) from a single HS; in comparison to the fluxes in
Figure 7 (plotted again in Figure 8 as dotted lines).
3.3. Hot Spot Emission
F1,pencil (a1) = I1(a1) cos a1,
3.4. Direction of Emission and Beaming Functions
The model is flexible in accommodating emission from an
offset magnetic dipole (see Figure 9): by subtracting the two
HS vectors m1 and m2 , we obtain a new vector along the chord
joining the two poles. The direction of this vector defines the
orientation of the offset magnetic-dipole axis (vertical blue line
in Figure 9). The contribution to the observed flux from each
HS can then be calculated by unitizing this vector and by
where sin a1 must be positive (otherwise it is reset to zero). An
overall emission pattern can then be realized by tuning the ratio
Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December Cappallo, Laycock, & Christodoulou
Figure 10. Antipodal configuration with the magnetic (green vectors) and spin
(red lines) axes offset by 90°.
(A color version of this figure is available in the online journal.)
Figure 8. Comparison of the simple fan and the simple pencil beams of
Figure 7 (dotted blue and dotted red line, respectively) to the beams with fluxes
proportional to a sine and cosine raised to the tenth power (solid lines,
respectively). Here, I1 = cos9 a1 and I1 = sin9 a1 in Equations (9) and(10),
respectively. Note the flattening of the trough near the magnetic axis (dashed
line) in the fan-beam sine function (solid blue line) and the steepening of both
beams around angles 0 and p 2 .
(A color version of this figure is available in the online journal.)
Figure 11. Flux from antipodal HSs with identical pencil beams. The dashed
lines correspond to the individual contributions, proportional to cos ai , from
each HS.
(A color version of this figure is available in the online journal.)
particular intensities used in the original work. So this model
serves as a test of the geometric features of our code, but it does
not test the beam functions that we commonly use in our
In the tests described in Sections 2.2 and 3.3 above, we used
some arbitrary geometric functions of αi for demonstration
purposes. But for detailed modeling of observed pulse profiles,
we rely on results of anisotropic radiative transfer calculations
and beaming calculations (Nagel 1981; Kaminker et al. 1982;
DeDeo et al. 2001). These results support intensity functions of
the forms (Equations(2) and(3) in Nagel 1981): Ii µ cos2 ai
(pencil); Ii µ sin2 ai (fan); and in some cases, even higher
powers, depending on choices for the mean-free-path functions
of the emitted photons.
Figure 9. Diagram of a NS (circle) with an offset dipolar magnetic field. The
red line denotes the rotation axis through the NS center. The green vectors
point to the HSs, and their difference defines the direction of the magnetic axis
(vertical blue line).
(A color version of this figure is available in the online journal.)
projecting it and its opposite vector along the LOS vector (as in
the models shown in Figures 16–18 below).
This configuration, along with more complicated beam
functions, was used successfully by Parmar et al. (1989) in the
detailed modeling of the pulse profile of the 42 s X-ray pulsar
EXO2030+375. We have reproduced this result with our code
and an offset magnetic field, but we had to also incorporate the
Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December Cappallo, Laycock, & Christodoulou
Figure 12. Flux from antipodal HSs with identical fan beams. The dashed lines
correspond to the individual contributions, proportional to sin ai , from each HS.
(A color version of this figure is available in the online journal.)
Figure 14. As in Figure 12, but the fluxes from the two fan beams are
proportional to (sin ai )10 .
(A color version of this figure is available in the online journal.)
Figure 13. As in Figure 11, but the fluxes from the two pencil beams are
proportional to (cos ai )10 .
(A color version of this figure is available in the online journal.)
Figure 15. Combination of (sin ai )10 and (cos ai )10 fluxes from two identical
antipodal HSs with Cp=0.7 and Cf=0.3 giving rise to four peaks in the
pulse profile. The dashed lines correspond to the individual contributions from
each HS.
(A color version of this figure is available in the online journal.)
4. Variation of Some Free Parameters and
Profile Features
(5)The compactness parameter rg/RNS (Figure 3) can also
generally be a free parameter (although in this paper we hold it
constant to its canonical value of 0.4135). The output of the code is
the total flux that reaches the observer as a function of phase at an
arbitrary resolution (currently set to 1000 points per period), as
well as a graphical pulse profile and an animation of the geometry
of the pulsar.
In the following subsections we describe various profile
features that can be produced by variations of some of these
free parameters. For simplicity, we assume an antipodal
For the sake of clarity, we begin by listing the free parameters
and the output products of our model (Table 1): (1)The location of
each HS is designated by a latitudinal angle γi and a longitudinal
angle θi (Figure 6). (2)Similarly, the location of the LOS is
specified by a latitude f and a longitude ξ (Figure 6), where ξ
represents the phase of the overall profile as the NS spin axis
remains fixed in time. (3)The intensities of the HSs Ii(αi) are
functions of the emission angles αi (Figure 2). (4)The coefficients
Cp and Cf control the pencil and beam contribution from each HS.
Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December Cappallo, Laycock, & Christodoulou
Figure 16. Pulse profile generated by two HSs emitting pencil-beam radiation
(Fi = I0 cos ai ). The individual HSs are shown in red and green, and their sum
is shown in black (see also Figure 5 for visualization of the geometry). The HSs
are asymmetric (f = 20°; γ1 = 70° and γ2 = −60°; and θ1 = 0 and θ2 = 90°).
(A color version of this figure is available in the online journal.)
Figure 18. Our model of the pulse profile of Cen X-3 that was originally
produced by Wang & Welter (1981) (Figure 7(j) in their paper). The model
uses pure fan emission with Fi = I0 sin ai from two identical HSs and our LOS
is inclined by i=31° to the pulsar’s spin axis, allowing for the secondary HS
(green dashed line) to become visible for a small fraction of the phase (see also
Figure 19 for visualization of the geometry). Model parameters are given in the
(A color version of this figure is available in the online journal.)
following are five examples of the inherent differences between
pencil and fan behavior from the two identical HSs and the
effects of using powers of the trigonometric functions for the
intensities of the beams.
4.1. Pencil Versus Fan
In Figure 11, the pencil-beam peaks are formed as each pole
moves across the LOS. Since the emission comes from a pure
cosine function, the total flux is constant whenever both poles
are in view, leading to the plateaus between the two peaks (see
also Beloborodov 2002).
In Figure 12, the fan-beam peaks are formed when the angle
between the magnetic-field axis and the LOS is 90°, with the
eclipse-like features occurring when a given pole travels across
the terminator on the far side of the NS. Note that the fan-beam
peaks are shifted in Figure 12 by a quarter of the phase in
comparison to the pencil profile of Figure 11. This shift can
account for quadruple-peaked structures in cases where the
profile is a combination of the two beams.
Figure 17. As in Figure 16, but for fan-beam emission (Fi = I0 sin ai ). Note the
eclipse-like behavior as the southern pole is briefly obscured when traveling
across the terminator (see also Figure 5 for visualization of the geometry, as
well as Wang & Welter 1981).
(A color version of this figure is available in the online journal.)
arrangement with both the LOS and the magnetic axis offset
initially by 90° from the spin axis (Figure 10). The model
parameters for this configuration are: f=0, implying that the
inclination of the LOS to the NS spin axis is i 0 = 90, where
i 0 º 90 - f ;
4.2. Narrowing of the Beam
In Figure 13, the geometry is the same as that in Figure 11,
but the cos ai function for each HS has been raised to the tenth
power. Note that the peaks are now significantly narrower and
that the plateaus between the peaks have vanished—when both
poles are visible, constant flux is seen only in the strict
Fi µ cos ai case. In this fashion, a geometric aspect of energy
γ1=γ2=0, θ1=0, and θ2=180°. This simple configuration is chosen because it can be visualized easily. The
Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December Cappallo, Laycock, & Christodoulou
dependence can be accommodated in the code because the
opening angle of the pencil beam is proportional to the square
root of the energy (e.g., Kaminker et al. 1982; Pavlov
et al. 1994; Ho & Lai 2001). Yet, other complicated processes
(e.g., self-absorption, scattering, and reprocessing of the
X-rays) that are energy-dependent and modify observed pulse
profiles cannot be accounted for in this model.
Similarly, in Figure 14 the sin ai function for the fan beam of
each HS has been raised to the tenth power. Again the profile
peaks are narrower, and, in addition, the vertical lines seen in
Figure 12 created by the terminator are here not as steep. This
makes sense as the positions of these lines in the pulse profile
are dependent only on the geometry, and with the narrowing of
the peaks there is less total flux near the maxima.
Figure 19. Visual representation of the geometry of Cen X-3 that produces the
pulse profile shown in Figure 18.
(A color version of this figure is available in the online journal.)
5.2. Wang & Wekter (1981) Profiles
Wang & Welter (1981) were the first to describe the eclipselike cutoffs that one sees in profiles dominated by fan-beam
emission (Fi = I0 sin ai ) from two identical HSs. Our model
reproduces this behavior using pure fan-beam radiation:
compare the sharp eclipse in Figure 17 (pure fan beam) to
the gradual dip in Figure 16 (pure pencil beam). The eclipse
and the dip are both due to the secondary HS traveling briefly
across the terminator. But the eclipse in Figure 17 occurs when
the total flux is near maximum, as the HS disappears from view
abruptly. We note that, without gravitational light-bending, this
eclipse would occur precisely at maximum flux.
Another test case is shown in Figure 18 in which we have
reproduced a model fit to the pulse profile of the source Cen
X-3 originally published by Wang & Welter (1981) (their
Figure7(j)). The model parameters for this profile are:
θ1=44° and γ1=60° (primary HS); θ2=220° and
γ2=−74° (secondary HS); f=59°, thus inclination of the
LOS to the NS rotation axis i 0 = 31; Cp=0 and Cf=1; and
Fi = I0 sin ai . The visual representation of this configuration is
shown in Figure 19. In the phase interval of approximately
0.3–0.5, the secondary HS is peaking out at the bottom of the
NS and this causes the pronounced plateau seen in the pulse
profile near maximum flux.
4.3. Four Peaks in the Profile
Figure 15 is an example showing a quadruple-peaked profile
that occurs when a combination of pencil and fan emission is
used from two identical antipodal HSs. In this model with
Cp=0.7 and Cf=0.3, the contribution from the pencil beams
is dominant and not only due to the particular choice of the
coefficients. For the orthogonal orientation used in this model,
the absolute maxima at phases 0 and 0.5 are produced by the
pencil beam of each HS as it crosses the LOS and the flux
tapers off gradually as the HS travels into the far side of the NS;
whereas the fan beam shuts off (comes up) abruptly as each HS
moves into (out of) the terminator of the NS. On the other hand,
both fan beams become fully visible at phases 0.25 and 0.75,
where the contributions of the pencil beams are minimal and
the total flux rises to local maxima of »2Cf Cp = 0.857.
5. Tests and Comparisons with Previous Results
Our code can reproduce several landmark results from the
literature when we adopt the corresponding input parameters.
Two specific test cases are described in this section.
5.1. Beloborodov (2002) Profiles
6. Discussion
Beloborodov (2002) defines four different classes of pulse
profiles, depending on the location of the LOS and the number
of poles that are visible in each case (the latter depends on the
inclination of the magnetic-field axis to the rotation axis). His
model assumes two antipodal, point-source HSs of equal
intensity I0 emitting beamed radiation. In our model, this is
duplicated by two antipodal HSs and pure pencil-beam
radiation (Fi = I0 cos ai ). The resulting pulse profiles are
identical to those published by Beloborodov (2002). For
example, compare our Figure 4(b) to Figure 4 (class III) in
that work.
In this paper, we have described a new code and its tests for
simulations of pulse profiles of pulsars. The code uses a wide
variety of geometries for the relativistic emission regions and
the dipolar magnetic field, as well as different orientations for
the observer. The parameter space is very large with typical
simulations fitting simultaneously 10 or more free parameters
to an observed profile. This flexibility allows us to fit the
observed pulse profiles of high-mass X-ray binary (HMXB)
pulsars and other XRBs with relative ease, but the results may
also be subject to degeneracies (different regions of the
parameter space effectively producing the same result; see,
Publications of the Astronomical Society of the Pacific, 129:124201 (11pp), 2017 December Cappallo, Laycock, & Christodoulou
e.g., the degenerate class IV flat profiles in Beloborodov 2002).
Our tests of the code have reproduced well-known results
previously published by other researchers in the field (Wang &
Welter 1981; Parmar et al. 1989; Beloborodov 2002).
The X-ray emission model of the new code is purely
geometrical in nature; i.e., there is no radiative transfer (cf.,
Nagel 1981; Kaminker et al. 1982; Pavlov et al. 1994; Ho &
Lai 2001) or input from physical properties of the emitting
regions (Ho 2007); nevertheless, it is capable of reproducing
many of the features observed in XRB pulse profiles; also the
profiles of the same object viewed in different energy bands can
be modeled in order to differentiate between profiles that have
strong energy and X-ray luminosity dependence.
We plan to use this code in conjunction with an automated
fitting algorithm in order to model the pulse profiles in a large
database of the 80 known HMXB pulsars in the Magellanic
Clouds (Coe et al. 2010; Coe & Kirk 2015; Klus 2015; Haberl
& Sturm 2016; Yang et al. 2017). Even if some degeneracies
are encountered in this type of analysis, such a large sample of
pulsars will certainly be broken down to various subclasses
each characterized by similar pulsar parameters (mass, radius,
spin, magnetic field) or profile shapes. In conjunction with
additional data such as X-ray spectra, the companions’ optical
spectra, and typical magnetic-field magnitudes such as those
proposed by Christodoulou et al. (2017b), these subclasses can
then provide information about the physical processes and the
pulsar geometries that are responsible for the various types of
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