Specific Features of Reflection of Radiation of a Compact Source Rotating about its Axis
aa r X i v : . [ a s t r o - ph . H E ] M a y Specific Features of Reflection of Radiation of a Compact SourceRotating about its Axis
A. V. Dementyev [email protected] ABSTRACT
Reprocessing of X-ray pulsar radiation by the atmosphere of a companionin a binary system may result in reflected pulse radiation with a period of thepulses equal to the period of the pulsar under suitable conditions. In this paperthe influence of the rotation of the source — the pulsar — about its axis on theparameters of thus reflected pulses is investigated. The binary system is modeledby the spherical reflective screen and the compact source uniformly rotating aboutits axis; the beam pattern (BP) of the source periodically runs over the surfaceof the screen. Irradiation of the screen by the pulses infinitely narrow in timeand by the rectangular pulses is considered. In this model parameters of thepulses reflected in some directions are calculated. The main conclusion providedby the consideration of this model is that the properties of reflected pulses —their profile, and the moments of reaching the observer — substantially dependon the correlation between the light speed and the speed of the BP passing overthe companion surface. The possibility of applying of the obtained results to theknown X-ray accretion-powered pulsars and rotation-powered pulsars in binarysystems is examined.
Subject headings: binaries: close — pulsars: general — X-rays: binaries
1. Introduction
An important way of obtaining information about stars in binary systems is observationof the reflection effect which involves reprocessing and reemission of the radiant energycoming from a star by the atmosphere of another star. If a binary system includes a neutronstar acting as a pulsar, with a companion occasionally getting into its beam pattern (BP), Department of Astrophysics, Faculty of Mathematics and Mechanics, Saint Petersburg State University(SPbSU), Universitetsky pr., 28, Stary Peterhof, Russia, 198504 < π and rotating like a lighthouse.In these binary systems the reflection effect looks as follows. With appropriate ori-entation of the pulsar BP the companion side facing the pulsar is occasionally exposed toX-ray radiation pulses — the BP runs over the surface of the companion. The interactionof this emission with the companion atmosphere results in reprocessing and reemission ofX-ray photons getting into the atmosphere. Some part of this energy is spent on heating theatmosphere and increasing the stationary radiation flux from the companion. Another partof this energy can be reemitted in the form of pulses with the period equal to the period ofpulsar rotation both in the X-ray and longer-wavelength bands of the spectrum (Basko et al.1974). Besides, penetration of high-energy photons (as well as pulsar wind particles) into thecompanion atmosphere can result in evaporation of the companion in close binary systems(Fruchter et al. 1988; Phinney et al. 1988). In this paper the formation of reflected pulse ra-diation is considered separately from other possible kinds of interaction between the pulsarradiation and the companion.The observation and interpretation of the reemitted pulses expand our possibilities ofstudying the properties of the source, companion and binary system in general (Avni & Bahcall1974). It would seem, besides, that the study of a quick-changing radiation flux can providemore information than the study of a stationary flux: in the first case we have a time-dependent function while in the second case we have one constant value only (we abstractfrom the orbital movement and the companion rotation about its axis — the periods ofpulsars are usually considerably less than the typical time of these motions). An exampleof using observations of reemitted pulses is determination of the mass of the neutron star inHercules X-1 system (Middleditch & Nelson 1976). 3 –Unambiguous interpretation of the results of observations of reemitted pulses in thebinary systems under consideration is likely to be a challenging task as it is necessary toconsider a whole number of factors influencing the properties of pulses (we do not discuss thetechnical possibility of observation of such pulses here). Firstly, this is the characteristics ofincident radiation: the kind of the spectrum and the beam pattern. Secondly, the structureof the companion atmosphere determining the way the reprocessing and reemission of thephotons getting into the atmosphere proceed. The entering of a considerable quantity ofhigh-energy photons and pulsar wind particles into the atmosphere results in its heating and,consequently, in its restructuring, which should be taken into account. Thirdly, radiationpropagation can be influenced by the gas-dynamic processes existing in a binary system;it especially concerns X-ray pulsars. Finally, the properties of reemitted pulses depend ongeometric parameters of the binary system.The present paper considers the last, geometric, factor. It is considered on the basis of asimple model of the binary system that does not concern the details of the reprocessing andreemission of the photons. This model consists of a spherical reflective screen that does notemit anything as it is. This screen is exposed to the radiation of the source rotating aboutits axis. The size of the source allows considering it a point source. Due to the location ofthe reflective sphere all the points of its surface in the line-of-sight of the source occasionallyget in the beam pattern of the source. Two kinds of the source BP are under consideration:in the first case of the BP every elementary area of the screen surface getting within itslimits is exposed by a pulse having infinitely narrow time spread while in the second case itis irradiated by a rectangular pulse. It is supposed that every elementary area of the surfaceof the sphere instantaneously reemits all the radiation reaching it so that the brightness ofthe area appears to be equal in all directions. The observer recording the reflected pulses isat such a great distance from the screen-source system that the screen has point dimensionsfor him. The model is described in detail in the next section.The main conclusion provided by the consideration of this model is that the properties ofreflected pulses — their profile, and the moments of reaching the observer — substantiallydepend on the correlation between the light speed and the speed of the BP passing overthe companion surface. In particular, at a certain correlation between these speeds themaximum amplitude of the reflected pulses is achieved (Dementyev 2014). Thus, it may beconcluded that the rotation of the source irradiating the companion atmosphere can have acertain influence on the reflected pulse radiation. This influence, however, is determined bythe geometric parameters of the binary system only and does not depend on the way thereprocessing and reemission of the photons getting into the atmosphere proceed. 4 –
2. The model
Let us specify the parameters of the screen – source – observer system described in theIntroduction.
Screen.
The reflective screen is spherical with radius R . The sphere has no radiation ofits own. We will assume the following reflecting properties of the surface of the sphere: everyelementary area of the surface of the sphere instantaneously reemits all the radiation fallingon it so that its brightness appears to be the same in all directions (lambertian source). Wewill relate the Cartesian coordinate system ( x, y, z ) with the sphere placing the beginning ofthis system in its center C (Fig. 1). We will also introduce the spherical coordinate system( r, θ, ϕ ) with the center in point C : x = r sin θ cos ϕ , y = r sin θ sin ϕ , z = r cos θ . (1) Source.
Let the linear size of the radiation source be such as compared to other typicaldimensions of the system that it may be considered a point source. The source is in θ = 0direction (i.e. on Cz axis) at the distance of r s from the center of the sphere so that r s = kR , k ≫ . (2)We will designate the location of the source with S . The source rotates about its axis AA ′ parallel with Cx with the constant angular velocity of ω = 2 π/P where P is the period ofrotation. The source rotation is counterclockwise if we look at the plane ( yz ) from the sideof the positive direction of Cx axis (Fig. 1).Regarding the beam pattern of the source we will first assume the following. We willconsider that the source radiation is concentrated within the limits of the plane angle α with the vertex in S point that remains perpendicular to the plane ( yz ). In addition, if weplace the angle α in the plane ( xz ), the cross-section of the sphere by this plane will becompletely within α (Fig. 1). We will also consider that within the BP limits the sourceemits uniformly. We neglect the dilution of radiation during its propagation from the sourceat the distance of r s − R to the point away from the source at the distance of r s . Observer.
The radiation reflected by the sphere is recorded at the distance r o from thecenter of the sphere, where r o ≫ r s ≫ R , (3)so the sphere is a point source for the observer. We will be interested in the points ofobservation lying in three possible directions: CO ( θ = 0), CO ( θ = π/ ϕ = π/ CO ( θ = π/ ϕ = 3 π/
2) — all of them located in the plane ( yz ). The transition from one 5 –point to another may be interpreted as observation of the binary system in different phasesof its orbital motion from one point. For the purposes of brevity these directions will befurther designated as O , O and O , respectively.We will be interested only in the relative changes of the radiation flux in the point ofobservation during the period abstracting from the properties of the radiation itself (spectralcomposition, absolute flux value, etc.) as well as from the details of the source emissionmechanism, radiation reprocessing by the reflective surface and radiation propagation in thespace. Thus, the parameters of the problem are r s , R and P , their correlation influencingthe properties of the reflected pulses recorded by the observer.As the BP of the source is a plane angle oriented as stated above, the radiation reachesevery elementary area of the surface of the sphere getting into the BP in the form of pulseshaving infinitely narrow time spread, their period being equal to P due to the source rotation.As the radiation does not reach different areas simultaneously and the distances from theareas to the observation points are not equal, the observer records the radiation reflectedfrom the sphere already in the form of pulses of finite duration as well as recurring with theperiod P (Avni & Bahcall 1974). Note 1.
It is usually conceived that BP of a pulsar consists of two identical beamssymmetrically located relative to the pulsar rotation axis. In this case we could also take apoint source with the BP consisting of two identical plane angles symmetrical relative to thesource rotation axis. Within the frames of our model this would just mean that the screenis exposed to radiation with the period of P/
2. Therefore, it is sufficient to confine ourselvesto consideration of the source with the BP consisting of one beam.
Note 2.
Consideration of the BP in the form of a plane angle means that the sphere isirradiated by pulses, their profile in time being described by the Dirac δ -function. Let theprofile of the reflected pulses be defined by the pulse function h ( τ ) here. If the source BPis such that a pulse with a certain s ( τ ) profile rather than a δ -like pulse falls on every areaof the sphere surface, the profile of the reflected pulses φ ( τ ) is found by convolution of h ( τ )and s ( τ ) (Avni & Bahcall 1974) φ ( τ ) = Z + ∞−∞ h ( τ ′ ) s ( τ − τ ′ ) dτ ′ . (4)Use of Equation (4) supposes that separate δ -like pulses are reflected from the screen surfaceindependently from one another, i.e. the reflection process is linear. In fact, the purpose ofthe present paper is to obtain the pulse function h ( τ ) for some cases. After calculation of h ( τ ) using Equation (4) we will find φ ( τ ) for the case of rectangular incident pulses. 6 –
3. Specific features of formation of the reflected pulses
Let us consider the screen–source system section by the plane ( yz ). We will introducethe Cartesian (˜ x, ˜ y ) and polar (˜ r, ˜ ϕ )˜ x = ˜ r cos ˜ ϕ , ˜ y = ˜ r sin ˜ ϕ (5)coordinate system in this plane with the center in point S . We will direct the S ˜ x axiscontrarily to the Cy axis and the S ˜ y axis contrarily to the Cz axis. The locus of (˜ x ˜ y ) planeor, what is the same, ( yz ) plane which were reached by the radiation at some moment oftime t is the Archimedean spiral determined by equation˜ r = c ( t − ˜ ϕ/ω ) , (6)where c is the speed of light (Bolotovski˘ı & Ginzburg 1972). Spiral (6) is actually the sectionof the wave front by (˜ x ˜ y ) plane.Let us consider two characteristic time intervals. The first of them is the time of passageof the distance equal to the sphere diameter by radiation: t = 2 R/c . (7)The second of them is the full time of BP passage over the surface of the sphere, i.e. thetime of the source turn by angle2 arcsin
R/r s = 2 arcsin 1 /k ≃ /k (8)(see Fig. 2) where condition (2) is taken into account. This time is equal to t = 2 /k π P = Pπk . (9)Below we will measure the time in the units of the period P and the distance in the unitsof the sphere radius R and will designate the dimensionless time through τ . We will alsointroduce the W = cP/R parameter which represents the dimensionless speed of radiationpropagation.Let t ≪ t , which means W ≪ πk . Then the arm of the spiral (6) is twisted at thedistance r s from the source (Fig. 3a). In this case it may be considered that a plane wavefront falls on the screen and passes over the screen with the speed of light in the directionopposite to the direction of Cz axis (Fig. 1).In the contrary case when t ≫ t , we have W ≫ πk . In this case the spiral armjust starts twisting at the distance r s from the source (Fig. 3b). The picture of the screen 7 – Sr s α R O O O y x zA ′ A C Fig. 1.— Scheme of arrangement of the spherical screen with the center in point C andradius R , the point source S and beam pattern of the source — plane angle α . The distancebetween the source and the center of the sphere is designated with r s . Line AA ′ is the axisof the rotation of the source; the arc with an arrow shows the direction of its rotation. SG C θ r s ˜ r ˜ y z ˜ xy θ max ˜ ϕB Fig. 2.— Section of the screen–source system by ( yz ) plane. Here CB = SG = R . 8 –irradiation can be represented as follows. The plane wave front passes over the screen inthe direction of Cy axis (Fig. 1), i.e. from right to left in accordance to Fig. 3b. Thespeed of motion of this front is v = ωr s ; in dimensionless values V = 2 πk . The direction ofpropagation of the radiation itself is perpendicular to the direction of the front motion.In case of intermediate values t ∼ t and, accordingly, W ∼ πk , the wave front passingover the screen can be probably no longer considered a plane one and its motion is no longerrectilinear.Thus, two extreme modes of radiation of the sphere can be identified depending on thecorrelation between two characteristic time intervals t and t (we can also speak about acorrelation between two speeds: W — the speed of light and V — the speed of passageof the source BP over the surface of the screen). It should be expected that the reflectedpulses reaching the observer will have different parameters depending on the particular screenirradiation mode. Note.
The particular mode of irradiation of the spherical screen does not depend on itsradius R . It is actually determined by the way the dimensionless Q value Q ≡ WV = cP πr s (10)correlates with the unit (Fig. 4). However, the parameters of the reflected pulses withthe same Q are different depending on the R value. Firstly, the bigger the radius of thesphere, the larger the reflective surface; therefore, the pulse amplitude may also appearlarger. Secondly, the pulse duration equal by order of magnitude t + t also depends on R according to Equations (7) and (9).
4. Method of calculation of the parameters of the reflected pulses
We will consider that individual reflected pulses do not overlap. In this case it is sufficientto calculate the change of the flux of the reflected radiation in the time interval equal toone period. Let us consider the small area dσ located on the reflective sphere so that thepolar angle of the spherical coordinate system of the place of its location is equal to θ andthe azimuthal angle is equal to ϕ . For the areas that can be reached by the radiation of thesource, the θ angle changes from 0 to θ max = arccos ( R/r s ) = arccos (1 /k ) (11)(Fig. 2) and the ϕ angle — from 0 to 2 π . We can represent the contribution of the radiationreflected by the area dσ over the period to the total flux recorded by the observer using the 9 – S (a) k = 10 W = 6 S (b) k = 10 W = 300 Fig. 3.— Wave front section by ( yz ) plane with two values of W parameter. The grey circleat the top of every figure is the screen section by the same plane. The arcs with arrow showthe direction of rotation of the source located in point S . . . . .
40 0 .
30 0 .
25 0 . . .
10 0 . . S Fig. 4.— Spiral (6) on which the possible points of its crossing with the center of thereflective sphere for different values of Q parameter are plotted. The word ”possible” meansthe assumption that the wave front passes through the center of the sphere (through point C ) ”without noticing” its surface; actually the radiation cannot get inside the sphere, ofcourse. The numbers near the respective points show the Q values for which the center ofthe sphere would be located in the point marked. In particular, for Figure 3a the parameter Q = 0 .
10 and for Figure 3b Q = 4 .
77. 10 – δ -function as follows: h o δ ( τ − τ so ) cos ψ s cos ψ o dσ . (12)Here h o is the radiation flux over the period that would be created by the area in the pointof observation if it were oriented perpendicular both to the direction towards the source andto the line of sight; τ so = τ so ( θ, ϕ ) is the moment of time when the radiation from this areais recorded by the observer. The ψ s angle is the angle of the radiation incidence on the areaunder consideration and the observer sees this area at ψ o angle.Considering that dσ = R sin θ dθ dϕ , the full h ( τ ) flux reflected from the semispherefacing the observer will be equal to h ( τ ) = h o R Z ϕ max ϕ min dϕ Z θ max δ ( τ − τ so ) cos ψ s cos ψ o sin θ dθ , (13)where ϕ min and ϕ max are the limits of change of the azimuthal angle on this semisphere.This formula means the following. To find the value of h flux at some moment of time τ itis necessary to integrate the integrand taken without the δ -function only for the values of θ and ϕ with which we have τ so ( θ, ϕ ) = τ (see in detail below). Identification of ψ s angle. The angle of radiation incidence on the area is the anglebetween the normal to the area and the direction towards the source (Fig. 5). Using the lawof cosines we getcos ψ s = r s cos θ − Rρ = r s cos θ − R p r s − r s R cos θ + R = k cos θ − √ k − k cos θ + 1 , (14)where the third equation follows from Equation (2). C Sr s R ρψ s θy x z D ˜ x ˜ y Fig. 5.— Identification of ψ s — angle of radiation incidence on the elementary area of thescreen surface. 11 – Identification of ψ o angle. Let the observer be located in O direction. This observersees the area under consideration at ψ o angle, its cosine being identified similarly to cos ψ s :cos ψ o = r o cos θ − R p r o − r o R cos θ + R ≈ cos θ , ( θ ∈ [0; θ max ] , ϕ ∈ [0; 2 π ]) . (15)The approximate equation here is obtained with account of condition (3). Given in paren-theses are limits of change of θ and ϕ of the areas reflecting the radiation that are in thedirect line of sight for observer O .For observers in O and O directions we similarly havecos ψ o ≈ sin θ sin ϕ , ( θ ∈ [0; θ max ] , ϕ ∈ [0; π ]) (16)and cos ψ o ≈ − sin θ sin ϕ , ( θ ∈ [0; θ max ] , ϕ ∈ [ π ; 2 π ]) . (17) Identification of the moment τ so . Let t s be the delay of the moment of radiation incidenceon some area dσ relative to the moment of incidence on the area for which θ = 0 (this areais the nearest to the source). The radiation reflected by the area dσ will also reach theobserver with a certain delay t o relative to the radiation reflected by the area nearest to thesource. Both t s and t o may appear either larger or less than 0. Thus, the radiation passesthe distance ( r s − R ) + c ( t s + t o ) + ( r o − R ) , (18)from the source to the observer in the direction O and( r s − R ) + c ( t s + t o ) + r o (19)from the source to the observer in the direction O or to the observer in the direction O .There is a specific middle term in both cases for each area. At the same time the sought-formoment of time expressed in the units of the period interesting for us is τ so = τ s + τ o . (20)First let us get the expression for τ o . As we suppose that the observer is at infinitely longdistance from the screen, the direction towards the observer from all dσ areas are parallel toone another. For the observer in the direction of O τ o ≡ τ o = dcP = RcP (1 − cos θ ) = 1 W (1 − cos θ ) , (21) 12 –where d is the length of the perpendicular dropped from the point with ( R, θ, ϕ ) coordinateson the plane perpendicular to Cz axis and passing through the point on this axis for which z = R . Similarly, for observers in the directions of O and O we have τ o ≡ τ o = τ o = − W | sin θ sin ϕ | . (22)To determine the τ s delay we must compare the moments of radiation incidence on therelevant areas of the sphere surface. We will measure off the time from the moment whenthe source BP projection on the plane (˜ x ˜ y ) is the semiaxis S ˜ x . In this case the moment ofradiation incidence on the area located in the neighbourhood of some point B of the plane(see Fig. 2) is equal t B = ˜ r/c + ˜ ϕ/ω . (23)Note that the reference time may be any other moment: then in the right part of Equation(23) there will appear some constant as another summand. But as we just need the differencebetween the moment of radiation incidence on the areas, this constant will reduce anywayin making up this difference.If the elementary area dσ is located in the neighbourhood of some point D not lying inthe (˜ x ˜ y ) plane (see Fig. 5), the moment of radiation incidence on this area is t D = ρ/c + ˜ ϕ D /ω , (24)where ˜ ϕ D angle is measured off from the S ˜ x axis to the projection of ρ segment on the (˜ x ˜ y )plane. Let point D have the Cartesian coordinates ( R sin θ cos ϕ, R sin θ sin ϕ, R cos θ ). Inaccordance with Equation (24) the time of radiation propagation from the source — point S to the dσ area — point D is τ D = ρR W + ˜ ϕ D π , (25)where ρ = p r s − r s R cos θ + R = R √ k − k cos θ + 1 . (26)We will determine the ˜ ϕ D angle from the triangle lying in ( yz ) plane, its vertexes being: 1)the point of location of the source S ; 2) the projection of point D on the ( yz ) plane; itscoordinates are (0 , R sin θ sin ϕ, R cos θ ); 3) some point on the positive S ˜ x semiaxis ; we willtake point G with the Cartesian coordinates ( x, y, z ) equal to (0 , − R, r s ) as this point (Fig.2). The sought-for angle is the angle at S vertex. According to the law of cosines we havecos ˜ ϕ D = − sin θ sin ϕ p k − k cos θ + cos θ + sin θ sin ϕ . (27) 13 –We will designate the moment of radiation incidence on the elementary area of the spheresurface nearest to the source through τ p (for it θ = 0). With account for Equations (25–27)this moment is equal to τ p = k − W + 14 . (28)Thus, τ s = τ D − τ p = √ k − k cos θ + 1 − k + 1 W + ˜ ϕ D π − , (29)where ˜ ϕ D is found as an arccosine of the expression (27). Calculation of the values of h ( τ ) . The values of the h ( τ ) function determined by theEquation (13) can be obtained as follows. On the intervals of θ and ϕ integration as well ason the interval of τ so change we will select some sets of discrete points0 = θ < θ < · · · < θ i < · · · < θ q − < θ q = θ max , (30) ϕ min = ϕ < ϕ < · · · < ϕ j < · · · < ϕ q − < ϕ q = ϕ max , (31) τ min = τ < τ < · · · < τ n < · · · < τ q − < τ q = τ max , (32)where τ min and τ max are the moments of arrival of the rising and falling edges of the pulse,respectively, and q , q , and q are some integer numbers that may also be identical. The τ min and τ max are initially unknown to us; therefore, we will select an interval of change τ so so that τ min and τ max should get to the interval a priori . After the calculation procedure of h ( τ n ) ≡ h n values the τ min and τ max moments are defined as the first and last moment oftime with which h n = 0. It is convenient to take the points (30–32) uniformly with steps∆ θ , ∆ ϕ , and ∆ τ respectively.The procedure of calculating the h ( τ ) flux values starts from assigning the value equalto 0 to h n values for all n . Then τ so is calculated for every pair of points θ i and ϕ j by theEquations (20–22, 29) and the interval ( τ n − ; τ n ] is found so that τ n − < τ so τ n . Afterthat the h o R cos ψ s ( θ i ) cos ψ o ( θ i , ϕ j ) sin θ i ∆ ϕ ∆ θ ∆ τ (33)value is added to the current h n value. Division by ∆ τ was needed here because the describedprocedure is actually numerical integration for interval ( τ n − ; τ n ] and we refer the value thusreceived to one point, namely to τ n . Calculation of the values of the convolution.
After finding the pulse function values h n we can find the φ ( τ ) profile of the reflected pulses resulting from irradiation of the spherewith pulses with arbitrary s ( τ ) profile using Equation (4). The φ ( τ k ) convolution values are 14 –calculated as usual (Bracewell 2000): φ ( τ k ) = q X n =1 h n s k − n . (34)As an example we will consider the rectangular profile of incident pulses s ( τ ) = ( /a , τ a < , τ < , τ > a . (35)
5. Irradiation of the sphere by parallel rays
Let us assume that the sphere is at an infinitely large distance from the source. Thenall the elementary areas on the sphere are exposed to radiation by parallel rays and for all ofthem ˜ ϕ D = π/
2. With such a mode of irradiation of the sphere the spiral arm (6) is twistedat the distance r s from the source ( Q ≪ θ max = π/ ψ s = cos θ (37)respectively. Then obviously τ s = τ o (38)for the observer in the O direction. With account for Equation (21) we have τ so = 2 W (1 − cos θ ) . (39)In this case 0 τ so W . (40)Thus, the observer will record the radiation that the sphere reflects over the period duringthe time 2 /W . In other words, the 2 /W value is the duration of the reflected pulse, withthe pulse incident on the sphere having infinitely narrow time spread.For the observer to see individual not overlapping reflected pulses their duration mustbe less than the period, i.e. 2 W Rc P . (41)The P/ (2 R/c ) value is the duty cycle of the reflected pulses. 15 –The expression (39) for τ so allows obtaining h ( τ ) in an explicit form. Namely, in theintegral (13) we will consider that ϕ min = 0, ϕ max = 2 π , will substitute cos ψ o , θ max and cos ψ s from the Equations (15), (36) and (37), respectively, and make the following substitution ofthe variable: u ≡ − cos θ ) /W . Then h ( τ ) = 2 πh o R W Z /W (cid:18) − W u (cid:19) δ ( τ − u ) du = (42)= 2 πf o R · (cid:18) W (cid:19) · (2 /W − τ ) . Here we used the well-known property of the δ -function. With 2 /W τ h ( τ ) = 0. For observers O and O in this case of irradiation of all the areas of the spherewith parallel rays we fail to get an explicit expression for h ( τ ) in the same way as τ so appearsto depend both on θ and ϕ .In accordance with Equation (42) the amplitude of the rising edge of the reflected pulsein the point of observation is h (0) = πh o R W = 2 πh o R P (2 R/c ) (43)— the bigger the R radius of the sphere, the larger the reflecting surface; thus, the larger theamplitude of the pulses. The h o value is the radiation flux over the period. The entire h o fluxreaches the observer with 2 R/c time that is less than or equal to the period in accordancewith Equation (41). The decreasing duty cycle of the pulses with unchanging h o flux and P period results in the fact that all this flux will take less time and, hence, the amplitude ofthe pulse is supposed to grow, which is shown by the Equation (43). According to Equation(41) the pulse duration depends on R . This means that the moments of arrival of the risingand falling edges of the pulse to the observer also depend on R .The parameters of the reflected pulses obviously depend on the sphere radius R bothfor the considered case of the irradiation of the sphere by parallel rays when Q ≪ Q . Besides, the influence of R value on the parameters of the pulsestakes place both for the observer in the O direction and for any other directions in whichthe reflected pulses are recorded.
6. Results
Fig. 6 presents profiles of the pulses reflected in O , O and O directions; the profileshave been plotted for two values of W parameter that match the different modes of the 16 –irradiation of the sphere (see Fig. 3). Moment τ = 0 on the time scale corresponds to arrivalof the radiation reflected by the sphere surface area that is the nearest to the source (forit θ = 0). In the upper figure showing the pulses recorded by the observer in O directionthe dotted line gives the pulse profile plotted according to Equation (42) with W = 6. Anunexpected specific feature of the profile of the pulse recorded in O direction with W = 300is its nonmonotonic growth — there is a small peak on the curve (see the lower figure).Note that for O and O directions the areas under the curves with any W values arethe same and approximately equal to 0 . O direction the areas under the curves arealso identical with any W and equal approximately 1 . π/ ≈ .
094 for any W .The Table 1 presents the following parameters of the reflected pulses: [ h/ ( h o R )] —the maximum value of the pulse profile (pulse amplitude); τ b — the moment of arrival ofthe pulse rising edge at the observer; τ max — the moment at which the pulse profile reachesthe maximum value; ∆ τ / — the width of the pulse measured at the level of the half ofits amplitude. The values of these parameters are given for the O , O and O directionsbeing considered; for all cases k = 10. The line at W = 6 with values in bold face shows thecorresponding values for the pulse profile plotted according to the expression (42).Let us discuss the obtained results. The profile of the reflected pulses is determined byelementary areas of the sphere surface which the observer sees at the moment. Figures 6 andthe data of the Table 1 show that a change of the sphere irradiation mode (change of thecorrelation between W and V ) has a substantial impact on the profile of the reflected pulsesas well as on the moments of their arrival at the observer. Let us imagine that W has thevalue ∼ W starts growing due to the growing period of rotation of source P withthe unchanged radius of the sphere R , the distance between the source and the sphere r s andfull source radiation flux. The amplitude of the reflected pulses will increase up to a certainmaximum value exceeding the initial value by more than an order; in this case the durationof pulses decreases. This increase of the pulse amplitude occurs only through redistributionof the moments of arrival of radiation of different areas at the observer. Having reached themaximum the pulse amplitude slowly decreased for the same reason.Another important conclusion from the received results is the following. Rotation of thesource irradiating the sphere results in the situation that the picture of the reflected pulseslooks different for O and O observers. In case of the untwisted spiral arm (Fig. 3b) thepulses arriving at these observers differ both by the profile shape and the arrival moments;in case of the twisted spiral arm (Fig. 3a) — by the arrival moments only. In this case the 17 – h ( τ ) / ( h o R ) τW = 6 W = 300 O − . . . . h ( τ ) / ( h o R ) τW = 6 W = 300 O − . . . . − . − .
01 40 h ( τ ) / ( h o R ) τ W = 6 W = 300 − . . . . O Fig. 6.— Profiles of the pulses reflected in O , O and O directions ( k = 10). 18 –difference between the moments of arrival of the pulses at O and O observers in accordancewith Equations (27) and (29) is determined by W and k values (the delay τ o is probablythe same for both observers, Eq.(22)). The data of the table show that at the values of W corresponding to the mode of the twisted spiral arm the difference between the moment ofpulse arrival at these observers does not depend on W and is equal approximately to 0.023(with k = 10). Thus, knowing this difference between the pulse arrival moments we can findthe companion radius R using the known P and r s . In all probability using the pulse arrivalmoments for other directions (apart from O , O and O ) we can also obtain both R and r s using the known period P . Note 1.
Note that if k increases at constant W , the difference between the moments ofpulse arrival at O and O observers disappears. This actually means that the source rotationstops influencing the properties of the reflected pulses. If O and O observers recorded theradiation of the pulsar in the Crab Nebula reflected by the Earth, they would obtain anidentical picture of reflected pulses. Note 2.
The spreading of the pulses incident on the sphere surface is evidently toresult in leveling of the differences between the properties of reflected pulses with differentirradiation modes. For example, Fig. 7 presents profiles of the pulses reflected in O directionwith the rectangular profile of the incident pulses; the duration of the incident pulses wastaken as follows: a = 0 .
7. Application to astronomical objects
As it was mentioned in the Introduction, the objects to which the results obtained in thepresent work can be applied are X-ray accretion-powered pulsars as well as rotation-poweredpulsars in binary systems. The reference to the relevant catalogs (Liu et al. 2006, 2007;ATNF Pulsar Catalogue 2014; Manchester et al. 2005; Cherepashchuk et al. 1996; Katysheva2005) shows that the observed values P and r s of the majority of these systems give Q ≪ Q for them using the known values of P and r s .We are not discussing the real possibility of emergence of reflected pulse radiation in thesesystems.The examples of interest for us among the high-mass X-ray binary systems (Liu et al.2006) are, in particular, objects 2S 0114+650, 4U 1538-52, and IGR J16320-4751.
2S 0114+650.
The pulsar period in this system is P = 9605 s (Bonning & Falanga Table 1. Parameters of the reflected pulses. [ h/ ( h o R )] max τ b τ max ∆ τ / W Q O O O O O O O O O O O O · · · · · · · · · · · · · · · · · · · · · · · ·
10 0.16 30.2 22.7 18.8 -0.001 -0.028 -0.051 -0.001 -0.028 -0.051 0.062 0.021 0.02520 0.32 58.1 46.1 33.3 -0.002 -0.009 -0.032 -0.002 -0.009 -0.032 0.029 0.010 0.01330 0.48 82.3 62.6 44.0 -0.002 -0.003 -0.026 -0.002 -0.003 -0.026 0.020 0.008 0.01035 0.56 92.6 64.6 48.1 -0.003 -0.002 -0.024 -0.003 -0.002 -0.024 0.018 0.008 0.00940 0.64 101.5 60.5 51.6 -0.003 -0.001 -0.023 -0.003 -0.001 -0.023 0.016 0.010 0.00863 1.00 127.3 56.0 61.2 -0.004 -0.000 -0.020 -0.004 0.006 -0.020 0.013 0.011 0.00885 1.35 133.5 58.8 64.4 -0.006 -0.000 -0.018 -0.006 0.007 -0.018 0.012 0.010 0.00890 1.43 133.4 59.4 64.7 -0.006 -0.000 -0.018 -0.006 0.008 -0.018 0.013 0.010 0.008100 1.59 131.4 60.0 64.8 -0.006 -0.000 -0.018 -0.006 0.008 -0.018 0.013 0.010 0.008150 2.39 110.1 58.6 61.4 -0.008 -0.000 -0.017 -0.008 0.009 -0.017 0.018 0.010 0.009300 4.77 90.0 55.7 51.5 -0.011 -0.000 -0.016 -0.002 0.010 -0.013 0.023 0.011 0.0113000 47.7 89.1 53.4 53.0 -0.015 -0.000 -0.016 -0.000 0.012 -0.012 0.024 0.011 0.011
20 –2005). Using the information given by Reig et al. (1996) — P orb ≃ . M NS ≃ . M ⊙ , the companion mass M c ≃ M ⊙ , the companion radius R ≃ . · cm, and using the 3rd Kepler’s Law r s = (cid:18) GP orb ( M NS + M c )4 π (cid:19) / , (44)we can find that for this object r s ≃ . · cm, Q ≃ .
8, and k ≃ . G — gravityconstant).
4U 1538-52.
According to Makishima et al. (1987) in this system P ≃
530 s, P orb ≃ . M NS ≃ . M ⊙ , M c ≃ M ⊙ , R ≃ . · cm. Therefore, for this object r s ≃ . · cm, Q ≃ .
4, and k ≃ . IGR J16320-4751.
In this system the pulsar period is P = 1303 s (Rodriguez et al.2006) while the orbital period is P orb = 8 .
96 days (Corbet et al. 2005) . Considering the factthat the pulsar companion is a supergiant of the spectral type BN0.5Ia (Coleiro et al. 2013),we assume M NS + M c ≃ M ⊙ . Then we obtain r s ≃ . · cm and Q ≃ . Q value isobtained for the object 4U 1626-67. According to Middleditch et al. (1981), P ≃ . P orb ≃ M NS + M c ≃ . M ⊙ , r s ≃ . · cm. Therefore, Q ≃ .
1. Levine et al.(1988) have evaluated the radius of the companion as follows: R ≃ . · cm; hence, k ≃ . Q ≪
1. The highest Q value is provided by the param-eters of the binary pulsar J0737-3039B: P ≃ . P orb ≃ . M NS + M c ≃ . M ⊙ (Lyne et al. 2006); in this case Q ≃ .
15. The next biggest Q value is obtained for the pulsarB1718-19: with r s ≃ . · cm (Lyne et al. 1993) we have Q ≃ .
8. Conclusion
The papers considering reprocessing and reemission/reflection of pulse radiation usu-ally assume that due to the geometrical factor the reemitted pulses have greater durationcompared to the incident ones by the order of
L/c , where L is the linear size of the reflec-tive screen, see, for example, Basko et al. (1974); Avni & Bahcall (1974). For our modelthis time is t , see Equation (7). The main conclusion of the present work is that if theirradiation of the screen with pulse radiation occurs due to the source rotation, it is alsonecessary to consider the increase of the duration of the reemitted pulses by a value equalto the time of passage of the source beam pattern over the surface of the screen — time t in our model, see Equation (9). If Q ≪ t only.In the opposite case, when Q ≫ t only. A significant fact is that in intermediate cases thereflected pulse duration may appear substantially lower than t + t . Moreover, the minimalpossible duration of the pulses appears to be less both than t and t . The decrease of theduration of the pulses with other conditions being equal results in growth of its amplitude.Another conclusion is that with any Q the conditions of the pulsar irradiation of theright and left (from the point of view of O observer) semispheres of the companion aredifferent. This provides a principally new opportunity to determine the pulsar companionradius in the binary system and the distance between the pulsar and the companion byanalyzing the parameters of the pulses reflected in different directions, which can be doneby observing the companion in the course of its orbital motion.The author is grateful to G. M. Beskin (SAO RAS) for drawing attention to the processesof interaction of radiation of the pulsars with their companions in binary systems. The authoris also appreciative of V. V. Ivanov (SPbSU) for discussion of geometrical specific features ofthe wave front of the rotating source and for helpful comments that improved the manuscript.The work has been performed within the frames of the Science Projects of SPbSU6.0.22.2010 and 6.38.669.2013. REFERENCES
This preprint was prepared with the AAS L A TEX macros v5.2.
24 – φ ( τ ) / ( h o R ) τW = 6 W = 3000 0 . . . . O Fig. 7.— Profiles of the pulses reflected in O direction with exposure of the sphere toradiation with rectangular pulses with the duration a = 0 . kk