Pressure Balance and Intrabinary Shock Stability in Rotation-Powered State Redback and Transitional Millisecond Pulsar Binary Systems

# Pressure Balance and Intrabinary Shock Stability in Rotation-Powered State Redback and Transitional Millisecond Pulsar Binary Systems

Zorawar Wadiasingh    Christo Venter    Alice K. Harding    Markus Böttcher    Patrick Kilian
###### Abstract

A number of low-mass millisecond pulsar (MSP) binaries in their rotation-powered state exhibit double-peaked X-ray orbital modulation centered at inferior pulsar conjunction. This state, which has been known to persist for years, has recently been interpreted as emission from a shock that enshrouds the pulsar. However, the pressure balance for such a configuration is a crucial unresolved issue. We consider two scenarios for pressure balance: a companion magnetosphere and stellar mass loss with gas dominance. It is found that the magnetospheric scenario requires several kilogauss poloidal fields for isobaric surfaces to enshroud the MSP as well as for the magnetosphere to remain stable if there is significant mass loss. For the gas-dominated scenario, it is necessary that the companion wind loses angular momentum prolifically as an advection or heating-dominated flow. Thermal bremsstrahlung cooling in the flow may be observable as a UV to soft X-ray component independent of orbital phase if the mass rate is high. We formulate the general requirements for shock stability against gravitational influences in the pulsar rotation-powered state for the gas-dominated scenario. We explore stabilizing mechanisms, principally irradiation feedback, which anticipates correlated shock emission and companion variability and predicts for the ratio of pulsar magnetospheric -ray to total shock soft-to-hard X-ray fluxes. This stability criterion implies an unbroken extension of X-ray power-law emission to hundreds of keV for some systems. We explore observational discriminants between the gas-dominated and magnetospheric scenarios, motivating contemporaneous radio through -ray monitoring of these systems.

stars: mass-loss, stars: magnetic field, pulsars: individual (J1023+0038, J1227–4853, J1723–2837, J2129–0429, J2215+5135, J2339–0533), X-rays: binaries, shock waves, accretion
\move@AU\move@AF\@affiliation

Astrophysical Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA \move@AU\move@AF\@affiliationCentre for Space Research, North-West University, Private Bag X6001, Potchefstroom 2520, South Africa

\move@AU\move@AF\@affiliation

Centre for Space Research, North-West University, Private Bag X6001, Potchefstroom 2520, South Africa

\move@AU\move@AF\@affiliation

Astrophysical Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

\move@AU\move@AF\@affiliation

Centre for Space Research, North-West University, Private Bag X6001, Potchefstroom 2520, South Africa

\move@AU\move@AF\@affiliation

Centre for Space Research, North-West University, Private Bag X6001, Potchefstroom 2520, South Africa

## 1 Introduction

The current decade has ushered in a new era for rotation-powered millisecond pulsars (MSPs) with radio, X-ray, and optical followup of unidentified Fermi Large Area Telescope sources yielding over 30 pulsar binaries111
Public+List+of+LAT-Detected+Gamma-Ray+Pulsars
in the Field. Recent population synthesis studies suggest the existence of order MSPs in the Galactic bulge and several hundred in each nearby globular cluster (2018arXiv180611215G). Moreover, the known population of MSPs is expected to surge enormously in the coming decade with the advent of the Square Kilometer Array (2015aska.confE..40K) and FAST (2009A&A...505..919S; 2011IJMPD..20..989N).

The subset of rotation-powered MSPs in detached binaries with low-mass companions are classified based on inferred companion mass , the “black widows” (BWs) with (1988Natur.333..237F) and “redbacks” (RBs) with (2011AIPC.1357..127R). For both classes, orbits are circularized with periods less than a day and inferred separation cm. The tidally-locked companions in RBs are bloated (compared a main-sequence star of similar mass) close to the Roche limit and are anisotropically heated, often with distinct day and night halves. In the standard evolutionary scenario of recycled MSPs, a low-mass X-ray binary accretion-powered spin-up phase (LMXB; 1982Natur.300..728A) precedes a pulsar rotation-powered spin-down state. The LMXB state can attain Eddington-scale X-ray luminosities erg s, precipitated by Roche lobe overflow (RLOF) and the formation of an accretion disk. A subset of neutron star LMXBs are the accreting millisecond X-ray pulsars (AMXP, cf. 2012arXiv1206.2727P, for a review) where the disk is truncated at the Alfvén radius  cm from the pulsar, the point at which MSP magnetospheric magnetic pressure balances accretion pressure of mass rate . For these millisecond spin periods , is similar to the small pulsar light cylinder radius few cm . Propellor states may exist if the disk inner Keplerian speed is smaller than the MSP corotation speed inside (e.g, 1975A&A....39..185I), i.e., when is larger than the corotation Keplerian radius.

Accretion-derived irradiation of the companion influencing mass loss has been invoked in long-term gigayear evolutionary models of LMXBs for their formation (1988Natur.334..225K; 1989ApJ...336..507R; 1989ApJ...343..292R; 1991Natur.351...39T; 2004A&A...423..281B), following the first suggestion of such “autoregulation” in the context of -disks by 1973A&A....24..337S. That is, emission from the accretion disk irradiating the companion bootstraps the mass loss and the accretion power. Such binary evolutionary tracks make simplifying assumptions about the poorly understood radiatively-driven winds or mass loss from the companion and the spectral energy distribution (SED) of the accretion luminosity. More recent work focused on BW and RB formation has found that irradiation feedback induces mass transfer cycles between rotation and accretion-powered states in the late-term evolution, with periods on the order of years, and predicts that RB companions should slightly under-fill their Roche lobe, the “quasi-Roche lobe overflow” model (qRLOF: 2014ApJ...786L...7B; 2015ApJ...798...44B). The existence of transitional systems and these models then imply conditions where the wind of the companion may be evaporative and supersonic rather than the more conventional higher-mass-rate subsonic RLOF, and also regimes intermediate between these two limits. Evaporative qRLOF would entail a wind from the high-energy tail of the Maxwellian in the photosphere or corona, that is sufficient to escape the low potential barrier of a companion slightly underfilling its Roche lobe.

Recently, some systems have been observed to transition between accretion and rotation-powered states (2009Sci...324.1411A; 2013Natur.501..517P; 2014ApJ...789...40B; 2015ApJ...800L..12R), persisting for years  s in one state preceding or following a transition. The transition itself may occur on a short timescale, shorter than a few weeks as sampled by typical observational cadences (2014MNRAS.441.1825B). In the disk phase, these transitional systems may exhibit complex X-ray phenomenology interpreted as propeller, sub-luminous or active/passive disk, or accretion states occasionally with coherent MSP spin pulsations similar to an AMXP (2014ApJ...795...72L; 2015ApJ...807...33P; 2015MNRAS.449L..26P; 2017ApJ...851L..34P). The X-ray persistent luminosities of transitional systems in disk states can be relatively low, erg s suggesting relatively low mass rates, g s accreted from the low-mass companion for a standard radiative efficiency. It has been advanced by 2015MNRAS.447.3034H that some subset of very faint X-ray binaries may be such transitional systems or AMXPs.

It is unknown if the companion fills its Roche lobe during the disk states in transitional MSPs. For many RBs in the rotation-powered state, the companions are known to be close but not quite filling their Roche lobe (e.g., 2015MNRAS.451.3468M; 2016ApJ...816...74B). However, small changes in the Roche filling factor or radiatively-driven wind physics may dramatically alter the mass loss rate in the transition between qRLOF and RLOF. The donor star need not significantly change its radius on short timescales associated with rotation-powered and accretion state changes if it is already nearly filling its Roche lobe.

In the pulsar state, radio or -ray magnetospheric pulsations of the MSP are often observable. We are not aware of any optical evidence of disks in the pulsar state, in contrast to the accretion state (e.g., 2013ATel.5514....1H). In this rotation-powered pulsar state, many BWs (2012ApJ...760...92H; 2014ApJ...783...69G) and RBs (e.g., 2015arXiv150207208R) exhibit persistent nonthermal and hard X-ray emission with photon indices typically (see Table 1). Thermal X-ray emission, besides that ascribed to the MSP polar caps (2011ApJ...742...97B), is absent. For J1023+0038 during its rotation-powered phase, no break in the power law was detected with NuSTAR up to at least keV (2014ApJ...791...77T). Similarly, J1723–2837 (2017ApJ...839..130K) and J2129–0429 (2018ApJ...861...89A; 2018arXiv180601312K) also exhibit no spectral cut-off at NuSTAR energies. Because of the rising spectra in a representation, the highest energies of the power laws dominate the energetics of this component.

Moreover, about nine systems exhibit orbital modulation of the persistent X-ray emission (see Table 1 of 2017ApJ...839...80W), with erg s, including emission in the NuSTAR band and even in some systems with inclinations far from edge-on (e.g., J1023+0038, 2010ApJ...722...88A; 2011ApJ...742...97B; 2014ApJ...791...77T). A possible tenth system omitted in 2017ApJ...839...80W is J1740–5340 (2003ApJ...584L..13F; 2010ApJ...709..241B). These X-ray orbital phase-folded light curves are often double-peaked, with a local minimum either near pulsar superior or inferior conjunction, which we denote SCDP (companion between pulsar and Earth) or ICDP (pulsar in front), respectively. Such ICDP orbital modulation is especially striking in J2129–0429 (2015arXiv150207208R; 2018ApJ...861...89A; 2018arXiv180601312K) and J1227–4853 (2015MNRAS.454.2190D), among others. In the disk state of J1023+0038, 2015ApJ...806..148B find no evidence of orbital modulation in either the low, high or flaring modes of the X-ray emission. Therefore the orbital modulation in the pulsar state is of a qualitatively different origin than X-rays in the disk state, and is, by definition causally associated with the stellar companion and its orbital timescale.

\twocolumngrid

Some scenarios for the persistent emission can be ruled out owing to its energetics. It may be shown that orbital energy extraction by any mechanism would yield too short an inspiral timescale if it is to entirely power the persistent X-ray emission, in disagreement with much smaller known constraints. The nonthermal ICDP modulated component is also difficult to explain as originating internally from the stellar companion. The minimum putative energy output is roughly erg, of order of the stellar gravitational binding energy. Even with conversion efficiency, this is much larger than the magnetic reservoir erg for kilogauss magnetic fields attainable in convection-dominated low-mass stars. Moreover, if the ICDP emission is powered by persistent companion-intrinsic magnetic activity, then it is unclear why there is no evidence for it in the disk state low-mode of J1023+0038 (2015ApJ...806..148B) where it may contribute of the observed flux in the soft X-ray band (2010ApJ...722...88A; 2011ApJ...742...97B). It is also unknown how such magnetically-powered activity would naturally yield persistent nonthermal high-energy ICDP modulation across many sources. Therefore, magnetic activity may only account for more transient phenomena. Furthermore, if there is no shock, the solid angle fraction of the pulsar wind intercepted by the companion is of order – this would then demand untenably large conversion efficiency of at around into the hard X-rays at the companion. Such pure wind conversion is also contradicted by relatively cool optically-derived photospheric temperatures K for RB companions. Therefore, the source of the persistent nonthermal X-ray emission is not proximate to the companion photosphere.

Phase-resolved X-ray hardness ratios of orbital modulation in ICDP systems exhibit a harder-when-brighter phenomenology (e.g., 2010ApJ...722...88A; 2011ApJ...742...97B; 2015MNRAS.454.2190D; 2015ApJ...801L..27H). This as well as NuSTAR power laws beyond 30 keV, rule out absorption as an origin of the orbital modulation (however, absorption may play a role in stability of a shock in some scenarios, as we explore in this paper). Moreover, absorption or obscuration of a putative disk emission by the companion or its wind neither yields double peaks nor modulation at the correct inferior-conjunction phasing. Some RBs in the pulsar state also exhibit large radio MSP eclipse fractions, of the orbit at low frequencies for RBs J1023+0038 and J2215+5135. Crucially, the pulsar is largely uneclipsed around pulsar inferior conjunction in eclipsing RBs (e.g., 2009Sci...324.1411A; 2013arXiv1311.5161A; 2016MNRAS.459.2681B; 2018JPhCS.956a2004M). These large orbitally phase-dependent eclipses, and lack thereof at pulsar inferior conjunction, are another feature unexpected if there exists a disk outside . Similar to the orbital modulation in the persistent X-ray emission, orbital-phase-dependent eclipses are causally associated with the companion. Note that radio eclipsing BWs such as B1957+20 appear to have qualitatively different eclipses than RBs such as J1023+0038; eclipses in BWs like B1957+20 are much shorter in duration and more regular (and similarly in BW J1810+1744, 2018MNRAS.476.1968P) around superior conjunction of the MSP.

An interpretation of the above phenomenology of ICDP systems is that an intrabinary pulsar termination shock accelerates electrons that rapidly cool principally via synchrotron radiation, similar to that surmised in the SCDP-type system BW B1957+20 (1990ApJ...358..561H; 1993ApJ...403..249A; 2012ApJ...760...92H) but with a shock curving around the MSP in ICDP systems. The shock subtends a solid angle from the pulsar much larger than the companion, with the total power budget is constrained by the pulsar spin-down power erg s. It may naturally account for the X-ray energetics and large orbital radio eclipses. The double-peaked light curves are putatively generated by Doppler-boosting of the synchrotron emission in a mildly relativistic flow along the termination shock with the X-ray double-peak modulation phase centering providing a discriminant of the shock orientation (2016arXiv160603518R; 2017ApJ...839...80W). The Doppler beaming may arise from MHD-like fast magnetosonic flows (e.g., 2002AstL...28..373B; 2002MNRAS.336L..53B; 2004MNRAS.349..779K), or kinetically by anisotropic particle distributions along a shear layer (2013ApJ...766L..19L; 2017ApJ...847...90L). The level of orbital modulation ascribed to Doppler-boosted shock emission is related to the binary inclination, among other factors, inhibiting identification of low-inclination systems if the MSP is not detectable in some epochs due to transitions to disk states. ICDP systems may thus involve an intrabinary shock oriented around the pulsar and well inside the pulsar Roche lobe. In this paper, we consider the energetics and stability of this configuration. The putative termination shock stagnation point is past the point of the companion within the MSP Roche lobe, yet still well outside the pulsar light cylinder since the pulsar mechanisms are operational.

We note that since the shock radiative power cannot exceed erg s, this limits the power-law extension to a few or tens of MeV. Physically, the maximum photon energy of the power-law extension is approximately set by the unknown maximum shock-accelerated electron/positron Lorentz factor , ignoring Doppler factors of order unity. Since the shock emission is putatively synchrotron emission, the maximum energy is roughly where G and is the post-shock magnetic field, expected to be on the order of a few Gauss (Eq. (5) in 2017ApJ...839...80W). Then, provided that . Such high Lorentz factors are generally accepted in pulsar wind termination shocks (e.g., 1996ApJ...457..253D; 2017hsn..book.2159S; 2017JPhCS.932a2050K).

In this paper, we suggest two scenarios for pressure balance for a putative shock curved around the pulsar. These are delineated as asymptotic limits of the plasma parameter where , and are the local plasma rest frame number density, temperature and magnetic field in the plasma arising from the companion, respectively. The companion plasma will not be everywhere either magnetically or gas-dominated, but for practicality we consider these two limits. Clearly, in the disk state but it is unclear if gas dominance persists in the pulsar state.

We first consider a strong companion magnetosphere in §2 (hereafter Scenario ) where everywhere prior to the shock and the companion wind gas pressure play no dynamically important role for the shock. We find that a sufficiently strong companion magnetic dipole moment will yield a curved quasi-hemispherical termination shock around the MSP, regardless of the orientation of the putative dipole moment, even for anisotropic pulsar winds, provided that the MSP spin and orbital axis are parallel. Scenario is also stable insofar as the companion magnetosphere is stable and the companion mass loss rate is low.

The other limit, Scenario , is examined in §3, where mass loss from the companion provides the pressure balance for the shock formation, i.e., the magnetic field plays no dynamically important role. The formation of a shock instead of a disk imposes constraints on the character of the companion wind and mass loss, and energetic arguments suggest the wind is gravitationally captured by the MSP in this scenario, conceivably by an advection-dominated-accretion-flow-like solution (ADAF). The ADAF premise and its observational consequences are examined in §3.4. However, in isolation, such a shock-ADAF configuration is inherently unstable to gravitational influences on dynamical timescales (2001ApJ...560L..71B), therefore stabilizing mechanisms ought to exist since observations demand metastability on at least few-year timescales for Scenario . Such potential mechanisms and their observational consequences, explored in §3.5, are almost certainly predicated on self-regulation for ICDP-state systems. For the case of irradiation feedback, this is conceptually different than such feedback in LMXBs which induces mass transfer cycles; in Scenario feedback on the self-excited wind stabilizes the shock until another process causes the system to transition to or from RLOF and disk states. While the irradiation flux is lower by a few orders of magnitude than in AMXPs, so is the companion mass loss rate. Irradiation feedback on the shock may also operate in SCDP-state BWs in the context of channeled particle heating rather than by photons as noted by 2017arXiv170605467S but such systems are not the focus of this work. The issue of internal companion dynamics and its influence on long-term stability is examined in §4. Finally, we discuss potential observational discriminants of the two scenarios in §5.

## 2 Scenario β≪1: Companion Magnetosphere Dominance

We consider the curved pulsar wind termination shock geometry as arising from a stellar companion magnetosphere. A strong field whose poloidal component is of order several kilogauss at the companion surface,  kG, is demanded for isobaric surfaces curved around the MSP in this scenario, implying a pulsar termination shock with curvature similar to the isobars.

### 2.1 General Considerations

Throughout this work, we assume the companion is close to Roche Lobe filling and its radius is approximated by the volume equivalent Roche radius of 1971ARA&A...9..183P,

 RvLa=234/3(1+q)−1/3, (1)

, where is the mass ratio. Assuming that the companion dipolar component dominates any multipolar components at large distances from the companion, a kilogauss scale is readily derivable from a pressure balance condition for the magnetopause for an isotropic pulsar wind, e.g., 1990ApJ...358..561H,

 B2c8π(Rca−rs)6=⟨S⟩c∼˙ESD4πcr2s (2)

where is the characteristic shock radius as measured from the MSP, and is the pulsar wind Poynting flux far outside . We define as the minimum required surface polar field for from Eq. (2),

 B0≡a22R3c√˙ESD2c (3) ≈6×102(˙ESD1035ergs−1)1/2(Pb2×104s)−2/3 ×(Mp1.7M⊙)−1/3(1+(3q/2)11.5)G,q≫1.

As we demonstrate in §2.2.12.2.2, the companion magnetosphere scenario calls for (i.e. several kilogauss surface fields) for isobaric surfaces that are appreciably curved around the MSP in the plane of the orbit. Note the scalings of Eq. (3) with and , which necessitate larger surface fields for lower-mass companions or shorter orbital periods. The isobaric surfaces are not only relevant for RBs with ICDP light curves, but also possibly for BWs and MSPs with synchronous but small quasi-degenerate companions with magnetic fields when .

There is evidence for large, perhaps localized, kilogauss surface fields in some M dwarfs (1985ApJ...299L..47S; 2009ApJ...692..538R; 2012LRSP....9....1R), brown dwarfs (2001Natur.410..338B) and T Tauri stars (2007ApJ...664..975J), but observational constraints of RB stellar companion fields are almost non-existent. The theoretical basis for large enduring poloidal fields is also undetermined in RB companions – convective dynamos are poorly understood even in the Sun, and in contrast to isolated M dwarfs of similar mass, RB companions are anisotropically irradiated, highly evolved, bloated and optically brighter. Yet, the synchronous orbital rotation is faster than axial rotation in isolated M dwarfs, therefore it may be plausible for strong large-scale (rather than localized) fields to arise. Indeed, if the convective dynamo ultimately extracts its energy from the orbit tidally (i.e., 1992ApJ...385..621A), then a kG poloidal field may be tidally replenished on timescales s without producing a large orbital period derivative violating observations. Note that the geometry of the putative poloidal component – whether it is aligned or skew with respect to the plane formed by the orbital momentum vector and line joining the two stars – is unknown. Due to this uncertainty, we explore arbitrary orientations.

From first principles, there are strict upper bounds on the putative poloidal field component which we treat as dipolar for practicality. Firstly, since the RB companions are tidally-locked, their dipolar fields are rotating with respect to the system barycenter. Analogous to pulsar spin-down, a repercussion is “orbital dipole radiation” which imparts a secular torque on the companion orbit. A precise estimate is rather involved even in the vacuum limit (e.g., 2016MNRAS.463.1240P) and in full generality also depends on the orientation of the dipole with respect to the orbital axis.

Note that the companion light cylinder is much larger than the binary separation, cm , so the system may be regarded as in the near zone, where a dipole field structure is a good approximation, for the present motivation of termination shock curvature. Moreover, orbital sweepback of companion magnetospheric field lines in the limit may be neglected. For a simpler order-of-magnitude estimate, we invoke the Larmor formula where is the orbital angular frequency; this is generally accurate within a factor of a few in comparison with force-free and dissipative MHD models (2006ApJ...648L..51S; 2012ApJ...754L...1K; 2014ApJ...793...97K). Then, the energy loss rate can be shown to be

 ˙T=−3239(GMpc2)2B2ccq2. (4)

The concomitant characteristic timescale for orbital evolution is where is the orbital kinetic energy. This rate may be compared to measured s which is attributed to the 1992ApJ...385..621A mechanism (which incidentally also posits a convective dynamo for the strong companion field; also see text following Eq. (27)) to yield the upper limit for ,

 Bc≪108(q7)1/2(a1011cm)−1/2(τmag1015s)−1/2G. (5)

In the vacuum limit, assuming there is no wind from the companion, there is an induced electric field on the companion surface with ,

 Eind ∼ 1.4(q1+q)2/3(Bc103G)(Mp1.7M⊙)1/3 (6) ×(Pb2×104s)−1/3G.

Analogous to the canonical pulsar case (e.g., 1969ApJ...157..869G), this electrostatic force on ionized hydrogen greatly exceeds the gravitational force where is the mass of the proton,

 qeEindFg∼1010(Bc103G)(Pb2×104s)q(1+q)2/3. (7)

Therefore the companion magnetosphere is plasma loaded, possibly with active currents corresponding to a global force-free MHD equilibrium. Moreover, complex current systems will also arise since pulsar wind Poynting flux distorts the companion magnetosphere, analogous to the solar wind distorting the Earth bow shock and magnetosphere. In this scenario, transitions between different force-free equilibria would manifest as bursts similar to those in the Sun (e.g., 2006A&A...451..319R; 2008A&A...488L..71T) and also invoked for magnetar bursts (2002ApJ...574..332T), with flare emission powered by reconnection and topological changes of currents and fields.

Irradiation of the companion by the pulsar -rays and shock emission induces mass loss and also fills the magnetosphere with plasma, which is contained by the magnetosphere until . This containment timescale must be at least as long as the pulsar persistence time . Assuming the companion mass loss is similar to that inferred in the AMXP/disk states of g s and that the magnetospheric reservoir is filled at a rate where is the isolated-star escape speed, we find a lower limit for ,

 Bc≳ 2×103(Pb2×104s)−4/3(q7)1/6(τp108s)1/2 × (Mp1.7M⊙)−1/6(|˙mc|1015gs−1)1/2G. (8)

Similarly, if mass loss exists via RLOF rather than a wind, then for ion thermal speed cm s and a higher rate g s,

 Bc≳ 3×102(Pb2×104s)−1(τp108s)1/2(Mp1.7M⊙)−1/2 × (q7)1/2(|˙mc|1016.5gs−1)1/2(cs106cms−1)G. (9)

Therefore, strong fields are also essential if there is significant mass loss during the pulsar state (see §3.2 for constraints) for magnetic dominance to be sustained.

A similar-in-magnitude constraint on to Eq. (5) may be established by noting that the energy of the putative poloidal field must be a small fraction of the gravitational binding energy, . This implies

 Bc≪108(Pb2×104s)−4/3(q7)−1/3(Mp1.7M⊙)1/3G. (10)

Another restriction is that gas dominance, , is required inside the star for the convective dynamo to exist. Estimating the density as the average value, for a pure hydrogen atmosphere, the temperature at the photosphere and the magnetic field as the surface value, we find,

 Bc≪ 7×106(Tc6000K)1/2(Pb2×104s)−1 ×(βc1)−1/2G. (11)

Areas of the photosphere may attain , as in the active Sun and magnetically active stars. At the photosphere, the number density is approximately where is the optical depth, is the pressure scale height, the continuum opacity and the isolated-star surface gravity. In Thomson electron scattering limit, dominant for very high plasma temperatures (1983psen.book.....C), we have cm g and is limited to kilogauss fields when and is moderately large,

 Besc≲ 1.4×103(Pb2×104s)−2/3(Mp1.7M⊙)1/6(q7)−1/6 ×(βc1)−1/2(τ1)1/2G. (12)

If the mean opacity follows a Kramer law, , then

 BKc≲ 2×102(κ0K1023cm2g−1)−1/4(Tc6000K)9/8 ×(Pb2×104s)−1/3(Mp1.7M⊙)1/12(q7)−1/12 ×(βc1)−1/2(τ1)1/4G. (13)

For a value of typical of bound and free-free absorption (1983psen.book.....C), apparently necessary for pressure balance are generally excluded. That is, moderately hot companions with K where a Kramer law dominates ought not to be found in ICDP RBs systems in the magnetospheric scenario for pressure balance. Pressure balance in the magnetospheric scenario also advocates for much lower opacities (at temperatures below where a Kramer law operates) due to bound-free transitions and metals (2005oasp.book.....G) to be the dominant sources of continuum opacity for the temperature range of interest in RB atmospheres. We also note that for the quiet Sun, in the photosphere and chromosphere (2017ApJ...850L..29B); if similar values are realized in RB companions, then cannot attain the kilogauss fields necessary for pressure balance.

The constraints Eqs. (5), (8)–(11) along with Eq. (3) for pressure balance form an allowed region in the and parameter space for this scenario’s tenability. Far from the shock in the companion magnetosphere, for ions is fulfilled for any reasonable plasma number density provided that the ion temperature K. This is readily found from pressure balance Eq. (2), where the magnetic field at the shock scales as , prior to any modification by MHD jump conditions (e.g., 1984ApJ...283..694K). Finally, we note that the electromagnetic forces on ions dominate any gravitational influences from the pulsar, i.e. is satisfied provided that,

 rs≫ GMpmpc3/2√2˙ESDqecs∼105(˙ESD1035ergs−1)−1/2 (14) × (Mp1.7M⊙)(cs106cms−1)−1(σ10−2)−1/2cm.

which is much smaller than . Therefore, gravitational influences of the MSP are negligible on the local plasma dynamics unless the ions are unjustifiably cold. Likewise, it can be shown that Coriolis influences are negligible on the local plasma dynamics. Gravitational influences could become consequential on the macroscopic (i.e. fluid description) plasma dynamics, but it depends on the details of the MHD equilibria and currents induced in the magnetosphere.

### 2.2 Isobaric Surfaces

Beyond the generic considerations above, we now explore pressure balance of a 3D dipolar companion field by a relativistic magnetized pulsar wind, analytically described as a Poynting flux. On radial length scales much larger than , or time-averaged over timescales much longer than the pulsar period, the pulsar wind is asymptotically radially outflowing and toroidal-field dominated. A simpler isotropic case considered in §2.2.1 preludes to the more complex anisotropic case in §2.2.2.

There are several caveats to our rudimentary considerations of pressure confinement of the pulsar wind or companion magnetosphere. The isobaric surface geometry describes where the extended shock structure ought to exist, but does not appraise any backreactions on the companion magnetosphere or pulsar wind. For instance, the companion magnetosphere will be severely distorted away from the vacuum dipole towards distinct non-potential force-free MHD equilibria. The gas pressure may also be dynamically important near the companion surface, i.e. where . Moreover, the termination shock itself is a significant region of conversion of the magnetic wind into particle energy, which will influence the global shock structures. Even in the hydrodynamic limit, there is some thickness to the overall shock structure: the pulsar termination shock, followed by an astropause and contact discontinuity (e.g., 2016A&A...586A.111S). In an MHD formalism with two or more fluid species (e.g., a pair plasma interacting with an electron-ion plasma), many different wave modes may be excited leading to complex interposing shock structures (2010adma.book.....G). Such rich complexity is exhibited in relativistic MHD simulations of pulsar winds (e.g., 2005A&A...434..189B; 2018arXiv180407327B), yet pressure balance/confinement of the pulsar wind remains a credible estimate of the global structure, particularly proximate to the shock apex. The termination shock head, where the particle acceleration occurs, is largely what is relevant for ICDP light curves in RBs (2017ApJ...839...80W), rather than peripheral regions of the shock structures. Therefore, we focus on such pressure surfaces and defer global MHD simulations to future studies.

Aside from pressure balance, the orientation of the companion dipolar magnetosphere will also influence the efficiency and locales of relativistic particle acceleration in the termination shock. How is rather unclear and is deferred to future kinetic studies. Large-scale dipolar fields may also introduce peculiar orbital phase-dependent polarization character on the synchrotron ICDP light curves.

#### 2.2.1 Isotropic Pulsar Wind

The companion dipolar field component in spherical polar coordinates with origin at the companion center is given by

 (15)

with dimensionless in units of and is a polar angle. Due to the lack of symmetry of the pressure balance condition for an arbitrarily oriented dipole, it is more convenient to work in Cartesian coordinates normalized to units of . We define the and as parallel with the orbital momentum vector and line joining the two stars, respectively, with the pulsar at and companion at the origin. The implicit isobaric surface of the companion magnetosphere and pulsar wind Poynting flux is given by the 3D generalization of Eq (2) with scalar field of pressure ,

 0=G(x,y,z)=∣∣B2∣∣8π−⟨S⟩c (16)

where for an isotropic pulsar wind

 (⟨S⟩c)iso=˙ESD4πca2|rp|2 (17)

and is the outward radial vector from the pulsar. For simplicity, we also impose the condition,

 ∇G⋅rp>0 (18)

that precludes multivaluedness of the pressure surface, i.e. physically, there is a single termination shock for the putatively radial pulsar wind. Beyond the boundary imposed by this condition, the interaction geometry is indeterminate but with the radial pulsar Poynting flux dominating far from the boundary locale.

For concreteness, consider a dipole whose axis is . Then, the isobaric surface of the companion magnetosphere and MSP Poynting flux may be shown to be implicitly defined by

 Gz= b2[(x−1)2+y2+z2](x2+y2+4z2) −64(x2+y2+z2)4\lx@stackrel!=0 (19)

where . We plot Eq. (19) in Figure 2.1 with color-coded values of . For smaller values of , the isobars are curved around the companion as expected. For larger values of , there is clear curvature of isobars around the MSP particularly near the magnetopause; this depiction is analogous to Fig 1 in 2017ApJ...839...80W. These larger values of are required for significant curvature of the shock head, particularly in the plane. Such geometric curvature is central to the X-ray orbital modulation observed in RBs, and in models of such emission the observed double-peak phase separation couples to the putative shock opening angle (2017ApJ...839...80W). Even larger are not shown, as they yield a total envelopment of the pulsar and may violate Eq. (11). Such envelopment, however, could lead to prolific reconnection events and flares behind the pulsar (i.e. . Some X-ray flares (2018arXiv180900215C) and mini radio eclipses of the MSP at pulsar inferior conjunction are observed in a some RBs (e.g., 2015ApJ...800L..12R), but not contemporaneously in the same system.

Note that in the peculiar case of Eq. (19), there is symmetry about the and axes. For a dipole with axis along , the surfaces (not shown) even exhibit azimuthal symmetry about . In general, there are no such symmetries for the isobaric surface for an arbitrarily oriented dipolar field for even an isotropic pulsar wind. One such skewed-dipole illustrative case is depicted in Figure 2.2.1. Close to the stagnation point when , the head of the shock region is approximately hemispherical, a consequence of the isotropic pulsar wind considered in this section. However, at locales of the boundaries defined by Eq. (18), there are clear asymmetries. Such asymmetries may account for the small phase offset from IC in some ICDP systems as well as apparent asymmetries about IC in ICDP light curves. This forms an alternative scenario to Coriolis effects of a companion wind invoked in the past (2016arXiv160603518R; 2017ApJ...839...80W) and for Scenario .

#### 2.2.2 Anisotropic Pulsar Wind

Soon after the discovery of pulsars, the Poynting flux of pulsar winds were widely recognized to be anisotropic (1969ApJ...158..727M; 1973ApJ...180..207M) and plasma-loaded (1969ApJ...157..869G). In the force-free limit of a plasma-filled magnetosphere, the anisotropy of the pulsar wind Poynting flux is contingent on the magnetic obliquity of the rotator and roughly varies between (aligned rotator) and to (orthogonal rotator) (1999A&A...349.1017B). Here, is the polar angle with respect to the spin axis, i.e., . Moreover, for the orbital scales of interest in BWs and RBs, the azimuthal anisotropies on the scale of may be neglected (i.e. we restrict to the far zone). For simplicity, we consider the pulsar spin axis aligned with orbital axis . Such alignment is expected from the formation/evolution recycling scenario for MSPs, as hinted for RB J2215+5135 and other MSPs (2014MNRAS.439.2033G; 2014ApJS..213....6J), and as known for other stellar contexts (2007A&A...474..565A; 2011MNRAS.413L..71W).

Using a sample of force-free MHD simulations which are appropriate for the gross global structure of the pulsar wind, 2016MNRAS.457.3384T analytically parameterized the dependence of asymptotic magnetized pulsar winds far outside as a sum of the wind structure of aligned and orthogonal rotators. Their semi-analytic construction is accurate to within to simulations for the differential Poynting flux averaged over azimuthal angles. From their prescription, we obtain a convenient expression of the azimuthally-averaged anisotropic differential (in solid angle) Poynting flux in terms of the observable after a modicum of algebra,

 (⟨S⟩c)aniso=2˙ESD4πca2|rp|2⟨B2⟩ϕC0sin2ϑ (20)

where

 ⟨B2⟩ϕ=12π∫2π0B(α,ϕ)2dϕ (21)

and

 B≈ B∥+B⊥ (22) B∥= [1+0.02sinγ+0.22(|cosγ|−1) −0.07(|cosγ|−1)4]|1−2α/π|sgn(cosγ) B⊥= (1+0.17|sin2α|−|1−2α/π|) ×sinϑcos(ϕ−π6).

Here, is related to the radial magnetic field in 2016MNRAS.457.3384T, with the azimuthal angle average, and the magnetic colatitude, . The constant is of order unity and normalizes the total Poynting flux integrated over solid angles,

 C0=∫π0⟨B2⟩ϕsin3ϑdϑ. (23)

Numerically, when , respectively.

The form of Eq. (20) allows for analogous nondimensionalization of the pressure balance condition Eq. (16) as Eq. (19) after some algebra. The condition Eq. (18) is more involved because of spatial derivatives of numerical integrals and is computed semi-analytically. Then, computation of implicit isobaric surfaces follows routinely. We do not consider the case, as the neutron stars in RBs are pulsars.

Figure 2.2.1 depicts computed isobaric surfaces for , comparable to the low magnetic obliquity inferred for RB J2215+5135 (2014ApJS..213....6J), and with an arbitrarily skewed companion dipole moment. There are several intriguing features worth highlighting, in comparison to the isotropic pulsar wind cases explored in §2.2.1. Firstly, for low values of , the isobaric surfaces are largely similar in form, since anisotropies are less pronounced in the plane of the orbit where . For larger values of , there is a dramatic shift in the topology of the surfaces principally due to the factor in Eq. (20) which guarantees a region of very low wind pressure along the spin axis . This leads to the pronounced “spin axis funnels”, some of which are disjointed from the shock head and tail in the plane owing to the condition Eq. (18). Moreover, for a skewed companion dipole moment, there are regimes of moderate where the spin axis funnels are only partially disjointed. Clearly, regimes may be also realized where is critical between a joined and unjoined topology. In this critical regime, magnetic reconnection and transient phenomena ought to be prolific.

In Figure 2.2.1, we depict computed isobaric surfaces with skewed companion dipole moment identical to that as Figure 2.2.1 but with . The topology of these surfaces is largely indistinguishable to the case, particularly for the shock head when which putatively governs the X-ray orbital modulation, implying an insensitivity of isobars with large disparities of pulsar . Indeed, there is negligible variance in the plane of the two cases where . Yet, the spin axis funnel for exhibits a much wider opening angle, due to the stronger anisotropy of the pulsar wind. This is suggestive that sporadic accretion may be easier for more orthogonal rotators.

For lower values of , it may be argued that these “spin axis funnels” are entirely spurious since they are disjointed from the principal isobaric surfaces near the companion and therefore current closure (in the force-free limit) is inhibited. Likewise, for larger values of , the funnels will be disrupted by the reflected back-flowing pulsar wind from the termination shock. Relativistic MHD simulations, and possibly kinetic ones as well, are required to assess the character of the funnels and shock structures as varies. Yet, relativistic MHD simulations of anisotropic pulsar wind shock interactions (e.g., 2004MNRAS.349..779K; 2018arXiv180407327B) indicate some reality to the funnel-like structures along the pulsar spin axis, as suggested by observations of the Crab plerion (2000ApJ...536L..81W; 2017hsn..book.2159S; 2017JPhCS.932a2050K). Speculatively, for larger values of , the surfaces are connected implying threading of the companion magnetosphere into the funnel which may be paths for sporadic accretion onto one or both poles of the MSP initiated by transitions of different force-free field configurations of the companion. Indeed, joined funnels may play a role in recently observed enhanced spin-down torques on J1023+0038 in a AMXP state (2016ApJ...830..122J) where the assumption of breaks down close to the MSP.

## 3 Scenario β≫1: Quasi-Hemispherical Gravitational Capture of Companion Mass Loss by the Pulsar

State transitions of some RBs to AMXP-like accretion disk states implies efficient angular momentum transport of the companion mass loss. Such disk states are evidently regimes of for the companion mass loss. Therefore, we are motivated to examine whether companion mass loss without dynamical influences of magnetic fields may yield a stable shock curved around the pulsar rather than a disk in the rotation-powered state. Without gravitational influences of the MSP, it is generally accepted that the companion wind overpowering the MSP wind is energetically untenable on long timescales. However, we suggest that if there is sufficient angular momentum loss of the companion wind, a shock curved around the MSP may be attainable. The stability of such a putative configuration is questionable, and we explore mechanisms that may provide stability.

Here, we assume the donor is near but not entirely Roche lobe filling in the rotational-powered state so that high mass loss rates g s are attainable without a disk as in conventional RLOF. The relatively high mass loss rates required by this scenario currently do not violate any observational constraints (see 3.2).

In this Section, we show that for evaporative winds from the companion, the existence of a shock implies a lower bound on the companion mass. This is a rather general result if angular momentum loss of the companion wind occurs far from the launching point which is putatively near the companion photosphere.

In the absence of a strong companion magnetosphere, a stipulation for a shock to exist bowed around the pulsar rather than a disk is that the wind characteristic circularization radius be small compared to the characteristic shock radius , for a companion wind with speed . The circularization radius is defined by where the specific angular momentum at the accretion radius is equal to that for a Keplerian orbit at radius , i.e. where is the orbital angular speed of the system and the MSP mass (1976ApJ...204..555S; 2002apa..book.....F). Therefore, is the lengthscale within which one may expect a disk to exist. This definition exhibits a strong scaling on the wind speed ,

 rcirca≈116(racca)4(1+qq)≈(vorbvw)8(1+qq)5, (24)

where is the orbital speed of the secondary. The ratio is the characteristic Rossby number of the secondary’s wind.

Parametrizing the stellar wind as a scaling of the isolated-star escape speed, , and casting the secondary stellar radius as a fraction of the characteristic volumetric Roche radius from Eq. (1), one arrives at being a simple function of and ratio ,

 rcirca≈3×10−3μ4q3(1+q)1/3. (25)

The typical thermal speed is cm s for K while an irradiation-induced evaporative wind speed may be on the order of the escape speed of the companion cm s for a typical RB secondary of mass and radius cm. This is an upper limit to , since for a star near the Roche limit may be substantially lower owing to the low potential barrier. Coincidentally, is also on the order of the escape speed from the entire system or the orbital speed of the companion.

Requiring for , since the putative shock exists past this point, then implies which clearly excludes some lower-mass RB companions; therefore this calls for or . This restriction is not very constraining due to the strong fourth-power dependence of in Eq. (25), requiring only a modest few to render . For instance, yields the constraint on the mass ratio . Irrespective of the actual balance of ram pressures in Eq. (31) below, the circularization constraint favors RBs () over more extreme-mass-ratio BWs for the existence of a shock enshrouding the pulsar. If the shock is constrained by other means, e.g., cooling breaks in hard X-rays, then an independent upper limit on is derivable.

### 3.2 Constraints on the Companion Mass Loss Rates

The mass loss rate from the companion intrinsically couples to and scenarios governing stability, as well as Eq. (8)-(9) for the magnetospheric scenario. Therefore, we briefly summarize constraints on the companion mass loss rates in BWs and RBs, which are generally much lower than that of Eddington-scale LMXBs.

The existence of isolated recycled radio MSPs above the pulsar death-line suggests that time-averaged mass loss rates could be substantial, of order Gyr g s, in many BWs and RBs if their evolutionary scenarios are similar. If near the shock and assuming the wind is gravitationally captured (cf. §3.3), then the condition with and yields,

 ˙mg≪˙ESDvorbc∼1017gs−1 (26)

not an implausible bound for typical RB parameters. Here, we define mass rate participating in the shock as a non-negligible fraction of the total companion mass loss rate . Alternatively, one can constrain the total mass loss rate energetics of evaporation, (1988Natur.334..227V; 1992MNRAS.254P..19S) where is the solid angle fraction of pulsar wind intercepted by the companion. For typical RB parameters this yields the upper bound g s.

Additionally, from the form of the total binary angular momentum and Kepler III, one may show that in the no-accretion shock scenario when ,

 ˙PbPb=3˙JJ+(−˙mc)mc(3−11+q) (27)

for idealized point masses (1924MNRAS..85....2J). Measurement of orbital period derivatives in BWs and RBs by timing the MSP pulsations in the radio or -rays yield erratic and often negative values of order s rather than secular changes expected from conservative () mass loss. The dominance of these nonsecular changes is interpreted in the 1992ApJ...385..621A framework, with the companion’s gravitational quadrupole moment changing due to a magnetically active convection in the companion outer layers or activity cycles, with a significant portion () of the companion mass possibly asynchronous. There is some evidence for such changing gravitational quadruple moments in B1957+20 (1994ApJ...436..312A), BW J2051-0827 (1998ApJ...499L.183S; 2001A&A...379..579D; 2011MNRAS.414.3134L; 2016MNRAS.462.1029S), RB J2339-0533 (2015ApJ...807...18P), and other MSPs (2018ApJS..235...37A) suggesting that mass loss is lower than the simple Jeans formulation g s. Anisotropic pulsar emission, well-motivated theoretically and observationally in the framework of offset dipoles (e.g., 1996A&AS..120C..49A; 2011ApJ...743..181H; 2015ApJ...807..130V; 2016ApJ...832..107B), may also cause quasi-cyclic wandering of residuals (1975ApJ...201..447H, Eq. (76)). Therefore, these measurements constitute an upper limit for the mass loss rate.

Likewise, utilizing Eq. (1) and again imposing , one may show that,

 12˙RLvRLv=˙JJ+(−˙mc)3mc(52−11+q). (28)

If and since there exists a critical mass loss rate such that , i.e. where the Roche potential radius switches between expansion and contraction. For a companion nearly filling its Roche lobe, a contracting Roche potential will drive mass loss towards the critical rate. Contrastingly, Roche radius expansion is only tenable via irradiation or ablation-driven mass loss beyond the critical rate when . Secular gravitational wave angular momentum loss, (1975ctf..book.....L) specifies a minimum critical mass loss rate ,

 |˙mGWc,crit|≈192G3M4p(1+q)25c5a4q3(3+5q) (29) ∼2×1015(Mp1.7M⊙)4(1011cma)4(7q)2gs−1

for , similar to the time-averaged Gyr evaporative rate.

Finally, a rudimentary lower limit may be estimated from radio eclipses of the radio MSP in BWs and RBs. After correcting for interstellar dispersion at uneclipsed orbital phases, excess delays near pulsar superior conjunction consistent with plasma dispersion generally imply the average dispersive free electron column density rises sharply from to cm, before total loss of radio emission in the eclipse (e.g. 1991ApJ...380..557R; 2001MNRAS.321..576S; 2009Sci...324.1411A; 2013arXiv1311.5161A; 2018MNRAS.476.1968P; 2018Natur.557..522M; 2018JPhCS.956a2004M), for the line-of-sight distance through the plasma. In the absence of any clumping, e.g., at the shock, and , this implies an isotropic mass loss rate for an ionization fraction . That is,

 |˙mc|≳ 1013X−1√F3λ(ΔDM2×1018cm−2) ×(Mp1.7M⊙)2/3(Pb2×104s)1/3gs−1. (30)

If the eclipse radius, which is a significant fraction of , is utilized rather than , then the bound for is larger by a factor (1994ApJ...422..304T). Long-term variations of the deepness of eclipses may be used as a proxy for variations in or the mass loss rate.

### 3.3 Gravitational Influences and Wind Angular Momentum Loss

As we discuss below, angular momentum loss of the companion wind is energetically essential for the ICDP shock state. The locale of such wind angular momentum loss is unknown. If it transpires far from the companion, the circularization radius constraints of §3.1 on the wind remain pertinent.

The stagnation point balancing the ram pressure of the isotropic and supersonic two-wind interaction is given by

 ˙ESD4πcr2s=|˙mc|vw4π(a−rs)2. (31)

This implies the well-known stagnation point formula in terms of the ratio of wind ram pressures (1990ApJ...358..561H),

 rsa=√Aw1+√Aw,Aw≡˙ESD/c|˙mc|vw. (32)

Anisotropic winds, as in §2.2.2, modify these expressions but not the following general conclusions which are pertinent to the shock nose. Using Eq. (32) at or , corresponding to the threshold of the shock orientation enshrouding the pulsar rather than the secondary, yields a lower limit on ,

 vw≳108.5(˙ESD1035ergs−1)(1016gs−1|˙mc|)cms−1, (33)

exceeding the typical evaporative anticipated from an RB by at least an order of magnitude. Moreover, any lower than the high value used above yields untenably larger values. There are also issues with energetics, with Eq. (33) implying , clearly unjustifiable in an MSP self-excited wind scenario. Therefore the companion wind requires an additional reservoir of energy to tap in order for the shock to wrap around the pulsar. This contrasts the situation of high-mass pulsar X-ray binaries where the massive companion wind readily dominates the energetics.

In the scenario, a resolution to the apparent contradiction of Eq. (33) is the influence of gravity of the MSP and angular momentum losses of the companion wind near or upstream of the shock. Two effects scale the ram pressure in quasi-spherical radial infall: a density enhancement nearer to the pulsar and Keplerian scaling of the fluid speed for the gravitationally-influenced mass rate participating in the shock . Accordingly by pressure balance, the shock stand-off scales as .

Viscosity and heating of the companion wind somewhere within the pulsar Roche lobe (with ) is a critical requirement for angular momentum losses in the flow. As in accretion disks, turbulent viscosity is a possible dissipative mechanism. The viscous timescale then ought to be comparable to the free-fall dynamical timescale s. This implies a kinematic viscosity of order