EDP Sciences
Free Access
Issue
A&A
Volume 592, August 2016
Article Number A52
Number of page(s) 8
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/201628116
Published online 22 July 2016

© ESO, 2016

1. Introduction

Pulsars are highly magnetized, rapidly rotating neutron stars (NS) that lose their energy principally via electromagnetic cooling, which results in their spin-down. The pulsar outflow comprises a low-frequency (ω = 2π/P, with P the pulsar spin period) Poynting flux-dominated component (also called the pulsar wave), a relativistic wind, and high-energy radiation from the magnetosphere. The energy of the outflow is believed to be dominated by Poynting flux close to the star, and by relativistic particles farther out (as observed in particular in the case of the Crab pulsar). However, the location of this transition remains a puzzle to the community (see, e.g., Kirk et al. 2009, for a review).

Pulsars are often observed to evolve in multiple systems, and evaluating the amount of energy absorbed by the companions is paramount to understanding their evolution and emission. The most common companions are white dwarfs (Savonije 1987; Manchester et al. 2005; Ransom et al. 2014) or M-dwarfs (Roberts 2013), but smaller objects such as planets, asteroids, or comets can also orbit pulsars (Wolszczan & Frail 1992; Thorsett et al. 1993; Bailes et al. 2011). The existence of asteroid belts around pulsars has been invoked by several authors to explain timing irregularities (Shannon et al. 2013), anti-glitches (Huang & Geng 2014), or burst intermittency (Cordes & Shannon 2008; Deneva et al. 2009; Mottez et al. 2013b).

Asteroids around pulsars could also be at the origin of fast radio bursts (FRBs). These brief radio signals (typically 5 ms at a given frequency) are dispersed in frequency (as for pulsar signals), but with a dispersion corresponding to cosmological distances (e.g., Champion et al. 2016, and references therein). Going against the mainstream scenarios involving the collapse of massive objects, Mottez & Zarka (2014) proposed that FRBs could be emitted by bodies orbiting extragalactic pulsars (see also Dai et al. 2016). Highly collimated waves would be produced by the magnetic wake of these objects immersed in the relativistic pulsar wind. The collimation would enable their detection even at distances of hundreds of Mpc. This model can naturally produce the repeating bursts reported by Spitler et al. (2016) in the presence of an asteroid belt.

For massive companions, the standard scenario stipulates that old neutron stars are spun up to periods <10 ms in close binary systems by transfer of mass and angular momentum by the low-mass companion (Alpar et al. 1982). This accretion process leads to the production of X-rays detected as low-mass X-ray binaries. In particular, the so-called black widow (dwarf companion with mass m ~ 0.002−0.07 M) and redback (M-dwarf companion with mass m ~ 0.1−0.4 M) pulsars illustrate the importance of energy absorption by companions on the evolution of binary systems. These systems have very low-mass secondaries and orbital periods of less than 10 h. Early studies pointed out that these pulsars should theoretically evaporate their companion by high-energy radiation (Ruderman et al. 1989) and/or by Roche lobe overflow. At the time of their discovery, it was assumed that black widows provided the missing link between low-mass X-ray binaries and isolated millisecond pulsars.

Since pulsar environments can evaporate white dwarves and M-stars orbiting them, it may seem surprising that smaller planets and asteroids could be maintained in a solid state at similar distances to the neutron star. In the present paper, we propose an explanation for their survival.

To date, it has been confirmed that two pulsars (PSR B1257+12 and PSR B1620-26) host planets. PSR B1257+12 hosts three planets at distances on the order of the astronomical unit (in the pulsar wind) with orbital periods in the range of days and weeks (Wolszczan & Frail 1992). The distances are therefore greater than in close binary systems (orbital periods in the range of hours) such as black widow and redback systems, which can provide an explanation to the stability of the planets. PSR B1620-26 is a neutron star-white dwarf binary, that was confirmed to host a Jovian mass companion orbiting at 23 AU (Sigurdsson et al. 2003). The distant orbit makes this system ill-suited for our framework, and could explain by itself why the companion has not been evaporated. On the other hand and if confirmed, the suspected belt of asteroids at a small distance from PSR 1931+24, with an orbital period ranging in minutes (Cordes & Shannon 2008; Deneva et al. 2009; Mottez et al. 2013a,b) is puzzling and would require an alternative justification for its survival. PSR J1719-1438 presents an orbiting candidate planet (Bailes et al. 2011). Its Jovian mass companion has a short orbital period of ~2.2 h, implying a compact orbital distance of ~0.95 R, and a minimum mean density of 23.3 g cm-3. This density suggests that it may be an ultra-low-mass carbon white dwarf rather than a planet. The evaporation timescale of this companion should also be investigated.

At distances within a few light-cylinder radii from the neutron star, there is a general agreement that most of the energy outflowing from the pulsar is under the form of the Poynting flux associated with the pulsar wave (Kirk et al. 2009). The pulsar wave is the electromagnetic structure created by the rotation of the neutron star magnetic field at the pulsar spin frequency (with an inclination angle i ≠ 0 relative to the spin axis). Outside the light-cylinder, this corresponds to a low-frequency spherical wave propagating at a velocity ~c and of wavelength equal to two light-cylinder radii (Deutsch 1955). On the scale of a companion radius, this wave can be considered a plane wave. Because the pulsar wave constitutes the main flux of energy out of the pulsar, we consider that it is potentially the main source of heat for pulsar companions. In the remaining sections, we investigate the physical parameters that control the efficiency of the pulsar companion heating by the pulsar wave.

2. Companion heating by induction: model

All numerical quantities are denoted QxQ/ 10x in cgs units unless specified otherwise.

2.1. Energy flux of the pulsar wind

The energy loss rate (or luminosity) of a pulsar with moment of inertia I, rotation period P, radius R, dipole magnetic field strength B, and corresponding period derivative reads (e.g., Shapiro & Teukolsky 1983) (1)The energy flux in the pulsar wind at distance a, large compared to the pulsar light cylinder radius, RL = cP/ (2π) ~ 4.8 × 108 cm P-3, can then be written (Arons & Tavani 1993) (2)where fp = ΔΩp/ (4π) is the fraction of the sky into which the pulsar wind is emitted. We note that this flux can also be expressed as Fw = cB2(1 + σB)/ (4πσB); the magnetization parameter is defined as σBB2/ [4πc2(nimi + κ±m±n±)], i.e., the ratio of the magnetic energy flux to the kinetic energy flux in the comoving wind frame. Here B, ni, mi, and n± and m± are respectively the magnetic field strength, the number density, the mass of ions, and pairs of the cold plasma, and κ± the pair multiplicity. In this work, we assume that the wind is Poynting-flux dominated (σB ≫ 1) in the region where the companion is located. It can be demonstrated that this is valid out to the termination shock for a supersonic, radially expanding ideal magneto-hydrodynamics (MHD) wind. In more realistic situations, the dissipation to kinetic energy could happen earlier, but all models predict a Poynting-flux dominated wind close to the star (e.g., Kirk et al. 2009).

The companion at a distance a can intercept a fraction f of this flux, provided that it falls in the wind beam and absorbs a flux, Fabs = fQabsFw, which can be written in terms of absorbed luminosity: (3)Here, Qabs is the energy absorption efficiency. Using the magnetization parameter σ defined above, this quantity can be decomposed into two components: the absorption of energy from the pulsar wave, characterized by an efficiency Qem, and the absorption of kinetic energy from the pulsar wind, of efficiency Qkin, (4)The next two sections are devoted to the evaluation of Qem ~ Qabs for a Poynting-flux dominated case when σB ≫ 1.

2.2. Absorption properties of a body in an electromagnetic wave

The absorption properties of a spherical body in an electromagnetic wave can be modeled by the Mie theory (van de Hulst 1981). Two parameters govern the regime in which absorption or scattering occurs: the ratio of the size of the body to the incident wavelength (5)and the complex refractive index of the medium N = Nr + Ni, where Nr and Ni are real.

2.2.1. Refractive index of the companion

Companions orbiting a 10 ms pulsar at a distance a = 1 R see an alternating magnetic field of amplitude B ≤ 10 G with a 100 Hz frequency1. The electric field induced by the pulsar wind (v ~ c) is E ≤ 105 V/m. This is less than the electric strength encountered in most materials (larger than 106 V/m). Therefore, the companions are not ionized by the electric field induced by the pulsar wave. Similar conditions are met for companions orbiting standard pulsars (P ~ 1 s) beyond the light cylinder. Therefore, the usual approximations and textbook data can be used to estimate the electromagnetic properties of the constituents of pulsar companions.

The refractive index can be calculated using the Maxwell equations and the material equation J = σE where J is the current density and σ is a scalar conductivity (Born & Wolf 1980). The dispersion equation reads (6)where ω = 2π/P, k is the wave number, and ϵ and μ the dielectric permittivity and magnetic permeability of the medium, respectively. The refractive index N defined by N2k2c2/ω2 can be written (7)Pulsars rarely spin faster than with a millisecond period, thus ω ≲ 6 × 103 s-1 (a strict lower bound on the spin is given by Haensel et al. 1999). For low electromagnetic frequencies (up to infrared frequencies), σ is real in any kind of material. For companions made of degenerate matter, the electrical conductivity can be estimated as σ ~ 1024 s-1ρ2/ 3T-1, for densities ranging from ρ ~ 106−12 g cm-3, and T the stellar temperature (Canuto & Solinger 1970). White dwarfs typically have ρ ~ 109 g cm-3 and T ~ 104−7 K. For T = 105 K, σ ~ 1022 s-1. To date, the planets orbiting pulsars that have been discovered could be either low-mass white dwarfs made of degenerate matter, or made of ordinary matter. For T = 105 K, σ ~ 1025 s-1. Smaller objects such as asteroids could be modeled as silicate rock or as ferrous metals. The conductivity of sea water is on the order of σ ~ 10 Mho m-1 ~ 1010 s-1. In metals, σ ~ 106−107 Mho m-1 ~ 1017 s-1. For solid rocks in the terrestrial crust, σ ~ 10-3 Mho m-1 ~ 106 s-1, but in some conductive belts (for instance between the African Rift and Namibia) σ ~ 10-1 Mho m-1 (Gough 1989).

In all materials, the permittivity ϵ is of order unity, rarely exceeding 100. For instance, water-free iron ore has ϵ ~ 5 and iron oxides ϵ ~ 2.

Given the above estimates, one can neglect the real part of N2 in our framework. The real and imaginary parts of N can then be extracted from Eq. (7): (8)For example, for a companion made of iron (or any other ferromagnetic material), μ> 103 and Nr ≫ 103. For solid rocks in the terrestrial crust, μ ~ 1 and Nr ~ 103 for a P = 1 s pulsar and Nr ~ 30 for a P = 1 ms pulsar.

For white dwarfs, the very large conductivity σ implies Nr ≫ 106.

thumbnail Fig. 1

Absorption coefficient Qabs as a function of the size ratio x = R/cP and of the refractive index Nr of the companion in the Metallic regime (N = Nr = −Ni). Overlayed are the contours of the analytical approximation, in black for Region 1 (xNr ≪ 1, Eq. (10)) and in white for Region 2 (xNr ≫ 1, Eq. (9)). The gray-shaded region corresponds to parameters outside the confidence range of the computation with Damie.

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2.2.2. Absorption coefficient

Here we compute the absorption coefficient Qabs, defined in Eq. (3) as the proportion of energy absorbed by the companion, relative to the flux of incident energy of the wave through the section of the companion. We perform a numerical computation of Qabs using a version of the Damie code, based on Lentz (1976)2. We have tested the program with real and complex values of N2, and compared the results with analytical approximations given in van de Hulst (1981) for various regions of the (x,N) parameter space. We find that numerical and analytical methods give consistent values within an error of <21%. This level of agreement is sufficient for our purpose as we are mostly interested in order-of-magnitude estimates, given the uncertainties of our other parameters.

Figure 1 shows the value of the absorption coefficient computed with Damie as a function of the size parameter x = R/ (cP) and of the refractive index N, in the so-called Metallic regime, where N = Nr = −Ni for values of N larger than one.

As expected, for large objects (x ≳ 1), Qabs ~ 1. This means that all the flux received from the pulsar wave is absorbed. The values x ≳ 1 correspond to planet-sized and white dwarf-sized objects (R ≥ 1000 km) in a millisecond pulsar wind (cP = 3000 km for P = 10 ms). For such objects, taking into account the Mie theory does not modify the heating rate.

On the other hand, for smaller objects with x< 1, the level of absorption is reduced by orders of magnitude. For instance, for a kilometer-sized asteroid orbiting a standard pulsar (P = 1 s), x = 3 × 10-6. If it is made of rocks, Nr ~ 30, and it can be read Qabs ~ 10-8. A value Qabs ~ 10-12 is even reached when Nr ~ 103.

For x ≪ 1, van de Hulst (1981) gives an analytical expression for the absorption coefficient (we note, however, the corrected typo in the numerator) (9)The absorption coefficient computed with Eq. (9) is plotted in Fig. 1 (white contours). For small values of x, this approximation is numerically unstable. An approximation for the parameter space where Nrx ≪ 1 called “Region 1” was proposed (van de Hulst 1981): (10)As shown in Fig. 1 (black contours), this approximation is valid for the smallest values of Nrx. The domain corresponding to xNr ≥ 1 was denoted “Region 2” by van de Hulst (1981). For xNr ≫ 1, Eq. (9) can be approximated as (11)Region 2 corresponds roughly to planets and white dwarfs, while Region 1 corresponds to planetesimals, asteroids, and comets3.

White dwarfs, with Nr ≫ 106, are hence highly reflective bodies, as confirmed by the Mie theory. Indeed, Qabs → 0 (Eq. (11)). If the pulsar outflow is under the form of a pulsar wave while it reaches the white dwarf, it should not absorb any energy and should not evaporate. This goes against the existence of black widow systems. It thus seems likely that the outflow is not Poynting-flux dominated as it reaches the white dwarf companion. One possible scenario is that the pulsar wave energy is dissipated by plasma heating, in the dense plasma surrounding the evaporating white dwarf.

3. Effects on the evaporation of companions

We discuss in this section the impact of Qabs on the evaporation rate of pulsar companions. A simple energy balance gives the evaporation timescale of the companions as (12)where tev is the minimum time of evaporation, Eabs is the total energy absorbed at time t, Erad the total radiated energy, Eg the gravitational binding energy, and Ec the cohesive energy4. Two regimes will be distinguished in this paper. For massive, degenerate companions the cohesive energy can be neglected (Sect. 3.1), while for low-mass companions it dominates (Sect. 3.2). Gravitational and cohesive energies are of the same order of magnitude for bodies about the mass of the Earth. In both cases, we neglect the radiative cooling, but we will include it in a more refined treatment in a future paper.

3.1. Ablation and evaporation of massive companions (degenerate and non-degenerate)

For massive companions, the dominant process for evaporation is the gravitational escape of the gas. The evaporation timescale, tev, results from the balance between the energy flux input from the pulsar wave and the rate of gravitational escape of the heated material (van den Heuvel & van Paradijs 1988). The luminosity available to drive the wind of a companion of mass M and radius R located at a distance a from the pulsar is then (1/ 2)v2 = Qabs(f/fp)(R/ 2a)2Lp. Assuming that the wind velocity is on the order of the escape velocity v = (2GM/R)1/ 2 from the surface of the companion, the gravitational binding energy of the companion reads (13)and the evaporation timescale For low values of M when the star is degenerate, the mass-radius relation follows roughly R/R = 0.013(1 + X)5/ 3(M/M)− 1/ 3, where X is the hydrogen fraction; the evaporation time can then be expressed(16)The spin-down luminosity is on the order of Lp ~ 1035erg s-1 for a typical millisecond pulsar with R = 10 km, P = 10 ms, B = 108 G, and on the order of Lp ~ 1031erg s-1 for a standard pulsar with P = 1000 ms, B = 1012 G (see Eq. (1)).

Table 1

Gravitational (tev,g) and cohesive (tev,c) evaporation timescales due to induction heating for various pulsar and companion parameters.

This equation is analogous to those in van den Heuvel & van Paradijs (1988). These estimates can be applied to PSR B1957+20, a benchmark black widow system suspected of ablating its companion (e.g., Phinney et al. 1988; Kluzniak et al. 1988; Arons & Tavani 1993; Callanan et al. 1995; Khechinashvili et al. 2000; Stappers et al. 2003; Reynolds et al. 2007; van Kerkwijk et al. 2011). PSR B1957+20 has a period of P = 1.61 ms, and its dwarf companion of mass M = 0.021 M orbits at a distance a = 2.1 R (Porb = 9.2 h), hence yr. However, as argued at the end of Sect. 2.2.1, the high refractive index of white dwarfs implies that Qabs ≪ 1, thus tev,WD ≫ 106yr. The conclusions derived in van den Heuvel & van Paradijs (1988), namely that the evaporation of the companion (strongly dependent on R and X) takes only a few million years, remain valid only if the absorbed energy is principally kinetic (Eq. (4)). The same conclusions apply for PSR J1719-1438 (period P = 5.7 ms), if its companion is indeed an ultra-low-mass carbon white dwarf with M = 0.015 × 10-3M, R = 4.2 × 104 km, and a = 0.95 R.

For planet companions made of ordinary matter, the mass/radius relationship simply reads M = (4π/ 3)R3ρ, where ρ is the average density. The evaporation time can then be expressed (17)This equation can be applied to the planets orbiting PSR 1257+12. The results are given in Table 1 for two values of the refractive index Nr that correspond to planets made of rocks and to more metallic planets. We find that the closest planet can survive a few million years. The two other planets can reach a billion years. This estimate neglects the thermal energy radiated by the planet; therefore, the evaporation time through Poynting flux absorption may be underestimated.

If the companion of PSR J1719-1438 is actually a non-degenerate planet with minimum mean density ρ = 23.3 g cm-3, then x ≫ 1, but with Nr ≲ 100, implying Qabs ≳ 10-2 and for f = fp = 1, which is compatible with the observation of the companion.

3.2. Evaporation of small non-degenerate companions

In the case of small objects like asteroids, Eq. (17) fails because the dominant process for evaporating the body is no longer gravitational escape but heating, melting, and evaporation.The cohesive energy can be expressed as (18)with K the cohesive factor that depends on the material (Kittel 1998). In particular, the cohesive factor of iron is Kiron = 7.4 × 1010 erg/g (and the orders of magnitude are similar for rock).

Equating with Eg, the cohesive energy starts to dominate for a mass-to-size ratio of M/R ≲ 1018 g cm-1. For ordinary matter, this constraint can then be expressed in terms of companion size as (19)In this regime, the evaporation timescale can be calculated (20)which leads to (21)For small objects, and when the pulsar energy is mostly in the form of Poynting flux, Qabs is low and has a strong influence on the evaporation time. The comparison of tev,g and tev,c shows, as expected, that when small bodies have melted and vaporized, the gravitational escape that follows is very fast.

thumbnail Fig. 2

Contours of evaporation timescales (in yrs) of non-degenerate pulsar companions (tev,g and tev,c, Eqs. (17) and (21)) as a function of the companion radius R and its distance a to the pulsar in solar radii, for pulsar spin periods and magnetic fields as indicated (left to right: P = 1,10,100 ms and B = 108,108,1012 G), for conductivity times magnetic permeability σμ = 106s-1 and density ρ = 1 g cm-3. The white dotted line delimits the region where the cohesive energy dominates over gravitational energy (for small companions) for evaporation. The white region indicates evaporation times tev< 10 yr and yellow regions tev> 1012 yr.

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Figure 2 shows the evaporation timescales of non-degenerate pulsar companions (tev,g and tev,c, Eqs. (17) and (21)), for three spin periods. For small companions, on the left-hand side of the white dotted line, the cohesive energy dominates over gravitational energy for evaporation. The green contours indicate evaporation timescales shorter than ~106 yr. A few typical examples are also given in Table 1 for companions made of rock (Cols. 8, 9) and for a more metallic composition (Cols. 10, 11).

The table shows that small non-degenerate bodies at one solar radius from a 1 ms pulsar evaporate in less than a few years. We note that for P = 1–100 ms and Nr ~ 100, the fastest evaporation occurs for R ~ 100 km, and not for the smallest companions. We also note also the strong dependency of the evaporation time on the pulsar period P. An asteroid at the same distance of a 10 ms pulsar can survive a few 107 yr. At 1 AU (200 R), it is definitively stable.

Figure 3 demonstrates that Nr, namely the chemical composition of the companion, has a strong influence on the evaporation timescale. An asteroid (R ~ 1 km) made of rock (Nr ~ 100 or μσ = 106 s-1) has an optimal lifetime. More resistive (Nr ~ 10), or more metallic asteroids (Nr ≳ 103) evaporate over a shorter timescale. For Nr ≲ 100, the distance a remains constant for small size ratios x (black line in Fig. 3). This stems from the change in the dependency of Qabs over x (and thus over R), as shown in Fig. 1 for small x and low Nr.

The comparison between Cols. 8, 9 and 10, 11 of Table 1 confirms that kilometer-size companions are more stable against evaporation when made of rock (μσ = 106 s-1) than for a more metallic refractive index μσ = 107 s-1. On the other hand, the table and Fig. 2 show that for R ≥ 100 km the evaporation timescale increases with the conductivity of the companion.

Mottez et al. (2013b) proposed that the intermittency of PSR 1931+24 can be explained by the existence of a stream of small bodies of kilometer or subkilometer sizes close to the pulsar (a = 0.14 R). We calculate that these asteroids would evaporate in about one Gyr. If the Mie theory were not taken into account, such asteroids would not survive a year. A billion years being a long timescale compared to the migration time in the pulsar wind magnetic field (10 000 yr, Mottez & Heyvaerts 2011), the evaporation caused by induction does not invalidate the model of Mottez et al. (2013b).

The last lines in Table 1 refers to planets discovered near PSR 1257+12 (Wolszczan 1994). Because of their size, the Mie theory still has an influence on the vaporization rate, with Qabs< 0.06. If it is made of rock, 1257+21a should evaporate in 105 yr. With a more metallic composition, it could last a few million years. The other two planets are stable for 108 yr, or more if they are metallic.

thumbnail Fig. 3

Influence of companion composition on evaporation timescale. The contours of tev = 106yr (solid lines) and tev = 1012 yr (dashed lines) are represented as a function of companion distance a and size R for P = 10 ms, B = 108 G, and an indicative (and conservative) density ρ = 1 g cm-3, and for various companion refractive indices as indicated in the legend (increasing from right to left): σμ = 104−14 s-1, corresponding to Nr = 101−6.

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thumbnail Fig. 4

Amplitude of the pulsar wind magnetic field for a) a surface dipole field l = 1,m = 1; b) quadrupole l = 2,m = 2; and c)l = 3,m = 3. Panels a) to c) are plotted with the same color scale. On these panels, the spiral structure is the pulsar wave. Comparison ratio a, see Eq. (22) for modes d) l = 2,m = 2; e)l = 3,m = 2; and f) l = 3,m = 3. Panels d) to f) are plotted with the same color scale.

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4. Effect of multipole magnetic field components

We have considered the case of a star with a dipole magnetic field. In terms of spherical harmonics analysis, it has longitudinal and azimuthal components l = 1, m = 0, and m = 1. Only the m = 1 component participates in the pulsar wave and produces a monochromatic frequency spectrum, but we know that neutron star magnetic fields can have multipole components l> 1. The components with azimuthal numbers m> 1 induce shorter wavelengths cP/m, i.e., 2πRL/m. In all the above inductive heating rates, a multipole term of azimuthal number m can be accounted for by replacing the ratio of the size of the body to the incident wavelength x by xm = mx. If they had the same amplitude, multipole m> 1 terms would contribute more efficiently to the companion heating than the m = 1 wave for small bodies where x ~ 1. Therefore, it is important to evaluate the contribution of multipole terms of the neutron star magnetic field to the pulsar wind.

Bonazzola et al. (2015) have computed the vacuum solution of the electromagnetic field surrounding a pulsar. The solutions have a complex structure at distances less than the light cylinder radius with an amplitude that decreases faster for higher values of l. At greater distances, it has the characteristic Parker spiral wave structure dominated by the azimuthal component Bφr-1. Beyond the light cylinder, the ratio of amplitudes of the various modes does not vary much on average. To illustrate this, we have plotted in Fig. 4 a few vacuum field solutions corresponding to the same maximum amplitude B0 (on the star surface). The amplitude B of the magnetic field in the equatorial plane is plotted up to distances r = 11RL for the dipole solution (panel a), for the quadrupole solution l = 2m = 2 (panel b)5, and for l = 3,m = 3 (panel c). For comparison of magnetic amplitudes, we define the ratio where the maximum is computed for any radius larger than R. This definition is intended to avoid a divergent ratio when the two functions reach their minimum for different values of R that would not be pertinent to what we want to compare. The ratios a2,2,a3,2, and a3,3 are plotted respectively in panels d)–f). We can see from a2,2 that beyond a distance 2RL, the contributions of l = 1,m = 1 (dipole) and l = 2,m = 2 modes with the same maximum magnetic field on the star’s surface are comparable, although larger with the dipole. For larger values of l, the dipole term is dominant by a factor exceeding 102. This means that the contribution of a large amplitude quadrupole term to the heating of a pulsar companion cannot be neglected, implying that companions of radius R/ 2 twice as small can absorb an energy comparable to that of a R-sized body in the dipole case. Higher multipole terms should not have an important influence in our example.

We note that the above computation does not take into account the contribution of the plasma to the evolution of the wave amplitude into the wind; it should be considered a first-order approximation.

5. Conclusion and discussion

In this paper, we have assessed the efficiency of the inductive heating of pulsar companions caused by the pulsar Poynting-flux electromagnetic wave. Inductive heating is generally considered to be the dominant heating process, for instance when the evaporation of black widow companions is considered (van den Heuvel & van Paradijs 1988). It is commonly assumed that the whole flux carried by the pulsar wave is absorbed by the companion. Taking into account the Mie theory, we have shown that this assumption fails for objects with radius R smaller than the pulsar wavelength cP, P being the pulsar spin period. The rate of absorption of the wave energy decreases by many orders of magnitude for small objects like asteroids or planetesimals. This is especially true with standard P = 1 s pulsars because of their long wavelength. One consequence is that asteroids, considered to be objects that should quickly evaporate, do not always do so even at very close distances (1 R) of a 1 s-period pulsar.

The conclusion regarding kilometer-sized asteroids at short distances from pulsars cannot be generalized to any kind of pulsar. For instance, bodies orbiting a P = 1 ms pulsar at 1 R would not last a year.

The behavior of small objects orbiting a pulsar is important because it can explain the intermittency of their radio-emissions. For instance, Cordes & Shannon (2008) proposed a model where small rocks (meter-sized) falling on a pulsar cause intermittency during the few seconds in which they evaporate. For metric objects, Qabs derived from the Mie theory is so low that these rocks should not evaporate.

The survival of small bodies around pulsars is a pre-requisite of the theory developed by Mottez & Zarka (2014) to explain the origin of FRBs. In that model, asteroids immersed in the pulsar relativistic wind produce collimated magnetic wakes, that can be observed at cosmological distances and lead to radio bursts similar to the observed signals. An asteroid belt could also be invoked to account for the repeating FRB reported by Spitler et al. (2016; see also Dai et al. 2016).

We have assumed in this paper that the wind was Poynting-flux dominated at the location of the companion. For objects made of ordinary matter, the consistency of our evaporation timescales compared to the observations seems to validate a posteriori this hypothesis. If a non-negligible fraction of the wind energy were kinetic, a heating process by high-energy particle irradiation would have to be considered. This process would evaporate most close-by companions, whatever their size, as has been calculated and observed for black widow and redback systems (e.g., Phinney et al. 1988; Kluzniak et al. 1988; Arons & Tavani 1993). In this case, the energy absorption coefficient Qkin can be taken as ~1 at first-order approximation. A more thorough evaluation of Qkin would require calculating the cascading interactions of particles, which is beyond the scope of this study. Models considering simultaneously inductive heating, sputtering by the wind, and blackbody X-rays will be developed in forthcoming papers. From Eqs. (4) and (21), one can then infer that in order to avoid heating by particles, the fraction of kinetic energy in the wind has to be lower than Qem, the absorption coefficient of the Poynting flux given by the Mie Theory (denoted Qabs by abuse of notation throughout this paper, see values in Fig. 1 and Table 1). This implies a magnetization of σB> 1/Qem−1. As Qem ≪ 1 over a large parameter space, as is demonstrated in this paper, a very low fraction of kinetic energy in the wind and thus a very high magnetization is required for the survival of objects close to the pulsar.

We note that we have also argued in this work that, because of their high refractive index to the pulsar wave, the observed ablation of white dwarfs and M-stars in back widow and redback systems can only be explained if the absorbed energy is principally kinetic.


1

The wave amplitude is much smaller than in a domestic transformer (where magnetic fields of 104 G are common).

3

From an electromagnetism point of view, one can understand these two regions as different regimes of penetration of the magnetic field inside the object. In Region 2, the inductive currents do not penetrate deep under the surface (the resistive skin-depth is shallow). For Region 1, the skin-depth is approximately the entire size of the object, and the magnetic field can be considered as roughly uniform over it.

4

Cohesive energy is the energy required to form separated neutral atoms in their ground electronic state from the solid at a given temperature under a given pressure (Kittel 1998).

5

A pulsar of period P4 with a pure quadrupole field might be confused, from an observational point of view, with a pulsar with a dipole field and a period P2 = 2P4.

Acknowledgments

This work was nurtured by Jean Heyvaerts’ notes on the Mie theory and by the inspiring discussions we had with him. His great insight and input were very much missed during the completion of the project. K.K. thanks S. Phinney for fruitful discussions that led to her involvement in this project. This work was supported by the Programme National des Hautes Énergies. K.K. acknowledges financial support from the PER-SU fellowship at Sorbonne Universités, from the Labex ILP (reference ANR-10-LABX-63, ANR-11-IDEX-0004-02), as well as from the NSF grant NSF PHY-1412261 and the NASA grant 11-APRA-0066 at the University of Chicago, and the grant NSF PHY-1125897 at the Kavli Institute for Cosmological Physics.

References

All Tables

Table 1

Gravitational (tev,g) and cohesive (tev,c) evaporation timescales due to induction heating for various pulsar and companion parameters.

All Figures

thumbnail Fig. 1

Absorption coefficient Qabs as a function of the size ratio x = R/cP and of the refractive index Nr of the companion in the Metallic regime (N = Nr = −Ni). Overlayed are the contours of the analytical approximation, in black for Region 1 (xNr ≪ 1, Eq. (10)) and in white for Region 2 (xNr ≫ 1, Eq. (9)). The gray-shaded region corresponds to parameters outside the confidence range of the computation with Damie.

Open with DEXTER
In the text
thumbnail Fig. 2

Contours of evaporation timescales (in yrs) of non-degenerate pulsar companions (tev,g and tev,c, Eqs. (17) and (21)) as a function of the companion radius R and its distance a to the pulsar in solar radii, for pulsar spin periods and magnetic fields as indicated (left to right: P = 1,10,100 ms and B = 108,108,1012 G), for conductivity times magnetic permeability σμ = 106s-1 and density ρ = 1 g cm-3. The white dotted line delimits the region where the cohesive energy dominates over gravitational energy (for small companions) for evaporation. The white region indicates evaporation times tev< 10 yr and yellow regions tev> 1012 yr.

Open with DEXTER
In the text
thumbnail Fig. 3

Influence of companion composition on evaporation timescale. The contours of tev = 106yr (solid lines) and tev = 1012 yr (dashed lines) are represented as a function of companion distance a and size R for P = 10 ms, B = 108 G, and an indicative (and conservative) density ρ = 1 g cm-3, and for various companion refractive indices as indicated in the legend (increasing from right to left): σμ = 104−14 s-1, corresponding to Nr = 101−6.

Open with DEXTER
In the text
thumbnail Fig. 4

Amplitude of the pulsar wind magnetic field for a) a surface dipole field l = 1,m = 1; b) quadrupole l = 2,m = 2; and c)l = 3,m = 3. Panels a) to c) are plotted with the same color scale. On these panels, the spiral structure is the pulsar wave. Comparison ratio a, see Eq. (22) for modes d) l = 2,m = 2; e)l = 3,m = 2; and f) l = 3,m = 3. Panels d) to f) are plotted with the same color scale.

Open with DEXTER
In the text

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