A MODEL OF WHITE DWARF PULSAR AR SCORPII

An outflow of relativistic particles from the open field line regions of the WD would impact the MD wind and drive a bow shock into it. Since the magnetization of the WD wind is not known, there may or may not be a reverse shock. In any case, since the accelerated particles are still within the magnetosphere of the WD, they would give rise to synchrotron radiation in the magnetic field of the magnetosphere.

The open field lines of the WD would be extruded by the ram pressure of the stellar wind during the period when they sweep the MD wind. When the polar cap faces the center of the MD, the effective opening angle of the open field lines reaches the maximum value (Figure 1), and the luminosity also reaches the peak value. We define the head position of the bow shock at this epoch as point "B." Since the magnetic pressure at B is larger than that at A, the magnetic pressure would push point B to be near the surface of the MD. Its distance to the center of the WD is $x\sim d-{R}_{\mathrm{MD}}\simeq 5.1\times {10}^{10}$ cm (${R}_{\mathrm{MD}}=2.5\times {10}^{10}$ cm is the radius of the MD) and the corresponding magnetic field is ${B}_{{\rm{B}}}={B}_{p}{(x/{R}_{\mathrm{WD}})}^{-3}\simeq 1860\,{\rm{G}}$. Since the synchrotron emission SED we model is an average one across different phases, we use an average magnetic field $\bar{B}=({B}_{{\rm{A}}}+{B}_{{\rm{B}}})/2\simeq 1200\,{\rm{G}}$ (${B}_{{\rm{A}}}$ is the magnetic field at A and is obtained according to its position; see Section 4.2) in our calculations.

For a relativistic electron of Lorentz factor ${\gamma }_{e}$, its radiation power is (Rybicki & Lightman 1979)

$$\begin{eqnarray}&&\dot{\epsilon }={\gamma }_{e}^{2}{\sigma }_{{\rm{T}}}{\bar{B}}^{2}c/6\pi =1.5\times {10}^{-9}{\gamma }_{e}^{2}\,\mathrm{erg}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 6 }$$

where ${\sigma }_{{\rm{T}}}$ is the Thomson cross section. Hereafter, the convention ${Q}_{x}=(Q/{10}^{x})$ is adopted in cgs units. Its cooling timescale can be calculated as ${t}_{\mathrm{cool}}={\gamma }_{e}{m}_{e}{c}^{2}/\dot{\epsilon }$. It is reasonable to assume that the relativistic electrons that give rise to synchrotron radiation obey a broken-power-law distribution, i.e.,

$$\begin{eqnarray}\displaystyle \frac{{{dn}}_{e}}{d{\gamma }_{e}}=\left\{\begin{array}{ll}C{\gamma }_{e}^{-p}, & {\gamma }_{m}\leqslant {\gamma }_{e}\leqslant {\gamma }_{c},\\ C{\gamma }_{c}{\gamma }_{e}^{-p-1}, & {\gamma }_{c}\lt {\gamma }_{e}\leqslant {\gamma }_{\max },\end{array}\right.\end{eqnarray} \tag{ 7 }$$

where $C\simeq (p-1){\gamma }_{m}^{p-1}{n}_{e}$, ${\gamma }_{m}$ is the typical Lorentz factor, ${\gamma }_{c}$ is the cooling Lorentz factor, ${\gamma }_{\max }$ is the maximum Lorentz factor, and n_e is the number density of the electrons. Then, the peak frequency of the corresponding synchrotron spectrum is

$$\begin{eqnarray}&&{\nu }_{m}=\displaystyle \frac{3{x}_{p}{q}_{e}\bar{B}{\gamma }_{m}^{2}}{4\pi {m}_{e}c}=2.5\times {10}^{9}{\gamma }_{m}^{2}\,\mathrm{Hz},\end{eqnarray} \tag{ 8 }$$

where ${x}_{p}\simeq 0.5$ (Wijers & Galama 1999). Inspecting the SED (Marsh et al. 2016, Figure 2), we find ${\nu }_{m}\simeq 5\times {10}^{12}\,\mathrm{Hz}$, which gives ${\gamma }_{m}=45$. The cooling Lorentz factor ${\gamma }_{c}$ can be estimated by equating ${t}_{\mathrm{cool}}({\gamma }_{c})$ with the mean dynamical time of the shock ${t}_{\mathrm{dyn}}\simeq \tfrac{45^\circ /2}{360^\circ }P$, where $45^\circ$ is the angle for the WD beam to sweep through before reaching the peak, and the half value reflects the mean angle, which defines the mean dynamical timescale. We then obtain ${\gamma }_{c}=73{(P/1.95\mathrm{minutes})}^{-1}$. The Lamor radius of a relativistic electron is ${R}_{L}={\gamma }_{e}{m}_{e}{c}^{2}/{q}_{e}\bar{B}$, and its acceleration timescale can be calculated as ${t}_{\mathrm{acc}}\simeq {R}_{L}/c$. Equating ${t}_{\mathrm{acc}}({\gamma }_{\max })$ with ${t}_{\mathrm{cool}}({\gamma }_{\max })$, one can obtain the maximum Lorentz factor of the shocked electrons as ${\gamma }_{\max }={(6\pi {q}_{e}/{\sigma }_{T}\bar{B})}^{1/2}=3.4\,\times \,{10}^{6}$. The corresponding maximum Lamor radius is ${R}_{L,\max }\,=4.8\times {10}^{6}\,\mathrm{cm}$.

Zoom In Zoom Out Reset image size

Assuming that the width of the emission shell is $\eta {R}_{\mathrm{MD}}$, the shell volume V is $2\pi (1-\cos ({\theta }_{{\rm{B}}}))\eta {R}_{\mathrm{MD}}^{3}$, where ${\theta }_{{\rm{B}}}$ is the half opening angle of the shell seen from the MD, and ${\theta }_{{\rm{B}}}\simeq 90^\circ -\arcsin ({R}_{\mathrm{MD}}/d)=71^\circ$ (Figure 1). The value of η may be inferred from the Lamor radius of the most energetic electron, i.e., $\eta \simeq 2{R}_{L,\max }/{R}_{\mathrm{MD}}=4\times {10}^{-4}$. Since ${\nu }_{m}\lt {\nu }_{c}$, the synchrotron emission is in the slow cooling regime, and the peak flux density is at ${\nu }_{m}$, which reads

$$\begin{eqnarray}&&{F}_{\nu ,\mathrm{peak}}=\displaystyle \frac{\sqrt{3}{q}_{e}^{3}\bar{B}}{{m}_{e}{c}^{2}}\displaystyle \frac{{n}_{e}V}{4\pi {D}_{L}^{2}},\end{eqnarray} \tag{ 9 }$$

where ${D}_{L}=116\,\mathrm{pc}$ is the distance of AR Sco to the observer. Observationally, the spectrum of AR Sco shows a peak flux density $\simeq 1.6\times {10}^{-24}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}$ (Marsh et al. 2016). We therefore obtain

$$\begin{eqnarray}{n}_{e} & = & 3.5\times {10}^{8}\,\left(\displaystyle \frac{{F}_{\nu ,\mathrm{peak}}}{1.6\times {10}^{-24}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}}\right)\\ & & \times {\left(\displaystyle \frac{\eta }{4\times {10}^{-4}}\right)}^{-1}{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 10 }$$

This gives an estimate on the parameter n_e of the emission region. The source of the emitting electrons remains not specified so far, which we will constrain next.

The Goldreich & Julian (1969) charge number density of the WD pulsar at position B is estimated as

$$\begin{eqnarray}&&{n}_{\mathrm{GJ}}=\displaystyle \frac{{\boldsymbol{\Omega }}\cdot {{\boldsymbol{B}}}_{{\rm{B}}}}{2\pi {q}_{e}c}=1.1\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 11 }$$

This is significantly lower than the required n_e in Equation (10). The primary electrons ejected from the polar cap may produce secondary electron–positron pairs through some pair production process (e.g., Zhang & Harding 2000). However, the pair multiplicity $\chi ={n}^{\pm }/{n}_{\mathrm{GJ}}$ is at most 10⁶, even for young radio pulsars (e.g., Arons 2009). This is not enough to account for Equation (10), which demands $\chi \sim {10}^{8}$. We therefore conclude that the synchrotron emitting electrons are not from the WD wind.

We now consider the possibility that the emitting electrons come from the stellar wind of the MD. Assuming that the wind velocity v is two times that of the escape velocity from the MD (stellar wind velocities are generally not much greater than the surface escape velocity; e.g., Wood 2004), i.e.,

$$\begin{eqnarray}&&v=2{v}_{\mathrm{esc}}=2\sqrt{\displaystyle \frac{2{{GM}}_{\mathrm{MD}}}{{R}_{\mathrm{MD}}}}=1.1\times {10}^{8}\,\mathrm{cm}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 12 }$$

the corresponding mass-loss rate of this stellar wind would be $\dot{M}=4\pi {R}_{\mathrm{MD}}^{2}{{vn}}_{e}{m}_{p}/\zeta =4.1\times {10}^{-11}{(\zeta /0.2)}^{-1}{M}_{\odot }\,{\mathrm{yr}}^{-1}$, where ζ denotes the fraction of electrons that are accelerated and n_e has been taken as $3.5\times {10}^{8}$ cm⁻³ as required in Equation (10). This value is within the range of mass-loss rate of MDs in the literature, ${10}^{-15}\mbox{--}{10}^{-10}{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (e.g., Mullan et al. 1992; Wood et al. 2001; Vidotto et al. 2014). It indicates that the wind from the MD could be a reasonable source for electrons.

The position of the balance point A (see Figure 1 and Section 3) can be found using the balance between the ram pressure of the wind and the magnetic pressure. We define the distance between A and the MD center as ${r}_{{\rm{A}}}$. The distance between A and the WD center may be roughly estimated as d. The relative velocity between the wind and the magnetic field lines is thus ${v}_{\mathrm{rel}}=\tfrac{2\pi }{P}d$. The condition that the magnetic pressure balances the wind ram pressure gives

$$\begin{eqnarray}&&\displaystyle \frac{{B}_{{\rm{A}}}^{2}}{8\pi }={\left(\displaystyle \frac{{r}_{{\rm{A}}}}{{R}_{\mathrm{MD}}}\right)}^{-2}\displaystyle \frac{1}{\zeta }{n}_{e}{m}_{p}{v}_{\mathrm{rel}}^{2},\end{eqnarray} \tag{ 13 }$$

where ${B}_{{\rm{A}}}={B}_{p}{(d/{R}_{\mathrm{WD}})}^{-3}$ is the magnetic field strength at point A, m_p is the proton mass, and the factor ${(\tfrac{{r}_{{\rm{A}}}}{{R}_{\mathrm{MD}}})}^{-2}$ takes into account the correction of the electron density n_e from point B to point A due to the radial expansion of the wind. Using Equations (10) and (13), one obtains

$$\begin{eqnarray}{r}_{{\rm{A}}} & = & 5.0\times {10}^{10}\,{\left(\displaystyle \frac{{F}_{\nu ,\mathrm{peak}}}{1.6\times {10}^{-24}\mathrm{erg}{\mathrm{cm}}^{-2}{{\rm{s}}}^{-1}{\mathrm{Hz}}^{-1}}\right)}^{1/2}\\ & & \times {\left(\displaystyle \frac{\eta }{4\times {10}^{-4}}\right)}^{-1/2}{\left(\displaystyle \frac{\zeta }{0.2}\right)}^{-1/2}\,\mathrm{cm}.\end{eqnarray} \tag{ 14 }$$

Another way to derive ${r}_{{\rm{A}}}$ is from the geometric relation in Figure 1. For the isosceles triangle defined by point A and the centers of the two stars, one has $\sin {\theta }_{{\rm{A}}}/{r}_{{\rm{A}}}=\sin (90^\circ -{\theta }_{{\rm{A}}}/2)/d$, which gives ${r}_{{\rm{A}}}=5.8\times {10}^{10}\,\mathrm{cm}$ for ${\theta }_{{\rm{A}}}\sim 45^\circ$. One can see that the values of ${r}_{{\rm{A}}}$ are consistent with each other using two methods, suggesting that the emitting electrons are indeed from the MD wind.

Using the electron distribution in Equation (7), one can calculate the synchrotron spectrum. The spectrum is characterized by a broken power law separated by three frequencies, the self-absorption frequency ${\nu }_{a}$, the injection frequency ${\nu }_{m}$, and the cooling frequency ${\nu }_{c}$ (Sari et al. 1998). The characteristic frequencies ${\nu }_{m}$ and ${\nu }_{c}$ are derived from ${\gamma }_{m}$ and ${\gamma }_{c}$, respectively. The spectrum in the spectral regime ${\nu }_{m}\lt \nu \lt {\nu }_{c}$ has $\nu {F}_{\nu }\propto {\nu }^{-(p-3)/2}$. According to the observed SED of AR Sco (Marsh et al. 2016), the source was also detected in the X-ray band by Swift X-Ray Telescope (even though not pulsed). The X-ray data constrain the value of p to be around 2.4. The self-absorption coefficient at frequency ν can be calculated as (e.g., Rybicki & Lightman 1979; Wu et al. 2003)

$$\begin{eqnarray}&&{\kappa }_{\nu }\simeq {c}_{p}\displaystyle \frac{{q}_{e}}{\bar{B}}{n}_{e}{\gamma }_{m}^{-5}{\left(\displaystyle \frac{\nu }{{\nu }_{m}}\right)}^{-5/3},\end{eqnarray} \tag{ 15 }$$

where ${c}_{p}=\tfrac{8{\pi }^{2}}{9\,{2}^{1/3}{\rm{\Gamma }}(1/3)}\tfrac{p\,+\,2}{p\,+\,2/3}(p-1)\simeq 5.2$, and ${\rm{\Gamma }}(1/3)$ is the Gamma function of argument 1/3. The self-absorption frequency can be solved when the optical depth ${\kappa }_{{\nu }_{a}}\eta {R}_{\mathrm{MD}}$ equals 1, which gives

$$\begin{eqnarray}{\left(\displaystyle \frac{{\nu }_{a}}{{\nu }_{m}}\right)}^{5/3} & = & 4.2\times {10}^{-5}\,\left(\displaystyle \frac{{F}_{\nu ,\mathrm{peak}}}{1.6\times {10}^{-24}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}}\right)\\ & & \times {\left(\displaystyle \frac{{\nu }_{m}}{5\times {10}^{12}\mathrm{Hz}}\right)}^{-5/2}.\end{eqnarray} \tag{ 16 }$$

One gets ${\nu }_{a}\simeq 2.4\times {10}^{-3}\,{\nu }_{m}$ for AR Sco. Figure 2 displays the comparison between our analytical model SED and the observed SED (Marsh et al. 2016). One can see that the model can well interpret the data. One interesting feature is that the observed flux at the thermal peak exceeds the model spectrum of the MD thermal emission. However, adding the contribution of the non-thermal synchrotron component (which extends all the way to the X-ray band), the total model flux matches the observations very well.