Constraining the Emission Geometry and Mass of the White Dwarf Pulsar AR Sco Using the Rotating Vector Model

We use the PPA data from Buckley et al. (2017), comprising observations on 2016 March 14 in the 340–900 nm range. We obtain the minimum PPA of the data set and define the corresponding time t₀ as the starting point to fold the data set. A few data points (∼3%) deviate from the average PPA versus ϕ curve, but we note that the folding is affected by the choice of t₀. We remedy this convention issue by generating smoothed PPA curves with a kernel density estimation and a Gaussian kernel technique. By inspecting the deviation between the smoothed curve and the folded data, we assign a new convention to the points with a large deviation by shifting these points by 180° (which is the intrinsic uncertainty of the convention assigned to the PPA, i.e., there is an ambiguity in the parallel and antiparallel directions). We then bin the data into 30 rotational phase bins. The predicted values of ψ from Equation (1) are discontinuous, since the arctangent function is discontinuous at ${\phi }_{\mathrm{disc}}=\arccos \left(\tan \zeta /\tan \alpha \right).$ Using this expression, we locate the discontinuities and shift the predicted PPA by 360° at these phases to finally obtain a smooth, continuous PPA model.

We adopt the convention of Everett & Weisberg (2001), letting ψ increase in the counterclockwise direction. Thus we define ψ' = −ψ. A Bayesian likelihood approach is employed to constrain the RVM using the optical PPA data. We define our "best fit" as the 50th percentile or median in the posterior distribution of the model parameters. We verified our code by obtaining and comparing independent RVM fits of radio pulsar data (Everett & Weisberg 2001), yielding consistent α and ζ values, within uncertainties.

For the Bayesian analysis, we employ a Markov Chain Monte Carlo technique (Foreman-Mackey et al. 2013) with 50 walkers, 40,000 steps, and a step burn-in of 14,000. We maximize the following likelihood function:

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{ln}p(y| \phi ,\alpha ,\beta ,{\phi }_{0},{\psi }_{0},f)\\ =\,-0.5{{\rm{\Sigma }}}_{n}[\displaystyle \frac{{\left({y}_{n}-H(\phi ,\alpha ,\beta ,{\phi }_{0},{\psi }_{0})\right)}^{2}}{{S}_{n}^{2}}+\mathrm{ln}(2\pi {S}_{n}^{2})],\end{array}\end{eqnarray} \tag{ 2 }$$

where ${S}_{n}^{2}={\sigma }_{n}^{2}+{f}^{2}{(H(\phi ,\alpha ,\beta ,{\phi }_{0},{\psi }_{0}))}^{2}$, y represents the data values, σ represents the uncertainties of the data (assumed to follow a Gaussian distribution), and H represents the model values. The factor f compensates for the case when the PPA uncertainties are underestimated. We assume uniform priors on the cosine of α and ζ (rather than the angles themselves). For $\cos \zeta$, this is an appropriate choice from Copernican arguments. For $\cos \alpha$, this convention is less justified, but nevertheless ultimately immaterial for the present context; we have verified there is no dramatic change in resulting uncertainties with a uniform prior choice on α rather than $\cos \alpha$.

The red curve in Figure 1 depicts the best-fit RVM⁶ to the polarization data, with yellow curves associated with a random selection of parameters from the posterior distribution of $\cos \alpha$ and $\cos \zeta$. The uncertainties for the best fit are taken to be the 68% probability around the median, i.e., the 16th and 84th percentile values. From our best fits of these quantities, we obtain $\alpha ={86\buildrel{\circ}\over{.} 6}_{-2\buildrel{\circ}\over{.} \,8}^{+3\buildrel{\circ}\over{.} \,0}$ and $\zeta ={60\buildrel{\circ}\over{.} 4}_{-6\buildrel{\circ}\over{.} \,0}^{+5\buildrel{\circ}\over{.} \,3}$. For the uncertainty parameter $\mathrm{ln}(f)$, only the maximum of $\mathrm{ln}(f)$ is constrained, and f is quite small, thus we conclude the data are described well by the model without the inclusion of this term.

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The correlation or degeneracy seen between ϕ₀ and ψ₀ in Figure 2 owes to the fact that these parameters translate the model horizontally and vertically; thus, a natural degeneracy exists, since the model is cyclic and a similar fit may be obtained for different choices of ϕ₀ and ψ₀. This is also the reason why small, disconnected contours could be eliminated from Figure 2 by constraining the priors of the nuisance parameters ϕ₀ and ψ₀. The large duty cycle leads to relatively small uncertainties on both α and ζ as compared to the case of known radio pulsars. The pulses of the latter typically have small duty cycles and thus relatively large uncertainties on α (the impact angle β = ζ − α is typically better constrained, given visibility requirements, but ζ may remain ill-constrained).

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Marsh et al. (2016) reported the mass ratio of AR Sco as 1/q = M_M/M_WD ≥ 0.35, assuming M_WD = 0.8M_☉ and M_M = 0.3M_☉ as a natural pairing for the system that is located at a distance of d = 116 ± 16 pc. They also measured the radial velocity of the M star, K₂ = 295 ± 4 k ${\rm{s}}{}^{-1}$, and obtained the following mass function

$$\begin{eqnarray}&&{m}_{\mathrm{WD}}{\sin }^{3}i{\left(\displaystyle \frac{q}{1+q}\right)}^{2}=\displaystyle \frac{{P}_{{\rm{b}}}{K}_{2}^{3}}{2\pi G}\approx {0.395}_{-0.0158}^{+0.0163}{M}_{\odot }.\end{eqnarray} \tag{ 3 }$$

They report no significant evidence of nonsinusoidal radial velocities, thus suggesting little overestimation of the true amplitude owing to irradiation. Based on the M star's emission line velocities, Marsh et al. (2016) infer q ≲ 2.86, perhaps close to the cited maximum (q ∼ 2.8) under the assumption that the M star is close to filling its Roche lobe. Moreover, from spectroscopic comparison to model atmospheres they conclude that the radius of the M star R_M ≈ 0.36R_⊙ using a mass⁷ of m_M ≈ 0.29M_⊙ via the volume-equivalent Roche radius approximation of Paczyński (1971),

$$\begin{eqnarray}&&\displaystyle \frac{1}{3}{\left(\displaystyle \frac{2G{m}_{{\rm{M}}}{P}_{{\rm{b}}}^{2}}{3{\pi }^{2}}\right)}^{1/3}\approx {R}_{{\rm{M}}}\approx 0.36{R}_{\odot }.\end{eqnarray} \tag{ 4 }$$

Finally, based on the stellar radius estimate and the stellar brightness, they quote a mass estimate of ${m}_{{\rm{M}}}\,\approx 0.3{(d/116\mathrm{pc})}^{3}{\text{}}{M}_{\odot }$. A more accurate parallax distance of d = 117.8 ± 0.6 pc was reported by Stiller et al. (2018), implying m_M ≈ 0.31M_⊙ when inverting the above mass–distance relation, suggesting general consistency with the estimate m_M ∼ 0.3M_⊙. Substitution of m_M ∼ 0.3M_⊙ into Equation (3) and demanding that m_WD is below the Chandrasekhar mass limit of 1.44M_⊙ yields the constraint i ≳ 47°.

The mass m_WD is generally⁸ insensitive to the value of m_M, as is readily apparent from the q ≫ 1 limit of Equation (3), and therefore also to any systematic errors associated with observational determinations of R_M and also the approximation employed in Equation (4). Constraints on m_WD may be obtained under the assumption ζ ∼ i, which is generally expected from formation/evolution and observed in other contexts (e.g., Albrecht et al. 2007; Watson et al. 2011). Note that the spin angular momentum of the WD is about two orders of magnitude inferior to the total orbital angular momentum. Since it is likely that the WD was spun up to its exceptionally low period of P = 117 s via past accretion episodes, the transfer of angular momentum would tend to align the WD spin to the orbit angular momentum. Allowing m_M be a free parameter, while propagating normally distributed uncertainties for K₂ and $\cos \zeta$ (see Figure 2) in Equation (3) yields a band of allowable values of {m_M, m_WD} depicted in Figure 3. Adopting R_M ≈ 0.36R_⊙ and solving the system of equations Equations (3) and (4) yields in ${m}_{\mathrm{WD}}={1.00}_{-0.10}^{+0.16}{M}_{\odot }$ for the median, with uncertainties for the 68% containment region of probability. This yields $q={3.45}_{-0.45}^{+0.66}$ that is somewhat in tension with Marsh et al.'s (2016) quoted q ≲ 2.86 derived from the velocity amplitudes of atomic emission lines relative to the M, but not meaningfully so, given the allowable range of uncertainties and systematics which may be present in the M star mass/radius estimate.

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If we allow misalignment of the spin and orbital axes $\zeta \ne i$, this may explain why the optical maximum does not occur at inferior conjunction of the WD (Buckley et al. 2017). Katz (2017) proposed that this mismatch may be explained either by dissipation in a bow wave or by misalignment between the WD spin axis and both the orbital axis and the WD's oblique magnetic moment as well as potential oblateness of the WD, causing precession of the spin axis. The latter would lead to variable heating of the companion surface along with a drift in the phase of the optical maximum, with a period of decades. Peterson et al. (2019) analyzed a century's worth of optical photometry on AR Sco but could not detect any precessional period as suggested by Katz (2017), although this period may be much longer for a larger angle of misalignment between the orbital plane and the WD spin axis, or a smaller oblateness.

We suggest that effective spin–orbit interaction may result in the precession of the WD pulsar spin axis (Damour & Ruffini 1974; Esposito & Harrison 1975) owing to the metric curvature of the companion, even if the WD is not oblate. While any precession in the system may be dominated by electromagnetic influences of the companion, the general relativity (GR) rate calculated below is the minimum rate in absence of any electromagnetic torques. In the framework of classical GR, the angular geodetic precession rate ω_p is derived by Barker & O'Connell (1975a, 1975b). The rate of precession is insensitive to m_WD and may be estimated independent of the assumption ζ ≈ i,

$$\begin{eqnarray}\begin{array}{rcl}{\omega }_{p} & = & \displaystyle \frac{{G}^{2/3}}{2{c}^{2}(1-{\varepsilon }^{2})}{\left(\displaystyle \frac{2\pi }{{P}_{{\rm{b}}}}\right)}^{5/3}{m}_{{\rm{M}}}^{2/3}\displaystyle \frac{3+4q}{{\left(1+q\right)}^{4/3}}\\ & \sim & \displaystyle \frac{4\times {10}^{-10}\,\mathrm{rad}\,{{\rm{s}}}^{-1}}{1-{\varepsilon }^{2}}\approx \displaystyle \frac{0.8\,\deg \,{\mathrm{yr}}^{-1}}{1-{\varepsilon }^{2}}.\end{array}\end{eqnarray} \tag{ 5 }$$

Since the orbital eccentricity ε is presumably close to zero, the numerical value in Equation (5) constitutes a lower limit to the expected precession rate. This rate is similar to pulsar systems where such precession has been detected (e.g., B1913+16 Weisberg et al. 1989; Cordes et al. 1990; Kramer 1998) over decade timescales. The observable scope of precession depends on the degree of misalignment; if ζ is significantly different from i, such precession (including that by oblateness of the WD) may be imprinted on the pulses and PPA swings of AR Sco and therefore changes in ζ values may be detectable on similar decade timescales as in pulsar binary systems. That is, from long-term time evolution of pulses and polarization data, the degree of misalignment may be estimated. Moreover, if well-characterized, such precession also affords an independent constraint on the component masses.

Takata & Cheng (2019) also note that a detailed comparison between model and measured PPA sweeps may constrain the orientation of the spin axis of the WD. In the future, we will study polarization data as a function of orbital phase ϕ_b to constrain the effects of precession that a varying α or ζ versus ϕ_b may point to. In addition, predictions from a full emission model should lead to predictions of the PPA evolution that may be fit to data to constrain the WD spin axis alignment with the orbital axis and/or precession in the system. It is hoped that contraints on precession may teach us more about the system's characteristics, analogous to the case of PSR J1906+0746 where observations over several years of this precessing pulsar revealed the average structure of the radio beam (Desvignes et al. 2019).

Using our fits for α and ζ, we can constrain the geometry of the emission region for the optical radiation. The derivation of the RVM (see Appendix of du Plessis et al. 2019) assumes that the observer samples emission that is tangent to local magnetic field lines and that the polarization vector is in the poloidal plane. This is a good approximation in either SR or curvature radiation (CR) scenarios, provided that the particle momentum parallel to the field is relativistic and the relativistic particles have small pitch angles.

At first sight, this assumption of small pitch angle η seems plausible since, for a large Lorentz factor γ, the pitch angle is $\eta \sim {\theta }_{\gamma }\sim 1/\gamma$ (see Equation (10) where we estimate that the particles radiating optical emission have γ ∼ 30–100 from the spectrum, depending on B and η, and that η ∼ 10⁻⁴ if γ ∼ 5). However, both γ and η evolve with distance as the particles move along the B-field lines. For example, Takata et al. (2017) solve an approximate form⁹ of the coupled set of equations that describe the evolution of γ and perpendicular momentum ${p}_{\perp }$, used earlier by Harding et al. (2005) in the context of neutron star pulsars. Takata et al. (2017, 2018) and Takata & Cheng (2019) assume that relativistic particles are injected from the companion and travel into the WD magnetosphere (along closed B-field lines, an assumption based on α ∼ 60° < 90°) before radiating significantly, since the SR timescale is much longer than the light-crossing timescale r/c at the companion position (r ∼ a). They then study a magnetic mirror effect in which the first adiabatic invariant $\mu \propto {p}_{\perp }^{2}/B$ is conserved and find that if the initial pitch angle is large enough ($\sin {\eta }_{0}\gt 0.05$, i.e., outside a loss cone set by initial conditions), the mirror effect operates and the pitch angles increase to ∼π/2 at the mirror point (thus violating the RVM assumption of tangential emission) as the particles move closer to the WD surface. In this case, significant emission occurs at the mirror point, since SR losses increase rapidly with B and η, before the particles return outward.

One can solve for the emission height at which the SR loss timescale equals the light-crossing timescale r/c:

$$\begin{eqnarray}&&\displaystyle \frac{r}{a}\approx 0.21\,{\gamma }_{50}^{1/5}{\mu }_{35}^{2/5}{\eta }_{0.1}^{2/5}\sim 0.03{R}_{\mathrm{LC}},\end{eqnarray} \tag{ 6 }$$

where γ₅₀ = γ/50, μ₃₅ = μ/10³⁵ G cm³, μ = 0.5B_sR_WD³ is the magnetic moment, and η_0.1 = η/0.1 is the pitch angle. According to the calculations of Takata et al. (2017), the mirror effect operates and the particles turn around at r ∼ 0.25a, for γ = 50, μ = 6.5 × 10³⁴ G cm³, and η = 0.1. However, if we choose plausible values of γ = 150 and μ = 2 × 10³⁵ G cm³ (which is closer to that implied by the estimated surface magnetic field B_s, see Equation (7)), then Equation (6) implies that r ∼ 0.34a > 0.25a. This suggests that ${p}_{\perp }^{2}/B$ is no longer an invariant and the mirror effect will not operate, but the particles will lose all their energy abruptly suffering catastrophic SR losses. Thus, the pitch angle may never attain very large values.

We thus note that small pitch angles (as assumed by the RVM) are plausible in two cases: (i) when some parameter choices exist where t_SR > t_cross at a height r that is a substantial fraction of a, so that the particles radiate all their energy before undergoing magnetic mirroring; (ii) when particles with small enough initial pitch angles (e.g., $\sin {\eta }_{0}\lesssim 0.05$) fall in the loss cone and are not impacted by the magnetic mirror. A complete model, solving the particle dynamics generally (for any η and γ, and not in the drift approximation) to predict the light curves, spectrum, and polarization properties of AR Sco will be able to address this issue more fully.

In a dipolar magnetosphere, multiple locations satisfy the viewing constraints derivable using our α and ζ fits within the framework of the RVM, which assumes emission to be tangential to the local B-field, irrespective of altitude, since a dipolar field is self-similar (additional spectral information may help constrain the actual emission heights, not just the emission directions). Depending on the current system, any nonzero subset of these locations could participate to produce the observed PPA curve. Thus, since the RVM variation is seen over the entire spin rotation of the WD, at least one of the multiple solutions must be realized (e.g., the model Potter & Buckley 2018b that only assumes inflowing particles would satisfy these constraints). We indicate this schematically in Figure 4, where we show an orthogonal rotator and four different tangents pointing in the observer direction for a particular ζ. Outflowing and inflowing particles with respect to the nearest magnetic pole are indicated by blue and red arrows, respectively. Blue arrows thus occur in the top half of the meridional slice (large magnetic longitude), and red ones in the bottom (small magnetic longitude). Dark and light colors are used to further distinguish between solutions. The meaning of colors remains identical in Figure 5, where we indicate constraints involving the magnetic colatitude θ_B and longitude ϕ_B (both defined with respect to the magnetic dipole moment axis ${\boldsymbol{\mu }}$). These constraints derive from the condition that the magnetic field tangents are sampled by the observer, for α = 87° and ζ = 60°. For θ_B, the solutions satisfy Equation (34) and its reflection given in Wadiasingh et al. (2018).

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The different panels in Figure 5 are 2D projections of the 3D plot in the leftmost corner. The top left panel indicates that the "red solutions" are located around ϕ_B ∼ 0 while the blue ones occur at ϕ_B ∼ π (i.e., in meridional and antimeriodional planes with respect to ${\boldsymbol{\mu }}$). In the top right panel, the light and dark colored lines coincide, indicating that the observer samples the same ϕ_B for each color during the course of one rotation of the WD, independent of θ_B. The red and blue solutions are of similar functional form, but offset by a factor of π in ϕ_B, as previously. The bottom right projection indicates that the light and dark red solutions have the same form (offset by a factor ∼π/2 in θ_B, and the same for the light and dark blue), but the red versus blue ones are mirror images of each other (being ϕ ∼ π out of phase), since they originate close to opposite magnetic poles. Thus, the observer would sample the light blue and dark red solutions (or light red and dark blue ones), offset by half a rotation in spin phase ϕ, even though they trace out the same range in θ_B.

The constrained spatial region that follows from our RVM fits suggests that a large portion of the magnetosphere, at multiple altitudes, must support relativistic charges. Depending on scenarios of particle acceleration, charges may be either ingoing or outgoing relative to the WD surface, halving the number of potential sites of emission, unless there are counterstreaming beams. In a more complete emission model beyond the geometric RVM, deriving constraints on the altitude of emission should be possible, thereby pinning down the precise location in the magnetosphere where emission arises at any phase.

Interestingly, the estimated polar cap opening angle is much lower (θ_PC/2π ∼ 1%) than the duty cycle ∼100%, implying that the two-star interaction may lead to a much larger opening angle, or that the emission comes from relatively high up, originating on flaring magnetic field lines.

In order to constrain the emission mechanism that might be responsible for the polarized optical radiation, let us estimate several relevant quantities and compare pertinent timescales and length scales. The magnetic field at the polar cap may be estimated assuming M_WD = 1.0M_☉ via

$$\begin{eqnarray}&&\begin{array}{l}{B}_{{\rm{p}}}\sim 8\times {10}^{8}\,{\rm{G}}\\ \quad \times \,{\left(\displaystyle \frac{{R}_{\mathrm{WD}}}{7\times {10}^{8}\mathrm{cm}}\right)}^{-3}{\left(\displaystyle \frac{P}{117{\rm{s}}}\right)}^{1/2}{\left(\displaystyle \frac{\dot{P}}{3.9\times {10}^{-13}{\rm{s}}{{\rm{s}}}^{-1}}\right)}^{1/2},\end{array}\end{eqnarray} \tag{ 7 }$$

from dipole spin-down, with R_WD the WD radius. Geng et al. (2016) calculates the magnetic field strength B_x at a distance x = a − R_WD ∼ 5 × 10¹⁰ cm above the stellar surface, where a is binary separation, obtaining ${B}_{{\rm{x}}}={B}_{{\rm{p}}}{(x/{R}_{\mathrm{WD}})}^{-3}\sim 2\,\times {10}^{3}\,{\rm{G}}$. We estimate the potential drop at the polar cap using (Goldreich & Julian 1969)

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Phi }}}_{\max }\sim 4\times {10}^{11}\,\mathrm{statvolt}\\ \quad \times \,{\left(\displaystyle \frac{{R}_{\mathrm{WD}}}{7\times {10}^{8}\mathrm{cm}}\right)}^{3}\left(\displaystyle \frac{{B}_{{\rm{p}}}}{8\times {10}^{8}\,{\rm{G}}}\right){\left(\displaystyle \frac{P}{117{\rm{s}}}\right)}^{-2}.\end{array}\end{eqnarray} \tag{ 8 }$$

This yields a maximum Lorentz factor of ${\gamma }_{\max ,{\rm{\Phi }}}\sim 3\times {10}^{8}$, although it is unlikely that the particle will be able to tap the full potential.

The corresponding SR, CR, and inverse Compton (IC) timescales ($\gamma /\dot{\gamma }$) are indicated in Figure 6, along with an acceleration (R_L/c) and crossing timescale (a/c), where R_L is the Larmor radius. To calculate the IC loss rate we used ${\dot{E}}_{\mathrm{IC}}\,=4{\sigma }_{{\rm{T}}}{cU}{\gamma }_{{\rm{e}}}^{2}{\gamma }_{\mathrm{KN}}^{2}/3({\gamma }_{{\rm{e}}}^{2}+{\gamma }_{\mathrm{KN}}^{2})$ (Schlickeiser & Ruppel 2010), where ${\gamma }_{\mathrm{KN}}=3\sqrt{5}{m}_{{\rm{e}}}{c}^{2}/8\pi {kT}$ and $U=2{\sigma }_{\mathrm{SB}}{T}^{4}{R}_{\star }^{2}/{{cR}}_{0}^{2}$ is the photon energy density. Here, m_e is the electron mass, σ_T is the Thomson cross section, σ_SB is the Stefan–Boltzmann constant, k is the Boltzmann constant, R_⋆ is the companion radius, and R₀ is the distance from the companion's center to the shock. We consider a "near" (B_p) and "far" (B_x) case. Since the IC cooling time is much larger than that of SR (even for small pitch angles), we can rule out IC.

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If we equate the SR and CR loss rates to solve for the average pitch angle η, this yields a maximum value of

$$\begin{eqnarray}&&\sin \eta \sim 4\times {10}^{-3}\left(\displaystyle \frac{\gamma }{3\times {10}^{8}}\right){\left(\displaystyle \frac{B}{2\times {10}^{3}{\rm{G}}}\right)}^{-1},\end{eqnarray} \tag{ 9 }$$

where e is electron charge and ρ_c ∼ 4R_WD/3θ_pc the curvature radius at the surface of the WD, with ${\theta }_{\mathrm{PC}}\sim {({\rm{\Omega }}{R}_{\mathrm{WD}}/c)}^{1/2}$. Thus, we see that SR dominates for all reasonable values of γ and for nonzero pitch angles, with CR only becoming relevant for $\gamma \gtrsim {\gamma }_{\max ,{\rm{\Phi }}}$ or for pitch angles $\eta \lesssim {10}^{-3}$.

Following Takata et al. (2018) and attributing the nonthermal pulsed X-ray emission to a power-law tail that is emitted by a power-law particle spectrum between γ_min and γ_max, we can infer constraints using the observed frequencies that correspond to the peak in the spectral energy density (${\nu }_{\mathrm{obs},\min }=2\times {10}^{13}\,\mathrm{Hz}$) and the highest pulsed X-ray photon (${\nu }_{\mathrm{obs},\max }\gtrsim 2\times {10}^{18}\,\mathrm{Hz}$):

$$\begin{eqnarray}&&{\gamma }_{\min }^{2}B\sin \eta \sim 5\times {10}^{6}\,{\rm{G}}\left(\displaystyle \frac{{\nu }_{\mathrm{obs},\min }}{2\times {10}^{13}\,\mathrm{Hz}}\right),\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\gamma }_{\max }^{2}B\sin \eta \gtrsim 5\times {10}^{11}\,{\rm{G}}\left(\displaystyle \frac{{\nu }_{\mathrm{obs},\max }}{2\times {10}^{18}\,\mathrm{Hz}}\right).\end{eqnarray} \tag{ 11 }$$

When enforcing that B < B_p, we obtain

$$\begin{eqnarray}&&{\gamma }_{\min }\gtrsim 30\left(\displaystyle \frac{{\nu }_{\mathrm{obs},\min }}{2\times {10}^{13}\,\mathrm{Hz}}\right){\left(\displaystyle \frac{\sin \eta }{{10}^{-5}}\right)}^{-1/2},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\gamma }_{\max }\gtrsim 8\times {10}^{3}\left(\displaystyle \frac{{\nu }_{\mathrm{obs},\max }}{2\times {10}^{18}\,\mathrm{Hz}}\right){\left(\displaystyle \frac{\sin \eta }{{10}^{-5}}\right)}^{-1/2},\end{eqnarray} \tag{ 13 }$$

for a fiducial value of $\sin \eta \sim {10}^{-5}$. (Imposing a lower limit of γ_min ≈ 5 formally constrains η ≳ 2 × 10⁻⁴.) These constraints are consistent with the assumptions made by Takata et al. (2017) who model the SR spectrum assuming γ_min = 50 and γ_max = 5 × 10⁶.

Let us calculate a characteristic length scale for both SR and CR:

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{SR}} & \approx & c\displaystyle \frac{\gamma }{{\dot{\gamma }}_{\mathrm{SR}}}={10}^{10}\,\mathrm{cm}\\ & & \times \,{\left(\displaystyle \frac{{\nu }_{\mathrm{obs},\min }}{2\times {10}^{13}\mathrm{Hz}}\right)}^{-1/2}{\left(\displaystyle \frac{{\rm{B}}}{8\times {10}^{8}{\rm{G}}}\displaystyle \frac{\sin \eta }{{10}^{-5}}\right)}^{-3/2}\end{array}\end{eqnarray} \tag{ 14 }$$

and

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\mathrm{CR}} & = & c\displaystyle \frac{\gamma }{{\dot{\gamma }}_{\mathrm{CR}}}\\ & = & {10}^{20}\,\mathrm{cm}{\left(\displaystyle \frac{{\nu }_{\mathrm{obs},\min }}{2\times {10}^{13}\mathrm{Hz}}\right)}^{-1}\left(\displaystyle \frac{{\rho }_{{\rm{c}}}}{5\times {10}^{10}\,\mathrm{cm}}\right).\end{array}\end{eqnarray} \tag{ 15 }$$

Since L_SR ≪ L_CR, it is apparent that SR should dominate over CR, even for very small values for η (see Figure 6).

Observations by Buckley et al. (2017) indicate that the degree of linear polarization of the optical data varies with orbital phase, but may reach up to 40%. This constraint should be exploited in an emission model, but is beyond the RVM's capabilities. For example, Takata & Cheng (2019) argue that a polarization degree that varies with orbital phase may be understood in the framework of different contributions of the SR and the thermal emission from the companion, since the latter may depolarize the observed emission. We expect that a model that includes emission from different emission heights that bunch in phase to form the peaks will not likely overpredict the maximum observed polarization degree of 40%. SR from a single particle in an ordered field may reach high values, but this will be lowered by contributions from several particles within the population of radiating particles emitting at different spatial locales. This is similar to the findings of Harding & Kalapotharakos (2017) who found large PA swings and deep depolarization dips during the light-curve peaks in all energy bands in their multiwavelength pulsar model that invoke caustic emission. Such a model may not be exactly applicable to the WD pulsar, although contributions from emitting particles at different altitudes may provide a blended polarization signature, thus lowering the polarization degree versus that expected from a single particle in an ordered field. The detailed characteristics depend on emission position and mechanism.

A constraint on B can be obtained by assuming that the cyclotron energy is below the SED peak (otherwise a break would be observable due to this threshold energy). At the WD surface, the cyclotron energy is 9 eV. When ${\nu }_{\mathrm{cycl}}={\nu }_{\mathrm{obs},\min }$, B ≲ 7 × 10⁶ G, implying that the emission site is at least ∼5R_WD above the stellar surface. Another constraint on B may be obtained by requiring the ratio of electromagnetic to kinetic particle energy density to be σ ≫ 1 (since the RVM results point to particles following the local field), but this limit is not very constraining due to the uncertain emission volume. On the other hand, the plasma density should be high enough to explain the observed flux of nonthermal emission. To estimate a lower limit for the plasma density (since the charge density may be much lower than the actual number density) we can calculate the Goldreich–Julian number density n_GJ (Goldreich & Julian 1969):

$$\begin{eqnarray}\begin{array}{rcl}| {n}_{\mathrm{GJ}}| & = & \displaystyle \frac{{\boldsymbol{\Omega }}\cdot {\boldsymbol{B}}}{2\pi {ec}}\\ & = & 4.7\times {10}^{5}\,{\mathrm{cm}}^{-3}{\left(\displaystyle \frac{{\rm{P}}}{117{\rm{s}}}\right)}^{-1}\left(\displaystyle \frac{{\rm{B}}}{8\times {10}^{8}\,{\rm{G}}}\right)\\ & = & 1.2\,{\mathrm{cm}}^{-3}{\left(\displaystyle \frac{{\rm{P}}}{117{\rm{s}}}\right)}^{-1}\left(\displaystyle \frac{{\rm{B}}}{2\times {10}^{3}\,{\rm{G}}}\right),\end{array}\end{eqnarray} \tag{ 16 }$$

with ${\boldsymbol{\Omega }}$ being the angular frequency. The number of particles needed to explain the SR spectrum may be estimated as follows. Let us focus on the peak frequency ν_peak = 0.29ν_c, with ${\nu }_{{\rm{c}}}\,=3e{\gamma }^{2}B\sin \eta /4\pi {m}_{{\rm{e}}}c$. We use $F({\nu }_{\mathrm{peak}}/{\nu }_{{\rm{c}}})=F(0.29)=0.924$ and assume a delta distribution for the steady-state particle spectrum ${dN}/{{dE}}_{{\rm{e}}}={N}_{0}\delta ({E}_{{\rm{e}}}-{E}_{{\rm{e}}}^{* })$, with ${E}_{{\rm{e}}}^{* }=\gamma {m}_{{\rm{e}}}{c}^{2}$ the energy needed to explain the peak of the spectrum at 0.29ν_c. From Figure 5 of Takata & Cheng (2019), we take $\nu {F}_{\nu }^{\mathrm{obs}}=8\,\times {10}^{-12}$ erg s⁻¹ cm⁻² at an observed energy of ∼0.02 eV. Using

$$\begin{eqnarray}&&{P}_{\nu }=\displaystyle \frac{\sqrt{3}{e}^{3}B\sin \eta }{{m}_{{\rm{e}}}{c}^{2}}F({\nu }_{\mathrm{peak}}/{\nu }_{{\rm{c}}})\end{eqnarray} \tag{ 17 }$$

and a rough estimate of the emitting volume $V\sim 2\pi (1\,-\cos {\theta }_{\mathrm{PC}}){a}^{3}/3\sim {10}^{30}$ cm³, we find

$$\begin{eqnarray}&&{N}_{0}\sim 5\times {10}^{36}\end{eqnarray} \tag{ 18 }$$

and thus¹⁰

$$\begin{eqnarray}&&{n}_{{\rm{e}}}\sim \displaystyle \frac{{N}_{0}}{V}\sim {10}^{6}\,{\mathrm{cm}}^{-3}.\end{eqnarray} \tag{ 19 }$$

This is lower than the estimate of Geng et al. (2016) who find n_e ∼ 4 × 10⁸ cm⁻¹, but still highly supra-Goldreich–Julian suggesting screening of E fields and electron-ion plasma sourced from the companion. On the other hand, since we have demonstrated that the PPA of the WD can be modeled by the RVM, this could mean that the emission of the AR Sco system originates from a dipole-like magnetosphere surrounding the WD (Buckley et al. 2017). Using observations of the linear flux, circular flux, and PPA observations, Potter & Buckley (2018b) show that the polarization is coupled to the WD spin period, agreeing with our conclusion, while models that place the emission site at the companion fail to explain the polarization signatures that are clearly coupled to the spin period of the WD. The "missing" particles noted by Geng et al. (2016) may either be supplied by the companion that may inject relativistic electrons into the WD magnetosphere, or less likely by pair cascades (probably needing severely nonpolar B-field structures) occurring in the WD magnetosphere, and does not per se argue for an emission location near a putative bow shock. Moreover, freshly injected particles from the companion may solve the issue of needing nonzero pitch angles for SR to dominate.