Simulations of a Conducting Sphere Moving through Magnetized Plasma II: Effect of Internal Magnetic Fields and Enhancement of Alfvénic Drag

Introduction

Since the discovery of radio emission and auroral signatures caused by Io cutting through Jupiter’s magnetosphere (Bigg 1964) and their first theoretical study (Goldreich & Lynden-Bell 1969), there has been considerable interest in observing similar interactions between magnetized astrophysical objects in binaries. These include planet-moon systems (Neubauer 1990; Bagenal & Dols 2020), star-planet systems (Li et al. 1998; Preusse et al. 2006; Lanza 2009; Laine & Lin 2011; Lai 2012; Lanza 2013; Strugarek et al. 2014; Saur 2018; De Colle et al. 2025; Lee & Owen 2025) and binary compact objects ([12]; Lai 2012; Most & Philippov 2022, 2023; Mahlmann & Beloborodov 2025). Such interactions may result in observable electromagnetic radiation and/or lead to long-term orbital evolution.

A related problem concerns an unmagnetized or weakly magnetized object moving in the more dominant magnetic field of its companion. In this case, the weaker companion can be treated as a perturber of the global background field, allowing for a clean local analysis of this object and its environment. Such an analysis began with Drell et al. (1965), who considered the model of a conductor (treated as a rectangular bar) moving with velocity $\mathbf{v}_0$ through a plasma with a perpendicular uniform magnetic field $\mathbf{B}_0$. Because of the sub-Alfvénic velocity ($v_0 < v_A$, where $v_A$ is the Alfvén speed of the background plasma), the conductor acts as an electrical generator that launches two tube-like Alfvén waves—termed Alfvén wings—that propagate outward along the background field lines, carrying power that drags the conductor. The concept of “Alfvén wings” has found application in the Jupiter-Io system (Neubauer 1980), where Alfvén waves could propagate back-and-forth between the two objects as though in a closed circuit. More recently, it has been applied more generically, such as in exoplanetary systems (e.g. Neubauer 1998; Strugarek et al. 2014; De Colle et al. 2025).

Many previous studies have examined the detailed dissipative mechanisms and radiative signatures of magnetic interactions. In planet-moon systems, such as Jupiter-Io, the power carried by Alfvén wings can produce aurorae and radio emissions (Neubauer 1990; Bagenal & Dols 2020). In star-planet systems, ohmic dissipation from the current carried by Alfvén wings may drive coronal activity or heating in either body (Li et al. 1998; Preusse et al. 2006; Lanza 2009; Laine & Lin 2011), and the flux tube connecting the two bodies may reconnect and flare out (Lai 2012; Lanza 2013). Similar processes may also operate in compact-object binaries ([12]; Lai 2012; Wang et al. 2016; Beloborodov 2021; Most & Philippov 2022, 2023). Recently, Sharma et al. (2026) proposed that Alfvén wings in merging neutron star binaries could lead to pulsar-like emission due to strong currents at the wing boundaries; Lyutikov (2026) followed up this proposal with kinetic simulation. These systems may also experience interactions not mediated by Alfvén wings. For instance, Mahlmann & Beloborodov (2025) suggested that in a binary neutron star system with aligned spins, turbulent heating could arise from the Kelvin–Helmholtz instability at the interface of their magnetospheres.

This paper is the second in a series, following Corso & Lai (2026) (hereafter Paper I), where we study numerically the interaction between a moving conductor and an (otherwise homogeneous) magnetized plasma. We con sider an ideal MHD model which is scale-independent, even if it may not capture the nuances of specific binary systems highlighted above. In Paper I, we focused on the case of an unmagnetized (i.e. zero internal magnetic field) conductor [see also De Colle et al. (2025); Sharma et al. (2026)], and calculated the drag force $F$ on the conductor. We introduced the dimensionless drag coefficient, $C_A$, such that the mechanical power, $P=Fv_0$, imparted by the flow on the conductor is given by

where $\rho_0$, $v_0$, and $v_{A0}=(B_0^2/4\pi\rho_0)^{1/2}$ are the background values for the density, velocity, and Alfvén speed, and $R$ is the radius of the conducting sphere. Using MHD simulations, we determined how $C_A$ depends on the Alfvén Mach number and the plasma $\beta$ parameter:

where $p_0$ and $B_0$ are the background values for gas pressure and the magnetic field. We validated the basic picture of the Alfvén wing model by demonstrating that $C_A$ is approximately constant for $M_A\lesssim1$. We also showed that slow magnetosonic waves provide a correction to the picture through their own wing-like features.

In this paper, we extend our previous investigation to the case where the conductor has an internal magnetic field $\mathbf{B}_{\mathrm{in}}$. Not only does this extend the range of applicability of our model to include the cases where the companion has its own magnetosphere, but we will show that the internal field can substantially increase the drag imparted on the conductor. As in Paper I, our primary goal is to obtain the general scaling relation for the drag coefficient that can be applied to different astrophysical and planetary contexts.

This paper is organized as follows: in Section § 2, we review the numerical method for our simulations and introduce the problem setup; in Section Section 3, we present our results for the sub-Alfvénic regime and model the power dissipation using a semi-analytical treatment; in Section § 4, we present our results for the super-Alfvénic regime; we conclude with Section § 5.

Problem Setup and Method

Our problem setup is illustrated in Figure 1. We consider a system composed of a conducting sphere of radius $R$ embedded in a plasma that is flowing and magnetized. Far from the conductor, the magnetic field approaches $\mathbf{B}_0=B_0\hat{\mathbf{x}}$, and the plasma velocity approaches $\mathbf{v}_0=v_0\hat{\mathbf{z}}$. For the present study, we add an intrinsic uniform magnetic field, $\mathbf{B}_{\mathrm{in}}=B_{\mathrm{in}}\hat{\mathbf{m}}$ (where $\hat{\mathbf{m}}$ is the unit vector), to the conductor. We simulate our system using the magnetohydrodynamic (MHD) code, PLUTO (Mignone et al. 2007). We close the MHD equations with an ideal equation of state with adiabatic index $\gamma=5/3$.

As in Paper I, we write the magnetic field as

and evolve $\delta\mathbf{B}(\mathbf{r},t)$ in our MHD simulation. The time-independent background field is given by

where $\mathbf{d}(\hat{\mathbf{n}})\equiv3(\hat{\mathbf{r}}\cdot\hat{\mathbf{n}})\hat{\mathbf{r}}-\hat{\mathbf{n}}$ is a vector representing a dipole field configuration with a magnetic moment aligned with $\hat{\mathbf{n}}$. This background field satisfies the asymptotic condition $\mathbf{B}_{\mathrm{bkg}}\to\mathbf{B}_0$ at $r\to\infty$ and the boundary condition at $r = R$, i.e. the radial component of the field is continuous across the conductor’s surface. We see in Eq. (4) that $\mathbf{B}_{\mathrm{in}}$ introduces a dipole contribution to the external field. We express the orientation of $\hat{\mathbf{m}}$ relative to $\hat{\mathbf{x}}$ (the direction of $\mathbf{B}_0$) using the polar angles $\theta_m$ and $\phi_m$, the standard spherical coordinate angles with respect to the $x$-axis pole, i.e.

With this general setup, the MHD flow outside the conductor and the associated drag coefficient $C_{\mathrm{A}}$ (defined in Eq. (1)) depend on $M_{\mathrm{A}}$, $\beta$, and $b_{\mathrm{in}} \equiv B_{\mathrm{in}}/B_0$, i.e.,

As detailed in Paper I, we compute the drag by evaluating total momentum flux across a spherical surface that encloses the conductor, and making sure that the result is independent of the radius of the surface.

We introduce a non-slip boundary condition, demanding $\mathbf{v} = 0$, at the conductor’s surface ($r = R$). Note that this stands in contrast to the case where the conductor is unmagnetized, where velocity slippage is allowed. When $B_{\mathrm{in}} \ne0$, the normal component of the magnetic field (which must be continuous across the surface) is nonzero. The non-slip condition is required to ensure that the tangential component of $\mathbf{E}$ vanishes at $r = R$ as a consequence of the induction equation,

From Faraday’s Law, this fixes the component of $\mathbf{B}$ normal to the surface, maintaining a frozen flux from the conductor itself. Thus, the fixed magnetic flux at the conductor’s surface creates a boundary layer akin to that caused by a finite viscosity. To accommodate this new boundary condition, we initialize the velocity field as follows:

such that $\mathbf{v}$ vanishes at $r = R$ and approaches $\mathbf{v}_0 = v_0\hat{\mathbf{z}}$ for $r \gg R$. Note that in Eq. (8), the spherical coordinates $(r, \theta, \phi)$ are defined with respect to the $z$-axis pole.

All simulations are run on a discretized grid using spherical coordinates $(r, \theta, \phi)$. The inner boundary of the computational domain is set at $r = R$, and the outer boundary at $r_{\mathrm{out}} \gg R$. We use a zero-gradient outflow condition at $r = r_{\mathrm{out}}$. We also include a damping zone where evolved variables are driven to their background values close to the outer boundary. For most simulations, the computational domain spans $r \in[R, 10R]$ and the full unit sphere, using $N_r \times N_\theta\times N_\phi= 200 \times64 \times128$ grid points, where the radial points are spaced logarithmically, such that the cell thickness spans $\Delta r \in[0.012R, 0.12R]$, and the angular points are spaced uniformly. The damping zone spans $r \in[8R, 10R]$. For cases where $b_{\mathrm{in}} > 10$, we extend the domain to $r_{\mathrm{out}} = 50R$, scaling the damping zone to $r \in[40R, 50R]$ and $N_r = 340$ (approximately preserving the value of $\Delta r$ close to the conductor’s surface). This extension is necessary to keep $r_{\mathrm{out}}$ sufficiently far from the effective radius of the conductor, which scales with $b_{\mathrm{in}}$. See Paper I for more details on our numerical methods.

Sub-Alfvénic Regime

In this section, we examine the sub-Alfvénic regime ($M_{\mathrm{A}} \lesssim1$), where Alfvén wings dominate the dynamics. This is the main focus of this paper and of Paper I. The transition to the super-Alfvénic regime will be discussed in § 4.

Numerical Results

We carry out two suites of simulations. In the first, we examine the effects of modifying the strength and orientation of the dipole. In these simulations, we fix $M_{\mathrm{A}} = 0.1$ and $\beta= 2$, and we consider the internal magnetic field in the range $b_{\mathrm{in}} \in[0, 100]$ and angular configurations: $\theta_m = 0^\circ, 90^\circ, 180^\circ$, all with $\phi_m = 0$; and $\theta_m = 90^\circ$, $\phi_m = 90^\circ$ (labeled $\theta_m = 90^\circ_{\perp}$). We further test the cases of $\theta_m = 30^\circ, 60^\circ, 120^\circ, 150^\circ$ for $b_{\mathrm{in}} = 5$. Note that we refer to $\theta_m = 0^\circ$ as the “aligned” case with respect to $\mathbf{B}_0$. In the binary context where the conductor is embedded in the dominant magnetic field of a primary object, this corresponds to anti-alignment between dipole moments of the two binary components. Our second suite of simulations examines the effect of varying $M_{\mathrm{A}}$. We consider both the aligned and anti-aligned cases, holding $b_{\mathrm{in}} = 5$ and $\beta= 2$, and we test values in the range $M_{\mathrm{A}} \in[0.01, 30]$. Each simulation is run until $C_{\mathrm{A}}$ reaches a constant (time-independent) value, indicating that the system has reached a steady state.

Figure 2 depicts key diagnostic quantities for our fiducial runs with $b_{\mathrm{in}} = 5$, and $\theta_m = 0^\circ, 90^\circ$, and $180^\circ$. We notice a stark contrast between the magnetic and velocity perturbations in the aligned and anti-aligned cases. In the anti-aligned case, where external field lines fully wrap around the conductor’s magnetosphere (where its magnetic field dominates over $\mathbf{B}_0$), the profiles of $\mathbf{v}$ and $\lvert\delta\mathbf{B}\rvert$ strongly resemble that of the unmagnetized case studied in Paper I, except that, with $b_{\mathrm{in}} = 5$ the conductor has a larger effective size due to its magnetosphere. By contrast, these profiles for the $\theta_m = 0, 90^\circ$ cases exhibit much larger disturbances. Comparing the profiles for $\mathbf{B}$, we can see that the Alfvén wings trace the open field lines that connect to the conductor.

We show the main result of our parameter survey in Figure 3 by plotting $C_{\mathrm{A}}$ as a function of $b_{\mathrm{in}}$ for all considered $\mathbf{B}_{\mathrm{in}}$ orientations. As shown in Figure 2, for the anti-aligned case, the main effect of finite $b_{\mathrm{in}}$ is the increased effective size of the conductor due to its magnetosphere. In general, the location of the magnetopause is determined by balancing the momentum flux $(p + \rho v_\perp^2 + B_\parallel^2/8\pi)$ across the contact surface, where $p$ is the gas pressure, $v_\perp$ is the gas velocity perpendicular to the surface, and $B_\parallel$ is the magnetic field parallel to the surface (e.g., (Shu 1992)). Neglecting the plasma density and pressure within the magnetosphere, the magnetopause has a radius given by

which, for our parameter regime ($M_{\mathrm{A}} \lesssim1$, $\beta\lesssim1$), reduces to

This expression is valid for $b_{\mathrm{in}} \gg1$. When $b_{\mathrm{in}} \lesssim1$, $r_{\mathrm{m}}$ becomes less than $R$ and is no longer relevant for the system. Thus, the conductor’s effective radius, which we label as $R_{\mathrm{eff}}$, is $R$ in the $b_{\mathrm{in}} \ll1$ limit, and proportional to $r_{\mathrm{m}}$ in the $b_{\mathrm{in}} \gg1$ limit. The exact connection of these two limits depends on the orientation of $\mathbf{B}_{\mathrm{in}}$. In the anti-aligned case, the effective radius is given by the location of the magnetic null points, where $\mathbf{B}_{\mathrm{bkg}}=0$ (see Eq. (4)). The system has dipole moment direction $\hat{\mathbf{m}}=-\hat{\mathbf{x}}$, and the null points are confined to the $x$-axis. With these constraints, we can solve for the radius satisfying $\mathbf{B}_{\mathrm{bkg}}=0$, giving

where the superscript “$(180^\circ)$” implies that the equation applies to the anti-aligned system. If we replace $R$ with $R_{\mathrm{eff}}^{(180^\circ)}$ in Eq. (1), we find that the drag coefficient for the anti-aligned magnetic configuration is

where $C_{\mathrm{A0}}\equiv C_{\mathrm{A}}(b_{\mathrm{in}}=0)\simeq1.2$ for the adopted fiducial parameters. The dashed curve in Figure 3 depicts Eq. (12), which closely reproduces our numerical results for the anti-aligned system.

More importantly, Figure 3 shows that deviation from anti-alignment can significantly enhance the drag coefficient. For the aligned case ($\theta_{\mathrm{m}}=0$) in the $b_{\mathrm{in}}\gg1$ limit, our numerical result can be approximated by

In this limit, $C_{\mathrm{A}}^{(0^\circ)}$ follows the same $b_{\mathrm{in}}^{2/3}\propto r_{\mathrm{m}}^2$ scaling as Eq. (12), albeit with a much larger prefactor, suggesting that the effective size of the magnetosphere also contributes to the drag force in this case. Despite this agreement in scaling, even for $b_{\mathrm{in}} \sim1$ the difference in $C_{\mathrm{A}}$ between the aligned and anti-aligned cases is about an order of magnitude. We discuss the origin of this difference in § 3.2. The enhancement over the anti-aligned case decreases with $\theta_{\mathrm{m}}$, but it remains robust up to $\theta_{\mathrm{m}} \sim120^\circ$. Despite strong dependence of $C_{\mathrm{A}}$ on $\theta_{\mathrm{m}}$, we only observe modest changes in its value when changing $\phi_{\mathrm{m}}$.

In Figure 4, we show the effect of varying $M_{\mathrm{A}}$ in the aligned and anti-aligned cases for $b_{\mathrm{in}} = 5$. For both orientations, the drag coefficient approaches a constant value (independent of $M_{\mathrm{A}}$) when $M_{\mathrm{A}} \lesssim0.1$. The flattening

of the $C_{\mathrm{A}}$–$M_{\mathrm{A}}$ curve supports the validity of the Alfvén wing model in the sub-Alfvénic regime. We consider the transition to the super-Alfvénic regime in § 4.

Analysis

Linked Flux

Figure 5 highlights the strong dependence of the drag coefficient on the relative orientation between $\mathbf{B}_{\mathrm{in}}$ and $\mathbf{B}_0$. How can we understand this dependence?

From Figure 2, we see that the most important difference between the aligned and anti-aligned cases is the magnetic flux (or the number of field lines) connecting the conductor to infinity. Since the Alfvén wings trace such field lines, it is natural to expect that the increase in this “linked flux” leads to the enhancement of $C_{\mathrm{A}}$.

Figure 6 shows some examples of the $\mathbf{B}_{\mathrm{bkg}}$ field lines for different $\theta_{\mathrm{m}}$, with $b_{\mathrm{in}} = 100$ and $b_{\mathrm{in}} = 1$. We see that, in general, there are three distinct regions: a closed field line region (around the conductor), an “outside” region (where the field lines do not intercept the conductor), and a “linked” region (where field lines from the conductor extend to a large distance and merge with $\mathbf{B}_0$). The closed region is confined approximately by the magnetic null points, where $\mathbf{B}_{\mathrm{bkg}} = 0$.

To better understand the connection between $C_{\mathrm{A}}$ and the linked field lines, we numerically measure the flux of the conductor’s linked field lines. Since the flow is sub-Alfvénic, we have $\delta B \ll B_{\mathrm{bkg}}$, so the background field is a suitable approximation to the field of the steady-state system. To perform the measurement, we subdivide the surface of the conductor into a domain of surface elements. From each surface element, we trace a $\mathbf{B}_{\mathrm{bkg}}$ field line, and if the field line escapes to infinity, we count its corresponding surface element as “linked.” We then integrate $\mathbf{B}_{\mathrm{bkg}}\cdot\hat{\mathbf{r}}$ over all linked surface elements to obtain their total flux, which we label $\Phi_{\mathrm{L}}$.

We show $\Phi_{\mathrm{L}}$ as a function of $b_{\mathrm{in}}$ for several values of $\theta_{\mathrm{m}}$ in Figure 7(a). For all curves, we find two clear limits, given by

These limits represent distinct physical pictures. In the $b_{\rm in}\lesssim1$ limit, which is depicted by the $b_{\rm in}=1$ cases in Figure 6 (bottom row), the background field dominates over the conductor’s own dipole field. As a result, even for misaligned cases (see the bottom-middle panel), all the conductor’s field lines open up. This means that $\Phi_{\rm L}$ is equal to the conductor’s total flux. We call this case “saturated” since all possible flux is linked.

By contrast, in the $b_{\rm in}\gg1$ limit, depicted by the $b_{\rm in}=100$ cases in Figure 6, the dipole is strong enough to create a magnetosphere that dominates over the background field. In this case, there are always some closed field lines, so $\Phi_{\rm L}$ cannot keep up with the conductor’s total flux. Instead, we can consider how many field lines approaching from infinity reach the conductor. Field lines are likely to link if they fall within the influence of the conductor’s magnetosphere. The linked flux, then, is roughly given by the number of $\mathbf{B}_0$ field lines passing through the cross-section of the magnetosphere, giving the $b_{\rm in}\gg1$ limit of Eq. (14).

To provide a fitting formula for $\Phi_{\rm L}$ at a given value of $\theta_m$, we may capture the above limits using the following expression,

where $q(\theta_m)$, $p(\theta_m)$ are fitting parameters that control the location and sharpness of the transition between the two regimes. These fits are shown in Figure 7(a).

In Figure 7(b), we show $\Phi_{\rm L}$ as a function of $\theta_m$ for several values of $b_{\rm in}$. We normalize the curves using $\Phi_0$, the flux value at $\theta_m=0^\circ$. For $b_{\rm in}\lesssim1$, the curves remain flat for low values of $\theta_{\mathrm{m}}$ due to the previously described saturation effect. Beyond a critical value of $\theta_{\mathrm{m}}$, however, the dipole is twisted enough that its field lines begin to close and $\Phi_{\mathrm{L}}$ decreases.

For $b_{\mathrm{in}} \gtrsim10$, we find that the fit

where

describes the data very well [see Figure 7(b)]. We are not able to derive this specific trigonometric relationship analytically.³ The $\cos(\theta_{\mathrm{m}}/2)$ dependence was also suggested in previous works ((Saur et al. 2013; Presa et al. 2026)).

Power

To see how the linked magnetic field affects the Alfvén power, consider the original picture of Alfvén wing (Drell et al. 1965). Suppose a “linked” magnetic field $B_{\mathrm{L}}$ connects a moving conductor with effective area $A$ and the background field. The Alfvén power is given by

In general, we replace $B_{\mathrm{L}}A$ by the linked flux and obtain

For $\theta_{\mathrm{m}} \simeq180^\circ$, although there is no linked flux in the unperturbed field $\mathbf{B}_{\mathrm{bkg}}$ (see Figure 6), the conductor perturbs the background magnetic field to induce an effective linkage. Thus $\Phi_{\mathrm{L}}$ may be replaced by an effective flux, $\pi(R_{\mathrm{eff}}^{(180^\circ)})^2B_0$, consisting of the background field lines warped around a sphere of radius $R_{\mathrm{eff}}^{(180^\circ)}$.

The corresponding power can be written as

where $\alpha_0$ is a constant. This implies (see also Eq. (12))

where the coefficient in the second line comes from our numerical simulation.

For the $b_{\mathrm{in}}\gtrsim10$ systems that are not strongly anti-aligned, combining Eqs. (16)–(17) with Eq. (19), we have

where $\alpha$ is a constant. This implies

where the coefficient in the second line comes from our numerical simulation (see Eq. (13)).

Note that Eqs. (21) and (23) apply to different limiting parameter regimes of $b_{\mathrm{in}}$ and $\theta_{\mathrm{m}}$, and have different physical origins: Eq. (21) arises from “induced” magnetic linkage between the conductor and the external magnetic field, while Eq. (23) arises from “intrinsic” magnetic linkage. Combining Eqs. (21) and (23), we obtain a general expression for the drag coefficient that is valid in various regimes:

In particular, for $b_{\mathrm{in}}\ll1$, Eq. (24) gives the correct $B_{\mathrm{in}}=0$ limit ($C_{\mathrm{A}}\simeq1.2$) studied in Paper I; for $b_{\mathrm{in}}\gtrsim10$, it recovers Eqs. (21) and (23). For $b_{\mathrm{in}}\sim1$, the $\cos(\theta_{\mathrm{m}}/2)$ factor does not apply [see Figure 7(b)], but Eq. (24) can still provide a reasonable approximation to the drag coefficient.

Super-Alfvénic Regime

To this point, we have been considering the sub-Alfvénic regime, where $M_{\mathrm{A}}\lesssim1$. In this case, the power is largely dissipated from the system in the form of Alfvénic disturbances propagating along magnetic field lines and can be described by Eq. (1), with the drag coefficient that depends weakly on $M_{\mathrm{A}}$ (see Eq. (6)) and can be approximated by Eq. (24). We now turn our attention to the super-Alfvénic regime, where $M_{\mathrm{A}}\gtrsim1$. This regime is commonly found in the context of solar-system planets interacting with solar wind (see (Kivelson & Russell 1995; Eastwood et al. 2015; Dong et al. 2019)). The parameters of our high $M_{\mathrm{A}}$ runs are already described in § 3.1.

In Figure 8, we show the magnetic field configuration in the steady state of our fiducial super-Alfvénic run (aligned, $b_{\mathrm{in}}=5$, $M_{\mathrm{A}}=3$, $\beta=2$). Rather than the Alfvén wing structure seen in the sub-Alfvénic runs, a series of shocks develop where the flow collides with the conductor. We highlight the features relevant to our system below:

• The undisturbed upstream flow is decelerated and compressed at the bow shock, which forms the outermost boundary of the interaction.

• Downstream of the shock, the plasma and open magnetic field lines pile up in the magnetosheath, where both thermal pressure and magnetic pressure are high, and the flow is deflected around the conductor.

• On the upstream face, the accumulated magnetic pressure of the sheath balances the conductor’s internal magnetic field at the magnetopause (a contact discontinuity), where the bulk flow stagnates, the magnetic field is strongly draped, and plasma flow is largely inhibited.

• Behind the magnetopause, field lines linked to the conductor form two lobes with oppositely directed magnetic fields. These lobes extend far downstream in a structure called the magnetotail. The lobes eventually become close enough that reconnection can occur between them, and a strong current sheet develops.

• Many of the conductor’s field lines are contained to a local magnetosphere between the lobes.

The above paradigm has been extensively studied, especially in the planetary context, so we refer the reader to the literature for further discussion of the above features (e.g. (Eastwood et al. 2015)). For our purposes, their identification serves both as a validation of our model’s capacity to reproduce known physics and as a qualitative demonstration that our model has reached a new regime.

We quantitatively investigate this regime change through measurements of $C_{\mathrm{A}}$, which are shown in Figure 4 (the sub-Alfvénic regime was discussed in § 3.1). In the $M_{\mathrm{A}}\gtrsim1$ regime, the splitting in $C_{\mathrm{A}}$ between aligned and anti-aligned cases collapses. In the aligned case, the drag force plummets, whereas it steadily increases in the anti-aligned case. Beyond $M_{\mathrm{A}}\sim$ a few, both trends converge to be approximately linear in $M_{\mathrm{A}}$ although we have not simulated $M_{\mathrm{A}} > 30$ cases in this paper to systematically explore this trend.

We propose that, in the super-Alfvénic regime, ram pressure overtakes the Alfvén wing model as the dominant drag contribution. In other words, we may consider the conventional drag coefficient, $C_{\mathrm{D}}$, defined by the expression,

where we have used $R_{\mathrm{eff}}$ as a stand-in for the conductor’s effective radius, accounting for the size of its magnetosphere (see Eq. (9)). When the drag force is dominated by ram pressure, we expect $C_{\mathrm{D}}$ to be of order unity and roughly constant. Equating Eq. (1) with Eq. (25) gives

This matches the trend observed in Figure 4 in the $M_{\mathrm{A}} \gg1$ limit. For example, using the parameters of Figure 4, which set $R_{\mathrm{eff}} \simeq r_{\mathrm{m}} \simeq1.4R$, we find $C_{\mathrm{D}} \simeq1.0$. A detailed exploration of how $C_{\mathrm{D}}$ depends on various parameters in the super-Alfvénic regime is beyond the scope of this paper.

Summary and Conclusions

We have performed 3D MHD simulations of magnetized flow past a magnetized spherical conductor and numerically measured the drag force on the conductor. We introduce an Alfvénic drag coefficient $C_{\mathrm{A}}$ (see Eq. (1)) to characterize the problem. In Paper I (Corso & Lai 2026), we considered a non-magnetic conductor and determined $C_{\mathrm{A}}$ as a function of the Alfvén Mach number $M_{\mathrm{A}}$ and the plasma beta $\beta$ (see Eq. (2)). Here, by introducing an internal magnetic field $B_{\mathrm{in}}$ to the conductor, we have extended the model to include two additional parameters: the relative strength of the magnetic field, $b_{\mathrm{in}} = B_{\mathrm{in}}/B_0$, and its orientation, defined through Eq. (5) by the angles $\theta_{\mathrm{m}}$ and $\phi_{\mathrm{m}}$. Alongside these new parameters, we have also extended our model by probing super-Alfvénic flows with $M_{\mathrm{A}} \gtrsim1$.

Our study has revealed new features for the power dissipation and drag force in different parameter regimes. Most importantly, we find that $C_{\mathrm{A}}$ is greatly enhanced from the $B_{\mathrm{in}} = 0$ value when $b_{\mathrm{in}} \gtrsim1$ and $\theta_{\mathrm{m}}$ is not too close to $180^\circ$ (see Figure 3). We show that this enhancement is largely driven by open magnetic field lines that are linked to the conductor. Power is efficiently dissipated by Alfvénic disturbances propagating along these field lines, which carry greater magnetic flux compared to the anti-aligned case ($\theta_{\mathrm{m}} = 180^\circ$) or non-magnetic case ($B_{\mathrm{in}} = 0$). We summarize these findings through the expressions for $C_{\mathrm{A}}$ in different parameter regimes (Eqs. (21) and (23)), and we provide an approximate general expression for $C_{\mathrm{A}}$ in Eq. (24).

We have also explored the effect of varying $M_{\mathrm{A}}$ on the drag coefficient. We find that while the sub-Alfvénic regime is characterized by Alfvén wings and the related scaling laws for $C_{\mathrm{A}}$, the super-Alfvénic regime exhibits a qualitatively different behavior. For $M_{\mathrm{A}} \gtrsim1$, the Alfvén wing picture breaks down, and the drag force is instead dominated by ram pressure. We summarize this effect through a new expression for $C_{\mathrm{A}}$ in Eq. (26).

Our numerical results may have important ramifications for magnetic interactions in binaries. For example, recent observations have revealed possible evidence that a close-in planet can induce flares on its host star through magnetic interactions ((Ilin et al. 2025; Revilla et al. 2026)); the observed flare power appears to exceed the prediction of Alfvén wing models, and alternative models are often invoked (e.g. (Lanza 2009, 2013; Strugarek 2016)). However, most Alfvén wing models to-date only consider the effective size of the planet’s magnetosphere when accounting for its internal magnetic field (e.g. (Saur et al. 2013)), corresponding to our result for anti-aligned systems (Eq. (20)). We have shown in this paper that the Alfvén wing power may be enhanced by at least an order of magnitude when the planet’s magnetic field is more aligned with the surrounding field from its host star (Eq. (22)). This enhancement may be sufficient to explain the flare power discrepancy and retain the viability of the Alfvén wing model. Regardless, a proper understanding of the observational signatures of the magnetic star–planet interactions requires a more sophisticated numerical treatment of the plasma and radiative processes. This is our intended direction for future work on such systems, and some efforts are already underway to this end (e.g. Lyutikov 2026).

Concerning binary magnetic interactions in general, we note that the current study is confined to the local region around the conductor, which is assumed to be weakly magnetized compared to its primary companion. Our model cannot capture more global effects that involve large-scale interactions in the system. As such, the local physical processes studied in this paper may be occurring in conjunction with some of the global processes (e.g. (Lai 2012; Strugarek 2016; Beloborodov 2021; Most & Philippov 2022, 2023; Mahlmann & Beloborodov 2025)). Nevertheless, our finding of the enhanced drag/power for the $\theta_{\mathrm{m}} \ne180^\circ$ systems is consistent with previous global studies of binary interactions. For example, Strugarek (2016) carried out simulations of magnetic stellar wind interacting with a magnetic planet and found that the aligned case exhibits much stronger interaction than the anti-aligned case. For coalescing binary neutron stars, Palenzuela et al. (2013a,b) and Most & Philippov (2020, 2022) carried out force-free electrodynamics simulations for a range of magnetic field configurations. For binaries with anti-aligned magnetic dipoles, which correspond to our “aligned cases”, and for more general mis-aligned dipoles, they found that a magnetic flux tube connects the two stars and gets twisted by their rotation, leading to quasi-periodic magnetic flares. In contrast, for binaries with aligned dipoles, a very different dissipation mechanism operates, which relies on Kelvin–Helmholtz-type instability at the magnetospheric interface, generating Alfvénic turbulence and escaping fast magnetosonic waves (Mahlmann & Beloborodov 2025).

Despite the generally broader coverage of the parameter space in this study compared to Paper I, this coverage is not complete. For example, although Paper I considered different values of $\beta$, we have held it fixed in this study for simplicity. Our expectation is that the effect of varying $\beta$ would be subdominant compared to the other parameters explored in this paper. Our model also currently assumes that the plasma is ideal, which neglects the effects of viscosity and resistivity. Their associated Reynolds numbers could serve as additional dimensionless parameters of our system and introduce new dissipation mechanisms not captured by our current model.

Acknowledgments

This research was supported in part by the NSF grant DGE-2139899. Claude Code was used to assist with code development in this research. We thank Nils Deppe, Chuanfei Dong, Lawrence Kidder, Elias Most, Anatoly Spitkovsky and Jim Stone for useful discussions.

Software: Claude Code [1], Matplotlib (Hunter 2007), NumPy (Harris et al. 2020), ParaView (Ayachit 2015).

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