© The Authors 2025

1. Introduction

A plasma cloud with a sufficiently strong internal magnetic field prevents a background magnetic field from penetrating inside. When moving, it represents a magnetic obstacle to the surrounding plasma flow. Here we consider a static spherical obstacle as a simplification dictated by an analytical approach. For an obstacle to maintain an approximately spherical static shape, it must contain a strong magnetic field (low-β plasma), or its shape must be held by other means (e.g., by gravitation and sufficient internal pressure). Such an obstacle can serve as an idealized prototype of interstellar clouds, stars (including pulsars) with their winds, supernova remnants, or coronal mass ejections. When the object moves supersonically, it creates a bow shock. Bow shocks are commonly observed in the Universe, for example, ahead of rapidly moving stars (e.g., Van den Eijnden et al. 2022; van den Eijnden et al. 2024). These bow shocks are usually detected in infrared wavelengths or optical emission lines, although they are also expected to emit radio waves (Martínez et al. 2021; Van den Eijnden et al. 2022). The dynamics of a magnetized cloud and external forces acting on it were investigated by Parker (1957).

In recent years, the motion of magnetic obstacles through the plasma-containing medium has attracted scientific attention, especially when applied to astrophysical phenomena. Barkov et al. (2020) investigated bow shocks produced by pulsars moving through the interstellar medium which were detected in the hydrogen H_α line emission. The morphology of these bow shock nebulae allowed them to probe the properties of the interstellar medium on scales below 0.01 pc.

Green et al. (2019) conclude that the Bubble Nebula (NGC 7635) is a bow shock created by the O star BD+60°2522 as the star moves through the interstellar medium. Bustard & Zweibel (2021) modeled the acceleration of a cold interstellar cloud in a multiphase interstellar medium. Similarly, Ko et al. (2018) studied a cold cloud, acting as a magnetic obstacle, within the interstellar medium. Their study examined bow shock formation around the cold cloud, whilst also analyzing cosmic rays and waves triggered in the surrounding region to determine the cloud’s acceleration. Wentzel (1966) summarized and critically evaluated the Petschek mechanism for magnetic field reconnection and dissipation, subsequently applying it to interstellar clouds. Fast reconnection may lead to rapid acceleration and formation of a bow shock around the clouds. Dempsey et al. (2020) state that “Orion fingers” are a system of dozens of bow shocks, with the wings of shocks pointing to a common system of origin, centered on a dynamically disintegrating system of several massive stars. The shock heads propagate with velocities of up to 300–400 km s⁻¹, but the formation and physical properties of the “bullets” leading the shocks are not known. One hypothesis is that they are fast moving jupiters that escaped from their parent stars. Thompson (2017) shows that a macroscopic superconductive dipole (a candidate for dark cosmic matter) with a mass of 10²⁰ g and 1 mm size would form a 100 km magnetosphere moving through the interstellar plasma. Cargill & Pneuman (1986) studied the properties of an isolated magnetized plasmoid in a nonuniform magnetic field, such as what arises in stellar atmospheres. In rare instances, shock parameters at the Earth’s bow shock can approach the Mach numbers predicted at supernova remnants. Madanian et al. (2021) report one such case with an Alfvénic Mach number M_A = 27.

Romashets et al. (2008b) and Romashets & Vandas (2019) solved the magnetic field in a plasma sheath around a parabolic obstacle. Their solution is applicable for both the parallel and perpendicular ambient magnetic fields. Subsequently, Corona-Romero & Gonzalez-Esparza (2013) solved the magnetohydrodynamic (MHD) bow shock around a spherical obstacle for a parallel magnetic field. Additionally, Romashets et al. (2007) and Romashets & Vandas (2019) estimate the force the magnetic field applies on toroidal and spherical magnetic obstacles, respectively. Using a vector potential for modeling the magnetic field in the sheath region provides not only a magnetic field but also the volume current density distribution in the region (Romashets et al. 2002; Romashets & Vandas 2005; Romashets et al. 2008a; Vandas & Romashets 2010).

The current study expands on the results of Romashets et al. (2007) by applying the methods presented by Romashets & Vandas (2019) to find a current-free magnetic field in a plasma sheath surrounding a spherical obstacle propagating through the ambient plasma. Additionally, we estimate the force derived from the interaction of the obstacle with the plasma sheath, for both homogeneous and non-homogeneous magnetic fields.

In this study, we examine the deceleration of a supersonic obstacle in more detail. Our results demonstrate that the effect is more significant than in a subsonic case, appearing even if the ambient magnetic field is homogeneous.

2. Draping of a uniform ambient magnetic field in the direction perpendicular to the obstacle’s trajectory

We consider a spherical obstacle with a radius r₀ and centered at the origin of a Cartesian coordinate system x, y, and z (Fig. 1). The obstacle moves supersonically in the minus x direction and creates a bow shock which is assumed to be spherical with the radius σ₁. It is shifted by the distance x_s and centered in such a way that the distance to the front of the obstacle is much smaller than to its rear part. For calculations, we set σ₁ = 2.5r₀ and x_s = r₀ (these parameters are used in Fig. 1). Our task is to find a disturbed magnetic field around the obstacle (in its magnetosheath, see Fig. 1) and a magnetic force causing deceleration of the object, which appears in the presence of the magnetic field perpendicular to the direction of propagation.

Fig. 1. Geometric situation of our system. The position of the bow shock is represented by the solid red line, while the spherical obstacle is shown by the solid blue line. The space between the obstacle and the bow shock represents the magnetosheath.

Spherical coordinates of the obstacle, r, θ, and φ, are related to Cartesian coordinates by

$$\begin{array}{cc}\hfill x& =rcos\theta =x_s+\sigma cos\tau ,\hfill \\ \hfill y& =rsin\theta cos\phi =\sigma sin\tau cos\phi ,\hfill \\ \hfill z& =rsin\theta sin\phi =\sigma sin\tau sin\phi .\hfill \end{array}$$(1)

Here we also introduce spherical coordinates σ, τ, and φ, centered at the bow shock. The coordinate φ is common to both spherical systems and is an azimuthal angle around the x axis. The undisturbed magnetic scalar potential of the background magnetic field (with the magnitude B₀) is

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_{\mathrm{IMF}}=B_{0}y=B_0\sigma sin\tau cos\phi ,\end{array}$$(2)

where the acronym IMF may denote the interstellar or interplanetary magnetic field. The normal component of the magnetic field at the obstacle, at r = r₀, should vanish (the first condition) and remain continuous at the bow shock, at σ = σ₁ (the second condition). So we looked for the scalar potential Ψ_MS in the magnetosheath in the following form

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\mathrm{MS}}& =B_0{\displaystyle \sum _{n=1}^N}{a}_n{r}_0{P}_n^1(cos\theta )cos\phi \hfill \\ \hfill & \times [1+\frac{n}{n+1}{\left(\frac{r_0}{r}\right)}^{2n+1}]{\left(\frac{r}{r_0}\right)}^n,\hfill \end{array}$$(3)

where a_n are coefficients and P_n¹ are the associated Legendre polynomials. The first condition,

$$\begin{array}{c}\hfill B_{r,\mathrm{IMF}}(r_0,\theta ,\phi )={\frac{\partial {\mathrm{\Psi }}_{\mathrm{IMF}}}{\partial r}|}_{r=r_0}=0,\end{array}$$(4)

is exactly met, which can easily be verified using Eq. (3). An appropriate N and coefficients a_n were searched to approximately fulfill the second condition,

$$\begin{array}{c}\hfill B_{\sigma ,\mathrm{IMF}}({\sigma }_1,\tau ,\phi )=B_{\sigma ,\mathrm{MS}}({\sigma }_1,\tau ,\phi ).\end{array}$$(5)

The left-hand side of Eq. (5) is

$$\begin{array}{cc}\hfill B_{\sigma ,\mathrm{IMF}}({\sigma }_1,\tau ,\phi )& ={\frac{\partial {\mathrm{\Psi }}_{\mathrm{IMF}}}{\partial \sigma }|}_{\sigma ={\sigma }_1}=B_{0}sin\tau cos\phi \hfill \\ \hfill & =B_{0}g(\tau )cos\phi ,\hfill \end{array}$$(6)

where we have formally introduced the function g(τ) = sin τ used below. The right-hand side of Eq. (5) is formally expressed as

$$\begin{array}{c}\hfill B_{\sigma ,\mathrm{MS}}({\sigma }_1,\tau ,\phi )={\frac{\partial {\mathrm{\Psi }}_{\mathrm{MS}}}{\partial \sigma }|}_{\sigma ={\sigma }_1}=B_{0}cos\phi {\displaystyle \sum _{n=1}^N}{a}_n{f}_n(\tau ).\end{array}$$(7)

We write Ψ_MS as a function of the bow-shock-centered coordinates, using

$$\begin{array}{cc}\hfill r& =\sqrt{{\sigma }^2+x_s^2+2\sigma x_{s}cos\tau },\hfill \end{array}$$(8)

$$\begin{array}{cc}\hfill cos\theta & =\frac{x_s+\sigma cos\tau }{\sqrt{{\sigma }^2+x_s^2+2\sigma x_{s}cos\tau }},\hfill \end{array}$$(9)

which follow from Eq. (1), and we get from Eq. (7)

$$\begin{array}{cc}\hfill f_n(\tau )& =n{P}_n^1\left(\frac{x_s+{\sigma }_{1}cos\tau }{\sqrt{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }}\right)\hfill \\ \hfill & \times \frac{{\sigma }_1+x_{s}cos\tau }{r_0}\hfill \\ \hfill & \times \{[1+\frac{n}{n+1}{\left(\frac{r_0^2}{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }\right)}^{\frac{2n+1}{2}}]\hfill \\ \hfill & \times {\left(\frac{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }{r_0^2}\right)}^{\frac{n-2}{2}}\hfill \\ \hfill & -\frac{2n+1}{n+1}{\left(\frac{r_0^2}{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }\right)}^{\frac{n+3}{2}}\}\hfill \\ \hfill & -P_n^{1\prime }\left(\frac{x_s+{\sigma }_{1}cos\tau }{\sqrt{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }}\right)\hfill \\ \hfill & \times [1+\frac{n}{n+1}{\left(\frac{r_0^2}{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }\right)}^{\frac{2n+1}{2}}]\hfill \\ \hfill & \times \frac{{\sigma }_1{x}_s{sin}^2\tau }{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }\hfill \\ \hfill & \times {\left(\frac{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }{r_0^2}\right)}^{\frac{n-1}{2}}.\hfill \end{array}$$(10)

The prime means a derivative of a function.

The coefficients a_n were searched via minimization

$$\begin{array}{c}\hfill {\displaystyle \underset{S}{\oint }}{[B_{\sigma ,\mathrm{MS}}({\sigma }_1,\tau ,\phi )-B_{\sigma ,\mathrm{IMF}}({\sigma }_1,\tau ,\phi )]}^2\mathrm{d}S=\min .\end{array}$$(11)

where the integration is over the bow shock. It leads to a system of linear equations for a_n, represented by the matrix equation

$$\begin{array}{c}\hfill FA=G\end{array}$$(12)

where F = (f_nm) is a matrix, and A = (a_n) and G = (g_n) are vectors, with elements

$$\begin{array}{c}\hfill g_n={\displaystyle {\int }_0^{\pi }}{f}_n(\tau )g(\tau )sin\tau \mathrm{d}\tau ,\end{array}$$(13)

$$\begin{array}{c}\hfill f_{\mathit{mn}}={\displaystyle {\int }_0^{\pi }}{f}_n(\tau )f_m(\tau )sin\tau \mathrm{d}\tau .\end{array}$$(14)

The solution of Eq. (12) gives a set of best-fitting coefficients a_n. We used N = 15 for all calculations. The magnetic field in the magnetosheath is given by B_MS = grad Ψ_MS. Figure 2 illustrates this case.

Fig. 2. Magnetic field lines around the obstacle when the IMF is perpendicular to the obstacle’s trajectory. Magnetic field lines are represented as white lines, the obstacle is illustrated as a white circle, and the background shows magnetic field magnitude contours.

3. Comparison with draping of a uniform ambient magnetic field along the obstacle’s trajectory

We proceeded in the same manner as the previous section. The undisturbed potential is now

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_{\mathrm{IMF}}=B_0(x-x_s)=B_0\sigma cos\tau .\end{array}$$(15)

We were looking for Ψ_MS in the following form

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\mathrm{MS}}& =B_0{\displaystyle \sum _{n=1}^N}{b}_n{r}_0{P}_n(cos\theta )\hfill \\ \hfill & \times [1+\frac{n}{n+1}{\left(\frac{r_0}{r}\right)}^{2n+1}]{\left(\frac{r}{r_0}\right)}^n,\hfill \end{array}$$(16)

where P_n are the Legendre polynomials. This form fulfills the first condition. For the second condition, we have

$$\begin{array}{c}\hfill B_{\sigma ,\mathrm{IMF}}({\sigma }_1,\tau ,\phi )=B_{0}cos\tau =B_{0}g(\tau )\end{array}$$(17)

(note that g(τ) here is different from that in Eq. (6)) and

$$\begin{array}{c}\hfill B_{\sigma ,\mathrm{MS}}({\sigma }_1,\tau ,\phi )=B_0{\displaystyle \sum _{n=1}^N}{b}_n{f}_n(\tau )\end{array}$$(18)

with

$$\begin{array}{cc}\hfill f_n(\tau )& =n{P}_n\left(\frac{x_s+{\sigma }_{1}cos\tau }{\sqrt{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }}\right)\frac{{\sigma }_1+x_{s}cos\tau }{r_0}\hfill \\ \hfill & \times \{[1+\frac{n}{n+1}{\left(\frac{r_0^2}{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }\right)}^{\frac{2n+1}{2}}]\hfill \\ \hfill & \times {\left(\frac{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }{r_0^2}\right)}^{\frac{n-2}{2}}\hfill \\ \hfill & -\frac{2n+1}{n+1}{\left(\frac{r_0^2}{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }\right)}^{\frac{n+3}{2}}\}\hfill \\ \hfill & -P_n^{\prime }\left(\frac{x_s+{\sigma }_{1}cos\tau }{\sqrt{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }}\right)\hfill \\ \hfill & \times [1+\frac{n}{n+1}{\left(\frac{r_0^2}{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }\right)}^{\frac{2n+1}{2}}]\hfill \\ \hfill & \times \frac{{\sigma }_1{x}_s{sin}^2\tau }{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }\hfill \\ \hfill & \times {\left(\frac{{\sigma }_1^2+x_s^2+2{\sigma }_1{x}_{s}cos\tau }{r_0^2}\right)}^{\frac{n-1}{2}}.\hfill \end{array}$$(19)

Figure 3 shows the magnetic field of the parallel case.

Fig. 3. Magnetic field lines around the obstacle when the IMF is parallel to the obstacle’s trajectory. Magnetic field lines are represented as white lines, the white circle denotes the obstacle, and the background shows magnetic field magnitude contours.

4. Combined case with presence of upstream parallel and perpendicular magnetic fields

The undisturbed potential is now

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\mathrm{IMF}}& =B_{0}cos{\varphi }_0(x-x_s)+B_{0}sin{\varphi }_{0}y\hfill \\ \hfill & =B_{0}cos{\varphi }_0\sigma cos\tau +B_{0}sin{\varphi }_0\sigma sin\tau cos\phi .\hfill \end{array}$$(20)

The angle ϕ₀ = 0 corresponds to the parallel case and ϕ₀ = π/2 to the perpendicular case. The scalar potential can be split into parallel and perpendicular parts, their coefficients each solved independently, and the resulting fields added. Figure 4 shows a magnetic field of an oblique upstream IMF. We examined how the condition at the bow shock is fulfilled in this case. The relative deviation |(B_σ, MS−B_σ, IMF)/B_σ, IMF| at the bow shock was mostly below 10⁻⁴, with the maximum being in the region of 2 × 10⁻³.

Fig. 4. Magnetic field lines around the obstacle when the IMF is inclined by 34° to the obstacle’s trajectory. Magnetic field lines are represented as white lines, the white circle denotes the obstacle, and the background shows magnetic field magnitude contours.

5. Combined case with a nonuniform ambient magnetic field

The undisturbed magnetic potential of the IMF is now

$$\begin{array}{c}\hfill {\mathrm{\Psi }}_{\mathrm{IMF}}=B_{0}cos{\varphi }_{0}x+B_{0}sin{\varphi }_{0}y+\frac{B_1}{2r_0}(x^2-y^2),\end{array}$$(21)

where B₁/r₀ represents the magnetic field component B_x rate of change in the x direction. In the spherical coordinates related to the obstacle, we have

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\mathrm{IMF}}& =B_{0}cos{\varphi }_{0}rcos\theta +B_{0}sin{\varphi }_{0}rsin\theta cos\phi \hfill \\ \hfill & +\frac{B_1}{4r_0}{r}^2[2-{sin}^2\theta (3+cos2\phi )].\hfill \end{array}$$(22)

We assume that the magnetosheath scalar potential is now in the following form:

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\mathrm{MS}}& =B_{0}cos{\varphi }_0{\displaystyle \sum _{n=1}^N}{b}_n{r}_0{P}_n(cos\theta )\hfill \\ \hfill & \times [1+\frac{n}{n+1}{\left(\frac{r_0}{r}\right)}^{2n+1}]{\left(\frac{r}{r_0}\right)}^n\hfill \\ \hfill & +B_{0}sin{\varphi }_0{\displaystyle \sum _{n=1}^N}{a}_n{r}_0{P}_n^1(cos\theta )cos\phi \hfill \\ \hfill & \times [1+\frac{n}{n+1}{\left(\frac{r_0}{r}\right)}^{2n+1}]{\left(\frac{r}{r_0}\right)}^n\hfill \\ \hfill & +\frac{B_1}{r_0}{\displaystyle \sum _{n=2}^N}{c}_n{r}_0^2{P}_n^2(cos\theta )cos2\phi \hfill \\ \hfill & \times [1+\frac{n}{n+1}{\left(\frac{r_0}{r}\right)}^{2n+1}]{\left(\frac{r}{r_0}\right)}^n\hfill \\ \hfill & +\frac{B_1}{r_0}{\displaystyle \sum _{n=1}^N}{d}_n{r}_0^2{P}_n(cos\theta )\hfill \\ \hfill & \times [1+\frac{n}{n+1}{\left(\frac{r_0}{r}\right)}^{2n+1}]{\left(\frac{r}{r_0}\right)}^n.\hfill \end{array}$$(23)

The sets of coefficients {a_n}, {b_n}, {c_n}, and {d_n} were calculated separately comparing the terms at B₀cos ϕ₀, B₀ sin ϕ₀cos φ, B₁/r₀, and B₁/r₀ cos2φ for the second condition. Due to symmetry, there is no necessity to keep z² in the scalar potential.

Figures 5–7 show the magnetic field of a parallel, perpendicular, and oblique upstream IMF with a gradient. The relative deviation |(B_σ, MS−B_σ, IMF)/B_σ, IMF| at the bow shock was again mostly below 10⁻⁴ in these cases, with the maximum being several percentage points.

Fig. 5. Magnetic field lines around the obstacle when ϕ0 = 0° and B1 = B0/10. Magnetic field lines are represented as white lines, the white circle denotes the obstacle, and the background shows magnetic field magnitude contours.

Fig. 6. Magnetic field lines around the obstacle when ϕ0 = 90° and B1 = B0/10. Magnetic field lines are represented as white lines, the white circle denotes the obstacle, and the background shows magnetic field magnitude contours.

Fig. 7. Magnetic field lines around the obstacle when ϕ0 = 34° and B1 = B0/10. Magnetic field lines are represented as white lines, the white circle denotes the obstacle, and the background shows magnetic field magnitude contours.

6. Magnetic field around the obstacle without a bow shock

For comparison, we consider a case in which a bow shock is not present. We assume a potential magnetic field which is tangential to the obstacle’s boundary, and it tends to the background field given by Eq. (21) toward infinity. Its scalar potential is

$$\begin{array}{cc}\hfill {\mathrm{\Psi }}_{\mathrm{IMF}}& =B_{0}cos{\varphi }_{0}r[1+\frac{1}{2}{\left(\frac{r_0}{r}\right)}^3]cos\theta \hfill \\ \hfill & +B_{0}sin{\varphi }_{0}r[1+\frac{1}{2}{\left(\frac{r_0}{r}\right)}^3]sin\theta cos\phi \hfill \\ \hfill & +\frac{B_1}{4r_0}{r}^2[1+\frac{2}{3}{\left(\frac{r_0}{r}\right)}^5]\hfill \\ \hfill & \times [2-{sin}^2\theta (3+cos2\phi )].\hfill \end{array}$$(24)

Figures 8–9 show the magnetic field for parallel and oblique cases, respectively.

Fig. 8. Magnetic field lines around the obstacle when ϕ0 = 0° and B1 = B0/10. A bow shock is not present. Magnetic field lines are represented as white lines, the obstacle is illustrated as a white circle, and magnetic field magnitude contours are shown in the background.

Fig. 9. Magnetic field lines around the obstacle when ϕ0 = 34° and B1 = B0/10. A bow shock is not present. Magnetic field lines are represented as white lines, the obstacle is illustrated as a white circle, and magnetic field magnitude contours are shown in the background.

7. Diamagnetic force on the obstacle

The diamagnetic force is given by

$$\begin{array}{c}\hfill \mathit{F}=-{\displaystyle \oint P_{\mathrm{mag}}}\mathit{n}\mathrm{d}S,\end{array}$$(25)

where integration is over the obstacle’s boundary. The P_mag is the magnetic pressure,

$$\begin{array}{c}\hfill P_{\mathrm{mag}}=\frac{B^2}{2{\mu }_0}=\frac{B_{\theta }^2}{2{\mu }_0}+\frac{B_{\phi }^2}{2{\mu }_0},\end{array}$$(26)

μ₀ is the magnetic permeability, and B_r is missing because it is zero at the obstacle’s boundary. The n is the normal unit vector to the boundary, pointing outward,

$$\begin{array}{c}\hfill \mathit{n}=(cos\theta ,sin\theta cos\phi ,sin\theta sin\phi ).\end{array}$$(27)

The force in the principal direction of the obstacle’s motion is

$$\begin{array}{cc}\hfill F_x& =-\frac{r_0^2}{2{\mu }_0}{\displaystyle {\int }_0^{2\pi }}{\displaystyle {\int }_0^{\pi }}[B_{\theta }^2(r_0,\theta ,\phi )+B_{\phi }^2(r_0,\theta ,\phi )]\hfill \\ \hfill & \times sin\theta cos\theta \mathrm{d}\theta \mathrm{d}\phi .\hfill \end{array}$$(28)

We calculated this force using magnetic field components for the fields derived in this paper, with the results summarized in Fig. 10. The force is scaled by

$$\begin{array}{c}\hfill F_s=\frac{\pi r_0^2{B}_0^2}{2{\mu }_0}.\end{array}$$(29)

Fig. 10. Diamagnetic force acting on the obstacle for different field gradients as a function of the background field angle ϕ0. Black lines are for B1 = 0 (without gradient), blue lines for B1 = B0/10, red lines for B1 = 2B0/10, and green lines for B1 = 3B0/10. Solid lines are for cases when the bow shock is present, and dashed lines are for cases without a bow shock.

A negative force indicates acceleration and a positive force indicates deceleration.

In Fig. 10, we compare the diamagnetic force with a different B₁ and the angle ϕ₀ acting on subsonic and supersonic obstacles, labeled 1 and 2 in Eq. (30). The difference between the two is approximately described as

$$\begin{array}{c}\hfill F_2=F_1+0.7F_s.\end{array}$$(30)

Using Eqs. (28) and (24), we get

$$\begin{array}{c}\hfill F_1=-\frac{2\pi r_0^2}{{\mu }_0}{B}_0{B}_{1}cos{\varphi }_0.\end{array}$$(31)

In principle, F₂ can also be calculated analytically, but the resulting formula is lengthy.

For the chosen position and size of the bow shock surface, a slowly moving spherical obstacle is accelerated at a higher rate. In a uniform magnetic field, the slow obstacle is neither accelerated nor decelerated. Conversely, the supersonic obstacle is decelerated in a uniform magnetic field, F_x < 0, and it is accelerated at a modest rate if B₁ is very large, approaching a value of 0.3B₀.

Formula (30) was found for our geometric situation where the front magnetosheath thickness is $\mathrm{\Delta }r=\frac{1}{2}{r}_0$. Farris & Russell (1994) presented the relationship

$$\begin{array}{c}\hfill \frac{\mathrm{\Delta }r}{r_0}=1.1\frac{{\rho }_1}{{\rho }_2},\end{array}$$(32)

based on measurements at the Earth’s bow shock (ρ₁ and ρ₂ are upstream and downstream densities at the bow shock, respectively). It follows that Δr should be smaller for larger Mach numbers (M), the field intensifies, and the diamagnetic force is greater.

8. Other forces acting on the obstacle

Here, we discuss drag and thermal forces and compare them with the diamagnetic force.

8.1. Drag force from the ambient plasma

We first assume a scenario where the obstacle is a solid elastic object. In this scenario, we also assume that the ambient plasma is frozen into the magnetic field such that it can be reflected from the sphere due to a collision-like process. Thus, the change of momentum of a single proton during the collision is

$$\begin{array}{cc}\hfill -\mathrm{\Delta }{p}_x& =-m_{\mathrm{p}}(v-v_{\mathrm{MS}})[cos(\pi -2\theta )-cos0)\hfill \\ \hfill & =-m_{\mathrm{p}}(v-v_{\mathrm{MS}})(-cos2\theta -1)\hfill \\ \hfill & =m_{\mathrm{p}}(v-v_{\mathrm{MS}})(1+cos2\theta )\hfill \\ \hfill & =2m_{\mathrm{p}}(v-v_{\mathrm{MS}}){cos}^2\theta .\hfill \end{array}$$(33)

Here, v and v_MS are the obstacle and ambient plasma velocities, respectively, and m_p is the proton mass. We consider a surface element dS at the sphere. The momentum change of one particle at dS is given above. The number of particles interacting with dS during the time interval dt is n(v − v_MS) dt cos θ dS. Therefore, the total momentum change, calculated as the momentum change of a single particle multiplied by the proportion of reflected protons (ξ), and the number density of protons (n ≈ ρ/m_p where ρ is the plasma density) colliding in time interval dt is

$$\begin{array}{cc}\hfill \mathrm{d}{p}_{\mathit{xS}}& =2m_{\mathrm{p}}\xi n{(v-v_{\mathrm{MS}})}^2{cos}^3\theta \mathrm{d}S\mathrm{d}t\hfill \\ \hfill & =2\xi \rho {(v-v_{\mathrm{MS}})}^2{cos}^3\theta \mathrm{d}S\mathrm{d}t\hfill \\ \hfill & \approx 2\xi \left[\frac{\rho }{{\rho }_0}\right]{\rho }_0{(v-v_0)}^2{cos}^3\theta ,\hfill \end{array}$$(34)

where $\left[\frac{\rho }{{\rho }_0}\right]$ stands for the density jump at the bow shock, and subscript 0 refers to upstream values. The total contribution is given by the integration over a hemisphere,

$$\begin{array}{cc}\hfill F_{\mathrm{d}}& =\frac{\mathrm{d}{p}_x}{\mathrm{d}t}\hfill \\ \hfill & =4\xi \left[\frac{\rho }{{\rho }_0}\right]{\rho }_0\pi r_0^2{(v-v_0)}^2{\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}}{cos}^3\theta sin\theta \mathrm{d}\theta \hfill \\ \hfill & =\xi \left[\frac{\rho }{{\rho }_0}\right]{\rho }_0\pi r_0^2{(v-v_0)}^2.\hfill \end{array}$$(35)

By comparing this with the commonly used formula for the drag force (F_d = C_dρ₀A(v − v₀)² where A is the cross-sectional area), we get the drag coefficient C_d ∼ ξρ/ρ₀. It is important to highlight that in order to keep quasi-neutrality of the ambient plasma, it should be ξ ≪ 1. In addition, ρ₀/ρ < 1, and this value decreases as M² increases. In this scenario, we can therefore conclude that F_d tends to be weaker than the diamagnetic force which increases in the sheath region.

This reflection-like effect on protons can be observed by in situ registers of the solar wind as well as in numerical simulations of planetary foreshocks (e.g., Blanco-Cano et al. 2009, and references therein). This kind of process is likely to be observable for quasi-parallel configurations of the magnetic field.

We assume a second scenario where the protons are frozen into the magnetic field and the plasma has an effective dynamic viscosity (η). In this scenario, the drag force F_D would arise from the viscous effect of the ambient plasma moving tangentially with respect to the obstacle’s boundary. In order to solve F_D for this case, we followed the approach by Corona-Romero et al. (2022):

$$\begin{array}{cc}\hfill F_{\mathrm{D}}& \approx S\eta \frac{\partial v_x}{\partial y}\hfill \\ \hfill & \approx 4\pi r_0^2\eta \sqrt{\mathfrak{R}}\frac{v-v_{\mathrm{MS}}}{r_0}.\hfill \end{array}$$(36)

To arrive at the last equation, the Reynolds number must be large (ℜ ≫ 1), since we assumed that the viscous interaction is limited to the boundary layer of width $\sim r_0/\sqrt{\mathfrak{R}}$. However, since ℜ = ρr₀|v − v_MS|/η, a large ℜ necessarily implies that η ≪ 1. Hence the effects of viscosity are only significant in a small region, which makes F_D weaker, as reported by Corona-Romero et al. (2022).

If we explore Eq. (36) inside the magnetosheath, we need to address the effects of the shock over η. For this purpose, we begin with the fact that the plasma is collisionless. Therefore the viscosity-like phenomenon (formally known as hybrid viscosity, see Subramanian et al. 1996) arises from MHD processes, rather than mechanical interactions such as collisions. Thus, we assume that viscosity in the magnetized plasma is expressed by the hybrid viscosity provoked by spontaneous irregularities (tangled structures) in the magnetic field,

$$\begin{array}{c}\hfill \eta \sim {10}^{-14}\frac{{\lambda }_1}{{\lambda }_2}\frac{T^{5/2}}{ln\mathrm{\Lambda }},\end{array}$$(37)

where Λ, λ₁, and λ₂ are the plasma parameter, the scale of irregularities, and the mean free path, respectively, and T is the temperature. The plasma parameter, expressed as its logarithm, is usually in the range of 10 to 20, and the scale of magnetic irregularities should be on the order of the distance between protons, that is, (ρ/m_p)^−1/3 = n^−1/3. Furthermore, we can approximate λ₂ through the Larmor radius ($m_{\mathrm{p}}\sqrt{2k_{B}T/m_{\mathrm{p}}}/|q|B$),

$$\begin{array}{c}\hfill \eta \sim {10}^{-15}\frac{|q|B}{n^{1/3}\sqrt{2m_{\mathrm{p}}{k}_B}}{T}^2,\end{array}$$(38)

with

$$\begin{array}{c}\hfill T=T_0\frac{[2\gamma M^2-(\gamma -1)][{(\gamma -1)}^2+2]}{{(\gamma +1)}^2{M}^2}.\end{array}$$(39)

Taking into account the presence of the bow shock and the subsonic nature of the flow in the sheath region, where v − c_s ≈ 0.5v_A (c_s and v_A denote sound and Alfvén speeds, respectively), and assuming that the maximum magnetic pressure in the magnetosheath is equal to the dynamic pressure, we can express F_D as

$$\begin{array}{c}\hfill F_{\mathrm{D}}\approx \pi r_0^2{C}_{\mathrm{D}}\frac{4F_s}{\pi r_0^2}=4C_{\mathrm{D}}{F}_s,\end{array}$$(40)

and this decelerating force is comparable with F_s when C_D = 1/4. The drag coefficient can be smaller than this value because the magnetic field is frozen in the ambient plasma. Only a fraction of plasma particles arriving at the cross-sectional area of the obstacle contribute to slowing it down, while other particles follow the modified magnetic field lines without “reflecting” from the boundary of the obstacle.

8.2. Force due to thermal pressure

We assume that the ambient plasma can be treated as a polytropic ideal gas. For fast shocks we know that the jump in thermal pressure across the shock front can be approximated by

$$\begin{array}{c}\hfill \frac{P_{\mathrm{T}}}{P_{0\mathrm{T}}}\approx \frac{2\gamma }{2\gamma +1}{M}^2.\end{array}$$(41)

Force due to thermal pressure is

$$\begin{array}{c}\hfill {\mathit{F}}_{\mathrm{P}}\approx V\mathrm{\nabla }{P}_{\mathrm{T}}.\end{array}$$(42)

Here, V is the volume of the spherical obstacle, and P_T is the thermal pressure. For the slow obstacle, the gradient of P_T is approximately ΔP_0T/(2r₀), since the pressure diminshes with distance. This approximation, combined with the shock effects on P_0T, leads to

$$\begin{array}{c}\hfill F_{\mathrm{P}}=-\frac{\mathrm{\Delta }{P}_{0\mathrm{T}}}{2r_0}\frac{2\gamma }{2\gamma +1}{M}^{2}V.\end{array}$$(43)

In contrast to the drag force, which decreases due to shock effects, the magnitude of F_P increases with M² and is accelerating. For the solar wind scenario, we can estimate F_P in terms of F_s. The ratio of magnetic pressure to thermal pressure in the ambient solar wind is ≈0.1. At the distance r₀, both magnetic and thermal pressures drop by five percent. This leads to

$$\begin{array}{c}\hfill F_{\mathrm{P}}\approx -\frac{4}{3}\pi r_0^3\frac{10F_s}{\pi r_0^2}\frac{0.05}{r_0}=-\frac{2}{3}{F}_s,\end{array}$$(44)

and this force can significantly contribute to acceleration.

9. Conclusions

We analytically derived the modification of a background magnetic field by a propagating obstacle, assuming a current-free field configuration. The ambient field may have an arbitrary orientation. The modified field acts on the obstacle by a diamagnetic force. For a subsonic obstacle, the diamagnetic force is only present when the background field has a gradient, and acts as an accelerating force. In the case of a supersonic obstacle, when a bow shock is present, the diamagnetic force decelerates the obstacle and has a smaller magnitude. The diamagnetic force is comparable in magnitude to other forces, specifically to the drag force and the force arising from the thermal pressure gradient in the background plasma.

The quantity σ₁ − x_s − r₀, referred to as the stand-off distance of the bow shock, mostly depends on the compression ratio or the density jump across the shock and can be estimated by empirical or analytical expressions. An empirical formula by Farris & Russell (1994) was used for Eq. (32). An analytic solution for this distance is given in Corona-Romero & Gonzalez-Esparza (2013) (their Eq. (7)).

The analytical results obtained for magnetic field modification by spherical obstacles moving at slow and high speeds can be applied to analytical consideration of their kinematics. Taking into account all the forces acting on the obstacle, its velocity profile can be calculated following a similar approach to that described by Romashets & Vandas (2001).

The assumption that the magnetic field in the magnetosheath is current-free (potential) is an approximation. However, observational evidence suggests that this is a logical simplification. Earth’s magnetic field represents an obstacle to the incoming solar wind, resulting in the formation of a bow shock and magnetosheath. Numerous measurements of the magnetic field in Earth’s magnetosheath have been obtained and potential models of its magnetic field configuration have been proposed. We recently performed a detailed comparison of these current-free models with measurements and found good agreement in the front (subsolar) region of the magnetosheath (Vandas et al. 2020; Vandas & Romashets 2024).

A spherical bow shock is another simplification, as a plasma sheet is expected in the tail region. Referring again to Earth’s magnetosheath and the preceeding discussion, a plasma sheet is indeed present in the tail region of Earth’s magnetosphere. Nevertheless, main dynamical interactions discussed in this paper occur in the front region, where the potential approximation of the magnetic field is acceptable.

Acknowledgments

This research was supported by NSF grant 2230363. M. V. acknowledges support from Grant 17-06065S by the Grant Agency of the Czech Republic and from the AV ČR Grant RVO:67985815. P. Corona-Romero is grateful for Investigadores por México-CONAHCYT (IxM – CONAHCYT Research Fellows) project 1045 Space Weather Service, financed by “Consejo Nacional de Ciencia y Tecnología” (CONAHCYT).

References

All Figures

	Fig. 1. Geometric situation of our system. The position of the bow shock is represented by the solid red line, while the spherical obstacle is shown by the solid blue line. The space between the obstacle and the bow shock represents the magnetosheath.
In the text

	Fig. 2. Magnetic field lines around the obstacle when the IMF is perpendicular to the obstacle’s trajectory. Magnetic field lines are represented as white lines, the obstacle is illustrated as a white circle, and the background shows magnetic field magnitude contours.
In the text

	Fig. 3. Magnetic field lines around the obstacle when the IMF is parallel to the obstacle’s trajectory. Magnetic field lines are represented as white lines, the white circle denotes the obstacle, and the background shows magnetic field magnitude contours.
In the text

	Fig. 4. Magnetic field lines around the obstacle when the IMF is inclined by 34° to the obstacle’s trajectory. Magnetic field lines are represented as white lines, the white circle denotes the obstacle, and the background shows magnetic field magnitude contours.
In the text

	Fig. 5. Magnetic field lines around the obstacle when ϕ0 = 0° and B1 = B0/10. Magnetic field lines are represented as white lines, the white circle denotes the obstacle, and the background shows magnetic field magnitude contours.
In the text

	Fig. 6. Magnetic field lines around the obstacle when ϕ0 = 90° and B1 = B0/10. Magnetic field lines are represented as white lines, the white circle denotes the obstacle, and the background shows magnetic field magnitude contours.
In the text

	Fig. 7. Magnetic field lines around the obstacle when ϕ0 = 34° and B1 = B0/10. Magnetic field lines are represented as white lines, the white circle denotes the obstacle, and the background shows magnetic field magnitude contours.
In the text

	Fig. 8. Magnetic field lines around the obstacle when ϕ0 = 0° and B1 = B0/10. A bow shock is not present. Magnetic field lines are represented as white lines, the obstacle is illustrated as a white circle, and magnetic field magnitude contours are shown in the background.
In the text

	Fig. 9. Magnetic field lines around the obstacle when ϕ0 = 34° and B1 = B0/10. A bow shock is not present. Magnetic field lines are represented as white lines, the obstacle is illustrated as a white circle, and magnetic field magnitude contours are shown in the background.
In the text

	Fig. 10. Diamagnetic force acting on the obstacle for different field gradients as a function of the background field angle ϕ0. Black lines are for B1 = 0 (without gradient), blue lines for B1 = B0/10, red lines for B1 = 2B0/10, and green lines for B1 = 3B0/10. Solid lines are for cases when the bow shock is present, and dashed lines are for cases without a bow shock.
In the text