Can Planet Nine Be Detected Gravitationally by a Subrelativistic Spacecraft?

Since Planet Nine is outside the heliopause of the solar wind, the spacecraft will encounter the ISM and inevitably experience a drag force due to collisions with gas particles and dust (Hoang et al. 2017; Hoang & Loeb 2017; Lingam & Loeb 2020). For a spacecraft moving at a subrelativistic speeds v through the ISM with a proton density n_H (and 10% ISM He by abundance), the ratio of the drag force to the gravitational force of Planet Nine equals

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{F}_{\mathrm{drag}}}{{F}_{\mathrm{grav}}} & = & \left(\displaystyle \frac{1.4{n}_{{\rm{H}}}{m}_{{\rm{H}}}{v}^{2}{A}_{\mathrm{sp}}}{{{GM}}_{\mathrm{pl}}{M}_{\mathrm{sp}}/{b}^{2}}\right)=\left(\displaystyle \frac{1.4{n}_{{\rm{H}}}{m}_{{\rm{H}}}{v}^{2}}{{\rho }^{2/3}{M}_{\mathrm{sp}}^{1/3}}\right)\left(\displaystyle \frac{{b}^{2}}{{{GM}}_{\mathrm{pl}}}\right)\\ & \simeq & 117.3\left(\displaystyle \frac{5{M}_{\oplus }}{{M}_{\mathrm{pl}}}\right){\left(\displaystyle \frac{b}{100\mathrm{au}}\right)}^{2}\left(\displaystyle \frac{{n}_{{\rm{H}}}}{1\,{\mathrm{cm}}^{-3}}\right)\\ & & \times \,{\left(\displaystyle \frac{v}{0.01c}\right)}^{2}{\left(\displaystyle \frac{\rho }{3{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-2/3}{\left(\displaystyle \frac{{M}_{\mathrm{sp}}}{1{\rm{g}}}\right)}^{-1/3},\end{array}\end{eqnarray} \tag{ 1 }$$

where M_pl is the mass of the planet, and b is the impact parameter at closest approach to Planet Nine. Equation (1) implies dominance of the ISM drag force over gravity for v ≳ 10⁻³c at b ≳ 100 au, assuming the typical spacecraft mass of ${M}_{\mathrm{sp}}=1\,{\rm{g}}$.

Equation (12) implies that the ratio of the forces decreases with increasing spacecraft mass as ${M}_{\mathrm{sp}}^{-1/3}$. The spacecraft mass required for dominance of the gravitational force over the drag force, ${F}_{\mathrm{grav}}\gt {F}_{\mathrm{drag}}$, is given by

$$\begin{eqnarray}&&\begin{array}{l}{M}_{\mathrm{sp}}\gt {M}_{\mathrm{sp},\mathrm{drag}}\equiv {\left(\displaystyle \frac{1.4{n}_{{\rm{H}}}{m}_{{\rm{H}}}{v}^{2}{b}^{2}}{{{GM}}_{\mathrm{pl}}{\rho }^{2/3}}\right)}^{3}\\ \quad \simeq \,1.6{\left(\displaystyle \frac{5{M}_{\oplus }}{{M}_{\mathrm{pl}}}\right)}^{3}{\left(\displaystyle \frac{b}{100\mathrm{au}}\right)}^{6}{\left(\displaystyle \frac{v}{{10}^{-3}c}\right)}^{6}\\ \qquad \times \,{\left(\displaystyle \frac{\rho }{3{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-2}{\left(\displaystyle \frac{{n}_{{\rm{H}}}}{1{\mathrm{cm}}^{-3}}\right)}^{3}{\rm{g}},\end{array}\end{eqnarray} \tag{ 2 }$$

where ${M}_{\mathrm{sp},\mathrm{drag}}$ is the critical value for which F_grav = F_drag. Equation (2) implies a strong dependence of the critical spacecraft mass on its speed v and the impact factor b. For a slow mission at v ∼ 10⁻³c, the critical mass is ${M}_{\mathrm{sp},\mathrm{cri}}\sim 1.6$ g, but it increases to ${M}_{\mathrm{sp},\mathrm{drag}}\sim {10}^{3}$ kg at v = 0.01c.

The spacecraft would inevitably get charged due to collisions with interstellar particles and the photoelectric effect induced by solar and interstellar photons (Hoang et al. 2017; Hoang & Loeb 2017). The frontal surface layer becomes positively charged through collisions with electrons and protons (i.e., collisional charging), with secondary electron emission being dominant for high-speed collisions (Hoang & Loeb 2017). The outer surface area is charged through the photoelectric effect by ultraviolet photons from the Sun.

The surface potential, U, increases over time due to collisions with the gas and achieves saturation when the potential energy is equal to the maximum energy transfer. One therefore obtains saturation at ${{eU}}_{\max ,{\rm{e}}}={m}_{e}{v}^{2}/2$ for impinging electrons and ${{eU}}_{\max ,{\rm{H}}}=2{m}_{e}{v}^{2}=4{{eU}}_{\max ,{\rm{e}}}$ for impinging protons (Hoang et al. 2015; Hoang & Loeb 2017). The corresponding maximum charge equals

$$\begin{eqnarray}&&\begin{array}{l}{Z}_{\mathrm{sp},\max }\sim \displaystyle \frac{{U}_{\max ,{\rm{H}}}(W/2)}{e}=\left(\displaystyle \frac{{m}_{e}{v}^{2}{\left({M}_{\mathrm{sp}}/\rho \right)}^{1/3}}{{e}^{2}}\right)\\ \ \simeq \,2.5\times {10}^{8}{\left(\displaystyle \frac{v}{0.01c}\right)}^{2}{\left(\displaystyle \frac{\rho }{3{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-1/3}{\left(\displaystyle \frac{{M}_{\mathrm{sp}}}{1{\rm{g}}}\right)}^{1/3},\end{array}\end{eqnarray} \tag{ 3 }$$

which implies a strong increase in the maximum charge with spacecraft speed.

The charged spacecraft would therefore experience a Lorentz force due to the interstellar magnetic field, ${F}_{B}={{eZ}}_{\mathrm{sp}}{{vB}}_{\perp }/c$, where B_⊥ is the magnetic field component perpendicular to the direction of motion. The ratio of the magnetic force to the gravitational force at an impact parameter b relative to Planet Nine is given by

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{F}_{\mathrm{mag}}}{{F}_{\mathrm{grav}}} & = & \left(\displaystyle \frac{{{eZ}}_{\mathrm{sp}}{{vB}}_{\perp }/c}{{{GM}}_{\mathrm{pl}}{M}_{\mathrm{sp}}/{b}^{2}}\right)=\left(\displaystyle \frac{{m}_{e}{v}^{3}{B}_{\perp }}{{ec}{\rho }^{1/3}{M}_{\mathrm{sp}}^{2/3}}\right)\left(\displaystyle \frac{{b}^{2}}{{{GM}}_{{pl}}}\right)\\ & \simeq & 6.6\left(\displaystyle \frac{5{M}_{\oplus }}{{M}_{\mathrm{pl}}}\right){\left(\displaystyle \frac{b}{100\mathrm{au}}\right)}^{2}\left(\displaystyle \frac{{B}_{\perp }}{5\,\mu {\rm{G}}}\right)\\ & & \times \,{\left(\displaystyle \frac{v}{0.01c}\right)}^{3}{\left(\displaystyle \frac{\rho }{3{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-1/3}{\left(\displaystyle \frac{{M}_{\mathrm{sp}}}{1{\rm{g}}}\right)}^{-2/3},\end{array}\end{eqnarray} \tag{ 4 }$$

where ${Z}_{\mathrm{sp}}={Z}_{\mathrm{zp},\max }$ is taken, and we adopt the typical magnetic field strength of B_⊥ ∼ 5 μG as inferred from analysis of the data from Voyager 1 and 2 (Opher et al. 2020). The equation above reveals dominance of the magnetic force over the gravitational force for b ≳ 100 au and v ≳ 0.0055c, assuming the typical spacecraft mass of ${M}_{\mathrm{sp}}=1\,{\rm{g}}$.

Using the above equation, one obtains the spacecraft mass required for F_grav > F_mag as follows:

$$\begin{eqnarray}&&{M}_{\mathrm{sp}}^{2/3}\gt {M}_{\mathrm{sp},\mathrm{mag}}^{2/3}\equiv \left(\displaystyle \frac{{m}_{e}{v}^{3}{B}_{\perp }}{{ec}{\rho }^{1/3}}\right)\left(\displaystyle \frac{{b}^{2}}{{{GM}}_{\mathrm{pl}}}\right),\end{eqnarray} \tag{ 5 }$$

which yields

$$\begin{eqnarray}\begin{array}{rcl}{M}_{\mathrm{sp},\mathrm{mag}} & \simeq & 5.7{\left(\displaystyle \frac{{M}_{\mathrm{pl}}}{5{M}_{\oplus }}\right)}^{-3/2}{\left(\displaystyle \frac{b}{100\mathrm{au}}\right)}^{3}\\ & & \times \,{\left(\displaystyle \frac{{B}_{\perp }}{5\mu {\rm{G}}}\right)}^{3/2}{\left(\displaystyle \frac{v}{0.01c}\right)}^{9/2}{\left(\displaystyle \frac{\rho }{3{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-1/2},\end{array}\end{eqnarray} \tag{ 6 }$$

where M_sp,mag is the critical value for which F_grav = F_mag. The above equation implies a strong dependence of the critical mass on the impact parameter and the spacecraft speed.

Next, we estimate the effect of the magnetic deflection on the time delay of the signal. Due to the Lorentz force, the charged spacecraft would move in a curved trajectory instead of a straight line from Earth to Planet Nine. The gyroradius of a charged spacecraft,

$$\begin{eqnarray}\begin{array}{rcl}{R}_{\mathrm{gyro}} & = & \displaystyle \frac{{M}_{\mathrm{sp}}{vc}}{{{eZ}}_{\mathrm{sp}}{B}_{\perp }}\simeq 2.5\times {10}^{11}\left(\displaystyle \frac{{M}_{\mathrm{sp}}}{1\,{\rm{g}}}\right)\\ & & \times \,\left(\displaystyle \frac{v}{0.01c}\right)\left(\displaystyle \frac{{10}^{9}}{{Z}_{\mathrm{sp}}}\right)\left(\displaystyle \frac{5\,\mu {\rm{G}}}{{B}_{\perp }}\right)\,\mathrm{au},\end{array}\end{eqnarray} \tag{ 7 }$$

provides the curvature of the spacecraft trajectory in the magnetic field.

The deflection in impact parameter from the target at a distance D ≪ R_gyro is given by

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}b & = & D\alpha \approx \displaystyle \frac{{D}^{2}}{{R}_{\mathrm{gyro}}}=\displaystyle \frac{{D}^{2}{{eZ}}_{\mathrm{sp}}{B}_{\perp }}{{M}_{\mathrm{sp}}{vc}}\\ & \simeq & {10}^{-6}{\left(\displaystyle \frac{D}{500\mathrm{au}}\right)}^{2}\left(\displaystyle \frac{{Z}_{\mathrm{sp}}}{{10}^{9}}\right)\left(\displaystyle \frac{1{\rm{g}}}{{M}_{\mathrm{sp}}}\right)\\ & & \times \,\left(\displaystyle \frac{{B}_{\perp }}{5\,\mu {\rm{G}}}\right)\left(\displaystyle \frac{0.01c}{v}\right)\,\mathrm{au},\end{array}\end{eqnarray} \tag{ 8 }$$

where $\sin \alpha \approx \alpha \approx D/{R}_{\mathrm{gyro}}$.

Due to the trajectory deflection, light signals will arrive at a slightly different time than expected without the deflection. The extra distance traversed by the spacecraft is ${\rm{\Delta }}x\,={({\rm{\Delta }}b)\tan \alpha \sim ({\rm{\Delta }}b)}^{2}/D$, where $\tan \alpha =({\rm{\Delta }}b)/D$. The time delay due to the magnetic deflection is

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\mathrm{mag}}=\displaystyle \frac{{\rm{\Delta }}x}{c}=\displaystyle \frac{{\left({\rm{\Delta }}b\right)}^{2}}{{Dc}}\sim {10}^{-12}{\left(\displaystyle \frac{{\rm{\Delta }}b}{{10}^{-6}\mathrm{au}}\right)}^{2}\left(\displaystyle \frac{500\,\mathrm{au}}{D}\right)\,{\rm{s}}.\end{eqnarray} \tag{ 9 }$$

The angular deflection of the spacecraft by the interstellar magnetic field is then given by

$$\begin{eqnarray}&&{\alpha }_{\mathrm{mag}}=\displaystyle \frac{{\rm{\Delta }}b}{D}\simeq 2\times {10}^{-9}\left(\displaystyle \frac{{\rm{\Delta }}b}{{10}^{-6}\,\mathrm{au}}\right)\left(\displaystyle \frac{500\,\mathrm{au}}{D}\right)\mathrm{rad}.\end{eqnarray} \tag{ 10 }$$

The time delay due to the gravitational effect was given by (Witten 2020)

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\mathrm{grav}}\simeq 7\times {10}^{-7}\left(\displaystyle \frac{{M}_{\mathrm{pl}}}{5{M}_{\oplus }}\right){\left(\displaystyle \frac{0.01c}{v}\right)}^{2}{\sinh }^{-1}\left(\displaystyle \frac{{vt}}{b}\right)\,{\rm{s}},\end{eqnarray} \tag{ 11 }$$

where the origin time t = 0 is associated with the closest approach to Planet Nine. The ISM drag force introduces a longitudinal time delay that exceeds Δt_grav by the factor given in Equation (1). The gravitational angular deflection was estimated by (Lawrence & Rogoszinski 2020)

$$\begin{eqnarray}&&{\alpha }_{\mathrm{grav}}\simeq 2.8\times {10}^{-11}\left(\displaystyle \frac{{M}_{\mathrm{pl}}}{5{M}_{\oplus }}\right)\left(\displaystyle \frac{100\,\mathrm{au}}{b}\right){\left(\displaystyle \frac{0.01c}{v}\right)}^{2}\mathrm{rad}\ll {\alpha }_{\mathrm{mag}},\end{eqnarray} \tag{ 12 }$$

which yields a decreasing deflection with spacecraft speed.

We quantify the rms fluctuations of the electron density due to the ISM turbulence on a scale L as (Draine 2011, p. 115)

$$\begin{eqnarray}\begin{array}{rcl}\delta {n}_{e} & \equiv & \langle {\left({\rm{\Delta }}{n}_{e}\right)}^{2}{\rangle }^{1/2}\simeq \left(\displaystyle \frac{6.4\times {10}^{-4}}{\,{\mathrm{cm}}^{3}}\right)\\ & & \times \,{\left(\displaystyle \frac{{C}_{n}^{2}}{5\times {10}^{-17}{\mathrm{cm}}^{-20/3}}\right)}^{1/2}{\left(\displaystyle \frac{L}{{10}^{14}\mathrm{cm}}\right)}^{1/3},\end{array}\end{eqnarray} \tag{ 13 }$$

where C_n² is the amplitude of the density power spectrum. Using measurements from Voyager 1 (Lee & Lee 2019), ${C}_{n}^{2}\,\approx {10}^{-2.79}{m}^{-20/3}\approx 7.52\times {10}^{-17}\,{\mathrm{cm}}^{-20/3}$ for Equation (13), yielding $\delta {n}_{e}\sim 0.0033{(L/500\mathrm{au})}^{1/3}$. With the mean electron density in the local ISM of ${n}_{e}\sim 0.04\,{\mathrm{cm}}^{-3}$ (see, e.g., Draine 2011, p.115), one obtains $\delta {n}_{e}/{n}_{e}\approx 0.08$.

Assuming $\delta {n}_{{\rm{H}}}/{n}_{{\rm{H}}}\sim \delta {n}_{e}/{n}_{e}$, one can calculate the ratio of the gravitational signal to noise induced by density fluctuations as follows:

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{({\rm{S}}/{\rm{N}})}_{{\rm{drag}}} & = & \displaystyle \frac{{F}_{{\rm{grav}}}}{ \langle {\left({\rm{\Delta }}{F}_{{\rm{drag}}}\right)}^{2}{ \rangle }^{1/2}}\\ & = & \displaystyle \frac{{{GM}}_{{\rm{pl}}}{M}_{{\rm{sp}}}/{b}^{2}}{1.4\delta {n}_{{\rm{H}}}{m}_{{\rm{H}}}{v}^{2}{A}_{{\rm{sp}}}}=\displaystyle \frac{{{GM}}_{{\rm{pl}}}}{{b}^{2}}\displaystyle \frac{{M}_{{\rm{sp}}}^{1/3}{\rho }^{2/3}}{1.4\delta {n}_{{\rm{H}}}{m}_{{\rm{H}}}{v}^{2}}\\ & \simeq & 0.085\left(\displaystyle \frac{{M}_{{\rm{pl}}}}{5{M}_{\oplus }}\right){\left(\displaystyle \frac{b}{100{\rm{au}}}\right)}^{-2}{\left(\displaystyle \frac{{M}_{{\rm{sp}}}}{1{\rm{g}}}\right)}^{1/3}\\ & & \times \,{\left(\displaystyle \frac{\rho }{3{\rm{g}}{{\rm{cm}}}^{-3}}\right)}^{2/3}\left(\displaystyle \frac{\delta {n}_{{\rm{H}}}}{0.1{n}_{{\rm{H}}}}\right){\left(\displaystyle \frac{{n}_{{\rm{H}}}}{1{{\rm{cm}}}^{-3}}\right)}^{-1}{\left(\displaystyle \frac{v}{0.01c}\right)}^{-2}.\end{array}\end{array}\end{eqnarray} \tag{ 14 }$$

Equation (14) implies the increase of the signal-to-noise ratio (S/N) with decreasing the impact parameter and spacecraft mass, but the S/N decreases rapidly with increasing spacecraft speed.

To detect the gravitational signal with (S/N)_drag ≳ 3, the minimum spacecraft mass must satisfy the following condition:

$$\begin{eqnarray}\begin{array}{rcl}{M}_{{\rm{s}}{\rm{p}}} & \gtrsim & {M}_{{\rm{s}}{\rm{p}},{\rm{f}}{\rm{l}}{\rm{u}}{\rm{c}}}={\left(3\times \displaystyle \frac{1.4\delta {n}_{{\rm{H}}}{m}_{{\rm{H}}}{v}^{2}{b}^{2}}{{GM}_{{\rm{p}}{\rm{l}}}{\rho }^{2/3}}\right)}^{3}{\left(\displaystyle \frac{({{\rm{S}}/{\rm{N}})}_{{\rm{d}}{\rm{r}}{\rm{a}}{\rm{g}}}}{3}\right)}^{3}\\ & \simeq & 4.8\times {10}^{3}{\left(\displaystyle \frac{5{M}_{\oplus }}{{M}_{{\rm{p}}{\rm{l}}}}\right)}^{3}{\left(\displaystyle \frac{b}{100{\rm{a}}{\rm{u}}}\right)}^{6}{\left(\displaystyle \frac{v}{0.01c}\right)}^{6}\\ & & \times \,{\left(\displaystyle \frac{\rho }{3{\rm{g}}{{\rm{c}}{\rm{m}}}^{-3}}\right)}^{-2}{\left(\displaystyle \frac{\delta {n}_{{\rm{H}}}}{0.1{n}_{{\rm{H}}}}\right)}^{3}{\left(\displaystyle \frac{{n}_{{\rm{H}}}}{1{{\rm{c}}{\rm{m}}}^{-3}}\right)}^{3}{\left(\displaystyle \frac{({{\rm{S}}/{\rm{N}})}_{{\rm{d}}{\rm{r}}{\rm{a}}{\rm{g}}}}{3}\right)}^{3}{\rm{g}},\end{array}\end{eqnarray} \tag{ 15 }$$

which indicates a steep increase of the required spacecraft mass with its speed, v, and the impact parameter, b.

Figure 2 (blue lines) shows the minimum spacecraft mass as a function of the spacecraft speed for the different impact parameters. For a mission of v = 0.001c that takes about ∼10 yr to probe Planet Nine, the spacecraft mass required to overcome the density fluctuations must increase from ∼10⁻³ g for b = 50 au to 20 g for b = 200 au. For v = 0.01c considered in Parkin (2018) that takes ∼1 yr to probe Planet Nine, the spacecraft mass must be larger than 10³ g for b > 50 au.

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Measuring the angular displacement of the spacecraft, as proposed by Lawrence & Rogoszinski (2020), would be particularly challenged by angular deflections from the fluctuating ISM magnetic field. The orientation of the magnetic field is not known in the Planet Nine region of interest. Let δB be the rms magnetic field fluctuations. The ratio of the gravitational signal by Planet Nine to the noise caused by the magnetic field fluctuations is then given by

$$\begin{eqnarray}\begin{array}{rcl}{({\rm{S}}/{\rm{N}})}_{{\rm{m}}{\rm{a}}{\rm{g}}} & = & \displaystyle \frac{{F}_{{\rm{g}}{\rm{r}}{\rm{a}}{\rm{v}}}}{ \langle {\left({\rm{\Delta }}{F}_{{\rm{m}}{\rm{a}}{\rm{g}}}\right)}^{2}{ \rangle }^{1/2}}\\ & = & \displaystyle \frac{{GM}_{{\rm{p}}{\rm{l}}}{M}_{{\rm{s}}{\rm{p}}}/{b}^{2}}{{eZ}_{{\rm{s}}{\rm{p}}}v\delta B/c}=\left(\displaystyle \frac{{GM}_{{\rm{p}}{\rm{l}}}{M}_{{\rm{s}}{\rm{p}}}^{2/3}}{{b}^{2}}\right)\left(\displaystyle \frac{ec{\rho }^{1/3}}{{m}_{e}{v}^{3}\delta B}\right)\\ & \simeq & 0.15\left(\displaystyle \frac{{M}_{{\rm{p}}{\rm{l}}}}{5{M}_{\oplus }}\right){\left(\displaystyle \frac{b}{100{\rm{a}}{\rm{u}}}\right)}^{-2}{\left(\displaystyle \frac{{M}_{{\rm{s}}{\rm{p}}}}{1{\rm{g}}}\right)}^{2/3}{\left(\displaystyle \frac{\delta B}{{B}_{\perp }}\right)}^{-1}\\ & & \times \,{\left(\displaystyle \frac{{B}_{\perp }}{5\mu G}\right)}^{-1}{\left(\displaystyle \frac{v}{0.01c}\right)}^{-3}{\left(\displaystyle \frac{\rho }{3{\rm{g}}{{\rm{c}}{\rm{m}}}^{-3}}\right)}^{1/3},\end{array}\end{eqnarray} \tag{ 16 }$$

where Z_sp was adopted from Equation (3), and we assumed ${(\delta B)}^{2}\sim {B}^{2}$ as measured by Voyager 1 (Burlaga et al. 2015).

To detect the gravitational signal in the presence of magnetic noise with (S/N)_mag ≳ 3, the minimum spacecraft mass is

$$\begin{eqnarray}\begin{array}{rcl}{M}_{{\rm{s}}{\rm{p}}} & \gtrsim & {M}_{{\rm{s}}{\rm{p}},{\rm{f}}{\rm{l}}{\rm{u}}{\rm{c}}}={\left(3\times \displaystyle \frac{{m}_{e}{v}^{3}\delta B}{ec{\rho }^{1/3}}\times \displaystyle \frac{{b}^{2}}{{GM}_{{\rm{p}}{\rm{l}}}}\right)}^{3/2}{\left(\displaystyle \frac{({{\rm{S}}/{\rm{N}})}_{{\rm{m}}{\rm{a}}{\rm{g}}}}{3}\right)}^{3/2}\\ & \simeq & 89.4{\left(\displaystyle \frac{{M}_{{\rm{p}}{\rm{l}}}}{5{M}_{\oplus }}\right)}^{-3/2}{\left(\displaystyle \frac{b}{100{\rm{a}}{\rm{u}}}\right)}^{3}{\left(\displaystyle \frac{v}{0.01c}\right)}^{9/2}{\left(\displaystyle \frac{\delta B}{{B}_{\perp }}\right)}^{3/2}\\ & & \times \,{\left(\displaystyle \frac{{B}_{\perp }}{5\mu {\rm{G}}}\right)}^{3/2}{\left(\displaystyle \frac{\rho }{3{\rm{g}}{{\rm{c}}{\rm{m}}}^{-3}}\right)}^{-1/2}{\left(\displaystyle \frac{({{\rm{S}}/{\rm{N}})}_{{\rm{m}}{\rm{a}}{\rm{g}}}}{3}\right)}^{3/2}\,\,{\rm{g}}.\end{array}\end{eqnarray} \tag{ 17 }$$

Figure 2 (red lines) shows the minimum spacecraft mass as a function of the spacecraft speed for the different impact parameters. For a mission of v = 0.001c that takes about ∼10 yr to probe Planet Nine, the spacecraft mass required to overcome the magnetic fluctuations must increase from ∼0.001 g for b = 50 au to 0.01 g for b = 200 au. For v = 0.01c considered in Parkin (2018) that takes ∼1 yr to probe Planet Nine, the spacecraft mass must be larger than 1–10³ g for b > 30–200 au.

Note that magnetic field fluctuations are not static, including Alfvén waves, which are time dependent. This means that the longer the journey is, the larger is the random walk that the spacecraft executes as a result of Alfvén waves. This implies a larger noise at spacecraft speeds since they take longer to traverse the vicinity of Planet Nine. Finally, during the passage through the heliosphere, the spacecraft would experience large drag and magnetic noise due to the solar wind.

In order to mitigate the effect of the drag force, one can increase the spacecraft mass or decrease the spacecraft speed (see Figure 1 and Equation (1)). This results in a larger energy cost to launch the spacecraft. A slower spacecraft takes a longer time to reach the Planet Nine region.

One can also design the spacecraft with a needle-like shape to reduce the frontal cross section. However, as shown in Hoang & Loeb (2017), the frontal surface area becomes positively charged and produces an electric dipole. The interaction of the moving dipole with the interstellar magnetic field causes the spacecraft to oscillate around the center of mass, exposing the long axis of the spacecraft to gas collisions and increasing the drag force.

To mitigate the effect of electromagnetic forces, an onboard electron gun could be added. However, this would put additional load on the spacecraft and increases the cost of launch.

Since the position of Planet Nine is not known, one could imagine sending a large array of spacecraft so that the impact parameter, b, will be minimized for one of them. Unfortunately, due to the fluctuations in the drag force, the spacecraft mass and energy cost must be larger for larger impact parameters b.