Discussion and results

In this section we consider dynamics of dust grains propelled from Earth reaching the nearby zone of the Jupiter’s moon Europa. It is well known that dust particles with sizes of the order of 10⁻⁴ cm can contain bacteria packed within this size (Portillo et al., Reference Portillo, Leff, Lauber and Fierer2013). It is clear that, for bacteria to survive, the temperature of dust particles during their motion must not significantly exceed T ≃ 300 K. Then, following an estimate presented by Osmanov (Reference Osmanov2025) the maximum velocity, υ _m , corresponds to a regime, when a drag power, Dρ _a πr ² υ _m ³ and the black body emission power 4πr ² σT ⁴ are of the same orders of magnitude

(1) $ v_m\simeq \left ( {4\sigma T^4}\over{\rho _a D}\right )^{1/3},$

where D represents a drag coefficient, σ is the Stefan-Boltzmann constant, and ρ _a represents the atmospheric mass density. Following (Brekke, Reference Brekke2013; Havens et al., Reference Havens, Koll and LaGow1952; Picone et al., Reference Picone, Hedin, Drob and Aikin2000) we consider Earth parameters as ρ _a ≃ (1.2×10⁻³ g/cm ³)exp(−H/7.04 km), where H is the altitude measured in km. By substituting the parameters at an altitude 150 km with D ≃ 1 we obtain a velocity υ _m ≃ 14 km/s, which exceeds the escape velocity υ _esc ≃ 11.2 km/s. This suggests that the dust particles can overcome Earth’s gravitational pull and potentially travel to other worlds. from energy conservation law one can show that at a distance of a several Earth radii, the velocity relative to Earth becomes υ ² _rel ≃ υ ² _m − υ ² _esc ≃ 8.4 km/s.

In the previous paper we have studied in detail the dynamics of dust particles propelled from the high altitudes of Earth.

In Figure 1 in polar coordinates we represent the schematic picture of velocity components and forces acting on the dust particles. Here $F_{rad}\simeq {{Lr^2}\over{4R^2c}}$ is the solar radiation force (Carroll and Ostlie, Reference Carroll and Ostlie2010), L ≃ 3.83 × 10³³ erg/s represents solar luminosity r denotes the radius of a spherical dust grain, c is the speed of light, F _g = GMm/R ² denotes the gravitational force, G is the gravitational constant, M ≃ 2 × 10³³ g represents the solar mass, m is the dust particles’ mass, R indicates its radial coordinate, F _d ≈ Dρπr ² υ ² is the drag force caused by the ambient, υ indicates the total speed of the grain, ρ ₀ ≃ 2m _p n ₀ and n ₀ ≃ 1 cm⁻³ are the mass density and the number density of a medium, and m _p represents the proton’s mass.

Figure 1.

The schematic picture of the components of the total velocity and forces acting on the dust particle from (Osmanov, Reference Osmanov2025 ).

Osmanov (Reference Osmanov2025) has found that dynamics of dust grains can be described by the following set of equations

(2) $\ddot \phi \simeq - 2\dot \phi {{\dot R} \over R} - {{3D{\rho _0}} \over {4\rho r}}\dot \phi {\left( {{{\dot R}^2} + {R^2}{{\dot \phi }^2}} \right)^{1/2}},$

(3) $\eqalign{ \ddot R \simeq& \; R{{\dot \phi }^2} - {{3D{\rho _0}} \over {4\rho r}}\dot R{\left( {{{\dot R}^2} + {R^2}{{\dot \phi }^2}} \right)^{1/2}} + \cr \quad\quad\quad & + {1 \over {{R^2}}}\left( {{{3L} \over {16\pi c\rho r}} - GM} \right), \cr}$

where ρ denotes the mass density of the dust.

In Figure 2 we show a trajectory of the dust particles (red) moving towards the Jupiter’s orbit (blue). The set of parameters is: M ≃ 2 × 10³³ g, L ≃ 3.83 × 10³³ erg/s, D = 1, ρ = 2 g/cm³, n ₀ = 1 cm⁻³ R(0) ≃ 1 AU, Ṙ(0) ≃ 0 km/s, and υ _φ ≃ 38.3 km/s. We set an initial velocity equal to 38.3 km/s where it has been taken into account that the velocity of a dust grains relative to Earth has the same direction as the Earth’s velocity relative to Sun (angle, α, between υ _tot and υ _φ (see Figure 1) equals zero).

Figure 2.

Here we represent a trajectory of a dust particle (red), reaching the Jupiter’s orbit (blue). The set of parameters is: M ≃ 2 × 10³³ g, L ≃ 3.83 × 10³³ erg/s, D = 1, ρ = 2 g/cm ³, n₀ = 1 cm ⁻³ R(0) ≃ 1 AU, Ṙ(0) ≃ 0 km/s, and υ_φ ≃ 38.3 km/s.

By numerically solving Equations (2, 3) across the full range of initial values of α = 0° to 360°, one obtains the average relative velocity of dust grains with respect to Jupiter as approximately u _rel ≃ 20.1 km/s. The impact of a dust particle on Europa’s surface results in maximum destruction when the angle of incidence – defined as the angle between the particle’s velocity vector and the surface – is 90°. On the other hand, it is empirically evident that T(θ) ≃ T(π/2)sin^3/2 θ (Pierazzo and Melosh, Reference Picone, Hedin, Drob and Aikin2000). After estimating T(π/2) ≃ υ ² _rel /(2c _v ), where c _v ≃ 1100 J/(kgK) is a specific heat of the dust material, and by assuming T(θ) ≡ T ₀ ≃ 300 K, one can straightforwardly derive a critical value of the incident angle, when the particle with bacteria inside will survive

(4) $ \theta \simeq \left ( {{2c_vT_0}\over{\upsilon _{rel}^2}}\right )^{2/3}\simeq 0.015\;rad\simeq 1^0,$

Therefore, out of the entire range of possible impact angles, only a very narrow fraction – approximately f ₁ = 1/360 ≃ 2.8 × 10⁻³ – would allow for the potential survival of bacteria. These dust particles will melt Europa’s ice surface and become embedded within it.

As discussed by Berera (Reference Berera2017); Osmanov (Reference Osmanov2025), the flux density of dust particles leaving Earth is ${{\cal F}_{{0}}}$ ≃ 1 cm⁻²s⁻¹, therefore, the total flux of particles is given by F ≃ 4πR _⊕ ² ${{\cal F}_{{0}}}$ ≃ 5 × 10¹⁸ s⁻¹, where R _⊕ ≃ 6400 km is the radius of Earth.

Numerical analysis shows that the dust grains reach the Jupiter’s orbit only for the following intervals of angles α = 0° − 130° α = 230° − 360°. Therefore, the corresponding fraction equals f ₂ ≃ 260/360 ≃ 0.72.

Another factor must be taken into account. For a particle to enter Jupiter’s zone, it must be there at the correct moment. As numerical analysis shows, particle’s angular position φ (see Figure 1) on the Jupiter’s orbit lies within the interval 83° − 135°, (for the case shown in Figure 2, φ ≃ 95°) which implies a probability of approximately f ₃ = (135−83)/360 ≃ 0.15 for arriving at the right time.

On average, dust particles are ejected isotropically from Earth in all directions. We now turn to estimating a maximum theoretical fraction of particles reaching a Jupiter zone. For the corresponding Hill’s radius (scale of the gravitational influence), one writes R _H = r _J (M _J /(3M _⊙))^1/3 ≃ 0.36 AU (Carroll and Ostlie, Reference Carroll and Ostlie2010), where M _J ≃ 2 × 10³⁰ g is Jupiter’s mass and r _J ≃ 5 AU is its orbital radius. Then, one can be approximate the fraction as follows: f ₄ ≃ R _H ²/(4r ² _av ) ≃ 1.4 × 10⁻³. On the other hand, recent Numerical simulations have found that 0.015% (f ₅ = 1.5 × 10⁻⁴) of particles entering the Jupiter’s gravitation zone, can fall onto the Europa’s surface.

Therefore, the total flux of dust grains on the Europa’s surface with potentially survived bacteria inside is given by

(5)

It should be noted that bacteria contained within dust particles that land on the surface undergo deactivation over a timescale of approximately 10000 years (Pavlov et al., Reference Pavlov, Cheptsov, Tsurkov, Lomasov, Frolov and Vasiliev2019). On the surface of Europa there are active zones – Chaos terrains, covering approximately 20% − 40% of the total surface (Schmidt et al., Reference Schmidt, Blankenship, Patterson and Schenk2011). On the other hand, based on detailed observations, it is estimated that the surface ice undergoes fracturing for approximately 10³ − 10⁵ years (Greenberg et al., Reference Greenberg, Geissler, Hoppa and Tufts2002). This provides a non-negligible probability that dust particles containing microbial life may reach the subsurface water before being deactivated by radiation.

Recent studies show that the age of Europa’s icy surface is at least 30 − 80 Myr (Vance and Brown, Reference Vance and Brown2005). Therefore, the total number of particles during the mentioned period is of the order of (3−8) × 10²³, which strongly suggests the likelihood of life being present in the subsurface ocean of Europa if the biological and biochemical conditions are compatible with Earth-originating life, which would require a new series of investigations to determine.