Gravitational slingshots with earth

To gain an understanding of the approximate change in energy that a transporting body receives through a random gravitational interaction with the Earth, we developed a Python code that randomly initializes and integrates the motions of particles from their points of closest approach to Earth in the past or future, computing the total change in energy over the interaction. The Python code created for this work used the open-source N-body integrator software REBOUNDFootnote ¹ to trace the motions of particles under the gravitational influence of the Earth-Sun system (Rein and Liu Reference Rein and Liu2012).

We initialize the simulation with the Sun, Earth and a volume of test particles surrounding Earth at 80 km from the Earth's surface, with near-zero gravitational binding energies from the Sun as appropriate for LPCs.Footnote ² The Sun and Earth define the ecliptic plane. For each test particle, we randomly pick an angle within the ecliptic plane between 0 and 2π, as well as a zenith angle between 0 and π. Using these two angles, we set each particle's position vector relative to Earth.

To ensure that each particle is at its distance of closest approach, we require the velocity vector to lie in the plane perpendicular to the position vector relative to Earth. For each particle, we pick a random angle between 0 and 2π to determine in which direction the velocity vector points within this plane. Using the angle within the plane perpendicular to the position vector, we construct each particle's velocity vector. At this point, we have fully initialized the 6D coordinates of each particle in both position and velocity.

In the first stage of the simulation, we integrate all of the particles backward in time. We use the IAS15 integrator in REBOUND to trace each particle from t = 0 to an earlier time − t _i (Rein and Spiegel Reference Rein and Spiegel2014), where t _i is an amount of time to sample either side of the closest approach to Earth. The only constraint on t _i is that it is a time interval at and above which the results of the simulation do not change, on the order of a few times the encounter period; in this case, t _i is on the order of a few days. We record the change in the speed at infinity for the incoming segment of the particle's trajectory, Δv _{∞, in}. In the second stage of the simulation, we integrate the particles with unbound initial conditions forward in time. We use IAS15 to integrate each particle from t = 0 to t _i. We record the change in the speed at infinity for the outgoing segment of the particle's trajectory, Δv _{∞, out}, and add it to theincoming and outgoing changes to find the change in speed at infinity for the entire encounter, Δv _∞.

We ran our Python code for 10⁵ particles, and the resulting distribution of Δv _∞ is shown in Fig. 1. Half of encounters resultin a positive change in energy, as expected from symmetry to time-reversal, and 95% of such encounters result in Δv _∞ ≤ 3 km s^−1;. This corresponds to objects with perihelion distances $\gtrsim 200\, {\rm AU}$, or LPCs.

Fig. 1.

Distribution of the gravitational Δv _∞ for random encounter geometries between Earth and LPCs marginally bound to the Sun near the Earth's surface.

Atmospheric drag

As the transporting object grazes the atmosphere, it encounters atmospheric drag, giving rise to constraints on its minimum size and minimum encounter altitude for it to escape.

Because the change in energy due to the gravitational slingshot is small relative to the initial kinetic energy, we approximate the transporting object's path as linear, summarized by the following expression for altitude:

(1)$$z\lpar x\rpar = R_{\oplus} + z_{\min} - \sqrt{R_{\oplus}^2 - x^2}\comma\; $$

where $R_{\oplus }$ is the radius of the Earth, z _min is the minimum altitude of the encounter and x is a distance parameter that fulfils z(0) = z _min and dx = v dt, where v is the instantaneous speed.

The density of the atmosphere as a function of altitude is ρ(z) = e^−z/8 km kg m⁻³, and the density of the transporting body is taken to be that of a typical cometary nucleus, ρ _tb = 600 kg m⁻³. The acceleration of the transporting body is given by the drag equation (Collins et al. Reference Collins, Jay and Marcus2010):

(2)$${{\rm d}v\over {\rm d}t} = -{3 \rho\lpar z\rpar C_{\rm D}\over 4 \rho_{{\rm tb}} L_{{\rm tb}}} v^2\comma\; $$

where C _D is the drag coefficient, set to the typical value of 2 and L _tb is the length of transporting body.

We use the expression given in Collins et al. (Reference Collins, Jay and Marcus2010) to estimate the yield strength Y _tb of the transporting body to be ~4300 Pa. The altitude at which the body begins to break up, $z_{\star }$, is given by the solution to the transcendental equation:

(3)$$Y_{{\rm tb}} = \rho\lpar z_{\star}\rpar v^2\lpar z_{\star}\rpar \comma\; $$

which yields, $z_{\star } \simeq 100\, {\rm km}$.

Expansion and slow-down

At z < 100 km, the transporting body expands according to the equation (Collins et al. Reference Collins, Jay and Marcus2010):

(4)$${{\rm d^2}L_{{\rm tb}}\over {\rm d}t^2} = {\rho\lpar z\rpar C_{\rm D}\over \rho_{{\rm tb}} L_{{\rm tb}}} v^2.$$

We note that equations (2)–(4) represent the ‘pancake’ or ‘liquid drop’ model, and become less accurate for km-sized impactors and larger, so our results regarding km-sized impactors should be regarded with this caveat (Register et al. Reference Register, Mathias and Wheeler2017).

We developed Python code that integrates along the path of the transporting body, using equations (1), (2) and (4) to continuously update v and L _tb, thereby deriving the total expansion of the object and change in speed as a function of minimum encounter altitude and size. The minimum size to guarantee a negligible expansion of ≲10% as a function of altitude is shown in Fig. 2 for the range of minimum encounter altitudes 20–80 km. The lower bound of 20 km was defined by the result that the minimum size was of order the altitude for lower altitudes. The upper bound, 80 km, was chosen because it is the highest altitude at which life has been detected as of yet (Imshenetsky et al. Reference Imshenetsky, Lysenko and Kazakov1978). It is important to note that the Imshenetsky et al. (Reference Imshenetsky, Lysenko and Kazakov1978) result should be treated with caution due to the lack of detail on how the system was sterilized and due to the fact that non-biological particles may resemble bacteria, and thus be mistaken for the latter (Wainwright et al. Reference Wainwright, Weber, Smith, Hutcheon, Klyce, Wickramasinghe, Narlikar and Rajaratnam2004; Smith Reference Smith2013).

Fig. 2.

Minimum diameter of transporting body to guarantee total expansion of <10% as a function of minimum encounter altitude. The red line indicates the result for an encounter velocity of 42 km s⁻¹, and the grey lines represent the upper and lower bounds, corresponding to the maximum and minimum encounter velocities of 72 km s⁻¹ and 12 km s⁻¹, respectively.

The change in speed for all successful encounters is $\lesssim \!10^{-5}\, {\rm m \, s}^{-1}$, which is reasonable considering the fact that significant slow-down will lead to runaway expansion. Such a small change in speed does not cause significant heating of the body.

Collection of microbial life

While the abundance of microbes in the upper atmosphere is poorly constrained (Burrows et al. Reference Burrows, Elbert, Lawrence and Pöschl2009), we use the Imshenetsky et al. (Reference Imshenetsky, Lysenko and Kazakov1978) detections to come up with an order-of-magnitude estimate. Imshenetsky et al. (Reference Imshenetsky, Lysenko and Kazakov1978) reported 31 colonies of microorganisms collected over four sounding rocket launches, each with ~30 s at the altitude range 48–77 km, travelling at ~0.75 km s⁻¹, with a detector of size ~5 × 10⁻³ m². The total detection volume for all four flights was then ~450 m³, giving an average colony number density of ~0.1 m⁻³ at the altitude range 48–77 km.

The total number of collected colonies is then estimated by the equation:

(5)$$\sim\!\!10^9\, {\rm colonies}\, \left({L_{{\rm tb}}\over 400 \, {\rm m}} \right)^2 \left({v\over 42 \, {\rm km \, s}^{-1}} \right)\left({\tau_{\min}\over 2 \, {\rm s}} \right)\comma\; $$

where τ_min is approximately the amount of time spent at z _min. This suggests a large number of collected colonies for typical values.

The collected microbes will experience accelerations of order 10⁵ g if they are accelerated over a distance of ~ 100 m. For microbes including Bacillus subtilis, Deinococcus radiodurans, Escherichia coli and Paracoccus denitrificans, a large proportion would survive accelerations of 4−5 × 10⁵ g, which are of interest for planetary impacts (Mastrapa et al. Reference Mastrapa, Glanzberg, Head, Melosh and Nicholson2001; Deguchi et al. Reference Deguchi, Tsuji and Horikoshi2011). We therefore assume that acceleration is not an important lethal factor for microbes picked up by the transporting body.

In addition, comet nuclei are porous, making it likely that some incident microbes become embedded several metres below the surface, providing protection from harmful radiation in interstellar space (Wesson Reference Wesson2010).

Number of exportation events

Francis (Reference Francis2005) estimates an LPC flux of F _LPC ~ 11 LPCs yr⁻¹ with q < 4 AU and $H \lesssim {\rm 11}$. Francis (Reference Francis2005) estimates H ~ 11 to correspond to a cometary diameter of L _H~11 ~ 1−2.4 km (Bailey and Stagg Reference Bailey and Stagg1988; Weissman Reference Weissman1990), but Fernandez and Sosa (Reference Fernandez and Sosa2012) estimate a cometary diameter of L _H~11 ~ 0.6 km. Weissman (Reference Weissman2007) calculates the impact probability of LPCs with the Earth to be $P_{\oplus } \simeq 2.2 \times 10^{-9}$ per comet per perihelion. While the cumulative size distribution of sub-km LPCs in uncertain, Vokrouhlicky et al. (Reference Vokrouhlicky, Nesvorny and Dones2019) estimate a cumulative power-law distribution of sizes with an index of ≃−1.5.

We estimate the total number of exportation events caused by Solar System bodies, N _S, over the age of the Earth, $T_{\oplus }$, given the cross-sectional area of the Earth, $A_{\oplus }$, and occurring between altitudes z ₁ and z ₂, to be:

(6)$$N_{\rm S} \sim F \; P_{\oplus} T_{\oplus} A_{\oplus}^{-1} \int^{R_{\oplus} + z_2}_{R_{\oplus} + z_1} \left({L_{{\rm tb}\comma \, \min}\over L_{H \sim11}} \right)^{-1.5} 2 \pi r \, {\rm d}r\comma\; $$

where the value of L _tb, min is computed as a function of altitude, z, with the encounter speed assumed to be 42 km s⁻¹. For z ₁ = 20 km and z ₂ = 80 km, N _S ~ 1−10.

We also consider ISOs, such as ‘Oumuamua (Meech et al. Reference Meech, Weryk, Micheli, Kleyna, Hainaut, Jedicke, Wainscoat, Chambers, Keane, Petric, Denneau, Magnier, Berger, Huber, Flewelling, Waters, Schunova-Lilly and Chastel2017; Micheli et al. Reference Micheli, Farnocchia, Meech, Buie, Hainaut, Prialnik, Schörghofer, Weaver, Chodas, Kleyna, Weryk, Wainscoat, Ebeling, Keane, Chambers, Koschny and Petropoulos2018). In the ~100 m size regime, the power law exponent for the cumulative size distribution is estimated to be ≃ −3 (Siraj and Loeb Reference Siraj and Loeb2019b). We can then express the total number of exportation events caused by ISOs, N _I, in terms of the size of ‘Oumuamua, L _O, and the timescale over which an ‘Oumuamua-size object is expected to collide with the Earth, $t_{\rm O}^{-1}$:

(7)$$N_{\rm I} \sim T_{\oplus} A_{\oplus}^{-1} t_{\rm O}^{-1} \int^{R_{\oplus} + z_2}_{R_{\oplus} + z_1} \left({L_{{\rm tb}\comma \, \min}\over L_{\rm O}} \right)^{-3} 2 \pi r \, {\rm d}r\comma\; $$

assuming that the composition of such objects are similar to LPCs and that the gravitational deflection by the Earth is small relative to the object's incoming energy.

‘Oumuamua's diameter is estimated to be L _O ≃ 100−440 m, based on Spitzer Space Telescope constraints on its infrared emission given its temperature (Trilling et al. Reference Trilling, Mommert, Hora, Farnocchia, Chodas, Giorgini, Smith, Carey, Lisse, Werner, McNeill, Chesley, Emery, Fazio, Fernandez, Harris, Marengo, Mueller, Roegge, Smith, Weaver, Meech and Micheli2018). The implied timescale for collisions of ‘Oumuamua-size ISOs with the Earth is t _O ~ 3 × 10⁷ years (Do et al. Reference Do, Tucker and Tonry2018). As a result, for z ₁ = 20 km and z ₂ = 80 km, N _I ~ 1−50.

Possibility of exportation events above 100 km

Life has not yet been detected above an altitude of 80 km. One obstacle to life in the thermosphere is that the temperature exceeds 400 K, a temperature at which no functional microbes have been confirmed to survive on Earth. However, if we extrapolate the Imshenetsky et al. (Reference Imshenetsky, Lysenko and Kazakov1978) results to $z \gtrsim 100\, {\rm km}$ by assuming that turbulent mixing makes the exponential scale height for microbial life equal to that of air, ≃ 8 km, breakup will not occur for icy objects, allowing for smaller and therefore more abundant transporting bodies. If we require that the total number of collected colonies $N_{\rm col} \gtrsim 10^3$, we can derive an expression for $L_{{\rm tb}\comma \, \min }^{\star }$ as a function of z:

(8)$$L_{{\rm tb}\comma \, \min}^{\star} \simeq {2\over 5} \sqrt{{{\rm e}^{\lpar z - 63 {\rm \, km}\rpar /8}\over \lpar v / 42 {\rm \, km \, s}^{-1}\rpar \lpar t / 2 {\rm \, s}\rpar }}.$$

The total number of $z \gtrsim 100 {\rm \, km}$ exportation events for Solar System bodies and ISOs, respectively, are,

(9)$$\eqalign{& N_{\rm S}^{\star} \sim N_{\rm S}\lcub z_1 = 100 {\rm \, km}\comma\; z_2 = \infty\comma\; L_{{\rm tb}\comma \, \min}^{\star}\rcub \comma\; \cr & N_{\rm I}^{\star} \sim N_{\rm I}\lcub z_1 = 100 {\rm \, km}\comma\; z_2 = \infty\comma\; L_{{\rm tb}\comma \, \min}^{\star}\rcub.}$$

We find $N_{\rm S}^{\star } \sim 3 \times 10^2 \!-\! 2 \times 10^3$ and $N_{I}^{\star } \sim 10^3\!-\!10^5$.