On Mautner-Type Probability of Capture of Intergalactic Meteor Particles by Habitable Exoplanets

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Article On Mautner-Type Probability of Capture of Intergalactic Meteor Particles by Habitable Exoplanets

Andjelka B. Kovaˇcevi´c

Department of Astronomy, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia; [email protected]

Received: 28 June 2019; Accepted: 10 July 2019; Published: 9 August 2019 First Version Published: 15 July 2019 (doi:10.3390/sci1020041) Second Version Published: 9 August 2019 (doi:10.3390/sci1020047)

Abstract: Both macro and microprojectiles (e.g., interplanetary, interstellar and even intergalactic material) are seen as important vehicles for the exchange of potential (bio)material within our solar system as well as between stellar systems in our Galaxy. Accordingly, this requires estimates of the impact probabilities for different source populations of projectiles, including for intergalactic meteor particles which have received relatively little attention since considered as rare events (discrete occurrences that are statistically improbable due to their very infrequent appearance). We employ the simple but yet comprehensive model of intergalactic microprojectile capture by the gravity of exoplanets which enables us to estimate the map of collisional probabilities for an available sample of exoplanets in habitable zones around host stars. The model includes a dynamical description of the capture adopted from Mautner model of interstellar exchange of microparticles and changed for our purposes. We use statistical and information metrics to calculate probability map of intergalactic meteorite particle capture. Moreover, by calculating the entropy index map we measure the concentration of these rare events. We further adopted a model from immigration theory, to show that the transient distribution of birth/death/immigration of material for the simplest case has a high value.

Keywords: intergalactic meteor particle; extrasolar planets; astrobiology

1. Introduction The Universe has been generating random events, which can be unpredictable and affect planetary environments in different ways. Probability distributions of such events are often observed to have power- law tails [1]. One of many examples include the natural transport of material within our Solar system. Molecules, excluding the most stable such as polycyclic aromatic hydrocarbons, are quickly (10–10,000 years) destroyed by e.g., the UV radiation of stars [2,3]. So, defensive vessels are required that would protect and transport them unaltered across space. Interestingly, recent observations accumulate evidence of interstellar, and even intergalactic transport of material. For example, observation of the first known Interstellar Object (ISO) 1I/2017 U1 Oumuamua by the Pan-STARRS telescope in October 2017 [4] has given a new impetus to a broad topic about the possibility of natural transport of solid material and complex molecules through interstellar space [5]. This object was the first macro–scale ISO observed. Almost two decades ago, Taylor et al. [6] discovered interstellar dust entering the Earth atmosphere. Ten years later, Afanasiev et al. [7] detected a hundreds of a micrometer size Intergalactic meteor particle (IMP). This IMP hit the Earth atmosphere with a hypervelocity 300 km s−1. However, only about 1% of meteors have velocities above 100 km s−1, and no previous meteor observations have confirmed velocities

Sci 2019, 1, 47; doi:10.3390/sci1020047 www.mdpi.com/journal/sci Sci 2019, 1, 47 2 of 16 of several hundred km s−1 [8]. Afanasiev et al. [7] calculated that IMP was 0.01 cm in size and its mass was about 7 × 10−6 g. This IMP was two orders of magnitude larger than common interstellar dust grains in our Galaxy [9,10]. Additionally, its spectral features were a typical for the materials being exposed to the temperatures of 15,000–20,000 K. Its radiant appeared to coincide with the apex of the motion of the Solar system toward the centroid of the Local Group of galaxies. The authors calculated that average density of IMP population in the Earth’s vicinity could exceed 2.5 × 10−31 g cm−3. If the extragalactic dust is uniformly distributed over the entire volume of Local Group, with above density, Afanasiev et al. [7] claimed that the total mass of the dust is about of 1% of the total mass of the Local Group. Moreover, their followup observations identified a dozen IMP candidates consistent with velocity and radiant estimates of identified IMP, which is in disagreement with the lack of any evidence from other optical meteor observatories (see e.g., [11]). Nevertheless, Afanasiev et al. [7] study shows that the existence of meteors of galactic velocities cannot completely be ruled out. Linking estimates that our Milky Way galaxy is home to ~1010 exoplanets [12] and above mentioned possible influx of IMP on our planet, raises the question of the probability of material migration in our Local Group, including the delivery of chemicals potentially involved in the emergence of life. Although the trade of viable organisms between planets in our system is an open question [13], the exchange of molecular species is much more likely [14]. It is believed that material exchange between rocky planets could amplify the chemical space within the planetary system [15]. On the chemical space we assume the property space spanned by all possible molecules and chemical compounds under a given set of construction principles and boundary conditions. Meteoroids (particles from 1 µm up to 1 cm), meteorites (1 cm up to 10 m), comet’s nuclei and asteroids were suggested as potential transfer vehicles [16–20]. Moreover, ISO objects [21] and even planets can serve as transporters of complex molecules through space. This is supported by the recent discovery of an extragalactic planetary companion around HIP 13044 in our Galaxy [22]. HIP 13044, a very metal-poor star on the red Horizontal Branch and a member of the Helmi stream, was probably bounded to the Milky Way several Gyr ago from a satellite galaxy. Because of the long galactic relaxation timescale, it is most likely that its planet (HIP 13044 b with mass of 1.25 mass of Jupiter) was not captured by any Milky Way star. Interstellar and intergalactic meteoroids, able to transfer molecules, are very difficult to observe. Therefore, we do not have reliable information about their population density and dynamics. Unlike them, interstellar dust grains (.1 µm in size) have been intensively studied for a long time [23]. Dust grains mainly flow with interstellar gas. For example, due to uneven distribution of brighter star in the Galaxy, dust grains can have speed s of 2 to 10 km s−1 relative to the gas due to the radiation pressure [24]. Betatron mechanism (i.e., acceleration happens when the particle drift motion is in resonance with changes in the induction electric field caused by the variable magnetic field) can accelerate them up to 30 to 100 km s−1. Finally, dust grains and gas can be expelled into the intergalactic space [25,26], particularly in galactic regions of violent star formation and death. If such particles end up in a thick cluster of galaxies, they can be destroyed by hot intergalactic gas in the cluster with temperature in the order of ten million kelvin. In contrast, a particle can reach another galaxy and survive. In order to establish the importance of IMPs as material transporters, we ought to resolve their physical-dynamical characteristics, and impact probabilities. In this study we consider the exoplanets in habitable zones (HZEP) as targets and IMPs as a projectile on a trajectory crossing their orbits. We focus on the random probability that an IMP, during its cruise through our Galaxy, will collide with some of HZEP. Sci 2019, 1, 47 3 of 16

2. Materials and Methods The statistical impact probability of IMPs with HZEP, can be equivalently stated as calculating probability of cooccurrence of a planet being in the habitable zone and hit by an IMP. As the geophysical properties of potentially habitable exoplanets are currently unknown, it is not possible to determine exact HZEP probabilities. Instead, we calculated HZEP probabilities as a probability density function with respect to observationally constrained planet properties (distances and radii), assuming a very optimistic geometrical HZ boundaries. The probablity density function was estimated by machine learning implementation of 2D kernel density estimate (KDE) in Python. It is well known that the impact probabilities of projectiles (of negligible dimensions, moving on − Keplerian orbits around Sun) with terrestrial planets in our system are .10 8 per orbital revolution [27]. Estimating impact probability for each exoplanet by counting the number of hits onto the collisional sphere with a good statistical accuracy, would require sample size of (~1012) projectiles or even larger for each of them [27]. To circumvent the computational problems associated with such large number of projectiles, we instead applied a probabilistic approach inspired by previous work on directed panspermia [28,29]. We modify the approach by adopting that particle travel along linear trajectory at the orbital velocity of 10−3c, and the radius of an exoplanet to estimate the capture probability. An eccentricity of IMPs trajectory depends on its perihelion distance and velocity. The larger its perihelion distance (or larger its velocity relative to the Sun), its trajectory would be less modified by gravitational acceleration, while its eccentricity would increase. Thus, very distant IMPs will traverse almost straight lines (i.e., eccentricities converging to infinity) (see [30]). For simplicity, we will assume that an IMP follows a straight line when entering into our Galaxy from position of our solar system. Trajectory of an IMP is within ecliptic plane and perhaps further studies should make use of 3D probes. Calculated HZEP probabilities are small due to small sample of HZEP confined in very large parametric space. Also, hit probabilities are small as a ratio of exoplanet cross section area and IMP’s letal area. The cooccurence of such rare events can not be calculated classically by their coupling, instead the information theory metrics must be employed. For this purpose we used Entropy index.

2.1. Exoplanet Data At the time of writing, more than 3000 exoplanets have been conﬁrmed (Extrasolar Planets Encyclopaedia, June 2019). The majority of detected planets are within distances of several tens to several thousands pc from Sun. Their host star ages are of hundreds of Myr to a few Gyr. For our calculation we needed radius and effective temperature of star, radius (in Jupiter radius RJ) and semimajor axis (in AU) of planet and distance from the Sun to planetary system (in pc), which yielded the sample of 171 HZEP. Since distance of our planet from the Sun is 4.84814 × 10−6 pc (i.e., 1 AU) we will assume that these distances are also from our planet. We note that the known population of exoplanets are not representative of the reality, since expected number of them is aabout 1010 [12].

2.2. Habitable Zone There are several prescriptions on how to deﬁne HZ and calculate its limits (see an excellent review in [31]). Usually, the circumstellar (HZ) is assumed to be an annular volume around the star where planets with orbits inside it may be expected to have liquid water on their surface. But for our purpose we calculated HZ as follows: based on polynomial expression for inner and outer limits of HZ given in [32], and effective stellar temperatures collected from EOD, we estimated the effective stellar ﬂuxes. Sci 2019, 1, 47 4 of 16

This prescription has been constructed for the stars which effective temperatures correspond to stellar masses between 0.1 and 1.4 M as they are in our sample. Then the HZ limits are given by [33] as

L !0.5 d = L AU (1) Seff where d is either inner or outer limit of HZ, L is the stellar luminosity, L is the luminosity of the present Sun, and Seff is effective stellar ﬂux. Then we estimated the geometric center (cHZ ) of HZ as arithmetic mean of inner and outer HZ limits and required that semimajor axis (a) of the planet satisﬁes following condition |cHZ − a| ≤ 0.5 (2)

This is very optimistic view. Up to now, the current number of potentially habitable exoplanets detected is about 49 (see http://phl.upr.edu/projects/habitable-exoplanets-catalog). This catalog also includes exoplanets up to 10 Earth masses and 2.5 Earth radii to include water-worlds, Mega-Earths, and the uncertainty of radius estimates. Today we have great uncertainty that any exoplanet is really habitable. We suspect that life could depend on many planetary characteristics that are simply not known for exoplanets. Therefore, our criterion is only used to select as large as possible sample of the best objects of interest for our study, not to strictly discriminate the habitable from non habitable worlds.

2.3. Probabilities Estimates A simple analogy with military operations theory [34], can aid in understanding the overall process. The calculation of the probability that a bullet impacts a target is reduced to estimating the hit probability into some region of a certain geometric shape. The basic interaction between weapon and target is given by damage function (or lethal area) D(r), which is the probability that the target is hit by a bullet if the relative distance between them (the miss distance) is r. The simple assumption that target is a planar ﬁgure with the density distribution of the position relative to the weapon PD(x,y). Then the probability of hit is determined by the double integral over the whole combat plane RR D(r)PD(x, y)d xd y. If the target was uniformly distributed within some large ∆y area then distribution of relative target positions 1 RR p 2 2 reduces to PD(x, y) = 1/∆y and double integral becomes ∆y D( x + y )d xd y. However, in our case ∆y 2 damage function would reduce to the area of exoplanet cross section (At) and ∆y = π(δy) where δy is a At resolution of target’s uncertainity position. Thus, the double integral would reduce to the ratio ∆y . This is an essence of estimate of impact probabilities by adopting [29] method. Note, that such deﬁned probability is dimensionless. Using the kernel density estimate (KDE) we obtain the probability distribution of the HZEP in the parameter space of the HZER radii and thier distances from Earth. Finally, we use information theory metric to evaluate the co-occurrence of two events: planet residence in the HZ and being hit by IMP.

2.3.1. Probability That Planet is within the HZ Here we describe the process of performing a Kernel density estimation (KDE), a statistical process for density estimation that a planet is within the HZ. We used the framework of multivariate nonparametric density estimation. In nonparametric statistics no stringent parametric assumptions are made on the underlying probability model that generated the data. The appeal of nonparametric methods is in their ability to reveal structure in data that might be omitted by classical parametric methods. Let X1, ... , Xn Sci 2019, 1, 47 5 of 16 be observation drawn independently from an unknown distribution P on Rp with the density f. Kernel density estimates an unknown underlying density f [35] as follows:

1 n fn(x, H) = ∑ KH(x − Xi) (3) n i=1

−0.5 −0.5 where KH = |H| K(H x), the matrix H, of dimension p × p is the matrix of smoothing parameters, it is symmetric and positive definite (i.e., xTHx > 0, ∀x ∈ Rn \{0}), and K : Rp → [0, ∞) is a probability density. In our case, Xi are two dimensional vectors containing the radii and distances HZE from Earth. Then given a set of data points, KDE interpolates and smooths probability density function on continuous surface using a given kernel. Due to the sparsity of existing data, KDE is particularly relevant in our calculation. The matrix H also controls the bandwidth of kernel (i.e., a norm of the matrix |H|). We used KDE implemented in machine learning package of Pyhton scikit-learn. KDE can be considered as a form of machine learning [36], where we want to estimate density but not to predict a new data given certain set of known data. In machine learning contexts, the hyperparameter tuning often is done empirically via a cross-validation approach. Since the radius of planets and their distances (to the Earth) vary over several order of magnitudes, for bandwidth estimate we used the cross validation least squares method. Then, the probability density function (PDF) is evaluated in a grid of points covering the region of analysis. KDE discretize user-defined evaluation space (or grid), computing the density in equally separated grid points. So, the evaluation space can be represented as a multi-dimensional matrix. We defined the separation between grid points, so that we have 800 points in each dimension of 2D parameter space defined by exoplanet distances and radii. The output is KDE of the same dimensions as defined grid [37,38].

2.3.2. Evaluating Collision Probability Here we consider the probability of capture of IMPs of negligible dimension within kinematical framework suggested by [29,39,40] for directed panspermia . A schematic overview is given in Figure1. This approach uses the proper motions of the targets, their distances (to the Earth) and the velocity of projectiles. Combining the positional uncertainty and dimension of the target object allows for calculating the probability of hitting. The positional uncertainty δy of the target at arrival time of projectile is given by following equation d2 δy = 1.5 × 10−13α (4) v where α is the resolution of proper motion of the target object, d is distance from the Earth (note we assume that particle trajectory passes within vicinity of our planet) and v is velocity of the projectile. We set the star proper motion resolution (uncertainiy) α = 10−5 arcsec yr−1 as suggested by [39,40]. This value is also in accordance with GAIA typical star proper motion uncertainty of 2 − 4 × 10−5 arcsec yr−1 [41]. Sci 2019, 1, 47 6 of 16

v=300 km/s �y/2

At Sun a[AU] HZ 1 AU

d [pc]

Figure 1. Scheme of an IMP impact with an exoplanet and parameters for calculating its probabilities. Under idealized conditions, a high velocity IMP passing in close proximity to the Earth and continues to move with roughly the same trajectory and speed toward an exoplanet whith cross section area At. The planet is located at distance d[pc] from the Earth and its orbit around host star has semimajor axis a[AU]. A fragment of HZ is given as solid curves. The ’lethal area’ of IMP is a sphere with diameter δy and is a consequence of a proper motion of the star. This schematic drawing is exaggerated for clarity and is not to scale.

The probability that projectile hits the target area At of radius rt is given by

2 2 At 25 rt v Pt = = 4.4 × 10 (5) π(δy)2 α2d4

For planetary system the area At may be even the width of HZ. For very large impactor initial velocities (v∞), as it is case of IMP, the impact parameter (deﬁned as the perpendicular distance between the velocity 2GM 0.5 direction of a projectile and the center of an object that the projectile is approaching) b = R(1 + 2 ) Rv∞ (where G is gravitational constant) converges to planet’s radius R, while the planet mass M does not play any role as it can be seen (i.e. the right parameter in the brackets is vanishing i.e., gravitational focusing is irrelevant). Thus, we will use the radius of an exoplanet to estimate At. Note that in Equation (4) we employ the Earth-exoplanet distance. However, the probability calculated with Equation (5) is valid not only for Earth-exoplanet direction but for any point on the sphere which radius is Earth-exoplanet distance and center in the exoplanet.

2.3.3. Information Theory Metrics In order to evaluate the statistical variability of hitting a planet within the HZ by an IMP, we utilize the information theory metrics of entropy (which is based on Shannon information entropy definition, [42]). The aim of this metric is to evaluate the co-occurrence of both planet residence in HZ and being hit by IMP. Entropy can be viewed as the expectation of log( f (X)) where f is the probability density function (PDF) of a random variable X, the uncertainty of the outcome of an event and the dispersion of the probabilities with which the events take place. Here, we first recall the definition of the Shannon entropy. Let we consider a set of planets S, which are a sample on a random vector variable X ∈ Rd, with an associated PDF describing their distribution. This PDF p(X = x) estimates the probability of an observation x on X, denoted as p(X). We evaluate the Shannon’s information entropy H [42] as Sci 2019, 1, 47 7 of 16

d H(X) = − ∑ p(xk) log p(xk) (6) k=1 The entropy H(X) provides a measure of the information content in the probability spectrum [43]. The joint Shannon entropy of a set of independent random variables is sum of their individual entropies. Conversely, the total entropy is a sum of conditional entropies. In this sense, let V = Vj|j = 1, ..m be a vector of probabilities obtained by concatenation of probability vectors estimated by Equations (3) and (5). For the planet being within the HZ and hit by an IMP are two independent events and their co-occurrence is given by

k − ∑j=1 Vj log Vj EI = (7) logk where k is the number of probability distributions used to create V (in our case it is two). Since Shannon’s entropy can take values H(X) ∈ [0, log(n)], the EI is greater or equal to zero, and normalized. The higher value of EI means the higher co-occurrence of the events. Also this metric can be seen as a reﬂection of the level of concentration of such events. Since Shannon entropy may be used globally, for the whole data or locally to evaluate entropy of probability density distributions around some points, the same is valid for EI as well. So this quality of Shannon entropy can be generalized to estimate importance of speciﬁc events, e.g., rare events [44].

3. Results Since the impact probabilities are calculated within the parameter space of planetary radii and their distances from Earth, we choose the same phase space for depicting HZ probability map. In Figure2 we show the probability map of habitability , obtained at each point of the plane determined by planetary radius and distance from the Earth. Because our results reﬂect the underlying exoplanets observational bias, we caution readers to keep this in mind when assessing the results. For example, the planetary size and especially the distance distribution observed are a consequence of the observational techniques and not the true spatial distribution of exoplanets. The increase of the planetary radius with distance is a direct consequence of the observational method and has no astrophysical basis. The results show that the hot zones of habitability is below 600 pc and 0.3 RJ , but the prominent spots are bellow distances of 400 pc, and planet’s radii of 0.2 RJ. There are large regions of parameter space where it is unlikely that planets resides within HZ because simply that part of parameter space is not populated by exoplanets in our sample. Next, we calculated the probabilities of exoplanets from our sample being hit by an IMP.We considered that an IMP enters our Galaxy within vicinity of our Solar system in the direction of the considered planet. We set a hypothetical velocity at 300 km s−1. The probability map of hitting is displayed in the form of stripes (see Figure3). The logarithmic scale is used for the color-coding which allows greater detail to be seen in regions where the probabilities are low. The ’hot zone’ is within 200 pc and it is spread evenly across the values of radii of exoplanets. However the probabilities are decreasing gradually with the distance. Sci 2019, 1, 47 8 of 16

1400

1200 6

1000 ] 27 − 800 4 Dist[pc] Prob[10 600

400 2

200

0 0.2 0.4 0.6 0.8 1.0 Rpl[RJ]

Figure 2. Probability that planet is located within the habitable zone. The contour levels correspond to the HZ probability as indicated on the color bars. Note the small values of probability, since we used non-parametric KDE and distances from Earth as a second dimension. Probability can take values from 0 to 1, where 0 represents regions of parameter space where it is unlikely that planets is within HZ and 1 vice versa.

7 10− 1400

9 10− 1200

11 10− 1000

13 10− 800 Prob Dist[pc] 10 15 600 −

17 400 10−

19 200 10−

21 10− 0.2 0.4 0.6 0.8 1.0 Rpl[RJ]

Figure 3. The probability that the exoplanets from our sample are hit by IMPs, given in the phase space of radius (x axis in Jupiter radii) and distance of planet from Earth (y axis in parsecs). The contour levels correspond to the probability of hitting as indicated on the color bars. Sci 2019, 1, 47 9 of 16

Figure4 reveals emerging pattern of areas with concentrated events of exoplanets within HZ being hit by IMP. Note that hotspots are determined by event density and not event counts. It is not given per any unit of time since the probabilities of a planet being hit by an IMP are dimensionless and Vj are probabilities. The unit of analysis is deﬁned from the grid of dimension of 800 × 800 points in phase space (i.e., the same grid as for KDE, see Section 2.3.1). Thus a hotspot can be created from very few small probabilities if they are concentrated in a small area. It also assists with identifying clusters of key populations of planets. The clustering is increasing toward the smaller distances, i.e., to Solar postion and the radii of planets are bellow 0.5 RJ. This feature can be ampliﬁed if the majority of planets bearing life are spread towards the inner part of the Galaxy, as suggested by [45].

6.4 1400

1200 4.8

1000 ] 9 800 − 3.2 Dist[pc] EI[10 600

400 1.6

200

0 0.2 0.4 0.6 0.8 1.0 Rpl[RJ]

Figure 4. The statistical map of co-occurrence of events that the exoplanets are in the HZ and being hit by IMPs, given in the phase space of radius (x axis in Jupiter radii) and distance of planet from Earth (y axis in parsecs). The contour levels correspond to the probability of hitting as indicated on the color bars. A logarithmic scale is used for the color-coding.

4. Discussion The lack of data to estimate the probability of an event accurately is not the only problem. In undersampled regime we may fail to detect all events with probabilities greater then zero. In general, estimating a distribution in this setting is difﬁcult. Contrary, estimating Shannon entropy Equation (6), is easier. In fact, in many cases, entropy can be accurately estimated with fewer sample. Even using relatively small sample of exoplanets, the data demonstrated that EI is a robust and sensitive measure for differentiating hot spots for IMP’s collision. The map is not based on any particular Sci 2019, 1, 47 10 of 16 assumption about the distribution of the exoplanets parameters, however we assumed the IMP entered into our Galaxy within the vicinity of our planet. The EI was stable in nearly the whole phase space deﬁned by exoplanet’s radius and its Earth distance except several aggregates within the range of distances 400–1200 pc and radii bellow 0.4 RJ. This region is characterized by a clear distribution and a sharp contrast to surrounding parameter space. Comparison with well known impact probabilities with terrestrial planets, which are not larger than − .10 8 per orbital period [27], indicates that our estimates of probabilities of IMP impacts with exoplanets within HZs are reasonable. Perhaps there are many mechanism which can increase IMP impact probabilities, but we will concentrate on only one due to the underlying observations [46]. Importantly, IMPs could decelerate in the vicinity of evaporating planets, whose outer layers are lifted into the surrounding space forming large clouds. For example, KIC 12557548 b is assumed to be a rocky planet more massive than Mercury, with a surface temperature of ~2100 K, which completes an entire orbit in just 16 h. These extreme dynamical and physical properties cause a continuous loss of material, forming an extended tail of dust following the planet in its orbital path [46]. An IMP, after collision with particles in such clouds, could be fragmented [47]. In such case, the probability of IMP fragments impacts would increase. Next, we consider the amount of possible organic content in the IMPs. Assuming it is a chondrite, probably carbonaceous, an estimated density of such chondrites can be between 2.3 and 2.5 g cm−3 [48]. Thus, the mass of 0.01 cm IMP could be 10−5 g which is relatively close to a mass inferred by [7]. Taking into account that about 2% of such bodies mass can be organic [49], we estimate that the mass of possible biotic content in IMP is about 10−6 g. It is possible organic content, since estimates assuming Earth origin, which is not the case since IMPs come from the extragalactic medium. There is a further question: is whether IMP mass mres satify biomass requirements stated by [28] in equation:

cres mbiom = mres (8) cbiom where mbiom is the amount of biomass constructed from mres of resource material, cres is a concentration of essential elements (C, H, N, O, P, S, Ca, Mg and K) in the resource material ( obviously, this series is for Earth biomass, which may not be the case), cbiom is the concentration of essential elements in a given biomass. −7 −1 −1 Here we used standard values for mbiom = 10 g [29], cres = 973.3 mg g , cbiom = 180.763 mg g [28]. −7 Plugging these values in Equation (8), the mres = 5.7 × 10 g is obtained. From comparison with the inferred mass of an IMP 10−5 − 10−3 g one can see that IMP is potentially viable vehicle for biological organic material transfer. The second question here is which and how complex organic molecules can be transported by IMP. To answer this question, it helps to recall that potentially organic, unidentified infrared emission (UIE) bands have been detected in planetary nebulae, reflection nebulae, HII regions, diffuse interstellar medium, and even in other galaxies [50]. Moreover, up to 20% of total luminosity of some active galactic nuclei is emitted in the UIE bands [51]. UIE bands detected in quasars [52] and in high-redshift galaxies [53] imply that complex organic species were widespread as early as 10 billion years ago. Thus, abiological synthesis of complex organics has been occurring through most of the Universe history. Different chemical species have been suggested as sources of the UIE bands: from polycyclic aromatic hydrocarbons (PAH) molecules [54], small carbonaceous molecules [55], hydrogenated amorphous carbon, soot and carbon nanoparticles [56], quenched carbonaceous composite particles [57], kerogen and coal [58], petroleum fractions [59], up to mixed aromatic or aliphatic organic nanoparticles [60]. Among these, the PAH hypothesis is the most popular, but has a number of difficulties and its validity has been questioned [61]. Sci 2019, 1, 47 11 of 16

To determine the largest space distance where material migration can occur on intergalactic scale, we can use as an upper limit for the velocity at which IMP ejected by galaxies may move- a recoil velocity produced by the ﬁnal stage of supermassive binary black holes coalescence. Numerical relativity simulations produce recoil velocities ~103 km s−1, even 5 × 103 km s−1 [62]. Moreover, Robinson et al. [63] using spectropolarimetric observations of E1821+643 found that the central supermassive black hole is moving with a velocity ~2100 km s−1 relative to the host galaxy, that implies a gravitational recoil following the merger of a supermassive black hole binary system. Thus, we can choose upper velocity limit of 3000 km s−1 which is 1% of light velocity. Consequently, exchange of material over 13 bilion years of Universe lifespan could happen over distances of 130 × 106 light years. The radius of observable Universe is about 45 × 109 light years [64], so a material transfer could occur within 0.22% of the observable Universe radius. Conselice at al. [65] found that the total number of galaxies is 2.8 ± 0.6 × 1012 in observable Universe, which means that 616 × 107 galaxies could exchange material. One can anticipate, high efﬁcient material transfer occurring within a galaxy cluster (containing between 100 and 1000 galaxies). Milky Way is not a member of any cluster, so the material exchange could probably occur only within our Local Group of galaxies.

Some Parallels with Biological Immigration Models Some recent studies on interplanetary transfer of photosynthesis organism [66] and simple life forms or bio material [67] employed analogies with Earth ecological models of ’immigration’ to qualitatively explore the possibility of interplanetary material immigration. Namely, in these analogies planets are seen as islands and even continents, and ‘immigration’ would essentially amount to transfer of lifeforms (or genetic material) via vehicles such as meteoroids. This enables us to make some qualitative description of IMPs immigration, if we assume that galaxies are ’continents’ and the material exchange could probably occur only within our Local Group of galaxies. For example, the large-scale distribution of the galaxies observed by VIPERS (see Figure 15 in [68]) shows quite clearly the abundance of structures, and the segregation of the overall galaxy population as a function of the local galaxy density. Thus the immigration of the material via IMPs is similar, to some extent, to a population linear model of birth-death-immigration with binomial catastrophes [69]. We can assume that we have a population of molecules on our planet, which can be terminated or give birth to other molecules and in addition there are immigrant molecules. The model considers a continuous time Markov chain N(t) : t ≥ 0 taking values on a countable set of integers N0 = 0, 1, 2 . . . and an initiator deﬁned as : ( iλ + ν i f j = i + 1, i ≥ 0 qij = i j i−j (9) (j)p (1 − p) γ + iµδi−1 i f j = 0, . . . , i, i ≥ 0 where δi−1 = 1, j = i − 1, and δi−1 = 0, j 6= i − 1. Thus, the transition of stochastic process N(t) from state i to state j occurs as follows: the upper branch resembles the individual up-leap associated with an immigration Poisson process at rate ν and the individual births (creation of molecules) occurring at rate λ. Moreover there exist two types of extinctions-natural death and random catastrophe mechanism. So the rate qi,i−1 = iµ is related to the individual death mechanism where the life time of molecule is terminated i j i−j after random time at rate µ. And second down rate occurs due to catastrophic events qij = (j)p (1 − p) γ. Catastrophes appear as a Poisson process at rate γ. Figure5 shows a transient distribution of birth, immigration, death ( π0(t)), as time evolves for state 0 while the birth rate is λ = 1, the immigration rate is ν = 1, the death rate is µ = 3, the survival probability is p = 0.5 and the catastrophe occurrence rate is γ = 0.000001. Sci 2019, 1, 47 12 of 16

0.5

0.4

0.3 ) ( 0 π

0.2

0.1

0.0 0 1 2 3 4 5 t

Figure 5. The transient distribution of birth, immigration, death process with binomial catastrophes for state 0. A transient distribution π0(t) is given on the x-axis is and the time in arbitrary units is given on y-axis.

5. Conclusions Here we considered the statistical impact probability of IMPs with HZEP. This problem can be equivalently stated as the problem of calculating probability of cooccurrence of a planet being in the habitable zone and hit by an IMP. As the geophysical properties of potentially habitable exoplanets are currently unknown, it is not possible to determine exact HZEP probabilities. Instead, we calculated HZEP probabilities as a probability density function with respect to observationally constrained planet properties (distances and radii), assuming a very optimistic geometrical HZ boundaries. The probablity density function was estimated by machine learning implementation of 2D KDE in Python. Instead of dynamical simulation approach which requires a large swarm of projectiles, we employed Mautner method to calculate probability that a planet is hit by an IMP. Due to high IMPs velocities their trajectories were taken as straight lines. Calculated HZEP and hit probabilities are small, indicating rare events. The former due to small sample of HZEP confined in very large parametric space and later as a ratio of lethal IMP’s area and exoplanet area. The cooccurence of such rare events can not be calculated classically by their coupling, instead the information theory metrics must be employed. For this purpose we used Entropy index. We calculated the entropy map, which clearly shows hot spots of IMP impacts with HZEP correlated with the dimension of planets. The clustering of these spots is increasing toward the Earth. It can be concluded that the Earth neighbourhoods with evidently larger entropy index are regions of possibly active and sustainable influx of IMPs, i.e. intergalactic material. This material is potentially biologicaly Sci 2019, 1, 47 13 of 16 viable, having in mind our estimate of IMP’s mass 10−5 g, under assumption it is a chondrite and its potential biological content mass of 10−6 g. We also utilized ’immigration’ concept from biology in the form of Markov chain to conclude that the possibility of immigration –birth–death process can be high and stable, even if the death rate is three times greater then immigration rate. In our study we employed available data from previous decades of exoplanets observations which were dedicated to the constraining the exoplanet population in the Milky Way on zeroth level (orbital parameters, radii and densities). However, the next generation of telescopes (e.g., James Webb Telescope, the European Extremely Large Telescope; Ariel) will provide data on the atmospheric properties of exoplanets. It is hoped that the calculations outlined in this article will be useful to make atlases (i.e., collection of maps) of interstellar and intergalactic transfer of material within our Galaxy, indicating possible hot spots of panspermia activity, because it builds on our empirical knowledge. The future observational evidence will improve the prediction both of the HZEP probabilites and IMPs influx probabilites, and thus our understanding of the habitability of the universe.

Acknowledgments: The author sincerely thank to the Referee for the constructive comments and recommendations which definitely improve the quality of the paper. This work is supported by project (176001) Astrophysical Spectroscopy of Extragalactic Objects. Funding: This research received no external funding Conflicts of Interest: The author declares no conflicts of interest.

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