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

Article
On Mautner-Type Probability of Capture of
Intergalactic Meteor Particles by Habitable
Exoplanets
Andjelka B. Kovačević
 Department of Astronomy, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade,
 Serbia; andjelka@matf.bg.ac.rs
 Received: 28 June 2019; Accepted: 10 July 2019; Published: 19 October 2019
 First Version Published: 15 July 2019 (doi:10.3390/sci1020040) 



 

 Second Version Published: 9 August 2019 (doi:10.3390/sci1020047)
 Third Version Published: 19 October 2019 (doi:10.3390/sci1030061)
 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 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 estimate the concentration
 of these rare events. We further adopted a model from immigration theory, to show that the time
 dependent distribution of single molecule immigration of material indicates high survivability of
 the immigrated material taking into account birth and death processes on our planet. At present
 immigration of material can not be observationally constrained but it seems reasonable to think that it
 will be possible in the near future, and to use it along other proposed parameters for life sustainability
 on some planet.

 Keywords: intergalactic meteor particle; extrasolar planets; astrobiology

1. Introduction
 The Universe generates 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 includes 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
to 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, i.e., 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

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atmosphere. Ten years later, Afanasiev et al. [7] detected an Intergalactic meteor particle (IMP) in the
100 µm range. 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 of several hundred km s−1 [8]. Afanasiev et al. [7] calculated that the 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 materials exposed to 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 the average density of the 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 contrasts the lack of
any evidence from other optical meteor observatories (see e.g., [11]). Nevertheless, Afanasiev et al. [7]
study also suggests 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 the 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]. However, study of [23] did not confirm existence of
this planet. 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 both star HIP 13044 and its planet (HIP
13044 b with mass of 1.25 mass of Jupiter) was captured by Milky Way.
 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 [24]. Dust
grains mainly flow with interstellar gas. For example, due to uneven distribution of brighter stars
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 [25]. Betatron mechanism (i.e., acceleration produced when the particle drift motion is in
resonance with changes in the induction electric field under a variable magnetic field of star) can
accelerate them up to 30 to 100 km s−1 . Besides this mechanism dust grains and gas can be expelled
into the intergalactic space [26,27], 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 projectiles on a trajectories crossing their orbits. We
focus on the random probability that an IMP, during its cruise through our Galaxy, will collide with
some HZEP.

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2. Materials and Methods
 The statistical impact probability of IMPs with HZEP, can be equivalently stated as calculating
the 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 discrete probability density function
with respect to observationally constrained planet properties (distances and radii), assuming a very
optimistic geometrical HZ boundaries. The discrete probability 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 [28]. 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 [28]. 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 [29,30]. We modify the approach by adopting that particles travel along linear
trajectory at the velocity of 10−3 c, and the radius of an exoplanet from exoplanet database to estimate
the capture probability. The eccentricity of an IMP’s 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 [31]).
For simplicity, we will consider the cases of IMPs following straight lines when entering into our
Galaxy from position of our solar system. The trajectory of an IMP in our calculation is within the
ecliptic plane. Our calculated HZEP probabilities are small due to small sample of HZEP confined
in very large parametric space. Also, hit probabilities are small since they are defined as ratio of
exoplanet cross section area and IMP’s lethal area ( defined in the subsection Probabilities estimates).
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 (EI) defined
in subsection Information theory metrics.

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 the star, the radius (in Jupiter radius RJ )
and semimajor axis (in AU) of the planet and the distance from the Sun to the planetary system (in
pc), which yielded a sample of 171 HZEP. Since the distance of our planet to 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 about
∼ 1010 [12].

2.2. Habitable Zone
 There are several prescriptions on how to define the HZ and calculate its limits (see an excellent
review in [32]). 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 a polynomial expression for inner and outer limits
of HZ given in [33], and the effective stellar temperatures collected from extrasolar data base, we
estimated the effective stellar radiation fluxes. This prescription has been constructed for stars with
effective temperatures corresponding to stellar masses between 0.1 and 1.4 M ( solar masses) as they
are in our sample. Their the HZ limits are given by [34] as

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 !0.5
 L
 L
 d= 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 ) as the
arithmetic mean of inner and outer HZ limits, and required that the semimajor axis (a) of the planet
satisfies the following condition

 |cHZ − a| ≤ 0.5 (2)

This is certainly a very optimistic view. 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 [35], 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 bullet
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
 D (r ) PD ( x, y)d xd y. If the target was uniformly distributed within some large ∆y area then
 RR
plane

 1
 RR p of relative target positions reduces to PD ( x, y) = 1/∆y and double integral becomes
distribution
∆y D ( x2 + y2 )d xd y. However, in our case damage function would be reduced to the area of an
 ∆y
exoplanet cross section (At ) and ∆y = π (δy)2 where δy is a resolution of target’s uncertainity position.
 At
Thus, the double integral would be reduced to the ratio ∆y . This is in essence the estimate of impact
probabilities by adopting [30] 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 an Exoplanet 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 the data that might be omitted by classical parametric
methods. Let X1 , . . . , Xn be observation drawn independently from an unknown distribution P on R p
with the density f. Kernel density estimates an unknown underlying density f [36] as follows:
 n
 1
 f n ( x, H ) =
 n ∑ K H ( x − Xi ) (3)
 i =1

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where K H = | H |−0.5 K ( H −0.5 x ), the matrix H, of dimension p × p is the matrix of smoothing parameters,
it is symmetric and positive definite (i.e. x T Hx > 0, ∀ x ∈ Rn \ {0}), and K : R p → [0, ∞) is a
probability density. In our case, Xi are two dimensional vectors containing the radii and distances of
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 [37], 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 the 2D parameter space defined by an exoplanet distance and radius.
The output is a KDE of the same dimensions as the defined grid [38,39].

2.3.2. Evaluating Collision Probability
 Here we consider the probability of capture of IMPs of negligible dimension within kinematical
framework suggested by [30,40,41] for directed panspermia . A schematic overview is given in Figure 1.
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 the following equation

 d2
 δy = 1.5 × 10−13 α (4)
 v
where α is the resolution of proper motion of the target object, d is distance from the Earth (note: we
assume that the particle trajectory passes within the vicinity of our planet) and v is the velocity of the
projectile. We set the star proper motion resolution (uncertainiy) α = 10−5 arcsec yr−1 as suggested
by [40,41]. This value is also in accordance with GAIA typical star proper motion uncertainty of
2 − 4 × 10−5 arcsec yr−1 [42].
 The probability that the 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 α2 d4
 For a planetary system the area At may be even the width of the HZ. For very large initial velocities
(v∞ ), as it is the case of IMPs, the impact parameter (defined as the perpendicular distance between
the velocity direction of a projectile and the center of an object that the projectile is approaching)
b = R(1 + 2GM Rv2∞
 )0.5 (where G is gravitational constant) converges to the planet’ 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 whose radius is the Earth-exoplanet distance and the center at the exoplanet.

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 v=300 km/s
 y/2

 At
 Sun
 a[AU]
 HZ
 1 AU

 d [pc]

 Figure 1. Schematic diagram of an IMP an impact trajectory with an exoplanet and the parameters
 used for calculating its probabilities. Under idealized conditions, a high velocity IMP passes 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 a distance d[pc] from the Earth and its
 orbit around a host star has a semimajor axis a[ AU ]. A fragment of the HZ of the target star is shown
 as solid curves. The ’lethal area’ of IMP is a sphere of a diameter δy and is a consequence of a proper
 motion of the star. As a star has real motion in 3D space, that motion projected on the line of sight of
 observer will be seen as proper motion. Naturally, planet will follow mother star and thus will have
 uncertainity in position. The diagram is exaggerated for clarity.

2.3.3. Information Theory Metrics
 In order to evaluate the statistical probability 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, [43]). The aim of this metric is to evaluate the co-occurrence of both, planet residence in
the HZ and being hit by an 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 [43] as
 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 [44].
The joint Shannon entropy of a set of independent random variables is the 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

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 − ∑kj=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 [45].

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
Figure 2 we show the probability map of exoplantes being in the HZ, obtained at each point of the
plane determined by planetary radius and distance from the Earth. Note that variable whether any
exoplanet belong to the Habitable Zone is not outcome of any survey since no one survey was designed
to hunt planets within Habitable zones. Consequently, this variable mitigates bias from the surveys
(see e.g., [46]).
 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 Figure 3). 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 from the Earth and it is spread evenly across the values of radii of exoplanets. However the
probabilities are decreasing gradually with the distance from the Earth.
 Figure 4 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 [47].

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 8

 1400

 1200
 6

 1000

 Prob[10−27 ]
 Dist[pc]

 800 4

 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 of its host star. 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 of the
 grid. Probability would have been larger between 0 (regions of parameter space where it is unlikely
 that planet is within HZ ) and 1 (parameter space regions where it is likely that planet is within HZ), if
 we had a much larger representative sample.

 10−7
 1400

 10−9
 1200

 10−11
 1000

 10−13
 Dist[pc]

 Prob

 800

 10−15
 600

 400 10−17

 200 10−19

 10−21
 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.

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 6.4
 1400

 1200
 4.8

 1000

 EI[10−9 ]
 Dist[pc]

 800
 3.2

 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 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 [28], indicates that our estimates of probabilities of IMP impacts with
exoplanets within HZs are reasonable.
 Perhaps there are many mechanisms which can increase IMP impact probabilities, but we will
concentrate on only one due to the underlying observations [48]. 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 [48]. An IMP, after collision with particles in such clouds,
could be fragmented [49]. In such case, the probability of IMP fragments impacts would increase.

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 Next, we consider the amount of possible organic content in the IMPs. Assuming it is
a carbonaceous chondrite, an estimated density of such chondrites can be between 2.3 and
2.5 g cm−3 [50]. 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 [51], we estimate
that the mass of possible biomolecular content in IMP is about 10−6 g. It is most likely to be organic
content rather then living cells, since IMPs are assumed to come from the extragalactic medium.
Another question: is whether IMP mass mres satify biomass requirements stated by [29] 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 for extragalactic particles), the cbiom is the concentration
of essential elements in a given biomass. Here we used standard values for mbiom = 10−7 g [30], cres =
973.3 mg g−1 , cbiom = 180.763 mg g−1 [29]. Plugging these values in Equation (8), a mres = 5.7 · 10−7
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 [52]. Moreover, up to 20% of total luminosity of some active galactic nuclei is emitted in
the UIE bands [53]. UIE bands detected in quasars [54] and in high-redshift galaxies [55] 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 [56], small carbonaceous molecules [57], hydrogenated amorphous carbon, soot and
carbon nanoparticles [58], quenched carbonaceous composite particles [59], kerogen and coal [60],
petroleum fractions [61], up to mixed aromatic or aliphatic organic nanoparticles [62]. Among these,
the PAH hypothesis is the most popular, but has a number of difficulties and its validity has been
questioned [63].
 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 an 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 [64]. Moreover, Robinson at
al. [65] 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 an upper velocity
limit of 3000 km s−1 which is 1% of light velocity. Consequently, exchange of material over about the
13 billion years of Universe age could happen over distances of 130 × 106 light years. The radius of
observable Universe is about 45 × 109 light years [66], so a material transfer could occur within 0.22%
of the observable Universe radius. Conselice at al. [67] estimated that the total number of galaxies is
2.8 ± 0.6 × 1012 in theobservable 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 organisms [68] and simple life
forms or non-living bio material [69] employed analogies with Earth ecological models of ’immigration’
to qualitatively explore the possibility of interplanetary material immigration. Namely, in these

Sci 2019, 1, 61 11 of 15

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 Fig. 15 in 70] 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 [71]. 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 after random time at rate µ. And second down rate occurs due to catastrophic events
qij = (ij) p j (1 − p)i− j γ. Catastrophes appear as a Poisson process at rate γ.
 Figure 5 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.

 0.5

 0.4

 0.3
 π0(t)

 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.

Sci 2019, 1, 61 12 of 15

 Time dependent distribution of single molecule immigration of material indicates high
survivability of the immigrated material taking into account birth and death processes on our planet.
At present immigration of material can not be observationally constrained but it seems reasonable to
think that it will be possible in the near future, and to use it along other proposed parameters for life
sustainability on some planet.

5. Conclusions
 Here we considered the statistical impact probability of IMPs with HZEP. This probability was
calculated as the probability of coccurrence 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 calculated HZEP and hit
probabilities are small comparable to the probabilities that terrestrial planets in our solar are hitted
by particle, indicating rare events. The former due to small sample of HZEP confined in very large
parametric space and later as a very small ratio of geometrical characteristics of IMP’s motion and an
exoplanet area.
 The entropy map, which clearly showed hot spots of IMP impacts with HZEP correlated with
the dimension of planets. The clustering of these spots is increasing toward the Earth, meaning
that the Earth neighbourhoods with evidently larger entropy index are regions of possibly active
and sustainable influx of IMPs, i.e. intergalactic material. Based on out estimates, this material is
potentially biologicaly viable, under the assumption it is a chondrite of a potential biological content
mass of 10−6 g. By utilized ’immigration’ concept borrowed from the Ecology, we conclude that
the possibility of immigration –birth–death process can be high and stable under ideal statistical
conditions, even if the death rate ( of any biotic material) is three times greater then immigration rate.
 While our data was constrained by the limited exoplanet available, 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 as well as follow up measurements of already
recorded relevant exoplanet data according to their appertures. It is hoped that the calculations
outlined in this article therefore 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. The future observations is therefore expected to 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 Referees for the constructive comments and
recommendations which definitely improved the quality of the paper. This work is supported by the project
(176001) Astrophysical Spectroscopy of Extragalactic Objects of Ministry of Education, Science and Technological
development of Republic Serbia.
Funding: This research received no external funding
Conflicts of Interest: The author declares no conflicts of interest.

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