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Open AccessArticlePost Publication Peer ReviewVersion 3, Revised

On Mautner-Type Probability of Capture of Intergalactic Meteor Particles by Habitable Exoplanets (Version 3)

Department of Astronomy, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia
Received: 28 June 2019 / Accepted: 10 July 2019 / Published: 19 October 2019
(This article belongs to the Special Issue Molecules to Microbes)
Version 3, Revised
Published: 19 October 2019
DOI: 10.3390/sci1030061
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Version 2, Revised
Published: 9 August 2019
DOI: 10.3390/sci1020047
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Version 1, Original
Published: 15 July 2019
DOI: 10.3390/sci1020040
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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 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 atmosphere. Ten years later, Afanasiev et al. [7] detected an Intergalactic meteor particle (IMP) in the 100 μ m 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 ~ 10 10 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 μ m up to 1 cm), meteorites (1 cm up to 10 m), comet’s nuclei and asteroids were suggested as potential transfer vehicles [16,17,18,19,20]. 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 μ m 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.

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 ( ~ 10 12 ) 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 R J ) 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 ∼ 10 10 [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
d = L L S eff 0.5 AU
where d is either inner or outer limit of HZ, L is the stellar luminosity, L is the luminosity of the present Sun, and S eff 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
c H Z a 0.5
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 plane D ( r ) P D ( x , y ) d x d y . If the target was uniformly distributed within some large Δ y area then distribution of relative target positions reduces to P D ( x , y ) = 1 / Δ y and double integral becomes 1 Δ y Δ y D ( x 2 + y 2 ) d x d y . However, in our case damage function would be reduced to the area of an exoplanet cross section ( A t ) and Δ y = π ( δ y ) 2 where δ y is a resolution of target’s uncertainity position. Thus, the double integral would be reduced to the ratio A t Δ 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 X 1 , , X n 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:
f n ( x , H ) = 1 n i = 1 n K H ( x X i )
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 H x > 0 , x R n \ { 0 } ), and K : R p [ 0 , ) is a probability density. In our case, X i 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 δ y of the target at arrival time of projectile is given by the following equation
δ y = 1.5 × 10 13 α d 2 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 A t of radius r t is given by
P t = A t π ( δ y ) 2 = 4.4 × 10 25 r t 2 v 2 α 2 d 4
For a planetary system the area A t 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 + 2 G M R v 2 ) 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 A t . 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.

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 R d , 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
H ( X ) = k = 1 d p ( x k ) log p ( x k )
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 = V j | 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
E I = j = 1 k V j log V j l o g k
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 R J , but the prominent spots are bellow distances of 400 pc, and planet’s radii of 0.2 R J . 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 V j 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 R J . 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].

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 R J . 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.
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 m r e s satify biomass requirements stated by [29] in equation:
m b i o m = m r e s c r e s c b i o m
where m b i o m is the amount of biomass constructed from m r e s of resource material, c r e s 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 c b i o m is the concentration of essential elements in a given biomass. Here we used standard values for m b i o m = 10 7 g [30], c r e s = 973.3 mg g 1 , c b i o m = 180.763 mg g 1 [29]. Plugging these values in Equation (8), a m r e s = 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 ~ 10 3 km s 1 , even 5 × 10 3 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 × 10 6 light years. The radius of observable Universe is about 45 × 10 9 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 × 10 12 in theobservable Universe, which means that 616 × 10 7 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 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 N 0 = 0 , 1 , 2 and an initiator defined as:
q i j = i λ + ν i f j = i + 1 , i 0 i j p j ( 1 p ) i j γ + i μ δ i 1 i f j = 0 , , i , i 0
where δ i 1 = 1 , j = i 1 , and δ i 1 = 0 , j 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 q i , i 1 = i μ 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 q i j = i j 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 ( π 0 ( 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 .
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.

Funding

This research received no external funding

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.

Conflicts of Interest

The author declares no conflicts of interest.

References

  1. Newman, M.E.J. Power Laws, Pareto Distributions and Zipf’s Law. Contemp. Phys. 2005, 46, 323–351. [Google Scholar] [CrossRef]
  2. Zolotov, M.Y.; Shock, E.L. Stability of Condensed Hydrocarbons in the Solar Nebula. Icarus 2001, 150, 323–337. [Google Scholar] [CrossRef]
  3. Fraser, H.J.; McCoustra, M.R.S.; Williams, D.A. The molecular universe. Astron. Geophys. 2002, 43, 10–18. [Google Scholar] [CrossRef]
  4. Meech, K.J.; Weryk, R.; Micheli, M.; Kleyna, J.T.; Hainaut, O.R.; Jedicke, R.; Wainscoat, R.J.; Chambers, K.C.; Keane, J.V.; Petric, A.; et al. A brief visit from a red and extremely elongated interstellar asteroid. Nature 2017, 552, 378–381. [Google Scholar] [CrossRef]
  5. Belbruno, E.; Moro-Martin, A.; Malhotra, R.; Savransky, D. Chaotic Exchange of Solid Material between Planetary Systems: Implications for Lithopanspermia. Astrobiology 2012, 12, 754–774. [Google Scholar] [CrossRef] [PubMed]
  6. Taylor, A.D.; Baggaley, W.J.; Steel, D.I. Discovery of interstellar dust entering the Earth’s atmosphere. Nature 1996, 380, 323–325. [Google Scholar] [CrossRef]
  7. Afanasiev, V.L.; Kalenichenko, V.V.; Karachentsev, I.D. Detection of an Intergalactic Meteor Particle with the 6-m Telescope. Astrophys. Bull. 2007, 62, 301–397. [Google Scholar] [CrossRef]
  8. Taylor, A.D.; Baggaley, W.J.; Bennett, R.G.T.; Steel, D.I. Radar measurements of very high velocity meteors with AMOR. Planet. Space Sci. 1994, 42, 135–140. [Google Scholar] [CrossRef]
  9. Pagani, L.; Steinacker, J.; Bacmann, A.; Stutz, A.; Henning, T. The ubiquity of micrometer-sized dust grains in the dense interstellar medium. Science 2010, 329, 1622–1624. [Google Scholar] [CrossRef]
  10. Hirashita, H.; Kobayashi, H. Evolution of dust grain size distribution by shattering in the interstellar medium: Robustness and uncertainty. Earth Planets Space (Special Issue: Cosmic Dust V) 2013, 65, 1083–1094. [Google Scholar] [CrossRef]
  11. Musci, R.; Weryk, R.J.; Brown, P.; Campbell-Brown, M.D.; Wiegert, P.A. An Optical Survey for Millimeter-sized Interstellar Meteoroids. Astrophys. J. 2012, 745, 161. [Google Scholar] [CrossRef]
  12. Swift, J.J.; Johnson, J.A.; Morton, T.D.; Crepp, J.R.; Montet, B.T.; Fabrycky, D.C.; Muirhead, P.S. Characterizing the Cool KOIs. IV. Kepler-32 as a Prototype for the Formation of Compact Planetary Systems throughout the Galaxy. Astrophys. J. 2013, 764, 105. [Google Scholar] [CrossRef]
  13. Meyer, C.; Fritz, J.; Migaiski, M.; Stöffler, D.; Artemieva, N.A.; Hornemann, U.; Moeller, R.; De VERA, J.P.; Cockell, C.; Horneck, G.; et al. Shock experiments in support of the Lithanopanspermia theory: The influence of host rock composition, temperature, and shock pressure on the survival rate of endolithic and epilithic microorganisms. Meteorit. Planet. Sci. 2011, 46, 701–718. [Google Scholar] [CrossRef]
  14. Berera, A. Space Dust Collisions as a Planetary Escape Mechanism. Astrobiology 2017, 17, 1274–1282. [Google Scholar] [CrossRef] [PubMed]
  15. Scharf, C.; Cronin, L. Quantifying the origins of life on a planetary scale. Proc. Natl. Acad. Sci. USA 2016, 113, 8127–8132. [Google Scholar] [CrossRef]
  16. Gladman, B.J.; Burns, J.A.; Duncan, M.; Lee, P.; Levison, H.F. The Exchange of Impact Ejecta Between Terrestrial Planets. Science 1996, 271, 1387–1392. [Google Scholar] [CrossRef]
  17. Zappalà, V.; Cellino, A.; Gladman, B.J.; Manley, S.; Migliorini, F. Asteroid Showers on Earth after Family Breakup Events. Icarus 1998, 134, 176–179. [Google Scholar] [CrossRef]
  18. Andjelka, B.K. On Mautner-Type Probability of Capture of Intergalactic Meteor Particles by Habitable Exoplanets. Science 2019, 1, 47. [Google Scholar] [CrossRef]
  19. Murad, E.; Williams, I.P. (Eds.) Meteors in the Earth’s Atmosphere: Meteoroids and Cosmic Dust and Their and their Interactions with the Earth’s Upper Atmosphere; Cambridge University Press: Cambridge, UK, 2002; pp. 1–319. [Google Scholar]
  20. Capaccioni, F.; Coradini, A.; Filacchione, G.; Erard, S.; Arnold, G. The organic–rich surface of comet 67P/Churyumov–Gerasimenko as seen by VIRTIS/Rosetta. Science 2015, 34, 1–4. [Google Scholar]
  21. Ginsburg, I.; Lingam, M.; Loeb, A. Galactic Panspermia. Astrophyical J. Lett. 2017, 868, L12–L21. [Google Scholar] [CrossRef]
  22. Setiawan, J.; Klement, R.; Henning, T.; Rix, H.W.; Rochau, B.; Rodmann, J.; Schulze-Hartung, T. A giant planet around a metal-poor star of extragalactic origin. Science 2010, 330, 1642–1644. [Google Scholar] [CrossRef] [PubMed]
  23. Jones, M.I.; Jenkins, J.S. (2014). No evidence of the planet orbiting the extremely metal-poor extragalactic star HIP 13044. Astron. Astrophys. 2014, 562, A129. [Google Scholar] [CrossRef]
  24. Draine, B.T. Interstellar Dust Grains. Annu. Rev. Astron. Astrophys. 2003, 41, 241–289. [Google Scholar] [CrossRef]
  25. Tarafdar, S.P.; Wickramasinghe, N.C. Effects of Suprathermal GrainsSolid state astrophysics. In Solid State Astrophysics, Proceedings of a Symposium Held at the University College, Cardiff, UK, 9–12 July 1974; Wickramasinghe, N.C., Morgan, D.J., Eds.; D. Reidel Publishing Company: Dodrecht, The Netherlands, 1976; pp. 249–260. [Google Scholar]
  26. Adam, R.; Ade, P.A.R.; Aghanim, N.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Banday, A.J.; Barreiro, R.B.; Bartolo, N.; Battaner, E.; et al. Planck intermediate results XLIII. Spectral energy distribution of dust in clusters of galaxies. Astron. Astrophys. 2016, 596, A104. [Google Scholar]
  27. Yamada, K.; Kitayama, T. Infrared Emission from Intracluster Dust Grains and Constraints on Dust Properties. MNRAS 2005, 57, 611–619. [Google Scholar] [CrossRef]
  28. Rickman, H.; Wisniowski, T.; Wajer, P.; Gabryszewski, R.; Valsecchi, G.B. Monte Carlo methods to calculate impact probabilities. Astron. Astrophys. 2014, 569, 1–15. [Google Scholar] [CrossRef]
  29. Mautner, M.N. Life in the Cosmological Future: Resources, Biomass and Populations. J. Br. Interplanet. Soc. 2005, 58, 167–180. [Google Scholar]
  30. Mautner, M.; Matloff, G.L. A technical and ethical evaluation of seeding nearby solar systems. J. Br. Interplanet. Soc. 1979, 32, 419–423. [Google Scholar]
  31. Engelhardt, T.; Jedicke, R.; Vereš, P.; Fitzsimmons, A.; Denneau, L.; Beshore, E.; Meinke, B. An Observational Upper Limit on the Interstellar Number Density of Asteroids and Comets. Astron. J. 2017, 153, 133. [Google Scholar] [CrossRef]
  32. Gallet, F.; Charbonnel, C.; Amard, L.; Brun, S.; Palacios, A.; Mathis, S. Impacts of stellar evolution and dynamics on the habitable zone: The role of rotation and magnetic activity. Astrophys. Astron. 2017, 597, A14. [Google Scholar] [CrossRef]
  33. Underwood, D.R.; Jones, B.W.; Sleep, P.N. The evolution of habitable zones during stellar lifetimes and its implications on the search for extraterrestrial life. Int. J. Astrobiol. 2003, 4, 289–299. [Google Scholar] [CrossRef]
  34. Kasting, J.F.; Whitmire, D.P.; Reynolds, R.T. Habitable Zones around Main Sequence Stars. Icarus 1993, 101, 108–128. [Google Scholar] [CrossRef] [PubMed]
  35. Binninger, G.C.; Castleberry, P.J., Jr.; McGrady, P.M. Mathematical Background and Programming Aids for the Physical Vulnerability System for Nuclear Weapons; Defense Intelligence Agency: Washington, DC, USA, 1978; p. 87. [Google Scholar]
  36. Wand, M.P.; Jones, M.C. Kernel Smoothing; CRC Monographs on Statistics & Applied Probability (60); Chapman & Hall: Boca Raton, FL, USA, 1995; pp. 1–208. [Google Scholar]
  37. Wang, Z.; Scott, D.W. Nonparametric density estimation for high-dimensional data—Algorithms and applications. Comput. Stat. 2019, 11, e1461. [Google Scholar] [CrossRef]
  38. Ivezić, Ž.; Connolly, A.J.; VanderPlas, J.T.; Gray, A. Statistics, Data Mining, and Machine Learning in Astronomy: A Practical Python Guide for the Analysis of Survey Data (Princeton Series in Modern Observational Astronomy); Princeton University Press: Princeton, NJ, USA, 2014; p. 533. [Google Scholar]
  39. Lopez-Novoa, U.; Sáenz, J.; Mendiburu, A.; Miguel-Alonso, J. An efficient implementation of kernel density estimation for multi-core and many-core architectures. Int. J. High Perform. Comput. Appl. 2015, 29, 331–347. [Google Scholar] [CrossRef]
  40. Mautner, M.N. Directed Panspermia. 2. Technological Advances Toward Seeding Other Solar Systems, and the Foundations of Panbiotic Ethics. J. Br. Interplanet. Soc. 1995, 48, 435–440. [Google Scholar]
  41. Mautner, M.N. Directed panspermia. 3. Strategies and motivation for seeding star-forming clouds. J. Br. Interplanet. Soc. 1997, 50, 93–102. [Google Scholar]
  42. Brown, A.G.A.; Vallenari, A.; Prusti, T. Gaia Data Release 2: Summary of the contents and survey properties. Astron. Astrophys. 2018, 616, A1. [Google Scholar]
  43. Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J. 1948, 27, 379–423. [Google Scholar] [CrossRef]
  44. Consolini, G.; Tozzi, R.; De Michelis, P. Complexity in the sunspot cycle. Astron. Astrophys. 2009, 506, 1381–1391. [Google Scholar] [CrossRef]
  45. Maszczyk, T.; Duch, W. Comparison of Shannon, Renyi and Tsallis Entropy used in Decision Trees. In Artificial Intelligence and Soft Computing ICAISC; Rutkowski, L., Tadeusiewicz, R., ZadehJacek, L.A., Zurada, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2008; pp. 643–651. [Google Scholar]
  46. Hosmer, D., Jr.; Lemeshow, S.; Sturdivant, R. Applied Logistic Regression; John Wiley & Sons: Hoboken, NJ, USA, 2013; pp. 1–479. [Google Scholar]
  47. Gowanlock, M.G.; Patton, D.R.; McConnell, S. A model of habitability within the Milky Way Galaxy. Astrobiology 2011, 11, 855–873. [Google Scholar] [CrossRef]
  48. Bochinski, J.J.; Haswell, C.A.; Marsh, T.R.; Dhillon, V.S.; Littlefair, S.P. Direct evidence for an evolving dust cloud from the exoplanet KIC 12557548 b. Astrophys. J. Lett. 2015, 800, L21–L27. [Google Scholar] [CrossRef]
  49. Napier, W.M. A mechanism for interstellar panspermia. Mon. Not. R. Astron. Soc. 2004, 348, 46–51. [Google Scholar] [CrossRef]
  50. Britt, D.T.; Consolmagno, G.J. Stony meteorite porosities and densities: A review of the data through 2001. Meteorit. Planet. Sci. 2003, 38, 1161–1180. [Google Scholar] [CrossRef]
  51. Sephton, M.A. Organic matter in ancient meteorites. Astron. Geophys. 2004, 45, 8–14. [Google Scholar] [CrossRef]
  52. Kwok, S. Complex organics in space from Solar System to distant galaxies. Astron. Astrophys. Rev. 2016, 24, 1–8. [Google Scholar] [CrossRef]
  53. Smith, J.D.T.; Draine, B.T.; Dale, D.A.; Moustakas, J.; Kennicutt, R.C., Jr.; Helou, G.; Armus, L.; Roussel, H.; Sheth, K.; Bendo, G.J.; et al. The mid-infrared spectrum of star-forming galaxies: Global properties of polycyclic aromatic hydrocarbon emission. Astrophys. J. 2007, 656, 770–791. [Google Scholar] [CrossRef]
  54. Lutz, D.; Sturm, E.; Tacconi, L.J.; Valiante, E.; Schweitzer, M.; Netzer, H.; Maiolino, R.; Andreani, P.; Shemmer, O.; Veilleux, S. PAH emission and star formation in the host of the z ∼ 2.56 QSO. Astrophys. J. Lett. 2007, 661, L25–L28. [Google Scholar] [CrossRef]
  55. Teplitz, H.I.; Desai, V.; Armus, L.; Chary, R.; Marshall, J.A.; Colbert, J.W.; Frayer, D.T.; Pope, A.; Blain, A.; Spoon, H.W.W.; et al. Measuring PAH emission in ultradeep spitzer IRS spectroscopy of high-redshift IR-luminous galaxies. Astrophys. J. 2007, 659, 941–949. [Google Scholar] [CrossRef]
  56. Léger, A.; Puget, J.L. Identification of the unidentified IR emission features of interstellar dust? Astron. Astrophys. 1984, 137, L5–L8. [Google Scholar]
  57. Bernstein, L.S.; Lynch, D.K. Small carbonaceous molecules, ethylene oxide and cyclopropenylidene: Sources of the unidentified infrared bands? Astrophys. J. 2009, 704, 226–239. [Google Scholar] [CrossRef]
  58. Hu, A.; Duley, W.W. Spectra of carbon nanoparticles: Laboratory simulation of the aromatic CH emission feature at 3.29 μm–3.29 μm. Astrophys. J. Lett. 2008, 677, L153–L156. [Google Scholar] [CrossRef]
  59. Sakata, A.; Wada, S.; Onaka, T.; Tokunaga, A.T. Infrared spectrum of quenched carbonaceous composite (QCC). II—A new identification of the 7.7 and 8.6 micron unidentified infrared emission bands. Astrophys. J. 1987, 320, L63–L67. [Google Scholar] [CrossRef]
  60. Papoular, R.; Conrad, J.; Giuliano, M.; Kister, J.; Mille, G. A coal model for the carriers of the unidentified IR bands. Astron. Astrophys. 1989, 217, 204–208. [Google Scholar]
  61. Cataldo, F.; Keheyan, Y.; Heymann, D. A new model for the interpretation of the unidentified infrared bands (UIBS) of the diffuse interstellar medium and of the protoplanetary nebulae. Int. J. Astrobiol. 2002, 1, 79–86. [Google Scholar] [CrossRef]
  62. Kwok, S.; Zhang, Y. Unidentified infrared emission bands: PAHs or MAONs? Astrophys. J. 2013, 771, 5. [Google Scholar] [CrossRef]
  63. Zhang, Y.; Kwok, S. On the viability of the PAH model as an explanation of the unidentified infrared emission features. Astrophys. J. 2015, 798, 37. [Google Scholar] [CrossRef]
  64. Lena, D.; Robinson, A.; Marconi, A.; Axon, D.J.; Capetti, A.; Merritt, D.; Batcheldor, D. Recoiling supermassive black holes: A search in the nearby Universe. Astrophys. J. 2015, 795, 146. [Google Scholar] [CrossRef]
  65. Robinson, A.; Young, S.; Axon, D.J.; Kharb, P.; Smith, J.E. Spectropolarimetric Evidence for a Kicked Supermassive Black Hole in the Quasar E1821+643. Astrophys. J. Lett. 2010, 717, L122–L126. [Google Scholar] [CrossRef]
  66. Gott, J.R., III; Jurić, M.; Schlegel, D.; Hoyle, F.; Vogeley, M.; Tegmark, M.; Bahcall, N.; Brinkmann, J. A Map of the Universe. Astrophys. J. 2005, 624, 463–484. [Google Scholar]
  67. Conselice, C.J.; Wilkinson, A.; Duncan, K.; Mortlock, A. The evolution of galaxy number density at z < 8 and its implications. Astrophys. J. 2016, 830, 83. [Google Scholar]
  68. Cockell, C.S. The Interplanetary Exchange of Photosynthesis. Origins Life Evol. Biosph. 2008, 38, 87–104. [Google Scholar] [CrossRef] [PubMed]
  69. Lingam, M.; Loeb, A. Enhanced interplanetary panspermia in the TRAPPIST-1 system. Proc. Natl. Acad. Sci. USA 2017, 114, 6689–6693. [Google Scholar] [PubMed]
  70. Scodeggio, M.; Guzzo, L.; Garilli, G.; Granett, B.R.; Bolzonella, M.; de la Torre, S.; Abbas, U.; Adami, C.; Arnouts, S.; Bottini, D.; et al. The VIMOS Public Extragalactic Redshift Survey (VIPERS). Full spectroscopic data and auxiliary information release (PDR-2). Astron. Astrophys. 2017, 609, 14. [Google Scholar]
  71. Kapodistria, S.; Phung-Duc, T.; Resing, J. Linear Birth/Immigration-Death Process with Binomial Catastrophes. Probab. Eng. Inf. Sci. 2015, 30, 79–111. [Google Scholar] [CrossRef]
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 A t . The planet is located at a distance d[pc] from the Earth and its orbit around a host star has a semimajor axis a [ A U ] . 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 δ y 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.
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 A t . The planet is located at a distance d[pc] from the Earth and its orbit around a host star has a semimajor axis a [ A U ] . 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 δ y 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.
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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.
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.
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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.
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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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.
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.
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Figure 5. The transient distribution of birth, immigration, death process with binomial catastrophes for state 0. A transient distribution π 0 ( t ) is given on the x-axis is and the time in arbitrary units is given on y-axis.
Figure 5. The transient distribution of birth, immigration, death process with binomial catastrophes for state 0. A transient distribution π 0 ( t ) is given on the x-axis is and the time in arbitrary units is given on y-axis.
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