Afr. Mat. https://doi.org/10.1007/s13370-018-0591-3

On probabilistic modeling and feasibility of collision between a randomly moving meteor and satellite Abd El-Moneim A. Teamah2 · Mohamed Abd Allah El-Hadidy1,2

Received: 14 August 2017 / Accepted: 12 April 2018 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Abstract The collision between a 3-Dimensional randomly moving meteor with the satellite is studied from a probability point of view. The satellite moves along slinky-turn-ellipse curve and starts its motion from a known point (X 0 , Y0 , Z 0 ). The meteor comes from a distant space with dynamic random movement. The main contribution of this paper is to find the conditions that make the expected value of the collision time is finite. In addition, we obtain two candidate utility functions namely the mean time to collision and the cumulative probability of collision. Also, we obtain the estimated distance between them at any time t by using Bayes estimation method. An example with numerical results is given to show the effectiveness of this method. Keywords Stochastic Euclidean distance · Bayesian approach · The mean time to collision · The cumulative · Probability of collision · Bayes estimator Mathematics Subject Classification 60K30 · 90B40

1 Introduction The world of comets is still surrounded with many secrets, up till now, all the details haven’t discovered yet. It is imperative that scientists spend seriously to uncover these secrets, because many of the comets pass close to the earth’s orbit. In July 1994, the first kind of a collision in our space occurred between the guilty stray and Jupiter, the largest planet in our solar system, and as a result of this violent impact an explosion releases as much energy as thousands of

B

Mohamed Abd Allah El-Hadidy [email protected] Abd El-Moneim A. Teamah [email protected]

1

Mathematics and Statistics Department, College of Science, Taibah University, Yanbu, Kingdom of Saudi Arabia

2

Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt

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hydrogen bombs. Astronomers have described this collision as if it had happened on earth, it would have destroyed every kind of sustainable life of plants, animals and the human species and the earth would be a bleak, barren planet. Because of the fear of repeating the same matter with our planet, the scientists introduce a new study on comets that pass close to the earth. Comets orbit in a very elongated orbits (elliptical orbit), and travel into space for billions of kilometers. When comets pass close to the solar system, it is affected by the gravitational force of the planets which make the comet slightly deviate from its orbit and travel into far space. One question has been raised on the scientists’ tongues and haunts their minds is: what would have happened, if one comet had collided with our planet? And how can we avoid the consequences of these comets while their orbits pass close to our planet orbit? Since the satellite move in a 3-dimension around the earth in a helix path, then in this work we study the first meeting time that probably the satellite meet the moving randomly stray meteor. This problem is similar as the search problem which has been studied before by many authors. Many papers in search theory studied the existence of finite search plan that give the expected value of the first meeting time. In the case of one dimension randomly moving target, this problem has been studied extensively in many variations, mostly by Beck et al. [1–3], MC Cabe [4], Mohamed [5], Fristedt and Heath [6], El-Rayes et al. [7] and Mohamed et al. [8]. On the other hand, Mohamed et al. [9] discussed this problem in the plane when the target moves with a conditionally deterministic motion. When the target starts its motion as parabolic spiral from a random point in the plane, Mohamed and El-Hadidy [10] studied this problem such that there is not any time information about the target’s position available to the searcher. For a 3-dimension randomly moving target, El-Hadidy [11] formulated a search model and obtained the conditions that make the expected value of the first meeting time between one of the searchers and a helix target is finite. Recently, El-Hadidy et al. [12] presented a new optimal search path that is called optimal slinky turn spiral search path. This path is more applicable to find a 3-dimensional randomly located target such as finding the wreckage which commentator underwater. They studied the proprieties of this search path from a geometric point of view. Also, in [13] El-Hadidy and El-Bagoury presented a new different optimal search model that finds a 3-dimensional randomly located target by a single searcher. They obtained the minimum expected value of the time for detecting the target, assuming trivariate normal distributed estimates of its position. For more recent works, the reader should see El-Hadidy et al. [14–17]. On the other hand, the prediction approach (Bayesian approach) that used in this paper is different from the prediction approach which has been studied before in Gao et al. [18], Yang et al. [19] and Bhimani et al. [20, 21]. Rather than, they implemented the new scheduling scheme by using some probabilistic model like stochastic markov model, they presented more sufficient and different optimization methods that helps the designer to know the effect of optimal parameters on the system performance changes. The main contribution of this paper is studying the existence of the collision between a 3-Dimensional randomly moving meteor with one of the satellites which revolve around the earth in a slinky turn ellipse path (from a mathematical point of view) as shown in Fig. 1. The satellite motion is independent of the meteor motion. The occurrence of a collision is required to find the collision time between the meteor and the satellite. In addition, we present a Bayesian approach that integrates and predicts the effects of the different information and the process model on the stochastic Euclidean distance distribution at known time t. This paper is organized as follows. Section 2 discusses the assumptions which model the problem. Section 3 obtains the conditions which make the expected value of the collision time between the meteor and the satellite finite. Section 3 presents the Bayesian filtering algorithm that accurately maintains and updates the distance state probability density function (PDF).

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On probabilistic modeling and feasibility of collision… Fig. 1 The slinky turn ellipse path of the satellite around the earth in revolution number i at time ti

Section 4 describes the highlights two utility candidate functions. Section 5 presents Bayes estimation for the stochastic Euclidean distance at time step k. An application has been given in Sect. 6 to demonstrate the applicability of this technique. Finally, the paper concludes with a discussion of the results and directions for future research.

2 Problem formulation Assuming that one of the meteorites movements started from a random point suddenly by the heat of the sun. This meteor will collide with one of the satellites moving around the earth as in Fig. 1. Figure 1 shows the movement of the satellite around the earth when it spins around the sun during its rotation number i after time ti in the trilateral dimension which is a slinky-turn-ellipse curve. Since the earth revolves around the sun in unclosed ellipse, then the path of the satellite seems like the motion of solar system relative to the sun as in Fig. 2 during specific time periods (see [22]). Let the satellite moves according to the model [12]: t [R + a cos(ωt)] cos t, 2 t y(t)  Yo + [R + a cos(ωt)] sin t 2 z(t)  Z o + ha sin(ωt),

x(t)  X o +

(1) (2) (3)

where a is the radius of the slinky revolution and ]ha − 1, ha + 1[ gives the vertical change of the curve. In addition, (X 0 , Y0 , Z 0 ) is the starting point of the searcher and t ∈ {0 ∪ R + } where R is the set of real numbers. When the time of the revolution equal 2π Eqs. (1–3) show that the satellite moves along slinky turn unclosed ellipse curve as in Fig. 1. The meteor is assumed to move randomly in the plane according to the process {S(t); t ∈ R + }, where R is the set of real numbers and S(t)  {S1 (t), S2 (t), S3 (t)} is a 3-Dimensional

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Fig. 2 The movement of the satellite around the earth at different times t1 , t2 , . . .

independent stochastic process. The collision time τ is a random variable valued in R + and t it is defined by τ  inf{t : D t  0}, where  D is the Euclidean distance between the meteor t and the satellite at time t. Since, D  (x(t) − S1 (t))2 + (y(t) − S2 (t))2 + (z(t) − S3 (t))2 is depend on S1 (t),S2 (t) and S3 (t), then we call it the stochastic Euclidean distance. We assume that, there is a known probability measure γ and it is describe the location of the meteor and the satellite. The objective is to obtain the condition that makes the collision be done and find two candidate utility functions namely the mean time to collision and the cumulative probability of collision.

3 Existence of a collision If the meteor passes a distance very close to the earth quickly above it, then the meteor may collide with some of the satellites that meet it. This is similar to the situation which occurred in 1992, where it was expected to collide a comet with earth planet but of Allah’s

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mercy, the comet did not bump the land. Also, the meteor was approached from the earth path with a distance not less than 600 thousand kilometers. A distance is caused a terror in the hearts of scientists, as the impact on the area earth’s atmosphere and the contents of satellites like the distance caused by the speed of the bullet in the hearing. The meteor has passed quickly, scientists had been unable to see him and filmed in our space, but only 2 days, so this meteor is the first large comet passes near the earth and we were able to see and photograph it. This makes us interested in finding the necessary conditions that show the existence of collision between a 3-Dimensional randomly moving meteor and the satellite. Theorem 1 The expected value of the collision time E τ between a 3-Dimensional randomly moving meteor with the satellite is finite if: ∞ ∞ ∞ P (Ψ1 (t) > −S 1 (t) , Ψ2 (t) > −S 2 (t) , Ψ3 (t) > −S 3 (t)) γ (d S 1 (t) d S 2 (t) d S 3 (t)); 0

0

0

0 ∞ ∞ P (Ψ1 (t) > −S 1 (t) , Ψ2 (t) > −S 2 (t) , Ψ3 (t) < −S 3 (t))γ (d S 1 (t) d S 2 (t) d S 3 (t)) ; −∞ 0

0

0 0 ∞ P (Ψ1 (t) > −S 1 (t) , Ψ2 (t) < −S 2 (t) , Ψ3 (t) < −S 3 (t))γ (d S 1 (t) d S 2 (t) d S 3 (t)) ; −∞ −∞ 0

∞ 0 0 P (Ψ1 (t) < −S 1 (t) , Ψ2 (t) < −S 2 (t) , Ψ3 (t) > −S 3 (t))γ (d S 1 (t) d S 2 (t) d S 3 (t)) ; 0 −∞ −∞

∞ ∞ 0 P (Ψ1 (t) < −S 1 (t) , Ψ2 (t) > −S 2 (t) , Ψ3 (t) > −S 3 (t)) γ (d S 1 (t) d S 2 (t) d S 3 (t)) ; 0 −∞

0

and 0 0 0 P (Ψ1 (t) < −S 1 (t) , Ψ2 (t) < −S 2 (t) , Ψ3 (t) < −S 3 (t)) γ (d S 1 (t) d S 2 (t) d S 3 (t)). −∞ −∞ −∞

are finite. Proof Since (X 0 , Y0 , Z 0 ) are independent random variables and independent of (S1 (t), S2 (t), S3 (t)) then we have: ∞ ∞ ∞ P(τ > t) ≤

p (xo + S1 (t) > t|xo  S1 (t), yo + S2 (t) > t|yo  S2 (t), z o + S3 (t) > t|z o  S3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t)) 0

0

0

0 ∞ ∞ p (xo + S1 (t) > t|xo  S1 (t), yo + S2 (t) > t|yo  S2 (t), z o + S3 (t) > t|z o  S3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t))

+ −∞ 0

0

0 0 ∞ p (xo + S1 (t) > t|xo  S1 (t), yo + S2 (t) > t|yo  S2 (t), z o + S3 (t) > t|z o  S3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t))

+ −∞ −∞ 0

∞ 0 0 p (xo + S1 (t) > t|xo  S1 (t), yo + S2 (t) > t|yo  S2 (t), z o + S3 (t) > t|z o  S3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t))

+ 0 −∞ −∞

123

A. A. Teamah, M. A. El-Hadidy ∞ ∞ 0 p (xo + S1 (t) > t|xo  S1 (t), yo + S2 (t) > t|yo  S2 (t), z o + S3 (t) > t|z o  S3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t))

+ 0 −∞

0

0 0 0 p (xo + S1 (t) > t|xo  S1 (t), yo + S2 (t) > t|yo  S2 (t), z o + S3 (t) > t|z o  S3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t))

+ ∞ −∞ ∞

Suppose that Ψ1 (t)  S 1 (t) − t, Ψ2 (t)  S 2 (t) − t, and Ψ3 (t)  S 3 (t) − t, then ∞∞∞ P (1 (t) > −S 1 (t) , 2 (t) > −S 2 (t) , 3 (t) > −S 3 (t))γ (d S 1 (t) d S 2 (t) d S 3 (t)) 0 0 0

+

0 ∞∞ P (1 (t) > −S 1 (t) , 2 (t) > −S 2 (t) , 3 (t) < −S 3 (t))γ (d S 1 (t) d S 2 (t) d S 3 (t))

−∞ 0 0

+

0 0 ∞ P (1 (t) > −S 1 (t) , 2 (t) < −S 2 (t) , 3 (t) < −S 3 (t))γ (d S 1 (t) d S 2 (t) d S 3 (t))

−∞ −∞ 0

∞ 0 0 P (1 (t) < −S 1 (t) , 2 (t) < −S 2 (t) , 3 (t) > −S 3 (t))γ (d S 1 (t) d S 2 (t) d S 3 (t))

+ 0 −∞ −∞

∞∞ 0 P (1 (t) < −S 1 (t) , 2 (t) > −S 2 (t) , 3 (t) > −S 3 (t))γ (d S 1 (t) d S 2 (t) d S 3 (t))

+ 0 0 −∞

li

0 0 0 P (1 (t) < −S 1 (t) , 2 (t) < −S 2 (t) , 3 (t) < −S 3 (t))γ (d S 1 (t) d S 2 (t) d S 3 (t)) ≤

+

2(i−1)π

the revolution number i take a time lvi and its length is given by li     2  l  x˙ + y˙ 2 + z˙ 2 dt. Also, the revolution number i − 1 take a time i−1 v and its length

is given by li−1  ≤

E (τ )

P (τ > t)dt,

i1 l −1 i v

−∞ −∞ −∞

where 2iπ 

∞ v 

∞ 

2(i−1)π  

  2   x˙ + y˙ 2 + z˙ 2 dt. Let Γi 

2(i−2)π ∞  i P (τ > t) 

i1

 i1

i

li −li−1 v ,

then

⎛∞∞∞    P(1 (t) > −S 1 (t), 2 (t) > −S 2 (t), 3 (t) > −S 3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t)) ×⎝ 0

0

0

0 ∞ ∞ P(1 (t) > −S 1 (t), 2 (t) > −S 2 (t), 3 (t) > −S 3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t))

+ −∞ 0

0

0 0 ∞ P(1 (t) > −S 1 (t), 2 (t) > −S 2 (t), 3 (t) > −S 3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t))

+ −∞ −∞ 0

∞ 0 0 P(1 (t) > −S 1 (t), 2 (t) > −S 2 (t), 3 (t) > −S 3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t))

+ 0 −∞ −∞

∞ ∞ 0 P(1 (t) > −S 1 (t), 2 (t) > −S 2 (t), 3 (t) > −S 3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t))

+ 0

0 −∞

0 0  0 +

P(1 (t) > −S 1 (t), 2 (t) > −S 2 (t), 3 (t) > −S 3 (t)) γ (d S 1 (t)d S 2 (t)d S 3 (t))

−∞ −∞ −∞

This problem is the first investigation of a studying the possibility of a collision between satellites and animated randomly meteors. This study is based on the estimated stochastic Euclidean distance between the meteor and the satellite at any time step k that make the collision may be done. Therefore; we have n-random variables that represent the dis-

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tance between the metro and the satellite at time step k denoted by Dkt ∈ R n D . We use Bayesian approach that integrates and predicts the effects of the different information and the process model on the distance distribution at known time. This approach is suitable for non-linear motion models, see Berger [23] and Stone [24]. Using Bayesian analysis, one can use state of knowledge (unknown quantity) about the meteor as a random variable. This variable is entirely expressed in the form of a probability density function (PDF). All the time we want to update the state of knowledge by using Bayes theorem through combining the new information in the form of a probabilistic measurement or observation with the previous PDF. As in Bourgault et al. [25], the analysis  starts by determining the prior probability density function (PDF) of D0t , that is p( D0t  h 0 ) ≡ p(D0t ) for the meteor and satellite state at time 0, given all available information including past experience and domain knowledge, respectively. For instance, this prior PDF could be in the form of a Normal distribution represented the prior coarse estimate of the parameter of interest. If nothing is known other than initial bounds on Dkt , then a least informative approach is to represent this knowledge by a uniform PDF, is used as the prior. Consequently,  once the prior distribution has been established, the PDF of the Dkt , is given by p( Dkt  h 1:k ) that can be constructed recursively, provided the sequences, h 1:k  {h 1 , h 2 , . . . , h k } of all provided information. It is clear that all information h 1:k are independent. The prior PDF of the distance at any time step k can be constructed recursively using the prediction and update equations alternatively. The update equation which computes the posterior density of this distance at time step k, is given by:    p( h k | Dkt ). p( Dkt  h 1:k−1 ) p( h k | Dkt ). p( Dkt  h 1:k−1 ) t  p( Dk h 1:k )  , (4)   t p( h k | h 1:k−1 ) p( h k | Dkt ). p( Dkt  h 1:k−1 )d Dk−1 where p( h k | Dkt ) represents the information likelihood given knowledge of the distance state, see Kassem and El-Hadidy [26]. In addition, we use Chapman Kolomgrov equation to predicate the PDF of the next time step k + 1 as follows:    t  t  t  D ). p( D t  h 1:k )d D t , p( Dk+1 h 1:k )  p(Dk+1 (5) k k k  t t  D ) is a probabilistic Markov model which maps the transition probability where p(Dk+1 k from a given previous state to a destination state at time k + 1. Note that if the motion model is invariant over the distance states, then the integral in (5) results in a convolution operation.

4 The probability of collision We assume that the collision event at time step k is Ck . Thus, the likelihood of collision   is given by: p(h k  Ck  Dkt ) and its complement is given by: p(C¯ k  Dkt )  1 − p(Ck  Dkt ). Given all previous observations h 1:k−1 , the conditional probability of no collision event occurring, denoted p(h 1:k  D¯ kt |h 1:k−1 )  qk , depends on how no collision likelihood and the latest distance PDF (5) overlap. Hence, if qk represents the conditional probability of failing collision for a specific observation step, then the joint probability of failing collision in all of the steps from 1 to k, denoted Q k  p(C¯ 1:k ), obtains from the product of all the qk ’s as follows: Qk 

k  w1

k 

 p C¯ w C¯ 1:w−1  qw  Q k−1 qk ,

(6)

w1

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A. A. Teamah, M. A. El-Hadidy Table 1 The different values of

std Π , α , std α and dˆ k k k k

dˆk

std Π k

αk

std α

0.8

0.01

0.01

0.85

0.02

0.02

200

0.9

0.03

0.03

150

0.95

0.04

0.04

100

0.99

0.05

0.05

50

k

250

¯ where C¯ 1:w−1 corresponds the set of information h 1:w−1 where all information equal to C. Therefore, we can deduce that the probability of collision in k-steps, is given by, Pk  1 − Q k .

(7)

It is also possible to compute the probability of collision on time step k, denoted by pk , as follows: pk 

k 

k



  p C¯ w C¯ 1:w−1 1 − p C¯ k C¯ 1:k−1  qw (1 − qk )  Q k−1 (1 − qk ) (8)

w1

w1

which in turn by summing over k provides a sequential method for evaluating Pk as, Pk 

k 

pk  Pk−1 + pk .

(9)

w1

For this reason, we will refer to Pk as the cumulative probability of collision to distinguish it from the conditional probability of collision at time step k which equals to 1 − qk . It is clear that, as k goes to infinity, the cumulative probability of collision increases towards one. With k increasing, the added probability of collision pk gets smaller and smaller as the conditional probability of collision 1 − qk gets a discounted by a continuously decreasing Q k−1 . The mean time to collision (MTTC) is the expectation of the number for steps required to the collision be done, that is, E(k) 

∞ 

kpk  MTTC

(10)

k1

The difficulty though in evaluating the MTTC lies in the fact that one must in theory evaluate pk for all k’s up to infinity.

5 Bayes estimation for the stochastic Euclidean distance at time step k The estimated stochastic Euclidean distance at time step k is the distance which has a probability of collision equal one. Thus, let u(Dkt ) be arbitrary function of Dkt , then the posterior expectation of u(Dkt ) is :    u(Dkt ) p( h k | Dkt ). p( Dkt  h 1:k−1 )d Dkt t t t t  E(u(Dk ) |h k )  u(Dk ). p(Dk |h 1:k )d Dk   t p( h k | Dkt ). p( Dkt  h 1:k−1 )d Dk−1 (11)

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Fig. 3 Likelihood of collision at time step k when std Πk  0.9 and αk  std αk  0.01 and dˆk  250, 200, 150, 100 and 50 unit of length

The Bayes estimator u ∗ (Dkt ) at time step k under a self is the posterior mean, which can be obtained from (11). The ratio of interest in (11) does not seem to take a closed form and so we must consider a method of approximation. Lindley [27] developed approximate procedures for evaluation of the ratio of integrals of the form:  u(Dkt ) exp[L(Dkt )]d Dkt  , (12) v(Dkt ) exp[L(Dkt )]d Dkt  where Dkt  (1 Dkt , 2 Dkt , . . . , m Dkt ), L[Dkt ]  log p(h k  Ck  Dkt ) (is the logarithm of the likelihood of collision), and v(Dkt )  p( Dkt  h 1:k ) is the prior density of Dkt . If w(Dkt )  u(Dkt ).v(Dkt ), then (11) yields the posterior expectation of u(Dkt ), (the self Bayes estimator), i.e.,

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Fig. 4 Likelihood of collision at time step k when dˆk  250, αk  std αk  0.01 and std Πk  0.8, 0.85, 9, 95 and 0.99

 E(u(Dkt ) |h k )



u(Dkt ). exp[L(Dkt ) + ρ(Dkt )]d Dkt  , exp[L(Dkt ) + ρ(Dkt )]d Dkt

where ρ(Dkt )  log(v(Dkt )). Equation (13) can be asymptotically approximated by: ⎡    ∂ 2 u(Dkt ) ∂u(Dkt ) ∂ρ(Dkt ) 1 t t E(u(Dk ) |h k )  ⎣u(Dk ) + +2 α t . β t .σαβ 2 α ∂ α Dkt ∂ β Dkt ∂ Dk ∂ Dk β   ∂ 3 L(Dkt ) ∂u(Dkt ) + .σ .σ . + O(n −2 ), ∂ α D t ∂ β D t ∂ ξ D t αβ ξ η ∂ η D t ξ

η

k

k

k

k

(13)

(14)

Dkt  Dˆ kt

where O(n −2 ) are terms of order n −2 or smaller, thus we can neglect them. In addition, α, β, ξ, η  1, 2, . . . , m and σαβ ; σξ η are the (α, β)th and (ξ, η)th elements of the variance— covariance matrix of Dkt , respectively. The linear approximation (14) is useful for the ratio of multidimensional integrals as in [28]. In this problem, we take m  1, where 1 Dkt  Dkt . Thus, (14) reduces to:   ∂ 2 u(Dkt ) ∂u(Dkt ) ∂ρ(Dkt ) ∂ 3 L(Dkt ) ∂u(Dkt ) 1 t t E(u(Dk ) |h k )  u(Dk ) + σ11 +2 . + . .σ11 . 2 (∂ Dkt )2 ∂ Dkt ∂ Dkt (∂ Dkt )3 ∂ Dkt (15)

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On probabilistic modeling and feasibility of collision…

Fig. 5 Likelihood of collision at time step k when std Πk  0.9 dˆk  250, and αk  std αk  0.01, 0.02, 0.03, 0.04 and 0.05

Thus, the estimated value of the stochastic Euclidean distance at time step k can be obtained from (15) at Dkt  Dˆ kt .

6 Application In this application, we consider a meteor is randomly moving in the space and have a stochastic distance d between them and a satellite which revolves around the earth. Thus, at time step k the signal power from the meteor and the satellite, A k , that received can be described by the following formula: A k 

k e−2αk dk , 16π 2 dk4

(16)

See Bourgault et al. [25], where αk is the transmission attenuation factor which is greatly affected by the size and density of the meteor in the space. The constant k accounts for other environmental factors such as background noise, temperature, etc. and should be a function of d. Let the probability of collision be a function of A k and the signal-to-noise ratio and  also let the standard likelihood of collision at time step k is std Πk  std p(h k  Ck  Dkt ) has

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Fig. 6 Likelihood of collision at time step k when std Πk  0.9 αk  std αk  0.01, 0.02, 0.03, 0.04, 0.05 and dˆk  250, 200, 150, 100, 50. std a value less than  one for a given amount of standard received signal power, Ak  or equal evaluated at dˆk , std αk . Consequently, from Bourgault et al. [25] and using (16), we get:

 dˆ 4 std ˆ Πk  p(h k  Ck  Dkt )  std Πk k4 e−2(αk dk − αk dk ) dk

(17)

Let the different values of std Πk , αk , std αk and dˆk be given by Table 1. In Fig. 3, the likelihood of collision at time step k is plotted as a function of d for various values of dˆk  250, 200, 150, 100 and 50 unit of length while the std Πk  0.9 and αk std αk  0.01 remain fixed. For each value of dˆk we found that the likelihood of collision Πk is vary from one value to another for dˆk . Whenever, the values of std Πk are changed and dˆk  250, αk std αk  0.01 remain fixed we found that Πk for the values of std Π  0.8, 0.85 and 0.9 are identical and different for the values 0.95, 0.99 as in Fig. 4. k For fixed std Πk  0.9, dˆk  250 and different values of αk std αk  0.01, 0.02, 0.03, 0.04 and 0.05 all likelihood of collision are equal when 260 ≤ dk ≤ 280 as in Fig. 5. For fixed value of std Πk  0.9 and different values of αk std αk  0.01, 0.02, 0.03, 0.04, 0.05,dˆk  250, 200, 150, 100, 50 we have different values of Πk , see Fig. 6. Also, Fig. 7 shows that Πk is affected by the different values of std Πk  0.8, 0.85, 0.9, .95, 0.99, dˆk  250, 200, 150, 100, 50 and fixed value of αk std αk  0.01. All values of Πk is equal to 0.75 at 260 ≤ dk ≤ 280 for different values of αk std αk and std Πk at dˆk  250 as in

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Fig. 7 Likelihood of collision at time step k when αk  std αk  0.01 std Πk  0.8, 0.85, 0.9, .95, 0.99 and dˆk  250, 200, 150, 100, 50.

Fig. 8. Figures 3, 4, 5, 6, 7 and 8 show that the probability of collision equal one at dˆk . The likelihood of collision is more influenced by the different values of std Πk , dˆk and has little effect by change the values of αk ,std αk .

7 Conclusion and future work The conditions which make the expected value of a collision time between a 3-Dimensional randomly moving meteor and a satellite have been given. A Bayesian approach has been presented to integrate and predict the effects of the different information and the process model on the stochastic Euclidean distance distribution between the meteor and the satellite at known time t. Bayes estimation for the stochastic Euclidean distance has been done to get the maximum value of the posterior expectation of (5). We show the effectiveness of this method by using an example with numerical results. The proposed model could be extended to the case that considers the combinations of movement of multiple satellites in the space.

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Fig. 8 Likelihood of collision at time step k when std αk  0.01 αk  std Πk  0.8, 0.85, 0.9, .95, 0.99 and dˆk  250, 200, 150, 100, 50.

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