2.3. Modeling of space debris

At present, there are several models of the objects fragmentation at hypersonic collision. Review of the probabilistic models of space debris is represented in monograph [5]. The main result of collision of two objects with masses M1 and M2 becomes formation of the large number of fragments of various shapes, masses, and dimensions. The following features are used to describe the effect of the collision:


Experiments show that hundreds and even thousands of space debris objects formed in collisions with the satellites. In 2009, the collision of a communications satellite Iridium with Russian satellite Kosmos-2251 resulted in about 600 shards that flew at an altitude from 500 to 1300 km [10].

Space debris poses a great danger to functioning spacecrafts because of the large relative velocities of convergence (Table 1). In recent years, collisions with space debris have killed several spacecrafts.

Given the fact that simulation of motion of space debris is performed under conditions of essential indeterminacy of the input data and using the processing algorithms of random


Table 1. Comparative analysis of the kinetic energy of objects [3].

events in large part, methods of probabilistic risk analysis and justification of preventive measures of damage reducing become most common for cosmic systems and technologies.

Δ, or a closer distance to a greater probability with a collision probability p<sup>c</sup> greater than the threshold pmin. Such convergences are about ≈15,000 per day. The archive is more than 20 years old. For each convergence, the following characteristics are stored in the ADC: the convergence time, trajectory and non-trajectory parameters of the objects convergence, the residuals at the moment of convergence and their probabilistic characteristics, the probability of collision. Briefly, the algorithm for the supporting ADC is described in the book [8]. There, a method for evaluat-

At present, there are several models of the objects fragmentation at hypersonic collision. Review of the probabilistic models of space debris is represented in monograph [5]. The main result of collision of two objects with masses M1 and M2 becomes formation of the large number of fragments of various shapes, masses, and dimensions. The following features are

• Nf (>m), Nf (>d) — the number of fragments with mass more than m, or dimensions more than d. This is one of the fundamental characteristics. Some assumption about fragments

• A/m(d) — the ratio of the square of the typical cross section to mass for fragments of different sizes. This parameter is related to the difference of shapes and materials of the colliding objects; it is necessary in the analysis of the evolution of SD to calculate the

• p(ΔV) — the statistical distribution of the incremental speed of fragments by their size and direction. As a result of collision, some of the energy goes to changing speed of fragments,

Experiments show that hundreds and even thousands of space debris objects formed in collisions with the satellites. In 2009, the collision of a communications satellite Iridium with Russian satellite Kosmos-2251 resulted in about 600 shards that flew at an altitude from 500 to

Space debris poses a great danger to functioning spacecrafts because of the large relative velocities of convergence (Table 1). In recent years, collisions with space debris have killed

Given the fact that simulation of motion of space debris is performed under conditions of essential indeterminacy of the input data and using the processing algorithms of random

Bullet weight: 7.9 g Mass: 14.5 t Mass of fragment: 40 g Muzzle velocity: 715 m/s Velocity: 90 km/h Relative speed:15 km/s Kinetic energy: 2 kJ Kinetic energy: 4.5 MJ Kinetic energy: 4.5 MJ

Kalashnikov KAMAZ with cargo Space debris

which leads to the spread of SD in some part of interplanetary space.

shape and weight are used to recalculate the mass values to dimension values;

ing various risk characteristics using ADC is also described. For more detail, see Ref. [9].

2.3. Modeling of space debris

206 Probabilistic Modeling in System Engineering

1300 km [10].

several spacecrafts.

used to describe the effect of the collision:

deceleration of fragments in the atmosphere;

Table 1. Comparative analysis of the kinetic energy of objects [3].

The movement of each element of the system of space objects can be divided into two components. First, the orbital object moves on a trajectory that can be represented in the elliptical form in the general case in the current time, which oriented in space in a certain way (osculating orbit). Second, the trajectory of the orbital object changes over time (generally, form and orientation are changing). Meanwhile, trajectories of motion of orbital objects change much slower than the position of orbital objects on these trajectories. Therefore, it is proposed to model changes of trajectories and identify the parts of the trajectories for the current moment in time, which are located from each other at a dangerous distance, from the point of view of possibility of collisions of orbital objects (nodes of mechanical conflicts). In other words, to simulate the nodes of mechanical conflicts, speed changing of which corresponds to speed changing of trajectories. For orbital objects, trajectories of which form a node conflict, it is necessary to determine the time intervals of their movement through the node conflicts without a significant investment of time (on the dangerous part of trajectory). Hence, the method of modeling system of orbital objects is based on the method of modeling the nodes of mechanical conflicts and the method of determining the time intervals of movement of the orbital object through a node of conflicts.

Tasks of the analysis of conflicts of orbital objects can be divided into two classes. In the first class, there are the tasks, where it is possible to analyze only the risk of collisions and not to predict specific orbital conflicts. Their solution is based on the consideration of the altitudelatitudinal density distribution of the orbital objects at a specific point in time.

In the second class, there are the tasks that demand prediction of orbital collisions. This prediction boils down to the prediction of convergence of pairs of orbital objects at a dangerous distance, from the point of view of their collision at possible deviations of objects from their calculated trajectories (these can be called dangerous or conflict convergences). In many cases, it is sufficient to predict only dangerous convergence of the orbital objects, and not to simulate the effects of conflicts, which change the trajectories of the colliding objects and form new orbital objects. Such tasks are solved when it is necessary to predict dangerous collisions for spacecraft, which can make the maneuver to avoid collisions. The task of prediction of dangerous convergence can be used as a base for models of near-Earth space contamination by orbital objects. The direct deterministic method is the most common. It is based on the formation of an archive of dangerous convergences of all possible pairs of orbital bodies at a specified time interval, which is included in the considered set of orbital objects (for each dangerous convergence, the passing time interval, the geometric characteristics of convergence and the probability of collision are determined).

The traditional method to predict dangerous convergence is based on modeling the movement of objects and analyzing the current distance between them. There is a difficulty in this method. The relative speed of orbital objects can be more than 10 km/s. Meanwhile, the convergence at a dangerous distance of several kilometers lasts less than 1 s. Therefore, the prediction of dangerous convergences requires modeling with a correspondingly small time step. At larger sizes of sets of orbital objects, it leads to significant time consumption.

An effective way to solving this problem is the implementation of the prediction of dangerous convergences in several stages. Each stage is the check of the possibility of dangerous convergence based on some rule or the simplified method of prediction.

It is shown how to predict the dangerous convergence between space objects and to justify preventive measures to reduce collision risk, solving tasks of modeling with random parame-

The Approach of Probabilistic Risk Analysis and Rationale of Preventive Measures for Space Systems…

http://dx.doi.org/10.5772/intechopen.74212

209

The approach to constructing a probabilistic mathematical model of a complex system based on the principles of functional integration of the models of elements and subsystems in a single integrated software and implemented algorithms to perform the simulation processes for

Modeling the motion of such satellites typically boils down to the obtainment of systems of ordinary differential equations of object motion and their integration by any method. The result is a dependence of the parameters of motion from time under given initial conditions. These equations are the form of representation of the laws of dynamics and kinematics, and

Considering the catalogs of space objects (such catalogs exist in Russia and the United States), it is possible to estimate their relative position and to forecast their movement. In particular, it

[1] Melrae Pictures, Space Junk 3D [Online image]: Retrieved January 11, 2017 from http://

[2] Paramonov NВ, Tokarev DA. Preliminary simulation of systems. Herald of MSTU

[3] Kozoriz FI, Skornyakov VA. Assessment of csollisions in the approach of the ISS to the

[4] Space environment (natural and artificial). Model of spatial and time distribution for

[6] Space track catalog of objects [Internet]. 2017. Available from http://www.space-track.org

[5] Nazarenko AI. Modeling of Space Debris. Moscow: IKI RAS; 2013. 216 p

is possible to assess the threat of convergence and even the collision of spacecrafts.

different input conditions and current state of the real system is given.

can be supplemented with the equations of control.

Address all correspondence to: paramonov\_n\_b@mail.ru

observed objects. Lesnoy vestnik. 2009;2:164-167

space debris in LEO. GOST-25645.167-2005

Moscow Technological University Mirea, Russia

www.spacejunk3d.com/

MIREA. 2015;4(9):165-170

[Accessed: January 11, 2018]

ters.

Author details

Nikolay Paramonov

References

There are three stages of checking the possibilities of dangerous convergence implemented for a given pair of orbital objects. In the first stage, the overlapping region of heights above the Earth's surface, where their trajectories pass is checked. The second stage is based on the fact that the conflict between orbital objects is possible only when their trajectories intersect. It is assumed that the orbital object cannot deviate from the position on the calculated trajectory more than a certain distance. Hence, in each moment of time, the orbital object may be within a sphere, which has the center of the calculated trajectory, and radius is Rcr. A pair of sections of trajectories will be called a node of the mechanical conflicts, if that trajectories are located on the distance L<Lcr from each other. If for a pair of trajectories of orbital objects there is defined a node of conflict, then the condition of the second stage is fulfilled. The third stage is based on the fact that the conflict convergence is possible during simultaneous motion of orbital objects on segments of trajectories, which form a node of conflict. In the third stage, the time intervals of orbital objects' motion through the node of conflicts are defined. If these time intervals overlap each other, then dangerous convergence is possible, and its probability can be calculated.

Assume [11] that the set of the cataloged orbital objects is considered as a multi-element mechanical system. There are some quasiregular components in the movement of the elements of this system. Meanwhile, the interactions of the elements of the system are not taken into account. Such restrictions allow the allocation of the node of conflict at the current time, which is formed by the dangerous parts of the trajectories of orbital objects k and l. This node of conflict restricts the dangerous part of the trajectory of each of these orbital objects. Considering the regularity of the motion parameters of the objects allows to simulate space debris as a combination of deterministic and probabilistic models.
