Probabilistic law of the distribution of the duration of a session of an artificial Earth satellite with a ground object
PART I. Preliminary information about the system and model.
Designing and calculating the ballistic characteristics of satellite systems for various purposes, modeling the processes of movement and functioning, presuppose a preliminary assessment of the possibilities of achieving the planned target effects by such systems. The purpose is currently reduced to information service in the widest sense of target objects (CO) by the onboard equipment of an artificial Earth satellite (AES). Where there are streams of information, there are always problems associated with its protection and ensuring information security, with all the ensuing consequences.
Long-term autonomous operation leads to a change in the design values of the ballistic characteristics, primarily the structure of satellite systems. In turn, this requires periodic correction of the values of a part of the characteristics of the system. In this case, it becomes necessary to pre-calculate them, which is possible if there are measured values of the motion parameters of both individual satellites of the systems and the system as a whole. Such measurements are possible in the presence of a network of ground measuring stations (PI) stationary or mobile (land, sea, air), equipped with appropriate measuring equipment and command radio. Here we consider a probabilistic approach to estimating and calculating the potential capabilities of satellite systems to influence information support on a global, planetary scale from the standpoint of the ballistic construction of the orbital part and the placement of objects, primarily measuring points, the ground part. Information security issues are not considered.
Each IP and DH is associated with a region of space (instantaneous service area (MLO)), if it enters the satellite, it can be served or it can perform maintenance of the central heating itself. The service process includes, in addition to direct measurements of ballistic and telemetric characteristics, reception and transmission of target information, communication with the satellite, transmission of the correction program to the board, bookmark of the work program (RP) for the operation of on-board special equipment (BSpA), taking into account the updated motion parameters and other operations.
A priori, the process of obtaining the required estimates cannot be determined deterministically, since the action of diverse natural and man-made factors leads to disturbance of parameters. The perturbations are primarily subject to the parameters of the satellite movement and, as a consequence, the characteristics of the processes of the BSAPA. Hence the need to study these phenomena as random, to establish the probability laws of the distribution of random events, quantities, functions of random arguments and random functions (fields).
The proposed work for the readers in the framework of the model of joint movement and functioning of the orbital and terrestrial parts of the system considers the interaction of a pair of satellites - IP. In this case, the model includes all the main factors that shape the process of functioning and generate randomness of the onset of important system events. From the set of system events, events of interaction of the satellite with ground-based PIs and target objects are distinguished.
The most important factors taken into account in the mathematical model of the motion of a system are:
- the rotation of the planet around its axis (the angular velocity is constant);
- the geographical location of the ground object (coordinates, radius of the MLO);
- the limited time of the interaction session (the time the object is in the MLO);
- ballistic characteristics of the satellite (flight altitude H, inclination of the orbit plane).
In the mathematical model, simplifying assumptions and assumptions are used (introduced):
- the planet has a spherical shape with a radius of the sphere R, the gravitational field of attraction is central;
- the influence of the atmosphere on the movement of orbital objects is not taken into account;
- the orbits of motion of the satellite are circular, the precession of the planes of the orbits of motion is not taken into account;
- routes of motion of the satellite are modeled by large circles of the sphere;
—MZO service areas — cones with apex in the center of the object, the intersection line of which with a sphere of radius R + H is projected onto the surface of the planet's sphere with a flat circle.
The basis of the model of the interaction of a satellite - IP is a geometric model of the elements of the system and the kinematic relations of their movement. In the three-dimensional space of the plane P of the orbits of each satellite, the systems are fixed, their lines of intersection with the planet's sphere are the paths. MLO ground object (the projection on the planet of the line of intersection of a cone with a sphere of radius R + H) is limited to a segment of a spherical surface, cut off by the plane K. The boundary of the MLO is a circle L of a small circle, "drawn" on the surface of the Earth and moves along with the Earth. It periodically intersects with the orbital plane of the AES, i.e. the MLO segment is intersected by the highway, which is also “drawn” on the surface of the Earth for each revolution of the satellite, again. On some turns, the route crosses the segment, and on some passes, bypassing it. The part of the route on the turn that crosses the segment from the entry point to the exit point lies inside the segment. The fact is that the passage of a route of one orbit by satellite takes 15-16 times less than the length of the day, that is, ≈ 1.5 hours. If the path crosses the segment, then the τ stay time of the satellite in the MLO is proportional to the length t of the chord of the circle L, τ = t / 2πR.
Indeed, the antenna of the measuring instrument PI, turning, accompanies the moving satellite from the moment it enters the zone until it leaves it.
Content statement of the problem
')
The task is as follows. Determine the probability that the plane P of the satellite orbit will cut the MLO segment at random position relative to the center of the MLO.
For definiteness of the model and convenience of reasoning, we introduce two coordinate systems: Cartesian and spherical (polar), the beginning of which is compatible with the center of the spherical planet. The position on the sphere of any point O1 in the spherical coordinate system will be determined by three coordinates:
- the length of the radius of the vector R;
- longitude λ;
- polar distance γ = π / 2 - φ.
The positive directions of reference coordinates are shown in Fig. one.
The axes oh, oy of the Cartesian coordinate system lie in the plane of the planet's equator, the axis oh coincides with its intersection line with the Greenwich meridian, the axis ou is turned to λ = π / 2 and passes through the meridian with eastern longitude λ = π / 2, the axis oz complements the system coordinates to the right system. The formulas for the transition from spherical to Cartesian coordinates and back have the form:
x = Rcosφcosλ; y = Rcos φ ssinà; z = Rsinfo;
R = [x 2 + y 2 + z 2 ] o.5 ; λ = arctg y / x; φ = arctan z / √ (x 2 + y 2 ).
In the Cartesian coordinate system, the equation of the plane K forming a segment on the planet's sphere can be given as
Ah + By + Cz + D = 0,
where A, B, C are the components of the vector n <3>, normal to the K plane, D ≠ 0.
The direction cosines of the vector n <3> = <A, B, C>, perpendicular to the plane K, are determined by the formulas:
cos (α) = A / N; cos (β) = B / N; cos (γ) = C / N; N = [A 2 + B 2 + C 2 ] o.5 ; p = D / N,
where p is the distance from the origin of the coordinates to the plane, with sign (N) = –sign (D).
A characteristic of a segment that has a circle L as a border, except for the position of its center, is the radius r of the circle L. This radius can be determined through the radius R of the sphere and the distance (p):
r = Rζc, where ζc = arccos (p / R).
In other words, a segment on a sphere can be given by the coordinates of the point O1 (x, y, z) - the center of the segment and the angle c, or the coordinates of the point O1 (x, y, z) and the distance p, or by the system: the equation of the sphere and the equation of the plane K .
Figure 1 Position of the satellite tracks with a slope of i large latitude of the upper point of the boundary of the MLO IP
The plane P passing through the center of the sphere can be defined in a similar way, i.e., the general equation of the plane
A1x + B1y + C1z + D1 = 0.
Obviously, for the plane P, the value D1 = 0. There is a more convenient way to model this plane for modeling. In the general case, the plane P is inclined to the plane of the Earth's equator, which coincides with the coordinate horizontal xy plane, at a certain angle i , called the inclination of the plane of the satellite orbit. The line of intersection of the planes of the orbit P and the horizontal coordinate xow of the plane is called the line of nodes (ascending and descending) of the orbit.
The position of this line in the equatorial plane is determined by the angle λ (geographic longitude), measured from the Greenwich meridian. At an arbitrary time, the line of nodes of the orbit of some satellite is characterized by a random value of the angle λ. Obviously, the range of possible values of λ is determined by the interval λ ∊ [0, 2π]. At an arbitrary point in time within this interval, it is impossible to specify a point that would have higher priority relative to all the others to be occupied by the satellite orbit node. There are no physical, mathematical or any other substantiations for this. Therefore, we will consider the probability distribution of the random longitude value λ for the line of orbit nodes in the interval [0, 2π] to be uniform .
The inclination angle i of the orbital plane to the equatorial plane will be considered nonrandom. In such a situation, for the plane P (i, λ) it is more convenient to use its representation through these angles, in the form:
x • tgλ - y + z • ctgi / cosλ = 0.
One of the possible values of the random variable λ will correspond to each random position in the orbital plane space of the satellite P (i, λ).
The passage of the satellite zone ground measuring point
Practical interest for the designer of the satellite system is to assess the possibility of conducting a communication session of the satellite and the PI at each turn of the trajectory.
Denote by symbol ℬ a random complex event consisting in the fact that for a random value of longitude λ, the plane (i, λ) passing through the center of the sphere of the planet will intersect the segment bounded by circle L centered on the radius-vector of point O1 (x, y z) Thus, in this part of the work, the task is to determine the probability of the occurrence of a random event ℬ. The definition of probability will be performed under the condition that λ takes one of its possible values. It is clear that the imposed condition is always satisfied. The methods of probability theory allow us to determine the probabilities of some events (complex) through known probabilities of other events that are related to them in a certain way, and the probabilities of which are specified or can be determined in one way or another.
Perform an analysis of the possibilities of implementing the event ℬ. Figure 1 illustrates the geometry of various situations with the objects considered earlier that favor the onset of a simulated random event ℬ and shows the conditions for the impossibility of its occurrence.
Figure 1 shows the upper hemisphere of the planet and the arcs of large and small circles are drawn on it. The closed curve MNN1M1 is the boundary of the MLO segment (circle L). The position of the center of the O1 segment (x, y, z) is characterized by a latitudinal φ angle and a longitudinal angle λ. The position of the plane P (i, λ) is determined by the angle of inclination i to the plane of the equator, which is deterministic and does not change during the simulation.
Analysis of possible situations for a random event ℬ.
We will distinguish situations that lead to different functional dependencies, generating a complex random event ℬ:
The first situation is φ + r m ≤ i. There are four positions of the arcs of large circles ANN2, BM1M2, CMB1, DN1A1, formed by the section of a sphere with the plane P (i, λ) at four values of λ, corresponding to such positions of the plane at which it touches the border of the segment (circle L) at four points N, M1 , N1, M. The arc KO1K1 is formed by the section of a sphere by a plane perpendicular to the xow plane in the meridian section and passing through the axis oz and the point O1.
In the cross section of the sphere of the planet by the equator plane, how are shown the lines of the nodes of the above-mentioned planes.
The corners are also shown:
- u1, u2 - between the lines of nodes and directions to the points of tangency of the circle L, measured by arcs AN and 1, respectively;
- φ1, φ2 - between the radius-vector of the center of the segment and the directions to the points of intersection of large circles, measured by arcs O1E and O1F in the meridian section;
- i is the angle of inclination of the plane (i, λ) to the xy plane.
Figure 2 Position of the satellite tracks with a slope of i less than the latitude of the upper point of the boundary of the MLO IP
The second situation φ is r m ≤ i ≤ φ + r m . The following figure 2 below shows two positions of the plane P (i, λ) for two values of λ, corresponding to its tangency of the circle L at the points M and M1.
Based on the analysis of the figures, it can be concluded that the realization of the random event ℬ will correspond to the hub point of the plane (i, λ) falling into two intervals AB and CD of the equatorial circle in the upper figure 1 and in one interval CB in the lower figure 2.
The upper figure corresponds to the interval of change of the angle i of the plane
φ + ζc ≤ i ≤ π - φo - ζc,
bottom picture - φ - ζc ≤ i ≤ φ + ζc.
We will call the intervals AB and CD (in the lower figure CB) the intervals of the plane P (i, λ) falling into the segment or, in short, the intervals of contact . For each fixed position and size ζc of a segment and angle i of inclination of the plane (i, λ), there are uniquely determined intervals of impact. The positions, sizes, number of intervals of hitting depend on the angle i, the coordinates of the center of the segment (φ, λ) and the radius c of the segment.
From the analysis performed, it follows that the random event ℬ is complex and can be represented through random simple events consisting in that the random value λ belongs to the intervals AB and CD for the situation of the upper pattern and the interval CB for the situation of the lower picture.
The third situation φ is r m > i. This situation is not considered in the work, since it does not correspond to the onset of a random event ℬ. The plane of motion of the satellite is inclined so low above the equator that MLO objects cannot intersect with it.
Representation of a complex random event ℬ simple random events .
We introduce the notation ℬ1 = (λ∊ ﺭ AB) and ℬ2 = (λ∊ ﺭ D), then ℬ = ℬ1 + ℬ2. Events ℬ1 and ℬ2 are incompatible and, therefore, under the condition that the distribution law of the node λ longitude is known, the relation (ℬ) = (ℬ1) + (2) will be satisfied.
Let the probability distribution of the random variable λ over a large circle in the xy plane, i.e. in the interval [0, 2 π ], given by some continuous function of the longitude of the node - the density of the probability distribution
φ λ (λ) = dF λ (λ) / dλ, where F λ (λ) is the distribution function of λ.
If on the numerical axis λ two arcs are marked by boundary points A, B and C, D, then the probability that λ falls on the intervals of arcs AB and CD is described as
(ℬ1) = ∫φ λ (λ) dλ; P (ℬ2) = ∫φ λ (λ) dλ is determined by the integrals in these designated limits, therefore,
P (ℬ) = ∫φ λ (λ) dλ + ∫ λ λ (λ) dλ, where the values of the limits of integration have not yet been determined. Thus, to determine the probabilities P (ℬ), it is necessary to obtain expressions for the interval boundaries characterized by the lengths of arcs λj, j = 1 (1) 4. It was noted earlier that λj = λj (R, φ, λo, i, ζc), j = 1 (1) 4. We find explicitly the expressions for the lengths of the intervals of the hit.
Let us turn to the analysis of Figure 1. From its consideration we can write down (we use the notation for the arc DD):
λ1 = λ = λ - ﺭ ; λ2 = λ = λ - ﺭ VK; λ3 = λC = λo - π + ﺭ VK; λ4 = λD = λo - π + ﺭ AK.
Ratios for performing numerical calculations
We use the theorems of spherical trigonometry and obtain calculation formulas for the necessary variables.
Here, arcs are defined from rectangular spherical triangles AEK and BFC:
ﺭ AK = arcsin (tg (φ + φ1) / tgi); ﺭ BK = arcsin (tg (φ - φ2) / tgi).
Unknown values φ2, φ1, λj = λj (R, φ, λ, i, ζc), j = 1 (1) 4 will be determined by means of known φ, i, ζc.
From the spherical triangle NEO1, by the sine theorem, we can write the ratio
sin ζc / sind = sin φ1 / sin90 °, where the angle is d = ے NEO1, from the relation we get sind = sin c / sin φ1.
By the cosine theorem from the spherical triangle AEK, we write the ratio
cosi = sindcos (φ + φ1), from which for sind we get another dependence sind = cosi / cos (φ + φ1).
We find two dependencies found in different ways and perform the transformation (we transfer functions to the left with the variable φ1)
sin ζc / sin φ1 = cosi / cos (φ + φ1) or cos (φ + φ1) / sin φ1 = cosi / sin c.
In the last expression, the unknown is the only variable φ1, which we managed to relate through known quantities, which allows us to perform its transformation and lead to a form convenient for its calculation
cos (φ + φ1) / sin φ1 = cos φoctg φ1 - sin φ and further cos φoctg φ1 - sin φ = cosi / sin ζc from where
ctg φ1 = (cosi / sin ζc + sinf) / cosf and finally,
φ1 = arctg (cos φ sin ζc / (cosi + sin φ sin ζc)).
By analogy, considering the spherical triangles OMF and BFK, we obtain the value
φ2 = arctg (cos φ sin ζc / (cosi - sin φ singin ζc)).
Substituting the obtained values of φ1 and φ2 into relations for the longitude λj, j = 1 (1) 4, we can write
λ1 = λ = λ - arcsin (tg (φ + arctg (cos φ sin ζc / (cosi + sin φ sin ζc))) / tgi);
λ2 = λB = λ - arcsin (tg (φ - arctg (cos φ sin ζc / (cosi - sin φ sin ζc))) / tgi);
λ3 = λC = λ - π + arcsin (tg (φ - arctan (cos φ sin ζc / (cosi - sin φ sining ζc))) / tgi);
λ4 = λD = λ - π + arcsin (tg (φ + arctan (cos φ sin ζc / (cosi + sin φ sin ζc))) / tgi).
Thus, all the necessary data for calculating the probability P (ℬ) are determined.
Consider an example . In applications, it is assumed that the law of longitude distribution λj is uniform in the interval [0, 2π]. Under this assumption, the distribution function F λ (λ) and the distribution density φ λ (λ) of the random variable λ are written as shown below figure 3:
The numerical characteristics of these laws of distribution are determined from the expressions: expectation m (λ) = π; sigma standard deviation (λ) = π / √3; dispersion D (λ) = π 2/3.For this case, the probability of an AES entering the IP service area is written in the form (taking into account the corresponding limits of integration and summation)
P (ℬ) = ∫φ λ (λ) dλ + ∫ λ (λ) dλ = 1 / 2π | λ 2j - λ 2j – 1 |, j = 1,2.
Substituting the values of the found integration limits and the found expressions λj into the last relation, we obtain the final expression for the probability that the satellite
enters the area of the measuring point P () = 1 / π ∑ (-1) k arcsin (tg (φ - (- 1) k arctg (cos φ sin ζc / (cosi - (- 1) k sin φ sin 2 ζ c))) / tgi), k = 1,2.
This ratio allows you to go from the fact that the satellite passes through the service area of a ground point to solve the problem of determining the residence time of the satellite in the area of the point depending on the numerical characteristics of the system’s movement (the longitudinal position of the ascending node of the orbit on the orbit to the PI area). This part of the work is planned to be published by the author in the near future.
Designing and calculating the ballistic characteristics of satellite systems for various purposes, modeling the processes of movement and functioning, presuppose a preliminary assessment of the possibilities of achieving the planned target effects by such systems. The purpose is currently reduced to information service in the widest sense of target objects (CO) by the onboard equipment of an artificial Earth satellite (AES). Where there are streams of information, there are always problems associated with its protection and ensuring information security, with all the ensuing consequences.
Long-term autonomous operation leads to a change in the design values of the ballistic characteristics, primarily the structure of satellite systems. In turn, this requires periodic correction of the values of a part of the characteristics of the system. In this case, it becomes necessary to pre-calculate them, which is possible if there are measured values of the motion parameters of both individual satellites of the systems and the system as a whole. Such measurements are possible in the presence of a network of ground measuring stations (PI) stationary or mobile (land, sea, air), equipped with appropriate measuring equipment and command radio. Here we consider a probabilistic approach to estimating and calculating the potential capabilities of satellite systems to influence information support on a global, planetary scale from the standpoint of the ballistic construction of the orbital part and the placement of objects, primarily measuring points, the ground part. Information security issues are not considered.
Each IP and DH is associated with a region of space (instantaneous service area (MLO)), if it enters the satellite, it can be served or it can perform maintenance of the central heating itself. The service process includes, in addition to direct measurements of ballistic and telemetric characteristics, reception and transmission of target information, communication with the satellite, transmission of the correction program to the board, bookmark of the work program (RP) for the operation of on-board special equipment (BSpA), taking into account the updated motion parameters and other operations.
A priori, the process of obtaining the required estimates cannot be determined deterministically, since the action of diverse natural and man-made factors leads to disturbance of parameters. The perturbations are primarily subject to the parameters of the satellite movement and, as a consequence, the characteristics of the processes of the BSAPA. Hence the need to study these phenomena as random, to establish the probability laws of the distribution of random events, quantities, functions of random arguments and random functions (fields).
The proposed work for the readers in the framework of the model of joint movement and functioning of the orbital and terrestrial parts of the system considers the interaction of a pair of satellites - IP. In this case, the model includes all the main factors that shape the process of functioning and generate randomness of the onset of important system events. From the set of system events, events of interaction of the satellite with ground-based PIs and target objects are distinguished.
The most important factors taken into account in the mathematical model of the motion of a system are:
- the rotation of the planet around its axis (the angular velocity is constant);
- the geographical location of the ground object (coordinates, radius of the MLO);
- the limited time of the interaction session (the time the object is in the MLO);
- ballistic characteristics of the satellite (flight altitude H, inclination of the orbit plane).
In the mathematical model, simplifying assumptions and assumptions are used (introduced):
- the planet has a spherical shape with a radius of the sphere R, the gravitational field of attraction is central;
- the influence of the atmosphere on the movement of orbital objects is not taken into account;
- the orbits of motion of the satellite are circular, the precession of the planes of the orbits of motion is not taken into account;
- routes of motion of the satellite are modeled by large circles of the sphere;
—MZO service areas — cones with apex in the center of the object, the intersection line of which with a sphere of radius R + H is projected onto the surface of the planet's sphere with a flat circle.
The basis of the model of the interaction of a satellite - IP is a geometric model of the elements of the system and the kinematic relations of their movement. In the three-dimensional space of the plane P of the orbits of each satellite, the systems are fixed, their lines of intersection with the planet's sphere are the paths. MLO ground object (the projection on the planet of the line of intersection of a cone with a sphere of radius R + H) is limited to a segment of a spherical surface, cut off by the plane K. The boundary of the MLO is a circle L of a small circle, "drawn" on the surface of the Earth and moves along with the Earth. It periodically intersects with the orbital plane of the AES, i.e. the MLO segment is intersected by the highway, which is also “drawn” on the surface of the Earth for each revolution of the satellite, again. On some turns, the route crosses the segment, and on some passes, bypassing it. The part of the route on the turn that crosses the segment from the entry point to the exit point lies inside the segment. The fact is that the passage of a route of one orbit by satellite takes 15-16 times less than the length of the day, that is, ≈ 1.5 hours. If the path crosses the segment, then the τ stay time of the satellite in the MLO is proportional to the length t of the chord of the circle L, τ = t / 2πR.
Indeed, the antenna of the measuring instrument PI, turning, accompanies the moving satellite from the moment it enters the zone until it leaves it.
Content statement of the problem
')
The task is as follows. Determine the probability that the plane P of the satellite orbit will cut the MLO segment at random position relative to the center of the MLO.
For definiteness of the model and convenience of reasoning, we introduce two coordinate systems: Cartesian and spherical (polar), the beginning of which is compatible with the center of the spherical planet. The position on the sphere of any point O1 in the spherical coordinate system will be determined by three coordinates:
- the length of the radius of the vector R;
- longitude λ;
- polar distance γ = π / 2 - φ.
The positive directions of reference coordinates are shown in Fig. one.
The axes oh, oy of the Cartesian coordinate system lie in the plane of the planet's equator, the axis oh coincides with its intersection line with the Greenwich meridian, the axis ou is turned to λ = π / 2 and passes through the meridian with eastern longitude λ = π / 2, the axis oz complements the system coordinates to the right system. The formulas for the transition from spherical to Cartesian coordinates and back have the form:
x = Rcosφcosλ; y = Rcos φ ssinà; z = Rsinfo;
R = [x 2 + y 2 + z 2 ] o.5 ; λ = arctg y / x; φ = arctan z / √ (x 2 + y 2 ).
In the Cartesian coordinate system, the equation of the plane K forming a segment on the planet's sphere can be given as
Ah + By + Cz + D = 0,
where A, B, C are the components of the vector n <3>, normal to the K plane, D ≠ 0.
The direction cosines of the vector n <3> = <A, B, C>, perpendicular to the plane K, are determined by the formulas:
cos (α) = A / N; cos (β) = B / N; cos (γ) = C / N; N = [A 2 + B 2 + C 2 ] o.5 ; p = D / N,
where p is the distance from the origin of the coordinates to the plane, with sign (N) = –sign (D).
A characteristic of a segment that has a circle L as a border, except for the position of its center, is the radius r of the circle L. This radius can be determined through the radius R of the sphere and the distance (p):
r = Rζc, where ζc = arccos (p / R).
In other words, a segment on a sphere can be given by the coordinates of the point O1 (x, y, z) - the center of the segment and the angle c, or the coordinates of the point O1 (x, y, z) and the distance p, or by the system: the equation of the sphere and the equation of the plane K .
Figure 1 Position of the satellite tracks with a slope of i large latitude of the upper point of the boundary of the MLO IP
The plane P passing through the center of the sphere can be defined in a similar way, i.e., the general equation of the plane
A1x + B1y + C1z + D1 = 0.
Obviously, for the plane P, the value D1 = 0. There is a more convenient way to model this plane for modeling. In the general case, the plane P is inclined to the plane of the Earth's equator, which coincides with the coordinate horizontal xy plane, at a certain angle i , called the inclination of the plane of the satellite orbit. The line of intersection of the planes of the orbit P and the horizontal coordinate xow of the plane is called the line of nodes (ascending and descending) of the orbit.
The position of this line in the equatorial plane is determined by the angle λ (geographic longitude), measured from the Greenwich meridian. At an arbitrary time, the line of nodes of the orbit of some satellite is characterized by a random value of the angle λ. Obviously, the range of possible values of λ is determined by the interval λ ∊ [0, 2π]. At an arbitrary point in time within this interval, it is impossible to specify a point that would have higher priority relative to all the others to be occupied by the satellite orbit node. There are no physical, mathematical or any other substantiations for this. Therefore, we will consider the probability distribution of the random longitude value λ for the line of orbit nodes in the interval [0, 2π] to be uniform .
The inclination angle i of the orbital plane to the equatorial plane will be considered nonrandom. In such a situation, for the plane P (i, λ) it is more convenient to use its representation through these angles, in the form:
x • tgλ - y + z • ctgi / cosλ = 0.
One of the possible values of the random variable λ will correspond to each random position in the orbital plane space of the satellite P (i, λ).
The passage of the satellite zone ground measuring point
Practical interest for the designer of the satellite system is to assess the possibility of conducting a communication session of the satellite and the PI at each turn of the trajectory.
Denote by symbol ℬ a random complex event consisting in the fact that for a random value of longitude λ, the plane (i, λ) passing through the center of the sphere of the planet will intersect the segment bounded by circle L centered on the radius-vector of point O1 (x, y z) Thus, in this part of the work, the task is to determine the probability of the occurrence of a random event ℬ. The definition of probability will be performed under the condition that λ takes one of its possible values. It is clear that the imposed condition is always satisfied. The methods of probability theory allow us to determine the probabilities of some events (complex) through known probabilities of other events that are related to them in a certain way, and the probabilities of which are specified or can be determined in one way or another.
Perform an analysis of the possibilities of implementing the event ℬ. Figure 1 illustrates the geometry of various situations with the objects considered earlier that favor the onset of a simulated random event ℬ and shows the conditions for the impossibility of its occurrence.
Figure 1 shows the upper hemisphere of the planet and the arcs of large and small circles are drawn on it. The closed curve MNN1M1 is the boundary of the MLO segment (circle L). The position of the center of the O1 segment (x, y, z) is characterized by a latitudinal φ angle and a longitudinal angle λ. The position of the plane P (i, λ) is determined by the angle of inclination i to the plane of the equator, which is deterministic and does not change during the simulation.
Analysis of possible situations for a random event ℬ.
We will distinguish situations that lead to different functional dependencies, generating a complex random event ℬ:
The first situation is φ + r m ≤ i. There are four positions of the arcs of large circles ANN2, BM1M2, CMB1, DN1A1, formed by the section of a sphere with the plane P (i, λ) at four values of λ, corresponding to such positions of the plane at which it touches the border of the segment (circle L) at four points N, M1 , N1, M. The arc KO1K1 is formed by the section of a sphere by a plane perpendicular to the xow plane in the meridian section and passing through the axis oz and the point O1.
In the cross section of the sphere of the planet by the equator plane, how are shown the lines of the nodes of the above-mentioned planes.
The corners are also shown:
- u1, u2 - between the lines of nodes and directions to the points of tangency of the circle L, measured by arcs AN and 1, respectively;
- φ1, φ2 - between the radius-vector of the center of the segment and the directions to the points of intersection of large circles, measured by arcs O1E and O1F in the meridian section;
- i is the angle of inclination of the plane (i, λ) to the xy plane.
Figure 2 Position of the satellite tracks with a slope of i less than the latitude of the upper point of the boundary of the MLO IP
The second situation φ is r m ≤ i ≤ φ + r m . The following figure 2 below shows two positions of the plane P (i, λ) for two values of λ, corresponding to its tangency of the circle L at the points M and M1.
Based on the analysis of the figures, it can be concluded that the realization of the random event ℬ will correspond to the hub point of the plane (i, λ) falling into two intervals AB and CD of the equatorial circle in the upper figure 1 and in one interval CB in the lower figure 2.
The upper figure corresponds to the interval of change of the angle i of the plane
φ + ζc ≤ i ≤ π - φo - ζc,
bottom picture - φ - ζc ≤ i ≤ φ + ζc.
We will call the intervals AB and CD (in the lower figure CB) the intervals of the plane P (i, λ) falling into the segment or, in short, the intervals of contact . For each fixed position and size ζc of a segment and angle i of inclination of the plane (i, λ), there are uniquely determined intervals of impact. The positions, sizes, number of intervals of hitting depend on the angle i, the coordinates of the center of the segment (φ, λ) and the radius c of the segment.
From the analysis performed, it follows that the random event ℬ is complex and can be represented through random simple events consisting in that the random value λ belongs to the intervals AB and CD for the situation of the upper pattern and the interval CB for the situation of the lower picture.
The third situation φ is r m > i. This situation is not considered in the work, since it does not correspond to the onset of a random event ℬ. The plane of motion of the satellite is inclined so low above the equator that MLO objects cannot intersect with it.
Representation of a complex random event ℬ simple random events .
We introduce the notation ℬ1 = (λ∊ ﺭ AB) and ℬ2 = (λ∊ ﺭ D), then ℬ = ℬ1 + ℬ2. Events ℬ1 and ℬ2 are incompatible and, therefore, under the condition that the distribution law of the node λ longitude is known, the relation (ℬ) = (ℬ1) + (2) will be satisfied.
Let the probability distribution of the random variable λ over a large circle in the xy plane, i.e. in the interval [0, 2 π ], given by some continuous function of the longitude of the node - the density of the probability distribution
φ λ (λ) = dF λ (λ) / dλ, where F λ (λ) is the distribution function of λ.
If on the numerical axis λ two arcs are marked by boundary points A, B and C, D, then the probability that λ falls on the intervals of arcs AB and CD is described as
(ℬ1) = ∫φ λ (λ) dλ; P (ℬ2) = ∫φ λ (λ) dλ is determined by the integrals in these designated limits, therefore,
P (ℬ) = ∫φ λ (λ) dλ + ∫ λ λ (λ) dλ, where the values of the limits of integration have not yet been determined. Thus, to determine the probabilities P (ℬ), it is necessary to obtain expressions for the interval boundaries characterized by the lengths of arcs λj, j = 1 (1) 4. It was noted earlier that λj = λj (R, φ, λo, i, ζc), j = 1 (1) 4. We find explicitly the expressions for the lengths of the intervals of the hit.
Let us turn to the analysis of Figure 1. From its consideration we can write down (we use the notation for the arc DD):
λ1 = λ = λ - ﺭ ; λ2 = λ = λ - ﺭ VK; λ3 = λC = λo - π + ﺭ VK; λ4 = λD = λo - π + ﺭ AK.
Ratios for performing numerical calculations
We use the theorems of spherical trigonometry and obtain calculation formulas for the necessary variables.
Here, arcs are defined from rectangular spherical triangles AEK and BFC:
ﺭ AK = arcsin (tg (φ + φ1) / tgi); ﺭ BK = arcsin (tg (φ - φ2) / tgi).
Unknown values φ2, φ1, λj = λj (R, φ, λ, i, ζc), j = 1 (1) 4 will be determined by means of known φ, i, ζc.
From the spherical triangle NEO1, by the sine theorem, we can write the ratio
sin ζc / sind = sin φ1 / sin90 °, where the angle is d = ے NEO1, from the relation we get sind = sin c / sin φ1.
By the cosine theorem from the spherical triangle AEK, we write the ratio
cosi = sindcos (φ + φ1), from which for sind we get another dependence sind = cosi / cos (φ + φ1).
We find two dependencies found in different ways and perform the transformation (we transfer functions to the left with the variable φ1)
sin ζc / sin φ1 = cosi / cos (φ + φ1) or cos (φ + φ1) / sin φ1 = cosi / sin c.
In the last expression, the unknown is the only variable φ1, which we managed to relate through known quantities, which allows us to perform its transformation and lead to a form convenient for its calculation
cos (φ + φ1) / sin φ1 = cos φoctg φ1 - sin φ and further cos φoctg φ1 - sin φ = cosi / sin ζc from where
ctg φ1 = (cosi / sin ζc + sinf) / cosf and finally,
φ1 = arctg (cos φ sin ζc / (cosi + sin φ sin ζc)).
By analogy, considering the spherical triangles OMF and BFK, we obtain the value
φ2 = arctg (cos φ sin ζc / (cosi - sin φ singin ζc)).
Substituting the obtained values of φ1 and φ2 into relations for the longitude λj, j = 1 (1) 4, we can write
λ1 = λ = λ - arcsin (tg (φ + arctg (cos φ sin ζc / (cosi + sin φ sin ζc))) / tgi);
λ2 = λB = λ - arcsin (tg (φ - arctg (cos φ sin ζc / (cosi - sin φ sin ζc))) / tgi);
λ3 = λC = λ - π + arcsin (tg (φ - arctan (cos φ sin ζc / (cosi - sin φ sining ζc))) / tgi);
λ4 = λD = λ - π + arcsin (tg (φ + arctan (cos φ sin ζc / (cosi + sin φ sin ζc))) / tgi).
Thus, all the necessary data for calculating the probability P (ℬ) are determined.
Consider an example . In applications, it is assumed that the law of longitude distribution λj is uniform in the interval [0, 2π]. Under this assumption, the distribution function F λ (λ) and the distribution density φ λ (λ) of the random variable λ are written as shown below figure 3:
The numerical characteristics of these laws of distribution are determined from the expressions: expectation m (λ) = π; sigma standard deviation (λ) = π / √3; dispersion D (λ) = π 2/3.For this case, the probability of an AES entering the IP service area is written in the form (taking into account the corresponding limits of integration and summation)
P (ℬ) = ∫φ λ (λ) dλ + ∫ λ (λ) dλ = 1 / 2π | λ 2j - λ 2j – 1 |, j = 1,2.
Substituting the values of the found integration limits and the found expressions λj into the last relation, we obtain the final expression for the probability that the satellite
enters the area of the measuring point P () = 1 / π ∑ (-1) k arcsin (tg (φ - (- 1) k arctg (cos φ sin ζc / (cosi - (- 1) k sin φ sin 2 ζ c))) / tgi), k = 1,2.
This ratio allows you to go from the fact that the satellite passes through the service area of a ground point to solve the problem of determining the residence time of the satellite in the area of the point depending on the numerical characteristics of the system’s movement (the longitudinal position of the ascending node of the orbit on the orbit to the PI area). This part of the work is planned to be published by the author in the near future.
Source: https://habr.com/ru/post/236901/
All Articles