Nowadays, humanity uses several different orbits to accommodate satellites. The greatest attention is riveted to the geostationary orbit, which can be used for "stationary" placement of a satellite over a particular point on the Earth. The orbit chosen for the operation of the satellite depends on its purpose. For example, satellites used to broadcast live television programs are placed in geostationary orbit. Many communications satellites are also in geostationary orbit. Other satellite systems, in particular those used for communications between satellite phones, are in low earth orbit. Similarly, the satellite systems used for navigation systems such as Navstar or the Global Positioning System (GPS) are also in relatively low earth orbits. There are countless other satellites - meteorological, research and so on. And each of them, depending on its purpose, receives a “registration permit” in a certain orbit.
Read also:
The specific orbit chosen for the operation of the satellite depends on many factors, among which are the functions of the satellite, as well as the territory served by it. In some cases, this may be an extremely low Earth orbit (LEO), located at an altitude of only 160 kilometers above the Earth, in other cases, the satellite is located at an altitude of more than 36,000 kilometers above the Earth - that is, in the geostationary orbit GEO. Moreover, a number of satellites do not use a circular orbit, but an elliptical one.
Earth's gravity and satellite orbits
As satellites turn in Earth orbit, they are slowly displaced from it due to the force of gravity of the Earth. If the satellites did not orbit, they would gradually fall to the Earth and burn up in the upper atmosphere. However, the very rotation of satellites around the Earth creates a force that repels them from our planet. Each of the orbits has its own calculated speed, which allows you to balance the Earth's gravity and centrifugal force, keeping the device in a constant orbit and preventing it from gaining or losing altitude.
It is quite clear that the lower the orbit of the satellite, the stronger the Earth's gravity affects it and the greater the speed required to overcome this force. The greater the distance from the Earth's surface to the satellite, the correspondingly lower speed is required to keep it in a constant orbit. An apparatus orbiting at a distance of about 160 km above the Earth's surface requires a speed of approximately 28,164 km / h, which means that such a satellite completes an orbit around the Earth in about 90 minutes. At a distance of 36,000 km above the Earth's surface, a satellite needs a speed of slightly less than 11,266 km/h to be in a permanent orbit, which makes it possible for such a satellite to orbit the Earth in about 24 hours.
Definitions of circular and elliptical orbits
All satellites revolve around the Earth using one of two basic types of orbits.
- Circular satellite orbit: when a spacecraft revolves around the Earth in a circular orbit, its distance above the earth's surface always remains the same.
- Elliptical Satellite Orbit: Rotating a satellite in an elliptical orbit means changing the distance to the Earth's surface at different times during one orbit.
Read also:
satellite orbits
There are many different definitions associated with different types of satellite orbits:
- Center of the Earth: When a satellite orbits the earth - in a circular or elliptical orbit - the satellite's orbit forms a plane that passes through the earth's center of gravity, or the center of the earth.
- Direction of movement around the Earth: The ways in which a satellite revolves around our planet can be divided into two categories according to the direction of this reversal:
1. Booster Orbit:
The revolution of a satellite around the Earth is called accelerating if the satellite rotates in the same direction as the Earth rotates;
2. Retrograde orbit:
The revolution of a satellite around the Earth is called retrograde if the satellite rotates in the opposite direction to the direction of rotation of the Earth.
- Orbit track: the path of the satellite's orbit is the point on the earth's surface, when flying over which the satellite is directly overhead in the process of orbiting around the earth. The track forms a circle, in the center of which is the Center of the Earth. It should be noted that geostationary satellites are a special case because they are constantly over the same point above the Earth's surface. This means that their orbit trace consists of a single point located on the Earth's equator. It can also be added that the path of the orbit of satellites rotating strictly above the equator stretches along this very equator.
These orbits are typically characterized by a westward shift in each satellite's orbital track as the Earth below the satellite rotates eastward.
- Orbital nodes: These are the points at which the orbit trace passes from one hemisphere to another. For non-equatorial orbits, there are two such nodes:
1. Ascending node:
This is the node at which the orbit trace passes from the southern hemisphere to the northern.
2. Descending Node:
This is the node at which the orbit trace passes from the northern hemisphere to the southern.
- Satellite height: When calculating many orbits, it is necessary to take into account the height of the satellite above the center of the Earth. This indicator includes the distance from the satellite to the surface of the Earth plus the radius of our planet. As a rule, it is considered that it is equal to 6370 kilometers.
- Orbital speed: For circular orbits, it is always the same. However, in the case of elliptical orbits, everything is different: the speed of the satellite in orbit changes depending on its position in this very orbit. It reaches its maximum at the closest approach to the Earth, where the satellite will have the maximum opposition to the planet's gravity, and decreases to a minimum when it reaches the point of greatest distance from the Earth.
- Climb angle: The elevation angle of the satellite is the angle at which the satellite is above the horizon. If the angle is too small, the signal may be blocked by nearby objects if the receiving antenna is not raised high enough. However, for antennas that are raised above an obstacle, there is also a problem when receiving a signal from satellites that have a low elevation angle. The reason for this is that the satellite signal then has to travel a greater distance through the earth's atmosphere and as a result is subject to more attenuation. The minimum allowable elevation angle for a more or less satisfactory reception is considered to be an angle of five degrees.
- Tilt angle: Not all satellite orbits follow the equatorial line - in fact, most low earth orbits do not follow this line. Therefore, it is necessary to determine the angle of inclination of the satellite orbit. The diagram below illustrates this process.
Satellite orbit inclination Other indicators related to the satellite orbit
In order for a satellite to be used to provide communications services, ground stations must be able to "monitor" it in order to receive a signal from it and send a signal to it. It is clear that communication with the satellite is possible only when it is in the visibility zone of ground stations, and, depending on the type of orbit, it can be in the visibility zone only for short periods of time. To make sure that communication with the satellite is possible for the maximum amount of time, there are several options that can be used:
- First option consists in using an elliptical orbit, the apogee point of which is exactly above the planned placement of a ground station, which allows the satellite to stay in the field of view of this station for a maximum period of time.
- Second option consists in launching several satellites into one orbit, and thus, at the time when one of them disappears from sight and communication with it is lost, another one comes in its place. As a rule, for the organization of more or less uninterrupted communication, it is required to launch three satellites into orbit. However, the process of changing one "duty" satellite to another introduces additional difficulties into the system, as well as a number of requirements for at least three satellites.
Definitions of circular orbits
Circular orbits can be classified according to several parameters. Terms such as Low Earth Orbit, Geostationary Orbit (and the like) indicate the identity of a particular orbit. A brief overview of the definitions of circular orbits is provided in the table below.
The trajectory of an artificial earth satellite (AES) is called its orbit.
An orbit is a flat curve of the 2nd order (a circle or an ellipse), in one of the foci of which is the center of mass that attracts the body. The satellite moves in a plane that retains its spatial orientation.
Two planes (orbital plane, equatorial plane), ellipse
G is the actual focus where the center of mass (Earth) is.
G' - imaginary focus.
S - satellite (somewhere in orbit)
r is the radius vector of the satellite (GS)
|r| - geocentric distance (number)
The X,Y,Z coordinate system is an absolute (star) coordinate system - it is a Cartesian coordinate system, fixed relative to the stars.
The Z axis is directed along the earth's axis of rotation and points north.
The OXY plane coincides with the equatorial plane.
P - perigee - the point of the orbit closest to the attracting center of mass.
A - apogee - the most distant point of the orbit from the attracting center of mass.
AP is the line of apsides - the line passing through the foci and connecting the apogee and perigee
Angle v is the true anomaly - the angle between the line of apsides and the radius vector
VN is the line of nodes - the line of intersection of the plane of the orbit with the plane of the equator.
B - the ascending node of the orbit - this is the point at which the orbit crosses the plane of the equator in the approach of the satellite from south to north
H - the descending node of the orbit is the point at which the orbit intersects the plane of the equator in the satellite's approach from north to south.
i - orbital inclination - the angle between the plane of the orbit and the plane of the equator.
omega - longitude of the ascending node - the angle between the positive direction of the abscissa (x-axis) and the line of angles towards the ascending node.
u is the satellite's latitude argument - this is the angle between the line of nodes and the radius vector
omegasmall - perigee argument - this is the angle between the line of nodes and the line of apses in the direction of perigee.
O - divides the apse in half, perpendicular to it to the orbit - C.
AO = a is the semi-major axis of the ellipse.
CO = b is the minor semiaxis of the ellipse.
e – eccentricity of the ellipse – shows the degree of compression of the ellipse.
e=sqrt(1-(a2/b2)) – compression ratio. 0=circle.
T - period of revolution - the time between two successive passages by the satellite of the same point of the orbit.
Types of satellite orbits
1. Polar orbits, i~90o; such satellites can be used to capture any point on the planet, but putting a satellite into such an orbit is difficult and very costly
2. Equatorial orbits i~0o; the planes of the orbit and the equator practically coincide. The poles and mid-latitudes cannot be removed.
3. Circular orbits. e=0. The same flight height, there will be one scale.
4. Stationary orbits. i~0, e=0; Equatorial and circular. The period of revolution of such satellites is equal to the period of revolution of the earth. Stationary relative to the surface of the earth.
5. The orbits are sun-synchronous. They tend to provide the same illumination of the earth's surface along the flight path of the spacecraft. The orbit parameters are chosen in such a way that the plane of the orbit rotates around the earth's axis, and the angle of the satellite turn is equal in sign and magnitude to the angular displacement of the earth around the sun.
6. Open, i.e. parabola or hyperbola instead of ellipse. Used to launch spacecraft.
Image types
An image is a function of two variables f(x,y) defined in some region C of the Oxy plane and having a known set of its values.
Black and white photo: f(x,y)>=0; 0<=x<=a; 0<=y<=b; где f(x,y) – яркость изображения в точке x,y; a – ширина кадра, b – высота.
Taking into account the features of the function f, the following classes of images are distinguished:
1. Halftone (gray) - B/W (grayscale) photography - the set of function values in the area C can be discrete f e (f0,f1,…,fn, n>1) or continuous (0<=f<=fmax}. Цветные изображения относятся сюда же, т.к. несколько монохромных цветовых компонент задают цвет (аналоговые, цифровые)
2. Binary (two-level) images. fe(0.1);
3. Linear - the image is a single curve or a set of them.
4. Bitmaps - the image is k points with coordinates (xi,yi), and the brightness is fi e ;
| 2 | | |
1. Perturbation of the focal parameter of the orbit
2. Perturbation of the orbital eccentricity
the result of integration is a trigonometric function with a period
3. Perturbation of the longitude of the ascending node of the orbit
4. Perturbation of the inclination of the orbit
5. Perturbation of the argument of the periapsis of the orbit
6. Time of orbital motion
assuming that j=1, then the draconian period is equal to the sidereal:
where
conclusions
1. Focal parameter
The change in the focal parameter is periodic. When passing the integration start point (the initial position of the spacecraft), the focal parameter returns the initial value, from which it can be concluded that the period of change of the focal parameter is equal to the orbital period of the spacecraft. At the expense of secular properties, the focal parameter does not have them, this can be seen from the dependence graph and from the formulas (the numerical deviation is due to the error of the numerical integration method).
This periodic parameter causes a change in the geometry of the orbit ellipse with the movement of the spacecraft along the orbit, but when the final complete revolution is reached, it returns to its original state. This indicates the invariance of the shape of the orbit over time.
2.Eccentricity
The eccentricity also changes periodically. It can be seen from the graph and the theoretical dependence that its change is described using the sum and products of trigonometric functions. The theoretical dependence quite adequately describes the dependence obtained by the numerical method. This gives us the right to define the period of change of this parameter as the orbital period of the spacecraft. Regarding secular changes, they are absent due to the dependence on the graph and the integration of the theoretical dependence after integration, we obtain a trigonometric function with a period of 2 (the deviation in numbers is due to the error of the numerical integration method).
Eccentricity, as a parameter of the shape of the orbit, is related to the focal parameter, and this suggests that this parameter confirms that the shape of the orbit does not change over time.
3.Longitude of ascending node
The longitude of the ascending node has a non-periodic character, since the spacecraft does not return the initial value when making a complete revolution. It has a wavy periodicity equal to the spacecraft revolution period, but goes downward per revolution. The presence of periodically repeating waviness is due to the presence in the formula of trigonometric functions with a period of 2. This parameter is, in fact, secular. After integrating the theoretical dependence, we get a specific value that depends on the number of revolutions. Again, the theoretical formulas adequately describe the change in this parameter.
This secular parameter shows that the orbit revolves around the Earth as the spacecraft moves along it; at the end of the revolution, it does not return to its initial position, but comes to some other one with a shift.
4. Orbital inclination
The inclination of the orbital plane is periodic. This conclusion can be drawn on the basis of model data and analytical dependence. The adequacy of the numerical and analytical data is evident. The theoretical formula and the dependency graph have trigonometric dependencies, which determines the periodicity. The inclination does not have secular properties due to the theoretical dependence, after integration of which we obtain zero and the numerical one, which shows the same effect.
From a physical point of view, this parameter shows us that the plane of the orbit periodically rotates relative to the plane of the equator.
5. Pericenter argument
The periapsis argument behaves both as a periodic parameter and as a secular parameter. The periodicity is due to the presence of trigonometric functions in the formula, and the secular ones are due to the fact that when the KA passes through a full revolution, the value before the passage does not coincide with the value after. The theoretical dependence clearly demonstrates the fact of secular change, since after its integration, an expression appears that depends on the number of revolutions.
From the point of view of the orbit, when the orbit is rotated relative to the point of Aries (GMT), the orbit also rotates in its own plane (precession of the line of apsides). Moreover, if the inclination is less than 63.4 0, then the precession occurs in the opposite direction of the spacecraft motion. This parameter must be taken into account primarily from the point of view of radio communication, otherwise at some point, when the radio communication zone was expected, the spacecraft will simply go into the shadow of the planet.
6. Time of orbital movement
Time depends linearly on the latitude argument. It is an independent parameter that grows all the time. We are more concerned with the circulation period.
The period of revolution is the time of a complete revolution of the spacecraft in its orbit.
The non-centrality of the Earth's gravitational field does not cause the semiaxes to change in the secular style, one hundred parameters j is approximately equal to 1 and from this it can be concluded on the basis of the theoretical formula and graph of the numerical method is approximately one, from which it follows that the draconian period of revolution is equal to the sidereal.
Kepler's laws
Kepler's laws are three empirical relationships intuitively selected by Johannes Kepler based on the analysis of Tycho Brahe's astronomical observations. Describe the idealized heliocentric orbit of the planet. Within the framework of classical mechanics, they are derived from the solution of the two-body problem by passing to the limit / → 0, where,,
The masses of the planet and the sun, respectively.
Kepler's first law (the law of ellipses):
Each planet in the solar system revolves around an ellipse with the sun at one of its foci. The shape of the ellipse and the degree of its similarity to a circle is characterized by the ratio, where is the distance from the center of the ellipse to its focus (half the interfocal distance), is the semi-major axis. The quantity is called the eccentricity of the ellipse. At, and, therefore, the ellipse turns into a circle.
Proof of Kepler's first law
Newton's law of universal gravitation states that "every object in the universe attracts every other object along a line connecting the centers of mass of objects, proportional to the mass of each object, and inversely proportional to the square of the distance between objects." This assumes that the acceleration a has a shape.
Recall that in polar coordinates:
We write in coordinate form:
Substituting and into the second equation, we get
which is simplified
After integration, we write the expression
for some constant, which is the specific angular momentum (). Let
The equation of motion in the direction becomes
Newton's law of universal gravitation relates the force per unit mass to
distance as
where G is the universal gravitational constant and M is the mass of the star.
As a result
This differential equation has a general solution:
for arbitrary integration constants e and θ0.
Replacing u with 1/r and setting θ0 = 0, we get:
We have obtained the equation of a conic section with eccentricity e and the origin of the coordinate system at one of the foci. Thus, Kepler's first law follows directly from Newton's law of universal gravitation and Newton's second law.
Kepler's second law (law of areas):
Each planet moves in a plane passing through the center of the Sun, and for equal periods of time, the radius vector connecting the Sun and the planet describes equal areas.
As applied to our solar system, two concepts are associated with this law: perihelion - the point of the orbit closest to the Sun, and aphelion - the most distant point of the orbit. Thus, from Kepler's second law it follows that the planet moves around the Sun unevenly, having a greater linear velocity at perihelion than at aphelion.
Every year in early January, the Earth moves faster as it passes through perihelion, so the Sun's apparent eastward movement along the ecliptic is also faster than the annual average. In early July, the Earth, passing aphelion, moves more slowly, therefore, the movement of the Sun along the ecliptic slows down. The law of areas indicates that the force that controls the orbital motion of the planets is directed towards the Sun.
Proof of Kepler's second law
By definition, the angular momentum L of a point particle with mass m and velocity v is written as:
where is the radius vector of the particle and the momentum of the particle. The area swept by the radius vector r in time dt from geometric considerations is
where is the angle between the r and v directions.
By definition
As a result we have
Differentiate both sides of the equation with respect to time
since the cross product of parallel vectors is zero. Note that F is always parallel to r because the force is radial, and p is always parallel to v by definition. Thus, it can be argued that L , and hence also
the speed of sweeping the area proportional to it is a constant.
Kepler's third law (harmonic law)^
The squares of the periods of revolution of the planets around the Sun are related as the cubes of the semi-major axes of the orbits of the planets. It is true not only for the planets, but also for their satellites.
where T1 and T2 are the periods of revolution of two planets around the Sun, aa1 and a2 are the lengths of the semi-major axes of their orbits.
Newton found that the gravitational pull of a planet of a certain mass depends only on its distance, and not on other properties such as composition or temperature. He also showed that Kepler's third law is not entirely accurate - in fact, it also includes the mass of the planet /
where M is the mass of the Sun, am1 and m2 are the masses of the planets.
Since motion and mass are related, this combination of Kepler's harmonic law and Newton's law of gravity is used to determine the masses of planets and satellites if their orbits and orbital periods are known.
Orbit parameters in the plane:
In celestial mechanics, this is the trajectory of a celestial body in the gravitational field of another body with a much larger mass (planets, comets, asteroids in the field of a star). In a rectangular coordinate system, the origin of which coincides with the center of mass, the trajectory can be in the form of a conic section (circle, ellipse, parabola or hyperbola). In this case, its focus coincides with the center of mass of the system.
Keplerian orbits
For a long time it was believed that the planets should have a circular orbit. After long and unsuccessful attempts to find a circular orbit for Mars, Kepler rejected this statement and, subsequently, using measurement data made by Tycho Brahe, formulated three laws (see Kepler's Laws) describing the orbital motion of bodies.
The Keplerian elements of the orbit are:
focal parameter, semi-major axis, periapsis radius, apoapsis radius - determine the size of the orbit,
eccentricity (e) - determines the shape of the orbit,
orbital inclination (i),
longitude of the ascending node () - determines the position of the plane of the orbit of a celestial body in space,
periapsis argument () - sets the orientation of the device in the orbit plane (often set the direction to the periapsis),
the moment of passage of a celestial body through the periapsis (To) - sets the time reference.
These elements uniquely define the orbit, regardless of its shape (elliptical, parabolic or hyperbolic). The main coordinate plane can be the plane of the ecliptic, the plane of the galaxy, the plane of the earth's equator, etc. Then the elements of the orbit are set relative to the selected plane.
The location of the orbit in space and the location of the celestial body in orbit.
Determining the orbits of celestial bodies is one of the tasks of celestial mechanics. To set the orbit of a satellite of a planet, an asteroid or the Earth, so-called "orbital elements" are used. Orbital elements are responsible for setting the basic coordinate system (reference points, coordinate axes), the shape and size of the orbit, its orientation in space and the time at which the celestial body is at a certain point in the orbit. There are basically two ways to set the orbit (in the presence of a coordinate system):
- using position and velocity vectors;
- using orbital elements.
Keplerian elements of the orbit
Other elements of the orbit
anomalies
Anomaly(in celestial mechanics) an angle used to describe the motion of a body in an elliptical orbit. The term " anomaly" was first introduced by Adelard Batsky when translating the astronomical tables of Al-Khwarizmi "Zij" into Latin to convey the Arabic term " al-heza" ("peculiarity").
True anomaly(marked in the figure ν (\displaystyle \nu ), also denoted T , θ (\displaystyle \theta ) or f) is the angle between the radius vector r body and direction to the periapsis.
Mean anomaly(commonly denoted M) for a body moving along an unperturbed orbit, - the product of its middle movement(average angular velocity per revolution) and the time interval after passing the periapsis. In other words, the mean anomaly is the angular distance from the periapsis to an imaginary body moving at a constant angular velocity equal to the average motion of the real body and passing through the periapsis simultaneously with the real body.
Eccentric anomaly(denoted E) is a parameter used to express the variable length of the radius vector r .
Addiction r from E and ν (\displaystyle \nu ) expressed by the equations
r = a (1 − e ⋅ cos E) , (\displaystyle r=a(1-e\cdot \cos E),) r = a (1 − e 2) 1 + e ⋅ cos ν (\displaystyle r=(\frac (a(1-e^(2)))(1+e\cdot \cos \nu ))),- a- semi-major axis of an elliptical orbit;
- e is the eccentricity of the elliptical orbit.
The mean anomaly and the eccentric anomaly are related through the Kepler equation.
Latitude argument
Latitude argument(denoted u) is an angular parameter that determines the position of a body moving along a Keplerian orbit. This is the sum of the commonly used true anomaly (see above) and the periapsis argument, forming the angle between the radius vector of the body and the node line. Counted from the ascending node in the direction of travel
