The trajectory of the center of mass of the artificial satellite from the earth from the suspension of the last stage of the rocket boost to the fall (or return to the ground) of the artificial satellite. It depends on the position of the orbital point and the orbital speed. The orbit is a complex curve slightly different from Kepler’s ellipse (see the two-body problem).
Kepler’s elliptical orbit is often used to describe the approximate motion of a satellite. Based on this, the orbital perturbation method can be used to obtain the accurate orbit solution and obtain an accurate satellite position and velocity prediction to meet the needs of satellite engineering.
When Kepler’s elliptical orbit, the satellite moves in Kepler’s elliptical orbit, satisfies the law of motion of the two-body problem.
As long as all six constants (orbital elements) are known, the motion of the satellite can be determined. The time it takes for a satellite to move a circle in an elliptical orbit is called the orbital period, and the length of the period is related to the semi-major axis.
Orbits with the same semimajor axis have the same period. By moving in an elliptical orbit, the satellite’s geocentric distance and speed are changing. The closest point P to the center of the Earth is the perigee, and the farthest point A is the apogee. The perigee and the apogee are collectively called arcs.
The sum of the geocentric distance between the perigee and the apogee is twice the semi-major axis. The speed of the satellite is related only to the distance to the center of the Earth, which satisfies the vitality formula (see spacecraft orbital speed).
The velocity is maximum at perigee and minimum at apogee. The Earth also rotates when the satellite is in orbit. When the satellite returns to the same point in the orbit, it does not necessarily return to the sky over the same area of the Earth (see subsatellite path orientation).
Due to the irregular shape of the Earth and the uneven mass distribution of the Earth’s gravitational field, the attraction of the satellites cannot be described by simple expressions and is often described by an infinite series expansion.
This series converges very slowly, indicating that Earth’s gravity is very complicated. This force is related only to the position of the satellite and is a conservative force. The gravitational acceleration received by the satellite is the directional derivative of the potential function. The expression of the bit function is:
In the formula, r, λ, and φ are the distance to the center of the Earth, the longitude of the center of the Earth, and the latitude of the center of the Earth in spherical coordinates that describe the position of the satellite; R e is the mean radius of the terrestrial equator; μ is the Earth’s gravitational constant; P n ( sin φ ) is the Legendre polynomial of order n of the independent variable sin φ , P nm ( sin φ ) is the order m – of order n of the Legendre polynomial; J n , J nm , λ nm are constants related to the shape and density distribution of the Earth. J 2 is × 1.08263 10 – . 3 , the other coefficients are 10 – . 6 orders of magnitude. Bit function terms can be divided into three categories:
① The first term of the bit function is the gravitational term of the spherical Earth. If there is only this element, the satellite’s orbit is Kepler’s elliptical orbit.
②The Legendre polynomial term is called the harmonic term. The harmonic term is related only to the latitude of the satellite and reflects the rotational symmetry of the Earth. The term J 2 indicates that the Earth is a spheroid and its equatorial radius is 21.4 kilometers greater than the polar radius.
The J 2 term is the leading term and is often called the Earth obliquity perturbation. J 3 term Earth’s north-south reflection asymmetry, the northern hemisphere than the southern hemisphere, the prominent recess of the Arctic and Antarctic, pear-shaped.
③The associated Legendre polynomial term is called the harmonic term. Tian Xie terms are related to the longitude and latitude of the satellite. For general satellite motion, the longitude value changes periodically and the effects cancel each other out. For geostationary satellites, especially geostationary satellites, the longitude changes very little and the influence of the field harmonic term is more apparent.
Point J 22 reflects that the Earth’s equator is also an ellipse, and the major axis of this ellipse is only 138 meters longer than the minor axis. The long axis is approximately 162° east longitude and 18° west longitude, and the short axis is approximately 72° east longitude and 108° west longitude. The disturbance of this element in the geostationary satellite orbit cannot be ignored.
The main orbital perturbation is that the actual orbit of the artificial Earth satellite is not Kepler’s orbit. Due to the influence of the perturbing force, the orbit of the satellite is more complicated. According to the perturbation theory, the orbital element is no longer constant.
According to the changing characteristics of the orbital elements, the orbital perturbation can be divided into long-term perturbation, long-period perturbation, and short-period perturbation (see spacecraft orbital perturbation). Long-term disturbance is proportional to time, which attracts special attention. The main long-term disturbances of the artificial orbit of Earth satellites are:
① The flattening of the Earth causes the orbital surface to rotate uniformly around the Earth’s axis of rotation, which is called precession of the orbital surface.
When the orbital inclination angle is less than 90°, the precession is clockwise as seen from the north pole; when it is greater than 90°, the precession direction is counterclockwise; when it is equal to 90°, it does not rotate. The angular rate of precession is related to the longitudinal axis, eccentricity, and inclination of the orbit.
② The flattening of the Earth causes the long axis of the ellipse to rotate uniformly in the orbital plane. The angular speed of rotation is expressed by the rate of change of the perigee argument. When the tilt angle is less than 63.4° or greater than 116.6°, the perigee argument increases uniformly. When it is between 63.4° and 116.6°, it decreases evenly. When it is equal to 63.4° or 116.6°, it does not rotate. 63.4° and 116.6° are called critical tilt angles.
③ The flattening of the earth causes long-term changes in the plane anomaly. The average angular velocity of the satellite moving on the ellipse is 360°/T and T is the period.
The mean angle of anomaly is the angle that the satellite turns when moving at a mean angular velocity after passing perigee, and is denoted by M. This is a theoretical angle, often used to replace the moment of perigee as one of the orbital elements. The long-term change of the mean anomaly is related to the size of the orbit, the eccentricity and the position of the perigee; the longer the satellite’s time of flight, the greater the change.
④ Atmospheric drag causes the orbital semi-major axis and eccentricity to be attenuated at the same time. This long-term perturbation is related to the satellite’s orbital life near Earth.
The long period perturbation and the short period perturbation cause the orbital elements to change periodically. It must also be taken into account when calculating the orbit accurately. Orbital perturbation brings problems in orbital period measurement and as a result there are several different useful periods.
For example, the intersection point period is the time interval from the ascending node until the ascending node passes again; the anomaly period is the time interval between the passage of the aircraft through two adjacent perigees; the sidereal period is the period calculated using the semi-long axis according to Kepler’s third law. These three periods are different from each other and can be converted to each other. Orbital perturbation complicates orbital calculations and some perturbations need to be avoided.
For example, the communications satellite “Lightning” of the Soviet Union was selected as the critical angle to avoid the movement of the apogee, so that the apogee would always be above the Soviet territory, or that it could maintain the domestic communications of the Soviet Union for a long time.
Sometimes people also use the perturbing force to get the necessary orbital change. For example, long-term rotation of the orbital surface is used to project a sun-synchronous orbit, and atmospheric drag is used to return the satellite to earth. Choosing the correct orbit according to the mission of the satellite is the main task of orbit design.
Practical orbits, such as sun-synchronous orbits, geostationary satellite orbits, polar orbits, and regression orbits, can be designed according to the law of orbital changes. To maintain orbit accuracy, the satellite must be equipped with an orbit control system to overcome the error in orbit and compensate for the influence of the perturbing force (see spacecraft orbit control system ).