Orbits of Artificial Earth Satellites

Original article in Chinese from 百度百科 (Baidu Wiki); translated into English on 2021-07-30.

The content below has been briefly edited for coherency, as well as to preserve only parts relevant to the Principia user.

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The orbit of an artificial Earth satellite is the path taken by the center of mass of the satellite in the time period between upper-stage engine cutoff and reentry. This orbit is determined by the position of, and velocity at, orbital insertion. It is a complex curve that deviates slightly from a Keplerian elliptical orbit (see the two-body problem). Thus, the Keplerian elliptical orbit is commonly used to roughly describe the motion of satellites. On this basis, orbital perturbation techniques can be used to produce more precise solutions of the orbit to accurately predict the position and velocity of the satellite to the satisfaction of mission requirements.

Relevant Theories

Elliptical Keplerian Orbits

The motion of a satellite along an elliptical Keplerian orbit is goverened by two-body mechanics, where only six constants (i.e., the orbital elements) need to be known to precisely define the satellite's motion. The time taken for a satellite to make a full revolution along the orbit is the orbit's period, whose length is related to the semimajor axis. Orbits with identical semimajor axes also have identical periods. While moving on an elliptical orbit, the satellite's distance from the center of the Earth as well as its velocity change constantly. The point closest to the center of the Earth is termed the perigee, and the point farthest away the apogee. These two points are also collectively known as apsides (sg. apsis). The distance between the perigee and the apogee is twice the length of the semimajor axis. The velocity of the satellite is related to its distance from the center of the Earth by the vis-viva equation. This velocity is greatest at perigee and lowest at apogee. While the satellite moves about its orbit, the Earth is also rotating. Thus, the satellite may or may not be at the same location above the Earth's surface every time it returns to the same position on its orbit.

Earth's Gravitational Field

Due to the irregular shape of the Earth and its nonuniform mass distribution, the gravitational influence it exerts on a satellite cannot be expressed in terms of simple expressions. Instead, it is commonly described using an infinite series expansion. This series converges very slowly, reflecting the complicated nature of Earth's gravitational field. The force on the satellite is only related to its position and is thus conservative. The gravitational acceleration on the satellite is the directional derivative of the potential function, which takes the form:

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In this equation, r, λ, and φ describe the satellite's distance from Earth's center, longitude, and latitude, respectively; R_e is Earth's average equatorial radius; μ is Earth's gravitational constant; P_n(sin φ) is the Legendre polynomial in terms of sin φ; and J_n, J_nm, and λ_nm are constants relating to Earth's shape and density. J₂ has a value of 1.08263×10^-3 and the other constants are on the order of 10^-6. There are three types of terms in the potential:

1. The first term is the gravitational attraction of a perfectly spherical body. If only this term is present, then the orbit of the satellite is Keplerian.
2. The Legendre polynomials, or the zonal harmonic terms. They are only related to the satellite's latitude, reflecting Earth's rotational symmetry. The J₂ term reflects that the Earth is a rotating oblate spheroid, with its equatorial radius 21.4 km longer than its polar radius. It is the most significant term. The J₃ term reflects that the Earth is not symmetric in a north-south direction, with the Southern Hemisphere larger than the Northern Hemisphere. The North Pole protrudes while the South Pole is recessed, resulting in a pear shape.
3. The associated Legendre polynomials, or the tesseral harmonic terms. They are related to both the satellite's latitude and longitude. Satellites generally experience periodic changes in latitude, canceling out perturbations. It is only with synchronous satellites, and especially stationary satellites, which only experience small changes in latitude, that the effects of the tesseral terms become more noticeable. The J₂₂ term reflects that the Earth's equator is elliptical, with its major axis longer than its minor axis by 138 m. The major axis is roughly aligned with 162°E and 18°W, and the minor axis is roughly aligned with 72°E and 108°W. The perturbation of this term on geostationary orbits is not negligible.

Major Sources of Orbital Perturbation

The actual orbits of artificial satellites are not Keplerian, but are rather relatively complex paths due to perturbations. According to perturbation theory, the orbital elements are no longer constant. Perturbations can be grouped into three types based on the characteristics of the changes in orbital elements: long-term perturbations, long-period perturbations, and short-period perturbations. Special attention should be brought to long-term perturbations, whose effects are proportional to duration. There are the following major types long-term perturbations:

1. The oblateness of the Earth causes the orbital plane to rotate about the Earth's axis of rotation. This is termed nodal precession. When viewed from the direction of the North Pole, an orbit with an inclination less than 90° will precess in the clockwise direction, an orbit with an inclination greater than 90° will precess counterclockwise, and an orbit with an inclination of exactly 90° does not precess. The rate of precession is related to the orbit's semimajor axis, eccentricity, and inclination.
2. The oblateness of the Earth causes the major axis of the orbit to steadily rotate within the plane of the orbit. This is termed apsidal precession and is measured in terms of the change in the argument of periapsis. When the inclination is less than 63.4° or greater than 116.6°, the argument of periapsis steadily increases. When the inclination is between 63.4° and 116.6°, the argument of periapsis steadily decreases. When the inclination is exactly equal to 63.4° or 116.6°, there is no precession. These two inclinations are known as critical inclinations.
3. The oblateness of the Earth causes long-term variations in the mean anomaly. The mean angular motion of a satellite along an elliptical orbit is 360°/T, where T is period. The mean anomaly, symbol M, is the angle swept out by the satellite starting from its perigee, if it were to move at the mean angular motion. This is a theoretical angle, commonly used in place of the time of perigee passage as one of the orbital elements. The variation in mean anomaly is related to the size, eccentricity, and perigee of the orbit, with greater variations resulting from longer time spent in orbit.
4. Atmospheric drag (note that this is not modeled by Principia) causes both the semimajor axis and eccentricity of an orbit to decay, which affects the lifespan of a low Earth orbit.

Periodic perturbations result in periodic changes in orbital elements, which must be taken into account when precisely computing orbits. These perturbations make it difficult to compute the period of an orbit, resulting in several types of periods. For example, the nodal period is the time period between two successive passes through the ascending node; the anomalistic period is the time period between two successive passes through the perigee; and the siderial period is computed from the orbit's semimajor axis using Kepler's third law. These three periods are distinct, but interconversion calculations exist.

The effects of perturbations complicate the computation of orbits and sometimes must be eliminated. For example, the Soviet Union's Молния ('lightning') satellites were placed at a critical inclination, fixing the position of their apogee. This ensured that the satellite's apogee stayed above the USSR, providing better communication coverage. Perturbations can also be used to achieve desired changes in orbits: For example, using nodal precession to design sun-synchronous orbits, or using atmospheric drag to deorbit a satellite. To preserve the accuracy of their orbits, satellites are equipped with maneuvering systems to correct for orbital insertion inaccuracies and counteract perturbations.

Retrograde Orbits

Retrograde orbits have inclinations greater than 90°. A satellite must be launched to the west to insert into this kind of orbit. This launch trajectory cannot take advantage of Earth's rotation velocity and in fact requires extra energy to be expended to cancel out Earth's rotation. Thus, such orbits are not generally used except for sun-synchronous orbits.

By carefully adjusting the height, inclination, and shape of a satellite's orbit, its nodal precession rate can be made to be 0.9856° per day to the east, which exactly matches the rate of Earth's revolution around the Sun. This is the extremely useful sun-synchronous orbit. Such an orbit allows a satellite to observe cloud cover and terrain at the same time and under the same lighting conditions and is commonly used for climate, resource, and reconnaissance satellites.

Equatorial Orbits

An equatorial orbit is characterized by an inclination of 0°. There are infinite such orbits, but the geostationary orbit is of especial importance. Since the velocity of a satellite depends on its height, with higher height resulting in a lower velocity and a longer period, it can be seen that an orbit moving west to east with an altitude of 35 786 km above Earth's surface has a period of exactly 23 hours, 54 minutes, and 4 seconds, or exactly one siderial day. This orbit is the geostationary orbit, where satellites appear to be fixed at a point in the sky when observed from the ground. Most major communication satellites are located in geostationary orbits, and three satellites evenly spaced in geostationary orbit are enough to provide global communication.

Polar Orbits

A polar orbit is an orbit with an inclination around 90°, named for the fact that it passes over the poles of the Earth. Such an orbit will pass over all of the Earth's surface.

Prograde Orbits

A prograde orbit has an inclination less than 90°. The vast majority of satellites in such orbits have low altitudes. Thus, it is also known as low Earth orbit. In the Northern Hemisphere, a launch into such an orbit is in the northeast direction. This allows the launch to take advantage of Earth's west-east rotational velocity, saving energy. The velocity saved can be computed from the equatorial rotational velocity, the launch azimuth, and the location of the launch site. It can be seen that the most energy savings are achieved by launching directly due east on the equator, with savings diminishing with increasing latitude.