INTRODUCTION
Orbital mechanics deals with the motion of bodies in space under the influence of gravity. This branch of physics was initially developed to explain the motion of the planets around the Sun and was then referred to as Celestial Mechanics. Although it is still used for that purpose, especially in the case of newly discovered asteroid and comets, it now particularly has the function of computing the orbits of man-made satellites moving around the Earth and occasionally around the Sun and other bodies in the solar system.
Johannes Kepler developed three laws to explain planetary motion, but it was not until Isaac Newton developed the basic law of gravity and the laws of motion that celestial mechanics was put of a solid foundation. Even after Albert Einstein developed his theory of relativity, Newton's laws were found to be accurate enough for all motion at speeds that we can achieve in space and in the gravitational fields of the planets. Only when we exceed 0.1 times the velocity of light (ie more than 30,000 km/sec) and/or go very close to the Sun do Newton's laws need to be modified.
Celestial mechanics can theoretically be applied to the motion of any bodies in space. However, because of the complexity involved it can only strictly be applied to the motion of two bodies about each other, and sometimes to three bodies when special conditions apply – such as a third body which has a much smaller mass than the other two bodies in the system, and through a technique called perturbations.
HISTORY
Celestial mechanics in essence introduced physics into astronomy and took it from mythology to mechanics.
Johannes Kepler (1571 - 1730) was the first to start this transformation with the introduction of three 'laws' into planetary motion. Laws that would allow calculation and prediction of these motions.
These laws are:
K2 relates to planetary velocity
K3 relates the orbital period to its size.
Isaac Newton discovered the law governing the general motion of any object under the influence of a general force, and he discovered the Universal law of Gravitation.
Newton's general law of motion states that dp/dt = Ftotal
In words this states that the time rate of change of momentum of a body is equal to the total force applied to that body.
Newton's law of gravitation states that Fgrav = G M1 M2 / r2
In words this states that the gravitational force between two bodies of mass Ma and M2 is proportional to the product of their masses and inversely proportional to the square of their separation.
Newton was then able to show that Kepler's three 'rules' were a consequence of these two more general laws of motion.
KEPLER'S FIRST LAW
K1 : Planets orbit the Sun in elliptical orbits with the Sun at one of the elliptical foci.
To construct an ellipse, hammer two nails into a white board at the positions labelled focus. Tie each end of a length of cord to these nails. With a pencil stretching the cord tight, draw a figure as the pencil is moved along the cord (both sides). This is an ellipse.
ECCENTRICITY OF AN ELLIPSE
The eccentricity of an ellipse is a measure of its elongation. It may vary from e=0 (circle) to e=1 (parabola) to values greater than one (hyperbola).
Note that if the orbiting body is in an ellipse around the Sun, the minimum and maximum distances of the orbit (from the central body) are called the perihelion and aphelion respectively. If the central body is the Earth, these same distances are called the perigee and apogee. The general term for these extremal distances are periapse and apoapse.
Ellipses may also be generated by the intersection of a cone with a plane, and thus are referred to as conic sections. A circle is a special case of an ellipse. A parabolic orbit is one in which the object has reached escape velocity. A hyperbola is not really an orbit but is the path followed by a high velocity object passing near to a star.
KEPLER'S SECOND LAW
K2: A line from the Sun to a planet sweeps out equal areas in equal times.
K2 says that the speed of an orbiting body increases the closer it gets to the body it orbits. This allows relative and sometimes absolute velocities of the planet or satellite to be calculated. Most of the solar system planets have low eccentricities and so do not experience large speed changes, but comets with large eccentricities show large speed changes as they come closer to the Sun.
KEPLER'S THIRD LAW
K3: The square of the period of a planet’s orbit is proportional to the cube of the size of the orbit.
The orbital period (P) is the time it takes the planet to complete one orbit around the Sun – eg from perihelion to perihelion.
The size of the orbit is usually taken as the length of the semi-major axis (a).
Another way of writing this is ( p2 / a3 = constant )
In graphical form we expect a straight line (left). The right hand graph shows planetary speed.
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If the orbits are not around the Sun (ie the Earth for artificial satellites or another star for exoplanets), then the constant of proportionality is different, but the relationship is still of the same form.
M is the mass of the central object in question. This equation is the backbone of the orbital mechanics for artificial satellites and also for calculating orbits of Exoplanets. Usually periods can be measured very accurately and this equation then gives the orbital size. G = 6.67 x 10-11 in SI units is the universal constant of gravitation, M is the mass of the central body and π = 3.14159... is the ratio of the circumference of a circle to its diameter.
Note that the orbital period only depends on ‘a’, and not ‘b’ or the eccentricity ‘e’ of the orbit.
POSITION IN SPACE
It is not sufficient to state that an orbit is an ellipse with a certain size and eccentricity. We must also know its orientation in space. This requires us to specify several angles of orientation. And these must be specified with respect to a reference plane and a reference direction.
REFERENCE PLANES
For the solar system the reference is defined by the plane in which the Earth orbits the Sun – this is termed the ecliptic.
For Earth-orbiting satellites the equatorial plane is used as the reference plane.
CELESTIAL COORDINATES AND REFERENCE DIRECTION
To specify the position of any object in the sky astronomers use ‘celestial coordinates’. These are similar to terrestrial coordinates. Declination is equivalent to latitude and Right Ascension to longitude. Where the ecliptic plane meets the equatorial plane is the reference direction γ for Right Ascension.
The celestial coordinate system is fixed with respect to the distant stars and so appears to rotate from an Earth observers viewpoint. Because the vernal equinox lies in both the ecliptic and equatorial planes it can be used for both coordinate systems.
ORBITAL ORIENTATION
Three angles are used to specify the orientation of an elliptical orbit.
RAAN – Right Ascension of the ascending node
ω - the argument of periapsis.
A 4th angle ν can be used to specify the position of the planet or satellite around the orbit.
ORBITAL ELEMENTS IN PRACTICE
If you wish to find the position of a satellite in Earth orbit there are several web sites that provide orbital elements for a wide range of objects in orbit. These are in a standardised form called Two-Line Elements (TLE for short). These specify the period of the orbit and thus its size [K3], its eccentricity, inclination , RAAN and argument of perigee. They also specify the time the object was last at perigee and other parameters related to orbital decay and some error checking characters. The format of these TLEs is shown below.
ISS (ZARYA) 1 25544U 98067A 08264.51782528 -.00002182 00000-0 -11606-4 0 2927 2 25544 51.6416 247.4627 0006703 130.5360 325.0288 15.72125391563537
It is not necessary to understand this format. You only need to download an orbital propagation program, specify your terrestrial coordinates and it will tell you what times the object will pass over your location.
ORBITAL REGIMES
In the solar system there are four classified orbital regimes distinguished by distance from the Sun.
Around the Earth the orbital regimes are classified according to altitude.
The transition between LEO and MEO is at the start of the Van Allen radiation belts. MEO is thus a region of high radiation, and any satellite in this volume of space must be constructed with radiation resistant electronics (such as the use of Gallium Arsenide semiconductors - instead of Silicon based components). GNSS satellites (GPS, GLONASS, Beidou/COMPASS, and Galileo) use this region in semisynchronous orbits (12 hour periods). The geosynchronous orbit (24 hour period) lies at the outer edge of the radiation belts and is very useful for communication and other satellites that do not need to be tracked by their ground stations. LEO is the most populated orbit regime and is useful for remote sensing satellites that produce high resolution images of the Earth and for communication satellites that require a low latency path to their ground stations.
Within each orbital regime there are several types of orbit, and there are some orbits that travel through more than one regime. They may be classified by orbital size, orbital inclination and ellipticity. Most Earth orbital satellites have reasonably low eccentricities, with the exception of transfer orbits (rocket bodies) and certain special purpose communication, military and scientific satellites. These include the Russian Molniya comsats.
MEO: Semisynchronous
GEO: Geostationary / Geosynchronous / Geo transfer orbits
HEO: Highly elliptical orbits, Molniya
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ORBITAL TYPES
| Inclination | Orbit |
|---|---|
| = 90 deg | Polar |
| 0 or 180 deg | Equatorial |
| 0 - 90 deg | Prograde |
| 90 - 180 deg | Retrograde |
LEO orbital type - Sun synchronous
A Sun-Synchronous orbit is a special type of retrograde orbit in LEO with inclinations of between 95 and 105 degrees that passes over any given place on the Earth's surface at the same time each day. This is important for remote sensing (Earth observation) and some LEO weather satellites as the images acquired have the same Sun angle and shadows for each overpass.
MEO orbital types
Middle Earth Orbit is normally a place to be avoided because of the high levels of radiation from the Van Allen belts. However, in MEO lies a semi-synchronous orbit (12 hour period) which is ideal for some navigation satellites (eg GPS) - shown opposite in 6 orbital planes (different RAAN) with i = 55 degrees.
GEO orbital types - Geosynchronous/Geostationary
A satellite in a circular orbit close to the Earth (LEO) travels at around 8 km/sec and completes about 16 orbits in one day. As the orbital altitude is raised the satellite travels more slowly and completes fewer daily .orbits. At an orbital height of 36,000 km the satellite orbits the Earth in the same time it takes the Earth to make one complete rotation (23 hr 56 m 4 s). The satellite thus appears to be stationary in the sky to a person on the Earth’s surface. For complete stationarity the satellite must have the following parameters:
a (orbital radius) = 42,000 km (36,000 ht + 6000 Earth radius)
e (eccentricity) = 0 (perfectly circular orbit)
i (inclination) = 0 (prograde equatorial orbit)
This is termed a geostationary orbit. Arthur C Clarke in 1946 was the first to point out the usefulness of this orbit for global communications.
Active maneuvres must be carried out (station-keeping) to keep a satellite in geostationary orbit. If either or both ‘e’ and ‘i’ are not exactly zero the satellite appears to wander around a point with a daily motion. It is then said to be in a geosynchronous orbit.
The diagram below shows the terrestrial surface coverage of a geostationary satellite located at longitude 0 deg (directly over the Gulf of Guinea). At the line of maximum theoretical coverage, the geosat will appear to be on the horizon for a sea level observer. This is obviously not usable in a practical sense. An elevation of greater than 10 degrees is required for imaging and communications. For accurate quantitative measurements not significantly affected by the Earth's atmosphere an elevation of greater than 30 degrees is really required.
GROUND TRACKS
The ground track of a satellite is a plot of all the points on the Earth’s surface that a satellite passes directly overhead in a given period of time. Below are the ground tracks of the International Space Station in the course of a day.
A satellite in LEO travels at around 8 km/second, and will typically complete 14 to 16 orbits of the Earth in a day. This ground track (below) of a Russian satellite in LEO also shows the position of the Sun and the daylight regions of the Earth’s surface. At this time the satellite is in darkness and so could not be seen visually from the Earth’s surface. Note that the position of the terminator (white line) is the position it had at the circled position of the spacecraft.
For a geosynchronous satellite the ground tracks are quite different and execute a closed geometrical figure over the course of 24 hours. If the satellite is truly geostationary this figure will be a point above at the longitude of the satellite. The image below shows the different tracks for a geosynchronous satellite with selected values of eccentricity &epsilon, inclination ι and argument of periapsis ω.
GETTING TO ORBIT
Circular orbit is achieved when a projectile launched horizontally from the top of a high mountain falls toward the Earth at the same rate that the Earth’s surface falls away below the projectile. Of course, there are no mountains anywhere high enough for this purpose. So a rocket carrying a satellite will normally be launched vertically and then turn in a series of maneuvres so that the satellite is placed in the required orbit with a velocity appropriate to the altitude of that orbit. For a circular orbit in LEO that velocity is around 7 to 8 km/second (~ 27,000 km/hr).
ORBITAL MANEUVRES
To change the apogee height of a satellite a rocket ‘burn’ must be made at perigee and vice versa. In the figure the rocket is initially in the red orbit. If the rocket is briefly ignited at perigee, in the direction of motion, the rocket velocity will increase but no immediate height change will occur. Instead the increased orbital energy will raise the apogee height and the rocket will move into the blue orbit. If the burn is in the opposite direction, the apogee height will decrease.
A Hohmann transfer orbit uses the same principles to transfer a spacecraft from Earth to Mars ( or another planet) with the least amount of energy (fuel). The spacecraft is initially put into an orbit around the Sun travelling with the Earth. Then a rocket burn is made to give the spacecraft just enough energy to raise its aphelion to that of the Martian orbit. A Hohmann transfer is the most efficient transfer, fuel wise, by which to move from one orbit to another.
The Earth’s gravity determines the bulk of the motion of a satellite. However, the smaller gravitational forces of the Sun and the Moon will change the orbit slightly. Even the fact that the Earth is not a perfect sphere causes a slow change in a satellite orbit.
The planet Neptune was discovered by the perturbing force it exerted on the orbit of Uranus. Three body motion cannot be solved exactly and continued refinement of orbits is done by approximation (osculating or kissing orbits) – to find the perturbed (true) orbit. Computers are essential.
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A special case of 3-body motion was solved by Lagrange in the 18th century where small bodies can orbit around 5 points (L1-L5) in the rotating system, because gravity is balanced by centrifugal force.
The Lagrange points are actually more like regions with little total force. Spacecraft have made use of L1 and L2 in particular, and execute halo orbits around these points to observe the Sun or deep space.
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ORBITAL SIMULATION PROGRAMS
Several orbital simulation programs are available to help get the feel of manuevering around in space.
Australian Space Academy