Orbits:
The direction a body travels in orbit can be direct, or prograde, in which the spacecraft moves in the same direction as the planet rotates, or retrograde, going in a direction opposite the planet's rotation. True anomaly is a term used to describe the locations of various points in an orbit. It is the angular distance of a point in an orbit past the point of periapsis, measured in degrees. For example, a spacecraft might cross a planet's equator at 10 degrees true anomaly. Nodes are points where an orbit crosses a plane. As an orbiting body crosses the ecliptic plane going north, the node is referred to as the ascending node; going south, it is the descending node.
To completely describe an orbit mathematically, six quantities must be calculated. These quantities are called orbital elements, or Keplerian elements. They are: (1) semi-major axis and (2) eccentricity, which are the basic measurements of the size and shape of the orbit's ellipse. Recall an eccentricity of zero indicates a circular orbit. The (3) orbit's inclination is the angular distance of the orbital plane from the plane of the planet's equator (or from the ecliptic plane, if you're talking about heliocentric orbits), stated in degrees: an inclination of 0 degree. means the spacecraft orbits the planet at its equator, and in the same direction as the planet rotates. An inclination of 90 degrees indicates a polar orbit, in which the spacecraft passes over the north and south poles of the planet. An inclination of 180 degrees indicates an equatorial orbit in which the spacecraft moves in a direction opposite the planet's rotation (retrograde). The (4) argument of periapsis is the angular distance of periapsis from the ascending node. Time of periapsis passage (5) and the celestial longitude of the ascending node (6) are the remaining elements. Generally, three astronomical or radiometric observations of an object in an orbit are enough to pin down each of the above six Keplerian elements.
The semi-major axis of an orbit is determined by the kinetic energy acquired by the rocket at burnout. This is equivalent to the burnout velocity. For low burnout velocities (below 25,000 ft/sec) the orbit is ballistic, meaning it does not escape the surface of the Earth. Burnout velocities above 25,000 ft/sec achieve stable orbit. At 35,000 ft/sec, the orbit reaches the distance of the Moon.
The amount of burnout velocity also determines the orbit type, an ellipse, a parabola or a hyperbolic path.
Satellites:
Satellites use a wide variety of orbits to fulfill their missions. The orbit chosen for a satellite is a compromise between the mission requirements, the capabilities of the rocket used to launch the satellite and orbital mechanics.
Low Earth Orbit:
Landsat 7 is an earth resources spacecraft which images the earth's surface in visible and infrared light. Therefore this satellite orbit is optimized for earth observation, i.e. close to the Earth's surface and moving so that it can cover the entire surface of the Earth in a short time. For this reason a near polar orbit of 700km, 98.8 inclination, 98 minute period is used which ensures that the satellite can (at least in theory) observe the entire globe. Several other features of this orbit make it especially useful for remote sensing satellites.
In theory an orbit should remain fixed in space whilst the earth rotates beneath the satellite. In reality the earth is slightly bulged and the effect of this bulge is to shift the point of perigee and the ascending node for any orbit which has an inclination other than 90. This effect is known as nodal regression, the result of which is that the plane of the orbit rotates or precesses.
However, this effect is used to advantage here to shift the orbit at exactly the same rate as the daily change in position of the sun over any point of the earth. So the satellite always passes over the earth on the sunlit part of its orbit at the same local time of day (for example at 9 am local time). This ensures that lighting conditions are similar (ignoring seasonal differences) for images taken of the same spot on the earth at different times. Additionally the orbit is resonant with the rotation period of the earth, meaning that the satellite passes over the same point on the earth at the same time of day at regular intervals (which may be daily or every 2 or more days depending on the resonance). In the case of Landsat there are 14.5 orbits per day or 29 orbits every 2 days.
Geosynchronous Orbits (GEO):
A geosynchronous orbit is an orbit which has an orbital period close to that of the earths rotation. A geostationary orbit is a special case of the geosynchronous orbit where inclination = 0 and the period is equal to the rotation period of the earth (approx 1436 minutes), corresponding to a circular orbit of approx. 35,700km altitude. A satellite in this orbit appears essentially stationary in the sky, which is why this orbit is used extensively for telecommunications & weather satellites. In reality lunar & solar gravitational influences perturb the satellites orbit, so that through the day the satellites position shifts slightly.
Below is shown the orbit of the TDRS-7 satellite, one of a series of NASA satellites which used to provide a near continuous communications link with the Space Shuttle, International Space Station & other spacecraft such as the Hubble Space Telescope.
Compared with the LEO orbit of Landsat, a much larger portion of the earth's surface is visible from the TDRS-7 spacecraft. The zone of visibility of the spacecraft has been highlighted by a cone. Approximately 40% of the earths surface can be viewed at any one time from geostationary altitude. Additionally, the spacecraft orbit is sunlight apart from a small zone which passes into the earths shadow. Actually, geostationary satellites only experience eclipses at two periods of the year - for a few weeks at a time at the spring and autumn equinoxes. The reason for this is simple. The earths rotation axis is inclined with respect to the ecliptic, hence the earth's shadow cone misses the plane of a zero inclination geostationary orbit apart from the times when the suns declination is close to zero. This occurs twice a year, once at the spring equinox and once at the autumn equinox.
As can be seen from this graphic a perfectly geostationary satellite stays over the same spot on the equator all day. However, if we were to look closely we would see that the satellite does appear to change position, generally describing a small figure of 8 or an arc due to the effect of lunar / solar perturbations dragging the satellite into a slightly elliptical, slightly inclined orbit. There are many non operational satellites in "graveyard" orbits slightly above or below a true geostationary orbit. Since the orbital period is slightly more or less than the earths rotation period these satellites appear to drift slowly around the earth.
Geosynchronous Transfer Orbits (GTO):
There are many rocket boosters which are observable in "transfer" orbits. These are the orbits used to transfer the satellite from an initial low earth orbit to the final orbit. The orbit used for transfer to geostationary orbit is named appropriately enough a "geostationary transfer orbit" (GTO). A standard GTO is an orbit which requires the minimum energy to reach geostationary altitude (A Hohmann transfer ellipse). The perigee corresponds to the altitude of the initial low earth orbit parking orbit, the apogee the geostationary orbit altitude and the inclination is usually the inclination of the initial parking orbit. At apogee the payload usually fires an on-board motor to circularize the orbit and adjust the inclination to zero. The GTO orbit of the Intelsat 4-2 rocket (An Atlas-Centaur) is shown below. This is a 600x35 km,700x28 km inclination orbit Note that the orbit is very elliptical. The perigee is in the southern hemisphere, so it is possible for observers over a narrow latitude range (centered on -28) to see this object at a range of only a few hundred kilometers. However, the satellite spends most of its time, due to the "equal areas" rule of orbital dynamics at high altitudes. The apparent trajectory of Intelsat 4-2 rk with respect to Alt / Azimuth is shown below. The rocket appears to trace out a hairpin loop - the shape is a combination of the orbit & the rotation of the earth as can be seen from the ground trace.
In recent years modified versions of the GTO orbit have been used. A supersynchronous orbit is one where the apogee is significantly greater than geosynchronous altitude. Why send the payload higher than the target orbit altitude? The reason is because the payload still has to adjust its inclination from the launch inclination (anything from 5 to 51) to 0. This maneuver is very expensive in terms of energy, much more so than an in plane change of orbital altitude. The energy required to do this maneuver decreases with orbital altitude, so it requires less fuel to perform this plane change at high altitude (e.g.. 60,000 or 70,000 km) and then descend to a geostationary orbit rather than do the plane change at geostationary height. Many supersynchronous transfer orbits also have very low perigee altitudes, to accelerate their decay and reduce the amount of debris in orbit. Subsynchronous transfer orbits also exist where the rocket only sends the payload part of the way to geostationary height, the payload then uses its own propulsion system to reach the final orbit.
Molniya orbits:
Many Russian cities are at high northern latitudes where it is impractical to use geostationary satellites for telecommunications since the satellite would appear either low on the horizon on not visible at all. To overcome this problem Molniya satellites are used for communications in these regions. The orbit used by these satellites is a 12h, high inclination elliptical orbits. The orbit of Molniya 3-47 is shown below. This is a 1470 x 38900 km, 63.4 inclination orbit. Again this orbit has special features which make it well suited for telecommunications. First the period is 12h, so there are 2 orbits per day.
As a result the ground track of the orbit repeats at the same time of day each day. Since the orbit is elliptical, the satellite spends most of it's time near apogee (where its velocity is slowest), so for 11h of each orbit the satellite is above the horizon for high northern latitudes. Additionally, for several hours per day the satellite moves only very slowly across the sky (as can be seen from the ground track), making it easy to follow with a communications antenna. There is a special reason for the 63.4 inclination. Normally, as described above for the sun-synchronous orbit of Landsat 7, the oblate shape of the earth causes a gradual shift of the orbits perigee along the orbit. As a result the perigee of this orbit would shift from the southern hemisphere into the northern hemisphere. However, the rate of shift depends on the inclination of the orbit and at certain inclinations the perigee does not move. For 63.4 inclination orbit with a perigee in the southern hemisphere the position of perigee remains fixed, which is why this inclination is used for Molniya orbits.
Mid Earth Orbit:
Mid earth orbit (MEO) is a term used to describe 12h period, medium inclination orbits generally used for Global Positioning Satellites. With a constellation of 24 appropriately spaced satellites in approx 20,000 km near circular orbits it is possible to ensure that at least 4 satellites are visible from any one location at any time to ensure reliable navigation using signals from these GPS satellites.
The orbit is resonant with the earths rotation period (2 orbits per day) so the orbit track repeats itself each day.
Hohmann Transfer Orbits:
To launch a spacecraft to an outer planet such as Mars, using the least propellant possible, first consider that the spacecraft is already in solar orbit as it sits on the launch pad. Its existing solar orbit must be adjusted to cause it to take the spacecraft to Mars. In other words, the spacecraft's perihelion (closest approach to the sun) will be Earth's orbit, and the aphelion (farthest distance from the sun) will intercept the orbit of Mars at a single point. This is called a Hohmann Transfer Orbit. The portion of the solar orbit that takes the spacecraft from Earth to Mars is called its trajectory.
To achieve such a trajectory, the spacecraft lifts off the launch pad, rises above Earth's atmosphere, and is accelerated in the direction of Earth's revolution around the sun to the extent that it becomes free of Earth's gravitation, and that its new orbit will have an aphelion equal to Mars' orbit. After a brief acceleration away from Earth, the spacecraft has achieved its new orbit, and it simply coasts the rest of the way. To get to the planet Mars, rather than just to its orbit, requires that the spacecraft be inserted into the interplanetary trajectory at the correct time to arrive at the Martian orbit when Mars will be at the point where the spacecraft will intercept the orbit of Mars. This task might be compared to throwing a dart at a moving target. You have to lead the aim point by just the right amount to hit the target. The opportunity to launch a spacecraft on a transfer orbit to Mars occurs about every 25 months.
To be captured into a Martian orbit, the spacecraft must then decelerate relative to Mars (using a retrograde rocket burn or some other means). To land on Mars, the spacecraft must decelerate even further (using a retrograde burn, or spring release from a mother ship) to the extent that the lowest point of its Martian orbit will intercept the surface of Mars. Since Mars has an atmosphere, final deceleration may be performed by aerodynamic braking, and/or a parachute, and/or further retrograde burns.
To launch a spacecraft to an inner planet such as Venus using the least propellant possible, its existing solar orbit must be adjusted so that it will take it to Venus. In other words, the spacecraft's aphelion will be on Earth's orbit, and the perihelion will be on the orbit of Venus. As with the case of Mars, the portion of this orbit that takes the spacecraft from Earth to Venus is called a trajectory. To achieve an Earth to Venus trajectory, the spacecraft lifts off of the launch pad, rises above Earth's atmosphere, and is accelerated opposite the direction of Earth's revolution around the sun (decelerated) to the extent that its new orbit will have a perihelion equal to Venus's orbit. Of course the spacecraft will end up going in the same direction as Earth orbits, just a little slower. To get to Venus, rather than just to its orbit, again requires that the spacecraft be inserted into the interplanetary trajectory at the correct time to arrive at the Venusian orbit when Venus will be at the point where the spacecraft will intercept the orbit of Venus. Venus launch opportunities occur about every 19 months.
Gravity Assist Trajectories:
The planets retain the vast majority of the solar system's angular momentum. It is this momentum that is used to accelerate spacecraft on so-called "gravity-assist" trajectories. It is commonly stated in newspapers that spacecraft such as Voyager and Galileo use a planet's gravity during a flyby to slingshot it farther into space. How does this work? In a gravity-assist trajectory, angular momentum is transferred from the orbiting planet to a spacecraft approaching from behind. Gravity assists would be more accurately described as angular-momentum assists.
Consider Voyager 2, which toured the Jovian planets. The spacecraft was launched on a standard Hohmann transfer orbit to Jupiter. Had Jupiter not been there at the time of the spacecraft's arrival, the spacecraft would have fallen back toward the sun, and would have remained in elliptical orbit as long as no other forces acted upon it. Perihelion would have been at 1 AU, and aphelion at Jupiter's distance of about 5 AU.
However, the spacecraft's arrival was carefully timed so that it would pass behind Jupiter in its orbit around the sun. As the spacecraft came into Jupiter's gravitational influence, it fell toward Jupiter, increasing its speed toward maximum at closest approach to Jupiter. Since all masses in the universe attract each other, Jupiter sped up the spacecraft substantially, and the spacecraft slowed down Jupiter in its orbit by a tiny amount, since the spacecraft approached from behind. As the spacecraft passed by Jupiter (its speed was greater than Jupiter's escape velocity), of course it slowed down again relative to Jupiter, climbing out of Jupiter's gravitational field. Its Jupiter-relative velocity outbound was the same as its velocity inbound. But relative to the sun, it never slowed all the way to its initial approach speed. It left the Jovian environs carrying an increase in angular momentum stolen from Jupiter. Jupiter's gravity served to connect the spacecraft with the planet's huge reserve of angular momentum. This technique was repeated at Saturn and Uranus.
The same can be said of a baseball's acceleration when hit by a bat: angular momentum is transferred from the bat to the slower-moving ball. The bat is slowed down in its "orbit" about the batter, accelerating the ball greatly. The bat connects to the ball not with the force of gravity from behind as was the case with a spacecraft, but with direct mechanical force (electrical force, on the molecular scale, if you prefer) at the front of the bat in its travel about the batter, translating angular momentum from the bat into a high velocity for the ball.
Gravity assists can be also used to decelerate a spacecraft, by flying in front of a body in its orbit, donating some of the spacecraft's angular momentum to the body. When the Galileo spacecraft arrived at Jupiter, passing close in front of Io in its orbit, Galileo experienced deceleration, helping it achieve Jupiter orbit insertion.
Three Body Problem:
The problem of determining the motion of three celestial bodies moving under no influence other than that of their mutual gravitation. No general solution of this problem (or the more general problem involving more than three bodies) is possible.
As practically attacked, it consists of the problem of determining the perturbations (disturbances) in the motion of one of the bodies around the principal, or central, body that are produced by the attraction of the third. Examples are the motion of the Moon around the Earth, as disturbed by the action of the Sun, and of one planet around the Sun, as disturbed by the action of another planet. The problem can be solved for some special cases; for example, those in which the mass of one body, as a spacecraft, can be considered infinitely small, and in the Lagrangian and Eulerian cases.
