[3.1] ELEMENTARY ORBITAL MECHANICS
[3.2] SATELLITE ORBITS
[3.3] INTERPLANETARY SPACECRAFT TRANSFER ORBITS
[3.4] GRAVITY ASSIST TRAJECTORIES
[3.5] GRAVITY AND PLANETS / TIDES
[3.6] GRAVITY IN THE SOLAR SYSTEM / LIBRATION POINTS / STELLAR MASSES
[3.7] LIBRATION POINTS, HALO ORBITS, & MANIFOLDS
[3.8] FOOTNOTE: LARRY NIVEN'S INTEGRAL TREES
[3.1] ELEMENTARY ORBITAL MECHANICS
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gravitational_force = constant * mass1 * mass2 / distance^2
_________________________________________________________________________
constant = 6.672E-11
The force of gravity continues to grow weaker as the distance from Earth increases, but it never stops completely. Earth, or any other celestial body in the Universe, is said to be at the bottom of a "gravity well".
gravitational_force = constant * mass_earth * mass_satellite / distance^2
Since force is equal to mass times acceleration, then:
gravitational_acceleration = constant * mass_earth / distance^2
gravitational_acceleration = 9.80665 / ( orbital_radius / earth_radius )^2
9.81 / ( ( 6,378,000 + 1,000,000 ) / 6,378,000 )^2
= 7.33 meters per second squared
centripedal_acceleration = orbital_radius * angular_velocity^2
9.81 / ( orbital_radius / earth_radius )^2
= orbital_radius * angular_velocity^2
1
orbital_radius^3 = ( 9.81 * earth_radius^2 ) * --------------------
angular_velocity^2
Radians per second can be converted to revolutions per second by dividing by
2PI, and then revolutions per second can be converted to revolutions per hour
by multiplying by 60 * 60 = 3,600. Orbital period is the inverse of
revolutions per hour, and so the conversion is:
2PI
angular_velocity = ------------------------
3,600 * orbital_period
orbital_radius^3
1
= ( 9.81 * earth_radius^2 ) * ------------------------------------
( 2PI / 3,600 * orbital_period )^2
3,600^2 * 9.81 * earth_radius^2
= --------------------------------- * orbital_period^2
2PI^2
3,600^2 * 9.81 * earth_radius^2
orbital_radius^3 = --------------------------------- * orbital_period^2
1,000^3 * 2PI^2
_________________________________________________________________________
orbital_radius^3 = constant * orbital_period^2
_________________________________________________________________________
Kepler's Third Law is true for any planetary orbit. In the case of an Earth
satellite, the value of the constant is 130,958.995,396 (using the more
precise value of 9.80665 for the acceleration of gravity instead of 9.81),
and the law can now be rearranged to give two simple formulas:
_________________________________________________________________________
orbital_radius = 5,078 * orbital_period^(2/3)
orbital_period = ( orbital_radius / 5,078 )^1.5
_________________________________________________________________________
[3.2] SATELLITE ORBITS
5,078 * 2^(2/3) = 8,060 kilometers
((6,378 + 19,000) / 5,078 )^1.5 = 11.2 hours
5,078 * 24^(2/3) = 42,250 kilometers
Geostationary orbits are very handy for some types of satellites, such as weather observation and, particularly, communications satellites. The orbital "slots" where a satellite may be placed in geostationary orbit are regulated and are traded as a commercial commodity by satellite companies. A satellite TV user has to focus his or her home dish antenna on a satellite in a particular orbital slot and then lock the dish in place.
By the way, the Earth's axis of rotation is at an angle to its orbit around the Sun, and so the orbits of geostationary satellites around the equator are at an angle to the Sun as well. This means that the Sun's pull tends to tug them out of position, and so they are fitted with small "station keeping" thrusters to keep them in their position. These thrusters do not need to be very powerful, since only a slight nudge is needed. Once the fuel for the thrusters is gone, the satellite will gradually drift out of its geostationary orbit and become useless.
Satellites that observe the Earth in detail for military, commercial, or scientific purposes are also usually in LEO, placed into orbits over the poles, so that as they spin around the Earth once every hour or so, the Earth rotates underneath them. In 24 hours, such a satellite will be able to scan the entire Earth.
In reality, while a purely polar orbit may be used for satellites that scan the Earth with radar, those that take pictures are not usually put into an orbit that takes them precisely over the poles. They are put into a slightly offset orbit at a specific altitude that has the interesting property of placing them over a particular location on the equator at the very same time every day in a particular time zone.
This is known as a "Sun synchronous" orbit, and it is done to ensure that the lighting conditions are as consistent as possible between observational passes. Changing shadow lengths and positions makes comparison of images from successive passes of a satellite difficult. If the shadows are the same on each orbit, then any changes are much more noticeable.
For example, Argentina's SAC-C scientific Earth observation satellite, launched late in 2000, was put into orbit at an altitude of 702 kilometers and an inclination of 98.2 degrees from the equator. The inclination is described as 98.2 degrees, rather than 81.8 degrees, to emphasize that the satellite has a "retrograde" orbit, that is, in the reverse direction to the Earth's rotation. SAC-C passes over the equator at 10:15 AM local time during each orbit, and probably flies over my head at northern mid-latitudes about 10:00 AM US Mountain Time every day.
Such an elliptical orbit demonstrates an interplay between kinetic and potential energy. At the lowest part of its orbit, the "perigee", the satellite is moving very fast. It has maximum kinetic energy and minimum potential energy. As it circles around the Earth and arcs to greater heights, it loses kinetic energy to gravitational potential energy and slows, eventually reaching a minium velocity at the very top of the ellipse, the "apogee". It now has minimum kinetic energy and maximum potential energy, in much the same way as a ball thrown upward reaches the top of its arc, floating for an instant before reversing its direction. Once the object in orbit reaches its apogee, it then falls back down along its orbit, picking up speed until it reaches the perigee again.
This behavior leads to a neat relationship: if a radius is drawn from the spacecraft to the center of the Earth, that radius will sweep out equal areas in equal intervals of time. This is "Kepler's Second Law". If the satellite is near its apogee, that radius is very long, and as it is moving slowly in a particular interval of time the area it sweeps out is a long, narrow slice. If the satellite is near its perigee, it is moving fast, and the area it sweeps out in the same amount of time is a short, wide slice with exactly the same area.
Elliptical orbits were particularly useful to the Soviet Union. The USSR was at a high latitude, and geosynchronous communications satellites tended to be too close to the horizon to be useful in the northern regions of the country. As a result, the Soviets launched communications satellites named "Molniyas" into elliptical orbits that took them high over the USSR, arcing overhead for long periods of time, and then falling back down around the other side of the Earth to make whip around quickly for another gradually slowing arc up over the Motherland.
This orbit became known as a "Molniya orbit". The Americans employed Molniya orbits for spy satellites used to listen in on Soviet communications. Although the modern Russian state seems to be more interested in geostationary communications satellites, some American "eavesdropping" satellites are still launched into Molniya orbits.
A more intuitive way to see this is to imagine two pins stuck into a piece of cardboard to act as focal points, with a loop of thread laid around the threads. A pen is used to stretch the loop taut and then trace a curve all the way around the two focal points. Obviously, if both focal points are in the same place, the pen will draw out a circle, but the farther apart the pins are placed, the more elliptical the curve becomes.
The longest distance from the edge of the ellipse to its center is called the "semi-major axis", while the shortest distance is unsurprisingly called the "semi-minor axis". The "eccentricity" is given as the ratio of the distance from the center to a focal point versus the length of the semi-major axis. This has a value of 0 for a circle, with the focal point at the center of the curve, and approaches a value of 1 as the curve becomes more elliptical.
After launch, a geostationary satellite is placed into a highly elliptical temporary "geostationary transfer orbit" with the apogee at 35,750 kilometers. Once the satellite reaches that altitude, it fires a rocket engine to "circularize" its orbit and not fall down to its perigee again.
An equatorial launch site is not the best choice for a polar orbit, since the launcher's upper stage must expend fuel to turn the spacecraft's orbit towards the poles. A high-latitude space center is better for launching such spacecraft, and a small space center has been built on Kodiak Island off of Alaska to support such launches.
The American Kennedy Space Flight Center at Cape Canaveral in Florida is between these two extremes, and so is equally useful, or equally flawed, for launching payloads into equatorial or polar orbits. In some cases, such as the early manned orbital spaceflights, the orbital plane is not particularly critical, and except for the equatorial orbital-velocity boost, there is no penalty or much advantage in using one launch site or another from the orbital dynamics point of view.
[3.3] INTERPLANETARY SPACECRAFT TRANSFER ORBITS
To send a space probe to another planet, the probe has to be blasted out of the Earth's gravity well at more than the escape velocity. However, it isn't enough to just point the launch rocket at the planet and blast away. Even the largest booster rocket can only burn for times measured in a total of minutes, while the planets are so far away that a probe will take months or years to reach them.
This means that the probe has to be launched on a trajectory that "leads" the motion of the target planet. The probe is launched out of the Earth's gravity well into an orbit around the Sun that arcs between the Earth's orbit and the target planet's orbit. This trajectory is called a "Hohmann transfer orbit", after German space pioneer Walter Hohmann, who published details of the concept in 1925.
To reach Mars, the probe must be given additional velocity, a "delta-vee" in the jargon, relative to the Earth's orbit around the Sun so that its transfer orbit will reach outward towards Mars. Counterintuitively, to reach Venus, the probe must be given a delta-vee that reduces its velocity relative to the Earth's orbit so that the transfer orbit will fall inward towards Venus. In other words, the launch vehicle actually slows down the spacecraft, at least relative to its orbit around the Sun.
Software is used to calculate transfer orbits. Such programs are by no means trivial to write. The problem is that there are many bodies in the Solar System, and a spaceflight engineer has to at the very minimum consider the interactions of the Earth, the target planet, and the Sun to properly calculate a transfer orbit. The gravitational interactions of three or more bodies are so complicated that they are computationally overwhelming, and to make matters worse such systems are "chaotic": Any slight change in initial conditions can quickly lead to wild variations in trajectory. This is known as the "N-body problem."
In practice, transfer orbits are calculated using simplified methods. For example, calculating the interactions of two bodies is an exercise in simple physics. The trajectories are defined by "Kepler's First Law", which states that motions of a planet or a spacecraft around the Sun or other larger mass is always in the form of a "conic section".
A conic section is the geometric figures that can be obtained by cutting through the side of a hollow cone and tracing out the edge of the cut. A circle is obtained by cutting through the cone at an angle perpendicular to the axis of the cone. Move the angle away from the perpendicular, the section becomes an ellipses that becomes more elongated as the angle increases, with its value of eccentricity increasing to approach 1.
Once the angle becomes exactly that of the slope of the cone, in which case the eccentricity has a value of 1, the ellipse becomes an open figure known as a "parabola". If the angle is increased further, the open figure widens into a succession of "hyperbolas".
In reality, the movements of celestial bodies in the Solar System are always either ellipses or hyperbolas. Circles and parabolas are "perfect" figures, corrupted by any variation in the cutting angle through the cone from the horizontal or vertical respectively, essentially teetering towards "imperfect" ellipses and hyperbolas with any change in circumstances, and so unlikely to occur in the somewhat messy real Universe.
Orbital planning software calculates one set of conics to define the spacecraft's trajectory at launch, calculates a second set of conics for the spacecraft's trajectory at its target, and tries to find one conic in each set that match up.
This is unsurprisingly a very precise operation. Although "aerobraking" was used to circularize the orbit of the American Mars Global Surveyor (MGS) probe launched in 1996, MGS used its rocket engine to allow it to be captured by Mars' gravity in the first place. The French CNES space agency is considering launch of a Mars probe in 2005 that will be the first to use aerocapture to enter orbit around another planet.
The astronaut's trip times for an Earth-Mars cycler would be several years, but some schemes fit the cycler with a high-efficiency rocket system to adjust the trajectory, permitting trip times as low as half a year.
[3.4] GRAVITY ASSIST TRAJECTORIES
This is a subtle idea. If a probe is sent on a flyby around a planet, it approaches the planet from one direction, arcs around it, and departs in another direction. As viewed from the planet, the probe will not leave the planet with any greater velocity than it arrived with. The probe has the same energy before and after the encounter, even if it speeds up during the flyby. The delta-vee is zero.
The trick is that the planet being used for the gravity assist is a moving object. If the probe swings behind the planet in its orbit, some of the velocity of the planet is transferred to the probe while the planet slows down slightly. However, since the velocity transfer between the two objects is inversely proportional to their mass, the probe's increase in velocity is substantial, while the planet's decrease in velocity is literally unmeasurable.
A probe can be swung around in front of a planet in its orbit to decrease velocity. A gravity assist can be regarded as similar to a "soft collision", with the spacecraft "bouncing" off a planet and obtaining a delta-vee thereby.
This "Grand Tour" trajectory was implemented on the Voyager 2 space mission, which was launched in 1977. The probe performed a flyby of Jupiter in 1979, slinging it on to a flyby of Saturn in 1981, a flyby of Uranus in 1986, and a flyby of Neptune in 1989. Voyager 2's scientific payback was unexcelled by any other single space mission.
Gravity assist trajectories have been used on many other deep-space missions. For example, the Galileo Jupiter orbiter was faced with being drastically cut back in the mid-1980s because the powerful booster rocket originally designed for it was cancelled and it had to use a much smaller rocket instead. Fortunately, the mission was saved by coming up with a "Venus-Earth-Earth Gravity Assist (VEEGA)" trajectory in which the probe swung past Venus once and Earth twice to obtain a gravity assist on each pass.
NASA's s "Stardust" comet sample-return mission, launched in February 1999, performed an Earth flyby in January 2001 to set it on course to pass near the comet Wild-2 in 2004, which has an orbit between Mars and Jupiter. The probe will use a glass-foam "aerogel" to obtain samples of materials in the comet's coma, and will then loop back to Earth, dropping off a capsule containing the samples that will parachute down into the Utah desert.
[3.5] GRAVITY AND PLANETS / TIDES
The first additional rule is that one body doesn't orbit around the other, they both orbit around the center of mass of the system. A spacecraft in orbit around the Earth has so little mass that its effect on the Earth is negligible, and so the spacecraft's orbit for all practical purposes goes around the center of the Earth. However, if two planet-sized bodies of the same size orbit each other, then they both orbit around a common point halfway between each other. This point is sometimes referred to as the "barycenter".
If they are less equal in size, the barycenter is their center of gravity. A
good example of this is the distant planet Pluto and its large moon Charon,
which are closer in size than any other planet and its moon in the Solar
System. Charon has a mass that is 15% that of Pluto's, and the distance
between the two is about 20,000 kilometers. The balance point P, as measured
from the center of Pluto, can be determined with the simple equivalence:
mass_pluto * P = ( 20,000 - P ) * mass_charon
mass_pluto * P + mass_charon * P = 20,000 * mass_charon
P * ( mass_pluto + mass_charon ) = 20,000 * mass_charon
P = 20,000 * mass_charon / ( mass_pluto + mass_charon )
= 20,000 * 0.15 / ( 1 + 0.15 ) = 2,610 kilometers
Although the sum of the gravitational force between the Earth and the Moon acts on a line through their centers, of course the gravitational force of each world acts on every part of the other. Since gravitational force decreases with the square of the distance, the Moon's force on the near face of the Earth is weaker than its force on the far face. This difference in force between the two faces is equivalent to a force between them that attempts to stretch the Earth into an ovoid, with its long axis pointed towards the Moon.
As the Earth is pretty solid, the tidal effect on the land masses is slight, but the tidal strain is enough to cause the oceans to shift around on a regular cycle. While the usual tidal change is a meter or two, some coastal estuaries funnel the tides and increase their height to up to 15 meters, and hydropower dams have been built in a few such locations. When the tide goes high, the dam opens to let the seawater into the estuary, spinning a power turbine on the way in. The dam is then closed, and when the tide drops, the dam opens to let the water back out into the ocean, spinning the turbine on the way back out again.
The Sun also contributes to tides, but the although the Sun is much more massive than the Moon and has a much greater gravitational force on the Earth, the Sun is much farther away. An analysis of tides as a differential gravitational force shows they fall off with the cube of distance between two masses, not the square as does gravity itself, so the Sun's tidal influence falls off much faster than its gravitational influence.
Higher tides do occur when the Sun and Moon are aligned on opposite sides of the Earth, causing their tidal effects to work together. These are known as "spring tides", meaning they "spring up" above normal, not because they happen in the spring -- they can happen in any season. When the Sun and Moon are at right angles relative to the Earth, their tidal effects work against each other, and the result are low or "neap" tides.
The effect of the Earth's tidal force on the Moon is much more dramatic and is easily seen by the fact that the Moon only presents one face to the Earth at all times. Over the eons, the Earth's tidal effect slowed the Moon's rotation until it became the same as the period of the Moon's orbit around the Earth. The Moon is said to be "tidally locked" to the Earth.
Incidentally, the Moon's tidal forces are very gradually slowing down the Earth's rotation as well. 900 million years ago, the Earth's day was only 18 hours long. The Moon was 7.5% closer to the Earth, and the lunar month was 23.4 days. The distance between the Earth and the Moon can now be precisely measured by bouncing a laser beam off a laser reflector left on the Moon by the Apollo manned lunar expeditions. These measurements show that tidal stresses set up by the Earth on the Moon are accelerating it, and it is moving farther away from the Earth at a rate of 3.8 centimeters per year.
[3.6] GRAVITY IN THE SOLAR SYSTEM / LIBRATION POINTS / STELLAR MASSES
It was once thought that the planet Mercury was tidally locked to the Sun, so that it kept one face eternally faced to the Sun and the other in darkness, one side burning and the other side freezing. This was found to be not exactly the case in the 1950s. It is in a "resonance" where it turns three times for every two orbits around the Sun.
This is known as a "metastable" state. If Mercury kept one face always to the Sun, it would be in a minimum energy state, but to get out of its 2:3 resonance state, it would have receive some sort of increase in rotational energy, possibly through a collision with another large body. Until something like this happens, Mercury's rotation will remain as it is, something like a stone that been caught on a ledge after falling halfway down a cliff instead of dropping all the way to the bottom.
Jupiter's gravitational influence led to the creation of the asteroid belt, since it prevented the formation of any single large body at that location. Interestingly, there are empty "bands" in the asteroid belt that correspond to asteroid orbits that would have a some sort of synchronization or "resonance" with that of Jupiter. Any asteroid in such an orbit is quickly pulled out of it by repeated tugs from Jupiter.
Another consequence of Jupiter's gravitational influence is that objects straying into the two locations in its orbit that are equally distant from the Jupiter and the Sun tend to stay there. These orbital positions are known as "libration points", or sometimes "Lagrangian points" after the 18th century French mathematician Louis Lagrange. Lagrange performed an analysis of the gravitational interactions of two bodies in orbit around each other to identify positions where objects would remain stationary relative to the bodies.
He actually discovered five such stable positions, now known as "L1" through "L5", roughly laid out as follows relative to two masses of the same size:
L1, L2, and L3 are only stable in the sense that an object balanced on a point is stable. L1 is located between the two bodies, and an object placed there will fall towards one or the other if disturbed. Objects at L2 and L3 are at a precise orbital radius around the two bodies where they keep pace with the orbits of the bodies. At a lower radius, the objects will have a faster orbit and move ahead, while at a higher radius, they will have a slower orbit and fall behind.
In contrast, the equidistant positions, L4 and L5, are stable in the sense that an object caught in a hole is stable. If any object caught at these locations is disturbed and begins to move out of them, the forces of the two bodies will tend to nudge it back into place. The L4 and L5 positions in Jupiter's orbit have trapped a number of asteroids, referred to as "Trojan asteroids".
This would seem implausible in the first place, since it's obvious there are no other "counter planets" such as a "Counter Mars" or "Counter Jupiter". The Earth is on the opposite side of the Sun relative to every other planet at some time or other, and no counter-worlds for Mars or Jupiter have been seen. Furthermore, a Counter Earth would be a bright object, and no deep space probe has ever spotted such a thing.
In reality, a Counter Earth is completely impossible, since even if such a world were magically created right now, the unbalanced interactions of all the other planets on the Earth and Counter Earth would disrupt the neat balance, possibly leading to a disastrous collision between the two planets.
This task is simplified if astronomers find a "double star" system, in which two stars orbit around each other. It is possible to precisely time how long the orbit takes through observation. It is also possible to determine how fast the stars are moving in their mutual orbit through changes in the color of their light emission, or "Doppler shift", which is discussed in a later chapter.
If the velocity of the stars in their mutual orbit is known and the period of the orbit is known, then the length of the circumference of the orbit can be determined by multiplying the velocity times the period, giving the length of the circumference of the orbit. Comparing the actual size of the orbit against its size as seen from Earth can give a good estimate of the distance to the double-star system.
Furthermore, knowing the size of the orbit and its period gives the combined
mass of the two stars, using Kepler's Third Law:
_________________________________________________________________________
orbital_period^2 = constant * orbital_radius^3
_________________________________________________________________________
_________________________________________________________________________
system_mass = constant * ( radius^3 ) / ( period^2 )
_________________________________________________________________________
For the case of the Earth-Sun system, this gives the system mass, the radius, and the period all as 1, and so the constant has to be 1 to get the right result. Of course, for the Earth-Sun system the mass of the Earth is negligible compared to that of the Sun.
Suppose a distant star system is discovered featuring two stars with an
orbital radius of 2.3 AU and an orbital period of 1.2 years. This gives the
combined mass of the two stars as:
( 2.3^3 ) / ( 1.2^2 ) = 8.45 solar masses
[3.7] LIBRATION POINTS, HALO ORBITS, & MANIFOLDS
Several solar observatories have been launched into the Sun-Earth L1 position, and in August 2001 NASA launched a spacecraft named "Genesis" that was placed there between Earth and Sun to capture particles emitted by the Sun for return to Earth. At the L1 position, these spacecraft have an excellent view of the Sun that is never blocked by the Earth, and are within easy communications range of the Earth.
In the summer of 2001, NASA launched the "Microwave Anisotropy Probe (MAP)", which was parked in the Sun-Earth L2 position, far beyond the Moon's orbit away from the Sun. There, the spacecraft began mapping low-level energy emissions from the Universe, deploying a shield to block out emissions from the Sun and Earth that might disrupt the measurements. Other astronomical spacecraft are now being designed to take up station at the L2 position, where they will have a view of deep space that also will never be obscured by the Earth, and which will be in relatively close communications range.
The solution to both these problems is what is known as a "halo" orbit, in which a spacecraft performs long, lazy curved loops around the libration point, almost as if the libration point were a celestial body.
A halo orbit is an N-body problem, involving interactions between the probe, the Sun, and the Earth. Early calculations of halo orbits involved making educated guesses, running them through a software simulation, and then using the result to get a better guess.
This approach was workable but crude. In the 1990s, improved software was developed that essentially mapped out entire ranges of three-body trajectories, known as "manifolds". Paths known as "dynamical channels" can be identified on the manifolds that chart out the course a spacecraft would follow on its own after given an initial push, analogous to the way a ball bearing would meander about on an uneven surface after being given a nudge one way or another.
The Genesis spacecraft, for example, was initially launched into a parking orbit around Earth. A short engine burn then sent it drifting towards the Sun, and three months later another brief burn put into a halo orbit around the L1 position. It is now performing a total of four halo orbits, each lasting six months.
After the end of the sampling mission, another burn will put the probe on a orbit that will send it all the way back to a halo orbit around L2, where MAP will be making its measurements, and then to a close Earth flyby, where Genesis will drop its sample package. The reason for this roundabout maneuver is to allow the package to be ejected during daylight hours. As it is to be recovered by helicopter to prevent contamination, it cannot be sent back at night.
Spaceflight engineers working with manifolds have envisioned them as very useful for probes to explore the Jupiter and Saturn systems. Both these planets have extensive sets of large moons that set up complicated manifolds. By exploiting dynamical channels along these manifolds, the probe could maneuver from moon to moon, remaining at each for as long as desired, and then changing position with a slight nudge from its rocket engine.
[3.8] FOOTNOTE: LARRY NIVEN'S INTEGRAL TREES
Voy had a giant planet in orbit around it in turn. Voy had once been a bigger, brighter star itself, but the stellar explosion that turned it into a hot stellar cinder stripped the planet of its gaseous outer layers, resulting in a ring or "torus" of gas that stretched completely around the planet's orbit around Voy.
This "Smoke Ring" was long-lasting and contained enough debris to support the evolution and propagation of life. Plants evolved that produced oxygen, eventually making the Smoke Ring an environment where humans could breathe and survive. When a mutiny stranded an interstellar expedition from Earth there, successive generations of humans colonized the Smoke Ring and gradually spread all the way around it.
The major landmarks in the Smoke Ring were the "Integral Trees", which were huge free-floating trees about a hundred kilometers long that were curved into an elongated "S" shape, similar to the integral symbol used in calculus. Humans tribes colonized the tips of the Integral Trees, which were covered with jungle-like "tufts" of greenery, while the intermediate parts of the tree were generally barren.
The orbit of an object is defined at its center of mass, which was at the center of an Integral Tree. Since the Integral Tree was an elongated object, tidal forces from the double star forced the Integral Tree to point toward Voy at the center of its orbit, crossways to the tree's motion.
An independent object orbiting below, or "south", of the center of mass of the Integral Tree would have a faster orbit, while an independent object orbiting above, or "north", of the center of mass of the Tree would have a slower orbit. However, as the Integral Tree was a solid object, the entire Tree orbited at the same speed. This meant that the south tuft of the Tree was moving slower than independent objects in the same orbital radius, and the north tuft of the Tree was similarly moving faster than independent objects in the same orbital radius.
At the south tuft, since the gases in the Smoke Ring were moving at their natural orbital speed but the tuft of the tree was moving more slowly, the tuft was in a perpetual howling wind blowing from behind the Tree's direction of motion. The southern end of the Tree curved in the direction of its motion to offer less resistance to this wind, while the tuft filtered debris from the wind to support the growth of the Tree.
Gravity was no longer completely balanced by centripedal acceleration, and the result was that the humans felt a net gravitational force that grew stronger the farther south they went. Of course, they had to stay on the inside of the curve of the tuft, since if they went below they could fall off into the Smoke Ring.
In a complementary fashion, the north tuft was moving faster than the gases at its orbital radius, and so the north tuft faced a similar howling wind but in the opposite direction, coming from in front of the Tree. As with the south tuft, it curved away from the wind and the tuft collected debris blown into it by the wind.
Centripedal acceleration was no longer balanced by gravity, and so the humans felt a net "artificial gravity" that grew stronger the farther north on the tuft they went. Once again, they had to stay on the inside of the curve, or they would be flung off into the Smoke Ring by their excess velocity.
The excess gravitational pull on the south tuft of course had to be balanced by the excess centripedal acceleration on the north tuft, or the Integral Tree would not have been in a stable orbit. This set up a force between the two ends of the Tree that increased as the Tree grew longer, and so limited its length. In fact, old Trees tended to break apart at the middle, with disastrous consequences for the human colonies in the tufts.
By the way, Smoke Ring materials unsurprisingly tended to accumulate in the L4 and L5 positions defined by the ring's parent planet and Voy.
The idea of making movies out of THE INTEGRAL TREES and THE SMOKE RING is appealing, though the special effects would be so difficult that an animated film might be the best approach, particularly since the human colonists had evolved into very tall and gangly forms in the low gravity and could not be played by any normal human actor. Unfortunately, the environment of the Smoke Ring is so alien and disorienting that most viewers would not be able to figure out what was going on, and it seems plausible that if shown on the wide screen it might cause vertigo and motion sickness.
From Vectorsite.net.
