[5.0] Gravity & Orbital Mechanics

v3.2.3 / chapter 5 of 14 / 01 aug 16 / greg goebel

* The most elegant and simple way to observe the action of gravity is to show how it defines the movements of objects in space. In fact, it was studies of the orbits of the planets that led Isaac Newton to devise his formula for gravity in the first place. This chapter gives a short introduction to "orbital mechanics".



* As established in a previous chapter, gravity is the only one of the four forces that operates over long distances. It is an attraction between two masses and its magnitude is given by:


   gravitational_force  =  constant * mass1 * mass2 / distance^2

-- where the constant has a value of 6.672E-11 in standard metric units and the distance is between the center of mass of the two objects. The precise value of the very small constant has been determined by painfully exact experiments, in which a pair of masses are balanced in a vacuum on a bar supported by a wire thread, and allowed to pivot towards larger fixed masses. This experiment was originally performed by the English scientist Henry Cavendish (1731:1810) in 1771, and has been repeated since then to ever greater levels of precision.

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".

* The workings of gravity can clearly considered by visualizing the motion of an artificial satellite in orbit around the Earth. The orbit of such a satellite is defined by the balance of the satellite's orbital velocity against the force of the Earth's gravitational attraction. This force is given by:

   gravitational_force  =  constant * mass_earth * mass_satellite / distance^2

The mass of the satellite is irrelevant. Double the mass of the satellite and the force of the Earth's attraction on it is doubled, but the amount of force required to get the increased mass to accelerate to a given velocity is doubled as well. Since force is equal to mass times acceleration, then:

   gravitational_acceleration  =  constant * mass_earth / distance^2

The acceleration of gravity at the surface of the Earth is 9.81 meters per second squared, and so this formula can be simplified further, to:

   gravitational_acceleration  =  9.80665 / ( orbital_radius / earth_radius )^2

The orbital radius is measured in meters from the center, not the surface, of the Earth. The radius of the Earth is 6,340,000 meters (6,340 kilometers). This means that the gravitational acceleration of a satellite in an orbit 1,000 kilometers above the surface of the Earth would be:

   9.81 / ( ( 6,378,000 + 1,000,000 ) / 6,378,000 )^2  
    = 7.33 meters per second squared

In a stable circular orbit, this force is balanced by the centripetal acceleration of the satellite due to its orbital velocity, which is given by:

   centripetal_acceleration  =  orbital_radius * angular_velocity^2

Balancing the two expressions gives:

   9.81 / ( orbital_radius / earth_radius )^2 
    = orbital_radius * angular_velocity^2

-- or:

    orbital_radius^3  =  ( 9.81 * earth_radius^2 ) * --------------------

This equation is a little easier to handle if angular velocity is converted into "orbital period", or the amount of time it takes for a satellite to make one orbit of the Earth, in hours. 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:

    angular_velocity  =  ------------------------
                          3,600 * orbital_period

Plugging this into the equation gives:


     =  ( 9.81 * earth_radius^2 ) * ------------------------------------
                                     ( 2PI / 3,600 * orbital_period )^2

         3,600^2 * 9.81 * earth_radius^2
     =  --------------------------------- * orbital_period^2

Giving the orbital radius in meters is also inconvenient, but that can be adjusted to give it in kilometers by dividing through by 1,000^3:

                         3,600^2 * 9.81 * earth_radius^2
   orbital_radius^3  =  --------------------------------- * orbital_period^2
                                1,000^3 * 2PI^2

This leads to a very simple relationship:


   orbital_radius^3  =  constant * orbital_period^2

This is known in general as "Kepler's Third Law". Kepler's three laws are a description of planetary orbits devised by the German astronomer Johannes Kepler (1571:1630) from observations of such orbits. They strongly influenced the development of Newton's law of universal gravitation. The other two laws are discussed in later sections of this chapter.

Kepler's Third Law is true for any planetary orbit. In the case of an Earth satellite, the value of the constant is 130,958.995396 (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

-- with the orbital period in hours and the orbital radius in kilometers.



* Kepler's Third Law can be used to calculate simple satellite orbits. For example, for a satellite to have a period of 2 hours, it will have an orbital radius of:

   5,078 * 2^(2/3)  =  8,060 kilometers

This corresponds to an altitude of 8,030 - 6,340 = 1,690 kilometers. Similarly, at an altitude of 19,000 kilometers, or three Earth radii, the satellite has a period of:

   ((6,378 + 19,000) / 5,078 )^1.5  =  11.2 hours

The fact that a higher orbit is a slower orbit, at least in terms of angular velocity and period, was confusing to astronauts performing the first space rendezvous operations in the mid-1960s. They would be trying to catch up with a target spacecraft, but simply firing their rockets directly ahead would push them into a higher orbit, where they would find themselves falling behind the target. They were all originally trained as pilots; it was nothing like flying an aircraft and was completely contrary to their instincts. They adjusted quickly, learning to direct their boost in an angle below the line to the target to modify their orbital direction as well as their speed.

An orbit with a period of 24 hours has an interesting property, in that this is the same as the rotation period of the Earth, and so the satellite "hangs" over one spot on Earth. There is a minor trick in calculating this because the motion of the Earth around the Sun makes the day about 4 minutes longer than it would be otherwise. The "sidereal" day relative to the stars is 23.93 hours. That means the geostationary altitude is:

   5,078 * 23.93^(2/3)  =  42,172 kilometers  

-- or an altitude of about 35,800 kilometers. Of course, since satellites orbit around the center of mass of the Earth, such a "geosynchronous" or "geostationary" orbit has to be over the equator, or the satellite will weave north and south of the equator during its 24-hour orbit. Geostationary orbits are very handy for some types of satellites, such as wide-area 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. Satellite TV users have to focus the 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 plane of the orbits of geostationary satellites around the equator are at an angle to the plane of the Earth's orbit as well. This means that the Sun's pull tends to tug the satellites 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 can be placed in variations on geostationary orbits. The Japanese "Michibiki" satellite, launched in 2010, was intended to augment the American Global Positioning System (GPS) satellite navigation network. The GPS satellites provide coded timing signals that allow a receiver to determine its location on Earth, but GPS signals are not always easy to pick up Japan's mountainous regions or skyscraper-studded cities like Tokyo. Michibiki was placed in an orbit that varied from above to below the geostationary distance, with the orbital plane at 45 degrees to the equator. Given this orbit, instead of hanging over the equator at a fixed location, it performed a daily figure-8 loop from Australia in the south to Japan in the north. The higher and smaller lobe of the figure-8 was over Japan, with the result that Michibiki remained high in the sky there for eight hours of its orbit, providing GPS augmentation signals to locations where GPS reception was otherwise spotty. Three more satellites are to be launched by 2018 to provide 24 hour a day coverage.

* Not all satellites are put into geostationary orbit, of course. 35,800 kilometers is a long way from Earth. Communications satellites that have to pick up signals from small ground transmitters that don't have the power to reach geostationary orbit are implemented as "constellations" of small satellites in fast low Earth orbits (LEO), with enough of the satellites in orbit to make sure one that when one goes down over the far horizon, another has come up over the near horizon to make sure communications remain constant.

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 strictly polar orbit might 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, with 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 positions and other lighting conditions 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; if shadows suddenly appear in a frame taken by a military reconnaissance satellite and weren't in the previous frame, it might mean that equipment has been moved into that location since the last pass over the target.

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, instead of 81.8 degrees, to emphasize that the satellite had a "retrograde" orbit, that is, in the reverse direction to the Earth's rotation, as opposed to "prograde" or in the same direction of the Earth's rotation. SAC-C passed over the equator at 10:15 AM local time during each orbit.

* The orbits described so far are circular, but there's also no reason that they can't be elliptical, dropping down close to the Earth at one part of its orbit, and then rising very high over the Earth at another part of its orbit.

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 minimum 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.

Incidentally, the terms "perigee" and "apogee" specifically refer to an orbit around the Earth. For an orbit around the Sun, the terms "perihelion" and "aphelion" are used; for an orbit around a star, the terms "periastron" and "apoastron" are used; and if the central body isn't specified, the terms "periapsis" and "apoapsis" are used.

In any case, the behavior of an object in an elliptical orbit 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 since 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.

Kepler's Second Law

Elliptical orbits were particularly useful to the Soviet Union. The USSR was at high latitudes, 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 with the name of "Molniya (Lightning)" 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 eavesdrop 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.

* Although everyone knows an ellipse is a "less than perfect" circle, it is formally defined as the points on a closed curve where the sum of the distances from two "focal points" within the curve to one of the points on the curve are a constant.

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. The semi-major axis is often used as the "average value" of an orbital radius, which makes sense since it's the average of the maximum and minimum distance to the central planet or other body.

the ellipse

Suppose that a hypothetical moon of some equally hypothetical large planet has an apoapsis of 7.5 million kilometers and a periapsis of 6 million kilometers. This is a fairly big orbit and so the diameter of the planet is disregarded. This gives the semi-major axis as:

    ( 7.5 + 6 ) / 2  =  6.75 million kilometers

The distance from the center of the orbit to the focal point where the planet resides is:

   7.5 - 6.75  =  0.75 million kilometers  = 750,000 kilometers

This gives the eccentricity of the orbit as:

   0.75 / 6.75  =  0.11

Similarly, suppose another moon of the same planet is said to have an orbit with an average value of 9 million kilometers and an eccentricity of 0.15. Interpreting the average value as meaning the semi-major axis, then the distance from the center to a focal point is:

   9 * 0.15  =  1.35 million kilometers 

This gives the apoapsis as:

   9 + 1.35  =  10.35 million kilometers

-- and the periapsis as:

   9 - 1.35  =  7.65 million kilometers

Of course, the orbital inclination of the moons relative to the planet's equator also has to be specified to define their orbits. Incidentally, moons that were created along with a planet usually have prograde, nearly circular orbits in the plane of the planet's equator, while moons that were captured later usually have strongly elliptical orbits that can be prograde or retrograde, at almost any inclination to the planet's equator.

How the moons were captured later is a bit of a puzzle, since if they fell in towards a planet from deep space and didn't hit it, by energy conservation they would simply swing back out to deep space from whence they came and not go into orbit around the planet. It is believed that planets with captured moons were surrounded by an envelope of gas and dust in their early days that imposed enough drag on the moons to capture them.

Also incidentally, there is a simple relationship between the semi-major axis, semi-minor axis, and distance from the center to a focal point (FP):

    semi-major_axis^2  =  semi-minor_axis^2  +  distance_to_FP^2

* The best place to launch an Earth satellite into a geostationary orbit is from a launch site near the equator, such as the European Space Agency's space center in Kourou, French Guiana, on the northern coast of the South American landmass. An equatorial launch not only minimizes the amount of fuel needed to bring the satellite into equatorial orbit, but the high linear velocity of the Earth's rotation at the equator gives a launcher an extra "boost" into space.

The Earth also has a slight "bulge" at the equator, with the equatorial diameter being about 42 kilometers more than the polar diameter, though this is a minor consideration in space launches. However, one implication of the irregular shape of our planet is that the standard value of the Earth's gravitational acceleration, 9.81 meters per second squared, varies a bit depending on location -- from about 9.83 at the North Pole to about 9.78 on the Equator. On top of Mount Everest, it actually drops down to almost 9.76.

After launch, a geostationary satellite is placed into a highly elliptical temporary "geostationary transfer orbit" with the apogee at 35,800 kilometers. Once the satellite reaches that altitude, it fires a rocket engine to "circularize" its orbit and not fall down to its perigee again. The concept of transfer orbits was described by the German space pioneer Walter Hohmann (1880:1945) in 1925, and so this is an example of a "Hohmann transfer orbit".

An equatorial launch site is the worst choice for a polar orbit, since the launcher's upper stage must expend fuel to turn the spacecraft's orbit towards the poles. The launcher will have to expend additional fuel to provide the change in velocity, or "delta vee", to get the payload into its proper orbit, and the weight of the additional fuel means a smaller payload weight.

A high-latitude space center is better for launching such spacecraft. The two main Russian launch centers are Baikonur in Kazakhstan in Central Asia and Plesetsk in northern Russia, northeast of Saint Petersburg; Baikonur is used to launch payloads into geostationary orbit, while Plesetsk is used to launch payloads into polar orbit. It is not impossible to launch a geostationary payload from high latitudes, nor is it impossible to launch a polar payload from low latitudes, but it is inefficient.

* As this explanation of satellite orbits shows, they are carefully planned and well defined. This brings up as a counterexample the memory of a thriller novel I read decades ago where the author had a spy satellite moving around sideways relative to its current orbit and then stopping over a particular target to take pictures. Many low-orbit spy satellites can in fact maneuver, but only to adjust their orbital paths to eventually take them over specific targets.

Obviously, the author was under the popular misconception that the force of gravity suddenly stops working when a spacecraft gets above the atmosphere. The reality is that the spacecraft is falling all the time, it's just that its forward velocity is great enough to allow it to fall over the horizon before it hits ground. If it stopped in orbit, it would drop like a brick. Considering that I don't remember much else about the novel, at least the author gave me a bit of unintentional humor to remember him by.



* Although the Earth's gravity well goes on forever in principle, that doesn't mean it takes infinite energy for a spacecraft to leave the Earth: one of the elementary but subtle truths of mathematics is that many curves that go on forever actually may be summed up to a specific value. If a spacecraft is launched at a high enough velocity, called an "escape velocity", it will have enough energy to ensure that it will never be pulled back into the Earth's gravity well again. The escape velocity for Earth is 11.2 kilometers per second.

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 interplanetary transfer orbit". To reach Mars, the probe must be given a delta-vee that increases its velocity 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 "nonlinear": 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 are always in the form of a "conic section".

conic sections

A conic section is one of 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 (eccentricity = 0) 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 ellipse (eccentricity < 1) that becomes more elongated as the angle increases. Once the angle becomes exactly that of the slope of the cone, the ellipse becomes an open figure known as a "parabola" (eccentricity = 1). For steeper angles up to the point where the cut is parallel to the axis, the open figure widens into a series of "hyperbolas" (eccentricity > 1).

The movements of celestial bodies are never precise circles or parabolas, these being "perfect" figures; in practice they are ellipses when they are in repeating orbits around a body, and hyperbolas when falling in from deep space on a nonrepeating orbit.

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.

* When the probe arrives at the target planet, it will generally have so much velocity that it will swing by the planet and continue on it orbit around the Sun. Traditionally, the probe has to fire its rocket engine to reduce its velocity enough to allow it to be captured by the planet's gravity. This requires a fair amount of fuel, and a more economical approach known as "aerocapture" has been considered. In this scheme, the probe is fitted with a heatshield and actually skims through the upper atmosphere of the planet, presuming it has one, to literally "burn off" the excess velocity using friction.

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. So far no spacecraft has successfully performed "aerocapture", or skimming a planet's atmosphere to reduce velocity enough to be captured by the planet's gravity.

* Spaceflight enthusiasts have used transfer orbit calculations to come up with an ingenious idea known as the "cycler". This is a space station that is in a permanent transfer orbit between, say, Earth and Mars. The cycler would pass close by Earth and later pass close by Mars on a regular schedule. Astronauts would be able to hitch a ride on the cycler station to go to Mars, and later pick up the cycler again to return to Earth. 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.



* A simple Hohmann interplanetary transfer orbit works well enough to send a probe to Mars or Venus, but sending a probe to the distant outer planets using such a trajectory would take far too much time. Space launch planners have developed an interesting and subtle way to "cheat" and greatly reduce the time and expense of such missions by sending the probe on a "slingshot" trajectory around a planet to obtain a "gravity assist".

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 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 a "soft collision", with the spacecraft "bouncing" off a planet and obtaining a delta-vee thereby.

* The concept of a gravity assist trajectory was pioneered in the 1960s by engineers at the US National Aeronautics & Space Administration's Jet Propulsion Laboratory (NASA JPL) in California. By a stroke of good luck, their work on gravity slingshots led them to realized that beginning in the late 1970s, all four outer gas giant planets would be strung out in a perfectly staggered formation beginning in the late 1970s, a configuration that only occurred once every 175 years. The new gravity assist technique could be used to slingshot a space probe from one planet to the next.

This "Grand Tour" trajectory was implemented on the Voyager 2 space mission, which was launched in 1977. The Voyager 2 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 not exceeded 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 canceled, and a much smaller rocket had to be used 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.

After one Galileo Earth flyby, JPL officials were startled to see protesters waving signs complaining about the way Galileo was robbing Earth of its orbital momentum. It quickly turned out the protesters were really just a gang of jokers who were being silly on purpose, partly to mock more earnest protesters who had previously objected to the launch of Galileo because its atomic-fueled electrical power units.

In an even more complicated session of gravity-assist flybys, the NASA MESSENGER Mercury orbiter probe, launched in 2005, spent the next five and a half years performing one Earth flyby, followed by two Venus flybys, and then three Mercury flybys before it was finally placed in Mercury orbit. The need to place the probe in orbit around the planet was the cause for the elaborate sequence of flybys.



* The orbital scenarios discussed so far have involved a great big planet and, in comparison, a very small spacecraft. The basic concepts aren't different when two planet-sized bodies are involved, but some additional rules come into play.

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 world Pluto and its large moon Charon, 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

The second additional rule is that the two bodies have "tidal" effects on each other, due to the differential effect of gravity across the diameter of a planet. The Moon and Earth have a strong tidal interaction on each other, manifested on Earth as the rise and fall of ocean coastal waters during the passage of a day.

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.

Since 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.

Tides also apply to the Earth's atmosphere, with atmospheric pressures varying (on the average) from minima to maxima in step with high and low tides. Oddly, the Sun has a greater effect on atmospheric tides than the Moon. This is because, as explained in a later chapter, warm gases tend to expand, and solar heating drives that expansion, magnifying the solar tidal effect.

Incidentally, while the explanation of tides given here follows that of a typical physics text, there is a faction that claims it is mistaken, that the Earth's tides are actually a result of the asymmetric rotation of the Earth around the Earth-Moon barycenter. The idea is a minority view and is not widely accepted.

* One of the interesting implications of the tidal forces on the Earth is that they cause precession of the Earth's spin axis. The Earth's spin axis is tilted at an angle of 23 degrees to the ecliptic, the plane of the planet's orbit around the Sun, while the Moon's orbit is in the plane of the ecliptic. The tidal forces of the Moon and Sun tug the spin axis of the Earth back towards the horizontal, but since the Earth is spinning this causes precession instead, with a period of about 26,000 years. At the present time, the Earth's spin axis points towards Polaris, the pole star, but 13,000 years from now it will point to a location near the bright star Vega.

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.

the Moon's constant face

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 laser reflectors 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.