## Intro

Pretty much everything in space that is not a beam of electromagnetic radiation or a torchship moves in an orbit. What is relevant to our interests is this class includes spacecraft, space stations, satellites, moons, and planets. Using orbits is critical for flying your spacecraft from planet A to planet B.

What is an orbit? That elliptical path traced by a secondary object (i.e., Luna) as it moves around a primary object (i.e., Terra).

If you jump out a window of a skyscraper, or out of an aircraft at high altitude, you will fall down. You will be in a wonderful state of free-fall…until you hit the ground.

An orbit is a clever way to constantly fall towards a planet but never hit the ground. Orbits are the sine qua non of space stations and communication satellites.

Pretty much all natural orbits are ellipses, though many look like circles to the naked eye (that was Kepler's valuable contribution to rocket flight). To the naked eye it also appears like the center of the secondary object's orbit is the center of the primary object, but this too is wrong. Ellipses have two focal points instead of one center, and one is parked on the barycenter of the primary-secondary system. It is just that the barycenter is typically quite close to the center of the primary object, if not it is generally beneath the primary object's surface.

The odd term "eccentricity" when applied to ellipses and orbits is basically a measure of how far apart the two focal points are. It is a number from 0.0 to less than 1.0. If the eccentricity is zero, the two focii are in the same spot, and the ellipse is a special kind of ellipse called, hang on to your hat, a circle. The higher the eccentricity, the more oval the ellipse becomes. Extreme ellipses are downright cigar shaped. If the eccentricity becomes equal to 1.0, the ellpse turnes into a parabola. Higher than 1.0 and it becomes a hyperbola, aka an hyperbolic escape orbit.

## Parking Orbits

These are orbits around a planet or moon where you park spacecraft, satellites, and space stations. As opposed to "transfer orbits" which are used by spacecraft traveling from one planet to another.

There are certain preferred orbits.

An equatorial orbit is a non-inclined orbit with respect to Terra's equator (i.e., the orbit has zero inclination to the equator, 180° inclination if retrograde). Most civilian satellites use such orbits. The United States uses Cape Canaveral Air Force Station and the Kennedy Space Center to launch into equatorial orbits.

An ecliptic orbit is a non-inclined orbit with respect to the solar system ecliptic.

An inclined orbit is any orbit that does not have zero inclination to the plane or reference (usually the equator).

A polar orbit is a special inclined orbit that goes over each pole of the planet in turn, as the planet spins below (i.e., the orbit is inclined 90° to the equator). Heinlein calls it a "ball of yarn" orbit since the path of the station resembles winding yarn around a yarn ball. The advantage is that the orbit will eventually pass over every part of the planet, unlike other orbits. Such an orbit is generally used for military spy satellites, weather satellites, orbital bombardment weapons, and Google Earth. The United States uses Vandenberg Air Force Base to launch into polar orbits. Google Earth uses data from the Landsat program, whose satellites are launched from Vandenberg.

Orbits around Terra (geocentric) are sometimes classified by altitude above Terra's surface (which is 6.37×103 km from Terra's center):

• Low Earth Orbit (LEO): 160 kilometers to 2,000 kilometers. At 160 km one revolution takes about 90 minutes and circular orbital speed is 8 km/s. Affected by inner Van Allen radiation belt.
• Medium Earth Orbit (MEO): 2,000 kilometers to 35,786 kilometers. Also known as "intermediate circular orbit." Commonly used by satellites that are for navigation (such as Global Positioning System aka GPS), communication, and geodetic/space environment science. The most common altitude is 20,200 km which gives an orbital period of 12 hours.
• Geosynchronous Orbit (GEO): exactly 35,786 kilometers from surface of Terra (42,164 km from center of Terra). One revolution takes one sidereal day, coinciding with the rotational period of Terra. Circular orbital speed is about 3 km/s. It is jam-packed with communication satellites like sardines in a can. This orbit is affected by the outer Van Allen radiation belt.
• High Earth Orbit (HEO): anything with an apogee higher than 35,786 kilometers. If the perigee is less than 2,000 km it is called a "highly elliptical orbit."
• Lunar Orbit: Luna's orbit around Terra has a pericenter of 363,300 kilometers and a apocenter of 405,500 kilometers.
• Ultra-Cautious Hill Sphere: 496,540 kilometers from surface of Terra (498,670 km from center)
• Long Term Stable Hill Sphere: 744,820 kilometers from surface of Terra (748,000 km from center)
• Ultimate Hill Sphere: exactly 1,489,630 kilometers from surface of Terra (1,496,000 km from center)

Geosynchronous Orbits (aka "Clarke orbits", named after Sir Arthur C. Clarke) are desirable orbits for communication and spy satellites because they return to the same position over the planet after a period of one sidereal day (for Terra that is about four minutes short of one ordinary day).

A Geostationary Orbit is a special kind of geosynchronous orbit that is even more desirable for such satellites. In those orbits, the satellite always stays put over one spot on Terra like it was atop a 35,786 kilometer pole (remember: 42,164 km from center of Terra). For complicated reasons all geostationary orbits have to be over the equator of the planet. In theory only three communication satellites in geostationary orbit and separated by 120° can provide coverage over all of Terra.

All telecommunication companies want their satellites in geostationary orbit, but there are a limited number of "slots" available do to radio frequency interference. Things get ugly when you have, for instance, two nations at the same longitude but at different latitudes: both want the same slot. the International Telecommunication Union does its best to fairly divide up the slots.

The collection of artificial satellites in geostationary orbit is called the Clarke Belt.

Note that geostationary communication satellites are marvelous for talking to positions on Terra at latitude zero (equator) to latitude plus or minus 70°. For latitudes from ±70° to ±90° (north and south pole) you will need a communication satellite in a polar orbit, a highly elliptical orbit , or a statite. Russia uses highly eccentric orbits since those latitudes more or less define Russia. Russian communication satellites commonly use Molniya orbits and Tundra orbits.

About 300 kilometers above geosynchronous orbit is the "graveyard orbit" (aka "disposal orbit" and "junk orbit"). This is where geosynchronous satellites are moved at the end of their operational life, in order to free up a slot. It would take about 1,500 m/s of delta V to de-orbit an old satellite, but only 11 m/s to move it into graveyard orbit. Most satellites have nowhere near enough propellant to deorbit.

Lagrangian points are special points were a space station can sit in a sort-of orbit. Lagrange point 1, 2, and 3 are sort of worthless, since objects there are only in a semi-stable position. The ones you always hear about are L4 and L5, because they have been popularized as the ideal spots to locate giant space colonies. Especially since the plan was to construct such colonies from Lunar materials to save on boost delta V costs. The important thing to remember is that the distance between L4 — Terra, L4 — Luna, and Terra — Luna are all the same (about 384,400 kilometers). Meaning it will take just as long to travel from Terra to L4 as to travel from Terra to Luna.

For a more exhaustive list of possible Terran orbits refer to NASA.

It is also possible for a satellite to stay in a place where gravity will not allow it. All it needs is to be under thrust. Which is rather expensive in terms of propellant. Dr. Robert L. Forward noted that solar sails use no propellant, so they can hold a satellite in place forever (or at least as long as the sun shines and the sail is undamaged). This is called a Statite.

If the planet has an atmosphere and the station orbits too low, it will gradually slow down due to atmospheric drag. "Gradually" up to a point, past the tipping point it will rapidly start slowing down, then burn up in re-entry. Some fragments might survive to hit the ground.

The "safe" altitude varies, depending upon the solar sunspot cycle. When the solar activity is high, the Earth's atmosphere expands, so what was a safe altitude is suddenly not so safe anymore.

NASA found this out the hard way with the Skylab mission. In 1974 it was parked at an altitude of 433 km pericenter by 455 km apocenter. This should have been high enough to be safe until the early 1980's. Unfortunately "should" meant "according to the estimates of the 11-year sunspot cycle that began in 1976". Alas, the solar activity turned out to be greater than usual, so Skylab made an uncontrolled reentry in July 1979. NASA had plans to upgrade and expand Skylab, but those plans died in a smoking crater in Western Australia. And a NOAA scientist gave NASA a savage I Told You So.

The International Space Station (ISS) orbited at an even lower at 330 km by 410 km during the Space Shuttle era, but the orbit was carefully monitored and given a reboost with each Shuttle resupply mission. The low orbit was due to the Shuttle carrying up massive components to the station.

After the Shuttle was retired and no more massive components were scheduled to be delivered, the ISS was given a big boost into a much higher 381 km by 384 km orbit. This means the resupply rockets can carry less station reboost propellant and more cargo payload.

If the planet the station orbits has a magnetic field, it probably has a radiation belt. Needless to say this is a very bad place to have your orbit located, unless you don't mind little things like a radiation dosage of 25 Severts per year.

There are known radiation belts around Terra, Jupiter, Saturn, Uranus and Neptune.

## Transfer Orbits

These are orbits used by spacecraft to travel from one planet to another. As opposed to "parking orbits" where spacecraft, satellites, and space stations circle a planet.

Spacecraft orbital manuevers are gone into in more detail elsewhere. But here are the basics.

ASTROGATION GLOSSARY

• Apoapsis In an orbit, the point of the orbit farthest from the astronomical body currently being orbited.
• Periapsis In an orbit, the point of the orbit closest to the astronomical body currently being orbited.
• Prograde In the direction of the spacecraft's trajectory, i.e., "forwards". Fun fact: since the trajectory is curved, prograde is actually at a tangent to the trajectory.
• Retrograde In the opposite direction of the spacecraft's trajectory, i.e., "backwards". 180 degrees from Prograde.

For instance, say that the good ship Polaris is given orders to travel from Terra to Mars. Refer to the diagram.

Terra's orbit is in light blue. The astronomical body being orbited is Sol. The orbit has a periapsis at point PA, and an apoapsis at point AA1. Terra is currently sitting on PA. Since the Polaris is orbiting Terra, the ship shares Terra's orbit.

By thrusting in a certain direction (burning prograde at periapsis) it is possible to raise the Polaris' orbital apoapsis from AA1 to AA2 (red arrow), that is, so that the apoapsis is at Mars (actually, where Mars will be in eight month's time, but we won't get into that). The periapsis will stay put at PA but the rest of the orbit will elongate.

The new orbit (the Transfer Orbit) is shown in green. The most important type of transfer orbit is called a Hohmann, but there are others.

The Polaris then coasts on its transfer orbit for the next 8.6 months. Both the Polaris and Mars will rendezvous at AA2 at that time. Because, like with all Hohmann transfers, the ship does not launch until the destination planet is in the proper position. That position being Mars 8.6 months prior in its orbit from point AA2.

At the rendezvous, if the Polaris does nothing, it will go sailing past Mars and continue on, as shown in the dark green path. But we don't want that. The Polaris wants to transform its orbit from the transfer orbit (green) into the Mars solar orbit (white ring). It's current periapsis is still at PA. By doing another burn in a certain direction (burning prograde at apoapsis) the periapsis can be raised from PA until it is on Mar's orbit. The Polaris will have matched the Martian solar orbit. This is called circularizing the orbit.

It is a bit more compilicated than that, but you get the main idea. Complications include the fact that the Polaris will end up orbiting Sol in the same orbit as Mars. It will then need another burn to change into an orbit around Mars. And another burn to actually land on Mars.

The initial burn at Terra is called the Trans-Martian Injection (TMI) burn. The second burn to circularize the orbit is called the Mars Orbit Insertion (MOI) burn. You change the planet name according to the target planet. For instance when the Polaris wants to go home, it does a Trans-Earth Injection (TEI) burn and an Earth Orbit Insertion (EOI) burn at arrival.

Another example: the proposed orbital transfers of the Orion Bomber.

• At A the Orion Bomber boosts into LEO (370 km) with solid rockets and Orion drive. The crew does a systems checkout.
• At B burns into a Hohmann transfer orbit (blue arc)
• At transfer apogee C it burns to circularize the orbit. The Orion Bomber is now in a 190,000 km circular orbit (green circle)
• At D burns to enter Patrol orbit (red ellipse). Orbit has a perigee of 190,000 km and apogee of 410,000 km (a 190,000-410,000 km Terran orbit). The orbital period is 18.9 days

Twenty bombers would be inserted into the Patrol orbit. With an orbital period of 18.9 days, this means one bomber would pass through perigee every 22.7 hours. One bomber carries 25 MIRVs, each containing three city-killer nuclear warheads. Thus would Mutual Assured Destruction be maintained.

## Circular Orbital Periods

How long it takes a space station or ship to make one orbit depends upon how massive the planet is and the altitude of the orbit. The mass of the station doesn't matter. The equations below are for a circular or near-circular orbit (low eccentricity) and where the mass of the planet is much larger than the mass of the space station (which is always the case unless the station is built out of stellar black holes or something). The equations for elliptical orbits are a bit more complicated.

An orbit with an orbital period exactly equal to the planetary rotation (one planetary "day") is highly prized for communication satellites.

OrbitalPeriod = (2 * π) * sqrt[ OrbitalRadius3 / (G * PlanetMass) ]

OrbitalVelocity = sqrt[ (G * PlanetMass) / OrbitalRadius ]

OrbitalRadius = cubeRoot[ (G * PlanetMass * OrbitalPeriod2) / (4 * π2) ]

μ = G * PlanetMass

μTerra = 3.99×1014

OrbitalPeriodTerra = 6.28318 * sqrt[ (6.37×106 + OrbitalAltitude)3 / 3.99×1014 ]

OrbitalVelocityTerra = sqrt[ 3.99×1014 / OrbitalRadius ]

OrbitalRadiusTerra = cubeRoot[ (3.99×1014 * OrbitalPeriodTerra2) / 39.478 ]

where:

OrbitalRadius = distance from station to center of planet (m)
OrbitalAltitude = distance from station to surface of planet (m)
PlanetRadius = distance from center of planet to planet's surface (m) (Terra = 6.37×106 m)
OrbitalPeriod = time it takes station to make one orbit around the planet (sec)
OrbitalVelocity = mean velocity of station in its orbit (m/s)
π = pi = 3.14159...
G = Newton's gravitational constant = 6.673×10-11 (N m2 kg-2)
PlanetMass = mass of planet (kg) (Terra = 5.98×1024 kg)
μ = standard gravitational parameter
sqrt[ x ] = square root of x
cubeRoot[ x ] = cube root of x

## Elliptical Orbital Periods

Just so you know, when it comes to planetary orbits and spacecraft trajectories, none of them are perfectly circular. It is just that so many of them are close enough to being a circle that a science fiction author can get away with using the above equations. We call them Kepler's laws of planetary motion because Kepler found that the equations worked if you assumed the planet orbits were ellipses (which are eccentric circles). Kepler's boss Tycho Brahe was dumped in the dust-bin of history because he stubbornly insisted that planet orbits were perfect circles.

And when you get to things like spacecraft transfer orbits, some are not even close to being circular.

What you have to do is use the orbiting object's Semi-Major Axis instead of OrbitalRadius.

Don't panic, it is easy to calculate. As long as you have the object's Periapsis and Apoapsis (in meters), which means the object's closest approach and farthest retreat from the planet it is orbiting. Those numbers are easy to find, for example see Wikipedia's entry for the Moon. Periapsis (called perigee) of 362,600 kilometers and Apoapsis (called apogee) of 405,400 kilometers, right in the data bar on the right. Don't forget to convert the values to meters for the equations, e.g., multiply 362,600 km by 1,000 to convert to 362,600,000 m.

Sometimes an orbit will actually be specified by the periapsis and apoapsis. For instance the Orion bomber's patrol orbit is described as a 190,000-410,000 km Terran orbit.

Given periapsis and apoapsis in meters, the Semi-Major Axis is:

SemiMajorAxis = (Periapsis + Apoapsis) / 2

Take the equation for OrbitalPeriod, replace OrbitalRadius with SemiMajorAxis, and you are good to go.

OrbitalVelocity unfortunately is a major pain. You see, the orbiting object moves at different speeds at different parts of its orbit. It moves fastest at periapsis and slowest at apoapsis. Only if the orbit is perfectly circular does the orbiting object always move at the same speed.

If you use the OrbitalVelocity equation replacing OrbitalRadius with SemiMajorAxis, you will get the Mean or Average orbital velocity.

If you want the orbital velocity at a specific point in the orbit, you will specify said point by its distance from the primary. The distance will be somewhere between periapsis and apoapsis, inclusive. Again it will be fastest at periapsis and slowest at apoapsis. The equation is:

OrbitalVelocity = sqrt[ (G * PlanetMass) * ( (2 / CurrentOrbitalRadius) - (1 / SemiMajorAxis) ) ]

This is the famous Vis Viva Equation, which comes in real handy to calculate delta-V requirements for various missions.

If for some reason you want to draw the orbit, it isn't too hard. As long as have a drawing program that can create an ellipse given a bounding box (The Gimp, Inkscape, Adobe Photoshop, Adobe Illustrator). First you calculate the semi-major axis and the semi-minor axis.

SemiMajorAxis = (Periapsis + Apoapsis) / 2

SemiMinorAxis = sqrt[ SemiMajorAxis2 - ( SemiMajorAxis - Periapsis )2 ]

Chose a convenient scale for your drawing program, like 1,000,000 meters equals one pixel. Draw the upper and lower sides of the box (red lines in diagram above) which are twice the length of the SemiMajorAxis in scale. Draw the left and right sides of the box (green lines) which are twice the length of the SemiMinorAxis in scale. That is the bounding box.

Draw a horizonal center line so it is equidistant from the top and bottom edges of the box. Draw a vertical line (blue in diagram above), and move it so it is one Periapsis scale length away from the right edge. Where these two lines cross is the location of the planet.

Move the box so the cross-hairs are on the planet image. Use the "draw ellipse" function of the drawing program such that the ellipse fits in the bounding box. That is the orbit. You can erase the bounding box now, you don't need it any more.

## Hill Sphere

It is greatly desired that satellites and space stations stay in stable orbits, because corporations and insurance companies become quite angry if hundred million dollar satellites or expensive space stations with lots of people are gravitationally booted into The Big Dark.

A good first approximation is ensuring that the orbiting object stays inside the parent's Hill Sphere. This is an imaginary sphere centered on the parent planet (the planet or moon the satellite is orbiting). Within the sphere, the planet's gravity dominates any satellites.

For first approximation you have three players: the space station (e.g., Supra-New York), the planet or moon it is orbiting (e.g., Terra), and the object the planet is orbiting (e.g., Sol) otherwise known as the planet's "primary".

The point is that Sol cannot gravitationally capture Supra-New York as long as all of the space station's orbit is inside Terra's Hill Sphere.

You can calculate the approximate radius of a planet's Hill Sphere with the following equation:

r ≈ a * cbrt( m / (3 * M) )

where:

r = Radius of Hill Sphere (kilometers)
a = Distance between the planet and its primary (kilometers)
m = mass of the planet (kilograms)
M = mass of the primary (kilograms)
cbrt(x) = cube root of x (the ∛x key on your calculator)

Actually you can use any desired unit of distance for r and a as long as you use the same for both. The same goes for units of mass for m and M.

This equation assumes that the planet is in a near-circular orbit. If it has some weird eccentric orbit the Hill Sphere link has the more complicated equation. The above equation also assumes that the mass of the station or sattelite is miniscule compared to the object it is orbiting. It further assumes that the mass of the primary is quite a bit bigger than the mass of the planet.

In practice, for long term stability, the station should not orbit its planet further than one-half the Hill sphere radius. No further than one-third the Hill sphere radius if you are ultra-cautious.

If you were interested in Lunar satellites, the planet would be Luna, the primary would be Terra, and a would be the distance between Terra and Luna.

Any object (like a spaceship) which enters a planet's Hill sphere but does not have enough energy to escape, will tend to start orbiting the planet. The surface of the Hill sphere is sometimes called the "zero-velocity surface" for complicated reasons.

## Tides

Most people only know tides with respect to the waters at the beach perodically rising and falling, with it having something to do with Luna orbiting Terra.

But it is much more general than that. And it can be much more deadly.

Tides can create tidal locking, which is why one face of Luna always faces Terra. In Isaac Asimov's FOUNDATION AND EMPIRE the planet Radole was tidally braked; with a too-hot sun-side, too-cold night-side, and a narrow bad of just-right temperature at the twilight zone (Asimov called them "ribbon worlds").

Tides can make planet elongated. Examples from science fiction include Jinx from Larry Niven's THE BORDERLAND OF SOL and the double planet Rocheworld from the eponymous novel by Robert L. Forward.

If a moon moves within the Roche limit of its primary, tides will rip it apart. That's how Saturn's ring was formed.

And if you get too close to a celestial object with a real fierce gravitation gradient, spaghettification is your doom.

### Spaghettification

Now that you've seen the horrible things capable of the tides of a massive planet or neutron star, you can just imagine how much worse it is around a black hole.

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