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.

GEO SPACE STATION

"Okay, T.K., look at it this way. Those three hundred people in LEO Base can get back to Earth in less than an hour if necessary; we'll have lifeboats, so to speak, in case of an emergency. But out there at GEO Base, it's a long way home. Takes eight hours or more just to get back to LEO, where you have to transfer from the deep-space passenger ship to a StarPacket that can enter the atmosphere and land. It takes maybe as long as a day to get back to Earth from GEO Base— and there's a lot of stress involved in the trip."

Hocksmith paused, and seeing no response from the doctor, added gently, "We can get by with a simple first-aid dispensary at LEO Base, T.K., but not at GEO Base. I'm required by my license from the Department of Energy as well as by the regulations of the Industrial Safety and Health Administration, ISHA, to set up a hospital at GEO Base."

He finished off his drink and set the glass down. "If building this powersat and the system of powersats that follow is the biggest engineering job of this century, T.K., then the GEO Base hospital's going to be the biggest medical challenge of our time. It'll be in weightlessness; it'll have to handle construction accidents of an entirely new type; it'll have to handle emergencies resulting from a totally alien environment; it'll require the development of a totally new area of medicine— true space medicine. The job requires a doctor who's worked with people in isolated places—like the Southwest or aboard a tramp steamer. It's the sort of medicine you've specialized in. In short, T.K., you're the only man I know who could do the job . . . and I need you."


Stan and Fred discovered that it took almost nineteen minutes just to get to Charlie Victor, Mod Four Seven. There were a lot of hatches to go through and a lot of modules to traverse. "Fred, if we don't find some faster way to move around this rabbit warren, a lot of people are going to be dead before we reach them," Stan pointed out, finally opening the hatch to Mod Four Seven.

Fred was right behind him through the hatch. "I'll ask Doc to see Pratt about getting us an Eff-Mu."

"What's that?"

"Extra Facility Maneuvering Unit. A scooter to anybody but these acronym-happy engineers."


Transporting was easy in zero-g, but getting through all the hatches while continuing to monitor his condition and maintain the positive-displacement IVs was difficult. It required almost a half hour to bring the man back to the med module.

From SPACE DOCTOR by Lee Correy (G. Harry Stine) 1981

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.

OrbitalRadius = OrbitalAltitude + PlanetRadius

OrbitalAltitude = OrbitalRadius - PlanetRadius

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
Example

The planet Mars has a mass of 6.4171×1023 kg and a mean radius of 3,390,000 m. What is the orbital period of a station at an altitude of 20,073,000 meters?

OrbitalRadius = OrbitalAltitude + PlanetRadius
OrbitalRadius = 20,073,000 + 3,390,000
OrbitalRadius = 23,463,000 m

OrbitalPeriod = (2 * π) * sqrt[ OrbitalRadius3 / (G * PlanetMass) ]
OrbitalPeriod = (2 * 3.14159) * sqrt[ 23,463,0003 / (6.673×10-11 * 6.4171×1023) ]
OrbitalPeriod = 6.28318 * sqrt[ 12,916,671,713,847,000,000,000 / 42,821,308,300,000 ]
OrbitalPeriod = 6.28318 * sqrt[ 301,641,220.84628 ]
OrbitalPeriod = 6.28318 * 17,367.82142
OrbitalPeriod = 109,125 seconds = 1,818.8 minutes = 30.3 hours

For this problem I used the altitude of the Martian moon Phobos, which has an orbital period of 30.312 hours. Our result of 30.3 is close enough for government work.

The planet Terra has a mass of 5.98×1024 kg and a mean radius of 6.37×106 m. What is the orbital altitude of a station with an orbital period of 5,559 seconds?

OrbitalRadius = cubeRoot[ (G * PlanetMass * OrbitalPeriod2) / (4 * π2) ]
OrbitalRadius = cubeRoot[ (6.673×10-11 * 5.98×1024 * 5,5592) / (4 * π2) ]
OrbitalRadius = cubeRoot[ (6.673×10-11 * 5.98×1024 * 30,902,481) / (4 * 9.86961) ]
OrbitalRadius = cubeRoot[ 12,331,492,891,637,400,000,000 / 39.47844 ]
OrbitalRadius = cubeRoot[ 312,360,186,766,179,210,728.69140725925 ]
OrbitalRadius = 6,785,031 m

OrbitalAltitude = OrbitalRadius - PlanetRadius
OrbitalAltitude = 6,785,031 - 6.37×106
OrbitalAltitude = 415,031 m = 415 km

For this problem I used the orbital period of the International Space Station. It has a mean orbital altitude of 404.55 km, which is close to our calculated 415 km. I presume the discrepancy is due to the fact that the ISS has an inclination of 51.64°, while Deimos has an inclination of about 1°.

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.

CALCULATING ORBITAL POSITIONS

A few times a year I find myself confronting a table full of numbers describing the orbits of things in the solar system, and cursing at myself because I've forgotten, again, what all these numbers mean and how to manipulate them to get the particular numbers I want. In particular, despite the fact that determining perihelion and aphelion distances from semimajor axis length and eccentricity could hardly be easier, I still always draw a blank. So I'm sitting down now to write a blog entry that will tell me what these numbers mean and how to use them to get the numbers I want! I'm posting it because I figure it'll be useful for some of you, too. In the following post, I'll show you why I was interested in getting these numbers today.

Here we go. The shape of an elliptical orbit is described by two parameters:

  • semi-major axis, a: one half of the ellipse's long axis

  • eccentricity, e: 0 for circular orbits; between 0 and 1 for ellipses

To compute other numbers describing the shape of the orbit, here's what you do:

  • Periapsis distance = a(1-e)

  • Apoapsis distance = a(1+e)

  • Orbital period = 2π√(a3/GM)

  • Orbital period (solar orbit, in years, with a in AU) = a1.5
    (and recall that 1 AU = 149.60×106 km)

To figure out where an object currently is in space requires a few more pieces of information, including inclination, longitude of ascending node, et cetera.

But because of the particular way in which orbital parameters are usually reported, it's actually quite difficult to use these elements to determine a body's current position in space in a way that makes intuitive geometrical sense to me. (I'm a geologist, dangit, not an orbital mechanic!) It's much faster and easier to make JPL's HORIZONS system crunch the numbers. (It makes me feel slightly better that when I asked this question on Twitter, all three Kuiper belt astronomers who follow me said they use HORIZONS too.) You can use HORIZONS via a telnet or email interface but if you don't have too many things to calculate it's easiest just to use the web interface. So:

  • Go to http://ssd.jpl.nasa.gov/horizons.cgi

  • Change the Target Body to the one you're interested in (click "change" and search on the name or number or provisional designation)

  • Change Observer Location to "@sun"

  • Go to Table Settings and check "Helio eclip. lon & lat" (or set the list to "18,20" to get heliocentric lat/lon and range)

  • Click Generate Ephemeris and look for "hEcl-Lon" and "hEcl-Lat," which are in degrees, and "delta," which is the range in AU.

Note: It defaults to a one-month time span with a time step of one day, so the output table will have about thirty entries in it, starting today. The time span permits you to choose a wide range of dates (for most Kuiper belt objects, I find it to cover 1600 to 2200), and the time step chooser permits you to pick calendar years and calendar months as well as increments of days, hours, minutes, or seconds.

HORIZONS can also be used to find the distance between Earth and any of these objects, too, obviously; and you can have it spit the results directly to a text file, which is very handy!

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.

EXAMPLE

What is the Hill sphere radius of Terra?

In this case, the planet is Terra, the primary is Sol, the mass of Terra (m) is 5.97×1024 kg, the mass of Sol (M) is 1.99×1030 kg, the distance between Terra and Sol (a) is 149.6 million kilometers (149,600,000 km).

r ≈ a * cubrt( m / (3 * M) )
r ≈ 149,600,000 * cubrt( 5.97×1024 / (3 * 1.99×1030) )
r ≈ 149,600,000 * cubrt( 5.97×1024 / 5.97×1030 )
r ≈ 149,600,000 * cubrt( 0.000001 )
r ≈ 149,600,000 * 0.01
r ≈ 1,496,000 kilometers

So the ultimate Hill sphere radius is 1.496 million kilometers from Terra's center. Luna is at 0.384 million kilometers, safely inside the Hill sphere. The implication is that all of Terra's stable satellites have an orbital period of less than seven months. Supra-New York would do well to stay inside.

The long term stable radius is 0.748 million kilometers from Terra's center (1/2 Hill sphere). The ultra-cautious radius is 0.499 million kilometers (1/3 Hill sphere).

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.

TIDAL FORCE

The tidal force is a force that stretches a body towards and away from the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomena, including tides, tidal locking, breaking apart of celestial bodies and formation of ring systems within the Roche limit, and in extreme cases, spaghettification of objects. It arises because the gravitational field exerted on one body by another is not constant across its parts: the nearest side is attracted more strongly than the farthest side. It is this difference that causes a body to get stretched. Thus, the tidal force is also known as the differential force, as well as a secondary effect of the gravitational field.

In celestial mechanics, the expression tidal force can refer to a situation in which a body or material (for example, tidal water) is mainly under the gravitational influence of a second body (for example, the Earth), but is also perturbed by the gravitational effects of a third body (for example, the Moon). The perturbing force is sometimes in such cases called a tidal force (for example, the perturbing force on the Moon): it is the difference between the force exerted by the third body on the second and the force exerted by the third body on the first.

Explanation

When a body (body 1) is acted on by the gravity of another body (body 2), the field can vary significantly on body 1 between the side of the body facing body 2 and the side facing away from body 2. Figure 4 shows the differential force of gravity on a spherical body (body 1) exerted by another body (body 2). These so-called tidal forces cause strains on both bodies and may distort them or even, in extreme cases, break one or the other apart. The Roche limit is the distance from a planet at which tidal effects would cause an object to disintegrate because the differential force of gravity from the planet overcomes the attraction of the parts of the object for one another. These strains would not occur if the gravitational field were uniform, because a uniform field only causes the entire body to accelerate together in the same direction and at the same rate.

Size and distance

The relationship of an astronomical body's size, to its distance from another body, strongly influences the magnitude of tidal force. The tidal force acting on an astronomical body, such as the Earth, is directly proportional to the diameter of that astronomical body and inversely proportional to the cube of the distance from another body producing a gravitational attraction, such as the Moon or the Sun. Tidal action on bath tubs, swimming pools, lakes, and other small bodies of water is negligible.

Figure 3 is a graph showing how gravitational force declines with distance. In this graph, the attractive force decreases in proportion to the square of the distance, while the slope relative to value decreases in direct proportion to the distance. This is why the gradient or tidal force at any point is inversely proportional to the cube of the distance.

The tidal force corresponds to the difference in Y between two points on the graph, with one point on the near side of the body, and the other point on the far side. The tidal force becomes larger, when the two points are either farther apart, or when they are more to the left on the graph, meaning closer to the attracting body.

For example, the Moon produces a greater tidal force on the Earth than the Sun, even though the Sun exerts a greater gravitational attraction on the Earth than the Moon, because the gradient is less. The Moon produces a greater tidal force on the Earth, than the tidal force of the Earth on the Moon. The distance is the same, but the diameter of the Earth is greater than the diameter of the Moon, resulting in a greater tidal force.

What matters is not the total gravitational attraction on a body, but the difference from one side to the other. The greater the diameter of the body, the more difference there will be from one side to the other.

Gravitational attraction is inversely proportional to the square of the distance from the source. The attraction will be stronger on the side of a body facing the source, and weaker on the side away from the source. The tidal force is proportional to the difference.

Effects

In the case of an infinitesimally small elastic sphere, the effect of a tidal force is to distort the shape of the body without any change in volume. The sphere becomes an ellipsoid with two bulges, pointing towards and away from the other body. Larger objects distort into an ovoid, and are slightly compressed, which is what happens to the Earth's oceans under the action of the Moon. The Earth and Moon rotate about their common center of mass or barycenter, and their gravitational attraction provides the centripetal force necessary to maintain this motion. To an observer on the Earth, very close to this barycenter, the situation is one of the Earth as body 1 acted upon by the gravity of the Moon as body 2. All parts of the Earth are subject to the Moon's gravitational forces, causing the water in the oceans to redistribute, forming bulges on the sides near the Moon and far from the Moon.

When a body rotates while subject to tidal forces, internal friction results in the gradual dissipation of its rotational kinetic energy as heat. In the case for the Earth, and Earth's Moon, the loss of rotational kinetic energy results in a gain of about 2 milliseconds per century. If the body is close enough to its primary, this can result in a rotation which is tidally locked to the orbital motion, as in the case of the Earth's moon. Tidal heating produces dramatic volcanic effects on Jupiter's moon Io. Stresses caused by tidal forces also cause a regular monthly pattern of moonquakes on Earth's Moon.

Tidal forces contribute to ocean currents, which moderate global temperatures by transporting heat energy toward the poles. It has been suggested that variations in tidal forces correlate with cool periods in the global temperature record at 6- to 10-year intervals, and that harmonic beat variations in tidal forcing may contribute to millennial climate changes. No strong link to millennial climate changes has been found to date.

Tidal effects become particularly pronounced near small bodies of high mass, such as neutron stars or black holes, where they are responsible for the "spaghettification" of infalling matter. Tidal forces create the oceanic tide of Earth's oceans, where the attracting bodies are the Moon and, to a lesser extent, the Sun. Tidal forces are also responsible for tidal locking, tidal acceleration, and tidal heating. Tides may also induce seismicity.

By generating conducting fluids within the interior of the Earth, tidal forces also affect the Earth's magnetic field.

From the Wikipedia entry for TIDAL FORCE
TIDE 1

      The ship lay on the sand beyond the roof. It was a No. 2 General Products hull: a cylinder three hundred feet long and twenty feet through, pointed at both ends and with a, slight wasp-waist constriction near the tail. For some reason it was lying on its side, with the landing shocks still folded in at the tail.
     Ever notice how all ships have begun to look the same? A good ninety-five percent of today’s spacecraft are built around one of the four General Products hulls. It’s easier and safer to build that way, but somehow all ships end as they began: mass-produced look-alikes.
     The hulls are delivered fully transparent, and you use paint where you feel like it. Most of this particular hull had been left transparent. Only the nose had been painted, around the lifesystem. There was no major reaction drive. A series of retractable attitude jets had been mounted in the sides, and the hull was pierced with smaller holes, square and round, for observational instruments. I could see them gleaming through the hull.
     The puppeteer was moving toward the nose, but something made me turn toward the stern for a closer look at the landing shocks. They were bent. Behind the curved transparent hull panels some tremendous pressure had forced the metal to flow like warm wax, back and into the pointed stern.
     “What did this?” I asked.
     “We do not know. We wish strenuously to find out.”
     “What do you mean?”
     “Have you heard of the neutron star BVS-l?”
     I had to think a moment. “First neutron star ever found, and so far the only. Someone located it two years ago, by stellar displacement.”
     “BVS-l was found by the Institute of Knowledge on Jinx. We learned through a go-between that the Institute wished to explore the star. They needed a ship to do it.
     They had not yet sufficient money. We offered to supply them with a ship’s hull, with the usual guarantees, if they would turn over to us all data they acquired through using our ship.”
     “Sounds fair enough.” I didn’t ask why they hadn’t done their own exploring. Like most sentient vegetarians, puppeteers find discretion to be the only part of valor.
     “Two humans named Peter Laskin and Sonya Laskin wished to use the ship. They intended to come within one mile of the surface in a hyperbolic orbit. At some point during their trip an unknown force apparently reached through the hull to do this to the landing shocks. The unknown force also seems to have killed the pilots.”
     “But that’s impossible. Isn’t it?”
     “You see the point. Come with me.” The puppeteer trotted toward the bow.

     I saw the point, all right. Nothing, but nothing, can get through a General Products hull. No kind of electromagnetic energy except visible light. No kind of matter, from the smallest subatomic particle to the fastest meteor. That’s what the company’s advertisements claim, and the guarantee backs them up. I’ve never doubted it, and I’ve never heard of a General Products hull being damaged by a weapon or by anything else.
     We rode an escalladder into the nose.
     The lifesystem was in two compartments. Here the Laskins had used heat-reflective paint. In the conical control cabin the hull had been divided into windows. The relaxation room behind it was a windowless reflective silver. From the back wall of the relaxation room an access tube ran aft, opening on various instruments and the hyperdrive motors.
     There were two acceleration couches in the control cabin. Both had been torn loose from their mountings and wadded into the nose like so much tissue paper, crushing the instrument paneL The backs of the crumpled couches were splashed with rust brown. Flecks of the same color were all over everything, the walls, the windows, the viewscreens. It was as if something had hit the couches from behind: something like a dozen paint-filled toy balloons striking with tremendous force.
     “That’s blood,” I said.
     “That is correct. Human circulatory fluid.”

(ed note: the protagonist has been forced by the president of General Products to fly the exact same mission that killed the Laskins in an attempt to figure out what can get through a General Products hull)

     Two hours’ to go—and I was sure they were turning blue. Was my speed that high? Then the stars behind should be red. Machinery blocked the view behind me, so I used the gyros. The ship turned with peculiar sluggishness. And the stars behind were blue, not red. All around me were blue-white stars.
     Imagine light falling into a savagely steep gravitational well. It won’t accelerate. Light can’t move faster than light. But it can gain in energy, in frequency. The light was falling on me, harder and harder as I dropped.
     I told the dictaphone about it. That dictaphone was probably the best-protected item on the ship. I had already decided to earn my money by using it, just as if I expected to collect. Privately I wondered just how intense the light would get.
     Skydiver had drifted back to vertical, with its axis through the neutron star, but now it faced outward. I’d thought I had the ship stopped horizontally. More clumsiness. I used the gyros. Again the ship moved mushily, until it was halfway through the swing. Then it seemed to fall automatically into place. It was as if the Skydiver preferred to have its axis through the neutron star.

     I didn’t like that.

     I tried the maneuver again, and again the Skydiver’ fought back. But this time there was something else. Something was pulling at me.
     So I unfastened my safety net—and fell headfirst into the nose.
     The pull was light, about a tenth of a gee. It felt more like sinking through honey than falling. I climbed back into my chair, tied myself in with Lhe net, now hanging face down, and turned on the dictaphone. I told my story in such nitpicking detail that my hypothetical listeners could not but doubt my hypothetical sanity. “I think this is what happened to the Laskins,” I finished. “If the pull increases, I’ll call back.”
     Think? I never doubted it. This strange, gentle pull was inexplicable. Something inexplicable had killed Peter and Sonya Laskin. Q.E.D.
     Around the point where the neutron star must be, the stars were like smeared dots of oil paint, smeared radially. They glared with an angry, painful light. I hung face down in the net and tried to think.

     It was an hour before I was sure. The pull was increasing. And I still had an hour to fall.
     Something was pulling on me, but not on the ship.
     No, that was nonsense. What could reach out to me through a General Products hull? It must be the other way around. Something was pushing on the ship, pushing it off course.
     If it got worse, I could use the drive to compensate. Meanwhile, the ship was being pushed away from BVS-l, which was fine by me.
     But if I was wrong, if the ship was not somehow being pushed away from BVS-l, the rocket motor would send the Skydiver crashing into eleven miles of neutronium.
     And why wasn’t the rocket already firing? If the ship was being pushed off course, the autopilot should be fighting back. The accelerometer was in good order. It had looked fine when I made my inspection tour down the access tube.
     Could something be pushing on the ship and on the accelerometer, but not on me? It came down to the same impossibility: something that could reach through a General Products hull.

     To hell with theory, said I to myself, said I. I’m getting out of here. To the dictaphone I said, “The pull has increased dangerously. I’m going to try to alter my orbit.”
     Of course, once I turned the ship outward and used the rocket, I’d be adding my own acceleration to the X-force. It would be a strain, but I could stand it for a while. If I came within a mile of BVS-l, I’d end like Sonya Laskin. She must have waited face down in a net like mine, waited without a drive unit, waited while the pressure rose and the net cut into her’ flesh, waited until the net snapped and dropped her into the nose, to lie crushed and broken until the X-force tore the very chairs loose and dropped them on her.
     I hit the gyros.
     The gyros weren’t strong enough to turn me. I tried it three times. Each time the ship rotated about fifty degrees and hung there, motionless, while the whine of the gyros went up and up. Released, the ship immediately swung back to position. I was nose down to the neutron star, and I was going to stay that way.

     Half an hour to fall, and the X-force was over a gee. My sinuses were in agony. My eyes were ripe and ready to fall out. I don’t know if I could have stood a cigarette, but I didn’t get the chance. My pack of Fortunados had fallen out of my pocket when I dropped into the nose. There it was, four feet beyond my fingers, proof that the X-force acted on other objects besides me. Fascinating.
     I couldn’t take any more. If it dropped me shrieking into the neutron star, I had to use the drive. And I did. I ran the thrust up until I was approximately in free fall. The blood which had pooled in my extremities went back where it belonged. The gee dial registered one point two gee. I cursed it for a lying robot.
     The soft-pack was bobbing around in the nose, and it occurred to me that a little extra nudge on the throttle would bring it to me. I tried it. The pack drifted toward me, and I reached, and like a sentient thing it speeded up to avoid my clutching hand. I snatched at it again as it went past my ear, and again it was moving too fast. That pack was going at a hell of a clip, considering that here I was practically in free fall. It dropped through the door to the relaxation room, still picking up speed, blurred and vanished as it entered the access tube. Seconds later I heard a solid thump.
     But that was crazy. Already the X-force was pulling blood into my face. I pulled my lighter out, held it at arm’s length and let go. It fell gently into the nose. But the pack of Fortunados had hit like I’d dropped it from a building.

     Well.

     I nudged the throttle again. The mutter of fusing hydrogen reminded me that if I tried to keep this up all the way, I might well put the General Products hull to its toughest test yet: smashing it into a neutron star at half light speed. I could see it now: a transparent hull containing only a few cubic inches of dwarf-star matter wedged into the tip of the nose.
     At one point four gee, according to that lying gee dial, the lighter came loose and drifted toward me. I let it go. It was clearly falling when it reached the doorway. I pulled the throttle back. The loss of power jerked me violently forward, but I kept my face turned. The lighter slowed and hesitated at the entrance to the access tube. Decided to go through. I cocked my ears for the sound, then jumped as the whole ship rang like a gong.
     And the accelerometer was right at the ship’s center of mass. Otherwise the ship’s mass would have thrown the needle off. The puppeteers were fiends for ten-decimal-point accuracy.

     I favored the dictaphone with a few fast comments, then got to work reprogramming the autopilot. Luckily what I wanted was simple. The X-force was but an X-force to me, but now I knew how it behaved. I might actually live through this.
     The stars were fiercely blue, warped to streaked lines near that special point. I thought I could see it now, very small and dim and red, but it might have been imagination. In twenty minutes I’d be rounding the neutron star. The drive grumbled behind me. In effective free fall, I unfastened the safety net and pushed myself out of the chair.
     A gentle push aft—and ghostly hands grasped my legs. Ten pounds of weight hung by my fingers from the back of the chair. The pressure should drop fast. I’d programmed the autopilot to reduce the thrust from two gees to zero during the next two minutes. All I had to do was be at the center of mass, in the access tube, when the thrust went to zero.
     I knew what the X-force was trying to do. It was trying to pull the ship apart.

     There was no pull on my fingers. I pushed aft and landed on the back wall, on bent legs. I knelt over the door, looking aft/down. When free fall came, I pulled myself through and was in the relaxation room looking down, forward into the nose.
     Gravity was changing faster than I liked. The X-force was growing as zero hour approached, while the compensating rocket thrust dropped. The X-force tended to pull the ship apart; it was two gee forward at the nose, two gee backward at the tail, and diminished to zero at the center of mass. Or so I hoped. The pack and lighter had behaved as if the force pulling them had increased for every inch they moved sternward.
     The back wall was fifteen feet away. I had to jump it with gravity changing in midair. I hit on my hands, bounced away. I’d jumped too late. The region of free fall was moving through the ship like a wave as the thrust dropped. It had left me behind. Now the back wall was “up” to me, and so was the access tube.
     Under something less than half a gee, I jumped for the access tube. For one long moment I stared into the three-foot tunnel, stopped in midair and already beginning to fall back, as I realized that there was nothing to hang on to. Then I stuck my hands in the tube and spread them against the sides. It was all I needed. I levered myself up and started to crawl.
     The dictaphone was fifty feet below, utterly unreachable. If I had anything more to say to General Products, I’d have to say it in person. Maybe I’d get the chance. Because I knew what force was trying to tear the ship apart.
     It was the tide.

     The motor was off, and I was at the ship’s midpoint. My spread-eagled position was getting uncomfortable. It was four minutes to perihelion.
     Something creaked in the cabin below me. I couldn’t see what it was, but I could clearly see a red point glaring among blue radial lines, like a lantern at the bottom of a well. To the sides, between the fusion tube and the tanks and other equipment, the blue stars glared at me with a light that was almost violet. I was afraid to look too long. I actually thought they might blind me.
     There must have been hundreds of gravities in the cabin. I could even feel the pressure change. The air was thin at this height, one hundred and fifty feet above the control room.
     And now, almost suddenly, the red dot was more than a dot. My time was up. A red disk leapt up at me; the ship swung around me; I gasped and shut my eyes tight.
     Giants’ hands gripped my arms and legs and head, gently but with great firmness, and tried to pull me in two. In that moment it came to me that Peter Laskin had died like this. He’d made the same guesses I had, and he’d tried to hide in the access tube. But he’d slipped… as I was slipping… From the control room came a multiple shriek of tearing metal. I tried to dig my feet into the hard tube walls. Somehow they held.
     When I got my eyes open the red dot was shrinking into nothing.

(ed note: later, back home, in the hospital)

     I was floating between a pair of sleeping plates, hideously uncomfortable, when the nurse came to announce a visitor. I knew who it was from her peculiar expression.
     “What can get through a General Products hull?” I asked it.
     “I hoped you would tell me.” The president (of General Products Inc.) rested on its single back leg, holding a stick that gave off green incense smelling smoke.
     “And so I will. Gravity.”
     “Do not play with me, Beowuif Shaeffer. This matter is vital.”
     “I’m not playing. Does your world have a moon?”
     “That information is classified.” The puppeteers are cowards. Nobody knows where they come from, and nobody is likely to find out.
     “Do you know what happens when a moon gets too close to its primary?
     “It falls apart.
     “Why?”
     “I do not know.”
     “Tides.”
     “What is a tide?”
     Oho, said I to myself, said I. “I’m going to try to tell you. The Earth’s moon is almost two thousand miles in diameter and does not rotate with respect to Earth. I want you to pick two rocks on the moon, one at the point nearest the Earth, one at the point farthest away.
     “Very well.”
     “Now, isn’t it obvious that if those rocks were left to themselves, they’d fall away from each other? They’re in two different orbits, mind you, concentric orbits, one almost two thousand miles outside the other. Yet those rocks are forced to move at the same orbital speed.”
     “The one outside is moving faster.”
     “Good point. So there is a force trying to pull the moon apart. Gravity holds it together. Bring the moon close enough to Earth, and those two rocks would simply ‘float away.”
     “I see. Then this ‘tide’ tried to pull your ship apart. It was powerful enough in the lifesystem of the Institute ship to pull the acceleration chairs out of their mounts.”
     “And to crush a human being. Picture it. The ship’s nose was just seven miles from the center of BVS-l. The tail was three hundred feet farther out. Left to themselves, they’d have gone in completely different orbits. My head and feet tried to do the same thing when I got close enough.

From NEUTRON STAR by Larry Niven (1966)
TIDE 2

(ed note: the rogue planet Bronson Alpha, about the size of the planet Uranus, is about to splatter Terra like a bug on a windshield. Our heroes have escaped Terra in their Space Ark, and hope to plant a colony on Bronson Beta. As Terra grows close to Bronson Alpha, there are tidal effects)

      NOW for an hour the passengers watched silently as Bronson Alpha swept upon the scene, a gigantic body, weird, luminous and unguessable, many times larger than Earth. It moved toward the Earth with the relentless perceptibility of the hands of a large clock, and those who looked upon its awe-inspiring approach held their breaths.
     Once again Hendron spoke. "What will take place now cannot be definitely ascertained. In view of the retardation of Bronson Alpha's speed caused by its collision with the moon, I have reason to believe that its course will be completely disrupted."

     Inch by inch, as it seemed, the two bodies came closer together. Looking at the screen was like watching the motion picture of a catastrophe and not like seeing it. Tony had to repeat to himself over and over that it was really so, in order to make himself believe it. Down there on the little earth were millions of scattered, demoralized human beings. They were watching this awful phenomenon in the skies. Around them the ground was rocking, the tides were rising, lava was bursting forth, winds were blowing, oceans were boiling, fires were catching, and human courage was facing complete frustration. Above them the sky was filled with this awful onrushing mass.
     To those who through the smoke and steam and hurricane could still pierce the void, it would appear as something no longer stellar but as something real, something they could almost reach out and touch. A vast horizon of earth stretched toward them across the skies. They would be able, if their reeling senses still maintained powers of observation, to see the equally tumultuous surface of Bronson Alpha, to describe the geography of its downfalling side. They would perhaps, in the last staggering seconds, feel themselves withdrawn from the feeble gravity of their own Earth, to fall headlong toward Bronson Alpha. And in the magnitude of that inconceivable manifestation, they would at last, numb and senseless, be ground to the utmost atoms of their composition.
     Tony shuddered as he watched. A distance, short on the screen—even as solar measurements are contemplated—separated the two planets. In the chamber of the hurtling Space Ship no one moved. Earth and Bronson Alpha were but a few moments apart. It seemed that even at their august distance they could perceive motion on the planet, as if the continents below them were swimming across the seas, as if the seas were hurling themselves upon the land; and presently they saw great cracks, in the abysses of which were fire, spread along the remote dry land. Into the air were lifted mighty whirls of steam. The nebulous atmosphere of Bronson Alpha touched the air of Earth, and then the very Earth bulged. Its shape altered before their eyes. It became plastic. It was drawn out egg-shaped. The cracks girdled the globe. A great section of the Earth itself lifted up and peeled away, leaping toward Bronson Alpha with an inconceivable force.

     The two planets struck.

     Decillions of tons of mass colliding in cosmic catastrophe.

     "It's not direct!" Duquesne shouted.
     Every one knew what he was thinking. Perhaps they were not witnessing complete annihilation. Perhaps some miracle would preserve a portion of the world.
     They panted and stared.
     Steam, fire, smoke. Tongues of flame from the center of the earth. The planets ground together and then moved across each other. It was like watching an eclipse.
     The magnitude of the disaster was veiled by hot gases and stupendous flames, and was diminished in awfulness by the intervening distances and by the seeming slowness with which it took place.
     Bronson Alpha rode between them and the Earth. Then— on its opposite side—fragments of the shattered world reappeared. Distance showed between them—widening, scattering distance. Bronson Alpha moved away on its terrible course, fiery, flaming, spread enormously in ghastly light.

From WHEN WORLDS COLLIDE by Philip Wylie and Edwin Balmer (1933)

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.

SPAGHETTIFICATION

In astrophysics, spaghettification (sometimes referred to as the noodle effect) is the vertical stretching and horizontal compression of objects into long thin shapes (rather like spaghetti) in a very strong non-homogeneous gravitational field; it is caused by extreme tidal forces. In the most extreme cases, near black holes, the stretching is so powerful that no object can withstand it, no matter how strong its components. Within a small region the horizontal compression balances the vertical stretching so that small objects being spaghettified experience no net change in volume.

Stephen Hawking described the flight of a fictional astronaut who, passing within a black hole's event horizon, is "stretched like spaghetti" by the gravitational gradient (difference in strength) from head to toe. The reason this happens would be that the gravity force exerted by the singularity would be much stronger at one end of the body than the other. If one were to fall into a black hole feet first, the gravity at their feet would be much stronger than at their head, causing the person to be vertically stretched. Along with that, the right side of the body will be pulled to the left, and the left side of the body will be pulled to the right, horizontally compressing the person. However, the term "spaghettification" was established well before this. Spaghettification of a star was imaged for the first time in 2018 by researchers observing a pair of colliding galaxies approximately 150 million light-years from Earth.

A simple example

In this example, four separate objects are in the space above a planet, positioned in a diamond formation. The four objects follow the lines of the gravitoelectric field, directed towards the celestial body's centre. In accordance with the inverse-square law, the lowest of the four objects experiences the biggest gravitational acceleration, so that the whole formation becomes stretched into a line.

These four objects are connected parts of a larger object. A rigid body will resist distortion, and internal elastic forces develop as the body distorts to balance the tidal forces, so attaining mechanical equilibrium. If the tidal forces are too large, the body may yield and flow plastically before the tidal forces can be balanced, or fracture, producing either a filament or a vertical line of broken pieces.

Examples of weak and strong tidal forces

In the gravity field due to a point mass or spherical mass, for a uniform rod oriented in the direction of gravity, the tensile force at the center is found by integration of the tidal force from the center to one of the ends. This gives F = (μ l m) / 4r3, where μ is the standard gravitational parameter of the massive body, l is the length of the rod, m is rod's mass, and r is the distance to the massive body. For non-uniform objects the tensile force is smaller if more mass is near the center, and up to twice as large if more mass is at the ends. In addition, there is a horizontal compression force toward the center.

For massive bodies with a surface, the tensile force is largest near the surface, and this maximum value is only dependent on the object and the average density of the massive body (as long as the object is small relative to the massive body). For example, for a rod with a mass of 1 kg and a length of 1 m, and a massive body with the average density of the Earth, this maximum tensile force due to the tidal force is only 0.4 μN.

Due to the high density, the tidal force near the surface of a white dwarf is much stronger, causing in the example a maximum tensile force of up to 0.24 N. Near a neutron star, the tidal forces are again much stronger: if the rod has a tensile strength of 10,000 N and falls vertically to a neutron star of 2.1 solar masses, setting aside that it would melt, it would break at a distance of 190 km from the center, well above the surface (neutron star typical radius is only about 12 km).

In the previous case objects would actually be destroyed and people killed by the heat, not the tidal forces - but near a black hole (assuming that there is no nearby matter), objects would actually be destroyed and people killed by the tidal forces, because there is no radiation. Moreover, a black hole has no surface to stop a fall. Thus, the infalling object is stretched into a thin strip of matter.

Inside or outside the event horizon

The point at which tidal forces destroy an object or kill a person will depend on the black hole's size. For a supermassive black hole, such as those found at a galaxy's center, this point lies within the event horizon, so an astronaut may cross the event horizon without noticing any squashing and pulling, although it remains only a matter of time, as once inside an event horizon, falling towards the center is inevitable. For small black holes whose Schwarzschild radius is much closer to the singularity, the tidal forces would kill even before the astronaut reaches the event horizon. For example, for a black hole of 10 Sun masses the above-mentioned rod breaks at a distance of 320 km, well outside the Schwarzschild radius of 30 km. For a supermassive black hole of 10,000 Sun masses, it will break at a distance of 3200 km, well inside the Schwarzschild radius of 30,000 km.

From the Wikipedia entry for SPAGHETTIFICATION

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