Orbital Mechanics

Apollo Command and Service modules

artist's concept of the THEMIS A, B, C, D, E spacecraft in orbit (aka ARTEMIS)

Orbital mechanics is the study of the motion of spacecraft moving under the influence of forces such as gravity and rocket thrust. Orbital mechanics is used for spacecraft trajectories, including orbital maneuvers, orbit plane changes, and orbital transfers, and is used to plan propulsive maneuvers for spacecraft missions.

Johannes Kepler developed laws of planetary motion to predict the motion of the planets around the Sun or the path of satellites around a planet, and his theories were confirmed when Isaac Newton developed his law of universal gravitation. The motion of objects in space are usually calculated from Newton's law of universal gravitation and of motion. Albert Einstein's general relativity is a more exact theory than Newton's laws for calculating orbits, and is sometimes necessary for greater accuracy. That said, Kepler's laws provide a good approximation of the path of a body in space under the influence of the gravity of another body.

Kepler's Laws

Kepler's 1st Law: The orbit of a body around a more massive body is an ellipse, with the more massive body located at one of the foci of that ellipse.
Kepler's 2nd Law: As the body moves in its orbit, the line from the more massive body to the body sweeps out equal areas in equal amounts of time.
Kepler's 3rd Law: For a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant.

As a mathematical formula the 3rd law is:

constant = a^3 / P^2

where:

a is the mean distance between the two masses
P is the mutual period of revolution of the two masses

Orbital Elements

There are six classic orbital elements (also known as Keplerian orbital elements) that are necessary for us to know about an orbit and a satellite's place in it. These elements help us describe: orbit size, orbit shape, orbit orientation, and orbit location. They also specify the part of the Earth the satellite is passing over at any given time and its Field of View (FOV), which is the angle that describes the amount of the Earth's surface the spacecraft can see at any given time. These six orbital elements are:

Semi-major Axis (a): Describes the size of the orbit, which is one-half of the major axis of the orbit.
Eccentricity (e): Specifies the shape of an orbit and is given by the ratio of the distance between the two foci and the length of the major axis. The eccentricity of a circular orbit is zero, and for an ellipse, it can range from zero to less than one.
Inclination (i): Angle between the plane of the equator and the orbital plane.
Longitude of the Ascending Node (O): It is the angle between the Sun and the intersection of the equatorial plane and the orbit on the first day of spring in the Northern Hemisphere. The day is called the vernal equinox. Looking down from above the North Pole, the right ascension of the ascending node is positive counter-clockwise.
Argument of Perigee (?): Angle between the ascending node and the orbit's point of closest approach to the Earth (perigee).
True Anomaly (v): True Anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the eccentric anomaly and the mean anomaly. True Anomaly represents the angle between the perigee and the vehicle in the orbit plane.

In principle once the orbital elements are known for a body, its position can be calculated forward indefinitely in time. However, in practice, orbits are affected or perturbed, by other forces than the gravity of a single more massive body, and thus the orbital elements change over time.

A two line element (TLE) format is the format used by the Combined Space Operations Center (CSpOC), as well as US NORAD and US NASA, to distribute Earth satellite orbital elements via https://www.space-track.org/. These TLEs are generated with an orbit determination process based on observations using a number of radar and electro-optical sensors. These TLEs are periodically updated, since they can be perturbed, so as to maintain a reasonable prediction capability on all objects in Earth orbit. Data for each satellite actually consists of three lines, the first line containing eleven characters for the satellite's name. This is followed by the standard two lines of elements. Tracking programs are usually unforgiving of anything that doesn’t fit the exact format. See AMSAT TLE format for more details.

Newton's Law

While Kepler's laws can describe planetary and satellite motion, the laws of orbital mechanics are Newton's law of universal gravitation and of motion, and the fundamental mathematical tool is Newton's differential calculus. Common assumptions that can be used to simplify calculations of orbital mechanics include non-interference from outside bodies, negligible mass for one of the bodies, and negligible other forces (such as from the solar wind, atmospheric drag, etc.). More accurate calculations can be made without these simplifying assumptions, but they are more complicated. The increased accuracy might not make a significant difference in the calculations. Also, as stated above, Albert Einstein's general relativity is an even more exact theory than Newton's laws for calculating orbits, but again the increased accuracy might not make a significant difference in the calculations.

Newton's law of universal gravitation provide a better explanation of the path of a bodies in space.

Newton's Law: every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

As a mathematical formula the law is:

F = G * m * M / (r^2)

where:

F is the force generated by gravitational attraction of the two bodies
G is Newton's gravitational constant equal to 6.67384 * 10^-11 m^3 / (kg * s^2)
m and M are the masses of the two bodies
r is the mean distance between the two bodies

Note that Kepler's 3rd law can be predicted from Newton's law of univeral gravitation. The gravitational force of a planet or satellite has to be equal to the centripetal force needed to keep it in an orbit. Therefore:

G * m * M / (r^2) = m * v^2 / r

Substituting in that the orbital velocity for a circular orbit is the distance traveled in the orbit divided by the time of the orbit or 2*pi*r/P where P is the period of revolution

v = (2 * pi * r / P)

you get:

G * m * M / (r^2) = m * (2 * pi * r / P)^2 / r

and after expanding and simplifying you get:

G * M / (4 * pi^2) = r^3 / P^2

where:

G is Newton's gravitational constant equal to 6.67384 * 10^-11 m^3 / (kg * s^2), and pi is the constant 3.14159
M is the mass of the more massive object, such as the Sun, that the less massive objects, such as the planets, are orbiting; another example would be the Earth, that the less massive object, such as a spacecraft or satellite, is orbiting.
r (or a) is the mean distance between the two masses
P is the mutual period of revolution of the two masses

So for bodies orbiting a particular more massive body, the mathematical formula is the same as Kepler's 3rd law.

Other useful equations for circular orbits that can be derived from these equations include:

v = sqrt(G * M / r)

P = 2 * pi * sqrt(r^3 / (G * M))

Orbital transfer

Transfer orbits are usually elliptical orbits that allow spacecraft to move from one orbit to another. At a minimum they require a rocket firing at the start, a rocket firing at the end, and sometimes one or more rocket firings in the middle. The Hohmann transfer orbit requires a minimal delta-v for only two rocket firings. Faster transfers may use any orbit that intersects both the original and destination orbits, at the cost of higher delta-v.

The delta-v required to start the Hohmann transfer can be calculated, assuming the satellites mass is negligible compared to the primary body's mass, the orbits are cicular, and that the impulse is instantaneous, with:

delta-v = ( sqrt( G * M / r1 ) ) * ( sqrt( 2 * r2 / ( r1 + r2 ) ) - 1 )

Similarly the delta-v required at the end of the Hohmann transfer can be calculated with:

delta-v = ( sqrt( G * M / r2 ) ) * ( 1 - sqrt( 2 * r1 / ( r1 + r2 ) ) )

where:

G is Newton's gravitational constant equal to 6.67384 * 10^-11 m^3 / (kg * s^2), and pi is the constant 3.14159
M is the mass of the more massive object, such as the Sun, that the less massive objects, such as the planets, are orbiting; another example would be the Earth, that the less massive object, such as a spacecraft or satellite, is orbiting.
r1 is the distance of the spacecrafts starting circular orbit from the primary body
r2 is the distance of the spacecrafts destination circular orbit from the primary body

Note that in the case where the destination isn't an orbit but an intended landing on a planet or moon, the end delta-v can be added to the descent and landing delta-v for a direct landing that doesn't orbit the planet or moon first.

These formulas tell you and instantaneous change in velocity (delta-v) that is needed. To approximate an appropriate rocket to do the maneuver, use the following force formula. Since the rocket firing will eject fuel mass the spacecraft mass will decrease so this is only a rough approximation.

delta-v = Ft * t / m

where:

Ft is the force of the rocket thrust
t is the time that the rocket fires for
m is the mass of the spacecraft including the rocket and fuel

For a better approximation of an appropriate rocket to do the maneuver, use the following rocket equation based formula. This formula accounts for the rocket firing ejecting fuel mass.

delta-v = Ft * t * ln ( m0 / mf ) / ( m0 - mf )

where:

Ft is the force of the rocket thrust
t is the time that the rocket fires for
ln is the natural logarithm function
m0 is the initial total mass, spacecraft including rocket and fuel mass or wet mass
mf is the final total mass after exhaust, spacecraft including rocket without fuel mass or dry mass

In addition to transfering from one orbit to another, it's sometimes desirable to move from one location in orbit to another body in a target orbit to either land or dock with it. To do this it is crucial for the spacecraft's starting delta-v to be triggered so the spacecraft will reach the destination orbit when the other body will be in the same location in the destination orbit. The angular alignment at the time of start between the spacecraft and the target body can be calculated with:

angle = pi * ( 1 - ( ( 1 / ( 2 * sqrt( 2 ) ) ) * sqrt( ( ( r1 / r2 ) + 1 )^3 ) ) )

where:

pi is the constant 3.14159
r1 is the distance of the spacecrafts starting circular orbit from the primary body
r2 is the distance of the spacecrafts destination circular orbit from the primary body

Escape velocity

The escape velocity from a body can be calculated with:

v = sqrt( 2 * G * M / r )

where:

G is Newton's gravitational constant equal to 6.67384 * 10^-11 m^3 / (kg * s^2)
M is the mass of of the star, planet, or moon that you're escaping from
r is the radius of the spacecraft from the center of the star, planet, or moon

During Apollo missions, news media sometimes said that the Apollo spacecraft reached escape velocity. Actually, the Apollo missions quite often reached just below escape velocity. The Apollo spacecraft didn't need to reach escape velocity because the Moon is a satellite of the Earth. Earth escape velocity is needed to go to other planets such as Mars or asteroids such as 1 Ceres, but not to reach high orbits around the Earth. By staying just under escape velocity the Apollo spacecraft remained under the influence of Earths gravity, and if its engines failed it would return to a location near the Earth where the rocket engines had been fired. However, by reaching close to escape velocity, rather than using a minimal delta-v Hohmann transfer orbit, Apollo missions were able to reach the Moon in approximately 3 days rather than 5 days. This allowed for less resources to be needed to sustain the crew, and less time for the spacecraft, it's power systems, and electronics to need to survive in the space environment.