In
astrodynamics
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the Newton's law of univ ...
, the patched conic approximation or patched two-body approximation is a method to simplify
trajectory
A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tra ...
calculations for
spacecraft
A spacecraft is a vehicle that is designed spaceflight, to fly and operate in outer space. Spacecraft are used for a variety of purposes, including Telecommunications, communications, Earth observation satellite, Earth observation, Weather s ...
in a multiple-body environment.
Method
The simplification is achieved by dividing space into various parts by assigning each of the ''n'' bodies (e.g. the
Sun
The Sun is the star at the centre of the Solar System. It is a massive, nearly perfect sphere of hot plasma, heated to incandescence by nuclear fusion reactions in its core, radiating the energy from its surface mainly as visible light a ...
,
planets
A planet is a large, rounded astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets by the most restrictive definition of the te ...
,
moons
A natural satellite is, in the most common usage, an astronomical body that orbits a planet, dwarf planet, or small Solar System body (or sometimes another natural satellite). Natural satellites are colloquially referred to as moons, a deriva ...
) its own
sphere of influence
In the field of international relations, a sphere of influence (SOI) is a spatial region or concept division over which a state or organization has a level of cultural, economic, military, or political exclusivity.
While there may be a formal a ...
. When the spacecraft is within the sphere of influence of a smaller body, only the
gravitational force
Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the sq ...
between the spacecraft and that smaller body is considered, otherwise the gravitational force between the spacecraft and the larger body is used. This reduces a complicated
n-body problem
In physics, the -body problem is the problem of predicting the individual motions of a group of astronomical object, celestial objects interacting with each other gravitationally.Leimanis and Minorsky: Our interest is with Leimanis, who first d ...
to multiple
two-body problem
In classical mechanics, the two-body problem is to calculate and predict the motion of two massive bodies that are orbiting each other in space. The problem assumes that the two bodies are point particles that interact only with one another; th ...
s, for which the solutions are the well-known
conic sections
A conic section, conic or a quadratic curve is a curve obtained from a Conical surface, cone's surface intersecting a plane (mathematics), plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is ...
of the
Kepler orbit
In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane i ...
s.
Although this method gives a good approximation of trajectories for
interplanetary spacecraft missions, there are missions for which this approximation does not provide sufficiently accurate results.
Notably, it does not model
Lagrangian point
In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium (mechanics), equilibrium for small-mass objects under the gravity, gravitational influence of two massive orbit, orbiting b ...
s.
Example
On an
Earth
Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
-to-
Mars
Mars is the fourth planet from the Sun. It is also known as the "Red Planet", because of its orange-red appearance. Mars is a desert-like rocky planet with a tenuous carbon dioxide () atmosphere. At the average surface level the atmosph ...
transfer, a
hyperbolic trajectory
In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the ...
is required to escape from Earth's
gravity well
A sphere of influence (SOI) in astrodynamics and astronomy is the oblate spheroid-shaped region where a particular celestial body exerts the main gravitational influence on an orbiting object. This is usually used to describe the areas in the ...
, then an
elliptic or
hyperbolic
Hyperbolic may refer to:
* of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics
** Hyperbolic geometry, a non-Euclidean geometry
** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...
trajectory in the Sun's sphere of influence is required to transfer from Earth's sphere of influence to that of Mars, etc. By patching these conic sections together—matching the position and velocity vectors between segments—the appropriate mission trajectory can be found.
See also
*
Two-body problem
In classical mechanics, the two-body problem is to calculate and predict the motion of two massive bodies that are orbiting each other in space. The problem assumes that the two bodies are point particles that interact only with one another; th ...
*
N-body problem
In physics, the -body problem is the problem of predicting the individual motions of a group of astronomical object, celestial objects interacting with each other gravitationally.Leimanis and Minorsky: Our interest is with Leimanis, who first d ...
*
Sphere of influence
In the field of international relations, a sphere of influence (SOI) is a spatial region or concept division over which a state or organization has a level of cultural, economic, military, or political exclusivity.
While there may be a formal a ...
* ''
Kerbal Space Program
''Kerbal Space Program'' is a 2015 Space flight simulation game, space flight simulation video game developed by Mexican studio Squad for Linux, macOS, Windows, PlayStation 4, PlayStation 5, Xbox Series X/S and Xbox One. In the game, players dir ...
'', a popular spaceflight simulator based on the patched conic approximation
References
Bibliography
* {{cite tech report , first=K. M. , last=Carlson , title=An Analytical Solution to Patched-Conic Trajectories Satisfying Initial and Final Boundary Conditions , work=Technical Memorandum , number=TM-70-2011-1 , institution=Bellcomm Inc. , date=1970-11-30 , format=pdf , url=https://ntrs.nasa.gov/api/citations/19710007291/downloads/19710007291.pdf
Astrodynamics