In astrodynamics, the patched conic approximation or patched two-body approximation is a method to simplify trajectory calculations for spacecraft 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, planets, moons) its own sphere of influence. When the spacecraft is within the sphere of influence of a smaller body, only the gravitational force 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 to multiple two-body problems, for which the solutions are the well-known conic sections of the Kepler orbits.

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 points.

Mathematical formulation

In the patched conic approximation, the spacecraft trajectory is modeled as a sequence of two-body problems. Within the sphere of influence of a body with gravitational parameterThe solution in each region is a conic section (elliptic, parabolic, or hyperbolic), which can be expressed using classical orbital elementsThis process is repeated at each sphere of influence boundary to construct an approximate multi-body trajectory.

Example: Earth-to-Mars transfer

On an Earth-to-Mars transfer, a hyperbolic trajectory is required to escape from Earth's gravity well, then an elliptic or hyperbolic 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.

Numerical example

A simple Earth-to-Mars patched conic calculation can be made by assuming circular, coplanar planetary orbits and a Hohmann transfer orbit between the mean orbital radii of Earth and Mars.The semi-major axisUsing the vis-viva equationThe heliocentric time of flight for the Hohmann transfer is half the period of the transfer ellipse

See also

• Two-body problem

• N-body problem

• Sphere of influence

• Kerbal Space Program, a popular spaceflight simulator based on the patched conic approximation

• Pseudostate trajectory model

References

Bibliography