Patched conic approximation

In astrodynamics, the patched conic approximation or patched two-body approximation¹² is a method to simplify trajectory calculations for spacecraft in a multiple-body environment.

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.³ Notably, it does not model Lagrangian points.

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 parameter ${\displaystyle \mu }$, the spacecraft motion satisfies:$${\displaystyle {\ddot {\mathbf {r} }}=-{\frac {\mu }{r^{3}}}\mathbf {r} }$$The solution in each region is a conic section (elliptic, parabolic, or hyperbolic), which can be expressed using classical orbital elements or state vectors. At the boundary between spheres of influence, the trajectory segments are "patched" by enforcing continuity of position and velocity:$${\displaystyle \mathbf {r} _{-}=\mathbf {r} _{+},\qquad \mathbf {v} _{-}=\mathbf {v} _{+}}$$For example, when departing a planet, the spacecraft follows a hyperbolic escape trajectory with hyperbolic excess velocity ${\displaystyle \mathbf {v} _{\infty }}$. This velocity is then used as the initial condition for the heliocentric trajectory:$${\displaystyle \mathbf {v} _{\text{helio}}=\mathbf {v} _{\text{planet}}+\mathbf {v} _{\infty }}$$This process is repeated at each sphere of influence boundary to construct an approximate multi-body trajectory.

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.

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.¹ Let$${\displaystyle r_{E}=1.496\times 10^{8}\ {\text{km}},\qquad r_{M}=2.279\times 10^{8}\ {\text{km}},\qquad \mu _{\odot }=1.327\times 10^{11}\ {\text{km}}^{3}/{\text{s}}^{2}}$$The semi-major axis of the heliocentric transfer ellipse is$${\displaystyle a_{t}={\frac {r_{E}+r_{M}}{2}}=1.888\times 10^{8}\ {\text{km}}}$$The circular heliocentric speeds of Earth and Mars are$${\displaystyle v_{E}={\sqrt {\frac {\mu _{\odot }}{r_{E}}}}=29.8\ {\text{km/s}},\qquad v_{M}={\sqrt {\frac {\mu _{\odot }}{r_{M}}}}=24.1\ {\text{km/s}}}$$Using the vis-viva equation, the spacecraft speed on the transfer ellipse at Earth's orbit is$${\displaystyle v_{t,E}={\sqrt {\mu _{\odot }\left({\frac {2}{r_{E}}}-{\frac {1}{a_{t}}}\right)}}=32.7\ {\text{km/s}}}$$For this tangential, prograde transfer, the required Earth departure hyperbolic excess speed is$${\displaystyle v_{\infty ,E}=v_{t,E}-v_{E}=2.94\ {\text{km/s}}}$$In the patched conic approximation, this quantity becomes the hyperbolic excess speed of the Earth departure hyperbola. If the spacecraft departs from a circular low Earth parking orbit with radius ${\displaystyle r_{p}=6,578\ {\text{km}}}$, then the required impulsive burn is approximately$${\displaystyle \Delta v_{E}={\sqrt {v_{\infty ,E}^{2}+{\frac {2\mu _{E}}{r_{p}}}}}-{\sqrt {\frac {\mu _{E}}{r_{p}}}}=3.61\ {\text{km/s}}}$$where ${\displaystyle \mu _{E}=398,600\ {\text{km}}^{3}/{\text{s}}^{2}}$. At Mars, the spacecraft speed on the transfer ellipse is$${\displaystyle v_{t,M}={\sqrt {\mu _{\odot }\left({\frac {2}{r_{M}}}-{\frac {1}{a_{t}}}\right)}}=21.5\ {\text{km/s}}}$$For this tangential transfer, the Mars arrival hyperbolic excess speed is the magnitude of the difference between Mars's circular speed and the transfer orbit speed$${\displaystyle v_{\infty ,M}=v_{M}-v_{t,M}=2.65\ {\text{km/s}}}$$This becomes the hyperbolic excess speed of the Mars arrival hyperbola. For insertion into a circular orbit 200 km above Mars, taking ${\displaystyle r_{p}=3,596\ {\text{km}}}$ and ${\displaystyle \mu _{M}=42,828\ {\text{km}}^{3}/{\text{s}}^{2}}$, the approximate capture burn is$${\displaystyle \Delta v_{M}={\sqrt {v_{\infty ,M}^{2}+{\frac {2\mu _{M}}{r_{p}}}}}-{\sqrt {\frac {\mu _{M}}{r_{p}}}}=2.10\ {\text{km/s}}}$$The heliocentric time of flight for the Hohmann transfer is half the period of the transfer ellipse:$${\displaystyle t=\pi {\sqrt {\frac {a_{t}^{3}}{\mu _{\odot }}}}=2.24\times 10^{7}\ {\text{s}}\approx 259\ {\text{days}}}$$