Fundamental lemma of calculus of variations

File:Porkchop plot.gif

Representative porkchop plot for the 2005 Mars launch opportunity. The horizontal axis is departure dates, and the vertical axis is arrival dates. A given contour represents a constant C₃ solution. The center of the porkchop is the optimal solution for the lowest C₃.

A porkchop plot (also pork-chop plot) is a chart that shows contours of equal characteristic energy (C₃) against combinations of launch date and arrival date for a particular interplanetary flight.

By examining the results of the porkchop plot, engineers can determine when launch opportunities exist (a launch window) that is compatible with the capabilities of a particular spacecraft. A given contour, called a porkchop curve, represents constant C₃, and the center of the porkchop the optimal minimum C₃. The orbital elements of the solution, where the fixed values are the departure date, the arrival date, and the length of the flight, were first solved mathematically in 1761 by Johann Heinrich Lambert, and the equation is generally known as Lambert's problem (or theorem).

Math

The general form of Characteristic Energy can be computed as:

$C_3=v_{\mathrm{\infty }}^2$

where

$v_{\mathrm{\infty }}$ is the orbital velocity when the orbital distance tends to infinity. Note that, since the kinetic energy is $\frac{1}{2}m{v}^2$, C₃ is in fact equal to twice the magnitude of the specific orbital energy ($ϵ$) of the escaping object.

Use

For the Voyager program, engineers at JPL plotted around 10,000 potential trajectories using porkchop plots, from which they selected around 100 that were optimal for the mission objectives. The plots allowed them to reduce or eliminate planetary encounters taking place over the Thanksgiving or Christmas holidays, and to plan the completion of the mission's primary goals before the end of the fiscal year 1981.

See also

• Launch window
• Specific orbital energy
• Orbit
• Parabolic trajectory
• Hyperbolic trajectory

References

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External links

• JPL Introduction to Porkchop plots

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