Parametric Analysis of Earth-to-Mars Trajectories - OTB

perform a parametric “sweep” of ballistic interplanetary trajectories from a user-defined Earth park orbit to encounter at Mars

This sunmission is a MATLAB script named e2m_sweep_otbthat can be used to perform a parametric “sweep” of ballistic interplanetary trajectories from a user-defined Earth park orbit to encounter at Mars. The user defines the initial departure calendar date, the sweep duration and the step size to use during the analysis. The software assumes that interplanetary injection occurs impulsively from a circular Earth park orbit. The software also assumes the arrival date of each sweep trajectory is fixed to the value provided by the user. The user can also select either the ascending or descending hyperbolic injection for the sweep.

The parametric sweep computed by this script can be envisioned as a horizontal or vertical “slice” through an interplanetary pork chop or contour plot.

The first part of this computer program computes the ballistic trajectory characteristics using a patched-conic, two-body Lambert solution for the transfer trajectory from Earth to Mars. The second part implements a simple shooting method that attempts to adjust the characteristics of the geocentric injection hyperbola while numerically integrating the spacecraft’s geocentric and heliocentric equations of motion and “targeting” to the three components of the heliocentric position vector of Mars at the user-defined arrival calendar date.

The spacecraft motion within the Earth’s sphere-of-influence (SOI) includes the Earth’s oblate gravity effect and the point-mass perturbation of the sun and moon. The heliocentric equations of motion include the point-mass gravity of the sun and the first seven planets of the solar system. The point-mass gravity of Mars is switched off when the spacecraft is within 25,000 kilometers of Mars. The Earth’s SOI distance is hard-wired in the software to a value of 925,000 kilometers.

The major computational steps implemented in this software are as follows:

• solve the two-body, patched-conic interplanetary Lambert problem for the energy , declination (DLA) and asymptote (RLA) of the outgoing or departure hyperbola
• compute the orbital elements of the geocentric departure hyperbola and the components of the interplanetary injection delta-v vector
• using the hyperbola computed in step two as an initial guess, perform geocentric orbit propagation from perigee of the geocentric hyperbola to the Earth’s sphere-of-influence
• perform an n-body heliocentric orbit propagation from the Earth’s sphere-of-influence to the user-defined arrival calendar date at Mars
• target to all three components of the heliocentric position vector of Mars at the arrival time by optimizing the v-infinity, right ascension and declination of the departure hyperbola

R. H. Gooding’s method is used to solve the two-body Lambert problem. This MATLAB script uses the fmincon algorithm from the Mathworks Optimization Toolbox to solve the system of nonlinear equations and optimize the characteristics of the geocentric departure hyperbola.

This MATLAB script reads JPL DE421 lunar and solar ephemerides in a machine-independent binary format (kernels) which are available from the SPICE web site and by anonymous ftp from ftp://ssd.jpl.nasa.gov/pub/eph/planets/bsp. These *.bsp ephemeris files are IEEE-Little Endian style of binary kernel. This is the binary form native to PC/Linux, PC/Windows and MAC/Intel machines. Additional information about JPL ephemerides can be found at http://naif.jpl.nasa.gov/naif/.

The e2m_sweep_otbscript uses routines from the MICE software suite to read and evaluate the JPL ephemeris file. Platform-specific MICE mex files, support functions and the binary ephemeris file (de421.bsp) are available at naif.jpl.nasa.gov/naif/toolkit_MATLAB.html. MICE is a MATLAB implementation of the SPICE library created by JPL.