P5A - From GTO to Mars

References

Paul A. Penzo: Mission Design for a Wright Brother's Centennial Mars Airplane Mission in 2003; AAS/AIAA Space Flight Mechanics Meeting, Clearwater, Florida; 23-26 January, 2000; Paper AAS 00-196; 12 pp.
Charles D. Brown: Spacecraft Mission Design; AIAA Education Series; ISBN 1-56347-041-1; 187 pp; 1992; Including: ORB:AIAA Mission Design Software
J.D. Giorgini et al.: JPL's On-Line Solar System Data Service; Bulletin of the American Astronomical Society, Vol 28, No. 3, p. 1158, 1996.
Stephen Wolfram: The Mathematica Book, 4th ed., 1999.

Preliminary Analysis of MEGA (Moon-Earth-Gravity Assist) proposal by P. Penzo (patched conic procedure for 6 elliptic and 3 hyperbolic sections around 4 centers of gravity with 4 changes of velocity and/or direction)

Penzo's dates:

Earth departure	Mars arrival	ToF[days]	C3[km^2/s^2](Penzo)
2003-05-31	2003-12-17	200	09
2005-08-04	2006-02-22	202	16
2005-09-02	2006-10-02	395	16
2007-09-13	2008-08-19	341	14
2009-10-18	2010-08-27	313	11
2011-10-30	2012-09-09	314	10

(6) Earth Mars Transfer

The following diagrams for C3 values have been calculated with JPL Horizons ephemeris data (osculating elements for longitude and true anomaly, vector data for radius and flight angle) and the analytic formulae used by the ORB package. According to Penzo's proposal, type 2 transfers only are considered for the years 2007, 2009, and 2011. The data have been calculated with intervals of 1 day. Mathemathica 4.1 is being used for calculations and production of diagrams.

Earth departure 2007

More detailed data for the departure date proposed by Penzo (2007-09-13) are shown in the following table.

Time of Flight [days]	Range Mars at arrival [km]	Longitude Mars at arrival[deg]	Inclination of Transfer Ellipse[deg]	True Anomaly of Earth on Transfer Ellipse[deg]	alpha=Flight direction change[deg]	C3 [km^2/s^2]	hyperbolic excess velocity [km/s]
334.	243263549.	196.474	-2.24109	352.361	2.32178	13.8398	3.72019
335.	243136337.	196.933	-2.17901	352.264	2.26718	13.7704	3.71085
336.	243007938.	197.392	-2.11876	352.166	2.21500	13.7061	3.70218
337.	242878361.	197.851	-2.06026	352.066	2.16518	13.6467	3.69415
338.	242747611.	198.311	-2.00342	351.965	2.11766	13.5920	3.68673
339.	242615697.	198.771	-1.94817	351.863	2.07242	13.5415	3.67988
340.	242482625.	199.232	-1.89444	351.759	2.02941	13.4952	3.67358
341.	242348403.	199.693	-1.84216	351.653	1.98860	13.4528	3.66780
342.	242213037.	200.155	-1.79125	351.546	1.94995	13.4140	3.66252
343.	242076537.	200.617	-1.74167	351.438	1.91346	13.3789	3.65771
344.	241938908.	201.080	-1.69335	351.328	1.87911	13.34710	3.65336
345.	241800160.	201.543	-1.64624	351.217	1.84686	13.3184	3.64944
346.	241660299.	202.007	-1.60029	351.104	1.81671	13.2929	3.64594
347.	241519334.	202.472	-1.55545	350.990	1.78866	13.2703	3.64284
348.	241377272.	202.937	-1.51167	350.874	1.76267	13.2505	3.64013
349.	241234122.	203.402	-1.46891	350.757	1.73876	13.2335	3.63779
350.	241089891.	203.868	-1.42714	350.638	1.71690	13.2190	3.63580
351.	240944589.	204.335	-1.38630	350.518	1.69707	13.2071	3.63416
352.	240798223.	204.802	-1.34637	350.397	1.67929	13.1976	3.63285
353.	240650802.	205.270	-1.30731	350.274	1.66351	13.1905	3.63187
354.	240502334.	205.738	-1.26908	350.149	1.64979	13.1856	3.63120
355.	240352828.	206.207	-1.23166	350.024	1.63799	13.1830	3.63083
356.	240202293.	206.677	-1.19501	349.896	1.62820	13.1825	3.63076
357.	240050737.	207.147	-1.1591	349.768	1.62035	13.1840	3.63098
358.	.239898170.	207.617	-1.12391	349.637	1.61443	13.1877	3.63148
359.	239744600.	208.089	-1.08942	349.506	1.61042	13.1933	3.63226
360.	239590037.	208.561	-1.05559	349.373	1.60828	13.2009	3.63330
361.	239434489.	209.033	-1.0224	349.238	1.60798	13.2104	3.63461
362.	239277967.	209.506	-0.989827	349.102	1.60950	13.2217	3.63617
363.	239120479.	209.980	-0.957856	348.964	1.61278	13.2349	3.63798

Earth departure 2009

Earth departure 2011

Preliminary Conclusions:

The optimum arrival dates (minimum of C3, calculated with ORB) are pretty near the dates proposed by Penzo. Since the Earth departure dates in Penzo's proposal are fixed by the Moon's position relative to the transfer velocity vector, these dependences of the different flight phases have to be considered as soon as all phases are investigated in more detail.

Penzo's table presents for C3 integer values only, therefore a comparison with ORB derived data is not really possible, although a fairly good correlation can be seen.

(5) Earth Departure - Alternative Heights of Perigee

deltaV3 = deltaV3a + deltaV3b

Note: All deltaV calculations are for infinite thrust only.

Penzo's case: hPerig = 300 km

Angle of asymptote: beta

deltaV3a[km/s]	hPerig = 300 km	hPerig = 1000 km	hPerig = 3000 km
hApog = 0.6*10^6 km	0.0610	0.0641	0.0723
hApog = 1.0*10^6 km	0.0365	0.0384	0.0433
hApog = 1.4*10^6 km	0.0260	0.0274	0.0309

Earth departure/Mars arrival: 2007-09-13/2008-08-19

Assumption: vHE = 3.7000 km/s

	hPerig = 300 km	hPerig = 1000 km	hPerig = 3000 km
vParab[km/s]	10.9259	10.3947	09.2199
deltaV3b[km/s]	0.6095	0.6389	0.7147
beta[deg]	35.5676	37.0768	40.8543

Earth departure/Mars arrival: 2009-10-18/2010-08-27

Assumption: vHE = 3.2200 km/s

	hPerig = 300 km	hPerig = 1000 km	hPerig = 3000 km
vParab[km/s]	10.9259	10.3947	09.2199
deltaV3b[km/s]	0.4646	0.4873	0.5461
beta[deg]	31.5704	32.9672	36.4964

Updated: 2004-05-15

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