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I came to the conclusion that a strictly Hohmann-trajectory is not always the cheapest one when the transfer is between two non-coplanar, elliptic (non-circular) orbits. In fact this is the case with an Earth-Mars-transfer.
Does anybody know how to determine such a trajectory? I read that it has to be a Hohmann-II trajectory, an extended version of the original Hohmann-trajectory. But I don't really understand why. I think, it has something to do with the nodes, because of lack of plane-changes in this case.
Who can help? :realllymad:
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I started discussing some of the principles of an elliptical transfer in the topic [http://www.newmars.com/forums/viewtopic.php?t=2442]orbital mechanics. I am not sure if this is the same thing as a Hohmann-II trajectory.
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I've read the thread and searched on the internet. What is explained in that thread is, as far as I can conclude, that what is called Lambert's Theorem in the book 'Orbital Mechanics' by Prussing and Conway.
I yet follow the procedure that is explained in the thread by first guessing some depart and arrival date, determining position vectors from the orbital elements, then determine some orbital elements (inclination, longitude ascending node) of the transfer, calculating minimum elliptical transfer time (parabolic transfer time) and transfer time at minimal semimajor axis. Then I let solve the computer Lamberts Theorem by input of the desired travel time.
But then my problem begins: What is the ideal trajectory (minimum delta-v)? It's certainly not a Hohmann, because of the inclination of the Mars-orbit, except when both departure and arrival are on the node-line. It seems that when somewhat lenghtened this is the cheapest, maybe a deep space manouvre is involved at aphelium (bi-elliptical orbit). But I can't find the right answer.
In some book where rough images where presented I come to the idea they let arrive the probe at one of the nodes (longitude 49 or 229 degrees from first point of Aries).
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So [http://scienceworld.wolfram.com/physics … eorem.html]Lambert's Theorem just computes the transfer time. Why do you calculate the parabolic travel time?
Do you want the minimum delta V for every day of the year? Obviously one could just wait in orbit until a Hophman transfer or “Lo road” is available. For a single elliptical transfer, numeric minimization should be easy. It might be nice to make some graphs for each day of the year plotting where in the orbit of mars we arrive at vs, the travel time and the delta v. Any Lay person could then look at these graphs to get an idea of the available flight options. Bi elliptical transfers could be interesting and would not be too difficult to optimize numerically.
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I calculate the parabolic transfer time, just to get the bottom-line transfer-time for an elliptical orbit. Then I check my determined transfer-time, by guessed date's, and solve Lambert's Theorem iteratively backward. Then I get the two angles, that characterize an Lambertian transfer, and can also calculate the velocity-vectors of departure and arrival. Simply subtracting the vectors from the velocity-vectors of the planets give the delta-v.
But your suggestion to put some things in graphics, I think that will be the best solution. But I'm really surprised that I can't find more information about that cheaper Hohmann-II orbits. I should think even NASA uses this orbits frequently.
BTW the web-site you mentioned contains really much interesting orbital mechanics information. Maybe when I surf through it I'll find the clue. :laugh:
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The numbers I calculated which, I think, are the best.
Optimization only involves injection delta-V, because at Mars aerobraking is used. I is directed to first point of Aries, J is rectangular on I in prograde direction in the ecliptical plane and K is rectangular to the plane.
Depart date from earth: march 19, 2016
Arrival at Mars: jan. 20, 2017
Vector of depart (million km):
-148.988I +0.135J
Velocity (km/s):
-0.5113I -29.9007J
Vector of arrival (million km):
+189.540I +97.912J -2.597K
Velocity (km/s):
-10.2037I +23.6014J +.7379K
Travel time is 307 days, angle around sun 207.4 degrees.
Hyperbolic departure velocity:
+0.2500I -2.6799J +0.8627K abs: 2.826 km/s
Hyperbolic arrival velocity:
-2.5787I -4.5943J -1.2409K abs: 5.413 km/s
Moving departure and/or arrival dates results in more hyperbolic departure delta-v.
The precision of the numbers is too much. I used this precision to avoid loss of significance due to much calculating during the process.
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