NASASpaceFlight.com Forum
General Discussion => Q&A Section => Topic started by: JohnFornaro on 04/29/2011 02:55 pm
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This thread is to discuss the orbital mechanics of the Earth-Moon System, which I understand to be a different problem than, say, the Earth-Mars System.
I have several questions already:
What are some good texts on Orbital Mechanics? Somewhere around here, somebody mentioned a text known by the acronym of it's authors, BMW, but I couldn't find that reference. Jim had given me a good reference, but I can't remember it.
As to Belbruno orbits to the Moon. These low energy orbits go way beyond the Moon, by maybe a million km or so, and then swing back to be captured by the Moon's gravity. How can it be that they go past the Moon?
Why is the Earth-Moon orbital problem different from other orbital problems?
How do you read an ephemeris table to determine the start and finish time of your orbit?
A Hohman transfer to the Moon is the "typical" path to the Moon, right?
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This is the BMW book - you can find it on either the UK or the US Amazon store:
http://www.amazon.co.uk/Fundamentals-Astrodynamics-R-R-Bate/dp/0486600610/ref=sr_1_fkmr1_1?ie=UTF8&qid=1304094511&sr=8-1-fkmr1
I have never understood why this volume has never been updated. As well as Earth orbital mechanics, trans-lunar and trans-planetary trajectories are discussed. It is an excellent volume and has been part of my library since it was first published.
These days Vallado's book is normally recommended, but depending on your maths it is not as accessible as Fundamentals of Astrodynamics:
http://www.amazon.co.uk/Fundamentals-Astrodynamics-Applications-Technology-Library/dp/0792369033/ref=sr_1_2?ie=UTF8&s=books&qid=1304094511&sr=8-2
I found it much cheaper to buy Vallado new via the US Amazon than through the UK store.
Good luck!
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This thread is to discuss the orbital mechanics of the Earth-Moon System, which I understand to be a different problem than, say, the Earth-Mars System.
I don't think there is such a thing as the Earth Mars system. There's the Earth system aka the Earth moon system, there's the Mars system and there's the solar system including both Earth and Mars. The idea appears to be to consider a dominant object, with a secondary object if it has significant influence.
I have several questions already:
What are some good texts on Orbital Mechanics? Somewhere around here, somebody mentioned a text known by the acronym of it's authors, BMW, but I couldn't find that reference. Jim had given me a good reference, but I can't remember it.
The acronym stands for Bate, Mueller, White, the book is called Fundamentals of Astrodynamics.
As to Belbruno orbits to the Moon. These low energy orbits go way beyond the Moon, by maybe a million km or so, and then swing back to be captured by the Moon's gravity. How can it be that they go past the Moon?
They swing back under the influence of the Earth's gravity (they haven't escaped), but by the time they do the orbit has been perturbed by the small but noticeable influence of the Sun's gravity. As a result the orbit doesn't return to its original perigee, but to a point at roughly moon orbit radius. If you time it right, the moon will be there at the right time as well and then if all the velocities and angles are lined up correctly you will be weakly captured by the moon's gravity. A further maneuver is necessary to enter a stable orbit.
Why is the Earth-Moon orbital problem different from other orbital problems?
Systems with two or three major gravitational influences (Sun, Earth, moon) are more complicated than systems with fewer influences.
How do you read an ephemeris table to determine the start and finish time of your orbit?
Can't answer that one.
A Hohman transfer to the Moon is the "typical" path to the Moon, right?
Hohman-like transfers are the cheapest central body transfer orbits and they take 3-4 days, which is a reasonable duration.
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This paper is great: http://cbboff.org/UCBoulderCourse/documents/LunarCyclerPaper.pdf
My favorite sentence is this one:
"The trajectories were propagated using Cowell's method and a 7th-Order, 10-cycle Runga-Kutta integration scheme although for some of the near-collision transfers, a variation of parameters method was selected, with mean anomaly as the fast variable, for use during lunar encounter."
I disagree with their use of commas, but other than that you've gotta appreciate all the cromulent words :) To Google!
This is an interesting hit: "A Comparison of Cowell's Method and a Variation-of-Parameters Method for the Computation of Precision Satellite Orbits: Phase Three Results" http://ipnpr.jpl.nasa.gov/progress_report2/XI/XIH.PDF
Unfortunately the references don't tell us what Cowell's Method *or* Variation-of-Parameters Method are...
This hit is better: "A Study of Perturbation Effect on Satellite Orbit Using Cowell’s Method" http://www.dlr.de/iaa.symp/Portaldata/49/Resources/dokumente/archiv4/IAA-B4-1308P.pdf
It both explains the concept and has references! Following them for a while I discover this book: "Introduction to Perturbation Methods" by Mark H. Holmes (1995) which seems to be an updated and simplified version of "Introduction to perturbation techniques" by Ali Hasan Nayfeh.
As for Runga-Kutta integration, I expect they actually mean http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods
Fun eh?
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Are you really asking about "Cis-Lunar Orbital Mechanics" rather than Lunar Orbital Mechanics?" That is, are you interested in trajectories that are not strictly orbits around the Moon? If so, you might want to change the thread title (by editing the subject of the first message in the thread).
Assuming "yes", it is a fascinating topic. In more general terminology it is a "three-body problem," or in cases where the Sun's gravity cannot be neglected, a "four-body problem." The fascination with these stems from the many cases where the behavior is "chaotic" in the sense that arbitrarily small changes in initial conditions can lead to large differences in eventual outcomes. Loosely, this makes it difficult to know if any numerically calculated solution was calculated with sufficient precision to approximate the exact result.
That makes predictions about non-maneuvering objects difficult, but for practical matters a maneuvering spacecraft would be able to make tiny course corrections and get "back on track" even when faced with an unanticipated perturbation en-route to its destination.
Still the "dynamical systems" approach is apparently the way to go:
http://forum.nasaspaceflight.com/index.php?topic=18070.0
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Wow. Thanks for the replies. Will be looking into BMW shortly. Changed the thread title to Cis-Lunar. Not sure of the terms, but I get it that Earth-Moon is a "system", but that Earth-Mars is not.
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Double wow. BMW was 75 cents on Amazon.
Also, there's no "cents" key on my 1391401 keyboard!
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This paper is great: http://cbboff.org/UCBoulderCourse/documents/LunarCyclerPaper.pdf
My favorite sentence is this one:
"The trajectories were propagated using Cowell's method and a 7th-Order, 10-cycle Runga-Kutta integration scheme although for some of the near-collision transfers, a variation of parameters method was selected, with mean anomaly as the fast variable, for use during lunar encounter."
Runge-Kutta is as you have found, a method of numerically solving differential equations, or in other words throwing brute force computing power at a problem that is too hard or impossible to solve in closed form using algebra and calculus.
Cowell's method is the direct approach to the orbit numerical integration problem, where you integrate the equations of motion of all the objects directly. In other words you use Newton's law of gravitation to calculate the acceleration of all the objects, then integrate the acceleration to velocity and velocity to position. This commonly not even called Cowell's method, as it has been reinvented probably thousands of times by thousands of different people, including myself.
To do this numerical integration, you can use Euler's method, another straightforward simple algorithm reinvented thousands of times. Say you know the vector position and velocity of an object, and can use universal gravity and perhaps other effects to find the acceleration. We proceed with time steps - one second is a pretty typical size for Earth-orbit problems. Having figured the acceleration, you multiply it by the size of the time step to get the change in velocity, and add it to the velocity vector. Then you multiply the velocity by the time step and add it to the position vector. Repeat thousands of times, and you have integrated the equations of motion.
You can use Runge-Kutta to perform the integrations required. This is just a refinement on the Euler method above, which can dramatically increase accuracy of one step, which you can then use to either get a more accurate answer or take larger steps.
This is contrasted with Encke's method, which integrates the difference between the actual trajectory of a body in a n-body problem with its reference two-body orbit. Cowell is simpler, but Encke may take less computing power to get the same accuracy when the orbit of the body you are interested in (say a spacecraft) is in deep space far from any planet. You use Kepler's equations to propagate the main two-body motion and just integrate the difference from that reference trajectory. Since the perturbations are smaller, you can take larger time steps and still maintain accuracy. The equations of motion themselves are interesting. You still basically have Newton's laws, but instead of gravity from each object attracting your spacecraft to itself, you effectively have the spacecraft being repelled from the reference trajectory, where the force grows stronger, not weaker, as the distance from the reference grows.
A third method, mentioned in the quote above, is variation of parameters (VOP). In this case, the orbital elements themselves are treated as a body. In a perfect two-body system, the orbit elements are constant. When we talk about perturbing an orbit, VOP takes this literally. Its equations of motion describe how things like the periapse distance and inclination change in the presence of a perturbation. The equations of motion are vastly complicated and not related to Newton's laws at all. Derived from, yes. But the periapse has no inertia as we think of it for a physical body. The advantage of this is that in certain cases, the integration can be done with algebra and calculus. For instance, the "main problem" in low-Earth orbital mechanics, which just considers the Earth as a spheroid and considers the point-mass and J2 effects on an orbit. We use this to design things like sun-synchronous orbits.
The other thing is that when the external effects are small and predictable, you can dramatically reduce the computation requirements. For instance, consider the problem of determining how long it will take a low-Earth orbit to decay. In this case you have to take into account the gravity of the sun, moon, perhaps other planets, solar radiation pressure, and atmospheric drag. This makes the equations of motion quite complicated, requiring Runge-Kutta or some other integrator to handle. It takes 1000 time steps per orbit to maintain good accuracy with Cowell's method, perhaps somewhat less perhaps with Encke, but VOP can take one time step every 10 orbits!
So to summarize, I have discussed three kinds of equations of motion:
*Cowell's (the direct, obvious approach using Newton's laws of motion)
*Encke's (Integrate the differences only between a reference and actual trajectory)
*Variation of Parameters (Equations of motion of the orbit elements themselves)
and three kinds of integrators
*Closed form integration - Completely accurate when possible, but almost never possible and very difficult when it is - requires lots of pencil and paper
*Euler's method - the obvious numerical integration method
*Runge-Kutta - a more complicated but more accurate numerical integration method.
In theory you can mix and match any equation of motion with any integrator.
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There are (ISTM) two areas of this worth lots of attention. The first is how to answer questions like, "If I start here and propel myself like this, where do I end up?" The second is, "If I want to start here and end up there, how should I propel myself?"
Obviously these are linked. Once we have a candidate transfer trajectory between here and there, we want to verify (using e.g. numerical methods) that the proposed propulsion really will effect the transfer. There's another linkage though, that involves iterative searches where we propose a particular transfer, find out that it misses our intended destination, but then we use a "continuation method" (IIUC) that indicates a hopefully better proposed transfer to test out next.
I don't think I'm alone in thinking that the "here" of greatest interest is LEO. Choosing the best "there" to study can be tough. Lunar orbit is certainly one possibility. But the dynamics of the situation have led a lot of people to focus on the "there" of halo orbits around the EML1 and EML2 points.
These orbits have what are termed stable manifolds. I'm getting in over my depth here, so someone please correct me if I'm wrong, but it seems the stable manifold for a halo orbit is the set trajectories which lead onto the orbit without requiring propulsion. I want to say the existence of the stable manifold makes one of these orbits an "attractor." No normal two-body orbit is like that! This seems to make them potentially great locations for e.g. space stations, orbital depots, or just generic rendezvous locations.
Understanding how the math of this plays out, and popularizing that understanding, are going to be important activities in which spaceflight enthusiasts should be fully engaged.
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So to summarize, I have discussed three kinds of equations of motion:
*Cowell's (the direct, obvious approach using Newton's laws of motion)
*Encke's (Integrate the differences only between a reference and actual trajectory)
*Variation of Parameters (Equations of motion of the orbit elements themselves)
and three kinds of integrators
*Closed form integration - Completely accurate when possible, but almost never possible and very difficult when it is - requires lots of pencil and paper
*Euler's method - the obvious numerical integration method
*Runge-Kutta - a more complicated but more accurate numerical integration method.
In theory you can mix and match any equation of motion with any integrator.
Thanks! Another term I've heard a lot and can't find a reference for is "Chebyshev polynomials". Any ideas?
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There are (ISTM) two areas of this worth lots of attention. The first is how to answer questions like, "If I start here and propel myself like this, where do I end up?" The second is, "If I want to start here and end up there, how should I propel myself?"
The first question is an (arbitrary complicated) equation of motion. The second question is a dynamic optimization problem. The solution is a function of n parameters (your control parameters) in an m space (your optimization problem). It's probably the you should look for different methods of the H-control optimization.
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Another term I've heard a lot and can't find a reference for is "Chebyshev polynomials". Any ideas?
You could start with Numerical Recipes (http://apps.nrbook.com/fortran/index.html). It's not the latest word in numerical techniques, but it gives nice introductions to many topics that, despite the title, go beyond cookbook formulations but also don't go off the theoretical deep end.
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Thanks!
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...If I want to start here and end up there, how should I propel myself?...
That's exactly the question I'm asking. Assuming one's "generic" rocket can put 10,000kg into a 400km orbit, what kind of payload can be landed at the crater rim on the north pole of the Moon? I believe that the simple answer is 1,00kg. Which gets me asking, what then would the 9,000kg EDS be like? What's the transfer orbit/trajectory like? What would be the optimizations? So...
Understanding how the math of this plays out, and popularizing that understanding, are going to be important activities in which spaceflight enthusiasts should be fully engaged.
...which is where I'm headed, for if I can figure this out, then I could explain it. On another thread, somebody opined that the knowledge of this field was "controlled", somehow, and Jim replied that the knowledge of rocketry is well out in the public domain, no conspiracy required. And I'd say, it's even better than that. With a song now going for 99 cents, and BMW offered for three quarters of that, truly this knowledge can be had for a song. The hard part is understanding the math.
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See what you can find on patched conics.
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This is contrasted with Encke's method, which integrates the difference between the actual trajectory of a body in a n-body problem with its reference two-body orbit. Cowell is simpler, but Encke may take less computing power to get the same accuracy when the orbit of the body you are interested in (say a spacecraft) is in deep space far from any planet.
Thanks for a good explanation.
To put into perspective, how much computing power? If you put a Quad Core onto it, using Cowell with 1 second iterations, how fast can you progress?
I suppose the measure is compute time / orbit time.
I just started a thread on orbits using slow SEP. Here you have an optimisation problem overlaid onto the Cowell algorithm. In other words, you have to run numerous Cowell algorithms to come to an optimum solution using some form of branch and bound - I assume?
What does NASA do if all the mathematicians have gone to Wall Street to devise casino algorithms?
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Thanks! Another term I've heard a lot and can't find a reference for is "Chebyshev polynomials". Any ideas?
Another kind of mathematical series that is useful for approximating classes of unknown functions (and the exact solution to the Chebyshev differential equations). Similar to how a Taylor series is comprised of terms that are elementary polynomials (x^n), a Fourier series of terms of cos(n*w_n*x), a Chebyshev series is in terms of the Chebyshev polynomials T_n(x), where these particular polynomials can be derived from either a recurrence relation or a trig identity. The first few Chebyshev polynomials have nice simple forms (e.g., T_0(x) = 1, T_1(x) = x, T_x(x) = 2x^2 - 1).
The Chebyshev series are rather well-behaved and in certain cases nicer to use than Taylor or Fourier series. WP and Mathworld (mathworld.wolfram.com) provide introductions.
Fun fact: they arise in practical cryogenics, e.g. liquid hydrogen, since some kinds of temperature sensors' response functions are well-described by such series, and IIRC NIST likes to use them for calibrations and hence traceability.
-Alex
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I'm using Chebyshev polynomials heavily in my PhD thesis to convert some rather complicated trigonometric functions into polynomial form.
Since everyone seems to be showing off their knowledge ;) I've always just thought of them as a resampled Fourier series (sampling density proportional to sqrt(1 - x^2)). They are often used for polynomial fitting as a way to approximate the min-max polynomial for a given function. Because of the increased sampling density at the periphery, a fit using Chebyshev polynomials is resistant to Runge's phenomenon.
What does NASA do if all the mathematicians have gone to Wall Street to devise casino algorithms?
It's not all speculation FWIW. Some of those people, like Arbitrageurs, actually help the markets to function better and are providing a legitimate service. I wouldn't mind a job like that!
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My jaw has been hanging open for the last several posts, as the various calculation techniques have been listed. The math is so out of my reach at the moment. But I have been reading a bit.
From "Celestial Mechanics and Astrodynamics",1964, Victor Szebehely, editior, p. 56, there is a 36 hour trajectory to the Moon suggested. It's seems like a lead and shoot trajectory. Lead the Moon by a certain amount, fire off the rocket, and hit (not land, necessarily), 36 hours later.
It prompted me to ask about the delta-vee penalty for flying to the Moon, basically as fast as possible.
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This seems like an appropriate thread.. anyone know the minimum required thrust/weight ratio to enter into lunar orbit?
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I've always just thought of them as a resampled Fourier series (sampling density proportional to sqrt(1 - x^2)). [ ...] Because of the increased sampling density at the periphery, a fit using Chebyshev polynomials is resistant to Runge's phenomenon.
Yeah, that's a sweet property and makes them worth using instead of the "I know, computers will let me brute-force it the simple way using a 20th-order Taylor series!" response. But they still experience Gibb's phenomena ... just taking a guess, that seems reasonable because Gibbs is obviously sorta fundamental in a Fourier-information sense. So Runge's is an inexact analogy?
-Alex
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This seems like an appropriate thread.. anyone know the minimum required thrust/weight ratio to enter into lunar orbit?
Why would there be one?
For low-lunar orbits, the orbital stability time due to mascons may put a lower bound on the practical T/W ratio, but that would be highly trajectory dependent. But for high orbits, or (IIRC) polar orbits?
-Alex
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Well clearly you have to deliver the required amount of delta-v on the right side of the Moon.. so there has to be a minimum burn duration. That's typically derived from thrust/weight of the engine.
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Well, now I have a few new questions. I'm trying to understand the way shadows are cast at the lunar poles.
Here's the scenario. There's a fairly detailed PDF of Whipple crater at the north pole, floating on these forums and the internets. I have parsed out the contour lines of that drawing and have created an AutoCAD R14 drawing, attached, showing the contours of that area in 3-D. At the top of the crater is a 400m mini-plateau, where I'd place a solar power generator. If you strike a line from the north pole at 5 degrees +/-, it shoots right down the wall of the crater to the "deep end", where supposedly ice is to be found. This is a tentative road location to the crater floor.
However, more or less due north, over the pole and beyond, is another plateu, which may shade my solar power generator every two weeks or so, depending on all the ecliptical conditions, which are fairly convoluted.
So, the Earth is tilted 23.44 degrees +/- from the ecliptic. The Moon is tilted 5.14 degrees from the ecliptic. Then, the Moon's axis is tilted 6.68 degrees from its orbital plane, and libration adds or subtracts about 1.5 degrees from that! Sheesh.
Knowing from the PDF where the peaks are, I'm trying to create two or four conditions to understand the two or four extreme conditions of solar incidence and thus figure out the shadows. This will also indicate the extent of the crater which is permanently in shadow.
I'm thinking it's a relatively simple exercise in geometry.
1. Place the Moon, Earth, and Sun at one axial extreme.
2. Investigate the shadow creation between the two peaks.
3. Place the Moon, Earth, and Sun at the other axial extreme.
4. Investigate that shadow creation between the two peaks.
Hopefully, this is the "easy" approach? It's not meant to predict dates, just to understand the limits of the shadows at the poles.
It's not orbital mechanix, I guess, but what is this investigation called?
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I think for this purpose, you can ignore the fact that the moon revolves around the earth (provided you're willing to ignore lunar eclipses). Just think of the moon as a body which rotates every 29.5 days with respect to the sun about an axis that is inclined 1.5424 degrees to the ecliptic (according to Wikipedia). If you're standing right at a lunar pole, then over the course of a 29.5-day lunar month, the sun will appear to execute a circle around you, its center rising as much as 1.5 degrees above the horizon and dropping as low as 1.5 degree below it. The directions in which you see the sun above the horizon will depend on the time of year.
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So speaking of Cis-Lunar Orbital Mechanics. I was troubling over a lunar mass-driver launching cargo to EML1. However it donned on me that launching a cargo from the lunar surface towards EML1 would put it the vector completely wrong to be captured into a halo orbit.
Could position of the mass-driver on the lunar surface, and aim point at launching(slightly leading the target) allow for miniscule vector change? Or will decent amount of burn be necessary to achieve a halo orbit around EML1?
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Not sure if this should be asked here or in historical spaceflight.
Does anyone know the actual velocities of the Apollo spacecraft (Moon and Earth relative) on crossing the boundary of the "Moon's sphere of influence" during both trans-Lunar and trans-Earth coast? If memory serves me right I think that they were of the order of a few hundred miles per hour, but I'm curious to know more accurate values.
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Ooh, that's a great question! Sorry I have no immediate answer.
Consider if you will the rotating frame of reference in which the Earth and Moon are at fixed locations. Would you be satisfied with finding the minimum velocity along the trajectory when viewed in that frame? I ask because the answer to that seems easy enough to estimate. We know the times Apollo took to traverse those trajectories, so we could recreate the trajectories in a CR3BP approximation using numerical analysis, (e.g. Runge–Kutta methods) and find the velocity minima. This approach wouldn't require explicitly calculating a sphere of influence....
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So speaking of Cis-Lunar Orbital Mechanics. I was troubling over a lunar mass-driver launching cargo to EML1. However it donned on me that launching a cargo from the lunar surface towards EML1 would put it the vector completely wrong to be captured into a halo orbit.
Could position of the mass-driver on the lunar surface, and aim point at launching(slightly leading the target) allow for miniscule vector change? Or will decent amount of burn be necessary to achieve a halo orbit around EML1?
Why not have the lunar surface mass driver propel the cargo onto a ballistic trajectory that leads onto a halo orbit, with no additional propulsive maneuver required? Parker showed trajectories like these can be found with transit times on the order of 100 days. The velocity required upon leaving the lunar surface might be higher, but wouldn't the payload simplicity make that worthwhile?
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Ooh, that's a great question! Sorry I have no immediate answer.
Consider if you will the rotating frame of reference in which the Earth and Moon are at fixed locations. Would you be satisfied with finding the minimum velocity along the trajectory when viewed in that frame? I ask because the answer to that seems easy enough to estimate. We know the times Apollo took to traverse those trajectories, so we could recreate the trajectories in a CR3BP approximation using numerical analysis, (e.g. Runge–Kutta methods) and find the velocity minima. This approach wouldn't require explicitly calculating a sphere of influence....
Thanks for the reply, that would be a great start. What frame of reference would you use? One of the things I was particularly intrigued to see was the difference in the values wrt Earth and the Moon.
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It's easy to choose the frame if we make the simplifying assumption that the Earth and Moon are each in circular orbits with the same period, centered at their combined gravitational barycenter. Since the orbits are circular it is possible to use the barycenter as the origin and rotate the reference frame such that the Earth and Moon do not move.
Kirk Sorensen provided for awhile a java based web app that used this approach, and it was highly informative. (It was at astrojava.com, I think, but it's gone now.) He seems to have mostly moved on from spaceflight to solving the global energy crisis, but his java source code if it were available would be a fine place to start.
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Pitty its no longer available, I wonder if another more basic approach would work?
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Pitty its no longer available, I wonder if another more basic approach would work?
Patched conic sections.
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Thanks.
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Hi, I was interested in understanding (in laymans terms) what was involved, from an orbital mechanics perspective, of landing something at the lunar poles.
* especially with regard to the relative difficulty of accessing poles and equator.
* and where the answer differs for manned and unmanned.
(I couldnt find this in the Q&A but perhaps I just did not know the terms to search for, in which case a link would be appreciated, and make sense to include here anyway)
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Regarding getting to the poles versus getting to the equator, here's how I like to think about it. Forget that figure-of-eight diagram we've all seen and try thinking of it this way....
The usual way to get a spacecraft from the earth to the moon on a fast (few days) trajectory is to first put the spacecraft into a highly-elliptical earth orbit, with an apogee somewhat beyond the moon. Departure from earth (or from low earth orbit) is timed so that the spacecraft will pass not far in front of the moon as the spacecraft nears apogee.
At apogee, the spacecraft is moving slowly with respect to earth -- maybe a couple of hundred meters per second. The moon, however, is orbiting at about 1 km/s. With respect to the earth, then the spacecraft is just kind of hanging there and the moon comes by and sweeps it up. From the moon's point of view, the spacecraft is zooming roughly toward it at about 1 km/s. That's at a distance where escape velocity is below 1 km/s, so the spacecraft must slow down with respect to the moon if it is to go into lunar orbit.
If the spacecraft is in the plane of the moon's orbit when it reaches the lunar vicinity, it will go roughly into an equatorial orbit about the moon. If left earth on an ellipse sightly inclined to the moon's orbit, then on reaching the moon it may be a few lunar radii above or below the plane of the moon's orbit. In this case, when it's swept up by the moon, it may pass over one of the lunar poles and go into a polar orbit. If it goes into a polar orbit, it won't pass behind the moon as seen from earth for the first few orbits. As the moon revolves about earth while the orientation of the spacecraft's orbit remains pretty much fixed, the spacecraft will eventually pass behind the moon as seen from earth. A quarter of a lunar month after arriving at the moon, it will be spending about half of each orbit behind the moon, if it's in a low orbit. This means, of course, that a polar orbit in general lines up for easy return to earth only twice a month, whereas in an equatorial orbit there's a departure opportunity on every orbit.
The other key point is that reaching the lunar poles does not require departing from a polar orbit around earth.
Robotic spacecraft might take chaotic trajectories involving multiple passes by the moon. This can allow them to use less propellant to get into lunar orbit, but the trip times are measured in months (Ed Belbruno, inventor of the technique, gave a FISO lecture about it (http://spirit.as.utexas.edu/~fiso/telecon/Belbruno_1-4-12/)). LCROSS used a trick like this to impact the lunar pole on a nearly-vertical trajectory.
Another difference between robotic and human missions could be a desire for a free-return trajectory for the human mission. I've read that for polar mission, free return requires quite a distant orbit about the moon, with a pericynthion of well over 10,000 km.
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Proponent, thanks for a very easily visualized explanation of the orbit, using the English language. Prompts me to ask a few questions myself:
I've attached P.324-325 of BMW.
r bar. First, is that how you say it?
2. r bar is a vector from the center of the Earth to the center of the Moon. It varies according to the eccentricity of the Moon's orbit. If you subtract the Moon's radius, and the Earth's radius, that's the distance from surface to surface at that time?
3. What's the value of r bar? What does knowing it enable you to calculate?
4. The Moon rotates from West to East around the Earth, from the Earth's POV. But the line of nodes rotates East to West, once every 18.6 years? Does this affect launch windows much? Particularly to the poles?
5. X sub-e is the Earth's vernal equinox. Is that the direction of the Earth's rotation around the Sun, or does it point to a star? It's supposed to be a fixed inertial frame, right? What does Z sub-e point to?
Now the hard one.
6. In launching from the Cape at 28.5 degrees to a parking orbit, where and when do you make the plane change so that you end up being grabbed by the Moon in a near polar orbit? Are any further inclination adjustments necessary once you get in LLO?
If you have the time and inclination, so to speak.
Thanks
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I've attached P.324-325 of BMW.
r bar. First, is that how you say it?
Well, that's how I say it, but I'm no authority.
2. r bar is a vector from the center of the Earth to the center of the Moon. It varies according to the eccentricity of the Moon's orbit. If you subtract the Moon's radius, and the Earth's radius, that's the distance from surface to surface at that time?
Sounds right to me, with the very minor proviso that neither the earth nor the moon is perfectly spherical, so the effective "radius" will vary a bit with orientation.
3. What's the value of r bar? What does knowing it enable you to calculate?
Well, for example, if you've got a spacecraft in cis-lunar space, you're going to need to know where the earth and the moon are in order to determine their gravitational pulls on the spacecraft.
4. The Moon rotates from West to East around the Earth, from the Earth's POV. But the line of nodes rotates East to West, once every 18.6 years? Does this affect launch windows much? Particularly to the poles?
As the line of nodes regresses, the moon maintains a fixed inclination of about 5o with respect to the ecliptic (the plane of the earth's orbit about the sun). That means that the inclination of the of the moon's orbit with respect to the earth's equator varies from about 18o to about 28o as the nodes regress. That obviously affects the trajectory one needs to reach the moon, but I'm not sure I'd say it affects launch windows as such. It seems to me it should be a little easier to get to the moon when its inclination matches the latitude of your launch site, but I don't think it's going to be a big deal. More about this later.
5. X sub-e is the Earth's vernal equinox. Is that the direction of the Earth's rotation around the Sun, or does it point to a star? It's supposed to be a fixed inertial frame, right? What does Z sub-e point to?
Xε is the direction from Earth to the sun at the moment spring begins in the northern hemisphere. Yε lies 90o from Xε in the plane of Earth's orbit. It points approximately opposite the Earth's velocity (if the Earth's orbit were perfectly circular, then it would be exactly opposite). Zε is perpendicular to both Xε and Yε.
Now the hard one.
6. In launching from the Cape at 28.5 degrees to a parking orbit, where and when do you make the plane change so that you end up being grabbed by the Moon in a near polar orbit? Are any further inclination adjustments necessary once you get in LLO?
Bear in mind that the inclinations of the trans-lunar trajectories for low lunar polar orbit and for low equatorial lunar orbit differ by an amount on the order of the lunar radius divided by the distance from the earth to the moon, i.e., by about (1700 km)/(380,000 km) = 0.045 radian = 0.26o. Anyway, it seems to me it would be optimal to take off from the Cape into the plane you want. So, you make the plane change during ascent to the parking orbit. But the plane of the trans-lunar orbit isn't all that important anyway. The important thing is to arrive in the vicinity of the moon at the right time and place. To be captured, it certainly helps to have your velocity on arrival lined up with the moon's velocity so that you velocity with respect to the moon is minimized. But since, in the earth-centric frame, you're moving slowly on arrival in the moon's sphere of influence, it's not that crucial -- most of the moon-centric velocity arises from the moon's motion around the earth.
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Thanks Proponent.
Those look like some pretty significant difficulties for manned polar missions. If I understand it:
(*) Difficulty of free return. (though I still dont really have any quantitative idea of how difficult, anyone care to speculate?)
(*) Lack of any-time return. You only have an option twice a month (unless your ascent vehicle can also take you all the way home without having to meet anything in lunar orbit)
On the other hand, robotic does not sound much harder. Anyone prepared to estimate the difference in difficulty of robotic equatorial vs polar missions as a factor of cost? :)
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Many thanks. BMW leaves many significant questions to the teacher to answer. I'm old fashioned in the sense that I ask the book to tell me everything; when it doesn't, I struggle.
5. X sub-e is the Earth's vernal equinox. ...
Xε is the direction from Earth to the sun at the moment spring begins in the northern hemisphere. Yε lies 90o from Xε in the plane of Earth's orbit. It points approximately opposite the Earth's velocity (if the Earth's orbit were perfectly circular, then it would be exactly opposite). Zε is perpendicular to both Xε and Yε.
This is not mentioned in BMW at all. I guess the authors figure everyone knows that already.
My understanding has always been that the Earth rotates counterclockwise around the Sun, seen from the Earth's north pole. Therefore -Yε is in the direction of the Earth's motion! This is starting to make sense! [Edit: -Yε !! Not enough attention to detail.]
4. The Moon rotates from West to East around the Earth, from the Earth's POV. But the line of nodes rotates East to West, once every 18.6 years? Does this affect launch windows much? Particularly to the poles?
As the line of nodes regresses, the moon maintains a fixed inclination of about 5o with respect to the ecliptic (the plane of the earth's orbit about the sun). That means that the inclination of the of the moon's orbit with respect to the earth's equator varies from about 18o to about 28o as the nodes regress. That obviously affects the trajectory one needs to reach the moon, but I'm not sure I'd say it affects launch windows as such. It seems to me it should be a little easier to get to the moon when its inclination matches the latitude of your launch site, but I don't think it's going to be a big deal. More about this later.
A key concept with lunar launch windows is the antipode, the point on the Earth's surface opposite the Earth-Moon vector. TLI must occur at or near the point where the orbit crosses the antipode. The antipode moves east-to-west along the Earth's surface, largely driven by the Earth's rotation but slightly counteracted by the moon's motion around the Earth. If the moon's declination is less than the launch site latitude, a due east launch (90 deg azimuth) will intersect the antipode twice per day. For Apollo these were called the Atlantic and Pacific injection opportunities. Apollo used variable-azimuth launch targeting (72-108 degrees, generally) to widen the launch window to around 2.5 hours.
EOR changes things a bit. Because the orbit plane is fixed, there's no way to widen the launch window - you get two injection opportunities per month, when the orbit plane crosses the antipode. And that only if the orbit inclination is greater than the moon's declination. So you don't want to do EOR for a lunar mission in an orbit of less than 28 degrees inclination. Otherwise there will be portions of the moon's nodal cycle where the antipode *never* crosses the orbit plane.
Thanks for this too, but it is harder to understand. For one thing BMW's discussion doesn't discuss the antipode at all.
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Those look like some pretty significant difficulties for manned polar missions. If I understand it:
(*) Difficulty of free return. (though I still dont really have any quantitative idea of how difficult, anyone care to speculate?)
Attached to this post (http://forum.nasaspaceflight.com/index.php?topic=13543.msg868568#msg868568) is a paper from the early Apollo era that discusses lunar free-return trajectories, including those of the polar variety.
But NASA dropped free-return trajectories after Apollo 12 and even after Apollo 13 didn't reinstate them. My guess is that they're not really necessary, except maybe for initial flights with a new systems.
(*) Lack of any-time return. You only have an option twice a month (unless your ascent vehicle can also take you all the way home without having to meet anything in lunar orbit)
Or unless you can change planes, as Jorge described for Constellation, or unless you stage at a Lagrange point. The catch is that at minimum delta-V, the transit time between the lunar surface and L1/2 is something like a day or two (and the lander's delta-V obviously needs to be larger than if you're staging in LLO). I would think that if you're going to and from a lunar base, this probably isn't such an issue. For sortie missions, in which a lightly-built lander might need to get the crew to safety when solar flare looms, it might be a bigger deal.
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My understanding has always been that the Earth rotates counterclockwise around the Sun, seen from the Earth's north pole.
Yes, although one usually says that the earth revolves rather than rotates about the sun.
Therefore -Xε is in the direction of the Earth's motion! This is starting to make sense!
Not following you here. At the moment of the vernal equinox, the direction of the earth's motion is approximately in the -Yε direction. Over the course of a year, the direction of motion changes, although the Zε component is always zero. The motion is in the -Xε direction sometime around the winter solstice. If the earth were moving in the -Xε direction at the vernal equinox, it would be moving straight away from the sun.
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The hardest part of following BMW is the notation. I'm used to a different one when doing dynamic optimization problems, and it's extremely difficult to switch. Imagine two languages that use the same words but have different meanings in each.
I highly recommend to start reading bit by the annexes, specially the notation one. And only then start reading.
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Thanks again.
My understanding has always been that the Earth rotates counterclockwise around the Sun, seen from the Earth's north pole.
Yes, although one usually says that the earth revolves rather than rotates about the sun.
Right. I knew that. Sloppy language.
Therefore -Yε is in the direction of the Earth's motion! This is starting to make sense!
Not following you here. [/quote]
Right. Went back to the illustration, and corrected my handwritten notes on that page. I pasted the wrong term. Edited my post while I was at it.
Turns out, the Constellation Aries (the ram) is more or less where -Xε points to at the vernal equinox. So, if you had a star finder linked to this constellation, is that how you'd keep your bearings on your inertial frame of reference?
And it has the words, "Don't Panic!" written in large friendly letters across its cover.
Hah! Your copy does.
I highly recommend to start reading bit by the annexes [appendices?], specially the notation one. And only then start reading.
Thanks, but Appendix C only reviews vector notation. Chapter 2.3 goes over the Classical Orbital Elements. So I'll just keep re-reading stuff.
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CxP's solution to both problems was a three-burn sequence for both LOI and TEI. LOI-1 placed the Orion/Altair stack in a highly elliptical, low inclination LLO (thus allowing economical free-return). LOI-2 was a plane-change burn at first apogee post-LOI-1, then LOI-3 circularized. For TEI, Orion would do basically the same thing in reverse.
I can see how that works nicely for landings at the poles, but what if you want to land at a latitude of, say, 45o? With LOI-3, Orion/LSAM circularizes into an orbit inclined at least 45o to the lunar equator, and the LSAM lands. Let's say we need to abort 7 days later, by which time the moon has rotated by about 90o. How do we get Orion and LSAM into the same plane, so they can rendezvous? Does Orion (or LSAM?) have the delta-V reserve for a big additional plane change?
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Turns out, the Constellation Ares (the ram) is more or less where -Xε points to at the vernal equinox.
You mean +Xε, right?
So, if you had a star finder linked to this constellation, is that how you'd keep your bearings on your inertial frame of reference?
In principle, you need to know just the directions to two objects whose co-ordinates are known. Then you can orient yourself. Even though the fundamental reference point, the vernal equinox, is in Ares, there's no particular reason you have to be able to see it.
Consider navigation on Earth, for example. In the western world, we generally take north as the fundamental direction, and measure bearings with respect to north. But if you had a device that pointed, say, east, that would do just as well -- once you know where east is, just turn 90o to your left and you're facing north.
EDIT: Grammar.
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You mean +Xε, right?
Yes, dammit! Apparently I know less about cutting and pasting than astrology...
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Yes, dammit! Apparently I know less about cutting and pasting than astrology...
:o
Emphasis mine.
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And about starfinders. Ok, so you pick two or more convenient stars, apparently on brightness? Not necessarily a astrological "constellation", but a constellation in that it's more than one star.
What stars do they pick, typically?
Also, I thought I remembered seeing a surplus rad hardened satellite starfinder for sale on Ebay once. Just took a googol and could only find terrestrial ones. What are the starfinders used on satellites called? Can you buy them?
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Yes, dammit! Apparently I know less about cutting and pasting than astrology...
:o
Emphasis mine.
Yeah. I've misplaced the sign twice now. And I'm studying the orbital stuff even as we speak!
When the Moon is in the second house,
And Jupiter aligns with Mars....
La la la la la-la la...
Age of Aquarius.....
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Many thanks. Here's some linx:
http://www.sodern.com/sites/en/ref/Hydra_50.html
http://www.ballaerospace.com/page.jsp?page=104
http://starbrite.jpl.nasa.gov/pds/viewInstrumentProfile.jsp?INSTRUMENT_ID=a-star&INSTRUMENT_HOST_ID=CLEM1
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Has there ever been any analysis of rendezvous during trans-lunar coast? I'm imagining a concept where the crew vehicle leaves Earth-orbit on a free return trajectory and then is met by another (presumably automated) vehicle along the way to the Moon.
The utility of this would come from the other vehicle providing the propulsion for lunar orbit insertion. That other vehicle could also be the lander, with added descent propulsion system propellant to handle the LOI burn.
For this scheme to work there would need to be a cis-lunar trajectory in which the lander could loiter; perhaps that's a continual free-return figure-eight? The crew vehicle would time its trans-lunar insertion so it met up with lander on one of its Earth passes....
Is this wacky? Does it overly constrain the crew vehicle launch or TLI timing?
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Has there ever been any analysis of rendezvous during trans-lunar coast? I'm imagining a concept where the crew vehicle leaves Earth-orbit on a free return trajectory and then is met by another (presumably automated) vehicle along the way to the Moon.
The utility of this would come from the other vehicle providing the propulsion for lunar orbit insertion. That other vehicle could also be the lander, with added descent propulsion system propellant to handle the LOI burn.
For this scheme to work there would need to be a cis-lunar trajectory in which the lander could loiter; perhaps that's a continual free-return figure-eight? The crew vehicle would time its trans-lunar insertion so it met up with lander on one of its Earth passes....
Is this wacky? Does it overly constrain the crew vehicle launch or TLI timing?
Read the t/Space PDF. Their tankers were to add propellants to the CEV in elliptical lunar orbit.
exploration.nasa.gov/documents/reports/cer_final/tSpace.pdf
Something like an ACES stage could possible do this if it were developed.
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Has there ever been any analysis of rendezvous during trans-lunar coast? [...] The utility of this would come from the other vehicle providing the propulsion for lunar orbit insertion.
Read the t/Space PDF. Their tankers were to add propellants to the CEV in elliptical lunar orbit.
exploration.nasa.gov/documents/reports/cer_final/tSpace.pdf
You are right, the t/Space concept is similar. The 50x50000 km lunar orbit it uses is entered with a LOI burn of 310 m/s. Perhaps a rendezvous there makes more sense than a rendezvous during trans-lunar coast.
I certainly admire the t/Space proposal for the boldness of its approach! The concept includes so many innovative ideas!
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Page 15 has a refuel in free-return-trajectory mission mode.
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Page 15 has a refuel in free-return-trajectory mission mode.
Thanks for pointing that out! That they considered it shows the idea isn't too wacky. The comparison chart on page 17 is quite helpful. Does the note that in this mode the CEV, "Requires synchronized launch with Tanker" indicate they are -- or are not -- envisioning something like a continual figure-eight loiter orbit for the tanker? It could be tough to arrange "launching within a few days of one another," if that's what they mean by "synchronized."
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Page 15 has a refuel in free-return-trajectory mission mode.
Thanks for pointing that out! That they considered it shows the idea isn't too wacky. The comparison chart on page 17 is quite helpful. Does the note that in this mode the CEV, "Requires synchronized launch with Tanker" indicate they are -- or are not -- envisioning something like a continual figure-eight loiter orbit for the tanker? It could be tough to arrange "launching within a few days of one another," if that's what they mean by "synchronized."
CEV's TLI burn would need to take place at a given time that the CEV and tanker would arrive at the same place in space headed to the moon. Tanker would have had it's TLI burn much earlier do to it's path for the weak stability boundary transfer.
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Maybe I missed it, but page 9 says:
This performance is possible by Tankers taking a Weak Stability Boundary
Transfer from LEO to an elliptical lunar orbit. Almost no deltaV is needed
to enter lunar orbit using WSBTs, but the round-trip transit is six months.
How were they avoiding boiloff of LH2?
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Maybe I missed it, but page 9 says:
This performance is possible by Tankers taking a Weak Stability Boundary
Transfer from LEO to an elliptical lunar orbit. Almost no deltaV is needed
to enter lunar orbit using WSBTs, but the round-trip transit is six months.
How were they avoiding boiloff of LH2?
That was my question I had back in 2005 when I read this.
I think that was one of the reasons this was not chosen.
So the WSBT's would work better for hypergolics.
For return or disposal of the tanker it is better to have the propellant transfer in elliptical orbit. However for the lander for a better performance ( down mass to surface ) it would be better to have the propellant transfer in LLO.
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MethaLox should be OK with a sunshield, once outside LEO.
Cheers, Martin
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Lunar mascons can bring a satellite in LLO down. I guess, in theory, there will be orbits in which they could be used to pump satellites into higher orbits.
Could such an effect be useful in terms of saving fuel when arriving at or departing Lunar orbit?
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Not strictly cis-lunar orbital mechanics but related. On the Apollo missions PTC (barbecue roll) was established when in cis-lunar space. I believe that the CSM X-axis was near perpendicular to the line of sight to the sun. Does anyone know the attitude more precisely and why it was chosen? Was the X-axis aligned approximately pointing Earth North or South?
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Was the X-axis aligned approximately pointing Earth North or South?
the difference is a matter of a few degrees
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Program Constellation selected a 3 burn LOI maneuver with highly elliptical intermediate Lunar orbit (with 24 hours orbital period).
If somebody need an electronic version of BMW book you are welcome to ask.
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Was the X-axis aligned approximately pointing Earth North or South?
the difference is a matter of a few degrees
Thanks Jim, that's what I suspected. Do you know if that was North or South or did it vary? Was the choice purely arbitrary?
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The usual way to get a spacecraft from the earth to the moon on a fast (few days) trajectory is to first put the spacecraft into a highly-elliptical earth orbit, with an apogee somewhat beyond the moon. Departure from earth (or from low earth orbit) is timed so that the spacecraft will pass not far in front of the moon as the spacecraft nears apogee.
At apogee, the spacecraft is moving slowly with respect to earth -- maybe a couple of hundred meters per second. The moon, however, is orbiting at about 1 km/s. With respect to the earth, then the spacecraft is just kind of hanging there and the moon comes by and sweeps it up. From the moon's point of view, the spacecraft is zooming roughly toward it at about 1 km/s. That's at a distance where escape velocity is below 1 km/s, so the spacecraft must slow down with respect to the moon if it is to go into lunar orbit.
If the spacecraft is in the plane of the moon's orbit when it reaches the lunar vicinity, it will go roughly into an equatorial orbit about the moon. If left earth on an ellipse sightly inclined to the moon's orbit, then on reaching the moon it may be a few lunar radii above or below the plane of the moon's orbit. In this case, when it's swept up by the moon, it may pass over one of the lunar poles and go into a polar orbit. If it goes into a polar orbit, it won't pass behind the moon as seen from earth for the first few orbits. As the moon revolves about earth while the orientation of the spacecraft's orbit remains pretty much fixed, the spacecraft will eventually pass behind the moon as seen from earth. A quarter of a lunar month after arriving at the moon, it will be spending about half of each orbit behind the moon, if it's in a low orbit. This means, of course, that a polar orbit in general lines up for easy return to earth only twice a month, whereas in an equatorial orbit there's a departure opportunity on every orbit.
The other key point is that reaching the lunar poles does not require departing from a polar orbit around earth.
I recently described a path to the lunar poles (http://hopsblog-hop.blogspot.com/2013/08/lunar-ice-vs-neo-ice.html). Still a work in progress. I believe what you've described is very similar to what I envision.
I am wondering if there are more formal descriptions of this route.
For reaching the lunar poles some people seem to imagine parking in a low inclination low lunar orbit and doing a 90 degree plane change from LLO. I believe this is what John DeLaughter envisions when he claims reaching the lunar poles takes 8 km/s (http://venera4.wordpress.com/2013/08/17/moon-first/). It bothers me when people want to do plane changes deep in a gravity well.
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The usual way to get a spacecraft from the earth to the moon on a fast (few days) trajectory is to first put the spacecraft into a highly-elliptical earth orbit, with an apogee somewhat beyond the moon. Departure from earth (or from low earth orbit) is timed so that the spacecraft will pass not far in front of the moon as the spacecraft nears apogee.
At apogee, the spacecraft is moving slowly with respect to earth -- maybe a couple of hundred meters per second. The moon, however, is orbiting at about 1 km/s. With respect to the earth, then the spacecraft is just kind of hanging there and the moon comes by and sweeps it up. From the moon's point of view, the spacecraft is zooming roughly toward it at about 1 km/s. That's at a distance where escape velocity is below 1 km/s, so the spacecraft must slow down with respect to the moon if it is to go into lunar orbit.
If the spacecraft is in the plane of the moon's orbit when it reaches the lunar vicinity, it will go roughly into an equatorial orbit about the moon. If left earth on an ellipse sightly inclined to the moon's orbit, then on reaching the moon it may be a few lunar radii above or below the plane of the moon's orbit. In this case, when it's swept up by the moon, it may pass over one of the lunar poles and go into a polar orbit. If it goes into a polar orbit, it won't pass behind the moon as seen from earth for the first few orbits. As the moon revolves about earth while the orientation of the spacecraft's orbit remains pretty much fixed, the spacecraft will eventually pass behind the moon as seen from earth. A quarter of a lunar month after arriving at the moon, it will be spending about half of each orbit behind the moon, if it's in a low orbit. This means, of course, that a polar orbit in general lines up for easy return to earth only twice a month, whereas in an equatorial orbit there's a departure opportunity on every orbit.
The other key point is that reaching the lunar poles does not require departing from a polar orbit around earth.
I am wondering if there are more formal descriptions of this route.
Still Googling....
http://www.youtube.com/watch?v=GCa_mHYK_Ik
A You Tube video of LRO's path. It seems to have a an apogee a little in front of and below the moon as Proponent and I imagine. Looks like it's hyperbolic orbit wrt moon is near polar and I'm guessing they did a burn nearly parallel to it's velocity vector for LOI.
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I'm guessing they did a burn nearly parallel to it's velocity vector for LOI.
That makes sense, since a parallel burn is much more efficient in changing velocity.
Of course, as Jorge pointed out previously (http://forum.nasaspaceflight.com/index.php?topic=24989.msg968934#msg968934), CxP envisioned a different approach, first entering a highly-elliptical lunar orbit of low inclination, then executing a cheap plane-change maneuver at apolune, and finally circularlizing at low altitude. I presume the point of this was to allow a fast free-return trajectory. Free returns exist for lunar polar orbit, but they are slow.
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Page 15 has a refuel in free-return-trajectory mission mode.
For some reason, I'm just reading this whole thread now. This is a great idea, a completely different mission mode. Very clever.
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Okay, here's a question:
Are there orbits which are shorter than the ~44 transit times from LEO to EML2 that Weak Stability Boundary-type trajectories typically have? I'm looking for something intermediate between WSB (at least 44 days, smallest I've seen?) and Hohmann transfers (which take just a few days? 3-6 days?).
At the one end, the delta-v required is about 3.2km/s, at the other it's 3.73-3.9km/s. Are there solutions in the middle between those two that only take, say, 10-20 days? I'm going to try to answer this question somehow, make a graph of delta-v versus delta-t for LEO to EML2 (or EML1) transfer.
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I'm guessing they did a burn nearly parallel to it's velocity vector for LOI.
That makes sense, since a parallel burn is much more efficient in changing velocity.
Of course, as Jorge pointed out previously (http://forum.nasaspaceflight.com/index.php?topic=24989.msg968934#msg968934), CxP envisioned a different approach, first entering a highly-elliptical lunar orbit of low inclination, then executing a cheap plane-change maneuver at apolune, and finally circularlizing at low altitude. I presume the point of this was to allow a fast free-return trajectory. Free returns exist for lunar polar orbit, but they are slow.
One of the requirements of CxP was that intermediate burns don't put crew in an orbit where periselene is below the Lunar crust, in case the subsequent burn fails. This was guaranteed by three burn manoeuvre.
IIRC, the relevant L2 document suggested that a two-burn was slightly lower dV.
Cheers, Martin
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I'm looking for something intermediate between WSB [...] and Hohmann transfers [...]. Are there solutions in the middle between those two that only take, say, 10-20 days?
FWIW I believe these transfers exist but finding them is computationally expensive. The general approach is to choose your destination orbit around the Lagrange point and then calculate trajectories (by running time backwards) that are on that orbit's stable manifold. Some locations on the manifold are "close" (in x,y,z,vx,vy,vz space) to the apogees of highly elliptical Earth orbits (or just plain high-Earth orbits), and by maneuvering where the trajectories are close you can patch them together.
Belbruno (January 4, 2012 FISO) suggested doing the patching with electric propulsion. I tried to think about this in a posting here:
I think the more delta-v you are willing to put into making the patch point work, the quicker you can arrive onto the manifold of the destination orbit.
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Are there orbits which are shorter than the ~44 transit times from LEO to EML2 that Weak Stability Boundary-type trajectories typically have?
Sure, the ones with a lunar flyby as advocated by Kirk Sorensen for example.