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Cis-Lunar Orbital Mechanics Q&A by JohnFornaro on 29 Apr, 2011 14:55
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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? -
#1 by Phillip Clark on 29 Apr, 2011 16:34
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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! -
#2 by mmeijeri on 29 Apr, 2011 16:43
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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.QuoteI 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.QuoteAs 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.QuoteWhy 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.QuoteHow do you read an ephemeris table to determine the start and finish time of your orbit?
Can't answer that one.QuoteA 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. -
#3 by QuantumG on 30 Apr, 2011 04:32
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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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#4 by sdsds on 30 Apr, 2011 07:49
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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 -
#5 by JohnFornaro on 30 Apr, 2011 15:36
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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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#6 by JohnFornaro on 30 Apr, 2011 15:45
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Double wow. BMW was 75 cents on Amazon.
Also, there's no "cents" key on my 1391401 keyboard! -
#7 by kwan3217 on 30 Apr, 2011 16:14
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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. -
#8 by sdsds on 30 Apr, 2011 22:03
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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. -
#9 by QuantumG on 30 Apr, 2011 22:14
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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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#10 by baldusi on 30 Apr, 2011 23:33
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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. -
#11 by Proponent on 01 May, 2011 03:26
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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. 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. -
#12 by QuantumG on 01 May, 2011 03:37
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Thanks!
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#13 by JohnFornaro on 01 May, 2011 15:17
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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...QuoteUnderstanding 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. -
#14 by Danny Dot on 01 May, 2011 16:10
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See what you can find on patched conics.
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#15 by alexterrell on 04 May, 2011 23:06
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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? -
#16 by alexw on 05 May, 2011 04:25
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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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#17 by madscientist197 on 05 May, 2011 07:49
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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! -
#18 by JohnFornaro on 06 May, 2011 12:55
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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. -
#19 by QuantumG on 25 May, 2011 09:00
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This seems like an appropriate thread.. anyone know the minimum required thrust/weight ratio to enter into lunar orbit?
To Google!
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.