Master the Complexity of Spaceflight

This – is an interplanetary transport network! And you think I can reach my home planet? J.: I mean, probably?! That’s the spirit! And this – is my probability tracer  solving tough spaceflight riddles. What’s the best path home,  optimizing fuel and time? That’s an insane  dynamical/combinatorial mixed problem. And as we solve it, you learn to use  manifolds, weak stability boundaries, periodic orbits, and many more spaceflight tools. All shown in ways you  probably haven’t seen before. And we start with building the tracer. I.. hope to get it right. Because my friend needs a rescue. I’m not judging, but how do you get here ..  running low on everything: the journey is over. Still, it’s a great example of what  to do when lost in outer space! Step one: What makes ‘getting home’ so hard? Imagine nothing else but you in a spaceship. Getting from here to there is peanuts. Fire and drift along a straight path. Let’s add a planet, and you swing by. Is it still a great path? Look into the future! And for this two-body constellation,  looking means solving analytic expressions. Which are pretty simple – give parameters,  and you get the Keplerian orbits. Now, you can’t see it, but the planet  moves around the gravitational center, too. Energy and momentum conservation. You will see it by adding more massive planets. But except for some motions, you rarely find these  simple mathematically closed-form expressions. Now we all have computers. So, there is an easy prediction method for  many masses: numerical time integration. The idea is simple. Take the current gravitational forces  to guess tiny velocity changes. And take the current velocities  to guess tiny location changes. You guessed it: Newton’s laws of motion. This stepwise computation is prone to  errors and takes your computer some time. Now, there is a tweak to speed up the  computation by splitting the simulation. The spaceship is lightweight compared to the  planets, and so is its impact on their motion. So, make a first run without the  spaceship and store the paths. And then, play back the planet’s motion in  the next run to get the spaceship’s reaction. That’s much faster in total if you  want to compute many spaceships anyway. Each new simulation only moves one spaceship  pulled by three gravitational forces. Sounds great? Here is the kicker. To see if a path is a winner, predict the future. Which, even for just one simulated spaceship,  gets harder the longer the journey. Here, I fire a little bit, and  my route zaps all over the place! With all the uncertainties about the real  world, longer simulations are hard to trust. But no need for thrust issues here. I can fire anytime to tune the path  and make a different winner, right? Absolutely, but the next question  is: what is the best firing? Say you are short on travel  time and propellant consumption. To get the optimal route: You have to search  along all possible firings at any time! That’s a continuum of decisions! Simply firing initially and then doing  nothing can’t be the whole story? Oh, and for my friend here,  there is an easy way home. Fire relative to planet Silk in prograde direction  at this point to slow down relative to the Sun. Wait a little bit. Reach planet Blue’s orbit. Fire retrograde relative to Blue to slow  down while falling into an orbit around Blue. From here on land! It’s that easy. And seriously, how else to get home. That’s a Hohmann transfer, which I  learned makes an ideal transfer orbit? The problem is, it’s too costly here. Propellant runs out during the  ‘Low Blue Orbit’ ‘injection burn.’ So, what’s the better sequence? Let’s compare all possible paths home,  considering all possible intermediate firings. Spoiler: The continuity of firing  possibilities actually helps. Spoiler spoiler: running special experiments,  low-energy routes will come up automatically. And don’t panic! I keep this playground solar system  flexible in sizes and masses. I want to see all local and global action at once. Because this system revolves around education. Step two: what’s wrong with rays? Forget about firing for now. That’s like spawning new rays. There is a problem with ray  tracing in the first place. I give an example. Say you perform a gravity  assist maneuver, aka swing-by. The speed before and after the swing-by  won’t change relative to the planet, but the path direction will. Relative to the Sun, this changes the speed and  the orbital energy: you can reach farther out. And here is the problem: After the swing-by,  the resolution of the ray package is lost. Now, honestly, this spreading is nothing bad. It’s great for route controlling. But it generates more non-sampled  space between the rays. To fill in the details, shoot more rays. But it doesn’t solve the unbalanced resolution. It’s easy to miss a winner. So why don’t we build not a ray  tracer but a probability tracer? Let’s push spaceship-position  probabilities over a computational grid. This gives a numerically equalized look,  and if a target is reachable, you see it! Step three: tracing spaceship probabilities First, let’s get this right: tracing  probabilities answers a different question. You don’t look for precise routes  but whether there are any routes. And then you know if it’s  worth reconstructing a winner. So, all I need is a probability  cloud that lives on a spatial grid. And the algorithm tunes grid values to show  whatever infinitely many spaceships do. Awesome, right? Well, that’s impossible. Think of two probability clouds overlapping. Nothing happens. They pass without interaction. And why should they? They represent infinitely  many independent spaceships. I hope you see the problem? Both clouds overlap with  different velocity vectors. So, which one tells the position-based  algorithm how to shift the grid values? Which vector wins? Or, for that matter, how to even  tell each cloud its velocity? It’s simple: A position-based description contains not enough space to handle  arbitrarily multi-valued velocities. The probability lives in position  and velocity space – the phase space. And don’t think of one magical phase  space size; it depends on the problem. I give you an example. These 1,000 particles make one point  in a 4,000-dimensional phase space. Each coordinate stores unique  information about some particle. You can make this point a blurred  blob to think collectively about infinitely many variations of this experiment. And here is the key. Instead of tracing probabilities in this  high-dimensional n-particle phase space, ask how likely to find any particle to be  in a given location and velocity range. You can chase the probability in the same  phase space as that of the spaceship. However, while the spaceship probability  passes without self-interaction, the fluid does, and the probability shows it. With enough particle interaction, they  struggle themselves to effectively two-dimensional behavior –  they equilibrate locally. And then, position-based fields make sense  – temperature, pressure, or probability. Reduce to non-self-interaction, and  you need a four-dimensional grid. So, the plasma or fluid side of the spectrum  are challenging for their own reasons. But they also come with their own  solution methods that we can steal. But it won’t be easy! Watch this! This simulation runs on the huge amount of  200 x 200 cells … by 200 cells … by 200 cells. 1.6 billion cells to get this blocky blöb? I was hoping for a full-blown statistical  analysis, but it’s getting out of reach! I mean, to investigate my  manifolds, I need a tracer. Don’t worry, things get worse. Imagine a spaceship in one  dimension within a potential well. The closer it gets to the  sides, the more it pushes back. The phase space has two dimensions:  one for position and one for velocity. And a probability density blob  reflects infinitely many spaceships. Now, look at this. As time passes, the probability mixes. Yet, per tracer, the probability  density stays constant. It doesn’t diffuse with neighbor values. The motion along position and  velocity compensate each other, and arbitrary phase-space volumes are conserved. So, this blob moves incompressibly, generating  infinite details without probability diffusion. It looks like diffusion if you  don’t look precisely, though. And that’s the problem. My algorithm just stores grid  values – it doesn’t look precisely. To update these, you trace back  where the probability came from. Which likely was somewhere between grid points. So you have to guess these values. This gives numerical softening or diffusion. Then, why bother with all this? Well, compared to ray tracing,  you equalize the resolution. Although, under the hood, it  uses structured ray segments. But the interpolation breaks the accuracy. So, is all my hope in statistical  astrodynamics doomed to diffuse? Thinking about it, doesn’t it look like randomly  firing – that I have to implement anyway? Also, I don’t need the  probability but the reachability. Do you get home – yes or no! Step 4: tracing spaceship reachability Careful! Numerical diffusion can look like random firing. But the probability blurs differently  – mainly along the phase space flow. Random firing, however, means acceleration, so it blurs along the velocity axis,  which then translates into position blurs. And this tiny distinction matters. We care about reachability but in phase space. You don’t want to reach planet  Blue’s location at any cost. You have a velocity window to reach as well. Now, there is a computational  shortcut to handle that. Think about reaching this spot  with the one-dimensional spaceship. Random firing makes many  spaceships reach the goal. But the fastest ones travel  on the blob’s boundary. They keep expanding the blob the  fastest by firing continuously. Here, for the fastest way, fire towards  the goal and then fire backwards. Now, there are electrical engines  built to fire continuously. Essentially, they generate a relatively  low thrust over a long period of time. In contrast, chemical engines give  higher thrust for shorter periods of time. So, when I later solve a  continuous firing problem, I just need to trace the boundary – great! And realizing this, I wanted to build  an exact grid-based boundary tracer. I put it in a longer version of this video. And I partly managed to make it work, but it looks  like a gap-free prediction implies diffusion. Making this whole grid thing is a debacle. And I get the hint: To get my  tracer, I have to sacrifice. Step 6: Building spray tracers. Back to scratch. I’m going to use rays –  with a statistical mindset. And I wanna use my grid – somehow! This is how I made it. Remember: ray segments plus interpolation  crushes accuracy, so I use non-stop tracers. And by tuning the phase point  updates, I control their accuracy. This is how it works: The time integration  shifts position and velocity via rules. Acceleration changes velocity,  which itself changes location. Take the simplest form. If you keep the current values constant  over a time step, the update is a product. Reduce the timestep, and you  reduce the error accumulation. But that’s inefficient. Higher-order methods, using  a mashup of test shots, give similar accuracy with  less computational cost. But you can do better. Remember how the probability moved in phase space? It conserved volume. At least when the system itself conserves energy. So, if you make your integrator share this  property, you get better long-time behavior. But how to make it volume-conserving? Look how the classical 1st order  method pushes this box over time steps. The simultaneous shifts in position  and velocity make the box bigger. So instead, make staggered shifts  along position and velocity. The trick is to split the update  in volume-conserving shear motions. Which here works since the acceleration  is independent of the velocity. It won’t stretch while shearing! And this so-called symplectic integration  gives great long-time behavior. Now, I’m not saying you should  always use symplectic integration. A non-symplectic integrator ramped up to high  accuracy fulfills symplecticity naturally. Both types converge to the  ground truth, just differently. And the more information about the system  you bake into the integrator, the better. Split in Kepler orbit and interplanetary interaction steps if you have weak  interactions in planetary scenarios. Make it fit your problem! Now, to let my tracers mimic spaceships, I  activate randomly firing and limited fuel. They also spawn offsprings to fill in the gaps. And I built a grid that limits the spaceship density in phase space and  so the computational time. There is my grid! 🙂 So the rays explore, and the grid prunes. That’s all great, but eventually,  you send millions of rays. Which is costly. And that’s where the statistical nature helps. Remember: predictions are hard, and  slight nudges make huge impacts. So, at least when getting first  inspiration for possible routes, why bother predicting accurately  when you fire randomly anyway. Numerical errors are easily overcompensated. Just make sure the less accurate  blob covers the exact one. You can artificially crank up the firepower  or take other creative global measures. Ultimately, modifying the blob’s global behavior  can be faster than making all tracers accurate. This cheaply generates an  over-optimistic blob evolution, but it definitely contains the winner route. Which means you drastically limit the phase space  portion where to look for the exact solution! Filtering impossible routes and optimizing  promising candidates is easily in post. Step 7: Testflight I’m not judging, but what on earth  is my friend doing near the Moon? Anyway, getting home is probably a cakewalk. Let’s compute the reachability for  a continuously firing engine first. Which I made comically strong  to highlight the maneuvers. And remember: we just need  to trace the phase space blob’s boundary to find the quickest way home. So, in the counting grid, I  can delete the tracers inside. And just for this experiment, I made  the spaceship ultra heat resistant. It allows for hefty aerobraking – and  reaching Blue’s position is here enough. And for the records, the projection of a phase space blob’s boundary looks like  a filled region in position space. And now, let’s run it. There you have it: boundary tracing  answers a very specific question. How to get home as quickly as  possible under continuous firing? That’s how! Fire prograde relative to Moon to screw  out of Moon’s gravitational significance. And once you fall down to Blue, I thought continuous retrograde firing eats  up the orbital energy most efficiently. Or maybe firing against the  angular momentum works better. Kind of like the inverse of  the maneuver for leaving Moon. But the solution is surprisingly elegant. Remember, I allowed for crashing into  Blue, and the code takes me at my words. So, it fires a little retrograde, but  there is an overlayed firing to the right. And it gets home, not in the first  encounter, but in the end much faster! The idea is: if you slowly chew up the angular  momentum and still miss at the first encounter, prepare yourself to at least  make it in the second attempt. And since the electrical engine can  only push so much, it needs a little bit more time and lever to do so, which  it gets by swinging to a higher altitude. So, being less efficient at the  first encounter gives more time to fight the angular momentum and crash home. Now, if this maneuver is so great,  why didn’t we leave Moon that way? Well, simply because we didn’t start in an  anti-crashing trajectory but in a circular orbit, for which screwing out is the  fastest way when firing continuously. But when landing, heat resistance allowed  for this rather “experimental” approach. Surprisingly, it still works when  holding the firing direction constant. This basically fakes a new potential  for a virtually non-firing spaceship. It’s like the classical rubber-sheet model tilted. And my codes found this recipe simply by  pushing millions of dumb tracers around. Anyway, the gridded boundary has some thickness. So, checking nearby tracers; you’re  quite flexible in getting home. And don’t forget it’s a  numerical global solution method. It gives a winner inspiration. The route gets better by fine-tuning later. Now that was fun, but … What about non-continuous, stronger firings? We could use stronger, continuous random  firing combined with a propellant cut-off logic. But tracing out all ‘technically  possible’ paths really bows up the blob. That’s why I add a little bit of  sanity by firing when it is beneficial. And to see this benefit, I shoot one-dimensional  spaceships into this potential well. Here, the phase space paths  quickly start to squeeze. This is because slower spaceships  spent more time in this well. They are accelerated for longer,  building up more velocity change. And since energy is here conserved, each  path is of constant energy per unit mass. By the way, in this reference  frame, the well doesn’t move. So velocity and kinetic energy  are relative to that well. Now, while firing, let’s say all  the reaction mass pushed by the engine is just one big mass pushed all at once. In fact, let’s make both  masses equal for simplicity. Then, firing splits the big  mass into two smaller masses, each following their own path, initially  differing by a fixed relative velocity. Obviously, to get the most bang for the  buck, make the split when you can jump as much energy levels as possible,  which is during the squeezing. There are the points of highest speed along  each path, where kinetic energy climbs easiest. The final speed-up or slow-down merely by firing  within a potential well is the Oberth effect. Now, if you start out at higher velocity,  the Oberth contribution shrinks. And this makes sense: passing so quickly, there is not enough time to accelerate  much and build up greater velocity changes. And so the squeezing fades  for higher initial speeds. It’s like leveling the well. When passing a rock instead of a  planet, there is less Oberth effect. On the other hand, taking your time is beneficial. That’s why the capture/escape path is  super sensitive to the Oberth effect. That’s the path that barely escapes. Nonetheless, along each path, it’s always  beneficial to fire at the fastest point. And you get an intuitive visualization  by shooting non-firing blobs. This makes it somewhat of  a squeezing detection tool. Since the phase space volume doesn’t change,  velocity-squeezing means position expansion. Apply it inside the well?! Once again, most squeezing  is at the fastest point. Now, the question is, how does it  translate to the two-dimensional case? It’s quite similar! First, let’s take paths that share  the same point of closest encounter. This connects them at their point of  highest velocity via a single burn. That’s great for later comparison. So, shooting-off blobs that have the same size at each path’s slowest point, you see the  highest stretching at the fastest point. Firing here splits the energy most effectively. So, in the diagrams, I compare firing  at the fastest and the slowest point. This is the Oberth effect. Now, you can tell; among the paths,  the parabola is most effective. And the reason simply comes down  to where the slowest point is. On an ellipse, it is the outmost point. And the more you fire at the fastest point, the  more it moves away, and the velocity lowers. Until you get a parabola with zero velocity at  infinity – that’s the route that barely escapes. Fire more, and you get hyperbolas  with non-zero velocity at infinity. And that’s it: Having non-zero velocity at the slowest points, ellipses and hyperbolas give you more kinetic  energy when firing on their worst side. This counters the Oberth effect. You can still see a huge absolute energy increase. But it would be huge already,  even without the Oberth effect. And, of course, arriving on a parabola,  the post-firing hyperbola won’t have the highest energy compared to hyperbolic arrivals. But the Oberth effect is “most effective.” So, in my simulation, I fire stronger aligned  with the velocity vector closer to the planet. And I hope millions of routes  show the best combination. You cannot always approach close to a parabola. Now, back to my friend: let’s get back to Moon! The blob quickly explores phase space, working on millions of possible  routes, and … I can’t see anything! So, just for learning, I  give you something better. Let’s sit on a Low Blue Orbit everywhere at once. And remembering the Oberth  effect, I will fire twice. One firing to get away from  Blue and another firing to get captured by the Moon – nowhere in between. On top, I vary the initial fire strength. Each of these rings are millions  of spaceships, colored by strength. Very soon, we hit Moon. And here, we could make the second  firing to get captured by the Moon. This family of routes is the Hohmann transfer  for a prograde or retrograde orbit around Moon. It’s pretty expensive fuel-wise  but quite fast: it’s the first hit. But let’s keep going. Every time the lines intersect with Moon,  a firing would conclude a family of routes. And there are a lot of them –  I cannot even show them all! But there is one type of family  that should draw your attention. These lines slowly approach Moon. Until some tracers get ballistically  captured by Moon without any firing. Now, seeing the winner route doesn’t tell  you why it is possible in the first place. And this is really cool. Let’s rerun the ballistic trajectory, and  I want you to focus on this section here. You see it? We shoot off on an ellipse, but instead  of sticking to it, we get pulled aside. Looks pretty much like the inverse of the  continuous firing solution we found before. But we don’t fire here! So where does it come from? Surprise, surprise: it’s the  Sun – nature fires for us! To get a first overview,  draw the potential energy! And you clearly see: close to Blue, most  of the local dynamics are governed by Blue. And the farther you go outwards, the more the collective gravitational  pull shifts towards the Sun. That’s why we don’t stick to the two-body ellipse. In this first swing of the journey, the Sun gets more relevant, and especially  the Blue-Moon interplay gets less relevant. And there is a simple model that “captures”  this: the restricted circular three-body model. It’s a great playground with already rich  dynamical structures that explain a lot. Now, to study the motion with the Sun always in  the same direction, co-rotate the reference frame. This change in perspective alters the form of the equations of spaceship motion when  written in co-rotating coordinates. By how much are they altered? Exactly as needed so that the generated  motion looks correct from “the outside.” I give you an example. Here, I look at a resting mass from a  moving and rotating reference frame. From that perspective, the mass  seems to get pushed around. So, computing in this accelerating reference  frame, you have to add some forces. Which combined push exactly as needed to  look like resting after back-transformation. So, these forces only pop up for kinematic  reasons – once we mess with the coordinates. You describe the same physical motion after  all – just from a fancy-dancy perspective. But, as it goes, some of this mess is beneficial. Here, the gravitational forces keep  both masses on circular orbits. And focus on the smaller mass here. Once co-rotating with the same rotational speed, the centrifugal force makes  the mass apparently force-free. In fact, you can compute the centrifugal  force anywhere by plugging in some position. So, we can see it as a downhill push coming from  a position-based potential scaled with the mass. And we can do the same with  the gravitational potential. Obviously, bake them into an effective potential. That’s the potential the spaceship sees  in this co-rotating reference frame. It’s still pushed by the Coriolis force, but the effective potential gives a hard  reachability limit for a given energy. And so, a non-firing blob never exceeds the  zero-velocity lines for a given energy level. You can lock the blobs to  certain energetic regions. This potential is a great predictive tool. And you can do even more. Apparently, you can sit at the stationary Lagrange points to rest  in the co-rotating system. Let’s isolate this point here and kick forward. The Coriolis force immediately pushes  to the right of the relative velocity. Eventually, we drift off. But if we start next to the Lagrange  point and shoot off just right, we can balance both forces to  orbit around the Lagrange point. In a way, this is how my code computes this orbit. The shooting method tweaks the kick-off to  nail down this orbit by rewarding periodicity. And for this symmetric orbit type it’s enough  to look for perpendicular axis crossings. Now, when there is one orbit, there are more. Tweaking not just the kick-off  velocity but the location as well – meaning the whole phase space  point – you land on a different orbit. Just enforce periodicity and some  distance measure towards the last orbit. Then you performed ‘numerical continuation’. You have a solution point of a problem – here a phase space point of an orbit – and  you find points on neighbor orbits. Now, tracing such an orbit numerically,  you can check it’s hard to stay on them. They are surrounded by regions in  phase space that, once you sit on them, lead you to the orbit or away from  it – stable or unstable manifolds. These are the structures I’m looking for. Why? Because they tell you in advance what you get. Take the one-dimensional spaceship  with a hill-shaped potential. Starting anywhere makes you follow the flow. Starting at the equilibrium leads you nowhere. This point is not an orbit, but let’s pretend it is equivalent as  in: once you sit on it, you stay on it. A stationary feature of the phase space motion. And the stable and unstable  manifolds are nothing more but curves that lead you to and from that point. Though you never reach it, it would  take an infinite amount of time. But they are helpful as they tell  you what happens close to them. You know in advance that once you ride close  to the stable manifold, you will curve away, following either side of the unstable manifold. That’s the skeleton of the phase space motion. Also, taking slightly different paths is great to generate some waiting time while  ending up at the same location. Both points are great for planning a spaceflight. Remember: to catch up with Moon, you need the  right position and velocity at the right time. And no matter how energetic you start out, you need to build up angular  momentum relative to Blue as well. That’s why the Hohmann transfer  comes with a strong second burn. But the manifolds tell us how to get it for free. Part of the stable manifold reaches down to Blue. You can follow close to it, having enough  energy but still low angular momentum. But looking at the unstable manifold,  we know there must be the right “curve away” bringing us up here in time –  having much more angular momentum. Under the hood, it’s the effective  potential and the Coriolis force pushing us. Also note: we curve away close to the  outside of the stable manifold tube. You can guess what happens being inside the tube. Now, having the angular momentum,  we need to approach Moon just right, which doesn’t even exist in the three-body  system where these manifolds come from. Well, closer to Blue and Moon, their restricted three-body system  with its manifolds is more relevant. So, hopping manifolds is the name of the game –  and we follow the stable manifold to the Moon. Now, approaching Moon statistically, some tracers  get temporarily captured for a number of cycles. And to get the essence of  this, go ahead systematically. Any passing trajectory has a point  of its first closest encounter. And so you start at an angle with a perpendicular  kick-off – here in prograde direction. If it wouldn’t be for Blue, you would then  make an ellipse with a predefined eccentricity. But since Blue spits in the soup, count the  number of revolutions around Moon until escape. For that, you check the Kepler energy sign of the  local spaceship-Moon system after each revolution. So, n-stable orbits make n revolutions  around Moon with a negative energy sign at the cut section – all without  making a full revolution around Blue. That’s the simulation cut-off time. Repeat this experiment for the  next points along the cut section, and you find changes in the n-stability. Now, let’s focus on 0- and 1-stable  orbits and their boundaries. These boundary points collected for all angles for the given eccentricity  make the weak stability boundary. And along this boundary, you find  points that will crash into Moon. Now, these sets of points  are great for two reasons: First, obviously, they tell you where to  be in phase space to get captured by Moon. This is similar to the effective  potential and manifold ideas. Second, while sometimes coinciding with  manifold points, they make a more general tool. Manifolds are great; they split energetic regions, making it easy to see where  stuff goes in and goes out. But they need orbits to emerge from. And the orbits need Lagrange  points to emerge from. And these Lagrange points here need the restricted  circular three-body system to emerge from. When you go with a restricted four-body system, the Lagrange points and the  emerging structures are no more. The three body sub-systems are  disturbed by the remaining mass. And so you don’t exactly rest in  the co-rotating reference frame. But there are still sensitive regions, and  the manifolds still make a great guide. The weak stability boundary – you can  build it for more general systems directly. You simply count revolutions  – however, they came about. It’s just time-dependent now. But honestly, if nature gently carries you close  to Moon, fire a little bit to safely stay with it. That’s how you blow up the n-stability. You jump from the 1-stability  for the eccentric approach to the above 10-stability for the circular orbit. These are the phase space  structures for low-energy transfer to the Moon – or anywhere else? Starting from a Low Blue  Orbit fire to get up here. Then, follow inside the stable manifold tube and  slip through the orbit into the unstable manifold. If we are lucky, it overlaps with  the stable manifold of planet Yellow. But we aren’t. They don’t even come close. Visualizing the speed by an upward shift,  you can guess the phase space distance. So, you look for a  cost-effective two-burn hop-over. That’s an intermediate Hohmann transfer. From here on, it is the inverted  sequence to a Low Yellow Orbit. This sequence of manifolds is part of  the interplanetary transport network. Now, as fancy as it looks, it’s hard to see if  it’s better than a classical Hohmann transfer. So, let me convince you with a much more intuitive  Blue-Moon example that such routes can be cheaper. Remember: Since, all we got  to do is getting up here, let’s get there via gravity assist by Moon itself. The Hohmann transfer would need a strong  retrograde burn to get captured by Moon. The low-energy ballistic transfer  gets you captured for free. It just takes some time. That’s the trade-off. And for a fair comparison, both missions  lead here to a 2-stable capture. Also, the Coriolis force and effective potential  work in our favor in the fourth quadrant as well. Here, the angular momentum keeps  building up in the right way. So you can expect similar routes to the Moon. You can also use inner routes. That’s when you hop on Blue-Moon’s  inner stable manifold – either via a Hohmann transfer or the low thrust spiral. This carries you up to the Moon. And to show what else is possible, the orbit  families we found are linked to other families. The planar Lyapunov orbits – the ones I have shown  before – bifurcate into Halo orbits and many more. Each having its own manifolds. It’s a mess. It’s helpful, nonetheless. Things seem cluttered from time to time, leaving you wondering if you ever make  it: there is always a way, always! Step 8: Time to get home. This is the problem: My friend makes an elliptic orbit at planet Silk  with enough propellant to reach planet Blue. But not enough to get into  a circular Low Blue Orbit. Oxygen is limited. And hull integrity is … we shouldn’t aerobrake. So, to lose energy, we need  clever swing-by’s, possibly fired. Speaking of firing: On the first elliptic passage – firing closest to the planet doesn’t make the best  departure angle relative to the Sun. Not all of the Oberth speed boost  goes in the backwards direction. That’s why the best firing point  makes an inefficient compromise. Simply waiting gives better departure  angles, and you can reach “farther down.” Then again, the planets must align for  great swing-bys – that’s a phasing problem. This is how I made it. I divided the space in an interplanetary  part and local planetary parts. These control the tracer resolutions. Now, the tracers without contact  didn’t make it to Blue in time. So, the winner route is hidden  in the uninvestigated shadows. These I explore with higher  resolution sampled from the recording. Eventually, I iterate along encounters. Now, at planet Blue, some tracers have enough  propellant to decelerate onto the goal orbit. That makes a complete route. And since tiny changes make a huge difference,  you get infinitely many routes per family. Simply pick a suitable route from an  efficient cluster, trading time vs. propellant. This route makes five revolutions around planet  Silk before leaving for the inner planets. Here, a swing-by at Blue  takes away a lot of energy. Still, it needs additional firing to  make the final arc hit home just right. If you can invest more travel time, take  this route and save even more propellant. The Blue-Yellow double swing-by  takes away even more energy. And then just wait until you catch up with Blue. Both variants are much better than the  instant and optimal Hohmann transfer. And as expected, since I encouraged  Oberth-promoting swing-bys, the winners are mostly composed  of locally efficient maneuvers. And if you color it not by cluster but  by interplanetary segment, you get this. The more swing-bys you consider, the easier it  is to get the final arc close to Blue’s orbit. Making it easier to arrive  on a low-energy manifold. By the way, this is another  riddle I’m trying to solve. Maybe you can help me with that  … if you enjoyed the video? Thank you for watching! Oberth and out.