[HN Gopher] Bi-elliptic transfer orbit maneuver
___________________________________________________________________
Bi-elliptic transfer orbit maneuver
Author : ColinWright
Score : 51 points
Date : 2024-12-08 14:16 UTC (3 days ago)
(HTM) web link (www.johndcook.com)
(TXT) w3m dump (www.johndcook.com)
| xnorswap wrote:
| This article seems very light on equations and calculations, and
| would be greatly enhanced by including them.
| daeken wrote:
| I have a simple implementation of a delta-V calculator for
| bielliptic transfers here if you want a quick look at the
| underlying math:
| https://github.com/daeken/OrbitalRingsBook/blob/main/space-t...
| -- the DeltaVForBiEllipticTransfer function. (The calculator
| widget in the book doesn't use it, but I was primarily using it
| for getting high orbits in an effort to minimize plane change
| delta-V for another calculation and this code inherited it.
| Also, you can actually play with the widget in action here,
| though again it's only using Hohmann transfers due to the low
| fuel requirements in play: https://ors.daeken.dev/space-
| travel.html)
|
| Happy to explain the math in more detail, but the code is -- I
| think -- pretty self-explanatory.
|
| Edit to add: this function doesn't calculate the initial lift
| to the higher orbit; that'd be the difference between the speed
| of your circular orbit and the perigee speed at the higher
| orbit.
| perihelions wrote:
| There wouldn't be much to add. In a one-mass system, the speed
| of a satellite at any point is a function solely of the radial
| distance _r_ : v^2 = 2/r - 1/a. The semimajor axis _a_
| characterizes the orbit; is positive for an elliptical orbit;
| and is the the arithmetic mean of the closest and farthest
| points on that ellipse--i.e. the periapsis and apoapsis.
|
| If you're at one of either the closest- or farthest- points, at
| distance _r_ , and the opposite one is _s_ , this reduces to
| (defun speed-at (r s) (sqrt (- (/ 2.0 r)
| (/ 2.0 (+ r s)))))
|
| And; if you have an orbit burn at one of these points that's
| co-linear with your direction of motion--something that simply
| expands or contracts the periapsis or apopasis, as is the
| question here--then velocity addition is a one-dimensional
| scalar sum. Concretely: if you're in an orbit defined by the
| peri- and apo- distances ( _r,s_ ), and you want an orbital
| maneuver, at _r_ , that either increases or decreases the
| opposite distance from _s_ to _s*_ , the Dv you need is
| (defun Dv (r s s*) (abs (- (speed-at r s)
| (speed-at r s*))))
|
| That's all the math needed for calculating delta-v's for the
| maneuvuers in the OP post. I.e., the Hohmann transfer between
| two circular orbits, of radius r1 and r2, is
| (defun Dv-hohmann (r1 r2) (+ (Dv r1 r1 r2) ;; at
| r1, raise apoapsis r1 -> r2 (Dv r2 r1 r2))) ;; at
| r2, raise periapsis r1 -> r2
|
| And the bielliptic transfer is a sequence of three of these. I
| think it's (defun Dv-bielliptic (r1 r2
| r-inter) (+ (Dv r1 r1 r-inter) (Dv
| r-inter r1 r2) (Dv r2 r-inter r2)))
|
| (I've left out the dimension-ful scaling factors; if you want a
| concrete calculation for a real object, you can scale
| everything throughout (inside the roots) by G*M, which is i.e.
| 398600 [km^3/s^2] for Earth).
| xnorswap wrote:
| I'm not sure why you think that numbers don't add anything,
| this example from wikipedia is helpful:
| https://en.wikipedia.org/wiki/Bi-elliptic_transfer#Example
|
| Including illustrative numbers on the diagram in the article
| would be helpful to show the difference.
| hoseja wrote:
| Bi-elliptic can also be better for large inclination changes. The
| catch is that coasting to the eccentric apogee takes way longer.
| perihelions wrote:
| That's a key use. A search keyword is "supersynchronous
| transfer"--the common case of (bielliptic) transfers to
| geosynchronous orbit, which first go above GEO for an
| inclination change.
|
| https://www.esa.int/ESA_Multimedia/Images/2022/10/Supersynch...
| koromak wrote:
| Oh boy something new to try in KSP. Funny I've never seen anyone
| do this, maybe the imperfect simulation means its never more
| efficient (in-game).
| montjoy wrote:
| I'm pretty sure it's done whenever there are inclination
| changes. Mechjeb had an option for it IIRC.
| radonek wrote:
| Absolutely. I've hardly ever done bieliptic transfer in KSP,
| but I stick to bieliptic plane changes for almost every big
| inclination change. Dv savings can be huge.
| natosaichek wrote:
| It should work in ksp. Its still an impulsive maneuver.
| xnorswap wrote:
| The transfer times are a killer.
|
| It's only more efficient when the ratio in orbits is around 12
| or higher, and then you have to go even higher than that to get
| more efficient, and takes at least twice as long to complete
| the transfer.
|
| I'm not sure there's even such a ratio in KSP, even Moho to
| Jool is less than that.
|
| So you'd need to be in a very artificial scenario for it to be
| more efficient, and the extra time would be tedious (even at
| full warp) compared to a regular transfer.
| idiotsecant wrote:
| Wow, it's completely non-intuitive that this is more efficient!
| Neat.
| daeken wrote:
| For an intuitive example of where this is more efficient,
| consider plane changes. The delta-V for a plane change is 2 *
| sin(angular_change/2) * speed
|
| The higher an orbit is, the slower you're going at apogee. If
| you're doing a large plane change, going even slower will save
| you a *ton* of delta-V, so you can transfer up to a much higher
| orbit, plane change at apogee, and then transfer down to your
| desired final orbit.
|
| That's by far the most useful place for bielliptic transfers,
| not just going from one orbit to another on the same plane.
| avmich wrote:
| In the limit, going from first space velocity V1 (low
| circular orbit) to second space velocity (parabolic speed) V2
| = sqrt(2) * V1 takes delta V = (sqrt(2)-1) * V1, then at
| infinity you do plane change to the opposite with expending
| zero delta V, then come back and return to the same orbit,
| only the opposite direction (largest speed change) spending
| again delta V = (sqrt(2)-1) * V1. Comparing this total of
| 0.82 * V1 to spending 2 * V1 if you do that on the spot -
| just reversing the orbital velocity in place - shows the
| benefit.
| GlenTheMachine wrote:
| The Hohmann transfer is formally the most fuel efficient two-burn
| transfer. Bi-elliptic transfers require three burns.
| bell-cot wrote:
| Which is an issue because re-starting a good-sized* rocket
| engine in 0g is non-trivial, and the penalty for failure tends
| to be "mission failed, vehicle lost".
|
| *Meaning "large enough for major orbital maneuvers".
| GlenTheMachine wrote:
| Correct.
|
| On the other hand, neither one is guaranteed to be optimal
| for continuous thrust maneuvers, which with electric prop are
| increasingly common.
| jgsteven wrote:
| For Earth orbiting satellites using their own on-board
| propulsion with storable propellant, hardly anyone actually
| does these transfer orbit maneuver sequences in just two-
| burns. (edit: For example, for geosynchronous transfer orbit)
| the delta-V is almost always broken up into 4 to 6 maneuvers
| of decreasing size (to improve targeting accuracy at the
| end). The launch vehicle upper stage usually does the first
| burn to go from the low circular orbit to the elliptical
| transfer orbit.
|
| For these onboard main engines, number of starts is not a
| concern.
|
| Back when some geosynchronous satellites used solid motors
| there would be one huge maneuver at apogee to get nearly all
| the way to GEO (e.g. the Hughes/Boeing 376 spinners). The
| uncertainty in performance on these maneuvers was quite large
| so correction maneuvers were always planned afterwards.
| tofof wrote:
| The direction change from first to second elipse does indeed
| produce the hook the author visualized. I'm not sure why he had
| trouble finding examples of this diagram already existing for
| suitably scaled orbits.
|
| https://www.researchgate.net/profile/Sergey-Zaborsky/publica...
| https://media.springernature.com/lw685/springer-static/image...
|
| From https://www.researchgate.net/figure/Scheme-of-bi-elliptic-
| tr... https://link.springer.com/article/10.1007/s40295-015-0043-3
| andrewflnr wrote:
| > The meaning of the dotted orange curve is different in this
| plot.
|
| I really think it would have been better to have two different
| colors for the two transfer orbits. Changing the meaning of part
| of your notation like that is needlessly confusing, and misses
| the chance to show the full shape of both transfer orbits as
| well.
| NotYourLawyer wrote:
| I wish he'd included some intuitive or hand waving explanation of
| why and when this is more efficient.
| mota7 wrote:
| The hand-waving explanation: The slower you're going, the
| easier (cheaper) it is to change direction. And for eliptical
| orbits, the outer-most part of the orbit is where you're going
| slow.
|
| So to make a drastic change in direction (aka, a very different
| orbit):
|
| 1. First burn to move far away from the center of the orbit so
| that you're going very slow.
|
| 2. Then burn to make a large change in direction (orbit).
|
| 3. Then wait until you cross your desired final orbit, and burn
| again to close it.
|
| The tradeoff is that these types of orbit change are very slow
| (because you want to be going very slow for the middle burn,
| which means you take ages to get there).
| simonjanssen wrote:
| The second image seems to be a ballistic capture leveraging weak
| stability boundaries in a 3 body problem.
|
| It's the kind of transfer used by many spacecrafts to get to the
| moon nowadays (hohmann transfers are only the most efficient in 2
| body problems).
|
| Further reading:
| https://en.wikipedia.org/wiki/Weak_stability_boundary#Applic...
|
| Edit: Also
| https://www.gg.caltech.edu/~mwl/publications/papers/lowEnerg...
___________________________________________________________________
(page generated 2024-12-11 23:01 UTC)