n-body gravitation: Lagrange points, Rectilinear Halos, & the Retrograde Square Moon Orbit...explore & share saves involving intriguing n-body dynamics

[AloE]

Exploring Lagrange Points & Rectilinear orbits...Let's start with the Moon Retrograde Square Orbit

Why explore this intriguing region of space in KSP with Principia & RSS?  

• The upcoming ESA/NASA/JAXA Lunar Gateway DSL station & SLS Artemis III+ missions &
• the 'mind puzzle' of navigating the 'shape' of forces in this region of space.

One simple way to start exploring the 'shape' of force potentials around the Moon gravity well in the Earth-Moon system is

the DRO 'Lunar Retrograde Square': an orbit from near L2 towards L5, back near L1 then towards L4, & falling back towards L2...

Update: Principia added an EML reference frame -- with the option to show equipotential lines for the sum of forces associated with gravity & rotation in the plane defined by the orbital of the Moon around Earth...

• These equipotential lines are shown for a single plane & for the 'current time' & do shift slightly with the Moon's distance from Earth
• i.e. the location of the equipotential lines will be slightly different when a craft actually reaches a given point on the Principia projected fuchsia or dashed paths...
• the shift is quite noticeable over time along the direction of an imaginary line say from L4 to L5. 
• a visual explanation of equipotential in a video by Scott Manley

 The EML frame offers an insightful way to visualize the dynamics shaping the Lunar Retrograde Square Orbit...

For example, for the 'square' orbit plotted in the MCI frame...

• see how a lower AP creates a more rounded 'corner' (e.g. see the AP on the L4 side of the orbit in the screenshots below);
• where as, traveling further towards L4 or L5 creates a sharper 'corner' (e.g. see the AP on the L5 side of the orbit in the screenshots below)...
• well, that is at least until you get close to a transfer to L4 or L5 where you will see a loop 'ear' corner instead.

Note: To help interpret the craft path, I have added to the screenshots in yellow text approximate L point information...this is _not_ shown in game. 

The above animation shows the 'Square' orbit over 7 years in MCI (Moon Centered Inertial) frame (fyi...with the small vessel in this save the 'square' path persisted for about 18 years with no station keeping).  Clarification: the Earth is traveling along the blue-ish arc counter-clockwise.

Summary of Earth-Moon system Challenges:

Challenge #1 (Beginner):  "Get a feel for the dynamics that shape the 'Retro Square' orbit: Make a 'Moon Pentagon' orbit"

Challenge #2 (Intermediate): 'Transfer to an Artemis III & Gateway Station type of Moon NRHO [pdf] (see animation & NASA graphic below)"

Challenge #3 (Intermediate): 'Transfers along the 'Lunar Retro Square' to halo orbits around L2, L1, L4, L5'

Challenge #4 (Advanced): "Moon L1 halo to Moon L2 halo transfer similar to the 2010 ARTEMIS-P1 mission (see NASA drawing below)"

Challenge #5 (Advanced):  "Create a 'Lunar Gateway (images post by NASA of the planned station) ' type station (module link sequence video NASA) (or a magic potatoroid mining station [Matt Lowne] ) in a long term 'Lunar Retro Square Orbit' (since a Moon NRHO might require more frequent station keeping) for ease of docked ship transfers to Moon NRHO, L2, L1, L4, L5" 

Challenge #6:  Create an even better n-body related challenge & share it!

Challenge #1:  "Get a feel for the dynamics that shape the 'Retro Square' orbit: Make a 'Moon Pentagon' orbit"

In the provided save, with a single less than 10s burn, turn the Retrograde "Moon Square" orbit into a "Moon Pentagon" orbit:

Inspired by the @maccollo "Principia Mun Retrograde Square Orbit" during the afternoon of an 'at home from school not so well day', my son & I decided to see if a related "Square Retrograde Moon" orbit could be created in Principia RSS.

Indeed, we found something similar that we could also manipulate purposefully with a single small burn at a specific time:

In this 1st case, turn the "Moon Square" orbit into a "Moon Pentagon" orbit as shown in the images with a single less than 10s burn like we did (yes, that's a hint, & improvements welcome).

We share this as an introduction of a thread to explore, discuss & share refined versions of this orbit as well as new saves of other intriguing orbits with the 'flight plan' of burns to reach them...

Discovery either by blind fortune dragging Principia maneuver burn sliders or intricate use of KSPTOT 1.6.9 welcome!

KSP version: 1.12.x

Required Mods:  Principia, RSS

Optional Mods: RealScaleBoosters, SMURRF, KOS

To load:

• download the "Moon Square Orbit" save
• create a  KSP 1.12.x GameData with Principia, RSS
• load KSP & create a new save game, find the folder with the name you gave to that newly created save game on your hard drive & copy into that folder the "MoonSquare.sfs"
• Load the MoonSquare save as normal from the usual in game menu (e.g. from the Space Center screen)
• Find the craft in the Tracking Station & select it to fly.
  • if you are unfamiliar with Principia here is a link to a post with some visuals & suggestions.

Metrics:

• Orbit Uniformity & Stability (both improved revisions as well as your own original orbits welcome)
• Quality & Ease of the flight plan (the easier/simpler to execute the better) in the save you share
  • discovery of a simple regular maintenance maneuver(s) would be a nice plus
• most importantly -- the fun factor: how much fun we all have exploring what you improve/create!

Objectives:

1) Fun (with Principia RSS ;-)

2) hands on play with fascinating orbits, one barely new existed, with saves someone else took the effort to make

3) improve a save: create & share Principia 'flight plans' of maintenance burn(s) that make a given orbit more uniform & stable

4) make a new save (with a simplified mod list): and share your own flight plan for some type of cool orbit...

• Flight Plan to make the "Moon Square": requires RealScaleBoosters added to GameData in addition to Principia & RSS

 

Related to Challenge #1: The recent SLS Artemis 1 mission made use of a Lunar distant retrograde orbit (DRO) [source link for animation below] (not square though ;-)  The Principia EML frame is similar to the left 'Earth-Moon Pulsating Rotating Frame': Challenge #2 (Intermediate): Transfer to a SLS Artemis III & Gateway Station Moon NRHO: NASA explains why a NRHO vs DRO or low lunar orbit for Gateway station: Spoiler Source:  https://www.nasa.gov/feature/a-lunar-orbit-that-s-just-right-for-the-international-gateway There is no shortage of options for how a spacecraft could orbit the Moon, but two in particular – low lunar orbit and distant retrograde orbit – are helpful for understanding why NRHO is the right fit for Gateway.   A spacecraft in low lunar orbit follows a circular or elliptical path very close to the lunar surface, completing an orbit every two hours. Transit between Gateway and the lunar surface would be quite simple in a low lunar orbit given their proximity, but because of the Moon’s gravity, more propellant is required to maintain the orbit. Therefore, low lunar orbit is not very efficient for Gateway’s planned long-term presence at the Moon – at least 15 years.   Meanwhile, a distant retrograde orbit provides a large, circular, and stable (or more fuel-efficient) orbit that circles the Moon every two weeks. However, what Gateway would gain in a stable orbit, it would lose in easy access to the Moon: the distant orbit would make it harder to get to the lunar surface.   A third option, NRHO, is just right for Gateway, marrying the upsides of low lunar orbit (surface access) with the benefits of distant retrograde orbit (fuel efficiency). Hanging almost like a necklace from the Moon, NRHO is a one-week orbit that is balanced between the Earth’s and Moon’s gravity. This orbit will periodically bring Gateway close enough to the lunar surface to provide simple access to the Moon’s South Pole where astronauts will test capabilities for living on other planetary bodies, including Mars. NRHO can also provide astronauts and their spacecraft with access to other landing sites around the Moon in addition to the South Pole.   The benefits of NRHO don’t end with surface access and fuel efficiency. NRHO will allow scientists to take advantage of the deep space environment for a new era of radiation experiments that will inspire a greater understanding of potential impacts of space weather on people and instruments. NRHO will also give Gateway a continuous line of sight, or “view”, of Earth, translating to uninterrupted communication between Earth and the Moon.   NASA is preparing to launch the Cislunar Autonomous Positioning System Technology Operations and Navigation Experiment, or CAPSTONE mission, that will use a 55-pound cube satellite to gather data on the characteristics of NRHO ahead of Gateway’s expected launch in late 2024. https://www.planetary.org/space-images/near-rectilinear-halo-orbits Challenge #3 (Intermediate): 'Transfers along the 'Lunar Retro Square' to halo orbits around L2, L1, L4, L5' Challenge #4 (Advanced): "Moon L1 halo to Moon L2 halo transfer similar to the 2010 ARTEMIS-P1 mission" 2010 "ARTEMIS-P1 - The First Earth-Moon Libration Orbiter": Additional videos: NASA Scientific Visualization Studio 2010 ARTEMIS at Lagrange: "In this version, the satellite trails are are constructed in a lunar-centric inertial coordinate system so the trails reveal the motion of the satellites relative to the Lagrange points in INERTIAL space (fixed with the distant stars)"

Some examples of additional orbits to explore:

Near Rectilinear Halo Orbit (NRHO) "KSPTOT LVD: (Tools -> Halo Orbit Constructor menu)" Arrowstar's post with details about this new tool (link)

Lissajous orbit  WMAP at Sun–Earth L2,[7]

 

Earth-Moon L points Visual Data

Mission ideas: place a station, gravitational wave 'telescope' (link), or comsat(s) at the stable Earth-Moon L4 & L5 Note: L3 is an unstable L point = requires small periodic maintenance burns.  Images: source source source

Edited July 10, 2023 by AloE
improve links & add images & challenge summary

[Arrowstar]

2 hours ago, AloE said:

In the provided save, with a single less than 10s burn, turn the "Moon Square" orbit into a "Moon Pentagon" orbit.

Inspired by the @maccollo "Principia Mun Retrograde Square Orbit" during the afternoon of an 'at home from school not so well day', my son & I decided to see if a related "Square Retrograde Moon" orbit could be created in Principia RSS.

Indeed, we found something similar that we could also manipulate purposefully with a single small burn at a specific time:

In this 1st case, turn the "Moon Square" orbit into a "Moon Pentagon" orbit as shown in the images with a single less than 10s burn like we did (yes, that's a hint, & improvements welcome).

We share this as an introduction of a thread to explore, discuss & share refined versions of this orbit as well as new saves of other intriguing orbits with the 'flight plan' of burns to reach them...

Discovery either by blind fortune dragging Principia maneuver burn sliders or intricate use of KSPTOT 1.6.8 PRC9, etc welcome!

To load:

• download the "Moon Square Orbit" save
• create a  KSP 1.11.2 GameData with Principia, RSS
• load KSP & create a new save game, find the folder with the name you gave to that newly created save game on your hard drive & copy into that folder the "MoonSquare.sfs"
• Load the MoonSquare save as normal from the usual in game menu (e.g. from the Space Center screen)
• Find the craft in the Tracking Station & select it to fly.
  • if you are unfamiliar with Principia here is a link to a post with some visuals & suggestions.

Metrics:

• Uniformity & Stability of the revised orbit &
• Quality & Ease of the flight plan (the easier/simpler to execute the better) in the save you share
  • discovery of a simple regular maintenance maneuver(s) would be a nice plus
• most importantly: how much fun we all have exploring what you improve/create!

Objectives:

1) Fun (with Principia RSS ;-)

2) hands on play with fascinating orbits, one barely new existed, with saves someone else took the effort to make

3) improve a save: create & share Principia 'flight plans' of maintenance burn(s) that make a given orbit more uniform & stable

4) make a new save (with a simplified mod list): and share your own flight plan for some type of cool orbit...

• Flight Plan to make the "Moon Square": requires RealScaleBoosters added to GameData in addition to Principia & RSS
• here are some further suggestions:

Some examples of additional orbits to explore:

Near Rectilinear Halo Orbit (NRHO)

Lissajous orbit  WMAP at Sun–Earth L2,[7]

Thanks for sharing KSPTOT!  I'm the author.  If anyone is interested in modeling fun 3/4/N body orbits in Launch Vehicle Designer (LVD), please be sure to grab the latest pre-release of KSPTOT, v1.6.8 PR9.   You'll also need to download and install the MATLAB Compiler Runtime R2021a.

Let me know if you have any questions!

 

[AloE]

Earth-Sun Lagrange point related Challenges:

Below are links to two KSP 1.12.2  saves that provide a flight plan example to explore the stable equilibrium region Earth-Sun L4. 

Challenges:

• create & share a similar flight plan to the stable equilibrium Earth-Sun L5 point
• Make a Trojan asteroid: Move a 'potatoroid' into an Earth-Sun L4 or L5 with a >100+ year stable trajectory

For further insights, NASA WMAP outreach wrote this concise non-technical page regarding L points (link)...based on this technical PDF that has the math equations (link).

This excellent 'topographic' view below from the WMAP page at the above link conceptually shows the rather large L4 & L5 regions for long term stable capture of a KSP craft or 'potatoroid'.

Saves made in KSP 1.12.2:

Orbiting Earth-Sun L4 with plane change example flight plan:

• Required Mods:  Principia, RSS
• L4viewPlaneChange.sfs

A flight plan to Earth-Sun L4:

• Note: this save Requires these Mods:  Principia, RSS, RealScaleBoosters, SMURRF
•  L4EarthSunFlightPlan02.sfs

Other Suggested missions to make & share in this thread:

• create a similar longterm (not ejected for at least 50 years) stable flight plan to the  "Closest to Earth" part of the L4 or L5 "hill" 
• find an interestingly shaped or timed stable 'orbit' around the L4 point

Earth-Sun L4

Enjoy!

Edited July 7, 2023 by AloE
add ESL frame screenshot

[AloE]

Principia has added a helpful new reference frame with the option to display equipotential lines for the sum of forces associated with gravity & rotation:

(for clarity, I have added in yellow text the 'approximate' locations of the L points...these are _not_ shown in game)

Because of the shape of the Moon's orbit around Earth, these equipotential lines do oscillate slightly with time in this frame relative to the projected craft path = fuchsia line...

perhaps most visibly in the L4-L5 axis...yellow bars help reveal the shift in the animation below...

so, keep in mind that the craft will end up over time, e.g. time warping the craft from AP to AP, appearing in a slightly different location 'of the equipotential lines' than suggested just by the fuchsia line simply because the equipotential lines are for the current time not the 'various future times' represented by the fuchsia line:

Useful for the explorations suggested in the prior posts.

Named 'EML' in the case of the Earth-Moon System, this new frame can help you to further refine the Retrograde Moon Square Orbit:

An example KSP 1.12.3 Principia RSS flight plan to improve the Retro-Square Orbit: FlightPlanImproveMoonRetroSquare.sfs

An example revised KSP 1.12.3 Principia RSS Moon Retro-Square:  BetterMoonSquare18years.sfs

Here is a second MCI (Moon Centered Inertial) frame animation showing the orbit over years 7 to 18, that continues the animation of years 0 to 7 shared in the original post.  

Clarification: the Earth is traveling along the blue-ish arc counter-clockwise.

 

Visualizing Sum of Gravity & Rotation Potential: Pink loop = path of the 'Moon Retro Square"

What Makes Lagrange Points Special Locations In Space (youtube) 

The Principia post announcing the new reference frame:

I've updated the original post with various NASA Artemis missions links. 

Posts with interesting revisions or new saves exploring related orbits are welcome, please share them via link(s)!

Edited July 8, 2023 by AloE
added links to the revised save files