Lunar Free Return Trajectory Simulation

Question

I have been trying to simulate a lunar free return trajectory using the state vectors for the Apollo spacecraft provided by JPL after they had performed their trans-lunar injection burn.

My understanding was that if something had gone wrong, the Apollo spacecrafts would just have swung around the moon and then back to Earth. However, in all of my simulations, disregarding which Apollo mission I'm using for the vectors, the spacecraft does not swing around the moon before coasting back to Earth, but enters a hyperbolic trajectory... Good bye brave astronauts :(

I'm wondering whether anyone could shed some light on whether this is because my algorithm is not powerful enough (tried super mega slow time-steps but you always get the same result), or if I have missed something?

Answer 1

This is a partial answer, hopefully it will lead to more discussion and a resolution.

I've found this animation of a free-return trajectory in what looks like earth-fixed "inertial" coordinates, at least the coordinates are not rotating with the Earth-Moon system.

Found at Robert A. Braeunig's Apollo 11's Translunar Trajectory and how they avoided the heart of the radiation belts.

Does your simulation aim for a spot in front of the leading edge of the moon?

This GIF is SLOW! Keep watching until complete!

Answer 2

Totally forgot about having asked this question, but fortunately I've been able to solve it. The were two problems: my integrator wasn't up for the task and you do have to include the Sun in a free return trajectory simulation as its gravitational influence on the Earth Moon system is not negligble, even on the time scale of days (at least with the integrator that I employed).

Here's the data I used for the scenario (Apollo 11, fetched from JPL Horizons), where g is the gravitational constant and dt is the time step. AUs for distance, years for time and solar masses for masses:

export default {
  name: 'Apollo 11 - Free Return Trajectory',
  g: 39.5,
  dt: 40e-7,
  distMax: 0.00713911058,
  distMin: -0.00713911058,
  distStep: 2.3797035266666667e-6,
  velMax: 0.5,
  velMin: -0.5,
  velStep: 5e-6,
  rotatingReferenceFrame: 'Earth',
  cameraPosition: 'Free',
  cameraFocus: 'Origo',
  freeOrigoZ: 16000,
  massBeingModified: 'Sun',
  primary: 'Earth',
  maximumDistance: { name: 'Moon to Earth * 10', value: 0.0256955529 },
  distanceStep: { name: 'Moon to Earth / 100', value: 0.0005139110579999999 },
  scenarioWikiUrl: 'https://en.wikipedia.org/wiki/Free-return_trajectory',
  masses: [
    {
      name: 'Earth',
      x: 0.4240363252016235,
      y: -0.9248449798862485,
      z: -1.232690294681233e-4,
      vx: 5.622675894714279,
      vy: 2.5745894556521574,
      vz: 3.8057228235271535e-4,
      trailVertices: 2e4
    },
    {
      name: 'Sun',
      x: 0.004494747940528018,
      y: 9.145777867796766e-4,
      z: -6.127893755128986e-5,
      vx: -1.7443876658803292e-4,
      vy: 0.002043973630637931,
      vz: -4.697196039923407e-6,
      trailVertices: 2e4
    },
    {
      name: 'Moon',
      x: 0.4220528422463315,
      y: -0.9230209264977778,
      z: 1.632323615688905e-5,
      vx: 5.486589374929882,
      vy: 2.420601498441581,
      vz: -0.014677846271227611,
      trailVertices: 2e4
    },
    {
      name: 'Apollo 11',
      x: 0.4240447232851519,
      y: -0.9247715402118077,
      z: -1.129301018611092e-4,
      vx: 4.395253850175561,
      vy: 3.8323649107803948,
      vz: 0.15792573886687206,
      trailVertices: 15e4
    }
  ]
};

export default [
  {
    m: 0.000003003,
    radius: 91.74311926605505,
    color: 'limegreen',
    name: 'Earth'
  },
  {
    m: 3.69396868e-8,
    radius: 22.93577981651376,
    color: 'grey',
    name: 'Moon'
  },
  {
    m: 0,
    radius: 1.2,
    color: 'limegreen',
    name: 'Apollo 11'
  },
  {
    m: 1,
    radius: 90000,
    type: 'star',
    color: 'yellow',
    name: 'Sun'
  }
];

And the result