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The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity of solving the effect of three bodies which all pull on each other while moving, a total of six interactions. Mathematician Richard Arenstorf while at NASA solved a special case of this problem, by simplifying the interactions to four, because, the effect of the spacecraft's gravity upon the motion of the vastly more massive Earth and Moon is practically non-existent. Arenstorf found a stable orbit for a spacecraft orbiting between the Earth and Moon, shaped like an '8'

http://en.wikipedia.org/wiki/Richard_Arenstorf

Was Arenstorf's solution purely analytical, or did he use numerical mechanisms? Is the '8' shape an optimal path, meaning the route on which the spacecraft would expand the least amount of energy? If yes, how was this requirement included in the derivation in mathematical form?

If anyone has a clean derivation for this problem, that would be great, or any links to books, other papers, etc.

Note: Apparently there was an earlier related mathoverflow question on this as well:

http://mathoverflow.net/questions/52489/on-the-non-rigorous-calculations-of-the-trajectories-in-the-moon-landings

Arenstorf's technical report is here

http://hdl.handle.net/2060/19630005545

The most clean, succint description of the equations I could find were here

http://farside.ph.utexas.edu/teaching/celestial/Celestial.pdf

http://www.uibk.ac.at/mathematik/personal/csomos/arenstorf_en.pdf

Restricted 3 Body Problem mentioned in this book (chapter 8)

http://ads.harvard.edu/books/1989fcm..book/

Some vpython (visual python) code can simulate Apollo 13 trip,

http://maths.anu.edu.au/comptlsci/Tutorial-Gravity/tutorial_apollo_link2.html

the result is the 8-orbit as seen here

enter image description here

Some explanation on the math, and superposition of gravities is also mentioned in this link.

http://maths.anu.edu.au/comptlsci/Tutorial-Gravity/tutorial_apollo.html

I have no idea yet how this would relate to Arenstorf's method, I believe in this code Earth and Moon are not moving, but I thought it'd be nice to share.

Regards,

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1  
Arenstorf actually found his orbits through numerics. There's a short discussion in Hairer/Norsett/Wanner with references. I'll post it if nobody beats me to it... –  J. M. Jul 31 '11 at 11:53
    
he used numerics.. interesting. thanks. Hairer/Norsett/Wanner discussion would be much appreciated. –  BB_ML Jul 31 '11 at 14:22
1  
I'm far away from my notes, so it might take me a while to write something. :( I was about to suggest looking at Fehlberg's "Runge-kutta type formulas of high-order accuracy and their application to the numerical integration of the restricted problem of three bodies" but there seems to be no digital copy of that available. Additionally, there's related work (in German) by Filippi that should interest you as well. –  J. M. Jul 31 '11 at 14:41
6  
This was crossposted to MO. In the future, please wait some time before posting your question in multiple fora, and when you do, provide links to the other posts - as you can imagine, it would be frustrating for someone to put time into answering your question here, only to see hear from you that you'd already gotten the solution elsewhere. –  Zev Chonoles Jul 31 '11 at 15:45
2  
See this as well... –  J. M. Jul 31 '11 at 15:49

1 Answer 1

In the comments, you were already informed that numerics had to be used so I will explain from that point on.

The figure 8 design isn't because it is an optimal path. It occurs due to the gravity of moon and the Earth. When the spacecraft comes within the sphere of influence (SOI) of the moon, the spacecraft is pulled towards it. If the spacecraft is moving at escape velocity, the moon will perturb the flight but the spacecraft won't do a fly by. With the current speed, the moon's gravity is enough to cause an orbital fly by. Upon exiting the moon's SOI, the spacecraft is being pulled in by the Earth. Since the trajectory of the fly by was throwing the spacecraft away from the moon, it crosses its original path but this is short lived since the Earth then pulls it back in. If the spacecraft would have picked up enough velocity from the orbital maneuver to be on a parabolic or hyperbolic trajectory, it could have escaped the pull of Earth and been sent out into space.

One way to determine test speeds in designing a flight is to find the Jacobi constant, $C$. For a given $C$, the zero velocity curves are determined. Since we wanted to reach the moon, $C\geq -1.6649$ which corresponds to an initial velocity of at least $10.85762$ but a velocity of $11.01$ is the required escape velocity from the Earth so the initial speed has to be less than $v_{esc}$.


We can derive the equations of motion only up to a point. I am skipping the basic derivation of the two body problem so we can move on to the crux of the matter. \begin{alignat*}{4} r_{12} & = \sqrt{(x_1 - x_2)^2} & \qquad & x_1 &&{}= \text{Is the } x \text{ location of } m_1\\ x_2 & = x_1 + r_{12} & & &&{}\phantom{=} \text{relative to the center of gravity.}\\ x_1 & = \frac{-m_2}{m_1 + m_2}r_{12} & & \pi_1 &&{}= \frac{m_1}{m_1 + m_2}\\ x_2 & = \frac{m_1}{m_1 + m_2}r_{12} & & \pi_2 &&{}= \frac{m_2}{m_1 + m_2}\\ 0 & = m_1x_1 + m_2x_2 \end{alignat*} We can describe the position of $m$ as $\mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}$ in relation to the center of gravity, i.e., the origin. \begin{align} \mathbf{r}_1 & = (x - x_1)\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}\\ & = (x + \pi_2r_{12})\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}\tag{1}\\ \mathbf{r}_2 & = (x - x_2)\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}\\ & = (x - \pi_1r_{12})\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}\tag{2} \end{align} Let's define the absolute acceleration where $\omega$ is the initial angular velocity which is constant. Then $\omega = \frac{2\pi}{T}$. $$ \ddot{\mathbf{r}}_{\text{abs}} = \mathbf{a}_{\text{rel}} + \mathbf{a}_{\text{CG}} + \mathbf{\varOmega}\times(\mathbf{\varOmega}\times\mathbf{r}) + \dot{\mathbf{\varOmega}}\times\mathbf{r} + 2\mathbf{\varOmega}\times\mathbf{v}_{\text{rel}} \tag{3} $$ where \begin{alignat*}{4} \mathbf{a}_{\text{rel}} & = \text{Rectilinear acceleration relative to the frame} & \quad & \mathbf{\varOmega}\times(\mathbf{\varOmega}\times\mathbf{r}) &&{}= \text{Centripetal acceleration}\\ 2\mathbf{\varOmega}\times\mathbf{v}_{\text{rel}} & = \text{Coriolis acceleration} \end{alignat*} Since the velocity of the center of gravity is constant, $\mathbf{a}_{\text{CG}} = 0$, and $\dot{\mathbf{\varOmega}} = 0$ since the angular velocity of a circular orbit is constant. Therefore, $(3)$ becomes: $$ \ddot{\mathbf{r}} = \mathbf{a}_{\text{rel}} + \mathbf{\varOmega}\times(\mathbf{\varOmega}\times\mathbf{r}) + 2\mathbf{\varOmega}\times\mathbf{v}_{\text{rel}} \tag{4} $$ where \begin{align} \mathbf{\varOmega} & = \varOmega\hat{\mathbf{k}}\tag{5}\\ \mathbf{r} & = x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}\tag{6}\\ \dot{\mathbf{r}} & = \mathbf{v}_{\text{CG}} + \mathbf{\varOmega}\times\mathbf{r} + \mathbf{v}_{\text{rel}}\tag{7}\\ \mathbf{v}_{\text{rel}} & = \dot{x}\hat{\mathbf{i}} + \dot{y}\hat{\mathbf{j}} + \dot{z}\hat{\mathbf{k}}\tag{8}\\ \mathbf{a}_{\text{rel}} & = \ddot{x}\hat{\mathbf{i}} + \ddot{y}\hat{\mathbf{j}} + \ddot{z}\hat{\mathbf{k}}\tag{9} \end{align} After substituting $(5)$, $(6)$, $(8)$, and $(9)$ into $(3)$, we obtain $$ \ddot{\mathbf{r}} = \left(\ddot{x} - 2\varOmega\dot{y} - \varOmega^2x\right)\hat{\mathbf{i}} + \left(\ddot{y} + 2\varOmega\dot{x} - \varOmega^2y\right)\hat{\mathbf{j}} + \ddot{z}\hat{\mathbf{k}}. \tag{10} $$ Newton's $2^{\text{nd}}$ Law of Motion is $m\mathbf{a} = \mathbf{F}_1 + \mathbf{F}_2$ where $\mathbf{F}_1 = -\frac{Gm_1m}{r_1^3}\mathbf{r}_1$ and $\mathbf{F}_2 = -\frac{Gm_2m}{r_2^3}\mathbf{r}_2$. Let $\mu_1 = Gm_1$ and $\mu_2 = Gm_2$. \begin{align} m\mathbf{a} & = \mathbf{F}_1 + \mathbf{F}_2\\ m\mathbf{a} & = -\frac{m\mu_1}{r_1^3}\mathbf{r}_1 - \frac{m\mu_2}{r_2^3}\mathbf{r}_2\\ \mathbf{a} & = -\frac{\mu_1}{r_1^3}\mathbf{r}_1 - \frac{\mu_2}{r_2^3}\mathbf{r}_2\\ \bigl(\ddot{x} - 2\varOmega\dot{y} - \varOmega^2x\bigr)\hat{\mathbf{i}} + \bigl(\ddot{y} + 2\varOmega\dot{x} - \varOmega^2y\bigr)\hat{\mathbf{j}} + \ddot{z}\hat{\mathbf{k}} & = -\frac{\mu_1}{r_1^3}\mathbf{r}_1 - \frac{\mu_2}{r_2^3}\mathbf{r}_2\\ \bigl(\ddot{x} - 2\varOmega\dot{y} - \varOmega^2x\bigr)\hat{\mathbf{i}} + \bigl(\ddot{y} + 2\varOmega\dot{x} - \varOmega^2y\bigr)\hat{\mathbf{j}} + \ddot{z}\hat{\mathbf{k}} & = \begin{aligned} - & \frac{\mu_1}{r_1^3}\Bigl[(x + \pi_2r_{12})\hat{\mathbf{i}} + \hat{\mathbf{j}} + \hat{\mathbf{k}}\Bigr]\\ - & \frac{\mu_2}{r_2^3}\Bigl[(x - \pi_1r_{12})\hat{\mathbf{i}} + \hat{\mathbf{j}} + \hat{\mathbf{k}}\Bigr] \end{aligned}\tag{12} \end{align} Now all we have to do is equate the coefficients. \begin{align} \ddot{x} - 2\varOmega\dot{y} - \varOmega^2x & = -\frac{\mu_1}{r_1^3}(x + \pi_2r_{12}) - \frac{\mu_2}{r_2^3}(x - \pi_1r_{12})\\ \ddot{y} + 2\varOmega\dot{x} - \varOmega^2y & = -\frac{\mu_1}{r_1^3}y - \frac{\mu_2}{r_2^3}y\\ \ddot{z} & = -\frac{\mu_1}{r_1^3}z - \frac{\mu_2}{r_2^3}z \end{align} We now have system of nonlinear ODEs. We can assume the trajectory is in plane and we do that by letting $z = 0$ so we only have two equations remaining: \begin{align} \ddot{x} - 2\varOmega\dot{y} - \varOmega^2x & = -\frac{\mu_1}{r_1^3}(x + \pi_2r_{12}) - \frac{\mu_2}{r_2^3}(x - \pi_1r_{12})\\ \ddot{y} + 2\varOmega\dot{x} - \varOmega^2y & = -\frac{\mu_1}{r_1^3}y - \frac{\mu_2}{r_2^3}y \end{align} Now to achieve the require trajectory we can use the true anomaly of $-90^{\circ}$, a flight path angle of $20^{\circ}$, and an initial velocity of $v_0 = 10.9148$ km/s. At this point, we have no other choice but to numerically integrate. Here is a code that will achieve the desired plot. However, in the actually apollo flight, they had mid course corrections so they arrived exactly on the other side of Earth.

Python

#!/usr/bin/env ipython

import numpy as np
import scipy
from scipy.integrate import odeint
from mpl_toolkits.mplot3d import Axes3D
from scipy.optimize import brentq
from scipy.optimize import fsolve
import pylab

me = 5.974 * 10 ** 24  #  mass of the earth
mm = 7.348 * 10 ** 22  #  mass of the moon
G = 6.67259 * 10 ** -20  #  gravitational parameter
re = 6378.0  #  radius of the earth in km
rm = 1737.0  #  radius of the moon in km
r12 = 384400.0  #  distance between the CoM of the earth and moon
M = me + mm
d = 200  #  distance the spacecraft is above the Earth
pi2 = mm / M
mue = 398600.0  #  gravitational parameter of earth km^3/sec^2
mum = G * mm  #  grav param of the moon
omega = np.sqrt(mu / r12 ** 3)
vbo = 10.9148
nu = -np.pi*0.5
gamma = 20*np.pi/180.0  #  angle in radians of the flight path

vx = vbo * (np.sin(gamma) * np.cos(nu) - np.cos(gamma) * np.sin(nu))
#  velocity of the bo in the x direction
vy = vbo * (np.sin(gamma) * np.sin(nu) + np.cos(gamma) * np.cos(nu))
#  velocity of the bo in the y direction

xrel = (re + d)*np.cos(nu)-pi2*r12
#  spacecraft x location relative to the earth
yrel = (re + 200.0) * np.sin(nu)

u0 = [xrel, yrel, 0, vx, vy, 0]

def deriv(u, dt):
    return [u[3],  #  dotu[0] = u[3]
            u[4],  #  dotu[1] = u[4]
            u[5],  #  dotu[2] = u[5]
            (2 * omega * u[4] + omega ** 2 * u[0] - mue * (u[0] + pi2 * r12) /
             np.sqrt(((u[0] + pi2 * r12) ** 2 + u[1] ** 2) ** 3) - mum *
             (u[0] - pi1 * r12) /
             np.sqrt(((u[0] - pi1 * r12) ** 2 + u[1] ** 2) ** 3)),
            #  dotu[3] = that
            (-2 * omega * u[3] + omega ** 2 * u[1] - mue * u[1] /
             np.sqrt(((u[0] + pi2 * r12) ** 2 + u[1] ** 2) ** 3) - mum * u[1] /
             np.sqrt(((u[0] - pi1 * r12) ** 2 + u[1] ** 2) ** 3)),
            #  dotu[4] = that
            0]  #  dotu[5] = 0


dt = np.linspace(0.0, 535000.0, 535000.0)  #  secs to run the simulation
u = odeint(deriv, u0, dt)

x, y, z, x2, y2, z2 = u.T

# 3d plot
fig = pylab.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot(x, y, z, color = 'r')

# 2d plot if you want
#fig2 = pylab.figure(2)
#ax2 = fig2.add_subplot(111)
#ax2.plot(x, y, color = 'r')

#  adding the moon
phi = np.linspace(0, 2 * np.pi, 100)
theta = np.linspace(0, np.pi, 100)
xm = rm * np.outer(np.cos(phi), np.sin(theta)) + r12 - pi2 * r12
ym = rm * np.outer(np.sin(phi), np.sin(theta))
zm = rm * np.outer(np.ones(np.size(phi)), np.cos(theta))
ax.plot_surface(xm, ym, zm, color = '#696969', linewidth = 0)
ax.auto_scale_xyz([-8000, 385000], [-8000, 385000], [-8000, 385000])
#  adding the earth
xe = re * np.outer(np.cos(phi), np.sin(theta)) - pi2 * r12
ye = re * np.outer(np.sin(phi), np.sin(theta))
ze = re * np.outer(np.ones(np.size(phi)), np.cos(theta))
ax.plot_surface(xe, ye, ze, color = '#4169E1', linewidth = 0)
ax.auto_scale_xyz([-8000, 385000], [-8000, 385000], [-8000, 385000])

ax.set_xlim3d(-20000, 385000)
ax.set_ylim3d(-20000, 80000)
ax.set_zlim3d(-50000, 50000)

pylab.show()

enter image description here

Matlab

script:

days = 24*3600;
G = 6.6742e-20;
rmoon = 1737;
rearth = 6378;
r12 = 384400;
m1 = 5974e21;
m2 = 7348e19;
M = m1 + m2;
pi_1 = m1/M;
pi_2 = m2/M;
mu1 = 398600;
mu2 = 4903.02;
mu = mu1 + mu2;
W = sqrt(mu/r12^3);
x1 = -pi_2*r12;
x2 = pi_1*r12;

d0 = 200;
phi = -90;
v0 = 10.9148;
gamma = 20;
t0 = 0;
tf = 6.2*days;
r0 = rearth + d0;
x = r0*cosd(phi) + x1;
y = r0*sind(phi);

vx = v0*(sind(gamma)*cosd(phi) - cosd(gamma)*sind(phi));
vy = v0*(sind(gamma)*sind(phi) + cosd(gamma)*cosd(phi));
f0 = [x; y; vx; vy];

[t,f] = rkf45(@rates, [t0 tf], f0);
x = f(:,1);
y = f(:,2);
vx = f(:,3);
vy = f(:,4);
xf = x(end);
yf = y(end);
vxf = vx(end);
vyf = vy(end);

df = norm([xf - x2, yf - 0]) - rmoon;
vf = norm([vxf, vyf]);

plot(x,y)

functions: 2 separate ones

function [tout, yout] = rkf45(ode_function, tspan, y0, tolerance)
a = [0 1/4 3/8 12/13 1 1/2];
b = [0 0 0 0 0
     1/4 0 0 0 0
     3/32 9/32 0 0 0
     1932/2197 -7200/2197 7296/2197 0 0
     439/216 -8 3680/513 -845/4104 0
     -8/27 2 -3544/2565 1859/4104 -11/40];
c4 = [25/216 0 1408/2565 2197/4104 -1/5 0];
c5 = [16/135 0 6656/12825 28561/56430 -9/50 2/55];
if nargin < 4
    tol = 1.e-8;
else
    tol = tolerance;
end
t0 = tspan(1);
tf = tspan(2);
t = t0;
y = y0;
tout = t;
yout = y';
h = (tf - t0)/100; % Assumed initial time step.

while t < tf
    hmin = 16*eps(t);
    ti = t;
    yi = y;
    for i = 1:6
        t_inner = ti + a(i)*h;
        y_inner = yi;
        for j = 1:i-1
            y_inner = y_inner + h*b(i,j)*f(:,j);
        end
        f(:,i) = feval(ode_function, t_inner, y_inner);
    end
    te = h*f*(c4' - c5');
    te_max = max(abs(te));
    ymax = max(abs(y));
    te_allowed = tol*max(ymax,1.0);
    delta = (te_allowed/(te_max + eps))^(1/5);
    if te_max <= te_allowed
        h = min(h, tf-t);
        t = t + h;
        y = yi + h*f*c5';
        tout = [tout;t];
        yout = [yout;y'];
    end
    h = min(delta*h, 4*h);
    if h < hmin
        fprintf(['\n\n Warning: Step size fell below its minimum\n'...
            ' allowable value (%g) at time %g.\n\n'], hmin, t)
        return
    end
end

second function:

function dfdt = rates(t,f)
r12 = 384400;
m1 = 5974e21;
m2 = 7348e19;
M = m1 + m2;
pi_1 = m1/M;
pi_2 = m2/M;
mu1 = 398600;
mu2 = 4903.02;
mu = mu1 + mu2;
W = sqrt(mu/r12^3);
x1 = -pi_2*r12;
x2 = pi_1*r12;
x = f(1);
y = f(2);
vx = f(3);
vy = f(4);
r1 = norm([x + pi_2*r12, y]);
r2 = norm([x - pi_1*r12, y]);
ax = 2*W*vy + W^2*x - mu1*(x - x1)/r1^3 - mu2*(x - x2)/r2^3;
ay = -2*W*vx + W^2*y - (mu1/r1^3 + mu2/r2^3)*y;
dfdt = [vx; vy; ax; ay];
end

enter image description here

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