Three-Dimensional Three-Body Problem

AM205 Final Project by ZiZi Zhang and Peter Chang

Due December 18, 2017

1 Introduction and Problem Motivation

Our project extends the circular restricted three-body problem, briefly introduced inHW3 Problem 6, into 3-dimensional space, i.e. (x, y, z) instead of (x, y), since the lossof one of three dimensions is a simplification of the actual motion. Accordingly, thevelocity in the three dimensions is u, v and w, so the state of the third body is representedin a 6-dimensional vector as [x, y, z, u, v, w]T. This is a classical problem in physics andmechanics that has been studied since the 1700s by Euler and Lagrange. Work on theproblem continues to present day, using capabilities of modern supercomputing that havehelped discover thousands of periodic orbits out of infinite possibilities. The equationsdescribed in Murray and Dermott’s Solar System Dynamics were extremely helpful, andhave been applied here. There has even been a novel written in Mandarin about theproblem, creatively titled The Three-Body Problem.

In our extension of the problem formulation, we positioned the system’s center of massat the origin, instead of placing the larger mass at the origin as we did in the homework,such that the two large bodies move in circular coplanar orbits about their center of mass(0, 0, 0). We also maintained a set of unit conversions that would allow for constants in theJacobi and Equations of Motion to equal 1. As we explored the ODEs and their numericalsimulations, we learned that there is a lot of richness in the 3-D space to explore, but alsomany computational limitations. So in our project, we focus primarily on the stability ofthe problem and performance of different numerical methods, and finding the Lagrangepoints, also using numerical methods to optimize the Jacobi.

2 Project Formulation

1. m1 and m2 are the masses of the two bodies, with m1 > m2. Define µ̄ as

µ̄ =m2

m1 + m2

1

and the new unit of mass as

µ = G(m1 + m2) = 1

Then, keeping the center of mass at the origin (0, 0, 0), we have:

(a) Body 1 with mass µ1 = Gm1 = 1− µ̄ at position (−µ2, 0, 0). If this mass is theEarth, the mass is 5.972 ∗ 1024 kg and the radius is re = 0.02.

(b) Body 2 with mass µ2 = Gm2 = µ̄ at position (µ1, 0, 0). If this is the Moon, themass is 7.34767309 ∗ 1022 kg and the radius is rm = 0.005.

(c) Total mass of the system is always 1, and total distance between the two bodiesis always 1, so n, the common mean motion, is also 1 in this unit system.

(d) In the Earth-Moon case, µ1 = 0.99 and µ2 = 0.01.

2. Body 3, the asteroid, is at position p(t) = (x, y, z), with velocity v(t) = (u, v, w).These six parameters form the vector in the system of differential equations.

3. The equations of motion, described by Murray and Dermott, are

∂2x∂t2 − 2n

∂y∂t− n2x = −

[µ1

x + µ2

r31

+ µ2x− µ1

r32

]∂2y∂t2 + 2n

∂x∂t− n2y = −y

[µ1

r31+

µ2

r32

]∂2z∂t2 = −z

[µ1

r31+

µ2

r32

]where r1, r2 are the distance from (x, y, z) to Body 1 and Body 2 respectively.

(a) r21 = (x + µ2)

2 + y2 + z2

(b) r22 = (x− µ1)

2 + y2 + z2

4. Alternatively, we can write the system in terms of the gradient of scalar function U,where

U =n2

2(x2 + y2) +

µ1

r1+

µ2

r2

∂2x∂t2 − 2n

∂y∂t

=∂U∂x

∂2y∂t2 + 2n

∂x∂t

=∂U∂y

∂2z∂t2 =

∂U∂z

2

5. The 3-D Jacobi integral to be conserved is

J(x, y, z, u, v, w) = 2U − ||v(t)||2

= n2(x2 + y2) +2µ1

r1+

2µ2

r2− u2 − v2 − w2

which is neither an energy integral nor an angular momentum integral since neitherof these quantities are conserved in the restricted three-body problem. However, inthe scalar function U, x2 + y2 is the centrifugal potential of the object, and µ1

r1+ µ2

r2is the gravitational potential. Their partial derivatives yield the centrifugal andgravitational forces.

6. Setting n = 1, the system of ODEs is then

x′ ≡ dxdt

= −12

∂J∂u

= u, y′ ≡ dydt

= −12

∂J∂v

= v, z′ ≡ dzdt

= −12

∂J∂w

= w

u′ ≡ dudt

= v +12

∂J∂x

, v′ ≡ dvdt

= −u +12

∂J∂y

, w′ ≡ dwdt

=12

∂J∂z

7. So finally, in condensed form, we have the system

x′

y′

z′

u′

v′

w′

=

uvw

v + x− µ1(x+µ2)

r31− µ2(x−µ1)

r32

−u + y− yµ1r3

1− yµ2

r32

− zµ1r3

1− zµ2

r32

3 Exploration of Various Numerical Integration Methods

In this section, we consider different numerical integration methods as applied to thissystem of ODEs over time range t = 0 to 10. Using the vode numerical integration methodin the scipy.integrate.ode function, we preliminarily simulate 20 trajectories with 20random initial conditions and time-step dt = 0.001, which takes <5 seconds. Again, theassumption is that the trajectory is linear between time 0 and 0.001, as assumed in thehomework assignment between time 0 and 0.02, but the time interval here is 20 timessmaller, significantly increasing the runtime. Figure 1 shows the 20 initial conditions asorange points with the Earth and Moon as blue points, and the 20 trajectories overlaid,and Figure 2 looks at a few of the trajectory shapes individually.

There are a few things to note about the types of trajectories we see here. Some ofthe trajectories that zoom off into outer space without interacting with the system of twobodies are representative of a class of initial conditions in which the initial placement

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of the satellite is too distant from the system, and the initial velocity is too high for thegravitational forces exerted to pull it into a periodic orbit. The combination of these sixparameters under the conservation of the Jacobi results in the satellite escaping out of thesystem and zooming away rapidly from the two bodies. Additionally, the first subplot inFigure 2 represents a trajectory that collides with one of the bodies, so after the collision,the satellite’s velocity is drastically altered and it also escapes the system.

:Fig 1: 20 Simulated Initial Conditions and Trajectories

:Fig 2: Individual Trajectories

3.1 Time Performance

We explored a variety of numerical integration methods to see which would performbest for this problem. Based on the documentation here, the ode numerical integration

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method in scipy.integrate.ode offers both the implicit Adams method for non-stiffproblems, the BDF method for stiff problems, the 5th-order Runge-Kutta method and8th-order Runge-Kutta method. Comparing runtimes of the routines, the Adams methodtakes shorter than the BDF method and RK methods, due to the lower order of calculations.However, the performance of the method is not only dependent on the runtime, but alsothe stability.

:Fig 3: Numerical Method Operating Times

3.2 Stability Performance

Now we compared the stability of the numerical methods, by generating a randominitial point p, then perturbing it by a small p + δp, where δp takes the form

δp =

δxδyδz000

Only the initial location is changed, not the initial velocities. Since perturbing the initial

direction of motion could drastically change the direction of the trajectory, the trajectorystarting at p0 would not be directly comparable to that starting at p0 + δp if δu, δv, δw werenot 0.

Plotting the Frobenius norm of the difference in the two solutions, | f (p0 + δp)− f (p0)| f ,generated by the four different methods versus time yields the following sample plots inFigures 4-7. Although the implicit Adams and BDF methods take the shortest time to run,

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:Fig 4: Initial Condition 1 Stability Comparison and Trajectory Comparison

:Fig 5: Initial Condition 2 Stability Comparison and Trajectory Comparison

:Fig 6: Initial Condition 3 Stability Comparison and Trajectory Comparison

:Fig 7: Initial Condition 4 Stability Comparison and Trajectory Comparison

they generally perform the worst, resulting in the largest output perturbation as notedwith Initial Conditions 2, 3 and 4.

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:Fig 8: Zero-Velocity Curves for Various Jacobi and Mass Ratio Values

4 Zero-Velocity Regions of the 3-Body Problem

Where does the boundary lie between initial conditions that result in a rogue trajectoryexiting the gravitational reaches of the system, and initial conditions that result in atrajectory orbiting the two bodies? Considering the set of zero-velocity surfaces plays animportant role in bounding the particle’s motion. Setting the velocity v′(t) = (u, v, w) to 0,the Jacobi becomes

Jc = n2(x2 + y2) +2µ1

r1+

2µ2

r2

Points (x, y, z) that satisfy the above equation for a value Jc fall onto the zero-velocitysurface. These regions compose the Hill’s stability region, inside of which the particleorbits the two bodies and cannot escape, while outside of which the particle has escapedthe system. In the two-dimensional plane (setting z = 0), the zero-velocity surfaces becomezero-velocity curves (ZVC), which are less oblong (i.e. more circular) as the value of theJacobi increases.

5 Lagrange Points

Related to the zero-velocity curves of stable regions in the problem are Lagrange points,an important topic in astrophysics. These points are the equilibrium points of the Jacobi

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integral where if a third body is introduced with zero velocity (in the co-rotating inertialframe) at one of these Lagrange points, it would remain in an equilibrium position. Thestability of these points may not always be sound however. In this part we find theseLagrange points derived through numerical optimization methods rather than in thealgebraic context normally used.

5.1 Co-Linear Lagrange Points

Again, the Jacobi integral is:

J(x, y, z, u, v, w) = x2 + y2 +2µ1

r1+

2µ2

r2− u2 − v2 − w2

= x2 + y2 +2µ1√

(x + µ2)2 + y2 + z2+

2µ2√(x− µ1)2 + y2 + z2

− u2 − v2 − w2

We can first simplify this Jacobi integral by noting that a Lagrange point must exist onthe plane z = 0. If a new body were introduced at a z 6= 0, both initial bodies would exerta gravitational force in the z-direction and so it may not be stable. We can also simplifythis by setting u = v = w = 0, according to the definition of the Lagrangian point.

We will start by searching for the Lagrange points that lie in the same line as the Earthand moon (y = 0). Setting up this simplified Jacobi, we get the following:

J(x, 0, 0, 0, 0, 0) = x2 +2µ1

|x + µ2|+

2µ2

|x− µ1|(1)

∂J∂x

= 2x− 2(x + µ2)µ1

|x + µ2|3− 2(x− µ1)µ2

|x− µ1|3(2)

Graphing the Jacobi shows that there are three equilibrium points in this co-linear axis.The gray lines in the graph represent the positions of Earth and the Moon and are pointswhere the Jacobi approaches infinite asymptotically:

Using numerical optimization methods to find the local minimum points gives usthree points at x = .84807871, 1.14676504,−1.00416661. By convention we will label theLagrange points, L1, L2, and L3 respectively:

L1 = (.84807871, 0, 0) (3)

L2 = (1.14676504, 0, 0) (4)

L3 = (−1.00416661, 0, 0) (5)

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:Figure 9: Plot of Jacobian vs x

:Figure 10: Positions of Respective Bodies

5.1.1 Comparison of Optimization Methods

With these local minima being so close to each other and with unstable area approachinginfinite at x = µ1 and x = µ2, choosing the correct optimization method was important toget good convergence.

Here we compared several different optimization algorithms to see which ones per-

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formed best in finding the Lagrangian points, by measuring the number of iterationsneeded to converge upon each point when initialized at specific values for each of therespective Lagrangian points. Certain algorithms also did not converge on all of theLagrangian points:

Iterations Needed (DNC- Did Not Converge)Algorithm L1 L2 L3BFGS 5 7 5Nelder-Mead 26 26 28Conjugate Gradient DNC DNC 4Powell DNC 2 2

We can see that the BFGS algorithm converged the quickest while also not sufferingfrom precision loss as was the common case in the Conjugate Gradient method. ThePowell algorithm had a faster convergence, but it was not able to accurately converge uponL1. Instead it converged upon L2. This is a flaw that could be expected due to the twoLagrangian points being near each other along the x-axis. For any future optimizationswithin this project, we will be using the BFGS algorithm.

5.2 L4 and L5

After finding the first Lagrangian points, we needed to find the other Lagrangian pointswhere y 6= 0. This becomes a more computationally intensive problem to solve since wemust now search a 2-dimensional space for the equilibrium points.

We start with the same baseline parameters of z = u = v = w = 0, but now keepingthe y term, we get a Jacobi of:

J(x, y) = x2 + y2 +2µ1√

(x + µ2)2 + y2+

2µ2√(x− µ1)2 + y2

In order to find the equilibrium points, we must satisfy the following two equations:

∂J∂x

= 2x− 2µ1(x + µ2)

((x + µ2)2 + y2)32− 2µ2(x− µ1)

((x− µ1)2 + y2)32= 0 (6)

∂J∂y

= 2y− 2µ1y

((x + µ2)2 + y2)32− 2µ2y

((x− µ1)2 + y2)32= 0 (7)

To solve this numerically using an optimization algorithm, we instead looked at thesquare value of the derivatives. Doing this casts all negative derivative values to positivevalues allowing us to use a minimization algorithm to find where the derivative is zero.As a result of the previous testing of optimization functions, we used the BFGS algorithmto find the minima for these two functions.

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:Figure 11: Equilibrium Curve with Respect to y

:Figure 12: Equilibrium Curve with Respect to x

Iterating over a wide range of initial (x,y) positions, we used BFGS optimization toindependently compute all points where

( ∂J∂x)2

= 0 and all points where( ∂J

∂y)2

= 0.

The two points at which both( ∂J

∂y)2

= 0 and( ∂J

∂y)2

= 0 are the final two Lagrangianpoints (Figure 13):

L4 = (.49, .866, 0) = (.49,

√3

2, 0) (8)

L5 = (.49,−.866, 0) = (.49,−√

32

, 0) (9)

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:Figure 13: Overlayed Curves

These two points, L4 and L5, each form an equilateral triangle with the Earth and theMoon (Figure 14).

:Figure 14: Positions of Respective Bodies

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6 Stability of Lagrange Points

Although all of these Lagrange points exist at equilibrium, not all of them are stable.Dependent on the gradient, small perturbations will either cause the new body to spin outof its position or will do nothing at all.

To test this, we introduced small perturbations to the bodies and simulated theirtrajectories for some time. These trajectories are mapped in the co-rotating frame, so Earthand the Moon are fixed at their constant positions (Earth is large blue dot, the Moon is thesmaller black dot):

L1 Trajectory L2 Trajectory

L3 Trajectory L4 Trajectory

L5 Trajectory

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From these simulations, we can see that objects placed at L1, L2, and L3 spin out ofposition when small perturbations are introduced while L4 and L5 remain stable.

The positions of L4 and L5 are known as Trojans in astrophysics and are positions atwhich space objects can often cluster do to the stability of these points. While here, anobject will minimize its potential energy.

In practice, however, even the unstable Lagrangian points are still useful. Satelliteshave been deployed at the L1 and L2 points in relation to the Earth and Sun. Satellites atL1 provide constant imaging of the Sun while satellites at L2 provide constant imaging ofthe cosmos. These satellites need stabilization in order to maintain these positions, but thiscost is very little relative to the benefits of having satellites there.

7 Conclusion

In this project we worked to expand the restricted three-body problem into 3 dimen-sions. From here we explored the various phenomena that are associated with such asystem and determined the best numerical methods for solving the respective problems.First, we simulated sallelite trajectories using different numerical integration methodsto solve the system of ODEs, and calculated the stability of such methods after a smallperturbation was introduced to the initial condition. Then, we solved the Jacobi for valuesof Jc to visualize the zero-velocity curves, representing the stability regions of orbits aroundthe two masses.

After that we explored the Lagrange points. Using optimization algorithms, we foundthe three Lagrange points in line with the Earth and the Moon as well as the two ”Tro-jan” Lagrange points. Stability analysis showed that the co-linear points are unstable toperturbations while the Trojans are stable.

These were our primary goals leading into the project and we were able to completethem satisfactorily.

7.1 Limitations of Models

One of the main limitations for our model is the computational complexity of consider-ing a 3-dimensional space. For example, we attempted to simulate the probability of anasteroid hitting either the Earth or Moon (as in a pset) but running a sufficient quantity ofsimulations to determine impact took much more computational intensity and runtimethan in the pset. This is a difficult trade-off since real-world simulations must be done in3-dimensions for accuracy but computation time grows substantially. However, based ona few preliminary simulations, it seems the collision probability is slightly lower in the3-D space versus the 2-D space, which is to be expected.

Additionally, because no closed-form solution exists for solving the 3-body problem,fully understanding the system requires simulation. However, it is practically impossibleto test all possible initial positions and velocities. Even modern supercomputers couldnot exhaust the infinite possible cases needed to conclusively describe the entire 3-body

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problem. So for the sake of our project, we were only able to test a tiny subset of thesepossible cases.

7.2 Further Questions

One thing we could further explore in this problem is how the locations of the Lagrangepoints change when dealing with two general bodies of mass m1 and m2. Mapping thesepositions as a function of m1 and m2 would be very interesting to look into.

This work on the 3-body problem also raised further interest in the n-body problem,which remains an unsolved problem in the scientific community. We explored interestingquestions in the 3-body problem alone and it would be very interesting to see how thesetranslate when more masses are added to the system.

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