Bringing your Analogue past into our Digital future

1.The Jacobi Integral
The Jacobi integral occurs within the circular restricted three body problem (CR3BP). In our case we are looking at the Earth moon system and a spacecraft which is presumed to have negligible mass. The two primary bodies, the Earth and the moon are, in this model rotating around there centre of mass. The model of the system is further restricted to the plane of the primary bodies. The model has two second order equations. The Jacobi integral is one of these equations.

Derivation of the Jacobi Integral
Looking to a system of three bodies, in this case the Earth, the Moon and a spacecraft. Making the masses of the Earth and the Moon into unity to give the Earth a mass of 1-mu and the Moon a mass of mu. The separation of the Earth and the Moon is also unified. We first consider the system in a fixed frame of reference. The coordinates of the Earth moon and spacecraft are
, and respectively. [ORBMOTION]

The equations of motion for the spacecraft are:

Changing our frame of reference into a rotating frame with axes x,y and z with the Earth and the Moon on the x axes and the centre of mass as the origin. The coordinates of the Earth and the Moon are set up to make them equal to the mass relation such that the Earth is at 1-mu and the Moon at mu along the x axis.
The new coordinates relate to the old via:

Differentiating these equations twice and substituting into Eqn 1 produces:

Simplifying Eqn 5 by multiplying the first equation by cos t then multiplying the second by sin t and adding. Multiple the first equation by -sin t and the second by cos t then add producing:

Define a function U as:

Using Eqn 7, Eqn 6 can be written as:
multiplying through produces:

Integrating provides us with the Jacobi equation, where C is the constant of integration otherwise known as the Jacobi constant

The basis of this report is to look into the uses of the Jacobi Integral as a tool to planning spacecraft trajectories. The Jacobi Integral is the first integral of motion of the restricted three body problem R3BP .[P1108PDF]. The Jacobi equation can be written in the form:

[ORBMOTION]
The R3BP has two integrals of motion, the Jacobi integral and the Natural energy integral. [3Body] This makes the R3BP completely integrable. The Jacobi integral represents the total energy of the R3BP and is otherwise called the integral of relative energy.[ORBMOTION]The Jacobi integral is constant for spacecraft on a given orbit. For a given spacecraft existing within the R3BP the Jacobi Integral represents the energy of that spacecraft. The total energy of the spacecraft is made up of the kinetic and potential energy. If the spacecraft in question had zero velocity the Jacobi Integral can be solved numerically. For the case when the spacecraft within the R3BP has zero velocity the Jacobi integral becomes equal to zero and the velocity terms are removed.

Further more looking at the R3BP system in a single rotating plane gives the Jacobi Integral in a useful and numerically solvable form. In this form the Jacobi constant C is the subject. The solution to the Jacobi equation in this from, Eqn 14, produces surfaces of zero velocity. When a spacecraft is on one of these surfaces there is a balance of energy and from this balance of energy the spacecraft has zero velocity. This means that the surfaces are boundaries to areas that cannot be entered or crossed unless the spacecraft is given extra energy. Within the surface the spacecraft is free to move. The Jacobi constant C determines the shape of the surface. Altering the Jacobi constant therefore determines whether the spacecraft can move between the local Earth region and the local Lunar region. High values of C prevent movement between the Earth and Lunar regions. As the value of C is reduced a gap opens between the regions. At a certain value Ct the gap is open just enough to allow a spacecraft through. Finding this value Ct will give the minimum energy situation.[ORBMOTION]

2.The Lagrangian points
The Lagrangian points are positions in a system of three bodies where the gravitational attraction of the two primary masses cancel the centripetal acceleration required to rotate with the two masses. They are the five stationary solutions to the circular restricted three body problem.
There are five Lagrangian points, every pair of Massive bodies has a set of five Lagrange points[EXIT]. Three of the Lagrange points are in line with the two bodies in a rotating frame of reference and two are at the apexes of equilateral triangles made by connecting the two bodies. If a particle is placed on any of these points it will remain in position relative to the two bodies. Joseph-Louis Lagrange (1736-1813) discovered the L4 and L5 points. Leonhard Euler (1707-1783) discovered the L1, L2 and L3 points.[EXIT]

The Lagrange points are also known as Libration points and L-points. They are labeled as L1-L5. All the L points exist in x,y plane. L1-L3 are saddle points with the potential curving upwards in one direction and downwards in the other direction. L4 and L5 are raised points with the potential falling away in all directions. Lagrangian points are solutions to the equations of motion of a spacecraft in the three body system. At the Lagrangian point the spacecrafts velocity in the x, y and z directions will be 0. There are five solutions to the equations the equations form a quintic polynomial which must be solved for x to get the positions of the L1 to L3 points. The L4 and L5 points are positioned at the top of equilateral triangles formed by the Earth and the Moon. [LPOINTS] Lagrangian points prove very useful and important in Lunar trajectory design [NASAL], [Lunar gate].

1.Lagrangian stability
Of the five points L4 and L5 are stable as long as the ratio of the masses is greater than 24.49. For our Earth moon system the ratio of masses is 81.36 so the L4 and L5 points are stable. The L4 and L5 points are stable even if an object on one of them is disturbed and begins to move away. Jules-Henri Poincare (1854-1912) developed some of the tools used to deal with celestial instability in the three body problem [EXIT]. As the object falls from the L point its velocity increases, this is when the Coriolis effect takes over and the object enters into an orbit around the L point. These orbits are known as Halo orbits as seen from Earth the trajectory appears to create a Halo around the Sun.[EXIT] Because of th stability of the L4 and l5 points asteroids and other space objects collect at thee points. The Sun Jupiter L4 and L5 points have a large collection of asteroids around them. These are known as Trojan asteroids. The other three points L1,L2 and L3 are unstable. [Lagrange].

2.Locating the Lagrangian Points
Along with analytical methods of locating the points the Jacobi equation can indirectly locate the points. This can be achieved using the Surfaces of zero energy, varying the value of C changes the surface shape and with certain values of C the points become apparent. For our purposes an important point to is the L1 point, located in between the Earth and the Moon. To find the L1 point using the Jacobi equation a value of C is used which gives a Surface of Zero Velocity that has just closed the gap between the Earth and the Moon. The point at which the Surface of Zero Velocity just closes is the L1 point.

(note all equations and pictures are missing from this copy, they will be added soon as I can. )