MontBlanc2012
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This post is the third part in a series focusing on the development of a Lambert Solver for trajectories in the vicinity of the Lagrange points of the restricted three-body model.
A Lambert Solver solves the two-point boundary problem of determining the trajectory of getting from point 'A' to point 'B' in a specified time 'dt'. Lambert Solvers exist are frequently used for solving trajectory planning problem in the standard restricted two-body problem (Keplerian motion) and an example of a solver can be found in IMFD's Course Program.
Whereas earlier parts focused primarily on the Circular Restricted Three Body Problem (CR3BP), this note shows that that earlier work can be extended to the Elliptical Restricted Three Body Problem (ER3BP). The CR3BP assumes that the primary and secondary gravitating bodies (e.g., the Earth and the Moon in the Earth-Moon restricted three body system) rotate about each other in precise circular orbits, while the ER3BP relaxes this assumption and assumes that the primary and secondary bodies move in precise elliptical orbits about each other. For many systems, this elliptical orbit assumption is nearly correct. Even for the Earth-Moon system, the motion of the Moon around the Earth is dominated by its near elliptical motion - even though the Sun does perturb that elliptical motion to a noticeable degree. But the introduction of orbital eccentricity in the model amounts to a significant complication that must be accounted for in the Lambert Solver.
The ER3BP represents something of a gold standard for Lagrange point orbit design, so the success of the Lambert Solver for this model is represents a significant milestone and provides the basis for some realistic trajectory design within Orbiter.
The story so far
So far, I've written two posts on the development of a Lambert Solver for Lagrange points:
Lambert Solve for Lagrange points
and
More on a Lambert Solver for Lagrange points
The first post introduced the possibility of building a Lambert Solver and demonstrated that in the linear approximation of the CR3BP high fidelity solutions to the two-point boundary problem could be found using simple linear algebra (and a pre-calculated very high-order variational integrator based on Gauss-Lobatto quadrature). The post was light on the mathematics - but the the supporting mathematics will be posted in this forum at a later date.
The second post extended this demonstration to the full, non-linear CR3BP model and showed that solution generated from the linearised equations of motion could be used to generate the corresponding solution of the full CR3BP model using straight-forward multivariate root-finding techniques. Again, more on the maths of this later.
And now we move on to a demonstration that we can do the same thing for the ER3BP. And this will complete the Lambert Solver prototyping and demonstration. The next step after this will be to link this Lambert Solver up to Orbiter (in something of an ad hoc fashion) to demonstrate that the Lambert Solver works as advertised.
The equations of motion in the ER3BP
Before going on, and to show how the orbital eccentricity changes things, I'm going to write down the equations of motion of the ER3BP model in detail. Here they are:
[MATH]\eta''(\nu)-2\,\kappa'(\nu)=\frac{1}{e\,\cos(\nu)+1}\,\frac{\partial\,\Omega}{\partial\,\eta(\nu)}[/MATH]
[MATH]\kappa''(\nu)+2\,\eta'(\nu)=\frac{1}{e\,\cos(\nu)+1}\,\frac{\partial\,\Omega}{\partial\,\kappa(\nu)}[/MATH]
where
[MATH]\Omega=\frac{\eta(\nu)^{2}}{2}+\frac{\kappa(\nu)^{2}}{2}+\frac{\mu_{2}}{\sqrt{\left(\eta(\nu)-\mu_{1}\right){}^{2}+\kappa(\nu)^{2}}}+\frac{\mu_{1}}{\sqrt{\left(\eta(\nu)+\mu_{2}\right){}^{2}+\kappa(\nu)^{2}}}[/MATH]
A few things about these equations:
1. This equations of motion work in a coordinate system know as rotating pulsating coordinates. These coordinates are set up that even though the Moon rotates around the Earth-Moon barycentre in an elliptical orbit, the position of the Moon (and of the Earth) are fixed. Moreover, it is set up so that the distance between the Earth and the Moon is equal to 1; the Moon and the Earth lie on the x-axis; and the origin of the coordinate system is located at the Earth-Moon barycentre (EMB).
2. The ER3BP uses a change of time coordinates. Instead of using ordinary clock time with which to measure speeds, accelerations and so on, it use the true anomaly of the Moon as a measure of time. This quantity is given the symbol [MATH]\nu[/MATH]. In the CR3BP, where the orbital eccentricity is zero, the true anomaly is related to ordinary clock time by a constant multiplicative factor.
3. The variables in the model are [MATH]\eta(\nu)[/MATH] and [MATH]\kappa(\nu)[/MATH]. [MATH]\eta(\nu)[/MATH] measures the distance along the x-axis in the rotating-pulsating coordinate system; and [MATH]\kappa(\nu)[/MATH] measures the distance along the y-axis in the the rotating-pulsating coordinate system. Motion in the y-direction is transverse motion in the orbital plane of the Earth-Moon system. (For this note, I've surpassed reference to motion in the 'out of the orbital plane' z-direction.)
4. [MATH]\mu_1[/MATH] and [MATH]\mu_2[/MATH] are defined as
[MATH]\mu_1 = \frac{m_1}{m_1+m_2}[/MATH]
[MATH]\mu_2 = \frac{m_2}{m_1+m_2}[/MATH]
where [MATH]m_1[/MATH] and [MATH]m_2[/MATH] are the masses of the Earth and Moon respectively.
5. [MATH]e[/MATH] is the orbital eccentricity of the Moon's orbit (and the Earth's!) around the EMB. If the eccentricity is zero, then
6. [MATH]\Omega[/MATH] is essentially the negative of the effective potential in the rotating-pulsating coordinates. It contains the terms related to centrifugal force, and the gravitational force due to the primary and secondary gravitating bodies.
7. The terms on the right-hand side are the 'force' functions. The orbital eccentricity appears only once in each equation as simple factors in these force functions.
This last item contains the essential point that I now wish to make: Whereas in the CR3BP, we have set the orbital eccentricity to zero so that the factors in the force functions are unity, in the ER3BP we know have to use the full expression for these factors which now contains an explicit reference to the cosine of the time variable, [MATH]\nu[/MATH]. Fortunately, adding this explicit time variable is very simple in the Lambert Solver scheme that I am using. All that needs be done is to evaluate the factor at small number of known time points. Only three things are needed to be able to do this:
a. the true anomaly at the start of the manoeuvre
b. the change in the true anomaly (i.e, dt) of the manoeuvre
c. the orbital eccentricity
Aside from this we can use our Lambert Solver to solver for the ER3BP as we did earlier for the CR3BP.
Orbital eccentricity shakes things up
So, let's look at the ER3BP solutions in the easy to calculate linear approximation. How does the orbital eccentricity change things? Well, if we return to the example of my previous notes, we can show that the trajectory solution of the two-point boundary problem in the ER3BP model looks like the following:
For comparison, in the linear approximation in the CR3BP we had:
In the CR3BP model, we were targeting a planar Lyapunov orbit passing through the terminal point 'A'. The approach orbit was regular; and the Lyapunov orbit (red) was well-defined - just a simple ellipse centred on L1.
On the other-hand, in the ER3BP model, the transfer problem to the same terminal end-point now looks far more jumbled. The general shape of the approach and final orbit is essentially the same, but the effect of the orbital eccentricity is to 'shake things up' a bit. More specifically, the orbital eccentricity induces periodic oscillations of the Lyapunov orbit that are not commensurate with the orbital frequency of the Lyapunov orbit itself. The result is, over time, to smear the Lyapunov orbit out from a simple line orbit to an orbital band.
We also examine the effect of orbital eccentricity in the full ER3BP. Now, the solution (blue solid line) is shown below.
Here the linear approximation orbit is also shown as the red-dotted line. This solution of the ER3BP is in contrast to the same set up in the CR3BP:
Again, we note that the effect of the orbital eccentricity is to smear out the Lyapunov orbit. However, the basic solutions to the two-point boundary problem is the same in both cases and differences are just a matter of detail
The bottom line
The take away from this post really is that moving over to the full, non-linear ER3BP presents no major complication in use of the Lambert Solver developed in previous notes.
The next step is to apply the Lambert Solver to a trajectory planning in Orbiter to demonstrate that the assorted coordinate transformations used in the ER3BP are manageable. (It should note that in preparing the graphs in these notes, I've taken a few short cuts in working through the coordinations in order to make my life a little easier. In applying all of this to a real Orbiter scenario, I'll work through the solutions and coordinate transformations properly.)
And if I am happy with that result of that exercise, I'll start translating all of this into something that others in this Orbiter community can use.
A Lambert Solver solves the two-point boundary problem of determining the trajectory of getting from point 'A' to point 'B' in a specified time 'dt'. Lambert Solvers exist are frequently used for solving trajectory planning problem in the standard restricted two-body problem (Keplerian motion) and an example of a solver can be found in IMFD's Course Program.
Whereas earlier parts focused primarily on the Circular Restricted Three Body Problem (CR3BP), this note shows that that earlier work can be extended to the Elliptical Restricted Three Body Problem (ER3BP). The CR3BP assumes that the primary and secondary gravitating bodies (e.g., the Earth and the Moon in the Earth-Moon restricted three body system) rotate about each other in precise circular orbits, while the ER3BP relaxes this assumption and assumes that the primary and secondary bodies move in precise elliptical orbits about each other. For many systems, this elliptical orbit assumption is nearly correct. Even for the Earth-Moon system, the motion of the Moon around the Earth is dominated by its near elliptical motion - even though the Sun does perturb that elliptical motion to a noticeable degree. But the introduction of orbital eccentricity in the model amounts to a significant complication that must be accounted for in the Lambert Solver.
The ER3BP represents something of a gold standard for Lagrange point orbit design, so the success of the Lambert Solver for this model is represents a significant milestone and provides the basis for some realistic trajectory design within Orbiter.
The story so far
So far, I've written two posts on the development of a Lambert Solver for Lagrange points:
Lambert Solve for Lagrange points
and
More on a Lambert Solver for Lagrange points
The first post introduced the possibility of building a Lambert Solver and demonstrated that in the linear approximation of the CR3BP high fidelity solutions to the two-point boundary problem could be found using simple linear algebra (and a pre-calculated very high-order variational integrator based on Gauss-Lobatto quadrature). The post was light on the mathematics - but the the supporting mathematics will be posted in this forum at a later date.
The second post extended this demonstration to the full, non-linear CR3BP model and showed that solution generated from the linearised equations of motion could be used to generate the corresponding solution of the full CR3BP model using straight-forward multivariate root-finding techniques. Again, more on the maths of this later.
And now we move on to a demonstration that we can do the same thing for the ER3BP. And this will complete the Lambert Solver prototyping and demonstration. The next step after this will be to link this Lambert Solver up to Orbiter (in something of an ad hoc fashion) to demonstrate that the Lambert Solver works as advertised.
The equations of motion in the ER3BP
Before going on, and to show how the orbital eccentricity changes things, I'm going to write down the equations of motion of the ER3BP model in detail. Here they are:
[MATH]\eta''(\nu)-2\,\kappa'(\nu)=\frac{1}{e\,\cos(\nu)+1}\,\frac{\partial\,\Omega}{\partial\,\eta(\nu)}[/MATH]
[MATH]\kappa''(\nu)+2\,\eta'(\nu)=\frac{1}{e\,\cos(\nu)+1}\,\frac{\partial\,\Omega}{\partial\,\kappa(\nu)}[/MATH]
where
[MATH]\Omega=\frac{\eta(\nu)^{2}}{2}+\frac{\kappa(\nu)^{2}}{2}+\frac{\mu_{2}}{\sqrt{\left(\eta(\nu)-\mu_{1}\right){}^{2}+\kappa(\nu)^{2}}}+\frac{\mu_{1}}{\sqrt{\left(\eta(\nu)+\mu_{2}\right){}^{2}+\kappa(\nu)^{2}}}[/MATH]
A few things about these equations:
1. This equations of motion work in a coordinate system know as rotating pulsating coordinates. These coordinates are set up that even though the Moon rotates around the Earth-Moon barycentre in an elliptical orbit, the position of the Moon (and of the Earth) are fixed. Moreover, it is set up so that the distance between the Earth and the Moon is equal to 1; the Moon and the Earth lie on the x-axis; and the origin of the coordinate system is located at the Earth-Moon barycentre (EMB).
2. The ER3BP uses a change of time coordinates. Instead of using ordinary clock time with which to measure speeds, accelerations and so on, it use the true anomaly of the Moon as a measure of time. This quantity is given the symbol [MATH]\nu[/MATH]. In the CR3BP, where the orbital eccentricity is zero, the true anomaly is related to ordinary clock time by a constant multiplicative factor.
3. The variables in the model are [MATH]\eta(\nu)[/MATH] and [MATH]\kappa(\nu)[/MATH]. [MATH]\eta(\nu)[/MATH] measures the distance along the x-axis in the rotating-pulsating coordinate system; and [MATH]\kappa(\nu)[/MATH] measures the distance along the y-axis in the the rotating-pulsating coordinate system. Motion in the y-direction is transverse motion in the orbital plane of the Earth-Moon system. (For this note, I've surpassed reference to motion in the 'out of the orbital plane' z-direction.)
4. [MATH]\mu_1[/MATH] and [MATH]\mu_2[/MATH] are defined as
[MATH]\mu_1 = \frac{m_1}{m_1+m_2}[/MATH]
[MATH]\mu_2 = \frac{m_2}{m_1+m_2}[/MATH]
where [MATH]m_1[/MATH] and [MATH]m_2[/MATH] are the masses of the Earth and Moon respectively.
5. [MATH]e[/MATH] is the orbital eccentricity of the Moon's orbit (and the Earth's!) around the EMB. If the eccentricity is zero, then
6. [MATH]\Omega[/MATH] is essentially the negative of the effective potential in the rotating-pulsating coordinates. It contains the terms related to centrifugal force, and the gravitational force due to the primary and secondary gravitating bodies.
7. The terms on the right-hand side are the 'force' functions. The orbital eccentricity appears only once in each equation as simple factors in these force functions.
This last item contains the essential point that I now wish to make: Whereas in the CR3BP, we have set the orbital eccentricity to zero so that the factors in the force functions are unity, in the ER3BP we know have to use the full expression for these factors which now contains an explicit reference to the cosine of the time variable, [MATH]\nu[/MATH]. Fortunately, adding this explicit time variable is very simple in the Lambert Solver scheme that I am using. All that needs be done is to evaluate the factor at small number of known time points. Only three things are needed to be able to do this:
a. the true anomaly at the start of the manoeuvre
b. the change in the true anomaly (i.e, dt) of the manoeuvre
c. the orbital eccentricity
Aside from this we can use our Lambert Solver to solver for the ER3BP as we did earlier for the CR3BP.
Orbital eccentricity shakes things up
So, let's look at the ER3BP solutions in the easy to calculate linear approximation. How does the orbital eccentricity change things? Well, if we return to the example of my previous notes, we can show that the trajectory solution of the two-point boundary problem in the ER3BP model looks like the following:
For comparison, in the linear approximation in the CR3BP we had:
In the CR3BP model, we were targeting a planar Lyapunov orbit passing through the terminal point 'A'. The approach orbit was regular; and the Lyapunov orbit (red) was well-defined - just a simple ellipse centred on L1.
On the other-hand, in the ER3BP model, the transfer problem to the same terminal end-point now looks far more jumbled. The general shape of the approach and final orbit is essentially the same, but the effect of the orbital eccentricity is to 'shake things up' a bit. More specifically, the orbital eccentricity induces periodic oscillations of the Lyapunov orbit that are not commensurate with the orbital frequency of the Lyapunov orbit itself. The result is, over time, to smear the Lyapunov orbit out from a simple line orbit to an orbital band.
We also examine the effect of orbital eccentricity in the full ER3BP. Now, the solution (blue solid line) is shown below.
Here the linear approximation orbit is also shown as the red-dotted line. This solution of the ER3BP is in contrast to the same set up in the CR3BP:
Again, we note that the effect of the orbital eccentricity is to smear out the Lyapunov orbit. However, the basic solutions to the two-point boundary problem is the same in both cases and differences are just a matter of detail
The bottom line
The take away from this post really is that moving over to the full, non-linear ER3BP presents no major complication in use of the Lambert Solver developed in previous notes.
The next step is to apply the Lambert Solver to a trajectory planning in Orbiter to demonstrate that the assorted coordinate transformations used in the ER3BP are manageable. (It should note that in preparing the graphs in these notes, I've taken a few short cuts in working through the coordinations in order to make my life a little easier. In applying all of this to a real Orbiter scenario, I'll work through the solutions and coordinate transformations properly.)
And if I am happy with that result of that exercise, I'll start translating all of this into something that others in this Orbiter community can use.
