MontBlanc2012
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In a recent post, Lambert Solver for Lagrange points I introduced the possibility of building a Lambert Solver tailored to trajectory planning in the vicinity of Lagrange points of, say, the Earth-Moon system or the Sun-Earth system.
That post focused on the calculation of trajectories using a linear approximation for of the CR3BP / ER3BP models and demonstrated that the linearised solution trajectories could be found readily. In this note, I'm going to demonstrate that the linear solution serves as good starting point to calculate the corresponding Lambert problem solution using the full, non-linear CR3BP model. (In a subsequent post, I'll move on to do the same in the full, non-linear ER3BP model which serves as the gold standard framework for trajectory planning near Lagrange points.)
The linear solution as a 'small-scale' trajectory solution
The linear approximation is generally used as starting point for much Lagrange point trajectory design because, if one is close enough to the Lagrange point, it is an exact representation of the dynamics near a Lagrange point. If we choose a Lambert problem that is 'small enough', we shouldn't gain very much in accuracy by using the full, non-linear CR3BP model.
To illustrate this, let's consider a small-scale problem where we start, say, just a few hundred kilometres away from the L1 Lagrange point of the Earth-Moon system. And let's solve the Lambert problem for the linear approximation models; and then use this solution to solve the same Lambert problem using the full CR3BP model. This set up is shown below:
As in the previous post, Lambert Solver for Lagrange points a vessel is initially located at the 'Start Point' and wishes to transfer to the point 'A' with a transfer time here of 28.83 days. (This is sightly longer than the transfer time of the earlier post because I've chosen a slightly different terminal point 'A'.)
There are actually two trajectories plotted on this graph - although it is hard to distinguish them. The first is the linear approximation solution (dotted red line); and the second is the one generated using the full CR3BP model (solid blue line) starting from the linear approximation. Although there are some minor differences between the two, it is clear that the trajectory solution of the full CR3BP is well approximated by the simple, linear approximation solution. And this reflects the fact that the trajectories are never much more than a few hundred kilometres from the Lagrange point so that the non-linear gravitational contributions are largely immaterial.
But what happens if we increase the problem scale by a factor of 10 say such that the satellite starts a few thousand kilometres from the Lagrange point rather than just a few hundred kilometres; and the distance of 'A' from the Lagrange point is also increased by a factor of 10.
Increasing the problem scale by a factor of 10
In this case, if we work through the linear approximation solution and the full CR3BP solutions again, we get the following trajectories:
Here, the linear approximation solution (dotted red line again) is just a re-scaled version of the first - as one might expect of a re-scaled linear solution. But the solution corresponding to the full, non-linear CR3BP solution (solid blue line) has noticeably diverged from the linear approximation solution. The differences aren't huge, but they are clearly there.
In order for the CR3BP solution to converge to a Lyapunov orbit, the transfer time has had to be reduced. Moreover, the shape of the Lyapunov is evidently distorted from a simple ellipse centred on the L1 Lagrange point. This distortions reflect the non-linear contributions to the gravitational field that become more material as one moves away from the Lagrange point.
Even with a scale of a few thousand km, the deviations from the linear approximation aren't that large. But what happens if we increase the scale of the problem by another factor of 10 (i.e., so that it is 100 times larger than the original problem)?
Increasing the problem scale by another factor of 10
Again, we can work through our linear approximations and full CR3BP solutions. and plot the results. And again, the linear approximation solution is just a re-scaled version of the earlier linear approximation trajectory solutions. But now there are significant changes to the full, non-linear CR3BP solution:
In order to transfer to the Lyapunov orbit, the transfer time has had to be shortened by about 1.5 days to 27.18 days. In turn, this has meant that the approach to the Lyapunov orbit now has to be quite different so that the initial velocity vector is materially different from the linear approximation solution. And the overall shape and size of the Lyapunov orbit has shrunk and shifted noticeably to the right in the diagram. The linear approximation no longer serves as a good trajectory model - although it still provides a starting point for calculating the full CR3BP trajectory solution.
What happens if we increase the scale of the problem even further? As with all things there are limits to how far we can use the linear approximation as a starting point for finding more accurate solution. At some point, the simple-minded multivariate Newton-Raphson approached used so far will break down and the root-finding algorithm will fail to find a solution. But, nonetheless, so long as the scale of your target orbit is around 30,000 km to 50,000 km, this approach seems to be reasonably robust.
So what's the take away
Basically this note demonstrates a few things:
1. If the scale of the Lambert problem that you are considering is small enough, the linear approximation solution provides a good model of the transfer trajectory;
2. For problem scales up to the 30,000 km to 50,000 km mark, there is a well-defined method for finding the Lamber problem transfer trajectory starting from the linear approximation and incorporating the full non-linear CR3BP model. This scale is probably big enough for most Lagrange point orbit insertions, so that's a promising start.
3. At sufficiently large-scale, the full CR3BP is absolutely required to calculate the transfer trajectory in the vicinity of Lagrange points.
(N.B. I'm aware that I haven't provided much by way of mathematical description of the algorithm. At some point, I'll 'back fill' by writing a number of more mathematically-oriented threads if all of this proves to be a useful framework for Lagrange point trajectory planning.)
That post focused on the calculation of trajectories using a linear approximation for of the CR3BP / ER3BP models and demonstrated that the linearised solution trajectories could be found readily. In this note, I'm going to demonstrate that the linear solution serves as good starting point to calculate the corresponding Lambert problem solution using the full, non-linear CR3BP model. (In a subsequent post, I'll move on to do the same in the full, non-linear ER3BP model which serves as the gold standard framework for trajectory planning near Lagrange points.)
The linear solution as a 'small-scale' trajectory solution
The linear approximation is generally used as starting point for much Lagrange point trajectory design because, if one is close enough to the Lagrange point, it is an exact representation of the dynamics near a Lagrange point. If we choose a Lambert problem that is 'small enough', we shouldn't gain very much in accuracy by using the full, non-linear CR3BP model.
To illustrate this, let's consider a small-scale problem where we start, say, just a few hundred kilometres away from the L1 Lagrange point of the Earth-Moon system. And let's solve the Lambert problem for the linear approximation models; and then use this solution to solve the same Lambert problem using the full CR3BP model. This set up is shown below:
As in the previous post, Lambert Solver for Lagrange points a vessel is initially located at the 'Start Point' and wishes to transfer to the point 'A' with a transfer time here of 28.83 days. (This is sightly longer than the transfer time of the earlier post because I've chosen a slightly different terminal point 'A'.)
There are actually two trajectories plotted on this graph - although it is hard to distinguish them. The first is the linear approximation solution (dotted red line); and the second is the one generated using the full CR3BP model (solid blue line) starting from the linear approximation. Although there are some minor differences between the two, it is clear that the trajectory solution of the full CR3BP is well approximated by the simple, linear approximation solution. And this reflects the fact that the trajectories are never much more than a few hundred kilometres from the Lagrange point so that the non-linear gravitational contributions are largely immaterial.
But what happens if we increase the problem scale by a factor of 10 say such that the satellite starts a few thousand kilometres from the Lagrange point rather than just a few hundred kilometres; and the distance of 'A' from the Lagrange point is also increased by a factor of 10.
Increasing the problem scale by a factor of 10
In this case, if we work through the linear approximation solution and the full CR3BP solutions again, we get the following trajectories:
Here, the linear approximation solution (dotted red line again) is just a re-scaled version of the first - as one might expect of a re-scaled linear solution. But the solution corresponding to the full, non-linear CR3BP solution (solid blue line) has noticeably diverged from the linear approximation solution. The differences aren't huge, but they are clearly there.
In order for the CR3BP solution to converge to a Lyapunov orbit, the transfer time has had to be reduced. Moreover, the shape of the Lyapunov is evidently distorted from a simple ellipse centred on the L1 Lagrange point. This distortions reflect the non-linear contributions to the gravitational field that become more material as one moves away from the Lagrange point.
Even with a scale of a few thousand km, the deviations from the linear approximation aren't that large. But what happens if we increase the scale of the problem by another factor of 10 (i.e., so that it is 100 times larger than the original problem)?
Increasing the problem scale by another factor of 10
Again, we can work through our linear approximations and full CR3BP solutions. and plot the results. And again, the linear approximation solution is just a re-scaled version of the earlier linear approximation trajectory solutions. But now there are significant changes to the full, non-linear CR3BP solution:
In order to transfer to the Lyapunov orbit, the transfer time has had to be shortened by about 1.5 days to 27.18 days. In turn, this has meant that the approach to the Lyapunov orbit now has to be quite different so that the initial velocity vector is materially different from the linear approximation solution. And the overall shape and size of the Lyapunov orbit has shrunk and shifted noticeably to the right in the diagram. The linear approximation no longer serves as a good trajectory model - although it still provides a starting point for calculating the full CR3BP trajectory solution.
What happens if we increase the scale of the problem even further? As with all things there are limits to how far we can use the linear approximation as a starting point for finding more accurate solution. At some point, the simple-minded multivariate Newton-Raphson approached used so far will break down and the root-finding algorithm will fail to find a solution. But, nonetheless, so long as the scale of your target orbit is around 30,000 km to 50,000 km, this approach seems to be reasonably robust.
So what's the take away
Basically this note demonstrates a few things:
1. If the scale of the Lambert problem that you are considering is small enough, the linear approximation solution provides a good model of the transfer trajectory;
2. For problem scales up to the 30,000 km to 50,000 km mark, there is a well-defined method for finding the Lamber problem transfer trajectory starting from the linear approximation and incorporating the full non-linear CR3BP model. This scale is probably big enough for most Lagrange point orbit insertions, so that's a promising start.
3. At sufficiently large-scale, the full CR3BP is absolutely required to calculate the transfer trajectory in the vicinity of Lagrange points.
(N.B. I'm aware that I haven't provided much by way of mathematical description of the algorithm. At some point, I'll 'back fill' by writing a number of more mathematically-oriented threads if all of this proves to be a useful framework for Lagrange point trajectory planning.)
