Introduction
Standard Keplerian physics, in which objects orbit around bodies in perfect ellipses or hyperbolas, is the mainstay of Orbiter mission planning. Its utility rests on the assumption that a spacecraft is, for the most part, either close to a planet (in which case the gravity of that planet dominates) or it is far away from the planet in which the gravity of the Sun dominates. In the case of Jupiter, Saturn and the other gas giants these systems can each be regarded as mini-solar systems in which the planet takes the role of the Sun, and the moons take the role of the planets. And generally, this dichotomous view of gravitational influence provides a very reasonable working model of the dominating gravitational influence on a spacecraft. Based on this principle, it is possible to calculate the trajectory needed to get from 'point A' to 'point B' with relative ease. The utility of tools such as TransX are working testament to the effectiveness of this principle.
However, there are situations in which this characterisation of Newtonian gravity in which only one gravitating body at a time influences the trajectory of a body breaks down - and breaks down really quite badly. These situations are going to be those in which a spacecraft spends a considerable amount of time in a condition that is neither particularly close to - or far from - a second gravitating body. A case in point is the libration points of the Earth-Moon system. The libration points of this system are locations where the gravity of the Earth and the Moon conspire to keep an object in an orbit that is most-decidedly non-Keplerian and keep the object perfectly balanced at a constant distance from the Earth and the Moon. To understand behaviour in the vicinity of these libration points, one is forced to use thee-body physics - rather than the more standard (and more straightforward) two-body Keplerian physics.
The purpose of this note, then, is to present a simple method by which to calculate a reference orbit for the libration points for the case where the primary and the secondary (e.g., the Earth and the Moon; or Jupiter and one of its moons) move about each other in an elliptical orbit.
The libration points
First, a little background on libration points. To start off, let's imagine that, relative to their common centre of mass, both the Earth (the 'primary' body) and the Moon (the 'secondary' body) move around each other in a circular orbit. Imagine that we are looking down from 'above' on the orbital plane of the Earth and the Moon. And imagine that we have set up a co-ordinate system that rotates with the Earth and the Moon (at a constant rate) such that the x-axis always passes through the centre of the Earth and the centre of the Moon. In that rotating reference frame, and only because it is rotating, two 'fictitious; forces - the centrifugal force and the Coriolis force - arise. Now there are five locations in the orbital plane of the Earth and the Moon where these fictitious forces - when taken together with the normal Newtonian gravitational forces arising from both the Earth and the Moon - exactly balance leading to no net force on any object placed there. In other words, in the rotating reference frame that co-rotates with the Earth and the Moon, there are five locations where an object placed 'at rest' in that rotating reference frame will remain at rest. These five points are known as the libration points and are given the names L1, L2, L3. L4 and L5.
In the normal (inertial) reference frame where the Earth and the Moon are seen to rotate around each other, these libration points are not fixed. What an observer sees when looking down on the Earth-Moon orbital plane is not five 'fixed' points but five orbits for which the distance from the Earth and the distance from the Moon remain constant. And because the Earth and Moon are presumed to orbit each other in a circular motion then each of the five libration 'points' prescribes a circle about the centre of mass of the system. In other words, all of the libration points are seen to be circular orbits - albeit rather special ones.
Above is a schematic of the location of the five libration points taken from Wikipedia. By convention, we call the three libration points that lie along the axis that passes through the centre of both the Earth and the Moon L1, L2 and L3. L1 is the libration point that that lies between the Earth and the Moon; L2 is the libration point that lies in the far side of the Moon; and L3 is the libration point that lies roughly on the opposite site of the Moon's orbit. L4 and the L5 complete the set of libration points with L4 lying 60 degrees in front of the Moon in its orbit around the Earth; and L5 lies 60 degrees behind the Moon in its orbit around the Earth. Of these five libration points, two are most often discussed - L1 and L2 - in large measure because of their proximity to the Moon, and because they are often seen as 'gateway' stations for getting to other places in the Solar System.
If we write down the equations of motion in the three-body system close to the libration points, we find that we have no way of writing down a 'closed-form' (i.e., easily to calculate) solution for those equations. It is this difficulty that makes the three-body problem considerably harder than the two-body problem for which a closed-form expression does exist. However, if we only want to consider solutions that are close to the Lagrange points, then we can derive analytical expressions. And that kind of analysis tells us some important things about the libration points.
First, the co-axial libration points - L1, L2 and L3 - are all unstable. If an object is placed there with even the slightest offset in position or velocity, it will start to move away from the libration point at a rate that grows exponentially with time. Think of trying to balance a knife vertically on a table, If one aligns it exactly vertically then one might imagine that it should be possible to balance it there. However, the slightest nudge or misplacement in any direction will result in the knife moving away from its balance point at an increasingly fast rate. Attempting to stay at one of the co-axial libration points is very much like trying to balance an knife upright on a table. It is simply impossible to remain at one of the libration points for any length of time without periodically exerting some 'balancing force' to drive the object back to the balance point. This process of exerting some periodic balancing force to drive the object back towards a libration point is known as 'station keeping'. Typically for the Earth-Moon system, station-keeping needs to be carried out every one or two weeks. If it is not done, then the spacecraft will be ejected from the libration point orbit after a period of three or four weeks.
Second, if a spacecraft is ejected from a libration point, then typically it will be ejected from the libration point orbit along something known as the 'unstable' manifold(s) - think 'orbit' or 'trajectory' here. For L1, L2 and L3 libration points, there are two unstable manifolds. For L1, one of those unstable manifold orbits points towards the Earth; and the other points towards the Moon. If one wants to leave L1 and head back towards the Earth, the natural thing to do is to stop station keeping and give the craft a slight nudge in the direction of Earth which will force exit from the unstable manifold that points towards Earth, On the other hand, if one wants to leave the L1 libration point and head to the Moon, one gives the spacecraft a slight nudge in the other direction which forces it to exit L1 along the other unstable manifold that heads of towards the Moon. For L2, one unstable manifold heads off towards the Moon, and the other heads off into deep space. For L3, one of the unstable manifolds heads off in the direction of Earth, and the other also heads of in the direction of deep space.
Third, just as there are unstable manifolds that lead away from the co-axial libration points, there are also a pair of stable manifolds that head in towards (and terminate at) the same libration points. These stable manifolds are special trajectories that, if you are on them and travelling along them with exactly the right speed and direction, gravity will automatically take you to (and stop you at) a libration points without any expenditure of delta-v. Once at a libration point, and if you wish to stay at that point, station-keeping will need to commence and continue for as long as you wish to remain there. And once station-keeping ends, the spacecraft will automatically be ejected along one of the two unstable manifolds.
Fourth, this simple picture of libration points is made considerably more complicated by the elliptical motion of the primary and secondary bodies; and gravitational perturbations from, say, non-spherical gravitational sources and other gravitating bodies (e.g., the Sun). The basic theory of libration points assumes that the primary and secondary move about each other in circular orbits. This theory is generally known as the Circular Restricted Three-Body Problem (or CRTBP for short). It is called 'Circular' because the primary and secondary are assumed to move around each other in perfect circles; it is called 'Restricted' because the third-body (the spacecraft) is assumed to have zero mass; and it is called 'the Three-Body Problem' because the theory takes into account the motion of three bodies - the primary, the secondary and the spacecraft. One can improve upon this basic circular theory (albeit at some considerable analytical and computational expense) if one moves over to a theory know as the Elliptical Restricted Three-Body Problem (ERTBP). This model is the same as the CRTPB except that the primary and secondary rotate around each other in perfect, unperturbed ellipses. For examining many of the libration points in the Solar System - of the Earth-Moon system, the Sun-Earth system, the Sun-Jupiter system or the Jupiter-Galilean moon systems, for example.
The location of the libration points in the ERTBP - and their use as reference orbits
Working out the equations of motion in the ERTBP is, frankly, a painful and rather technical exercise in classical mechanics involving four separate co-ordinate transformations to get standard result that one can find in textbooks. In this note, I'm not going to bother focusing on the derivation of these equations. Instead, it is more appropriate to focus on the resulting equations of motion and what one can do with them. And below, what I want to do is to focus on:
From the analytical solution to the ERTBP, we can calculate the position of a libration point using the following algorithm:
First, calculate the following quantities:
[MATH]gm = GM_1 + GM_2[/MATH]
[MATH]\mu_1 = GM_1 / gm[/MATH]
[MATH]\mu_2 = GM_2 / gm[/MATH]
[MATH]COM = Q_{1}\,\mu_{1}+Q_{2}\,\mu_{2}[/MATH]
[MATH]COV = P_{1}\,\mu_{1}+P_{2}\,\mu_{2}[/MATH]
where [MATH]GM_1[/MATH] is the standard gravitational constant for the primary body in SI units; [MATH]GM_2[/MATH] is the standard gravitational constant for the secondary body in SI units; [MATH]Q_1[/MATH] and [MATH]P_1[/MATH] are the position and velocity of the primary in Orbiter's global coordinate reference frame (information that can be extracted from Orbiter using the 'oapi.get_globalpos(hObj)' and 'oapi.get_globalvel(hObj)' API commands.
Next, calculate the following:
[MATH]\boldsymbol {\mathbf {r}} = Q_ {2} - Q_ {1}[/MATH]
[MATH]\boldsymbol {\mathbf {v}} = P_ {2} - P_ {1}[/MATH]
[MATH]\boldsymbol {e} = \frac {v^{2}} {gm}\, \boldsymbol {r} - \frac{\boldsymbol {r}.\boldsymbol {v}} {gm}\, \boldsymbol {v} - \frac{\boldsymbol {r}} {r}[/MATH]
[MATH]a = \frac {gm} {2\, gm/r - v^{2}}[/MATH]
[MATH]\nu = \arccos (\frac {\boldsymbol {e}.\boldsymbol {r}} {e\, r})[/MATH]
such that if [MATH]\mathbf {r} . \mathbf {v} < 0[/MATH] then [MATH]\nu = 2\,\pi-\nu[/MATH].
Here, [MATH]\mathbf {r}[/MATH] is the position of the secondary with respect to the primary in Orbiter's global reference frame; [MATH]\mathbf {v}[/MATH] is the velocity of the secondary with respect to the primary in Orbiter's global reference frame; [MATH]\boldsymbol {e}[/MATH] is the eccentricity vector of the secondary in its elliptical orbit around the primary; [MATH]a[/MATH] is the semi-major axis of the elliptical orbit; and [MATH]/nu[/MATH] is the true anomaly of the secondary in its orbit around the primary.
Once these basic preliminary calculations have been completed, one next calculates the location of the libration point in the rotating reference frame of the ERTBP:
[MATH]Q_{x}=\alpha\, a\,\left(1-e^{2}\right)\,\frac{\cos(\nu)}{1+e\,\cos(\nu)}[/MATH]
[MATH]Q_{y}=\alpha\, a\,\left(1-e^{2}\right)\,\frac{\sin(\nu)}{1+e\,\cos(\nu)}[/MATH]
[MATH]P_{x}=-\alpha\,\sqrt{\frac{gm}{a\,\left(1-e^{2}\right)}}\,\sin(\nu)[/MATH]
[MATH]P_{y}=+\alpha\,\sqrt{\frac{gm}{a\,\left(1-e^{2}\right)}}\,(e+\cos(\nu))[/MATH]
where [MATH]Q_x[/MATH] and [MATH]Q_y[/MATH] are the x and y co-ordinates of the libration point in the rotating reference frame ([MATH]Q_z = 0[/MATH] is defined to be zero in this reference frame); and where [MATH]P_x[/MATH] and [MATH]P_y[/MATH] are the x and y co-ordinates of the velocity vector of the libration point in the same reference frame (again, [MATH]P_z = 0[/MATH] is defined to be zero on this reference frame).
these equations introduce a new parameter, [MATH]\alpha[/MATH], which is specific to the libration point and gravitating bodies of interest. If one is interested in the L1 libration point, to find [MATH]\alpha[/MATH] one needs to solve the following equation:
For the Earth-Moon system, and using Orbiter's default values for the masses of the Earth and Moon, it is possible to show that:
If one is interested in the L2 libration point, to find [MATH]\alpha[/MATH] one needs to solve the following equation:
For the Earth-Moon system, and again using Orbiter's default values for the masses of the Earth and Moon, it is possible to show that:
And if one is interested in the L3 libration point, to find [MATH]\alpha[/MATH] one needs to solve the following equation:
For the Earth-Moon system, and using Orbiter's default values for the masses of the Earth and Moon, it is possible to show that:
Having calculated the position of the libration point in the rotating reference frame using the above algorithm, the final step is to convert this position back to Orbiter's global reference frame. To this, we first calculate the unit vectors of the rotating reference frame in Orbiter's (non-rotating) global coordinate system.
Calculate:
[MATH]\hat{\boldsymbol{x}}=\boldsymbol{e}/e[/MATH]
[MATH]\hat{\boldsymbol{z}}=\boldsymbol{r}\times\boldsymbol{v}[/MATH]
[MATH]\hat{\boldsymbol{z}}=\hat{\boldsymbol{z}} / \sqrt{\hat{\boldsymbol{z}}.\hat{\boldsymbol{z}}}[/MATH]
[MATH]\hat{\boldsymbol{y}}=\hat{\boldsymbol{z}}\times\hat{\boldsymbol{x}}[/MATH]
So that, finally, we calculate the position of the libration point in Orbiter's global coordinate system as:
This algorithm is quite general and is the exact analytical result (aside, that is, from the numerical calculation of [MATH]\alpha[/MATH]) for the location of the co-axial libration points - L1, L2 and L3 - for any choice of primary and secondary bodies. H
Because the above may appear somewhat complicated, here is a lua code snippet that goes through exactly this calculation for the Earth-Moon L2 point:
A reference orbit
Because one can calculate the position of a libration point for any time (or MJD) the above provides a way of calculating a reference orbit or that libration point. In particular, one can use this calculation as a measure of how far one (and fast) has moved away from a libration point and, therefore, provide some information on how to make corrections to one's trajectory in order to move back towards the libration point - i.e., how to perform station keeping.
As an example of how this information can be used, attached to this post is an Orbiter scenario and lua script. The scenario places a Delta Glider at the Earth-Moon L2 'point'. On the right-hand side of the screen, the lua script runs continuously in the background and provides information about the location of the Delta-Glider with respect to the reference orbit of the Earth-Moon L2 point. The scenario starts with the Delta Glider placed about 0.5 metres from the L2 point and to with 0.1 mm/s is stationary with respect to the L2 point. Over the course of one lunar orbit, the Delta Glider slowly slips further and further away from the L2 point as it is progressively evicted from L2 along that point's unstable manifold. Unless station keeping is performed, by around week 4 the Delta Glider is well on its way to escape from the Earth-Moon system. The updated display provides information about position and velocity of L2 in the reference frame of the Delta Glider's orbit osculating two-body orbit around the Moon. This is the same reference frame used by Orbit MFD (and most other MFDs) if the Moon is referenced. Rather than use x, y and z co-ordinates, it provides information in terms of 'forward', 'inward' and 'plane' - i.e., the same co-ordinate system used by IMFD's Delta Velocity program. The scenario starts out with 10 m/s worth of RCS. This should be enough to keep the Delta Glider 'on station' for at least one month - so long as one is judicious in using that fuel.
Some further examples
This same algorithm for calculating the location of libration points can be extended to any choice of primary and secondary. For example, it is possible to place a craft at one of the libration points of the Jupiter-Europa system. Below, is a screenshot of the view from the Jupiter-Europa L2 point.
Here, the view is towards Europa. Jupiter is directly behind Europa - and, so long as the craft remains behind Europa, Jupiter will always remain behind Europa. The MFD on the right-hand side is IMFD's Map Program. It shows the Delta Glider will be evicted from the L2 position (and in quite short-order). The trajectory traces out a good approximation to the unstable manifold for that Lagrange point. Interestingly, the unstable manifold passes within about 70 km of Europa's surface. This raises the possibility that a sensible Europa approach strategy may be to approach the Jupiter-Europa L2 point along one of the stable manifolds then, once there, give a small nudge to exit L2 along the unstable manifold leading to Europa. This will automatically take one down almost to Europa's surface. And at Europa periapsis, execute a 430 m/s retrograde burn to enter Low Europa Orbit.
Equally, we could place a craft at the Jupiter-Io L2 point. A screenshot of this scenario is given below:
Again, the unstable manifold take the craft down to Io's surface. As with all L2 points, the primary (Jupiter) is directly behind the secondary (Io). Io acts, then, as kind of radiation shield for any body in the vicinity of Jupiter. If we let the spacecraft, drift 'off station' a little, then we can see Jupiter energy from behind Io:
Here, you can see Jupiter appearing from behind Io. In passing, one notes that the apparent size of Io neatly matches the apparent size of Jupiter so that Io, just (and only just) covers the disk of Jupiter. A rather interesting curiosity.
Standard Keplerian physics, in which objects orbit around bodies in perfect ellipses or hyperbolas, is the mainstay of Orbiter mission planning. Its utility rests on the assumption that a spacecraft is, for the most part, either close to a planet (in which case the gravity of that planet dominates) or it is far away from the planet in which the gravity of the Sun dominates. In the case of Jupiter, Saturn and the other gas giants these systems can each be regarded as mini-solar systems in which the planet takes the role of the Sun, and the moons take the role of the planets. And generally, this dichotomous view of gravitational influence provides a very reasonable working model of the dominating gravitational influence on a spacecraft. Based on this principle, it is possible to calculate the trajectory needed to get from 'point A' to 'point B' with relative ease. The utility of tools such as TransX are working testament to the effectiveness of this principle.
However, there are situations in which this characterisation of Newtonian gravity in which only one gravitating body at a time influences the trajectory of a body breaks down - and breaks down really quite badly. These situations are going to be those in which a spacecraft spends a considerable amount of time in a condition that is neither particularly close to - or far from - a second gravitating body. A case in point is the libration points of the Earth-Moon system. The libration points of this system are locations where the gravity of the Earth and the Moon conspire to keep an object in an orbit that is most-decidedly non-Keplerian and keep the object perfectly balanced at a constant distance from the Earth and the Moon. To understand behaviour in the vicinity of these libration points, one is forced to use thee-body physics - rather than the more standard (and more straightforward) two-body Keplerian physics.
The purpose of this note, then, is to present a simple method by which to calculate a reference orbit for the libration points for the case where the primary and the secondary (e.g., the Earth and the Moon; or Jupiter and one of its moons) move about each other in an elliptical orbit.
The libration points
First, a little background on libration points. To start off, let's imagine that, relative to their common centre of mass, both the Earth (the 'primary' body) and the Moon (the 'secondary' body) move around each other in a circular orbit. Imagine that we are looking down from 'above' on the orbital plane of the Earth and the Moon. And imagine that we have set up a co-ordinate system that rotates with the Earth and the Moon (at a constant rate) such that the x-axis always passes through the centre of the Earth and the centre of the Moon. In that rotating reference frame, and only because it is rotating, two 'fictitious; forces - the centrifugal force and the Coriolis force - arise. Now there are five locations in the orbital plane of the Earth and the Moon where these fictitious forces - when taken together with the normal Newtonian gravitational forces arising from both the Earth and the Moon - exactly balance leading to no net force on any object placed there. In other words, in the rotating reference frame that co-rotates with the Earth and the Moon, there are five locations where an object placed 'at rest' in that rotating reference frame will remain at rest. These five points are known as the libration points and are given the names L1, L2, L3. L4 and L5.
In the normal (inertial) reference frame where the Earth and the Moon are seen to rotate around each other, these libration points are not fixed. What an observer sees when looking down on the Earth-Moon orbital plane is not five 'fixed' points but five orbits for which the distance from the Earth and the distance from the Moon remain constant. And because the Earth and Moon are presumed to orbit each other in a circular motion then each of the five libration 'points' prescribes a circle about the centre of mass of the system. In other words, all of the libration points are seen to be circular orbits - albeit rather special ones.
Above is a schematic of the location of the five libration points taken from Wikipedia. By convention, we call the three libration points that lie along the axis that passes through the centre of both the Earth and the Moon L1, L2 and L3. L1 is the libration point that that lies between the Earth and the Moon; L2 is the libration point that lies in the far side of the Moon; and L3 is the libration point that lies roughly on the opposite site of the Moon's orbit. L4 and the L5 complete the set of libration points with L4 lying 60 degrees in front of the Moon in its orbit around the Earth; and L5 lies 60 degrees behind the Moon in its orbit around the Earth. Of these five libration points, two are most often discussed - L1 and L2 - in large measure because of their proximity to the Moon, and because they are often seen as 'gateway' stations for getting to other places in the Solar System.
If we write down the equations of motion in the three-body system close to the libration points, we find that we have no way of writing down a 'closed-form' (i.e., easily to calculate) solution for those equations. It is this difficulty that makes the three-body problem considerably harder than the two-body problem for which a closed-form expression does exist. However, if we only want to consider solutions that are close to the Lagrange points, then we can derive analytical expressions. And that kind of analysis tells us some important things about the libration points.
First, the co-axial libration points - L1, L2 and L3 - are all unstable. If an object is placed there with even the slightest offset in position or velocity, it will start to move away from the libration point at a rate that grows exponentially with time. Think of trying to balance a knife vertically on a table, If one aligns it exactly vertically then one might imagine that it should be possible to balance it there. However, the slightest nudge or misplacement in any direction will result in the knife moving away from its balance point at an increasingly fast rate. Attempting to stay at one of the co-axial libration points is very much like trying to balance an knife upright on a table. It is simply impossible to remain at one of the libration points for any length of time without periodically exerting some 'balancing force' to drive the object back to the balance point. This process of exerting some periodic balancing force to drive the object back towards a libration point is known as 'station keeping'. Typically for the Earth-Moon system, station-keeping needs to be carried out every one or two weeks. If it is not done, then the spacecraft will be ejected from the libration point orbit after a period of three or four weeks.
Second, if a spacecraft is ejected from a libration point, then typically it will be ejected from the libration point orbit along something known as the 'unstable' manifold(s) - think 'orbit' or 'trajectory' here. For L1, L2 and L3 libration points, there are two unstable manifolds. For L1, one of those unstable manifold orbits points towards the Earth; and the other points towards the Moon. If one wants to leave L1 and head back towards the Earth, the natural thing to do is to stop station keeping and give the craft a slight nudge in the direction of Earth which will force exit from the unstable manifold that points towards Earth, On the other hand, if one wants to leave the L1 libration point and head to the Moon, one gives the spacecraft a slight nudge in the other direction which forces it to exit L1 along the other unstable manifold that heads of towards the Moon. For L2, one unstable manifold heads off towards the Moon, and the other heads off into deep space. For L3, one of the unstable manifolds heads off in the direction of Earth, and the other also heads of in the direction of deep space.
Third, just as there are unstable manifolds that lead away from the co-axial libration points, there are also a pair of stable manifolds that head in towards (and terminate at) the same libration points. These stable manifolds are special trajectories that, if you are on them and travelling along them with exactly the right speed and direction, gravity will automatically take you to (and stop you at) a libration points without any expenditure of delta-v. Once at a libration point, and if you wish to stay at that point, station-keeping will need to commence and continue for as long as you wish to remain there. And once station-keeping ends, the spacecraft will automatically be ejected along one of the two unstable manifolds.
Fourth, this simple picture of libration points is made considerably more complicated by the elliptical motion of the primary and secondary bodies; and gravitational perturbations from, say, non-spherical gravitational sources and other gravitating bodies (e.g., the Sun). The basic theory of libration points assumes that the primary and secondary move about each other in circular orbits. This theory is generally known as the Circular Restricted Three-Body Problem (or CRTBP for short). It is called 'Circular' because the primary and secondary are assumed to move around each other in perfect circles; it is called 'Restricted' because the third-body (the spacecraft) is assumed to have zero mass; and it is called 'the Three-Body Problem' because the theory takes into account the motion of three bodies - the primary, the secondary and the spacecraft. One can improve upon this basic circular theory (albeit at some considerable analytical and computational expense) if one moves over to a theory know as the Elliptical Restricted Three-Body Problem (ERTBP). This model is the same as the CRTPB except that the primary and secondary rotate around each other in perfect, unperturbed ellipses. For examining many of the libration points in the Solar System - of the Earth-Moon system, the Sun-Earth system, the Sun-Jupiter system or the Jupiter-Galilean moon systems, for example.
The location of the libration points in the ERTBP - and their use as reference orbits
Working out the equations of motion in the ERTBP is, frankly, a painful and rather technical exercise in classical mechanics involving four separate co-ordinate transformations to get standard result that one can find in textbooks. In this note, I'm not going to bother focusing on the derivation of these equations. Instead, it is more appropriate to focus on the resulting equations of motion and what one can do with them. And below, what I want to do is to focus on:
- the location of the librations in the ERTBP;
- translate those location of those points back into Orbiter's global reference frame;
- sketch the way in which those analytical calculations can be used to identify the relative position of a spacecraft with respect to a libration point; and
- indicate how that relative position information can be used to perform some basic (and somewhat manual ) station-keeping to keep a spacecraft close to a libration point indefinitely
From the analytical solution to the ERTBP, we can calculate the position of a libration point using the following algorithm:
First, calculate the following quantities:
[MATH]gm = GM_1 + GM_2[/MATH]
[MATH]\mu_1 = GM_1 / gm[/MATH]
[MATH]\mu_2 = GM_2 / gm[/MATH]
[MATH]COM = Q_{1}\,\mu_{1}+Q_{2}\,\mu_{2}[/MATH]
[MATH]COV = P_{1}\,\mu_{1}+P_{2}\,\mu_{2}[/MATH]
where [MATH]GM_1[/MATH] is the standard gravitational constant for the primary body in SI units; [MATH]GM_2[/MATH] is the standard gravitational constant for the secondary body in SI units; [MATH]Q_1[/MATH] and [MATH]P_1[/MATH] are the position and velocity of the primary in Orbiter's global coordinate reference frame (information that can be extracted from Orbiter using the 'oapi.get_globalpos(hObj)' and 'oapi.get_globalvel(hObj)' API commands.
Next, calculate the following:
[MATH]\boldsymbol {\mathbf {r}} = Q_ {2} - Q_ {1}[/MATH]
[MATH]\boldsymbol {\mathbf {v}} = P_ {2} - P_ {1}[/MATH]
[MATH]\boldsymbol {e} = \frac {v^{2}} {gm}\, \boldsymbol {r} - \frac{\boldsymbol {r}.\boldsymbol {v}} {gm}\, \boldsymbol {v} - \frac{\boldsymbol {r}} {r}[/MATH]
[MATH]a = \frac {gm} {2\, gm/r - v^{2}}[/MATH]
[MATH]\nu = \arccos (\frac {\boldsymbol {e}.\boldsymbol {r}} {e\, r})[/MATH]
such that if [MATH]\mathbf {r} . \mathbf {v} < 0[/MATH] then [MATH]\nu = 2\,\pi-\nu[/MATH].
Here, [MATH]\mathbf {r}[/MATH] is the position of the secondary with respect to the primary in Orbiter's global reference frame; [MATH]\mathbf {v}[/MATH] is the velocity of the secondary with respect to the primary in Orbiter's global reference frame; [MATH]\boldsymbol {e}[/MATH] is the eccentricity vector of the secondary in its elliptical orbit around the primary; [MATH]a[/MATH] is the semi-major axis of the elliptical orbit; and [MATH]/nu[/MATH] is the true anomaly of the secondary in its orbit around the primary.
Once these basic preliminary calculations have been completed, one next calculates the location of the libration point in the rotating reference frame of the ERTBP:
[MATH]Q_{x}=\alpha\, a\,\left(1-e^{2}\right)\,\frac{\cos(\nu)}{1+e\,\cos(\nu)}[/MATH]
[MATH]Q_{y}=\alpha\, a\,\left(1-e^{2}\right)\,\frac{\sin(\nu)}{1+e\,\cos(\nu)}[/MATH]
[MATH]P_{x}=-\alpha\,\sqrt{\frac{gm}{a\,\left(1-e^{2}\right)}}\,\sin(\nu)[/MATH]
[MATH]P_{y}=+\alpha\,\sqrt{\frac{gm}{a\,\left(1-e^{2}\right)}}\,(e+\cos(\nu))[/MATH]
where [MATH]Q_x[/MATH] and [MATH]Q_y[/MATH] are the x and y co-ordinates of the libration point in the rotating reference frame ([MATH]Q_z = 0[/MATH] is defined to be zero in this reference frame); and where [MATH]P_x[/MATH] and [MATH]P_y[/MATH] are the x and y co-ordinates of the velocity vector of the libration point in the same reference frame (again, [MATH]P_z = 0[/MATH] is defined to be zero on this reference frame).
these equations introduce a new parameter, [MATH]\alpha[/MATH], which is specific to the libration point and gravitating bodies of interest. If one is interested in the L1 libration point, to find [MATH]\alpha[/MATH] one needs to solve the following equation:
[MATH]\alpha=\frac{\mu_{1}}{\left(\alpha+\mu_{2}\right){}^{2}}-\frac{\mu_{2}}{\left(\alpha-\mu_{1}\right){}^{2}}
[/MATH]
For the Earth-Moon system, and using Orbiter's default values for the masses of the Earth and Moon, it is possible to show that:
[MATH]\alpha=0.83691519487205988712[/MATH]
If one is interested in the L2 libration point, to find [MATH]\alpha[/MATH] one needs to solve the following equation:
[MATH]\alpha=\frac{\mu_{1}}{\left(\alpha+\mu_{2}\right){}^{2}}+\frac{\mu_{2}}{\left(\alpha-\mu_{1}\right){}^{2}}[/MATH]
For the Earth-Moon system, and again using Orbiter's default values for the masses of the Earth and Moon, it is possible to show that:
[MATH]\alpha=1.1556821114336216547[/MATH]
And if one is interested in the L3 libration point, to find [MATH]\alpha[/MATH] one needs to solve the following equation:
[MATH]\alpha=-\frac{\mu_{1}}{\left(\alpha+\mu_{2}\right){}^{2}}-\frac{\mu_{2}}{\left(\alpha-\mu_{1}\right){}^{2}}[/MATH]
For the Earth-Moon system, and using Orbiter's default values for the masses of the Earth and Moon, it is possible to show that:
[MATH]\alpha = -1.0050626399593038526[/MATH]
Having calculated the position of the libration point in the rotating reference frame using the above algorithm, the final step is to convert this position back to Orbiter's global reference frame. To this, we first calculate the unit vectors of the rotating reference frame in Orbiter's (non-rotating) global coordinate system.
Calculate:
[MATH]\hat{\boldsymbol{x}}=\boldsymbol{e}/e[/MATH]
[MATH]\hat{\boldsymbol{z}}=\boldsymbol{r}\times\boldsymbol{v}[/MATH]
[MATH]\hat{\boldsymbol{z}}=\hat{\boldsymbol{z}} / \sqrt{\hat{\boldsymbol{z}}.\hat{\boldsymbol{z}}}[/MATH]
[MATH]\hat{\boldsymbol{y}}=\hat{\boldsymbol{z}}\times\hat{\boldsymbol{x}}[/MATH]
So that, finally, we calculate the position of the libration point in Orbiter's global coordinate system as:
[MATH]\boldsymbol{Q}=Q_{x}\,\hat{\boldsymbol{x}}+Q_{y}\,\hat{\boldsymbol{y}}+COM[/MATH]
[MATH]\boldsymbol{P=}P_{x}\,\hat{\boldsymbol{x}}+P_{y}\,\hat{\boldsymbol{y}}+COV[/MATH]
[MATH]\boldsymbol{P=}P_{x}\,\hat{\boldsymbol{x}}+P_{y}\,\hat{\boldsymbol{y}}+COV[/MATH]
This algorithm is quite general and is the exact analytical result (aside, that is, from the numerical calculation of [MATH]\alpha[/MATH]) for the location of the co-axial libration points - L1, L2 and L3 - for any choice of primary and secondary bodies. H
Because the above may appear somewhat complicated, here is a lua code snippet that goes through exactly this calculation for the Earth-Moon L2 point:
Code:
-- get the handle for the spacecraft
ves = vessel.get_focushandle()
-- get the handles for the Earth and the Moon
earth = oapi.get_objhandle("Earth")
moon = oapi.get_objhandle("Moon")
-- define a set of constants relevant to the Earth-Moon L2 libration points
GM1 = 398600440157821.0 -- gravitational constant for the Earth (SI units)
GM2 = 4902794935300.0 -- gravitational constant for the Moon (SI units)
GM = GM1 + GM2
MU1 = GM1 / GM
MU2 = GM2 / GM
alpha = 1.155682111433621 -- the libration parameter for the Earth-Moon L2 point
-- get the current location of the vessel
q = oapi.get_globalpos(ves)
p = oapi.get_globalvel(ves)
-- get the current location of Earth
q_ear = oapi.get_globalpos(earth)
p_ear = oapi.get_globalvel(earth)
-- get the current location of the Moon
q_mon = oapi.get_globalpos(moon)
p_mon = oapi.get_globalvel(moon)
-- calculate the weighted average position and velocity of the Earth and Moon
com = vec.add( vec.mul( q_ear, MU1 ), vec.mul( q_mon, MU2 ) )
cov = vec.add( vec.mul( p_ear, MU1 ), vec.mul( p_mon, MU2 ) )
-- calculate the position of the Moon relative to the Earth
r = vec.sub( q_mon, q_ear )
v = vec.sub( p_mon, p_ear )
-- calculate some quantities that are used multiple times in the ensuing calculations
vsq = vec.dotp( v, v)
rln = vec.length( r )
rv = vec.dotp( r, v)
-- calculate:
-- 'e' - the eccentricity vector of the Moon relative to Earth
-- 'ecc' - the eccentricity
-- 'a' - the semi-major axis
-- 'nu' - the mean anomaly
e = vec.sub( vec.sub( vec.mul( r, vsq / GM ), vec.mul( v, rv / GM) ), vec.mul( r, 1.0 / rln ) )
ecc = vec.length(e)
a = GM / (2.0 * GM / rln - vsq)
nu = math.acos( vec.dotp(e, r) / ecc / rln)
if rv < 0 then
nu = 2.0 * math.pi - nu
end
-- calculate the unit vectors of a dextral reference frame aligned with the Moon's
-- orbital plane and orbital orientation:
-- 'xhat' - a unit vector pointing in the direction to the Moon's orbital periapsis
-- 'zhat' - a unit vector point normal to the Moon's orbital plane
-- 'yhat' - the third unit vector to complete the trio
xhat = vec.unit( e )
zhat = vec.unit( vec.crossp( r , v ) )
yhat = vec.unit( vec.crossp( zhat, xhat ) )
-- calculate some more intermediate values
k1 = a * (1.0 - ecc * ecc )
k2 = math.sqrt( GM / k1 )
cnu = math.cos(nu)
snu = math.sin(nu)
k3 = 1.0 + ecc * cnu
-- calculate the position of the Lagrange point in the dextral reference frame:
-- 'qx' - the position of the Lagrange point in the 'xhat' direction
-- 'qy' - the position of the Lagrange point in the 'yhat' direction
-- 'qz' = 0 by definition
-- 'px' - the speed of the Lagrange point in the 'xhat' direction
-- 'py' - the speed of the Lagrange point in the 'yhat' direction
-- 'pz' = 0 by definition
qx = alpha * k1 * cnu / k3
qy = alpha * k1 * snu / k3
px = -alpha * snu * k2
py = alpha * (ecc + cnu) * k2
-- caluclate the position of the Lagrange point in Orbiter's global reference frame
l2qa = vec.mul( xhat, qx )
l2qb = vec.mul( yhat, qy )
l2qc = vec.add( l2qa, l2qb )
l2q = vec.add( l2qc, com )
-- calculate the velocity of the Lagrange point in Orbiter's global reference frame
l2pa = vec.mul( xhat, px )
l2pb = vec.mul( yhat, py )
l2pc = vec.add( l2pa, l2pb )
l2p = vec.add( l2pc, cov )A reference orbit
Because one can calculate the position of a libration point for any time (or MJD) the above provides a way of calculating a reference orbit or that libration point. In particular, one can use this calculation as a measure of how far one (and fast) has moved away from a libration point and, therefore, provide some information on how to make corrections to one's trajectory in order to move back towards the libration point - i.e., how to perform station keeping.
As an example of how this information can be used, attached to this post is an Orbiter scenario and lua script. The scenario places a Delta Glider at the Earth-Moon L2 'point'. On the right-hand side of the screen, the lua script runs continuously in the background and provides information about the location of the Delta-Glider with respect to the reference orbit of the Earth-Moon L2 point. The scenario starts with the Delta Glider placed about 0.5 metres from the L2 point and to with 0.1 mm/s is stationary with respect to the L2 point. Over the course of one lunar orbit, the Delta Glider slowly slips further and further away from the L2 point as it is progressively evicted from L2 along that point's unstable manifold. Unless station keeping is performed, by around week 4 the Delta Glider is well on its way to escape from the Earth-Moon system. The updated display provides information about position and velocity of L2 in the reference frame of the Delta Glider's orbit osculating two-body orbit around the Moon. This is the same reference frame used by Orbit MFD (and most other MFDs) if the Moon is referenced. Rather than use x, y and z co-ordinates, it provides information in terms of 'forward', 'inward' and 'plane' - i.e., the same co-ordinate system used by IMFD's Delta Velocity program. The scenario starts out with 10 m/s worth of RCS. This should be enough to keep the Delta Glider 'on station' for at least one month - so long as one is judicious in using that fuel.
Some further examples
This same algorithm for calculating the location of libration points can be extended to any choice of primary and secondary. For example, it is possible to place a craft at one of the libration points of the Jupiter-Europa system. Below, is a screenshot of the view from the Jupiter-Europa L2 point.
Here, the view is towards Europa. Jupiter is directly behind Europa - and, so long as the craft remains behind Europa, Jupiter will always remain behind Europa. The MFD on the right-hand side is IMFD's Map Program. It shows the Delta Glider will be evicted from the L2 position (and in quite short-order). The trajectory traces out a good approximation to the unstable manifold for that Lagrange point. Interestingly, the unstable manifold passes within about 70 km of Europa's surface. This raises the possibility that a sensible Europa approach strategy may be to approach the Jupiter-Europa L2 point along one of the stable manifolds then, once there, give a small nudge to exit L2 along the unstable manifold leading to Europa. This will automatically take one down almost to Europa's surface. And at Europa periapsis, execute a 430 m/s retrograde burn to enter Low Europa Orbit.
Equally, we could place a craft at the Jupiter-Io L2 point. A screenshot of this scenario is given below:
Again, the unstable manifold take the craft down to Io's surface. As with all L2 points, the primary (Jupiter) is directly behind the secondary (Io). Io acts, then, as kind of radiation shield for any body in the vicinity of Jupiter. If we let the spacecraft, drift 'off station' a little, then we can see Jupiter energy from behind Io:
Here, you can see Jupiter appearing from behind Io. In passing, one notes that the apparent size of Io neatly matches the apparent size of Jupiter so that Io, just (and only just) covers the disk of Jupiter. A rather interesting curiosity.
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