The Tisserand Plot - representing elliptical orbits and gravitational encounters

Keithth G

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Introduction
This is the second in a series of three (or, possibly, four) posts aimed at providing the theoretical background for designing transfer manoeuvres that are relevant for the reduction of delta-V requirements for both:

a. the 'Begin Game' problem - i.e., using gravity assists to boost the spacecraft's kinetic energy so as to reduce the overall mission delta-V requirements (e.g. leaving Earth and setting up a transfer trajectory to, say, Jupiter or Saturn);

b. the 'End Game' problem - i.e, using gravity de-assist to shed kinetic energy upon arrival at the target planetary system in order to slow the spacecraft sufficiently to permit final orbit insertion around one of the moons).

Such transfer manoeuvres are generally called VILTs ("V-ininity Leveraged Transfers"). They form the basis of most modern real-world mission planning. Without them, many of the missions currently underway (or on the drawing board) simply wouldn't be possible.

The first post in this series was titled "A very body post" (http://www.orbiter-forum.com/showthread.php?t=36407. It serves as a repository of useful equations for elliptical motion in which quantities of interest are calculated in terms of [MATH]r_a[/MATH] (the apoapsis radius) and [MATH]r_p[/MATH] (the periapsis radius).

In this thread, I will focus on the representation of ellipses in the Tisserand plot - which is a useful graphical technique for calculating an optimal series of transfer manoeuvres for both the Begin Game problem and the End Game problem. I will also show how a gravitational encounter with, say, the Earth transfers the elliptical orbiting its representation in the Tisserand Plot.

The third of this sequence of posts will use the Tisserand plot to calculate a transfer sequence that via a gravitational encounter with the Earth 'kicks' a spacecraft from a relatively low energy orbit about the Sun into a much higher energy orbit, one that is capable of reaching Jupiter. This is the classic Earth Gravity Assist (EGA) scheme in which the gravity assist with the Earth is used to save fuel on missions sending spacecraft to Jupiter.


Representing elliptical orbits
As most will already know, to describe the motion of a spacecraft moving in an elliptical orbit around a massive body, for example the Sun, one needs six parameters. In the 'state vector' representation, we have three quantities that represent the position of the spacecraft; and three that represent the velocity of the spacecraft at that position. In the 'orbital elements' representation, we have the semi-major axis, the orbital eccentricity, the orbital inclination, the longitude of the ascending node, the argument of periapsis and the mean anomaly. We are free to choose which six quantities we use, but in three dimensions we must always have six.

In a lot of preliminary orbital design work, it is convenient to assume that all bodies move in the same orbital plane (e.g., the ecliptic). For the most part, this is a fair description of the motion of the planets around the Sun, or indeed the motion of moons around Jupiter or Saturn. To keep life simple, we will make this assumption here.

If all relevant motion lies in the same orbital plane, then we can immediately eliminate two variables from our set of six quantities needed to describe elliptical motion. That leaves us with four. Two of the remaining four specify the overall size and shape of the ellipse; one is needed to specify the orientation of the ellipse; and the final quantity tells us 'where' we are on the ellipse at any point in time. Now, if we aren't particularly interested in knowing 'where' we are on the ellipse; and if we aren't interested in knowing the orientation of the ellipse, then we are just left with the two quantities that, together, specific the size and shape of the ellipse.

Now, we have considerable freedom to choose these last two quantities. In this note, we shall use [MATH]r_a[/MATH], the apoapsis radius, and [MATH]r_p[/MATH], the periapsis radius to characterise the size and shape of an elliptical orbit. If one reads enough papers on this subject, one may find references to a similar representation of elliptical orbits using the orbital period and semi-major axis instead. This is an equally valid representation of elliptical orbits, but for our purposes the [MATH]r_a ~ r_p[/MATH] representation is better since with it, it is easier to identify resonant orbits and to see how the orbit is affected by a prograde or retrograde burn at apoapsis or periapsis. Moreover, as the post "A very boring post" showed, many quantities of interest can be expressed quite simply using these two quantities. Certainly, some information is 'lost' because the 'orientation' and 'phasing' information has been thrown away. But even so, it still provides a detailed picture of how a complicated series of transfer manoeuvres needs to be connected to achieve the desired goal (e.g., getting a velocity boost to reach Jupiter or Saturn, or for designing a series of gravity de-assists aimed at reducing orbital energy prior to orbital insertion around a Galilean moon, say).


The Tisserand Plot
The Tisserand Plot is a very efficient way of representing elliptical orbits and encounters with gravitational bodies such as the Earth that give rise to gravity assists (or gravity de-asissts). The Tisserand Plot is nothing more than a two-dimensional graph with [MATH]r_a[/MATH] on one axis; and with [MATH]r_p[/MATH] on the other axis. Any point such that [MATH]r_a \ge r_p \ge 0[/MATH] on this graph is a valid Keplerian orbit. An example of a Tisserand Plot (taken from someone else's presentation on the subject) is given below:



This graph plots valid orbits around the Sun. The black dots are the (presumed circular) orbits of the planets. The two red dots identify Hohmann transfer orbits. The first identifies a Hohmann transfer orbit from Earth to Jupiter; and the second identifies another Hohmann transfer orbit from Jupiter to Uranus.


Gravitational encounters and the Tisserand Plot
In standard two-body Keplerian physics, if a spacecraft is set in orbit in some elliptical orbit about the Sun, say, then, barring close encounters with a planet, the spacecraft will remain in exactly the same Keplerian orbit. On the Tisserand Plot, the graphical representation of this is that the spacecraft's orbit is represented by an unchanging point.

What happens if there is a gravitational encounter with a planet? The standard two-body patched conic approximation to this encounter is that when the spacecraft enters the sphere of influence of the planet, the description of the trajectory shifts from an ellipse around the Sun to a hyperbolic approach around the planet. Once the spacecraft moves out of the planet's sphere of influence, the description shifts back to an ellipse around the Sun and, so long as there are no further gravitational encounters, the spacecraft will thereafter remain forever more in that orbit. The gravitational encounter, then, is represented by a 'shift' in the point identifying the elliptical orbit to another point in the Tisserand Plot representing a new elliptical orbit.

However, the shift in one elliptical orbit to another is not arbitrary. In particular, by virtue of the conservation of energy, the hyperbolic excess velocity is conserved. What does this mean? If the spacecraft encounters a planet with a certain hyperbolic excess velocity, [MATH]v_\infty[/MATH], then it departs the planet with exactly the same hyperbolic excess velocity. Moreover, should the spacecraft ever encounter the planet a second time it will again approach the planet with exactly the same hyperbolic excess velocity. So, in pure ballistic encounters with a planet, and even if the spacecraft gets a velocity 'kick' from the encounter, the hyperbolic excess velocity of any subsequent encounter will remain unchanged. If you think about this for long enough, this is a surprising result. But it happens to be true.

The result of this is that when a spacecraft encounters a planet (or moon) its orbit is transformed to a new ellipse that lies on a line of constant hyperbolic excess velocity, [MATH]v_\infty[/MATH]. Another Tisserand Plot, this time of the Jovian system, is given below.



The black, blue, red and green lines are the lines of constant hyperbolic velocity for Io, Europa, Ganymede and Callisto respectively. Gravitational encounters with these moons, then, is represented by a shift in the point describing the orbital ellipse (in this case, around Jupiter) to a new point that must lie somewhere along the curved lines of constant hyperbolic velocity.

OK, so how far does the point move along these lines? Well, that depends on a couple of things. First, it depends on whether or not the encounter is prograde or retrograde. If it is prograde, the shift will be from left to right, say; and if it is retrograde, the shift along the curve will be from in the other direction from right to left. [N.B., without checking, I'm not actually sure if the prograde shift is from left to right or from right to left. But either way, the retrograde shift is in the opposite direction.] Second, it depends on how close an encounter the spacecraft has with the planet/moon. The closer the encounter with the moon/planet, the greater the magnitude of the shift. The exact magnitude of the shift has to be calculated by applying the equations of "A very boring post", however. Of course, because there is a minimum approach altitude that the spacecraft can attain, there is an upper bound on the amount of shift that any single gravitational encounter can achieve. If one wishes to achieve a grater shift along the same line of constant hyperbolic velocity then one would have to set up two or more encounters with the same moon/planet.

In the next post, and for a specific example, I will show how the equations of "A very boring post" can be used to calculate the Tisserand Plot coordinates of the orbit resulting from the gravitational encounter - and, in son doing, show how one can construct an EGA manoeuvre. However, before doing that, it is worth pointing out a few things about some of the lines that appear on the above Tisserand Plot of the Jovian system. In addition to the lines of constant hyperbolic velocity, there are a series of diagonal lines on the graph. These are lines of constant resonance. "What the hell is that?", I hear you say. Well, if one wishes to have a series of encounters with a planet/moon, one must ensure that one leaves the planet/moon on a resonant orbit. A resonant orbit is one that is in say a 3:1 ratio of the orbital period of the spacecraft to the planet/moon. In the case of Ganymede orbiting Jupiter, a 3:1 resonance with Ganymede means the spacecraft will orbit Jupiter once for exactly three orbits of Ganymede around Jupiter. And on Ganymede's third passage around Jupiter, in a 3:1 resonance, the spacecraft will encounter Ganymede for a second time. Equally, if one wishes to have a third encounter with Ganymede then, at the end of the second encounter, one needs to be in a new resonant orbit with Ganymede (e.g., a 2:1 resonance) in order for that third encounter to take place. If orbits are not resonant, then there are no guarantees that the spacecraft will encounter the moon again (at least, not in a reasonable timescale). Consequently, resonant orbits are important for repeat encounters with the same moon.

In additional to the diagonal lines of constant resonance, there are also a grid of horizontal and vertical lines. These lines look a bit like a stretched piece of graph paper. To explain what these are, consider what happens if one executes a prograde or retrograde burn either at apoapsis or periapsis. If one executes a burn at periapsis, this will keep the periapsis constant but will raise or lower the apoapsis. On the Tisserand Plot, this burn is represented by a shift horizontally to the left or right on the Tisserand Plot. Note, that in making this shift one jumps between lines of constant hyperbolic velocity. Equally, if one executes a burn at apoapsis, the orbital apoapsis remains constant but periapsis is raised or lowered. On the Tisserand Plot, this apoapsis burn is represented by a vertical shift in the orbit. Again, in executing the burn one jumps between lines of constant hyperbolic velocity. The grey grid lines, then, represent the amount of delta-V required to achieve a shift of the orbit in the horizontal or vertical direction at any point in the graph.

Finally, there are a series of dots on the lines of constant hyperbolic velocity. These provide a rough scale of the maximum extent one can move along a line of constant hyperbolic velocity in any single (ballistic) gravitational encounter.


The key elements of the Tisserand Plot
So, in summary, the key elements of the Tisserand Plot are as follows:

1. By throwing information about 'phasing' and orientation, one can represent elliptical orbits as a single point on a two-dimensional graph.

2. In a ballistic gravitational encounter, a spacecraft's orbit will move along lines of constant hyperbolic velocity. But in any one encounter, there is a maximum extent to which an orbit can move along that line.

3. In order to have a series of encounters with the same moon, the orbit after the gravitational encounter needs to be a resonant orbit. This restricts relevant orbits to lie along a series of diagonal lines on the Tisserand Plot

4. One can move between lines of constant hyperbolic velocity by executing prograde/retrograde burns at apoapsis or periapsis.

5. Finally, one can also switch 'focus' between one moon and another at the intersection points of lines of constant hyperbolic velocity for one moon, with those of another moon.

Taken together, the Tisserand Plot very efficiently summarises the possible 'moves' one can make in the Begin Game and End Game problems. It is then possible to systematically to search through these possible moves to find an optimal set of moves.

For example, using the Tisserand Plot for the Jovian system, one can answer the question: what is the optimal sequence of burns and gravitational encounters needed to enter into low orbit around Europa, say, starting from some initial orbit around Jupiter such that the total sequence of moves reduces the final Europa approach speed to less that 800 m/s; and takes less than two years with at least two flybys of Callisto and Ganymede on the way.


Next steps
Of course, using the Tisserand Plot to answer mission design questions of that complexity is a complex programming challenge. But the Tisserand Plot provides the tools with which that type of question can be answered.

In the next post in this series, we'll use the principles of the Tisserand Plot to do do something that is a little easier - namely design a basic Earth Gravity Assist sequence for insertion into a Hohmann transfer from Earth to Jupiter.
 
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