Introduction
This is the third in a series of posts aimed at providing tools with which to design transfer manoeuvres that minimise the delta-V requirements for:
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).
In particular, in this post, we will focus on designing an Earth Gravity Assist manoeuvre.
The first post in this series was titled "A very boring 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 the apoapsis radius and the periapsis radius.
The second post in this series was titled "The Tisserand Plot - representing elliptical orbits and gravitational encounters" (http://www.orbiter-forum.com/showthread.php?t=36461). This post introduced the Tisserand Plot as a graphical device for representing elliptical orbits and the effect on those orbits of gravitational encounters with some gravitating body. The purpose of the Tisserand Plot is to simplify the task of searching for the optimal sequence of gravitational encounters that satisfies certain criteria - e.g., least fuel cost - subject to total mission time, say, being less than a certain value.
This, 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, in fact, 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.
Although the Tisserand Plot can tell us a lot about the EGA scheme needed to generate the required energy 'kick' to get to Jupiter, it can't tell us everything that we need to know to set up the EGA manoeuvre within Orbiter. So, the next post in this series, the fourth, will go through the somewhat laborious exercise of using the equations set out in the first post, "A very boring post", to calculate the timing (or 'phaising') of the specific sequence of manoeuvres that need to be executed in order to perform the EGA. In this sense the Tisserand Plot should be seen as a graphical means of identify the skeletal components of a mission - but having done that, some additional mathematical 'grunt' work still needs to be done to put 'flesh on the bones'. Putting the 'flesh on the bones' basically involves leveraging the equations set out in the first post, "A very boring post".
Of course, the Tisserand Plot can be used to design far more complicated missions than the (relatively) straightforward EGA manoeuvre. At a later stage, I will set out how the same graphical technique can be used to design missions to the moons of Jupiter and Saturn.
But now down to business….
Constructing the Tisserand Plot
The previous post in this series introduced the Tisserand Graph. The key points to remember were as follows:
1. By throwing information about 'phasing' and orientation, one can represent elliptical orbits as a single point on a two-dimensional graph - with the orbital apoapsis on the x-axis; and the orbital periapsis on the y-axis.
2. In a ballistic gravitational encounter with a body - i.e., a planet or a moon, a spacecraft's orbit will move along lines of constant hyperbolic excess velocity. But in any one encounter, there is a maximum extent to which an orbit can move along that line (because one assumes that nose-diving through the surface of the encountered planet or moon is not a feasible manoeuvre).
3. In order to have a series of encounters with the same moon, the orbit after the gravitational encounter needs to be close to 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.
To construct the Tisserand Plot for the EGA manoeuvre we need to do three things. First, we need to know how to construct lines of 'constant hyperbolic excess velocity'. Second, we need to know how to construct lines corresponding to resonant orbits - in particular, we need to construct the 2:1 resonance line in which the spacecraft executes one orbit of the Sun while the Earth executes two orbits around the Sun. Third, we need to know what manoeuvre we need to perform (at aphelion) to move between two different lines of constant hyperbolic excess velocity so that the spacecraft receives the required energy 'kick' during its gravity assist encounter with Earth.
In using the Tisserand Plot, we make a number of simplifying assumptions:
1. Bodies providing gravity assists - e.g., the Earth in its motion around the Sun; or Europa in its orbit around Jupiter - move in circular orbits. Of course, in reality, orbits are elliptical but we simplify the maths enormously if we assume motion is circular. And for all bodies other than, say, Mercury (which has a rather large orbital eccentricity) this is a reasonable assumption that doesn't introduce major errors into the calculation.
2. All bodies in the Solar System move in the same orbital plane. Again, this is a idealisation of a more complex situation, but as with the assumption of uniform circular motion it provides a good starting point for analysis.
3. We will model gravitational encounters between the spacecraft and gravitating bodies using a 'linked conics' model. Most people who use Orbiter will have heard of the 'patched conics' model. The 'patched conics model patches various 2-body trajectories (circular, elliptical, parabolic or hyperbolic) at the so-called 'sphere of influence. In the linked conics model, the sphere of influence is assumed to have zero radius, hence an alternative name for the model: the 'zero radius sphere of influence patched conics' model. This is a bit of a mouthful, so the more succinct name for it is the 'linked conics' model. In this scheme, motion is nearly always elliptical (around the central gravitating body - e.g., the Sun) except when there is a gravitational encounter with some other body (e.g., the Earth) whereupon it is assumed that there is an instantaneous rotation of the velocity vector of the spacecraft about some angle sending the spacecraft on another elliptical orbit around the primary.
4. It is useful to work in dimensionless units. For example, if we wish to explore gravitational encounters with Earth, we work in units of distance and time for which the orbital radius of the Earth in its motion about the Sun is unity; and for which its orbital speed of Earth around the Sun is also unity. We do all of our calculations in these units and when we wish to convert back to more conventional units, we just need to remember to multiply all distances by the Earth's orbital radius (i.e., 1 AU); and to multiply all speeds by Earth's mean orbital speed (i.e., 29.78 km/s). Again, we introduce this little 'trick' to simplify the mathematical expressions as much as possible.
Calculating the lines of constant hyperbolic velocity
An important first step in using the Tisserand Plot is to learn how to construct lines of constant hyperbolic velocity. From the post "A very boring post", we have an expression for the hyperbolic excess velocity of a spacecraft encountering a planet or moon:
[MATH]v_{\infty} = v_{c}\,\sqrt{3-\frac{2\,r_{c}}{r_{a}+r_{p}}-2\,\sqrt{\frac{2\,r_{a}\,r_{p}}{r_{c}\left(r_{a}+r_{p}\right)}}} [/MATH]
where [MATH]v_c[/MATH] is the mean orbital speed of the gravitating body; [MATH]r_c[/MATH] is the mean orbital radius of the gravitating body; [MATH]r_a[/MATH] is the apoapsis of the spacecraft in its orbit around the primary; and [MATH]r_p[/MATH] is the periapsis of the spacecraft in its orbit around the primary. If we work in our dimensional units, we set [MATH]r_c \to 1[/MATH] and [MATH]v_c \to 1[/MATH] so that our expression simplifies to:
[MATH]v_{\infty} = \sqrt{3-\frac{2}{r_{a}+r_{p}}-2\,\sqrt{\frac{2\,r_{a}\,r_{p}}{\left(r_{a}+r_{p}\right)}}} [/MATH]
with [MATH]0 < r_p < 1[/MATH] and [MATH]r_a > 1[/MATH]. We can now invert this equation and solve for [MATH]r_a[/MATH] as a function of [MATH]r_p[/MATH] and [MATH]v_\infty[/MATH]. If we do this, we derive the following expression for [MATH]r_a[/MATH]:
[MATH]r_{a}(v_{\infty})=\frac{-r_{p}\,v_{\infty}^{4}+6\,r_{p}\,v_{\infty}^{2}+4\,r_{p}^{2}-9\,r_{p}-2\,v_{\infty}^{2}+6}{-8\,r_{p}+v_{\infty}^{4}-6\,v_{\infty}^{2}+9}+4\,\sqrt{\frac{r_{p}^{2}\,v_{\infty}^{2}+r_{p}^{4}-3\,r_{p}^{2}+2\,r_{p}}{\left(-8\,r_{p}+v_{\infty}^{4}-6\,v_{\infty}^{2}+9\right){}^{2}}}[/MATH]
such that [MATH]r_{p,min} \le r_p \le r_{p,max}[/MATH] and [MATH]0 \le v_\infty < \sqrt{3}[/MATH] and with
[MATH]\begin{array}{cc} r_{p,min}(v_{\infty})=\{ & \begin{array}{cc} \frac{-v_{\infty}^{2}+2v_{\infty}-1}{v_{\infty}^{2}-2v_{\infty}-1} & 0\leq v_{\infty}\leq1\\ 0 & 1<v_{\infty}\le\sqrt{3} \end{array}\end{array}[/MATH]
and
[MATH]r_{p,max}(v_{\infty})=\begin{array}{cc} \{ & \begin{array}{cc} 1 & 0\leq v_{\infty}\leq\sqrt{2}-1\\ \frac{1}{8}\left(v_{\infty}^{4}-6v_{\infty}^{2}+9\right) & \sqrt{2}-1<v_{\infty}\leq\sqrt{3} \end{array}\end{array}[/MATH]
Now, if you are thinking that this is a horrible mess of an expression, you would be right. To help those that might be intimidated by this algebraic mess, I've written a small 'user-defined function' in an excel spreadsheet which can be found as a zipped file at the bottom of this post. (Note again that in that spreadsheet calculations are carried out in dimensionless units as defined above.)
With this expression, and for any given value of [MATH]v_\infty[/MATH] (dimensionless units), we can calculate these curves of constant hyperbolic velocity. This calculation is carried out in the 'zipped' spreadsheet at the end of this post - and a few of these curves are plotted in the graph:
Here, I have chosen a few values for the hyperbolic excess velocity ([MATH]v_\infty[/MATH]) and plotted the corresponding curves for those values. The values of ([MATH]v_\infty[/MATH] chosen are 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30 and so on. This choice is somewhat arbitrary but, taken together, they give an idea of how these curves look.
[Aside: what do we mean by a hyperbolic excess velocity of 0.05 in dimensionless units? Suppose, for example, we wish to consider a gravitational encounter of a spacecraft with the Earth. In more normal units, the mean orbital speed of the Earth is around 29.78 km/s. So, in this case, a dimensionless speed of 0.05 corresponds to a hyperbolic excess velocity of 29.78 x 0.05 km/s = 1.489 km/s. If instead, we were considering a gravitational encounter with, say, Callisto, which has a mean orbital speed of 8.204 km/s around Jupiter, the same value of 0.05 corresponds to a hyperbolic excess velocity of 0.410 km/s.]
And as a reminder of the significance of these lines of constant hyperbolic excess velocity, suppose that the spacecraft starts in an orbit around some body (e.g., the Sun). On the Tisserand Plot, this orbit can be represented as a point that lies on one of these lines of constant hyperbolic velocity (as calculated for some other body - e.g., the Earth, moving in a circular orbit around the Sun). So long as the spacecraft doesn't encounter the second gravitating body (e.g., the Earth), the spacecraft will remain at exactly the same point on the Tisserand Plot forever (i.e., it will be on the same unperturbed elliptical orbit). If, however, the spacecraft encounters the second gravitating body (e.g., the Earth), physics requires that after the encounter, the spacecraft will have moved to a new elliptical orbit around the primary (e.g., the Sun) which, on the Tisserand Plot, will be represented by another point lying on the same line of constant hyperbolic velocity. And so long as all subsequent encounters with the second body (e.g., the Earth) are ballistic - i.e., the spacecraft's propulsion system is not used - the spacecraft's orbit must forever lie on the same line of hyperbolic excess velocity.
How, then, does one move between lines of constant hyperbolic excess velocity? Well, the basic mechanism is to use the spacecraft's propulsion system - but not just arbitrarily. But we'll come onto that in a bi.
Constructing the lines of resonance
Before (finally) constructing the EGA manoeuvre, we need to be able to add one more set of lines to our Tisserand Plot. In addition to the lines of constant hyperbolic velocity, it is also useful to construct lines of resonance. These lines are groups of orbits that have a simple multiple orbital period of the orbital period of the secondary body (e.g., the Earth). For example, we could consider those orbits whose orbital period is twice the orbital period of the Earth (a 2:1 resonance); or we could choose those orbits that have an orbital period that three times the orbital period of the Earth (a 3:1 resonance). We could even consider those orbits whose orbital period is 1.5 times the Earth's orbital period (a 3:2 resonance). And so on.
And why do this? In most cases, we want to design orbits so that having left the secondary body (e.g., the Earth) on some elliptical orbit around the Sun, we want to make sure that we encounter the same body again after a specified number of orbits. To do this, we need to exploit these orbital resonances by making sure that our departure orbit lies (roughly) on one of these lines of resonance on the Tisserand Plot.
Fortunately, these lines are simple to construct - they are just straight lines with the equation:
[MATH]r_p = 2 n^{2/3}-r_a[/MATH]
where, again, we are working in dimensionless units and where [MATH]n[/MATH] is the resonance that we are interested in. For example, for the 2:1 resonance, we would set [MATH]n=2[/MATH]; and for the 3:2 resonance, we would set [MATH]n=1.5[/MATH]. And so on.
On the graph below, some of these lines of resonance have been drawn:
Design the EGA manoeuvre - 2:1 resonance
Now, finally, we are in a position to design the EGA sequence that exploits a 2:1 resonance. Consider the following graph:
This is, in fact, the EGA sequence (2:1 resonance) as depicted on a Tisserand Plot. For convenience, I've converted back to more standard units for which distances are measured in Astronomical Units (AU) and velocities are measured in 'km/s'. Let's go through the components of this graph one by one.
1. We start the EGA sequence at the point 'A'. Why there? Well, we know that we want to try and exploit the 2:1 resonance, so we know that after our initial hyperbolic escape from Earth, our Earth departure orbit needs to lie close to the 2:1 resonance. Now the usual way of escaping the Earth would be via an escape burn that resulted in our spacecraft leaving Earth in a 'prograde' orientation. This means that the perihelion of the resulting orbit around the Sun is at the Earth's orbital radius which means on the Tisserand Plot that [MATH]r_p = 1\,AU[/MATH]. These two facts place our initial departure orbit from Earth at the point 'A'. We immediately note from the Tisserand Plot that our initial orbital aphelion after escaping Earth must be around 2.2 AU.
2. Using the equations from "A very boring post", we can quickly calculate that this initial departure orbit must have a hyperbolic excess velocity of around 5.2 km/s. Now this isn't anywhere near enough speed to get us to Jupiter. Typically, for a direct transfer to Jupiter from Earth, we would need a hyperbolic excess velocity of around 8.9 km/s (or more). In this note, I'm going to assume that we are going to target a hyperbolic excess velocity of around 9.1 km/s (slightly higher than the minimum 8.9 km/s - just to give us a bit of leeway in our planning). The next thing we can do is draw the line of constant hyperbolic excess velocity corresponding to [MATH]v_\infty = 9.1\,km/s[/MATH] on the Tisserand Plot.
3. The point where we want to end up after the EGA manoeuvre is at position 'C' on the Tisserand Plot. What is this position? It is, in fact, another elliptical orbit, but one with a perihelion close to (but slightly less than) Earth's orbital radius of 1 AU and with an aphelion of around 5.2 AU, the mean orbital radius of Jupiter around the Sun. This point, then, represents a Hohmann-like transfer orbit from Earth to Jupiter.
4. The final thing that we need to consider is: how do we get from our initial starting orbit 'A', to our destination orbit 'C'? The most simple way of doing this is to move from the orbit 'A' to the orbit 'B'; and then from orbit 'B' to the orbit 'C'. What does moving from orbit 'A' to orbit 'B' entail? On the Tisserand Plot, this is vertical displacement of our orbit. The aphelion remains constant but the orbital perihelion is lowered from 1.00 AU to around 0.90 AU. From basic orbital mechanics, we know that the way to execute this manoeuvre is to perform a retrograde burn at aphelion. And how big a retrograde burn is this? Since we know the apoapis and periapsis of our 'before' and 'after' orbits, this is an easy calculation to carry out using the equations of "A very boring post" and it works out that the required retrograde burn is around 0.55 km/s. To be sure, this is a substantial burn but the effect of this burn is to move us from a line of constant hyperbolic velocity of 5.2 km/s to a line of constant hyperbolic velocity of 9.1 km/s. This means that when we next encounter the Earth, we will get a velocity 'kick' of 3.9 km/s - with just enough energy to get us all the way to Jupiter.
5. And what about the move from orbit 'B' to orbit 'C'. On the Tisserand Plot, any move along a line of constant hyperbolic excess velocity corresponds to a gravitational encounter with the secondary body (in this case, the Earth). Roughly two years after the spacecraft's initial departure from Earth and a deep space manoeuvre at aphelion, the spacecraft will again encounter Earth. On its approach to Earth, the spacecraft will have orbit 'B'. But the hyperbolic encounter with Earth will transform the orbit to orbit 'C'. Again, just using the equations of "A very boring post", one can calculate that the perigee altitude of the encounter with Earth needs to be around 500 km - just a few hundred kilometres above the top of the atmosphere. Not quite a grazing encounter with Earth, but close!
Design the EGA manoeuvre - 3:2 resonance
In a way, there is nothing particularly special about the 2:1 resonance. It would be quite feasible to consider exploiting another resonance - e.g., the 3:2 resonance. The Tisserand Plot for this manoeuvre is shown below:
The logic of the construction of this manoeuvre is exactly the same as before. We start at orbit 'A' leaving Earth with an initial aphelion of around 1.60 AU. Our departure hyperbolic excess velocity is around 3.60 km/s. At aphelion, we execute a retrograde burn - this time of around 1.10 km/s - lowering our perihelion to around 0.84 AU and in so doing transferring to an orbit that will have a hyperbolic excess velocity of 9.1 km/s when the spacecraft next encounters Earth.
All good, so far. However, if one goes through the exercise of calculating the perigee required at Earth encounter to move from orbit 'B' to orbit 'C' one quickly calculates that the closest approach to Earth needed is 2000 km below the surface of the Earth. Sadly, then, an EGA manoeuvre based on a 3:2 resonance is infeasible!
EGA manoeuvres - a score card and fact sheet.
Based just on the Tisserand Plot and the equations of "A very boring post", one can detail a considerable amount of information about the EGA manoeuvre. Here are some tables that have been put together quickly that summarise the information that can be extracted readily from this analysis:
and
Next steps
The Tisserand Plot is a largely graphical means for analysing a sequence of manoeuvres. Although it is based on a number of simplifying assumption and throws away a lot of detail regain the exact timing ('phasing') of some of the manoeuvres being examined, it clarifies what is going on while still providing the basis for some considerable quantitative information. This quantitive information can be used to eliminate some options and elucidate the general shape of those that are considered optimal.
However, the information from the Tisserand Plot isn't quite enough to form the basis for execution a mission. To do that, two further steps are required. Having sifted through the chaff to find a mission that is close to optimal, one needs to tun that mission design through a process that re-inserts information about the exact timing of the sequence of events envisaged in the mission design. Doing this will be the subject of the next post.
The final step is to run the resulting mission through a 'high fidelity' model. If one for working for NASA or the ESA a higher fidelity model would involve using a full n-body integrator to optimise the mission design. However, in Orbiter-land, it is generally sufficient to use TransX and IMFD to do the same thing. So at this point, we should be in a position to 'fly' the mission. Flying the EGA manoeuvre may also be the subject of a subsequent post.
This is the third in a series of posts aimed at providing tools with which to design transfer manoeuvres that minimise the delta-V requirements for:
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).
In particular, in this post, we will focus on designing an Earth Gravity Assist manoeuvre.
The first post in this series was titled "A very boring 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 the apoapsis radius and the periapsis radius.
The second post in this series was titled "The Tisserand Plot - representing elliptical orbits and gravitational encounters" (http://www.orbiter-forum.com/showthread.php?t=36461). This post introduced the Tisserand Plot as a graphical device for representing elliptical orbits and the effect on those orbits of gravitational encounters with some gravitating body. The purpose of the Tisserand Plot is to simplify the task of searching for the optimal sequence of gravitational encounters that satisfies certain criteria - e.g., least fuel cost - subject to total mission time, say, being less than a certain value.
This, 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, in fact, 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.
Although the Tisserand Plot can tell us a lot about the EGA scheme needed to generate the required energy 'kick' to get to Jupiter, it can't tell us everything that we need to know to set up the EGA manoeuvre within Orbiter. So, the next post in this series, the fourth, will go through the somewhat laborious exercise of using the equations set out in the first post, "A very boring post", to calculate the timing (or 'phaising') of the specific sequence of manoeuvres that need to be executed in order to perform the EGA. In this sense the Tisserand Plot should be seen as a graphical means of identify the skeletal components of a mission - but having done that, some additional mathematical 'grunt' work still needs to be done to put 'flesh on the bones'. Putting the 'flesh on the bones' basically involves leveraging the equations set out in the first post, "A very boring post".
Of course, the Tisserand Plot can be used to design far more complicated missions than the (relatively) straightforward EGA manoeuvre. At a later stage, I will set out how the same graphical technique can be used to design missions to the moons of Jupiter and Saturn.
But now down to business….
Constructing the Tisserand Plot
The previous post in this series introduced the Tisserand Graph. The key points to remember were as follows:
1. By throwing information about 'phasing' and orientation, one can represent elliptical orbits as a single point on a two-dimensional graph - with the orbital apoapsis on the x-axis; and the orbital periapsis on the y-axis.
2. In a ballistic gravitational encounter with a body - i.e., a planet or a moon, a spacecraft's orbit will move along lines of constant hyperbolic excess velocity. But in any one encounter, there is a maximum extent to which an orbit can move along that line (because one assumes that nose-diving through the surface of the encountered planet or moon is not a feasible manoeuvre).
3. In order to have a series of encounters with the same moon, the orbit after the gravitational encounter needs to be close to 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.
To construct the Tisserand Plot for the EGA manoeuvre we need to do three things. First, we need to know how to construct lines of 'constant hyperbolic excess velocity'. Second, we need to know how to construct lines corresponding to resonant orbits - in particular, we need to construct the 2:1 resonance line in which the spacecraft executes one orbit of the Sun while the Earth executes two orbits around the Sun. Third, we need to know what manoeuvre we need to perform (at aphelion) to move between two different lines of constant hyperbolic excess velocity so that the spacecraft receives the required energy 'kick' during its gravity assist encounter with Earth.
In using the Tisserand Plot, we make a number of simplifying assumptions:
1. Bodies providing gravity assists - e.g., the Earth in its motion around the Sun; or Europa in its orbit around Jupiter - move in circular orbits. Of course, in reality, orbits are elliptical but we simplify the maths enormously if we assume motion is circular. And for all bodies other than, say, Mercury (which has a rather large orbital eccentricity) this is a reasonable assumption that doesn't introduce major errors into the calculation.
2. All bodies in the Solar System move in the same orbital plane. Again, this is a idealisation of a more complex situation, but as with the assumption of uniform circular motion it provides a good starting point for analysis.
3. We will model gravitational encounters between the spacecraft and gravitating bodies using a 'linked conics' model. Most people who use Orbiter will have heard of the 'patched conics' model. The 'patched conics model patches various 2-body trajectories (circular, elliptical, parabolic or hyperbolic) at the so-called 'sphere of influence. In the linked conics model, the sphere of influence is assumed to have zero radius, hence an alternative name for the model: the 'zero radius sphere of influence patched conics' model. This is a bit of a mouthful, so the more succinct name for it is the 'linked conics' model. In this scheme, motion is nearly always elliptical (around the central gravitating body - e.g., the Sun) except when there is a gravitational encounter with some other body (e.g., the Earth) whereupon it is assumed that there is an instantaneous rotation of the velocity vector of the spacecraft about some angle sending the spacecraft on another elliptical orbit around the primary.
4. It is useful to work in dimensionless units. For example, if we wish to explore gravitational encounters with Earth, we work in units of distance and time for which the orbital radius of the Earth in its motion about the Sun is unity; and for which its orbital speed of Earth around the Sun is also unity. We do all of our calculations in these units and when we wish to convert back to more conventional units, we just need to remember to multiply all distances by the Earth's orbital radius (i.e., 1 AU); and to multiply all speeds by Earth's mean orbital speed (i.e., 29.78 km/s). Again, we introduce this little 'trick' to simplify the mathematical expressions as much as possible.
Calculating the lines of constant hyperbolic velocity
An important first step in using the Tisserand Plot is to learn how to construct lines of constant hyperbolic velocity. From the post "A very boring post", we have an expression for the hyperbolic excess velocity of a spacecraft encountering a planet or moon:
[MATH]v_{\infty} = v_{c}\,\sqrt{3-\frac{2\,r_{c}}{r_{a}+r_{p}}-2\,\sqrt{\frac{2\,r_{a}\,r_{p}}{r_{c}\left(r_{a}+r_{p}\right)}}} [/MATH]
where [MATH]v_c[/MATH] is the mean orbital speed of the gravitating body; [MATH]r_c[/MATH] is the mean orbital radius of the gravitating body; [MATH]r_a[/MATH] is the apoapsis of the spacecraft in its orbit around the primary; and [MATH]r_p[/MATH] is the periapsis of the spacecraft in its orbit around the primary. If we work in our dimensional units, we set [MATH]r_c \to 1[/MATH] and [MATH]v_c \to 1[/MATH] so that our expression simplifies to:
[MATH]v_{\infty} = \sqrt{3-\frac{2}{r_{a}+r_{p}}-2\,\sqrt{\frac{2\,r_{a}\,r_{p}}{\left(r_{a}+r_{p}\right)}}} [/MATH]
with [MATH]0 < r_p < 1[/MATH] and [MATH]r_a > 1[/MATH]. We can now invert this equation and solve for [MATH]r_a[/MATH] as a function of [MATH]r_p[/MATH] and [MATH]v_\infty[/MATH]. If we do this, we derive the following expression for [MATH]r_a[/MATH]:
[MATH]r_{a}(v_{\infty})=\frac{-r_{p}\,v_{\infty}^{4}+6\,r_{p}\,v_{\infty}^{2}+4\,r_{p}^{2}-9\,r_{p}-2\,v_{\infty}^{2}+6}{-8\,r_{p}+v_{\infty}^{4}-6\,v_{\infty}^{2}+9}+4\,\sqrt{\frac{r_{p}^{2}\,v_{\infty}^{2}+r_{p}^{4}-3\,r_{p}^{2}+2\,r_{p}}{\left(-8\,r_{p}+v_{\infty}^{4}-6\,v_{\infty}^{2}+9\right){}^{2}}}[/MATH]
such that [MATH]r_{p,min} \le r_p \le r_{p,max}[/MATH] and [MATH]0 \le v_\infty < \sqrt{3}[/MATH] and with
[MATH]\begin{array}{cc} r_{p,min}(v_{\infty})=\{ & \begin{array}{cc} \frac{-v_{\infty}^{2}+2v_{\infty}-1}{v_{\infty}^{2}-2v_{\infty}-1} & 0\leq v_{\infty}\leq1\\ 0 & 1<v_{\infty}\le\sqrt{3} \end{array}\end{array}[/MATH]
and
[MATH]r_{p,max}(v_{\infty})=\begin{array}{cc} \{ & \begin{array}{cc} 1 & 0\leq v_{\infty}\leq\sqrt{2}-1\\ \frac{1}{8}\left(v_{\infty}^{4}-6v_{\infty}^{2}+9\right) & \sqrt{2}-1<v_{\infty}\leq\sqrt{3} \end{array}\end{array}[/MATH]
Now, if you are thinking that this is a horrible mess of an expression, you would be right. To help those that might be intimidated by this algebraic mess, I've written a small 'user-defined function' in an excel spreadsheet which can be found as a zipped file at the bottom of this post. (Note again that in that spreadsheet calculations are carried out in dimensionless units as defined above.)
With this expression, and for any given value of [MATH]v_\infty[/MATH] (dimensionless units), we can calculate these curves of constant hyperbolic velocity. This calculation is carried out in the 'zipped' spreadsheet at the end of this post - and a few of these curves are plotted in the graph:
Here, I have chosen a few values for the hyperbolic excess velocity ([MATH]v_\infty[/MATH]) and plotted the corresponding curves for those values. The values of ([MATH]v_\infty[/MATH] chosen are 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30 and so on. This choice is somewhat arbitrary but, taken together, they give an idea of how these curves look.
[Aside: what do we mean by a hyperbolic excess velocity of 0.05 in dimensionless units? Suppose, for example, we wish to consider a gravitational encounter of a spacecraft with the Earth. In more normal units, the mean orbital speed of the Earth is around 29.78 km/s. So, in this case, a dimensionless speed of 0.05 corresponds to a hyperbolic excess velocity of 29.78 x 0.05 km/s = 1.489 km/s. If instead, we were considering a gravitational encounter with, say, Callisto, which has a mean orbital speed of 8.204 km/s around Jupiter, the same value of 0.05 corresponds to a hyperbolic excess velocity of 0.410 km/s.]
And as a reminder of the significance of these lines of constant hyperbolic excess velocity, suppose that the spacecraft starts in an orbit around some body (e.g., the Sun). On the Tisserand Plot, this orbit can be represented as a point that lies on one of these lines of constant hyperbolic velocity (as calculated for some other body - e.g., the Earth, moving in a circular orbit around the Sun). So long as the spacecraft doesn't encounter the second gravitating body (e.g., the Earth), the spacecraft will remain at exactly the same point on the Tisserand Plot forever (i.e., it will be on the same unperturbed elliptical orbit). If, however, the spacecraft encounters the second gravitating body (e.g., the Earth), physics requires that after the encounter, the spacecraft will have moved to a new elliptical orbit around the primary (e.g., the Sun) which, on the Tisserand Plot, will be represented by another point lying on the same line of constant hyperbolic velocity. And so long as all subsequent encounters with the second body (e.g., the Earth) are ballistic - i.e., the spacecraft's propulsion system is not used - the spacecraft's orbit must forever lie on the same line of hyperbolic excess velocity.
How, then, does one move between lines of constant hyperbolic excess velocity? Well, the basic mechanism is to use the spacecraft's propulsion system - but not just arbitrarily. But we'll come onto that in a bi.
Constructing the lines of resonance
Before (finally) constructing the EGA manoeuvre, we need to be able to add one more set of lines to our Tisserand Plot. In addition to the lines of constant hyperbolic velocity, it is also useful to construct lines of resonance. These lines are groups of orbits that have a simple multiple orbital period of the orbital period of the secondary body (e.g., the Earth). For example, we could consider those orbits whose orbital period is twice the orbital period of the Earth (a 2:1 resonance); or we could choose those orbits that have an orbital period that three times the orbital period of the Earth (a 3:1 resonance). We could even consider those orbits whose orbital period is 1.5 times the Earth's orbital period (a 3:2 resonance). And so on.
And why do this? In most cases, we want to design orbits so that having left the secondary body (e.g., the Earth) on some elliptical orbit around the Sun, we want to make sure that we encounter the same body again after a specified number of orbits. To do this, we need to exploit these orbital resonances by making sure that our departure orbit lies (roughly) on one of these lines of resonance on the Tisserand Plot.
Fortunately, these lines are simple to construct - they are just straight lines with the equation:
[MATH]r_p = 2 n^{2/3}-r_a[/MATH]
where, again, we are working in dimensionless units and where [MATH]n[/MATH] is the resonance that we are interested in. For example, for the 2:1 resonance, we would set [MATH]n=2[/MATH]; and for the 3:2 resonance, we would set [MATH]n=1.5[/MATH]. And so on.
On the graph below, some of these lines of resonance have been drawn:
Design the EGA manoeuvre - 2:1 resonance
Now, finally, we are in a position to design the EGA sequence that exploits a 2:1 resonance. Consider the following graph:
This is, in fact, the EGA sequence (2:1 resonance) as depicted on a Tisserand Plot. For convenience, I've converted back to more standard units for which distances are measured in Astronomical Units (AU) and velocities are measured in 'km/s'. Let's go through the components of this graph one by one.
1. We start the EGA sequence at the point 'A'. Why there? Well, we know that we want to try and exploit the 2:1 resonance, so we know that after our initial hyperbolic escape from Earth, our Earth departure orbit needs to lie close to the 2:1 resonance. Now the usual way of escaping the Earth would be via an escape burn that resulted in our spacecraft leaving Earth in a 'prograde' orientation. This means that the perihelion of the resulting orbit around the Sun is at the Earth's orbital radius which means on the Tisserand Plot that [MATH]r_p = 1\,AU[/MATH]. These two facts place our initial departure orbit from Earth at the point 'A'. We immediately note from the Tisserand Plot that our initial orbital aphelion after escaping Earth must be around 2.2 AU.
2. Using the equations from "A very boring post", we can quickly calculate that this initial departure orbit must have a hyperbolic excess velocity of around 5.2 km/s. Now this isn't anywhere near enough speed to get us to Jupiter. Typically, for a direct transfer to Jupiter from Earth, we would need a hyperbolic excess velocity of around 8.9 km/s (or more). In this note, I'm going to assume that we are going to target a hyperbolic excess velocity of around 9.1 km/s (slightly higher than the minimum 8.9 km/s - just to give us a bit of leeway in our planning). The next thing we can do is draw the line of constant hyperbolic excess velocity corresponding to [MATH]v_\infty = 9.1\,km/s[/MATH] on the Tisserand Plot.
3. The point where we want to end up after the EGA manoeuvre is at position 'C' on the Tisserand Plot. What is this position? It is, in fact, another elliptical orbit, but one with a perihelion close to (but slightly less than) Earth's orbital radius of 1 AU and with an aphelion of around 5.2 AU, the mean orbital radius of Jupiter around the Sun. This point, then, represents a Hohmann-like transfer orbit from Earth to Jupiter.
4. The final thing that we need to consider is: how do we get from our initial starting orbit 'A', to our destination orbit 'C'? The most simple way of doing this is to move from the orbit 'A' to the orbit 'B'; and then from orbit 'B' to the orbit 'C'. What does moving from orbit 'A' to orbit 'B' entail? On the Tisserand Plot, this is vertical displacement of our orbit. The aphelion remains constant but the orbital perihelion is lowered from 1.00 AU to around 0.90 AU. From basic orbital mechanics, we know that the way to execute this manoeuvre is to perform a retrograde burn at aphelion. And how big a retrograde burn is this? Since we know the apoapis and periapsis of our 'before' and 'after' orbits, this is an easy calculation to carry out using the equations of "A very boring post" and it works out that the required retrograde burn is around 0.55 km/s. To be sure, this is a substantial burn but the effect of this burn is to move us from a line of constant hyperbolic velocity of 5.2 km/s to a line of constant hyperbolic velocity of 9.1 km/s. This means that when we next encounter the Earth, we will get a velocity 'kick' of 3.9 km/s - with just enough energy to get us all the way to Jupiter.
5. And what about the move from orbit 'B' to orbit 'C'. On the Tisserand Plot, any move along a line of constant hyperbolic excess velocity corresponds to a gravitational encounter with the secondary body (in this case, the Earth). Roughly two years after the spacecraft's initial departure from Earth and a deep space manoeuvre at aphelion, the spacecraft will again encounter Earth. On its approach to Earth, the spacecraft will have orbit 'B'. But the hyperbolic encounter with Earth will transform the orbit to orbit 'C'. Again, just using the equations of "A very boring post", one can calculate that the perigee altitude of the encounter with Earth needs to be around 500 km - just a few hundred kilometres above the top of the atmosphere. Not quite a grazing encounter with Earth, but close!
Design the EGA manoeuvre - 3:2 resonance
In a way, there is nothing particularly special about the 2:1 resonance. It would be quite feasible to consider exploiting another resonance - e.g., the 3:2 resonance. The Tisserand Plot for this manoeuvre is shown below:
The logic of the construction of this manoeuvre is exactly the same as before. We start at orbit 'A' leaving Earth with an initial aphelion of around 1.60 AU. Our departure hyperbolic excess velocity is around 3.60 km/s. At aphelion, we execute a retrograde burn - this time of around 1.10 km/s - lowering our perihelion to around 0.84 AU and in so doing transferring to an orbit that will have a hyperbolic excess velocity of 9.1 km/s when the spacecraft next encounters Earth.
All good, so far. However, if one goes through the exercise of calculating the perigee required at Earth encounter to move from orbit 'B' to orbit 'C' one quickly calculates that the closest approach to Earth needed is 2000 km below the surface of the Earth. Sadly, then, an EGA manoeuvre based on a 3:2 resonance is infeasible!
EGA manoeuvres - a score card and fact sheet.
Based just on the Tisserand Plot and the equations of "A very boring post", one can detail a considerable amount of information about the EGA manoeuvre. Here are some tables that have been put together quickly that summarise the information that can be extracted readily from this analysis:
and
Next steps
The Tisserand Plot is a largely graphical means for analysing a sequence of manoeuvres. Although it is based on a number of simplifying assumption and throws away a lot of detail regain the exact timing ('phasing') of some of the manoeuvres being examined, it clarifies what is going on while still providing the basis for some considerable quantitative information. This quantitive information can be used to eliminate some options and elucidate the general shape of those that are considered optimal.
However, the information from the Tisserand Plot isn't quite enough to form the basis for execution a mission. To do that, two further steps are required. Having sifted through the chaff to find a mission that is close to optimal, one needs to tun that mission design through a process that re-inserts information about the exact timing of the sequence of events envisaged in the mission design. Doing this will be the subject of the next post.
The final step is to run the resulting mission through a 'high fidelity' model. If one for working for NASA or the ESA a higher fidelity model would involve using a full n-body integrator to optimise the mission design. However, in Orbiter-land, it is generally sufficient to use TransX and IMFD to do the same thing. So at this point, we should be in a position to 'fly' the mission. Flying the EGA manoeuvre may also be the subject of a subsequent post.
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