In this post, I present some general musing about Belbruno & Topputo's "Earth-Mars transfer with ballistic capture" paper. These musings are a consequence of thinking about how to achieve capture by any of the Galilean moons 'on the cheap' with as low a delta-v budget as possible. Much of this thinking is captured in the thread entitle "Entering into orbit around Callisto using the Oberth Effect (revisited)". In that thread, the delta-v cost of entering a circular orbit around Europa from a Hohmann transfer from Earth to Jupiter (with excess hyperbolic velocity upon arrival at the Jovian system of 5.35 km/s) was steadily whittled away from something that started out close to 6.0 km/s and ended up costing an anticipated 1.8 km/s. Of that residual 1.8 km/s, roughly 1.0 km/s was needed to achieve capture by Jupiter by executing an "Oberth burn" at Ganymede periapsis; and roughly 0.6 km/s was needed for circularisation of orbit around Europa once Europa capture had been achieved - thereby leaving just 0.2 km/s with which to effect capture by Europa once capture by Jupiter had been achieved. The thread went on to muse about the possibility of using the stable manifolds of the Sun-Jupiter L1 point as a way of side-stepping this Jupiter capture delta-v cost by relying upon ballistic 3-body dynamics. And in turn, musings about the 3-body dynamics led me to thinking anew about the much publicised Belbruno & Topputo paper which examined a similar situation for transfer from Earth to Mars.
The Belbruno paper
In 2014, Belbruno & Topputo published a paper entitled "Earth-Mars transfer with ballistic capture". This was widely promoted as being a new discovery of a ballistic trajectory from Earth to Mars that would lead to capture by Mars and save up to 25% in delta-v. If one wants to read this paper, one can find it here:
http://arxiv.org/pdf/1410.8856v1.pdf
So, what is the ballistic mechanism that Belbruno & Topputo propose? The key feature of their trajectory design is utilisation of the Sun-Mars L1 point as a means of arriving at Mars with a zero hyperbolic excess velocity. To do this, they propose a trajectory that has three parts to it:
1. An Earth escape/Mars transfer orbit burn. For the most part, this looks like a standard Hohmann transfer burn - and it has a very similar delta-v cost to a more standard Hohmann transfer burn - i.e., around 3.0 km/s.
2. A Sun-Mars L1 stable manifold injection burn. The L1 point has what is known as a 'stable manifold'. This is a trajectory in space which, if you are on it, will automatically take you to the L1 libration point and terminate there with a very low hyperbolic excess velocity. The stable manifold injection burn transitions from the Earth escape trajectory to the L1 stable manifold. Belbruon & Topputo estimated the delta-v cost of this transition to be around 2.0 km/s - a not inconsiderable (and most decidedly non-ballistic) delta-v cost. At this point, the ship is effectively captured by Mars.
3. Upon the inevitable arrival at the Sun-Mars L1 libration point, the ship is given a slight 'nudge' inwards towards Mars. Like a ball rolling down from the top of a hill, the ship then falls in towards Mars in a near parabolic orbit. Normal orbital manoeuvres can then be used to manipulate the orbit the appropriate eccentricity and altitude.
Comparison with a Hohmann transfer
So, why is this trajectory design claimed to be better than the standard Hohmann transfer? Well, with a normal Hohmann transfer, although the Earth escape/Mars transfer orbit burn is essentially the same, the ship arrives at Mars with a hyperbolic excess velocity of around 2.7 km/s. So, as the argument goes, spending 2.0 km/s to ride the L1 stable manifold to capture is cheaper than shedding the 2.7 km/s of hyperbolic excess velocity upon arrival. In fact, at first blush, one might think that one is 25% better off.
However....
In most cases, one would normally want to shed the 2.7 km/s of hyperbolic excess velocity by executing an Oberth burn at a low altitude periapsis at Mars. With a burn of around 0.7 km/s, one could should all of the hyperbolic excess velocity leaving one in much the same state as if one had arrived at Mars on the L1 stable manifold - i.e., on the very edge of being captured by Mars. So, rather than spend the 2.0 km/s needed to ride the Sun-Mars L1 stable manifold, one could instead use 0.7 km/s at Mars periapsis and achieve the same state - a saving using standard Hohmann transfer techniques of 1.3 km/s.
In other words, if delta-v considerations alone are important, a standard Hohmann transfer would seem to be better.
Riding the stable manifold
Its worth taking a moment at this stage in thinking about the L1 stable manifold and what happens when you are on it. Now the stable manifold is just a trajectory that starts out looking like a standard 2-body elliptical orbit when you are a long way from Mars (with an aphelion and perihelion lower than Mars orbital radius). But as one approaches Mars along it, that elliptical orbit is magically converted into a roughly circular orbit that closely matches Mars' orbit. In other words, in proximity to Mars, there is an energy exchange between Mars and the ship that transfers energy to the ship leaving it with more energy than it otherwise would.
However, Mars is not a big planet. In fact, in relation to the Sun it is positively miniscule. So, the amount of energy that it can transfer to the ship via 3-body effects as the ship moves along the stable manifold is limited. As it turns out, although the aphelion and perihelion are lifted along the trajectory, they are not lifted by much. This means that the stable manifold never passes close by to Earth. In turn, this means that any ship wishing to ride this stable manifold has to travel a long way out towards Mars' orbit in order to get on it. And, in turn, this means a substantial delta-v burn to escape Earth just to get out to the stable manifold. And once there, there is another substantial burn (2.0 km/s) to adjust speed and direction to match those needed to be on the manifold.
Altogether, not very ballistic - and not very cheap. (But it does take a long time.)
So what is all the fuss about?
In the main, the Belbruno paper has been subject to a lot of hype. The paper itself does not claim the things that the press has claimed on its behalf. If one is free to select the date of travel to Mars, then one can always choose a date on which a standard Hohmann transfer to Mars will (in delta-v terms) outperform Belbruno's 'ballistic' trajectory.
However, one has to be very specific about the date of travel for the Hohmann transfer. On other dates, the Hohmann transfer is not a Hohmann transfer at all - and can become extremely expensive. On the other hand, the cost of getting out to the stable manifold doesn't changed very much. And so it offers a much broader set of dates on which it becomes feasible to fly to Mars. This alone makes it worth of some attention.
The Belbruno paper
In 2014, Belbruno & Topputo published a paper entitled "Earth-Mars transfer with ballistic capture". This was widely promoted as being a new discovery of a ballistic trajectory from Earth to Mars that would lead to capture by Mars and save up to 25% in delta-v. If one wants to read this paper, one can find it here:
http://arxiv.org/pdf/1410.8856v1.pdf
So, what is the ballistic mechanism that Belbruno & Topputo propose? The key feature of their trajectory design is utilisation of the Sun-Mars L1 point as a means of arriving at Mars with a zero hyperbolic excess velocity. To do this, they propose a trajectory that has three parts to it:
1. An Earth escape/Mars transfer orbit burn. For the most part, this looks like a standard Hohmann transfer burn - and it has a very similar delta-v cost to a more standard Hohmann transfer burn - i.e., around 3.0 km/s.
2. A Sun-Mars L1 stable manifold injection burn. The L1 point has what is known as a 'stable manifold'. This is a trajectory in space which, if you are on it, will automatically take you to the L1 libration point and terminate there with a very low hyperbolic excess velocity. The stable manifold injection burn transitions from the Earth escape trajectory to the L1 stable manifold. Belbruon & Topputo estimated the delta-v cost of this transition to be around 2.0 km/s - a not inconsiderable (and most decidedly non-ballistic) delta-v cost. At this point, the ship is effectively captured by Mars.
3. Upon the inevitable arrival at the Sun-Mars L1 libration point, the ship is given a slight 'nudge' inwards towards Mars. Like a ball rolling down from the top of a hill, the ship then falls in towards Mars in a near parabolic orbit. Normal orbital manoeuvres can then be used to manipulate the orbit the appropriate eccentricity and altitude.
Comparison with a Hohmann transfer
So, why is this trajectory design claimed to be better than the standard Hohmann transfer? Well, with a normal Hohmann transfer, although the Earth escape/Mars transfer orbit burn is essentially the same, the ship arrives at Mars with a hyperbolic excess velocity of around 2.7 km/s. So, as the argument goes, spending 2.0 km/s to ride the L1 stable manifold to capture is cheaper than shedding the 2.7 km/s of hyperbolic excess velocity upon arrival. In fact, at first blush, one might think that one is 25% better off.
However....
In most cases, one would normally want to shed the 2.7 km/s of hyperbolic excess velocity by executing an Oberth burn at a low altitude periapsis at Mars. With a burn of around 0.7 km/s, one could should all of the hyperbolic excess velocity leaving one in much the same state as if one had arrived at Mars on the L1 stable manifold - i.e., on the very edge of being captured by Mars. So, rather than spend the 2.0 km/s needed to ride the Sun-Mars L1 stable manifold, one could instead use 0.7 km/s at Mars periapsis and achieve the same state - a saving using standard Hohmann transfer techniques of 1.3 km/s.
In other words, if delta-v considerations alone are important, a standard Hohmann transfer would seem to be better.
Riding the stable manifold
Its worth taking a moment at this stage in thinking about the L1 stable manifold and what happens when you are on it. Now the stable manifold is just a trajectory that starts out looking like a standard 2-body elliptical orbit when you are a long way from Mars (with an aphelion and perihelion lower than Mars orbital radius). But as one approaches Mars along it, that elliptical orbit is magically converted into a roughly circular orbit that closely matches Mars' orbit. In other words, in proximity to Mars, there is an energy exchange between Mars and the ship that transfers energy to the ship leaving it with more energy than it otherwise would.
However, Mars is not a big planet. In fact, in relation to the Sun it is positively miniscule. So, the amount of energy that it can transfer to the ship via 3-body effects as the ship moves along the stable manifold is limited. As it turns out, although the aphelion and perihelion are lifted along the trajectory, they are not lifted by much. This means that the stable manifold never passes close by to Earth. In turn, this means that any ship wishing to ride this stable manifold has to travel a long way out towards Mars' orbit in order to get on it. And, in turn, this means a substantial delta-v burn to escape Earth just to get out to the stable manifold. And once there, there is another substantial burn (2.0 km/s) to adjust speed and direction to match those needed to be on the manifold.
Altogether, not very ballistic - and not very cheap. (But it does take a long time.)
So what is all the fuss about?
In the main, the Belbruno paper has been subject to a lot of hype. The paper itself does not claim the things that the press has claimed on its behalf. If one is free to select the date of travel to Mars, then one can always choose a date on which a standard Hohmann transfer to Mars will (in delta-v terms) outperform Belbruno's 'ballistic' trajectory.
However, one has to be very specific about the date of travel for the Hohmann transfer. On other dates, the Hohmann transfer is not a Hohmann transfer at all - and can become extremely expensive. On the other hand, the cost of getting out to the stable manifold doesn't changed very much. And so it offers a much broader set of dates on which it becomes feasible to fly to Mars. This alone makes it worth of some attention.
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