Entering into orbit around a Callisto using the Oberth Effect (revisited)

[Keithth G]

Hi, RGClark

Earlier you asked:

Could we dispense with the Ganymede gravity assist by going deeper into Jupiter's gravity well? Our objective is to wind up at Europa. Perhaps doing capture burn at Europa's distance then performing circularization burn could give lower total delta-v. I doubt it though. Otherwise they would have done it this way in the first place.

But perhaps we could go a deeper distance in to do the capture burn and then circularize at Europa's orbital distance. Would this result in a lower total delta-v?

I've calculated the total delta-V requirement for doing a capture (Oberth) burn around Jupiter at a radius [MATH]r_0[/MATH], followed by a circular orbit insertion burn around Europa with [MATH]71000\,km \leq r_0 \leq 700000 \,km[/MATH]. This calculation covers the range from the surface of Jupiter out to Europa's orbital radius. I've assumed a hyperbolic excess velocity of around 6.5 km/s and no gravity assists. The resulting delta-V requirement is plotted as the following graph.



The graph shows that the total delta-V requirement to achieve a circular orbit around Europa without gravity assists is between 5.1 km/s and 5.6 km/s - depending on the orbital radius at which the capture burn is executed. It is also easy to see that the best place to do the capture burn is as close to the Jovian surface as possible. With a grazing encounter of the Jovian surface, and without gravity assists, the minimum delta-V requirement is 5.1 km/s.

This minimum of 5.1 km/s is, of course, much higher than the approximately 2.0 km/s that your Europa mission planners budgeted for in their calculations using gravity assists. Roughly speaking, then, one is 3.1 km/s better off with gravity assists than without.

Clearly, this is why the mission planners relied so heavily on gravity assists: without gravity assists, the delta-V requirements of the mission would have been completely impracticable given current budget and technology constraints.

As dgatsoulis has just demonstrated here, http://www.orbiter-forum.com/showthread.php?t=35916, gravity assists are an effective and practicable method for reducing total mission delta-V requirements.

 

[RGClark]

Hi, RGClark

Earlier you asked:



I've calculated the total delta-V requirement for doing a capture (Oberth) burn around Jupiter at a radius [MATH]r_0[/MATH], followed by a circular orbit insertion burn around Europa with [MATH]71000\,km \leq r_0 \leq 700000 \,km[/MATH]. This calculation covers the range from the surface of Jupiter out to Europa's orbital radius. I've assumed a hyperbolic excess velocity of around 6.5 km/s and no gravity assists. The resulting delta-V requirement is plotted as the following graph.



The graph shows that the total delta-V requirement to achieve a circular orbit around Europa without gravity assists is between 5.1 km/s and 5.6 km/s - depending on the orbital radius at which the capture burn is executed. It is also easy to see that the best place to do the capture burn is as close to the Jovian surface as possible. With a grazing encounter of the Jovian surface, and without gravity assists, the minimum delta-V requirement is 5.1 km/s.

This minimum of 5.1 km/s is, of course, much higher than the approximately 2.0 km/s that your Europa mission planners budgeted for in their calculations using gravity assists. Roughly speaking, then, one is 3.1 km/s better off with gravity assists than without.

Clearly, this is why the mission planners relied so heavily on gravity assists: without gravity assists, the delta-V requirements of the mission would have been completely impracticable given current budget and technology constraints.

As dgatsoulis has just demonstrated here, http://www.orbiter-forum.com/showthread.php?t=35916, gravity assists are an effective and practicable method for reducing total mission delta-V requirements.

It's not likely to make a major difference but Dgatsoulis assumes a lower hyp. excess of 5.35 km/s:

Since the journey is a VEGA trajectory, meaning the spacecraft will encounter Earth at the second sling after Venus, I will base this calculation on a theoretical perfect Hohmann transfer from Earth to Jupiter, arriving exactly at Jupiter's aphelion to minimize the encounter velocity. For this calculation I will consider Jupiter and Earth in coplanar orbits, (as if we are picking it up after the broken plane maneuver).
At its apoapsis (TrL=194.2° wrt ecliptic) Jupiter has a velocity of 12.44 km/s and is at a distance of 816.3e6 km from the center of the Sun.
When Earth is exactly on the opposite side of the Sun (TrL=194.2°-180°=14.2°), it has a velocity of 29.67 km//s and is at a distance of 149.5e6 km.
A Hohmann trajectory around the Sun with these characteristics (ApD = 816.3e6 km, PeD 149.5e6 km, SMa = 482.9e6 km) has an Aphelion velocity of 7.09 km/s
So the hyperbolic excess velocity at the arrival is 12.44-7.09=5.35 km/s

What would the required delta-v be then?

Bob Clark

 

[Keithth G]

RGClark

You are correct in having guessed that the change in hyperbolic excess velocity does not change the results significantly. Here are the updated results using an updated value of the hyperbolic excess velocity of 5.35 km/s.



The 'Oberth' capture burn is very efficient at shedding process. It doesn't take much of a change in the delta-V of that burn to shed an extra 1.15 km/s or so of hyperbolic excess velocity.

If we just focus on the 'optimal' result of executing the Oberth burn at a grazing altitude above Jupiter, we see that the total delta-V requirement is just 4.994 km/s. This is the sum of contributions from three burns:

1. The 'Oberth' burn at first Jovian periapsis - 0.324 km/s (quite small, given that this burn is sufficient to her 5.35 km/s of hyperbolic excess velocity)

2. A second burn at the Jovian apoapsis of 0.345 km/s (needed to raise orbital periapsis up to Europa's orbital radius)

3. A Europa orbit insertion and circularisation burn of 4.325 km/s.

It is clear that the third of these burns is easily the most substantial. The reason is that after the first two burns, the craft's orbit is still highly eccentric. This means that its encounter velocity at Europa is very high (5.436 km/s - if I have done my calculations correctly. For the avoidance of doubt, this is the encounter speed when the craft is close to but still outside Europa's gravitational well).

With the standard Delta Glider, executing a 4.325 km/s orbit insertion and circularisation presents no major problem. In fact, if the fuel is available, it will take just 153s to execute over which time. At a Europa periapsis velocity of around 6.0 km/s, this means that, relative to Europa, the Delta Glider will have moved along its trajectory close to 900 km.

For more realistic spacecraft, however, a 4.325 km/s burn is very substantial. I am guessing a little here, but I suspect that the power of the Delta Glider's RCS system is roughly comparable to what a mission probe might be able to achieve. If so, then the estimated burn time increases to around 10,000 seconds - or close to three hours - over which time the craft will have moved upwards of 60,000 km along its trajectory. Given that Europa's diameter is around 3,000 km, the craft will take around 20 Europa's diameter to slow down. Clearly, this is going to cause a few problems.

So, based on current technology, it seems that gravity de-assists are going to be an essential part of any realistic plan to enter into orbit around the Jovian moons - both to reduce to the mission delta-V requirements and to reduce the size of burns to something that current engine technology can manage more easily.

 

[RGClark]

...
The 'Oberth' capture burn is very efficient at shedding process. It doesn't take much of a change in the delta-V of that burn to shed an extra 1.15 km/s or so of hyperbolic excess velocity.
If we just focus on the 'optimal' result of executing the Oberth burn at a grazing altitude above Jupiter, we see that the total delta-V requirement is just 4.994 km/s. This is the sum of contributions from three burns:
1. The 'Oberth' burn at first Jovian periapsis - 0.324 km/s (quite small, given that this burn is sufficient to shed 5.35 km/s of hyperbolic excess velocity)
2. A second burn at the Jovian apoapsis of 0.345 km/s (needed to raise orbital periapsis up to Europa's orbital radius)
3. A Europa orbit insertion and circularisation burn of 4.325 km/s.
It is clear that the third of these burns is easily the most substantial. The reason is that after the first two burns, the craft's orbit is still highly eccentric. This means that its encounter velocity at Europa is very high (5.436 km/s - if I have done my calculations correctly. For the avoidance of doubt, this is the encounter speed when the craft is close to but still outside Europa's gravitational well).

Thanks for the calculation. Those first two burns will give an elliptical orbit with periapsis at Europa's orbital distance. Then we can arrange this so that it swings repeatedly by very close to Europa. Eventually it will be captured by Europa. This will greatly reduce the delta-v required.

Bob Clark

 

[RGClark]

Thanks for the calculation. Those first two burns will give an elliptical orbit with periapsis at Europa's orbital distance. Then we can arrange this so that it swings repeatedly by very close to Europa. Eventually it will be captured by Europa. This will greatly reduce the delta-v required.

Bob Clark

I understand Orbiter can do three-body simulations. Then this should be something Orbiter can simulate. Assuming the spacecraft swings very close to Europa on that first elliptical orbit, how long will it it take to be captured by Europa?

Bob Clark

 

[Keithth G]

RGClark

Orbiter most certainly can do three-body simulations. Actually, Orbiter's core n-body integration engine is surprisingly accurate and there is no reason to believe that it could model close encounters with Europa (or any other body for that matter) with some considerable fidelity. If one were to develop the right planning tools, one could:

1. develop a highly accurate flight plan for repeated ballistic encounters with, say, Europa;

2. implement that within Orbiter with very low mid course correction costs; and

3. be confident that that flight plan would compare very favourably with that produced by, say, JPL's more rigorous flight planning software.

However, Orbiter currently lacks the right planning tools for this job. Neither TransX nor IMFD are quite up to this task - although 'dgatsoulis' has recently jerry-rigged a workable (but not altogether highly accurate) method using IMFD's Map program. Presently, and jus for fun, I am working on some more accurate planning software that can develop these repeated encounter flight plans with considerably greater fidelity. But it will take a little more time before I have a working prototype to showcase.

In the interim, the best that I can offer to answer your question is some 'back of the envelope' patched conic approximations that provide some approximate solutions to the repeated ballistic encounter problem. I will need to spend a bit more time working through those patched conic calculations before I can answer your question.

 

[RGClark]

In doing a web search I found this is asking the same thing as "ballistic capture" or "low energy transfers". These have already been done for actual spacecraft at the Moon and recent research shows it is possible for high altitude Mars orbits.

Bob Clark

 

[RGClark]

The GravitySimulator page apparently can do such simulations. See the link here:

Re: Gravitational capture by Europa?
Reply #2 - 08/05/15 at 14:12:39
http://www.orbitsimulator.com/cgi-bin/yabb/YaBB.pl?num=1438803846/2#2

To make the cases that make it into orbit around Europa easier to see, in the "Camera A" frame change the drop down menu that has "Europa" as default to "Stationary".
You might also want to check the "Labels" option to see which moons are which and be able to follow each of separate cases in their trajectories.


Bob Clark

 

[Keithth G]

RGClark

Interesting stuff. Although I suspect that if one works hard enough, one could find a truly ballistic trajectory that leads to Europa capture. And, yes, this would require making use of a series of Lagrange points, i.e., requiring considerable familiarity with '3-body physics'.

However, the purpose of this thread was to explore the utility of a series of straightforward gravity de-assists in reducing mission delta-V requirements - as used by, say, the Messenger mission to Mercury. Although not truly ballistic in that, at some point, one or more burns will be needed to achieve capture, the total amount of fuel consumed can be reduced very significantly.

These more traditional gravity de-assists can be understood in terms of a series of 2-body encounters (aka 'a patched conic' approximation). No knowledge of 3-body physics is required. Given the state of mission planning tools with Orbiter 2010, avoiding 3-body physics is probably advisable at this stage. A longer term goal, perhaps, may be to examine 3-body physics with Orbiter. But one has to learn how to walk before one can run.

 

[Keithth G]

RGClark

Earlier, you asked:

Assuming the spacecraft swings very close to Europa on that first elliptical orbit, how long will it take to be captured by Europa?

The short answer to this question is that if:

1. you approach the Jupiter system with a hyperbolic excess velocity of, say, 5.35 km/s; and

2. you execute a 324 m/s retrograde burn at Jupiter periapsis; and then a further 345 m/s prograde burn at Jupiter apoapsis (as per earlier posts); and

3. you rely solely upon ballistic encounters with Europa to achieve capture

then it will take a very, very long time (approximately a few decades) to reduce your speed enough to be captured by Europa. To be captured more quickly, one has to augment the sequence of gravity de-assists with those around Callisto and/or Ganymede as well.


Why?

The gravity de-assists work by transferring some of the spacecraft's kinetic energy during an encounter with a moon. The efficiency of that energy transfer is very much dependent on the approach speed of the spacecraft with respect to the moon. At high approach speeds, very little kinetic energy is transferred whereas at low approach speeds very much more kinetic energy is transferred.

Below is a plot of the change in orbital speed (with respect to Jupiter) of a spacecraft following a close encounter with Europa. This is plotted as function of the spacecraft approach speed. If the spacecraft's approach speed (relative to Europa) is, say, 5,500 m/s then the change in the spacecraft's orbital speed relative to Jupiter drops by just 30 m/s. If however, the approach speed is 1,000 m/s, the change in the speed of the craft is around 800 m/s.



Unfortunately, Europa lies a long way inside Jupiter's gravity well. After the first two burns - the 324 m/s and then the 345 m/s, the resulting Europa approach speed is around 5,500 m/s.Consequently, on the next encounter with Europa (some 250 days later), the approach speed has reduced to 5,470 m/s. And then on the second encounter with Europa, the orbital speed is reduced a further 30 m/s so that on the third encounter with Europe (another 250 days later), the approach speed has fallen to 5,440 m/s. And so, on. Some decades later, the encounter speed has lowered to the point where orbital speed can start to be shed rapidly.

A better approach
A better way of approaching Europa (using gravity de-assists) is, first, to use Callisto to brake the spacecraft. Callisto approach speeds are closer to 3,000 m/s - so that, typically, that approach speed will be reduced by 200 m/s on the first encounter; 300 m/s on the second encounter; 500 m/s on the third encounter and then the bulk of the rest in the fourth encounter.



Once one has completed this sequence with Callisto, one will the lower one's orbit to commence the process again with Ganymede.

And, finally, when sufficient orbital energy has been shed via encounters with Ganymede, the orbit will be lowered to Europa where after a few more encounters (this time with Europa), capture by Europa becomes possible.

A much faster strategy
Although conceptually more complicated, this sequence of gravity de-assists involving Callisto, Ganymede and the Europa is a much faster way of entering into orbit around Europa using ballistic (or near ballistic) encounters.

In fact, whether by design or accident, this is the strategy used to good effect by 'dgatsoulis' in his recently posted solution to the "Europa Challenge". Although dgatsoulis' solution may not have been entirely ballistic, he managed orbit insertion using gravity de-assist in under one year - a considerable improvement over the projected 'few decades' estimated for the 'Europa only' gravity de-assist sequence.

 

[dgatsoulis]

In fact, whether by design or accident, this is the strategy used to good effect by 'dgatsoulis' in his recently posted solution to the "Europa Challenge".

Not by (my) design nor by accident. The initial seeds were given in this and the "Callisto Challenge" threads.
In the Europa Challenge, I just wanted to see if I could use my newly found understanding of the Jovian system, to"climb down the ladder".

The Jovian system is something that always fascinated me, but in Orbiter, there were few things that I could do with it.
TransX's patched conics and (mostly) my one dimensional approach to figuring out the delta-v requirement for a mission was hindering me. Especially when I was using many linear calculations in parallel, trying to figure out the fuel from start to finish, just by hand.
In some of my best sessions in Orbiter, I never even left the launch-pad.
(Now bare in mind that my maths is at high-school level, plus whatever I've picked up from Orbiter).

This thread was a real eye-opener, taking things to the next level.

Regarding a "European" mission, I'd gladly go for Europa head on, but the velocity will always be too high and Europa's "stopping power" will always be too low.

Although dgatsoulis' solution may not have been entirely ballistic, he managed orbit insertion using gravity de-assist in under one year - a considerable improvement over the projected 'few decades' estimated for the 'Europa only' gravity de-assist sequence.

Not entirely ballistic for sure. I did have a -not so small- 2000 m/s delta-v budget, which was after the JOI and Callisto encounter burns. Not very realistic.

2000 m/s could be realistic for an "all included" mission; the launcher's burn gets you to Jupiter -whatever the trajectory, as long as it has a low encounter velocity (between 5.5 and 6.5 km/s) - everything else is up to the spacecraft.

I am more than willing to try different strategies.
The 3 burns solution has one more elephant in the room. Jupiter's radiation belts. The setup in the Callisto Challenge will not work.

An a la Juno approach will work, going in an almost polar trajectory, otherwise kiss you spaceraft's electronics goodbye. This means a very hefty plane alignment burn after the JOI.

So we go in almost polar, get captured,aim for an outer moon and work our way down.
Callisto could work. I don't really know for sure with such a low budget, have to try it.

How about we avoid the bulk of Jupiter's radiation?
Ganymede is the best bet for a direct approach. At 3 times Europa's mass (but 2/3 of the density), you get a lot more stopping power.
A tangential flyby, accompanied by a JOI burn somewhere near periganymedion (I always wanted to use that word) and a sequence of flybies.
After that it's a simple matter of "climbing down the ladder" again. This time, just one step.

My Orbiter time is very limited these days, but I will try to set up a couple of realistic approaches to Jupiter. One polar with a perijove just above the clouds and one equatorial, with the same excess velocity, aimed for the edge of Ganymede's orbit (Ganymede being there of course).

Let's see which one does best.

 

[RGClark]

...
Not entirely ballistic for sure. I did have a -not so small- 2000 m/s delta-v budget, which was after the JOI and Callisto encounter burns. Not very realistic.
2000 m/s could be realistic for an "all included" mission; the launcher's burn gets you to Jupiter -whatever the trajectory, as long as it has a low encounter velocity (between 5.5 and 6.5 km/s) - everything else is up to the spacecraft.
I am more than willing to try different strategies.
The 3 burns solution has one more elephant in the room. Jupiter's radiation belts. The setup in the Callisto Challenge will not work.
An a la Juno approach will work, going in an almost polar trajectory, otherwise kiss you spaceraft's electronics goodbye. This means a very hefty plane alignment burn after the JOI.
...

How much would be the total delta-v, after arriving at the Jovian system, for the Europa orbital capture including the JOI and the Callisto/Ganymede burns?


Bob Clark

 

[Keithth G]

How about we avoid the bulk of Jupiter's radiation?
Ganymede is the best bet for a direct approach. At 3 times Europa's mass (but 2/3 of the density), you get a lot more stopping power.
A tangential flyby, accompanied by a JOI burn somewhere near periganymedion (I always wanted to use that word) and a sequence of flybies.
After that it's a simple matter of "climbing down the ladder" again. This time, just one step.

I've calculated the cost of using Ganymede/Callisto periapsis to achieve JOI. Using the same assumptions of a Jovian arrival speed of 5.35 km/s and the same target apojove (25 million km) as in previous posts, I calculate that:

1. At periganymedion (good word!), one needs a retrograde burn of 860 m/s. After this burn, one can use ballistic encounters to climb down the ladder to Europa, starting with Ganymede.

2. At pericallistion, one needs a retrograde burn of 1,127 m/s. Again, one uses ballistic encounters to climb down the ladder to Europa, but this time starting from Callisto - a process that may actually be faster than starting from Ganymede because of the lower encounter velocities.

In comparison, a "Callisto Challenge" style JOI requires a 324 m/s retrograde burn at perijove; and then a 472 m/s prograde burn at apojove - a total delta-v requirement of 796 m/s. Consequently, executing 'Approach 1' above is just 64 m/s more costly than the direct perijove Oberth burn.

Given the radiation issues, it is pretty clear I think that executing the JOI burn at Ganymede is the best option. And given that the there is only a spread of 64 m/s between that and the "Callisto Challenge" strategy , I think this also rules out a polar approach too. But I will be interested to see what your simulations show.


So, how much does it (should it) cost to get into a circular orbit around Europa?

Well, there is the 860 m/s JOI burn at Ganymede. And after one has used ballistic encounters to step down to Europa's orbit (which should require essentially zero delta-v), the Europa orbit capture and circularisation procedure will cost at least a further 839 m/s, the minimum cost is going to be 1,700 m/s. Depending on the cost of plane changes, and how much energy one sheds through ballistic encounters prior to orbit insertion, I would say that the total mission delta-v costs should come in just under 2,000 m/s.

If one isn't interested in circularising one's orbit around Europa, then mission delta-v costs would fall to around 1,200 m/s.

P.S. I'll go through in detail how the JOI delta-v costs were calculated in an ensuing post. I may easily have made a mistake and it would be useful if others could review.

 

[RGClark]

...So, how much does it (should it) cost to get into a circular orbit around Europa?
Well, there is the 860 m/s JOI burn at Ganymede. And after one has used ballistic encounters to step down to Europa's orbit (which should require essentially zero delta-v), the Europa orbit capture and circularisation procedure will cost at least a further 839 m/s, the minimum cost is going to be 1,700 m/s. Depending on the cost of plane changes, and how much energy one sheds through ballistic encounters prior to orbit insertion, I would say that the total mission delta-v costs should come in at just under 2,000 m/s.

If one isn't interested in circularising one's orbit around Europa - just in Europa capture - then mission delta-v costs would fall to around 1,200 m/s.

P.S. I'll go through in detail how the JOI delta-v costs were calculated in an ensuing post. I may easily have made a mistake and it would be useful if others could review.

---------- Post added at 12:46 PM ---------- Previous post was at 06:57 AM ----------

[Ooops. While writing this post, I noticed an error in my earlier calculations. This post presents the corrected values. Whereas earlier, I claimed that 860 m/s at Ganymede periapsis would be sufficient to put the ship into an elliptical orbit around Jupiter with an apojove of 25 million km, I now calculate 956.8 m/s is required - i.e., am additional 96.8 m/s.].

I'm trying to get a comparison to the delta-v needed in the report I cited, as indicated in the graphic:

If you add up the delta-v's in this graphic from "Broken-Plane" to "Europa Insertion", you get 1,990 m/s. If I'm reading this correctly, this is to take it to an elliptical orbit about Europa.

In that case it would be most appropriate to compare it to your 1,200 m/s + 96.8 m/s = 1,296.8 m/s it takes to do Europa capture, without circularizing.

Then your estimates would be less than that suggested by the report I cited, and the total delta-v to go all the way to landing would still be in the doable range. In fact, it should still be doable if the total delta-v up to and including circularization is in the 2,000 m/s range.


Bob Clark

 

[Keithth G]

RGClark

I would interpret the numbers slightly differently. I think that the "Broken Plane" cost of 200 m/s is a deep space manoeuvre that takes place along the Hohmann transfer from Earth to Jupiter needed to align the craft with Jupiter's orbital plane. In my calculations, I've assumed that the craft is already aligned with Jupiter's orbital plane. To compare apples with apples, I think you need to exclude the "Broken Plane" cost.

Let's look, then, at the other costs more closely:

JOI
In my calculations, I assume that upon arrival in the Jovian system, the spacecraft executes one retrograde burn at the periapsis of the slingshot encounter with Ganymede. My amended estimate was 957 m/s to achieve Jovian capture and to put the craft in a elliptical orbit with an apojove of 25 million km.

In the table that you have provided, the orbital insertion cost appears to broken into two parts: a first 710 m/s retrograde burn at periapsis of the slingshot encounter with Ganymede and then a second burn at some other point in the orbit of 270 m/s. This is a total of 980 m/s - not very different from my estimate of 957 m/s. In fact, the difference is small enough - just 23 m/s to say that the two are essentially equivalent.

Why do they split this burn into two parts? Frankly, I don't now - but I calculate that the minimum burn needed at Ganymede to achieve capture by Jupiter is 700 m/s. My guess is that the first burn ensures capture. And there is a subsequent second burn to save the mission some time. Also, if I put into my calculations a 980 m/s, then the apoapsis of the ensuing orbit drops to 22.9 million km - also with a substantial time saving of 100 days.

In addition, to this they allow 50 m/s to accommodate unforeseen orbital perturbations. I've made no allowance for this, and again, to have an apples-for-apples comparison, I should add 50 m/s to my estimates.


Ballistic manoeuvres to drop down to Europa
After JOI, what follows in their mission design is a sequence of ballistic encounters, first with Ganymede and then with Europa to achieve a Europa Transfer Orbit (ETO). Similarly, I assume a sequence of ballistic encounters.

Here, then, we agree.


Final Europa orbit capture and circularisation
The table quotes a combined figure of 801 m/s to achieve low orbit around Europa. This allows for a series of corrections to line up finally with Europa prior to orbit insertion and then the orbit insertion to achieve low orbit.

Earlier, I claimed that the minimum needed to do this was 839 m/s. I must have been tired last night because I've recalculated it again this morning, and the actual (minimum) figure is 575 m/s. This is the cost of entering a circular orbit at Europa's surface starting from a stationary position at infinity.

Although I haven't done the calculations in any detail, my guess was that at Europa orbital insertion one would have an excess hyperbolic velocity (wrt to Europa) of greater than zero, one would need to spend somewhere between 200 m/s and 400 m/s more to achieve orbit insertion and circularisation. In total, then my guesstimate would be a total cost for Europa orbit capture and circularisation of between 775 m/s and 975 m/s. The tables value of 801 m/s sits more or less right in the middle of this guesstimate range. In any event, any difference here is likely to be small ~ 100 - 200 m/s.

The bottom line
Overall, the two sets of estimates of mission costs are essentially the same. If they differ, the it is only by a few hundred m/s at most.

There are two 'irreducible' costs involved: the first is the JOI burn(s) of around 975 m/s. To this figure, we should an extra 50 m/s to allow for assorted orbital perturbations - a total, then of 1,025 m/s. If we are to rely on a Hohmann transfer from Earth to Jupiter, then I see no way of avoiding this delta-v cost. And this cost accounts for half of the mission costs.

There is also an absolutely irreducible amount of 575 m/s needed to achieve low Europa orbit. Summing these two values means that we have an absolutely minimum mission requirement of 1,600 m/s.

On top of this, it is inevitable that we will need to spend a little more delta-V setting up the ETO and shedding any residual hyperbolic excess velocity (wrt Europa) during the final orbit insertion sequence. Their estimate is around 226 m/s to do this. I guessed somewhere between 200 m/s and 400 m/s. If I were to refine my calculations, I wouldn't be surprised if I ended up with a number that was towards the bottom end of the range.

Overall, there is no material difference in delta-v costs between my calculations and theirs. This is not surprising since the two mission designs are performing essentially the same sequence of manoeuvres.

 

[RGClark]

...
There are two 'irreducible' costs involved: the first is the JOI burn(s) of around 975 m/s. To this figure, we should an extra 50 m/s to allow for assorted orbital perturbations - a total, then of 1,025 m/s. If we are to rely on a Hohmann transfer from Earth to Jupiter, then I see no way of avoiding this delta-v cost. And this cost accounts for half of the mission costs.
There is also an absolutely irreducible amount of 575 m/s needed to achieve low Europa orbit. Summing these two values means that we have an absolutely minimum mission requirement of 1,600 m/s.
...

It might be possible to reduce the Jupiter orbital insertion delta-v as well by ballistic capture. Belbruno and Topputo figured out how to do this at Mars saving 25% off the fuel requirements, so it might also be possible at Jupiter:

A New Way to Reach Mars Safely, Anytime and on the Cheap.
Ballistic capture, a low-energy method that has coasted spacecraft into lunar orbit, could help humanity visit the Red Planet much more often.
By Adam Hadhazy | December 22, 2014

Belbruno worked out how to let the competing gravities of Earth, the sun and moon gently pull a spacecraft into a desired lunar orbit. All three bodies can be thought of as creating bowl-like depressions in spacetime. By lining up the trajectory of a spacecraft through those bowls, such that momentum slackens along the route, a spacecraft can just "roll" down at the end into the moon's small bowl, easing into orbit fuel-free. "It's a delicate dance," Belbruno says.
Unfortunately, pulling off a similar maneuver at Mars (or anywhere else) seemed impossible because the Red Planet's velocity is much higher than the Moon's. There appeared no way to get a spacecraft to slow down enough to glide into Mars' gravitational spacetime depression because the "bowl," not that deep to begin with, was itself a too-rapidly moving target. "I gave up on it," Belbruno says.
However, while recently consulting for the Boeing Corp., the major contractor for NASA's Space Launch System, which is intended to take humankind to Mars, Belbruno, Topputo and colleagues stumbled on an idea: Why not go with the flow near Mars? Sailing a spacecraft into an orbital path anywhere from a million to even tens of millions of kilometers ahead of the Red Planet would make it possible for Mars (and its spacetime bowl) to ease into the spacecraft's vicinity, thus subsequently letting the spacecraft be ballistically captured.
http://www.scientificamerican.com/article/a-new-way-to-reach-mars-safely-anytime-and-on-the-cheap/

Earth--Mars Transfers with Ballistic Capture.
Francesco Topputo, Edward Belbruno
(Submitted on 27 Oct 2014)
http://arxiv.org/abs/1410.8856

Bob Clark

 

[Keithth G]

RGClark

It might be possible to reduce the Jupiter orbital insertion delta-v as well by ballistic capture. Belbruno and Topputo figured out how to do this at Mars saving 25% off the fuel requirements, so it might also be possible at Jupiter:

Yes, I agree. But, as yet, I don't know how to do that.

---------- Post added 08-13-15 at 03:17 AM ---------- Previous post was 08-12-15 at 02:41 PM ----------

RGClark

Just as a quick addition to my earlier reply:

It occurs to me that the 'obvious' thing to do is to exploit the dynamics of the Sun-Jupiter L1 Lagrange point. The Sun/Jupiter pair is, perhaps, the best representation of the Elliptical Restricted Three Body Problem (ERTBP) that we have: perturbations from other bodies, e.g., its moons and Saturn, are relatively small. And since a lot of theoretical work has been done on the ERTBP, applying that to the Sun/Jupiter Lagrange points should be reasonably straightforward. In contrast, the dynamics of the Earth-Moon Lagrange points are heavily distorted - making life considerably harder (not impossible, just harder).

The Sun/Jupiter Lagrange point is interesting in this respect because of its 'inbound' stable manifolds and 'outbound' unstable manifolds. If one can 'hop on to' a stable manifold somewhere near Earth, then you automatically get a free ride all the way to L1. Now, L1 is unstable, so shortly after having arrived, you will be ejected again, along an unstable manifold. But you have some choice about your exit path, so long as you execute a (very small) burn to transition from an inbound stable manifold to an outbound stable manifold of your choosing. Now, if the outbound stable manifold intersects with either Ganymede or Callisto then you can another free carry all the way down to a grazing encounter with one of those moons. And from there, we now know how to step down the ladder to Europa using a sequence of gravity de-assists. In do doing, one completely side-steps the circa 1,000 m/s burn needed for JOI. This saving, may come, however at an increased cost on departure from Earth in setting up the transition to the inbound stable manifold.

At the moment, I haven't taken the time to map out the stable and unstable manifolds. I've done this analytically in the past for the Circular Restricted Three Body Problem (CRTBP) - where the Sun and Jupiter are assumed to move in circular paths. Extending that 'know how' to the ERTBP shouldn't be a problem. This together with a quality numerical integrator should be sufficient to do some useful work here.

 

[dgatsoulis]

I don't see how the Sun-Jupiter L1 point could help in this case. Any savings would be dwarfed by either the Oberth effect or the moon gravity de-assists methods.

Of course, I could be very wrong about this, but here's what my intuition and "common sense" tell me:
(Yes, it does occur to me that when one uses the "common sense" contradiction, one is probably wrong).

The main problem is the transfer trajectory. Too eccentric to be of much help.

With the exception of very few and rare trajectories that get to Jupiter via Mars, most of the time you have -in a best case scenario- an Earth-Jupiter Hohmann transfer, whatever method you choose to get there (multi-slings of Earth-Venus or just Earth with DSMs or even direct E-J). -Remember, we are looking to minimize the encounter velocity.

Even the rare Earth-Mars-Jupiter trajectories (once about every 40 years or so -next one coming up in 2031), only slightly improve the direct Hohmann transfer, as far as the encounter velocity goes.

The S-J L1 point orbits the Sun at a distance of about (very roughly rounded) 50 million kilometers (less) in the direction of the Sun, sitting on the Sun-Jupiter line and moving along its path with the same velocity as Jupiter.

An Earth to S-J L1 Hohmann trajectory has an encounter velocity with it, of roughly 0.5 km/s less than the encounter velocity with Jupiter itself.
But the L1 point has no "gravity" of its own.
So in order to enter the manifold you need a substantial burn, far more expensive than the other methods.

I am not saying that this is completely accurate (for example, at 50 million km away from Jupiter, you get a hefty pull from it), but still, an encounter velocity of ~5 km/s with the L1 point doesn't seem as a delta-v saving method.

Of course, I'd love to be shown wrong about this.

Just my :2cents:

 

[Keithth G]

What is absolutely true is that there is a set of trajectories that, if on that trajectory will have you terminate at a Lagrange 'point' as the inevitable dynamical consequence of being on that trajectory. If you are on one of these trajectories, i.e., on the stable manifold of the Lagrange point, you will get a free carry to that point.

Now the Lagrange point is really only a 'point' in rotating coordinates. In normal inertial space, the L1 'point' is a circular orbit around the Sun about 50 million km from Jupiter towards the Sun. A body at this location will orbit around the Sun with the same orbital period as Jupiter. A trajectory on the stable manifold automatically lifts you, then, from something that is not a circular orbit, and via a series of 3-body interactions with Jupiter will lift your orbital perihelion and circularise your orbit. Strange - but true.

Where does the energy come from to do this? Well, from Jupiter itself, of course. The ability of trajectories to carry out this 'lifting' process is one of the main reasons why there is so much interest in doing this. It would seem to remove the need to carry out a JOI burn at Jupiter - as per a Hohmann transfer - because the trajectory automatically converts an earlier non-circular orbit into a circular orbit that closely matches Jupiter's orbital speed.

I think the only way of exploring this is to map out stable manifold of the S-J L1 point, set this up in Orbiter and demonstrate that if you are on the manifold that you will indeed end up with a circular orbit with the same orbital period as Jupiter. The maths to do this isn't as simple as Kepler 2-body physics, unfortunately, but I'll try to find the time to do this.

 

[Keithth G]

As promised, attached is a scenario file which has a Delta Glider (roughly) on the stable manifold - starting at about 5 years before arrival at the stable manifold.

This scenario starts with the Delta Glider in an elliptical orbit around the sun with an eccentricity around 0.2 and with solar apoapsis considerably below Jupiter's orbital radius. The Scenario can be run forward for around 7 years (or more) and ends up with the Delta Glider being 'captured' by Jupiter. There is no need for the viewer to do anything: capture, in this instance, is inevitable and requires no intervention.

This scenario was set up up working out the linearised form of the equations of motion for the Circular Restricted Three Body Problem about the Sun-Jupiter L1 point. Having done this, the stable manifold close to L1 was identified. Then, taking a point close to L1 on this manifold, the trajectory was integrated back in time five years to identify an appropriate starting position for the attached scenario. The numerical integrator used was based on the 4th order explicit symplectic integrator detailed in an earlier thread in the "Maths & Physics" section of this forum.

Because, the calculations I have used assumed the mean circular motion of Jupiter around the Sun whereas Jupiter's motion is elliptical, the approach to the L1 'point' is not perfect - but good enough to show how the basic orbit lifting mechanism works.

I suggest that if you want to view the scenario, you load into Orbiter, make a cup of coffee, and watch the orbit trajectory unfold at 100,000X time acceleration. It will probably take up to 20 minutes or so for the scenario to unfold.