Banishing transfer windows

[Hotel26]

The following is intended as an invitation to a discussion, rather than a simple "Question":

In brief, what are your thoughts on the science for plotting any particular transfer, to start NOW (at one planetary body) and END (at another within the same system (Kerbol)) AS SOON AS reasonably^[1] possible?

[1] ignoring diminishing returns

Edited May 7, 2020 by Hotel26

[Hotel26]

I've only just started experimenting with some first ideas.

So, to give an illustration, in this Map View, the next transfer window is in 323d.  This plan seems to achieve an encounter with Eve in 369d.

I have 7.6 km/s dV available.  Not yet sure what the 3 maneuvers add up to in the budget^[1].

The "subway map" says the Eve transit is 168d.  Hmm.  Much longer in space but an earlier arrival by 122d.

[1] 7,600 - 448 (inclination) - 1,662 - 991 - 1,625 (capture) = 2,874 remaining

 

Edited May 8, 2020 by Hotel26

[purpleivan]

Back in the day (2014/15) I entered a few challenges that had the goal of "get to planet X in the shortest time" and as the saying goes, the short distance between two points is a straight line.

At least as straight as you can get it with finite fuel.

This was my transfer from Kerbin to Eve.

But to do that

1. You needed a huge launcher.

2. You're not coming back... not on the same vehicle anyway.

This was the first vehicle I lobbed Eve's way.

Ah yes, the wonderful (if unrealistic) days of a parachute being able to brake your descent to Eve while hitting at 20km/s.

First attempt made it there in 9 days.

Not content with that I thought "hmm... how about topping that stack off with a nice big ion drive section" (pre nerf days).

You tend to get places quckly at 36km/s

Such as to Eve in 5 days.

All of this completely impossible in the current game as the moment you touched Eve atmosphere at that kind of speed you'd vaporise. So you'd need to decelerate massively to survive entry to Eve, which means a much bigger vehicle making it to the planet, which in turn means much less velocity on the trip there.

However you can still get some pretty spectacular transit times with the current atmospherics and ion drive specs, such as Moho to Eeloo in 228 days.

The challenge allowed vehicles to be cheated to Moho, but the transfer to Eeeloo is still done the tried and tested way.

Start with a gigantic, massively staged rocket.

Point it across the system in something close to a straight line.

Wait eons for ions, in this case to do much of the braking.

Then touch down and checking your time (started on Y7 D278)

But again... Val'sdidn't make it home from this trip.

[theJesuit]

Playing with kerbalism this is a fair question as radiation and time spent cooped up in the cans is a factor in mission success, especially beyond Eve or Duna.

Basically I look at where I want to go and the practical time to get there by fastest way to leave time before the next transfer window home.  Then, if the slower time to get there leaves enough time to do all the visiting, science, or whatever mission objective I have, that's what I aim do.

The craft that arrives has to have enough delta vee for a fast as possible return trip usually by having refueled drop boosters.  I look for a round trip to Jool being about 6 years.

Peace.

[Hotel26]

On 5/8/2020 at 2:02 AM, purpleivan said:

This was my transfer from Kerbin to Eve.

 

Thank you, purpleivan, and may I remark:

1. that the professional quality of your photo-shoot is as exquisitely superb as usual!  (always a joy)
2. your effort and craft in retrieving images from the Archives is impressive and is appreciated, as always!
3. furthermore, your documented efforts in getting from A to B in a straight line are breathtakingly inspirational!

                                                                                                                             

I now wish to contribute the following to the discussion for everyone's perusal:

 

				Transfer Windows
		Moho	Eve	Kerbin	Duna	Dres	Jool	Eeloo
		102	262	426	801	2217	4845	7268
Moho	102		167	134	117	107	104	103
Eve	262	167		681	389	297	277	272
Kerbin	426	134	681		910	527	467	453
Duna	801	117	389	910		1254	960	900
Dres	2217	107	297	527	1254		4087	3190
Jool	4845	104	277	467	960	4087		14533
Eeloo	7268	103	272	453	900	3190	14533

 

First row and column list the orbital periods of the planets themselves.

The intersection of row and column gives the period of the respective Transfer Window (in days) between the origin and destination bodies.

I draw attention particularly to the quantities neighboring the blank diagonal.

This is something to think about!

 

Edited May 10, 2020 by Hotel26

[Hotel26]

So, here's an idea.  What if the majority of asteroids in the orbit of Moho are roughly in the ecliptic (unlike Moho itself (7 deg incline))...?  Making them quite a bit easier (than Moho) to get to...??

5 of them, equally-spaced in that orbit, would be just 20d apart.

Enquiring minds now wish to know.

(See also Deep Space Relay Networks)

Edited May 8, 2020 by Hotel26

[Hotel26]

So, to sketch out what I surmise so far:

1. when you are late for a transfer to an inner body, you begin by diving deeper than its orbital altitude
2. ditto to an outer body, you begin by ballooning out past its orbital altitude
3. I presume, but am not so sure, that when you are early for a transfer, you can initially balloon out from your own altitude for an inner destination, and inward for an outer destination

What I also am completely unsure about is when and whether to adjust the ejection angle to effectively burn (partially) Radial In and Out in some cases.  The difference seems to be that Prograde/Retrograde has the most profound effect on the other side of your orbit, whereas Radial In/Out makes an immediate difference closer to where you are.

This is where I would really like to hear ideas!

It may also be that the sum of community expertise here is that the solution is to "wing it" by intuition and fiddling with maneuver nodes?  In any case, I've spent 5 years wondering about the mechanics of this but have not seen anywhere in writing any kind of summation of the heuristics of plotting such a transfer -- and I feel there must be other kerboanuts in the same situation who would benefit from some sharing of knowledge on this subject.

Edited May 9, 2020 by Hotel26

[Laie]

On 5/7/2020 at 1:51 PM, Hotel26 said:

In brief, what are your thoughts on the science for plotting any particular transfer, to start NOW (at one planetary body) and END (at another within the same system (Kerbol)) AS SOON AS reasonably^[1] possible?

The word "reasonably" is doing a lot of work here.

Personally, I'm quite interested in alternate transfers, trading travel time against deltaV. However, I usually have a dV budget to start with, or a deadline I'd like to meet. I assume that this defines what is still "reasonable", and what isn't. Otherwise we're at @purpleivan's straight line.

Here's the last faster-than-usual transfer I've done. Duna-Kerbin on Y2D025 -- about 300 kerbin days behind the last opportunity, and 600 days ahead of the next one. It's for a vessel that pushes cargo to Duna on every standard transfer opportunity, and returns unburdened in the time between.

• 1460m/s to a sub-Eve PE
• 860m/s to Kerbin-Like AP and encounter
• not visible: ~15m/s plane change an final adjustments
• that's a total of 2340m/s to Kerbin
• 1100m/s to capture into 180km Kerbin parking orbit.

Because of reasons, aerobraking at Kerbin is out of the question. Factoring in capture at Kerbin, going through two transfer orbits is cheaper than a single direct transfer.

Edited May 9, 2020 by Laie

[Hotel26]

34 minutes ago, Laie said:

The word "reasonably" is doing a lot of work here

The objective I had in mind is not so much a Faster Transfer Time as an Earlier Arrival.  I think, in general, such trips are likely to spend LONGER time in space.

In general, I think torch flights are more applicable to interstellar and not of so much interest here.  For one thing, this strategy does not work well when the target is on the opposite side of the sun and probably not well at all when the target is far behind.

Your flight plan above was very reasonable.  Thank you, Laie.

Edited May 9, 2020 by Hotel26

[Laie]

4 hours ago, Hotel26 said:

The objective I had in mind is not so much a Fast Transfer Time as an Earlier Arrival.  I think, in general, such trips are likely to spend LONGER time in space.

...as in my case, where I want to have the tug back at Kerbin in time for the next ordinary Kerbin-Duna transfer.

For that, I simply lowered my PE around the Sun, then put a maneuver at the PE to bring AP down to Kerbin's altitude. It's not hard to find the right PE where things work out to a Kerbin encounter.

This method has two benefits: it's pretty easy to plan without any tools, and it minimizes dV at capture. The latter may be of no interest if you can count on aerocapture. In that case, just picking the desired arrival time from a porkchop plot may be the best choice.... well, let's say "the easiest".

My transfer returns me to Kerbin much sooner that would be strictly necessary, by over 300 days -- it stands to reason that there should be a solution that requires less propellant and is still on time. I wouldn't know how to systematically look for it, though.

Edited May 9, 2020 by Laie

[Hotel26]

1 hour ago, Laie said:

For that, I simply lowered my PE around the Sun, then put a maneuver at the PE to bring AP down to Kerbin's altitude. It's not hard to find the right PE where things work out to a Kerbin encounter

In my case, I lowered the PE below the target; there I lowered the AP below the target; then later raised it to the target.  I think I had much further to catch up, so rather than go very deep, I just spent time on the inside fast-track.

I think the hard limit is dV.  I'm sending light parcels by Express Post and am using a single Dawn engine in transit (after a very hefty initial boost into the initial transfer injection).  So I have pretty generous dV available and it's just a matter of how to use it.

Here's an interesting challenge.  Leave Duna for Kerbin with Kerbin exactly opposite.  I'm going to give this a shot.  (We ought to use a common vehicle with the same dV, perhaps?  May I propose SENTINEL, if you are interested at all?)

Edited May 9, 2020 by Hotel26

[Hotel26]

I gave this a very conservative try.  Recall, if you will, that the objective is TO depart NOW in order to ARRIVE EARLIER (than one would  by first waiting for the Transfer Window).  Anything earlier is a success (with the biggest penalty being a lot of time in space for any Kerbals riding along).

I used a SENTINEL vehicle mounted atop a Zephyr booster for this mission and the Zephyr made it all the way from Transfer Injection (2,448 m/s) to the final 67 m/s dV to be performed (a 2m ion burn to complete the Kerbin capture).

The Map View screen shot below was taken just after the first Course Adjustment, which was a 0.1dg re-inclination.  The green Commnet line indicates the position of SENTINEL (LHS) and Kerbin (RHS).

Four course adjustment maneuvers (2-5 in the list below) were performed during the voyage:

1. Transfer Injection (LDO): 2,448 m/s @277dg prograde, lowering Kerbolar Pe to 4,861 Mm
2. after 170d: AN:0.1dg 14.5 m/s [current SENTINEL position]
3. after 20d @Pe:4,787, lower Ap:9,690 (Eve orbit) [mustard m.n.]
4. after 88d: @9,690 Mm 1,962 m/s to raise Pe:9,689 (circularization) [purple m.n.]
5. after 68d: @9,690 Mm 878 m/s to raise Ap:13,873 (Kerbin) for intercept [green m.n.]
6. after 222d: 889 m/s for capture with Zephyr exhaustion with 67 m/s remaining for the capture (completed with a 2m ion burn) and with 15.3 km/s dV in the tank

Total transit time: 568d

Conventional Kerbin-Duna transit is 300d.

At SENTINEL launch, the transfer window was showing a wait time of 297d amounting to a total elapsed duration of 597d.

Conclusion A: is (unsurprisingly) that my trip above was only marginally successful (30d) in arriving earlier.  Ascending to Eve orbital altitude and plotting the rendez-vous there was a big mistake with the penalty of the very slow co-periodic rotation that orbital neighbors exhibit.  (See Chart 1 above.)  Nevertheless, a distinct advantage of dispelling the Tyranny of the Transfer Window is when one wishes to send a whole flotilla of equipment; it can all be shipped independently, flight by flight, at the convenience of Mission Control.

Conclusion B: the idealized algorithm of lowering Kerbolar Pe below the target to a point such that one can immediately there lower Ap for intercept is a specific simplification of the more General Algorithm.

The General Algorithm (for the objective of transferring from an outer body to an inner body) is likely to be to descend to a particular Kerbolar Pe and then circularize there to remain on the inside fast-track for a chosen time.  Then re-raise the Ap to the target for the final rendez-vous.  This is analogous to the operations one performs in planetary orbit in order to rendez-vous with another vehicle: get into a circular orbit with a different orbital period and then choose the transfer point.

The reason is because lowering the Pe does not, by itself, change the focal point of the route as needed.  Also, the expense to travel at greater speed is only enjoyed at that point.  And lastly, deepening the Pe becomes prohibitively more expensive deeper in the gravity well for diminishing returns (not in speed, but in desired course change).

The idea then is, I propose, to get on the fast track, utilize the speed there to a) reduce time travel and to b) position for the return maneuver to target altitude for intercept.  As the wait time for the transfer window increases, the ideal depth for the Periapsis decreases to some minimal practical point (e.g. 4Gm), at which point the Periapsis needs to be circularized into a segment with length that exactly suits the point at which the intercept transition should begin.

That minimal practical point (maximum depth) should further be constrained by the vehicle's dV limit, but which should be used to the max with regard to some proper reserve for miscalculation/mishap.  (Experience should be the teacher.)  This is because there is no point coming home with unused fuel in the tank!

I shall make a Second Try at this!!

 

 

Edited May 10, 2020 by Hotel26

[Laie]

9 hours ago, Hotel26 said:

The General Algorithm (for the objective of transferring from an outer body to an inner body) is likely to be to descend to a particular Kerbolar Pe and then circularize there to remain on the inside fast-track for a chosen time.  Then re-raise the Ap to the target for the final rendez-vous.

The general approach may well be general, but requires two expensive maneuvers in solar orbit. In my use case, that turns out to be more expensive than my special solution of dropping do a deeper PE, doing only one other maneuver, and arriving 360 days sooner than strictly necessary.

Oberth, I guess: first around Duna, and then from being at a (comparatively) low solar PE.

Still surprising, though; 360days is ~85% of a Kerbin year. I'd really have thought that this could be utilized, but cannot find a way.

360d is also more than one Eve year, so I expect one should be able to pick up an assist of some kind somewhere along the way; I expect that this would make quite a difference. Didn't even try, though.

[Klapaucius]

On 5/8/2020 at 6:02 AM, purpleivan said:

Back in the day (2014/15) I entered a few challenges that had the goal of "get to planet X in the shortest time" and as the saying goes, the short distance between two points is a straight line.

At least as straight as you can get it with finite fuel.

This sounds like very cool challenge to resurrect or redo in the current KSP version.

[purpleivan]

4 hours ago, Klapaucius said:

This sounds like very cool challenge to resurrect or redo in the current KSP version.

Yes... more of a challenge now too as you can't just slam into an atmosphere at 30+km/s and expect to survive.

[RoninFrog]

On 5/9/2020 at 10:39 PM, Hotel26 said:

Total transit time: 568d

I like how many crafts you have near Moho that didn't make the capture burn. 

Edit:  Wait a sec, what are those things?  Death Stars?

Edited May 20, 2020 by RoninFrog

[Hotel26]

2 hours ago, RoninFrog said:

On 5/10/2020 at 1:39 PM, Hotel26 said:

Total transit time: 568d

I like how many crafts you have near Moho that didn't make the capture burn. 

Very, very observant!!!!  (I very much like that quality, by the way, so you are now A Friend For Life...  congratulations, sincerely.)

Worst disaster.  I sent about a thousand tons and 99.44% of it had to be jettisoned, including the guys.  Now and forever Lost In Space.

Basically, the only things that achieved capture were about a dozen ion-powered sats that baled out and got there alone!  The Moho window came up first in this save and I was ill-prepared.  I've been to Moho before and I got complacent.  I don't think I'll ever live it down.

Edited May 21, 2020 by Hotel26

[magnemoe]

On 5/7/2020 at 1:51 PM, Hotel26 said:

The following is intended as an invitation to a discussion, rather than a simple "Question":

In brief, what are your thoughts on the science for plotting any particular transfer, to start NOW (at one planetary body) and END (at another within the same system (Kerbol)) AS SOON AS reasonably^[1] possible?

[1] ignoring diminishing returns

You have porkshops in mechjeb, I have used that to do orion pulse nuclear burns of above 100 km/s. 
With this level of dV its no reason to wait, retrograde is no issue, you still have to aim ahead of target and you travel time to inner planets still depend on their position. 
One exception don't be Icarus, your trajectory might get very close to the sun, now in KSP this is not an problem unless you jump to your ship, but here you might want to either wait launching or go slower so you don't get as close. 

Note my first probe to Moho had required 8 km/s to get into orbit, my dV was 2 km/s. I selected the 8km/s landing option. 

[Zhetaan]

On 5/8/2020 at 6:28 AM, Hotel26 said:

[Helpful Chart of Planet Periods]

First row and column list the orbital periods of the planets themselves.

The intersection of row and column gives the period of the respective Transfer Window (in days) between the origin and destination bodies.

I draw attention particularly to the quantities neighboring the blank diagonal.

This is something to think about!

I would like to caution you that the intersection doesn't quite describe the period of the transfer window so much as it describes the time between transfer windows.  Period of the transfer window could be mistaken to mean the transit time.

Incidentally, this quantity has a name; it's called the synodic period.  That is normally encountered in astronomy and is used to calculate conjunctions, but since the timing for a transfer window is arguably a conjunction with different alignment parameters, the term was used for that, as well.

The reason for the increase in quantities nearer to the blank diagonal has to do with the fact that the blank diagonal relates to a planet's transfer window to itself.  However, there's more to it than that.  The synodic period only gives the time between transfer windows, and even then it is an approximation because it does not account for eccentricity, inclination, or other factors.  It is based completely in the relationship between two bodies' orbital periods as they co-orbit about a third body:  thus, for example, the Mun and Minmus also have a synodic period, as do the moons of Jool, but Ike and the Mun do not--at least, not in terms of a first-order relation.

However, what this means is that the increase in synodic period as your planet of arrival approaches the planet of departure is not limited to the planet itself, but rather to anything that shares in the planet's orbit.  For example, if Lagrange points were possible in this game, the 910-day synodic period for a window between Kerbin and Duna is the same for a window between Kerbin and Duna's L5--the difference is that your time of departure must change.

However, there is a more interesting implication:  the synodic period doesn't only increrase, but its increase approaches infinity as your planet of arrival approaches your planet of departure.  Though not obvious, this should make intuitive sense:  if you can imagine two planets whose orbits are mutable enough that they can be made to approach one another, then as the two planets' orbital speeds become more similar because their orbits get closer to one another, the slower each planet appears to move relative to the other.  Put in a different way, if you find yourself chasing someone on a race track and you are the faster runner, then the faster the person you're chasing goes, the longer it will take you to catch up and overtake.

But you must remember that the synodic period doesn't care about the actual transfer; it merely considers the time between an arbitrary alignment and the next similar alignment in terms of the orbital period and without respect to what the alignment is.  This means that the period can be used to predict not only infinite time between windows for a transfer from a planet to itself (there is no 'best' time to go from Kerbin to Kerbin), but also infinite time between windows for a transfer from a planet to anything sharing its orbit.  To go back to the runner analogy, if the other runner runs with your same speed, then you will never catch up; neither will the runner catch you.

This has interesting implications for co-orbital bodies and simulated Lagrange points:  infinite time between transfer windows does not imply that the destination is the same as the origin, but rather implies that there simply are no transfer windows from that point of departure to that destination.  Stock does not have any examples of this, but Outer Planets Mod has a co-orbital set of moons in Polta and Priax.  I believe that one of the popular replacement solar system planet packs has a body that is co-orbital with the starting world.  The trick to travelling from one to the other is to use a phasing orbit, but on the other hand, there is no preferred, more-efficient choice of departure time, so you can go whenever you like and expect it to cost the same in fuel as at any other time.

[Hotel26]

10 hours ago, Zhetaan said:

the synodic period

Thank you.

                                                         

Consider the case in which a traveler wishes to depart planetary body A to travel to p.b. B at a time in which the phase angle is 180 deg.  I'd say that a "torch burn" strategy is not a good solution as it would start with direct acceleration toward the sun.  Granted, the sun won't impede your path, but that's exactly the point: in a strong gravitational field, a direct-target burn will not achieve the path you think you really want.

So the biggest puzzle for me is whether Radial burn components have any strategic place in the science of Immediate Departures?  I understand the difference in effect of, say, Radial In and Retrograde.  A shorter, faster transit through the inner radii may be the name of the game for most cases, but radial burns make more immediate modifications to the local path.  Maybe these make more sense therefore only when the target is "close", which is exactly the way rendez-vous maneuvers work once in the target vicinity?

Edited May 23, 2020 by Hotel26

[Pds314]

I created a challenge for doing this with Duna and Tylo. For a value of "now" being year 1 day 1.

The Tylo one was kinda a flop with only one entry by EveMaster, but the Duna one got pretty competitive.

As for "not doing diminishing returns," well it's kinda difficult to know what you mean by that. If you spend double the delta-V but get there in half the time, is that diminishing returns?

The porkchop plotter will give you transfers well outside minimum energy trajectories. Take note, though, that there's no law of nature that says it's never faster to wait.

For example, if it's Year 1, day 208, and you want to go to Duna without aerobraking, that'll be 1712 Delta-V to get there by Year 2, day 65.

But if you have 1712 Delta-V, the fastest possible route that I can see is actually to wait until year 1, day 219, then spend 1712 Delta-V to get there by Year 2, day 51. You gained 14 Kerbin days by leaving 11 Kerbin days later.

Somewhat paradoxically, the lowest energy transfer to Duna on day 1 is actually not a single orbit, but waiting over two years in orbit for Duna to line up.This method is however quite slow, and waiting for Hohmann, would actually be faster.
 

Edited May 23, 2020 by Pds314

[Pds314]

7 hours ago, Hotel26 said:

Thank you.

                                                         

Consider the case in which a traveler wishes to depart planetary body A to travel to p.b. B at a time in which the phase angle is 180 deg.  I'd say that a "torch burn" strategy is not a good solution as it would start with direct acceleration toward the sun.  Granted, the sun won't impede your path, but that's exactly the point: in a strong gravitational field, a direct-target burn will not achieve the path you think you really want.

So the biggest puzzle for me is whether Radial burn components have any strategic place in the science of Immediate Departures?  I understand the difference in effect of, say, Radial In and Retrograde.  A shorter, faster transit through the inner radii may be the name of the game for most cases, but radial burns make more immediate modifications to the local path.  Maybe these make more sense therefore only when the target is "close", which is exactly the way rendez-vous maneuvers work once in the target vicinity?

Regarding radial components.

Assuming impeding objects are not an issue (burning up due to getting too close to the sun come to mind), your burn shouldn't have large radial components relative to your initial planetary orbit. It will definitely have large radial components relative to the direction of planetary travel, especially if you care about getting there faster than waiting for Hohman transfers, and not, e.g. waiting 15 orbits for a delayed minimum energy transfer.

[Hotel26]

3 hours ago, Pds314 said:

Assuming impeding objects are not an issue (burning up due to getting too close to the sun come to mind), your burn shouldn't have large radial components relative to your initial planetary orbit. It will definitely have large radial components relative to the direction of planetary travel, especially if you care about getting there faster than waiting for Hohman transfers, and not, e.g. waiting 15 orbits for a delayed minimum energy transfer.

I don't think this is true.  The key to transfer window calculations is co-relative rotation speed.  The more similar the orbital speed of the destination is to the origin orbital speed, the longer the transfer window will be.  To "circumvent" this, one must temporarily find a transit orbital altitude with a significantly higher or lower orbital speed in order to quicken the relative motion.  This, therefore, is about spending energy to get to a more distant orbit than a minimal Hohmann transfer would entail.  And orbital altitude changes are all about the usual prograde/retrograde acceleration.  It's less effective at short-range (in which the radial burn works better) and it's also limited in the long-range by diminishing returns, at least in the case of "catching-up", in which one must descend deeper and deeper into the solar gravity well.

Where two orbital positions are closer, the relative direction of the mutual gravity field is closer and effecting both orbits more equivalently.  This becomes more like a straight-line problem,  The caveat on torch acceleration is exactly the curved nature of the gravitational field: if you cannot outperform the gravitational effect, particularly its curvature, you will not win.  This is why the final step of rendez-vous, the "straight shot", has to be performed reasonably close to the target.

It may be that "lagging", to shorten the closure with a trailing inner destination, may be easier than the inverse, because the steepness of the gravitational field lessens inversely proportional to the square of distance going outward.

You spend a lot of energy to deepen your solar periapsis to catch up with your target, and your speed increases proportionally to the energy you spend, but the time you spend on that arc is shorter, and the pay-off of being on the faster track (for a shorter time) is diminishing(???).  This is why I wonder if a multi-segment approach in descending to an altitude and then somewhat reducing the apoapsis below the target's altitude is more efficient than simply picking a very low periapsis, descending to it, and then immediately reducing the apoapsis to an intersection with the target's expected position.

If I drew a graph it would look like a U versus a V.  (And I doubt that's clear without more explanation.)

??? about this I wonder, because it's not actually orbital speed that matters, but orbital angular speed...

I think there is a pretty simple break-even point between radial cheating (transfer window is actually pretty close) and pro/retrograde cheating.

                                                                                      

To categorize and summarize:

1. typical Hohmann transfer involves waiting until the transfer window and then performing a) transfer injection and then b) arriving and performing the capture burn (total: 2)
2. suggested elsewhere above: a) transfer injection to burn to an interim altitude for faster transit, b) arriving at that interim altitude, burn to arrive at the destination orbit, c) capture burn (total: 3)
3. my scenario: a) injection burn for interim altitude, b) circularization at interim altitude to spend time at an accelerated angular closure, c) departure burn for arrival at destination altitude, d) capture burn (total: 4)

0, 1 (choice of interim altitude) and 2 degrees of freedom (choose interim altitude and time spent there), respectively.

Think in terms of orbiting Kerbin and you are at 75km and your target is 1,200 km away at 70km...  first thing you do is blow out to a much wider orbit to lag behind, yeah?  and when you get to that new apoapsis, check the rendez-vous time and if it's still too far away, circularize to lag even harder!

                                                                                      

Case 1 above is the way to get there with the minimal work.

Case 2 above is how to work harder to get there sooner.

Case 3 is arguably additional, unnecessary work to gain the convenience of plotting the rendez-vous intercept from co-circular orbits.  In the Kerbin example, I gave, one could think, "why not just increase the apoapsis far enough to get a harmonic intercept -- and save a burn?"

But at this point we have to consider an upper limit on how much extra work we can perform to get there sooner.  (Else we could just use a near-infinite burn to get started followed by a similar near-infinite burn to arrive and stop.)  And that's where I think Case 3 may come into play.  You may simply have to consider your maximum dV available to settle on some higher interim altitude (including the cost of the extra burn) that gets you there sooner -- but within your total budget.

And this is what makes this problem intriguingly more interesting than a walk in the park.

Edited May 23, 2020 by Hotel26

[Hotel26]

The above was a whole cartload of "thinking out loud" and I should condense it.  But I want to record one more thought: I think the idea of "circularizing" at a lower altitude could be shudderingly expensive.  The problem is that the altitude you can afford might not offer an immediate adjustment of the apoapsis to match the target.  This is where a radial burn component might be the extra degree of freedom that clinches the deal.