Progress in Theoretical Rocket Cost Optimization

[Cunjo Carl]

All set and ready to go!

Post still under construction! Large posts never seem to save consistently or work first try for me, so I'm editing it all in piece by piece. I work slowly these days, so it may be a few days before it's all together, fixed for readability and proofed. Thanks for the patience!

Also, I'm never sure where to put the mathy stuff. Feel free to move the post around as desired! This felt like a general thing, so I went with it.

 

Progress in Theoretical Rocket Cost Optimization

                   •   What's the most bang for your buck?
 

dV/dC is the bang for your buck in (m/s)/(/ton). Everything on the right hand side is a materials property, situation or design choice for your rocket. Knowing these things, we can compute the optimum design choices! There's some notable caveats involved- there's no doubt this is a first order solution. Still, you gotta start somewhere!

Terms Above: (T = Dry mass fraction of the fuel tanks. E = TWRrocket/TWRengines. P = Stage payload fraction. C = Launch cost per ton up to this point . Cf cost of Fuel per ton . Ce = Cost of Engines per ton.)
 

Results

When designing a stage, we need to chose how big this stage will be relative to the next one. This size difference is called the payload fraction, P. A big P means lots of small stages, and a small P means fewer big stages. The figure below shows how P effects the look, cost and mass efficiency of a rocket (which is oddly triangular and not quite to scale). Each of the options below are made to push the same final payload the same amount of deltaV, so from a design perspective all of the rockets below are doing the same job. Each black line on the rockets represents the engines between stages, and because it's the most mass efficient the middle rocket should be considered as the 'normally best option'. (Figure 1 below)

Using this idea, the very most efficient rockets we can design with a single kind of engine look like the ones below. Notice the cost efficient option on the right! (Figure 2 below)

Though this same concept will apply in rockets which have more than one kind of engine, we'll stick with one kind for this to keep things simple and so we know that what we're talking about is inherent to rocket design, not our engine choices.

Regardless, P is a very convenient variable for us to use when trying to optimize a rocket. The amount of engines our stage brings along is fixed because they need to push our required TWR, meanwhile the rest of the mass is free to be either payload or fuel tanks, and how much of each we bring winds up being described through P. Since the top equation describes our cost efficiency in terms of P, we're able to use it to find find and compare the most cost-optimal engines and configurations in different situations! This is the best 1:1 comparison we can make (to date) without plugging your whole travel itinerary into a solver (which is also an effective solution). The graph below shows the cost difference between the best possible configurations of three engines (Poodle, Skipper and Dart) in terms of their cost per ton per-deltaV versus what I've been calling the accumulated launch cost, or in other words how much money we've spent to get to where we are in the mission. The chart shows that to be the most cost effective at TWR=1.7, the first few stages are best as Poodles, then one or two as Skippers, and finally we would switch to Darts deep into the mission. Of course all this would make for a silly rocket, but it makes for a great thought experiment to guide our intuition, because we can ask 'why?' and relate things back to the rockets' properties. So, why?

The Poodle is by far the cheapest (best cost per thrust), so when we're in 'big dumb rocket' mode near the beginning it's the perfect choice. That said, despite its wonderful Isp, at a TWR of 1.7 its high drymass actually causes it to be _less_ mass efficient than the other options. By the time we're 5,000 Funds/tonne into a mission, the cost of our components begin to matter less and their mass efficiency matters more so the higher TWR and decent price of the Skipper makes it the best choice next. However, eventually the ridiculously expensive Dart finally comes into its own when 20,000 Funds/tonne into the mission mass efficiency is all that matters. (Figure 3 below)

Now we've had a chance to look at where all of this is going, let's go back to the beginning and discuss how we came to these conclusions and how we can use them (or things very similar) to inform our rocket design. To start with, a ton of conversational stuff that need saying!

 

The Talky Stuff

Spoiler

 

Hooray!
I finally got some good headway on the cost problem I figured I'd share! I've been poking at the problem for a long time, and whacked my head into hilariously many promising-looking dead ends, so I feel lucky in the end that my intuition kept dragging me along and nagging me to stick with it- there are some interesting an unexpected effects to see! I finally realized the right way to pose the problem about a month ago, and it's been smooth sailing since. The trick came in the form of abstracting away large portions of reality, and adding them back in later as boundary conditions and correction factors. This trick is used fairly commonly in some sciences, such as in chemical thermodynamics. It's a bit of a surprise in this context! But it makes sense now in retrospect.

For those who happen to be familiar with my personal story, the length of the post is probably a surprise! I've been working on it for ages, the last two weeks in particular, but I'm glad to finally get it out there. I'll probably take a well-needed break from typing (couple weeks) after answering the first couple rounds of responses.
 

State of the Work
There's still plenty of work to be done, but these opening steps are to the point where we can see a number of fascinating interrelationships involved in rocket cost optimization. Naturally, the amount of work left to do is much greater than what's been done so far. The way I see it, after bumping my head on every tree in the forest I've finally blazed the trail to base camp so now we get to try climbing the mountain.
 

The Cow in the Room
I know megathreads of doom aren't generally the best reading, but my intended audience here is the crazy few who will read it anyways and probably prefer the detail. I've tried to organize this post so that cherry picking through it would be easy. And, though I've tried to write succinctly, there's been a lot to say. Thanks in advance for taking the time to check through! Another cow in the room is the information flow with some of the results being up front. It just kinda felt right,.
 

A Practical Launch System
For people looking to push orange tanks into orbit, this post may not be for you! This said, for a quick, easy, cost efficient rocket I can recommend 4xKickBacks surrounding a TwinBoar+fuel through (detachable) structural pylons as a first stage and a half. It works great!
 

Why use theory?
There's a ton of more practical ways to make a cost efficient rocket than diving into theory. For example, there's been a few great KSP challenges, which leverage intuitive guess-and-check, and the industry apparently uses FEA solvers. However, this post is all about theory! Though it may not be the most practical, by looking at the elements of the theory and how they come together we can understand not just _what_ the best rocket is but _why_ it's the best rocket. Though understanding is the main goal (and there will be some great surprises along the way) another side benefit of a solid theoretical model is it makes programming numerical approaches waaay simpler, often with faster results, better starting conditions and more meaningful/fundamental readouts. No guarantee here, but it's a hope. Mostly though, it's to allow us to see how the different forms of efficiency (TWR, Isp, Cost, drymass) interplay together.
 

Novelty
Looking at the form of the solution, I believe it's unique. And, as far as I've seen in my limited experience, I believe it's novel. That's not to say "it's the most amazing thing ever!", but rather that all the tools and conclusions we can make from here will be treading new ground.
 

Premise / Starting Assumptions / Where's the SRBs?
This will be looking at staged rockets. Though SSTOs are fascinating in their own right, optimizing them is centrally dependent on minimizing parasitic drag, which is good fun but also a tricky moving target in KSP. For our staged rocket, there's several things we could add to our model without a lot of extra mathematical work, but they'll be omitted from the main derivation for simplicity: SRBs, LFBs, Asparagus, Mixed stages, decouplers and other utilities. I've tacked on a very rough version of the math for them near the end. Also for mathematical simplicity, everything will be assumed to launch in vacuum without launch losses. However, these could be estimated through experiment in KSP or with a spreadsheet launch solver using the equations in this post (||||||).
 

Nomenclature Notes
First, some nomenclature, sorry! By necessity, I'll be using some terms out of their ordinary context, so I'm mentioning them here. Though it's backwards from KSP, I'll be referring to the biggest stage of the rocket as the first stage, aka. stage 1, and the smallest stage as stage 5. From here, I'll use the term payload to refer to the what's being pushed by any stage, whether it's the final payload or not. Likewise, 'launch cost' will refer to the amount we spent to get to wherever we're talking about, regardless of whether we're on final payload or not. Finally, 'Effective Isp' will be conceptually the same as the one used in aviation, but will appear very different in this context. In this context, it's synonymous with mass efficiency.
 

Expected Simplifications in Limits
When first attacking a problem, it's nice to figure out what answers we'd expect in certain extremes (where the problem simplifies nicely and intuitively), so we have a way to double check our work as we go and can expect what terms will show up. If we have a 5 stage rocket, and we're trying to optimize one of the final stages (say stage 4) we'd expect the _cost_ efficiency of that stage to be mostly determined by  its _mass_ efficiency. After all, the main cost by that point was the cost required to launch it to where it is, so mass efficiency there is key (seen later as Ispeff). Also in the limit of fuel and engines costing the same, we would expect the cost efficiency problem to simplify to a mass efficiency as well (seen later as Ce-Cf). On the other hand, in the limits of right-at-launch and cheap fuel, we would expect the required engine's thrust per cost to play a major roll (seen later in the form of E*Ce).

Things that don't work
There's a ton of wonderfully intuitive ways to attack this problem which sadly don't pan out. For example, we could consider that our rocket has a cost per payload = Sum(stageCosts)/massPayload , and create various systems for describing stage costs in terms of their fuels, engines and payload fractions. While great for tabulating, this setup has a key weakness that it doesn't allow you to compare the value of two rockets with different deltaV. In other words, if I have two identically priced rockets one with 12000m/s and the other with 12001m/s, what's the difference in value between the two? Unfortunately, because deltaV and cost share an exponential relationship, it's very difficult to meaningfully compare the two without choosing an arbitrary deltaV or energy goal (eg. C3). There's several ways to try rolling with the punches such as requiring precomputed optimal stage sizes, removing the deltaV discrepancy by adding extra TWR, and measuring the deltaV difference with the cost and ISP of a reference engine. But all the options (that I found) wind up covering up some interesting or important aspect of cost optimization.
 

Result: The working equation
There's a few final forms of the cost equation. Just to make it explicit, the most useful one for maths is the working equation (eq. 46). It can be integrated to relate rocket cost to deltaV, which is kinda handy. On the other hand, because its inverse eq 45 has a beautiful linear nature, I used it to make most of the charts. In eq. 45, straight lines represent rockets and stages possible in the real world, and the tangents of the curved lines represent real rockets/stages as well (this is explained more later in the more results section). Also good to note, a real stage will be represented as a given length along any of these lines. All this said, even though they're not physically possible, I like to think of those continuous curved lines as the equations for what a rocket _wants_ to be, in the absence of our terrible limitation that it must be built out of real world materials. The cow wants to be spherical!

 

Major Assumptions

Spoiler

 

We're going to make two major assumptions called 'uniform' and 'continuous' that I think are easiest to talk about now. Uniform means that each stage, rocket, or differential slice we're talking about has the same properties (like engine(s), TWR, payload fraction, etc) all the way throughout it. Continuous means that our equations will describe the rocket at any scale: rocket, stage, or even a differential slice. These assumptions can be 'undone' later by integrating and applying certain boundary conditions. The differential slice in particular isn't a real thing, but they can be added together to make a real stage in a way that's mathematically very convenient. When made together, I've taken to calling the assumptions the 'homogeneous' rocket . That said, a much sillier but also fitting name would be the 'bolognese' rocket instead. It's rather perfect because bolognese is a discrete thing (meat) made homogeneous and sliced thin for easy use. It's also a double meaning for 'fake' or 'stupid', which I think will be a fun thing for potential nay-sayers of the assumption.

The figure below shows how this assumption works in its main context of creating the concept of mass efficiency, and is used later to extend the concept to differential slices of rocket. Both of the rockets shown below are identical (by definition) from all engineering standpoints, except the homogeneous one on the right is differentiable by its nature, which makes it a useful model for math. From a conceptual standpoint, if you were to watch the homogeneous rocket burning, you would see it slowly and continuously burning away from the bottom to the top, like a magical caseless solid motor. (Figure 4 below)

 

 

Quick and Easy Derivation

Spoiler

 

The 00 equations are a hammy run through of the derivation which work well enough for the conceptually curious but leave some admitted gaps for the rigorously demanding. Still, I think it's a good in-between relative to the 50 equation romp that is the real thing! Probably the most important thing to note during the derivation is how the mass of our differential slice of rocket cancels out when we consider dV / dCost. We can imagine it as making a rocket equation that works directly in terms of cost rather than math. Descriptions for each step below.

Terms:
Cx = Cost per ton of x
mx = mass of x
r = rocket (including payload)
p = payload
f = fuel (including tanks)
e = engines

00-1. Let's start with making an example to have in mind for this 'derivation', let's say we've spent 1000Funds/tonne to get a fully fueled space craft into LKO on a transfer stage. We'd like to optimize that transfer stage and use it to know what our craft's final cost per ton will be after decoupling the transfer stage. In other words, in the beginning the cost was 1000funds/t, let's call it C0, what will the cost per ton be after the transfer burn? What's the change? We'll call the change per ton deltaC, and then this times the mass of our payload (deltaC*mp) would be the total change in cost caused by our transfer stage.

The total change in cost caused by our transfer stage should be made by 3 things summed together: Launch cost, fuel cost and engine cost. Specifically: 1. The 1000Funds/tonne (C0) times the starting mass of the transfer stage without its payload (massRocket - massPayload).  2. Adding in the cost of the fuel, which will be Cf*mf. Here, we know the mass of fuel (including tanks) will be everything in the rocket that's not engines or payload so this becomes Cf*(mr-mp-me). 3. And finally, we add the cost of the engines Ce*me.

00-2. Algebra! Pull out a mr-mp for use later. Note that the term me/mr = TWRcraft/TWRengines, and is something set when we choose our engines and TWR. Notice how this equation separated into two sections: mr-mp on the left, which describes the size of the transfer stage, and everything in the parenthesis on the right which describes design properties of the stage not directly affecting its size. Don't mind the leading 1/mp. It will cancel out soon.

00-3. Let's imagine the cost of a differential uniform slice of this stage by bringing mr-mp -> dm (see major assumptions section for the continuous assumption). It's mathematically very hard to compare the values of entire stages, but the differential slices can be easily compared. This equation now shows us the cost per mass of a differential slice of rocket, but what's the deltaV of this slice?

00-4. To find the deltaV per differential slice, we'll need a version of the rocket equation which is aware of both the exhaust mass and the drymass associated with it. As engineers, we just care about how much total mass was lost and how much deltaV we got for it. Let's use this to make a correction factor for the standard exhaust-mass-centric rocket equation to take these things into account. If we were to count all discarded mass (rather than just exhaust) the rocket equation would like deltaV = -X*ln(mpayload / mrocket) where X is some constant. It's much simpler than the standard one! However, to be useful it must be equivalent to the actual rocket equation's deltaV = -g0Isp*ln(mdry/mwet) in all cases, so next we'll find the value of X which allows that.

00-5. Let's find this X and call it g0*Ispeff. The only tricky part is calculating the rocket equation's mdry = mrocket - mburnedFuel. The mass of burned fuel is the mass of the fuel tanks (mf) times their wet mass fraction ((mFullTanks - mEmptyTanks)/mFullTanks). Since the fuel tanks are everything in the rocket that's not payload or engines we can substitute mf = mr - mp - me . After plugging these in, we isolate X (aka g0*IspEff), and we're set!

00-6. Let's use g0*IspEff to find the deltaV of our differential slice of rocket. To derive the rocket equation, we use conservation of momentum to say that the total momentum before and after using some stage is the same. Before the mass was mp+dm, traveling at V velocity. After, the rocket will be just mp mass, but will be traveling at V+dV velocity, meanwhile the spent mass will be traveling at V-Ve,eff  , where Ve,eff equals the g0Ispeff we just found. This makes sense, because the Ve,eff is just an average of the actual exhaust velocity (Ve=g0*Isp) and the 0m/s drymass we'll be dropping after the stage burns out. This average must be a special logarythic average cognizant of the nature of the rocket equation, and it is because of how we defined it. It's just an average though.

00-7. Algebra!

00-8. Leibnitz rule to victory. Notice how (talking about the inverse of our equation here) dV/dC is the bang for your buck, or in other words the cost efficiency. And, it's made of dV/dm the mass efficiency and dm/dC the cost per mass. All of the properties involved are materials properties and design parameters of the rocket. So, by maximizing dV/dC (or minimizing the inverse dC/dV) we can optimize the rocket parameters (most importantly mp/mr = P) based on the material properties.

It's all pretty easy going in this context! There's a ton of little nit-pickable tricks and troubles being swept under the rug here, and showing they're ok by purely conceptual reasoning would be quite tricky. That's why there's the full derivation, which circumvents the troubles by going around the long way... the reeeaally long way.

 

 

Full Derivation

Spoiler

 

Summary of the Math
First we'll recap mass efficiency, Ispeff. Then we'll find a way to discuss cost that allows us to talk about any stage of the rocket without reference to the others besides their total cost. After that, we'll go full spherical cow and describe differential slices of a rocket in a way that its predictions can always be related back to the discrete rockets of our real world. The correct form of these differential equations are the ones which will integrate to make the correct answer every time. So, we'll find the correct deltaVs and costs for an n stage rocket, bring them together, and then find the derivative which will produce these. Finally, we'll check our work by using the differential equations to once again describe a real rocket.

Terms

 

 

Calculating the deltaV of a stage

1. The mass of a whole rocket is the sum of its payload, fuel and engines. "Payload" is used to describe whatever a stage is pushing, whether it's the final payload or not.

2. Dividing all terms by mr puts them into their fraction forms. These are much more convenient as they can describe rockets of all sizes.

3. Rearrangement for later substitution

4. A new equation: Rocket drymass is the same as wet mass, but with the full fuel tanks (mf) replaced with empty fuel tanks (T*mf).

5. The difference between a full and empty rocket is mf-T*mf. This is rearranged for later convenience.

6. Divide both sides by the rocket mass to put this into the fraction form.

7. F isn't useful later, so we substitute it out with eq. 3. Then isolate the term on the left.

8. Rearrange to the preferred form. Speaking relative to wet mass, the dry mass of a rocket would be T (fuel tank drymass fraction) at its minimum. This is added to the drymass of P (payload) and E (engines), but we need a (1-T) term on these to not double count them with the T from earlier.

9. Plug it in to the rocket equation to get deltaV!

 

Calculating Mass Efficiency, also the deltaV and payload fraction of an n-stage rocket.

10. The defining equation for Ispeff, the mass efficiency.

11. DeltaV for a rocket must be equal no matter how we calculate it, so eq.9 = eq.10 . This is the easiest form of the Ispeff equation, and it can be maximized with respect to P to maximize mass efficiency of the stage it represents. This said, Ispeff can represent the mass efficiency of any stage, optimized or not. The nuances can be found in steps 00-4 through 00-6 in the quick derivation.

12. Showing that Ispeff remains constant whether we're talking about stages or rockets made of n equivalent stages. This is a requirement for anything defining rocket mass efficiency, and Ispeff (and those things related to it by a simple factor) are unique in satisfying it. In earlier works, the Ispeff for rockets made of different kinds of stages is also shown to be sum(ln(Pi)*Ispeff,i)/sum(ln(Pi)). Back to our rocket, if it has n stages each with a given deltaV, the whole rocket will have n*deltaV total. A rocket of n stages, each with payload fraction P has a total P^n payload fraction between the final payload and the initial full rocket.

13. Restate and rearranging eq.12

14. Rearranging to isolate P^n

15. Substituting P^n with b^-n and simplifying to the final form used in a substitution later along.

 

Calculating the cost of a single stage in convenient terms.

16. For this, "Launch Cost" describes the cost to put a thing wherever it is, such as:    in orbit, around the Sun, or 20m off the launch pad. "Payload" is used to describe whatever a stage is pushing, whether it's the final payload or not. The launch cost of each subsequent stage in a rocket is the launch cost of the stage before it (payload included), plus the cost of the fuel and engines in that stage. Note that we don't care how it got there, just how much it cost.

17. Substitute cost terms ($) with their cost per ton terms times mass (m*C).

18. Substitute the not-very-usefull fuel mass term (mf) with mr - mp - me. This is equivalent to F=1-P-E substitution from earlier, we're just leaving it in masses this time so a few later equations would have a simpler real world meaning.

19. Redistribute. The real world meaning of this equation would be: The cost to put the payload where it is = the mass of the whole rocket times the cost per ton as if it were all fuel including the initial launch cost. Then there's two correction terms, one to say the payload isn't fuel, and the other to say that the engines cost a different amount than fuel. Though a little silly, this form of equation winds up being a necessary abstraction to drastically simplify later math.

20. Subtract mp*C0 from both sides. Now we're talking about the change in launch cost caused by this stage.

21. Simplify

22. Divide both sides by mp to make the equation describe rockets of any size. mr/mp = 1/P   mp/mp = 1   me/mp = (me/mr)/(mp/mr) = E/P

23. Substitute the P terms with the convenience variable b=1/P to simplify things later. b describes how much bigger the full rocket is than just its payload, normal values being between 2-5. I call it relative stage size. This is the final form of this equation, describing the change in launch costs per ton of payload caused by a single stage. It can be described as:
  "Everything that's not payload" (b-1)
  "Costs the initial launch cost, plus the cost of fuel" *(C0+Cf)
  "But the engines..." + E*b
  "Cost a different amount we need to correct for." *(Ce-Cf)

Extend the cost equation eq. 23 to multistage rockets, starting with a 2 stager where both stages have identical properties.

24. The change in launch costs between the second payload and the initial rocket (C2-C0) is the change in launch costs of the second stage (C2-C1) + the change in launch cost of the first stage (C1-C0).

25. If we imagine eq.23 is a function with respect to C0, we can say f(C0) = (C1-C0) = the change in launch cost of the first stage,    and f(f(C0)) = (C2-C1) = the change in launch costs of the second stage. So, in total, we have C2-C0 = f(f(C0)) + f(C0)

26. Simplify the mess. Notice the equation has remained largely the same as eq 23. That's the reason for this silly form!

27. Generalize to the nth stage. I originally wrote this equation directly from a picture of an n-stage rocket, and I'll try to add it in, but this can also be seen from the math if you do the simplification in eq. 25 yourself.

 

Simplifying the sum

That sum(i=0 -> n of b^i) term is a gnarly one that won't let us use calculus easily, so we'll take the time to convert it into something convenient in standard closed form. I originally did this by finding the patterns in b=2, b=3 and b=4, and then finding the pattern to the pattern. From there I back calculated what the derivation must have been, so if it looks like I pull some of the following steps from nowhere, I probably did.

28. sum(i=0 -> n of b^i) shown in terms of its elements

29. Magic. Multiply by 1 = (b-1)/(b-1). Pay attention to the (b-1) term on top...

30. ...The b term shifts the whole pattern forward by one, while the 1 leaves the pattern in its original place...

31. ...The only elements not canceled out are the very last (b^(n+1)) and the very first (b^0 = 1)

32. It would be really nice to wrestle that b^(n+1) term down into b^n so we can consolidate terms with the b^n in eq. 27. Let's turn the -1 into -(b-1)/(b-1) and see what happens.

33. The double negative on the 1 turned it into a +1, which cancels the -1 from the other term, leaving us with (b^(n+1)-b)/(b-1) . Now we just factor out a b from the top to get the extremely convenient equation shown.

34. Why wait? Plug it into eq. 27!

35. Factor out the (b^n-1). This term describes the mass of the entire rocket relative to its payload. This is a very convenient way to know the cost of a uniformly designed rocket given its size (b^n), TWR (E), and payload fraction (b/(b-1)) = (1/(1-P)) . This final term (b/(b-1)) which will later be (1/(1-P)) is significant and would have been difficult to guess about without doing one of the derivations.

 

 

Bringing the equations together, applying the continuous and uniform assumptions, and making the form differential

36-38. Restating equations 9,10,11 and 15 here for convenience.  and 38 (being used for convenience) as a substitution slightly later.

39. Substituting the b^n term from eq. 35 with eq. 37 . Now we can talk about the cost of a real (albeit uniformly designed) rocket with respect to its deltaV. Ispeff is being used as a convenience variable here, and will be substituted out later.

40. Bring in the "continuous" assumption to turn the n-stage rocket into a differential slice of a rocket. See the major assumptions section for more on this.

41. Oh no! There's a differential in an exponent! Fortunately, the linearization of e^x-1 = x is exact when done on a differential. For more on this see the Taylor and Maclaurin expansions.

42. Substitute e^(dV/g0Ispeff)-1 = dV/g0Ispeff

43. Divide both sides by dV.

Final Forms

Make the final rearrangements. Equations 44-47 are all useful forms of the equation. Eq. 46 is the working form of the equation for use in solvers. Eq. 45 is used to make the plots because of its nice linear relationship.

44. Substitute Ispeff (convenience variable) with eq. 38. This form is optimized with respect to P and used to make the graphs.

45. Noticing eq. 44 takes the form y=mx+b, factor it out as such. This is the form used to make the 'mass optimal' lines given pre-computed mass optimal values for P. Here we're using Ispeff for convenience once again for visual clarity.

46. The working equation. It's main use would be by choosing a set P and integrating until the area (the deltaV) = g0*Ispeff*ln(P). However, it can also be useful by maximizing this differential form for P at each value of C. Then the area under the reulting curve from C0 -> Cn is the deltaV of the best possible configuration of a stage with continuously varying properties. The result is of course non-physical, but still interesting.

47. A Rearrangement to emphasize the aspects of cost efficiency shared with the mass-optimal solution (First two groups) and then the aspects that are different from the mass optimal solution (Final group)

And, one more form. We can swap out P for 1-E-F, and 1-P for E+F thanks to eq. 2. This makes a form which may be more intuitive in some circumstances, where rather than discussing a stage's size in terms of how big its payload is, we discuss it in terms of how big the engines and fuel tanks are relative to the whole thing. It looks a lot simpler in many ways! Although I think it's a little less useful.

 

Double check the equations work

48-54. Just showing the proof in the pudding that the working equations can be integrated across C0 -> Cn to find deltaV, and then back calculating past the uniform/continuous assumptions to the single stage case. Undoing your work is always a good dummy check, and I figured I'd include it. This does also mean that the costs and deltaVs provided by the equations will always be correct, and the tricky assumptions only cause inadequacies in optimization and simple graphing. As a note, the simultaneous requirement could also be deltaV = -n*g0*Isp*ln(T+(1-T)*(E+P)) for a given constant P and the integer n describing the number of stages in question.

 

 

 

 

 

Further Results

Spoiler

 

So what can we do with it? As I've mentioned, math from here is best done with the working equation (eq. 46) which describes dV/dC because it can be integrated across a cost range to give the deltaV you'd get for spending that price. Plotted directly though, it's difficult to read, because the lines are all of the form 1/x! Eq. 45 for dC/dV plots much more nicely because it reduces to straight lines for stages with (realistically) constant properties. Seen below is 3 different engines being compared in this way across a range corresponding to maybe the first 3 stages of a craft. I purposefully chose 3 similar engines because they can actually be alternatives for eachother, so it's interesting to see how they compare. This said, comparing very different engines like SRBs and the LV-N would make for a much more dramatic plot!

In the plot the lines represent cost so lower is better. The slopes of the lines are determined by the inverse of the mass efficiency (1/g0Ispeff) and the Y intercept is determined by their cost on the launch pad divided by their mass efficiency. An SRB would have a very low Y intercept and a very steep slope. These can be seen in the 'engine plots' section.

 

When zoomed out though, trying to compare similar engines is difficult (see below) so we'll try a different kind of plotting in a moment.

 

The difference between two engines in dC/dVplots really nicely even deep into a mission, so I think they're a good way to go. To interpret the graph, for now, we can be happy with knowing that a lower value of Cost/deltaV is always better for our rocket, so go with the lowest ! That said, I believe the values of these differences are actually useful and comparable even at different points on the x-axis (different C -> different lengths into the mission) based on the following shaky reasoning: The values of dC/dV (see above) grow roughly linearly with C so the _relative_ differences between engines are inversely proportional to C. Relative differences in dC/dV are proportional to relative differences in its inverse dV/dC. Finally, to find a deltaV for a stage, the integration bounds for the stage in dV/dC are roughly linearly proportional to C. Put all together, the change in area (deltaV) caused by the differences shown in the plot below will be proportional to the differences * 1/C * 1 * C . It's more than a bit shaky, I know, and I'll try to link it more explicitly later. For now, I'm happy with the 'lower is better' explanation, but I figure I'd mention where I was trying to go with it.

 

So I've been making a big deal that even for a single engine the most mass efficient approach isn't necessarily the most cost effective when we're early in the mission. The following plot shows this effect and how it changes as the mission progresses (and launch costs increase). The effect is in the 2-20% range for most liquid engines in the early mission. The setup of engine/TWR/fuel-cost is the same here as in the above plots.

 

 

The plots above show curves of the best possible costs, but only straight lines are possible for real stages to have. Fortunately any straight line tangent to the above curves will describe a real stage. Since stages have a fixed cost, the line representing a single stage will have a fixed width in the x-axis (C), and early in the mission these lines can be quite short (only a few hundred Funds/ton in width), meanwhile the curvature is often very gentle and broad being as much as 20,000-ish funds/ton across. While it's not always true, in quite a few cases the curvature really isn't a big deal, so we can consider our real stages to be pretty close to what's on the curve.

Below is a plot showing rockets made of real stages (straight lines) with different payload fractions (P). Notice how these different payload fractions are better than eachother at different points in the mission (different accumulated launch costs). Bringing them all together, the straight lines will form the shape of the curve from above! This particular example is just using a dummy engine.

So in the plot above, the lower P values are useful for their lower intercept (cheaper cost on the launch pad), and the higher P values are good for their lower/better slopes (more mass efficient). Let's take a look at how these two aspects vary in the plot below. The initial dC/dV is the intercept, and IspEff is the inverse of the slope. Since they have different units, they're just being compared via arbitrary 'happiness points' . In some cases the optima for these two things almost line up, so there's not a lot of difference between the most mass efficient design and the most cost efficient design. This can be seen with the Poodle a couple plots up. In other cases though, the optima are spaced quite broadly like here (also see the Dart a couple plots up). I've been finding that the optima seem to be the most 'different' (= narrowest and most spaced) when E = TWRcraft/TWRengine = ~ 1/6 . Not sure why yet.

Anyways, from the plot below we can see why there's only a certain range of P values that's worth using regardless of the conditions. Below we can see that, P values beneath .12 start to actually increase initial dC/dV (the Y intercept) while also raising slope in the expected way,   and P values above .29 will actually start to reduce mass efficiency (raising the slope) while also raising the intercept in the expected way. Only P values between the two optima below can every be useful. The plot below is also just using a dummy engine.

 

So what's the best P value (Payload fraction)? As we've been seeing in the above two plots, we need to choose the best one for where we are in the mission, and that value can vary heavily. Using the Dart as an example once again, early in the mission 0.18 would be best, and very late in the mission 0.42 is best. This is super noticeable! Put in another way, early in the mission each stage should be 5.6 (= 1/0.18) times bigger than the stage it pushes, and late in the mission each stage should only be 2.4 (= 1/0.42) times bigger than the stage it pushes. Since 5.6 ~ 2.4^2, this means the stages are packed twice as tightly late in the mission compared to early on even when we're using the same engine all the way through! These values are for the Dart, and it's normal for some engines to be less pronounced in this effect. It's also normal for us to change to new kinds of engine during a mission which muddies things somewhat, but the effect is still present and can be profound in magnitude.

But what the heck do we call it?

 

 

 

 

 

 

Engine Plots

Spoiler

 

These are a few plots put in as a first-blush example of the of what could be done with the equations.

I have to say, there's really no useful surprises going on with these plots below. For the most part, the SRBs and Twinboar are good engines at launch, and the Poodle's good for everything else! That's of course a bit of an oversimplification, but it holds pretty well. There is one surprise though! It happens not to affect anything, but it is interesting to notice...

So just from the glance, it seems that the Thumper actually becomes a better choice than Kickback at TWR>1.9 or so. This is pretty important, because with SRBs you really want to fly off the pad to not have heavy launch losses. With my setups, I typically like TWR~2.4 or so, but I know there's different thoughts on the matter. So we might expect that for high TWR launches we should always use thumper! But, the super common trick with kickbacks is to launch with the kickbacks acting just as strap on boosters for a twin boar, in which case they're only responsible for less than TWR=1.8 worth of acceleration because the twin boars are helping as well!
So TwinBoar+Kickback > TwinBoar+Thumper > Thumper > Kickback > TwinBoar .

In other words, the TwinBoar+Kickback combo is a special combo greater than the sum of its parts.

 

 

Behaviour in limits

Spoiler

 

Finally, we should double check the behaviour in limits, make sure it meets our earlier expectations and see if there's any interesting tidbits.

56. In the limit of C0 and Cf being small (ie we're on the launch pad and fuel is cheap), we can expect the cost of engines to dominate the equation.

57. In the same limit, rememberring that E=me/mr=TWRcraft/TWRengine, we can get an interesting form showing that on the launch pad our bang for buck has a very convenient term of g0*Isp*TWR/CostPerMass . Since I learned what Isp and TWR were I've been wanting to multiply them together, so it's wonderful to have this excuse!

58. In the same limit, since TWR and Ce are both in terms of mass, the masses can cancel and we're left g0*Isp*Thrust/Cost. This completes the search for how Thrust per Cost relates to engine performance!

59. In the limit of C0 >> Cf or Ce (ie very late in a mission), the bang for buck simplifies to being entirely determined by IspEff, regardless of cost of the engines being used.

 

Discussion - Handling the Assumptions

Spoiler

 

Troubles faced from here
The work has been a great success for its main goal of gaining more understanding of what mechanisms play into optimizing a rocket's cost. When handled as stated, the equations will always provide the correct cost and deltaV for a rocket, however there are still some notable weak points when it comes to the optimization. It's natural to try to patch in the weak points and extend things farther, so here's the weak points, what they do, and how we might approach them.

Troubles of the Uniform Assumption
There's two aspects of a real rocket that can allow us to improve upon the 'optimum' recommended by the equations shown. The first, and simpler, is the concept of 'dilution'. When optimizing a rocket made of many stages, we can often eke a bit of extra efficiency by making the more efficient stages bigger (lower P) and the less efficient stages smaller (greater P) in order to dilute away the bad effects of the inefficient stages. This can be done even when it makes both stages individually less efficient. This is seen in both mass optimization and cost optimization. For small perturbations, the effect can be handled with linear optimization techniques. For large perturbations, I believe it can only be handled numerically. Though this is a possible way to eke a bit of extra optimization, it does have a drawback. We normally use less efficient stages (perhaps with high TWR, good gimballing) for a reason (eg. we need them to land), and by decreasing the size of the less efficient (landing) stage, we may be shooting ourselves in the foot from a design standpoint. The place where I've found this crop up most often in a useful way is that in KSP, large engines have slightly better efficiency (and slightly fuel costs) which will promote us to make the big early stages even larger than we would expect from the simple optimization using the formulas and graphs shown here. The effect can also be seen in simple mass efficiency in the mixed stage Ispeff equation Rocket_IspEff = sum(ln(P_i)*IspEff_i)/sum(ln(P_i)).  Normal values for the mass optimum have been stages increasing/decreasing in size by ~10-100m/s per stage and gaining a couple percent overall efficiency for the rocket.

The second and trickier aspect of optimizing beyond the uniform assumption is that Ce-Cf (engine cost) is assumed constant and governs our expectation that making fewer-and-bigger stages will save on engine costs in the first stages of a rocket. This is true, but because engine costs (and payload fractions) will vary, it's hard to tell exactly how much is warranted. This can be handled in various ways, but the one's I've found so far are a little too tabular for my liking. I think putting some work into the continuous assumption first would be best. It may simplify/complicate this a bit. Left to my own devices, this is the thing I'll poke at next.

Troubles of the Continuous Assumption
The main issue with the continuous assumption is that if we want to integrate the equations into a real rocket we need to use an extra condition during the integration to apply the requirement that rockets can't change their properties mid stage. We can do this by simply fixing the design variables (like P) to be constant, but in this case determining exactly what value is optimal (for P in particular) requires iteration at present. This condition isn't necessary to compare engines, but it is for building a whole rocket from theory in a way we could optimize. Also, it's easy to estimate the most optimum P by grabbing a value from the curves at the end of the More Results section. The exact optimum though can't be found in this way. However, because the differential equations for cost and deltaV are actually very simple (1/a)dy = dx/(x+b) I'm hoping to formulate a way to 'solve' the system of equations in a graphical form with some transforms or alternatively find an approximate but acceptable linearization to make a closed form solution. If neither pans out, this aspect would probably need to be left computational which would be a real shame.

 

Handling SRBs, LFBs and the like

Spoiler

 

Fortunately, it's fairly easy to incorporate SRBs, LFBs , asparagus and the like. Things like their TWR and cost per mass would just be calculated in the normal ways from the following (TWR = Thrust/Mass). As a note, I'm doing these without my notebook handy, so I'm just doing them fresh from memory. I'm 98% sure everything will be right, but this is much more likely to have a mistake than the rest.

SRBs can be handled as follows:
Thrust = Thrust
Engine Mass = SRB drymass
Engine Cost = Empty cost
Fuel Mass = full mass - dry mass
Fuel Cost = full cost - empty cost
T = 0 (all of the fuel tank is already incorporated into the engine)
          P = 1-E-F
         (E+F) </= E*(fullMass/dryMass)      ... The fuel tanks can be underfilled but not over filled, so E+F must be < or = E*(fullMass/dryMass)
Min P = 1-E*(fullMass/dryMass)

LFBs are discontinuous and must be handled in 2 ways. When the LFB is being used with a partially filled tank and no extra fuel, it should be handled just as an SRB. When an LFB is being used with additional fuel it should be handled as follows:
Thrust = Thrust
Engine Mass = dryMass - (fullMass - dryMass)*T/(1-T)       ...  The mass of the engine is the dry mass minus the mass of the tanks. The tanks are the mass of the burnable fuel (full-dry) times T/(1-T)
Engine Cost = Full Cost - Cf*(fullMass - dryMass)/(1-T)
Fuel Cost/Mass = Cf
          P = 1-E-F
          Minimum Initial Fuel Mass per Engine = (fullMass - dryMass)/(1-T)           .... Then divide this by the mass per engine to get...
          F/E > [fullMass - dryMass)/(1-T)]   /  [ dryMass - (fullMass - dryMass)*T/(1-T) ]  = 1/(  (dryMass/(fullMass-dryMass))*(1-T) - T  )
         (E+F) >/= E*(1+ 1/((dryMass/(fullMass-dryMass))*(1-T) - T) )
Max P = 1-E*(1+ 1/((dryMass/(fullMass-dryMass))*(1-T) - T) )

Decouplers and other extra sources of mass can be handled like engines. They can have a X = mx/mr  and a Cx = $x/mx, and otherwise their path through the derivation just follows the engines till the end, winding up as an extra term like: E*(1/(1-P))*(Ce-Cf) + X*(1/(1-P))*(Cx-Cf) .   and IspEff = Isp*ln(T+(1-T)*(E+X+P))/ln(P)

Asparagus can be handled by tracking the free TWR being provided by each stage starting at the top of the rocket and working down. If a stage provides free TWR, the amount it provides the next stage down is TWR*P, so the amount the next stage will need to provide itself is (TWR_next - P*TWR) . This sets that stage's E. Then combined Isps occur in the standard way.

 

 

Real World Applications
Not many. Yet at least!
Applying this to KSP is fairly straightforward due to the way the game ignores (I believe smartly) many aspects of rocket science. The exception to this is fuel cost for small tanks (2.25t and less), which becomes  scale dependent, but in a predictable way (I think). Applying these equations to the industry is even more complicated. Many terms taken to be constant in this work are actually very scale sensitive in real life. An example would be T, the fuel tank dry mass fraction, which in real life scales roughly with square-cube law (the surface area to volume ratio). Fortunately, many of these correction factors can be plugged in directly to the working equation (eq. 46) without refactoring any of the derivation, it's mostly a question of how jumbled it will get. I very much doubt this will change anything in the industry, but for us arm chair rocket scientists who value understanding over the bottom dollar, I think it's still very worth while!

Further Work
Too many things!

Errata
Have you found a typo in the math, or a confusing explanation that could use some work? Let me know! I'm going to try editting the main post to make this as accessible as possible. Please note that for health reasons my ability to type is very limited, so changes may take a while, but thanks in advance for bringing things to my attention.

Code
I can make the solver code available if desired.
 

A note on the term "Effective Isp"

Spoiler

 So, it's been mentioned in an earlier thread that the term effective Isp is already in use in aviation, so I probably shouldn't use it. I've thought it over carefully, and I think I'll stick with it on the basis that the effective Isp here and the effective Isp in aviation (jet engines) describe a very similar concept just in different contexts. In jet engines, the term Isp is used to describe how fuel is burned with incoming air to make a jet of exhaust that would have a certain Isp, but then energy is bled out (with a turbine) and used to push surrounding air (with a bypass fan) which increases the effective mass flow and effectively increases the impulse. This use of energy doesn't technically have an Isp any more than the fuel burning for energy in a prop plane's internal combustion engine does, but mathematically (and conceptually) we can handle it as if it did! Similarly, when our rocket dumps spent tanks and engines, it's not really dropping a 0 m/s exhaust stream, but we can treat it mathematically is if it were! In both cases all's we're trying to do is relate the total mass lost to the total impulse created despite the greater system doing something complicated.

Plus, what else would we call it?

 

And finally,

Thanks again for giving a glance, and I hope it's been interesting to ponder! It's always fun to see how these concepts we've learned intuitively really come together.

 

Edited September 30, 2018 by Cunjo Carl

[Ajiko]

Seems interesting, but what the conclusions?

Seems Skipper+Poodle pair must have top cost efficiency for LfOx rockets stages, right?

 

Edited September 27, 2018 by Ajiko

[Sharpy]

4 minutes ago, Ajiko said:

Seems interesting, but what the conclusions?

Conclusion is " Post still under construction! " and this promises some quite useful tables of what engines, what TWR on them per stage etc. Currently it's not quite "edible for general public".

 

@Cunjo Carl: Consider an option that appears rather frequently in the literature: heavy crappy lowest stage ("big dumb rocket"), and better, more efficient, more expensive engines on subsequent stages.

[Cunjo Carl]

11 hours ago, Ajiko said:

Seems interesting, but what the conclusions?

Seems Skipper+Poodle pair must have top cost efficiency for LfOx rockets stages, right?

The conclusion is that the most cost efficient engine and stage size can actually depend heavily on your TWR and on where you are in your mission, specifically on how many funds you've spent up to this point.

Most people already have an intuitive feel for this because KSP is fantastic for building intuition along that line, so this doesn't change any of the quintessential cost-effective rocket recipes like Kickbacks&Twinboar -> Skipper or Poodle -> Terrier -> Spark. It does explain 'why' though, which let's us figure do things like figure out exactly what the TWR cutoff is between Poodle or Dart being the better engine... Apparently above TWR=0.75 Dart's a better choice than Poodle if you're extremely deep into a mission.  Then, above TWR=2.5, skipper would become the better choice. If you happen to be interested in more, here's the old v1.4 mass efficiency plots! https://drive.google.com/open?id=1TQUZmyqrDSCHopF3RR1x-hTuCoQZ-O0U

Sharpy's right that the end idea is that we'll probably make plots/charts/programs based on this cost analysis as well, but for now the post is mostly for the chart and plot makers!

 

10 hours ago, Sharpy said:

Conclusion is " Post still under construction! " and this promises some quite useful tables of what engines, what TWR on them per stage etc. Currently it's not quite "edible for general public".

 

@Cunjo Carl: Consider an option that appears rather frequently in the literature: heavy crappy lowest stage ("big dumb rocket"), and better, more efficient, more expensive engines on subsequent stages.

Totally agreed about the big dumb booster! I got back into this project (which is mostly for KSP) thinking about the recently proposed Pyrios booster's F1-B engines, which are a _less_ efficient but cheaper version of the old Saturn F1 engine. The juxtapose of less efficient but more modern is a fun one. Taking it a step farther, the once-proposed Sea Dragon is the pinnacle of big dumb booster'ing, and I've always wondered at exactly what point do we go from smart money-conscious design to just wasting money with inefficient boosters fizzling away fuel and dragging around the spent tanks.

The effect that really struck me from all this was that a big dumb booster actually has a cost incentive to be even bigger than its most mass efficient setup, so in KSP rather than using the standard 2000m/s per stage they might be more cost efficient at 2300m/s per stage (numbers from a hat). They're happy to trade out lost efficiency from being oversized (and pushing empty fuel tanks) in order to make next stage a little smaller than it would otherwise need to be. This is even the case if the next stage isn't any more expensive! That's one aspect of the big dumb booster concept that really surprised me. Naturally if the next stage is pricier, the effect becomes even greater. So to answer the question 'how big?' the answer is apparently 'really big!'

 

[Mad Rocket Scientist]

This looks like an excellent post. It might take me a while to read it though. Are you using making history? I'd be curious to see whether the wolfhound is balanced in terms of cost.

[drhay53]

As is typical for me, I went for the Monte Carlo like approach. The discrete nature of rocket building makes it a little harder so I had to learn something new to do it.

I have tables of engines and fuel tank properties (mostly for SSTU), including SRBs, and I use simulated annealing to minimize the cost that will successfully launch a payload of a given mass. I have grown to love simulated annealing as an algorithm in place of brute force combinatorics in a large discrete parameter space.

Launch vehicle is considered valid if dV > 3600 and sea-level TWR is between 1.2 and 1.3. 

I commend you for the work in your post, but I'm a better empiricist than theorist

[Cunjo Carl]

19 hours ago, drhay53 said:

As is typical for me, I went for the Monte Carlo like approach. The discrete nature of rocket building makes it a little harder so I had to learn something new to do it.

I have tables of engines and fuel tank properties (mostly for SSTU), including SRBs, and I use simulated annealing to minimize the cost that will successfully launch a payload of a given mass. I have grown to love simulated annealing as an algorithm in place of brute force combinatorics in a large discrete parameter space.

Launch vehicle is considered valid if dV > 3600 and sea-level TWR is between 1.2 and 1.3. 

I commend you for the work in your post, but I'm a better empiricist than theorist

I happen to be working in particle physicist at the moment so Monte Carlo is near and dear to me as well! I'm not familiar with simulated annealing, I'll be sure to give it a look.

Have you been finding that TWR 1.2-1.3 is the most practical for your launchers? I'd been having a hard time getting the SaturnV style stroll to work for me in KSP.

 

On 9/27/2018 at 10:03 PM, Mad Rocket Scientist said:

This looks like an excellent post. It might take me a while to read it though. Are you using making history? I'd be curious to see whether the wolfhound is balanced in terms of cost.

I'd be happy to do a comparison with Wolfhound, I'm a little curious as well. Could you tell me the TWR you'd be interested in?

I think, along the same lines as you, I find the 'balance' of the new engines to be a bit of a misnomer when it comes to the Wolfhound , so it'll be interesting to see if its cost winds up being balanced.

[drhay53]

5 hours ago, Cunjo Carl said:

I happen to be working in particle physicist at the moment so Monte Carlo is near and dear to me as well! I'm not familiar with simulated annealing, I'll be sure to give it a look.

Have you been finding that TWR 1.2-1.3 is the most practical for your launchers? I'd been having a hard time getting the SaturnV style stroll to work for me in KSP.

I'm an experimental astrophysicist myself (supernova cosmology to be exact) though at the moment I work on HST and JWST data from a technical standpoint rather than mostly research. Once I learned python back in graduate school I found that solving a problem quickly with a monte carlo or empirical approach was usually good enough for me outside of work

I like a low TWR for liftoff because it "feels" the least "wasteful". I haven't optimized that this is true, but it seems like it's the right combination of engine cost with necessary fuel; too much TWR and you're probably overspending on the engine, and a cheaper option should be available. That's only true if the engines in your save are well balanced in terms of ISP, thrust, and cost though. I also like the low TWR because it works well with mods like GravityTurn for the ascent profile; typically my launchers are SSTO (occasionally requiring SRBs) and the circularization burn is less than 100 m/s. I feel like a low TWR and a small circularization burn is more "realistic". 

Since I've been playing with SSTU, I've noticed I frequently don't need SRBs for typical launches, but as you're going through the tech tree, the annealing frequently picks SRBs to punch up that initial TWR while using a more efficient engine once they come off. I don't simulate the ISP atmospheric curve in my procedure, but I do keep track of how long the burns are and the rate of fuel consumption, so the code knows how much sea level TWR will be there when the SRBs come off. I require it to be > 1, which in practice means it's a bit higher than that since usually the SRBs burn for 30-45 seconds and the ISP and thrust have come up a bit.

My philosophy for this particular code was "don't let perfect be the enemy of good enough"  With SSTU and my code I can build a launcher for any payload in < 10 minutes and I don't have to build a collection of payload-rated subassemblies. I did that once and it took me weeks. And then I wanted to change mods for my next career and most of my launchers didn't work. So I wrote this to be flexible about the options available in any particular save and so that I could build the most cost-efficient launcher for any payload super fast.

 

Edited September 29, 2018 by drhay53

[Mad Rocket Scientist]

I just finished reading the post, very impressive work. I don't yet know enough calculus to follow the derivation, but the results are very satisfying, and align with my intuition and the common sense advice on the forums. It's super interesting how the idea of using more expensive engines only when going very far shows up in the graphs.

Where does the NERV cross the poodle's line at 0.8 TWR? And does it cross it faster at 0.2 TWR?

On 9/29/2018 at 10:22 AM, Cunjo Carl said:

I'd be happy to do a comparison with Wolfhound, I'm a little curious as well. Could you tell me the TWR you'd be interested in?

I think, along the same lines as you, I find the 'balance' of the new engines to be a bit of a misnomer when it comes to the Wolfhound , so it'll be interesting to see if its cost winds up being balanced.

I'd be most interested in 0.8 TWR or thereabouts, but I wonder how it would look at 1.2 TWR or higher.

I agree, but I was surprised to see how well balanced the vector was for cost. (Only justifiable for reusable first stages)

Edited September 30, 2018 by Mad Rocket Scientist

[Cunjo Carl]

19 hours ago, Mad Rocket Scientist said:

I just finished reading the post, very impressive work. I don't yet know enough calculus to follow the derivation, but the results are very satisfying, and align with my intuition and the common sense advice on the forums. It's super interesting how the idea of using more expensive engines only when going very far shows up in the graphs.

Where does the NERV cross the poodle's line at 0.8 TWR? And does it cross it faster at 0.2 TWR?

I'd be most interested in 0.8 TWR or thereabouts, but I wonder how it would look at 1.2 TWR or higher.

I agree, but I was surprised to see how well balanced the vector was for cost. (Only justifiable for reusable first stages)

Thanks! And thanks for taking the time to read through.

It's true about the vector- I also really appreciate how it fills a funny niche of having the best thrust per size in the game. It was the first time I'd ever exploded a rocket from overheating while going up!

(plots in the spoiler)

Spoiler

 

 

 

So the interplay between Poodle and NERV seems to be an interesting one. Above a TWR ~.75, Poodle actually winds up being more mass efficient, and since it's already much cheaper, there's no real benefit for NERV above there. NERV winds up becoming useful for late missions with TWR=0.5, and early mission (ie. circularizing to LKO) with TWR=0.2 . At this low TWR=0.2, Dawn then becomes cost efficient for the late mission, or more likely for me, any time the NERV is too big and clunky! Dawn is much more _mass_ efficient than NERV everywhere beneath a TWR~~0.5, but it's also far and away more expensive. For that reason, it seems to mostly be useful for very late missions or times when a smaller engine is unavoidable.

Something important to note, the cost problems of NERV or Dawn can be heavily offset if they're used on dockable/refillable craft with a mothership. In these cases, if they happen to better on paper late mission, they'll actually be better anytime.

So it turns out Wolfhound is just massively better than Poodle in every way. It's actually really disappointing! It has better TWR, Isp and Cost per thrust than the Poodle, so on paper there's no cases where you'd ever want to use the old stand by, given the choice. It wasn't until I plugged the values into my spreadsheets that I noticed the cost per thrust of Wolfhound is only 4.48F/kN, which is easily the best for any liquid fueled engine in the game! It even edged out the previous cheapest Reliant at 4.58, so Poodle didn't stand a chance (5.2F/kN). Out of some puckish curiosity, I plotted Poodle and Wolfhound alongside Thud (representing KSP's worst medium-sized LFO engine), and well, I think the plot almost speaks for itself! Wolfhound is as much better than Poodle as Poodle's better than Thud!

I of course have no idea, but I've always wondered if they designed the specs for new engines like Wolfhound so that certain of their missions would work out conveniently. That's all I can imagine, at least? Or someone just had a really big crush on the AJ10!

[Mad Rocket Scientist]

Wow, looks like it's better than the poodle in every case, and better than the NERV in a surprising number of cases. How does it compare to the terrier?

[Cunjo Carl]

48 minutes ago, Mad Rocket Scientist said:

Wow, looks like it's better than the poodle in every case, and better than the NERV in a surprising number of cases. How does it compare to the terrier?

Terrier is slightly worse than Poodle in all three categories (Isp, TWR and Cost/thrust), so it'll always wind up being slightly less cost effective than Poodle on a per-ton-of-payload basis. That means that Wolfhound is better than Terrier to an even greater degree than it's better than Poodle! As far as I can see, Wolfhound is the simply the best by a wide margin for TWR=~0.7 to TWR=~3 in vacuum.

I haven't run the numbers, but just looking at its stats, Skiff is pretty fantastic looking as a launch engine (used as a stage-and-a-half with Kickback boosters), being almost as good on the launch pad as the two present reigning champions Skipper and TwinBoar, while it manages to handily outclass them in TWR and Isp above 8km when the kickbacks would separate.

The engines from the original game have a lot of interesting niches and nuances, which I think we've all found over the years. As far as I can see (and saying this as someone who hasn't played with the new engines), on paper it looks like Kickback, Skiff and Wolfhound could outclass everything else from the launch pad up, assuming TWR>0.7 .  It's a bit of a dull conclusion, and to be honest, I think it's one of the big reasons I never adopted the new engines, despite having been a been a big fan of the other new parts and features. For others though, I'm glad they make for some exciting missions.

[Nich]

Could the wolfhound be good because of economics of scale?  Also if you are using the wolfhound you have an enormous payload.  Perhaps you need to be more efficient with your payload.  I have always wished I had versions of the terrier and poodle that were 70% smaller.

[Cunjo Carl]

1 hour ago, Nich said:

Could the wolfhound be good because of economics of scale?  Also if you are using the wolfhound you have an enormous payload.  Perhaps you need to be more efficient with your payload.  I have always wished I had versions of the terrier and poodle that were 70% smaller.

Could be, the economics of scale definitely appear in the engines, where larger versions are typically a little bit better on the whole than smaller versions. The most direct example is Poodle/Terrier or Mamoth/Mainsail, but it's definitely a trend in KSP that bigger is in general a bit better. That said, despite them sharing a niche, Wolfhound (2.5tonne) is dramatically better than Rhino (9tonne) in almost all conditions they'd normally be used in, so I don't think economics of scale quite cover it here...

(** Here using economics of scale to refer to larger objects being cheaper per mass rather than the other meaning of things being cheaper by the dozen)

As for payload efficiency, it's definitely super important. But fortunately, the whole kit'n'kaboodle up at the top is expressly to find the best possible payload efficiency, so that's all covered! It's been a long time coming.

I also wish we could tweak the scale of the engines. I'd totally want a little baby twin boar for trips to Eve!

Edited October 2, 2018 by Cunjo Carl

[farmerben]

Outstanding post!  I'm slowly figuring out the answers to my first few questions.  I'll have to reread the entire thing again sometime.  Thanks.