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Optimizing for cost


Sivert3

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With the career mode being bounded by costs, it is obviously very

beneicial to be sending up rockets that give the heaviest payload for

the least amount of funds into orbit. I realize this doesn't really fit

well into the game as the gameplay isn't balanced for this kind of

optimizing rocketry.

The first hurdle is to define what is cost efficient, and after a bit of

testing I've ended up with a definition of cost efficiency of delta v *

payload / cost and I think that is a reasonable metric. It gives a

number that compares well for large and small rockets.

Onward to the point. I've been cooking up a python script for

calculating the most cost efficient configuration of a two stage rocket

where both stages use the same type of engine, for a set target delta v,

payload and TWR. It's not usable for designing rockets as two stages of

the same engine type isn't efficient for getting to orbit. But anyway

here are two examples.

Two stages with LV-T30 engines lifting a command pod to orbit:

VEsBw6p.png

This diagram shows how much fuel the lower stage (red line) of a two

stage rocket with a given fuel in the upper stages needs to have in

order to yield 4500 delta v. The payload carried is 0.84 t (a command

pod), the TWR is at least 1.5 for both stages, and the engine used is

the LV-T30. The cost efficiency of each particular configuration is the

green line.

From the graph the peek efficiency is obtained by using 4 tonnes of fuel

in the upper stage and 6 tonnes of fuel in the lower stage. Resulting in

a cost efficiency of 0.65 and a total cost of about 4500 * 0.84 / 0.65 =

5815 funds. This is off from the real cost due to few error sources and

it's only based on the engine, fuel and fuel tank costs.

Two stages with LV-T30 engines lifting 5 tonnes to orbit:

M065CY8.png

All the parameters for this is the same as the previous graph except for

payload which is 5 tonnes. Peek efficiency is reached with 7 tonnes of

fuel in the upper stage and 25 tonnes of fuel in the lower stage. Since a

single LV-T30 can't lift all that, the bottom stage has 5 of them. The

efficiency is about 1.26 which is almost twice that of the previous

example, meaning that if you lift up 5 tonnes to orbit with the previous

rocket (requiring 6 launches) it will cost almost twice as much as one

launch with this.

I'm thinking about looking further into this and particularly comming up

with ways to compute cost effective configuration for different kinds of

stagings and engines. I'm not aiming for an optimal solution to this, as

I don't think that is feasable (or sensible). Maybe throw up a

webservice at one point future. What do you think?

Edited by Sivert3
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You're overthinking it.

If you really want something to be cost-efficient, it should be fully recoverable. Most of the cost of a rocket is in the hardware, not the fuel, so if you're throwing away the hardware that's a huge amount of Roots down the drain. Multi-stage rockets, obviously, lose at least one entire stage, so you should be trying for a single-stage design and figuring out how to keep from losing THAT as well.

So try this instead:

> Make a simple SSTO rocket. Mainsails are good for this, as they have a decent ISP and more than enough thrust, but any engine with an ISP of ~350ish should be enough. Stick enough fuel tanks on top to get an initial TWR of around 1.7-1.8. (As it loses fuel you'll quickly reach the optimum TWR of 2.2ish.)

> Put parachutes on it. Small girders on the side make a good place to mount them, since you'll be using the top to carry your payload.

> Put a small probe core on it, along with a few batteries and a few of the smallest solar panels. (That way, the probe core doesn't run out of power, especially if you're using SAS on the booster.)

> Give it something to land on. Four XL girders, pointed downwards, work just fine for flat terrain. (On slopes they tend to "walk", which doesn't break anything but is just annoying.)

This design gives enough delta-V to get to a nice 75km circular orbit, release your payload, and then safely deorbit afterwards. Since it's a circular orbit, you can deorbit whenever you want, which means you can aim to land at/near KSC (greatly increasing your recovery percentage). All you'll lose is the cost of the fuel, but that's negligible compared to the rocket itself.

This concept is very scaleable; I've made giant rockets massing twenty or thirty thousand tons (21 linked 5m KW Rocketry stacks) capable of lifting payloads of several thousand tons. The fact that I can land these back near KSC for recovery makes a tremendous difference in my finances. Here's a 9-stack version:

RPjQfab.png

Obviously, SSTO spaceplanes are even better, but that requires a pretty large tech investment first and has significant limitations on payload size.

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However rocket SSTO uses a lot of fuel.

An recoverable core with SRB is more effective as the core only need around 3000 m/s, current favorite is skipper +2+2 of the large SRB, 2 at 100% trust and 2 at 70%.

Skipper with 1.5 orange tank, TWR without SRB is below 1 at launch who is common for real rockets with SRB.

Spaceplanes for crew, resupply and small payloads.

If you use kethane or other mining its an good idea to launch large interplanetary ships with SRB and a booster engine like a mainsail at bottom docking port.

At worst case two liquid boosters with crossfeed.

you want to reach orbit pretty dry and refuel with free fuel before continue.

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However rocket SSTO uses a lot of fuel.

An recoverable core with SRB is more effective as the core only need around 3000 m/s, current favorite is skipper +2+2 of the large SRB, 2 at 100% trust and 2 at 70%.

Skipper with 1.5 orange tank, TWR without SRB is below 1 at launch who is common for real rockets with SRB.

That's very similar to the vessel I designed today. However, mine also has 16 basic jets to take it to ~15km before I even touch off the boosters. All four boosters at 100%. Ultimately, it consumes 1.5 orange tanks and can deliver one full orange tank (or any similar weight). Can it be that the jets effectively only carry their own weight?

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However rocket SSTO uses a lot of fuel.

An recoverable core with SRB is more effective as the core only need around 3000 m/s, current favorite is skipper +2+2 of the large SRB, 2 at 100% trust and 2 at 70%.

If you're throwing away the SRBs, then that's more money down the drain, and carrying them to orbit and back is wasteful for obvious reasons. Besides which, SRBs are inherently inefficient; the best possible ascent follows the terminal velocity curve. Go too fast and you're wasting too much energy fighting drag, and go too slowly and you're wasting too much energy fighting gravity. Get it right and you can save a few hundred m/s on an ascent. Without the ability to adjust your thrust, SRBs just don't do very well at that. They're still good for that initial burst when you're trying to get up to speed, but not for the rest of the ascent, so I stopped using them long ago.

Obviously, following a precise velocity curve is difficult in stock; I used to write down the various benchmarks, so that I'd know how fast to be travelling at altitudes of 1km, 3km, 6km, etc. But ever since I started using the Engineer mod, this became much easier since it has the ratio of your current velocity to the local terminal velocity as one of its data points.

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The largest SRB produces a bit more impulse than a Mainsail with 200X-16 and 200X-8 tanks, for less than 20% the cost. That SRB costs about 160% of the amount of LF+O that does the same job with a pre-existing engine, but when you factor in the need for more tanks and engines, as well as even more LF+O to lift the LF+O you're adding, disposable SRB's become quite cost effective.

You can tune the thrust to avoid going too far over or under terminal velocity, and if you use enough, you can avoid even firing your LF engines until about 10km, about when you start your turn. That makes it a lot easier to keep all tanks and LF engines, where the bulk of the money is, in stages that make it to orbit so they can be returned for a refund.

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SSTO rockets do indeed use tons of fuel. But have a look at the costs of fuel vs. parts, especially with ships that have a lot of parachutes, wings, or electronic bits on them. While I lack an exhaustive mathematical analysis, I have a fair amount of confidence that it's much cheaper to build a big reusable rocket than a smaller expendable one - especially if you do what I did and adopt a strategy to improve recovery payouts (or just land really precisely).

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You can tune the thrust to avoid going too far over or under terminal velocity

Except that terminal velocity changes a lot as you go up, so a thrust setting that's fine at low altitude might be totally unacceptable at high, or vice versa. It's just a bit more efficient to use liquid engines, in that regard, even if you use multiple SRB types with varying thrust profiles and timings. Yes, SRBs are dirt cheap by comparison on a thrust-vs-thrust basis, but if you're wasting a bunch of delta-V on the way up due to an inefficient ascent then you're having to use quite a bit more engine power/fuel than you should to compensate. And if you're doing that, the exponential nature of the rocket equation starts to kill you.

Plus, it's a LOT easier to tweak the ascent of a liquid engine when you use a different payload than the one it was initially designed for. I've got a set of four liquid SSTO boosters for various payload mass ranges, but within those ranges are a lot of room. For instance my basic booster, the Space Hippo 1a, can lift anything up to about 50 tons into a circular orbit. But in terms of throttle settings, there's a big difference between a 50-ton payload and a 10-ton one; if you were using SRBs to do the same thing, you'd have to do all sorts of adjustments to get even close to an optimal flight curve.

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If you're throwing away the SRBs, then that's more money down the drain, and carrying them to orbit and back is wasteful for obvious reasons. Besides which, SRBs are inherently inefficient; the best possible ascent follows the terminal velocity curve. Go too fast and you're wasting too much energy fighting drag, and go too slowly and you're wasting too much energy fighting gravity. Get it right and you can save a few hundred m/s on an ascent. Without the ability to adjust your thrust, SRBs just don't do very well at that. They're still good for that initial burst when you're trying to get up to speed, but not for the rest of the ascent, so I stopped using them long ago.

Obviously, following a precise velocity curve is difficult in stock; I used to write down the various benchmarks, so that I'd know how fast to be travelling at altitudes of 1km, 3km, 6km, etc. But ever since I started using the Engineer mod, this became much easier since it has the ratio of your current velocity to the local terminal velocity as one of its data points.

Yes, however SRB give far more dV/ funds than rocket fuel, yes its unrealistic but fuel cost might include ground service and other costs too.

My default medium launcher in 0.235 used an 24x4 with 1.75 of the 3.75 meter tank. it could lift 20 ton to LKO, with four SRB it could lift 40. Did not use multiple trusts on SRB.

Now to make an SSTO capable of 40 ton I would need to double the size of the launcher.

It would cost me (5184+1425)*1,75 =11566. Four KD25K (largest SRB) with decoplers and seperatrons is (1800+50+700)*4=10200. Yes I could save another 1500 with mounting the SRB on each other. Another benefit is that the up front and failiure cost of the SRB assisted launch is the half.

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Another benefit is that the up front and failiure cost of the SRB assisted launch is the half.

However, with an SSTO rocket there's less need of a "failure cost", because there's no danger of catastrophic collisions at all during the ascent. No separating stages crashing into your rocket and knocking the engine off, for instance. If you're playing with quickload/revert disabled, this makes a huge difference to your bottom line.

Honestly, each approach has its own pluses and minuses. One of the reasons I prefer the liquid method is that I like building huge payloads (thousand-ton space stations, 600-ton rovers...) and a liquid approach is much more scaleable due to the low part counts. Even with chutes, landing gear, etc. my 9-stack design (~500 ton payload) only has about 250 parts. Being able to keep part counts low is essential, at least until we get the 64-bit Unity engine.

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@Spatzimaus: I don't think making a single stage to orbit rocket using liquid rocket engines is going to be cost effective. I don't have a proof for this, but I do have an example.

Let's say we want to push 5 tonnes into orbit, for sake of simplicity I'm going to limit myself the LV-T30 liquid engine, it's stats are not the best for this purpose but lies somewhere in the middle. By doing a bit of math I have calculated that I need 74 tonnes of fuel and 7 rocket engines to pull it off in one stage.

I build an SSTO from those figures and sure enough the math checks out.

O5KUVoOl.png

How cost effective is it? If I land the whole thing on the launch pad the only expense would be the fuel. Which is 77 tonnes * 91.8 funds/tonne = 7068 funds. The efficiency is thus 4500 dv / 7068 funds * 5 tonnes = 3.18. (Note that the 2 stage liquid rocket efficiency calculations in my previous post did not take account to recoverd costs in the upper stage, so you can't really compare this to the 1.2 figure derived there)

After making a few modifications to my python script I've made it calculate the cost efficiency of a 2 stage rocket with the LV-T30 in the upper stage and the RT-10 solid rocket engine in the bottom stage. The resulting graph of upper stage fuel to cost efficency is thus.

5XJHhLb.png

Which promises 5 tonnes to orbit at an efficiency of 5.65 using a dual engine upper stage with 19 tonnes of fuel and 7 solid rocket engines in the lower stage. Of course these numbers are useless if they can't be demonstrated to hold in practice. To make the construction simpler I chose a different set from the same graph: dual LV-T30 with 18 tonnes of fuel in the upper stage and 8 RT-10 Solid rocket engines in the bottom with a calculated efficiency of about 5.41.

iEXqPYxl.png

And sure enough these numbers do check out.

8YhO9lpl.png

The cost of this rocket if recovered on the launch pad is the cost of 18 tonnes of fuel and the whole lower stage. Which is 18 * 91.8 + 8 RT-10 + 8 Struts + 2 stack sepparators = 5388 funds. And the efficiency is 4500 / 5388 * 5 = 4.17. Which is about 30% off from the calculated value. There are multiple sources of error present, wich accounts for the inaccurate result. But all in all it's about 24% cheaper than the SSTO. Which proves that a rocket using a stage of solid rocket engines can be cheaper.

I did not experience a stage of Solid Rockets as being hard to get right. I made a wild guess of thrust at 50% and it followed an accent that was slightly to slow in the start of the burn and a bit to fast near the end of the burn. Because the thrust is constant you're bound to have this kind of behaviour with solid rockets, but doesn't make them unusable. Just slightly less optimal.

Edited by Sivert3
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Except that terminal velocity changes a lot as you go up, so a thrust setting that's fine at low altitude might be totally unacceptable at high, or vice versa.

Which is balanced by the reducing mass of the SRBs increasing your TWR as you ascend.

Spaceplanes and vertical SSTOs stomp all over this, though. I can put fuel into orbit for less than √1 per unit.

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Sivert3, this is all great data but could you run a plot with a more interesting payload? 5t and that is too small for anything outside Kerbin's system so what do you make of the 10 - 50t range? As far as I've been able to check (quite a bit but not enough to publish) SRBs stop being cost-effective for anything but small loads, making SSTO rockets quite attractive.

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However, with an SSTO rocket there's less need of a "failure cost", because there's no danger of catastrophic collisions at all during the ascent. No separating stages crashing into your rocket and knocking the engine off, for instance. If you're playing with quickload/revert disabled, this makes a huge difference to your bottom line.

Its not that hard to ensure no colisions. Separatrons are rather afordable at 50 funds a pop. For the largest solid boosters you need 2 separatrons per booster cluster at or just above the centerpoint and it will ensure they never impact the core rocket they are attached too. I have sub assemblies setup for my solid booster needs. one is just a single booster premounted on a decoupler with a pair of separatrons prepositioned and the other is my quadpack arrangement. Quadpack is the same booster on decoupler but has 3 additional boosters straped too it, left right and directly oppisit the decoupler with the separatrons mounted on the outside pair. A pair of struts top and botom compleats the package to ensure things dont wobble in flight. I've never had a booster impact the next stage in this arrangement that wasn't caused by an extreme control input at the moment of separation.

If the ship has alot of rotational control authority and I'm in the middle of a hard turn as I drop the boosters I can slam the falling boosters with the rocket and break something. Any other time the boosters drop safely even if I'm flying horizontal fairly low in the atmosphere with one booster directly above the core. The separatrons have enough kick to lift it up long enough for the main rocket to get out of the way before gravity pulls it back down. With proper positioning you can safely drop solids even without seperatrons as long as your still near vertical or in thin atmosphere when separation happens, the separatrons just make it a sure thing.

My own experience has been that its significantly cheeper to get any significant payload to orbit useing a disposable solid first stage with a recoverable core that gets the rest of the way to orbit and then returns. To make a pure liquid SSTO ends up with a significantly more massive vessle to return and a smaller mass fraction to orbit. While you technicly recover more of the craft with a SSTO its a much more expensive craft and more sensitive to landing errors. you need more engiens to lift the extra fuel and potentialy more parachutes (asumeing not useing a powered landing) adding cost that is subject to the % return loss. If you can nail the runway/launchpad every time then this does not apply but if you miss by 100-50km that extra cost can add up and you still have extra fuel costs. The solids on the other hand are dirt cheep, most of the cost is the fuel and actualy recovering them is a negligable return. The separator is normally more valuable than the depleted booster.

And on the subject of ineffiency due to not being able to precisely follow terminal velocity and lower ISP rockets, I'd argue that its not a huge deal. Yes most pure solid stages end up going faster than terminal velocity and loose some effiency to drag. However if you loose out on 10-15% effective DV due to drag you gain it back by the fact that the solid was significantly cheeper than the liquid alternitive for the same impulse. You come out ahead on ISP for the same reason, the cost per dV on the solids is just that much lower than for the equivelent in liquid fuel you just come out ahead even with drag losses added in.

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...My own experience has been that its significantly cheeper to get any significant payload to orbit useing a disposable solid first stage with a recoverable core that gets the rest of the way to orbit and then returns....

As with my question to Sivert, can you say what you mean by 'significant payload', as I'm finding SRBs don't add significantly to payloads over about 10 or 20t. I'd love more information and examples for this, whether I'm right or wrong, because I'm about to start writing a tutorial on exactly this subject. It would be nice to get it right!

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As far as maximizing efficiency goes, I find that almost universally everything should be at 100% thrust - if you go over terminal velocity at a low altitude when at this level, it means that each engine involved could be lifting more fuel, so you can make the stage bigger or reduce the number of engines. This includes SRBs - if 100% thrust is too strong, then you actually need (brace yourselves):

LESS BOOSTERS.

There, I said it. The greatest blasphemy in the entire history of the KSP community. But in some cases, it is actually true. ;P

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As with my question to Sivert, can you say what you mean by 'significant payload', as I'm finding SRBs don't add significantly to payloads over about 10 or 20t. I'd love more information and examples for this, whether I'm right or wrong, because I'm about to start writing a tutorial on exactly this subject. It would be nice to get it right!

I have multiple examples from 0.235, where strapping SRB on SSTO rockets let me close to double the payload:

24x4 bases system 20 to 40 ton with 4 SRB, 7 24x4 system 120 ton to 200 ton with 12 SRB.

Rarely lift so heavy payloads in 0.24 so my default is a skipper, two orange tanks and 4 SRB. who does 10 ton to LKO,

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If you want to go hardcore on this you also have to throw ROI into the mix. Sure, solution "A" might be $200,000 cheaper than solution "B" when all is said and done, but when solution "A" means coughing up $4,000,000 for a Duna mission and "B" means $1,000,000 it also means that you can run 4 "B" missions for the price of 1 "A" mission while your precious funds are tied up in a 2 year mission. If each of these missions can generate $300,000 in income then it's penny-wise and pound foolish.

Cost effective is not always the most economic solution.

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As with my question to Sivert, can you say what you mean by 'significant payload', as I'm finding SRBs don't add significantly to payloads over about 10 or 20t. I'd love more information and examples for this, whether I'm right or wrong, because I'm about to start writing a tutorial on exactly this subject. It would be nice to get it right!

How does an orange tank to orbit strike your fancy(36t)? just loaded this up in a normal mode save with science/funds cheated in for the test.

Javascript is disabled. View full album

Solid booster

cost 77,008

Recovery at 1.7km 97.9% 40,755

Cost to deliver orange tank to orbit 36253

Liquid booster SSTO

cost 156,956

Recovery at 1.8km 97.9% 116,549

cost to deliver orange tank to orbit 40,407

Still a 10% or so saveings for a 36 ton payload. I probably could have goten to orbit with a tad more fuel on the solid if I'd tweekd the clusters down 5-10% thrust. Solids were max thrust and mainsail was max thrust once it fired stage 2 and was going a bit faster than I liked at solid burnout. Liquid SSTO was full throttle all the way, peaked at about 102% atmospheric efficiency. Incidently solid variant was going 8m/s once the chutes deployed, only needed a tiny thrust to softland. Liquid variant on the same amount of chutes was going 15m/s and needed a fair bit more babying to land safely. Liquid variant is 1mainsail core with trilateral symmetry on the side tanks, mainsails on them as well.

Edited by merendel
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For a medium payload example I put in the numbers for a 50 tonnes to orbit rocket in my python script with the Rockomax Mainsail engine in the upper stage and the SRB-KD25k solid rocket in the lower stage. Resulting in a suggested 126 tonnes of fuel, 2 mainsails and 12 KD25k rockets with a calculated efficiency of 6.8.

qyXcvDzl.png

And at this point it should come as no suprise that it indeed reaches orbit with 50 tonnes of cargo.

WQ2qXhEl.png

Payload delivery cost = 126 * 81.9 + 12 KD25k + 24 struts + 24 sepratron = 34,128

Efficiency = 4500 / 34 128 * 50 = 6.6

This is assuming you recover 100% of the upper stage.

For a more beefy rocket I put in the numbers for a 500 tonne payload with the S3 KS-25x4 Engine Cluster instead of the mainsail. The suggested configuration is 1401 tonnes of fuel, 10 engine clusters and 98 KD25k solid rockets. With an efficiency of 7.3. The interesting thing here is that when I calculated the efficiency of the equivalent SSTO it's around 6-7.

Efficiency of merendel's rockets

To get a more comparable numbers I've calculated the actual cost efficiencies of the two rockets.

SRB Rocket

Payload: 36 tonnes

At Launch cost: 77,008

Payload cost: 12,800.

Recovered funds: 40,755

Efficiency: 4500 / 23,453 * 36 = 6.9

SSTO Rocket

Payload: 36 tonnes

At Launch cost: 156,956

Payload cost: 12,800

Recovered funds: 116,549

Efficiency: 4500 / 27,607 * 36 = 5.9

I must say these are rather impressive results. Nice work.

SSTO vs SRB assisted rockets

What we can learn from these results is that SSTO and SRB assisted rockets are more or less equal in costs. The SRB has one advantage over the SSTO, the up front investment is around half. Which also halfes the cost of a total failure. But this is also the weaknes of the SRB. I spent around 10 complete rockets assemblies, and a few launch pads trying to get the SRBs from colliding into the core. The most troubleling being the Sepratrons destroying the core fuel tank. Ouch! Proper design and a bit of experience mitigates these risks greatly.

Scaleability of rockets

The rocket equation is normaly regarded as tyrant and exponential. But the truth is that it has two sides in KSP, one tyrant and one lenient one. The tyrant is met when you try to increase the delta v of a rocket, especially when met with the requirement of being able to take off the ground. More fuel needs more engines to lift it. And the size and costs increases exponentially.

The lenient side (at least in KSP) is when you try to increase the payload of a rocket that can already reach orbit. Doubeling the payload is as simple as doubeling the fuel and engines in each stage. Which net aproximantly double the costs. But since the rocket is bigger, it's possible to take advantage of bigger engines with better efficiencies. Which makes the rocket slightly cheaper per tonne of payload.

There is also the demonic side of the equation. Namely when you try to increase the delta v of a single stage where you have a thrust to weight ratio that must be met. Which is the case for an SSTO rocket. When you increase the delta v of such a rocket you meet a brick wall where the amount of fuel and engines needed explode into infinity. This is the reason why a SSTO using only solid rockets isn't possible.

Thus depending on how you're trying to scale your rocket the size and costs will either scale proportionally, exponentially or be hyperbolic.

Edited by Sivert3
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Here's a mostly solid fuel launch of a 40t fuel station to LKO I did a while ago for a challenge: link

It could be optimised some more, and has no recovery - but still has under 1000Kr/tonne launcher cost. Going to larger payloads than this would have me wanting some liquid engines to provide vectored thrust.

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I don't have the game in front of me at the moment, but I've had some cost effective lifters that use a full size 3.75m tank and a mainsail, ringed by either 6 or 8 of the big S1 SRBs. I don't bother recovering the SRBs, but it's fairly time and cost effective to recover the central core. By having the mainsail going at the same time as the SRBs, you have the ability to stay in the sweet spot in terms of terminal velocity.

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I agree that a pure solid SSTO isnt possible. No mater how you slice it you'd never be able to achieve orbit as you'd burn out before you could get high enough so you'd reenter atmo. The big booster with just a stayputnik technicly has the dV to reach orbit you just cant make it achieve a stable one. It might work with carrying a couple smaller SRB's for use at AP to try and achieve and orbit but even if it works it isnt what I'd call a funtional payload. Even scaleing wouldnt help much as mass fraction is already about as good as its possible to get.

I have done a nearly pure solid to orbit but I used 2 stages on it. a cluster of the big SRB's in the central core and a bunch of my quadpacks straped on as initial stages. Could get orange tanks to orbit with only a 909 attached to finish the circularization, didnt take much fuel out of the payload at all.

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I've long had the common habit of building ships in 3 components/stages. 1. Launch. 2. Achieve orbit. 3. Actual mission craft. Recovering the launch stage would be quite a challenge, as it is usually quite massive and ejected while still in atmo, where it's automatically eliminated once it gets more than 2.5kms from the active part of the ship, so I don't try. Since economics were added, I just make that 2nd stage capable of de-orbiting and soft-landing after delivering the mission to orbit, which recoups a portion of the original cost. It's a compromise which doesn't require massive changes to my construction style and existing, tested designs.

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