Effect of TWR, TVR, and DVR on Orbital Launch Efficiency (deltaV)

[Anquietas314]

I'm curious about a further question: say you're on the lip of a crater, infinitely deep, and you won't reach orbital velocity by the time you get to the other side. Should you follow this grazing approach, or should you fall into the crater a bit (thrusting due sideways) and recover later on?

(disclaimer: speculation) I think that would depend on how wide the crater was... though, if you can't reach orbital velocity by the time you get to the other side of the crater at the lip, why would you expect things to improve by going deeper into the crater?

[numerobis]

My speculation is that you'll do better by following the straight line from lip to lip, than following the curve that maintains constant altitude. But I'm not sure.

[Anquietas314]

My speculation is that you'll do better by following the straight line from lip to lip, than following the curve that maintains constant altitude. But I'm not sure.

Hmm yeah I guess that might make sense. That would need to be one big (and deep - but that's assumed) crater for the planet's curvature to really matter though. I think the bigger ones on Mun might be the sort of proportion you'd want, but I don't think those are deep enough to pull it off.

[Torquemadus]

Yes, but if you have low TWR (which in general, should be cheaper in terms of funds, since it requires fewer engines), you cannot wait until you reach apoapsis to burn horizontal, since you cannot accelerate fast enough. Thus, you probably have to start burning horizontally before, and once you reach apoapsis, then pitch up to the angle necessary to keep "time to apoapsis" as close to zero seconds as possible

Also, yeah it is scary. Sometimes you pass over a hill by only a few meters Gets the blood pumping...

This is an accurate description of how I perform this kind of ascent. Blood pumping indeed, and I wouldn't want it any other way!

I've tended to view the ascent as consisting of two manoeuvres. The first manoeuvre is the near horizontal take off burn that places me on a sub-orbital trajectory. The second is the circularisation burn.

The take off burn is the part of the manoeuvre that is most hurt by low TWR, as I will inevitably waste some fuel fighting gravity. The circularisation burn is an orbit change manoeuvre, and is most hurt by factors such as dry mass and specific impulse. Thrust affects the efficiency of the circularisation burn, but carrying heavier engines around also increases my dry mass, which hurts efficiency.

In practical game terms, adding extra engines to my Mun lander increases it's dry mass, which has a knock-on effect on the size and cost of the launch vehicle I need to get it there. I play on custom difficulty, so I tend to launch missions that are very marginal in terms of mass.

[Bryce Ring]

Nice work.

I'm curious about a further question: say you're on the lip of a crater, infinitely deep, and you won't reach orbital velocity by the time you get to the other side. Should you follow this grazing approach, or should you fall into the crater a bit (thrusting due sideways) and recover later on?

correcting would change your trajectory (AP), raise it.

I have a feeling that this would probably only be useful if that AP point is exactly where you wish your final AP to raise from.

Curious how it would effect the Dv expended if it were other than there.

this has been an interesting development of discussion. I only wish I could understand all the Math involved. (only did Veggie math (Minimum 2Units math) at High school )

[Sean Mirrsen]

Why on earth would i have to provide this? That would defeat the whole point of non-dimensionalization! The player can calculate what their TVR is for the engines they are using and the body they are departing from.

It might be more useful to use ISP and then have separate plots for each body in KSP, but it is definitely more convenient to have one chart, which can be used to generate all those graphs for all those bodies in KSP, and in RSS, and for any new planets that are made etc...

Besides, the definition of TVR teaches us something about the physics: that a craft with ISP=400 on a planet with V_orb = 1000 m/s will have the same efficiency as craft with ISP=800 on a planet with V_orb = 500 m/s.

Except there's no orbital velocity for a given planet as such, there's only the velocity required to maintain whichever orbit you're trying to reach.

Also, the usefulness is what I'm getting at here. The chart by itself is a curiosity. In order to be useful as a piece of gameplay advice, it has to be useful. TVR is harder for a player to calculate for a given craft and situation than an I_sp/TWR-based chart is to generate for a given planet, with the advantage that the latter is useful for any ship on that given planet, with no further calculations necessary - I_sp and TWR numbers are readily available even without Mechjeb/KER et al.

That's not the definition of DVR. DVR is the ratio of deltaV your craft carries if it were floating in space in the absence of any gravitational field to the amount of deltaV required to get into orbit using an impulse burn i.e. minimum possible. It could easily have been the case that carrying more deltaV would effect efficiency and require higher TWR. Regardless, it is good to know and show it doesnt.

You... basically said the same thing. Ratio of the delta-V you have, versus the delta-V you need, plus fluff.

We are talking past each other. I agree, they will need less mass of fuel-- that is what increasing ISP does. However, knowing this, players are likely to decrease the mass of their fuel accordingly (!) such that they will have the same total deltaV, not realizing that they need more deltaV to get into orbit due to reduced efficiency (assuming they also adjusted their engines to maintain the same TWR)....

I was just correcting you, you said they'd need more fuel, not more dV. I did say I agreed that it's a rarely acknowledged notion.

That is true but not relevant at all. My model requires inputs of TWR, TVR, and DVR. The players can get these numbers from their craft and use it to assess whether their design has enough deltaV.

It might not be possible to get any arbitrary combination of TWR, TVR, and DVR from the parts available in KSP, as you pointed out. The player either (a) supplies these numbers using current design to assess current design, ( uses the results to generally guide TWR selection (i.e. keep it 1.4 < TWR < 1.6) or © uses the results to try to improve design, but has to cleverly tradeoff TWR, TVR, and DVR to improve efficiency since adding removing one part might improve TWR, but reduce TVR and/or DVR...

Therein lies the problem, again. DVR has no effect on your efficiency chart, yet it obviously has an effect on the TWR curve, therefore it has an effect on the time spent on the ascent, therefore it has an effect on your chosen definition of efficiency. Consider two craft with the same TWR and I_sp, but different DVR. Say, each has a starting TWR of 1.5, but one has a DVR of 2, and another has a DVR of 5. I can't be assed to dredge up the correct numbers at this point, but let's assume it means the latter craft has twice as much fuel for its mass. It means that by the time the first craft reaches orbit, it will be dry - its TWR will be maximum - say, 6. Assuming the same dry mass/wet mass ratio between the two craft, the second craft will still be pushing half-full tanks, having an effective TWR of somewhere around 3.5-4. This changes the time spent on ascent, and changes the TWR curve, which should have an impact on efficiency the same way high ISP does.

My formula assumes nothing about the craft, except its TWR, TVR, and DVR. If you can get those values, then the model will predict results accurately. For the case you described above, where the only mass of the ship is fuel, then DVR = infinity since m_wet/m_dry = infinity since m_dry = 0. Thus, in order for my model to "make" those assumptions, I will have to input DVR = infinity. Otherwise, it assumes realistic values for DVR.

Assumes nothing about the craft, yet "realistic values for DVR"? What does "realistic values for DVR" even mean, in this context? DVR depends entirely on the design of the rocket, a dedicated lifter will have just enough dV to reach orbit, while a lifter-pusher stage will have extra dV for orbital maneuvers, and an SSTO will have enough to go up, maneuver, and get back down. That your formula produces results independent on DVR, especially per my response to the previous quote segment, would seem to point to an error in your formula somewhere. Edited December 27, 2014 by Sean Mirrsen

[numerobis]

TVR is interesting and at first glance surprising (though the explanation makes sense). I don't get Sean Mirrsen's objection: TVR depends on the specifics of your spacecraft and where you landed, so no lookup table can be ever be complete, but it is easy to calculate.

DVR I don't get at all. If you have enough deltaV, you can make orbit; if you don't, you can't. If your equations told you that bringing extra fuel and compensating for its mass by adding thrust would overall reduce the deltaV requirement, you'd be looking for the error in your equations, because that's nonsensical. So it's a check on things being reasonable, but it doesn't deserve being the first plot or even any more than a footnote.

In these plots you're not taking account of sidereal rotation speed. Eeloo and Gilly give you more than 10% of orbital velocity for free, Minmus and Dres more than 5%. I wonder how you'd fit that detail in nicely.

[GoSlash27]

TVR is interesting and at first glance surprising (though the explanation makes sense). I don't get Sean Mirrsen's objection: TVR depends on the specifics of your spacecraft and where you landed, so no lookup table can be ever be complete, but it is easy to calculate.

Agreed.

DVR I don't get at all. If you have enough deltaV, you can make orbit; if you don't, you can't. If your equations told you that bringing extra fuel and compensating for its mass by adding thrust would overall reduce the deltaV requirement, you'd be looking for the error in your equations, because that's nonsensical. So it's a check on things being reasonable, but it doesn't deserve being the first plot or even any more than a footnote.

Also agreed. I prettied up arkie's table and made it more user- friendly. While I was at it, I removed the reference to DVR.

In these plots you're not taking account of sidereal rotation speed. Eeloo and Gilly give you more than 10% of orbital velocity for free, Minmus and Dres more than 5%. I wonder how you'd fit that detail in nicely.

Yeah, that's what I said.

So I tried a launch from Eeloo to check it and (don't ask me why) I performed about 1% over what the model said I would.

I'm guessing that sidereal rotation doesn't help you at the bottom end and doesn't save much at the top end.

Beats me,

-Slashy

[numerobis]

Oh, just calculated. Sidereal rotation means the apparent gravity at Eeloo's surface is about .8% less than the actual gravity. So I guess it's safe to ignore the low-speed effects and simply subtract out the sidereal rotation from the overall cost.

[arkie87]

Except there's no orbital velocity for a given planet as such, there's only the velocity required to maintain whichever orbit you're trying to reach.

Also, the usefulness is what I'm getting at here. The chart by itself is a curiosity. In order to be useful as a piece of gameplay advice, it has to be useful. TVR is harder for a player to calculate for a given craft and situation than an I_sp/TWR-based chart is to generate for a given planet, with the advantage that the latter is useful for any ship on that given planet, with no further calculations necessary - I_sp and TWR numbers are readily available even without Mechjeb/KER et al.

I'll triple what Numerobis and Slashy said here.

You... basically said the same thing. Ratio of the delta-V you have, versus the delta-V you need, plus fluff.

if DVR affected efficiency, then the relationship would be more complicated. It could be the case that increasing DVR decreases efficiency faster than DVR increases deltaV, such that, you could never get into orbit. It's good that isnt the case...

I was just correcting you, you said they'd need more fuel, not more dV. I did say I agreed that it's a rarely acknowledged notion.

I said fuel and deltaV. You are splitting hairs, or, i think you need to re-read what i said.

Therein lies the problem, again. DVR has no effect on your efficiency chart, yet it obviously has an effect on the TWR curve, therefore it has an effect on the time spent on the ascent, therefore it has an effect on your chosen definition of efficiency. Consider two craft with the same TWR and I_sp, but different DVR. Say, each has a starting TWR of 1.5, but one has a DVR of 2, and another has a DVR of 5. I can't be assed to dredge up the correct numbers at this point, but let's assume it means the latter craft has twice as much fuel for its mass. It means that by the time the first craft reaches orbit, it will be dry - its TWR will be maximum - say, 6. Assuming the same dry mass/wet mass ratio between the two craft, the second craft will still be pushing half-full tanks, having an effective TWR of somewhere around 3.5-4. This changes the time spent on ascent, and changes the TWR curve, which should have an impact on efficiency the same way high ISP does.

Therein lies the issue. DVR does not effect TWR, and TVR doesnt affect TWR either. They are all independent variables. In practice, when designing a rocket, if a player adds fuel, he/she reduces TWR. Thus, the player has two choices (a) add more engines to maintain same TWR or ( look up efficiency with new (reduced) TWR. The model doesnt care how you arrive at the given TWR, TVR, and DVR, it just provides the efficiency if you manage to obtain it.

Assumes nothing about the craft, yet "realistic values for DVR"? What does "realistic values for DVR" even mean, in this context? DVR depends entirely on the design of the rocket, a dedicated lifter will have just enough dV to reach orbit, while a lifter-pusher stage will have extra dV for orbital maneuvers, and an SSTO will have enough to go up, maneuver, and get back down. That your formula produces results independent on DVR, especially per my response to the previous quote segment, would seem to point to an error in your formula somewhere.

I think the confusion here is that it takes the same amount of deltaV, not the same amount of fuel. DeltaV is more "dimensionless" in that respect, since it scales with the mass of the rocket.

Also, before you were arguing that it's obvious that efficiency should be independent of DVR... now you are saying there is an error in my model since it shows this independence?

And DVR is... payload? As you defined it, it's literally "ratio of how much fuel you carry to how much fuel you need to actually make orbit". With TWR and I_sp collectively defining the rocket as a whole, of course DVR has no impact - excess fuel reduces TWR instead, but if that is assumed constant then changing DVR mostly just changes the size of the rocket.

[numerobis]

arkie, I am having trouble understanding why you're fighting tooth and nail to defend this DVR definition, which you've proved has no effect on anything (other than if you don't have enough deltaV you're dead).

[arkie87]

arkie, I am having trouble understanding why you're fighting tooth and nail to defend this DVR definition, which you've proved has no effect on anything (other than if you don't have enough deltaV you're dead).

I am not really defending it.. it's just the third independent variable...

but it's also good to know it doesnt effect deltaV (it obviously effects fuel though).

[Sean Mirrsen]

I said fuel and deltaV. You are splitting hairs, or, i think you need to re-read what i said.

Am doing nothing of the sort. Observe what you said:

It is important to know that if you increase ISP, you efficiency drops, and so while you might gain more deltaV, you will also need more deltaV to get into orbit. If a player didnt know this, they might improve their engine ISP, and reduce FMR (amount of fuel) to compensate to maintain the same total deltaV they had before, without realizing they will need more fuel to get into orbit..

Note the last sentence please.

Therein lies the issue. DVR does not effect TWR, and TVR doesnt affect TWR either. They are all independent variables. In practice, when designing a rocket, if a player adds fuel, he/she reduces TWR. Thus, the player has two choices (a) add more engines to maintain same TWR or ( look up efficiency with new (reduced) TWR. The model doesnt care how you arrive at the given TWR, TVR, and DVR, it just provides the efficiency if you manage to obtain it.

DVR is dependent on FMR, because a rocket gains its delta-V budget by carrying fuel. Again, for starting conditions you are correct - a rocket can have almost any combination of those parameters, as they are independent. Look again please at my example:

Two rockets, standing on the surface of, say, Tylo, side by side. Each has the same TWR (1.5), and TVR (aiming for same orbit, same I_sp), with the only difference being DVR. The smaller rocket has exactly enough dV to get it to its target orbit, so its DVR is at the border of the "fault value", as low as possible. The larger rocket has a lot more dV than it needs to just get to orbit - its DVR is at 3, or 4.

The rockets start simultaneously accelerating towards the orbital trajectory. The smaller rocket loses its fuel mass fraction faster because it has less fuel. It gains TWR faster, and arrives at the target orbit ahead of the larger rocket, having spent less time getting to that point - with I_sp of the rockets being the same, and throttle constant, this means it will have used less fuel per mass, ergo less overall dV, to get there - it's more efficient.

If we add a third rocket, that instead of having extra DVR has extra I_sp, wouldn't the change in the efficiency due to the lower FMR for the same DVR of the more efficient engine effectively mean the same thing as I just described, per your formula? Exactly what effect does your formula attribute to an increase in I_sp that lowers efficiency?

Also, before you were arguing that it's obvious that efficiency should be independent of DVR... now you are saying there is an error in my model since it shows this independence?

I was extrapolating from just the definition of your term before, not from what it actually means. The binary "dropoff" is fairly obvious, and until I looked at the mechanics of why an increase of efficiency could mean a decrease of "efficiency", DVR having no overall effect made sense. The thing is, DVR isn't just "scale". You can have two rockets of vastly different size that have the same TWR, same I_sp, and same DVR for where they're going. The unused delta-V budget affects the mass of the "useless" part of the rocket, and if you're being strict enough about efficiency as to notice that increasing engine efficiency can decrease overall efficiency for the same ascent profile, then you should notice it.

[Anquietas314]

The rockets start simultaneously accelerating towards the orbital trajectory. The smaller rocket loses its fuel mass fraction faster because it has less fuel. It gains TWR faster, and arrives at the target orbit ahead of the larger rocket, having spent less time getting to that point - with I_sp of the rockets being the same, and throttle constant, this means it will have used less fuel per mass, ergo less overall dV, to get there - it's more efficient.

At first I thought this made sense, but after some thought in order to hit a given TWR, your total amount of thrust from engines would depend on how much mass (fuel) you had in the first place. If it takes the same mass ratio for both rockets to get to orbit, then their end TWR should be exactly the same, which strongly suggests their TWR would be identical the whole way up.

I haven't done any calculations to back this up though, this is purely my intuition based on playing the game for (at least) 3 years.

EDIT: It's entirely possible I've misunderstood something.

I do at least know this is wrong (emphasis mine):

this means it will have used less fuel per mass, ergo less overall dV, to get there - it's more efficient

The deltaV calculation (assuming you mean the rocket equation) takes TWR throughout the flight into account, since it's based solely on wet and dry mass plus engine I_SP.

Edited December 28, 2014 by armagheddonsgw

[Sean Mirrsen]

At first I thought this made sense, but after some thought in order to hit a given TWR, your total amount of thrust from engines would depend on how much mass (fuel) you had in the first place. If it takes the same mass ratio for both rockets to get to orbit, then their end TWR should be exactly the same, which strongly suggests their TWR would be identical the whole way up.

I haven't done any calculations to back this up though, this is purely my intuition based on playing the game for (at least) 3 years.

You should compare the end states of each rocket. The rocket with only as much dV as it needed to get to the orbit, arrives at the orbit empty. Its mass is now its dry mass, and its TWR has hit the theoretical maximum for its engines. If it, you know, still had fuel to have any TWR at all. The rocket with more dV than it needs to get to the orbit, arrives at the orbit with partially full tanks. Its mass is not its dry mass, and its TWR is below its theoretical maximum, because TWR is dependent on engine power and craft mass, and in this case craft mass includes extra fuel. For a rocket, "delta-V budget" is synonymous with fuel, at least until those QVP thrusters start working. So the more delta-V you have remaining after a burn, the more mass you have remaining - and the lower your TWR at that point.

Edited December 28, 2014 by Sean Mirrsen

[Anquietas314]

You should compare the end states of each rocket. The rocket with only as much dV as it needed to get to the orbit, arrives at the orbit empty. Its mass is now its dry mass, and its TWR has hit the theoretical maximum for its engines. If it, you know, still had fuel to have any TWR at all. The rocket with more dV than it needs to get to the orbit, arrives at the orbit with partially full tanks. Its mass is not its dry mass, and its TWR is below its theoretical maximum, because TWR is dependent on engine power and craft mass, and in this case craft mass includes extra fuel.

Okay, but is it not possible that the dry mass in the larger ship is simply a smaller portion of its total mass compared to the small rocket (e.g. due to engines etc being a smaller ratio of mass)?

[Anquietas314]

Sean: Hmm okay. That made sense mostly. Your answer appears to have disappeared though (forum issue? deleted?), but it contained an example that demonstrated you could have different end TWRs for the two ships, with different DVRs. There was a part (which I can't quote - message missing) that mentioned that with the same fuel flow for both ships, the second ship would burn twice as much fuel in the same amount of time, which is obviously nonsense. The rest of it was fine though.

arkie/Sean (mostly arkie): I had a look at the model in the PDF and decided to expand the terms in the efficiency calculation (in particular, tbar* and FMR). The result was that the efficiency did not depend on DVR whatsoever. I think that probably suggests the efficiency measure is inadequate though.

// tbarstar used to avoid confusion with * for multiplication
efficiency = 1 / (TVR*ln(1 / (1 - FMR * tbarstar))

DVR = TVR * ln(1 / (1 - FMR))
FMR = (m_wet - m_dry) / m_wet

tbar = t / t0
t0 = (FMR * TVR * v0) / (g * TWR)
tbar = t(g * TWR) / (FMR * TVR * v0)
tbarstar = tstar(g * TWR) / (FMR * TVR * v0)

efficiency = 1 / (TVR * ln(1 / (1 - FMR * tstar(g * TWR) / (FMR * TVR * v0))))
// No FMR or other mass-based terms present, besides TWR, hence DVR independent
= 1 / (TVR * ln(1 / (1 - tstar(g * TWR) / (TVR * v0))))

It's of course entirely possible I've made an error somewhere

Edited December 28, 2014 by armagheddonsgw
Removed redundant expansions

[Sean Mirrsen]

Okay, but is it not possible that the dry mass in the larger ship is simply a smaller portion of its total mass compared to the small rocket (e.g. due to engines etc being a smaller ratio of mass)?

Hmm. Okay, so I guess it's possible that I'm not a rocket scientist.

There was a post here where I wanted to provide an example of what I was thinking of, however after several revisions I couldn't get to the result I wanted to achieve.

/me goes to experiment in KSP proper to see if he can get better results with a non-abstracted system.

edit: @armagheddonsgw: the example I used in that post had an error, which is why I deleted it so I could review it. I appreciate someone taking a second look at the formulas, but at the moment it appears I made a mistake in some of my own... well, "calculations" is too generous a word.

Edited December 28, 2014 by Sean Mirrsen

[Anquietas314]

There was a post here where I wanted to provide an example of what I was thinking of, however after several revisions I couldn't get to the result I wanted to achieve.

The example that I saw (one 0.5 mass engine vs 2 smaller engines with 0.25 total mass) was perfectly adequate: same TVR, same start TWR, different DVR, different end TWRs after expending the same amount of deltaV.

EDIT: The "edit" feature exists for a reason dude

Edited December 28, 2014 by armagheddonsgw

[Sean Mirrsen]

The example that I saw (one 0.5 mass engine vs 2 smaller engines with 0.25 total mass) was perfectly adequate: same start TWR, different DVR, different end TWRs after expending the same amount of deltaV.

Er, no. The example had a ship of 1.5 tons with a dry mass of .5, and a ship of 3 tons with a mass of 1 after the burn (.5 tons of engine, .5 tons of fuel). Both ships used 1 ton of fuel per engine, however with the same starting TWR the latter ship's thrust is 6 tons-force, therefore its final TWR is also 6. I got caught up in my own verisimilitude and made an error.

The example is possible in principle, but not for ships with the same I_sp (or without, say, staging). At least not in abstracted/idealized circumstances as the formula is dealing with, which is why I want to see if a more complex system introduces any new variables.

Edited December 28, 2014 by Sean Mirrsen

[Anquietas314]

Er, no. The example had a ship of 1.5 tons with a dry mass of .5, and a ship of 3 tons with a mass of 1 after the burn (.5 tons of engine, .5 tons of fuel). Both ships used 1 ton of fuel per engine, however with the same starting TWR the latter ship's thrust is 6 tons-force, therefore its final TWR is also 6.

Hmm. Okay, in that case, perhaps the model's efficiency measure is perfectly fine and this is just a very unexpected result?

If the ships have different I_SPs then TVR is different anyway, which does affect efficiency.

[arkie87]

Am doing nothing of the sort. Observe what you said:

Note the last sentence please.

DVR is dependent on FMR, because a rocket gains its delta-V budget by carrying fuel. Again, for starting conditions you are correct - a rocket can have almost any combination of those parameters, as they are independent. Look again please at my example:

It is important to know that if you increase ISP, you efficiency drops, and so while you might gain more deltaV, you will also need more deltaV to get into orbit. If a player didnt know this, they might improve their engine ISP, and reduce FMR (amount of fuel) to compensate to maintain the same total deltaV they had before, without realizing they will need more fuel to get into orbit..

Ok, so what i meant is that they will need more fuel to get into orbit than what they had rationed after reducing deltaV/fuel due to increasing Isp... simple misunderstanding.

Two rockets, standing on the surface of, say, Tylo, side by side. Each has the same TWR (1.5), and TVR (aiming for same orbit, same I_sp), with the only difference being DVR. The smaller rocket has exactly enough dV to get it to its target orbit, so its DVR is at the border of the "fault value", as low as possible. The larger rocket has a lot more dV than it needs to just get to orbit - its DVR is at 3, or 4.

Ok, let's analyze this example.

For Tylo, V_orbital @ sea level is 2170 m/s.

To have DVR = 1 @ Isp = 300, the m_wet/m_dry = 2.09, and FMR = 0.5216.

To have DVR = 4 @ Isp = 300, the m_wet/m_dry = 19.09, and FMR = 0.9476.

Thus, the second craft carries a higher fraction of fuel with it. However, before the fuel is burnt, it is basically a payload (it doesnt matter that it is fuel).

The rocket equation states that:

deltaV_expended(t) = Isp * g0 *ln(m(t=0)/m(t))

To burn the same amount of deltaV, since Isp and g0 are constant, all that matters is the ratio between initial mass, m(t=0), and the mass at the end of the burn, m(t). The FMR does not effect deltaV. Thus, the TWR at the end of both burns (and, in fact, throughout the entire burn) will be the same, since the ratio of the initial and final masses will be the same.

If we add a third rocket, that instead of having extra DVR has extra I_sp, wouldn't the change in the efficiency due to the lower FMR for the same DVR of the more efficient engine effectively mean the same thing as I just described, per your formula? Exactly what effect does your formula attribute to an increase in I_sp that lowers efficiency?

The reason TVR effects efficiency was given in the OP, but is as follows:

Let's assume a craft is designed for DVR = 1 on Tylo. With a low Isp engine (say 300), an FMR of about 0.5 will be needed to achieve this, and so, TWR will nearly double during the flight. Thus, even if one starts out with TWR = 1, by the end, TWR might be around 2, and so, average TWR will be between 1 and 2. Since increasing TWR results in increased efficiency, low Isp engines do not require as high initial TWR, since their TWR will naturally increase significantly over the flight.

However, with a high isp engine (say 4200), for DVR = 1 on Tylo, a FMR of nearly 0.05 is needed. During the burn, only 5% of the mass will be lost, and so, TWR will icrease by the same factor. Thus, the average TWR will stay around 1, and so, an initial higher TWR is needed for the same efficiency.

Edited December 28, 2014 by arkie87

[LethalDose]

So, there's something that's been bugging me about these equations, and I think I've finally figured out what it is.

Equation 2 equates sin Theta to an expression involving the difference between g and what the author calls "centripetal lift". However, the centripetal lift can exceed the local gravity (explicitly at the moment the vessel reaches prograde velocity sufficient to circularize an orbit at that altitude). This creates a value for sin(theta) that results in a negative theta value.

Basically, the model states the vessel should be pointed towards the ground during part of the burn, doesn't it? And, for escape velocity, it may be for a substantial part of the burn. If acceleration were held constant, it would be for (1-2^-0.5)*100 = 29% of the burn. It's going to be less than that, but not by a lot.

I fail to see how spending any thrust in the same direction as gravity is going to lead to the most efficient burn to escape...

[arkie87]

So, there's something that's been bugging me about these equations, and I think I've finally figured out what it is.

Equation 2 equates sin Theta to an expression involving the difference between g and what the author calls "centripetal lift". However, the centripetal lift can exceed the local gravity (explicitly at the moment the vessel reaches prograde velocity sufficient to circularize an orbit at that altitude). This creates a value for sin(theta) that results in a negative theta value.

Basically, the model states the vessel should be pointed towards the ground during part of the burn, doesn't it? And, for escape velocity, it may be for a substantial part of the burn. If acceleration were held constant, it would be for (1-2^-0.5)*100 = 29% of the burn. It's going to be less than that, but not by a lot.

I fail to see how spending any thrust in the same direction as gravity is going to lead to the most efficient burn to escape...

You would be absolutely correct if this work integrated up to escape velocity. But this work is concerned with the most efficient way to get up to orbital velocity, and so, integration stops after orbital velocity is reached i.e. the instant theta = 0 (since once you reach orbital velocity, you no longer need to spend any thrust fighting gravity, and can aim prograde, regardless of TWR).

[GoSlash27]

So, there's something that's been bugging me about these equations, and I think I've finally figured out what it is.

Equation 2 equates sin Theta to an expression involving the difference between g and what the author calls "centripetal lift". However, the centripetal lift can exceed the local gravity (explicitly at the moment the vessel reaches prograde velocity sufficient to circularize an orbit at that altitude). This creates a value for sin(theta) that results in a negative theta value.

Basically, the model states the vessel should be pointed towards the ground during part of the burn, doesn't it? And, for escape velocity, it may be for a substantial part of the burn. If acceleration were held constant, it would be for (1-2^-0.5)*100 = 29% of the burn. It's going to be less than that, but not by a lot.

I fail to see how spending any thrust in the same direction as gravity is going to lead to the most efficient burn to escape...

At higher velocities, pointing horizontally results in a negative θ WRT prograde.

Since I want to achieve a desired apoapsis anyway, I just let it sling me up there and circularize on arrival.

*edit* ^ what he said.

Best,

-Slashy

Edited December 28, 2014 by GoSlash27