Calculate maximum payload mass to orbit

[Rdivine]

So i've been playing KSP for 3 years. I'm sure many of us use the subassembly to our advantage.

For me, when i have to put a payload into an orbit, i simply drag a launch vehicle underneath it. However, designing a launch vehicle isn't easy as you have to calculate the maximum payload mass it can carry into low kerbin orbit(LKO).

Up until now, i've been using my own method of calculating a maximum payload mass a launch vehicle can carry.

Here's my method:

 -In this example, i'm using a simple, 2 stage Atlas V 501 rocket for demonstration.

1. According to the wiki, a rocket needs to have about 3400m/s-3500m/s of Δv to reach LKO. I'm calculating the payload mass using 3500m/s as launch conditions may not be favorable all the time.

2. Since a rocket needs 3500m/s of Δv, we can find the total Δv in the rocket using basic calculation.

 is the initial mass of the rocket.

 is the final mass of the rocket.

 is the specific impulse of the engine ( in seconds).

3. To measure the Δv for multiple stages, we can calculate the Δv of the stages individually, and adding them up together. The equation for the Atlas-V 501 is as follows:

Δv = (Isp of first stage engines)*(9.81)*ln(mass of whole rocket/mass of rocket after stage 1 is empty) + (Isp of second stage engine)*(9.81)*ln(mass of rocket after first stage is jettisoned/mass of rocket after 2nd stage burns out)

Now we run into a problem. The payload mass we are to calculate is included in the masses of the rockets! So we add a variable, x.

Δv = (Isp of first stage engines)*(9.81)*ln(mass of whole rocket+x/mass of rocket after stage 1 is empty+x) + (Isp of second stage engine)*(9.81)*ln(mass of rocket after first stage is jettisoned+x/mass of rocket after 2nd stage burns out+x)

Take into account the mass of the fairings and when it is jettisoned! The fairings were jettisoned after the first stage so i substracted 3.5tons for the second part of the equation.

4. Since we need about 3500m/s of Δv to reach LKO, we equate the equation to 3500.

3500 = (Isp of first stage engines)*(9.81)*ln(mass of whole rocket+x/mass of rocket after stage 1 is empty+x) + (Isp of second stage engine)*(9.81)*ln(mass of rocket after first stage is jettisoned+x/mass of rocket after 2nd stage burns out+x)

5. Now, we solve for x. Solving for x in an equation with a natural logarithm takes time, so i tend to key the equation into online tools to help me calculate, such as wolfram alpha.

For the Atlas-V 501, i've substituted in the appropriate masses into the equation :

3500 = 302*9.81*ln((104.8+x)/(40.8+x)) + 310*9.81*ln((21.3+x)/(5.3+x))

6. So in order to achieve 3500m/s of Δv, the payload mass have to be 21.76 tons.

However, since the specific impulse of the engines are taken from the average of the highest and lowest, and launch conditions may vary, there is a 10% margin of error in the calculation, so the actual range is between 19.58 - 23.94 tons.

Fair enough, i was able to get a payload of 23 tons into orbit, corroborating the calculations!

 

However, one of the biggest flaws of my method is that it requires alot of mental work. Is there any other way of calculating the maximum payload mass without mods or trail and error? If you have any opinions, suggestions and improvements please share them below!

Edited December 16, 2015 by Rdivine

[Qwarkk]

If you're willing to mod your game, Kerbal Engineer Redux is the answer. Its a mod which calculates all of this for you as you construct your craft.

It's a must have mod in my opinion.

[Rdivine]

12 minutes ago, Qwarkk said:

If you're willing to mod your game, Kerbal Engineer Redux is the answer. Its a mod which calculates all of this for you as you construct your craft.

It's a must have mod in my opinion.

Hmm, i'm looking for ways to solve this without mods. Plus, i doubt my computer can handle so many mods already installed in the game  

[Nathair]

51 minutes ago, Rdivine said:

Hmm, i'm looking for ways to solve this without mods. Plus, i doubt my computer can handle so many mods already installed in the game  

I'm with Qwarkk on this one, KER is a top priority mod for me. If you have to swap something else out to fit it in, I would do so. Try it and see!

[Xavven]

Hm, I haven't tried rearranging the rocket equation to solve for payload but I think it's possible, since I have an equation to calculate "bingo fuel", the minimum fuel required for a certain amount of delta-v, required to return home. Maybe I'll try to work it out if I have time this evening. It's going to probably have a couple of terms in it that look similar to ln(3500/[Isp*9.81])-1 as that's what happens when you move the natural log from one side of the equation to the other.

Alternatively, you could put the rocket equation in a spreadsheet and use Excel's "solver" function (it's now called "goal seek" in later versions) to find the answer. It's on the ribbon under the Data tab on the What-If Analysis button.

By the way, here's a modified formula that will calculate delta-v when you have more than one engine:

http://wiki.kerbalspaceprogram.com/wiki/Tutorial:Advanced_Rocket_Design#Multiple_engines

 

When GoSlash gets here he'll probably have the equation for you already anyway

Edited December 16, 2015 by Xavven

[OhioBob]

2 hours ago, Rdivine said:

However, since the specific impulse of the engines are taken from the average of the highest and lowest

I've used a method similar to yours to compute the payload capacity of various designs, but I've found that averaging the ISP for all stages isn't very accurate.  The best results seem to come when averaging the ISP for all engines that ignite on the ground, and using the vacuum ISP for any subsequent stages that are air lit.  The air thins out fast enough that the ISP penalty fades away quickly and doesn't need to be considered for upper stages.

The calculation only gives an estimated payload capacity.  The only way I've found to determine an accurate value is to test launch the vehicle with ballast to simulate a payload.  Adjust the amount of ballast until you find the maximum that the rocket can lift.  This method can be a time consuming grind, but doing things right and accurately is rarely always fast and easy.

 

Edited December 16, 2015 by OhioBob

[Pecan]

The equation you want is the reverse rocket equation and, yes, Slashy amongst others has published a thread all about it http://forum.kerbalspaceprogram.com/index.php?/topic/92716-the-quotreverse-rocket-equationquot-explained/#comment-1581147

Edited December 16, 2015 by Pecan

[Kerbol Macrosystems]

7 hours ago, Rdivine said:

However, since the specific impulse of the engines are taken from the average of the highest and lowest,

You might be able to increase your accuracy by using the log mean of the Isp [log mean = (a-b)/ln(a/b)].  The log mean is useful for dealing with the exponential function of atmospheric pressure vs. height.

This should yield an exact average performance if: 1)you take the average of Isp at stage ignition and stage cut-off and 2)your vertical rise rate is constant.  Practically your vertical rate is not going to be constant, but this should help shore up your guess for launch stages.

[GoSlash27]

Rdivine,

 Yes, you can calculate the "throw weight" of a booster assembly, but there's no way I know to make the process easy.

 For maximum efficiency, you need to mathematically design a booster for a desired payload. I've got a write- up on the procedure here:

The booster assembly is then acceptable for that payload or less.

For a booster assembly that already exists, the problem gets hairier. You need to make sure that each individual stage meets it's requirement both in DV and t/w. Whichever criteria for any stage is the lowest is going to be your payload limit for the entire stack. The weakest link in the chain, so to speak.

 If you are going to have a prefab family of boosters, it is much easier to do it the first way. You can design them from the outset so that "this one is rated at 1 tonne, that one at 2 tonnes" and so forth. If you go through the headache of reverse engineering a preexisting stack, you'll just come up with an awkward payload rating that won't be nearly as handy.

Me *personally*... I don't use standardized lifters, but I do use standardized criteria for my designs. They are all 2 stage designs to orbit. Upper stage has .7g minimum acceleration and 1,700 m/sec DV, both calculated as vacuum. Lower stage has 1.4g minimum acceleration (sea level thrust) and 1,800 m/sec DV (50% atmosphere Isp) with fins as required. If I were to design a standardized set of lifters, I would stick to these criteria.

Best,

-Slashy

 

 

 

Edited December 17, 2015 by GoSlash27

[Temstar]

42 minutes ago, GoSlash27 said:

They are all 2 stage designs to orbit. Upper stage has .7g minimum acceleration and 1,700 m/sec DV, both calculated as vacuum. Lower stage has 1.4g minimum acceleration (sea level thrust) and 1,800 m/sec DV (50% atmosphere Isp) with fins as required.

This, this is good design. Rule of thumb in multi-stage rocket design is that for the best efficiency each stage should contribute about equal amount of dela-V to the mission.

It's easier to do in real life since stages are made to order, in KSP it harder since we only have limited range of parts to build our rockets from. For example:



This two stage LV has a all solid first stage which contributes about 1100m/s of delta-V. If this was real life you'll tell the SRB maker to make you a longer, fatter SRB that gives you 1750m/s of delta-V in a single tube, and while they're at it setup proper grain profile so the thrust reduces during the burn and your vehicle TWR doesn't go up too much.

You could simulate this in KSP by bundling even more SRBs than I have here and reduce their thrust to control the TWR, but it's probably easier to just make the upper stage bigger to pick up the slack.

[Pecan]

I just SSTO these days, it's so easy now to chuck a single-stage rocket into orbit and such fun to watch any crew/pax as they fall back to Kerbin in a ball of fire.  Sometimes I let them know it'll land ok, but most of the time ... nah - let them scream.

[Crzyrndm]

23 minutes ago, Temstar said:

setup proper grain profile so the thrust reduces during the burn and your vehicle TWR doesn't go up too much.

May be of interest (although it doesn't solve the small SRB issue...):https://github.com/Crzyrndm/VariableThrustLimiter/releases

[Xavven]

Alright, I had some fun with math tonight and here's what I've got:

Original Tsiolkovsky Rocket Equation:

∆v = ln([m₀+x]/[m₁+x]) * I_sp * 9.81

m₀ = Initial mass except payload

m₁ = Dry mass except payload

x = Payload

Now to algebra our way out of this. We're going to get x on one side, but first we need to get that yucky natural log out.

Step 1: Convert the logarithmic expression into an exponential one

∆v = ln([m₀+x]/[m₁+x]) * I_sp * 9.81

∆v / ( I_sp * 9.81) = ln([m₀+x]/[m₁+x])

(m₀+x)/(m₁+x) = e^( ∆v / ( I_sp * 9.81))

Step 2: Substitution to make this look less evil. Trust me on this.

I propose we call e^( ∆v / ( I_sp * 9.81) ) = B

"B" stands for "Big f'ing term"

(m₀+x)/(m₁+x) = B

Step 3: Some crazy stuff I don't know why.

m₀+x = B * (m₁+x)

Distribute the right side

m₀+x = Bm₁ + Bx

x = Bm₁ + Bx - m₀

Step 4: Isolate X

x - Bx = Bm₁ - m₀

Step 5: Factor for x on the left side and solve

x * (1 - B ) = Bm₁ - m₀

x = (Bm₁ - m₀ )/ (1 - B )

Step 6: Substitution to get our evil back

Remember that e^( ∆v / ( I_sp * 9.81) ) = B

x = ( e^( ∆v / ( I_sp * 9.81) ) * m₁ - m₀ ) / (1 - e^( ∆v / ( I_sp * 9.81) ))

 

So I plugged this into Excel and holy crap, it worked! Now this is only for a single stage with one type of engine. You would have to have additional columns in your spreadsheet with copies of this formula for each stage, and it would only tell you how much payload each stage could handle while delivering the specified delta-v.  Keep in mind there are other factors, like "how much delta-v should I put in each stage?" and "what should my minimum TWR be for each stage?" It's too complicated for me, personally to solve for payload with multiple stages in a single equation because your initial equation goes from this:

∆v = ln([m₀+x]/[m₁+x]) * I_sp * 9.81

to this:

∆v = ln([m₀+x]/[m₁+x]) * I_sp1 * 9.81 + ln([m₀+m₂+x]/[m₀+m₃+x]) * I_sp2 * 9.81

See what I'm getting at? That's a nightmare!

EDIT: Oh god, my formula got interpreted as smileys! How do I turn this crap off?

Edited December 17, 2015 by Xavven

[Rdivine]

2 hours ago, Pecan said:

I just SSTO these days, it's so easy now to chuck a single-stage rocket into orbit and such fun to watch any crew/pax as they fall back to Kerbin in a ball of fire.  Sometimes I let them know it'll land ok, but most of the time ... nah - let them scream.

 

19 minutes ago, Xavven said:

Alright, I had some fun with math tonight and here's what I've got:

So I plugged this into Excel and holy crap, it worked! Now this is only for a single stage with one type of engine. You would have to have additional columns in your spreadsheet with copies of this formula for each stage, and it would only tell you how much payload each stage could handle while delivering the specified delta-v.  Keep in mind there are other factors, like "how much delta-v should I put in each stage?" and "what should my minimum TWR be for each stage?" It's too complicated for me, personally to solve for payload with multiple stages in a single equation because your initial equation goes from this:

∆v = ln([m₀+x]/[m₁+x]) * I_sp * 9.81

to this:

∆v = ln([m₀+x]/[m₁+x]) * I_sp1 * 9.81 + ln([m₀+m₂+x]/[m₀+m₃+x]) * I_sp2 * 9.81

See what I'm getting at? That's a nightmare!

EDIT: Oh god, my formula got interpreted as smileys! How do I turn this crap off?

Ah, i've tried rearranging the terms with respect to x. However, it takes lots of time and sometimes i make mistakes when changing the logarithms, so i've stuck without changing the terms because it's easier for me  

 

2 hours ago, Pecan said:

I just SSTO these days, it's so easy now to chuck a single-stage rocket into orbit and such fun to watch any crew/pax as they fall back to Kerbin in a ball of fire.  Sometimes I let them know it'll land ok, but most of the time ... nah - let them scream.

I smell evil  

 

3 hours ago, GoSlash27 said:

Rdivine,

 Yes, you can calculate the "throw weight" of a booster assembly, but there's no way I know to make the process easy.

 For maximum efficiency, you need to mathematically design a booster for a desired payload. I've got a write- up on the procedure here:

The booster assembly is then acceptable for that payload or less.

For a booster assembly that already exists, the problem gets hairier. You need to make sure that each individual stage meets it's requirement both in DV and t/w. Whichever criteria for any stage is the lowest is going to be your payload limit for the entire stack. The weakest link in the chain, so to speak.

 If you are going to have a prefab family of boosters, it is much easier to do it the first way. You can design them from the outset so that "this one is rated at 1 tonne, that one at 2 tonnes" and so forth. If you go through the headache of reverse engineering a preexisting stack, you'll just come up with an awkward payload rating that won't be nearly as handy.

Me *personally*... I don't use standardized lifters, but I do use standardized criteria for my designs. They are all 2 stage designs to orbit. Upper stage has .7g minimum acceleration and 1,700 m/sec DV, both calculated as vacuum. Lower stage has 1.4g minimum acceleration (sea level thrust) and 1,800 m/sec DV (50% atmosphere Isp) with fins as required. If I were to design a standardized set of lifters, I would stick to these criteria.

Best,

-Slashy

 

 

 

I do use preexisting lifters as i love making replicas of real life lifters. However i will give your method a try!

 

4 hours ago, Kerbol Macrosystems said:

You might be able to increase your accuracy by using the log mean of the Isp [log mean = (a-b)/ln(a/b)].  The log mean is useful for dealing with the exponential function of atmospheric pressure vs. height.

This should yield an exact average performance if: 1)you take the average of Isp at stage ignition and stage cut-off and 2)your vertical rise rate is constant.  Practically your vertical rate is not going to be constant, but this should help shore up your guess for launch stages.

Indeed i have given this method a trial, but through multiple tests of rocket engines, it is better to take the Isp as slightly lower than the mean( because the rocket hangs around within 3-10km longer than it does from 10-40km. )

 

[Warzouz]

I don't get the point of to OP. Without mods (especially KER) or "mental work", there is no way to know what your dV is, or even TWR. KSP is just "stick parts and fly it until it gets to LKO !"

 

At 3500m/s you're not at the maximum payload for a rocket. You can go to LKO with 3200m/s, so you can turn those 300m/s fuel into payload.

Then rocket drag can change the "maximum payload" a lot. You may need more fuel to orbit a draggy/large payload. As there is no way to evaluate drag, you have to guess.

 

For my part, I use a rule of thumb.

- 3200m/s for a streamlined payload

- 3300m/s for a reasonably unstreamlined payload

- 3400m/s for a streamlined payload with launch stage recovery near KSP

- TWR between 1.35 and 1.6 (ideally 1.45 for my personal taste)

- I've pre-build LKO stages sub-assemblies each 25T of payload (15,30,50,75,100,125,150,200,300,400,600T)

- I can adjust the launchers to fit my need. But as the only thing I save is fuel, I don't care too much.

That's saves me a lot of time an money, in career. (you can check the Cygnus SSTO rockets in my signature for more details).