How can I calculate the optimal dV values/ratio for a 2-stage rocket, when accounting for lift?

[boosters++]

Hi all,

I want to build an efficient 2-stage orbital rocket.

Given that:

I have a first stage engine better for use in atmosphere, and a second stage engine better for use in vaccuum.

I have a fine amount of control (through say, Procedural Tanks mod) over the amount of fuel each stage of the rocket will have (and the dV calculations per stage for atmo/vac through MJ and KER)

I will be using MechJeb, so the flight path will consist of two burns: A: ascent to desired apopapsis; and B: circularization burn.

I have taken math through calculus II/integral calculus.

I have the following questions:

1. How can I figure out the delta-V needed by the first stage to just reach the desired apopapsis?

2. How do I calculate the dV "savings" (compared to an ascent in vacuum) due to lift if I am using FAR (or, presumably, the 1.0 aero model)?

3. What are the optimal turn shape, final flight path angle, and altitude at which to begin a (mechjeb-approximated) gravity turn?

4. Assuming my second stage is now headed to my desired apoapsis (in vacuum), how do I figure out the dV I need for solely the second circularization burn?

Thanks in advance!

Edited April 22, 2015 by boosters++

[Xannari Ferrows]

1. That would be difficult. Your equation changes every single second, as the atmosphere get's less dense, and your engines gain/lose ISP.

2. I'm not sure what you mean by "savings", but I can give you loses.

End velocity loses = Gravity loses per second • (Seconds spent ascending(1/(G•target body's gravitational acceleration²))) + 2 • 1/Aerodynamic loses per meter² • Number of meters traveled

Feel free to correct me if I'm wrong in that.

3. This is heavily dependent on your TWR. In layman's terms; higher means turn more, and earlier.

4. Your current speed subtracted from the circular orbital speed of the planet. Without knowing the circular orbital speed, it is a whole lot of math. I'm not exactly sure of the formula, but trust me. The former is much easier.

[OhioBob]

I don't think there is an easy way to calculate the things you are asking for because it gets very complicated when there is an atmosphere. I attempted to determine the ideal 2-stage design by performing a series of launch simulations. With a simulation I could break the launch down into many small time steps so I could frequently update all the changing variables. I reported the results of my findings in this thread. Those simulations were based on pre-1.0 stock aerodynamics. I just recently switched to FAR and started to modify my simulations to see if I could find a FAR-optimized design, however I've put a hold on that. Now that version 1.0 is only days away, I'm planning to ditch FAR and try out Squad's new aero model.

Edited April 23, 2015 by OhioBob

[Admac]

As far as Delta V specifics, the answer is going to be more vague than you like, since there are a huge number of variables;

-Your TWR Profile

-Your aerodynamic profile (Needle vs pancake as it were)

-Efficiency of your ascent path (A function of the first two modified by your piloting skill)

Barring somehow getting higher resolution aerodynamic information this is super close to unsolvable at a resolution better than about a 100ish meters of DV.

The ancient rule of Kerbal ascent is "For a 75km LKO Orbit use 4500 DV in Stock, 3500 DV in FAR" with the condition you need a pretty aerodynamic launcher for FAR to not do hilarious things.

I often like to get my stage 1 to get me over 12km altitude because that's the Max Q point for most ascents, which is usually a hair under 1600 M/s^2 DV, but that's as much a personal preference thing. SSTO launchers are viable and 5+ stages to orbit are a pain but marginally more efficient (As far as mass fraction goes, probably not for price) Kerbal has a pleasingly forgiving low orbital speed, so double digit precision is rarely required.

Aerodynamic lift in rockets is a small enough of an effect to be ignored in FAR unless you are strapping wings to your rocket, which is terrifyingly inefficient, and should be avoided.

One thing you said that makes me nervous is you mentions stage 1 for altitude then stage 2 for circularization. In a real aerodynamic model, your gravity turn starts pretty early (First 20 seconds) and depending on your launch profile it could be anything from a 200-1200 M/s Circ burn. (High TWR means you will have a bigger circ burn)

Speaking of Max Q and TWR, a rule of thumb I use to develop my launch profiles is to start my turn once my speed is 100m/s, Keep my angle to 75 degrees until I hit Max Q, then aggressively turn to about 15 degrees and hold that heading until my apoapsis tops 80ish Km, then coast to circ burn.

[FleshJeb]

Lots and lots of differential equations. ;-)