Tutorial:Advanced Rocket Design

By Vincent McConnell and Kosmo-not

Introduction:

Getting to learn basic rocket science for a space game like Kerbal Space program can be very important to the success of building rockets that can perform a desired job. In this guide, we will be covering things like calculating the full Delta-V of your ship, explaining how to perform transfer maneuvers, getting Thrust to Weight Ratio, calculating the Peak G-force experienced during a particular burn, also calculating Delta-V needed for a full-Hohmann transfer and much more.

Delta-V

$\Delta v$ (change in velocity) is the bread and butter of rocket science. It is probably the most important thing to know about your rocket because it determines what your rocket is capable of achieving. Among the several things we will explain in this basic tutorial, $\Delta v$ is most likely the most useful thing you will apply to Kerbal Space Program while building a rocket. To find the $\Delta v$ of your rocket -- each stage at a time -- we have to sum up the part masses of every single part of the stage. When summing up fuel tank masses, it may be easier to write them like this on your paper:

Full Mass: x
Dry Mass: x

The reason for this is that it will be easier to calculate full mass and empty mass. So, simply sum up your entire stage mass.

The next important part of this set of calculations is to find your engine's specific impulse . Specific impulse is a measure of how fuel efficient an engine is (the greater the Specific Impulse, the more fuel efficient it is). For example, the non-vectoring stock engine has a vacuum specific impulse of 370 s. So here, we must apply the Tsiolkovsky Rocket Equation. More informally known as "The Rocket Equation".

It states: $\Delta v=9.81{\frac {m}{s^{2}}}\cdot I_{sp}\cdot \ln \left({\frac {m_{1}}{m_{2}}}\right)$

$m_{1}$ = total mass of the stage (including subsequent stages), $m_{2}$ = dry mass of the stage, $9.81{\frac {m}{s^{2}}}$ is to convert the I_sp (see also Terminology about I_sp)

So go ahead and sum up your stage's full mass with fuel. Then, go ahead and sum up the mass minus the fuel (this can be done by just adding up the 'dry mass' where given). Input these into the equation in the place of $m_{1}$ and $m_{2}$. So, we will show a quick example, here:

File:Screenshot7ex.png

Example rocket

Stage 3 (TMI, Mun lander, Return):

Full mass:	$3.72kg$
Dry mass:	$1.72kg$
$I_{sp}$:	$400s$
$\Delta v$:	$3027.0{\frac {m}{s}}$

Stage 2 (Kerbin orbit insertion)

Full mass:	$7.27kg$
Dry mass:	$5.27kg$
$I_{sp}$:	$370s$
$\Delta v$:	$1167.8{\frac {m}{s}}$

Stage 1 (Ascent):

Full mass:	$38.52kg$
Dry mass:	$14.52kg$
$I_{sp}$:	$350s$ (estimated due to atmospheric flight)
$\Delta v$:	$3349.9{\frac {m}{s}}$

Total $\Delta v$: $7544.6{\frac {m}{s}}$

Note: To calculate the $I_{sp}$ for multiple engines with different $I_{sp}$ values, you need to take the weighted average of the specific impulses relative to thrust. The equation looks like this: ${\frac {I_{sp_{1}}\cdot thrust_{1}+I_{sp_{2}}\cdot thrust_{2}+\dots }{thrust_{1}+thrust_{2}+\dots }}$

This will give you the correct $I_{sp}$ to use for your $\Delta v$ calculation.

Calculating transfer maneuvers

The next part of this tutorial is how to perform a transfer maneuver. This kind of action is called a Hohmann Transfer and it requires two burns at opposite points in an orbit. Adding velocity will boost our apoapsis higher. We would then simply wait until we hit our newly established Apoapsis and then add more velocity to boost our Periapsis to circularize. Or, we could drop our orbit by subtracting velocity by burning retro-grade.

We can also apply some $\Delta v$ calculations to find out how much thrust we will need to perform this maneuver. We will break this burn up into impulses. For example purposes, we will start at a 100Km orbit and then boost into a 200Km orbit. Both circularized. The formula for the first burn is the following:

$\Delta v_{1}={\sqrt {\frac {\mu }{r_{1}}}}{\Bigg (}{\sqrt {\frac {2r_{2}}{r_{1}+r_{2}}}}-1{\Bigg )}$

This is the formula for the final burn in the transfer:

$\Delta v_{2}={\sqrt {\frac {\mu }{r_{2}}}}{\Bigg (}1-{\sqrt {\frac {2r_{1}}{r_{1}+r_{2}}}}{\Bigg )}$

Where:
$\mu$= Gravitational Parameter of Parent Body. (3530.461 km³/s² for Kerbin).
$r_{1}$= The Radius of our first orbit. (100 km in this case).
$r_{2}$= The Radius of our second orbit. (200 km in this case).

This formula will give us our velocity for the burn in km/s (multiply by 1000 to convert it into m/s). It's important to make sure that you will have the $\Delta v$ in the stage to make this burn. Again, you can do that by using the $\Delta v$ calculations above.

Calculating fuel flow

Next, we will explain how to calculate fuel flow in mass to see how much fuel a burn uses up in a specific amount of time.

If we know the $\Delta v$ needed for the burn and the total mass of the rocket before the burn, we can calculate how much fuel is required to complete the burn.

First, we calculate the mass of the rocket after the burn is complete. To do this, we use the Tsiolkovsky Rocket Equation, inputting the initial mass and $\Delta v$ of the burn. We can then solve the equation for the final mass after the burn. The difference between these two masses will be used to determine the length of time that is needed to complete the burn.

The equation for mass flow rate of fuel, given $I_{sp}$ and thrust, is:
${\dot {m}}={\frac {\frac {thrust}{9.81{\frac {m}{s^{2}}}}}{I_{sp}}}$
where ${\dot {m}}$ is the mass flow rate of fuel consumed (in seconds)

Dividing the difference between initial mass and final mass for the burn by the mass flow rate of fuel, we arrive at how many seconds are required.

Note: The mass flow rate of fuel can be converted into the consumption rate of the fuel units used in KSP (Liters, I presume). The conversion ratio is 1 mass unit per 200l of fuel.

Orbital velocity

Rather easy is the formula to calculate the orbital velocity of an orbit. This assumes circular orbit or the velocity of a specific point in an orbit. For this, we simply do this calculation:

${\sqrt {\frac {\mu }{r}}}$

Where:
$\mu$ = Gravitational Parameter of parent body. (km³/s²)
$r$ = radius of orbit. (km)

If we input the radius of the orbit in Kilometers, our orbital velocity will come out in Kilometers per second. In a 100km orbit, our radius will be 700km. Meaning our velocity will be ~2.2458 kilometers per second (km/s), or 2245.8 m/s.

Delta-v map

A $\Delta v$ map consists of approximate amounts of $\Delta v$ needed to get from one place (whether it is on the ground or in space) to another. The $\Delta v$ values we have for our $\Delta v$ map are approximate and include a fudge factor (in case we slip up on our piloting). Our map is as follows:

Launch to 100km Kerbin orbit:	4700 m/s
Trans-Munar Injection:	900 m/s
Landing on the Mun:	1000 m/s
Launch from Mun and return to Kerbin:	1000 m/s
Total $\Delta v$:	7600 m/s

If we design our rockets to have 7600 total $\Delta v$, and the acceleration of the launch stages are adequate, we can have confidence that our rocket is able to land on the Mun and return to Kerbin. A rocket with a little less $\Delta v$ can accomplish this goal, but it is less forgiving of less efficient piloting.

Thrust to Weight Ratio

Calculating Thrust to Weight Ratio is only three very simple steps.

It is important to know the thrust to weight ratio of your rocket to ensure your rocket will actually liftoff. If your TWR is less than 1, you can bet that you won t make an inch in altitude when starting from the launch pad. The minimum optimal TWR to have for your rocket at launch is 2.2.

The formula for this is simply the thrust of all of your current stage engines divided by the weight (mass * 9.81 m/s²) of your ship, fully fueled. At the same time, this will give you the minimum G-force you can expect on the current stage. Your peak G-force will occur instantly before fuel depletion. The way to calculate this is to simply divide thrust by the dry mass of your stage+the fully fueled stages above it.

In conclusion: This guide will hopefully have helped with designing your rockets to allow you to get the job done -- whatever it may be -- with no test flights first. We hope this guide has been helpful to new and continuing KSP pilots alike.