Chapter 1. Calculating and Getting into Orbit

Here we have a spacecraft with a full tank. What can it do?

_images/ship1.jpg

Let’s rehash the basics: any maneuver requires thrust in a given direction, and this thrust adds velocity. The added velocity is called delta-v (𝛥v). You can see it on a graph as a difference between velocity vectors. We don’t usually need to calculate them: in map view you can see it if you hover over or open a maneuver node.

_images/tn_pic-chap1-maneuver_node2.jpg

By pulling the markers you edit the direction and the size (𝛥v) of maneuver impulse vector.

To know how much 𝛥v you need for a maneuver, say, to escape from Kerbin and intercept Duna, you may create a maneuver node and play with it, or see a 𝛥v map. The map shows most maneuvers needed to travel between planets and moons.

Let’s say we want to perform a 1000 m/s maneuver. Will the fuel be enough? The 𝛥v fromula, or Tsiolkovsky rocket equation, is this:

\Delta v = v_\text{e} \ln \frac {m_0} {m_1}

Where ve = g Isp. In other words, you find exhaust speed by multiplying standard surface gravity (9.8 m/s) by your engine’s specific impulse, which you can find in engine properties. m0 is start mass and m1 is end mass.

If we use all the fuel, we take full mass for m0 and dry mass for m1. If we sum all the parts of this ship, dry mass will be 6 tonnes and full mass will be 10.32 tonnes. Specific impulse of the 909 engine is 390 s. Put these figures in the equation, get the result: 1877 m/s.

If you need to know the fuel (propellant) required for a certain 𝛥v, here’s another form of the same equation:

P = m_1 (e^{\frac {\Delta v} {g \times I_\text{sp}}} - 1)

Now, knowing all that, we can assemble a simple rocket to send a pod in the sky.

Excercise

Let’s send an Mk2 command pod with a big RSC tank, just to increase payload for an excercise. Then I add a big tank and a Poodle engine.

_images/ship2.jpg

A ship with a huge RSC tank to imitate payload.

_images/ship3.jpg

The same ship with Poodle engine and a liquid fuel tank.

Note that if you add a tank and an engine, both of them are included in m1 and m0 parameters, and Isp changes too. If we change the engine, we’ll have to recalculate everything again. I made a spreadsheet in OpenOffice to make things easier and avoid confusion. But if you add stages things start become more complicated, the process is long, and you might ask yourself: is there a tool to calculate everything automatically? Yes, there is Flight Engineer Redux by cybutek. After you add a plugin part to a ship, it will show you live stats on its every stage: mass, 𝛥v, thrust, cost. MechJeb –the famous autopilot mod– also has a window for that, although without the ability to calculate for other planets.

Let’s see the ships with stats:

_images/tn_pic-chap1-stats2.jpg
_images/tn_pic-chap1-stats1.jpg

If we add the orange tank and Mainsail engine, the ship should almost get into orbit. We can add the smallest wide tank to be safe.

_images/tn_pic-chap1-stats3.jpg

Is a Mainsail powerful enough here? This is another important value to consider: thrust to weight ratio, TWR:

TWR = \frac {thrust} {weight}

If thrust equals weight, TWR is 1.0, and the rocket can just hover. Lower TWR means the rocket falls down, but slower than free fall. Any value greater than 1 will let you take off.

But the lower TWR is, the longer it takes to get into orbit, and the more fuel you use. On takeoff TWR saves you much fuel. TWR has to be about 2 for comfortable takeoff. TWR of 3 and greater is already too much, because you start losing energy to atmospheric drag. Having higher TWR at higher altitudes is ok, because the atmosphere gets thinner and gives less drag as your rocket rises. For maneuvers in orbit, TWR can be as low as 0.5 (lower values will lead to losses due to Oberth effect). For takeoff and landing in other planets you need TWR between 1 and 3, when compared to that planet’s own surface gravity. More than 4 is impractical: any mistake will be fatal as you run out of fuel rapidly.

Advantages of Staging

Just look at the pictures and notice: the single-stage rocket has the most powerful engine and two huge tanks, yet low TWR and can’t even get into orbit: 𝛥v is lower than 4500. The two-staged rocket on the right has a bit more 𝛥v than necessary and good TWR in both stages.

_images/tn_pic-chap1-too_low.jpg

Too low TWR even with Mainsail, and not enough 𝛥v to get into orbit.

_images/tn_pic-chap1-stats3.jpg

Two stages make a rocket with good both TWR and 𝛥v.

Staging helps you reduce the m parameters of the rocket equation, but you have to add more engines, which counters this effect. I tried a rocket with 6 stages, and in the end it was worse (heavier) than a two-staged one.

For lateral stages there’s a famous technique called asparagus staging, using fuel pumps. You can look it up on KSP Wiki or in a Youtube tutorial.

Bad Design Examples

In the delta-v-map, the budget for getting into orbit is 4550 m/s. Here are some of my overoptimization attempts. These rockets look very “optimal” both in terms of TWR and 𝛥v, but can’t get into orbit, even though I’m an experienced pilot. They fall short by 100-200 m/s. If you look at the figures, you’ll notice the low TWR of all the stages but the first.

_images/tn_pic-chap1-mistake1.jpg

Notice: the upper stage does a lot of work (1300 m/s), starts yet in upper atmosphere, but has extremely low TWR of 0.37.

_images/tn_pic-chap1-mistake2.jpg

Better last stage (TWR = 1.1), but it starts at even lower altitude, and loses speed due to drag.

_images/tn_pic-chap1-mistake3.jpg

Due to drag, the apoapsis starts to lower, and you can’t even do all the acceleration work before falling down.

Delta-V Cheat Sheet

KSP 𝛥v map is a good cheat sheet to keep around. I flew enough, and can assure you the values are quite correct, and include safety margins for human imprecision. It means if you are experienced, you’ll fly up to these figures, but for any orbit corrections afterwards you need some 100-200 m/s more.

_images/tn_pic-chap2-delta-v-map.png

Delta-V budget map.

The figures in the 𝛥v chart are a fair estimation. I couldn’t get into orbit better than at 4550 m/s, so it’s a good estimation of a human pilot. If you feel inexperiencend and want error margins, add 100-300 m/s to that. MechJeb autopilot can do 200 m/s better than human.

After getting to orbit, you’ll still need some fuel for maneuvers. For those matters you can have an orbital stage with 300 m/s in the tank left and TWR=0.5.

This is all for our initial chapter. Read my next part in a week. Fly on budget and enjoy flying!

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See also

One Giant Leap
a photo gallery of Apollo program (with comments in Russian)