Revision as of 03:36, 21 October 2014

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門外漢のためのKSP数学

筆者がKSPを始めたのは数ヶ月前。他のプレイヤー達が口にするTWR、ISP、Δv、Ap、離心率などなど数学や物理の専門用語を私は全て理解しようと努力しました。本チュートリアルではこれらの数値が何を意味しているか、なぜ重要なのかを履修経験の無い人向けに短時間で理解できるよう説明していきます。

物理について一言

軌道力学の大部分は直感的にイメージすることが出来ます。プレイヤーの中に宇宙へ行ったことがある人は恐らくいないので、物体がどう動くかのイメージは過去の経験に基づくことになります。歴史的にも物理学では経験によるイメージが大きな争点になってきました。because we had to think in ways that were different from our intuition to really understand how planets moved. 多くのチュートリアルではニュートンの功績を研究することを奨励していますが、本チュートリアルでは基本的な概念のみ解説していきます。本チュートリアルでは厳密な正確さではなく直感的なわかりやすさを念頭に置いていることを理解して置いてください。

速度・速さ・摩擦・加速

まず車を時速70マイル(約113km/h)で運転しているとしましょう。ギアをニュートラルにしたら車は減速してやがて止まるでしょう。速度を維持するためにアクセルを吹かし続けないといけないことを私たちは直感的にイメージできます。これは摩擦が働くとわかっているからです。車の例では地面との摩擦や空気抵抗が働きます。アクセルが前進させる力だとしたら摩擦は後退させる力と考えることが出来ます。ニュートラルでは摩擦の後退させる力のみが残り、車はいずれ停止します。

宇宙空間では摩擦を発生させる地面も空気もありません。そのため後退させる力も働きません。星が一切無い仮想の宇宙空間では車は永遠にどこまでも直進し続けるでしょう。これは力が加わらない限り静止している物は静止し続け、運動しているものは同じ運動を続ける、というニュートンの法則のひとつから来ています。地球上では摩擦の後退させる力とアクセルの前進させる力が働いています。

「速度」は、「速さ」と移動方向の両方をまとめて指しています。そのため同じ「速さ」で旋回していても「速度」は変化しているということになります。Think about making a 90-degree turn in a car, you have to push the gas pedal down during the turn. 加速は速度の変化を指しているだけです。そのため物理学で「加速している」といえば、速さが変化しているか、移動方向が変化しているか、その両方を指しています。 Note for the interested readers: There is an important mathematical relationship between position, velocity, and acceleration. In fact, the exploration of this relationship by Isaac Newton and Gottfried Leibniz lead to the invention of what we math people now call Calculus.

軌道力学

→ 参照： Orbit

Kerbal Space Programをプレイするとなると、かなりの時間を軌道に費やすことになるでしょう。軌道上での動きを理解できるように、「思考実験」をしておきましょう。屋外で実演しても構いません。ひもが付いたボールを想像して下さい。ひもを持ちグルグルと素早く振り回したとします。するとボールはあなたの周りに「軌道」を描きます。KSPの場合は、ひもの代わりに重力が宇宙船を天体に引き寄せようとします。

完全な円の軌道にいる宇宙船に視点を移すと、宇宙船の速さは変わらずずっと一定です。(進行方向が変わり続けているため速度は変化しています。)天体は真下に引き寄せ続けていますが、宇宙船も十分な速さで直進しようとするため円を描いて飛行します。実際には宇宙船も落下し続けていますが、それと同じだけ遠ざかり続けているのです。Readers are encouraged to convince themselves of this, it is easier to see in a highly eccentric orbit than in a circular orbit, more on that later. In KSP, as long as you are in a well behaved orbit (not on an escape trajectory, not going to crash into the surface, completely above the atmosphere), your orbit will never change. You will keep moving in the same path forever. In the real world, it's not quite that simple, but that is outside the scope of this explanation.

Orbital Velocity

Given a fixed, stable orbit, your velocity (the speed you are moving and direction you are moving in) depends only on your position in that orbit. Understanding this is incredibly important, so here is some explanation. In the image to the right, look at any point on the bigger shape (the ellipse). If you don't burn your engines, your speed when you are at that point will be the same as when you return to that point after one orbital period (the amount of time your orbit takes to complete one full revolution).

Eccentricity

Eccentricity is a number to describe the 'shape' of our orbit. The closer our eccentricity is to 0, the more circular our orbit is. Elliptical orbits have eccentricity somewhere between 0 and 1. In the image to the right, the smaller, circular orbit would have an eccentricity of 0, and the larger orbit would have eccentricity around 1/2 (very rough number). Eccentricity of 1 or greater is what we call a parabolic or hyperbolic orbit. All that means is that you will not stay in orbit, you will escape the planet's sphere of influence and never come back without accelerating back towards it somehow.

Maneuvers

→ 参照： Basic maneuvers

Say we are at point A in the figure to the right, and we point the nose of our rocket in the direction of the arrow (prograde, the direction we are moving) and burn our engines for a while. In reality, this burn takes a certain amount of time, and our position changes during that time. However, that is some much more complicated math, so we are going to pretend that the burn starts and ends instantly. Basically, we pretend that we accelerate by a certain amount at point A. Since we are still at point A, when we complete one orbital period, we will be back at point A. However, the opposite side of our orbit will move away from us, making our orbit more eccentric in this case. The important take-away here is that we accelerate at point A to change our velocity. This changes the shape of our orbit. Say our original velocity was 10,000 m/s (10 kilometers per second), and our new speed is 10,200 m/s. Our velocity changed by 200 m/s, and this is our Δv!

What does all this mean?

Okay, by now if you are still reading, you are probably starting to think "How does all this stuff help me go to space!?" Well, we are going to start talking about that right now!

Why is Δv so important?

At this point, we have a rough idea of what Δv is, and Δv is probably the most important thing to understand. Note that above, when we talked about how fast our rocket was moving, or how it's orbit changed, we said nothing about the mass of the rocket. We don't know if we are talking about a tiny satellite, a spaceplane, or a huge rocket, but we know that how our orbit changes is only dependent on how our velocity changes! This is why we talk about Δv so much, because no matter what rocket you build, it takes the same amount of Δv to go from point A to point B in space. Furthermore, we know it takes about 4600 Δv to get into orbit around Kerbin, so if we know the Δv our ascent stage generates, we know if it will get us to space!

Determining Δv

Since this is an article for people without mathematical backgrounds, we are not going to look at the formula for calculating Δv. There are great tutorials explaining the equations for all of this, and readers are encouraged to consult them for a more rigorous understanding. However, most people see big, complicated equations and they stop reading, whether it comes from some post-traumatic stress left over from school, or being generally uninterested in mathematics, and that is okay. Here, we are going to simply look at what Δv depends on, that is, what effect does building a rocket in one way or another impact Δv?

There are mods that will tell you what your stage's Δv is, and I personally use one of them, but since this article is about vanilla KSP we will leave them out of this discussion. There are also mods that will do your entire take-off, gravity turn, and all your orbital maneuvers for you. While these can be fun, I personally do not believe in using them outside of sandbox mode for experimental purposes, since the point of playing the game is learning for me.

Thrust and Thrust-to-Weight Ratio(TWR)

→ 参照： Thrust-to-weight ratio

Thrust is the amount of force (how much 'push') your engine is generating. Remember the example of the car going down the road where the engine pushes the car forward and friction pushes the car backwards? The same thing happens with rockets. Thrust is basically how hard the rocket is being pushed up from the surface of Kerbin.

We all know that heavy things are harder to pick up than lighter things. If you don't believe me, go lift a piece of paper off the ground over your head, and then do the same with a piece of furniture, like a couch. The couch is much harder to pick up. The same thing is true in rocket science, heavier rockets are harder to pick up (lift off) than lighter rockets! This is why TWR is so important, the rocket's engines are pushing the rocket up, and gravity is pushing the rocket back down. If you have ever arm-wrestled, you know that the person who pushes harder is going to win. The rocket's weight is how hard gravity is pushing down. Therefore, the rocket's thrust must push harder than gravity, or you are not going to space. A TWR less than one means that gravity is going to win. A TWR over one means the rocket is going to win, and a TWR of exactly one means the rocket will not hover in place. However, once you are in a stable obit, you no longer need a TWR over one to change your velocity.

Side-note for interested and advanced readers

TWR changes during the flight of a rocket. As you burn more fuel, you lose mass, and your TWR increases since your weight decreases. TWR also depends on what planet/moon you are on since each celestial body has different gravity. Therefore, the same rocket with the same amount of fuel as a lower TWR on Eve than it does on Kerbin.

So, what engine should my rocket use, and what is engine ISP?

→ 参照： Specific impulse

Let's start with engine ISP. Basically, it tells you how fuel-efficient your engine is. An engine with a higher ISP will give you more Δv for the same amount of fuel as an engine with lower ISP. However, you also need to look at how much thrust they generate, and determine if you need a lot of Δv in a small amount of time, or less Δv all at once.

Example

The LV-909 Liquid Fuel Engine engine has ISP of 300 in atmosphere or 390 in space. The Rockomax "Mainsail" Liquid Engine has ISP of 320 in atmosphere or 360 in space. This doesn't seem like much of a difference. However, the mainsail can generate a thrust of 1500, while the LV-909 can only generate a thrust of 50. Therefore, the LV-909 will accelerate your rocket more slowly. Sometimes, you need to generate a lot of thrust very quickly (like when you are trying to go from the ground to orbit), so an engine with lower ISP but higher thrust may be better

What about SRBs?

→ 参照： SRB

My first ship to successfully orbit Minmus had around 50 SRB's on it. I do not recommend this approach. First off, as of .24, we have to pay for parts, so efficiency is important. Second, it was very difficult to get enough struts on the ship to get it to stay in one piece. Third, it was very difficult to steer, so my gravity turn was very inefficient. I only mention this because a common part of the learning curve for new players is to add more SRB's and more struts when we have trouble reaching orbit.

SRB's add a good amount of thrust, but also add weight. Therefore, the more you add, the less of a benefit you are getting. Also, since they only have one setting, which is to burn until empty, they are best used for ascent stages only, since we need more control when we are in space. Therefore, my recommendation is that if you are having trouble getting into orbit, putting a few SRB's on the side of your rocket as a first stage may help. As you unlock more parts in career mode, and get a better feel for getting into orbit, you may or may not continue to use them.

Final Notes

Hopefully this tutorial has given you a decent primer on what all the math really means. I highly recommend experimenting with different rocket designs, reading more rigorous mathematical explanations, and continuing to learn, since this is just a baseline to get you started.