Now we have some idea of how we launch our rocket and how the particles behave. It's time to see if we can make it to space! Escaping from a planet isn't easy, just try to jump that far and you'll see for yourself.
We know we've reached space when we've reached our planets escape velocity. This is the point where our rocket's kinetic energy is equal to the potential energy of the planet, or:
\(\frac{1}{2}m_{rocket}v_{esc}^2 = -\gamma \frac{m_{rocket}M}{r}\)
Where M is the mass of our planet, r is the radius of the planet and \(\gamma\) is the gravitational constant of our planet. Stock these around a little and voilà:
\(v_{esc} = \sqrt{\gamma\frac{2M}{R}}\)
At this point we would not fall back to the earth and see our rocket die a fiery death if we turned our engine off!
With a little help from some fuel, we would like to launch our rocket, which we lovingly named Samal Naga after the legend of the dragon trapped in the Milky Way. Our rocket isn't nearly as heavy as a dragon the size of a galaxy, at only 1100 kg. Turns out this is a good thing for our wallets, as a heavier rocket means more fuel, and fuel isn't exactly free. In addition, more fuel means an even heavier rocket, which means even more fuel, which means an even heavier rocket, which means- okay you get the picture. We want to bring enough fuel to last us the whole journey, but not so much that we struggle to launch.
We also need to account for the change in fuel, as we have to spend most of it when launching. The change in fuel depends on several factors:
- Fuel consumption
- The rocket thrust force
- The initial rocket mass
The initial rocket mass is constant, but the fuel consumption and rocket thrust force will change over time, and we need to take this into account.
How does the fuel consumption change? We know how many particles, n, that are in the engine and we know when they will (probably) escape. Thus we know how many particles escape over a certain time interval! If we multiply this by the mass of an \(H_2\) particle we get the total mass that escapes:
Amount of fuel consumed = \(m_H\cdot\frac{n}{dt}\).
What about the thrust force? We recall that a force can be described as \(\vec{F} = m\vec{a} = \frac{d\vec{p}}{dt} \), and that \(\vec{p} = m\vec{v}\). We know the mass of the particles and we use our knowledge of statistics to say something about the velocity perpendicular to the ground (as that's the direction that gives us thrust upwards). We know that the momentum that escaped from our engine transfers force to the rocket, but with only one engine we would generate very little momentum. Instead, we will imagine that we have a number, N, of smaller engines, and if we add the momentum from all these together, we get the total momentum. Thus, the thrust force is:
\(\vec{F}= \frac{m\vec{v}}{dt}\cdot N\)
Now that we know how to describe how the fuel changes, and therefore how our mass changes, let's have a look at the movement of the launched rocket. How can we know where it is when it reaches escape velocity? We certainly can't afford to launch another rocket is we lose this one.
We recall that we know the force on the rocket and can therefore find the acceleration:
\(\vec{F} = m\vec{a} \Longleftrightarrow \vec{a} = \frac{\vec{F}}{m}\)
Keep in mind that the total force is both the thrust force and the gravitational force, and that the gravitational force, \(g = \gamma\frac{M}{R^2}\), decreases as the distance increases.
As the acceleration changes, so does the velocity and position of the rocket. How can we keep track of this change?
Normally, if we know the position, we can differentiate this and get the velocity, and if we differentiate the velocity we can find the acceleration:
\(x''(t) = v'(t) = a(t)\)
So, it stands to reason that going from acceleration to position we need to integrate. We decide not to do this by hand, and instead utilize the Euler Cromer method, which step by step finds the next velocity and position by utilizing the acceleration and the previous velocity and positions.
\(\vec{a}_{i} = \frac{\vec{F}}{m} \\ \vec{v}_{i+1} = \vec{v}_i + \vec{a}_{i} \: dt \\ \vec{r}_{i+1} = \vec{r}_i + \vec{v}_{i+1} \: dt \)
Here dt is a small time-step, think of it as taking a small step forward in time for every calculation we do.
As you can see, there are many things to consider, and if something goes wrong we won't have another rocket to attempt launch with :(
That's why we need to simulate our launch before takeoff!
While looking at an endless stream of Greek letters is nice, for the simulation we need to use some actual numbers.
First, our home planet is a rock planet with a mass of \(6.808\cdot 10^{24}\) kg and a radius of \(6853.98\) km, making it slightly larger than Earth at \(5.972\cdot 10^{24}\) kg and \(6371\) km. Since these values differ from the Earth, so will gravity:
\(g = \gamma\frac{M}{R^2}=9.67 \: m/s^2\)
Gravity differs depending on how far away from the planet you are, so we have to take into account that as we move further away from the planet, the gravitational force will decrease.
Let's also find the escape velocity:
\(v_{esc} = \sqrt{\gamma\frac{2M}{R}} = 11481.85 \: m/s\)
Our speed, however, isn't constant, as we are accelerating upwards. Therefore, we need to constantly check to see when we've reached the escape velocity.
We have to decide how many engines to bring and how big they are, as well as how big the escape hole at the bottom should be. The temperature inside the engine is also an important factor, and should realistically lie close 3000 K, as this is the burning temperature of hydrogen. Our simulation won't be entirely realistic here, as we set the temperature to 10000 K, but you do what you gotta do, right?
We estimate that we need around 20-30 tons of fuel, because we will be using a lot just to launch, and we need some extra if we want to control how our rocket moves in space.
After inserting these parameters into our computer, we observe our rocket backing up 600 km straight into our planet. This took around 0 minutes.
Huh.
We try a couple of different parameters, but at best we are left standing completely still. (And at worst, we were moving through our planet and then another 6 million kilometers backwards. This is not the recommended way to travel in space, and should be avoided at all costs.)
At this point we decided to hit up headquarters at NASA, and our bros hooked us up with some better coordinates.
We've reached escape velocity after 571 seconds, or a little over 9 minutes. (And we are going the right way!)
We are where we want to be- out in space and ready to explore our solar system! What exactly is here? Any planets? Will we meet any aliens on the way? Find out in the next blog post!