Seems simple enough? As usual, it's not that easy. We may be able to send off our rocket from coordinates we choose, at a suitable time, taking into account the forces from our star and planets, but we cannot account for every single thing. A black hole could pop up at any time and kill us all. Or maybe not, but there are more things in a solar system than just planets and stars!
For one, we know our planets have moons, but we aren't taking these into account. Hopefully they are small enough compared to the planets, and far enough away that they won't make a difference, but we won't know this until our dear rocket is already in space. Another concern is asteroids. We definitely can't keep track of them all, as they are not always easy to spot. They can potentially come close enough to our rocket to push it somewhat off course, which can make a small or big impact on its trajectory.
Therefore, we aren't going to let out rocket cruise until it hits our destination planet, instead we will keep track of our whereabouts using the software we developed in the previous three posts. This way, if we notice we aren't where we want to be, we can give the rocket a boost and hopefully steer it back on course.
With that out of the way, we launch! After 104.86 years since we started simulating our orbit, our planets are at their optimal position and we launch from \([x = 6854\cos(230^{\circ}), y= 6854\cos(230^{\circ})]\) at the equator. (Written in polar coordinates, where \(r\) is our home planet's radius written in kilometers) We have a planned boost when our space craft is close to our destination planet, but we know we're probably going to have to give it a few more on the way, to make sure we are heading in the right direction. For now, though our space craft cruises away, not a care in the world.
Our rocket is mostly held together by duct tape and prayers, and is therefore inexpensive. We therefore choose to take that extra cost of building a new one in case we miss. Let's start by following our plan:
...And its back to the drawing board. We end up in an orbit further in, when we want to reach a planet further out! We also end our journey when we are way off course, so any boost we could have given our rocket wouldn't have saved us, as the destination planet would be far away by the time we reached the rendezvous point.
When launching the rocket, we are using software provided us by our guys at NASA, which can be a bit tricky to combine with our simulated path. When we integrate, we work with small time steps, while the NASA software wants the input to be in years. That leaves a lot of room for error when translating a boost "after 450 time steps" to a boost after x amount of years. Especially because several of our boosts are very close after one another, we get a severe drop in accuracy when we do the conversion. Another possible source of error is that we have now worked in several different frames of reference, where we have started measuring our positions at different points in time. The orbits are simulated over a 120 year period, but we launch after 105 years, and when we launch, we consider the time of launch to be t=0. Confusing? IT IS.
We cobbled together a new rocket, using whatever we could find in our basement, and launched again. We noticed that braking in the beginning of our journey brought us further in, so we tried the opposite approach: giving an actual boost. This is more true to a Hohmann transfer, where you give a boost in the beginning, and coast for the remainder of the trip. Reality is a lot more complicated, so we had to continually adjust our course to stay on the planned trajectory.
Even though we had to completely discard our plan when it came to boosting, we chose to stay on the planned path. That's because the hardest part of travelling from one planet to another is to end up in the same place at the same time. So even though we didn't do a proper Hohmann transfer, we concluded that a Hohmann orbit was still our safest choice!
We got pretty close by boosting and braking through the whole journey, which wasn't even close to fuel efficient, as Hohmann transfers are supposed to be. We ended up burning about 16000 kg of fuel, out of the 20000 kg we had left after launch. There won't exactly be any return journey with that little fuel left, so we're grateful that this is an unmanned mission.
We also notice that our journey became a bit shorter, as we are actually closest to the planet after 4 years. Unfortunately, "close" in space isn't always close enough.
We are 266 million kilometers off, and it's time to throw in the towel and ask for help. This way we BARELY reach our destination with a little help from NASA. You rock guys, thanks.
Below we've included a plot of our orbit as seen from above, with the destination planet at the origin (finally out of the giant solar system point of view).
It's hard to spot, but the spacecraft orbits the planet several times over a years time. (We know this for sure when we calculate the period further down!) When looking at the orbits over 10 years, we can see that they don't overlap. The most important thing for now is that the rocket stays in orbit and don't just fly away, which would have meant that it wasn't caught by the gravitational pull from the destination planet.
When we arrive, we do one orbit and let our spacecraft measure its position and velocity relative to the planet. We use this to find a variety of information:
- We can tell how far our spacecraft is from (the center of) the planet by subtracting our distance from our star, \(\vec{r}_{*,s}\), from the distance between the star and the planet, \(\vec{r}_{*,p}\). (It's the same method we used here when changing the frame of reference! Nothing fancy, just basic vector addition).
- We can find the radial component of the spacecraft's velocity (meaning the part of the velocity vector pointing towards the star) with respect to the planet, which can be found from the velocity \(\vec{v}\) and the position vector \(\vec{r}\). We make a unit vector \(\hat{r} = \frac{\vec{r}}{|\vec{r}|}\), and dot this with the velocity like this: \(v_r = \vec{v} \cdot \hat{r}\)
- We can find the angular velocity (how fast we rotate relative to the planet) by dividing the distance along the orbit by the time it took to travel that distance. This involves a lot of approximations, some of them here when deriving Keplers second law. A short recap: we measure the distance over a very small time, which gives us something close to a triangle, where two of the sides are our position vectors at two points in time. Using the length of the vectors, we can find the last side of the triangle by using Pythagoras, which will be the distance travelled \(\Delta S\). The angular velocity is the given by \(v_{\theta} = \frac{\Delta S}{\Delta t}\)
And from this we can find out a whole lot about our orbit! Recall one of our earlier posts where we talked about ellipses, and qualities of elliptical orbits? It's time to once again use this knowledge.
Previously, we've explored different qualities of ellipses, such as defining the semi-major axis and the semi-minor axis, as well as the eccentricity and what this tells us about the ellipse. We can use the information about the movement of our rocket in order to find this information for our rocket.
First, we are interested in the periapsis and the apoapsis of the planet. These correspond to the Earth's aphelion and perihelion, but those terms are explicitly for usage around the Sun. We can simply measure these from the absolute value of the position where we are the closest to and the furthest from the planet we orbit.
Periapsis: 0.00134 AU
Apoapsis: 0.00815 AU
We can use these distances to find the length of the semi-major axis, \(a\), if we add them together and divide the answer by two (as \(a\) is defined from the center of the ellipse to the furthest point).
\(a = ?0.00475\)
The length between the center of the ellipse (where the semi-minor axis crosses the semi-major axis) to the planet is equal to the eccentricity multiplied by the semi-major axis. We can find this length, which we refer to as c, by subtracting the semi-major axis from the apoapsis, meaning that we can find the eccentricity, e:
\(e=\frac{c}{a}= 0.71743\)
When the eccentricity is 1 we have a perfect circle, and our value of 0.71743 indicates that our orbit should be elliptical, just not too elliptical.
We can now use the expression for eccentrticity \(e=\sqrt{1-\frac{b^2}{a^2}}\) in order to find the semi-minor axis, b. We rewrite and voila:
\(b = a\sqrt{1-e^2}=0.00331\)
Lastly, we use \(a\) in order to find the orbital time period, \(P\), which we know from Keplers third law:
\(P=2\pi\sqrt{\frac{a^3}{G(m_p+m_s)}}=0.2347\)
where \(a\) is the semi-major axis, \(m_p\) is the mass of the planet, \(m_s\) is the mass of the spacecraft, and \(G\) is the gravitational constant. It seems as if it takes the spacecraft approximately 0.23 years to orbit the planet, which matches well with what we saw when we let it orbit the planet for a year.
As the above calculations greatly depends on the periapsis and apoapsis, we decided to check if we calculated them correctly.
After plotting their respective positions, we can conclude that they look reasonable!
Now that we are in orbit, it's time to make some preparations for the next step in our journey: landing! Stay tuned as we are getting to know our planet better before landing in the next post.