# Orbits are hard, Okay?

Begin LogEntry_0023 StarDate 197759.60

Okey, so the last time we had a look at our orbits, our helpful calculator robots told us that our orbits were sufficiently close to what they would be in reality, even though we only looked at the star's gravitational pull on the planet, and not the planet's gravitational pull on the star.

Now we have to remedy this!

This however makes our calculations significantly harder to calculate for our robots with pencil and paper in a timely fashion, so we will limit our calculations to the heaviest planets in our system. This is firstly to not have to wait a very very long time for the calculations to be done, secondly to not waste too much of our pencils and paper, but thirdly and most critically, the smaller planet's effect on the sun is minimal and can be ignored.

To perform these calculations we have to reframe our solar system in terms of it's center of mass. If you've never heard of center of mass, it's actually a quite intuitive concept, it is the average position of all mass in the system. Okay now that I write it out, it's not that obvious from just that statement.

Okay, let's see if we can put this in better terms, let's imagine we have a planet \(A \) with mass \(m_A \), and a planet \(B\) with the mass \(m_B \), the center of mass between these two planets will be somewhere on the straight line between them. Let's call this center of mass \(\vec{R}\), and \(M\) as the total mass of the system, \(m_A + m_B \), then the center of mass can be found with the formula \(\vec{R} = \vec{r}_{A} \frac{m_A}{M} + \vec{r}_{B}\frac{m_B}{M}\).

In these illustrations we have called the center of mass \(C\), and we can see how it's located on the line between planet \(A\) and \(B\).

This now allows us to find the center of mass easily between a planet and the star, we can then reframe our numerical integration loop in relation to this, and then do all the calculations in relation to the center of mass.

We do this by calculating the center of mass for each timestep, and then subtracting it from the current position of the planet and the current position of the star. We also have to take into account the gravitational pull of the planet on the star, since this is what causes it to move. Once we managed to get all of this working, we simulated the orbits over a period of 5 full orbits around the star for the chosen planet, and we ended up with this plot:

Now we can barely see the movement of the star, so here's a plot of the star's movement over the 5 orbit period:

We can clearly see that the star is moving around the shared center of mass, but it's orbit is significantly smaller than the orbit of the planet. Oh and do make make note that this plot is only of the movement of the center of mass of our star, it does not take into account the size of the star. So let's give that a look!

Our star has a radius of approximately 551 337km, whilst and astronomical unit (AU) is 149 597 871km, we can now easily find the radius of our star in AU: \(r_{star} [AU] = \frac{r_{star} [km]}{AU[km]}\)

Calculating this tells us that \(r_{star} = 0.003685 \) AU, we can see from figure 2 that the orbit is a bit more than 0.004 AU from the center of mass, so in our case the center of mass \(\vec{R}\) is actually outside the star.

Now our star is a bit smaller than the star in our ancestors solar system, it's radius is about 79% of the radius of the sun in in the ancestral system, and about our star has about 81% of the ancestral sun's mass.

It is actually quite common for the center of mass to be located inside the interior of the star, this is due to how large a star usually is compared to it's planets.

Now we can look at the energy of our system, we calculated the energy from an equation we managed to find in the library, however it's origin and derivation was lost. We did attempt to work it out ourselves, but we kept ending up with some terms that would not cancel and are yet to be able to figure out how it is derived.

Oh well, the equation states that the total energy of the two body system as seen from the center of mass frame is: \(E = \frac{1}{2} \hat{\mu}v^{2} - \frac{GM\hat{}\mu}{r}\)

Where \(\hat{\mu}\) is the reduced mass and is defined as \(\frac{m_1 m_2}{m_1 + m_2}\), \(M\) is the total mass.

Plotting this gives us:

Here we can see that the energy fluctuates, but it fluctuates periodically. We're using a symplectic integration method, which should on average conserve the energy of the system.

We also decided to run this with the 4 heavies planets in our solar system, it gave us these plots:

Here we see that the energy again fluctuates periodically in the same manner, but if we look closely at the energy between planet 2 and the sun, we can see that it has some very minor periodic fluctuations, within it's periodic fluctuations, caused by the smaller planets.

For some reason our robots did not have a protocol for verifying these results, but we think we've done it right.

End LogEntry_0023