The solar carousel

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StarDate 197721.19

Now that we've successfully (or at least close enough) launched a rocket into orbit, it's time to find out where we'd like to go. Ok, so maybe we should have done that a bit earlier, but you know what they say: hindsight's twenty-twenty. Anyway, all the planets in the solar system are orbiting around the star with us, meaning we should really try to simulate where they'll be at any given time to know how to get there. This all starts with a few simplifications.

First of all, we'll be using Newtonian gravity as the only force in our system. This doesn't take in to account things like gravitational precession, but should work plenty fine any way. Also, since our star is a lot more massive than any of our planets, it seems a safe assumption that we can ignore the gravitational effects between the planets and that the sun is fixed at the origin. Well, at least for the time being.

With this being the case, there is an easy solution to finding the orbits. Luckily the two body problem has an analytical solution which can be found through some mathematical trickery with the Newtonian expression for gravity in it's vector form \(\vec{F} = G \frac{m_1 m_2}{r^2} \frac{\vec{r}}{r}\). This expression, through a change of basis and some substitutions, gives rise to conical sections describing the path of our planets. What we get out is an expression for the distance between a planet and our star as a function the angle from periapsis with our planet rotating counter clockwise. This function is \(r(f) = \frac{a(1 - e^2)}{1 + \cos(f)}\)[1], where f is the angle from periapsis, e is the eccentricity of our orbit and a is the semi-major axis.

This may need some more explanation before we run through how we made or robots calculate and plot the function we've found for it. To show what will be what in our final figure, we made this nice litte sketch:

Bildet kan inneholde: tekst, blå, skrift, linje, sirkel.

The only thing this sketch doesn't show us is eccentricity, which quite simply is how unlike a circle our elliptical orbit is. An eccentricity of 0 be an elliptical orbit with the foci in the same location, more commonly known as a circle. This  measure is defined as \(e = \sqrt{1 - \frac{b^2}{a^2}}\) where a and b are the semi-major and semi-minor axis respectively.

Bildet kan inneholde: tekst, sirkel, plott, linje, diagram.

This gives us a nice plot of how our orbits will look, but not any information about how the move through space with regards to time. For this, we will have to do some simulations which we will hopefully be able to detail in our next log entry.

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[1] AST2000 Lecture Notes - Part B https://www.uio.no/studier/emner/matnat/astro/AST2000/h20/undervisningsmateriell/lecture_notes/part1b.pdf

Publisert 22. sep. 2020 16:54 - Sist endret 22. sep. 2020 16:54