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StarDate 197743.30

We know there are planets here in our solar system, as you would expect seeing as we're living on one of these lonely dusty balls in the middle of nowhere for now. What we're wondering is whether or not the planets in this solar system would be visible from far away solar systems like our home. We can once again use our tiny calculation robots to help us find out if it would be possible to see the dimming effect of our largest planet passing the star.

To do this, we start off by introducing some simplifications. To begin with we will say that the maximum amount of relative dimming with regards to our stars normal flux is going to be the proportion of the cross-sectional area of the star that the planet's cross-sectional area can cover. This is given by the fraction \(\frac{\pi r^2}{\pi R^2}\) which we can simplify to get \(\frac{r^2}{R^2}\)where \(r\) is the radius of the planet which is eclipsing the star, and \(R\) is the radius of the star. Seeing as these two bodies both are unimaginably far away from any observer in any other solar system, it's a good approximation to say that they are both just as far away from any observer as each other.

Now that we have an amount the observed flux will diminish by, we can move on to our next problem. As the first part of the planet starts to cross the first part of the star, we will have a phase of increasing eclipsing of the star, where the transiting planet goes from not dimming the star at all to the maximal amount of dimming. This can be quite complicated to simulate, so let's not bother. This may seem like a silly choice, but there are two important factors that make this a sensible thing to do. First of all, this transitionary phase is not what we're here to study, meaning we won't gain anything from bothering to simulate it, and we will eventually add Gaussian noise to the light curve we generate, which will probably drown out any effect this transition would have had anyway. We therefore choose to say that at a given point in time, we go from no dimming to the maximum dimming which we calculated using the method from earlier.

Now, for one last aspect of our simulation that we'll also be simplifying; time. Over time, as our planet traverses it's path around our star, it's distance from the star and the time a transit takes will vary slightly. This will also be ignored, as this too is not important. What we're interested in is simply how much our planet's transit effects the flux, which we've already simplified to be indipendant of the distance between the planet and star. This means simply simulating the transit on an arbitrary timescale from 0 to 1 is just as good as using any actual timing, so why make it more complicated than it has to be?

This now leaves us with a simulation that only uses the radius of the star and the transiting planet to generate a light curve. The instructions we fed to or calculating robots only told them to generate a long list of ones, and subtract the relative dimming from a subset of the list in the middle of it. This gave us a nice graph of a step-function. The only thing that remained then was to add some Gaussian noise. We added some noise as a Gaussian distribution with an average of 0 and a standard deviation of 20% relative flux. From this our robots drew the following plot:

Bildet kan inneholde: tekst, linje, plott.

This plot does not show us a whole lot of dimming, but at least it shows us that our theory of the necessary noise drowning out our transit was correct. In other words, a direct use of the transit method, even on our systems largest planet as shown in the plot, will not allow for observers in other solar systems to discover that there are planets here. Hopefully another method like Radial Velocity will be more successful, but that remains to be seen. Hopefully we will be able to simulate that and include our findings in our next entry.

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Publisert 30. sep. 2020 20:21 - Sist endret 30. sep. 2020 20:21