In my last post about KIC 8462852 I made a point about the difference between the opacity and density of debris blocking the light from the oddly behaving star. Picture a sphere with a uniform distribution of debris in its interior. The density of the material is the same at any point inside the sphere, but its opacity — from the point of view of someone trying to look through the sphere — will depend on which part of the sphere is intersected by the line of sight.

This matters considerably, in my view. I will argue that an opacity distribution capable of explaining the day-792 dip, with all of its characteristics, cannot be produced by an object with a 3D density distribution expressed as a function of the distance from a point in space. Instead, it would have to be produced by material distributed over a largely flat surface.

The day-792 dip is peculiar in a lot of ways and very different to an ordinary U-shaped or V-shaped exoplanetary transit. It’s gigantic and it lasts many days. It’s smooth and builds up very gradually. It peaks briefly, and then gradually subsides in a slightly asymmetric way. Because of these characteristics, the day-792 light curve would have to be produced by a transit whose opacity distribution in the direction of the orbit is approximated by the following chart.

This opacity distribution was obtained by optimizing a transit model that produces a very good fit for the day-792 light curve — a transit that happens to look a lot like a faint Niven ring. Granted, that’s not the only possible model. Different constraints and parameters could produce different object shapes. I contend, however, that the average opacity distribution in the direction of the orbit should follow essentially the same pattern for any transit models that are plausible. That’s because of the very peculiar characteristics of the aforementioned light curve.

Such an opacity distribution can be approximated by the following formula.

We want to figure out if there’s a 3D density distribution that could produce a debris opacity distribution like the one I’ve just outlined. I haven’t come up with a closed-form mathematical solution to this problem, but we can explore it with simulations.

The following R code produces n points inside a sphere of radius 5. Points are produced using rejection sampling of a density-by-distance function.

generatePoints = function(n, densityFunction, maxRadius = 5) { maxRadiusSq = maxRadius ^ 2; resultTable = data.frame(x = numeric(n), y = numeric(n), z = numeric(n)); for(i in 1:n) { repeat { xyz = runif(n = 3, min = -maxRadius, max = maxRadius); x = xyz[1]; y = xyz[2]; z = xyz[3]; dSq = x^2 + y^2 + z^2; if(dSq <= maxRadiusSq) { distance = sqrt(dSq); p = densityFunction(distance); if(runif(n = 1, min = 0, max = 1) < p) { break; } } } resultTable[i,] = c(x, y, z); } return(resultTable); }

First, let’s test the simulation with a uniform distribution, as a sanity check.

uniformDF = function(distance) { 1.0 } points = generatePoints(5000, uniformDF);

We can visualize the resulting opacity distribution as the distribution of points along any one axis. We will use the X axis, as follows.

hist(points$x, breaks = 20, main = "Opacity from uniform density");

The resulting histogram looks like this:

This is clearly the opacity distribution we would expect from a sphere containing uniformly distributed debris.

Now, let’s try a density decay distribution similar to what we’re expecting for opacity and check how it behaves.

satDF = function(distance) { 1.0 / (1.0 + 20 * distance) } points = generatePoints(5000, satDF); hist(points$x, breaks = 20, main = "Opacity from saturation decay density");

It does flatten the round shape a little, but it’s nowhere near what we expect. Let’s try an exponential density distribution.

expDF = function(distance) { exp(-(2 * distance)) } points = generatePoints(5000, expDF); hist(points$x, breaks = 40, main = "Opacity from exponential density");

It’s closer to what we want, but it doesn’t quite match the target distribution. Let’s take a look at a comparison between our expected opacity distribution and the simulated one.

I have analyzed other types of density distributions (gaussian, exponential of exponential, and others). The result is always the same: Our expected opacity distribution cannot be reproduced. Mathematically, the opacity distribution that explains the day-792 light curve would have to be produced by debris that sits roughly on top of a plane or an orbit.