# Why are asteroids with zero orbital inclination rare?

This is a plot of orbital inclination ($$i_p$$) vs. semi-major axis ($$a_p$$) of 96944 asteroids in the Main Belt, done by Piotr Deuar. Some structure can be seen in this diagram; clumps are asteroidal families and vertical empty spaces are Kirkwood gaps due to mean-motion resonances with the planets (mosly Jupiter).

Recently I've noticed that there are fewer points towards the ecliptic in the first $$\small{\frac 1 2}$$ degree of orbital inclination. Is this due to some selection bias? or Is it because of some kind of resonance that eliminates asteroids with orbital inclinations too close to zero?

• Oh this is a cool question! This is more of an example of rounding error than selection bias, and presumably it went away when their orbits were better measured. – uhoh Mar 11 at 3:58

I'll propose that it can be understood trivially.

What would the inclination distribution of circles randomly distributed in three dimensions about some point? We could generate them by distributing the normals to their orbital planes uniformly on a unit sphere, and call the midplane or the plane defined by $$\theta=\pi/2$$ the ecliptic. Orbits with an inclination $$i=0$$ will have their normals pointing "up" ($$\theta=0$$) and those orbiting the "wrong way" with $$i=\pi$$ will likewise have ($$\theta=\pi$$).

Then we have $$dn/d\theta = \sin(\theta)$$ which is zero for orbits near the ecliptic and increases linearly at first.

Fall off at 10-15 degrees is because the solar system is not random but is both a creature and creation of angular momentum.

If you take that data and project it all back to the inclination axis, I predict that you'll see a roughly linear increase from zero. If there's a dead zone very close to zero, that's because the planets are scattered a few degrees in inclination and have a propensity to mix things up.

• Can you clarify a bit? First derivative only shows the slope of density, not the actual value at that point. – Carl Witthoft Mar 11 at 17:28
• @CarlWitthoft $dn / d\theta$ is a one-dimensional density; it's the number per unit inclination but integrated over all $\phi$. The integral or total number $n$ is the solid angle of the whole unit sphere i.e. all normals for all possible circular orbits, and the value is $4 \pi$. If we were counting asteroids rather than possible orbits, then $n$ would be the total number of asteroids. – uhoh Mar 11 at 18:04
• Mmmm I also would be very gratefull if the explanation was a little more detailed (maybe with some visual representation), sorry for asking, its just that I'm not sure to fully understand the reasoning behind this. Also your argument seems to point to an bias/effect unrelated to an actual physcial mechanism but in the last sentence you introduce an idea for a mechanism. Could you please clarify also the connection between both? – Swike Mar 11 at 19:34
• @Swike There aren't any details to give. Here's an image of dots on a sphere. A dot on the north pole would in this case represent a circle on the equator. There can be only one. A dot 10 degrees from the pole would represent a circular orbit with inclination of 10 degrees, there can be many. One at 20 represents i=20 and there can be twice as many. I'll try to modify that to make a diagram for this, but if you look at that 2D rectangular plot and shift it up or down by 90 degrees you can see there will be a minimum at the equator. – uhoh Mar 12 at 0:37
• OK, that explanation of $\frac{dn}{d\theta}$ makes sense. – Carl Witthoft Mar 12 at 13:04