<p>Orbital eccentricities cannot be normally distributed around zero, since the minimum eccentricity is zero.</p>
<p>Thus, in the same way that the Maxwell-Boltzmann distribution for particle <em>speeds</em> is zero at zero speed and peaks at finite speed, then no asteroids have exactly zero eccentricity and we would expect a peak at higher eccentricities.</p>
<p>The analogy can be extended further by asking what distribution might be expected for a &quot;thermal distribution&quot; of eccentricities - a distribution where the energies of the orbits follow the Boltzmann distribution and have been allowed to randomise by some unspecified dynamical or formation processes (e.g. <a href="https://arxiv.org/abs/1902.00019#:%7E:text=This%20is%20referred%20to%20as,theoretical%20investigations%20of%20binary%20stars.&amp;text=Thus%2C%20if%20a%20thermal%20eccentricity,subsequent%20long-term%20dynamical%20processing." rel="nofollow noreferrer">Geller et al.
2019 </a>). In those case the density of eccentricities should go as
<span class="math-container">$$n(e) \propto e^2$$</span>
and there should be more high eccentricity objects than low eccentricity objects.</p>
<p>Of course asteroids are not expected to be represented by this random distribution, since their dynamical histories likely have them being perturbed away from very low eccentricity orbits induced by the drag of a gas/dust disc in the early solar system.
Nevertheless, any initial perturbation moves them away from zero eccentricity and subsequent random perturbations might be expected to make them head towards the thermal distribution. I would guess that objects in the top end of the thermal distribution must also then just get selected out - falling into the Sun, hitting planets or getting ejected out of the main asteroid belt.</p>

CC BY-SA 4.0
2020-12-22T09:21:19.320Z
2020-12-22T09:21:19.320Z
ProfRob
<p>Regarding this excellent question by Swike: <a href="https://astronomy.stackexchange.com/questions/35471/why-are-asteroids-with-zero-orbital-inclination-rare/40518#40518">Why are asteroids with zero orbital inclination rare?</a>, I recently proposed that orbits with zero inclination are rare as a natural result of the statistical distribution of orbital revolution axes extended onto the celestial sphere.</p>
<p>In the below <a href="https://www.researchgate.net/figure/The-orbital-element-distributions-of-real-and-modelled-asteroidsThe-distributions-of_fig1_51437374" rel="nofollow noreferrer">graph</a> we see a similar sparsity of orbital eccentricities near zero. If asteroid orbital eccentricities were normally distributed about zero, wouldn't we expect them to be densest here?  What is the explanation?</p>
<p><a href="https://i.sstatic.net/tQvDi.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/tQvDi.jpg" alt="enter image description here" /></a></p>

<asteroids><orbital-mechanics><eccentric-orbit>
Why are asteroids with circular orbits rare?
Connor Garcia
# Why are asteroids with circular orbits rare?

(Asked by: username_0 on 2020-12-21)

Regarding this excellent question by Swike: [Why are asteroids with zero orbital inclination rare?](https://astronomy.stackexchange.com/questions/35471/why-are-asteroids-with-zero-orbital-inclination-rare/40518#40518), I recently proposed that orbits with zero inclination are rare as a natural result of the statistical distribution of orbital revolution axes extended onto the celestial sphere.

In the below [graph](https://www.researchgate.net/figure/The-orbital-element-distributions-of-real-and-modelled-asteroidsThe-distributions-of_fig1_51437374) we see a similar sparsity of orbital eccentricities near zero. If asteroid orbital eccentricities were normally distributed about zero, wouldn't we expect them to be densest here? What is the explanation?

[![enter image description here](https://i.sstatic.net/tQvDi.jpg)](https://i.sstatic.net/tQvDi.jpg)

**Tags:** asteroids, orbital-mechanics, eccentric-orbit


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## 1 answers

### Answer by username_1 on 2020-12-22. Score: 3

Orbital eccentricities cannot be normally distributed around zero, since the minimum eccentricity is zero.

Thus, in the same way that the Maxwell-Boltzmann distribution for particle *speeds* is zero at zero speed and peaks at finite speed, then no asteroids have exactly zero eccentricity and we would expect a peak at higher eccentricities.

The analogy can be extended further by asking what distribution might be expected for a "thermal distribution" of eccentricities - a distribution where the energies of the orbits follow the Boltzmann distribution and have been allowed to randomise by some unspecified dynamical or formation processes (e.g. [Geller et al.
2019](https://arxiv.org/abs/1902.00019#:%7E:text=This%20is%20referred%20to%20as,theoretical%20investigations%20of%20binary%20stars.&text=Thus%2C%20if%20a%20thermal%20eccentricity,subsequent%20long-term%20dynamical%20processing.) ). In those case the density of eccentricities should go as
$$n(e) \propto e^2$$
and there should be more high eccentricity objects than low eccentricity objects.

Of course asteroids are not expected to be represented by this random distribution, since their dynamical histories likely have them being perturbed away from very low eccentricity orbits induced by the drag of a gas/dust disc in the early solar system.
Nevertheless, any initial perturbation moves them away from zero eccentricity and subsequent random perturbations might be expected to make them head towards the thermal distribution. I would guess that objects in the top end of the thermal distribution must also then just get selected out - falling into the Sun, hitting planets or getting ejected out of the main asteroid belt.


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