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There are a lot of objects of different sizes in the Kuiper belt. Are objects in the Kuiper belt stable? Is there any chance in the future that one of the larger objects would accumulate enough critical mass to start the process of becoming a proper planet (i.e., clearing its neighboorhood)?

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The Kuiper Belt is fairly large (with inner and outer radii $\sim30\text{ AU}$ and $\sim50\text{ AU}$), but it does not contain much mass. The total mass is likely somewhere between $0.01$ and $0.1$ Earth masses[1], [2], which is essentially a mass range from five times the mass of Pluto to the mass of Mars. Now, it was likely much more massive in the past - its origin and mass loss are tied to the evolution of the orbits of ice giants in the early Solar System (see in particular the Nice model) - but we're talking about the formation of planets in the future. At present, there simply isn't enough mass to go around.

Let's talk about "clearing the neighborhood". The IAU's definition of a planet merely states, as one of its three criteria, that a planet must have

cleared the neighbourhood around its orbit.

Wikipedia gives several quantitative measures of whether or not an object has cleared its neighborhood. The quantity $\Pi$ (Margot (2015) is given by $$\Pi(m,a)=\frac{m}{M^{5/2}a^{9/8}}k$$ where $m$ is the mass of the body (Earth masses), $M$ is the mass of the star (solar masses), $a$ is the radius of its semimajor axis (AU), and $k\simeq807$. A body is a planet if $\Pi>1$. If the mass of this hypothetical planet, $m_H$, is $0.01$ Earth masses, then no orbit in the range $30\text{ AU}<a<40\text{ AU}$ gives me $\Pi(.01M_{\oplus},a)>1$. If $m_H$ is $0.1$ Earth masses, then almost all $a$ in the same range give me $\Pi(0.1M_{\oplus},a)>1$. That said, it is extremely unlikely that a body could amass even a small fraction of the total mass in the Kuiper Belt - remember the inner and outer boundaries!

The quantity $\Lambda$ (Stern & Levison (2002)) is given by $$\Lambda(m,a)=\frac{\mu^2}{a^{3/2}}\times\left[\text{Terms accounting for the properties of the body's orbit}\right]$$ where $\mu$ is the ratio of the mass of the body to the mass of the star. The author's list a value on the order of $\Lambda=3\times10^{-3}$ for Pluto; if we assume that the orbital terms are about the same for this body, we find $\Lambda(0.01M_{\oplus},a)\sim0.075$ and $\Lambda(0.1M_{\oplus},a)\sim7.5$, with the cutoff being $\Lambda=1$. Again, it is not possible for the body to accumulate the total mass of the Kuiper Belt, so we can safely assume that $\Lambda\ll1$.

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