Disassociation energies for the finite-density N-body problem

Abstract

This paper considers the energy required for collections of finite-density bodies to undergo escape under internal gravitational interactions alone. As the level of the system energy is increased, there are different combinations of components that can escape, until the total energy becomes positive, when the entire system can undergo mutual disruption. The results are also defined for bodies modeled as a continuum. These results provide rigorous constraints for the disruption of rubble-pile asteroids when only considering gravitational interaction effects, with the energy provided by rotation of an initial collection of the system. These issues are considered for discrete particles in the N-body problem and for size distributions of discrete particles in the continuum limit.

Appendix: Size distribution functions

Consider a cumulative size distribution of the form $\mathcal{N}_{\alpha }(r) = \frac{A_\alpha }{r^\alpha }$ for $2 \le \alpha \le 3$. Associated with this distribution is a maximum and minimum grain radius, $r_1$ and $r_0$, respectively. The function $\mathcal{N}_{\alpha }(r)$ is the cumulative number of particles with radius between r and the maximum size $r_1$. The term $A_\alpha$ is initially chosen to agree with the observed number of largest boulders, $\mathcal{N}_1$, such that $\mathcal{N}_{\alpha }(r_1)=\mathcal{N}_1$. With this interpretation, the nominal form for the function is:

$$\begin{aligned} \mathcal{N}_{\alpha }(r)= & {} \mathcal{N}_1 \left( \frac{r_1}{r}\right) ^\alpha . \end{aligned}$$

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The cumulative distribution is the integral of a cumulative density function $n_\alpha (r)$, defined as:

$$\begin{aligned} \mathcal{N}_{\alpha }(r)= & {} \int _{r}^{r_1} n_\alpha (r) \ \hbox {d}r. \end{aligned}$$

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This definition establishes that $n_\alpha (r) = - \frac{d\mathcal{N}_{\alpha }}{\hbox {d}r}$, leading to the cumulative density function

$$\begin{aligned} n_\alpha (r)= & {} \frac{\alpha \mathcal{N}_1 \ r_1^\alpha }{r^{\alpha +1}}. \end{aligned}$$

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A density distribution function that integrates to unity can also be defined, denoted as $\bar{n}_\alpha (r)$:

$$\begin{aligned} \bar{n}_\alpha (r)= & {} \frac{n_\alpha (r)}{\int _{r_0}^{r_1} n_\alpha (r) \ \hbox {d}r}. \end{aligned}$$

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Carrying out this computation yields

$$\begin{aligned} \bar{n}_\alpha (r)= & {} \frac{\alpha r_1^\alpha r_0^\alpha }{(r_1^\alpha - r_0^\alpha ) r^{\alpha +1}}. \end{aligned}$$

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There are several quantities of interest that can be defined and calculated with a power law size distribution. A few of them are reviewed here, in addition to stating some key results used in the paper.

Mean grain radius The mean grain radius is defined as

$$\begin{aligned} \bar{r}= & {} \int _{r_0}^{r_1} r \bar{n}_\alpha (r) \ \hbox {d}r \end{aligned}$$

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$$\begin{aligned}= & {} \frac{\alpha r_1 r_0}{\alpha -1}\frac{ r_1^{\alpha -1} - r_0^{\alpha -1}}{r_1^\alpha - r_0^\alpha }. \end{aligned}$$

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Thus, if $r_0 \ll r_1$, the mean radius is $\bar{r} \sim \frac{\alpha }{\alpha -1} r_0$.

Surface area of grains The total surface area of a collection of grains is computed as

$$\begin{aligned} {SA}_T= & {} \int _{r_0}^{r_1} 4\pi r^2 {n_\alpha }(r) \ \hbox {d}r \end{aligned}$$

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$$\begin{aligned}= & {} 4\pi \mathcal{N}_1 \alpha r_1^\alpha \int _{r_0}^{r_1} r^{1-\alpha } \hbox {d}r. \end{aligned}$$

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If $2 < \alpha \le 3$, this can be integrated to find

$$\begin{aligned} {SA}_T= & {} \frac{4\pi \mathcal{N}_1 \alpha }{\alpha -2} r_1^2 \left[ \left( \frac{r_1}{r_0}\right) ^{\alpha -2} - 1\right] , \end{aligned}$$

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and if $\alpha = 2$, the total surface area equals

$$\begin{aligned} {SA}_T= & {} 8\pi \mathcal{N}_1 r_1^2 \ln \left( \frac{r_1}{r_0}\right) . \end{aligned}$$

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For either case, if $r_0 \ll r_1$, the total surface area becomes arbitrarily large.

Volume of grains The total volume of grains can be found by

$$\begin{aligned} {V}_T= & {} \int _{r_0}^{r_1} \frac{4\pi }{3} r^3 {n_\alpha }(r) \ \hbox {d}r \end{aligned}$$

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$$\begin{aligned}= & {} \frac{4\pi \ \alpha \mathcal{N}_1 }{3} r_1^\alpha \int _{r_0}^{r_1} r^{2-\alpha } \ \hbox {d}r. \end{aligned}$$

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If $2 \le \alpha < 3$, the total volume equals

$$\begin{aligned} {V}_T= & {} \frac{4\pi }{3} \frac{\alpha \mathcal{N}_1 r_1^3}{3-\alpha } \left[ 1 - \left( \frac{r_0}{r_1}\right) ^{3-\alpha }\right] . \end{aligned}$$

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If $\alpha = 3$, the total volume equals

$$\begin{aligned} {V}_T= & {} 4\pi \mathcal{N}_1 r_1^3 \ln \left( \frac{r_1}{r_0}\right) . \end{aligned}$$

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For $\alpha < 3$, one can take the limit $r_0\rightarrow \infty$ without any singularity. For $\alpha =3$, however, this leads to an infinite mass.

Total self-potential Finally, the total self-potential of a size distribution, assuming spherical grains, is computed as

$$\begin{aligned} \mathcal{U}_\mathrm{Self}= & {} -\frac{3\mathcal{G}}{5} \left( \frac{4\pi \rho _g}{3}\right) ^2 \int _{r_0}^{r_1} r^5 n_\alpha (r) \ \hbox {d}r \end{aligned}$$

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$$\begin{aligned}= & {} -\frac{3\mathcal{G}}{5} \left( \frac{4\pi \rho _g}{3}\right) ^2 {\alpha \mathcal{N}_1 r_1^\alpha } \int _{r_0}^{r_1} r^{4-\alpha } \ \hbox {d}r. \end{aligned}$$

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The integral is defined for the whole interval of $2\le \alpha \le 3$, yielding

$$\begin{aligned} \mathcal{U}_\mathrm{Self}= & {} -\frac{3\mathcal{G}}{5} \left( \frac{4\pi \rho _g}{3}\right) ^2 \frac{\alpha \mathcal{N}_1 r_1^5}{5-\alpha } \left[ 1 - \left( \frac{r_0}{r_1}\right) ^{5-\alpha }\right] . \end{aligned}$$

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Across the entire interval, the limit $r_0\rightarrow 0$ can be taken.