IAO 2018, Теория, Група α — Задача 5. αβ-5. Облакът на Оорт

The fact that every century we observe from 10 to 20 bright comets means that approximately once every 6 years 8 months ($\tau\approx2.1\cdot10^8\ \mathrm s$) two comet bodies collide in the outer Oort cloud, resulting in one of them rushes into the inner regions of the solar system. Collisions occur because all these bodies move chaotically with typical velocities less than the second cosmic velocity (otherwise they would leave this area) and more than half of the first cosmic velocity (otherwise they would move into the inner cloud areas). To estimate the average their velocities, we take the first cosmic velocity for the average distance $(R+r)/2=0.5\ \mathrm{ly}$:

where $M$ is the mass of the Sun, $R$ and $r$ are the outer and inner radii of the cloud. (One can also use the formula $u\sim R^{-1/2}$, 0.5 ly is about 30,000 times larger than the distance from the Earth to the Sun, therefore, the speed of the Earth in its orbit should be divided by the square root of 30,000. So we obtain $u\approx170\ \mathrm{m/s}$.

Let us find how the frequency of collisions is related to the number of comet bodies in the Oort cloud. If there are $N$ comet bodies in total, then their concentration in space is

(The fact $r^3\ll R^3$ was taken into account). We will use the collision model in which each comet body moves relative to the others with the same average velocity $u$. During the time $\tau$, the comet body will cover the distance $u\cdot\tau$ and in its movement will capture the volume

where $d$ is the average diameter of the comet body, we will consider it equal to 2.5 km. If another comet body appears in this volume, then a collision occurs, and the probability of such an event is equal to

Since any of the $N$ comet bodies can collide, the total number of collisions will be equal to:

Each collision leads to the fact that with some probability $\beta$ our body will be sent to the inner regions of the Solar System. This probability can be estimated from different models, let us estimate it as an approximate ratio of solid angles:

Thus, in order that one body goes to the inner regions of the solar system, it is necessary to fulfill the condition

Calculations give: $N=2.6\cdot10^{16}$,

with an accuracy of one significant digit: $N\approx3\cdot10^{16}$.

If the concentration of the comet bodies is $n$ (bodies per $\mathrm{m^3}$), then each body has a volume of space $1/n$, and, respectively, the average distance will $(1/n)^{1/3}$.

This is roughly the minimum distance between the Earth and Venus.

To estimate the total mass of the comet bodies, we assume that their average density is equal to the density of ice ($\rho\approx900\ \mathrm{kg/m^3}$), and the average volume is $v=1/6\ \pi d^3$. Thus, the average mass of one comet body is:

and the sum of the mass of all the bodies:

This is one hundred masses of Jupiter or one-tenth of the mass of the Sun.