The Kanagy clan makes its home on a distant planet of mass $M_p$ with $k$ moons. Suppose the moons are identical with mass $m$. Furthermore, these moons share a common circular orbit on an orbital plane. The circular orbits are at distance $L$ from the center of the planet and the moons are evenly spaced.
- What is the speed $v(k)$ of the lunar orbits for a fixed, but finite, value of $k$?
- Suppose we hold $km=M_m$ fixed as $k \rightarrow \infty$, what is the limiting value of $\lim_{k \rightarrow \infty}v(k)$ in this context?
My ideal solution to this problem addresses $(1.)$ by vector analysis paired with the equation of motion for constant speed circular motion. Often cases $k=2$ or $k=3$ are given as homework problems in first semester university physics. I've found an expression for this in a previous attempt, but I'd rather not include it here for fear of biasing the reader. Next, the solution continues to $(2.)$, when I attempted to compute the limit directly it was rather involved. However, by intuition, I know the answer should easily derive from Newton's Law of Gravitation as follows: $$ \frac{M_mv^2}{L}= \frac{GM_mM_p}{L^2} \qquad \Rightarrow \qquad v = \sqrt{\frac{GM_p}{L}} $$
Given the preceding discussion, show show $v(k) \rightarrow \sqrt{\frac{GM_p}{L}}$ as $k \rightarrow \infty$. Alternatively, prove my intuition is incorrect.
Incidentally, I gave this as a bonus problem on a final exam in my university physics course. I had a student pretty well solve $(1.)$, but $(2.)$ I've not yet cracked. It is assumed that classical mechanics applies to this problem and any relativistic effects may be neglected.
The centre of mass of the system is at the planet, which we can thus assume is stationary. If we number the moons $0$ to $k-1$, moon $0$ sees an angle $\theta_j = \dfrac{\pi}{2} - \dfrac{j \pi}{k}$ between moon $j$ and the planet, and the distance from moon $0$ to moon $j$ is $2 L \sin(j \pi/k)$. So the component in the direction of the planet of the gravitational acceleration of moon $0$ due to moon $j$ is $\cos(\theta_j) G m/(2 L \sin(j \pi/k))^2 = G m/(4 L^2 \sin(j \pi/k))$. The net gravitational acceleration (directed toward the planet) due to all the other moons is then $$ \sum_{j=1}^{k-1} \dfrac{G m}{4 L^2 \sin(j \pi/k)} = \sum_{j=1}^{k-1} \dfrac{G M_m}{4 k L^2 \sin(j \pi/k)} > \sum_{j=1}^{k-1} \dfrac{G M_m}{4 L^2 \pi j}$$ But that goes to $+\infty$ as $k \to \infty$. So we must also have $v(k) \to \infty$.