Why does conservation of angular momentum explain why planet orbits lie in a plane?

Question

I just learned about sector velocity and how you can manipulate the equation to show that angular momentum $L$ is conserved by $L/2M$. I know that sector velocity is constant and therefore angular momentum is constant. I have made it that far but I still can't see why a constant angular momentum would explain the coplanar concept.

Answer 1

The angular momentum is given by a vector product $\vec{L}=m[\vec{r}\times\vec{v}]$. One o the properties of the vector product is that the result is orthogonal to both factors. I.e. both $\vec{r}$ and $\vec{v}$ should be orthogonal to $\vec{L}$. Or you can say that both $\vec{r}$ and $\vec{v}$ lie in the plane orthogonal to $\vec{L}$. As $\vec{L}$ is conserved this plane is conserved.

Answer 2

It's because dust, gases, proto-planets, and other objects in solar systems have existed for sufficient time at a high enough density to interact strongly.

Many orbits exist initially, but any non-coplanar or highly elliptical orbits will eventually intersect other existing orbits. These intersections result in a collision, causing transfer of momentum. The only possible set of non-intersecting orbits is in the plane perpendicular to the total angular momentum.

Every time there is a collision, some material might move into the orthogonal plane and some might move into a different one. But the different one will (eventually) experience another collision. The system evolves to a state where almost all the non-coplanar material is removed.

Unlike solar systems, globular clusters and the Oort cloud are examples of objects where the there is no single orbital plane. There is still a single average angular momentum vector, but the density of the material in these objects are much lower. So the interactions haven't had sufficient time to remove material not in the orthogonal plane.

Why should non coplanar orbits intersect? Many non-coplanar ellipses do not intersect at all, and this is even without counting the fact that the objects needs to both be at the intersection point at the same time for a collision to occur.

That depends on the number of orbits and the cross-section of the orbit. If you draw 10 random orbits there may be no interaction (at least for a while). But if you draw billions and they have a cross-section, you're going to have trouble avoiding interactions. The early solar system started as dust particles that could interact with neighbors not only gravitationally but also through electrostatic forces.

Here's a less mechanistic way to think about it. If the rate of interaction is low, aggregation doesn't happen. All the particles remain separate. As the solar system is no longer a large cloud of dust, we know the interactions must have occurred.

Aggregation also requires inelastic collisions. Every perfectly inelastic collision produces an output particle with the momentum average of the input particles. The more adhesion, the more particles that the momentum is averaging over. If you have something that has a large enough fraction of the whole (like a planet), it is not a surprise that it has momentum in the same direction as the average of the whole.

To get planets moving in wildly different inclinations during initial formation would require some mechanism for there to be different populations of precursors that can interact and adhere with themselves to form the planet, but not interact with the other population to cause the angular momenta to be averaged out.