Orbital Mechanics Angular Momentum

[FQVBSina_Jesse]

Hello all,

I have a question regarding the precise definition of angular momentum in orbital motion.

I see one definition says angular momentum h, position, r, and radial velocity, r_dot, are related as follows:
h = r x r_dot.

However, I also see one definition that says h is related to r and tangential velocity as follows:
h = r²θ_dot = r* (r*θ_dot)
Where θ_dot is the angular velocity, which makes rθ_dot the tangential velocity.

How come both definitions can be correct at times?

 

[FQVBSina_Jesse]

I don't think your first definition makes sense. $\vec r \times \vec {\dot r} = 0$ since $\vec r$ and $\vec {\dot r}$ are in the same direction.
What would make sense is angular momentum $\vec L =m \vec r \times \vec v$.

It is exactly as written in the book.
Actually, I just figured it out...
In an orbit, the r is defined from the center of the inertial frame. So r2 is rotated from r1 but the root of the vector is still at the inertial center. That means delta_r is a tangential vector that connects r1's vector head to r2's vector head. Making r_dot actually in the direction of the angular velocity.

 

[Dale]

I see one definition says angular momentum h, position, r, and radial velocity, r_dot, are related as follows:
h = r x r_dot.

You may want to check the source of this definition a little more carefully. Here $r$ should be position and $\dot{r}$ should be velocity (not radial velocity) and the expression should be $r \times m \dot{r}$