Angular momentum conservation - elliptical motion By equating $F_{\rm gravitation}$ to $F_{\rm centripetal}$, the formula for the velocity of an object traveling along a certain orbital becomes $\sqrt{\frac{GM}{r}}$. The formula for angular momentum can be written as $L=mvr$. In elliptical motion, for example, when earth travels further away from the sun, the radius increases. Therefore, writing the formula of $L=mvr$ as $L=\sqrt{\frac{GM}{r}}mr$, it can be seen that as $r$ increases angular momentum should increase. However, this shouldn't happen because no net torque was applied.
 A: 
the formula for the velocity of an object traveling along a certain orbital
becomes $\sqrt{\frac{GM}{r}}$.

The equation
$$v=\sqrt{\frac{GM}{r}}$$
is valid only for circular orbits.
More generally (i.e. for elliptical, parabolic and hyperbolic orbits)
this formula needs to replaced by the vis-viva equation
$$v=\sqrt{GM\left(\frac{2}{r}-\frac{1}{a}\right)}$$
where $a$ is the semi-major axis of the orbit.
The circular orbit is a special case of this.
Only in this case we constantly have $r=a$,
and the the formula above reduces to $v=\sqrt{\frac{GM}{a}}$.

Therefore, writing the formula of $L=mvr$ ...

Again, this equation in this simplified form is valid only for circular orbits.
More exactly, it needs to be written as a vector equation
$$\vec{L}=m\vec{v}\times\vec{r}$$
where $\times$ is the cross product, or
$$L=mvr\sin(\gamma)$$
where $\gamma$ is the angle between $\vec{v}$ and $\vec{r}$.
The circular orbit is a special case of this.
Only in this case we constantly have $\gamma=90°$,
and then the formula above reduces to $L=mvr$.
