kinematics - Acceleration vector for an object moving in a elliptic trajectory

I am interested in deriving what the radial and tangential components of the acceleration vector should be for an object following an elliptical trajectory centered on the origin, in which the relation between the speed $v$ (module) is related to the distance of the object to the origin ($r$) by $v=kr^\beta$ where $k$ and $\beta$ are constants. I tried to find information online, but much of the information about elliptical motion is devoted to objects subject to gravitation.

I know that the components of acceleration in polar coordinates are

$$a_r = \frac{d^2r}{dt^2}-r \left(\frac{d \theta}{dt}\right)^2$$ and $$a_\theta = r\frac{d^2\theta}{dt^2}+2 \frac{dr}{dt} \frac{d\theta}{dt}$$ but I do not know how to implement the constraint $v=kr^\beta$ there or whether that is the way to go.

Answer

Take the ansatz

$$\begin{align} \vec{r}(\theta(t)) = \begin{pmatrix} \rho_1\cos(\theta(t))\\ \rho_2\sin(\theta(t)) \end{pmatrix} \end{align}$$ with a yet unknown scalar smooth function $\theta$.

Solve the ode

$$k \bigg(\left|\vec{r}\big(\theta(t)\big)\right|\bigg)^{\beta} = |\vec{r}'(\theta(t))| \cdot\dot\theta(t)$$ for $\theta$ or explicitely $$\dot\theta(t) = \frac{k\cdot\left|\vec{r}(\theta(t))\right|^{\beta}}{|\vec{r}'(\theta(t))|}$$ Then use the found $\theta$ to calculate $$\vec{a}(t)=\frac{d}{dt}\left(\vec{r}'(\theta(t)) \dot\theta(t)\right).$$