Acceleration of a particle on a logarithmic spiral in cartesian coordinates?

Question

I have a logarithmic spiral where the length of vector $r(\varphi)$ is given by the following equation: $$r(\varphi)=R_0 e^{-a\varphi}$$ a and $R_0$ are known, as well as the constant speed $v_0$. I want to find out the acceleration in cartesian coordinates $\vec a(\varphi)= \begin{pmatrix} \ddot x \\ \ddot y \end{pmatrix}$ Does somebody maybe have an idea on how you could do this? I know that there has to be acceleration in cartesian coordinates because the velocity is tangent to the logarithmic spiral which changes direction. I also know that$$x(\varphi)=\cos(\varphi)R_0e^{-a\varphi}$$ and $$y(\varphi)=\sin(\varphi)R_0e^{-a\varphi}$$, but to create the time derivatives, I think that I need to know how to calculate $\varphi(t)$, so that I can make $t$ my independant variable. Does somebody maybe know how to do this, maybe using the given velocity? Or can somebody maybe give a hint whether this task can even be solved? Thank you a lot for any ideas and replies!