Loss of energy via EM waves by a charged particle in a uniform magnetic field

A charge moving in a uniform magnetic field travels in a circular path and objects moving in circular paths have a constant acceleration. It is well known that accelerating charges produce electromagnetic waves. So eventually the particle in this setup must move towards the centre of its circular path.

I tried to calculate the time for this to occur and got a rather quite strange answer. Here’s the derivation:

Now from the Larmor formula:
$$P=frac{q^2 a^2}{6pi epsilon_0 c^3}$$

Hence the total energy of the system over time can be formulised as:

$$E(t)=E_i-int_0^t P dt$$

Now differentiating wrt t:
$$frac{d}{dt}left(E(t)=E_i-int_0^t P dtright)Rightarrow E'(t)=-P$$

Since the charged particle has only kinetic energy, $E'(t)=frac{d}{dt}left(0.5mv^2right)=mav$

Now inputing and some algebra:
$$mav=-frac{q^2 a^2}{6pi epsilon_0 c^3}Rightarrow -6pi epsilon_0 c^3mv=q^2 a Rightarrow int_0^tfrac{-6pi epsilon_0 c^3m }{q^2}dt=int_{v_0}^vfrac{dv}{v} $$

This produces the answer;

$$v=v_0 expleft(frac{-6pi epsilon_0 c^3m }{q^2}tright)$$

Now calculating the half life of the speed,

$$t_{frac{1}{2}}=left(frac{ln(2)q^2}{6pi epsilon_0 c^3m }right)$$

which for an protonis $2.46 cdot 10^{-27}$ which is too short. Can someone explain why this is happening? Because in cyclotrons protons can whizz around for longer than that.