Fun With Irodov!(1)

An electron moving with an acceleration $a$ loses its energy due to radiation in accordance to

$\dfrac {dE}{dt} = -\dfrac {2e^2}{3c^3} a^2$

The time during which the energy of the electron performing harmonic motion with frequency $f=5\times 10^5$ s $^{-1}$ decrease by 10 times is given by $T.$

Find $\lfloor 10^9 T\rfloor .$

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Details -

$(1)$ $e$ is the charge of an electron $=1.6 \times 10^{-19}$ C.

$(2)$ $c$ is speed of light in vacuum $=3\times 10^8$ ms $^{-1}.$