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We present for the neutron-star low-mass X-ray binary 4U 1636$-$53, and forthe first time for any source of kilohertz quasi-periodic oscillations (kHzQPOs), the two-dimensional behaviour of the fractional rms amplitude of the kHzQPOs in the parameter space defined by QPO frequency and photon energy. We findthat the rms amplitude of the lower kHz QPO increases with energy up to$\sim12$ keV and then decreases at higher energies, while the rms amplitude ofthe upper kHz QPO either continues increasing or levels off at high energies.The rms amplitude of the lower kHz QPO increases and then decreases withfrequency, peaking at $\sim 760$ Hz, while the amplitude of the upper kHz QPOdecreases with frequency, with a local maximum at around $\sim 770$ Hz, and isconsistent with becoming zero at the same QPO frequency, $\sim1400$ Hz, in allenergy bands, thus constraining the neutron-star mass at $M_{NS} \leq 1.6M_{\odot}$, under the assumption that this QPO reflects the Keplerian frequencyat the inner edge of the accretion disc. We show that the slope of the rmsenergy spectrum is connected to the changing properties of the kHz QPOs indifferent energy bands as its frequencies change. Finally, we discuss apossible mechanism responsible for the radiative properties of the kHz QPOsand, based on a model in which the QPO arises from oscillations in aComptonising cloud of hot electrons, we show that the properties of the kHzQPOs can constrain the thermodynamic properties of the inner accretion flow.

The amplitude of the kilohertz quasi-periodic oscillations in 4U 1636$-$53 in the frequency-energy space