The hot plasma filling galaxy clusters emits copious X-ray radiation .
The classic unheated and unperturbed cooling flow model predicts dramatic cooling rates and an isobaric X-ray spectrum with constant differential luminosity distribution .
The observed cores of clusters ( and groups ) show instead a strong deficit of soft X-ray emission : dL _ { x } / dT \propto ( T / T _ { hot } ) ^ { \alpha = 2 \pm 1 } .
Using 3D hydrodynamic simulations , we show that such deficit arises from the tight self-regulation between thermal instability condensation and AGN outflow injection : condensing clouds boost the AGN outflows , which quench cooling as they thermalize through the core .
The resultant average distribution slope is \alpha \simeq 2 , oscillating within the observed 1 < \alpha < 3 .
In the absence of thermal instability , the X-ray spectrum remains isothermal ( { { { { \alpha \mathrel { \mathchoice { \lower 2.5 pt \vbox { \halign { \cr } $ \displaystyle \hfil > $% \cr$ \displaystyle \hfil \sim$ } } } { \lower 2.5 pt \vbox { \halign { \cr } $ \textstyle \hfil > % $ \cr$ \textstyle \hfil \sim$ } } } { \lower 2.5 pt \vbox { \halign { \cr } $ \scriptstyle \hfil > % $ \cr$ \scriptstyle \hfil \sim$ } } } { \lower 2.5 pt \vbox { \halign { \cr } $% \scriptscriptstyle \hfil > $ \cr$ \scriptscriptstyle \hfil \sim$ } } } } 8 ) , while unopposed cooling drives a too shallow slope , \alpha < 1 .
AGN outflows deposit their energy inside-out , releasing more heat in the inner cooler phase ; radially distributed heating alone induces a declining spectrum , 1 < \alpha < 2 .
Turbulence further steepens the spectrum and increases the scatter : the turbulent Mach number in the hot phase is subsonic , while it becomes transonic in the cooler phase , making perturbations to depart from the isobaric mode .
Such increase in d \ln P / d \ln T leads to \alpha \approx 3 .
Self-regulated AGN outflow feedback can address the soft X-ray problem through the interplay of heating and turbulence .