Synchrotron Radiation Damping in LHC and Longitudinal Bunch Shape

In LHC each proton loses about 7 keV per turn due to synchrotron radiation, leading to radiation damping of the synchrotron oscillation. This can have both positive and negative consequences. On the one hand it can counteract the emittance increase due to unwanted RF noise or intra-beam scattering or on the other hand it may lead to a reduction in the Landau damping and increased intra-beam scattering if the bunches shrink significantly in size. Therefore - neglecting other effects - we analyze analytically the damping time scale, the equilibrium bunch shape for t > infinity and establish an analytic formulation for the development of any given initial bunch shape with time. The latter will be applied to display the time-development of an initially Gaussian bunch during an LHC coast of 24 h, showing a considerable contraction of the bunch length of up to a factor 2.5 without opposing effects. Finally we analyze the possibilities for conserving the bunch shape. We find that we can stabilize the bunch length and shape - compensating the synchrotron radiation damping - by injecting (white) phase noise with either 0.5º rms applied incoherently to all the eight 400 MHz RF cavities, or 0.17º coherently. This value of white phase noise can also be considered as the maximum value that can be allowed from the total RF system.