We investigate how extra central light in the surface brightness profiles of cusp ellipticals relates to the profiles of ellipticals with cores. Cusp elliptical envelopes are formed by violent relaxation in mergers acting on stars in progenitor disks, while their centers are structured by dissipational starbursts. Core ellipticals are formed by subsequent merging of (now gas-poor) cusp ellipticals, with the fossil starburst components combining to preserve a compact component in the remnant (although the transition is smoothed). Comparing hydrodynamical simulations and observed profiles, we show how to observationally isolate the relic starburst components in core ellipticals. We demonstrate that these survive re-mergers and reliably trace the dissipation in the initial gas-rich merger(s). The typical degree of dissipation is a strong function of stellar mass, tracing observed disk gas fractions. We find a correlation between dissipation and effective radius: systems with more dissipation are more compact. The survival of this component and scattering of stars into the envelope naturally explain high-Sersic index profiles characteristic of massive core ellipticals. This is also closely related to the kinematics and isophotal shapes: only systems with matched starburst components from their profile fits also reproduce the observed kinematics of boxy/core ellipticals. We show that it is critical to adopt physically motivated profiles when attempting to quantify how much mass has been scoured or scattered out of the inner regions by binary black holes. Estimates of scoured mass ignoring multi-component structure can be strongly biased, potentially explaining observed systems with large inferred core masses in apparent conflict with core-scouring models.
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