tokens: and Fig. 6 , we have shown that the radial density profile of the ejecta is well described by a power-law function of the velocity v with an exponent 6. This distribution is shallower than those of expanding envelopes in normal SNe (from n 9 to 12 as assumed in our model), which is clearly a consequence of the additional energy injection. The density distribution of freely expanding ejecta is a key to distinguishing existing models of SLSNe and other extraordinary SNe. In the following, we discuss how ejecta with such density structure form in the presence of the central energy source. For simplicity, we restrict ourselves to the Newtonian limit. We consider the following idealistic case. In normal SNe, the strong blast wave driven by a point explosion is the only way to transport the explosion energy to the outermost layer. As a result, only a small fraction of the energy is deposited in the outer envelope, while the inner ejecta possess the bulk of the energy, leading to a steep kinetic energy distribution (Matzner McKee 1999 ). In contrast to normal SNe, the central engine deposits an energy much larger than the supernvova ejecta for a time-scale much longer than the expansion time-scale of the ejecta. In our setting, the additional energy is injected at a constant rate L . Therefore, if the energy is continuously transferred throughout all the layers of the ejecta without loss or stagnation, the kinetic energy flux v 3 of the ejecta powered by the energy injection at a radius R 0 roughly proportional to the energy flux . The presence of relativistic flows leaking from the hot bubble makes the efficient energy transport possible. The flows can directly bring and deposit the additional energy throughout different layers. Thus, the following relation, is expected to hold for SN ejecta with a sufficiently long-term energy supply. In other words, the kinetic luminosity of the flow is constant. This relation gives the density distribution immediately after the ejecta are affected by the energy injection. We denote the time when the ejecta following this relation are created by t 0 . The ejecta are not freely expanding at this time. As the ejecta expand to the surrounding space, the density distribution would gradually be modified. Our goal is to derive the density distribution at the freely expanding stage. We regard the radial velocity v as a Lagrangian coordinate and derive the density distribution as a function of v . We consider a concentric shell with inner and outer velocity coordinates v and v v . The inner and outer boundaries are located at R R 0 ( v ) and R R 0 ( v v ) at t 0 . The width l s of the shell is initially given by, where the last expression is obtained for a sufficiently small v . The boundaries travel at the velocities v and v v with time and reach R R 0 ( v ) v ( t t 0 ) and R R 0 ( v v )( v v )( t t 0 ) at t . For t much longer than t 0 , the initial radius R 0 ( v ) can be neglected and the inner and outer radii are
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