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We revisit the first type self-similar solutions for ultrarelativistic shockwaves produced by explosions propagating into cold external medium whosedensity profile decreases with radius as $\rho\propto r^{-k}$. The first typesolutions proposed by Blandford and McKee (hearafter BM solution) conforms tothe global conservation of energy and applies when $k<4$. They have been foundto be invalid when $k>17/4$ because of the divergence of total energy containedin the shocked fluids. So far no attention has been paid to the particlenumber. We use the BM solution to calculate the total particle number traversedby the shock and find it diverges when $k>3$. This is inconsistent with thefinite particles in the surrounding medium. We propose a possible solution when$k>3$ based on the conservation of particle number and discuss its implicationfor the second type solutions.

Revisiting first type self-similar solutions of explosions containing ultrarelativistic shocks