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A growing planet embedded in a protoplanetary disk induces three-dimensionalgas flow, which exhibits a midplane outflow that can suppress dust accretiononto the planet and form global dust substructures (rings and gaps). Becauseanalytic formulae for the planet-induced outflow are useful for modeling itsinfluences on local and global dust surface densities and planet accretion, wederive the analytic formulae that describe the morphology and velocity of theplanet-induced outflow. We first perform three-dimensional, nonisothermalhydrodynamical simulations of the gas flow past a planet, which enables us tointroduce a fitting formula describing the morphology of the outflow. We thenderive an analytic formula for the outflow speed using Bernoulli's theorem. Wesuccessfully derived a fitting formula for the midplane outflow morphology (theshape of the streamline), which is valid when the dimensionless thermal massfalls below $m\lesssim0.6$. The obtained analytic formulae for the outflow,such as the maximum outflow speed and the velocity distributions of the outflowin the radial and vertical directions to the disk, show good agreement with thenumerical results. We find the following trends: (1) the maximum outflow speedincreases with the planetary mass and has a peak of $\sim$30--40$\%$ of thesound speed when the dimensionless thermal mass is $m\sim0.3$, corresponding toa super-Earth mass planet at 1 au for the typical steady accretion disk model,and (2) the presence of the headwind (namely, the global pressure force actingin the positive radial direction of the disk) enhances (reduces) the outflowtoward the outside (inside) of the planetary orbit. The planet-induced outflowof the gas affects the dust motion when the dimensionless stopping time of dustfalls below ${\rm St}\lesssim\min(10m^2,0.1)$.

Analytic description of the gas flow around planets embedded in protoplanetary disks