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Using a kinetic model for the ions and adiabatic electrons, we solve a steadystate, electron-repelling magnetic presheath in which a uniform magnetic fieldmakes a small angle $\alpha \ll 1$ (in radians) with the wall. The presheathcharacteristic thickness is the typical ion gyroradius $\rho_{\text{i}}$. TheDebye length $\lambda_{\text{D}}$ and the collisional mean free path of an ion$\lambda_{\text{mfp}}$ satisfy the ordering $\lambda_{\text{D}} \ll\rho_{\text{i}} \ll \alpha \lambda_{\text{mfp}}$, so a quasineutral andcollisionless model is used. We assume that the electrostatic potential is afunction only of distance from the wall, and it varies over the scale$\rho_{\text{i}}$. Using the expansion in $\alpha \ll 1$, we derive ananalytical expression for the ion density that only depends on the iondistribution function at the entrance of the magnetic presheath and theelectrostatic potential profile. Importantly, we have added the crucialcontribution of the orbits in the region near the wall. By imposing thequasineutrality equation, we derive a condition that the ion distributionfunction must satisfy at the magnetic presheath entrance --- the kineticequivalent of the Chodura condition. Using an ion distribution function at theentrance of the magnetic presheath that satisfies the kinetic Choduracondition, we find numerical solutions for the self-consistent electrostaticpotential, ion density and flow across the magnetic presheath for severalvalues of $\alpha$. Our numerical results also include the distribution of ionvelocities at the Debye sheath entrance. We find that at small values of$\alpha$ there are substantially fewer ions travelling with a large normalcomponent of the velocity into the wall.

Solution to a collisionless shallow-angle magnetic presheath with kinetic ions