Gaussian line-of-sight velocity distribution ($p(v_{\mathrm{los}})=(\sqrt{2\pi }{\sigma }_0{)}^{-1}{e}^{-v_{\mathrm{los}}^2/(2{\sigma }_0^2)}$)

For an isothermal collisionless tracer without streaming or an escape cutoff, integrate its three-dimensional Maxwell-Boltzmann velocity distribution over the two transverse velocities. The remaining marginal is the displayed normal distribution. Its width and mean do not depend on position. Integrating along the line of sight therefore only multiplies the marginal by the projected surface density; the normalized observed velocity distribution remains the same Gaussian at every projected radius.