We point out that an accurate in situ determination of the earth matter density $\rho$ is possible in neutrino factory by placing a detector at the magic baseline, $L=\sqrt{2}\pi /G_F{N}_e$ where $N_e$ denotes electron number density. The accuracy of matter density determination is excellent in a region of relatively large ${\theta }_{13}$ with fractional uncertainty $\delta \rho /\rho$ of about 0.43%, 1.3%, and $\lesssim 3\%$ at $1\sigma$ CL at ${sin}^{2}2{\theta }_{13}=0.1$, ${10}^{-2}$, and $3\times {10}^{-3}$, respectively. At smaller ${\theta }_{13}$ the uncertainty depends upon the $CP$ phase $\delta$, but it remains small, 3%–7% in more than $3/4$ of the entire region of $\delta$ at ${sin}^{2}2{\theta }_{13}={10}^{-4}$. The results would allow us to solve the problem of obscured $CP$ violation due to the uncertainty of earth matter density in a wide range of ${\theta }_{13}$ and $\delta$. It may provide a test for the geophysical model of the earth, or it may serve as a method for a stringent test of the Mikheyev-Smirnov-Wolfenstein theory of neutrino propagation in matter once an accurate geophysical estimation of the matter density is available.

• Received 5 December 2006

DOI:https://doi.org/10.1103/PhysRevD.75.073013