Pulsar radio emission mechanism I : On the amplification of Langmuir waves in the linear regime

Observations suggest that in normal period radio pulsars, coherent curvature radiation is excited within 10$\%$ of the light cylinder. The coherence is attributed to Langmuir mode instability in a relativistically streaming one-dimensional plasma flow along the open magnetic field lines. In this work, we use a hot plasma treatment to solve the hydrodynamic dispersion relation of Langmuir mode for realistic pulsar parameters. The solution involves three scenarios of two-stream instability viz., driven by high energy beams, due to longitudinal drift that leads to a separation of electron-positron distribution functions in the secondary plasma and due to cloud-cloud interaction causing spatial overlap of two successive secondary plasma clouds. We find that sufficient amplification can be obtained only for the latter two scenarios. Our analysis shows that longitudinal drift is characterized by high growth rates only for certain multi-polar surface field geometry. For these configurations, very high growth rates are obtained starting from a few tens of km from the neutron star surface, which then falls monotonically with increasing distance. For cloud-cloud overlap, growth rates become high starting only after a few hundred km from the surface, which first increases and then decreases with increasing distance. A spatial window of up to around a 1000 km above the neutron star surface has been found where large amplitude Langmuir waves can be excited while the pair plasma is dense enough to account for high brightness temperature.