Definition of lifetime due to wave-particle interactions for the pitch angle of 90 degrees

One of very important, but still it is not enough the investigated problems in the theoretical physics of the Earth's magnetosphere is the definition of lifetime of the charged particles due to wave-particle interactions. Therefore for the pitch angle of $90$ degrees as mathematical model is offered the ordinary differential equation (ODE) for the analytical description of a perpendicular differential flux of the charged particles in the Earth's magnetosphere which depends on time $t$ and several parameters. Using the analytical solution of the ODE, the new simple formula for definition of lifetime due to wave-particle interactions for the pitch angle of $90$ degrees for different geophysical conditions is received. For calculation under this formula it is used correlated observation of enhanced electromagnetic ion cyclotron waves and dynamic evolution of ring current energetic ($5$–$30$ keV) proton flux collected by Cluster satellite near the location $L = 4.5$ during March 26–27, 2003, a nonstorm period ($Dst>-10$ nT. In addition are found the perpendicular coefficients of the particle loss function, the particle source function and the pitch angle diffusion. For the first time the modeling dependences of lifetime due to wave-particle interactions for the pitch angle of $90$ degrees from the local time $\mathrm{LT}$ and the geomagnetic activity $Kp$-index are received. Mathematical statement of a problem in the general view (the system of two ODEs of the first order with two boundary conditions) is offered, when parameters depend from time, which allows to define the lifetime due to wave-particle interactions for the pitch angle of $90$ degrees numerically.