Wave Vector of H⁺ Band EMIC Waves and Corresponding Electron Resonance Energy Determinations by 4 MMS Satellites

Electromagnetic ion cyclotron (EMIC) waves are electromagnetic emissions with frequencies below the ion cyclotron frequencies. These waves are an important agent for interacting with ring current and relativistic electrons. The wave vector of the EMIC waves is crucial parameter to determine the electron minimum resonance energy. However, such parameter is hard to determine from a singular satellite measurement by using the 3D electric and magnetic fields due to the large error of the axial electric field measurement. To estimate the EMIC wave vector from theory, an ion composition is assumed, and the cold plasma dispersion relation is often assumed, which may be not realistic. By using four satellites, we can directly estimate the three components of the wave vector from observation. The principle of this technique is as follows. The wave vector k, position difference between any two satellites $\Delta \mathbf{r}$ , and phase difference of the EMIC wave spectrum between the two satellites $\Delta \varphi$ at a given time and a given frequency satisfy the condition $k\cdot\Delta r=\Delta\varphi$ . With four satellites and first two largest components of magnetic fields, there are 12 independent measurements of $\Delta \mathbf{r}$ and $\Delta\varphi$ to determine the wave vector k by solving the overdetermined equation. We use MMS satellite data from 2015 to 2022 and find 284 H ⁺ band EMIC wave events. Then, we determine their wave vectors and then evaluate the electron minimum resonance energy. We find that the wavenumber of the EMIC waves ranges from 10 ⁻⁶ to 10 ⁻⁴ m ⁻¹ and the electron minimum cyclotron resonant energy ranges from 10 keV to 10 MeV. The wave normal angle, wavenumber, and electron resonant energy of the 284 events are analyzed and statistically studied. Their power weighted values are also investigated.