Measurement principle

Langmuir probes have been widely used as diagnostic instruments for both laboratory and space plasma. Figure on the right shows the representative current-voltage (I-V) characteristic curve of a Langmuir probe, where three operational regions of electron saturation, electron retardation and ion saturation are separated by the plasma potential, \(V_p\), and the floating potential, \(V_f\). Mott-Smith and Langmuir 1926 presented the orbital-motion-limited (OML) theory, which deals with collisionless electron and ion trajectories surrounding a spherical or a cylindrical probe. This approach provides a quantitative understanding of the cross-sections for electron and ion collection. Traditional Langmuir probe designs sweep through a range of voltages to obtain the I-V characteristic curve, from which the plasma parameters including ion density, electron density and electron temperature can be derived. However, sweeping takes time and makes this approach unsuited for high spatial resolution measurements. The m-NLP design uses four cylindrical probes biased at different fixed voltages within the electron saturation region such that the currents to these four probes can be sampled at a much higher rate, resulting in high-resolution plasma density observations. In general, an expression for electron saturation current of a Langmuir probe (planar, cylindrical or spherical) is given by the OML theory as following [Bekkeng et al. 2010, Jacobsen et al. 2010]: \begin{equation} I_{e}=CI_{e0}\left ( 1 + \frac{eV}{k_{B}T_{e}} \right )^{\beta}\label{eq:1} \end{equation} where \(C = 2/\sqrt{\pi}\) for a cylindrical geometry and \(C = 1\) for planar and spherical geometries. \(I_{e0} = N_{e}Ae\sqrt{\frac{k_{B}T_{e}}{2\pi m_{e}}}\), is the current to the probe at the plasma potential, \(N_e\) is the electron density, \(A\) is the probe surface area and \(m_e\) is the electron mass. \(V\) is the probe bias potential with respect to the plasma potential, i.e. \(V = V_b - V_p\). It is noted that the equation used above is valid under the assumption of a non-drifting, collisionless and non-magnetized plasma. At the altitude of the NorSat-1 orbit all three assumptions are typically fulfilled since

1. the thermal speed of the electrons is much larger than the speed of the spacecraft relative to the plasma;
2. the mean free path of the electrons is far greater than both the probe radius and the scale length of the electric potential distribution around the probe;
3. the Larmor radius is much larger than the probe radius. The current collection expression is a function of the applied bias potential relative to the plasma potential.

The probe bias is relative to a common electrical ground, which is connected to the conductive parts of satellite surface so the signal ground is as same as the floating potential of the payload platform. The measurement method of the m-NLP system handles at least two probes biased at two different voltages over the plasma potential to determine the absolute electron density as: \begin{equation} N_{e}=\frac{1}{KA}\sqrt{\frac{\Delta\left(I_{c}\right)^{2}}{\Delta V_{b}}} \end{equation} where \(K\) is a constant given by \(\frac{e^{3/2}}{\pi}\sqrt{\frac{2}{m_{e}}}\), \(\Delta\left(I_{c}\right)^{2}\) is the difference in the square of collected currents and \(\Delta V_{b}\) is the difference in the probe biases. A key feature of the m-NLP technique is an ability to determine the electron density without the need to know the plasma potential or electron temperature. Under certain conditions, the m-NLP is capable of monitoring the spacecraft potential and its variations as described in Bekkeng et al. 2013.

A study of data analysis techniques for the m-NLP instrument can be found in Hoang et al. 2018.