Derivations of the Total Radiation Belt Electron Content

We present multiple derivations of the Total Radiation Belt Electron Content (TRBEC), an indicator of the global number of electrons that instantaneously occupy the radiation belts. Derived from electron flux measurements, the TRBEC reduces the spatial information into a scalar quantity that concisely describes global aspects of the system. This index provides a simple, global, and long-term assessment of the radiation belts that enables systematic analysis. In this work, we examine the TRBEC using the adiabatic invariants of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mfenced><mml:mrow><mml:mi>μ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:msup><mml:mi>L</mml:mi> <mml:mo>∗</mml:mo></mml:msup> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mrow> </mml:math> which has been used in previous articles as this coordinate system removes reversible adiabatic effects. We then introduce a new expression to compute the TRBEC using the non-adiabatic coordinates of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mfenced><mml:mrow><mml:mi>E</mml:mi> <mml:mo>,</mml:mo> <mml:msub><mml:mi>α</mml:mi> <mml:mrow><mml:mi>e</mml:mi> <mml:mi>q</mml:mi></mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msup><mml:mi>L</mml:mi> <mml:mo>∗</mml:mo></mml:msup> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mrow> </mml:math> , relevant in the contexts of energetic electron precipitation, chorus, and hiss scattering where adiabatic invariant quantities are no longer conserved. From both expressions of the TRBEC we demonstrate that an erroneous factor of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:msup><mml:mrow><mml:mo>(</mml:mo> <mml:mrow><mml:mn>2</mml:mn> <mml:mi>π</mml:mi></mml:mrow> <mml:mo>)</mml:mo></mml:mrow> <mml:mn>3</mml:mn></mml:msup> </mml:mrow> </mml:mrow> </mml:math> that appeared in previous works using the adiabatic derivation led to an overestimate of the reported electron populations. In addition, we quantify electron loss in the outer radiation belt <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mfenced><mml:mrow><mml:mn>3.5</mml:mn> <mml:mo><</mml:mo> <mml:msup><mml:mi>L</mml:mi> <mml:mo>∗</mml:mo></mml:msup> <mml:mo><</mml:mo> <mml:mn>5</mml:mn></mml:mrow> </mml:mfenced> </mml:mrow> </mml:mrow> </mml:math> via a case study using the Van Allen Probes data over a 20-day period from March 2013 specifying particle populations both in terms of the aforementioned adiabatic and non-adiabatic variables. The total number of electrons in the outer radiation belt reached upwards of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:mrow><mml:mn>1</mml:mn> <mml:msup><mml:mn>0</mml:mn> <mml:mn>28</mml:mn></mml:msup> </mml:mrow> </mml:mrow> </mml:math> electrons at the peak of the storm, a rest mass of roughly 10 g.