Electron Kinetic Entropy across Quasi-Perpendicular Shocks

We use Magnetospheric Multiscale (MMS) data to study electron kinetic entropy per particle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi mathvariant="normal">e</mi></msub></semantics></math></inline-formula> across Earth’s quasi-perpendicular bow shock. We have selected 22 shock crossings covering a wide range of shock conditions. Measured distribution functions are calibrated and corrected for spacecraft potential, secondary electron contamination, lack of measurements at the lowest energies and electron density measurements based on plasma frequency measurements. All crossings display an increase in electron kinetic entropy across the shock <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><msub><mi>S</mi><mi mathvariant="normal">e</mi></msub></mrow></semantics></math></inline-formula> being positive or zero within their error margin. There is a strong dependence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><msub><mi>S</mi><mi mathvariant="normal">e</mi></msub></mrow></semantics></math></inline-formula> on the change in electron temperature, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><msub><mi>T</mi><mi mathvariant="normal">e</mi></msub></mrow></semantics></math></inline-formula>, and the upstream electron plasma beta, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>β</mi><mi mathvariant="normal">e</mi></msub></semantics></math></inline-formula>. Shocks with large <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><msub><mi>T</mi><mi mathvariant="normal">e</mi></msub></mrow></semantics></math></inline-formula> have large <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><msub><mi>S</mi><mi mathvariant="normal">e</mi></msub></mrow></semantics></math></inline-formula>. Shocks with smaller <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>β</mi><mi mathvariant="normal">e</mi></msub></semantics></math></inline-formula> are associated with larger <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><msub><mi>S</mi><mi mathvariant="normal">e</mi></msub></mrow></semantics></math></inline-formula>. We use the values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><msub><mi>S</mi><mi mathvariant="normal">e</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><msub><mi>T</mi><mi mathvariant="normal">e</mi></msub></mrow></semantics></math></inline-formula> and density change <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><msub><mi>n</mi><mi mathvariant="normal">e</mi></msub></mrow></semantics></math></inline-formula> to determine the effective adiabatic index of electrons for each shock crossing. The average effective adiabatic index is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⟨</mo><msub><mi>γ</mi><mi mathvariant="normal">e</mi></msub><mo>⟩</mo><mo>=</mo><mn>1.64</mn><mo>±</mo><mn>0.07</mn></mrow></semantics></math></inline-formula>.