Derivation of temperature anisotropy from shock jump relations: Theory and observations

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Abstract
The MHD Rankine‐Hugoniot (RH) relations for shock waves in a collisionless plasma with bi‐Maxwellian distribution functions are considered. By introducing the pressure anisotropy in the RH relations, the number of unknowns (a total of 9) becomes one more than the total number of conservation equations. However, it is possible to use the observed quantities on both sides of a shock to study the anisotropy changes across the shock. The relationship of the anisotropy change across the shock is derived as a function of the ratio of magnetic fields m(= B′/B), the shock normal angle θBn and the plasma betas β and β′ (primes are downstream values). Since for low‐β, quasi‐perpendicular (Q‐⟂) laminar shocks, m and θBn can be determined fairly accurate from observations, the reliability of the anisotropy change deduced is mostly dependent on the accuracy of the measurements β and β′. The flow velocity and shock speed which consist of large uncertainties do not enter into our derivations. Other physical parameters such as the various Mach numbers, and the jumps of all other quantities across the shock can be calculated. We have applied the results to six low‐β Q‐⟂, laminar bow shock crossings with temperature anisotropy measured in the magnetosheath. It is found from these test cases that the predicted pressure anisotropy agrees well with the observed one in the magnetosheath.