Rankine**–**Hugoniot relations (RH) connect the upstream and downstream plasma states. They allow us to determine the magnetic compression, the density compression, and the plasma heating as functions of the Mach number, shock angle, and upstream temperature. RH are based on the conservation laws in the hydrodynamical form. In collisionless shocks, the ion distributions behind the shock transition are determined by ion dynamics in the macroscopic fields of the shock front. The ion parameters upon crossing the shock are directly related to the magnetic compression and the cross-shock potential. For given upstream parameters, RH provide the magnetic compression. If there is no substantial overshoot, an analytical estimate provides the cross-shock potential as a function of the magnetic compression and the Mach number. Numerical tracing of ions across a shock profile with the derived parameters provides the ion pressure, which is in good agreement with the combination of the two theoretical approaches.

## I. INTRODUCTION

At a magnetized collisionless shock, the plasma and magnetic field parameters experience sharp changes: the density, temperature, and magnetic field increase, while the flow velocity along the shock normal drops. Within the magnetohydrodynamic (MHD) description, the shock structure is not resolved, and the upstream (before the shock transition) and the downstream (behind the transition) parameters are related via Rankine**–**Hugoniot (RH) relations, which are nothing but the conservation laws and jump/continuity conditions for the electromagnetic field.^{1} The conserved density, momentum, and energy are formulated in terms of the hydrodynamic variables in the region where the distributions are gyrotropic but possibly anisotropic.^{2–13} Initially, isotropic incident ion distributions become non-gyrotropic after crossing the shock front,^{14–17} because of the strong ion gyration. The non-gyrotropic distributions gradually gyrotropize due to the kinematic gyrophase mixing.^{18,19} This process occurs in low- and high-Mach number shocks as well.^{19–21} The shock transition, measured as the distance from the point where the magnetic field starts to increase from the upstream value to the maximum magnetic field magnitude in the overshoot, is of the order of the ion inertial length,^{22–25} so that ions become demagnetized inside the transition layer. They are mostly affected by the cross-shock electric field, and the relation of the ion pressure at the downstream edge of the transition can be derived analytically, which, together with the pressure balance, gives the relation between the Mach number, the overshoot strength, and the cross-shock potential.^{26–28} In the shocks with only a weak overshoot, the relation between the magnetic compression and the Mach number should be the same as obtained from RH. The magnetic compression can be related to the Mach number using Rankine**–**Hugoniot relations^{8} (Sec. II). The same magnetic compression and Mach number can be related using the analysis of the ion kinetics in the thin ramp^{26–28} (Sec. III). The latter includes also the cross-shock potential as an additional parameter. Therefore, comparing two independent relations would provide an estimate of the cross-shock potential. In the present paper, we check the estimate by applying the adjustable test particle tracing^{29} and comparing the numerically derived pressure with the pressure required by RH.

## II. HYDRODYNAMIC RH

Let the shock normal be in the *x*-direction, while *x*–*z* be the coplanarity plane. Let *ρ* be the density, $V$ be the hydrodynamic velocity, *p* be the pressure (which is taken isotropic for simplicity), *ε* be the internal energy density, ** B** be the magnetic field, and

**be the electric field. Denoting upstream and downstream by the subscripts**

*E**u*and

*d*, RH are obtained from the coplanarity theorem and the following conservation laws (see, e.g., Ref. 8):

These equations should be completed with an equation of state, that is, a relation between *ε* and *p*. For an isotropic monatomic gas $\epsilon =(3/2)p$. Usually, RH are used to derive the downstream parameters for given shock angle *θ*, Alfvénic Mach number *M*, and upstream *β*, defined as follows:

where $Vu=Vux$. Alternatively, it can be used for determination of *M* using given *θ*, *β*, and $Bd/Bu$. It is typically done by numerically solving the above equations after suitable reduction. In the shock vicinity, ion distributions are non-gyrotropic,^{15,16,30} and the above expressions for the momentum and energy fluxes are not valid. Accordingly, the hydrodynamic RH are valid only sufficiently far from the shock, where the distributions are isotropic or at least gyrotropic**–**anisotropic, in which case modifications should be invoked.^{2–4,7,9,10,12} If a shock is approximately planar and stationary, then the total pressure should be constant throughout the shock,

Here, $\rho =nmp$, and *V _{x}* and

*p*are expressed in terms of the exact distribution function $f(v,x)$,

_{xx}and *m _{p}* is the proton mass. Ion deceleration in the shock results in the drop of the dynamic pressure $\rho Vx2$. The reduction of this pressure should be compensated by the increase in the kinetic pressure

*p*, that is, ion heating, and the increase in the magnetic pressure. The deceleration occurs due to the cross-shock potential, while substantial heating is due to the gyration of the transmitted ions.

_{xx}^{27,31,32}At higher Mach numbers, reflected ions contribute substantially.

^{14,33,34}Larger Mach numbers mean large magnetic compression and stronger drop of the dynamic pressure, while heating increase lags behind, which results in magnetic overshoots.

^{35}In low-Mach number shocks, overshoots are small or negligible, and the magnetic field nearly levels off at the downstream value just upon the ramp crossing. This means that Eqs. (3) and (11) can be expected to be consistent.

## III. KINETIC RH

The velocity and pressure in the pressure balance (11) can be estimated from the analysis of ion dynamics in the shock ramp.^{26} The ramp width is of the order of the ion inertial length or smaller.^{22,24,36–39} Ions become demagnetized inside the ramp, that is, an ion completes only a small part of the whole gyration and the main effect is the deceleration by the cross-shock potential, which determines the velocity of the ion along the shock normal upon crossing the ramp. Here, we use a simplified version of the pressure balance at the ramp for quasi-perpendicular low-*β* shocks,^{27}

where $sNIF=2e\phi /mpVu2$ is the normalized cross-shock potential in the normal incidence frame (NIF). In NIF, the upstream flow velocity is along the shock normal. Neglecting the difference between the magnetic compression at the ramp and the magnetic compression farther downstream, Eq. (12) allows us to estimate the cross-shock potential. As an example, we choose the following parameters: $\theta =70\xb0$, *M* = 3.3, $\beta i=\beta e=0.1$, which gives the magnetic compression of $Bd/Bu=2.8$, for the isotropic hydrodynamic RH. The corresponding cross-shock potential is estimated as *s _{NIF}* = 0.3. Once the external parameters, $M,\theta ,\beta $, are chosen and the internal parameter,

*s*, is estimated, consistency can be verified using the adjustable test particle analysis.

_{NIF}^{40,41}

## IV. ION TRACING

The adjustable test particle analysis uses ion tracing in a model shock front, numerical calculation of the moments of the ion distribution function, substitution into Eq. (8), comparison with the initially chosen profile, variation of the parameters, if needed, and performing the analysis anew, until reasonable agreement is achieved. The coordinates are chosen so that *x* is along the shock normal, pointing toward downstream, and *x*–*z* is the coplanarity plane. The *z*-component of the magnetic field profile is given by the expression,^{40}

The parameter *D* gives the shock width, and overshoot is not included. The analysis is done in normalized variables. The normalization length is the upstream ion convective gyroradius, $Vu/\Omega u$, where $\Omega u=eBu/mpc$. Velocities are normalized with *V _{u}*, time is normalized with $\Omega u\u22121$, and pressure is normalized with $\rho uVu2$. The cross-shock electric field and the non-coplanar magnetic field shapes are related to the two most convenient shock frames, the de Hoffman

**–**Teller frame (HT), in which the upstream and downstream plasma velocities are along the upstream and downstream magnetic field vectors, respectively, and the normal incidence frame (NIF), in which the upstream plasma velocity is along the shock normal. In HT, the motional electric field $Ey(HT)=0$ vanishes identically, while in NIF it is constant $Ey(NIF)=VuBu\u2009sin\u2009\theta /c$ throughout the shock, while

*E*= 0 in both frames. The shape of the cross-shock electric field $Ex(HT)$ in the applied model is given by the expression,

_{z}where the coefficient *K _{E}* is determined from the cross-shock potential,

and is one of the model parameters. In the dimensionless form

The noncoplanar magnetic field is chosen in a similar form,

while

For $Vu/\u2009cos\u2009\theta \u226ac$, the transformation between the two frames is non-relativistic, which means that the magnetic field can be considered the same. The parameter

is also the model parameter, which determines the coefficient *K _{B}*. The parameters for the ion tracing are given in Sec. III:

*M*= 3.3, $Bd/Bu=2.8,\u2009\theta =70\xb0,\u2009\beta i=0.1$. We choose

*s*= 0.1, as a typical value for heliospheric shocks,

_{HT}^{42}while

*s*is varied starting at the value

_{NIF}*s*= 0.3. The width is $D=Vu/M\Omega u$. The incident ion distribution is Maxwellian,

_{NIF}with $vT/Vu=\beta i/2/M$. We also assume $\beta e=\beta i$.

Figure 1 shows the results of the best adjustment achieved with *s _{NIF}* = 0.35. The black curve shows the normalized model magnetic field $|B|/Bu$. The blue curve shows the normalized $|B|/Bu$ derived from (8), where

*V*and

_{x}*p*of ions are determined numerically, while for electrons we use the adiabatic law $p/n5/3=const$. The derived magnetic field has a long train of oscillations but converges to the chosen value of $Bd/Bu$ well beyond the ramp. The amplitude of the magnetic oscillations is small, $max(|B|/Bd)\u22121\u22480.16$. The same figure shows the reduced distribution function,

_{xx}which illustrates the ion gyration and gradual gyrophase mixing. The reduced distribution function $f(vx,x)$ is derived tracing 80 000 initially Maxwellian distributed ions across the shock and catching ions on a two-dimensional grid in the phase space $(x,vx)$, using the staying time method.^{43} The overall agreement of the hydrodynamic RH, the analytical approximation, and the test particle analysis is very good. Figure 2 shows the three eigenvalues of the temperature tensor,

and the total temperature defined as $Ttot=13\u2211iTii$. The shown values are normalized on $mpVu2$. Strong non-gyrotropy behind the ramp is gradually smoothed out due to the gyrophase mixing. The latter is slow since the shock angle is large and *β _{i}* is small.

^{29}Real shocks are not exactly planar and stationary, either because of rippling or intrinsic time dependence, or due to the some level of waves/turbulence. This is not included in the test particle analysis here. Weak deviations from planarity and stationarity would not affect substantially the overall energy redistribution but could make the oscillations of the temperature and of the magnetic field smaller. For the chosen shock parameters, ion reflection is negligible, because of the low magnetic compression, low cross-shock potential, and small ratio $vT/Vu$. An example of a shock where the above MHD and kinetic RH approach can be expected to be a reasonable approximation is shown in Fig. 3. The figure shows the magnetic field magnitude of the Earth bow shock for the crossing, which was observed on 2020‐11-21 by the magnetospheric multiscale (MMS), probe 1. The magnetic field is normalized on the upstream value. The Mach number of the shock, as given at https://zenodo.org/record/6343989#.YugILy8Rokw, is $M\u22483.2$, close to that of Fig. 1. The shock angle $\theta \u224864\xb0$ is slightly smaller, as well as the corresponding magnetic compression. The magnetic compression and the overshoot are quite similar to those of the derived magnetic field (Fig. 1, blue curve).

One may expect that the agreement of the hydrodynamic RH, the analytical approximation, and the test particle analysis would become worse for higher Mach numbers, since the overshoot and undershoots amplitudes increase with the increase in the Mach number.^{35,45–47} An example of a higher Mach number shock is shown in Fig. 4 This Earth bow shock crossing was observed by MMS1 on the same day. For theoretical analysis, we will use a more moderate set of parameters: *M* = 4.3, $Bd/Bu=3,\u2009\theta =60\xb0,\u2009\beta i=\beta e=0.1$. From the kinetic relation one has *s _{NIF}* = 0.29. For the numerical ion tracing, we again increase a little the potential to

*s*= 0.35. Figure 5 shows the results of the test particle analysis. The derived magnetic field does not converge to the model magnetic field. Further variations of

_{NIF}*s*do not improve convergence. The too high derived magnetic field means that the magnetic pressure is too high, that is, ion heating is insufficient. The reduced distribution does not show any reflected ions although significant ion reflection is expected for this Mach number. Since reflected ions have high gyration energies, their presence would substantially increase ion heating. This emphasizes necessity of ion reflection in high-Mach number shocks.

_{NIF}In this case, we can expect that the overshoot would affect ion motion. We can try to improve the analysis by adding an overshoot and undershoot approximately at the positions where these are seen in Fig. 5 in the derived profile (blue). Figure 6 shows the results of the best adjustment at *s _{NIF}* = 0.44. The obvious improvement is that the overshoot and undershoot in the derived field (blue) occur close to the positions of the modeled overshoot and undershoot (black). Now, ion reflection is present, which agrees with what is expected at this Mach number. Ion reflection occurs in the overshoot region and is clearly a combined effect of the electrostatic deceleration and magnetic deflection,

^{44}since the cross-shock potential alone is too low to stop the ions. The contribution of the reflected ions makes the difference and provides better convergence to the downstream magnetic field. It is also responsible for smoother oscillations of the temperature eigenvalues, as seen in Fig. 7, as well as for faster gyrotropization (convergence of the maximum and medium eigenvalues) and lower anisotropy (larger minimum eigenvalue). The potential at the overshoot maximum is $sm\u2248sNIF(max\u2009|B|/Bd)\u22480.61$. The theoretically proposed relation,

^{27}

gives in this case $max\u2009|B|\u22483.9$, which is not far from the value $max\u2009|B|\u22484.1$ of the model magnetic field.

Note that the anisotropy is large even far from the shock transition, while the MHD RH were used for isotropic case. Using the anisotropic modifications of RH,^{12} it is possible to show that for this $Tmin/Tmax$ the downstream magnetic compression of $Bd/Bu=3$ cannot be achieved. In real shocks, downstream fluctuations of the magnetic or instabilities reduce anisotropy. The above analysis used direct calculation of *p _{xx}*. We shall move one step forward and replace this

*p*in the pressure balance equation with $p=13\u2211ipii$, thus forcibly including isotropization. Note that the present case gives the far downstream value of $Ttot=13(Tmin+Tmid+Tmax)=0.11$, while RH require

_{xx}*T*= 0.15 (which is equal to

_{tot}*T*in this case). The best convergence for forced isotropy is achieved for

_{max}*s*= 0.46. It is shown in Fig. 8. The time-dependent and stationary model (13), even with the added overshoot and undershoot, cannot provide isotropization. Figure 4 shows a series of large amplitude magnetic oscillations after the undershoot. The magnetic field in these oscillations goes out of the coplanarity plane, which may speed up isotropization of the downstream ion distribution. Possible time dependence and non-planarity would also enhance the isotropization. We leave this issue for further studies. The cross-shock potential at the overshoot is $sm\u22480.64$, and the theoretically predicted maximum magnetic field in the overshoot is $max\u2009|B|\u22484$.

_{NIF}## V. DISCUSSION AND CONCLUSIONS

We combined the MHD derived Rankine**–**Hugoniot relations with the kinetic relation at the ramp, proposed theoretically.^{27} This allows us to estimate the cross-shock potential at the ramp by comparing the RH required temperature with the temperature ions have due to the gyration onset upon crossing the ramp. The estimate was originally proposed for shocks with negligible ion reflection, where the contribution to the temperature comes from directly transmitted ions only. It was, thus, expected that the potential estimate is applicable for subcritical and moderately supercritical shocks. In such shocks, the overshoot is also small. The estimated cross-shock potential was further used to complete a stationary planar shock model used for test particle analysis. Tracing ions in the model shock front allows us to derive the ion pressure. Requiring the validity of the pressure balance throughout the shock, we derived the magnetic field, which would be consistent with the parameters chosen for the numerical analysis. For given shock angle *θ*, Mach number *M*, and upstream *β*, the magnetic compression $Bd/Bu$ used in the model is prescribed by the MHD RH. By varying the cross-shock potential, convergence of the derived magnetic field to the model magnetic field was achieved well beyond the ramp. The best fit cross-shock potential is only slightly higher than the estimated potential, which shows the consistency of the MHD RH and ion dynamics in the macroscopic fields of the shock front.

A higher Mach number shock is expected to have a larger overshoot, and ion reflection should also play a more important role. Note that the overshoot is not due to reflected ions but because of the stronger drop of the dynamic pressure $nmpVx2$ of the bulk flow across the shock. The same procedure as above shows that in this case, the estimate of the cross-shock potential from MHD RH is not correct. Indeed, what happens at the ramp depends on the fields there, while MHD RH refer to the quiet plasma states sufficiently far from the shock transition. An attempt to achieve convergence of the derived magnetic field to the model field fails, since no ion reflection occurs and ion heating is not sufficient. Inclusion of an overshoot and undershoot in the model resulted in moderate ion reflection, which does not spoil the shock but provides the right addition of ion heating. The numerically adjusted cross-shock potential at the overshoot appears to be in good agreement with the theoretical prediction,^{27} so in this case also the MHD RH and ion dynamics in the shock front agree well. Estimate of the cross-shock potential, however, is not so straightforward as for the no-overshoot case and requires some more efforts. The role of the cross-shock potential is twofold: a higher cross-shock potential decreases heating of the directly transmitted ions but increases the number of reflected ions and, thus, increases their contribution to the temperature. Ion reflection occurs in the tail of the incident distribution and is, therefore, rather sensitive to the potential, so that fine tuning is necessary. The role of the overshoot is to increase the effective magnetic compression at the shock crossing and to enhance the magnetic deflection of the ions crossing the shock. Both effects act toward enhancement of ion reflection, together with the cross-shock potential. We found that ions are reflected non-specularly in the region of the overshoot maximum. The position of the overshoot and its width are determined by the pressure oscillations of the gyrating ions. In the numerical analysis these parameters are important: a too wide overshoot would cause unphysical strong reflection, a too thin overshoot is not efficient in reflecting ions. In the test particle analysis, there was some sensitivity of ion reflection to the ramp width too, apparently due to separate parameters for the ramp and overshoot widths.

Present test particle analysis does not include mechanisms of gyrotropization (work is in progress), and the obtained anisotropies of temperature are quite high, $Tmin/Tmax=0.2$, since the dominant heating is in the direction perpendicular to the magnetic field. Figure 9 shows the magnetic compression $Bd/Bu$ predicted by anisotropic MHD RH for *M* = 7, $\theta =60\xb0$, and $\beta p+e=0.2$, as a function of the anisotropy ratio $T\u2225/T\u22a5$. Here, $\u2225$ and $\u22a5$ refer to the downstream magnetic field direction. It is seen that for $T\u2225/T\u22a5=0.2$ and *M* = 4.3 the compression ratio of $Bd/Bu=3$ cannot be achieved. Forcing isotropy by replacing *p _{xx}* with the 1/3 of the trace of the pressure tensor, we were able to further correct the adjusted cross-shock potential so that the numerically determined distributions reasonably agree with the model.

To summarize, we were able, for the first time, to bridge over the two theoretical approach: MHD RH connecting the asymptotic uniform regions and ion distributions formed due to the ion dynamics in the macroscopic fields of thin shock transition. For shocks with weak ion reflection and weak overshoots, the proposed analytical expressions may be directly compared to RH. For shocks, where ion reflection is substantial and overshoots affect ion motion significantly, an intermediate step of ion tracing in a model shock front is required. The results are in good agreement with the theoretical predictions.

## ACKNOWLEDGMENTS

The work was partially supported by the European Union's Horizon 2020 research and innovation program under Grant Agreement No. 101004131 (SHARP). The MMS data files reside at https://lasp.colorado.edu/mms/sdc/public/.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Michael Gedalin:** Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.