# Density jump as a function of magnetic field strength for parallel
collisionless shocks with anisotropic upstream pressure

Antoine Bret1,2
1ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real,
Spain
2Instituto de Investigaciones Energéticas y Aplicaciones Industriales, Campus
Universitario de Ciudad Real, 13071 Ciudad Real, Spain E-mail:
<EMAIL_ADDRESS>
(Last updated -; in original form -)

###### Abstract

The properties of collisionless shocks are frequently assessed in the
magnetohydrodynamics (MHD) model. Yet, in a collisionless plasma, an ambient
magnetic field can sustain a stable anisotropy in the upstream or the
downstream, resulting in a departure from the MHD predicted behavior. We
present a model allowing to derive the downstream anisotropy, hence the shock
density jump, in terms of the upstream quantities. For simplicity, the case of
a parallel shock in pair plasma is considered. Contrary to previous works
where the upstream was assumed isotropic, here the upstream anisotropy
$A=T_{\perp}/T_{\parallel}$ is a free parameter. The strong sonic shock regime
is formally identical to the isotropic upstream case. Yet, for intermediate
sonic Mach numbers, a variety of behaviors appear as a result of the
anisotropy of the upstream.

###### keywords:

MHD -– plasmas -– shock waves.

††pubyear: 2023††pagerange: Density jump as a function of magnetic field
strength for parallel collisionless shocks with anisotropic upstream
pressure–Density jump as a function of magnetic field strength for parallel
collisionless shocks with anisotropic upstream pressure

##  1 Introduction

Shock waves are fundamental processes which display a sudden jump of density,
temperature, etc. in the medium where they propagate. The amplitude of these
jumps can be determined assuming that matter, momentum and energy are
conserved before and after the shockwave. When binary collisions are frequent,
these conservation laws take a simple form as the pressure upstream and
downstream can be considered isotropic. In the absence of such collisions, the
possible existence of shock waves was debated up to the early 90s (Sagdeev &
Kennel, 1991).

The reason for the debate was that not only collisions isotropize flow, but
they are also the microscopic agent which mediates the very existence of the
shockwave. In this respect it was found that shock waves could exist in the
absence of collisions only in a plasma, where collective electromagnetic
effects can substitute collisions as an agent mediating the shock (Sagdeev,
1966). Such shock waves have been called “collisionless shocks”. Their very
existence is now firmly established by _in situ_ measurements of the earth bow
shock (Bale et al., 2003; Schwartz et al., 2011).

Collisionless shocks are ubiquitous in astrophysical environments. Their
ability to accelerate particles (Blandford & Ostriker, 1978) makes them good
candidates to explain the origin of high energy cosmic rays, fast radio bursts
(Zhang, 2020; Sironi et al., 2021) or gamma gay bursts (Piran, 2004). Because
the spectral index of shock accelerated particles depends on the shock density
ratio (Blandford & Ostriker, 1978), this quantity is of prime relevance for
the physics of the aforementioned phenomena. As mentioned previously, the
earth bow shock is another example of collisionless shock. More generally,
shock detections in the solar system are routine (David et al., 2022), and
some instances of anisotropic upstream have been reported (Fraschetti &
Giacalone, 2020).

From the theoretical point of view, the absence of collisions requires dealing
with collisionless shocks at the kinetic level, making use of the Vlasov
equation. Since this equation is notoriously difficult to address, the
properties of collisionless shocks are frequently deduced from
magnetohydrodynamics (MHD), even though MHD, as the plasma counterpart of
fluid dynamics, assumes collisions, hence, isotropy of pressures (Goedbloed et
al. (2010) ch. 2 and 3, Thorne & Blandford (2017) §13.2).

Therefore, one needs to ask the question: to which extent can pressure be
considered isotropic in a collisionless plasma? In the absence of external
magnetic field, the Weibel instability ensures isotropy, at least on time
scales longer than the inverse growth rate, since it states that an
anisotropic collisionless plasma is unstable (Weibel, 1959; Silva et al.,
2021). Yet, an external magnetic field can stabilize an anisotropy in a
collisionless plasma. This has been beautifully proved by _in situ_
measurements in the solar wind (Bale et al., 2009; Maruca et al., 2011;
Schlickeiser et al., 2011). The instabilities limiting the anisotropy are the
firehose and the mirror instability (Gary, 1993). While in the limit of zero
external field they impose isotropy, they allow for a window of stable
anisotropies in the presence of a magnetic field.

Note that Fig. 1-top of Bale et al. (2009) shows than even though the firehose
and mirror thresholds constraint the solar wind measurements, most of them
have indeed $T_{\perp}\sim T_{\parallel}$. As stated in the caption of the
figure, this results “largely from Coulomb collisions”, which are eventually
not completely absent. In fact, the bottom plot of the same figure shows that
the “collisional age” of such measurements is higher. Within the framework of
the present purely collisionless study, Coulomb collisions are absent so that
the two collisionless instability thresholds are the only factors constraining
the anisotropy.

As a consequence, MHD may not be well adapted to the study of collisionless
shock waves in a magnetized environment. Various authors worked out the MHD
conservation equations for anisotropic pressures (Hudson, 1970; Erkaev et al.,
2000; Double et al., 2004; Gerbig & Schlickeiser, 2011). Yet, the degree of
anisotropy in the upstream and the downstream is a free parameter in these
equations. Granted, the upstream anisotropy may well be considered an input
parameter on which depends the downstream quantities. After all, shock waves
analysis usually results in an expression of the downstream quantities in
terms of the upstream ones. But a theory leaving the downstream anisotropy as
a free parameter is incomplete.

Recently, a model was developed allowing to compute the downstream anisotropy
in terms of the (isotropic) upstream parameters (Bret & Narayan, 2018, 2019,
2022). For simplicity, this model considered pair plasmas, where the parallel
and perpendicular temperatures are the same for both species. In contrast, in
an electron/ion plasma, species can be heated differently at the front
crossing (Guo et al., 2017, 2018), which would require dealing with four
different temperatures (two for the electrons and two for the ions). Note that
preliminary results seem to indicate that the model can be applied to
electrons/ions plasma, provided the electron mass is replaced by the ion mass
in some key parameters (Shalaby, 2022).

Since collisionless shocks in space may well have an anisotropic upstream
(Vogl et al., 2001; Bale et al., 2009), the goal of the present article is to
extend the model to the case of an anisotropic upstream.

Figure 1: System considered. In this parallel configuration, the magnetic
field does not change from the upstream to the downstream.

The system considered is pictured on figure 1. The upstream and downstream
quantities are labelled with subscripts “1” and “2” respectively. As in Bret &
Narayan (2018), the shock is parallel, with an external field parallel to the
direction of motion of the shock. Since in a parallel shock, the upstream and
downstream fields are the same (Kulsrud, 2005), the magnetic field is labelled
$\mn@boldsymbol{B}_{0}$. There lies the interest of this setup to study
collisionless effects effects: because in MHD the fluid and the field are
decoupled for a parallel shock, any change to the shock when the field
changes, is a kinetic effect.

For a purely collisionless plasma, the Vlasov equation imposes gyrotropy,
namely, equality of the temperatures perpendicular to the field (see Landau &
Lifshitz (1981), §53). This is why we only consider two temperature here
instead of three. Admittedly, gyrotropy could be broken, for example, by
electric fields developing during the growth of the instabilities involved in
this work. Yet, we consider in the sequel the state of the downstream far from
the front, and long after the instabilities developed. There and then, the
Vlasov equation should enforce gyrotropy.

This article is structured as follows: the method implemented to solve the
problem is explained in section 2. The conservation equations for anisotropic
pressures, and the dimensionless variables used here, are presented in section
3. In section 4 we express the restrictions set on the upstream parameters by
the requirement that it be stable. Then from sections 5 to 7 we solve our
model for an anisotropic upstream, before we reach our conclusions in section
8.

##  2 Method

The method implemented here is the same than that of Bret & Narayan (2018).
For completeness of the present article, it is now briefly recalled.

If the journey of the plasma through the shock were completely adiabatic, the
double adiabatic relations of Chew et al. (1956) would apply,

$\displaystyle T_{2\parallel}$ $\displaystyle=$ $\displaystyle
T_{1\parallel}\left(\frac{n_{2}B_{1}}{n_{1}B_{2}}\right)^{2},$ (1)
$\displaystyle T_{2\perp}$ $\displaystyle=$ $\displaystyle
T_{1\perp}\frac{B_{1}}{B_{2}}.$

For a parallel shock where $B_{1}=B_{2}$, they would result in

$\displaystyle T_{2\parallel}$ $\displaystyle=$ $\displaystyle
T_{1\parallel}\left(\frac{n_{2}}{n_{1}}\right)^{2},$ (2) $\displaystyle
T_{2\perp}$ $\displaystyle=$ $\displaystyle T_{1\perp}.$

Yet the shock crossing is not adiabatic since in a shock, there is an entropy
increase from the upstream to the downstream. It was then hypothesized in Bret
& Narayan (2018) that the relations above hold for the perpendicular
temperature, but not for the parallel one. The reason for this is that the
plasma compression when crossing the front can be pictured as if it were
compressed between 2 converging virtual walls parallel to the shock front,
that is, perpendicular to the motion. As a result of such an anisotropic
compression, the parallel temperature increases111By more than the
$(n_{2}/n_{1})^{2}$ factor of Eq. (2). while the perpendicular one is left
unchanged. Note that these heuristic prescriptions have been fully confirmed
by PIC simulations in Haggerty et al. (2022).

This scheme allows to define 2 stages of the kinetic history of the plasma
from the upstream to the downstream.

  * •

“Stage 1” is the state of the plasma right after the front crossing, where its
perpendicular temperature has been conserved. As shall be checked in Section
5, this prescription, together with the conservation equations, allows for a
full characterization of Stage 1.

  * •

Depending on its properties, Stage 1 can be stable or unstable. If stable,
then it represents the end state of the downstream. If unstable, it migrates
to “Stage 2”, defined as follow.

    * –

If Stage 1 was mirror unstable then Stage 2 lies on the mirror instability
threshold.

    * –

If Stage 1 was firehose unstable, then Stage 2 lies on the firehose
instability threshold.

Again, in both cases, the conservation equations allow for a full
characterization of Stage 2.

The shock front is eventually treated as a discontinuity, which means we only
consider the state of the plasma far downstream, once it has reached a steady
state. This steady state is Stage 1 if it is stable, or Stage 2 if Stage 1 is
unstable.

##  3 Conservation equations

The conservation equations for anisotropic pressures were derived in Hudson
(1970) and studied in Erkaev et al. (2000); Double et al. (2004); Gerbig &
Schlickeiser (2011). For a parallel shock, $m$ being the mass of the
particles, they take a simple form,

$\displaystyle n_{1}V_{1}$ $\displaystyle=$ $\displaystyle n_{2}V_{2},$ (3)
$\displaystyle n_{1}mV_{1}^{2}+P_{\parallel 1}$ $\displaystyle=$
$\displaystyle n_{2}mV_{2}^{2}+P_{\parallel 2},$ (4) $\displaystyle
m\frac{V_{1}^{2}}{2}+\frac{P_{\parallel 1}}{n_{1}}+U_{1}$ $\displaystyle=$
$\displaystyle m\frac{V_{2}^{2}}{2}+U_{2}+\frac{P_{\parallel 2}}{n_{2}},$ (5)

where on both sides, the internal energy $U$ is given by

$U=\frac{P_{\parallel}+2P_{\perp}}{2n}.$ (6)

In Eqs. (4,5), the pressure term only involves the parallel pressure since it
is the pressure component pushing the plasma through the front (see for
example the discussion in Feynman et al. (1963), §4-3).

The problem is then solved in terms of the dimensionless variables,

$\displaystyle r$ $\displaystyle=$ $\displaystyle\frac{n_{2}}{n_{1}},$
$\displaystyle A_{i}$ $\displaystyle=$ $\displaystyle\frac{T_{\perp
i}}{T_{\parallel i}},$ $\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{B_{0}^{2}/8\pi}{n_{1}mV_{1}^{2}/2},$
$\displaystyle\chi_{\parallel 1}^{2}$ $\displaystyle=$
$\displaystyle\frac{mV_{1}^{2}}{P_{\parallel 1}/n_{1}}.$ (7)

The $\chi_{\parallel 1}$ parameter is close to the upstream sonic Mach number.
It only misses the adiabatic index $\gamma$. Yet, it is preferable to leave
this index out of the definition of any dimensionless number because our model
switches continuously from a system with 3 degrees of freedom to a system with
1 degree (Bret, 2021).

As shall be checked in section 5.1, our model requires $\chi_{\parallel
1}>\sqrt{3}$ for Stage 1 to have a positive entropy jump.

The $\sigma$ parameter is widely used in PIC simulations of collisionless
shocks like Haggerty et al. (2022). It is related to the upstream Alfvén Mach
number $\mathcal{M}_{A1}$ through

$\mathcal{M}_{A1}=\frac{V_{1}}{c_{A1}}=\frac{1}{\sqrt{\sigma}},$ (8)

where the Alfvén speed $c_{A1}$ reads

$c_{A1}=\sqrt{\frac{B_{0}^{2}/4\pi}{n_{1}m}}.$ (9)

##  4 Instability restrictions on upstream parameters

The range of upstream parameters is restricted by the requirement that the
upstream be both firehose and mirror stable. The firehose instability is
triggered for an upstream anisotropy $A_{1}$ fulfilling (Gary, 1993)222See
Gary & Karimabadi (2009); Schlickeiser (2010) for the case of pair plasma.

$A_{1}\equiv\frac{T_{\perp 1}}{T_{\parallel 1}}<1-\frac{1}{\beta_{\parallel
1}},$ (10)

with

$\beta_{\parallel 1}=\frac{n_{1}k_{B}T_{\parallel 1}}{B_{0}^{2}/8\pi},$ (11)

where $k_{B}$ is the Boltzmann constant. The mirror instability is triggered
for too strong an upstream anisotropy fulfilling

$A_{1}>1+\frac{1}{\beta_{\parallel 1}}.$ (12)

Since the upstream has to be stable, Eqs. (10,12) restrict the upstream phase
space parameter to consider. In terms of the dimensionless parameters (3), we
find

$\beta_{\parallel 1}=\frac{2}{\sigma\chi_{\parallel 1}^{2}},$ (13)

so that Eqs. (10,12) read

$1-\frac{\sigma\chi_{\parallel 1}^{2}}{2}<A_{1}<1+\frac{\sigma\chi_{\parallel
1}^{2}}{2}.$ (14)

In MHD, the density jump of a parallel shock does not vary with $\sigma$
Kulsrud (2005). Since the present work eventually aims at studying the
deviations from MHD in collisionless shocks, it is advantageous to use
$\sigma$ as the main variable and recast criteria (14) in terms of it. The
upstream is then stable if $\sigma>\sigma_{us}$ (subscript “us” for _u_
pstream _s_ table), with

$\sigma_{us}\equiv\left\\{\begin{array}[]{l}\frac{2}{\chi_{\parallel
1}^{2}}(1-A_{1}),~{}\mathrm{when}~{}A_{1}<1~{}(\mathrm{firehose}~{}\mathrm{stability}),\\\
\frac{2}{\chi_{\parallel
1}^{2}}(A_{1}-1),~{}\mathrm{when}~{}A_{1}>1~{}(\mathrm{mirror}~{}\mathrm{stability}).\end{array}\right.$
(15)

For $A_{1}=1$, any $\sigma$ is allowed since the upstream is always stable.

##  5 Analysis of Stage 1, with $T_{\perp 2}=T_{\perp 1}$

As explained in section 2, we assume the plasma arrives in the downstream with
the same perpendicular temperature as the upstream. This is our Stage 1. It
can be characterized by the conservation equations Eqs. (3-5). Setting
$T_{\perp 2}=T_{\perp 1}$ and $P=nk_{B}T$, they read,

$\displaystyle n_{1}V_{1}$ $\displaystyle=$ $\displaystyle n_{2}V_{2},$ (16)
$\displaystyle n_{1}mV_{1}^{2}+n_{1}k_{B}T_{\parallel 1}$ $\displaystyle=$
$\displaystyle n_{2}mV_{2}^{2}+n_{2}k_{B}T_{\parallel 2},$ (17) $\displaystyle
m\frac{V_{1}^{2}}{2}+\frac{3k_{B}T_{\parallel 1}+2k_{B}T_{\perp 1}}{2}$
$\displaystyle=$ $\displaystyle m\frac{V_{2}^{2}}{2}+\frac{3k_{B}T_{\parallel
2}+2k_{B}T_{\perp 1}}{2}.$ (18)

Eq. (16) can be used to eliminate $V_{2}$ everywhere. Then $T_{\parallel 2}$
can be expressed from Eq. (17). Replacing in Eq. (18) gives for
$r=n_{2}/n_{1}$,

$(r-1)(\chi_{\parallel 1}^{2}(r-2)+3r)=0.$ (19)

The first factor yields the trivial solution $r=1$ corresponding to the
absence of any jump. The second factor gives

$r=\frac{2\chi_{\parallel 1}^{2}}{\chi_{\parallel 1}^{2}+3},$ (20)

formally identical to the isotropic case reported in Bret & Narayan (2018),
replacing $\chi_{1}$ by $\chi_{\parallel 1}$. Noteworthily, it does not depend
on the upstream anisotropy $A_{1}$.

In the strong sonic shock limit $\chi_{\parallel 1}\rightarrow\infty$, Eq.
(20) gives $r=2$, consistent with a shock in 1 dimension. This is expected
since in Stage 1 we freeze the perpendicular degrees of freedom, leaving only
1 free degree, namely, the parallel one.

The downstream anisotropy $A_{2}$ in Stage 1 is then given by

$A_{2}=A_{1}\frac{4\chi_{\parallel 1}^{2}}{\chi_{\parallel
1}^{4}+2\chi_{\parallel 1}^{2}-3}.$ (21)

Figure 2: Stability diagram for the upstream and the downstream. The plasma is
stable within the shaded area.

The stability diagram for the firehose and mirror instabilities is pictured in
figure 2. According to Eq. (21), the vertical location of Stage 1 in this
diagram only relies on the values of the upstream parameters $A_{1}$ and
$\chi_{\parallel 1}$. Therefore, these are the parameters which determines
whether Stage 1 can be mirror or firehose unstable. The ultimate answer to the
stability question eventually relies on the field strength $\sigma$, but the
choice between these 2 instabilities only relies on $A_{1}$ and
$\chi_{\parallel 1}$. This feature will be key to analyse the end result of
our model is section 7.

###  5.1 Entropy of Stage 1 - Restriction on $\chi_{\parallel 1}$

As is the case in a fluid, the requirement of a positive entropy jump at the
shock sets a lower limit on $\chi_{\parallel 1}$. In Bret & Narayan (2018) we
found it was $\sqrt{3}$. We shall now see that this lower limit remains the
same here.

Expression (21) for the anisotropy in Stage 1 allows to compute the entropy
jump between the upstream and the downstream in Stage 1.

Following Bret & Narayan (2018), we consider a bi-Maxwellian of the form

$f(\mn@boldsymbol{v})=\frac{n}{\pi^{3/2}\sqrt{a}b}\exp\left(-\frac{v_{x}^{2}}{a}\right)\exp\left(-\frac{v_{y}^{2}+v_{z}^{2}}{b}\right),$
(22)

where $a=2k_{B}T_{\parallel}/m$ and $b=2k_{B}T_{\perp}/m$. The entropy density
reads

$S=-k_{B}\int f\ln fd^{3}v=\frac{1}{2}k_{B}n[3+\ln(\pi^{3}ab^{2})-2\ln n].$
(23)

After some straightforward algebra, the difference of entropy per particle
$S/n$ between the upstream and the downstream reads

$\displaystyle\Delta s\equiv\frac{S_{2}}{n_{2}}-\frac{S_{1}}{n_{1}}$
$\displaystyle=$
$\displaystyle\frac{k_{B}}{2}\ln\left(\frac{A_{1}}{A_{2}}\frac{T_{\perp
2}^{3}}{T_{\perp 1}^{3}}\right)-2\ln r$ (24) $\displaystyle=$
$\displaystyle-\frac{k_{B}}{2}\ln\left(\frac{A_{2}}{A_{1}}r^{2}\right),$

since $T_{\perp 2}=T_{\perp 1}$ in Stage 1. From the expression (21) of
$A_{2}$ we find that the entropy jump is the same as for the $A_{1}=1$ case
studied in Bret & Narayan (2018). Therefore, as in Bret & Narayan (2018),
$\Delta s<0$ for $\chi_{\parallel 1}<\sqrt{3}$, so that our model is
physically meaningful only for $\chi_{\parallel 1}>\sqrt{3}$.

Having characterized Stage 1 we turn to its firehose and mirror stability
analysis.

###  5.2 Firehose stability of Stage 1

We now clarify the conditions on $\sigma,A_{1},\chi_{\parallel 1}$ for
firehose instability of Stage 1. The $\beta_{\parallel 2}$ parameter for the
downstream is given by

$\beta_{\parallel 2}=A_{1}\frac{2r}{\sigma A_{2}\chi_{\parallel 1}^{2}}.$ (25)

When substituting $r$ and $A_{2}$ by their expression (20) and (21), we find

$\beta_{\parallel 2}=\frac{1}{\sigma}\frac{\chi_{\parallel
1}^{2}-1}{\chi_{\parallel 1}^{2}},$ (26)

identical to the result of Bret & Narayan (2018).

The criteria for firehose stability reads

$A_{2}>1-\frac{1}{\beta_{\parallel 2}},$ (27)

that is,

$\chi_{\parallel 1}^{2}\frac{\sigma(\chi_{\parallel
1}^{2}+3)+4A_{1}}{\chi_{\parallel 1}^{4}+2\chi_{\parallel 1}^{2}-3}>1.$ (28)

This equation yields the critical $\sigma\equiv\sigma_{cf}$ below which Stage
1 is firehose unstable,

$\sigma_{cf}=1-\frac{4A_{1}}{\chi_{\parallel
1}^{2}+3}-\frac{1}{\chi_{\parallel 1}^{2}}.$ (29)

###  5.3 Mirror stability of Stage 1

The criteria for mirror stability reads

$A_{2}<1+\frac{1}{\beta_{\parallel 2}},$ (30)

that is,

$\chi_{\parallel 1}^{2}\frac{\sigma(\chi_{\parallel
1}^{2}+3)-4A_{1}}{\chi_{\parallel 1}^{4}+2\chi_{\parallel 1}^{2}-3}<-1.$ (31)

This equation allows to define a critical $\sigma\equiv\sigma_{cm}$ under
which Stage 1 is mirror unstable,

$\sigma_{cm}=\frac{4A_{1}}{\chi_{\parallel 1}^{2}+3}+\frac{1}{\chi_{\parallel
1}^{2}}-1=-\sigma_{cf}.$ (32)

Figure 3: Respective position of $\sigma_{us}$, defining the minimum field
required to stabilize the upstream, and $\sigma_{cf},\sigma_{cm}$, defining
the minimum field required to sustain a stable Stage 1 in the downstream.
Figure 4: Difference $\sigma_{cm}-\sigma_{us}$ in terms of $(\chi_{\parallel
1},A_{1})$ where $\sigma_{cm}$ is given by Eq. (32) and $\sigma_{us}$ by Eq.
(15) with $A_{1}>1$.

###  5.4 Condition for the need of Stage 2

In the present model, if Stage 1 is unstable, the downstream switches from
Stage 1 to Stage 2.

If Stage 1 is firehose unstable, that is, $A_{2}<1-\beta_{\parallel 2}^{-1}$,
Stage 1 will switch to Stage 2 on the firehose threshold if
$\sigma<\sigma_{cf}$.

If Stage 1 is mirror unstable, that is, $A_{2}>1+\beta_{\parallel 2}^{-1}$,
Stage 1 will switch to Stage 2 on the mirror threshold if
$\sigma<\sigma_{cm}$.

In addition, the $\sigma$-range is limited by the requirement of a stable
upstream, as explained in section 4.

All of these constraints are summarized in figure 3. For Stage 2 to play any
role, such a picture obviously requires that the minimum field stabilizing the
upstream be smaller than $\sigma_{cf}$ or $\sigma_{cm}$.

For $A_{1}<1$, $\sigma_{cf}$ is relevant and it is easily proved that
$\sigma_{us}<\sigma_{cf}$ is always true.

Yet, for $A_{1}>1$, $\sigma_{cm}$ is relevant and $\sigma_{us}<\sigma_{cm}$ is
not always fulfilled. Figure 4 pictures the difference
$\sigma_{cm}-\sigma_{us}$ in terms of $(\chi_{\parallel 1},A_{1})$. The
difference is found positive only for $A_{1}>\frac{1}{2}(\chi_{\parallel
1}^{2}+3)$. As a consequence, for such values of the upstream anisotropy
$A_{1}$, there is no need for Stage 2. An example of such a case will be given
in section 7.

##  6 Analysis of Stage 2

According to the premises of our model, the downstream switches to Stage 2 in
case Stage 1 is unstable. From the expression (21) of the downstream
anisotropy in Stage 1, it is easily proved that $A_{1}<1\Rightarrow A_{2}<1$.
Hence, if $A_{1}<1$, Stage 1 can only be firehose unstable. On the contrary,
if $A_{1}>1$, then Stage 1 can only be mirror unstable. We need therefore to
study both options, namely, “Stage 2 firehose”, where the downstream lies on
the firehose instability threshold, and “Stage 2 mirror”, where it lies in the
mirror instability threshold.

###  6.1 Stage 2 firehose

Imposing Stage 2 to lie on the firehose instability threshold means imposing
$A_{2}=1-1/\beta_{\parallel 2}$ in the conservation equations (3-5). From

$U_{2}=\frac{P_{\parallel 2}+2P_{\perp 2}}{2}=\frac{n_{2}k_{B}T_{\parallel
2}+2n_{2}k_{B}T_{\perp 2}}{2},$ (33)

and

$A_{2}=\frac{T_{\perp 2}}{T_{\parallel
2}}=1-\frac{B_{0}^{2}/8\pi}{n_{2}k_{B}T_{\parallel 2}},$ (34)

we get

$U_{2}=\frac{1}{2}\left(3k_{B}T_{\parallel
2}-\frac{B_{0}^{2}/4\pi}{n_{2}}\right).$ (35)

Substituting $U_{2}$ by the expression above in Eq. (5) and solving the system
for the density ratio gives a 2nd order polynomial for the density ratio $r$,

$r[(2A_{1}+3)r-5]+\chi_{\parallel 1}^{2}[r(r+\sigma-5)+4]=0.$ (36)

The anisotropy is given by

$A_{2}=\frac{2r-\chi_{\parallel
1}^{2}[r(\sigma-2)+2]}{2\left[(r-1)\chi_{\parallel 1}^{2}+r\right]}.$ (37)

The polynomial equation (36) gives 2 solutions. Yet, one of them yields a
nonphysical, namely negative, value for the anisotropy $A_{2}$. The physical
density jump that has $A_{2}>0$ reads

$rS2_{f}=\frac{\sqrt{\Delta_{f}}-(\sigma-5)\chi_{\parallel
1}^{2}+5}{2\left(2A_{1}+\chi_{\parallel 1}^{2}+3\right)},$ (38)

with

$\Delta_{f}=25-2\chi_{\parallel
1}^{2}(16A_{1}+5\sigma-1)+(\sigma-9)(\sigma-1)\chi_{\parallel 1}^{4}.$ (39)

###  6.2 Stage 2 mirror

Imposing Stage 2 to lie on the mirror instability threshold means imposing
$A_{2}=1+1/\beta_{\parallel 2}$ in the conservation equations (3-5). Following
the same path than for Stage 2 firehose also gives a 2nd order polynomial for
the density ratio $r$,

$\chi_{\parallel 1}^{2}[(r(5-r+\sigma)-4)]+r[5-(2A_{1}+3)r]=0,$ (40)

with the anisotropy

$A_{2}=\frac{\chi_{\parallel
1}^{2}(r(\sigma+2)-2)+2r}{2\left[(r-1)\chi_{\parallel 1}^{2}+r\right]}.$ (41)

The polynomial equation (40) gives 2 solutions. Again, one of them yields a
nonphysical, namely negative, value for the anisotropy $A_{2}$. The physical
density jump reads

$rS2_{m}=\frac{\sqrt{\Delta_{m}}+(\sigma+5)\chi_{\parallel
1}^{2}+5}{2\left(2A_{1}+\chi_{\parallel 1}^{2}+3\right)},$ (42)

with

$\Delta_{m}=25-\chi_{\parallel
1}^{2}(32A_{1}-10\sigma-2)+(\sigma+9)(\sigma+1)\chi_{\parallel 1}^{4}.$ (43)

###  6.3 Some comments on Stage 2

To which extent can we consider that if Stage 1 is unstable, then Stage 2
remains on the marginal stability threshold, without migrating further into
the stability region? As previously stated, Bale et al. (2009) concluded that
weak Coulomb collisionality is responsible for the concentration of solar wind
measurements around $T_{\perp}=T_{\parallel}$. Possibly collisionless
relaxation (Kadomtsev & Pogutse, 1970; Lapenta et al., 2009; Gedalin, 2015)
could also play a role on longer time scales, which goes beyond the scope of
the present work. For the case of an isotropic upstream, the numerical study
of Haggerty et al. (2022) found that Stage 2, when triggered by Stage 1
instability, fulfils the marginal stability condition.

While the key assumption for Stage 1 is that the perpendicular temperature is
conserved, no such assumption is made on any temperature for Stage 2. There,
the only constraint on the downstream is to lie on the firehose or mirror
stability threshold. For the case of an isotropic upstream, both temperatures
have been explicitly and separately studied in Bret (2021), showing how in
Stage 2, both of them are eventually a function of $\sigma$.

The density jump for Stage 2 is derived from Eqs. (3-5). These equations
consider the field is homogenous and aligned with the flow, which is why it
does not appear in these expressions. Clearly, during the growth of the mirror
or the firehose instability, the total field does not have to be homogenous
nor aligned with the flow. Yet, as stated in the introduction and at the end
of Section 2, we consider the state of the downstream long after the
instabilities developed, once it has reached a steady state. At this stage,
the downstream has reached marginally stability and the macroscopic magnetic
field is back in line with the flow. Indeed, the simulations of Haggerty et
al. (2022) show that marginal stability is reached within a few
$c/\omega_{p}$’s behind the front333Where $\omega_{p}$ is the plasma frequency
and $c$ the speed of light. See for example figure 4(b) of Haggerty et al.
(2022)..

Figure 5: Summary of the various cases offered by the model. Upper part:
$A_{2}>1$ and the jump is given by Stage 1 or Stage 2 mirror. Intermediate
shaded region: the jump can only be given by Stage 1 since
$\sigma_{cm}>\sigma_{us}$. Lower part: $A_{2}<1$, the jump is given by Stage 1
or Stage 2 firehose. The red crosses indicate the parameters featuring figure
6.

##  7 Putting Stages 1 and 2 together

Having computed all the density jumps for Stage 1 and Stage 2 we can now
provided a full picture of the shock density jump in terms of
$\sigma,A_{1},\chi_{\parallel 1}$.

Figure 5 pictures the various cases offered by the model in terms of
$(A_{1},\chi_{\parallel 1})$. In the upper part, $A_{2}>1$ and the jump is
given by Stage 1 or Stage 2 mirror. In the intermediate shaded region, the
jump can only be given by Stage 1 since $\sigma_{cm}>\sigma_{us}$. In the
lower part, $A_{2}<1$, the jump is given by Stage 1 or Stage 2 firehose.

(a)

(b)

(c)

Figure 6: Plots $r(\sigma)$ for the values of $\chi_{\parallel 1}$ and $A_{1}$
marked by the red crosses on figure 5.

Figure 6 presents plots $r(\sigma)$ for the values of $\chi_{\parallel 1}$ and
$A_{1}$ marked by the red crosses on figure 5.

Panel (a) shows cases pertaining to the upper part of figure 5, with
$A_{2}>1$. The curves only start for values of $\sigma$ corresponding to a
stable upstream. From Eq. (15), this value can be quite high for large
anisotropies. The curves present a segment where the jump is the one for Stage
2 mirror [Eq. (42)], since the field is still too weak to stabilize Stage 1.
Then the flat part at large $\sigma$ is the jump of Stage 1 [Eq. (20)]. The
jump in the Stage 2 mirror phase is an increasing function of $\sigma$. The
strong shock case $\chi_{\parallel 1}\rightarrow\infty$ cannot be explored in
this regime because, as visible in figure 5, this limit always ends up leaving
this region of the phase space444Any point pertaining to the upper left region
of figure 5 (the one labelled “$A_{2}>1$, Stage 1 or Stage 2 mirror”),
eventually leaves this region if $\chi_{\parallel 1}$ is increased while
$A_{1}$ is kept constant..

Panel (b) shows cases pertaining to the shaded part of figure 5, where the
field required to stabilize the upstream is larger than the minimum field
required to mirror stabilize the downstream with $A_{2}>1$. The jump is
therefore uniquely given by Stage 1 for all relevant $\sigma$´s. The
requirement of a stable upstream still sets a minimum value of $\sigma$ below
which a solution would be un-physical.

Finally, panel (c) shows cases pertaining to the lower part of figure 5, with
$A_{2}<1$. Here, if unstable, Stage 1 is firehose unstable. In this regime,
studying the strong shock limit $\chi_{\parallel 1}\rightarrow\infty$ is
meaningful since it can be reached without leaving this region. In this
respect, $r(\sigma)$ has been plotted for $\chi_{\parallel 1}=100$ (not shown
by a red cross on figure 5). As evidenced by looking at Eqs. (38,39), this
limit for Stage 2 firehose cannot depend on $A_{1}$. In addition, the jump for
Stage 1 does not depend on $A_{1}$ either (see Eq. 20). As a result, the
strong shock limit is independent of the upstream anisotropy $A_{1}$. It is
eventually identical to the strong shock result for the isotropic upstream
case described in Bret & Narayan (2018).

##  8 Conclusion

Since MHD is frequently used to assess the properties of collisionless shocks,
it is important to study to which extent this strategy is pertinent (Bret,
2020). In this respect, parallel shocks are of particular interest since
within the MHD framework, the fluid and the field are decoupled Kulsrud
(2005). As a consequence, the density jump of a parallel shock in MHD does not
depend on the field strength.

In a previous work (Bret & Narayan, 2018), it was found that due to kinetic
effects, the density jump of parallel collisionless shocks can definitely vary
with the field strength. For strong enough a field, the density jump can go
down to 2, instead of 4, the value predicted by MHD. The model presented was
successfully checked through Particle-In-Cell (PIC) simulations in Haggerty et
al. (2022). Although pair plasmas were considered in these works, preliminary
results (Shalaby, 2022) seem to indicate that the model holds for electron/ion
plasmas, provided the ions mass is used to define the dimensionless parameters
(3).

An isotropic upstream has been assumed hitherto. The present work is devoted
to extend the model to the case of an anisotropic upstream.

The kinetic history of the plasma as it crosses the shock front is decomposed
into 2 stages. First, the perpendicular temperature of the plasma is
conserved. This is Stage 1. If it is stable, then this is the end state of the
downstream. If Stage 1 is firehose unstable, it migrates to Stage 2 firehose,
marginally firehose stable. If Stage 1 is mirror unstable, it migrates to
Stage 2 mirror, marginally mirror stable.

For an isotropic upstream with $A_{1}=1$, it was found that Stage 1 can only
be firehose unstable. Considering $A_{1}\neq 1$ brings the possibility that
Stage 1 be also mirror unstable. It also restricts the minimum field strength
to be considered since the upstream has to be both mirror and firehose stable.
The analysis splits the phase space parameters into 3 regions pictured in
figure 5.

The upper part of figure 5 has the downstream anisotropy $A_{2}$ always larger
than unity. There, for moderate values of the field, the jump is given by
Stage 2 mirror. Then, as soon as the field is strong enough to stabilize Stage
1, the jump is the one of Stage 1. Figure 6(a) pictures some curves
$r(\sigma)$ in this region.

In the middle part of figure 5, namely the shaded area, the downstream
anisotropy $A_{2}$ is always larger than unity. Therefore, if unstable, Stage
1 can only be mirror unstable. Yet, in this region, the smallest field
required to stabilize the upstream is larger than the smallest field required
to mirror-stabilize Stage 1. As a consequence, Stage 2 never occurs in this
region. Figure 6(b) pictures some curves $r(\sigma)$ in this region.

Finally, in the lower part of figure 5, the downstream anisotropy $A_{2}$ is
always lower than unity. If unstable, Stage 1 can only be firehose unstable.
In many respects, this case is formally identical to the isotropic case : at
large $\sigma$, namely $\sigma>\sigma_{cf}$, the jump is given by Eq. (20),
identical to the isotropic result. For $\sigma<\sigma_{cf}$, the field is too
weak to stabilize Stage 1 and the jump is given by Eq. (38, still quite close
to the $A_{1}=1$ case since here, $A_{1}\in[0,1]$. In addition, the critical
sigma $\sigma_{cf}$ is also formally identical to its $A_{1}=1$
value555Compare the present Eq. (29) with Eq. (2.11) of Bret & Narayan
(2018).. The strong sonic shock limit $\chi_{\parallel 1}\rightarrow\infty$
necessarily pertains to this case, since this region is the only one of figure
5 from which one cannot escape when taking this limit. Figure 6(c) pictures
some curves $r(\sigma)$ in this region.

As explained in section 5.1, the present model requires $\chi_{\parallel
1}>\sqrt{3}$ to provide a positive entropy jump. From the definition of
$\chi_{\parallel 1}$ given by Eq. (3), we find $\mathcal{M}_{\parallel
1}=\chi_{\parallel 1}\gamma_{\parallel}^{-1/2}$, where $\gamma_{\parallel}$ is
the parallel adiabatic index and $\mathcal{M}_{\parallel 1}$ the parallel
upstream sonic Mach number. Note that in an anisotropic collisionless plasma,
the adiabatic index is 3 in the direction parallel to the field, and 2 in the
perpendicular directions (Abraham-Shrauner, 1967). Hence, the model can
describe shocks with $\mathcal{M}_{\parallel 1}>1$, for a parallel adiabatic
index $\gamma_{\parallel}=3$.

Future works should contemplate testing these results through PIC simulations,
like those of Bret & Narayan (2018) have been tested in Haggerty et al.
(2022).

## Acknowledgements

A.B. acknowledges support by Grant PID2021-125550OB-I00 from the Spanish
Ministerio de Ciencia e Innovación. Thanks are due to Federico Fraschetti for
valuable inputs.

## Data Availability

The calculations presented in this manuscript were performed using
_Mathematica_. The Notebook files will be shared on reasonable request to the
corresponding author.

## References

  * Abraham-Shrauner (1967) Abraham-Shrauner B., 1967, Journal of Plasma Physics, 1, 361
  * Bale et al. (2003) Bale S. D., Mozer F. S., Horbury T. S., 2003, Phys. Rev. Lett., 91, 265004
  * Bale et al. (2009) Bale S. D., Kasper J. C., Howes G. G., Quataert E., Salem C., Sundkvist D., 2009, Phys. Rev. Lett., 103, 211101
  * Blandford & Ostriker (1978) Blandford R., Ostriker J., 1978, Astrophysical Journal, 221, L29
  * Bret (2020) Bret A., 2020, ApJ, 900, 111
  * Bret (2021) Bret A., 2021, Physics of Plasmas, 28, 082107
  * Bret & Narayan (2018) Bret A., Narayan R., 2018, Journal of Plasma Physics, 84, 905840604
  * Bret & Narayan (2019) Bret A., Narayan R., 2019, Physics of Plasmas, 26, 062108
  * Bret & Narayan (2022) Bret A., Narayan R., 2022, Journal of Plasma Physics, 88, 905880320
  * Chew et al. (1956) Chew G. F., Goldberger M. L., Low F. E., 1956, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 236, 112
  * David et al. (2022) David L., Fraschetti F., Giacalone J., Wimmer-Schweingruber R. F., Berger L., Lario D., 2022, ApJ, 928, 66
  * Double et al. (2004) Double G. P., Baring M. G., Jones F. C., Ellison D. C., 2004, The Astrophysical Journal, 600, 485
  * Erkaev et al. (2000) Erkaev N. V., Vogl D. F., Biernat H. K., 2000, Journal of Plasma Physics, 64, 561
  * Feynman et al. (1963) Feynman R., Leighton R., Sands M., 1963, The Feynman Lectures on Physics. No. v. 2 in The Feynman Lectures on Physics, Pearson/Addison-Wesley
  * Fraschetti & Giacalone (2020) Fraschetti F., Giacalone J., 2020, MNRAS, 499, 2087
  * Gary (1993) Gary S. P., 1993, Theory of Space Plasma Microinstabilities. Cambridge Atmospheric and Space Science Series, Cambridge University Press, doi:10.1017/CBO9780511551512
  * Gary & Karimabadi (2009) Gary S. P., Karimabadi H., 2009, Physics of Plasmas, 16, 042104
  * Gedalin (2015) Gedalin M., 2015, Journal of Plasma Physics, 81, 905810603
  * Gerbig & Schlickeiser (2011) Gerbig D., Schlickeiser R., 2011, The Astrophysical Journal, 733, 32
  * Goedbloed et al. (2010) Goedbloed J., Keppens R., Poedts S., 2010, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas. Cambridge University Press
  * Guo et al. (2017) Guo X., Sironi L., Narayan R., 2017, ApJ, 851, 134
  * Guo et al. (2018) Guo X., Sironi L., Narayan R., 2018, ApJ, 858, 95
  * Haggerty et al. (2022) Haggerty C. C., Bret A., Caprioli D., 2022, Monthly Notices of the Royal Astronomical Society, 509, 2084
  * Hudson (1970) Hudson P. D., 1970, Planetary and Space Science, 18, 1611
  * Kadomtsev & Pogutse (1970) Kadomtsev B. B., Pogutse O. P., 1970, Phys. Rev. Lett., 25, 1155
  * Kulsrud (2005) Kulsrud R. M., 2005, Plasma physics for astrophysics. Princeton Univ. Press, Princeton, NJ
  * Landau & Lifshitz (1981) Landau L., Lifshitz E., 1981, Course of Theoretical Physics, Physical Kinetics. Elsevier, Oxford
  * Lapenta et al. (2009) Lapenta G., Markidis S., Kaniadakis G., 2009, Journal of Statistical Mechanics: Theory and Experiment, 2009, 02024
  * Maruca et al. (2011) Maruca B. A., Kasper J. C., Bale S. D., 2011, Phys. Rev. Lett., 107, 201101
  * Piran (2004) Piran T., 2004, Reviews of Modern Physics, 76, 1143
  * Sagdeev (1966) Sagdeev R. Z., 1966, Reviews of Plasma Physics, 4, 23
  * Sagdeev & Kennel (1991) Sagdeev R. Z., Kennel C. F., 1991, Scientific American, 264, 106
  * Schlickeiser (2010) Schlickeiser R., 2010, The Open Plasma Physics Journal, 3, 1
  * Schlickeiser et al. (2011) Schlickeiser R., Michno M. J., Ibscher D., Lazar M., Skoda T., 2011, Phys. Rev. Lett., 107, 201102
  * Schwartz et al. (2011) Schwartz S. J., Henley E., Mitchell J., Krasnoselskikh V., 2011, Phys. Rev. Lett., 107, 215002
  * Shalaby (2022) Shalaby M., 2022, In preparation
  * Silva et al. (2021) Silva T., Afeyan B., Silva L. O., 2021, Phys. Rev. E, 104, 035201
  * Sironi et al. (2021) Sironi L., Plotnikov I., Nättilä J., Beloborodov A. M., 2021, Phys. Rev. Lett., 127, 035101
  * Thorne & Blandford (2017) Thorne K., Blandford R., 2017, Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics. Princeton University Press
  * Vogl et al. (2001) Vogl D. F., Biernat H. K., Erkaev N. V., Farrugia C. J., Mühlbachler S., 2001, Nonlinear Processes in Geophysics, 8, 167
  * Weibel (1959) Weibel E. S., 1959, Phys. Rev. Lett., 2, 83
  * Zhang (2020) Zhang B., 2020, Nature, 587, 45