X-ray Line Diagnostics of Ion Temperature at Cosmic-Ray Accelerating Collisionless Shocks

A novel collisionless shock jump condition is suggested by modeling the entropy production at the shock transition region. We also calculate downstream developments of the atomic ionization balance and the ion temperature relaxation in supernova remnants (SNRs). The injection process and subsequent acceleration of cosmic-rays (CRs) in the SNR shocks are closely related to the formation process of the collisionless shocks. The formation of the shock is caused by wave-particle interactions. Since the wave-particle interactions result in energy exchanges between electromagnetic fields and charged particles, the randomization of particles associated with the shock transition may occur with the rate given by the scalar product of the electric field and current. We find that order-of-magnitude estimates of the randomization with reasonable strength of the electromagnetic fields in the SNR constrain the amount of the CR nuclei and ion temperatures. The constrained amount of the CR nuclei can be sufficient to explain the Galactic CRs. The ion temperature becomes significantly lower than in the case of no CRs. To distinguish the case without CRs, we perform synthetic observations of atomic line emissions from the downstream region of the SNR RCW~86. Future observations by {\it XRISM} and {\it Athena} can distinguish whether the SNR shock accelerates the CRs or not from the ion temperatures.


Introduction
Collisionless shocks of supernova remnants (SNRs) are invoked as the primary sources of Galactic cosmic-rays (CRs); however, the production process of CRs is an unsettled issue despite numerous studies reported. The most generally accepted and widely studied mechanism for CR acceleration is the diffusive shock acceleration (DSA, Bell 1978;Blandford & Ostriker 1978). In the DSA mechanism, we assume energetic particles around the shock, and the particles go into bouncing back and forth between the upstream and downstream regions by scattering particles. The particle scattering results from interactions between plasma waves and the particles. The maximum energy of the accelerated particles depends on the magnetic field strength and turbulence (e.g. Lagage & Cesarsky 1983a;Lagage & Cesarsky 1983b). To explain the energy spectrum of the CR nuclei observed around the Earth, the maximum energy of the accelerated protons should be at least 10 15.5 eV (so-called the knee energy). The knee energy can be achieved in the DSA mechanism by the magnetic field strength of > ∼ 100 µG which is larger than the typical strength of ∼ 1 µG seen in the interstellar medium (ISM, Myers 1978;Beck 2001). Bell (2004) pointed out that the upstream magnetic field is amplified by the effects of a back reaction from the accelerated protons themselves. This amplification is called the Bell instability, whose growth rate is proportional to the CR energy density. Observations of nonthermal X-ray emissions around the SNR shocks imply the existence of amplified magnetic fields in the downstream region (e.g., Vink & Laming 2003;Bamba et al. 2005;Uchiyama et al. 2007). Hence, in a modern scenario of the CR acceleration, the SNR shock is assumed to inject a considerably large amount of CRs ( > ∼ 10 % of the shock kinetic energy), and their effects on the background plasma are regarded as one of the most important issues. Since the wave-particle interactions also give the formation of the collisionless shock, the injection of energetic particles, subsequent acceleration by the DSA mechanism, and the amplification of the magnetic field are closely related to the formation process. Although many kinetic simulations studying collisionless shock physics are reported (e.g., Ohira 2013; Ohira 2016b; Ohira 2016a; Matsumoto et al. 2017;Caprioli et al. 2020;Marcowith et al. 2020), self-consistent treatment of the collisionless shock, including these effects, is currently incomplete due to the limitation of too short simulation time compared to actual SNR shocks.
When the SNR shock consumes its kinetic energy to accelerate energetic, nonthermal particles, the downstream thermal energy can be lower than the case of adiabatic shock without the CRs (e.g., Hughes et al. 2000, Helder et al. 2009Morlino et al. 2013b;Morlino et al. 2013a;Morlino et al. 2014;Hovey et al. 2015;Hovey et al. 2018;Shimoda et al. 2015;Shimoda et al. 2018b). Thus, we observe small downstream ion temperatures if the SNR shocks efficiently accelerate the CR nuclei (protons and heavier ions). In the near future, spatially resolved high energy-resolution spectroscopy of the SNR shock regions will be achieved by the micro-calorimeter array with Resolve (Ishisaki et al. 2018) onboard XRISM (Tashiro et al. 2020) and with the X-IFU onboard Athena (Barret et al. 2018) providing precise line diagnostics of plasmas to represent the effect of the CR acceleration. Note that observations of γ-ray emissions possibly provide the amount of the CR proton from luminosities. However it may be challenging to determine the amounts of the CR nuclei individually. Thus, in this paper, we study the shock jump conditions of ions, including the effects of CR acceleration. To distinguish the case without the CRs by the future Xray spectroscopy, we calculate temporal evolutions of the downstream ionization structure and downstream ion temperatures resulting from the Coulomb interactions. From the calculations of the downstream values, we also perform synthetic observations of atomic lines, including effects of downstream turbulence. Since the turbulence affects the line width by the Doppler effect, it is non-trivial whether the observed line width reflects the intrinsic ion temperature.
Typical example of the missing thermal energy measurement was provided at the SNR RCW 86 (Helder et al. 2009). The RCW 86 is considered as the remnant of SN 185 (Vink et al. 2006), so its age is ∼ 2000 yr. The current radius is ∼ 15 pc, and the shock velocity is ∼ 3000 km s −1 with an assumed distance of 2.5 kpc (Yamaguchi et al. 2016), where we estimate the angular size as ∼ 40 arcmin. Since 2000 yr×3000 km s −1 ≃ 6 pc < 15 pc, the blast wave has already decelerated. The mean expansion speed should be ∼ 10 9 cm s −1 if the radius becomes ∼ 10 pc within the time of ∼ 1 kyr. Such high expansion speed can be maintained during 1 kyr if the progenitor star explodes in a wind-brown cavity created by the progenitor system. Broersen et al. (2014) studied this scenario by comparing the X-ray observations and hydrodynamical simulations. They concluded that the progenitor star of SN 185 exploded as a Type-Ia supernova inside the wind-cavity. The progenitor system may be a binary system consisting of a white-dwarf and donor star. In the present day, the RCW 86 shows Hα filaments everywhere (Helder et al. 2013). The Hα emission means that the shock is now propagating into a partially ionized medium (Chevalier et al. 1980). In the case of a stellar wind by a massive star, the ionization front precedes the front of the swept-up matter (e.g., Arthur et al. 2011). Thus, the forward shock of the RCW 86 currently may propagate in the medium not swept up by the wind. In this paper, we suppose such a scenario for the RCW 86 and perform synthetic observations of the X-ray atomic lines.
This paper is organized as follows: In section 2, we review a physical model of the temporal evolutions of the downstream ionization balance and the ion internal energies. The ion temperatures are derived from the equation of state. Section 3 provides shock jump conditions as initial conditions for the downstream temporal evolution. We introduce shock jump conditions usually supposed in the SNRs and a novel condition given by modeling the entropy productions of the ions due to the wave-particle interactions. The latter includes the effects of the CR acceleration, magnetic-field amplification, and ion heating balance. The results of the downstream temporal evolutions are summarized in section 4. In section 5, we perform synthetic observations of atomic lines, including the effects of the downstream turbulence. Finally, we summarize our results and prospects.

Physical Model of downstream ionization balance and ion internal energies
Here we review a physical model of the temporal evolutions of the downstream ionization balance and ion internal energy. Let V be a fluid parcel volume. The parcel contains a mass of M . We assume that species within the parcel always have the Maxwell velocity distribution function with a temperature of Tj, where the subscript j indicates the species j. Then, the internal energy Ej and pressure Pj of the species j can be written as where Nj , nj , and k are the total number of the species j, the number density of the species j (nj = Nj /V ), and Boltzmann constant, respectively. The adiabatic index is γ = 5/3. From the first law of thermodynamics, we obtain where dQj/dt is the external energy gain or loss per unit time of the species j; we discuss it later. Defining the internal energy per unit volume εj ≡ Ej/V and the external energy gain or loss per unit time per unit volumeqj ≡ V −1 dQj/dt, we rewrite the equation (3) as where ρ ≡ M/V is the total mass density. In this paper, we suppose the case of young SNRs and approximate their dynamics by the Sedov-Taylor model (Sedov 1959;Vink 2012). Then, we approximate the downstream velocity profile as where r is the radial distance from the explosion center and rc is the compression ratio, respectively. The radius of the SNR and the shock velocity are given by, respectively, where we have assumed the ambient density structure around the SNR is uniform. The dimensional constants R0 and t0 are characterized by the combination of the explosion energy of the supernova, the structure of the ejecta, and the ambient density structure. The actual values of R0 and t0 are not used in our model calculation; we only use V sh /R sh = (2/5)t −1 . The temporal evolution of the mass density along the trajectory of the fluid parcel is derived from the continuous equation as To calculate ρ and εj along the trajectory of the fluid parcel, we introduce the position of the fluid parcel atr(t) that is derived from the differential equation of Defining the time t * when the fluid parcel currently at r(t) = r crosses the shock, i.e.,r(t * ) = R sh (t * ), we obtain where we regard rc = const. When we observe the atomic line emissions from the fluid parcel at r =r(tage), where tage is the age of the SNR, the crossing time is derived as Thus, by introducing t ′ = t − t * , the temporal evolution of the downstream internal energy and the mass density are written as, respectively, Integrating the differential equation of εj from t ′ = 0 to t ′ = tage − t * (r) with the shock jump conditions given by V sh (t * ), we obtain the spatial profile of the downstream internal energy at the observed time t = tage. The age is known for a historical SNR (e.g., SNR RCW 86, SN 1006, Tycho's SNR, Kepler's SNR). The shock velocity at the current time can be estimated from the proper motion of the shock. To calculate the rate of the Coulomb interactions (see below), the number density needs. The density is evaluated from the surface brightness of the X-ray or Hα emissions, for instance.
Here we consider the energy source or sink termqj . The charged particles exchange their momenta and energies via the Coulomb collision. Although the exchange is negligible during the shock transition, the effect becomes important for the long-time evolution in the downstream region. The energy exchange rate is given by (e.g., Spitzer 1962;Itoh 1984) q j,Col = m (nj εm − nmεj) zm 2 zj 2 ln Λ 5.87AmAj where Aj and ln Λ are the particle mass in atomic mass units and the Coulomb logarithm. In this paper, we fix lnΛ = 30 for simplicity. For atoms, the energy transfer due to the ionization or recombination may be given bẏ where we introduce the notation j = {Z, z} to represent the species with an atomic number Z and ionic charge state z, respectively (e.g., Z = 2 and z = 1 indicate He +1 or He II). The subscript 'e' indicates the electron. The electron-impact ionization rate per unit time per particle (s −1 cm 3 ) is RZ,z(Te), and the recombination rate per unit time per particle is KZ,z(Te). In this paper, we omit the the calculation of t * , we use the compression ratio given by V sh (tage), but for the other cases, we calculate the shock jump conditions using V sh (t * ) given by the calculated t * .
charge-exchange reactions and the ion impact ionization for simplicity and consider ten atoms H, He, C, N, O, Ne, Mg, Si, S, and Fe with the solar abundance (Asplund et al. 2009). The atomic data used in this paper are the same as Shimoda & Inutsuka (2021): The ionization cross-sections are given by Janev & Smith (1993) for H, and Lennon et al. (1988) for the others. The fitting functions for those data are given by International Atomic Energy Agency. 2 Table 1 summarises the literature on the recombination rates. Those data are fitted by the Chebyshev polynomials with twenty terms. For the hydrogen-like atoms, the fitting function is given by Kotelnikov & Milstein (2019). The electron number density is given by the charge neutrality condition as and the total number density n is given by For electrons, by following Shimoda & Inutsuka (2021), the radiative and ionization losses are given bẏ where IZ,z is the first ionization potential of the species j = {Z, z} (we omit the inner shell ionization). The radiation power WZ,z includes the bound-bound, free-bound, free-free, and two-photon decay. For the continuum components, the formula given by Gronenschild & Mewe (1978) (free-free and two-photon decays) and Mewe et al. (1986) (free-bound) are used. For the bound-bound component, the radiation power per particle is given by where the emitted photon energy is the subtraction of the upper energy level Eu and the lower energy level E l , E ul = Eu − E l . The collisional excitation rate per unit time per particle (s −1 cm 3 ) is given by (e.g., Osterbrock & Ferland 2006) where g l is the statistical weight of the lower level. The collision strength is where f lu is the oscillator strength, and E ul,Ryd is the photon energy given in the Rydberg unit. The g l is the statistical weight of the lower level. The averaged Gaunt factor is g. The value of the averaged Gaunt factor is around unity and determines the detailed temperature dependence of the excitation rate. The precise data of the excitation rate (orḡ) are, however, not available yet. The following fitting function (Mewe 1972) g(Te) = 0.15 + 0.28 log where χ = E ul /kTe, is used for the neutral atoms, whilē g = 1 is assumed for the ionized atoms. Note that the cooling function mainly depends on the ionization structure rather thanḡ. For the oscillator strength and energy levels, the data table given by the National Institute of Standards and Technology 3 are used. For the calculation of the radiative cooling rate, it is sufficient to consider only the allowed transitions from the ground state. We obtain the net radiation power and thus the net radiative loss by integrating the photon frequency. Here is a summary of the energy source or sink term:qj =qZ,0 for the neutral atoms,qj =q j,Col +qZ,z for the ions, andqj =q j,Col +qe for the electrons.
Since the SNR shock may heat the plasma faster than the Coulomb collisions due to the wave-particle interactions in the plasma, the ionization state of atoms can significantly deviate from the ionization equilibrium. Thus, we simultaneously solve the atomic rate equations Note that in our formulation, the velocity distribution function of the species is always assumed to be the Maxwellian.

Shock jump conditions
Here we give the initial conditions for the temporal developments of the downstream ionization balance and temperature relaxation by considering shock jump conditions. We introduce the conditions usually supposed in the SNR shocks from analogs of collisional shocks and the novel condition given by modeling the energy exchange between electromagnetic fields and particles.
3.1 collisional shock model (Model 0, 1, and 2) For the pre-shock gas (denoted by the subscript '0'), we set Tj,0 = T0 = 3 × 10 4 K with assuming the collisional ionization equilibrium and temperature equilibrium. In this condition, the fraction of the neutral atoms is ∼ 1.3 × 10 −2 in the number. For the downstream values (denoted by the subscript '2') of the charged particles, assuming a negligibly small magnetic field at the upstream region (or a parallel shock), we consider the total flux conservation laws as where the total pressure is P = j Pj, and the total internal energy is ε = j εj , respectively. The mass density of the species j is ρj = mj nj , where mj is the particle mass, and the total mass density is ρ = j ρj. The compression ratio rc and total pressure jump xc are derived as where Ms ≡ v0/ γP0/ρ0 is the sonic Mach number defined by the total pressure and mass density. For each species j, we assume the flux conservation laws as ρj,0v0 2 + Pj,0 = ρj,2v2 2 + P2 ρj,0 ρ0 , where we have assumed that the downstream ion velocities are the same as each other (vj,2 = v2), and that the downstream internal energy εj,2 = (ρj,0/ρ0)ε2 and pressure Pj,2 = (ρj,0/ρ0)P2 are proportional to the upstream kinetic energy ρj,0v0 2 /2. The downstream temperature of the species j, kTj,2 = Pj,2/nj,2 = (P2/ρ2)mj, is derived as where v ′ 0 = v0 −v2 is the upstream velocity measured in the downstream rest. In the strong shock limit with γ = 5/3, we obtain the relation of (3/2)kTj,2 = mjv ′ 0 2 /2 indicating that the upstream coherent motion of the particles is completely randomized due to the shock transition. The temperature ratio of the species j to k is equal to their ion mass ratio, Tj/T k = mj/m k . This corresponds that the widths of the Maxwell velocity distribution function of each species are the same. The neutral particles do not form the shock structure because they do not interact with the electromagnetic fields. Then, for the neutral particles, we approximately adopt εj,2 = ρj,0v0 2 /2 + εj,0. We will refer this collisional shock model to Model 0. Table 1. Literature on the recombition rate. The superscript * denotes that we use the Mewe's formula for the radiative recombination (Mewe et al. 1980a;Mewe et al. 1980b).
Nahar ( Nahar (1995) To investigate the effects of the electron heating around the shock transition region, we parameterize the energy exchange between protons and electrons as where the subscript 'p' denotes the proton. The degree of equilibrium is represented by the parameter feq that is related to the temperature ratio as follows Introducing β ≡ Te,2/Tp,2, we obtain We consider the cases of β = 0.01 (Model 1) and β = 0.1 (Model 2). Although the electron might exchange its internal energy with other ions, we omit this possibility for simplicity. Complete treatments of the electron heating around the shock may need to solve the nature of electromagnetic fields and wave-particle interactions in detail, and this issue is unsettled yet (e.g., Ohira & Takahara 2007;Ohira & Takahara 2008;Rakowski et al. 2008;Laming et al. 2014).

collisionless shock model (Model 3, 4, and 5)
Here we consider another way of giving a shock transition with the CR acceleration. We assume that a part of shock kinetic energy is consumed for the generation of the CRs and the amplification of the magnetic field. The generated magnetic field is assumed to be disturbed (not an ordered field). In this model, we consider the randomization of the particles incoming from the far upstream region at the shock transition region. The randomization is quantified by the entropy. We notice that the 'randomization' results in a more isotropic particle distribution downstream than the pre-shock one (measured in the shock rest frame). In the collisional shocks, it may be called as the 'thermalization', however, the particle distribution may deviate from the Maxwellian in the collisionless shocks. We use the 'randomization' for both collisional and collisionless shocks in the following.
Conservation laws of total mass and momentum flux can be written as where the generated (turbulent) magnetic-field strength is δB. We regard that the field with δB has a coherent length scale (injection scale of turbulence) much larger than the Larmor radius of the thermal particles with a velocity of ∼ v0 and that the turbulence cascades to the smaller scale. The disturbances associated with the field are assumed to randomize the thermal particles by the wave-particle interactions. The CR pressure of the species j is defined as Pcr,j and the total CR pressure is Pcr = j Pcr,j. The net momentum flux of escaping CRs is Fesc < ∼ ρ0v0 3 /3c. We neglect the total flux and the flux of each species j in this article (Fesc = 0 and Fesc,j = 0). For each species denoted by the subscript j, we give the flux conservation laws as ρj,0v0 2 + Pj,0 = ρj,2v2 2 + Pj,2 where we assume a contribution of the species j for the magnetic field amplification and nonthermal pressure is proportional to the upstream kinetic energy ρj,0v0 2 /2. The compression ratios of the species j are equal (v2,j = v2). From these conservation laws, we can derive the relation between the compression ratio rc,j = ρj,2/ρj,0 = ρ2/ρ0 ≡ rc and the jump of internal energy xc,j ≡ εj,2/εj,0 as or where ξB ≡ δB 2 /(4πρ0v0 2 ), ξcr ≡ Pcr/(ρ0v0 2 ), and Ms,j = v0/ γPj,0/ρj,0 is the sonic Mach number defined by the pressure and density of the species j. Thus, once another relation between rc and xc,j is found, we can derive the shock jump condition with given ξB and ξcr. As usual, the energy flux conservation is considered by modeling the magnetic field amplification and the injection rate of nonthermal particles. Since we focus on the downstream ion temperature, we consider the randomization process of thermal ions rather than modeling the behavior of nonthermal particles. Thus, we consider the entropy production of the thermal particles explicitly. The entropy of the species j per unit mass is defined as where dQj is the energy transferred from electromagnetic fields to the internal energy of the species j due to the shock transition, and Mj = Njmj is the total mass of the species j within the fluid parcel. 4 Note that dQj = dEj + Pj dV indicates only the increment of the internal energy rather than the total kinetic energy of the thermal particles (a sum of the bulk motion and the random motion). The upstream total kinetic energy of the thermal particles is divided into δB and Pcr. Substituting dQj = dEj +PjdV to the equation (43), and using the relation of dεj = d(ρjej ) = ejdρj + ρjdej, where ej ≡ Ej/Mj , we can derive the change of the internal energy per unit volume as Note that we have presumed that Nj is constant during the shock transition. Thus, we obtain the entropy jump before and after the shock transition, ∆sj = sj,2 − sj,0 as Then, the jump conditions are derived by estimating ∆sj independently from the equation (45). Since the SNR shock is expected to be formed by the wave-particle interactions, the transferred energy in total ∆Qj may be around ∼ Jj · E∆tj, where Jj is the electric current of species j. The electric field measured in the comoving frame of the ions is E. ∆tj is a time taking the shock transition. We estimate each value as Jj ∼ qj Nj ṽj , E ≡ |E| ∼ ( ṽj /c)δB, and ∆tj ∼ mj c/qj δB, where qj is the electric charge of the species j, ṽj = v0 + 2kT0/mj is the mean kinetic velocity of the species j, and c is the speed of light, respectively. The transition time scale is assumed to be comparable with an inverse of the cyclotron frequency. In a hybrid simulation solving the particle acceleration (e.g., Ohira 2016b), the shock jump seems to occur at a very small length scale despite a significant amplification of turbulent magnetic fields at the 'upstream' region (it may correspond to a shock precursor region in our situation). We regard that the randomization of particles resulting in the shock transition mainly occurs at such a very small length scale. Thus, we assume the entropy production due to the shock transition as where we suppose Tj ∼ Tj,2. Substituting the equition (46) to the equation (45), we obtain the relation between rc and xc,j as We solve this equation setting Pcr, δB and Ms,j = v0/ γPj,0/ρj,0 with the equation (41) to derive xc,j in the case of the proton by regarding that the most abundant ions form the shock structure. Then, the compression ratio rc is derived from the equation (41) by using the derived xc,p. The downstream pressures of the other species j are derived from the equation (42) by using the derived compression ratio rc. Note that if we supposed small δB and Pcr, the resultant downstream values would be different from the case of the collisional shock (Model 0) reflecting the different randomization process. In this paper, we consider the most efficiently accelerating CR shock feasible. In such a situation, the CR pressure is a practical function of δB because of the energy budget of the shock. The upstream kinetic energy is divided into the thermal energy, the magnetic field, and the CRs. The fraction of the thermal energy is given by the entropy production. The fraction of the magnetic field is treated as a free parameter. Thus, the remaining energy is divided into the CRs. The left panel of figure 1 shows f = f (xc,j ) (upper part), rc = rc(xc,j) (middle part), and ∆sj (xc,j)/∆sj,ncr (lower part) for the proton with γ = 5/3. We set parameters as ξcr = 0.5 (purple line), ξcr = 0.3 (black line), and ξcr = 0 (green line) with fixed values of v0 = 4000 km s −1 , T0 = 3 × 10 4 K, and 1/ √ ξ B = v0/(δB/ √ 4πρ0) = 3. The entropy jump ∆sj,ncr for the case without the CRs (Model 0, and thus the case of the usual collisional shock) is derived from (γ − 1) mj∆sj,ncr = ln xc,j,ncr − γ ln rc,ncr, where xc,j,ncr and rc,ncr are given by Model 0. The right panel of figure 1 shows the sets of ξcr, xc,j, and rc satisfying f = 0. The function f (xc,j) shows two solutions for a given δB depending on Pcr. Although we don't have a precise explanation about these two solutions that may require a full understanding of the ion heating by the kinetics theory, we may be able to interpret them from resultant downstream values. Let us consider the case of ξcr = 0 in which ∆sj/∆sj,ncr ≈ 1 around each solution. We will refer to the solution giving xc,j/Ms,j 2 ≈ 0.17 and rc ≈ 1.27 as 'solution A', while we will refer to the other solution giving xc,j/Ms,j 2 ≈ 1.28 and rc ≈ 8.31 as 'solution B'. The resultant temperature (Tj,2/T0 = xc,j/rc ≈ 0.1mv0 2 /γkT0) is almost the same as each other. This means that the speed of particles' random motion is almost the same as each solution. On the other hand, the difference in the compression ratios indicates that the speed of particles' bulk motion is significantly different from each other. In a collisional shock in the strong shock limit, the downstream temperature satisfies (3/2)kT2 = mv ′ 0 2 /2, where v ′ 0 = v0 − v2 is the upstream velocity measured in the downstream rest frame and we use γ = 5/3. This might mean that since our shock consumes its energy for the generation of the nonthermal components, the random motion speed measured in the downstream rest frameṽ ′ R ≡ 3kTj,2/mj should be equal or smaller than v ′ 0 = v0 − v2 for the solution representing the shock transition (i.e.ṽ ′ R /v ′ 0 ≤ 1). Solution A gives the speed asṽ ′ R /v ′ 0 ≃ 2.3, while solution B givesṽ ′ R /v ′ 0 ≃ 0.6. Hence, solution B may correspond to the shock transition. Solution A should be rejected because it does not satisfy the energy flux conservation law.
When ξcr becomes large, the two solutions approach with each other, coinciding at ξcr ≃ 0.3 (multiple roots), and finally, the solution vanishes. The multiple roots (ξcr = 0.3) giveṽ ′ R /v ′ 0 ≃ 0.7 and ∆sj/∆sj,ncr ≃ 0.93. Thus, the multiple roots may represent the shock transition giving the maximum Pcr feasible in our shock model. In this article, we set the maximum ξcr to compare the no CR cases with the case of extremely efficient CR acceleration. The maximum ξcr is derived from the multiple roots of f = 0 with given ξB.
For the case of v0 = 4000 km s −1 and T0 = 3 × 10 4 K with given 1/ √ ξB = 3, we obtain the maximum acceptable CR production ξcr ≃ 0.3, ∆sj/∆sj,ncr ≃ 0.92-0.95 depending on mj, rc ≃ 3.29, and kTp,2 ≃ 14.4 keV. Note that in the case of Model 0 (the usual collisional shock case), we obtain rc = 4.00 and kTp,2 = 31.3 keV. The fraction of the CRs ξcr = 0.3 seems to be reasonable for the SNR shocks as sources of Galactic CRs. From the subtraction of the energy fluxes of the thermal particles at the far upstream and downstream, we can regard that roughly 50 % of the upstream energy flux is transferred to the nonthermal components. The fraction of magnetic pressure 1/ √ ξB = 3 corresponds to magnetic-field strength of δB ≃ 611 µG(v0/4000 km s −1 )(np,0/1 cm −3 ) 1/2 which is consistent with estimated strength from X-ray observations of young SNRs (e.g., Vink & Laming 2003;Bamba et al. 2005;Uchiyama et al. 2007). Thus, our parameter choice of 1/ √ ξB = 3 can be reasonable to adopt our model to the young SNR shocks.
Here we consider about the choice of the maximum ξcr. In the case of the collisional shock formed by the hardsphere collisions, for example, the collisions result in one of the most efficient randomizations of particles. Thus, the collisional shock can 'easily' dissipate its kinetic energy within the mean collision time. In the collisionless plasma, such efficient randomization process is absent. The particles in the plasma tend to behave as 'nonthermal' particles resulting in a generation of electromagnetic disturbances by themselves. The collisionless shock is formed by the self-generated disturbances so that almost all particles become thermal particles. Although the number of the nonthermal particles is very smaller than the number of the thermal particles, the efficient randomization caused by the nonthermal particles is required to form the collisionless shock. Our choice of the maximum ξcr corresponds that the effect is minimized per one nonthermal particle. 10 100  Figure 2 shows the maximum ξcr derived from f = 0 as a function of 1/ √ ξB for Ms,p = 197. The fraction ξcr drops around 1/ √ ξB < ∼ 3 but is flattened for 1/ √ ξB > ∼ 3. This depletion of the maximum ξcr is qualitatively obvious in terms of the energy budget of the shock; the upstream kinetic energy is divided into the thermal components, Pcr and δB. The fraction of δB is a given parameter. The fraction of the thermal components and the maximum fraction of Pcr are derived from the entropy production. We will refer to this model with 1/ √ ξB = 3 and the maximum ξcr as Model 3.  Figure 3 shows the results of downstream ion temperatures divided by 2Z (i.e., the particle mass in atomic unit) for Model 0 and Model 3 with v0 = 4000 km s −1 and T0 = 3 × 10 4 K. The reduced temperatures kTZ,z/2Z of Model 3 do not depend on the particle mass, indicating that the temperature ratios between the ions are equal to their ion mass ratio. Such mass-proportional ion temperatures are observed at SN 1987A (Miceli et al. 2019). The temperature jump Tj,2/T0 is given by xc,j/rc ∼ Ms,j . The relation of xc,j/rc ∼ Ms,j is also implied by the condition of f = 0. Thus, Model 3 predicts that the ion temperature ratio is given by the mass ratio, similar to the case of Model 0. Note that Model 1 and Model 2 give ion temperatures almost the same as Model 0. On the other hand, kTZ,z/2Z of Model 3 is smaller than the case of Model 0 by a factor of 2 due to the generations of the nonthermal components.
The existence of more than two solutions is usually seen in the CR accelerating shock model (e.g., Drury & Voelk 1981;Vink et al. 2010;Vink & Yamazaki 2014, and references therein). The unphysical solution like Solution A of our model, which does not satisfy the energy flux conservation law, also exists in previous studies. The essential difference between our model and previous studies is the treatment of the randomization in the shock transition. In the previous studies, the randomization process, which determines the downstream thermal energy, may be implicitly chosen to satisfy the flux conservation laws with assumed parameters (Pcr, energy flux of escaping CRs, etc.). Vink & Yamazaki (2014) also derived a critical sonic Mach number Macc = √ 5 below which the particle acceleration should not occur. In our model, a similar sonic Mach number may be derived from conditions of ∆sj ≤ ∆sj,ncr and Tj,2 ≤ Tj,2,ncr, where Tj,2,ncr is given by Model 0 (the equation 32). The former states that the generated entropy in the collisionless shock should be smaller than in the case of collisional shocks. The latter states that the downstream temperature should be smaller than the case of adiabatic, collisional shocks without the CRs. Note that the entropy and temperature must be determined independently to derive the density or pressure in thermodynamics. In other words, the conditions ∆sj ≤ ∆sj,ncr and Tj,2 ≤ Tj,2,ncr are independent with each other. From the equations (48) and (46), and using the relation of mjv0 2 /kT0 = (ρj,0/ρ0)γMs 2 , we can derive Tj,2,ncr T0 ≥ Tj,2 T0 ≥ ρj,0 ρ0 γ(γ − 1)Ms 2 ( ṽj /v0) 2 ln (Tj,2,ncr/T0) − (γ − 1) ln rc,ncr , where Tj,2,ncr T0 = ρj,0 ρ0 γMs 2 rc,ncr 1 − 1 rc,ncr and rc,ncr is given by the equation (27). The critical Mach number Ms,acc is given when the equal sign of the inequality holds. Regarding ṽj ≃ v0 and ρj/ρ0 ≃ 1 for simplicity, we obtain the numerical value of Ms,acc ≃ 16.34 above which we can find sets of ξcr and ξB satisfying the inequality (49). The larger critical Mach number than that derived by Vink & Yamazaki (2014) may be due to the difference in the assumed randomization process. However, the value of Ms,acc may also depend on the Alfvén Mach number, whose effects are not studied in this paper. When the sonic Mach number decreases due to a shock deceleration, the effects of the mean magnetic field at the far upstream region can be important. The shocks with a lower Mach number are seen in the solar wind at coronal mass ejection events, clusters of galaxy, and so on. In predictions of the accelerated paritcles amount in such cases, we shoud include the pre-existing ordered magnetic field to the flux conservation laws and evaluation of J · E term, differing from the current approach. We will study a general critical Mach number with more elaborate models in future work.
Finally, we parameterize the electron heating for the case of the extremely efficient CR acceleration as where ε p,2,Model3 and ε e,2,Model3 are the internal energy calculated by Model 3. Here we have supposed an additional energy transfer: the internal energy of the thermal protons is transferred to the thermal electrons. The fraction of the transferred internal energy is written by the temperature ratio of β = Te,2/Tp,2 as feq = β(ne,0/np,0) − (ε e,2,Model3 /ε p,2,Model3 ) β(ne,0/np,0) + 1 .
(52) Laming et al. (2014) pointed that the electron temperature can be significantly large (β ∼ 0.1) when the shock accelerates the CRs efficiently. We calculate the cases of β = 0.01 (Model 4) and β = 0.1 (Model 5) in this paper. Table 2 shows a summary of our shock models.

evolution track of the downstream ionization balance and temperatures
Here we show the results of the ionization balance and temperature relaxation in the downstream region, omitting the effects of the expansion (dV /dt = 0) as a reference. For convenience, we introduce dτ = ndt, where n is the total number density, so that where we use ρ =const. Figures 4 (for He, C, and N), 5 (for O, Ne, and Mg), and 6 (for Si, S, and Fe) show nZ,z/nZ and kTZ,z for Model 0 with the shock velocity of v0 = 4000 km s −1 , where nZ = z nZ,z is the total number density of the atoms with the atomic number Z. Note that τ = ndt ≃ nt because of the small neutral fraction. Here we display the ion temperatures for the most abundant species among its ionic charge.
The evolution tracks of nZ,z/nZ and TZ,z for other models are not so different from the case of Model 0. In the case of a higher electron temperature (Model 1 and Model 2), the ions are quickly ionized. Figure 7 shows the electron temperatures for Model 0, 1, 2, and 3 with v0 = 4000 km s −1 . The relation between Model 4 and Mode 1 (Model 5 and Model 2) is similar to that of Model 3 and Model 0. The ionization balance nZ,z/nZ becomes the same in each model after the electron temperature coincides. Note that the electron temperature increases within a column density scale of ntV sh ∼ 10 14 cm −2 (nt/10 6 cm −3 s)(V sh /4000 km s −1 ). This col-  logτ (cm -3 s) umn density scale is comparable to the size of the Hα emission region (e.g., Shimoda & Laming 2019). Therefore, to study the electron heating at the shock, the Hα observation may be better than the X-ray line observations. In the case of a lower ion temperature due to the production of Pcr and δB (Model 3), the temperature equilibrium is achieved at a smaller nt (e.g., the temperature of Fe is equal to the proton's at nt ≃ 2 × 10 11 cm −3 s) because the relaxation time of the Coulomb collision depends on T 3/2 (Spitzer 1962). Note that the lower electron temperature results in a lower ionization state at given τ ≃ nt.  When the effects of the expansion become important, we cannot characterize the evolution only by τ ≃ nt and we should introduce parameters to describe the expansion of SNRs and the observed position r/R sh (tage). Here we set ρ0 = 4.08 × 10 −2 mp, tage = 1836 yr, and V sh (tage) = 3000 km s −1 for example. This parameter set will be used in comparisons of our Model to the SNR RCW 86 (discuss later in Sect. 5). Figure 8 shows the downstream ionization structure of He, C, N (top panels), O, Ne, Mg (middle panels), Si, S, and Fe (bottom panels). The fluid parcel currently at r/R sh (tage) = 0.8 crossed the shock at the time of t * when the shock velocity was V sh (t * ) = 5094 km s −1 for Model 0, 1, and 2 (4565 km s −1 for Model 3, 4, and 5). Since the compression ratio depends on whether the CRs exist, the shock transition time t * , shock velocity V sh (t * ), and Te(t * ) are different from each model for the fluid parcel currently at r/R sh (tage). The evolution of the ionization balance is similar to the case of the plane-parallel shock until t ′ ∼ 10 9 -10 10 s ∼ tage. The cooling due to the expansion becomes important at t ′ ∼ tage. The ion temperatures decrease before the ions are well ionized due to the expansion (decreasing of the density, ion temperature, and electron temperature). Figure 9 shows the electron temperatures for Model 0 (black dots), 1 (purple dots), 2 (green dots), 3 (red dots), 4 (orange dots), and 5 (blue dots).

Synthetic Observations
In this section, we perform synthetic observations of the shocked plasma considering the effects of turbulence for the case of the SNR RCW 86. Since we do not calculate the overall spectrum of the emitted photons which needs enormous calculations about emission lines, we mainly es-timate the line shape.
The SNR RCW 86 is one of the best targets for the study of the CR injection via the ion temperatures because the shells of the SNR show different thermal/nonthermal features from position to position (Bamba et al. 2000;Borkowski et al. 2001;Tsubone et al. 2017). The RCW 86 is considered as a historical SNR of SN 185 (Vink et al. 2006). Thus, we set tage = 1836 yr. Along the northeastern shell of the RCW 86, the dominant X-ray radiation changes from thermal to synchrotron emission (Vink et al. 2006). The thermal emission-dominated region is referred to 'E-bright' region, and the synchrotron one is referred to 'NE' region. The ionization age at NE is estimated as τ = (2.25 ± 0.15) × 10 9 cm −3 s though this estimate potentially contains errors due to the lack of the thermal continuum emissions (Vink et al. 2006). The E-bright region is fitted by two plasma components: (i) τ = (6.7±0.6)×10 9 cm −3 s, and (ii) τ = (17 ± 0.5) × 10 9 cm −3 s. Both E-bright and NE show clear O VII Heα and Ne IX Heα line emissions. From the width of the synchrotron-emitting region (NE), the magnetic-field strength is estimated as ≈ 24 ± 5 µG (Vink et al. 2006). Yamaguchi et al. (2016) measured proper motions around these regions (not exactly the same regions) as v0 = 720 ± 360 km s −1 (E-bright), v0 = 1780 ± 240 km s −1 (upper part of NE referred to 'NE b '), and v0 = 3000 ± 340 km s −1 (lower part of NE referred to 'NE f '). In the case of Model 3, the fractions of CR pressure ξcr become ξcr,720 ≃ 0.14 for v0 = 720 km s −1 , ξcr,1780 ≃ 0.24 for v0 = 1780 km s −1 , and ξcr,3000 ≃ 0.28 for v0 = 3000 km s −1 , respectively. If we simply suppose ρ0 = (τ /tage)mp and adopt 1/ √ ξB = 3, the CR pressure and δB of each region becomes Pcr,720 ∼ 0.2 keV cm −3 and δB720 ∼ 51 µG, Pcr,1780 ∼ 0.3 keV cm −3 and δB1780 ∼ 55 µG, and Pcr,3000 ∼ 1.1 keV cm −3 and δB3000 ∼ 93 µG, where we adopt τ = 12.0 × 10 9 cm −3 s for the E-bright region as an average of the two components and τ = 2.25 × 10 9 cm −3 s for the NE region, respectively. If we adopt v0 = 360 km s −1 for the E-bright region, we obtain ξcr,360 ∼ 2.9 × 10 −2 . The thermal-dominated E-bright region results from the higher density than the density at the NE region. The magnetic-field strength δB is the almost same as one another. Note that Vink et al. (2006) estimated the electron density at the E-bright region as ∼ 0.6-1.5 cm −3 from the emission measure with assuming the volume of the emission region. Our model predicts the downstream density as rcρ0/mp ≈ 0.55 cm −3 for the E-bright region with v0 = 720 km s −1 that is consistent with the previous estimate. For the NE region, the number density is not well constrained because of the lack of the thermal continuum component. Thus, our choice of model pa-  rameters can be consistent with the observations of the RCW 86. In the following, we apply our model to the NE region setting the parameters as tage = 1836 yr, V sh (tage) = 3000 km s −1 , and ρ0/mp = τ /tage = 4.08 × 10 −2 cm −3 , where τ = 2.25 × 10 9 cm −3 s is used. We suppose that the downstream region from r = R obs = 0.8R sh (tage) to r = R sh (tage) is observed. Then, our model supposes that the expansion follows the Sedov-Taylor model during a time of ∆t ≥ tage − t * (R obs ) = {1 − (R obs /R sh ) rc } tage ≈ 0.6tage, where rc = 4 and the equation (11) are used. If the RCW 86 expands with a velocity of ∼ 10 9 cm s −1 on average before entering the Sedov-Taylor stage, we effectively assume the radius at the transition time of t0 ≈ 0.6tage as R0 ∼ 10 9 cm s −1 × 0.6tage ∼ 11 pc. Then, the radius at the current time is R sh (tage) ∼ R0(1/0.6) 2/5 ∼ 13.5 pc which can be consistent with the actual radius of ∼ 15 pc (the distance is assumed as 2.5 kpc). Figure 10 shows the radial profile of the electron temperature at t = tage for Model 0 (black solid line), Model 1 (purple dots), Model 2 (green broken line), Model 3 (red solid line), Model 4 (orange dots), and Model 5 (blue broken line). To reproduce the bright O VII Heα, a relatively high electron temperature is preferred in terms of the excitation (∼ 1 keV, see also Vink et al. 2006), though it is degenerating by the number density uncertainty. Note that the excitation rate is C l,u ∝ exp(−E ul /kTe)/ √ Te and E u,l ≃ 0.574 keV for O VII Heα. Thus, we mainly consider Model 2 (β = Te,2/Tp,2 = 0.1 without the CRs) and Model 5 (β = 0.1 with the CRs). Model 5 predicts kTe(r) ≃ 0.5 keV ≃ E u,l therefore the predicted O VII Heα line would be the brightest among the models. Figure 11 shows the radial profile of the oxygen abundance nZ,z/nZ for Model 2 (left panel) and Model 5 (right panel). The O VII abundance (orange) is large.
Oxygen Note that the other models (e.g., Model 0) also result in a large O VII abundance. Model 2 predicts the smaller abundance of O VII than the case of Model 5 because the higher electron temperature results in a faster ionization. The temperature of O VII is approximately kTZ,z(r) ≈ 250 keV × [r/R sh (tage)] for Model 2 and kTZ,z(r) ≈ 140 keV × [r/R sh (tage)] for Model 5.
We estimate the line emission as follows: the observed specific intensity per frequency Iν at the sky position X from the center of the SNR is calculated as where L = √ R sh − X 2 . The position along the line of sight is Z so that r = √ X 2 + Z 2 . vZ (r) = (Z/r)v(r) is the line of sight velocity. The probability distribution function of the turbulence G is assumed to be a Gaussian as where v turb is a typical turbulent velocity and KZ ≡ (1/2)mZ v turb 2 . Note that wt is the variable for the integration. In this paper, we assume that the intensity of the turbulence is proportional to the proton sound speed as v turb (r) = δ γkTp(r)/mp. Supposing the incompress- g olumn density, Model 5 I ν , Model 5 Column density, Model 2 I ν , Model 2 ible turbulence is driven in the downstream region (see Shimoda et al. 2018a), we calculate the case of δ = 0.5 and the case without the turbulent Doppler broadening δ = 0 for a comparison. The emissivity of the line is given by where we have neglected the cascade from the higher excitation levels. The line profile function is defined as where ν ′ 0 is the frequency of the line measured in the rest frame of the atom. Then, we obtain where MZ,z 2 ≡ KZ/kTZ,z = (γδ 2 /2)(mZ/mp)(Tp/TZ,z). The line shape is broadened by the bulk Doppler effect (1 + vZ /c) and the turbulent Doppler effect 1 + MZ,z 2 . Figure 12 shows Iν (X )/Iν(0.8R sh ) at the line center for Model 5 (solid lines) and Model 2 (dots). We also display profiles of the column density of O VII (green). The difference between the column density profile and the intensity profile results from the excitation. The spatial variation of the electron temperature is relatively less important in this case because the excitation rate depends on exp(−E ul /kTe)/ √ Te that is not so sensitive on Te unless kTe ≪ E ul . Figure 13 shows  . 13. The calculated O VII Heα line with δ = 0.5 for Model 5 (blue solid line) and Model 2 (green solid line). We also display the results of Model 0 (black dots), Model 1 (purple dots), Model 3 (red dots), and Model 4 (orange dots). We assume the distance of the RCW 86 is 2.5 kpc and the observed area is 0.2R sh × 0.2R sh , where R sh = 15.27 pc.
Model 5 (blue solid line) and Model 2 (green solid line) derived from IνdX with δ = 0.5. We assume the distance of the RCW 86 as 2.5 kpc (Yamaguchi et al. 2016) and the observed area as 0.2R sh ×0.2R sh , where R sh = 15.27 pc. We also display the results of Model 0 (black dots), Model 1 (purple dots), Model 3 (red dots), and Model 4 (orange dots). The results show a good agreement with the observed photon counts ∼ 0.15 counts s −1 keV −1 (Vink et al. 2006). Table 3 shows a summary of the calculated O VII Heα line. The derived temperatures reflect the effects of the efficient CR acceleration. From the comparison of δ = 0.5 to δ = 0, the turbulent Doppler broadening results in the higher observed temperatures by a factor of ∼ 1.05. The degree of the broadening can be estimated as 1 + MZ,z 2 ≈ 1.1 for δ = 0.5 with approximating Tp/TZ,z ≈ mp/mZ. Since the observed line consists of multiple temperature populations, and since a higher temperature population less contributes around the line center, a lower temperature population is accentuated around the line center. The contribution of the higher temperature population appears far from the line center like a 'wing'. If we measure the temperature using the full width at the e-folding scale, the difference in the derived temperatures becomes large. Hence the observed FWHM is smaller than that expected from 1 + MZ,z 2 .
The RCW 86 also shows bright Ne IX Heα however, our model predicts a faint Ne IX Heα emission (the intensity is smaller than a tenth of O VII Heα intensity). The line intensity also depends on the ion abundance. In this paper, we use the solar abundance that reflects the condition of our galaxy ≃ 4.6 Gyr ago. Moreover, De Cia et al. (2021) found large variations of the chemical abundance of the neutral ISM in the vicinity of the Sun over a factor of 10 (Si, Ti, Cr, Fe, Ni, and Zn they analyzed). Their findings  imply that the gaseous matter is not well mixed. The predicted faint Ne IX-Heα might reflect a different abundance pattern from the solar abundance pattern. 160.6 keV (10.0 keV) 153.8 keV (9.62 keV) 4 160.6 keV (10.0 keV) 154.2 keV (9.63 keV) 5 157.7 keV (9.85 keV) 152.5 keV (9.53 keV) Figure 14 represents the line shape with 5 eV resolution for δ = 0.5. We additionally show O VII Lyα, Ne IX Heα, and Ne X Lyα. Since the widths of the particle distribution function are almost the same as each other for nt ∼ 10 9 -10 11 cm −3 s, the observation of lines at higher photon energy is better to resolve the line width. Note that the observed O VII Heα and Ne IX Heα are bright compared to the continuum emission (Vink et al. 2006). The energy resolution of XRISM's micro-calorimeter Resolve is sufficient to distinguish whether the SNR shock accelerates the CRs (Model 5) or not (Model 2).

Summary and Discussion
We suggest the novel collisionless shock jump condition, which is given by modeling each ion species' entropy production at the shock transition region. As a result, the amount of the downstream thermal energy is given. The magnetic-field amplification driven by the CRs is assumed. For the given strength of the amplified field, the amount of the CRs is constrained by the energy conservation law. The constrained amount of the CRs can be sufficiently large to explain the Galactic CRs. The ion temperature is lower than the case without the CRs because the upstream kinetic energy is divided into the CRs and the amplified field. The strength of the filed around the shock transition region is assumed to be 1/ √ ξB = v0/(δB/ √ 4πρ0) ≃ 3.
Downstream developments of the ionization balance and temperature relaxation are also calculated. Using the calculated downstream values, we perform synthetic observations of atomic lines for the SNR RCW 86, including the Doppler broadening by the turbulence. Our model predictions can be consistent with the previous observations of the SNR RCW 86, and the predicted line widths are sufficiently broad to be resolved by the XRISM's microcalorimeter. Future observations of the X-ray lines can distinguish whether the SNR shock accelerates the CRs or not from the ion temperatures.
Our shock model constrains the maximum fraction of the CRs depending on the shock velocity, the upstream density, and the sonic Mach number (see figure 2). Since the SNR shock decelerates gradually, we can predict a history of the CR injection and related nonthermal emissions, especially the hadronic γ-ray emissions. Although the injection history of the CRs is essential to estimate the intrinsic injection of the CRs into our galaxy per one supernova explosion, this issue currently remains to be resolved (e.g., Ohira et al. 2010;Ohira & Ioka 2011). The injected CRs will contribute to the dynamics of the ISM as a pressure source, leading to a feedback effect on the star formation rate, for example (e.g., Hopkins et al. 2018;Girichidis et al. 2018;Shimoda & Inutsuka 2021). The origin of γ-ray emissions in the SNRs is also unsettled, whether the hadronic origin or leptonic origin (Abdo et al. 2011, but see Fukui et al. 2021). We will study them in a forthcoming paper.
For distinguishing the case of extremely efficient CR acceleration (Model 3) from the case of no CRs (Model 0), a comparison of the FWHM to other values is required in general (e.g., the difference between the ionization states, the shock velocity, and so on). The FWHM of Model 3 becomes smaller than Model 0 at a given shock velocity and nt, and abundant ions of Model 3 tend to be less ionized than in the case of Model 0 because of the lower electron temperature. The lower electron temperature and lower ionization states of Model 3 may result in a different photon spectrum from the case of other Models, especially the equivalent widths, recombination lines, Augér transitions due to the inner shell ionization, and so on. We will attempt further investigations by calculating the overall photon spectrum in future work.
The line diagnostics of the thermal plasma of young SNRs on the effect of CR acceleration will be a good science objective for the XRISM mission (Tashiro et al. 2020), which will provide high-resolution X-ray spectroscopy. Since the micro-calorimeter array is not a distributed-type spectroscope like grating optics on Chandra and/or XMM-Newton, the Resolve onboard XRISM (Ishisaki et al. 2018) can accurately measure the atomic-line profiles in the Xray spectra from diffuse objects like SNRs. The XRISM will have the energy resolution of 7 eV (as the design goal), and the calibration goals on the energy scale and resolution are 2 eV and 1 eV, respectively (Miller et al. 2020). Therefore, the line broadening values from multiple elements with/without CRs in figure 3 can be distinguished by XRISM. Another importance of XRISM is the wider energy coverage, with which atomic lines not only from light elements (C, N, O, etc.) but also Fe will be measured. So, the intensity of the turbulence demonstrated in section 5 will be constrained with XRISM. The preparation for the instruments (Nakajima et al. 2020;Porter et al. 2020) and in-orbit operations (Terada et al. 2021;Loewenstein et al. 2020) are proceeding smoothly for the launch in 2022/2023, and several young SNRs, including RCW86 are listed as the target during the performance verification phase of XRISM. 5 We expect to verify our predictions observationally soon.