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At least 541 records · Page 30
All sourcesNASA NTRSDOE OSTI
DOE OSTI2015

Collisionless Weibel shocks and electron acceleration in gamma-ray bursts

A study of collisionless external shocks in gamma-ray bursts is presented. The shock structure, electromagnetic field, and process of electron acceleration are assessed by performing a self-consistent 3D particle-in-cell simulation. In accordance with hydrodynamic shock systems, the shock consists of a reverse shock (RS) and forward shock separated by a contact discontinuity. The development and structure are controlled by the ion Weibel instability. The ion filaments are sources of strong transverse electromagnetic fields at both sides of the double shock structure over a length of 30–100 ion skin depths. Electrons are heated up to a maximum energy ϵ{sub ele}≈√(ϵ{sub b}), where ϵ is the energy normalized to the total incoming energy. Jet electrons are trapped in the RS transition region due to the presence of an ambipolar electric field and reflection by the strong transverse magnetic fields in the shocked region. In a process similar to shock surfing acceleration for ions, electrons experience drift motion and acceleration by ion filament transverse electric fields in the plane perpendicular to the shock propagation direction. Ultimately, accelerated jet electrons are convected back into the upstream.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

PHYSICAL CONDITIONS OF CORONAL PLASMA AT THE TRANSIT OF A SHOCK DRIVEN BY A CORONAL MASS EJECTION

We report here on the determination of plasma physical parameters across a shock driven by a coronal mass ejection using white light (WL) coronagraphic images and radio dynamic spectra (RDS). The event analyzed here is the spectacular eruption that occurred on 2011 June 7, a fast CME followed by the ejection of columns of chromospheric plasma, part of them falling back to the solar surface, associated with a M2.5 flare and a type-II radio burst. Images acquired by the Solar and Heliospheric Observatory/LASCO coronagraphs (C2 and C3) were employed to track the CME-driven shock in the corona between 2–12 R{sub ⊙} in an angular interval of about 110°. In this interval we derived two-dimensional (2D) maps of electron density, shock velocity, and shock compression ratio, and we measured the shock inclination angle with respect to the radial direction. Under plausible assumptions, these quantities were used to infer 2D maps of shock Mach number M{sub A} and strength of coronal magnetic fields at the shock's heights. We found that in the early phases (2–4 R{sub ⊙}) the whole shock surface is super-Alfvénic, while later on (i.e., higher up) it becomes super-Alfvénic only at the nose. This is in agreement with the location for the source of the observed type-II burst, as inferred from RDS combined with the shock kinematic and coronal densities derived from WL. For the first time, a coronal shock is used to derive a 2D map of the coronal magnetic field strength over intervals of 10 R{sub ⊙} altitude and ∼110° latitude.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

SIMULATIONS AND THEORY OF ION INJECTION AT NON-RELATIVISTIC COLLISIONLESS SHOCKS

We use kinetic hybrid simulations (kinetic ions-fluid electrons) to characterize the fraction of ions that are accelerated to non-thermal energies at non-relativistic collisionless shocks. We investigate the properties of the shock discontinuity and show that shocks propagating almost along the background magnetic field (quasi-parallel shocks) reform quasi-periodically on ion cyclotron scales. Ions that impinge on the shock when the discontinuity is the steepest are specularly reflected. This is a necessary condition for being injected, but it is not sufficient. Also, by following the trajectories of reflected ions, we calculate the minimum energy needed for injection into diffusive shock acceleration, as a function of the shock inclination. We construct a minimal model that accounts for the ion reflection from quasi-periodic shock barrier, for the fraction of injected ions, and for the ion spectrum throughout the transition from thermal to non-thermal energies. This model captures the physics relevant for ion injection at non-relativistic astrophysical shocks with arbitrary strengths and magnetic inclinations, and represents a crucial ingredient for understanding the diffusive shock acceleration of cosmic rays.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Amplification of turbulence through multiple planar shocks

We study the amplification of isotropic, incompressible turbulence through multiple planar, collisional shocks, using analytical linear theory. There are two limiting cases we explore. The first assumes shocks occur rapidly in time such that the turbulence does not evolve between shocks. Whereas the second case allows enough time for turbulence to isotropize between each shock. For the latter case, through a quasi-equation-of-state, we show that the weak multishock limit is agnostic to the distinction between thermal and vortical turbulent pressures, like an isotropic volumetric compression. When turbulence does not return to isotropy between shocks, the generated anisotropy—itself a function of shock strength—can feedback on amplification by further shocks, altering choices for maximal or minimal amplification. In addition for this case, we find that amplification is sensitive to the shock ordering. As a result, we map how choices of shock strength can impact these amplification differences due to ordering, finding, for example, shock pairs which lead to identical mean postshock fields (density, temperature, pressure) but maximally distinct turbulent amplification.

Astrophysical fluid dynamics↗
DOE OSTI2021

Shock Propagation and Associated Particle Acceleration in the Presence of Ambient Solar-Wind Turbulence

The topic of this review paper is on the influence of solar wind turbulence on shock propagation and its consequence on the acceleration and transport of energetic particles at shocks. As the interplanetary shocks sweep through the turbulent solar wind, the shock surfaces fluctuate and ripple in a range of different scales. We discuss particle acceleration at rippled shocks in the presence of ambient solar-wind turbulence. This strongly affects particle acceleration and transport of energetic particles (both ions and electrons) at shock fronts. In particular, we point out that the effects of upstream turbulence is critical for understanding the variability of energetic particles at shocks. Moreover, the presence of pre-existing upstream turbulence significantly enhances the trapping near the shock of low-energy charged particles, including those near the thermal energy of the incident plasma, even when the shock propagates normal to the average magnetic field. Pre-existing turbulence, always present in space plasmas, provides a means for the efficient acceleration of low-energy particles and overcoming the well known injection problem at shocks.

, solar wind↗
DOE OSTI2016

Diffusive cosmic ray acceleration at relativistic shock waves with magnetostatic turbulence. II. Influence of a finite downstream medium

The diffusive acceleration of relativistic cosmic rays at parallel shock waves with magnetostatic turbulence and a finite size of the downstream medium is investigated. For ultrarelativistic shock speeds with Lorentz factor Γ{sub 1}≫1, both the differential momentum spectrum at the shock and the volume-integrated momentum spectrum are power-law distribution functions with different spectral indices as compared to the case of an infinitely extended downstream medium. However, the spectral differences are only modest as compared to the case of nonrelativistic shocks. The behavior of the momentum spectrum of shock-accelerated particles depends sensitively on the relativistic shock wave Peclet number G(p)=τ{sub D}(p)/τ{sub C}, i.e., the ratio between the diffusion and convection timescales of cosmic rays to propagate from the shock position to the downstream boundary z {sub 0}. For large values of G(p)≫1 the free-escape boundary has no influence on the effectiveness of particle acceleration, still providing a flat momentum power-law spectrum of the accelerated particles. In the opposite case of small Peclet numbers G(p)≪1 at all momenta, the momentum spectrum at the shock steepens to the greater spectral index ξ{sub 0}=3−s+(3.18/Γ{sub 1}), whereas the volume-integrated momentum spectrum flattens by the same factor 2−s for its power-law spectral index, where s denotes the spectral index of the downstream power spectrum of magnetostatic turbulence. This effectiveness of relativistic shocks in generating flat power-law momentum spectra irrespective of the Peclet number G(p) differs completely from the behavior of nonrelativistic shocks.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

THE INTERACTION OF TURBULENCE WITH PARALLEL AND PERPENDICULAR SHOCKS: THEORY AND OBSERVATIONS AT 1 au

Shocks are thought to be responsible for the amplification of turbulence as well as for generating turbulence throughout the heliosphere. We study the interaction of turbulence with parallel and perpendicular shock waves using the six-coupled-equation turbulence transport model of Zank et al. We model a 1D stationary shock wave using a hyperbolic tangent function and the Rankine–Hugoniot conditions for both a reduced model with four coupled equations and the full model. Eight quasi-parallel and five quasi-perpendicular events in the WIND spacecraft data sets are identified, and we compute the fluctuating magnetic and kinetic energy, the energy in forward and backward propagating modes, the total turbulent energy, the normalized residual energy, and the normalized cross helicity upstream and downstream of the observed shocks. We compare the observed fitted values upstream and downstream of the shock with numerical solutions to our model equations. The comparison shows that our theoretical results are in reasonable agreement with observations for both quasi-parallel and perpendicular shocks. We find that (1) the total turbulent energy, the energy in forward and backward propagating modes, and the normalized residual energy increase across the shock, (2) the normalized cross helicity increases or decreases across the shock, and (3) the correlation length increases upstream and downstream of the shock, and slightly flattens or decreases across the shock.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

STRUCTURE, PROPAGATION, AND EXPANSION OF A CME-DRIVEN SHOCK IN THE HELIOSPHERE: A REVISIT OF THE 2012 JULY 23 EXTREME STORM

We examine the structure, propagation, and expansion of the shock associated with the 2012 July 23 extreme coronal mass ejection. Characteristics of the shock determined from multi-point imaging observations are compared to in situ measurements at different locations and a complex radio type II burst, which according to our definition has multiple branches that may not all be fundamental-harmonic related. The white-light shock signature can be modeled reasonably well by a spherical structure and was expanding backward even on the opposite side of the Sun. The expansion of the shock, which was roughly self-similar after the first ∼1.5 hr from launch, largely dominated over the translation of the shock center for the time period of interest. Our study also suggests a bow-shock morphology around the nose at later times due to the outward motion in combination with the expansion of the ejecta. The shock decayed and failed to reach Mercury in the backward direction and the Solar Terrestrial Relations Observatory B ( STEREO B ) and Venus in the lateral directions, as indicated by the imaging and in situ observations. The shock in the nose direction, however, may have persisted to the far outer heliosphere, with predicted impact on Dawn around 06:00 UT on July 25 and on Jupiter around 23:30 UT on July 27 by a magnetohydrodynamic model. The type II burst shows properties generally consistent with the spatial/temporal variations of the shock deduced from imaging and in situ observations. In particular, the low-frequency bands agree well with the in situ measurements of a very low density ahead of the shock at STEREO A .

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

Particle Acceleration in Two Converging Shocks

Observations by spacecraft such as ACE , STEREO , and others show that there are proton spectral “breaks” with energy E {sub br} at 1–10 MeV in some large CME-driven shocks. Generally, a single shock with the diffusive acceleration mechanism would not predict the “broken” energy spectrum. The present paper focuses on two converging shocks to identify this energy spectral feature. In this case, the converging shocks comprise one forward CME-driven shock on 2006 December 13 and another backward Earth bow shock. We simulate the detailed particle acceleration processes in the region of the converging shocks using the Monte Carlo method. As a result, we not only obtain an extended energy spectrum with an energy “tail” up to a few 10 MeV higher than that in previous single shock model, but also we find an energy spectral “break” occurring on ∼5.5 MeV. The predicted energy spectral shape is consistent with observations from multiple spacecraft. The spectral “break,” then, in this case is caused by the interaction between the CME shock and Earth’s bow shock, and otherwise would not be present if Earth were not in the path of the CME.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Planar Collisionless Shock Simulations with the Semi-implicit Particle-in-cell Model FLEKS

This study investigates the applicability of the semi-implicit particle-in-cell code FLexible Exascale Kinetic Simulator (FLEKS) to heliospheric shock simulations. We examine one- and two-dimensional local planar shock simulations, initialized using MHD states with upstream conditions representative of plasmas in the hypersonic, β ∼ 1 regime, for both quasi-perpendicular and quasi-parallel configurations. The refined algorithm in FLEKS proves robust, enabling accurate shock simulations with a grid resolution on the order of the electron inertial length d e . Our simulations successfully capture key shock features, including shock structures (foot, ramp, overshoot, and undershoot), upstream and downstream waves (fast magnetosonic, whistler, Alfvén ion-cyclotron, and mirror modes), and non-Maxwellian particle distributions. Crucially, we find that at least two spatial dimensions are critical for accurately reproducing downstream-wave physics in quasi-perpendicular shocks and capturing the complex dynamics of quasi-parallel shocks, including surface rippling, shocklets, short, large-amplitude magnetic structures, magnetic reconnection, and jets. Furthermore, our parameter studies demonstrate the impact of mass ratio and grid resolution on shock physics. This work provides valuable guidance for selecting appropriate physical and numerical parameters for shock simulations using a semi-implicit PIC method, paving the way for incorporating kinetic shock processes into large-scale collisionless plasma simulations with the MHD-AEPIC model.

plasma astrophysics↗
DOE OSTI2017

The Dynamics of Very High Alfvén Mach Number Shocks in Space Plasmas

Astrophysical shocks, such as planetary bow shocks or supernova remnant shocks, are often in the high or very-high Mach number regime, and the structure of such shocks is crucial for understanding particle acceleration and plasma heating, as well inherently interesting. Recent magnetic field observations at Saturn’s bow shock, for Alfvén Mach numbers greater than about 25, have provided evidence for periodic non-stationarity, although the details of the ion- and electron-scale processes remain unclear due to limited plasma data. High-resolution, multi-spacecraft data are available for the terrestrial bow shock, but here the very high Mach number regime is only attained on extremely rare occasions. Here we present magnetic field and particle data from three such quasi-perpendicular shock crossings observed by the four-spacecraft Cluster mission. Although both ion reflection and the shock profile are modulated at the upstream ion gyroperiod timescale, the dominant wave growth in the foot takes place at sub-proton length scales and is consistent with being driven by the ion Weibel instability. The observed large-scale behavior depends strongly on cross-scale coupling between ion and electron processes, with ion reflection never fully suppressed, and this suggests a model of the shock dynamics that is in conflict with previous models of non-stationarity. Thus, the observations offer insight into the conditions prevalent in many inaccessible astrophysical environments, and provide important constraints for acceleration processes at such shocks.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2023

Multi-Spacecraft Observations of Shocklets at an Interplanetary Shock

Interplanetary (IP) shocks are fundamental building blocks of the heliosphere, and the possibility to observe them in situ is crucial to address important aspects of energy conversion for a variety of astrophysical systems. Steepened waves known as shocklets are known to be important structures of planetary bow shocks, but they are very rarely observed related to IP shocks. We present here the first multi-spacecraft observations of shocklets observed by upstream of an unusually strong IP shock observed on 3 No v ember 2021 by several spacecraft at L1 and near-Earth solar wind. The same shock was detected also by radially aligned Solar Orbiter at 0.8 AU from the Sun, but no shocklets were identified from its data, introducing the possibility to study the environment in which shocklets developed. The Wind spacecraft has been used to characterize the shocklets, associated with pre-conditioning of the shock upstream by decelerating incoming plasma in the shock normal direction. Finally, using the Wind observations together with ACE and DSCOVR spacecraft at L1, as well as THEMIS B and THEMIS C in the near-Earth solar wind, the portion of interplanetary space filled with shocklets is addressed, and a lower limit for its extent is estimated to be of about 110 R E in the shock normal direction and 25 R E in the directions transverse to the shock normal. Using multiple spacecraft also reveals that for this strong IP shock, shocklets are observed for a large range of local obliquity estimates (9° –64°).

plasmas↗
DOE OSTI2014

Thermophysical properties of multi-shock compressed dense argon

In contrast to the single shock compression state that can be obtained directly via experimental measurements, the multi-shock compression states, however, have to be calculated with the aid of theoretical models. In order to determine experimentally the multiple shock states, a diagnostic approach with the Doppler pins system (DPS) and the pyrometer was used to probe multiple shocks in dense argon plasmas. Plasma was generated by a shock reverberation technique. The shock was produced using the flyer plate impact accelerated up to ∼6.1 km/s by a two-stage light gas gun and introduced into the plenum argon gas sample, which was pre-compressed from the environmental pressure to about 20 MPa. The time-resolved optical radiation histories were determined using a multi-wavelength channel optical transience radiance pyrometer. Simultaneously, the particle velocity profiles of the LiF window was measured with multi-DPS. The states of multi-shock compression argon plasma were determined from the measured shock velocities combining the particle velocity profiles. We performed the experiments on dense argon plasmas to determine the principal Hugonoit up to 21 GPa, the re-shock pressure up to 73 GPa, and the maximum measure pressure of the fourth shock up to 158 GPa. The results are used to validate the existing self-consistent variational theory model in the partial ionization region and create new theoretical models.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2022

Comparing the shock sensitivity of insensitive energetic materials

We present a tabletop method to study the shock sensitivity of plastic-bonded explosives that are considered shock insensitive using high dynamic range optical emission spectroscopy with laser-launched km/s flyer plate impacts (2, 3, and 4 km/s), which measures the spectral radiance (the emission spectrum vs a calibrated intensity standard) every 0.8 ns in the nanosecond and microsecond regimes. The explosives were TATB (1,3,5 trinitro, 2,4,6 triamino benzene), FOX-7 (1,1-diamino-2,2-dinitroethylene), LLM-105 (2,6-diamino-3,5-dinitropyrazine-1-oxide), and NTO (nitrotriazolone), all with 20% Sylgard 182 binder. Time resolving emission from shocked explosives produces a unique fingerprint at each shock pressure, and the emission temporal profile can be used to understand each material's shock response and the underlying mechanisms within the framework of the well-known shock ignition and growth models. Ignition was characterized by the emission intensity during nanosecond hot spot ignition. The growth process, which occurs after the shock has unloaded, was characterized by the time derivative of microsecond emission intensity. Singular-value decomposition was used to determine a characteristic spectral radiance over the entire time range, and this radiance was fit to a graybody model to determine a characteristic temperature Tc. Even though these insensitive explosives have similar shock sensitivities, our method can clearly distinguish their shock response. FOX-7 and NTO were the most sensitive, with FOX-7 having the greatest shock ignition sensitivity, growth rate, and Tc. LLM-105 was much less sensitive than FOX-7 and NTO, and TATB consistently showed the lowest sensitivity by any measure.

Akhtar, Meysam (ORCID:0000000243897837)↗
DOE OSTI2013

SELF-REGULATED SHOCKS IN MASSIVE STAR BINARY SYSTEMS

In an early-type, massive star binary system, X-ray bright shocks result from the powerful collision of stellar winds driven by radiation pressure on spectral line transitions. We examine the influence of the X-rays from the wind-wind collision shocks on the radiative driving of the stellar winds using steady-state models that include a parameterized line force with X-ray ionization dependence. Our primary result is that X-ray radiation from the shocks inhibits wind acceleration and can lead to a lower pre-shock velocity, and a correspondingly lower shocked plasma temperature, yet the intrinsic X-ray luminosity of the shocks, L{sub X}, remains largely unaltered, with the exception of a modest increase at small binary separations. Due to the feedback loop between the ionizing X-rays from the shocks and the wind driving, we term this scenario as self-regulated shocks. This effect is found to greatly increase the range of binary separations at which a wind-photosphere collision is likely to occur in systems where the momenta of the two winds are significantly different. Furthermore, the excessive levels of X-ray ionization close to the shocks completely suppress the line force, and we suggest that this may render radiative braking less effective. Comparisons of model results against observations reveal reasonable agreement in terms of log (L{sub X}/L{sub bol}). The inclusion of self-regulated shocks improves the match for kT values in roughly equal wind momenta systems, but there is a systematic offset for systems with unequal wind momenta (if considered to be a wind-photosphere collision).

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

CONSTRAINING STELLAR FEEDBACK: SHOCK-IONIZED GAS IN NEARBY STARBURST GALAXIES

We investigate the properties of feedback-driven shocks in eight nearby starburst galaxies using narrow-band imaging data from the Hubble Space Telescope. We identify the shock-ionized component via the line diagnostic diagram [O III] (λ5007)/Hβ versus [S II] (λλ6716, 6731) (or [N II] (λ6583))/Hα, applied to resolved regions 3-15 pc in size. We divide our sample into three sub-samples: sub-solar, solar, and super-solar, for consistent shock measurements. For the sub-solar sub-sample, we derive three scaling relations: (1) L{sub shock}∝SFR{sup 0.62}, (2) L{sub shock}∝Σ{sub SFR,{sub HL}} {sup 0.92}, and (3) L{sub shock}/L{sub tot}∝(L{sub H} /L{sub ☉,{sub H}}){sup –0.65}, where L{sub shock} is the Hα luminosity from shock-ionized gas, Σ{sub SFR,{sub HL}} the star formation rate (SFR) per unit half-light area, L{sub tot} the total Hα luminosity, and L{sub H} /L{sub ☉,{sub H}} the absolute H-band luminosity from the Two Micron All Sky Survey normalized to solar luminosity. The other two sub-samples do not have enough number statistics, but appear to follow the first scaling relation. The energy recovered indicates that the shocks from stellar feedback in our sample galaxies are fully radiative. If the scaling relations are applicable in general to stellar feedback, our results are similar to those by Hopkins et al. for galactic superwinds. This similarity should, however, be taken with caution at this point, as the underlying physics that enables the transition from radiative shocks to gas outflows in galaxies is still poorly understood.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

How bad or good are the external forward shock afterglow models of gamma-ray bursts?

The external forward shock models have been the standard paradigm to interpret the broadband afterglow data of gamma-ray bursts (GRBs). One prediction of the models is that some afterglow temporal breaks at different energy bands should be achromatic; that is, the break times should be the same in different frequencies. Multiwavelength observations in the Swift era have revealed chromatic afterglow behaviors at least in some GRBs, casting doubts on the external forward shock origin of GRB afterglows. In this paper, using a large sample of GRBs with both X-ray and optical afterglow data, we perform a systematic study to address the question: how bad or good are the external forward shock models? Our sample includes 85 GRBs up to 2014 March with well-monitored X-ray and optical light curves. Based on how well the data abide by the external forward shock models, we categorize them into five grades and three samples. The first two grades (Grade I and II) include 45 of 85 GRBs. They show evidence of, or are consistent with having, an achromatic break. The temporal and spectral behaviors in each afterglow segment are consistent with the predictions (the “closure relations”) of the forward shock models. These GRBs are included in the Gold sample. The next two grades (Grade III and IV) include 37 of 85 GRBs. They are also consistent with having an achromatic break, even though one or more afterglow segments do not comply with the closure relations. These GRBs are included in the Silver sample. Finally, Grade V (3/85) shows direct evidence of chromatic behaviors, suggesting that the external shock models are inconsistent with the data. These are included in the Bad sample. We further perform statistical analyses of various observational properties (temporal index α, spectral index β, break time t{sub b}) and model parameters (energy injection index q, electron spectral index p, jet opening angle θ{sub j}, radiative efficiency η{sub γ}, and so on) of the GRBs in the Gold sample, and derive constraints on the magnetization parameter ϵ{sub B} in the forward shock. Overall, we conclude that the simplest external forward shock models can account for the multiwavelength afterglow data of at least half of the GRBs. When more advanced modeling (e.g., long-lasting reverse shock, structured jets, arbitrary circumburst medium density profile) is invoked, up to >90% of the afterglows may be interpreted within the framework of the external shock models.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Symmetry Determining Equations of the Rankine-Hugoniot Equations for Variable Velocity Shock Waves

The “constant velocity piston” problem (Fig. 1), also known as the “piston problem,” is a standard model for a one dimensional, in our case linear, symmetric shock wave moving through an inviscid, perfect gas. The model can be divided into two regions - a perturbed section on the left and an unperturbed section on the right - by a moving shock wave moving left to right. Both the perturbed and unperturbed sections, i.e. the shocked and unshocked regions, respectively, obey the Eulerian conservation equations; however, at the exact location of the shock, there is a mathematical discontinuity not satisfied by the Euler equations. To ensure continuity and conservation of certain quantities when crossing between the unshocked and shocked regions, we evoke a series of equations derived from the Eulerian conservation equations, called the Rankine-Hugoniot equations, or “jump” equations as it is often referred to in the literature on the topic. The classical constant-velocity piston problem assumes the piston features a constant driving velocity (among many other willing suspensions of belief required in the pursuit of a first principles equation model); consequent to this assumption is a constant-velocity shock and a constant-velocity shocked flow state. However, using Lie Group Theory (LGT), also known as symmetry analysis, we can attempt to reinterpret the model with a shock wave of variable velocity in time and space. An extension of the model in this way opens up the possibility for obtaining new analytical solutions to the piston problem for certain shock velocity models. In this report, we use LGT to derive the symmetry determining equations (SDEs), whose solutions are Lie groups, which permit analytical solutions. In the future, we can then use the SDEs to define constraint equations on the shock velocity model and what the successive solutions to the Euler equations might be based off such constraints. This report is structured as follows: Section 2 provides a brief derivation of the Rankine-Hugoniot (“jump”) equations; Section 3 gives an overview of Lie group theory; Section 4 derives the SDEs of the jump equations; Section 5 derives the Euler conservation equations for fluids; and Section 6 presents concluding remarks and opportunities for future studies.

42 ENGINEERING↗