Maximum Expansion Velocity Research Articles

Effect of microstructure on functional force and air explosion performance of CL-20-based aluminum-containing explosive

A solvent–non-solvent approach was used to create a CL-20-based aluminum-containing explosive with a composite structure. The SEM results demonstrate that the aluminum powder is equally embedded on the surface of the CL-20 crystal in the CL-20/Al explosive with a composite structure but the aluminum powder in the mechanically mixed CL-20/Al explosive is agglomerated and spread unevenly. The maximum expansion velocity of the composite structure CL-20/Al explosive was 17.74% higher than that of the mechanically mixed explosive in the cylinder test, and the acceleration ability of the CL-20/Al explosive with a composite structure to the cylinder wall is superior to that of the mechanically mixed explosive. In the air explosion test, the shock wave overpressure and impulse of the CL-20/Al explosive with a composite structure are superior to those of the mechanically mixed explosive, but as the distance increases, the gap between the two explosives gradually narrows until it is flat. The results of interfacial particle velocity show that the length of the reaction zone and duration of CL-20/Al explosives with composite structures are longer than those of mechanically mixed structures, and the interfacial particle velocity of CL-20/Al explosives with composite structures is higher than that of mechanically mixed structures.

The unusual 3D distribution of NaCl around the asymptotic giant branch star IK Tau

Context. Sodium chloride (NaCl) is a diatomic molecule with a large dipole moment, which allows for its detection even at relatively low abundances. It has been detected towards several evolved stars, including the asymptotic giant branch (AGB) star IK Tau, around which it is distributed in several clumps that lie off-centre from the star. Aims. We aim to study the 3D distribution of NaCl around the AGB star IK Tau and determine the abundance of NaCl relative to H2 for each of the clumps. Methods. First, a new value for the maximum expansion velocity was determined. The observed ALMA channel maps were then de-projected to create a 3D model of the distribution of NaCl. This model was then used as input for the radiative transfer modelling code Magritte. We determined the NaCl abundances of each of the clumps by comparing the observations with the results of the Magritte simulations. Results. We derive an updated value for the maximum expansion velocity of IK Tau: υexp = 28.4 ± 1.7 km s−1. A spiral-like shape can be discerned in our 3D distribution model of NaCl. This spiral lies more or less in the plane of the sky, with the distribution flatter in the line-of-sight direction than in the plane of the sky. We find clump abundances of between 9 × 10−9 and 5 × 10−8 relative to H2; the relative abundance is typically lower for clumps closer to the star. Conclusions. This is the first time de-projection has been used to understand the 3D environment of molecular emission around an AGB star and to calculate the fractional abundance of NaCl in clumps surrounding the star.

Real-time spatiotemporal characterization of mechanics and sonoporation of acoustic droplet vaporization in acoustically responsive scaffolds.

Phase-shift droplets provide a flexible and dynamic platform for therapeutic and diagnostic applications of ultrasound. The spatiotemporal response of phase-shift droplets to focused ultrasound, via the mechanism termed acoustic droplet vaporization (ADV), can generate a range of bioeffects. Although ADV has been used widely in theranostic applications, ADV-induced bioeffects are understudied. Here, we integrated ultra-high-speed microscopy, confocal microscopy, and focused ultrasound for real-time visualization of ADV-induced mechanics and sonoporation in fibrin-based, tissue-mimicking hydrogels. Three monodispersed phase-shift droplets-containing perfluoropentane (PFP), perfluorohexane (PFH), or perfluorooctane (PFO)-with an average radius of ∼6 μm were studied. Fibroblasts and tracer particles, co-encapsulated within the hydrogel, were used to quantify sonoporation and mechanics resulting from ADV, respectively. The maximum radial expansion, expansion velocity, induced strain, and displacement of tracer particles were significantly higher in fibrin gels containing PFP droplets compared to PFH or PFO. Additionally, cell membrane permeabilization significantly depended on the distance between the droplet and cell (d), decreasing rapidly with increasing d. Significant membrane permeabilization occurred when d was smaller than the maximum radius of expansion. Both ultra-high-speed and confocal images indicate a hyper-local region of influence by an ADV bubble, which correlated inversely with the bulk boiling point of the phase-shift droplets. The findings provide insight into developing optimal approaches for therapeutic applications of ADV.

Experimental investigation of the influence of the panel stiffness on the behaviour of a wedge under slamming

In this paper, slamming experiments at constant velocity are presented. The hydraulic shock test-rig enables fast experiments at quasi-constant velocity during a water entry of a wedge. This approach provides results more suited for validation of numerical models. The objective of this paper is to discuss the influence of panel stiffness on pressure and strain, for an aluminum wedge with a deadrise angle of 30° and for various velocity levels. Information on wet surface expansion velocity is inferred from the detection of the threshold pressure on the placed sensors. The experimental evolution of the maximum pressure and wet surface expansion velocity is in good agreement with analytical results and can be predicted by semi-analytical formulas.

Morpho-kinematic Modeling of the Expanding Ejecta of the Extremely Slow Nova V1280 Scorpii

Knowledge of the morphology of nova ejecta is essential for fully understanding the physical processes involved in nova eruptions. We studied the 3D morphology of the expanding ejecta of the extremely slow nova V1280 Sco with a unique light curve. Synthetic line profile spectra were compared to the observed [O iii] λλ4959, 5007 and [N ii] λ5755 emission line profiles in order to find the best-fit morphology, inclination angle, and maximum expansion velocity of the ejected shell. We derive the best-fitting expansion velocity, inclination, and squeeze as km s−1, deg, and using [O iii] line profiles, and km s−1, deg, and using the [N ii] λ5755 line profile. A high inclination angle is consistent with the observational results showing multiple absorption lines originating from clumpy gases, which are produced in dense and slow equatorially focused outflows. Based on additional observational features such as optical flares near the maximum light and dust formation on V1280 Sco, a model of internal shock interaction between slow ejecta and fast wind proposed for the γ-ray emission detected in other novae seems to be applicable to this extremely slow and peculiar nova. Increasing the sample size of novae whose morphology is studied will be helpful in addressing long-standing mysteries in novae such as the dominant energy source to power the optical light at the maximum, optical flares near the maximum, clumpiness of the ejecta, and dust formation.

Breaking Orion's Veil with fossil outflows

Context. The role of feedback in the self-regulation of star formation is a fundamental question in astrophysics. The Orion Nebula is the nearest site of ongoing and recent massive star formation. It is a unique laboratory for the study of stellar feedback. Recent SOFIA [C II] 158 μm observations have revealed an expanding bubble, the Veil shell, that is powered by stellar winds and ionization feedback. Aims. We have identified a protrusion-like substructure in the northwestern portion of the Orion Veil shell that may indicate additional, highly directional feedback mechanisms. Our goal is to investigate the origin of the protrusion by quantifying its possible driving mechanisms. Methods. We use the [C II] 158 μm map of the Orion Nebula obtained with the upGREAT instrument on board SOFIA. The spectral and spatial resolution of the observations are 0.3 km s−1 and 16", respectively. The velocity-resolved [C II] observations allow us to construct position-velocity (pv) diagrams to measure the morphology and the expansion velocity of the protrusion. For the morphology, we also use new observations of 12CO and 13CO J = 2-1 (to trace molecular gas), Spitzer 8 μm observations (to trace the far-ultraviolet-illuminated surfaces of photodissociation regions), and Hα observations (to trace ionized gas). For the kinematics, we perform a line-profile analysis of [C II], 13CO, and 12CO at 12 positions, covering the entire protrusion. To quantify the stellar feedback, we estimate the mass of the protrusion by fitting the dust thermal emission. We compare the kinetic energy with the stellar wind of θ1 Ori C and the momentum of the outflows of massive protostars to investigate the driving mechanism of the protrusion. Results. The pv diagrams reveal two half-shells expanding at velocities of +6 km s−1 and +12 km s−1. We find that the protrusion has a diameter of ~1.3 pc with a ~45 M⊙ shell expanding at +12 km s−1 at the northwestern rim of the Veil. The thickness of the expanding shell is ~0.1 pc. Using the mass in the limb-brightened shell and the maximum expansion velocity, we calculate the kinetic energy and the momentum of the protrusion to be ~7 × 1046 erg and 540 M⊙ km s−1, respectively. We consider three possible origins for this protrusion: fossil outflow cavities created by jets and outflows during the protostellar accretion phase, preexisting "clumpiness" in the OMC-1 core, and the stellar wind during the main-sequence phase. Based on the energetics and the morphology, we conclude that the northwestern part of the preexisting cloud was locally perturbed by outflows ejected from massive protostars in the Trapezium cluster. This suggests that the protrusion of the Veil is the result of mechanical rather than radiative feedback. Furthermore, we argue that the location of the protrusion is a suitable place to break Orion's Veil owing to the photo-ablation from the walls of the protrusion. Conclusions. We conclude that the outflows of massive protostars can influence the morphology of the future H II region and even cause breakages in the ionization front. Specifically, the interaction of stellar winds of main-sequence stars with the molecular core preprocessed by the protostellar jet is important.

A Study of the Characteristics of Plasma Generated by Infrared Pulse Laser-Induced Fused Silica

When high energy infrared laser pulses are incident on fused silica, the surface of the fused silica is damaged and a laser-induced plasma is produced. Based on the theory of fluid mechanics and gas dynamics, a two-dimensional axisymmetric gas dynamic model was established to simulate the plasma generation process of fused silica induced by a millisecond pulse laser. The results show that the temperature of the central region irradiated by the laser is the highest, and the plasma is first produced in this region. When the laser energy density is 1.0 × 104 J/cm2 and the pulse width is 0.2 ms, the maximum expansion velocity of the laser-induced plasma is 17.7 m/s. Under the same experimental conditions, the results of the simulation and experiment are in good agreement. With an increase in pulse width, the plasma expansion rate gradually decreases.

Comparative visualized investigation of impact-driven high-speed liquid jets injected in submerged water and in ambient air

This paper is a comparative study on the characteristics of high-speed liquid jets injected in surrounding water and air using shadowgraph technique. One of the main objectives is to investigate the effects of liquid’s physical properties, used to generate the high-speed liquid jets, on jet generation’s characteristics. Moreover, comparative investigations on effects of those liquid jets after injected in water and air are reported. The high-speed liquid jets were generated by the impact of a projectile launched by a horizontal single-stage power gun. The impact-driven high-speed liquid jets were visualized by shadowgraph technique and images were recorded by a high-speed digital video camera. The process of impact-driven high-speed liquid jet injection in air and water, oblique shock waves, jet-induced shock waves, shock waves propagation, the bubble behavior, bubble collapse-induced rebound shock waves and bubble cloud regeneration were clearly observed. It was found that different properties of liquid (surface tension and kinematic viscosity) affect the jet maximum velocity and shape of the jet. Bubble behaviors were only found for the jet injected in water. From the shadowgraph images, it is found that the maximum average jet velocity, expansion and contraction velocities of bubble in axial direction increase when the value of the multiplied result of surface tension by kinematic viscosity increases. Therefore, surface tension and kinematic viscosity are the significant physical properties that affect characteristics of high-speed liquid jets.

Monocrystalline Silicon Plasma Expansion Induced by Millisecond Laser

Ablation of monocrystalline silicon to atmospheric environment induced by a millisecond pulse of an Nd:YAG optical laser with a wavelength of 1064 nm is studied. Shadowgraphy is applied to study the process of monocrystalline silicon plasma expansion for the laser energy density of 955.4–2736.0 J/cm2. It is shown that the outer boundary of the plasma plume diffuses outside with time. Plasma expansion occurs in both axial and radial directions, but the velocity of plasma expansion in the radial direction is smaller than in the axial direction. Two centerlines of the laser action are symmetric. The maximum expansion velocity of 162.1 m/s is reached with the laser energy density of 2376.0 J/cm2, and a laser-supported combustion wave is generated in this case. In contrast to monocrystalline silicon under the action of a short-pulse laser, the millisecond laser action can make the plasma expansion velocity increase for the second time. A material splash can be observed from the expansion images in the case of a high laser energy density.

Parameters of type IIP SN 2012A and clumpiness effects

The explosion energy and the ejecta mass of a type IIP supernova (SN IIP) derived from hydrodynamic simulations are principal parameters of the explosion theory. However, the number of SNe IIP studied by hydrodynamic modeling is small. Moreover, some doubts exist in regard to the reliability of derived SN IIP parameters. The well-observed type IIP SN 2012A will be studied via hydrodynamic modeling. Their early spectra will be checked for a presence of the ejecta clumpiness. Other observational effects of clumpiness will be explored. Supernova parameters are determined by means of the standard hydrodynamic modeling. The early hydrogen Halpha and Hbeta lines are used for the clumpiness diagnostics. The modified hydrodynamic code is employed to study the clumpiness effect in the light curve and expansion kinematics. We found that SN 20012A is the result of the explosion of a red supergiant with the radius of 715 Rsun. The explosion energy is 5.25x10^50 erg, the ejecta mass is 13.1 Msun, and the total Ni-56 mass is 0.012 Msun. The estimated mass of a progenitor, a main-sequence star, is 15 Msun. The Halpha and Hbeta lines in early spectra indicate that outer ejecta are clumpy. Hydrodynamic simulations show that the clumpiness modifies the early light curve and increases the maximum velocity of the outer layers. The pre-SN 2012A was a normal red supergiant with the progenitor mass of about 15 Msun. The outer layers of ejecta indicate the clumpy structure. The clumpiness of the external layers can increase the maximum expansion velocity.

Dynamic Ductility and Fragmentation for Aluminum Alloy Using Electromagnetic Ring Expansion

In order to investigate the effects of ductility enhancement for aluminum alloy under the high-velocity EMF condition, the electromagnetic ring expansion test is carried out. Through a use of Photron digital high-speed video camera, the sequence of deformation and failure in the expanding ring are determined. Subsequently, the variation of expansion velocity with time is measured. In particular, this work reveals the effects of expansion velocity on the ductility, necking characteristic, fracture behavior of Al 5052-O and 6061-T4. In addition, the dynamic ductility of Al 5052-O under electromagnetic ring expansion experiment is compared with that in split Hopkinson tension bar (SHTB) test and quasi-static tensile case. It is shown that the number of necks and fragments increase as the maximum expansion velocity increases. Furthermore, the uniform strain and fracture strain augment with increasing expansion velocity, which is more evident for Al 5052-O. The fracture surfaces of ring fragments are characterized by the larger and deeper dimple-like structure at higher expansion velocity. Compared with the SHTB tests at the same strain rate, the ductility of Al 5052-O under electromagnetic ring expansion is better. The ductility enhancement results from not only the inertia effect but also the changes in material constitutive behavior.

Aortic tissue Doppler imaging after tetralogy of Fallot repair: beyond aortic diameters

Purpose: Structural abnormalities of the aortic wall have been described in tetralogy of Fallot (ToF) and could be responsible for aortic stiffening leading to aortic dilatation. We aimed to study if aortic root stiffness indices assessed by tissue Doppler imaging (TDI) could be related to the aortic diameter late after ToF repair. Methods: We included adult (≥ 18 years) patients (pts) with surgically repaired ToF followed at a terciary care center from January till December 2012. We defined two groups based on the Cornell data-based z-score formulae for two-dimensional transthoracic echocardiographic diameter of the aortic root at the level of the sinuses of Valsalva (AoZ), in parasternal long-axis view: group 1 - with Ao dil (AoZ > 2) and group 2 - without Ao dil (AoZ ≤ 2). Aortic root M-mode systolic and diastolic diameters and TDI variables were measured, including systolic maximum anterior wall expansion velocity (SAo), early (EAo) and late (AAo) diastolic retraction velocity and time to peak systolic (TAo). Strain, distensibility and stiffness index of the aortic root were determined noninvasively. Results: We included 43 pts (mean age 32±11 years; 58% female); mean follow-up time since ToF repair was 22±8 years. In 21 pts (49%) an aortopulmonary (AP) shunt was done prior to complete repair, with a median interval of 3 years. In 15 pts complete repair used a transannular patch. In 10 pts (23%) we identified Ao dil (mean age 32±11 years; 5 males). Only 1 pt had moderate aortic regurgitation. Both group 1 and 2 were similar concerning age, gender, body surface area, age at ToF repair and follow up time after repair. Also, no differences could be found in systemic blood pressure, pulse pressure, and TDI variables. There was a trend to a positive correlation between the aortic root z-score and the aortic stiffness index, whereas a trend to a negative correlation was found for the aortic strain and distensibility. The aortic stiffness index was significantly increased in group 1 (14.8±6.3 vs 8.5±6.8, p=0.019) and in univariate and multivariate analysis, the aortic stiffness index was an independent predictor of aortic dilatation (odds ratio 1.30, 95% confidence interval 1.02-1.65, p=0.033). Conclusions: TDI can assess the aortic elastic properties noninvasively in this subset of pts and the aortic stiffness index may be a predictor of aortic dilatation late after ToF repair.

Morpho-kinematical modelling of Nova Eridani 2009 (KT Eri)

Modelling the morphology of a nova outburst provides valuable information on the shaping mechanism in operation at early stages following the outburst. We performed morpho-kinematical studies, using {\sc shape}, of the evolution of the H\alpha\ line profile following the outburst of the nova KT Eridani. We applied a series of geometries in order to determine the morphology of the system. The best fit morphology was that of a dumbbell structure with a ratio between the major to minor axis of 4:1, with an inclination angle of 58$^{+6}_{-7}$ degrees and a maximum expansion velocity of 2800$\pm$200 km/s. Although, we found that it is possible to define the overall structure of the system, the radial density profile of the ejecta is much more difficult to disentangle. Furthermore, morphology implied here may also be consistent with the presence of an evolved secondary as suggested by various authors.

OPTICAL MORPHOLOGY, INCLINATION, AND EXPANSION VELOCITY OF THE EJECTED SHELL OF NOVA MONOCEROTIS 2012

The morphology of the ejected shell of the He/N Nova Monocerotis 2012 outburst was studied in detail. Synthetic line profile spectra were compared to the [O {\sc iii}] 4959,5007 \AA\ emission line profiles in order to find the best fit morphology, inclination angle and maximum expansion velocity of the ejected shell. The simplest morphology was found to be that of a bipolar structure with an inclination angle of 82$\pm6$ degrees and a maximum expansion velocity of 2400$^{+300}_{-200}$ km/s (at day 130 after outburst). Such a high degree of shaping is un-expected for a system with a main sequence star (as suspected from the systems colors). The degree of shaping may be disentangled with resolved optical imaging. Furthermore, these results may be confirmed with radio imaging which is expected to follow the same gross features of the outburst as the optical band and the high inclination implied here can be corroborated with a 7.1 hour period which has been suggested to arise from partial eclipses of extended emission by an accretion disk rim.

Effect of Cross Section Size on Ductility and Fragmentation of Copper Ring at High Strain Rate Loading Conditions

In the present work, different copper ring samples geometries with a small aspect ratio of 0.5 or 1 were subjected to an experimental study under electromagnetic expanding ring test conditions. The experimental multiple ring tests were performed under similar loading conditions, that is, the applied maximum expansion velocities covered a range from 128 to 147 m/s (7.4 103s-1) for all ring samples geometries, with the exception of rings with a cross section of 1 mm x 0.5 mm. For these rings, the averaged maximum expansion velocity was higher and equal to 195 m/s (1.2 104s-1). The results of experimental investigations revealed a minor influence of the applied cross section sizes on ductility of copper rings, whereas its fragmentation seems to be dependent on a ring cross section area.

Low-Resolution Spectroscopy of the Recurrent Nova T Pyxidis at its Early Stage of the 2011 Outburst

We present our observational results of the recurrent nova T Pyx at its early stage of the 2011 outburst, using a low-resolution spectrograph ($ R$$ \approx$ 400) attached to a 28 cm telescope. The total number of nights of our observation was 11, among which 9 nights were during the pre-maximum stage. As a result, we obtained a detailed evolutional feature of this recurrent nova on the way to its maximum light. At first, on the earliest three nights (25–21 days before maximum), broad and prominent emission lines, such as Balmer series, He I, He II, N II, N III, and O I together with the P Cyg profile, are seen on the spectra. The blueshifted absorption minima of H$ \alpha$ yields a maximum expansion velocity of approximately 2200 km s$ ^{-1}$ , and the velocity gradually decreases. Then, helium and nitrogen lines are weakened day by day. After that (18 days before maximum light), Fe II (multiplets) lines emerge on the spectra. These lines are then strengthened day by day, and the P Cyg profiles also become more prominent. Accordingly, the expansion velocities turn to gradual increase. In addition, during the pre-maximum stage, the nova spectral type of T Pyx is thought to evolve from the He$ /$ N type to the Fe II one.

Nonlinear pulsations of red supergiants

Excitation of radial oscillations in population I (X=0.7, Z=0.02) red supergiants is investigated using the solution of the equations of radiation hydrodynamics and turbulent convection. The core helium burning stars with masses 8M_odot <= M <= 20M_odot end effective temperatures T_eff < 4000K are shown to be unstable against radial pulsations in the fundamental mode. The oscillation periods range between 45 and 1180 days. The pulsational instability is due to the kappa-mechanism in the hydrogen and helium ionization zones. Radial pulsations of stars with mass M < 15M_odot are strictly periodic with the light amplitude Delta M_bol <= 0.5 mag. The pulsation amplitude increases with increasing stellar mass and for M > 15M_odot the maximum expansion velocity of outer layers is as high as one third of the escape velocity. The mean radii of outer mass zones increase due to nonlinear oscillations by <= 30% in comparison with the initial equilibrium. The approximate method (with uncertainty of a factor of 1.5) to evaluate the mass of the pulsating red supergiant with the known period of radial oscillations is proposed. The approximation of the pulsation constant Q as a function of the mass-to-radius ratio is given. Masses of seven galactic red supergiants are evaluated using the period-mean density relation.

The morphology of the expanding ejecta of V2491 Cygni (2008 N.2)

Determining the evolution of the ejecta morphology of novae provides valuable information on the shaping mechanisms in operation at early stages of the nova outburst. Understanding such mechanisms has implications for studies of shaping for example in proto-Planetary Nebulae. Here we perform morpho-kinematical studies of V2491 Cyg using spectral data to determine the likely structure of the ejecta and its relationship to the central system and shaping mechanisms. We use shape to model different morphologies and retrieve their spectra. These synthetic spectra are compared with observed spectra to determine the most likely morphology giving rise to them, including system inclination and expansion velocity of the nova ejecta. We find the best-fitting remnant morphology to be that of polar blobs and an equatorial ring with an implied inclination of 80+3−12 degrees and an maximum expansion velocity of the polar blobs of 3100+200−100 km s−1 and for the equatorial ring 2700+200−100 km s−1. This inclination would suggest that we should observe eclipses which will enable us to determine more precisely important parameters of the central binary. We also note that the amplitude of the outburst is more akin to the found in recurrent nova systems.

324 Multimodal imaging assessment of arterial stiffness in patients with repaired aortic coarctation versus normal subjects

Despite successful surgical correction, morbidity of patients with coarctation of the aorta(CoAo) is high. The increased large arteries stiffness may be related to their impaired prognosis. The aim of the study is to asses the relationship between aortic stiffness parameters (2D echo), aortic wall velocities (tissue Doppler imaging, TDI) and carotid stiffness parameters (e-tracking) in pts. with repaired CoAo. Study group consisted of 23 pts. with repaired CoAo (mean age 28,6 ± 9,9 yrs) and 20 matched healthy subjects. Aortic elasticity was assessed using M-mode evaluation of aortic diameters, then calculating accepted indexes: aortic strain (Ao Strain), distensibility (Ao Dis), stiffness index (Ao SI). Systolic maximum expansion velocity (SW) of aortic wall was assesed using TDI.E-tracking determined parameters were: pulse wave velocity (PWV), stiffness index (β), elastic modulus (Ep), arterial compliance (AC). When correlating (table) the parameters derived from the 3 methods, statistical significant association (*) was obtained for Ao M-mode derived stiffness parameters and SW. Only PWV correlated with Ao Dis and Ao Strain, while PWV and β correlated with SW. Ao Dis Ao Strain Ao SI SW PWV -,343(*) -,349(*) 0,01 -,388(*) AI -0,12 -0,16 0,08 -0,2 Beta -0,22 -0,22 0,09 -,373(*) Ep -0,26 -0,26 -0,03 -0,29 AC 0,17 0,13 0,08 -0,12 SW ,554(**) ,578(**) -,494(**) Precoarctational bed arteries stiffness parameters assessed by different techniques showed partly concordant results. While M-mode derived parameters of aortic elasticity are time-consuming, they could be replaced in daily practice by aortic wall TDI evaluation. Carotid stiffness indexes measured by e-tracking are complementary to aortic stiffness parameters.

Dependence of the acceleration of the outer edge of a stellar envelope on supernova explosion energy

Based on numerical simulations of a supernova explosion, we investigate the shock-wave breakout through the stellar surface. The computations have been performed in a wide range of explosion energies and presupernova masses. The results are compared with the classical Gandelman-Frank-Kamenetsky self-similar solution. We have determined the dependence of an arbitrary coefficient in the self-similar solution on the explosion energy and presupernova structure. The derived analytical approximation formula for this coefficient can be used to estimate the supernova explosion energy from such parameters of the ejected envelope determined from astronomical observations as its maximum expansion velocity and the density distribution along its outer edge. The formula may prove to be also useful in studying the X-ray and gamma-ray bursts that accompany the shock-wave breakout through the surface of compact presupernovae.