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The standard model of long-duration gamma-ray bursts (GRBs) holds that these explosions are ultra-relativistic (initial γ greater than approximately 100), which is invoked to explain the so-called "compactness" problem: absent this ultra-relativistic expansion, the ejecta would be optically thick to pair production at typical peak spectral ...
The energy of an ultrarelativistic particle is almost completely due to its kinetic energy = (). The total energy can also be approximated as E = γ m c 2 ≈ p c {\displaystyle E=\gamma mc^{2}\approx pc} where p = γ m v {\displaystyle p=\gamma mv} is the Lorentz invariant momentum .
This equation states that the kinetic energy (E k) is equal to the integral of the dot product of the momentum (p) of a body and the infinitesimal change of the velocity (v) of the body. It is assumed that the body starts with no kinetic energy when it is at rest (motionless).
Total energy is the sum of rest energy = and relativistic kinetic energy: = = + Invariant mass is mass measured in a center-of-momentum frame. For bodies or systems with zero momentum, it simplifies to the mass–energy equation E 0 = m 0 c 2 {\displaystyle E_{0}=m_{0}c^{2}} , where total energy in this case is equal to rest energy.
Looking at the above formula for invariant mass of a system, one sees that, when a single massive object is at rest (v = 0, p = 0), there is a non-zero mass remaining: m 0 = E/c 2. The corresponding energy, which is also the total energy when a single particle is at rest, is referred to as "rest energy".
Thus, the ratio of the kinetic energy to the absolute temperature of an ideal monatomic gas can be calculated easily: per mole: 12.47 J/K; per molecule: 20.7 yJ/K = 129 μeV/K; At standard temperature (273.15 K), the kinetic energy can also be obtained: per mole: 3406 J; per molecule: 5.65 zJ = 35.2 meV.
When this happens, the electron may couple to an excited energy state of the nucleus and take the energy of the nuclear transition directly, without an intermediate gamma ray being first produced. The kinetic energy of the emitted electron is equal to the transition energy in the nucleus, minus the binding energy of the electron to the atom.
According to the assumptions of the kinetic theory of ideal gases, one can consider that there are no intermolecular attractions between the molecules, or atoms, of an ideal gas. In other words, its potential energy is zero. Hence, all the energy possessed by the gas is the kinetic energy of the molecules, or atoms, of the gas.