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10. Star Formation Driven Feedback

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10. Star Formation Driven Feedback 10. Star formation driven feedback Stars can output energy in three different channels: radiation neutrino emission mass flow. The energy output in radiation can be obtained by integrating the luminosity of a star over its lifetime. Typically, a star of 53.5 52.5 Solar mass emits about 10 erg in radiation, while the corresponding numbers for 10 M⊙ and 100 M⊙ stars are ∼ 10 erg and ∼ 1053.7 erg, respectively. A star also emits ∼ 1056 neutrinos when it becomes a white dwarf or a red giant, and the total energy involved is of the order 1052 erg. Not including contributions from supernova explosions. Of particular interest is the kinetic energy that is loaded in supernova ejecta and stellar winds, because such energy may be effectively injected into the ISM, affecting the gas properties and star formation in galaxies. We now examine how much mass-loaded kinetic energy is expected from individual stars. 10.5.1 Mass-Loaded Kinetic Energy - Stellar Winds Some stars can lose substantial amounts of mass via stellar winds during their late evolutionary stage. Mass loss can be calculated from stellar evolution theory: 10.5.1 Mass-Loaded Kinetic Energy - Stellar Winds Conventionally, a stellar wind is specified by its mass-loss rate, M ̇ , and its terminal velocity, v∞. The kinetic energy of the wind, sometimes called the wind ‘luminosity’, is: The challenge is then to relate both M ̇ and v∞ to the global properties of stars, such as luminosity, effective temperature, mass and radius. Based on about 300 stars of all spectral types and luminosities, Waldron (1984) suggests the following relations: −1 −1 −1 where M ̇ is in M⊙ yr , v∞ in kms and Lwind in ergs . −1 The typical wind velocity for supergiant OB stars is about 2000kms , and the corresponding kinetic energy is Ekin ∼ 50 −1 2 4×10 (Mwind/10M⊙)(v∞/2000kms ) erg. This energy is comparable to the kinetic energy output from a supernova. 10.5.1 Mass-Loaded Kinetic Energy - SN explosions For supernova explosions, the typical mass-loaded kinetic energy can be estimated by modeling in detail supernovae observed in the local Universe: the mass-loaded kinetic energies are quite similar for Type Ia and Type II SNe 10.5.1 Mass-Loaded Kinetic Energy - SN explosions As an approximation we may write: For Type Ia SNe, the ejecta masses are in a small range around Mejecta = 1M⊙, - progenitors are C/O white dwarfs near the Chandrasekhar mass limit (1.4M⊙) and such SNe do not leave any remnants behind. In this case, the initial velocity of the ejecta is ∼ 104 kms−1. In contrast, the ejecta mass of Type II SNe can change substantially from object to object, as their progenitors can cover a large mass range and the remnant mass depends on the details of the properties of the progenitor. In this case, the initial velocity is difficult to predict. Fortunately, in applications to galaxy formation and evolution, only the total kinetic energy is relevant, because here the energy in the ejecta is transferred into a gas component that has mass much larger than that of the ejecta itself. 8.6 Evolution of Gaseous Halos with Energy Sources In the presence of energy sources, gas can be heated through non-gravitational processes, such as radiation from stars and AGN, stellar explosions and stellar winds. In the presence of source terms, gasdynamics is described by the following set of mass, momentum and energy conservation equations: where Sm, Smom and Se are the changes per unit time in mass density, momentum density, and energy density due to the sources. 8.6 Evolution of Gaseous Halos with Energy Sources Smom = Smvinj, with vinj the local mean velocity of the injected material, and: 2 where C = n HΛ is the cooling rate per unit volume, H is the (radiative) heating rate per unit volume, and θinj ≡ kBTinj/µmp is the local mean thermal velocity of injected material. If all three source terms are known, the above set of equations can be solved together with the Poisson equation for the gravitational potential Φ. 8.6.1 Blast Waves - Self-Similar Model Blast wave: a large amount of energy is released locally during a short period of time, and the disturbance produced in the medium propagates as a shock wave. Instantaneous release of energy ε0 from a source of negligible size in a uniform medium with density ρ0. Assuming the medium is an ideal fluid and that the mass ejected from the explosion is negligible. This assumption is valid when the mass swept by the expanding shock wave is much larger than the initial mass of the ejecta. In the energy-conserving phase of the evolution, i.e., when the shock wave is at a stage where the total radiated energy is much smaller than ε0, the only time scale involved is t, the age of the explosion, and the only length scale is rsh(t), the radius of the shock wave at time t. In this case, we expect the problem to admit ‘self-similar’ solutions, meaning that any dimensional quantity Q(r,t) at radius r (from the center of the ex- plosion) and time t can be written as Q(r,t) = QchQ(r/R), where Qch is the characteristic value of Q, obtained by combining rsh, t and ρ0 (or ε0). For instance, the gas density, velocity and pressure at (r,t) can be written, respectively, in the following forms: 8.6.1 Blast Waves - Self-Similar Model Once rsh(t) is known, the evolution is completely determined by the forms of the single-variable functions D (λ ), V (λ ) and P (λ ). Since the evolution of rsh(t) should be completely determined by ε0 and ρ0, and since no combination of t, ε0 and ρ0 can give a dimensionless quantity, the most general form of rsh(t) is: Thus, under the assumption of self-similarity, the time dependence of rsh is completely deter- mined by dimensionality considerations. The expansion speed of the shock front is therefore which shows that the shock becomes weaker as it expands. Once vsh is reduced to a level comparable to the sound speed of the ambient medium, the shock disappears. 8.6.1 Blast Waves - Self-Similar Model The above dimensionality analysis can be extended to cases where ρ0 is a power law of r, and ε0 is a power law of t: The value of A in Eq.(8.162) and the forms of D, V and P can all be obtained by solving the fluid equations. In spherical symmetry, the mass, momentum and energy equations are: where we have ignored any heating or cooling. As we have seen in §8.1.1, the last equation can be replaced by the following entropy equation, 8.6.1 Blast Waves - Self-Similar Model Under the assumption of similarity, all quantities depend on r and t only through the com- bination λ = r/rsh(t), and Eqs. (8.166), (8.167) and (8.169) are reduced to the following set of ordinary differential equations: where a prime denotes a derivative with respect to λ , and η = 2/5 is the power of the time- dependence of rsh(t). These equations can be integrated from λ = 1 to λ = 0 subject to the jump conditions (8.49) and (8.50) at λ = 1. Note that these jump conditions are obtained by an observer moving with the shock wave. For an observer at rest with the ambient medium, v1 = −vsh, v2 = vb − vsh, where vb is the rest-frame velocity of the flow just behind the shock front. For strong shocks, the jump conditions are: 8.6.1 Blast Waves - Self-Similar Model An extra condition is required in order to specify the constant A in Eq. (8.162). This condition can be obtained by integrating the energy equation (8.168) over the entire space. Since rv = 0 at both r = 0 and r = ∞, only the first term in Eq. (8.168) contributes to the integration: Outside the shock radius, rsh, the velocity v = 0 and the pressure is a constant P0. The integration from rsh to infinity is then equal to: 8.6.1 Blast Waves - Self-Similar Model For γ = 5/3, this gives A ≈ 1.15. The corresponding density, pressure and velocity profiles are shown in Fig. 8.9. 8.6.1 Blast Waves - Supernova Remnants We can now discuss in more detail the different evolutionary stages of a blast wave. As a concrete example, we consider supernova remnants, which are produced by supernova explosions at the late evolutionary stages of relatively massive stars. The typical energy output of such an explosion is about 1051 erg, released in seconds. Almost all this energy is initially in the form of kinetic energy of the ejecta; the integrated photon luminosity of a supernova is a factor ∼ 100 smaller. However, the supernova ejecta creates a shock that becomes radiative, and by the time the supernova blast wave fades away, almost 95 percent of its energy has been radiated away. Blast wave consists of four well-defined stages: 1) first the ejected mass exceeds the swept-up ambient mass, and to lowest order the ejecta undergo free expansion. 2) once the mass of the swept-up material becomes comparable to the mass of the ejecta, the blast-wave enters the adiabatic (or Sedov) phase, in which the evolution is self-similar as long as the ambient medium is homogeneous. 3) Eventually the radiative losses from the interior of the blast-wave become significant, and the supernova remnant enters the third, radiative stage of its evolution.
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