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. 4) Finally, once the interior pressure becomes comparable to that of the ambient medium, the supernova remnant merges with the ISM. 8.6.1 Blast Waves - 1st phase
At the beginning of a supernova explosion, when the gas swept up by the shock is still smaller than the mass of the ejecta,
Mejecta, the remnant is in free expansion, with a constant velocity vsh, and its radius increases as rsh = vsht.
The free expansion continues until the mass swept by the shock is comparable to Mejecta:
2 Since the total explosion energy is ε0 = (1/2)Mejectav , the free-expansion phase terminates at a time
51 where ε51 ≡ ε0/10 erg, and nH is the number density of hydrogen nuclei in the medium. At this time the size and velocity of the remnant are
8.6.1 Blast Waves - 2nd phase
At t > tf, the mass swept by the shell is larger than Mejecta and so the expansion of the shell is decelerated. As long as radiative energy loss is negligible, i.e. in the adiabatic (or Sedov) phase, the evolution of the remnant is given by the similarity solution discussed above [Eqs. (8.162) and (8.163)], with
As a crude approximation, this phase ends when the radiative energy loss is about 1/2 of the thermal energy, or 1/4 of the total energy: 8.6.1 Blast Waves - 2nd phase
To gain some insight into the problem, let us approximate the cooling function by a power-law function of temperature:
5 7 −23 3 For gas with Solar metallicity, λ ≈ −2/3 in the temperature range 10 K < T < 10 K (see Fig. 8.1), and Λ0 ≈ 2 × 10 erg cm −1 7 s for T0 = 10 K. Since post-shock gas is the densest and coolest just behind the shock, we expect that most of the energy loss occurs there.
The post-shock temperature near rsh is
where we have used the self-similar solutions (8.159) and (8.173). Inserting this and Eq. (8.185) into Eq. (8.184), and using γ = 5/3, we have
8.6.1 Blast Waves - 2nd phase
2 2 2 2 Replacing n H(r) by its rms value within rsh,n H ≈⟨n H⟩=ζn H,1, where nH,1 is the number density of hydrogen in the pre-shock gas, and ζ ≈ 2.29 for the adiabatic (Sedov) solution, we finally obtain
If we define trad by floss(trad) = 1/4, then for fully ionized gas with nHe = nH/12,
where we have used the values of A, λ, T0 and Λ0 given above. Note that vsh(trad) is almost independent of ε51 and nH. 8.6.1 Blast Waves - 3rd phase
For t > trad, the adiabatic model is no longer valid. The problem should then be solved by including the cooling term, −(γ − 1) C /P, to the right-hand side of the energy equation (8.169). This in general introduces extra scales and so the problem no longer admits self-similar solutions.
At t = tsp ≫ trad, however, simple arguments can again be used to give some useful results. During this late stage, the pressure inside the shock is negligible, and the remnant resembles a ‘snowplow’, in which ambient gas is swept up by the inertia of the moving shell.
3 In this case, the evolution of the remnant is governed by momentum conservation: Mshvsh = constant. Since Msh ∝ r (most −3 1/4 gas is swept up), we have rsh/t ∝ vsh ∝ r , and thus rsh ∝ t . Numerical calculations give a somewhat different result, rsh ∝ t . This is due to the fact that the internal pressure is not entirely negligible, which slightly increases the momentum. 8.6.1 Blast Waves
We have seen that a supernova explosion can accelerate the gas in its surrounding. An interesting question is, what fraction of the total explosion energy is transformed into kinetic energy of the gas (the rest is radiated away). The final kinetic energy in the remnant is given by the mass and velocity at the time when it fades into the interstellar medium: Ekin 2 = (1/2)Mfadev . If we denote the mass and velocity at the onset of the ‘snowplow’ phase by Msp and Msp, then
where we have used that momentum is conserved during the ‘snowplow’ phase, and assumed that another 1/4 of the total energy is radiated away between trad and tsp .
−1 −1 If we take vfade = 10 km s (the typical velocity dispersion of the ISM) and vsp = 100kms (half the value at trad), then fkin ∼ 0.05. Although this number is fairly uncertain, it is clear that only a relatively small fraction of the total explosion energy is ultimately transferred to kinetic energy. 8.6.1 Blast Waves - Supernova Heating
So far we have only considered the effect of a single blast wave on the surrounding gas. In reality, there may be several or many explosions confined to a small region. In this case, individual shocks can overlap and be thermalized, thereby heating the gas. How much of the initial explosive energy can be thermalized depends on the time scale of thermalization relative to the cooling time scale.
Suppose that the thermalized energy is a fraction fth of the explosive energy ε0. In this case the volume heating rate due to explosions can formally be written as
where
is the mass injection rate per unit volume (with n ̇ blast the rate of explosions per unit volume), and
is the effective temperature of the ejected material. 8.6.1 Blast Waves - Supernova Heating
The value of fth may be estimated by assuming that thermalization occurs at a time when the volume filling factor of the supernova remnants approaches unity. The volume filling factor of supernova remnants can be defined as
′ ′ ′ where νSN(t) is the supernova rate per unit volume at time t and VSNR(t −t) is the volume of a supernova remnant at the age t ′ −t. With the model described above, one can estimate PSNR as a function of t for given νSN(t), and thus identify the time tov at which PSNR = 1. The value of fth then follows from
′ ′ Here εSNR(tov −t) is the sum of the kinetic and thermal energy of a supernova remnant at an age of tov −t, and we have 51 assumed that each supernova releases a total of 10 ergs. If νSN(t) is very high, such as in starbursts, the individual bubbles can start to overlap before the supernova remnants have reached their radiative stages, and fth can be close to unity. If, on the other hand, νSN(t) is low, then bubbles only overlap when the supernova remnants are already in the snowplow stage and have already radiated away most of their energy. In this case fth ∼ fkin ∼ 0.05. 8.6.2 Winds and Wind–Driven Bubbles
In some cases, the energy injection from a source occurs over an extended period of time, rather than instantaneous. Examples in this category include stellar winds driven by the radiation pressure of stars, and galactic winds driven by multiple supernova explosions associated with extended periods of star formation. Blastwave model, in which energy injection is assumed to be an explosion, is no longer valid.
In order to understand how a long-lasting wind propagates and interacts with its surrounding medium, let us consider the following idealized case. Suppose that at time t = 0 a point source begins to blow a spherically symmetric wind with some terminal velocity vw and mass-loss rate dMw/dt. The power of the wind is given by the mechanical luminosity,
For a steady wind, both vw and dMw/dt are constant, but in general they may depend on time. We assume the wind to be cold, so that the sound speed in the wind is much smaller than the terminal velocity, and that the source is embedded in an ambient cold medium of constant density, ρ0. Throughout we also adopt γ = 5/3. Our task is to obtain the structure of the interaction between the wind and the ambient medium. 8.6.2 Winds and Wind–Driven Bubbles
The dynamical system in consideration should consist of four distinct zones:
Region I (r < r1): the hypersonic wind. In this region the wind is propagating with the terminal velocity vw.
Region II (r1 < r < rc): hot, shocked wind. This is material that is part of the wind, but that has been shocked.
Region III (rc < r < r2): a shell of shocked interstellar gas. This is material from the ambient medium that has been shocked by the wind.
Region IV (r > r2): the ambient medium which has not yet been affected by the wind.
Thus, in this case there are two shocks, one at r1 and the other at r2. 8.6.2 Winds and Wind–Driven Bubbles
During the early stages of the evolution, radiative losses are everywhere negligible. In this case, if the wind is assumed to be steady, so that Lw is a constant, the only dimensionless variable composed of Lw, ρ0, the radial coordinate r and time t is
with A a constant of order unity whose exact value remains to be determined. We expect that the problem admits self-similar solutions, and so all quantities should depend on r and t only through λ . In this case, the fluid equations to be solved reduce to the set of ordinary differential equations (8.170) - (8.172) with η = 3/5. With the jump conditions given in Eq. (8.173), this set of equations can be integrated numerically (see Weaver et al.,
1977) to obtain the structure of the gas distribution inside r2.
Unlike the blastwave solution, the gas density drops rapidly to zero at a radius rc = 0.86r2, where there is a contact discontinuity separating the swept-up gas from the shocked wind. At this radius the velocity is v(rc) = 0.86v2, where v2 = dr2/dt, and the pressure is P(rc) = 0.59ρ0v2.
Note that v(rc) is the velocity of the gas at r = rc, which should not be confused with r ̇ c, the velocity with which the contact discontinuity propagates. Since region III admits a self-similar solution with rc = 0.86r2, we have that r ̇ c = v2.
8.6.2 Winds and Wind–Driven Bubbles
In region II, the gas is approximately isobaric, because the temperature of the shocked wind is so high that the time for a sound wave to cross the region is much smaller than the age of the system. The density in this region is roughly uniform.
Both the gas density and pressure can be estimated from the jump conditions at r1:
− 2 where ρ(r1 ) = (dMw/dt)/(4πr1 vw) is the density of the freely propagating wind just interior to the shock radius r1. Note that r1 may depend on time, and so both the density and pressure are time dependent.
γ −4/5 In order to determine r1, we use the adiabatic condition d(P/ρ )/dt = 0. Since P ∼ P(rc) ∝ t , we have
With the boundary condition that v(rc) = 3rc/5t, the solution of the above equation is
2 On the other hand, in the region r1 ≤ r ≪ rc, the flow is nearly steady, and so v ∼ (vw/4)(r1/r) , where we have used the jump + condition v(R 1 ) = vw/4. Matching this velocity with the above solution and using the fact that r ≪ r , we get r ∼ (44/25)1/2r3/2/(v t)1/2.
Inserting this into the expression of P in Eq. (8.201) and matching the pressure thus obtained with P(rc ), we obtain A ≈ 1/5 0.87. Note that r2/r1 ∝ t , so that the shock at r2 propagates faster than that at r1. 8.6.2 Winds and Wind–Driven Bubbles
At some later stage, the cooling in the swept-up gas becomes important so that it collapses into a thin shell, while the cooling of the shocked wind is still negligible. Thus, the system then consists of a thin expanding shell enclosing and driven by a hot bubble, whose internal energy is much larger than its kinetic energy. At this stage, the time for sound waves to cross the hot bubble is still small compared to the age, so that the entire region is approximately isobaric. Since the volume of region I is much smaller than that inside r2, the pressure is related to the internal energy simply by
Assuming the shell to be infinitesimally thin and the mass in the bubble negligible, the momen- tum equation for the shell can be written as 8.6.3 Supernova Feedback and Galaxy Formation
In general, the full set of fluid equations (8.155) - (8.157) has to be solved, in order to study how a wind is generated by energy and mass sources, and how it evolves with time. Here we consider a simple model to demonstrate qualitatively how supernova explosions may drive galactic winds and affect star formation in galaxies. The wind is assumed to be spherical and steady, propagating in a static, spherically symmetric potential, Φ. Under these assumptions, the fluid equations (8.155) - (8.157) can be combined to give two first order differential equations for the fluid velocity, v , and the adiabatic sound speed of the gas, w:
2 where Vc ≡ r(dΦ/dr) specifies the shape of the gravitational potential well, and the injected gas is assumed to have an 1/2 initial isothermal sound speed, wi ≡ (kBTi/µmp) , with Ti the initial temperature of the injected gas. The quantities A and B are given by
where ρ ̇ inj is the mass injection rate per unit volume. As one can see, the sonic point, where v = w, is a critical point at which v2(r) and w2(r) are not smooth functions of r. Note also that Eqs. (8.208) and (8.209) are invariant under the transformation v → −v , so that they describe both outflow and inflow (depending on the boundary conditions). 8.6.3 Supernova Feedback and Galaxy Formation
In some applications, the source term, ρ ̇ inj, is non-zero only within a confined region near the center of the potential well (i.e. that of a dark matter halo), where stars form. In such cases, one can separate a halo into an inner heating base and an outer region. The equations describing the flow in the outer region are Eqs. (8.208) and (8.209) with B = 0. For supersonic flows, which are most relevant to the large-scale outflows from galactic systems, the boundary conditions 2 2 can be set at the radius r1 where v is slightly above w . If radiative cooling is negligible in the flow, so that A = 0, the properties of the flow are determined by the gas temperature relative to the halo gravitational potential at the heating base.
Depending on whether the sound speed near r1, w1, is bigger or smaller than Vc(r1)/sqrt(2), the outflow results in either a galactic wind or a hot corona. √ Numerical integrations of Eqs. (8.208) and (8.209) show that the wind can reach a bulk velocity of about 2.5w1 (Efstathiou, 2000). √ Thus, if w1 >∼ Vesc/ 2.5, where Vesc is the escape velocity from the central part of the halo, the wind may escape; otherwise
a hot corona is produced. The importance of radiative cooling is specified by A(r1), which is roughly the ratio between the 2 2 flow time (r/v) and the cooling time (ρw /Λn H) near r1. If A(r1) ≫ 1 so that cooling is effective, the outgoing gas can cool and, via thermal and hydrodynamical instabilities (see §8.5), may form cold clouds that either leave the galaxy (if w1 >∼ √ √ Vesc/ 2.5), or fall back (if w1 < Vesc/ 2.5) 8.6.3 Supernova Feedback and Galaxy Formation
The gas temperature at the heating base is determined primarily by the intensity of the supernova heating, which depends on the supernova rate per unit volume, and the cooling rate of the shocked gas.
If these rates were independent of halo mass so that the temperature at the heating base does not depend strongly on Vc,
then the ratio w1/Vesc would be larger for less massive halos, making outflows easier to generate in halos of lower mass.
In order to quantify the conditions for gas removal from dark matter halos, Dekel & Silk (1986) considered a simple
model in which the star formation rate (the mass that turns into stars per unit time) in a galaxy is assumed to be M ̇ ⋆ = Mg/
(εSF tff), where Mg = fgM is the mass of cold gas in the galaxy, εSF is a constant specifying the star formation efficiency, and tff ≡ [3π/(32Gρ)]−1/2 is the free-fall time, with ρ the mass density. With the assumption that the star formation rate is a constant, the number of supernova remnants at time t can be written
as NSN(t) = µSNM ̇ ∗t, where µSN is the number of supernovae corresponding to a unit mass of stars that have formed. The total energy in the supernova remnants can then be written as
with εSNR(τ) the total energy of a supernova remnant at the age τ. 8.6.3 Supernova Feedback and Galaxy Formation
In the adiabatic phase, εSNR(τ) = (1 − floss)ε0 with floss defined in Eq. (8.184). At later times, the energy content of a supernova −2 remnant is roughly εSNR(τ) ∼ 0.22ε0[rsh(τ)/rsh(trad)] (Cox, 1972). Dekel & Silk (1986) argued that gas can be removed from a
dark matter halo if the total energy of the su- pernova remnants at time tov, when PSNR ∼ 1, is larger than the binding energy of the gas:
1 2 E(tov) > MgVc . This defines a critical value for Vc:
so that gas removal occurs in halos with Vc < Vcrit. Using the results for the evolution of super- nova remnants described −1 above, Dekel & Silk (1986) found Vcrit ∼ 100 km s . 8.6.3 Supernova Feedback and Galaxy Formation
If the energy input from star formation is equal to the binding energy of the cold gas, the star formation rate, M ̇ ⋆, is given
2 by E0M ̇ ⋆ = (M ̇ g −M ̇ ⋆)Vc /2, where E0 measures the energy feedback per unit mass of formed stars, and the right-hand side is
a crude estimate of the binding energy of the cold gas. Solving for M ̇ ⋆ we obtain
2 where V0 = 2E0.
Based on the discussion presented above, we have V0 ∼ Vcrit. In this simple model, the star formation efficiency in halos 2 with Vc ≪ Vcrit is reduced by a factor proportional to Vc . It is necessary to suppress the efficiency of star formation in low-mass halos in order to explain the observed galaxy luminosity function at the faint end in the CDM scenario of galaxy formation. Star formation feedback through supernova explosions provides an appealing mechanism.
Unfortunately, the details regarding this feedback mechanism have yet to be quantified. It is still unclear how effective the energy feedback from star formation is coupled to the gas. Some numerical simulations show that the coupling is rather poor so that much of the feedback energy can escape from a galaxy without affecting the bulk of the gas (e.g. Mac Low & Ferrara, 1999a). Furthermore, the evolution of supernova remnants in real star forming regions is expected to be much more complicated than that given by the simple model described above, so that the fraction of supernova energy available to drive a potential galactic wind is also uncertain. 10.5.2 Gas Dynamics Including Stellar Feedback
The general description of the evolution of gaseous halos in terms of the fluid equations needs to include the source terms that describe the gas consumption due to star formation, as well as the energy and mass injection due to stellar evolution. Assuming that the injected material is well mixed with the interstellar gas so that the total gas can be considered as a single fluid, the fluid equations are given by Eqs. (8.155) - (8.157).
We now describe the source terms Sm, Smom, Se, which are the changes per unit time in mass density, momentum density, and energy density, that are due to the consumption of gas by star formation and the injection of gas by stellar winds and supernovae. For simplicity, we consider scales that are much larger than individual stars, so that the velocities of the injected material average out. In other words, we assume that the kinetic energy of the injected material has already been
thermalized, which allows us to set Smom = 0. 10.5.2 Gas Dynamics Including Stellar Feedback
Let ψ(x,t) describe the local specific star formation rate (per unit mass), then for a given IMF, φ(m), we can write:
is the local specific rate of mass feedback, with f(ej)(m,t)dt the fraction of the star’s initial mass m that is returned to the ISM in the time interval [t , t + dt ].
(ej) (ej) If we assume that most of the mass is ejected from a star at the end of its lifetime, then f (m,t) ≈ δ(t −τm)f (m), with τm the lifetime for a star of mass m, and
where f(ej)(m) is the energy per unit mass that is ejected by a star with mass m at the end of its life. 10.5.2 Gas Dynamics Including Stellar Feedback
For supernova explosions where f (ej) is independent of m, the rate of energy injection per volume reduces to:
where RIa(x) and RII(x) are the local specific rates for Type Ia and Type II SNe, respectively.
For a given IMF, we can use the mass return, the supernova rates and the energy feedback to calculate both Sm and Se.
The stellar mass loss provides most of the feedback mass but little energy, while supernova explosions provide most the kinetic energy but relatively little mass.