–1– 15. Snrs, STELLAR WIND BUBBLES, and the HOT ISM 15.1

– 1 –

15. SNRs, STELLAR WIND BUBBLES, AND THE HOT ISM

15.1. Blast Wave Dynamics

15.1.1. Equation of motion

Energy injection by stars—H II regions, stellar winds, supernovae—leads to supersonically expanding gas in the ISM. These structures are termed blast waves. Here we derive an approximate equation of motion for the blast wave, allowing for momentum injection by the central star. We assume that the medium into which the blast wave advances is spherically symmetric, and we assume that magnetic and gravitational forces are negligible.

Let Rs(t) be the radius of the shock at time t, and let R0 = Rs(t) + ²(t) be a constant radius just outside Rs(t) (obviously, R0 can be both constant and outside Rs for only a short period of time if ² is small, as we assume). Let M be the mass of gas inside Rs; it grows both because the shock is sweeping up interstellar gas and because the star is injecting mass. Let M0 be the mass inside R0; it changes only because of mass injection by the central star,

M˙ 0 = M˙ in. (1)

Let v¯ be the mean magnitude of the velocity in the blast wave. The total magnitude of the momentum is M0 Mv¯ = v dM, (2) Z0 so that d M0 dv Mv¯ = dM + M˙ v , (3) dt dt in in Z0 where vin is the magnitude of the velocity of the injected gas. We then have d M0 1 ∂p Mv¯ = − dM + M˙ v , (4) dt ρ ∂r in in Z0 R0 ∂p = − 4πr2dr + M˙ v , (5) ∂r in in ZRi R0 2 2 ˙ = − 4πR0p(R0) − 4πRi p(Ri) + 8π prdr + Minvin, (6) ZRi £ ¤ where Ri ¿ R0 is the radius at which momentum in injected into the ﬂow by the star. We now 2 make three approximations: (1) the shock is strong, so that p(R0) is negligible; (2) 4πRi p(Ri) is negligible, since Ri is small; and (3) R0 Rs 1 prdr ' p¯ rdr = pR¯ 2, (7) 2 s ZRi Z0 where 1 Rs p¯ ≡ pdV (8) V Z0 – 2 – is the mean pressure in the blast wave. With these approximations, the equation of motion for the blast wave reduces to d Mv¯ ' 4πR2p¯ + M˙ v . (9) dt s in in This equation becomes exact for a thin shell: If almost all the mass were concentrated in a thin shell at Rs, then the remaining mass in the interior would have to be very hot in order to exert a signiﬁcant pressure, so that the interior pressure would be uniform.

15.1.2. Self-similar solutions

For a point explosion (a supernova) or for a stellar wind in a spherically symmetric medium, there is no length or time scale. As a result, the flow is self-similar: the flow at any time is a scaled version of the flow at another time: R v(r, t) = s f(λ), (10) t ρ(r, t) = ρ0g(λ), (11) R 2 p(r, t) = ρ s h(λ), (12) 0 t µ ¶ where Rs(t) is the radius of the shock and λ ≡ r/Rs(t). Note that this does not apply if there is a scale in the problem—e.g., if ρ0 ∝ exp(−Rs/h). Real supernova remnants (SNRs) and stellar wind bubbles go through several stages, most of which can be approximated by a self-similar solution. For example, the self-similar solution for a point explosion (the model for an SNR) requires that the swept-up interstellar mass greatly exceed the mass ejected by the supernova, and that the shock be highly supersonic. If radiative losses from the shocked gas are negligible, the blast wave is termed a Sedov-Taylor blast wave after the individuals who discovered the similarity solution. If radiative losses are dominant, then one can have either a pressure-driven snowplow or a momentum- conserving snowplow.

The thermal and kinetic energies in a self-similar blast wave are

Rs p(r, t) 4πρ R5 1 E = 4πr2dr = 0 s h(λ)λ2dλ, (13) th γ − 1 (γ − 1) t2 Z0 i i µ ¶ Z0 Rs 1 4πρ R5 1 E = ρv2 4πr2dr = 0 s g(λ)f 2(λ)λ2dλ. (14) k 2 2 t2 Z0 µ ¶ Z0

We have distinguished the value of the ratio of speciﬁc heats in the interior of the blast wave, γi, from that at the shock, γ, in order to treat the case of radiative shocks below. Since Ek and Eth 5 2 scale as Rs/t , it follows that we can express Rs(t) in terms of the total energy E = Eth + Ek, E R5 ∝ t2. (15) s ρ µ 0 ¶ – 3 –

Introducing the dimensionless constant ξ, we have

ξE 1/5 R = t2/5. (16) s ρ µ 0 ¶ This is the fundamental solution for the dynamics of a blast wave. We shall assume that ρ0 is −kρ a constant (although the solution is readily generalized to the case in which ρ0 ∝ Rs ). If the energy scales as a power of the time, E ∝ tηE , then 2 + η R ∝ tη with η = E . (17) s 5 It follows that the shock velocity is dR R d ln R ηR v = s = s s = s . (18) s dt t d ln t t

We can use the equation of motion for the blast wave to derive an approximate value of the coeﬃcient ξ, thereby completing the solution. We make one additional approximation: that the mass is swept up into a thin shell, so that the mean velocity of the shocked gas is approximately equal to the post-shock velocity 2 v¯ ' v = v . (19) ps γ + 1 s 5 2 Together with the results from self-similarity (vs = ηRs/t, Rs = ξEt /ρ0, and ηE = 5η − 2), this suﬃces to determine the solution.

Let the thermal energy be a fraction β of the total, so that

3 4πRs Eth = βE = p.¯ (20) 3(γi − 1) Inserting these results into the equation of motion for the blast wave (eq. 9), we ﬁnd

d 4πρ R3 2η R (γ − 1)βE 0 s s = 4πR2 i . (21) dt 3 γ + 1 t s 4πR3/3 µ ¶ s Evaluating the derivative and making use of the self-similarity of the solution (vs = ηRs/t and 5 2 Rs = ξEt /ρ0), we ﬁnd 9(γ − 1)(γ + 1) ξ = i β. (22) 8πη(4η − 1)

We also have that the total energy is E = Ek +Eth. With the approximation that the mean velocity of the shell is vps, we have 1 4πR3ρ 2 2 E ' s 0 v2 + βE. (23) 2 3 γ + 1 s µ ¶ Using the self-similarity again, this gives 8πη2 1 = ξ + β. (24) 3(γ + 1)2 – 4 –

Together with equation (22), this yields our approximate result for ξ:

− 3(γ + 1)2 (γ + 1)(4η − 1) 1 ξ ' 1 + . (25) 8πη2 3η(γ − 1) · i ¸

Supernova remnants

When the blast wave has swept up a mass large compared to the mass ejected in the supernova explosion, an SNR can be modeled as a point explosion. There are two cases: energy-conserving and radiative.

Case 1: Sedov-Taylor blast wave. This is an energy-conserving blast wave, so that ηE = 2 5 0 ⇒ η = 5 and γi = γ. For γ = 3 , equation (25) gives ξ = 50/9π = 1.77, within 15% of the 2 exact value, ξ = 2.026. Similarly, we ﬁnd β = 3 , whereas the exact value is β = 0.717. Case 2: Radiative blast wave. If the shock is radiative, then the rate of energy loss is given by dE 1 = −4πR2 ρ v3 . (26) dt s 2 0 s µ ¶ 5 2 Inserting vs = ηRs/t and Rs = ξEt /ρ0 into this expression and using equation (17), we ﬁnd

3 ηE = 5η − 2 = −2πη ξ. (27)

Since the outer shock is radiative, we set γ = 1; on the other hand, the interior is hot and 5 approximately non-radiative, so that γi = 3 . Our approximate result for ξ (eq. 25) gives − 3 4η − 1 1 ξ = 1 + . (28) 2πη2 η µ ¶ Combining this with equation (27), we ﬁnd

3η2 5η − 2 = − , (29) 5η − 1

1 2 which has two solutions, η = 4 and η = 7 .

1 1/4 Case 2a: Momentum-conserving snowplow, η = 4 . For Rs ∝ t , the total momentum scales as R R4 Mv ∝ R3 s ∝ s = constant, (30) s s t t µ ¶ so that the total momentum is constant. Correspondingly, equation (22) implies that

Eth = βE = 0 since ξ is ﬁnite. – 5 –

2 2 4 Case 2b: Pressure-driven snowplow, η = 7 . For η = 7 , we have ηE = 5η − 2 = − 7 :

R ∝ t2/7 ⇒ E ∝ t−4/7 ∝ R−2. (31)

The interior thus expands adiabatically, since

pV 5/3 ∝ EV 2/3 ∝ ER2 = constant. (32)

A pressure-driven snowplow evolves into a momentum-conserving snowplow only after the interior undergoes radiative cooling.

Stellar wind bubbles

OB stars produce powerful stellar winds with velocities in excess of 1000 km s−1. The rate at which they inject energy into the surrounding ISM is given by the wind luminosity, 1 L = M˙ v2 . (33) w 2 w w When a stellar wind impacts the ambient ISM, it produces two strong shocks: one heats the wind to temperatures in excess of 107 K and one advances into the ambient ISM. Bubbles with non- radiative interiors can be treated by the same formalism as SNRs; radiative bubbles require the inclusion of the M˙ in term in the equation of motion.

Case 1: Adiabatic bubble. The energy in an adiabatic bubble is E = Lwt. We assume that 3 5 the wind luminosity is constant, so that ηE = 1 and hence η = 5 . For γ = γi = 3 , equation (25) gives ξ = 200/111π = 0.574. We therefore obtain

L 1/5 R ' 0.895 w t3/5. (34) s ρ µ ¶ The exact solution has a coeﬃcient of 0.88, so our approximate solution is quite good.

Case 2: Bubble with radiative outer shock. If the outer shock is radiative but the interior is 5 not, then γ = 1 and γi = 3 . Since the gas is swept up into a thin shell, our solution should be exact. Note that the net rate of increase of the energy in the bubble is dE 1 = L − 4πR2 ρ v3. (35) dt w s 2 0 s For a self-similar solution, the second term must be constant in order to scale with the ﬁrst 3 term. We observe that this will be true provided η = 5 : the exponent is unchanged from the adiabatic case. Integrating with respect to time and inserting the self-similar form of the solution, we ﬁnd L t E = L t − 2πη3ξE ⇒ E = w . (36) w 1 + 2πη3ξ – 6 –

Evaluating ξ from equation (25), we ﬁnd ξ = 5/4π, so that 50 E = L t : (37) 77 w In contrast to the case of an explosive blast wave, a bubble with a radiative outer shock loses only a small fraction of the energy (27/77=35%) and therefore is only slightly smaller than an adiabatic bubble, L 1/5 R = 0.763 w t3/5. (38) s ρ µ 0 ¶ This agrees with the similarity solution of Weaver et al. (1977).

Case 3: Radiative bubble. If both the shock and the interior are radiative, then the equation of motion (9) reduces to d 2Lw Mv¯ = M˙ wvw = , (39) dt vw

where we have speciﬁed the injection as being due to a wind (M˙ in → M˙ w, vin → vw). Since the outer shock is radiative, the swept-up gas is in a thin shell and v¯ = vs = ηRs/t. Integrating 4 2 1 this equation leads to Rs ∝ t , so that η = 2 . Evaluating the constants, we ﬁnd

3L 1/4 R = w t1/2, (40) s πρ v µ 0 w ¶ in agreement with Steigman et al (1975).

15.2. Supernova Remnants

Supernovae (SNe) inject about 1051 erg of energy into the ISM. Type II and Type Ibc SNe are the products of massive stars, and inject O(10 M¯) of mass, much of it in the form of heavy elements, into the ISM. Type Ia SNe result from white dwarfs in binaries being driven over the Chandrasekhar mass limit, and inject about 1.4 M¯ of heavy elements into the ISM. Supernova remnants (SNRs) create hot gas in the ISM, drive turbulence, and accelerate cosmic rays. In the ideal case of a SNR in a uniform medium, an SNR goes through 4 well-deﬁned stages of evolution:

1. Free expansion, or ejecta-dominated;

2. Sedov-Taylor: non-radiative blast wave;

3. Radiative blast wave: pressure driven snowplow;

4. Merger with ISM. – 7 –

This simple picture is complicated by the fact that young SNRs often ﬁnd themselves immersed in circumstellar matter. Massive progenitors have stellar wind bubbles, so that after the SNR expands beyond the circumstellar material, it goes into a very low density medium that is surrounded by a dense shell of gas. To complicate matters further, SNRs accelerate cosmic rays, which can aﬀect the dynamics; the ambient medium is often very inhomogeneous; and in addition supernovae can occur in clusters.

15.2.1. Ejecta-dominated SNRs

3 So long as the mass of the ejecta is large compared with the swept-up mass 4πρ0Rb /3, most of the ejecta expand freely so that Rb ' vejt and

1/2 1 2 4 E51 −1 E = Mejvej ⇒ vej = 1.00 × 10 km s . (41) 2 M /M¯ µ ej ¶ During this stage, there are actually two shocks, the blast wave shock advancing into the ambient medium and a reverse shock moving back into the ejecta. The ejecta are cold due to adiabatic expansion, and the reverse shock transmits the information that the free expansion must halt due to the ambient medium.

The ejecta-dominated stage lasts until the swept-up mass is comparable to Mej, which occurs at a radius 1/3 M /M¯ R = 1.9 ej pc. (42) swept n µ 0 ¶ −2 −3 If the SNR is expanding into a low-density stellar wind bubble with n0 ∼ 10 cm , this radius can be quite large, −2 −3 1/3 Mej 10 cm Rswept ' 19 pc. (43) 10M¯ n µ 0 ¶

15.2.2. Sedov-Taylor stage

Once the swept-up mass dominates the mass inside the blast wave and before radiative losses become important, the SNR enters the Sedov-Taylor stage. We obtained approximate estimates for the dynamics of this stage above. With the exact value of ξ = 2.026 for a blast wave in a uniform medium, one ﬁnds

2.026E 1/5 E 1/5 R = t2/5 = 0.314 51 t2/5 pc, (44) b ρ n yr µ 0 ¶ µ 0 ¶ 2 R E 1/5 v = b = 1.23 × 105 51 t−3/5 km s−1, (45) b 5 t n yr µ 0 ¶ – 8 –

3 µv2 E 2/5 T = b = 2.09 × 1011 51 t−6/5 K. (46) b 16 k n yr µ 0 ¶

Uniﬁed solution. For a given density distribution in the ejecta, ρej(r), which is determined by the density distribution in the pre-SN star (Matzner & McKee 1999), the evolution of an SNR in a uniform medium is determined by three dimensional quantities, E, Mej, and ρ0. These in turn deﬁne unique scales for the length, mass and time,

5/6 M 1/3 R M R ∝ ej , M ∝ M , t ∝ ST ∝ ej . (47) ST ST ej ST 1/3 ρ0 vej 1/2 µ ¶ E ρ0 It is therefore possible to describe the evolution of all non-radiative SNRs with a single solution (Truelove & McKee 1999; erratum, 2000). For example, an analytic approximation for the blast wave radius for uniform ejecta is 2.01(t/t ) ST (t ≤ t ), R (t) 3/2 2/3 ST b ' [1 + 1.72(t/tST) ] , (48) R  2/5 ST  1.42 t − 0.254 (t ≥ t )  tST ST h ³ ´ i where  5/6 Mej tST = 0.495 , (49) 1/2 1/3 E ρ0 M 1/3 R = 0.727 ej . (50) ST ρ µ 0 ¶ Similar results can be given for the location and velocity of the reverse shock in the ejecta. These results are useful since many of the observed historical SNRs, such as Cas A and Tycho, are at an intermediate stage of evolution in which the ejecta mass is comparable to the swept-up mass. However, real SNRs are more complicated, and may be expanding into non-uniform circumstellar media (see Borkowski et al 1996 for such a model of Cas A).

Effect of cosmic rays. SNRs are believed to be the source of Galactic cosmic rays. About 10% of the SNR energy must go into cosmic rays in order to account for their observed intensity. Cosmic ray acceleration can have a significant effect on the evolution of SNRs since they have a 4 5 softer equation of state (γ = 3 for relativistic particles instead of γ = 3 for non-relativistic ones) and because they can leak ahead of the shock. In Tycho’s SNR, X-ray observations show that the blast wave radius, Rb, the radius of the contact discontinuity between the ejecta and the shocked ISM, Rc, and the radius of the reverse shock are in the proportion 1 : 0.93 : 0.71 (Warren et al 2006). An upper limit on Rc/Rb can be obtained by assuming that the shock does not decelerate. 5 Then, for γ = 3 , the compression is 4 and

4π 4π R 3 1/3 4ρ (R3 − R3) = ρ R3 ⇒ c = = 0.909. (51) 3 0 b c 3 0 b R 4 b µ ¶ – 9 –

As the blast wave decelerates, the pressure declines and the the shocked ISM expands, so that

Rc/Rb declines further below the observed value. Warren et al. suggest that this discrepancy is due to energy being drained from the gas into the cosmic rays. They did not ﬁnd any evidence for cosmic ray acceleration at the reverse shock, consistent with the expectation that the magnetic ﬁeld there should be very weak due to the expansion.

15.2.3. Radiative SNRs

Hot gas with normal abundances cools primarily to due collisionally excited line emission. The cooling function of hot, collisionally ionized gas can be approxmated by

Λ ' 1.6 × 10−19T −1/2 erg cm3 s−1 for 105 K . T . 107.5 K. (52)

The total radiative luminosity of an SNR is

2 3 3 −1/2 3 −1 9/2 L ∝ n0Rb Λ(Tb) ∝ Rb Tb ∝ Rb vb ∝ Rb . (53) As a result, radiative cooling sets in rather abruptly when the cooling time is comparable to the age of the SNR. At this point, the evolution approximates that of a pressure-driven snowplow 2/7=0.286 (Rb ∝ t ). Cioﬃ et al. (1988) carried out numerical calculations and found an approximate solution for the radiative stage that joins smoothly onto the ST evolution at a time tPDS:

4 t 1 3/10 R = R − , (54) b PDS 3 t 3 · PDS ¸ − 4 t 1 7/10 v = v − , (55) b PDS 3 t 3 · PDS ¸ where

4 3/14 −4/7 tPDS = 1.33 × 10 E51 n0 yr, (56) 2/7 −3/7 RPDS = 14.0E51 n0 pc, (57) 1/7 1/14 −1 vPDS = 413n0 E51 km s . (58)

The time tPDS is deﬁned so that radiative losses result in the formation of a dense shell at a time 2.718tPDS. The solution is approximately valid for t . 35tPDS. Note that this numerical solution allows for the cooling of the hot gas in the interior, which was neglected in the analytic theory of the pressure-driven snowplow.

15.2.4. Merger with the ISM

The SNR stops expanding into the ISM when its expansion velocity becomes comparable to the sound speed. It actually overshoots somewhat, and then begins contracting as the interior gas – 10 – continues to cool. If we adopt the criterion that the SNR merges with the ISM when its expansion velocity is βC0, where β is a number of order unity and C0 is the isothermal sound speed of the ambient ISM, then the PDS solution gives

− 4t 7/10 v ' v merge = βC , (59) b PDS 3t 0 µ PDS ¶ 1/14 1/7 10/7 E51 n0 ⇒ tmerge ' 153 tPDS, (60) Ã βC0,6 !

−1 where C0,6 ≡ C0/(10 km s ).

15.2.5. Eﬀect of interstellar clouds

We have mentioned several complications that occur in real SNR evolution: ejection of circumstellar matter by the progenitor (or by previous SNe), processing of the ambient ISM by the progenitor, and the dynamical eﬀect of cosmic ray acceleration. Another eﬀect is that the ISM is inhomogeneous. In fact, the visible emission from SNRs such as the Cygnus Loop often arises from the denser parts of the ambient medium. A shock driven into a cloud has a post shock pressure that is comparable to that behind the blast wave–i.e., there is appoximate dynamical pressure balance:

2 2 ρcvsc ' ρ0vb (61)

(McKee & Cowie 1975; Klein et al. 1994; MacLow et al 1994). If the clouds occupy only a small fraction of the volume, then the blast wave propagates in the intercloud medium, and the clouds have a relatively small dynamical eﬀect.

15.3. Stellar Wind Bubbles

Prior to exploding as supernovae, massive stars inject energy into the ISM via powerful stellar winds. Just as in the case of the ejecta-dominated stage of SNR evolution, there are two shocks, one advancing into the ISM (radius Rb) and one decelerating the stellar wind (radius Rsw). The 2 wind shock occurs when the ram pressure of the wind, ρwvw, is comparable to the pressure behind 2 the blast wave shock, ρ0vb . There are four main stages for bubbles created by fast stellar winds, just as for SNRs:

1. Wind-dominated. This is important only at high densities. SNR analog: ejecta-dominated.

2. Non-radiative injection. This is the classical case analyzed by Castor et al (1975) and Weaver et al. (1977). – 11 –

a) Non-radiative outer shock, so that E = Lwt. The SNR analog is the Sedov-Taylor blast wave.

b) Radiative outer shock, so that E = (50/77)Lwt. The SNR analog is the pressure-driven snowplow. However, whereas the pressure-driven snowplow diﬀers signiﬁcantly from the Sedov-Taylor solution, the bubble with a radiative outer shock is dynamically very similar to the bubble with a non-radiative shock.

3. Radiative bubble: when the wind shock is radiative, the bubble is momentum conserving. The SNR analog is the momentum-conserving snowplow.

4. Pressure-conﬁned bubble (PCB): The bubble ceases expanding when it comes into pressure equilibrium with the ISM. Note that it is possible for bubbles to go from (2a) or (2b) directly to a PCB; it is not necessary to go through stage (3). The SNR analog is merger with the ISM.

The evolution of bubbles driven by slow stellar winds is quite diﬀerent: they begin life as radiative bubbles and can evolve into adiabatic winds. This distinction, and the criterion separating fast and slow winds, are discussed in Koo & McKee (1992).

If the ambient ISM is homogeneous, the dominant stage of evolution is stage (2a), non-radiative injection with a radiative outer shock. Assuming that stellar winds are homogeneous, Howarth & −6 6 1.69 Prinja (1989) determined that the mass loss rate for OB stars is M˙ w ' 5.5 × 10 [L/(10 L¯)] . However, recent studies have suggested that OB star winds are very clumpy, so that estimates of the mass loss rate based on emission, such as those of Howarth & Prinja, are too high by a factor of order 3 − 10 (Bouret et al. 2006; Fullerton et al. 2006). In any case, a star that loses mass at −6 −1 −1 a rate of 10 M¯ yr at a velocity of 2000 km s (which is typical) has a wind luminosity of 1.3×1036 erg s−1. The radius and expansion velocity of a wind bubble with a radiative outer shock is

L 1/5 R = 27 w, 36 t3/5 pc, (62) b n 6 µ 0 ¶ 1/5 L − v = 16 w, 36 t 2/5 km s−1, (63) b n 6 µ 0 ¶ 6 36 −1 6 where t6 ≡ t/(10 yr). A star with Lw ∼ 10 erg s has a lifetime of about 4 × 10 yr, so it −3 would expand to a radius of about 62 pc in a medium of density n0 = 1 cm . Castor et al (1975) included evaporation of gas from the inner edge of the swept-up shell, and found that the gas in the −3 6 interior has a typical density n0 ∼ 0.02 cm and a temperature T ∼ 10 K. However, the medium around OB stars is far from homogeneous, and mass will be injected into the interior of the bubble due to photoevaporation, leading to signiﬁcant radiative losses and probably preventing the bubble from expanding to the radius given above (McKee et al 1984). – 12 –

15.4. Supershells and Superbubbles

OB stars are generally concentrated in associations, so that they act collectively instead of individually. OB associations create large volumes of hot gas, called superbubbles, surrounded by large shells of H I, called supershells or galactic “worms” (Heiles 1984). Observations of extragalactic supershells do not always show evidence for a stellar association, however, and it is possible that supershells are also created by unusually energetic supernovae or by the impact of high-velocity H I clouds with the disk of a galaxy.

Superbubbles undergo several stages of evolution (McCray & Kafatos 1987):

1. Initially, the superbubble is driven by the combined eﬀect of many stellar winds.

2. After about 3×106 yr, supernovae begin to occur and go on for about 50×106 yr, the lifetime of the lowest mass (8M¯) and therefore longest-lived SN progenitor star. Note that this time is long compared to the lifetime of an O star (m∗ & 20M¯, lifetime . 9 Myr). In terms of the number of SN progenitors, N∗, the radius and expansion velocity of a superbubble (assumed to have non-radiative injection, but a radiative outer shock) are

1/5 N∗ E R = 200 51 t3/5 pc, (64) b 40 n 7 µ 0 ¶ 1/5 N∗ E − v = 12 51 t 2/5 km s−1. (65) b 40 n 7 µ 0 ¶ 3. When the superbubble radius expands to the thickness of the gaseous disk, it begins to expand primarily in the vertical direction. Heiles (1990) termed such superbubbles “breakthrough” bubbles, and pointed out that it took considerably more energy for a bubble to actually break out of the disk entirely. Koo & McKee (1992) estimated that it takes about 50 and 1000 SNe, respectively, for breakthrough and breakout bubbles; they pointed out that the actual number could be larger if radiative losses from the superbubble interior are important.

15.5. The Three-Phase Model of the ISM

Field, Goldsmith & Habing (1969) developed the two-phase model for Galactic H I, in which warm and cold H I are in pressure equilibrium. The three-phase model (McKee & Ostriker 1977; hereafter MO) incorporates the hot gas produced primarily by SNRs into a global model of the ISM. The central hypothesis of the three-phase model is that the hot gas is pervasive, and determines the thermal pressure of the cold and warm phases. Remarkably enough, 30 years after the theory was developed, the ﬁlling factor of the hot gas remains uncertain. – 13 –

15.5.1. Filling factor of hot gas

Cox & Smith (1974) determined the “porosity” of the ISM due to SNRs and showed that it could be signiﬁcant. Let dQ(t) be the probability that a given point in the ISM is inside an SNR of age t to t + dt, and let S be the SN rate per unit volume. Then Sdt is the number of SNRs per unit volume of age t to t + dt and dQ = SV (t)dt, where V (t) is the volume of an SNR at age t. The expected number of SNRs younger than t encompassing a given point is then

t Q(t) = S V (t0)dt0. (66) Z0 Note that the integral is just the four-volume of the SNR. The spatial ﬁlling factor of SNRs younger η than t is just Q if Q ¿ 1. If the SNRs expand as R ∝ t for t ≤ tm, then

4πSR3 tm t 3η V t Q(t ) = m dt = m m . (67) m 3 t 3η + 1 Z0 µ m ¶ To get the total ﬁlling factor of SNRs, QSNR, one must include the contraction phase as well. This depends on the cooling rate of the SNR interiors. The maximum contraction rate is about the sound speed of the ambient medium, which one can show (McKee 1990) leads to QSNR = 1.36Q(tm). The results of Cioﬃ et al (1988) then yield

1.26 S−13E51 QSNR = 0.78 0.11 1.36 , (68) n0 P0,4

−13 −3 −1 where S−13 ≡ S/(10 pc yr ). There are two diﬀering estimates for QSNR:

- MO took S−13 > 1, neglecting any correlations among SNRs. A contemporary estimate of the SNR rate is 3.8 × 10−11 core-collapse SN pc−2 yr−1 at the solar radius (McKee & Williams 1997); if these occur within a disk thickness of 300 pc (which is generous, since the scale height of OB stars is small), then S−13 ' 1.3. Since n0 ∼ 0.3 and P0,4 ∼ 0.3 from Field et al, and E51 ∼ 1, MO concluded that QSNR > 1: SNRs will overlap, and one must consider a model in which the hot gas is pervasive. Their model leads to QSNR ' 0.6.

- Slavin & Cox (1993) excluded SNe in clusters, since they make superbubbles and Heiles had shown that superbubbles have a filling factor ∼ 0.1; as a result, they estimated S−13 = 0.4. They also included a magnetic field of 5 µG, yielding a total ambient pressure P0,4 = 0.9. (They did not allow for the fact that the random component of the field has a reduced pressure, however.) Finally, they took E51 = 0.75. They concluded that QSNR = 0.18.

There are two ways of looking at this: It is clear that the theoretical estimates are very uncertain, so the fact that they give QSNR = 0.4 § 0.2 is encouraging agreement. On the other hand, there is a dramatic difference between a model in which hot gas is in isolated pockets filling about 20% of the volume and one in which it pervades the ISM and fills about 60% of the volume. – 14 –

15.5.2. Basics of the three-phase model

MO determined the properties of the ISM by balancing energy, mass and ionization in the hot gas, which they termed the HIM (hot ionized medium).

Energy balance. What happens to the energy injected into the ISM by SNRs? There appear to be four possibilities:

- It is radiated by the hot gas. This is the assumption of MO, and it still appears to be consistent with observation.

- It is radiated away by shocks in the cold and warm gas. This is the picture advocated by Slavin & Cox, and is also consistent with observation.

- The energy is advected into the halo by a “galactic fountain” (Shapiro & Field 1976). One can show that individual SNe are incapable of driving a galactic fountain, but large superbubbles can deposit energy in the halo. Observations suggest that only a small fraction of the SN energy goes into the halo, since the diﬀuse X-ray luminosity of spiral galaxies is small compared to the energy injection rate by SNe (E˙ ' 1042 erg s−1 for an SN rate of 1 every 30 years).

- The energy is advected away in a Galactic wind. This is impossible: at a temperature of 6 −1 about 10 K, the wind would have to have a mass loss rate of about 30M¯ yr in order to carry away the energy injected by SNe, which is far too large (Chevalier & Oegerle 1979).

MO described SNRs by their porosity; that is, SNRs of age t and size R correspond to a porosity Q. A critical value of the porosity is that at which radiative cooling sets in, Qc. MO determined this by calculating when half the energy would have been radiated by the hot gas,

Qc 1 βn2ΛdQ = SE, (69) 2 Z0 where β > 1 is the enhancement of the cooling over its equilibrium value due to inhomogeneities (hn2i > hni2), evaporation of cold gas by hot gas, and by non-equilibrium ionization. They adopted β ' 10.

Mass balance. If the porosity is small, then hot gas is created by shock-heating warm gas to high temperatures. On the other hand, if the porosity is large, then the SNRs will expand primarily into the hot gas, heating it to very high temperatures. Most of the mass of the ISM is in condensations that are much cooler and that will tend to evaporate into the hot gas. As a result, the bulk of the mass of hot gas is actually gas that has been heated by thermal conduction. MO balanced this rate of creating hot gas with an approximation of the eﬀects of radiative cooling: they assumed that a dense shell would form at Qc, made up of half the interior gas; for 1 > Q > Qc, the ambient hot gas would be swept up into the shell. They assumed that the mass of the hot – 15 –

1 gas would be regulated so as to make Qc ∼ 2 : it could not be much smaller, since the porosity is predicted to grow to large values if the medium is initially dominated by warm gas; and it cannot be & 1 since SNRs would overlap, heating the HIM further, increasing the rate of evaporation, and thereby causing SNRs to cool earlier (i.e., reducing Qc). Thermal evaporation of warm gas by hot gas plays a central role in the MO model. Such a process is known to be important in the solar corona, and direct observation of hot gas in the ISM suggests that it should be important there also. However, there has been very little direct evidence for cloud evaporation. High-resolution spectroscopic observations of gas in the Local Bubble shows that highly ionized gas has narrow line widths, suggesting that it is produced by photoionization (Welsh & Lallement 2005).

Ionization balance. An important implication of a large ﬁlling factor for the hot gas is that ionizing photons can travel much farther, producing a diﬀuse photoionized gas: the WIM (warm ionized medium). All HI clouds should be surrounded by WIM envelopes. MO adopted a spectrum of cold clouds based on optical absorption line studies by Hobbs (1974). His results showed that −3 dNcl/d ln a ∝ a , where a is the cloud thickness. MO assumed that the clouds are spherical; for clouds of constant density, this implies a mass spectrum d ln N m m cl ∝ = constant. (70) d ln m a3 They found that the radius of the WIM envelope was generally larger than the radius of the CNM cloud. Since the rate of evaporation depends on the cloud size, that means that the cloud evaporation rate is tied to the ionization balance. They estimated that the production rate of ionizing photons in the disk from isolated B stars, hot white dwarfs, and SNRs was

−15 −3 −1 ²UV ' 2 × 10 ionizing photons cm s . (71)

Ionization balance implies (2) 2 ²UV = α (nwxw) fw, (72) where nw is the density of the WIM, xw its ionization, and fw its ﬁlling factor. Finally, the optical depth nw(1 − xw)σHa is of order unity.

15.5.3. Properties of the three phases

MO adopted the following parameters from observation: A cloud size spectrum ∝ a−3; an −15 −3 −1 ionizing photon rate ²UV = 2 × 10 photons cm s ; a cloud temperature T (CNM)=80 K; a WIM temperature T (WIM)=8000 K; and a mean density n¯ = 1 cm−3. They assumed that the cloud mass spectrum had a minimum mass set by the condition that the WIM mass be half the cloud mass, since smaller clouds would be mostly or entirely photoionized and subject to more rapid destruction. They assumed that the upper limit on the cloud mass spectrum was set by – 16 –

gravitational instability, and they estimated a maximum radius of 10 pc. These parameters remain reasonable today.

Solving the equations of energy, mass and ionization balance, MO then predicted:

HIM Typical (i.e., median, not average) values are

−3 −3 5 −3 nh = 3.5 × 10 cm , Th = 4.5 × 10 K, p/k = 3700 K cm . (73)

WIM

−3 −3 nw = 0.25 cm , new = 0.17 cm , fw = 0.23, (74) 2 −6 −3 haWIMi = 2.1 pc, hnei = 0.0065 cm , hnei = 0.04 cm . (75)

The mean free path to a WIM cloud is predicted to be 12 pc.

WNM This phase was predicted to occur only in regions of low pressure:

−3 −3 nWNM = 0.16 cm , p/k = 1300 cm K, fWNM = 0.1 − 0.2. (76)

CNM −3 nc = 42 cm , fc = 0.024. (77) Note that this estimate of the CNM density appeared prior to the publication of A Hitchhiker’s Guide to the Galaxy. The radius of the smallest CNM cloud is predicted to be 0.38 pc and the mean free path to a CNM cloud is 88 pc.

15.5.4. Comparison with observation

- Turbulence: Extending an argument by Spitzer, MO showed that supernovae could generate the observed level of turbulence in the ISM, provided that the hot gas is pervasive. Since then it has been shown that the outer disks of galaxies are also turbulent, even though SNR stirring is much less important, so this is not a clear success of the model.

- Pressure: MO assert that the pressure of the HIM determines the pressure in the H I phases. Jenkins & Tripp (2001) infer a pressure of 2040 cm−3 K. However, as Wolfire et al. (2003) point out, they adopted a temperature of 40 K in the CNM, lower than observed; the actual pressure should therefore be higher. The observations are thus in reasonable agreement with the prediction, but it should be noted that a two-phase model of the ISM (Wolfire et al 2003) does an equally good job of predicting the thermal pressure without considering the hot gas at all. The three-phase model predicts a spectrum of pressure fluctuations, which have been observed, although with a somewhat different spectrum. – 17 –

- Cloud size spectrum. The cloud sizes predicted by the model are in reasonable agreement with observation. However, their geometry is generally not: while some spherical clouds have been observed, Heiles and Troland have inferred that clouds are typically sheet-like.

- WNM. The MO model signiﬁcantly underpredicts the amount of WNM, ∼ 4%, including the H I in the WIM, compared with observed value in the plane of about 16%. As yet, there is still not a quantitative theory that predicts the relative proportions of WNM and CNM.

- WIM: The prediction of the properties of the WIM is one of the major successes of the three-phase model. The model is in reasonable agreement with both observations of pulsar dispersion measures and of the Hα emission. Slavin et al. (2000) showed that emission from old SNRs can account for the observed level of emission in many directions in the Galactic halo; B stars and hot white dwarfs appear to provide a smaller contribution to the emission.

- Soft X-ray background: Slavin et al. (2000) have shown that old SNRs can account for the galactic component of the soft X-ray background. This is consistent with the three-phase model, but the calcuations did not rely on it. The three-phase model predicts that most of the X-ray emission from the hot gas will be absorbed in the disk of the Galaxy, which is consistent with observations of low X-ray luminosities of other disk galaxies.

- Filling factor of hot gas. As remarked above, this important interstellar parameter remains quite uncertain. It should be noted that analyses of interstellar absorption lines often ﬁnd that a signﬁcant fraction of the line of sight is not occupied by either warm or cold gas.

- Superbubbles. MO did not consider the fact that OB stars generally occur in associations and therefore produce correlated energy injection. This reduces the eﬀective value of the SN rate per unit volume.

- O VI. The O VI absorption lines, ﬁrst observed by the Copernicus satellite, were taken as a prime observational conﬁrmation of the existence of large amounts of hot gas, both by Cox & Smith and by MO. However, while the MO model predicts a total column of O VI comparable to what is observed, it predicts that it should be distributed in a larger number of components. In addition, as remarked above, the line widths of O VI in the Local Bubble are too narrow to be consistent with thermal evaporation.

15.6. REFERENCES

Shu, Vol 2, Ch 17

Tielens, Ch 12

Ostriker & McKee 1988, Rev. Mod. Phys.: Approximate analytic dynamics of blast waves.

Cox, ARAA 43: Review of the diﬀuse ISM