CLEARING THE GAS FROM GLOBULAR CLUSTERS AND DWARF SPHEROIDALS WITH CLASSICAL NOVAE

The stars in a galactic globular cluster (GC) lose mass during their evolution, injecting material into the GC that accumulates over time to make an intracluster medium (ICM) of mass M_ICM. The most robust mechanism for clearing this matter is ram pressure stripping during the GC's passage through the galactic disk every ∼10⁸–10⁹ yr, implying M_ICM ∼ 100–1000 M_☉ (Tayler & Wood 1975). However, searches for the ICM have yielded upper limits (Birkinshaw et al. 1983; Roberts 1988) or detections at levels more than 10 times smaller than implied by this mechanism (Smith et al. 1976; Freire et al. 2001; van Loon et al. 2006, 2009). Recent searches for dust in GCs confirm this paucity (Lynch & Rossano 1990; Knapp et al. 1995; Origlia et al. 1996; Penny et al. 1997; Hopwood et al. 1999; Matsunaga et al. 2008). Spitzer observations place upper bounds on dust masses 10–100× below than expected (Barmby et al. 2009), and, where detections are made (Boyer et al. 2006, 2008), the values are much lower than expected.

It is unclear if the diversity in GC escape velocities and the possibility for fast stellar winds (Dupree et al. 2009) can explain all of these observations by simply having the material directly leave the cluster. A more robust solution for all GCs would be a more frequent ICM clearing mechanism than disk passages. Though many mechanisms such as pulsar winds (Spergel 1991), stellar collisions (Umbreit et al. 2008), ram pressure stripping by galactic halo medium (Priestley et al. 2011) are discussed, one appears to be robust (but often neglected): ICM clearing from classical novae (CNe) explosions (Scott & Durisen 1978). Scott & Durisen (1978) showed that CNe had explosion energies, E_nova, and recurrence rates adequate to unbind the accumulated ICM, but the astrophysical uncertainties (at that time) made definitive statements about this mechanism difficult. We show here that the improved knowledge of mass loss from the stellar population coupled with better constraints on the CNe rates and energetics reveals that CNe will clear the ICM and do so with a frequency that explains the ICM observations.

The mass loss rate from the old stellar population in a GC, $\dot{M}_{\rm ICM}$, is now observationally well constrained. The main-sequence turnoff mass is ≈0.78–0.88 M_☉ (Dotter et al. 2010; Thompson et al. 2010) in an old (>10 Gyr) GC, and white dwarfs (WDs) are born with an average mass of ≈0.53 ± 0.01 M_☉ (Kalirai et al. 2009) implying ≈0.3 M_☉ of material placed into the ICM.³ We estimate the WD birth rate from the calculated specific evolutionary flux for a Salpeter initial mass function (IMF) of 1.7 × 10⁻¹¹ L⁻¹_☉ yr⁻¹ (from Buzzoni et al. (2006), where a steeper of shallower IMF only changes this number by ±20%) giving $\dot{M}_{\rm ICM} \approx 5\times 10^{-12}\, M_\odot \ {\rm yr}^{-1}\ L_\odot ^{-1}$. Assuming that each CNe clears the accumulated ICM, the typical ICM mass is M_ICM ≈ 0.03 M_☉ for our fiducial (see Section 2) CNe rate of 20 yr⁻¹ per 10¹¹ L_☉. The ICM binding energy in a GC of mass M_GC and half-mass radius R_h is

$$\begin{equation} E_{\rm b}\approx 10^{45} \ {\rm erg}\left(M_{\rm GC}\over 5\times 10^5\, M_\odot \right)\left(M_{\rm ICM}\over 0.03 \,M_\odot \right)\left(1 \ {\rm pc}\over R_h\right), \end{equation} \tag{ 1 }$$

similar to the CNe energy release, E_nova, as first highlighted by Scott & Durisen (1978).

These arguments motivate our reconsideration of the dynamics of CNe shell evolution in Section 3, where we show that the nova shells escape the cluster before suffering significant cooling. In Section 4, we apply this model to all galactic GCs, and discuss the possibility that the current ICM knowledge can constrain the underlying population of binaries that cause CNe, the cataclysmic variables (CVs). We conclude in Section 5 and mention the impact that CNe will have on the ICM in the dwarf spheroidal galaxies of the Milky Way's halo.

CNe result from the unstable thermonuclear ignition of accumulated hydrogen on a WD in a mass-transferring binary (Gehrz et al. 1998). The rapid nuclear burning drives the ejection of most of the accumulated material, leading to outflows of M_ej ≈ 10⁻⁴–10⁻⁵ M_☉ at velocities of V_ej ≈ 1000 km s^-1 (see Starrfield et al. 1972; Prialnik 1986). The typical kinetic energy is best inferred from observations late in the CNe evolution, typically >10 yr later (Cohen & Rosenthal 1983; Cohen 1985; Downes & Duerbeck 2000), where V_ej ≈ 700–1200 km s^-1 is measured for the shell, motivating a fiducial CNe explosion energy of

$$\begin{equation} E_{\rm nova}={M_{\rm ej} V_{\rm ej}^2 \over 2}=10^{45} \ {\rm erg}\left(M_{\rm ej}\over 10^{-4}\,M_\odot \right)\left(V_{\rm ej}\over 1000 \ {\rm km \ s^{-1}}\right)^{2}, \end{equation} \tag{ 2 }$$

which we use throughout this paper. The calculated accumulated masses depend on the mass accretion rate, WD core temperature, and accreted metallicity (Prialnik & Kovetz 1995; Townsley & Bildsten 2005; Yaron et al. 2005) and, for a "typical" case yields M_ej ≈ 3(10) × 10⁻⁵ M_☉ for M_WD = 1.0(0.6) M_☉, consistent with the observed range (Gehrz et al. 1998). This variation in CNe ejecta masses motivates our later consideration of the impact of a range of E_nova.

The CNe rate in GCs is not known, as only two CNe have been observed in galactic GCs (see historical discussions in Shara et al. 2004; Shafter & Quimby 2007), and three in extragalactic GCs (one of M87, Shara et al. 2004; and two of M31, Shafter & Quimby 2007, Henze et al. 2009, Pietsch et al. 2010). Shafter & Quimby (2007) discussed this challenge, suggesting that the CNe rate per unit stellar mass in GCs may only be slightly higher than in the field, but is likely not much lower. Another way to make an estimate is to take advantage of the explicit connection between CNe and the observed CV population. Townsley & Bildsten (2005) showed that the majority of galactic CNe arose from CVs with P_orb < 8 hr undergoing the conventional mass-transfer scenario of disrupted magnetic braking (see Howell et al. 2001). The measured CNe in distant galaxies, 20 yr⁻¹ per 10¹¹ L_☉ (Güth et al. 2010) then implied a CV population per unit mass comparable to that seen in our local galactic disk.

The combination of Hubble Space Telescope and Chandra observations of GCs is revealing their CV populations (Edmonds et al. 2003; Dieball et al. 2005, 2007; Bassa et al. 2008; Dieball et al. 2010; Huang et al. 2010), allowing for much better raw statistics on their incidence. However, these CV populations appear to be a combination of those from primordial binaries and dynamical interactions (Pooley & Hut 2006; Stefano & Rappaport 1994), and may, in the end, depend on the GC's central stellar density, ρ₀. Haggard et al. (2009) discuss the situation in ω Cen, where they conclude a lower CV incidence per unit mass than in the field. Given this current diversity of answers at present, we simply adopt the field CNe rate of 20 yr⁻¹ per 10¹¹ L_☉ (Güth et al. 2010) for our work.⁴

We now show that the kinetic energy injected by a CNe, E_nova, is adequate to unbind the ICM accumulated since the prior explosion. Although the local ICM structure will influence the shell evolution, our aim here is to show the ease of unbinding the ICM. We define a characteristic (and uniform) ICM number density n₀ ≡ 3M_ICM/m_p4πR³_h ≃ 0.1 cm⁻³, an overestimate since some ICM mass will be beyond R_h. We treat the evolution using a three-stage model (Spitzer 1978), starting with the brief free-expansion phase that lasts the time

$$\begin{equation} t_I = 120 \ {\rm yr} \left(\frac{M_{\rm {ej}}}{10^{-4}\,M_\odot }\right)^{5/6} \left(\frac{E_{\rm nova}}{10^{45}\ \rm {erg}}\right)^{-1/2} \left(\frac{n_0}{\rm {cm^{-3}}}\right)^{-1/3}, \end{equation} \tag{ 3 }$$

needed to sweep up the M_ej of ICM, creating a reverse shock. At this point, the evolution transitions into the Sedov–Taylor phase and ends when the radiative cooling of the post shock material becomes significant.

The simple case of a strong explosion in a homogeneous medium is given by the Sedov–Taylor solution (Zel'dovich & Raizer 2002). The radius of the shock front, R_s, the velocity of the shock front, V_s, and the temperature immediately behind the shock, T_s, are, respectively,

$$\begin{equation} R_s(t) \approx 2.3\ {\rm pc}\left(\frac{E_{\rm nova}}{10^{45}\ \rm {erg}}\right)^{1/5} \left(\frac{n_0}{\rm {cm^{-3}}}\right)^{-1/5} \left(\frac{t}{10^4 \ \rm {yr}}\right)^{2/5}, \end{equation} \tag{ 4 }$$

$$\begin{equation} V_s(t) \approx 92 \rm \;km\;s^{-1}\left(E_{\rm nova}\over 10^{45} {\rm \;erg}\right)^{1/5}\!\!\left(n_0 \over {\rm cm}^{-3} \right)^{-1/5}\!\!\left(t \over 10^4 \ {\rm yr}\right)^{-3/5}, \end{equation} \tag{ 5 }$$

$$\begin{equation} T_s(t) = 1.1\times 10^5\mbox{ K}\left(E_{\rm nova}\over 10^{45} {\rm \;erg}\right)^{2/5}\!\!\left(n_0 \over {\rm cm}^{-3}\right)^{-2/5}\!\!\left(t \over 10^4 \ {\rm yr} \right)^{-6/5}, \end{equation} \tag{ 6 }$$

where t is the time since explosion. These numbers show that the shell will be moving at a speed in excess of the GC escape velocity, V_esc ≈ 30 km s^-1, even 10⁴ yr after the explosion, consistent with our energetic arguments. The time it takes the shell to escape the cluster, t_esc, is simply the time for the shell to reach R_h with V_s > V_esc. These are shown by the solid lines in Figure 1, where we have used the Plummer model (Binney & Tremaine 2008) to estimate V_esc.

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The only remaining question is whether the remnant radiates and cools prior to escape. To answer that, we calculate the cooling of the remnant using collisional ionization equilibrium cooling rates for a range of metallicities from Gnat & Sternberg (2006). The mass of the shell is heavily concentrated near the shock front; therefore, we approximate the remnant as being spatially isothermal at the temperature just behind the shock, T_s(t). From Cox (1972), the energy loss rate is

$$\begin{equation} \frac{d E}{d t}(T,n)=-\int _0^{R_s}{ \Lambda (T) n^2(R) 4\pi R^2 dR}, \end{equation} \tag{ 7 }$$

where Λ is the cooling function (in erg cm³ s⁻¹), so Λn² is the energy loss rate per unit volume. We adopt Cox (1972)'s compaction parameter

$$\begin{equation} \lambda =\left(\frac{4}{3}\pi R_s^3\right)^{-1}\int _0^{R_s}{\left(\frac{n(R)}{n_0}\right)^24\pi R^2 dR}, \end{equation} \tag{ 8 }$$

which has the value λ ≃ 2.3 for a strong shock. This yields

$$\begin{equation} \frac{d E}{d t}=-\Lambda n_0^2 \left(\frac{4}{3}\pi R_s^3(t)\right)\lambda, \end{equation} \tag{ 9 }$$

an equation that gives the energy as a function of time, defining t_cool, the time it takes for half the energy to be lost. We also compute the longer non-equilibrium cooling times for the case where the gas cools on a timescale shorter than the recombination times of the various ions, causing a relative overionization as compared to the equilibrium case. We therefore distinguish cooling times that are calculated from the equilibrium cooling efficiencies, t_cool,eq, and two types of non-equilibrium cooling efficiencies—isobaric, t_cool,ib, and isochoric, t_cool,ic. The upper curves in Figure 1 show the different cooling times for a range of R_h for M_GC = 5 × 10⁵ M_☉ (and ICM particle densities ranging from n₀ = 0.06to4 cm⁻³ implied by a fixed M_ICM = 0.03 M_☉). The isobaric and isochoric cooling times are similar, so we only plot t_cool,ic. All of these times are much longer than the escape time, making it clear that the shell will escape in the Sedov phase and radiative cooling will be an important energy loss mechanism only after the shell leaves the GC.

We now apply our model of ICM accumulation to the Milky Way GCs (Harris 1996). We assume that each GC has accumulated M_ICM ≈ 0.03 M_☉ from stellar mass loss since the last clearing event, and that the characteristic particle density, n₀, is uniform. For each GC, we compute the equilibrium cooling time, the shell radius at the cooling time, R_cool, and the shell's velocity, V_h, when it was at the half-mass radius (i.e., V_h = V_s when R_s = R_h). We plot these values in Figure 2, showing that V_h > V_esc in all cases, and that cooling occurs when R_s ≫ R_h. Escape is the outcome in this simple formulation.

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Since we do not expect the CNe to be strictly periodic, we performed a Monte Carlo simulation for 10⁸ yr allowing for variation in CNe recurrence times and drawing the nova energies from a log-uniform distribution ranging from (2.5–10) × 10⁴⁴ erg. For each GC, we calculate the expected number of CVs (assuming 1 CV per 10⁴ M_☉, see footnote 4), and generate a nova explosion history assuming an average CNe recurrence time in the range (3.3–6.7) × 10⁵ yr, centered on the recurrence time calculated from the CNe rate, 5 × 10⁵ yr.

We calculate how large an effect these variances will have by calculating how often a GC is in a state where M_ICM > 0.03 M_☉. During the 10⁸ yr, there is a 12% chance that the GC will have twice the fiducial M_ICM, a 4% chance for 3×, and a 0.4% chance to have 5× the fiducial ICM built up—a prediction nearly independent of GC parameters. Shell escape can become difficult for some galactic GCs if M_ICM is increased by a factor of 10. Figure 2 looks qualitatively the same in that case, but the line representing V_h = V_esc would now cross between the triangles on the y-axis and the x-axis range would shrink to 6–14.

The maximum quiescent time was ∼10× the average time between novae, allowing for a very large buildup of M_ICM, potentially to values where E_b > E_nova, forcing us to consider the possibility that some GCs may, due to the randomness of the process, experience a "runaway" accumulation. We determine the probability for this outcome by computing the average time it takes for this condition to be reached and assuming that the GC has uniform probability of being at any point in its 10⁸ yr orbital period (longer orbital periods will increase the chance of seeing a runaway). For ρ₀ = 10³ L_☉ pc⁻³ and R_h = 2.0 pc, we find that GCs with M_GC = (1.5–5) × 10⁵ M_☉ have an even chance of experiencing a runaway, where the exact value depends on the nova energy distribution. This probability increases very quickly with mass, so more massive clusters are much more likely to undergo a runaway, while less massive GCs will almost never have one. This probability is also sensitive to ρ₀, with more collapsed clusters much more likely to undergo runaways. Although we refer to it as a "runaway," it may not be unmitigated mass accumulation, as the asymmetries of the ICM will allow the CN to unbind some material. However, the stochastic nature of this process inhibits our ability to make a secure prediction for a specific GC.

There are also some instances (e.g., M15) where the shell escape time, t_esc, is so long that it does not escape before another CN occurs. The outcomes from these colliding remnants are more complex, either leading to a net heating of the ICM that creates a wind like solution (Scott & Durisen 1978) or more intense cooling that keeps the material bound. In either case, novae clearing is most effective in the smaller, less dense clusters, where the shell safely escapes before another nova goes off and the GC is in no danger of a runaway scenario.

There have been few positive ICM measurements in GCs. Barmby et al. (2009) derive M_ICM upper limits of 0.04–1.4 M_☉ from dust mass upper limits. These upper limits are (with the exception of M2) consistent with observed M_ICM values from Freire et al. (2001), van Loon et al. (2006), and Grindlay & Liller (1977). We collect the four reported M_ICM measurements in Table 1, along with other useful GC parameters and timescales we derive. The ranges in t_esc and t_cool come from taking our fiducial M_ICM = 0.03 M_☉ in the clusters for one bound (the upper bound on t_cool and the lower bound on t_esc) and the measured M_ICM for the opposite bounds. Selection effects will bias likely detections toward GCs with higher M_ICM values. According to our calculations, the high central densities and masses of these clusters place them in the runaway regime.

We have assumed that the mass lost from stellar evolution is a smooth process at the rate of $\dot{M}_{\rm ICM} \approx 5\times 10^{-12}\, M_\odot \ {\rm yr}^{-1}\ L_\odot ^{-1}$. However, stochastic effects in the mass loss should be examined as well. We start by considering the most extreme case, where the mass is lost in a sudden event (e.g., He core flash, thermal pulse on the AGB). Since the WD birth rate is one-tenth the CNe occurrence rate, such an event would occur on an average timescale of 10t_rec. Though certainly the case that the CNe immediately following this event would not be adequate to unbind it (both because of energetic arguments and relative filling fractions), such a stellar ejection "chunk" would be shocked, on average, by ≈10 CNe. Whether this proves sufficient to unbind it depends on the location and time to cool between shocks, which is beyond the scope of our work here.

An additional possibility is wind-driven mass loss that is so strongly dependent on the stellar luminosity (e.g., $\dot{M}\propto L^\alpha$), that it makes $\dot{M}_{\rm ICM}$ time dependent on the t_rec timescale. As an example, consider mass loss near the tip (L_TRGB) of the red giant branch (RGB). It takes 3 × 10⁶ yr to evolve from L = L_TRGB/2 to L = L_TRGB (Salaris & Cassisi 1997), implying ≈30 stars in these final stages for a 5 × 10⁵ M_☉ cluster. Let us consider the amount of mass one of these stars could lose during the final t_rec of its evolution to the expected total mass loss along the RGB, ∼0.2 M_☉ (Origlia et al. 2007). For typical values of α ∼ 1–3 (McDonald et al. 2009; Mészáros et al. 2009), this ratio is less than 10⁻³. To get a substantial mass loss in the last t_rec requires α>100, basically becoming equivalent to the sudden event model already discussed. So, we do not expect any of the current wind models or observations to create a time dependent $\dot{M}_{\rm ICM}$ on the CNe recurrence timescale.

We have reconsidered the Scott & Durisen (1978) model of using CNe explosions to clear the accumulated ICM in galactic GCs, allowing for a more secure analysis of its efficacy. Using estimated CNe rates and the known stellar mass loss rates, we predict that novae clearing is highly effective in the less massive and less centrally condensed galactic GCs (those unlikely to undergo a runaway), leading to a typical ICM buildup of ≈0.03 M_☉ that is consistent with observational upper limits. For those few GCs with positive ICM measurements, the M_ICM values are higher than our predictions, though still one to two orders of magnitude less than the ∼300 M_☉ predicted from galactic plane crossing. We discuss in Section 4 whether this could simply be the result of the stochastic nature of CNe explosions and the large number of GCs observed, or from a systematic dependence on M_GC or ρ₀. In either case, it is clear that CNe have a large deleterious impact on the ability for the GC to build up its ICM.

A very robust mechanism to clear the ICM is Type Ia supernovae (SNe). The simplest assumption (Pfahl et al. 2009) is to use the rate expected from the mass, (5.3 ± 1.1) × 10⁻¹⁴ yr^-1 M⁻¹_☉ (Sullivan et al. 2006), yielding an average time between events of ≈2 × 10⁷ yr in a 10⁶ M_☉ GC, shorter than a galactic plane crossing. If CNe are ineffective in the intervening time, then the ICM mass would reach M_ICM ≈ 100 M_☉, but with a binding energy orders of magnitude less than the 10⁵¹ erg of the SNe. Such outcomes may be more prevalent in massive GCs around elliptical galaxies, since there is no plane crossing to clear out the accumulated ICM. When M_ICM ≈ 100 M_☉, the explosion would occur in a density of n₀ ≈ 100 cm^-3, much larger than expected in an elliptical galaxy, ≈10⁻² cm^-3 (Stewart et al. 1984). Application of our work in Section 2 finds that the velocity of such a remnant would be V_s ≈ 130 km s^-1 at a time of 2700 yr, when it reaches R_h = 1 pc. The cooling time is orders of magnitude longer, and so this shell will easily escape, and be quite bright (due to the high density). Less than a few per cent of extragalactic GCs have shown emission lines consistent with these velocities (Chomiuk et al. 2008), raising the question as to whether these could be the first indicators of the expected Ia explosions in GCs. If such a remnant is detectable for 10⁴ yr, then we would expect to see one such case amongst all the GCs in a large elliptical galaxy like M87.

Novae clearing can also be applied to the newly discovered ultrafaint (only ≈10⁴ M_☉ in stars, potentially so low that only a few CVs may be present, see footnote 4) dwarf spheroidals (dSph's; Belokurov et al. 2007; Simon & Geha 2007), where ICM deficits have been reported (Bailin & Ford 2007).⁵ Just as in GCs, these dSph have a mass-losing stellar population. However, the lower escape velocities (2–5 km s^-1 from the ≈10⁶ M_☉ of dark matter in a half-mass radius of 100 pc) may allow the stellar winds to directly leave the dSph. In cases where the mass accumulates, the ICM binding energy is <10⁴⁵ erg for M_ICM = 0.03 M_☉, so that a single CNe can be effective. Though no challenge energetically, the 100 pc dimension means that the escape timescale is so long (⩾10⁵ yr) that much of the ICM would still be present when the next nova occurs. These systems are therefore much closer to the original limit discussed by Scott & Durisen (1978), which predicts a wind leaving the system at the expected $\dot{M}_{\rm ICM}$, but with a kinetic energy flux given by E_nova and the CNe recurrence time. Such a wind would have a terminal velocity of ≈60 km s^-1 (independent of the dSph mass), and a density at 100 pc of ∼10⁻⁷ cm^-3 for a dSph with 10⁴ M_☉ of old stars.

We thank Matt van Adelsberg, David Chernoff, Aaron Dotter, Craig Heinke, David Kaplan, Michael Shara, Ken Shen, and Jay Strader for discussions. We also thank the anonymous referee for their useful comments. This work was supported by the National Science Foundation under grants PHY 05-51164 and AST 07-07633.