NATURE VERSUS NURTURE: LUMINOUS BLUE VARIABLE NEBULAE IN AND NEAR MASSIVE STELLAR CLUSTERS AT THE GALACTIC CENTER

Figure 1(a) shows that the location of the Pistol Star, which is indicated by the red cross, is slightly displaced to the northwest of the center of the nebula. The nebula itself appears compressed along the northern and western edges and has east–west and north–south dimensions of 1.4 × 1.2 pc. The nearest distance in projection between the compressed northern and western edges of the nebula and the Pistol Star is ∼0.4 pc, whereas the projected distance to the eastern and southern edges is ∼0.8 pc. Dust emission from the nebula peaks along the northwest edge at all wavelengths; however, as can be seen in Figure 1(a), the region of peak emission migrates from the north to the west with increasing wavelength. Unlike a typical H ii region (Salgado et al. 2012), we find that the emission from the nebula appears nearly identical at each wavelength, which confirms its shell-like morphology. Normalized intensity line cuts through the nebula at each wavelength are shown in Figure 3. Figures 3(b) and (c) reveal the multi-wavelength similarities of the filaments and structures within the nebula. We also note that on the eastern edge of the nebula there is a faint "ridge" of enhanced emission resembling a bow shock that possibly indicates an interaction with the winds of another nearby star.

Zoom In Zoom Out Reset image size

Paschen-α contours (Dong et al. 2011) overlaid on the 31.5 μm intensity map (Figure 4(a)) show that the ionized gas emission from the nebula closely traces the dust along the northern and western edges. The total Paschen-α flux from the nebula is ∼4.6 × 10⁻¹⁰ erg cm⁻² s⁻¹. We note that Paschen-α emission is more sensitive to density variations than dust emission since $I_{P\alpha }\propto n_e^2$ and I_Dust∝n_e. As a first order correction to relate the two emission maps we "square-root" the Paschen-α intensity. The "square-root" Paschen-α intensity map closely traces the dust emission throughout the entire nebula which suggests the entire nebula is ionized. Since the Pistol Star has a luminosity and effective temperature of 3.3 × 10⁶ L_☉ (Mauerhan et al. 2010) and ∼12,000 K (Najarro et al. 2009) it will not be able to ionize the entire nebula; therefore, we require the ionizing flux from the nearby hot, Quintuplet Cluster stars to fully ionize the nebula (Figer et al. 1999b).

Zoom In Zoom Out Reset image size

We derive the 19.7/37.1 μm dust temperature map of the nebula (Figure 4(b)) assuming that the emission is optically thin and can be approximated by a blackbody power-law modified by an emissivity of the form ν^β. Since the Pistol Star is oxygen-rich, the dust in the nebula is likely composed primarily of silicate-type grains; therefore, we adopt a power-law index of β = 2 which is consistent with the emissivity curve of silicates at wavelengths of λ > 19 μm (Draine 2011). This leads to temperatures ranging from 100–180 K throughout the nebula, with an average of 150 ±10 K, assuming a photometric error of 20%. The temperature peaks along filaments southeast of the projected location of the Pistol Star. There is also a slight temperature gradient that gradually decreases from the northeast (T_d ∼ 160K) edge to the southwest edge (T_d ∼ 140K).

The optical depth at 37.1 μm for optically thin emission can be expressed as

$$\begin{equation} \tau _{37.1}= \frac{F_{37.1}}{\Omega _p\,B_\nu (T_d)}, \end{equation} \tag{ 1 }$$

where F_37.1 is the 37.1 μm flux of the dust per pixel solid angle, Ω_p, and B_ν(T_d) is the Planck function at dust temperature, T_d. We produce a 37.1 μm optical depth map (Figure 4(c)) using Equation (1) and applying the values from the 19.7/37.1 μm temperature and the 37.1 μm intensity maps. The 37.1 μm optical depth peaks along the northwest edge of the nebula, with the largest value of τ₃₇ ∼ 10⁻³, and decreases to the southeast to values of τ₃₇ ∼ 3 × 10⁻⁴. The average 37.1 μm optical depth throughout the nebula is ∼5 × 10⁻⁴. We find that the compressed edges of the nebula at the north and west are consistent with the regions of increased column density, which suggests that its morphology is influenced externally by winds from the nearby carbon-dominated Wolf–Rayet (WC) stars in the Quintuplet Cluster (Figer et al. 1999b) and/or density gradients in the ambient medium. This claim is reinforced by the regions of low column density at the south and east of the nebula that are on the opposite side of the projected location of the Quintuplet Cluster. Figure 4(d) shows the fractional column density integrated over the 8 octants about the Pistol Star and overlaid on the 37.1 μm optical depth map. We observe a slight north–south asymmetry in the column density in a sense that ∼13% more dust lies in the four southern octants of the nebula.

Under the assumption that the dust in the nebula is solely heated by the Pistol Star and is in radiative equilibrium we can estimate a theoretical dust temperature, T_d, to compare with the observed temperatures. By balancing the power input from the Pistol Star and output by the dust grains we obtain the following expression for assumed spherical grains:

$$\begin{equation} \pi a^2Q_*(a)\, F_*=4 \pi a^2\,Q_{\mathrm{dust}}(T_d,a)\,\sigma _{\mathrm{SB}}\,T_d{}^4, \end{equation} \tag{ 2 }$$

where Q_dust is the dust emission efficiency averaged over the Planck function of dust grains of size a and temperature T_d, Q_* is the dust absorption efficiency averaged over the incident radiation field, and F_* is the incident flux at the location of the dust. For an a = 50 Å sized silicate-type grains and a 12,000 K Kurucz model star with a log surface gravity of 4.5 and two times the solar metallicity (Kurucz 1993) as the heating source we have $Q_\mathrm{dust}(T_d, a)=6.5\times 10^{-8} ({a}/{50\,{\rm \mathring{\rm{A}}}})\,T_d^2$ and Q_*(a) ∼ 0.04(a/50 Å) (Draine 2011). Assuming a distance of 0.7 pc from the star to the nebula and a total source luminosity of 3 × 10⁶ L_☉ (Mauerhan et al. 2010) we derive a theoretical equilibrium temperature of 88 K, which is significantly cooler than the observed 150 K. We note that since both Q_dust and Q_* are approximately proportional to the grain size, a, the derived equilibrium temperature is not sensitive the grain size. The radiation from the star itself therefore does not provide enough energy to heat the dust in the nebula to the observed temperature.

When including the radiation from the Quintuplet Cluster stars, which we approximate as a 35,000 K blackbody source with a total luminosity of 2 × 10⁷ L_☉ (Figer et al. 1999b) at a distance of 2.0 pc from the center of the nebula, we derive a dust equilibrium temperature of 130 K for the 50 Å sized silicate grains. The equilibrium temperature is slightly cooler than the observed temperature derived from the 19/37 flux ratio, which may be partially due to uncertainties in the stellar models; however, we argue that the discrepancy arises from the stochastic heating of very small grains (VSGs), an effect that enhances the flux at shorter wavelengths. Our calculations therefore show that the observed dust temperatures are consistent with radiative heating by both the Pistol Star and Quintuplet Cluster stars, with the QC stars dominating the heating (85%).

We find that the observed emission from the nebula at λ < 10 μm is a factor of ∼5 greater than the flux predicted by a blackbody of temperature T_dust = 130 K multiplied by the silicate dust emission efficiency Q_Sil(λ, a) (Draine & Li 2001) fit to the observed FORCAST intensities (Table 2). Again, we attribute the enhanced intensity at the shorter IR wavelengths to the presence of very small (a < 100 Å), transiently heated grains in the nebula. VSGs can reach temperatures much greater than equilibrium-heated large grains since they have small heat capacities that result in large temperature spikes after absorbing single photons. The emissivity of small grains can be characterized as

$$\begin{equation} j_\nu =\int da \frac{dn}{da} \int dT \left(\frac{dP}{dT}\right)_a \sigma _\mathrm{abs}(\nu \mathrm{, }\,a)B_\nu (T), \end{equation} \tag{ 3 }$$

where n is the dust density and dP/dT is a probability distribution function with P(T) being the probability that a grain will have a temperature less than or equal to T. For large grains in radiative equilibrium the probability distribution function is simply a delta function at the equilibrium temperature. The steady state probability distribution function for small grains, which is much broader than the large grain function, can be solved for analytically (Guhathakurta & Draine 1989; Draine & Li 2001).

We utilize the DustEM (Compiègne et al. 2011) code to model the SEDs from the northern, central, southern, and full regions of the nebula. The apertures used to extract the fluxes are overlaid on the 31.5 μm image of the nebula, as in Figure 5(a). DustEM can model the emission from small, transiently heated grains and uses the formalism in Desert et al. (1986) to derive the temperature probability distribution function. In the models, we assume the dust is heated by both the Pistol Star and the hot stars in the Quintuplet Cluster, the latter of which also dominates the contribution of ionizing photons (Figer et al. 1999b).

Zoom In Zoom Out Reset image size

Since the three-dimensional spatial distribution of the Quintuplet Cluster stars with respect to the Pistol is not known we treat the cluster's radiation field in the DustEM models in two different ways: first, as arising from a point source-like distribution having a radiation field that declines as r⁻², and second, as an extended distribution that produces a plane-parallel field. The distance between the Quintuplet Cluster and the Pistol Star must be greater than the projected separation (∼1.5 pc) since the back side of the nebula, the side farthest from the observer, has been decelerated more than the front side due to interactions with the winds from the nearby WC stars (Figer et al. 1999b). We therefore adopt a factor of $\sqrt{2}$ to account for the distance projection, implying a three-dimensional separation of ∼2 pc. For the plane-parallel field model we assume the entire nebula is heated by a radiation field identical to as the one expected at the center of the nebula in the r⁻² case (d = 2.0 pc). Assuming the separation distance of 2.0 pc we find that the Pistol Star contributes an average of ∼15% to heating the dust. The best DustEM model fits to each nebular region, assuming the plane-parallel field, are shown in Figures 5(b)–(e) with the minimum and maximum sizes of the small grains, a_{min, SG} and a_{max, SG}, respectively. The r⁻² QC field case provides nearly identical results. The fitting parameters for both models are summarized in Table 3.

Table 3. DustEM Fitting Parameters

	d*	dC	L*	T*	LC	TC	amin	amax
	(pc)	(pc)	(L☉)	(K)	(L☉)	(K)	(Å)	(Å)
Pistol (PP)	0.5–0.7	2.0	3.3 × 106	12000	3 × 107	35000	10	25–60
Pistol (r−2)	0.5–0.7	1.8–2.7	3.3 × 106	12000	3 × 107	35000	10	26–40
LBV3	0.7	14	4 × 106	12000	6 × 107	35000	$170^{+430}_{-160}$	$210^{+1490}_{-170}$

Download table as:  ASCIITypeset image

We find that the nebula is composed of a distribution of VSGs with sizes ranging from 10 to  ∼ 35 Å. In addition to a gradient of decreasing flux from the north to the south of the nebula we also observe a decreasing F_8μm/F_25.2μm ratio. The decreasing F_8μm/F_25.2μm ratio in the plane parallel field case suggests the presence of increasingly larger grains towards the central and southern regions of the nebula. Since the total dust mass, M_Tot, is proportional to the dust flux we also observe a north–south gradient of decreasing mass in the plane-parallel radiation field case. A nearly identical north–south gradient is observed in the 37.1 μm optical depth map (Figure 4(c)). We derive a total dust mass of 0.03 M_☉ and an integrated IR luminosity of 5.2 × 10⁵ L_☉.

We interpret the radiation field of the Quintuplet Cluster as plane-parallel since the Pistol Star is a member and the other cluster members are distributed within a region having an extent comparable to the distance separating them from the Pistol Star. The gradient of decreasing mass from the north to the south regions of the nebula from the models is also consistent with the observed 37.1 μm optical depth gradient. As mentioned previously, treating the Quintuplet Cluster as a point-like source with an r⁻² field does not result in significant quantitative and qualitative differences from the plane-parallel case.

DustEM fits were attempted with the addition of an independent population of large, equilibrium-heated dust grains with a = 1000 Å. The fits indicate that in the central and southern regions of the nebula a population of large grains that comprises ∼25% of the total dust mass may be present. The peak flux contribution of the large grain population, however, is more than an order of magnitude smaller than that of the VSG population at the same wavelength. This suggests that even if large grains exist in the nebula, the total mass and IR luminosity are dominated by the VSGs.

The dust emission from the LBV3 nebula shown in Figure 1(b) appears circularly symmetric about LBV3. Although the nebula exhibits a different morphology from the compressed, asymmetric Pistol nebula they share a similar size scale: the LBV3 nebula has a diameter of ∼1.3 pc. The consistent limb-brightening at all wavelengths indicates that the nebula has a shell-like morphology. Figure 6(a) shows the normalized and azimuthally averaged radial intensity profiles at 25.2, 31.5, and 37.1 μm centered on LBV3. The intensity throughout the nebula is not perfectly symmetric; Figure 1(b) shows that the intensity from the northeast quadrant is greater than the rest of the cavity by a factor of ∼2. The enhanced emission may be due to external heating contributions from other hot stars in the region or the asymmetric ejection of material from LBV3. Due to the low signal-to-noise ratio it is difficult to accurately map out the two-dimensional temperature structure of the nebula to resolve this issue. Given the relative isolation of LBV3, we expect that LBV3 itself is the dominant source of heating of the dust in the nebula.

Zoom In Zoom Out Reset image size

Paschen-α contours (Dong et al. 2011) overlaid on the 31.5 μm intensity map (Figure 6(b)) reveal that the emission from the ionized gas closely traces the dust around the entire nebula, including the enhanced emission from the northeast quadrant. The dust and ionized gas are therefore likely confined to the same physical region within the nebula. The total Paschen-α flux from the nebula is ∼2 × 10⁻¹⁰ erg cm⁻² s⁻¹, which is a factor of 2.3 less than that of the Pistol nebula.

The uniformity of the Paschen-α emission and its strong similarities to that of the dust strongly suggest that the nebula is primarily ionized by a central stellar source. Since there are currently no direct radio observations of the nebula we scale the 6 cm flux of the Pistol nebula by the ratio of the Paschen-α fluxes from the nebulae to estimate the free-free emission at 6 cm from the LBV3 nebula. Lang et al. (1997) find the 6 cm emission from the Pistol nebula to be 0.5 Jy, which implies from our assumptions that the 6 cm emission is 0.22 Jy for the LBV3 nebula.

The observed ratio of the Pistol nebula Paschen-α emission to that of the LBV3 nebula implies that ∼2 × 10⁴⁸ Lyman-continuum photons s⁻¹ are required to ionize the gas in the LBV3 nebula assuming it exhibits same electron temperature as that of the Pistol nebula, T_e = 3600 K (Lang et al. 1997). Adopting an electron temperature of 7000 K, a value similar to typical H ii regions, only changes the Lyman-continuum flux estimate by less than 20%.

With a luminosity and effective temperature of 4 × 10⁶L_☉ (Mauerhan et al. 2010) and 12, 000 K, respectively, LBV3 is unable to provide the ionizing flux to satisfy the observations. This suggests that the Arches Cluster, which produces ∼4 × 10⁵¹ Lyman-continuum photons s⁻¹ (Figer et al. 2002) and is located ∼10 pc away in projection from LBV3, is ionizing the nebula. Assuming the distance between LBV3 and the Arches Cluster is $\sqrt{2}\times 10$ pc we derive that the cluster contributes ∼2.5 × 10⁴⁸ Lyman-continuum photons s⁻¹ to ionizing the nebula, which is consistent with required ionizing flux. Although the Quintuplet Cluster does not produce as many ionizing photons as the Arches it is closer in projection to the nebula and may also provide a significant fraction of the ionizing photons. Assuming a distance of $\sqrt{2}\times 7$ pc between the QC and the nebula we find that it contributes ∼10⁴⁸ Lyman-continuum photons s⁻¹.

From the observed 19.7 and 37.1 μm fluxes of the nebula we derive a dust temperature of 105 ± 8 K assuming the silicate emissivity power law with an index of β = 2. Given a dust temperature of 105 K and the 37.1 μm intensity profile we derive the radial 37.1 μm optical depth profile shown in Figure 6(c). We find that τ₃₇ peaks at the edges of the nebula with a value of ∼4.5 × 10⁻⁴ and that it averages ∼3.5 × 10⁻⁴ over the entire nebula. The optical depth profile suggests that the dust is arranged in a shell-like morphology.

We can estimate the equilibrium temperature expected for the dust in the nebula from Equation (2), assuming that the heating is dominated by LBV3. For 0.005 Å size silicate-type grains and a heating source with an effective temperature of 12,000 K, a total luminosity of 4 × 10⁶L_☉, and at a distance from the star of 0.7 pc, we derive a theoretical dust equilibrium temperature of 93 K. As with the Pistol nebula, the estimated dust equilibrium temperature may be slightly cooler than the observed temperature derived from the 19/37 flux ratio due to the presence of VSGs.

Including the heating from the Arches Cluster, which is approximated as a 35,000 K blackbody with a luminosity of 6.5 × 10⁷ L_☉ (Figer et al. 2002) at a presumed distance of 14 pc from the nebula, we find that the dust equilibrium temperature rises to 98 K. This dust temperature estimate is more consistent with the observed temperature than in the case with only central heating by LBV3, which suggests that, while the dust heating is dominated by LBV3 (70%), there is an additional radiative heating contribution from the Arches Cluster. This scenario is consistent with the nebula being externally ionized by the Arches Cluster since the LBV3 star has an effective temperature of 12,000 K and is unable to provide enough ionizing photons.

We utilize the DustEM code to model the observed SED of the dust emission from the nebula and assume the nebula is symmetric about LBV3 and heated by both LBV3 and the Arches Cluster. LBV3 and the Arches Cluster are approximated as the same 12,000 K Kurucz model star as for the Pistol model and a 6.5 × 10⁷ L_☉ blackbody source (Figer et al. 2002) with an effective temperature of 35,000 K, respectively. Figure 6(d) shows three DustEM fits to the emission from the nebula at 19.7, 25.2, 31.5, and 37.1 μm: the best fit, the 1σ maximum grain size cutoff upper limit fit, a⁺, and the 1σ maximum grain size cutoff lower limit fit, a⁻. The 1σ upper limit of the minimum grain size cutoff is 600 Å. Also shown in 6 d is the background continuum flux at 8 μm detected by Spitzer/IRAC. Owing to the non-detection of this background in the mid-IR, and to the "flatness" of the LBV3 SED, we are unable to tightly constrain the grain size distribution of the nebula. The dust mass and integrated IR luminosity derived from the best-fit DustEM model are ${\sim }0.021^{+0.011}_{-0.003}$ M_☉ and $8_{-0.5}^{+1.5} \times 10^4$ L_☉, respectively. The fitting parameters and results of the model are summarized in Table 3.

Dust production around massive stars poses a difficult challenge to explain, given their greater luminosities and effective temperatures. We suggest that the formation scenario of the dust in the LBV nebulae is that proposed by Kochanek (2011): dust is formed during a short outburst phase (∼1000 yr) of the LBV with extreme mass-loss rates (∼10⁻³ M_☉) and dense winds to shield dust formation regions from UV photons that would otherwise photoevaporate the small grains. In their quiescent state LBVs have too high an effective temperature (∼15,000 K) and too low a mass loss rate (∼10⁻⁵ M_☉ yr⁻¹) for dust grains to form or survive. LBVs may also exist in cool states where they expand and exhibit effective temperatures of ∼7000 K; however, the soft UV photons will still photoevaporate small grains and thus impede dust formation. We therefore require a high mass-loss rate to both initiate the collisional growth of the dust grains as well as produce a dense wind to shield the grains. From the momentum conservation of the radiatively driven winds, $\dot{M}v_\infty \sim \tau _V L/c$, we see that for L = 10⁶ L_☉ and v_∞ = 100 km s⁻¹ $\dot{M}$ must be ≳ 10^−3.5 M_☉ for τ_V ≳ 1. Since the swept-up mass is negligible, the nebulae are therefore a reflection of the history of the mass-loss rate from the LBVs.

From the maximum grain size it is possible to approximate the mass-loss rate during grain formation, assuming the grains grow by accretion of gas particles with mass m_g (Kochanek 2011). Assuming 100% sticking probability, the collisional growth rate can be expressed as

$$\begin{equation} \frac{d N}{d t}=\frac{\dot{M}X_g \,\pi a^2 \,v_t}{4 \pi R^2 \,v_w\,m_g}, \end{equation} \tag{ 5 }$$

where $\dot{M}$ is the mass-loss rate, X_g is the mass fraction of the condensible dust species, v_t is the thermal velocity of the particles, and v_w is the velocity of the stellar winds. The number of collisions with the gas particles can be related to the grain size by N = ((4/3)πa³ρ_b/m_g), where ρ_b is the bulk density of a grain (∼3 gm cm⁻³), and we can then derive the grain growth rate as

$$\begin{equation} \frac{d a}{d t}=\frac{\dot{M}X_g\,v_t}{16 \pi R^2\,v_w\,\rho _b}. \end{equation} \tag{ 6 }$$

Since the gas will cool as it expands, v_t can be expressed in terms of R, the radius, and R_f, the grain formation radius (∼10¹⁵ pc), as v_t(R) = v_t0(R/R_f)⁻ⁿ, where n = 2/3 for cooling by adiabatic expansion (Kochanek 2011). After integrating, the grain size as a function of radius can be expressed as

$$\begin{equation} a(R)\approx \frac{\dot{M}X_g \,v_{t0}}{16 \pi R_fv_w^2\rho _b\,(1+2/3)}\left[1-\left(\frac{R_f}{R}\right)^{1+2/3}\right]. \end{equation} \tag{ 7 }$$

Assuming that the grains stop growing and reach their maximum size when the growth rate drops to 10% of the initial rate, or at a distance of R ≃ 3R_f, we find that

$$\begin{eqnarray} a_\mathrm{max} &\sim & 60\,\left(\frac{\dot{M}}{10^{-3}\,M_\odot \,{\rm yr}^{-1}}\right)\,\left(\frac{X_g}{0.003}\right)\, \nonumber\\ && \times \left(\frac{v_{t0}}{1 \, \mathrm{{\rm km\, s}^{-1}}}\right)\,\left(\frac{100 \,\mathrm{{\rm km \,s}^{-1}}}{v_w}\right)^2\,{\rm \mathring{\rm{A}}}. \end{eqnarray} \tag{ 8 }$$

We suggest that the Pistol Star produced silicate grains with sizes ranging from 10–60 Å during the outburst that formed the nebula. The selection of this size distribution is based on the parameters of the SED fit to the southern region of the nebula (Figure 5) which appears to be freely expanding with little or no interaction with the winds from the nearby, northern WC stars.

We estimate a mass-loss rate of ∼10⁻³ M_☉ yr⁻¹ during the nebula's dust forming phase from Equation (8), assuming that a_max = 60 Å, X_g = 0.003, v_t0 = 1 km s⁻¹, R_f ∼ 10¹⁵ pc, v_w = 100 km s⁻¹ and ρ_b = 3 g cm⁻³. Although this mass-loss rate implies the duration of the dust production is about the dynamical timescale of the nebula, given a total nebular mass of ∼9 M_☉, we note that in our derivation of the grain growth rate we assumed 100% sticking probabilities for the grain collisions and ignored the possible exhaustion of the condensible species. This mass-loss rate should therefore be treated as a lower limit.

The gradient of decreasing maximum grain size from the south to the north of the Pistol nebula (Figure 5(e)) suggests that the grains are being sputtered by the high velocity v_w ∼ 2000 km s⁻¹ winds from the nearby WC stars north of the nebula. A grain encountering the WC star winds of velocity, v_WC, is sputtered at a rate of

$$\begin{equation} \frac{d N}{d t}=2 \pi \,a\,v_{{\rm WC}}\,n_H\,A_i\,Y_i(E_i), \end{equation} \tag{ 9 }$$

where a is the grain size, n_H is the hydrogen density of the wind, A_i is the sputtering ion abundance, and Y_i(E_i) is the sputtering yield as a function of sputtering ion's kinetic energy, E_i. The rate the grain size decreases can then be given by

$$\begin{equation} \frac{d a}{d t}=\frac{m_g}{2 \rho _b}v_{{\rm WC}}\,n_H\,A_i\,Y_i(E_i), \end{equation} \tag{ 10 }$$

where m_g is the mass of the sputtered atom and ρ_b is the bulk density of the grain.

With the analytically derived sputtering yields from Tielens et al. (1994) we find that the total change in grain size due to sputtering from the WC winds over 10⁴ yr is ∼20 Å at the north of the nebula, assuming it is at a distance of 1 pc from the WC stars (a factor of $\sqrt{2}$ times the projected distance). A more detailed sputtering calculation involving the decelerating expansion velocity of the nebula (Figer et al. 1999b) and the decreasing separation between the nebula and the WC stars yields a grain size difference of ∼15 Å.

Our calculated grain size change due to sputtering is consistent with the difference in the maximum grain sizes we observe between the northern and the southern regions of the nebula (Figures 5(c) and (e)). The preferential sputtering at the north will lead to a deficit of grains in that region, which is what we observe based on the fractional column density around the Pistol Star (Figure 4(d)).

The geometry of our interpretation of the nebula and WC wind interaction agrees with the observed compression at the north of the nebula. Assuming the WC stars decelerate the projected northern portion of the nebula as it does the back side, we expect a northern expansion velocity of ∼10 km s⁻¹ (Figer et al. 1999b). From the observed initial and final nebula expansion velocities we can approximate the deceleration of the northern edge and derive the expected distance from the Pistol Star after 10⁴ yr. The initial expansion velocity, v_N0, is approximated as the Pistol Star wind velocity (100 km s⁻¹; Figer et al. 1999b; Mauerhan et al. 2010) and the final expansion velocity, $v_{N0}^{\prime }$, is the observed value of 10 km s⁻¹ (Figer et al. 1999b). We therefore predict the northern edge to be ∼0.5 pc from the Pistol Star, which is consistent with the observations.

Using conservation of momentum we derive the following expression for the total mass of the WC winds, M_w, required to decelerate the northern edge:

$$\begin{equation} M_{w}=M_{p,N}\left(\frac{v_{N0} - v_{N0}^{\prime }}{v_w-v_{N0}^{\prime }}\right)= M_{p,N}\times 0.045. \end{equation} \tag{ 11 }$$

M_{p, N} is the mass of the northern half of the nebula, and v_w is the WC wind velocity. Assuming that M_{p, N} ∼ 4.5 M_☉, half of the observed total nebula mass, the northern edge must encounter ∼0.2 M_☉ of particles from the WC winds. The mass flux from the WC winds through the solid angle of the nebula is ∼0.12 × 10⁻⁴ M_☉ yr⁻¹ given a WC mass-loss rate of 10⁻⁴ M_☉ yr⁻¹; therefore, over 10⁴ yr we approximate the total WC wind mass swept up by the northern edge of the nebula to be ∼0.12 M_☉, which agrees with the M_w derived from Equation (11).

Unlike the Pistol nebula, the LBV3 nebula freely expands into its surroundings since there are no nearby stars with strong winds to interact with. Due to the lack of near and mid-IR detections of the nebula we are unable to tightly constrain its grain size distribution; however, the distribution of our model fit of the southern Pistol nebula SED (∼10–60 Å) lies within the 1-σ constraints derived from the DustEM model fits to the LBV nebula SED. The sputtered distribution of the northern Pistol nebula (10–25 Å) does not fall within the constraints (Figure 6(c)). Given the similar nature of the stars themselves (Mauerhan et al. 2010), we may therefore treat the LBV3 nebula as an identical twin to the Pistol nebula with the differences in their appearances only due to the influence of the surrounding environment. Besides the sputtering and compression from external winds, the same evolutionary arguments made for the Pistol can be applied to LBV3.

There are multiple explanations for the non-detection of nebular emission surrounding qF362: the star has already ejected a nebula that is too diffuse to detect, it has undergone an outburst with a mass-loss rate too low to form grains, or it is less evolved than the Pistol and LBV3 and has yet to go through an outburst phase. Although we cannot test all the cases we can rule out and comment on the feasibility of several of them given the history of H-recombination line observations.

We first address the possibility that qF362 may have already formed a nebula that is now too diffuse for detection. From the expression for the optical depth of a dusty shell, $\tau _v=({M\kappa _V}/{4\pi v_{{\rm exp}}^2 t^2})$, where M is the mass of the shell and κ_V (∼100 cm² g⁻¹) is the visual opacity of the dust, we can estimate the timescale, t, for detectability. Assuming that the nebula has properties similar to the LBV3 and Pistol nebulae (M ∼ 6 M_☉ and v_exp ∼ 70 km s⁻¹) and that we can detect the nebula until τ_V = 0.01, the average optical depth of the Pistol nebula, it should be observable for ∼1.5 × 10⁴ yr. If indeed the LBV lifetime is ∼10⁴ yr we rule out the possibility of qF362 having formed already formed such a nebula.

The recent HST/NICMOS Paschen-α observations (Dong et al. 2011) taken in 2008 June reveal that, unlike the Pistol and LBV3, qF362 is not producing a strong Paschen-α emission line; however, from 1999 May spectral observations of qF362 in the H and K bands, Najarro et al. (2009) found that it had similar H-recombination line strengths to that of the Pistol Star. This implies that qF362 is either expanding and cooling or entering an outburst phase of increased mass loss.

If it is currently undergoing dust formation we would expect the flux at 8 and 19.7 μm of the optically thick dust shell surrounding the star to be ∼170 and ∼40 Jy, respectively, assuming it has a radius of 10¹⁵ cm, the formation radius, and a temperature of 1500 K. We conclude that dust is unlikely to be forming since a source with a 19.7 flux of 40 Jy would be easily detectable given our integration time and sensitivity. The 8 μm Spitzer/IRAC only detects a 2.2 Jy source at the location of qF362 (Stolovy et al. 2006) which is more consistent with a star having a luminosity and effective temperature identical to that of the Pistol.