Ram-pressure Stripping of a Kicked Hill Sphere: Prompt Electromagnetic Emission from the Merger of Stellar Mass Black Holes in an AGN Accretion Disk

Post-mass loss, parcels of gas within the Hill sphere are at the pericenter of a new eccentric orbit. The Hill sphere gas will self-collide on timescales >O(t_orb) (O'Neill 2009). Supersonic collisions occur deepest in the Hill sphere (r = R₁ < R_H), where the collisional Mach number (${ \mathcal M }$) is

$$\begin{eqnarray}&&{ \mathcal M }\approx \displaystyle \frac{1}{2}\displaystyle \frac{1}{{\left(r/{R}_{{\rm{H}}}\right)}^{1/2}}\displaystyle \frac{{q}^{1/3}}{r{\left({r}_{g}\right)}^{1/2}}\left(\displaystyle \frac{c}{{c}_{s}}\right)\left(\displaystyle \frac{{\rm{\Delta }}M}{{M}_{b}}\right),\end{eqnarray} \tag{ 3 }$$

where c_s is the gas sound speed and ${\rm{\Delta }}v/{v}_{{\rm{K}},\mathrm{Hill}}\approx {\rm{\Delta }}M/2{M}_{b}$ where ${v}_{{\rm{K}},\mathrm{Hill}}(r)=c/\sqrt{{r}_{g,b}}$ is the gas Keplerian velocity with r_g,b = GM_b/c². For q ≥ 10⁻⁴, supersonic collisions can extend to R₁ ∼ R_H. If ${ \mathcal M }\gg 1$, (r ≪ R_!) the postshock temperature is $T\sim (5/16){{ \mathcal M }}^{2}{T}_{\mathrm{disk}}$ from Rankine–Hugoniot jump conditions, where T_disk is the model disk temperature at the merger radius. If ${ \mathcal M }\sim 1+\epsilon$, (r ∼ R₁), then T ∼ (1 + )T_disk. The collisions at R₁ < R_H generate energy E₁ ≈ (3/2)M_Hill(R₁/R_H)³(k_B/m_H)T_disk on orbital timescales around M_f.

In the subsonic collision zone (R₁ < r < R_H − ΔR_H), ${ \mathcal M }\lt 1$ and the average gas temperature is T ∼ (1/3)(m_H/k_B) (ΔMΔv₂/M_b)², where Δv₂ is the velocity difference between [R₁, R_H]. Subsonic collisions contribute E₂ ≈ (1/2)M_Hill(ΔMΔv₂/M_b)². Gas between R_H − ΔR_H and R_H, with total mass ΔM_Hill, collides with gas orbiting the SMBH at velocity differential Δv(r) ≈ (c/2r(r_g)^1/2)(q/3)^1/3, yielding an average uniform temperature of T = (1/3)(m_H/k_B)(Δv(r))² and energy E₃ ≈ (1/2)ΔM_HillΔv(r)².

The luminosity of the self-collisions above are limited by the timescales on which the gas collides, which could be $\gg {t}_{\mathrm{orb}}$ (O'Neill 2009). The most luminous electromagnetic (EM) contribution postmerger occurs as the kicked BH remnant attempts to carry its original Hill sphere gas with it. This is because (1) it involves most of M_Hill and (2) the timescale of the disk response is short (R_H/v_kick) compared to most self-collision timescales. Physically, once the Hill sphere gas collides with an equivalent mass of disk gas (in time R_H/v_kick), much of the original Hill sphere gas is lost via ram-pressure stripping. As a result, energy ${E}_{\mathrm{kick}}=1/2{M}_{H}{v}_{\mathrm{kick}}^{2}\,\sim$ ${10}^{47}\mathrm{erg}{({M}_{\mathrm{Hill}}/1{M}_{\odot })({v}_{\mathrm{kick}}/{10}^{2}\mathrm{km}{{\rm{s}}}^{-1})}^{2}$ is dissipated via shocks at temperature T ∼ O(10⁵)K ${({v}_{\mathrm{kick}}/{10}^{2}\mathrm{km}{{\rm{s}}}^{-1})}^{2}$ over a duration O(R_H/v_kick). Here we assume an adiabatic shock; however, the timescales for high-mass SMBHs, especially for merging binaries on large orbits, imply nonadiabatic processes will be important. For such circumstances, our estimates represent upper limits.

Photons liberated by gas shock heating can be scattered inside the disk before escape. If the merger occurs where gas pressure dominates, then the disk atmospheric density is $\rho (z)={\rho }_{0}\exp (-z/H)$ where ρ₀ is the midplane density, z is the vertical distance, and H is the disk height. The mean free path length of photons in the disk is ℓ = 1/(κρ), where κ is the disk opacity. Total path length L traveled by photons to the disk surface is L = Nℓ where N = H²/ℓ² is the average number of steps. Therefore, L = H²/ℓ and the mean photon travel time is t_Hill = L/c. Assuming constant dissipation per unit optical depth, the disk surface temperature (T_eff) is T_eff ≈ ((3/8)τ + 1/(4τ))^−1/4T₀ where τ = κΣ/2 and T₀ is the midplane temperature (Sirko & Goodman 2003). Now we consider two illustrative AGN disk models where R_H ≥ H and R_H < H. This allows us to quantify EM signatures at the order-of-magnitude level.

Consider a kicked M_b = 65 M_⊙ binary located at r = 10³r_g from an M_SMBH = 10⁹ M_⊙ SMBH, so R_H ∼ 3r_g. For a Thompson et al. (2005) disk model at radius r = 10³r_g, H/r ∼ 10⁻³, so H ∼ r_g and R_H > H in this example. The volume of gas in the Hill sphere is

$$\begin{eqnarray}&&{V}_{\mathrm{gas}}=\displaystyle \frac{4}{3}\pi {R}_{{\rm{H}}}^{3}-\displaystyle \frac{2}{3}\pi {\left({R}_{{\rm{H}}}-H\right)}^{2}[3{R}_{{\rm{H}}}-({R}_{{\rm{H}}}-H)].\end{eqnarray} \tag{ 4 }$$

Here, V_gas/V_Hill ∼ 0.5, so M_Hill = V_gasρ ∼ 0.8M_⊙.

The ram-pressure stripping of the kicked Hill sphere gas releases shock energy ${E}_{\mathrm{kick}}=1/2{M}_{\mathrm{Hill}}{v}_{\mathrm{kick}}^{2}\,\sim$ ${10}^{47}\mathrm{erg}({M}_{\mathrm{Hill}}/1{M}_{\odot })({v}_{\mathrm{kick}}/{10}^{2}\,\mathrm{km}\,{{\rm{s}}}^{-1})$ at $T\sim ({m}_{{\rm{H}}}/{k}_{B}){v}_{\mathrm{kick}}^{2}\,\sim$ $O({10}^{5})({v}_{\mathrm{kick}}/{10}^{2}\,\mathrm{km}\,{{\rm{s}}}^{-1}){\rm{K}}$ over the timescale O(R_H/v_kick) ∼ 6mo, yielding a UV luminosity ∼10⁴¹ erg s⁻¹. If disks similar to the Thompson et al. (2005) model are found around lower-mass SMBHs, the timescale R_H/v_kick drops considerably and the luminosity can reach ∼10⁴²erg/s around M_SMBH ∼ 10⁶M_⊙ (see below).

An M_b = 65 M_⊙ binary located at r = 10³ r_g from an M_SMBH = 10⁸ M_⊙ SMBH has a Hill sphere of radius R_H ∼ 6r_g. In a Sirko & Goodman (2003) disk model at r = 10³ r_g, H/r ∼ 10⁻², so H ∼ 10 r_g and R_H < H. Therefore, V_gas = V_Hill and M_Hill = V_Hill ρ ∼ 1.5M_⊙. The photon diffusion length in the disk is $\langle L\rangle =\langle H{\rangle }^{2}/{\ell }$ where $\langle H\rangle$ spans [H − R_H, H], or [2.1,10]r_g in this example. For an exponential atmosphere with z ∼ H, $\langle L\rangle =\langle H{\rangle }^{2}/{\ell }\sim [0.4,22]\times {10}^{17}\,\mathrm{cm}$ and the photon travel time spans ${t}_{\mathrm{Hill}}=\langle L\rangle /c$ or t_Hill ∼ [15 days, 2.3 yr]. For τ₀ ∼ 10⁴, T_eff ∼ 0.1T₀. So, while UV/optical photons emerge from the kick-shock on a timescale O(month) postmerger, the reprocessed optical signature is smeared out over several months with luminosity ≤10⁴¹ erg s⁻¹. Upper limits on follow-up of LIGO/Virgo search volumes therefore constrains (H, ρ) in AGN disks.

The temperature of an unperturbed thin AGN disk is T(r) = T_max r^−3/4, where ${T}_{\max }\sim 6\times {10}^{5}\,{({M}_{\mathrm{SMBH}}/{10}^{8}{M}_{\odot })}^{1/4}$ ${(\dot{m}/{\dot{M}}_{\mathrm{Edd}})}^{1/4}{\rm{K}}$ and ${\dot{M}}_{\mathrm{Edd}}$ is the Eddington accretion rate. For R_H > H, T_eff can be >T_max if τ₀ is small. The emitting area of the kicked Hill sphere is (R_H/R_disk)² ∼ 5 × 10⁻⁷ smaller than the disk (if R_disk ∼ 10⁴ r_g). The kicked hotspot has a temperature dependent only on ${v}_{\mathrm{kick}}^{2}$. Thus, a kicked merger in an AGN disk around a low-mass SMBH can generate a short-lived luminous hotspot that could dominate continuum emission in the UV or optical bands.

Figure 1 shows (black curves) the ratio of rest-frame g-band optical flux from the hotspot to that emitted by the AGN disk and (blue curves) the corresponding Galaxy Evolution Explorer NUV flux ratio. The curves assume a kicked M_b = 65 M_⊙ BH merger (v_kick ∼ 10² km s⁻¹) in a disk around an M_SMBH = 10⁶ M_⊙ SMBH accreting at $[0.01,0.1]\,{\dot{M}}_{\mathrm{Edd}}$, with R_H > H so flux escapes. Figure 2 is as Figure 1, but for M_SMBH = 10⁹M_⊙. Horizontal dashed lines show the detectability threshold of a 5% flux change. Figures 1 and 2 show the postmerger Hill sphere kick can be a substantial fraction of (or even dominate) g-band/NUV emission from the AGN disk if a_b ≥ 10³r_g. ZTF and Large Synoptic Survey Telescope (LSST) can easily detect such changes (>5%) in the g band for well-detected sources. If R_H < H, the hotspot is obscured. The curves in Figures 1 and 2 drop by a factor O(T₀/T_eff) or −1.2 in log flux ratio for T₀/T_eff ∼ 2. Mergers in lower accretion rate AGN disks with R_H < H may be detectable for small T₀/T_eff if ${\dot{M}}_{\mathrm{Edd}}=0.01$.

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2.5.1. Asymmetry in BLR Line Profiles