A POSSIBLE GENERATION MECHANISM FOR THE IBEX RIBBON FROM OUTSIDE THE HELIOSPHERE

In a plane-parallel geometry (Figure 2(a)), the ENA intensity I(θ) (cm² s sr keV) ⁻¹ at an angle θ to the vector z normal to the interface is derived by integrating the omnidirectional ENA emissivity j_ENA(s) = σ_cxs v_pLBs n_pLBs n_H/ΔE (cm³ s keV) ⁻¹ due to the CX reaction H⁺ + H⁰ → H⁰ + H⁺, corrected for ENA extinction inside the LB and for obscuration in the LIC, over the line of sight s = z/cos θ. Here σ_cxs, v_pLBs, and n_pLBs denote, respectively, the CX cross section, velocity, and density of the suprathermal protons in the LB (all functions of energy), and n_H is the density of neutral H in the interface layer, which depends on the details of interface structure (ionization front/shock wave/evaporative boundary, see Slavin 1989; Slavin & Frisch 2002).

Current estimates of the properties of the LIC and LB suggest that they can be near pressure equilibrium. The thermal LB pressure is p_LB/k = 2n_eLBT_LB ∼ 10⁴ K cm⁻³ (k = Boltzmann constant) assuming that the density of the thermal protons is n_pLBt ∼ n_eLB = 0.005 cm⁻³ and T_LB = 10⁶ K. Reassessment of the LB conditions, considering that some of the soft X-ray background may come from the solar wind, suggests p_LB/k = 8700 K cm⁻³ (Welsh & Shelton 2009). On the other hand, the total (thermal + magnetic) LIC pressure p_LIC/k = (n_HLIC + 2 n_eLIC) T_LIC + B²/8πk, for n_eLB and T_LB as above, is in the range 4200–6800 K cm⁻³. Here B denotes the magnitude of the local interstellar magnetic field, estimated to be 3.1–3.8 μG from the equipartition between the thermal and magnetic energy densities (Frisch & Slavin 2006). An upper limit of 3.8 μG was also inferred from the distances at which the Voyagers crossed the termination shock (Ratkiewicz & Grygorczuk 2008), although fields up to 5.5 μG have recently been suggested (Opher et al. 2009). Based on this, we assume approximate equilibrium between the LIC and LB, with the LIC's edge at a distance z₀ from the Sun.

The density n_H(z) of the neutral H evaporating from the LIC follows from the continuity equation

$$\begin{eqnarray} &&{\rm div}({n_{\rm H}\, {\bm v}_{\rm H} }) = - \gamma _{e{\rm ff}}\, n_{\rm H}, \end{eqnarray} \tag{ 1 }$$

where γ_eff = σ_cxtv_pLBtn_pLBt +  ε_imp(T_LB)n_eLB +  γ_ph is the effective ionization rate (s⁻¹) of H resulting from CX, electron impact ε_imp(T_LB)n_eLB, and photoionization γ_ph, and $\bm v$_H is the average speed of H atoms. In a static case for an isotropic Maxwellian, the velocity component normal to the interface v_Hz = (8 k T_LIC/(π m_H))^1/2, where m_H is the proton mass. For T_LIC = 6300 K, v_Hz = 2.9 km s⁻¹. In hydrodynamic solutions (Cowie & McKee 1977), the speed of evaporating medium attains and even exceeds the local sound speed, which changes from ∼9 km s⁻¹ on the LIC side to >100 km s⁻¹ in the LB at T_LB = 10⁶ K. Therefore, we explore v_Hz values from ∼2.9 to ∼25 km s⁻¹ using v_pLBt = 140 km s⁻¹ and, at this speed, σ_cxt = 2.9 × 10⁻¹⁵ cm² (Lindsay & Stebbings 2005). With ε_imp(10⁶ K) = 5.3 × 10⁻⁸ cm³ s⁻¹ (Olsen et al. 1994), the CX and electron impact ionization rates for H are 2.0 × 10⁻¹⁰ and 2.7 × 10⁻¹⁰ s⁻¹, respectively. The photoionization rate, estimated using the radiation density from Slavin & Frisch (2008), is γ_ph < 10⁻¹³ s⁻¹ and therefore negligible. Assuming at the LB–LIC interface n_H (z ≤ z₀) = n_{H LIC}, Equation (1) yields n_H(z) = n_{H LIC} exp[−γ_eff (z − z₀)/v_Hz] with v_Hz/γ_eff = ∼ 40–360 AU.

With this model, the expected ENA intensity I(θ) results from integration of j_ENA over an infinite line of sight and accounting for ENA loss by ionization through CX and electron impact inside the LB. This is expressed by

$$\begin{eqnarray} I\left(\theta \right) &=& \frac{1}{{4\pi \Delta E}}\exp ({ - \tau _0 \sec\theta })\nonumber\\ &&\times\frac{{n_{{\rm H}{\rm LIC}} n_{{\rm pLBs}} v_{{\rm Hz}} v_{{\rm pLBs}}^{\rm 2} \sigma _{{\rm cxs}} \sec\theta }}{{v_{{\rm pLBs}} \gamma _{{\rm eff}} + n_{{\rm pLBt}} v_{{\rm Hz}} ({\varepsilon _{{\rm imp}} + v_{{\rm pLBs}} \sigma _{{\rm cxs}} })\sec\theta }}, \end{eqnarray} \tag{ 2 }$$

where $\exp (- \tau _0 \sec \theta)$ is the dimming of the ENA intensity by the LIC. The LIC extinction thickness of ENAs

$$\begin{eqnarray} \tau _0 &=& ({\sigma _{{\rm cxs}} n_{{\rm pLIC}} + \sigma _{{\rm A62}} n_{{\rm He}^ + {\rm LIC}} + \sigma _{{\rm E2}} n_{{\rm H LIC}} + \sigma _{{\rm E10}} n_{{\rm He}{\rm LIC}} })z_0 \nonumber\\ &&\sim \sigma _{{\rm cxs}} n_{{\rm p}{\rm LIC}} z_0 \end{eqnarray} \tag{ 3 }$$

arises primarily from CX of ENAs by LIC protons and secondarily from CX with He⁺ (σ_A62 = 4.5 × 10⁻¹⁷ cm²) and ionization from interactions with H⁰ and He⁰ (σ_E2 = 2.3 × 10⁻¹⁸ cm² and σ_E10 = 3.6 × 10⁻¹⁷ cm²; both the notation and quoted 1.1 keV values of the cross sections σ_A62, σ_E2, σ_E10 are from Barnett 1990). We use the LIC He I density n_HeLIC = 0.015 cm⁻³ derived by Ulysses (Witte et al. 2004). In calculating τ_0, $n_{{\rm He}^ + {\rm LIC}}$ was chosen to always satisfy the abundance condition H/He = 10 for any value of the unknown n_pLIC.

From Equation (2), I(θ) attains its maximum value at θ < 90^o, resulting in a circular arc like the ribbon. We introduce the density ratio f of suprathermal-to-thermal protons (i.e., n_pLBs = f n_pLBt). Assuming f ≪ 1 and that the suprathermal energy distribution has a ∼1 keV effective width and peaks near 1.1 keV, one obtains from Equation (2) the ENA intensity profiles near Earth shown in Figure 3(a) for f ranging from 0.36% to 1.3%. In Figure 3(a), the contrast ratio I_max/I(0°) of maximum intensity I_max to that at 0° is ∼2–2.6, similar to the observed values (see McComas et al. 2009b and Figure 2 in Fuselier et al. 2009). We note that I(θ) depends linearly on the a priori unknown f. Nevertheless, the contrast ratios derived in Figure 3(a) could only be obtained for the LIC extinction thickness τ₀ < ∼0.15–0.2, irrespective of adjusting other parameters including f and in particular of the effective outflow velocity v_Hz of the H atoms for which the best fitting value for v_Hz is in the range 7–25 km s⁻¹. Therefore, even for the assumed very low values of n_pLIC = 0.01–0.03 cm⁻³, the distance to the edge of LIC is very small, 260–740 AU. Another feature is that I_max lies in the narrow range θ ∼ 75°–82° instead of ∼70°, and is generally insensitive to variation in individual parameters. The difference between the simulated and measured opening half-angle of the ribbon suggests that the plane-parallel interface may need to be replaced with a curved (spherical section) model.

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If the edge of the LIC is curved and convex toward the Sun (e.g., a local bay of the LB—Figure 2(b)), then ENA emission is brightest (for τ₀ small enough) when viewing close to tangent to the curved surface. We seek a value of the radius of curvature R such that the maximum of emission occurs at θ ≈ 70°, in agreement with the IBEX results; actually we assume I(θ) tends to 0 at θ → 75°, thus R = sin75°/(1 − sin75°) z₀ = 28.3 z₀.

Here, the IBEX line of sight is defined by angle θ from the vector between the Sun and the closest point of the LB, which is also the axis of symmetry of the model. The ENA intensity from the LB is given by the integral of j_ENA over the distance s along the line of sight, corrected for the ENA extinction both inside the LB and LIC:

$$\begin{eqnarray} I\left(\theta \right) &=& \frac{1}{{4\pi \Delta E}}\exp[ { - \tau _{{\rm LIC}} ({0,s_{{\rm near}} })}]\nonumber\\ &&\times\!\int_{s_{{\rm near}} }^{{\rm s}_{{\rm far}} }\!\! {j_{{\rm ENA}} (s)\exp [ { - \tau _{{\rm LB}} ({s_{{\rm near}},s})} ]\,ds} \,({\rm cm}^2 \,{\rm s}\,{\rm sr}\,{\rm keV})^{ - 1}{.}\nonumber\\ \end{eqnarray} \tag{ 4 }$$

Here s_near and s_far, both functions of θ, denote distances to the near edge and far edge (= infinity in practice) of a spherical bay. τ_LB(s_near, s) and τ_LIC(0, s_near) are the ENA extinction thicknesses over the path from s to s_near in the LB and then from s_near to the Sun in the LIC, respectively. While τ_LIC(0, s_near) is directly proportional to s_near, τ_LB(s_near, s) is an integral of the varying CX and electron impact probabilities over the flight path. These come from Equation (1), which we solve assuming spherical symmetry.

The I(θ) profiles resulting from Equation (4) show that a contrast ratio I_max/I(0°) of ∼2–3 can be achieved for larger extinction thicknesses, up to τ₀ ∼ 0.5 (Figure 3(b)), for values of f and v_Hz similar to the plane-parallel model (0.375% ≤ f ≤ 1.8%, 7 km s⁻¹ ≤ v_Hz ≤ 25 km s⁻¹). The resulting shortest distances z₀ from the Sun to the LIC edge are correspondingly larger, 390 AU ≤ z₀ ≤ 1100 AU (R = 28.3 z₀ = 11 × 10³–31 × 10³ AU) for the cases shown in Figure 3(b), which include also an often-quoted value of n_pLIC = 0.04 cm⁻³ (Izmodenov et al. 2003). The model with a curved interface is not only more adjustable through R, but also numerically sensitive to the opening angle of the ribbon. For example, a decrease of the half-opening angle by 5° compared with Figure 3(b) results in a contrast ratio of 2.2 even at τ₀ = 0.5 and n_pLIC = 0.01 cm⁻³, which implies z₀ = 1800 AU and R = 29000 AU. If this idea is correct, IBEX will provide an independent measure of the distance to the boundary of the LIC and of its curvature.