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Appendix A: MmD with an absorbed extended continuum

We consider a line profile where absorption is present. We write where F_ce represents the flux from an extended source of continuum, which is only macrolensed as the emission line flux F_ee, F_cc ≠ 0 is the flux from the source of continuum compact enough to be microlensed, and τ_cc (τ_ce) is the optical depth of the absorber in front of cc (ce). Using these relations in Eqs. (3) and (4) with A = Mμ, we obtain In these equations, A′ and M′ are parameters tuned to satisfy constraints on F_M and F_Mμ. The constraint F_Mμ = 0 at wavelengths where F_cce^−τ_cc = 0 and F_ee ≠ 0 can only be satisfied with M′ = M. If M is correctly estimated, F_Mμ = F_cce^−τ_cc ≥ 0 independently of the existence of a nonmicrolensed extended continuum.

In the absence of a nonmicrolensed extended continuum (i.e., F_ce = 0), making F_M = 0 at wavelengths where F_ee = 0 and τ_cc = τ_ce = 0 (i.e., in the unabsorbed continuum) gives A′ = A. In the presence of a nonmicrolensed extended continuum, making F_M = 0 in the unabsorbed continuum gives A′ ≠ A. In this case, F_M = F_ce (e^−τ_ce − e^−τ_cc) at wavelengths where the continuum is absorbed and F_ee = 0, which can result in either positive or negative values for F_M whether τ_ce<τ_cc or τ_ce > τ_cc, respectively. Making F_M ≥ 0 allows the detection of the extended continuum in the latter case (τ_ce > τ_cc), but A′ = A is derived only if e^−τ_ce = 0 at some wavelengths where F_ee = 0 and e^−τ_cc ≠ 0. Without absorption (i.e., τ_ce = τ_cc = 0), such a nonmicrolensed extended continuum would remain unnoticed.

Note that μ′ = A′/M, determined by making F_M = 0 in the unabsorbed continuum, can be seen as the average microamplification of the whole continuum source, while μ = A/M refers to the microamplification of the compact continuum source. They are related by (A.5)

Appendix B: Equivalent width in the absorption trough

In order to understand how microlensing affects the EW, we can write Eq. (5), with more explicit expression of F_λ and of the continuum flux ℱ. We assume that the latter is constant over the emission line and the absorption trough and is well approximated by the continuum flux measured in the vicinity of the line. We consider a broad emission line characterized by a flux F_λ,E, and a continuum originating from two regions, a compact region cc, compact enough to be microlensed, and a more extended region ce, which is not microlensed. We can specialize Eq. (5) in the presence of absorption. For a lensed image i, we obtain (B.1)where μ_i is the amplitude of micromagnification of the compact continuum cc, and τ_cc, τ_ce, and τ_E are the optical depths of the gas absorbing the compact continuum, the extended continuum, and the broad emission line, respectively. Contrary to Appendix A, we make explicit the fact that the emission line can be absorbed (i.e., F_ee = e^−τ_EF_λ,E).

Equation (B.1) clearly shows that if F_λ,ce = F_λ,E = 0 (i.e., no extended continuum or emission line), and if the continuum does not vary over the range [λ₁,λ₂] (i.e., F_λ,cc ≈ F_cc), then the EW is independent of the amplitude of microlensing μ_i. In the following, we compare the difference of equivalent widths ΔEW = EW_D − EW_A, between image D, characterized by microlensing μ, and image A, unaffected by microlensing, in the presence of flux from the emission line and/or from an extended continuum region.

Appendix B.1: Compact continuum+emission line

We first consider the situation where there is only the compact source of continuum F_cc and the emission line over the wavelength range [λ₁,λ₂]. We then have F_λ,ce = 0 in (B.1). Assuming that the continuum flux does not vary over the range [λ₁,λ₂] such that F_λ,cc = F_cc, we can rewrite (B.1): The difference of equivalent width, ΔEW = EW_D − EW_A, can be derived from the above equation specialized for the pair of images, i.e., (B.4)It results from this equation that when μ > 1, one may only observe EW_D > EW_A.

Appendix B.2: Compact + extended continua

We now consider that there is no flux from the emission line (F_λ,E = 0) over the wavelength range [λ₁,λ₂], but two sources of continuum: a compact continuum microlensed by a factor μ_i and an extended continuum F_ce which is not microlensed. Assuming that the continuum flux is constant over the range [λ₁,λ₂] such that F_λ,cc = F_cc and F_λ,ce = F_ce, we obtain where we defined ℱ_i = μ_iF_cc + F_ce, the flux in the unabsorbed continuum.

The difference of equivalent width ΔEW = EW_D − EW_A, is then (B.7)We clearly see that a difference of equivalent width is only detected if there is a difference of opacity between the absorbers covering the extended and the compact continua, i.e., τ_cc ≠ τ_ce. If μ > 1, we find ΔEW > 0 (EW_D > EW_A), and if τ_cc > τ_ce, and conversely ΔEW < 0 (EW_A > EW_D) if τ_cc<τ_ce.

Appendix B.3: Compact + extended continua + emission line

The situation where the source of light under the absorption is the sum of a compact and an extended continuum and of an emission line, directly results from the two previous cases. If we define the flux of the unabsorbed continuum of image A and D, ℱ_A = F_cc + F_ce, and ℱ_D = μF_cc + F_ce, then we can write the following difference of equivalent width ΔEW = EW_D − EW_A between images D and A:(B.8)

From this equation, we see that when μ > 1 and τ_cc > τ_ce (i.e., when the compact continuum is more absorbed than the extended continuum), ΔEW > 0. Conversely, ΔEW < 0 can only be obtained if τ_cc<τ_ce, i.e., when the compact continuum is less absorbed than the extended continuum. Positive value of ΔEW may also occur depending on the absorption rate of the different components, and on the relative contribution of the extended continuum and of the emission line.

© ESO, 2015