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In general, we have that the deection angle resulting from an arbitrary mass dis- tribution can be computed through the numerical integration of the projected mass density over the lens plane according to equation1.23. Analytical expressions of the deection angle (or equivalently of the deection potential), when available, are how- ever preferred as these often yield to straightly derivable quantities like, for example, an instantaneous calculation of the amplication factors. Regarding this point, an interesting class of GL models are the axially symmetric lens models that are charac- terized by Σ(ξ) = Σ(|ξ|). Indeed, for such models, we have that the deection angle is simply given by ˆ α(ξ) = 4GM (|ξ|) c2_|ξ|2 ξ (1.40) where M(ξ) = 2πRξ 0 Σ(ξ

0_)ξ0_dξ0 _{is the projected mass enclosed in a circle of radius}

ξ centred around the axis of symmetry. Consequently, equation 1.40 allows us to easily derive the analytical solutions of the deection angle for all models satisfying Σ(ξ) = Σ(|ξ|). A simple model that fulls the latter condition while enabling the basic properties of galaxies to be modelled is the singular isothermal sphere (SIS) lens model. This model supposes that the stars constituting the lensing galaxies behave like particles in an ideal gas of constant temperature (i.e. it is isothermal) so as to account for the at rotation curves that are observed in those galaxies. From the equation of state of an ideal gas, we have that the pressure exerted by a spherical cloud of gas of radius r at constant temperature T is given by P (r) = ρ(r)kBT /mp where kB

is the Boltzmann constant and where mp is the mean mass of the gas particles (i.e.

16_{δ(x) =} 1, if x = 0 0, otherwise

stars in our case). The velocity dispersion of the stars along the line-of-sight towards the galaxy being related to T using the Virial theorem as σ2

v ≈ kBT /mp. Since we

suppose the cloud of gas and the galaxies to be in hydrostatic equilibrium, we further have that dP (r)/dr = −GM(r)ρ(r)/r2 _{such that}

ρ(r) = σ 2 v 2πGr2, Σ(ξ) = σ_v2 2Gξ, M (ξ) = πσ_v2 G ξ. (1.41)

The attentive reader will notice here that we made a distinction between M(r), the mass enclosed in a sphere of radius r and M(ξ), the projected mass enclosed within the impact parameter of radius ξ. Substituting the latter value within equation1.40 yields a deection angle for the SIS model of

ˆ α(ξ) = 4πσ 2 v c2 ξ |ξ|, (1.42)

that is a constant deection angle pointing towards the center of mass of the deector. The SIS lens model we just described, despite being based on a physical model of the kinematics of the galaxies, suers from two main drawbacks, namely the fact that its projected mass density diverges as ξ → 0, whence the reason why this model is termed `singular', and its inability to model asymmetric deectors that are otherwise commonly observed amongst galaxies. The SIS model would eectively correspond to cases where the deector is a spiral galaxy that is viewed nearly face-on which hence only covers a limited fraction of the observations. More complex models are hence required in order to accommodate for the observed asymmetry as well as to break the singularity at Σ(ξ = 0) or equivalently at κ(θ = 0). Kormann et al.(1994) concurrently solved both issues by introducing the non-singular isothermal ellipsoid (NSIE) lens model through a straight generalization of the dimensionless surface mass density of the SIS model as given by

κ(θ) = θe √ f p θ₁2+ f2_θ2 2 + θ2c = θe √ f ζ (1.43)

where f is the ratio of the minor axis to the major axis of the isodensity contours of the mass distribution (0 < f ≤ 1), θc is the core radius of this mass distribution

and where θeis a scaling factor termed the angular Einstein radius. The limiting case

where f = 1, θc= 0 corresponding then to the dimensionless mass distribution of the

SIS model while κ(0) has now a nite value if θc> 0, whence the non-singularity of this

class of model. The non-singular isothermal sphere (NSIS) lens model corresponding then to the case where f = 1, θc > 0 while the singular isothermal ellipsoid (SIE)

corresponds to the case in which we have f < 1, θc = 0. The derivation of the

normalized deection angle associated to the NSIE model can be found inKeeton & Kochanek(1998) and results in

α(θ) = θe √ f 2f0  arctanh  f0θ1 ζ + f θc  e1+ arctan  f0θ2 ζ + θc/f  e2  (1.44) where f0 ₌ p₁

− f2 _{is dened as the eccentricity of the isodensity contours of the}

mass distribution and where e1 ≡ (1, 0), e2 ≡ (0, 1).

A last degree of realism can be added by considering the stretching of the lensed images owing to the presence of a massive object in the vicinity of the deector (Kovner 1987). This external shear being then straightly given by

αγ(θ) = γR(ω)α(θ) = γ  cos ω _{− sin ω} sin ω cos ω  α(θ), (1.45)

Figure 1.20: Illustration of the NSIEg lens model parameters projected on the sky with coordinates (θ1, θ2) and origin given by the center of mass of the deecting

galaxy D. The source and image position being respectively given by S and I, as usual. The NSIEg lens model is characterized by an isothermal mass distribution having an ellipse shape with an axis ratio of f. An isodensity contour being depicted here as a gray ellipse having a semi-major axis corresponding to the scale factor θe,

called the Einstein radius. The singularity of the SIE model is solved by introducing a cut-o in the density prole corresponding to the angular radius of the core, θc,

depicted here as a lled gray circle. These parameters allow us to compute the scaled deection angle(s) α(θ) associated with a given source position θs. The presence of an

external massive object has as an eect to introduce a shear of the image(s) translating into a counter clockwise rotation of the scaled deection angle corresponding to the shear orientation ω and into a scaling of α(θ) according to the shear strength γ. The resulting deection angle αγ(θ) then provides the nal image position through

Table 1.3: Parameters of the non-singular isothermal ellipsoid lens model in presence of an external shear (NSIEg). Image(s) positions, θ, can be retrieved from the lens equation using the scaled deection angle provided in equations1.44 and 1.45. The position of the deector is here supposed to stand at the origin of the coordinate system.

Parameter Description

θs Source position

θe Scaling factor (angular Einstein radius)

θc Angular size of the radius of the core of the deector

f Axis ratio of the isodensity contour of the mass distribution of the deector (0 < f ≤ 1) where f0 ₌p₁

− f2 _{is the associated}

eccentricity

γ Shear intensity

ω Shear orientation

where γ is a unitless constant representing the shear strength and where R(ω) is a rotation matrix whose parameter ω is the shear orientation. We summarized the parameters of the NSIE lens model in presence of an external shear (NSIEg) in Table 1.3as well as in gure1.20.