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Problem Set #4 Elliptical lensing potential

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Ast 8110
Problem Set #4
Please do all the calculations in parts (a)-(c) by hand,
not using Mathematica, or other symbolic-manipulation software.
SISell lensing potential model.
Consider a commonly used elliptical lensing potential that has the form
Ψ(R) = bR[1 + γ cos(2θ)],
where b and γ are constants, θ is a polar angle in the lens plane, and R is a distance in the lens
plane, from the lens center. For this potential
(a) Derive the expression for the deflection angles in x and y directions.
(b) Derive the expressions for κ, γ1 and γ2, making sure to simplify these as much as possible.
What is the relation between κ and γ =
q
γ
2
1 + γ
2
2
?
(c) Find the expression for det[A], the determinant of the magnification matrix, simplifying it
as much as possible.
(d) On the same plot, plot the critical curve and caustic for SISell using the parameter set
(b, γ) = (1, 0.05).
(e) On the same plot, plot the critical curve and caustic for SISell using the parameter set
(b, γ) = (1, 0.15).

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