Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 72 / 3 / iv / a

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- Solution a

Solution

It is useful to rewrite the local lensing convergence as

κ (θ) = θ_{E} \frac{θ _{c}^{2} + θ ^{2} /2}{( θ ^{2} + θ _{c}^{2} ) ^{3/2}} .

For positive core radius, the requested limits are

κ (θ) \sim \frac{θ _{E}}{2∣ θ ∣} ⟶ 0 (∣ θ ∣ \to \infty), κ (0) = \frac{θ _{E}}{θ _{c}} < \infty.

A singular isothermal sphere lens instead has

κ (θ) = θ_{E} / (2∣ θ ∣)

, which diverges at the centre. The finite core of the softened isothermal lensing potential regularizes this central lensing convergence; at large angular radius the two models have the same leading profile.

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