388 Gravitational lensing
Iνdoes not change, since the total number of photons stays constant as well as
their frequency. The amplification factorμis thus given by the ratio
μ=d(θ)
d(β)=
1
detA(θ), (14.40)
with
A(θ)=dβ
dθ(
Aij=dβi
dθj=δij−+,ij)
, (14.41)
(where+,ij = ∂i∂j+) which is the Jacobi matrix of the corresponding lens
mapping given by equation (14.31). Notice that the amplification factorμcan be
positive or negative. The corresponding image will then havepositive or negative
parity, respectively.
For some values ofθ,detA(θ)=0 and thusμ→∞. The points (or the
curve)θin the lens plane for which detA(θ)=0 are defined ascritical points
(or critical curve). At these points the geometrical optics approximation used so
far breaks down. The corresponding points (or curve) of the critical points in the
source plane are the so calledcaustics.
The matrixAijis often parametrized as follows.
Aij=(
1 −k−γ 1 −γ 2
−γ 2 1 −k+γ 1)
(14.42)
withγ 1 =^12 (+, 11 −+, 22 ),γ 2 =+, 12 =+, 21 andγ =(γ 1 ,γ 2 ).Wehave
therefore
detAij=( 1 −k)^2 −γ^2 (14.43)
andγ=
√
γ 12 +γ 22 ,
trAij= 2 ( 1 −k). (14.44)The eigenvalues ofAijarea 1 , 2 = 1 −k±γ.
In the next paragraphs, we study how small circles in the source plane are
deformed. Consider a small circular source with radiusRaty, bounded by a
curve described by
c(t)=y+(
Rcost
Rsint)
( 0 ≤t≤ 2 π). (14.45)The corresponding boundary curve of the image is
d(t)=x+A−^1(
Rcost
Rsint)
. (14.46)
Inserting the parametrization (14.42) one finds that the image is an ellipse centred
onxwith semi-axes parallel to the main axes of A, with magnitudes
R
| 1 −κ±γ|