Gravitational lens equation: critical solutions and magnification near folds and cusps
We study approximate solutions of the gravitational lens equation and corresponding lens magnification factor near the critical point. This consideration is based on the Taylor expansion of the lens potential in powers of coordinates and an introduction of a proximity parameter characterising the cl...
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Головна астрономічна обсерваторія НАН України
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| Цитувати: | Gravitational lens equation: critical solutions and magnification near folds and cusps / A.N. Alexandrov, S.M. Koval, V.I. Zhdanov // Advances in Astronomy and Space Physics. — 2012. — Т. 2., вип. 2. — С. 184-187. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1193972017-06-07T03:05:49Z Gravitational lens equation: critical solutions and magnification near folds and cusps Alexandrov, A.N. Koval, S.M. Zhdanov, V.I. We study approximate solutions of the gravitational lens equation and corresponding lens magnification factor near the critical point. This consideration is based on the Taylor expansion of the lens potential in powers of coordinates and an introduction of a proximity parameter characterising the closeness of a point source to the caustic. Second-order corrections to known approximate solutions and magnification are found in case of a general fold point. The first-order corrections near a general cusp are found as well. 2012 Article Gravitational lens equation: critical solutions and magnification near folds and cusps / A.N. Alexandrov, S.M. Koval, V.I. Zhdanov // Advances in Astronomy and Space Physics. — 2012. — Т. 2., вип. 2. — С. 184-187. — Бібліогр.: 12 назв. — англ. 2227-1481 http://dspace.nbuv.gov.ua/handle/123456789/119397 en Advances in Astronomy and Space Physics Головна астрономічна обсерваторія НАН України |
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We study approximate solutions of the gravitational lens equation and corresponding lens magnification factor near the critical point. This consideration is based on the Taylor expansion of the lens potential in powers of coordinates and an introduction of a proximity parameter characterising the closeness of a point source to the caustic. Second-order corrections to known approximate solutions and magnification are found in case of a general fold point. The first-order corrections near a general cusp are found as well. |
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Article |
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Alexandrov, A.N. Koval, S.M. Zhdanov, V.I. |
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Alexandrov, A.N. Koval, S.M. Zhdanov, V.I. Gravitational lens equation: critical solutions and magnification near folds and cusps Advances in Astronomy and Space Physics |
| author_facet |
Alexandrov, A.N. Koval, S.M. Zhdanov, V.I. |
| author_sort |
Alexandrov, A.N. |
| title |
Gravitational lens equation: critical solutions and magnification near folds and cusps |
| title_short |
Gravitational lens equation: critical solutions and magnification near folds and cusps |
| title_full |
Gravitational lens equation: critical solutions and magnification near folds and cusps |
| title_fullStr |
Gravitational lens equation: critical solutions and magnification near folds and cusps |
| title_full_unstemmed |
Gravitational lens equation: critical solutions and magnification near folds and cusps |
| title_sort |
gravitational lens equation: critical solutions and magnification near folds and cusps |
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Головна астрономічна обсерваторія НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/119397 |
| citation_txt |
Gravitational lens equation: critical solutions and magnification near folds and cusps / A.N. Alexandrov, S.M. Koval, V.I. Zhdanov // Advances in Astronomy and Space Physics. — 2012. — Т. 2., вип. 2. — С. 184-187. — Бібліогр.: 12 назв. — англ. |
| series |
Advances in Astronomy and Space Physics |
| work_keys_str_mv |
AT alexandrovan gravitationallensequationcriticalsolutionsandmagnificationnearfoldsandcusps AT kovalsm gravitationallensequationcriticalsolutionsandmagnificationnearfoldsandcusps AT zhdanovvi gravitationallensequationcriticalsolutionsandmagnificationnearfoldsandcusps |
| first_indexed |
2025-07-08T15:48:09Z |
| last_indexed |
2025-07-08T15:48:09Z |
| _version_ |
1837094346023763968 |
| fulltext |
Gravitational lens equation: critical solutions
and magni�cation near folds and cusps
A.N.Alexandrov1, S.M.Koval2∗, V. I. Zhdanov1
Advances in Astronomy and Space Physics, 2, 184-187 (2012)
© A.N.Alexandrov, S.M.Koval, V. I. Zhdanov, 2012
1Astronomical Observatory, Taras Shevchenko National University of Kyiv, Observatorna str., 4, 04053, Kyiv, Ukraine
2National University of Kyiv-Mohyla Academy, Skovorody str., 2, 04655, Kyiv, Ukraine
We study approximate solutions of the gravitational lens equation and corresponding lens magni�cation factor
near the critical point. This consideration is based on the Taylor expansion of the lens potential in powers of
coordinates and an introduction of a proximity parameter characterising the closeness of a point source to the
caustic. Second-order corrections to known approximate solutions and magni�cation are found in case of a general
fold point. The �rst-order corrections near a general cusp are found as well.
Key words: gravitational lensing, methods: analytical
introduction
Main equation of gravitational lens theory (eq.
(1) below) sets a relation between the angular posi-
tion y of the point source and the observable position
x of its image [12]. The main interest is related to
critical points of the two-dimensional lens mapping,
i. e. the values of xcr where Jacobian of the lens map-
ping vanishes: J (~xcr) = D (y1, y2)/D (x1, x2)|~xcr
=
0. In its turn, the image of set of critical points is a
set of caustic curves. Each caustic typically appears
as a closed smooth curve with the so-called cusps at
some isolated points. Regular points of a caustic as
singularities of di�erential mapping are called folds.
When a point source crosses the fold caustic, the two
critical images either emerge or disappear. The crit-
ical images of the point source approach the critical
curve and their brightness tends to in�nity when the
source comes close to fold. In the vicinity of a cusp
we have three critical images with in�nite bright-
ness, but only two images disappear after crossing
the cusp. The details can be found in [12].
The singular properties of caustic points play a
key role in explanation of some qualitative features
and quantitative characteristics of the gravitational
lensing phenomenon. For example, the qualitative
picture of quadruple lensing can be modelled by a
singular isothermal ellipsoid [4]. Speci�cally, rela-
tive positions of four images and their brightness de-
pend on position of the source with respect to the
caustic [9, 10]. Another example is related to the
so-called strong microlensing events, which are inter-
preted as a crossing of a microcaustic by an extended
source. In this case astronomical observations give
us a chance to get some information about size of
the source and distribution of the brightness on its
surface [5, 8, 11].
The well known approximate solutions of the lens
equation and expressions for magni�cation of each
image obtained in the lowest approximation [6, 7, 12]
have a sense of asymptotic relations, which are per-
formed the better, the closer the source is located
relative to the caustic.
In the case of a fold caustic, the �rst-order cor-
rections for approximate coordinates of the critical
images were found in [2]; corrections for the mag-
ni�cation of separate images were obtained in pa-
per [10]. Note that it is possible to observe only a
total brightness of all microimages during a strong
microlensing event. In this case, the �rst-order cor-
rections for magni�cation of two critical images are
mutually cancelled. The second-order corrections for
image coordinates, as well as for the magni�cation,
were found in papers [1, 3]. Besides, it was demon-
strated on example of the strong microlensing event
in image C of gravitational lens system Q2237+0305
that the second-order corrections can be statistically
signi�cant. It was made under the simplifying as-
sumption that there is no continuous matter near
the line of sight [1, 3]. Because of importance of ac-
counting dark matter, we generalize expressions for
the second- order corrections near the fold in the
present paper.
Concerning the cusp caustic, the �rst-order cor-
rections were considered in [4], but some expressions
in that paper require revisions. Moreover, calcula-
tions are missing logical conclusions, they were left
on some intermediate stage. Therefore, the second
part of our paper is dedicated to looking for com-
plete and more compendious expressions in the �rst-
order approximation for the coordinates of images
∗seregacl@gmail.com
184
Advances in Astronomy and Space Physics A.N.Alexandrov, S.M.Koval, V. I. Zhdanov
and magni�cation near the cusp.
lens equationsnearcritical point
The normalized lens equation has the form:
~y = ~x− ~∇Φ(~x) , (1)
where Φ (~x) is the lens potential. This equation re-
lates every point ~x = (x1, x2) of the image plane
to the point ~y = (y1, y2) of the source plane. In
the general case, there are several solutions ~X(l) (~y)
of the lens equation (1) that represent images of a
point source at ~y; we denote the solution number by
the index in parentheses.
Potential Φ(~x) satis�es equation ∆Φ = 2k, where
k (~x) is the density of continuous matter on the line
of sight normalized on the so-called critical den-
sity. The magni�cation factor of each separate im-
age is K(l) (~y) = 1
/∣∣∣J (
~X(l) (~y)
)∣∣∣, where J (~x) ≡
|D (~y)/D (~x)| is the Jacobian of the lens mapping
(1).
Recall that, critical curves of mapping (1) are de-
termined with equation J (~x) = 0. Caustics are im-
ages of critical curves obtained with mapping (1).
The stable critical points of a two-dimensional map-
ping can be folds and cusps only.
Using standard approach to examine neighbour-
hood of the caustic, potential near the point pcr of
the critical curve can be approximated with the Tay-
lor polynomial. Let this point be the coordinate ori-
gin. We suppose that eq. (1) maps pcr onto the co-
ordinate origin of the source plane. Then, we rotate
synchronously the coordinate systems until the ab-
scissa axis on the source plane becomes tangent to
the caustic at the origin; the quantity |y2| de�nes
locally the distance to the caustic and y1 is a dis-
placement along the tangent.
With a su�cient accuracy, the lens equations
have the following form:
y1 = 2 (1− k0)x1 + a1x
2
1 − a2x
2
2 + 2b2x1x2+
+ c2x
3
1 − 3c1x1x
2
2 − d1x
3
2 + 3d2x
2
1x2 + g1x
4
2 + ...
y2 = b2x
2
1 − b1x
2
2 − 2a2x1x2 + d2x
3
1−
− 3d1x1x
2
2 + c2x
3
2 − 3c1x2x
2
1 + f3x
4
2 + ... (2)
Here k0 = k (0) is the matter density at the origin
and the following notations are:
a1 = −Φ,111/2; a2 = Φ,122/2; b1 = Φ,222/2;
b2 = −Φ,112/2; c1 = Φ,1122/6; c2 = −Φ,2222/6;
d1 = Φ,1222/6; d2 = −Φ,1112/6; g = −Φ,12222/24;
f = −Φ,22222/24.
When density k is constant, then a1 = a2 = a,
b1 = b2 = b, c1 = c2 = c, d1 = d2 = d. Parameter d2
will not appear in the following formulae; therefore
we put d1 = d.
approximate formulae
near fold caustic
One of the approaches for �nding critical solu-
tions of eq. (1) involves an expansion of the image
coordinates into series in powers of some parameter
t, which demonstrates proximity to the caustic [1]-
[3]. If we put yi = t2ỹi, then, as it was shown in
[1]-[3], the critical solutions of eq. (1) are analytical
functions of parameter t, and x1 = t2x̃1, x2 = tx̃2,
where x̃1 (t), x̃2 (t) are zero-order functions. Putting
these expressions into Taylor expansion of eq. (1),
and restricting our solutions to second-order terms
inclusive, we get the following equations:
ỹ1 = 2 (1− k0) x̃1 − a2x̃
2
2 + t
(
2b2x̃1x̃2 − dx̃32
)
+
+ t2
(
a1x̃
2
1 − 3c1x̃1x̃
2
2 + gx̃42
)
,
ỹ2 = −b1x̃
2
2 + t
(
−2a2x̃1x̃2 + c2x̃
3
2
)
+
+ t2
(
b2x̃
2
1 − 3dx̃1x̃
2
2 + fx̃42
)
. (3)
After performing calculations, it is enough to put
t = 1 and thus return to the initial variables yi and
xi.
A condition that initial point pcr is a fold is that
b1 6= 0. Without losing generality of our approach,
we assume that b1 < 0. When density k is constant,
then a1 = a2 = a, b1 = b2 = b, c1 = c2 = c. There-
fore, the system (3) includes four additional parame-
ters in comparison with previous case of [1]-[3] where
k (~x) ≡ 0.
We seek solutions of equations (3) accurate within
second-order terms in a form: x̃1 = x10 + x11t +
x12t
2, x̃2 = x20 + x21t + x22t
2. Imposing notations
R2 = a22 + b1b2, σ = 1 − k0 and ε = ±1 we �nd
the following expressions in the zero-order approxi-
mation:
x10 =
1
2σ
(ỹ1 − a2ỹ2/b1) , x20 = ε
√
ỹ2/|b1|. (4)
Two signs of parameter ε correspond to two critical
solutions. The �rst approximation gives:
x11 = − ε
2b21σ
2
√
ỹ2/|b1|
{
b1R
2ỹ1−
−
[
a2R
2 − (b1d+ a2c2)σ
]
ỹ2
}
, (5)
x21 =
−a2b1ỹ1 +
(
a22 − c2σ
)
ỹ2
2b21σ
. (6)
185
Advances in Astronomy and Space Physics A.N.Alexandrov, S.M.Koval, V. I. Zhdanov
Concerning the second-order approximation for the
�rst coordinate we found:
x12 =
M1ỹ
2
1 +M2ỹ1ỹ2 −M3ỹ
2
2
8b41σ
3
, (7)
where
M1 = b21
(
3a2b1b2 + 2a32 − a1b
2
1
)
, (8)
M2 = 2b1
[
b21
(
a1a2 − 2b22 − 3c1σ
)
− b1
(
7a22b2−
−(b2c2 + 6a2d)σ)− 4a22
(
a22 − c2σ
)]
, (9)
M3 = b21
[
a1a
2
2 − 4a2b
2
2 + (4b2d− 6a2c1)σ + 4gσ2
]
+
+ b1
[
−11a32b2 + (16a22d+ 6a2b2c2)σ − (4a2f +
+6c2d)σ
2
]
− 6a2
(
a22 − c2σ
)2
. (10)
And for the second coordinate:
x22 = ε
√
ỹ2/|b1|
N1ỹ2 +N2ỹ1 +N3ỹ
2
1
/
ỹ2
8b31σ
2
, (11)
N1 = −5a22R
2 + 10
(
a2b1d+ a22c2
)
σ−
−
(
5c22 + 4fb1
)
σ2, (12)
N2 = 6b1
[
a2R
2 − (b1d+ a2c2)σ
]
, (13)
N3 = −b21R
2. (14)
In its turn, for the Jacobian of the lens mapping,
calculated in points where images are situated, we
found:
J = tJ0 + t2J1 + t3J2, (15)
J0 = 4εσ
√
|b1| ỹ2, J1 = 4
R2 − c2σ
b1
ỹ2, (16)
J2 = ε
√
ỹ2/|b1|
S1ỹ2 + S2ỹ1 −N3ỹ
2
1
/
ỹ2
2b21σ
, (17)
S1 = −11a22b1b2 + 4a1a2b
2
1 + 30a2b1dσ−
− 7
(
a22 − c2σ
)2 − 4b1 (3b1c1 + b2c2 + 3fσ)σ, (18)
S2 = 2b1
[
3a32 + 5a2b1b2 − 2a1b
2
1
−3 (a2c2 + b1d)σ] . (19)
Take notice that formula for J1 was found in [10]. Fi-
nally, for the total magni�cation factor of two critical
images, we obtained:
Kcr =
1
2
Θ (y2)
σ
√
|b1| y2
[
1 + Py2 +Qy1 −
κ
4
y21
y2
]
, (20)
P = 2κb2/b1 − T
/
8b31σ
2, (21)
T = b1
[
19a22b2 − 4a1a2b1 − (30a2d+ 12b2c2 −
−12b1c1)σ + 12fσ2
]
+ 15
(
a22 − c2σ
)2
, (22)
Q =
3a32 − 2a1b
2
1 + 5a2b1b2 − 3 (a2c2 + b1d)σ
4b21σ
2
,
(23)
κ =
R2
2 |b1|σ2
. (24)
In comparison with the formulae that were found un-
der assumption of k = 0, we shown that all functional
dependencies on the coordinates yi remain the same.
Only expressions of coe�cients in terms of deriva-
tives of potential have changed.
firstapproximationnearcusp
We assume that the origin of coordinates in eq. (2)
is a cusp: b1 = 0. In this case, parameter of
proximity is introduced by the following relations:
y1 = t2ỹ1, y2 = t3ỹ2, x1 = t2x̃1, x2 = tx̃2. It can
be shown that coordinates of image x̃i (with param-
eterization proposed above) are analytical functions
of t. To return to initial coordinates, it is enough to
put t = 1. We can �nd from formulae (2), accurate
within �rst order terms, that the lens equations near
cusp caustic are
ỹ1 = 2σx̃1 − ax̃22 +
(
2bx̃1x̃2 − dx̃32
)
· t,
ỹ2 = −2ax̃1x̃2 + cx̃32 +
(
bx̃21 − 3dx̃1x̃
2
2 + fx̃42
)
· t,
(25)
where a = a2, b = b2, c = c2.
We looked for solutions in the form: x̃1 = x10 +
tx11, x̃2 = x20 + tx21. The basis for solutions con-
struction is a cubic equation for x20:
Cx320 − aỹ1x20 − σỹ2 = 0, (26)
where C = cσ − a2.
Equation (26) has one or three real roots depend-
ing on the sign of expression Q =
ỹ22σ
2
4C2
− a3ỹ31
27C3
, one
real root when Q > 0 and three real roots when
Q ≤ 0. And explicit expressions for solutions of
eq. (26) are given with Cardano formulae.
For the �rst coordinate in zero order approxima-
tion, we found:
x10 =
1
2σ
(
ỹ1 + ax220
)
. (27)
186
Advances in Astronomy and Space Physics A.N.Alexandrov, S.M.Koval, V. I. Zhdanov
We do not present intermediate formulae for the �rst
order corrections in form that repeats results of [4].
Instead, we give �nal and simpli�ed expressions at
once, which can be checked using substitution into
eq. (25). Hence, we have:
x21 =
B1ỹ1x
2
20 +B2ỹ2x20 + Cbỹ21
4σCE
, (28)
x11 =
CB1ỹ2x
2
20 +A1ỹ
2
1x20 +A2ỹ1ỹ2
4σC2E
. (29)
Here the following notations are imposed:
E = aỹ1 − 3Cx220, (30)
B1 = 6σabc− a3b− 4σa2d− 6σ2cd+ 4σ2af, (31)
B2 = σ
(
5a2b− 10σad+ 4σ2f
)
, (32)
A1 = σa
(
5bc2 − 10acd+ 4a2f
)
, (33)
A2 = a4b−2σa2bc+σ2
(
6bc2 − 10acd+ 4a2f
)
. (34)
For Jacobian components J̃ = t2 (J0 + tJ1) we
found the following expressions:
J0 = −2E, (35)
J1 =
I1
(
3Cx220 + aỹ1
)
ỹ2 + I2x20ỹ
2
1
CE
, (36)
where
I1 = a2b+ σ (10ad− 6bc)− 4σ2f, (37)
I2 = 16a3d−8a2bc−σ2
(
6acd− 3bc2 + 4a2f
)
. (38)
The magni�cation factor of each image in the �rst
approximation is given by the expression:
K =
1
|J |
=
1
t2
1
|J0 + tJ1|
=
1
t2 |J0|
(
1− t
J1
J0
)
. (39)
While �nding last equality, we took into account that
|tJ1/J0| < 1 (for small values of parameter t).
results and conclusions
The obtained formulae (7)-(19) represent expres-
sions of the second-order corrections for image coor-
dinates and Jacobian near fold caustic in the case of
general eq. (1). Formulae (20)-(24) describe the to-
tal magni�cation of two critical images in the second-
order approximation with respect to proximity to the
caustic. It is important to note that the functional
dependence on the coordinates yi and on �tting pa-
rameters remain the same, as in the case of k (~x) ≡ 0.
All the di�erences are in expressions for the �tting
parameters; these expressions have four additional
constants when a continuous matter is distributed
near the line of sight. The same situation will be with
formulae for the magni�cation factor of extended
sources [1, 3] provided that we correspondingly re-
place coe�cients P,Q, κ and take into account that
σ 6= 1. Coe�cients that are discussed in the present
paper play a role of adjustable parameters in mod-
elling observable light curves. Speci�cally, taking
into account a continuous matter does not change
anything in previous treatment of the strong magni-
�cation event in Q2237+0305 [1, 3]. Explicit depen-
dencies of coe�cients (21)-(24) on the derivatives of
potential Φ(~x) will be important in case of modelling
de�ector mass distribution.
In the last section we obtained the �rst-order cor-
rections for the image coordinates and the Jacobian
near a cusp caustic (28)-(38). Some inaccuracies of
paper [4] have been corrected, and explicit expres-
sions of the corrections are found in terms of the
potential expansion parameters and the roots of the
cubic equation (26).
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