a new approach to gravitational lensing by spherical mass

A new approach to gravitational lensing by sphericalmass profiles

27th Texas Symposium on Relativistic Astrophysics

Roger HurtadoLeonardo Castaneda

Observatorio Astronomico NacionalUniversidad Nacional de Colombia

December 12, 2013

Roger Hurtado (Dallas, TX) Axisymmetric lenses 27th Texas Symposium 1 / 17

Summary

Summary

The aim of this work is to explain a method to find the properties of a spher-ically lens through a first order differential equation that depends on the massdistribution of the lens.The idea is to express the surface mass density as a combination of a decreasingfunction of a dimensionless coordinate on the lens plane, more its derivative.This method implies that it is not necessary to solve the Poisson equation tofind the deflection potential, and thereby, the observables of the lens are founddirectly and the lens equation can be expressed immediately in terms of the lensparameters.The method is tested with the most common spherical lens models, e.g., SingularIsothermal Sphere, Non-Singular Isothermal Sphere and NFW profile.


Convergence and Lens equation

Suppose a spherically symmetric mass profile lying at a distance DOL, actingas a gravitational lens on the light emitted by a source at a distance DOS fromus, and assume that the distance between lens and source is DLS. The surfacemass density

Σ(x) = 2

∫ ∞0

ρ(x, z)dz, (1)

suppose thatΣ(x) ∝ [f(x) + g(x)] (2)

where f(x) and g(x) are monotonically decreasing function, defined on interval(0,∞). Since Σ(x) ≥ 0, or f(x) + g(x) ≥ 0, we can make f(x) ≥ g(x) andf(x) > 0. Since Σ(x) may be divergent at origin, we impose the condition

limx→0

x2f(x) = 0. (3)

In turn, the convergence is defined by

κ(x) =1

2C[f(x) + g(x)] , (4)

where C depends on the distances which are functions of the cosmological model,and the physical parameters of the lens.



Poisson equation

It relates the convergence and the deflection potential of the lens ψ(x), whichfor a spherically symmetric mass distribution is

1

x

∂

∂x

(x∂ψ(x)

∂x

)= 2κ(x), (5)

thus

α(x) =1

C

1

x

∫ x

0

x′ [f(x′) + g(x′)] dx′. (6)

Since f(x) is a decreasing function ∂xf(x) ≤ 0, we can write the g(x) functionas a combination of g(x) = f(x) + x∂xf(x), so

α(x) =1

C

1

x

∫ x

0

x′ [2f(x′) + x′∂xf(x′)] dx′ =1

Cxf(x). (7)

This implies that for a spherically symmetric mass profile, whose mass densitycan be written in the form of Eq. (2), the deflection angle is proportional tothe function f(x).



Lens equation

The lens equation for a spherically symmetric situation takes the one dimen-sional form

y(x) = x− α(x) = x|1− 1

Cf(x)|. (8)

Inserting g(x) in Eq. (4), the function f(x) satisfies the following equation

xdf(x)

dx+ 2f(x)− 2Cκ(x) = 0 , (9)

and the initial condition Eq. (3).Thus the problem is reduced to solve the first-order ordinary differential equa-tion (9) for f(x).


Magnification and Shear

Magnification

The magnification is defined as the ratio between the solid angles of the imageand the source, namely

µ(x) =[yx∂xy(x)

]−1

(10)

from Eq. (8), this is

µ(x) =C2

|C − f(x)||C − g(x)|. (11)

Eq. (11) shows that magnification has two singularities in f(x) = C and g(x) =C and therefore its curve has two asymptotes at these points. These points inthe lens plane are the critical points.Noting that the magnification Eq. (11) can be written in terms of the conver-gence κ and shear γ(x), which mesasures the distortion of images

µ(x) =(

[1− κ(x)]2 − γ(x)2

)−1

(12)



Shear

This implies that

γ(x) = [1− κ(x)]2 −

[1− 1

Cf(x)

] [1− 1

Cg(x)

], (13)

and from Eq. (4), the shear is

γ(x) =1

2C[f(x)− g(x)] = − 1

2Cx∂xf(x). (14)

This implies that the shear is tangential to the function f(x).The critical curves are those points x in the lens plane where the lens equationcan not be inverted, which satisfy

[1− κ(x)]2 − γ(x)2 = 0, (15)

orκ(x) + γ(x) = 1, or κ(x)− γ(x) = 1, (16)

but, from Eq. (4) and (14)

κ(x) + γ(x) =1

Cf(x), or κ(x)− γ(x) =

1

Cg(x). (17)



Critical and Caustic curves

The critical curves are the level countours of the f(x) and g(x) functions

f(xc1) = C, or g(xc2) = C. (18)

If g(x) > 0 there are two critical circles.The caustics curves are the corresponding locations in the source plane of thecritical curves through the lens equation, that is

y(xc1) = 0, or y(xc2) = − 1

Cx2c2∂xf(xc2). (19)

with ∂xf(x) ≤ 0 en x ∈ (0,∞). Thus, the caustic curves will be a point and acircle concentric with the lens.In general, the image multiplicity depends on the source position with respectto the caustic circle.


f(x) for SIS, NIS and NFW profiles SIS profile

Singular Isothermal Sphere profile

A spherical model widely used in the gravitational lensing theory is the singularisothermal sphere (SIS), whose convergence is given by

κS(x) =2πσ2

c2DLS

DOS

1

x, (20)

where σ is the one-dimensional velocity dispersion. The differential equationfor the SIS profile in terms of the function fS(x), is

xdfS(x)

dx+ 2fS(x)− 1

x= 0, (21)

thus

fS(x) =1

x, (22)

with

CS =c2

4πσ2

DOS

DLS. (23)

Now,gS(x) = 0. (24)


f(x) for SIS, NIS and NFW profiles NIS profile

The critical curves are found when

x = C−1S , (25)

and the caustic curve will be y(xc) = 0, therefore this model only produces twoimages.To find the deflection angle, make the product x with fS(x)/CS

αS(x) = C−1S . (26)

The shear can be found through Eq. (14)

γS =1

2CS

1

x. (27)



Non-Singular Isothermal Sphere profile

One generalization of the SIS model is frequently used with a finite core x0,called the non-singular isothermal sphere (NIS). In this case, the convergenceis given by

κN (x) =2πσ2

c2DLS

DOS

2x20 + x2

(x20 + x2)3/2

. (28)

The fN (x) function associated to this profile is

fN (x) =1

(x20 + x2)1/2

, (29)

and

gN (x) =x2

0

(x20 + x2)3/2

. (30)

The critical curves are found when

xc1 = (C−2S − x2

0)1/2 and xc2 = x2/30

(C−2/3S − x2/3

0

)1/2

(31)



The associated caustics are

yc1 = 0 and yc2 =(C−2/3S − x2/3

0

)3/2

(32)

Now, the deflection angle is

αN (x) = C−1S

x

(x20 + x2)1/2

. (33)

The shear for this distribution is

γN (x) =1

2CS

x2

(x20 + x2)3/2

. (34)


f(x) for SIS, NIS and NFW profiles NFW

Navarro, Frenk & White profile

The convergence of the NFW profile is given by

κ(x) = − 1

2C(1− x2)

(1− 2

(1− x2)1/2

ArcTanh

[(1− x)

1/2

(1 + x)1/2

]), (35)

with the definition of the constant

C =Σcr

4ρ0rs. (36)

The differential equation that constrains the function fDM(x) is

df(x)

dx+

2

xf(x)+

1

x(1− x2)

(1− 2

(1− x2)1/2

ArcTanh

[(1− x)

1/2

(1 + x)1/2

])= 0, (37)



whose solution is

f(x) =1

x2

(ln(x

2

)+

2

(1− x2)1/2

ArcTanh

[(1− x)1/2

(1 + x)1/2

]). (38)

Therefore g(x), is

g(x) = − 1

x2

(ln(x

2

)+

x2

1− x2+

2(1− 2x2

)(1− x2)

3/2ArcTanh

[(1− x)1/2

(1 + x)1/2

]). (39)

Figura: An horizontal line C de-termines the critical curves whencrossed with the functions. Notethat in x > 1.3182... the functiong(x) < 0.



The lens equation is

α(x) =1

Cx

(ln(x

2

)+

2

(1− x2)1/2

ArcTanh

[(1− x)1/2

(1 + x)1/2

]), (40)

Figura: Lens equation by a NFWprofile.



Figura: Behavior of the ra-dius of the critical and causticcurves whre the lens equationtakes its maximum values as afunction of C.



Conclusions

Through the Eq. (9) we can explore all the properties of the sphericallenses.

The main result is the use of a new function which depends on the lensproperties and the lens problem is described by a first order differentialequation which encondes all information about lensing observables.

With this method we can find the lens properties in a direct way, forexample the critical and caustic curves, which are found by an equationthat relates the function and the parameter C which contains all thephysical information of the lens and also is a function of the cosmologicalmodel.

The next step is to study through the differential equation the families ofmodels that depend on certain parameters contained in the convergence.


a new approach to gravitational lensing by spherical mass

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