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1. INTRODUCTION
In the study of gravitational lenses (GL) volumetric distribution of mass density of the deflector (ρ), can be projected onto a plane perpendicular to the line of vision between the observer and the light source. This plane is called the lens plane, according to Narayan, R. and Bartelmann, M. (1997). The lens is considered thin and the volumetric distribution of lens mass is replaced by a plane on which the surface mass density is (Σ), constituting the so-called approximation of the thin lens, in accordance with Schneider, P. et al. (1992).
In general, GL require a few basic elements such as: surface mass density (Σ), lens equation, deviation angle (α), deflection potential (Ψ) and time delay (∆t), which comprise basic tools for their application in the study of some lens systems in astrophysics. These analytical expressions are specified when applied to a model of mass volume distribution in a particular galaxy.
Some observational data shows that astronomical objects that act as lenses are modeled in diverse ways, according to Cohen, A. S. and Hewitt, J. N. (2000). In systems of gravitational lenses per galaxy, we observationally mea- sure parameters such as: dispersion speed of the material particles that make up the lens (σ p ), angular position of the images (θ), red shifts of lens (z L ), source (z S ) and time delay (∆t); with these it becomes possible to study and assume models in said systems.
The distances from the source to the lens and from the lens to the observer are ∼
1pc. In this way, systems that constitute the lenses, such as
deflecting galaxy, source and observer, are too far away, which causes light to
travel in free space most of the time, this light only locally becoming deviated
when it passes through the lens. We must therefore model the universe through which
the ray of light should pass; to do this we need to set cosmological parameters,
Used, for example, by the authors Adler et al.
(1975), Ciufolini et al. (1995),
Foster et al. (1994), Kenion (1995): vacuum density (Ω
v
); matter density (Ω
m
); softness parameter (
By applying the properties of GL to a specific system, using observational values and setting cosmological parameters, it is possible to study different galactic models.
In our case we will assume a model of a galaxy which is elliptical in its volumetric distribution of mass (ρ), with which we determine the analytical expressions of GL, used, e.g., by Brainerd et al. (1996) and Golse et al. (2002). Here we follow Hjorth et al. (1997) and Molina et al. (2006), who propose a distribution of volumetric mass density useful for models of elliptical galaxies acting as gravitational lenses.
2. LENS ELEMENTS AND ELLIPTIC MODEL IN THE GALAXY DEFLECTOR
2.1 Lens elements
In GL literature, the approximation of a flat lens is characterized by a surface mass density given by the projection operator, according to Miranda, C., Molina, U and Viloria, P. (2014),
where
Fig. 1 Illustration of a gravitational lens system. The angular separations of the source and the image from the optic axis as seen by the observer are β and θ, respectively. The angular diameter distances between the observer and the source, the observer and the lens, and the lens and the source are DS , DL and DLS , respectively, by Narayan et al. (1997)
when x’ < x. The quantity κ defined as
The deflection potential of the lens is according to Narayan et al. (1997),
The time delay ∆t between two light beams detected by an observer is given by Molina et al. (2006) and Narayan et al. (1997),
where z L is the redshift of the deflecting galaxy, D L is the angular diameter distance of observer-lens, D S is the angular diameter distance of observer- source, D LS is the angular diameter distance of lens-source.
The expression (4), contains the geometric delay described in equation (2), and the gravitational potential given by equation (3).
Furthermore, the relationship between source position (β),
positions of images (
which is called the lens equation. Equations (2), (3), (4) and (5) make up the group of basic elements for a GL study.
2.2 Elliptical galaxy model deflector
For our study, we modeled the distribution of lens mass as an elliptical galaxy, following Hjorth and Kneib (1997), who proposed a distribution of volumetric density which is useful for elliptical galaxy models acting as gravi- tational lenses. With these distributions we find the analytical expressions of GL. This volumetric distribution of mass is,
The model of deflecting galaxy contains a central nucleus with radius a, a free-form parameter acting as scale b (b > a), and a volumetric density of fixed mass in the nucleus ρ0. Introducing the volumetric density given in equation (6), in the projection operator defined in equation (1) and making the change of variable, z 2 = r 2 − R 2 (see Fig. 1), we obtain the surface mass density of the lens. The distance z is typically much smaller than the distances between observer and lens and between lens and source, so after developing the integrals the surface mass density of the lens takes the following form,
where
To find the analytical expressions of the deviation angle and the deflection potential, first we change R = ξ0x, in equation (7), in such a way that the surface density of the flat lens is,
where
where we choose a scale factor such as,
The scale factor is set by the dispersion speed of the components of the
deflecting galaxy σ
p
(this is explained in the following paragraphs). We observe that the
scale factor is expressed in terms of the central radius a, and
the adimensional parameter
In the time delay expressed in equation (4), we see the arbitrary scale factor ξ 0 , which we choose for the distribution of elliptical mass as shown in equation (10). The central volumetric density ρ 0 is not a quantity that is observationally measurable in a gravitational lens system, but it is possible to establish it by knowing the rate of dispersion σ p of the matter components of the deflecting galaxy. To do this, through the expression proposed by Hjorth and Kneib (1997),
it is possible to obtain the dispersion rate.
In this expression (11) we see that if the dispersion velocity of the matter components of the deflecting galaxy is known, the central volumetric density can be set, and with this, the scale factor (10) is also set.
2.3 Deviation angle
Expressed in terms of impact parameter R = ξ 0 x and using the convergence factor in equation (9), the deviation angle (2) takes the form,
which is given in terms of central radius a and adimensional
parameter
2.4 Deflection Potential
We find the deflection potential of the lens that we are considering by re- placing equation (9) in equation (3) and writing in terms of impact parameter R = ξ 0 x, we obtain,
given that n > 1. We note that if this adimensional parameter approximates one, that is, n ≈ 1, and if we select a set scale, the deflection potential becomes zero, ψ(R) ≈ 0.
2.5 Time delay
Differentiating the square of the deviation angle for two images,
Equation (13) allows us to determine the time delay between two images, where a the radius of the nucleus of lens mass distribution is and n is the adimensional parameter.
Furthermore, we have defined two new functions: h(n, a) and g(n, a), de- fined below, depend on how the nucleus radius a is set and and on how much these two new functions vary from the adimensional parameter n.
Function h(n, a) is defined in the form,
Similarly, g(n, a) is defined as,
h(n, a) and g(n, a), are normalized so that when they are introduced into the time delay equation (14), their units are expressed in seconds.
3. APPLICATION OF THE PROPOSED GRAVITATIONAL LENS MODEL TO GALACTIC LENS B0218 + 357
The expressions articulated in the previous section are completely general and can be applied to any gravitational lens system. In our case, by way of example, we choose to apply them to the B0218 + 357 lens system, to analyze the effectiveness of the results we obtain.
Some researchers, such as, notably, Wucknitz et al. (2004) who, in their work Models for the Lens and Source of B0218 + 357 determine the Hubble constant H0, and discuss different models for the B0218 + 357 galactic lens. In addition to this, in order to obtain an estimate of the Hubble constant,
A.D. Biggs et al. (1999) model the B0218 + 357 system using the lens model described by Kormann et al. (1994) as a Singular Isothermal Ellipsoid (SIE) mass. More information on the morphology of the received images can be found in the work of C. Spingola et al. (2015). Observational data for the B0218 + 357 lens system deposited in the CASTLES Survey, according to Cohen, A. S., and Hewitt, J. N., (2000) is summarized in table 1 below: the dispersion speed σp, the angular positions of two images θ 1 and θ 2 , the redshifts for lens z L and the source z S ; and the difference in time delay between the two images ∆t, as follows:
Table 1 Data observed in B0218 +357 lens system
| σ p | θ 1 | θ 2 | z L | z S | ∆t |
| 150km/s | 40mas | 290mas | 0.68 | 0.94 | 10.5±0.2days |
For our case, we choose cosmological parameters within the range of values most
accepted in literature; we thus rely on the work of several authors, and among these
point especially to: Kessler et al. (2009),
Boughn and Crittenden (2001), Bartelmann et al. (1997), Weinberg (1972), Grogin et al.
(1996)) and others. The parameters we choose for our elliptical lens
model are: Hubble Constant,
According to Dyer, C. C. (1973) and Wucknitz et al. (2004), by using, for the B0218
+ 357 galactic lens system, the previous cosmological parameters and the following
values: cross section
With these quantities, we can find the elements of the proposed lens model.
3.1. Scale factor, surface mass density of the lens, deviation angle and deflection potential.
Because there are two images in the B0218 + 357 lens system, the previous
calculations on the two impact parameters R
1
= 264·58pc and R
2
= 1918·21pc allow us to set the radius
of the nucleus in the range of values of these two parameters
(R
1
and R
2
). In this work, we give the radius of the nucleus ap- proximately the
value of the minor impact parameter, expressing the radius of the nucleus
approximately as a = 264pc, which corresponds
to the impact parameter with least deviation. To facilitate analysis we define
the adimensional quantity λ as
Furthermore, the scale factor in equation (10), is in terms of the adimen- sional parameter n, that is,
being pc = 3.086 × 1016 m. The surface mass density of the lens, expressed in equation (8) is then found to be,
where for this particular lens, the impact parameter satisfies the condition 1 ≤ λ ≤ 7.25 and n > 1. When the value of n is fixed, expression (19) allows us to estimate the surface mass density of the lens as a function of λ, in the given interval.
After making the replacements required by our proposed system, the de- viation angle represented in equation (12), takes the form,
in which we know that 1 ≤ λ ≤ 7.25 and n > 1.
At the same time, the deflection potential expressed in equation (13), takes the new form,
which depends upon the scale factor represented in equation (18) and the following conditions: 1 ≤ λ ≤ 7.25 and n > 1.
3.2. Time Delay Model
By using the values obtained from the B0218 + 357 lens system, we reduce the time delay stated in equation (14) is reduced to,
Quantities h and g, and equations (15) and (16), depend only upon the adimensional parameter n. This allows us to establish values for the time delay. We take advantage of the fact that the time delay between the two images is measured observationally, as shown in table 1 (10.5 days), and a series of different values of n are explored in equation (22), until observed value is reached. The time delay of equation (22), for some values of n, is as follows:
When we compare the observational time delay ∆t = 10.5 ± 0.2days, as shown in table 1, with equation (22), we see that for the B0218 + 357 lens system, the adimensional parameter has a range of certainty of 2 ≤ n ≤ 2.2. We can appreciate the most approximate value to n = 2.1 in Table 2.
Table 2 Time delay values, for some values of n, in the B0218 +357 lens system, according to equation (22).
| n | 2 | 2.1 | 2.2 |
| ∆t(days) | 9.67 | 10.3 | 10.9 |
Establishing the value of parameter n allowed us to estimate the geometric parameters of the elliptical lens, a and b, and afterwards find the numerical values for the other basic elements such as the surface density of the lens, the deviation angle, the scale factor and the deflection potential, as indicated in the following section.
3.3. Estimation of deviation angle, deflection potential and lens equation
Given that for the B0218 + 357 lens system, the approximate radius of the nucleus
is, the adimensional parameter is a = 264pc,
the cross section is
Using as our premise equation (19), taking into account that the value of the adimensional parameter is n = 2.1, the surface density diminishes as the impact parameter increases, and because this is in the interval 1 ≤ λ ≤ 7.25, the surface mass density of the lens for the proposed system is estimated in the range of 0.32kg/m 2 ≤ Σ ≤ 32kg/m 2 .
In accordance with the equation (20), and taking into account that the parameter λ is in the range 1 ≤ λ ≤ 7.25, we deduce that deviation angle oscillates between 120mas and 220mas.
In accordance with equation (18), and with the values of adimensional parameter n, represented in section 3.1, we can establish an approximate value of the scale factor, that is, ξ 0 = 260pc. Furthermore, as the values of parameter λ oscillate between 1 ≤ λ ≤ 7.25, then the values of the deflection potential, according to equation (21), oscillate between 150mas and 930mas.
4. CONCLUSIONS
This work is based on a volumetric mass distribution which describes a fast
relaxation scenario, similar to a model of an SIS isothermal sphere, according to
Kenion, I. R. (1995), where the mass of
the lens is considered spherically symmetric. The model of elliptical density
contains a central nucleus with radius a, mass density
ρ
0
in the central nucleus and free-form parameter a, a mass
density in the central core ρ
0
and also a free shape parameter b (b >
a). From this volumetric mass distribution we find new analytical
expressions of the lens elements. These new elements are equations for: lens surface
density, deviation angle, deflection potential and time delay. These expressions
depend upon the impact parameter of the images and on geometric lens elements
a and b, related by the adimensional parameter
The analytical expressions of the surface mass density of the lens, deviation angle, deflection potential and time delay used in this work to describe our proposed model can be used to analyze any other system of galaxy lens. Our equations are quite general and for their application to study a specific lens system only require the observational measurements indicated in Table 1.
To implement the results of section (2), one basically needs a system of lenses per galaxy whose mass density distribution fits the elliptical model we describe in this paper, the analytical expressions found here being a good point of departure for further research. In this work the proposed gravitational lens model is applied specifically to the B0218 + 357 lens system, using observational values shown in Table 1. To do this, we have used the cosmological parameters most widely accepted in literature, used, e.g., by authors Bartelmann, M., at al. (1997), Boughn et al. (2001), Foster et al. (1994), Kessler et al. (2009), Schneider et al. (1992),such as the Hubble constant, vacuum density, matter density and softness parameter contained in the angular dia- metric distances of observer-lens, observer-source and lens-source. This allows us to find the approximate radius of the nucleus of the deflecting galaxy, the angular diametrical distances, the scale factor synthesized in equation (18), and afterwards, adjust the adimensional parameter n.
The adimensional parameter n is adjusted through the time delay
represented in equation (22), and compared to the observed time delay
∆t
obs
= 10.5days, shown in Table
1. To do this, it was necessary to write the impact parameter R, in terms
of the nucleus radius ; in the form of
Because the impact parameter is expressed in terms of the radius of the nucleus, we were able, once we have set cosmological parameters, determine the theoretical time delay given in equation (22), depending only on the adimensional parameter n. Thus, when we compare the theoretical time delay given in equation (22), with the observationally measured time delay in Table 1, we find that the range of values of the adimensional parameter n is 2 ≤ n ≤ 2.2.
Finally, as evidenced in section (3.3), by adopting the approximate value n = 2.1 for the adimensional parameter and the interval for the impact parameter 1 ≤ λ ≤ 7.25, we were able to estimate numerical values for the surface density of the lens, deviation angle, scale factor and deflection potential in our proposed system.
