Difference between revisions of "Point gravitational lenses"

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(Problem 3: deflection of light near the Sun)
(Problem 9: source of finite size)
 
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When light is deflected in optical lens, its deflection angle is proportional to impact parameter. In contrast, gravitational field of gravitational lens decreases with distance, and so does deflection angle.
 
When light is deflected in optical lens, its deflection angle is proportional to impact parameter. In contrast, gravitational field of gravitational lens decreases with distance, and so does deflection angle.
 
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Considering opaque lenses and small angles, the light ray deflection angle is inversely proportional to impact parameter.
 
Considering opaque lenses and small angles, the light ray deflection angle is inversely proportional to impact parameter.
 
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Two main conclucions can be drawn from these differences in deflection angle dependencies for optical and gravitational lenses. First, unlike optical lens, gravitational lens has no focal point. Indeed, rays, which passed through the gravitational lens, can not intersect in one point, since rays with larger impact parameters would cross the axis of the system at a greater distance from lens, than rays with smaller impact parameters $p$.
 
Two main conclucions can be drawn from these differences in deflection angle dependencies for optical and gravitational lenses. First, unlike optical lens, gravitational lens has no focal point. Indeed, rays, which passed through the gravitational lens, can not intersect in one point, since rays with larger impact parameters would cross the axis of the system at a greater distance from lens, than rays with smaller impact parameters $p$.
 
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Instead, gravitational lens has focal semiaxis. Moreover, opacity of the lens leads to the existence of shadow area behind the lens, which location is determined by the rays, passed close to the lenses edge.
 
Instead, gravitational lens has focal semiaxis. Moreover, opacity of the lens leads to the existence of shadow area behind the lens, which location is determined by the rays, passed close to the lenses edge.
 
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Second, in contrast to optical lens, the intensity of registered radiation after passing the gravitational lens is infinite not in focal point, but on entire focal semiaxis.  However, this singularity exists only in small angles approximation and dissapears if we account for finite size of the source.
 
Second, in contrast to optical lens, the intensity of registered radiation after passing the gravitational lens is infinite not in focal point, but on entire focal semiaxis.  However, this singularity exists only in small angles approximation and dissapears if we account for finite size of the source.
 
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It's important to understand the notion of ''small angles'' in the context of gravitational lensing. Rays near the axis of optical lens have small deflection angles, so that their impact parameter is small. Small deflection angle of light ray in gravitational lens means that impact parameter is large (much larger, than gravitational radius of the lens, to be more precise).</p>
 
It's important to understand the notion of ''small angles'' in the context of gravitational lensing. Rays near the axis of optical lens have small deflection angles, so that their impact parameter is small. Small deflection angle of light ray in gravitational lens means that impact parameter is large (much larger, than gravitational radius of the lens, to be more precise).</p>
 
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=== Problem 7: multiple images  ===
 
=== Problem 7: multiple images  ===
Show that when the gravitational lens is placed between source and observer in the general case the two images of the source would be observed. How are the images placed relative to the lens and observer?<div class="NavFrame collapsed">
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Show that when the gravitational lens is placed between source and observer in the general case the two images of the source would be observed. How are the images placed relative to the lens and observer?
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=== Problem 8: Einstein ring  ===
 
=== Problem 8: Einstein ring  ===
How should source, gravitational lens and observer be placed relative to each other in order to observe the Einstein ring? Calculate the radius of the ring.<div class="NavFrame collapsed">
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How should source, gravitational lens and observer be placed relative to each other in order to observe the Einstein ring? Calculate the radius of the ring.
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Considering the case of the finite size source, we can first investigate the image of the disk source of radius $R$. Angle distance between the images of two uttermost point of the disk  is $\beta$. Moreover, since the problem is axisymmetric, the Einstein ring would have finite width. Since its angular width is $\beta$, it's easy to obtain, that width is equal to the radius of the source projected on the lens plane.
 
Considering the case of the finite size source, we can first investigate the image of the disk source of radius $R$. Angle distance between the images of two uttermost point of the disk  is $\beta$. Moreover, since the problem is axisymmetric, the Einstein ring would have finite width. Since its angular width is $\beta$, it's easy to obtain, that width is equal to the radius of the source projected on the lens plane.
 
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In a more precise manner this can be considered as follows. Let the boundary of the source to be described by $\tilde\rho_S(\varphi)$. If its size is so that $\tilde\rho_S(\varphi)\ll\tilde l$, we can obtain
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In a more precise manner this can be considered as follows. Let the boundary of the source to be described by $\tilde\rho_S(\varphi)$. If its size is so that $\tilde\rho_S(\varphi)\ll\tilde{l}$, we can obtain
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$\vec p_{1,2} \simeq \vec \tilde{\rho_S} \left\{\frac 1 2 \pm {\frac{\tilde{l}}{ \tilde{\rho_S}}}\right\}.$
\vec p_{1,2} \simeq \vec \tilde \rho_S \left\{{1\over 2} \pm {\tilde l \over \tilde \rho_S}\right\}.
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For disk of the projected radius $\tilde R_S$ Einstein ring has finite width of  
 
For disk of the projected radius $\tilde R_S$ Einstein ring has finite width of  
 
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$$
\Delta p = p_1 - p_2 \simeq \tilde R_S,
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\Delta p = p_1 - p_2 \simeq \tilde{R_S},
 
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with the mean radius $\tilde l$, as in the case of point source.</p>
 
with the mean radius $\tilde l$, as in the case of point source.</p>
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=== Problem 10: source of finite size  ===
 
=== Problem 10: source of finite size  ===
 
Qualitatively consider the general situation, when a source of finite size, lens and observer are not on one line. Estimate the angular sizes of the observed images.
 
Qualitatively consider the general situation, when a source of finite size, lens and observer are not on one line. Estimate the angular sizes of the observed images.
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=== Problem 12: double Einstein ring  ===
 
=== Problem 12: double Einstein ring  ===
 
Recently the exceptional phenomenon was observed using the Hubble space telescope: the double Einstein ring, formed by the influence of the gravitational field of the galaxy on the light from two other more distant galaxies. What conditions are necessary for the observation of this phenomenon?
 
Recently the exceptional phenomenon was observed using the Hubble space telescope: the double Einstein ring, formed by the influence of the gravitational field of the galaxy on the light from two other more distant galaxies. What conditions are necessary for the observation of this phenomenon?
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These expression are valid when $x>x_{\min}$, while $\mu = 0$ for $x<x_{\min}$.
 
These expression are valid when $x>x_{\min}$, while $\mu = 0$ for $x<x_{\min}$.
 
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When observer is located on the source-lens axis, the amplification tends to infinity $\mu \to\infty$. Thus, unlike optical lens, gravitational lens produces image on any distance from it. When moving away from the lens the intenity increases as $\sqrt {D_d}$.</p>
 
When observer is located on the source-lens axis, the amplification tends to infinity $\mu \to\infty$. Thus, unlike optical lens, gravitational lens produces image on any distance from it. When moving away from the lens the intenity increases as $\sqrt {D_d}$.</p>
 
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Latest revision as of 00:36, 23 April 2014

Problem 1: optical lens vs. gravitational lens

Compare the dependence of refraction angle $\hat{\alpha}$ on impact parameter $p$ for optical and gravitational lenses.


Problem 2: deflection angle: Newronian approach

Obtain the formula $\hat{\alpha}=r_g/p$ for refraction of light ray using the Newtonian theory.


Problem 3: deflection of light near the Sun

Calculate the angle of refraction of light in the gravitational field of the Sun.


Problem 4: refractive index

Propagation of light in gravitational field could be considered as propagation in a medium. Calculate the effective refractive index for such a medium.


Problem 5: shadow area

Determine the dependence of a ray's shifting from the axis of symmetry after the refraction on a nontransparent lens. Find the region of shadow and estimate its size, considering the Sun as a lens.


Problem 6: scales

What scales of angles and distances determine the position of the images of the light source after the passage through the gravitational lens? Consider two cases: 1) the source and the lens are at cosmological distances from the observer; 2) the distance from the observer to the lens is much smaller than the distance to the source.


Problem 7: multiple images

Show that when the gravitational lens is placed between source and observer in the general case the two images of the source would be observed. How are the images placed relative to the lens and observer?


Problem 8: Einstein ring

How should source, gravitational lens and observer be placed relative to each other in order to observe the Einstein ring? Calculate the radius of the ring.


Problem 9: source of finite size

How would the Einstein ring change if we take into account the finite size of the source? Estimate the space characteristics of the observed image assuming that the radius of the lens is much smaller than the raius of the lens.


Problem 10: source of finite size

Qualitatively consider the general situation, when a source of finite size, lens and observer are not on one line. Estimate the angular sizes of the observed images.


Problem 11: angular shift of the Einstein ring

Calculate the angular shift of the Einstein ring from the circle of the gravitational lens. Estimate it, considering the Sun as a lens. Is this value observable?


Problem 12: double Einstein ring

Recently the exceptional phenomenon was observed using the Hubble space telescope: the double Einstein ring, formed by the influence of the gravitational field of the galaxy on the light from two other more distant galaxies. What conditions are necessary for the observation of this phenomenon?


Problem 13: brightness amplification

Calculate the energy amplification coefficient for images produced by the gravitational lens. Determine its peculiarities. Compare with an optical lens.