2016年IOAA理论第10题-引力透镜望远镜 - 版本历史

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创建页面，内容为“{{需要解答}} ==英文题目== (T10) Gravitational Lensing Telescope Einstein's General Theory of Relativity predicts bending of light around massive bodies. For…”

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==英文题目==
(T10) Gravitational Lensing Telescope
Einstein's General Theory of Relativity predicts bending of light around massive bodies. For simplicity,
we assume that the bending of light happens at a single point for each light ray, as shown in the figure.
The angle of bending, 𝜃b , is given by

$$\theta_b =\frac{2R_{sch}}{r}$$

where 𝑅sch is the Schwarzschild radius associated with that gravitational body. We call 𝑟, the distance of
the incoming light ray from the parallel 𝑥-axis passing through the centre of the body, as the “impact
parameter”.
[[文件:IOAA2016T10.jpg|缩略图]]
A massive body thus behaves somewhat like a focusing lens. The light rays coming from infinite distance
beyond a massive body, and having the same impact parameter 𝑟, converge at a point along the axis, at a
distance 𝑓𝑟
from the centre of the massive body. An observer at that point will benefit from huge
amplification due to this gravitational focusing. The massive body in this case is being used as a
Gravitational Lensing Telescope for amplification of distant signals.

(T10.1) Consider the possibility of our Sun as a gravitational lensing telescope. Calculate the shortest
distance, 𝑓min, from the centre of the Sun (in A. U.) at which the light rays can get focused.

(T10.2) Consider a small circular detector of radius 𝑎, kept at a distance 𝑓min centered on the 𝑥-axis and
perpendicular to it. Note that only the light rays which pass within a certain annulus (ring) of
width ℎ (where ℎ ≪ 𝑅⊙) around the Sun would encounter the detector. The amplification factor
at the detector is defined as the ratio of the intensity of the light incident on the detector in the
presence of the Sun and the intensity in the absence of the Sun.

Express the amplification factor, 𝐴m, at the detector in terms of 𝑅⊙ and 𝑎.

(T10.3) Consider a spherical mass distribution, such as dark matter in a galaxy cluster, through which
light rays can pass while undergoing gravitational bending. Assume for simplicity that for the
gravitational bending with impact parameter, 𝑟, only the mass 𝑀(𝑟) enclosed inside the radius
𝑟 is relevant.

What should be the mass distribution, 𝑀(𝑟), such that the gravitational lens behaves like an ideal optical
convex lens?

[[分类:引力透镜]]

@@ 第2行： / 第2行： @@
 ==英文题目==
 (T10) Gravitational Lensing Telescope
 Einstein's General Theory of Relativity predicts bending of light around massive bodies. For simplicity,
 we assume that the bending of light happens at a single point for each light ray, as shown in the figure.
@@ 第39行： / 第40行： @@
 What should be the mass distribution, 𝑀(𝑟), such that the gravitational lens behaves like an ideal optical
 convex lens?
 [[分类:引力透镜]]

@@ 第6行： / 第6行： @@
 The angle of bending, 𝜃<sub>b</sub> , is given by
-$$\theta_b =\frac{2R_{sch}}{r}$$
+<nowiki>$$\theta_b =\frac{2R_{sch}}{r}$$</nowiki>
 where 𝑅<sub>sch</sub> is the Schwarzschild radius associated with that gravitational body. We call 𝑟, the distance of
 the incoming light ray from the parallel 𝑥-axis passing through the centre of the body, as the “impact
 parameter”.
-[[文件:IOAA2016T10.jpg|缩略图]]
 A massive body thus behaves somewhat like a focusing lens. The light rays coming from infinite distance
 beyond a massive body, and having the same impact parameter 𝑟, converge at a point along the axis, at a