# Point gravitational lenses

### 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.

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.