Gravitational Lensing

Gravitational Lensing

General relativity predicts that mass curves spacetime, bending the path of light. When light from a distant source passes a massive object at impact parameter $b$, it is deflected by:

$\alpha = \frac{4GM}{c^2 b} = \frac{2 r_s}{b} \quad \text{(radians)}$

This is twice the Newtonian prediction. The factor of 2 arises because GR accounts for both the spatial and temporal curvature of spacetime.

The 1919 Eclipse Measurement

In 1919, Arthur Eddington's expedition measured starlight deflection during a solar eclipse. For light grazing the Sun's surface ($b = R_\odot = 6.96 \times 10^8$ m):

$\alpha = \frac{4GM_\odot}{c^2 R_\odot} \approx 1.75 \text{ arcseconds}$

This matched GR's prediction and made Einstein world-famous. Newton's theory predicted only 0.875 arcseconds.

Einstein Ring

When source, lens, and observer are perfectly aligned, light bends symmetrically to form a ring. The angular radius of the Einstein ring is:

$\theta_E = \sqrt{\frac{4GM}{c^2} \cdot \frac{D_{LS}}{D_L D_S}}$

For collinear geometry with $D_{LS} = D_S - D_L$, the physical ring radius projected at the lens plane is:

$r_E = D_L \, \theta_E = \sqrt{\frac{4GM \, D_L (D_S - D_L)}{c^2 D_S}}$

Your Task

Implement three functions. All physical constants must be defined inside each function body.

• deflection_angle(M, b) — returns deflection in radians
• deflection_angle_arcsec(M, b) — converts radians to arcseconds (multiply by $180/\pi \times 3600$)
• einstein_ring_radius(M, D_L, D_S) — returns physical Einstein ring radius in meters