Gravitational Lensing

Gravitational Lensing

Einstein's general relativity predicts that mass bends the path of light. A massive object — a star, galaxy, or cluster — acts as a gravitational lens, distorting and magnifying the images of background sources.

The Einstein Radius

When a source, lens, and observer are perfectly aligned, the source appears as a ring (an Einstein ring) with angular radius $\theta_E$:

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

where:

• $D_L$ = angular diameter distance to the lens
• $D_S$ = angular diameter distance to the source
• $D_{LS}$ = angular diameter distance from lens to source

Point-Source Magnification

For a point source at angular position $\beta$ from the lens axis, the total magnification is:

$\mu = \frac{u^2 + 2}{u \sqrt{u^2 + 4}}, \quad u = \frac{\beta}{\theta_E}$

At perfect alignment ($\beta = 0$, $u = 0$), $\mu \to \infty$. At $u = 1$, $\mu \approx 1.34$.

Scales

Your Task

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

• einstein_radius_rad(M_kg, D_L_m, D_S_m, D_LS_m) — returns $\theta_E$ in radians
• einstein_radius_arcsec(M_kg, D_L_m, D_S_m, D_LS_m) — returns $\theta_E$ in arcseconds
• lensing_magnification(beta_rad, theta_E_rad) — returns $\mu$

Use $G = 6.674 \times 10^{-11}$ N m² kg⁻², $c = 2.998 \times 10^8$ m/s. For arcseconds: multiply radians by $180/\pi \times 3600$.