Skip to main content
Sandbox Physics

Optics 020 · Ray worlds, boundaries, and natural optics

Gravitational Lens Telescope

A deep-space point-mass lens maps a finite source into an Einstein annulus or two parity-reversed arcs, connects both stationary rays to an observer, warps an angular reference grid, and turns uniform relative motion into a symmetric microlensing light curve.

Interactive modelGravitational Lens Telescope
Einstein angle θE\theta_E0.500.50
Total point-source magnification μtot\mu_{\mathrm{tot}}50%50\%
Image-root invariant θ+θ−\theta_+\theta_-0.00π0.00\pi
Model regimevalid model regime\text{valid model regime}

Physics tutorial

Audit the physics of Gravitational Lens Telescope

BackgroundA point mass changes the stationary light directions without acting like an ordinary glass lens. In the thin-lens limit the angular mapping is β=θ−θE2θ∣θ∣2\boldsymbol\beta=\boldsymbol\theta-\theta_E^2\frac{\boldsymbol\theta}{|\boldsymbol\theta|^2}. One off-axis point source therefore has two images with opposite parity; exact axial alignment expands the degenerate image direction into a ring.

Why it mattersHow can image positions, arcs, and brightness reveal an invisible lens mass?

Start with the essentials

Focus question
How can image positions, arcs, and brightness reveal an invisible lens mass?
One-sentence intuition
The natural angular scale is θE=4GMc2DLSDLDS\theta_E=\sqrt{\frac{4GM}{c^2}\frac{D_{LS}}{D_LD_S}}. Image separation reveals this combination of mass and distances; magnification adds information but is sensitive to alignment and source extent.

Core mathematical model

Point-mass lens equation

β=θ−θE2θ\beta=\theta-\frac{\theta_E^2}{\theta}

The two roots are stationary image directions; their dimensionless product is exactly minus one.

Einstein angular radius

θE=4GMc2DLSDLDS\theta_E=\sqrt{\frac{4GM}{c^2}\frac{D_{LS}}{D_LD_S}}

At alignment the source, lens, and observer symmetry turns the image solution into a circular Einstein ring.

Point-source total magnification

μtot(u)=u2+2uu2+4\mu_{\mathrm{tot}}(u)=\frac{u^2+2}{u\sqrt{u^2+4}}

The observed flux sums the absolute magnifications of positive- and negative-parity images.

Common difficulties

Drawing one bent ray and calling it an Einstein ring

Typical misconceptionA ring is merely a glow effect around the lens mass, independent of source alignment and lens mapping.

Better mental modelThe ring is the axisymmetric set of image solutions at alignment. A finite source maps to an annulus, and off-axis motion continuously deforms it into parity-distinct arcs or images.

Run the experiment

  1. 01

    Scene 1: Einstein ring

    Set alignment near zero and compare the mapped inner and outer source boundaries with the highlighted Einstein angular ruler.

    What to observe: Perfect alignment produces an annular finite-source image; slight misalignment immediately gives two parity-distinct mapped boundaries.
  2. 02

    Scene 2: Multiple-image arcs

    Move the extended source off axis and follow both stationary spatial rays to the unequal gold and rose image boundaries.

    What to observe: One image lies outside the Einstein radius with positive parity, while the fainter inner image has reversed parity.
  3. 03

    Scene 3: Microlensing light curve

    Open the microlensing scene and change impact parameter while comparing peak height, symmetry, and far-time baseline.

    What to observe: Uniform straight relative motion produces a symmetric achromatic point-source light curve whose peak grows as impact parameter shrinks.