---
title: "From a bent light ray to two stellar images"
description: "Derive the point-mass lens equation, locate its two images, and calculate their parity and combined magnification."
canonical_url: "https://sandboxphysics.com/stories/microlensing-planets/from-bent-light-to-two-images/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/microlensing-planets/from-bent-light-to-two-images/"
last_modified: "2026-11-03"
---

[A dark planet bends starlight](https://sandboxphysics.com/stories/microlensing-planets/)

AC23 · L1

# From a bent light ray to two stellar images

Sandbox Physics · 2026-11-03

A single point-mass lens makes two images because light can pass on either side of it. A geometric equation tells us where those images appear. Differentiating that map tells us how much apparent area each one occupies.

## Before you begin

Read How a dark planet makes a distant star brighten. This chapter uses similar triangles, quadratic equations and the derivative as a local rate of change; the area-map step explains the derivative it needs.

## Define the angles before following the ray

Put the observer at the left, the lens plane a distance $D_L$ away, and the source plane a distance $D_S$ away. The lens-source separation is $D_{LS}$. These are positive lengths. In the nearby Galactic setting we use ordinary Euclidean distances, so the last distance is their difference. Cosmological lensing instead needs the appropriate angular-diameter distances.

Measure sky angles from the direction of the lens. The true source direction is $\beta$; an observed image direction is $\theta$. On a line through the lens, rightward angles are positive. An image to the left has a negative angle even when the source is on the right.

The weak-field deflection of light passing a compact mass $M$ at impact distance $b$ is the following input from general relativity. Here $G$ is the gravitational constant and $c$ is the speed of light. The ray stays far outside the lens’s gravitational radius.

$$
\widehat\alpha=\frac{4GM}{c^2b},\qquad b\simeq D_L|\theta|.
$$

The deflection grows for a closer passage. The small-angle relation converts image angle to distance in the lens plane.

Reading: Bozza et al., VBMicroLensing, arXiv:2410.13660v1; A&A 694 A219 (2025)

## Use the triangle to find the image

Trace the observed ray back from the telescope to the lens plane, then follow its deflected direction to the source plane. Similar triangles relate the displacement in the source plane to the deflection. The thin-lens approximation places the bending at one plane; the true gravitational field is spread out along the path.

$$
\beta=\theta-\frac{D_{LS}}{D_S}\frac{4GM}{c^2D_L\theta}.
$$

The signed denominator points the deflection toward the lens on either side. A negative image angle is part of the same equation.

Group the distances and mass into an angular scale, the Einstein radius $\theta_E$. It is the radius of the image ring when source, compact lens and observer line up. This is an angle, despite the word radius.

$$
\theta_E^2=\frac{4GM}{c^2}\frac{D_{LS}}{D_LD_S},\qquad u=\frac{\beta}{\theta_E},\quad x=\frac{\theta}{\theta_E}.
$$

Dividing every angle by the Einstein radius leaves a dimensionless geometry.

$$
u=x-\frac1x,\qquad x^2-ux-1=0,\qquad x_\pm=\frac{u\pm\sqrt{u^2+4}}2.
$$

The quadratic has one positive and one negative root. The two paths arrive from opposite sides of the lens.

Reading: Bozza et al., VBMicroLensing, arXiv:2410.13660v1; A&A 694 A219 (2025)

## Worked example with a half-radius offset

Let $\theta_E=1.00\,\mathrm{mas}$ and $\beta=0.500\,\mathrm{mas}$. One milliarcsecond is one thousandth of an arcsecond. The dimensionless source offset is one half. Substituting it into the quadratic gives

$$
x_+=1.28078,\qquad x_-=-0.780776.
$$

The images appear 1.28078 milliarcseconds to one side and 0.780776 milliarcseconds to the other.

The image separation is $2.06155\,\mathrm{mas}$. It is larger than twice the Einstein radius. This is a useful check: the two-image separation is $\theta_E\sqrt{u^2+4}$, which approaches twice the Einstein radius as the source approaches perfect alignment.

At exact alignment the direction of the line is no longer unique. Every direction around the lens supplies a solution, and a point source has an ideal ring. A finite star fills an annular area; the next chapter adds its light across the full disk.

Reading: Bozza et al., VBMicroLensing, arXiv:2410.13660v1; A&A 694 A219 (2025)

## The area map supplies the brightness

Imagine a tiny rectangular patch on the source. The lens stretches it differently along the radial direction and along the ring. A derivative measures either local stretch: how a small change at the image maps to a change at the source. Multiplying the two stretches gives the signed area ratio.

$$
\lambda_r=1+\frac1{x^2},\qquad\lambda_t=1-\frac1{x^2},\qquad J=\lambda_r\lambda_t=1-\frac1{x^4}.
$$

The radial and tangential factors describe the source area per image area. Their product is the Jacobian determinant.

An image’s signed magnification is $1/J$. The sign records orientation. A negative image is mirrored, not a negative amount of light. Its observed contribution is the absolute value. Sum the contributions of both images to get the point-source magnification:

$$
A(u)=\frac{u^2+2}{|u|\sqrt{u^2+4}}.
$$

Use this expression for a point source with nonzero offset. A finite disk is integrated separately.

For the half-radius example, the two contributions are $1.59141$ and $0.591410$. Their sum is $2.18282$. The outer image has positive orientation; the inner image is mirrored. Far from the lens the total approaches one.

Reading: Bozza et al. (2018), VBBinaryLensing finite-source methods · Bozza et al., VBMicroLensing, arXiv:2410.13660v1; A&A 694 A219 (2025)

## What the ring scale can tell us about mass

The Einstein radius combines mass and distance. A larger mass makes a larger ring, but changing $D_LD_S/D_{LS}$ also changes which mass gives the same angular scale. Measuring the ring or an astrometric shift therefore supplies a relation between mass and distance.

A light-curve timescale introduces relative angular speed $\mu_{\mathrm{rel}}$: $t_E=\theta_E/\mu_{\mathrm{rel}}$. A small lens moving slowly can share an event length with a larger lens moving faster. In chapter three we combine source size and parallax to obtain additional scales.

The useful habit is to ask which quantity an observation measures directly. Image angles give angular geometry. A duration gives an angular scale divided by a speed. A mass in physical units needs the remaining information.

Reading: Dong et al., A Saturn-mass microlensing object, arXiv:2601.00057v1; Science

## Exercises and explanations

1. Keep $u=0.5$ but double the Einstein radius. What happens to image positions and magnification?

   Read the explanation

   Both image angles double because their dimensionless roots stay fixed. The magnification remains $2.18282$. Angular scale and dimensionless area ratio play different roles.

2. An inner image has a negative Jacobian. Does it dim the outer image by subtraction?

   Read the explanation

   It contributes positive light. The negative sign says that its orientation is reversed. Add the absolute magnifications when the telescope collects incoherent stellar light.

3. A shorter event is proposed as proof of a smaller mass. Name a change that keeps mass fixed and shortens the event.

   Read the explanation

   Increase the relative proper motion while keeping the geometry fixed. From $t_E=\theta_E/\mu_{\mathrm{rel}}$, doubling that angular speed halves the timescale.

## Try the prediction

### [Recognize the two-image limit](https://sandboxphysics.com/labs/gravitational-lens-telescope/?mode=binary)

1. Choose the smooth-event preset and reduce the companion mass ratio. Predict which image stays outside the Einstein ring.
2. Move the source along its path and compare the total brightening with the point-lens expression. A finite source and a remaining companion account for small departures.

[Previous: How a dark planet makes a distant star brighten](https://sandboxphysics.com/stories/microlensing-planets/a-dark-planet-brightens-a-star/) [Next: What happens when a stellar disk crosses a caustic?](https://sandboxphysics.com/stories/microlensing-planets/a-stellar-disk-crosses-a-caustic/)

## Sources and calculation scope

1. [Bozza et al. (2018), VBBinaryLensing finite-source methods](https://doi.org/10.1093/mnras/sty1791)

   Sections 2–4 and equations 14–20, 25–26 explain contour area, error refinement and radial source profiles. The pinned LGPL code remains a separately downloaded offline reference.

2. [Bozza et al., VBMicroLensing, arXiv:2410.13660v1; A&A 694 A219 (2025)](https://arxiv.org/abs/2410.13660v1)

   Read equations 1–16 and section 6 before interpreting binary roots, parity, critical curves or a missing image pair. This is a methods reading, not a replication of its benchmark suite.

3. [Dong et al., A Saturn-mass microlensing object, arXiv:2601.00057v1; Science](https://arxiv.org/abs/2601.00057v1)

   Fixed author version, Figure 2 and supplement S5–S11: finite-source scale, ground/Gaia parallax, alternative blending and solution weights. arXiv identifies the paper as published in Science. This guide reads the evidence; it does not rerun that event.