---
title: "Gravitational Lens Telescope Interactive Lab | Sandbox Physics"
description: "How can light bent around an unseen mass reveal how heavy it is? From the images it makes of a star behind it."
canonical_url: "https://sandboxphysics.com/labs/gravitational-lens-telescope/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/gravitational-lens-telescope/"
last_modified: "2026-11-03"
---

Optics 020 · Weighing a lone black hole

# Gravitational Lens Telescope

**How can light bent around an unseen mass reveal how heavy it is?** From the images it makes of a star behind it. The mass bends the star's light into two images, one each side of a circle called the Einstein ring, whose angular size grows with the square root of the mass. The images' spacing and brightness, with the distances, give the mass. In 2025 Hubble weighed a lone, dark black hole of about 7 solar masses from the shift these images give the star's light.

Drag a black hole across the line of sight to a star. The star brightens and splits into two images, a ring or arcs, and their spacing gives the mass.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

[A lone black hole](https://sandboxphysics.com/labs/gravitational-lens-telescope/) [A star with a companion](https://sandboxphysics.com/labs/gravitational-lens-telescope/?mode=binary)

Model assumptions

The lens is a single point mass and the light is bent by the weak-field result of general relativity. The event follows OGLE-2011-BLG-0462 as Sahu et al. (2025) measured it: an Einstein radius of 5.496 mas crossed in 273.1 days, so that lens and star move apart at 7.35 mas a year, a closest approach of 0.00271 Einstein radii, and a lens of 7.15 solar masses 1.52 kpc away in front of a star 7.1 kpc away. The star stays at that distance for every setting. Its default radius of 2.88 solar radii gives the 0.0038 mas diameter found by Sahu et al. (2022). The lens moves in a straight line: the Earth’s yearly motion, which made the parallax measurable, is left out, so the light curve is symmetric about closest approach. The star is a uniform disc without limb darkening, and no light from other stars is blended in. The images are found by tracing 720 points round the star’s edge through the lens equation. Their areas give the brightening, which is exact for a uniform disc, and the light-weighted centre of both images gives the shift. The mass from the images uses each image’s light centre and the two brightnesses. It is exact for a point-like star and for a complete ring, whose radius comes from the outer and inner edges, but reads low when the images are wide arcs. In the side scene, distances across the line of sight are drawn about 28 million times those along it, and the stars and the black hole are enlarged; the sky image uses true angles.

Physics tutorial

## Weigh a black hole that gives no light

**Background** On 2 June 2011 the OGLE and MOA surveys both announced that a star towards the centre of the Galaxy was brightening. It peaked at 369 times its normal brightness, and the rise and fall took months. Nothing could be seen in front of it. Kailash Sahu and colleagues then imaged the star with the Hubble Space Telescope for eleven years and found that it had also moved slightly on the sky. In 2025 they put the unseen lens at 7.15 ± 0.83 solar masses, 1.52 kpc away: the first black hole found alone, without a companion star.

**Why it matters** A black hole on its own gives no light and pulls on nothing we can watch. But gravity bends the light of any star that happens to lie behind it, by an amount set by the mass. If the bending can be measured, the black hole can be weighed without ever being seen.

Start with the essentials

- **Focus question**: How do the brightening and the images of a background star tell how heavy an invisible lens is?
- **One-sentence intuition**: Light from the star passes the black hole on both sides and reaches the telescope as two images, one outside and one inside a circle of angular radius $\theta_{\mathrm{E}}$, the Einstein radius. For OGLE-2011-BLG-0462 it is 5.5 mas, about the width of a coin seen from 700 km. The product of the two images’ distances from the lens is always $\theta_{\mathrm{E}}^{2}$, and $\theta_{\mathrm{E}}^{2}$ grows in proportion to the mass, so the images and the distances together give the mass.

### Core mathematical model

#### The Einstein radius grows with the square root of the mass

$$
\begin{gathered}\theta_{\mathrm{E}}=\sqrt{\kappa M\pi_{\mathrm{LS}}},\quad\kappa=\frac{4G}{c^{2}\,\mathrm{AU}}=8.144\,\frac{\mathrm{mas}}{M_{\odot}}\\\pi_{\mathrm{LS}}=1\,\mathrm{AU}\left(\frac{1}{D_{\mathrm{L}}}-\frac{1}{D_{\mathrm{S}}}\right)\end{gathered}
$$

A lens of mass $M$ at distance $D_{\mathrm{L}}$ in front of a star at $D_{\mathrm{S}}$ bends the star’s light into a ring of angular radius $\theta_{\mathrm{E}}$ when the three line up. $\pi_{\mathrm{LS}}$ is the difference in parallax of lens and star. Turned round, $M=\theta_{\mathrm{E}}^{2}/(\kappa\pi_{\mathrm{LS}})$: Sahu et al. (2025) found the mass from $\theta_{\mathrm{E}}=5.496\,\mathrm{mas}$ and the parallax measured in the light curve.

#### Every star behind a point mass has two images

$$
\begin{gathered}\beta=\theta-\frac{\theta_{\mathrm{E}}^{2}}{\theta}\\\theta_{\pm}=\frac{\beta\pm\sqrt{\beta^{2}+4\theta_{\mathrm{E}}^{2}}}{2},\quad\theta_{+}\theta_{-}=-\theta_{\mathrm{E}}^{2}\end{gathered}
$$

A star at angle $\beta$ from the lens appears at the angles $\theta$ that solve the lens equation. There are always two, $\theta_{+}$ outside the Einstein ring on the star’s side and $\theta_{-}$ inside it on the far side. When the star lies right behind the lens, every direction round it works and the images merge into a ring.

#### Brightness and the brightness ratio

$$
\begin{gathered}A=\frac{u^{2}+2}{u\sqrt{u^{2}+4}},\quad u=\frac{\beta}{\theta_{\mathrm{E}}}\\r=\frac{A_{+}}{\lvert A_{-}\rvert}=\left(\frac{\theta_{+}}{\theta_{-}}\right)^{4}\;\Rightarrow\;\theta_{\mathrm{E}}=\frac{\Delta\theta}{r^{1/4}+r^{-1/4}}\end{gathered}
$$

Bending stretches each image without changing its surface brightness, so a larger image is brighter. $A$ is the total brightening of a point-like star at $u$ Einstein radii from the lens; it grows without limit as $u$ falls. The two images’ brightness ratio $r$ and separation $\Delta\theta$ are enough to find $\theta_{\mathrm{E}}$ without knowing where the dark lens is.

#### The star’s light appears to move

$$
\delta=\frac{u}{u^{2}+2}\,\theta_{\mathrm{E}},\qquad\delta_{\max}=\frac{\theta_{\mathrm{E}}}{2\sqrt{2}}\ \text{at}\ u=\sqrt{2}
$$

When the images are too close to split, their combined light still sits off the star’s true place by $\delta$ (Sahu et al. 2022, Eq. 6). It is largest at $u=\sqrt{2}$, long after the brightening has faded, and it is proportional to $\theta_{\mathrm{E}}$. Hubble measured this shift.

### Common difficulties

#### The black hole is not seen at all

**Typical misconception** The images show the black hole itself, outlined by bent light.

**Better mental model** All the light in the images comes from the background star. The black hole only bends it; its own size, about 21 km for 7 solar masses, is far too small to show, and Sahu et al. found no light from the lens.

#### Brighter does not mean closer to us

**Typical misconception** The star brightens because the lens brings it closer or focuses it like a magnifying glass at one point.

**Better mental model** The star stays where it is. Its images are stretched round the lens, and since bending keeps surface brightness the same, the larger images carry more light. The brightening depends only on how far the lens is from the line of sight, in Einstein radii.

#### The light curve alone does not give the mass

**Typical misconception** How long the event lasts tells how heavy the lens is.

**Better mental model** The 273-day timescale is the Einstein radius divided by how fast lens and star move apart on the sky, and that speed is not known in advance. The size of the images, or the shift of their light, gives the Einstein radius itself; the distances or the parallax then give the mass.

#### The two-image formula needs a small star

**Typical misconception** The separation and brightness ratio give the exact mass whatever the star.

**Better mental model** The formula is exact for a point-like star. When the star is a sizeable fraction of the Einstein radius, the images become arcs whose light centres move, and the formula drifts: 9 % low in the arcs preset. A full ring gives the radius again from its two edges.

### Run the experiment

1. 01

   #### See the two images

   Keep the default preset, Two images of one star.

   **What to observe:** The sky image shows the star as an outer image beyond its true place and a fainter inner image on the other side of the black hole, 11.0 mas apart. Together they are 4.63 times brighter than the star alone, and the separation and brightness ratio give 7.15 solar masses.

2. 02

   #### Drag the black hole across the line of sight

   Drag the black hole sideways in the scene, or move the date slider.

   **What to observe:** Closer to the line, both images brighten and the separation shrinks towards the diameter of the Einstein ring, 11.0 mas. The light curve and the two plots of shift and separation move their cursor with you.

3. 03

   #### Watch the peak

   Load Closest approach: 370 times brighter.

   **What to observe:** The star is 370 times brighter, close to the 369 recorded in 2011. The images sit almost on the Einstein ring, on opposite sides, and their combined light is back at the star’s true place: the shift is only 0.007 mas.

4. 04

   #### Put a big star behind it

   Load A big star behind it: Einstein ring.

   **What to observe:** With a star 1000 times the Sun’s radius right behind the black hole, the images join into a ring 11.0 mas across, 16.8 times brighter. A larger star is brightened less at its peak, because its parts lie at different distances from the lens.

5. 05

   #### Move the big star slightly off

   Load Slightly off: two arcs.

   **What to observe:** The ring breaks into two arcs. Their light centres are no longer where a point-like star’s images would be, so the two-image formula reads 6.49 solar masses, 9 % low.

6. 06

   #### Try a neutron star

   Load A neutron star: smaller ring, shorter event.

   **What to observe:** At 1.4 solar masses the Einstein ring is 2.43 mas in radius, the images are 5.0 mas apart and the event lasts 121 days across one Einstein radius. In 2022 one team put the lens at 1.6 to 4.4 solar masses, most likely a neutron star or a light black hole; later Hubble images favoured a black hole of about 7.

Physical conclusion

Sixty days before closest approach the star shows as two images 11.0 mas apart, 4.63 times brighter in all; their separation and brightness ratio give 7.15 solar masses. At closest approach the star is 370 times brighter and the images sit almost exactly on opposite sides of the Einstein ring. A neutron star of 1.4 solar masses at the same place would make images 5.0 mas apart and an event lasting 121 days instead of 273. No telescope split this event’s images: Hubble measured the shift of the star’s combined light instead, which peaks at 1.94 mas about 386 days from closest approach.

## Related experiments and articles

From bent starlight to a planetary light curve · Reading path

### From bent starlight to a planetary light curve

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a dark planet makes a distant star brighten**](https://sandboxphysics.com/stories/microlensing-planets/a-dark-planet-brightens-a-star/)
2. [Article **From a bent light ray to two stellar images**](https://sandboxphysics.com/stories/microlensing-planets/from-bent-light-to-two-images/)
3. [Article **What happens when a stellar disk crosses a caustic?**](https://sandboxphysics.com/stories/microlensing-planets/a-stellar-disk-crosses-a-caustic/)
4. [Article **What can a microlensing light curve tell us about its lens?**](https://sandboxphysics.com/stories/microlensing-planets/read-a-light-curve-without-weighing-too-soon/)
5. [Article **Read a dark planet’s mass from Earth and space**](https://sandboxphysics.com/stories/microlensing-planets/read-ground-space-and-roman-evidence/)
6. [Article **Fit the published OGLE-2005-BLG-390 photometry**](https://sandboxphysics.com/stories/microlensing-planets/refit-the-published-ogle390-files/)

Astronomy and gravity · Reading path

### Gravity leaves a trace

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **The stars that weighed the Milky Way’s dark centre**](https://sandboxphysics.com/stories/astronomy-gravity/galactic-centre-stellar-orbits/)
2. [Lab **Radial Velocities, Active Stars and Binary Masses** Connect projected orbits to mass](https://sandboxphysics.com/labs/binary-star-mass/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)