<p>I did the annotation of that figure for the press release, so let me start by apologizing for the poor explanation, and then try to dig deeper into what's going on :) (although you already seem to pretty much get it right).</p>
<h3>Lens models</h3>
<p>The big orange galaxies all belong to a foreground cluster called WHL0137-08. For known point masses, there is an analytical solution to how exactly the mass deflects and magnifies light from background sources, but galaxies are not points, and we don't know their exact masses. Nevertheless, based on the observed distribution of light, assumed stellar mass-to-light ratios and stellar mass-dark matter halo mass relations, as well as assumed halo density profiles, we can estimate numerically how light from a background source travels through/around the cluster, and hence how it's magnified.</p>
<p>This is called a <em>lens model</em>, and because of these not always too well-constrained assumptions, there are quite a few different models. This is why the mass of Earendel is only constrained to within an order of magnitude, namely 50–500 <span class="math-container">$M_\odot$</span>.</p>
<h3>Caustic lines</h3>
<p>A lens model thus returns a magnification factor for each 2D point in the vicinity of the cluster. In certain regions called &quot;caustic lines&quot;, or just <em>caustics</em>, which lie along 1D lines, the magnification increases tremendously (in principle to infinity).</p>
<p>In the picture you see such a caustic, drawn in red. It bends smoothly around a handful of galaxies in the upper left corner, but where it comes close to the galaxy a little below the center, it bends around that one. The reason we're able to see Earendel is that it happens to lie almost exactly on top of the caustic.</p>
<p>Figure 2 from the paper (which isn't yet available online, but I think it's ok to share here with proper credits) showing the caustics predicted from four different lens models:</p>
<p><a href="https://i.sstatic.net/m0zFU.jpg" rel="noreferrer"><img src="https://i.sstatic.net/m0zFU.jpg" alt="earendel" /></a>
<sup>Credit: NASA, ESA, Brian Welch (JHU), Dan Coe (STScI).</sup></p>
<p>The red curve that I used in the image you post is the &quot;Light-Traces-Mass model&quot; (LTM; <a href="https://ui.adsabs.harvard.edu/abs/2015ApJ...801...44Z" rel="noreferrer">Zitrin et al. 2015</a>). To understand the effect of the uncertainties in the model, the posterior distribution was sampled and used to generate critical curves from each resultant parameter set. Those are the thin brown lines close to the red. In 80/100 of such model runs, Earendel is within 0.1 arcsec of the line, with a maximum of 0.4&quot;.</p>
<h3>Mirror images</h3>
<p>If an objects lies close to, but not on top of, a caustic, its light will be bent around it and reach us from two different directions. Close to Earendel lies a small star cluster (called &quot;1.1&quot; in the paper). Because of the way the caustic bends, 1.1 is mirrored not twice but <em>three</em> times, indicated in cyan. In fact there's a second star cluster called &quot;1.7&quot; that I didn't mark, to keep it simple, but you see it as slightly enhanced dots of light along the red arc.</p>
<h3>Arcs</h3>
<p>So, what <em>is</em> that red arc? That's the host galaxy — called WHL0137-zD1, but nicknamed the &quot;Sunrise Arc&quot; — of both the star clusters and Earendel. This is a typical feature of gravitational lenses, distorting background galaxies to banana-shaped arcs along directions which are more or less tangential to the cluster. In fact, the Sunrise Arc is one of the longest known, being 15 arcseconds long.</p>
<h3>Magnification</h3>
<p>The term &quot;magnification&quot;, usually denoted <span class="math-container">$\mu$</span>, is used both for the increase in <em>area</em> and the increase in <em>brightness</em>. Usually, if you zoom in on an image of some object, the brightness is conserved so the surface brightness decreases. But with lensing, surface brightness is conserved, so the total brightness increases.</p>
<p>Stars are much too small to be spatially resolved by our telescopes, however, so they are seen as point sources. In principle their sizes do increase when they're lensed, and in principle they may also be elongated, but they are so small that they're still way below our resolution, and will always be seen as point sources. Hence, for stars the magnification just refers to the increase in brightness.</p>
<p>If Earendel is 10× the diameter of our Sun, its angular extent seen by us would be <span class="math-container">$\sim10^{-10}\,\mathrm{arcsec}$</span>. Even if Earendel is magnified <span class="math-container">$10^5$</span> times — the estimated maximum according to the lens models — it's still of the order of ten microarcsec, unresolvable by both Hubble and James Webb. This resolution is actually within reach for interferometry, but that would require Earendel to be bright in the much longer wavelengths required for that technique, which it isn't.</p>

CC BY-SA 4.0
2022-04-04T02:40:46.040Z
thank you for taking the time to write that explanation. A caustic in optics usually refers to a 1D bright line where the rays tend to pile up.  In the swimming-pool analogy, this is caused by a quadratic ridge on the surface. Bright points where these caustics meet would then correspond to two ridges meeting to form a circular mound (convex lens) on the water surface. I gather these stellar images can be likened to looking at something through a wavey water surface where (minus) the gravitational potential corresponds to the surface height.
Roger Wood
CC BY-SA 4.0
2022-04-04T02:44:51.367Z
I'm still unsure if the 'magnification' corresponds to 1D cylindrical lensing or if there are places where the gravitational potential is quadratic along two axes and it's more like a true focus?  Is that why the star isn't stretched?  Does the question make sense?
Roger Wood
CC BY-SA 4.0
2022-04-04T06:06:15.333Z
@RogerWood checking previous questions and answers here will show that in the context of various gravitational lensing cases the term "magnification" has been used in discussions of both a spatial and a brightness effects, and in cases where the object is recognizable and magnified in both directions (though usually quite distorted) and predominantly in one direction. I think you can also ask a new question trying to get to the bottom of how "magnification" is used in these contexts if you like.
uhoh
CC BY-SA 4.0
2022-04-04T06:08:18.467Z
@RogerWood this just skims the surface and illustrates the need for a more global question: [What exactly is it that is being magnified 50 times in this gravitational lensing observation?](https://astronomy.stackexchange.com/q/20865/7982)
uhoh
CC BY-SA 4.0
2022-04-04T07:26:19.250Z
@RogerWood As uhoh says, the term magnification refers both to the increase in size and in brightness. The cool thing about lensing is that it increases the size without decreasing the surface brightness. However, Earendel is a point source, so it cannot be stretched. Or, more correctly, it is stretched, but it is still a point source since it's way, way below our resolution. I edited the answer to include a discussion on this.
pela
CC BY-SA 4.0
2022-04-04T17:06:46.157Z
@pela super answer, thanks for elaborating on magnification. Am I right in saying that the magnification on the caustic corresponds to us being at the focal point of a cylindrical lens and it would be exceedingly rare for anything like a spherical lens to ever occur?
Roger Wood
CC BY-SA 4.0
2022-04-04T20:47:35.920Z
@RogerWood I'm unsure what you mean by "cylindrical" and "spherical" lens. Do you mean like a cylinder seen from the side, and a glass ball? I'm not very good at optics, so someone else might provide a better answer…
pela
CC BY-SA 4.0
2022-04-04T21:20:43.040Z
@pela I think RogerWood means this https://i.sstatic.net/qgBL9.png where the angle of deflection locally varies ~linearly in either one direction (cylindrical lens-like) or roughly similarly in orthogonal directions (spherical lens-like) (from https://www.opticsforhire.com/blog/5-applications-of-cylindrical-lenses-design)
uhoh
CC BY-SA 4.0
2022-04-05T06:09:28.363Z
@pela I added an illustration to my original question
Roger Wood
CC BY-SA 4.0
2022-04-05T10:16:19.430Z
@RogerWood That's an amazing illustration (and yes, making illustrations is more time-consuming than most viewers realize), but if I understand yours, and uhoh's, illustrations correctly, then in this terminology gravitational lenses are actually not too far from spherical lenses. Clusters are of course quite inhomogeneous, but single galaxies can be rather symmetric. If you look at something like the [Cosmic Horseshoe](https://en.wikipedia.org/wiki/Cosmic_Horseshoe), it's very close to being what I imagine you mean by a spherical lens.
pela
CC BY-SA 4.0
2022-04-05T12:57:20.720Z
It is so cool that we have the person who created the referenced image onboard as one of our contributors. It is so uncool that I can only give a +1.
David Hammen
CC BY-SA 4.0
2022-04-05T13:59:47.653Z
@DavidHammen Ha ha thanks :)
pela
CC BY-SA 4.0
2022-04-05T20:56:16.317Z
@pela - I second David Hammen's comment!
Roger Wood
CC BY-SA 4.0
2022-04-05T21:10:46.390Z
@pela Yes, 'spherical' lens is a bit ambiguous in this context. As a lens, a black hole has perfect spherical symmetry, but an optician would definitely not call it a spherical lens.  For a black hole, the field and hence the deflection goes as r^-2 with distance from the center. In contrast, the deflection for an optician's spherical lens goes as r^+1.  The former gives rise to the characterisitic arcs or rings.  The latter brings everything to a focus at a single point.  (assuming the deflections are all very small)
Roger Wood
CC BY-SA 4.0
2022-04-05T22:16:47.417Z
@RogerWood Thanks :) Okay, I understand now. Yes, a gravitational lens doesn't really have a focus, like a normal lens does, so you're right in your previous comment that it would be exceedingly rare for a spherical lens to ever occur. Or rather, it doesn't have a focal _point_, but instead has a "focal line". If you want to play around with creating caustics, [this Python script](https://galileo-unbound.blog/2021/04/05/the-lens-of-gravity-einsteins-rings/) might be helpful.
pela
CC BY-SA 4.0
2022-04-08T02:29:45.600Z
@pela thanks. I'm not a Python person, but that's a really good article
Roger Wood
CC BY-SA 4.0
2022-04-08T07:06:23.970Z
@RogerWood Fair enough :)
pela
CC BY-SA 4.0
2022-04-08T07:07:47.150Z
@RogerWood Btw if you think I managed to answer your questions, I'd be happy if you could mark the answer as solved, so that future readers know that it did :)
pela
CC BY-SA 4.0
2022-04-08T15:43:00.287Z
@pela oops - that was an oversight - super answer!!!
Roger Wood
CC BY-SA 4.0
2022-04-08T20:56:59.863Z
@RogerWood Thanks a lot :)
pela
CC BY-SA 4.0
2022-04-08T23:18:57.900Z
Obligatory [Dr. Becky video](https://www.youtube.com/watch?v=VChgsXbIgdw)
uhoh
CC BY-SA 4.0
2022-04-09T14:21:02.313Z
@uhoh Very cool video!
pela
CC BY-SA 4.0
2022-04-03T21:58:06.820Z
2022-04-07T14:43:36.293Z
2022-04-07T14:43:36.293Z
pela
<p><a href="https://i.sstatic.net/PmzEV.png" rel="noreferrer"><img src="https://i.sstatic.net/PmzEV.png" alt="Hubble image of the distant star 'Earendel'" /></a></p>
<p>&quot;<em>Close-up of the tiny region where Earendel happened to fall right on top of the narrow line where the magnification increases by (tens of) thousands of times. A cluster of many stars is seen slightly offset from the line, resulting in a much smaller magnification but instead being mirrored by gravity. Credit: NASA/ESA/Brian Welch (JHU)/Dan Coe (STScI)/Peter Laursen (DAWN)</em>&quot; <a href="https://phys.org/news/2022-03-twinkle-astronomers-farthest-star.html" rel="noreferrer">from phys.org</a>.</p>
<p>The photo has been annotated by way of an explanation, but it's still not very clear to me what's going on.</p>
<p>There is a well-defined faint diagonal arc of light with several 'lumps' in it. Three lumps are labeled &quot;Mirrored star cluster&quot; and one is labeled 'Earendel'. The center of the arc is far beyond the top left corner of the image.  This is presumably where the lensing galaxy cluster is centered. Do those bright objects in the top left have anything to do with this?</p>
<p>Why does it appear as an arc with blobs in it? Is it because the lens itself is a bit bumpy?</p>
<p>Then what about the red &quot;magnification line&quot;? This seems to be associated with the bright foreground(?) galaxy just below the center of the picture. The line forms a horseshoe loop around it. Is this an estimated line along which the field from the background galaxy cluster and the field from the foreground galaxy conspire to focus along two dimensions (more like a regular lens) and create a brighter more-familiar image of any object that happens to fall on that line?</p>
<p>Looking forward to a good explanation!</p>
<p><strong>[Edit to augment @pela's excellent answer below]</strong>
<a href="https://i.sstatic.net/3toEw.png" rel="noreferrer"><img src="https://i.sstatic.net/3toEw.png" alt="Earendel: ray-tracing illustrating caustic" /></a></p>
<p>Cross section and ray-tracing to illustrate caustic formed as rays are bent by the gravitational field from the foreground galaxy cluster. The lens is effectively cylindrical  -  rather like a long, thin, lumpy sausage bent around the galaxy cluster.</p>
<p>(<em>BTW I spent far too long making this picture!</em>)</p>

CC BY-SA 4.0
2022-04-03T20:37:44.590Z
Phys.org links to Nature Astronomy (paywalled) [A highly magnified star at redshift 6.2](https://www.nature.com/articles/s41586-022-04449-y) I haven't found a preprint of it yet.
uhoh
CC BY-SA 4.0
2022-04-06T00:00:09.063Z
Related: ["_WHL0137-LS_"](https://en.wikipedia.org/wiki/WHL0137-LS), Wikipedia (also known as "_Earendel_").
Nat
<star><hubble-telescope><gravitational-lensing><galaxy-cluster><earendel-whl0137-ls>
Is there a better explanation of this picture showing the very distant star "Earendel"?
Roger Wood
# Is there a better explanation of this picture showing the very distant star "Earendel"?

(Asked by: username_0 on 2022-04-03)

[![Hubble image of the distant star 'Earendel'](https://i.sstatic.net/PmzEV.png)](https://i.sstatic.net/PmzEV.png)

"*Close-up of the tiny region where Earendel happened to fall right on top of the narrow line where the magnification increases by (tens of) thousands of times. A cluster of many stars is seen slightly offset from the line, resulting in a much smaller magnification but instead being mirrored by gravity. Credit: NASA/ESA/Brian Welch (JHU)/Dan Coe (STScI)/Peter Laursen (DAWN)*" [from phys.org](https://phys.org/news/2022-03-twinkle-astronomers-farthest-star.html).

The photo has been annotated by way of an explanation, but it's still not very clear to me what's going on.

There is a well-defined faint diagonal arc of light with several 'lumps' in it. Three lumps are labeled "Mirrored star cluster" and one is labeled 'Earendel'. The center of the arc is far beyond the top left corner of the image. This is presumably where the lensing galaxy cluster is centered. Do those bright objects in the top left have anything to do with this?

Why does it appear as an arc with blobs in it? Is it because the lens itself is a bit bumpy?

Then what about the red "magnification line"? This seems to be associated with the bright foreground(?) galaxy just below the center of the picture. The line forms a horseshoe loop around it. Is this an estimated line along which the field from the background galaxy cluster and the field from the foreground galaxy conspire to focus along two dimensions (more like a regular lens) and create a brighter more-familiar image of any object that happens to fall on that line?

Looking forward to a good explanation!

**[Edit to augment @username_3's excellent answer below]**
[![Earendel: ray-tracing illustrating caustic](https://i.sstatic.net/3toEw.png)](https://i.sstatic.net/3toEw.png)

Cross section and ray-tracing to illustrate caustic formed as rays are bent by the gravitational field from the foreground galaxy cluster. The lens is effectively cylindrical - rather like a long, thin, lumpy sausage bent around the galaxy cluster.

(*BTW I spent far too long making this picture!*)

**Tags:** star, hubble-telescope, gravitational-lensing, galaxy-cluster, earendel-whl0137-ls

> **Comments:**
> **username_1**: Phys.org links to username_2ure Astronomy (paywalled) [A highly magnified star at redshift 6.2](https://www.nature.com/articles/s41586-022-04449-y) I haven't found a preprint of it yet.  
> **username_2**: Related: ["_WHL0137-LS_"](https://en.wikipedia.org/wiki/WHL0137-LS), Wikipedia (also known as "_Earendel_").  

---

## 1 answers

### Answer by username_3 on 2022-04-03. Score: 84 | Accepted answer

I did the annotation of that figure for the press release, so let me start by apologizing for the poor explanation, and then try to dig deeper into what's going on :) (although you already seem to pretty much get it right).

### Lens models

The big orange galaxies all belong to a foreground cluster called WHL0137-08. For known point masses, there is an analytical solution to how exactly the mass deflects and magnifies light from background sources, but galaxies are not points, and we don't know their exact masses. Nevertheless, based on the observed distribution of light, assumed stellar mass-to-light ratios and stellar mass-dark matter halo mass relations, as well as assumed halo density profiles, we can estimate numerically how light from a background source travels through/around the cluster, and hence how it's magnified.

This is called a *lens model*, and because of these not always too well-constrained assumptions, there are quite a few different models. This is why the mass of Earendel is only constrained to within an order of magnitude, namely 50–500 $M\_\odot$.

### Caustic lines

A lens model thus returns a magnification factor for each 2D point in the vicinity of the cluster. In certain regions called "caustic lines", or just *caustics*, which lie along 1D lines, the magnification increases tremendously (in principle to infinity).

In the picture you see such a caustic, drawn in red. It bends smoothly around a handful of galaxies in the upper left corner, but where it comes close to the galaxy a little below the center, it bends around that one. The reason we're able to see Earendel is that it happens to lie almost exactly on top of the caustic.

Figure 2 from the paper (which isn't yet available online, but I think it's ok to share here with proper credits) showing the caustics predicted from four different lens models:

[![earendel](https://i.sstatic.net/m0zFU.jpg)](https://i.sstatic.net/m0zFU.jpg)
Credit: NASA, ESA, Brian Welch (JHU), Dan Coe (STScI).

The red curve that I used in the image you post is the "Light-Traces-Mass model" (LTM; [Zitrin et al. 2015](https://ui.adsabs.harvard.edu/abs/2015ApJ...801...44Z)). To understand the effect of the uncertainties in the model, the posterior distribution was sampled and used to generate critical curves from each resultant parameter set. Those are the thin brown lines close to the red. In 80/100 of such model runs, Earendel is within 0.1 arcsec of the line, with a maximum of 0.4".

### Mirror images

If an objects lies close to, but not on top of, a caustic, its light will be bent around it and reach us from two different directions. Close to Earendel lies a small star cluster (called "1.1" in the paper). Because of the way the caustic bends, 1.1 is mirrored not twice but *three* times, indicated in cyan. In fact there's a second star cluster called "1.7" that I didn't mark, to keep it simple, but you see it as slightly enhanced dots of light along the red arc.

### Arcs

So, what *is* that red arc? That's the host galaxy — called WHL0137-zD1, but nicknamed the "Sunrise Arc" — of both the star clusters and Earendel. This is a typical feature of gravitational lenses, distorting background galaxies to banana-shaped arcs along directions which are more or less tangential to the cluster. In fact, the Sunrise Arc is one of the longest known, being 15 arcseconds long.

### Magnification

The term "magnification", usually denoted $\mu$, is used both for the increase in *area* and the increase in *brightness*. Usually, if you zoom in on an image of some object, the brightness is conserved so the surface brightness decreases. But with lensing, surface brightness is conserved, so the total brightness increases.

Stars are much too small to be spatially resolved by our telescopes, however, so they are seen as point sources. In principle their sizes do increase when they're lensed, and in principle they may also be elongated, but they are so small that they're still way below our resolution, and will always be seen as point sources. Hence, for stars the magnification just refers to the increase in brightness.

If Earendel is 10× the diameter of our Sun, its angular extent seen by us would be $\sim10^{-10}\,\mathrm{arcsec}$. Even if Earendel is magnified $10^5$ times — the estimated maximum according to the lens models — it's still of the order of ten microarcsec, unresolvable by both Hubble and James Webb. This resolution is actually within reach for interferometry, but that would require Earendel to be bright in the much longer wavelengths required for that technique, which it isn't.

> **Comments:**
> **username_0**: thank you for taking the time to write that explanation. A caustic in optics usually refers to a 1D bright line where the rays tend to pile up.  In the swimming-pool analogy, this is caused by a quadratic ridge on the surface. Bright points where these caustics meet would then correspond to two ridges meeting to form a circular mound (convex lens) on the water surface. I gather these stellar images can be likened to looking at something through a wavey water surface where (minus) the gravitational potential corresponds to the surface height.  
> **username_0**: I'm still unsure if the 'magnification' corresponds to 1D cylindrical lensing or if there are places where the gravitational potential is quadratic along two axes and it's more like a true focus?  Is that why the star isn't stretched?  Does the question make sense?  
> **username_1**: @RogerWood checking previous questions and answers here will show that in the context of various gravitational lensing cases the term "magnification" has been used in discussions of both a spatial and a brightness effects, and in cases where the object is recognizable and magnified in both directions (though usually quite distorted) and predominantly in one direction. I think you can also ask a new question trying to get to the bottom of how "magnification" is used in these contexts if you like.  
> **username_1**: @RogerWood this just skims the surface and illustrates the need for a more global question: [What exactly is it that is being magnified 50 times in this gravitational lensing observation?](https://astronomy.stackexchange.com/q/20865/7982)  
> **username_3**: @RogerWood As username_1 says, the term magnification refers both to the increase in size and in brightness. The cool thing about lensing is that it increases the size without decreasing the surface brightness. However, Earendel is a point source, so it cannot be stretched. Or, more correctly, it is stretched, but it is still a point source since it's way, way below our resolution. I edited the answer to include a discussion on this.  
> **username_0**: @username_3 super answer, thanks for elaborating on magnification. Am I right in saying that the magnification on the caustic corresponds to us being at the focal point of a cylindrical lens and it would be exceedingly rare for anything like a spherical lens to ever occur?  
> **username_3**: @RogerWood I'm unsure what you mean by "cylindrical" and "spherical" lens. Do you mean like a cylinder seen from the side, and a glass ball? I'm not very good at optics, so someone else might provide a better answer…  
> **username_1**: @username_3 I think RogerWood means this https://i.sstatic.net/qgBL9.png where the angle of deflection locally varies ~linearly in either one direction (cylindrical lens-like) or roughly similarly in orthogonal directions (spherical lens-like) (from https://www.opticsforhire.com/blog/5-applications-of-cylindrical-lenses-design)  
> **username_0**: @username_3 I added an illustration to my original question  
> **username_3**: @RogerWood That's an amazing illustration (and yes, making illustrations is more time-consuming than most viewers realize), but if I understand yours, and username_1's, illustrations correctly, then in this terminology gravitational lenses are actually not too far from spherical lenses. Clusters are of course quite inhomogeneous, but single galaxies can be rather symmetric. If you look at something like the [Cosmic Horseshoe](https://en.wikipedia.org/wiki/Cosmic_Horseshoe), it's very close to being what I imagine you mean by a spherical lens.  
> **username_4**: It is so cool that we have the person who created the referenced image onboard as one of our contributors. It is so uncool that I can only give a +1.  
> **username_3**: @DavidHammen Ha ha thanks :)  
> **username_0**: @username_3 - I second username_4's comment!  
> **username_0**: @username_3 Yes, 'spherical' lens is a bit ambiguous in this context. As a lens, a black hole has perfect spherical symmetry, but an optician would definitely not call it a spherical lens.  For a black hole, the field and hence the deflection goes as r^-2 with distance from the center. In contrast, the deflection for an optician's spherical lens goes as r^+1.  The former gives rise to the characterisitic arcs or rings.  The latter brings everything to a focus at a single point.  (assuming the deflections are all very small)  
> **username_3**: @RogerWood Thanks :) Okay, I understand now. Yes, a gravitational lens doesn't really have a focus, like a normal lens does, so you're right in your previous comment that it would be exceedingly rare for a spherical lens to ever occur. Or rather, it doesn't have a focal _point_, but instead has a "focal line". If you want to play around with creating caustics, [this Python script](https://galileo-unbound.blog/2021/04/05/the-lens-of-gravity-einsteins-rings/) might be helpful.  
> **username_0**: @username_3 thanks. I'm not a Python person, but that's a really good article  
> **username_3**: @RogerWood Fair enough :)  
> **username_3**: @RogerWood Btw if you think I managed to answer your questions, I'd be happy if you could mark the answer as solved, so that future readers know that it did :)  
> **username_0**: @username_3 oops - that was an oversight - super answer!!!  
> **username_3**: @RogerWood Thanks a lot :)  
> **username_1**: Obligatory [Dr. Becky video](https://www.youtube.com/watch?v=VChgsXbIgdw)  
> **username_3**: @username_1 Very cool video!  

---

