<p><em>Gravitational lensing is everywhere!</em> because it falls off so slowly with <span class="math-container">$r$</span>:</p>
<p><span class="math-container">$$\Delta \phi \approx \frac{4GM}{c^2r_0}.$$</span></p>
<p>That's the first order term. For a nice derivation see Viktor Toth's <a href="https://www.vttoth.com/CMS/physics-notes/306-the-bending-of-light-by-gravity" rel="noreferrer">The bending of light by gravity</a> &quot;...square rooted expression in the denominator can be simplified to first order...&quot;</p>
<p>Wikipedia's <a href="https://en.wikipedia.org/wiki/Gravitational_lens#Description" rel="noreferrer">gravitational lensing</a> breaks it up into three general categories</p>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Strong_gravitational_lensing" rel="noreferrer">strong lensing</a></li>
<li><a href="https://en.wikipedia.org/wiki/Weak_gravitational_lensing" rel="noreferrer">weak lensing</a></li>
<li><a href="https://en.wikipedia.org/wiki/Gravitational_microlensing" rel="noreferrer">microlensing</a></li>
</ul>
<p><strong>Questions:</strong></p>
<ol>
<li>What's the largest angle that light has been &quot;seen to bend&quot; by gravity? (of one object by a separate object)</li>
<li>Was it large enough that a higher order term in this expansion was at all significant?</li>
</ol>
<p>I'm going to arbitrarily exclude seeing the accretion disk behind a black hole or other strong local phenomenon like that, so I've added &quot;of one object by a separate object&quot; to the title, otherwise the <a href="https://commons.wikimedia.org/wiki/File:Black_hole_-_Messier_87_crop_max_res.jpg" rel="noreferrer">now famous Event Horizon Telescope image</a> seems like a winner. (To my understanding the dimmer part across the top is light from the accretion disk <em>behind</em> the black hole &quot;bent over the top and towards us&quot; <a href="https://astronomy.stackexchange.com/q/39386/7982">Is the angular size of the black hole in the movie &quot;interstellar&quot; completely overblown?</a>) As an aside, certainly the first order equation above would no longer apply in this case.</p>

CC BY-SA 4.0
2022-01-30T21:20:56.887Z
This question makes little sense, citing the deflection angle of a _point mass_. Extended mass distributions have deflection laws that don't look like point masses anywhere where significant deflection occurs. An isothermal sphere, for example, has constant deflection angle. So "higher order terms", being a bad choice of name, are important for almost all gravitational lenses. Besides, it's not clear whether the first question asks about the largest physical deflection angle (i.e. kink in light ray), or the largest apparent displacement of an object seen (i.e. largest Einstein ring observed).
ntessore
CC BY-SA 4.0
2022-01-30T22:12:35.987Z
@ntessore If you don't like Viktor Toth's use of "first order" in the derivation I've linked to, you can take it up with them. Of course this is for a point object, I've simply used the equation to illustrate why "*Gravitational lensing is everywhere!* because it falls off so slowly with $r$" Now check the question again; this asks is about the largest deflection angle that has been *observed* rather than calculated. So the mathematical aspect will not be the focus of answers.
uhoh
CC BY-SA 4.0
2022-01-30T23:45:58.327Z
Fair enough, although the way the question is written, it sounds as if you are asking about strong lensing (*exclude [...] strong local phenomenon like that*). There you need large, extended masses to get large deflection angles, which don't deflect like point masses where they are deflecting significantly. So the premise of the question seems odd.
ntessore
CC BY-SA 4.0
2022-01-31T00:38:56.680Z
@ntessore what can I do, I explain what I know. I don't think the question "What's the largest angle that light has been "seen to bend" by gravity? (of one object by a separate object)" is premised on anything particularly strange at all. I think the part you are discussing is better termed *background* rather than premise.
uhoh
CC BY-SA 4.0
2024-02-08T04:46:20.933Z
As of 2009, this was the largest cluster lens with 55" Einstein ring:   THE LARGEST GRAVITATIONAL LENS: MACS J0717.5+3745 (z = 0.546)  https://iopscience.iop.org/article/10.1088/0004-637X/707/1/L102/pdf
eshaya
CC BY-SA 4.0
2024-02-08T06:22:45.167Z
*"I'm going to arbitrarily exclude seeing the accretion disk behind a black hole or other strong local phenomenon like that"* So you only want lensing by an extended body (a galaxy or cluster) of a more distant body? How about a black hole deflecting light from a normal star orbiting it? Or must the light source not be gravitationally bound to the lensing body?
PM 2Ring
CC BY-SA 4.0
2024-02-08T07:29:25.660Z
@PM2Ring your example is certainly light deflection "...of one object by a separate object..." We don't see naked accretion disks just floating around all by themselves.
uhoh
CC BY-SA 4.0
2024-02-08T08:56:59.760Z
In comments: "It seems that observation of light from the accretion disk behind a black hole is a good candidate, why not take a moment and post something like that as an answer?" In the Q: "I'm going to arbitrarily exclude seeing the accretion disk behind a black hole or other strong local phenomenon like that". Confused.
ProfRob
CC BY-SA 4.0
2024-02-08T15:41:07.157Z
Sorry, I'm confused by your last response. I didn't mention accretion discs, which I understand you want to exclude. Are you saying that you're permitting other light sources that are gravitationally bound to the lensing body?
PM 2Ring
CC BY-SA 4.0
2024-02-08T18:43:20.057Z
@PM2Ring I'm explaining the scope of what I excluded, and why your example would not be excluded. The accretion disk is being constantly created by the black hole, so for this question it does not count as a separate object. The black hole and the start are definitely separate objects. There are no constraints based on how "extended" something is.
uhoh
CC BY-SA 4.0
2024-02-08T18:44:51.810Z
@ProfRob I'm deleting the 2nd comment - it was made 10 months after the original post and it seems I responded without re-reading my question. It was made at 06:13 local time and so I was probably deep pre-coffee.
uhoh
CC BY-SA 4.0
2024-02-09T12:20:19.800Z
@uhoh That deflection equation is quite adequate for $r_0>100r_s$, but it's easy to improve on it, as I mention [here](https://astronomy.stackexchange.com/a/49874/16685). The precise deflection equation requires an elliptic integral, but that only takes a few lines in Sympy (or mpmath or Sage).
PM 2Ring
CC BY-SA 4.0
2024-02-14T05:28:35.130Z
@PM2Ring 22 hours left on the bounty (plus the invisible 24 hour grace period). Any further thoughts? It wouldn't have to be the exact angle of the absolutely certain largest instance, if there's arguably a class of phenomena that are the largest, then mentioning a big one of those I think would be sufficiently informative.
uhoh
CC BY-SA 4.0
2024-02-14T08:31:33.910Z
@uhoh Sorry, I can focus on the mathematical aspects, but I don't know much about potential record-breaking lensing observations.
PM 2Ring
<observational-astronomy><gravitational-lensing>
What's the largest angle that light has been "seen to bend" by gravity? (of one object by a separate object)
uhoh
# What's the largest angle that light has been "seen to bend" by gravity? (of one object by a separate object)

(Asked by: username_0 on 2021-03-31)

*Gravitational lensing is everywhere!* because it falls off so slowly with $r$:

$$\Delta \phi \approx \frac{4GM}{c^2r\_0}.$$

That's the first order term. For a nice derivation see Viktor Toth's [The bending of light by gravity](https://www.vttoth.com/CMS/physics-notes/306-the-bending-of-light-by-gravity) "...square rooted expression in the denominator can be simplified to first order..."

Wikipedia's [gravitational lensing](https://en.wikipedia.org/wiki/Gravitational_lens#Description) breaks it up into three general categories

* [strong lensing](https://en.wikipedia.org/wiki/Strong_gravitational_lensing)
* [weak lensing](https://en.wikipedia.org/wiki/Weak_gravitational_lensing)
* [microlensing](https://en.wikipedia.org/wiki/Gravitational_microlensing)

**Questions:**

1. What's the largest angle that light has been "seen to bend" by gravity? (of one object by a separate object)
2. Was it large enough that a higher order term in this expansion was at all significant?

I'm going to arbitrarily exclude seeing the accretion disk behind a black hole or other strong local phenomenon like that, so I've added "of one object by a separate object" to the title, otherwise the [now famous Event Horizon Telescope image](https://commons.wikimedia.org/wiki/File:Black_hole_-_Messier_87_crop_max_res.jpg) seems like a winner. (To my understanding the dimmer part across the top is light from the accretion disk *behind* the black hole "bent over the top and towards us" [Is the angular size of the black hole in the movie "interstellar" completely overblown?](https://astronomy.stackexchange.com/q/39386/7982)) As an aside, certainly the first order equation above would no longer apply in this case.

**Tags:** observational-astronomy, gravitational-lensing

> **Comments:**
> **username_1**: This question makes little sense, citing the deflection angle of a _point mass_. Extended mass distributions have deflection laws that don't look like point masses anywhere where significant deflection occurs. An isothermal sphere, for example, has constant deflection angle. So "higher order terms", being a bad choice of name, are important for almost all gravitational lenses. Besides, it's not clear whether the first question asks about the largest physical deflection angle (i.e. kink in light ray), or the largest apparent displacement of an object seen (i.e. largest Einstein ring observed).  
> **username_0**: @username_1 If you don't like Viktor Toth's use of "first order" in the derivation I've linked to, you can take it up with them. Of course this is for a point object, I've simply used the equation to illustrate why "*Gravitational lensing is everywhere!* because it falls off so slowly with $r$" Now check the question again; this asks is about the largest deflection angle that has been *observed* rather than calculated. So the mathematical aspect will not be the focus of answers.  
> **username_1**: Fair enough, although the way the question is written, it sounds as if you are asking about strong lensing (*exclude [...] strong local phenomenon like that*). There you need large, extended masses to get large deflection angles, which don't deflect like point masses where they are deflecting significantly. So the premise of the question seems odd.  
> **username_0**: @username_1 what can I do, I explain what I know. I don't think the question "What's the largest angle that light has been "seen to bend" by gravity? (of one object by a separate object)" is premised on anything particularly strange at all. I think the part you are discussing is better termed *background* rather than premise.  
> **username_2**: As of 2009, this was the largest cluster lens with 55" Einstein ring:   THE LARGEST GRAVITATIONAL LENS: MACS J0717.5+3745 (z = 0.546)  https://iopscience.iop.org/article/10.1088/0004-637X/707/1/L102/pdf  
> **username_3**: *"I'm going to arbitrarily exclude seeing the accretion disk behind a black hole or other strong local phenomenon like that"* So you only want lensing by an extended body (a galaxy or cluster) of a more distant body? How about a black hole deflecting light from a normal star orbiting it? Or must the light source not be gravitationally bound to the lensing body?  
> **username_0**: @PM2Ring your example is certainly light deflection "...of one object by a separate object..." We don't see naked accretion disks just floating around all by themselves.  
> **username_4**: In comments: "It seems that observation of light from the accretion disk behind a black hole is a good candidate, why not take a moment and post something like that as an answer?" In the Q: "I'm going to arbitrarily exclude seeing the accretion disk behind a black hole or other strong local phenomenon like that". Confused.  
> **username_3**: Sorry, I'm confused by your last response. I didn't mention accretion discs, which I understand you want to exclude. Are you saying that you're permitting other light sources that are gravitationally bound to the lensing body?  
> **username_0**: @PM2Ring I'm explaining the scope of what I excluded, and why your example would not be excluded. The accretion disk is being constantly created by the black hole, so for this question it does not count as a separate object. The black hole and the start are definitely separate objects. There are no constraints based on how "extended" something is.  
> **username_0**: @username_4 I'm deleting the 2nd comment - it was made 10 months after the original post and it seems I responded without re-reading my question. It was made at 06:13 local time and so I was probably deep pre-coffee.  
> **username_3**: @username_0 That deflection equation is quite adequate for $r_0>100r_s$, but it's easy to improve on it, as I mention [here](https://astronomy.stackexchange.com/a/49874/16685). The precise deflection equation requires an elliptic integral, but that only takes a few lines in Sympy (or mpmath or Sage).  
> **username_0**: @PM2Ring 22 hours left on the bounty (plus the invisible 24 hour grace period). Any further thoughts? It wouldn't have to be the exact angle of the absolutely certain largest instance, if there's arguably a class of phenomena that are the largest, then mentioning a big one of those I think would be sufficiently informative.  
> **username_3**: @username_0 Sorry, I can focus on the mathematical aspects, but I don't know much about potential record-breaking lensing observations.  

---

## 0 answers

