<p>If I understand you well, your reasoning is that, since redshift can be translated to distance, then there should be a difference in redshift from the front side and the back side. But galaxies are held together by self-gravity and do not take part in the expansion of the Universe. Rather, the uncertainty in redshift comes from internal motion of the gas and stars (and to a lesser degree from measuring uncertainties).</p>
<p>Nevertheless, this uncertainty <em>is</em> related to the thickness of the disk, but in another way than you think.</p>
<h3>Galactic dynamics</h3>
<p>Stars move up and down in the gravitational potential of the galaxy, as illustrated here:</p>
<p><a href="https://i.sstatic.net/wjVSskdY.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/wjVSskdY.png" alt="galactic-potential" /></a></p>
<p><sup>Exaggerated path of an individual star through the galactic disk of NGC 7331. Note that the star is drawn as orbiting in the opposite direction relative to the spiral pattern than usually seen. This is probably unintentional, but not impossible; the phenomenon is called &quot;leading spiral arms&quot; (as opposed to <em>trailing</em> spiral arms which is the standard). Credit (unearthed by @PM Ring): <a href="https://www.nbcnews.com/id/wbna40787453" rel="nofollow noreferrer">Mikhail Medvedev</a> / <a href="https://www.spitzer.caltech.edu/image/ssc2004-12a-morphology-of-our-galaxys-twin" rel="nofollow noreferrer">NASA/JPL-Caltech/M. Regan (STScI), and the SINGS Team</a>.</sup></p>
<p>You can imagine that stars moving in this way will have their greatest speeds <span class="math-container">$v_z$</span> perpendicular to the galactic orbital plane when they cross the plane, while <span class="math-container">$v_z$</span> vanishes when the distance <span class="math-container">$z$</span> from the plane takes it minimum/maximum. These maxima defines the thickness of the disk.</p>
<p>Hence, the lower and upper errors in the observed redshift come from the stars crossing the plane, whereas the central value comes from stars at the extreme positions.</p>
<p>But since the gravitational potential determines both the speed and the maximum <span class="math-container">$|z|$</span> that a star will reach, the uncertainty in redshift and hence velocity <em>is</em> related to the thickness. To estimate the thickness in this way, you can solve the <a href="https://en.wikipedia.org/wiki/Poisson%27s_equation" rel="nofollow noreferrer">Poisson equation</a>.</p>
<h3>Hoag's Object</h3>
<p>Spiral galaxies general have thicknesses that are 1–2 orders of magnitude smaller than their diameter (e.g. the thickness of the Milky Way is ~0.7 kpc whereas its diameter is ~30 kpc). Ring galaxies are typically formed due to a head-on collision with another galaxy. Such collisions tend to increases the speeds of the stars, so their thicknesses increase, making them more torus-shaped. However, in the particular case of Hoag's Object, there actually isn't any sign of another galaxy that could have cause this, so I think its formation is unsettled.</p>
<p>Observations show that the mass of the ring in Hoag's Object is roughly 1/3 of the central galaxy (<a href="https://ui.adsabs.harvard.edu/abs/1987ApJ...320..454S/abstract" rel="nofollow noreferrer">Schweizer et al. 1987</a>; <a href="https://ui.adsabs.harvard.edu/abs/2011MNRAS.418.1834F/abstract" rel="nofollow noreferrer">Finkelman et al. 2011</a>; <a href="https://ui.adsabs.harvard.edu/abs/2013MNRAS.435..475B/abstract" rel="nofollow noreferrer">Brosch et al. 2013</a>). However, solving the Poisson equation is much more complicated for a torus than for a disk, unless some approximations are made, e.g. modeling it as an infinitely thin ring (<a href="https://ui.adsabs.harvard.edu/abs/1992ApJ...394..248W/abstract" rel="nofollow noreferrer">Woodward et al. 1992</a>) or a thick disk (<a href="https://ui.adsabs.harvard.edu/abs/2009A%26A...507..573H/abstract" rel="nofollow noreferrer">Huré &amp; Pierens 2009</a>). <a href="https://ui.adsabs.harvard.edu/abs/2018MNRAS.476.3269B/abstract" rel="nofollow noreferrer">Bannikova 2018</a> developed a model for Hoag's Object where, <em>assuming</em> a perfect torus, i.e. with the height of the ring equal to its width, the dynamics is consistent with the observed structure (i.e. size of the central galaxy, gap, and inner and outer diameter of the ring). This hints at a torus-like shape.</p>
<p>However, as pointed out by Peter Erwin in the comments, Hoag's Object does in fact exhibit spiral patterns inside the ring. This hints at a much flatter geometry, since it's difficult to imagine how a spiral structure would form and be maintained in a torus (see also <a href="https://astronomy.stackexchange.com/a/58885/5264">his answer</a> for more arguments for a thin ring).</p>

CC BY-SA 4.0
2024-11-09T10:23:35.300Z
I don't think Bannikova's model is very relevant, since it's hard to imagine how you'd get spiral waves in a torus.
Peter Erwin
CC BY-SA 4.0
2024-11-09T10:32:25.713Z
Also, I don't thank that "Peng 1988" reference is very relevant. It doesn't use kinematic information at all, and no one other than Peng seems to have found it useful.
Peter Erwin
CC BY-SA 4.0
2024-11-09T21:27:34.783Z
@PeterErwin I did find it a bit suspicious that Peng's paper(s) don't seem to have gained much traction, but the work itself didn't raise any flags with me. But do you disagree that an uncertainty in redshift comes from internal kinematics and hence correlates with the thickness?
pela
CC BY-SA 4.0
2024-11-10T17:05:29.577Z
Could be... the quoted redshift and uncertainty comes from a 1974 optical absorption-line spectrum, without much in the way of information about the instrumental uncertainties. (NED gives the SDSS heliocentric redshift as $12718 \pm 3$ km/s, which is obviously not including any internal kinematics.)
Peter Erwin
CC BY-SA 4.0
2024-11-10T20:09:06.260Z
@PeterErwin Right… I thought 50 km/s didn't sound too off for a galaxy more massive than the Milky Way.
pela
CC BY-SA 4.0
2024-11-10T20:55:37.913Z
It looks like that image was created by Mikhail Medvedev  https://www.nbcnews.com/id/wbna40787453 using a public domain NASA image (from the Spitzer Space Telescope) of the galaxy NGC 7331  https://commons.wikimedia.org/wiki/File:NGC_7331_Ssc2004-12a1.png
PM 2Ring
CC BY-SA 4.0
2024-11-10T21:59:32.913Z
Nice work, @PM2Ring, I added the credit!
pela
CC BY-SA 4.0
2024-11-10T22:05:49.633Z
@PeterErwin I'm not sure you'd need spiral waves. Couldn't a superposition of up-down motion and elliptical orbits do the job?
pela
CC BY-SA 4.0
2024-11-11T09:52:48.120Z
@pela Sorry, you've lost me -- "need spiral waves" for what? Do what job?
Peter Erwin
CC BY-SA 4.0
2024-11-12T11:36:54.670Z
@PeterErwin Ah sorry, I think I misinterpreted your first-most comment. I thought you meant that a torus would consist of star spiraling around, forming a "tube", but you're referring to the fact that spiral patterns are seen in Hoag's ring, right? If that's what you mean, that's a good point that I hadn't thought of. I'll edit some more.
pela
CC BY-SA 4.0
2024-11-12T14:20:23.243Z
@pela Yes, that's it: the fact that we see spiral structure in the ring makes more sense if the ring is thin than if it's a thick torus.
Peter Erwin
CC BY-SA 4.0
2024-11-12T20:44:41.670Z
I wonder what's with the downvote, though. Was my answer that useless?
pela
CC BY-SA 4.0
2024-11-13T10:05:27.390Z
"I wonder what's with the downvote". I see both answers gained same votes (4). From that It seems to me more data is needed to make more definite conclusion. Thanks for your effort!
Martian2020
CC BY-SA 4.0
2024-11-13T12:27:03.753Z
@Martian2020 You're welcome, but I am pretty convinced by Peter Erwin's arguments, meaning that, to the best of our knowledge, it is _not_ torus-shaped but more [aerobie](https://www.google.com/search?q=aerobie)-shaped. So I think you can safely mark his answer as "Accepted" (regarding my downvote, clicking between the up and down arrows reveals that it has 5 ups and 1 down, for a total of 4).
pela
CC BY-SA 4.0
2024-11-08T13:00:02.990Z
2024-11-12T14:36:11.627Z
2024-11-12T14:36:11.627Z
pela
<p>We cannot <em>directly</em> measure the thickness of the ring. As ProfRob points out in his comments, you cannot generally use velocity measurements (redshifts) within an individual galaxy, since what you are measuring is the combination of the (single) cosmological redshift for the entire galaxy and various Doppler shifts for motions within the galaxy.</p>
<p>We can, however, make some plausible arguments suggesting the ring is probably quite thin. For example, the visible ring in optical light sits in the inner part of a more extended gas ring, seen in HI emission from atomic hydrogen (e.g., <a href="https://ui.adsabs.harvard.edu/abs/2013MNRAS.435..475B/abstract" rel="noreferrer">Brosch et al. 2013</a>). The gas ring can be seen to be rotating in a manner consistent with circular rotation (and with the rotation of the optical ring), except near its outer edge, where there are distortions consistent with a modest warp. All of this is consistent with the rotation of a thin gas disk.</p>
<p>In addition, the stellar ring shows spiral structure (and the weak non-circular motions in the H<span class="math-container">$\alpha$</span> kinematics in the stellar rings are consistent with spiral structure; <a href="https://ui.adsabs.harvard.edu/abs/2011MNRAS.418.1834F" rel="noreferrer">Finkelman et al. 2011</a>), which is very plausible for a thin disk-like system,[1] but very hard to imagine for a thick torus.</p>
<p>By analogy with the disks of spiral galaxies, the thickness is probably <span class="math-container">$\sim$</span> 1/10 of its extent, or no more than about <span class="math-container">$\sim 1$</span> kpc.</p>
<p>[1] In general, you need a thin disk to get gravitational instabilities that general spiral waves. This is true for galactic disks, <a href="https://www.mpia.de/news/science/2016-12-disk-spiral" rel="noreferrer">protoplanetary disks</a>, and even <a href="https://science.nasa.gov/resource/staggering-structure/" rel="noreferrer">Saturn's ring system</a>.</p>

CC BY-SA 4.0
2024-11-09T10:35:55.573Z
2024-11-09T10:51:15.947Z
2024-11-09T10:51:15.947Z
Peter Erwin
<p>I've read <a href="https://en.wikipedia.org/wiki/Hoag%27s_Object" rel="nofollow noreferrer">wiki: Hoag's Object</a> and I've tried web search for redshift differences, but could not find info on Hoag's Object thickness.</p>
<blockquote>
<p>Redshift  12740±50 km/s</p>
</blockquote>
<p>As its thickness is not mentioned in wiki, I suspect &quot;±50&quot; is error margin, not a number that allows to calculate thickness (also 100 kms is in the order of 1000 kiloparsecs - twenty times its diameter of 45.41 kiloparsecs).</p>
<p>Any info/ideas on whether the object is thin like <a href="https://en.wikipedia.org/wiki/Milky_Way" rel="nofollow noreferrer">Milky Way</a> or torus shaped?</p>

CC BY-SA 4.0
2024-11-08T07:54:46.147Z
You cannot use varations in redshift within a galaxy to infer variations in distance using Hubble's law.
ProfRob
CC BY-SA 4.0
2024-11-08T08:46:15.860Z
@ProfRob, why? I've noted in my post for this galaxy error margin is too large. Because of that in general for any galaxy? Or what else?
Martian2020
CC BY-SA 4.0
2024-11-08T09:02:55.270Z
Because the range of velocities within a galaxy is unrelated to their distance.
ProfRob
CC BY-SA 4.0
2024-11-08T21:02:21.440Z
But not unrelated to the *redshift*.
pela
<galaxy>
Is Hoag's Object torus shaped (what's its thickness)?
Martian2020
# Is Hoag's Object torus shaped (what's its thickness)?

(Asked by: username_0 on 2024-11-08)

I've read [wiki: Hoag's Object](https://en.wikipedia.org/wiki/Hoag%27s_Object) and I've tried web search for redshift differences, but could not find info on Hoag's Object thickness.

> Redshift 12740±50 km/s

As its thickness is not mentioned in wiki, I suspect "±50" is error margin, not a number that allows to calculate thickness (also 100 kms is in the order of 1000 kiloparsecs - twenty times its diameter of 45.41 kiloparsecs).

Any info/ideas on whether the object is thin like [Milky Way](https://en.wikipedia.org/wiki/Milky_Way) or torus shaped?

**Tags:** galaxy
> **Comments:**
> **username_1**: You cannot use varations in redshift within a galaxy to infer variations in distance using Hubble's law.  
> **username_0**: @ProfRob, why? I've noted in my post for this galaxy error margin is too large. Because of that in general for any galaxy? Or what else?  
> **username_1**: Because the range of velocities within a galaxy is unrelated to their distance.  
> **username_2**: But not unrelated to the *redshift*.  

---

## 2 answers

### Answer by username_3 on 2024-11-09. Score: 5 | Accepted answer

We cannot *directly* measure the thickness of the ring. As ProfRob points out in his comments, you cannot generally use velocity measurements (redshifts) within an individual galaxy, since what you are measuring is the combination of the (single) cosmological redshift for the entire galaxy and various Doppler shifts for motions within the galaxy.

We can, however, make some plausible arguments suggesting the ring is probably quite thin. For example, the visible ring in optical light sits in the inner part of a more extended gas ring, seen in HI emission from atomic hydrogen (e.g., [Brosch et al. 2013](https://ui.adsabs.harvard.edu/abs/2013MNRAS.435..475B/abstract)). The gas ring can be seen to be rotating in a manner consistent with circular rotation (and with the rotation of the optical ring), except near its outer edge, where there are distortions consistent with a modest warp. All of this is consistent with the rotation of a thin gas disk.

In addition, the stellar ring shows spiral structure (and the weak non-circular motions in the H$\alpha$ kinematics in the stellar rings are consistent with spiral structure; [Finkelman et al. 2011](https://ui.adsabs.harvard.edu/abs/2011MNRAS.418.1834F)), which is very plausible for a thin disk-like system,[1] but very hard to imagine for a thick torus.

By analogy with the disks of spiral galaxies, the thickness is probably $\sim$ 1/10 of its extent, or no more than about $\sim 1$ kpc.

[1] In general, you need a thin disk to get gravitational instabilities that general spiral waves. This is true for galactic disks, [protoplanetary disks](https://www.mpia.de/news/science/2016-12-disk-spiral), and even [Saturn's ring system](https://science.nasa.gov/resource/staggering-structure/).


---

### Answer by username_2 on 2024-11-08. Score: 5

If I understand you well, your reasoning is that, since redshift can be translated to distance, then there should be a difference in redshift from the front side and the back side. But galaxies are held together by self-gravity and do not take part in the expansion of the Universe. Rather, the uncertainty in redshift comes from internal motion of the gas and stars (and to a lesser degree from measuring uncertainties).

Nevertheless, this uncertainty *is* related to the thickness of the disk, but in another way than you think.

### Galactic dynamics

Stars move up and down in the gravitational potential of the galaxy, as illustrated here:

[![galactic-potential](https://i.sstatic.net/wjVSskdY.png)](https://i.sstatic.net/wjVSskdY.png)

Exaggerated path of an individual star through the galactic disk of NGC 7331. Note that the star is drawn as orbiting in the opposite direction relative to the spiral pattern than usually seen. This is probably unintentional, but not impossible; the phenomenon is called "leading spiral arms" (as opposed to *trailing* spiral arms which is the standard). Credit (unearthed by @PM Ring): [Mikhail Medvedev](https://www.nbcnews.com/id/wbna40787453) / [NASA/JPL-Caltech/M. Regan (STScI), and the SINGS Team](https://www.spitzer.caltech.edu/image/ssc2004-12a-morphology-of-our-galaxys-twin).

You can imagine that stars moving in this way will have their greatest speeds $v\_z$ perpendicular to the galactic orbital plane when they cross the plane, while $v\_z$ vanishes when the distance $z$ from the plane takes it minimum/maximum. These maxima defines the thickness of the disk.

Hence, the lower and upper errors in the observed redshift come from the stars crossing the plane, whereas the central value comes from stars at the extreme positions.

But since the gravitational potential determines both the speed and the maximum $|z|$ that a star will reach, the uncertainty in redshift and hence velocity *is* related to the thickness. To estimate the thickness in this way, you can solve the [Poisson equation](https://en.wikipedia.org/wiki/Poisson%27s_equation).

### Hoag's Object

Spiral galaxies general have thicknesses that are 1–2 orders of magnitude smaller than their diameter (e.g. the thickness of the Milky Way is ~0.7 kpc whereas its diameter is ~30 kpc). Ring galaxies are typically formed due to a head-on collision with another galaxy. Such collisions tend to increases the speeds of the stars, so their thicknesses increase, making them more torus-shaped. However, in the particular case of Hoag's Object, there actually isn't any sign of another galaxy that could have cause this, so I think its formation is unsettled.

Observations show that the mass of the ring in Hoag's Object is roughly 1/3 of the central galaxy ([Schweizer et al. 1987](https://ui.adsabs.harvard.edu/abs/1987ApJ...320..454S/abstract); [Finkelman et al. 2011](https://ui.adsabs.harvard.edu/abs/2011MNRAS.418.1834F/abstract); [Brosch et al. 2013](https://ui.adsabs.harvard.edu/abs/2013MNRAS.435..475B/abstract)). However, solving the Poisson equation is much more complicated for a torus than for a disk, unless some approximations are made, e.g. modeling it as an infinitely thin ring ([Woodward et al. 1992](https://ui.adsabs.harvard.edu/abs/1992ApJ...394..248W/abstract)) or a thick disk ([Huré & Pierens 2009](https://ui.adsabs.harvard.edu/abs/2009A%26A...507..573H/abstract)). [Bannikova 2018](https://ui.adsabs.harvard.edu/abs/2018MNRAS.476.3269B/abstract) developed a model for Hoag's Object where, *assuming* a perfect torus, i.e. with the height of the ring equal to its width, the dynamics is consistent with the observed structure (i.e. size of the central galaxy, gap, and inner and outer diameter of the ring). This hints at a torus-like shape.

However, as pointed out by Peter Erwin in the comments, Hoag's Object does in fact exhibit spiral patterns inside the ring. This hints at a much flatter geometry, since it's difficult to imagine how a spiral structure would form and be maintained in a torus (see also [his answer](https://astronomy.stackexchange.com/a/58885/5264) for more arguments for a thin ring).

> **Comments:**
> **username_3**: I don't think Bannikova's model is very relevant, since it's hard to imagine how you'd get spiral waves in a torus.  
> **username_3**: Also, I don't thank that "Peng 1988" reference is very relevant. It doesn't use kinematic information at all, and no one other than Peng seems to have found it useful.  
> **username_2**: @PeterErwin I did find it a bit suspicious that Peng's paper(s) don't seem to have gained much traction, but the work itself didn't raise any flags with me. But do you disagree that an uncertainty in redshift comes from internal kinematics and hence correlates with the thickness?  
> **username_3**: Could be... the quoted redshift and uncertainty comes from a 1974 optical absorption-line spectrum, without much in the way of information about the instrumental uncertainties. (NED gives the SDSS heliocentric redshift as $12718 \pm 3$ km/s, which is obviously not including any internal kinematics.)  
> **username_2**: @PeterErwin Right… I thought 50 km/s didn't sound too off for a galaxy more massive than the Milky Way.  
> **username_4**: It looks like that image was created by Mikhail Medvedev  https://www.nbcnews.com/id/wbna40787453 using a public domain NASA image (from the Spitzer Space Telescope) of the galaxy NGC 7331  https://commons.wikimedia.org/wiki/File:NGC_7331_Ssc2004-12a1.png  
> **username_2**: Nice work, @PM2Ring, I added the credit!  
> **username_2**: @PeterErwin I'm not sure you'd need spiral waves. Couldn't a superposition of up-down motion and elliptical orbits do the job?  
> **username_3**: @pela Sorry, you've lost me -- "need spiral waves" for what? Do what job?  
> **username_2**: @PeterErwin Ah sorry, I think I misinterpreted your first-most comment. I thought you meant that a torus would consist of star spiraling around, forming a "tube", but you're referring to the fact that spiral patterns are seen in Hoag's ring, right? If that's what you mean, that's a good point that I hadn't thought of. I'll edit some more.  
> **username_3**: @pela Yes, that's it: the fact that we see spiral structure in the ring makes more sense if the ring is thin than if it's a thick torus.  
> **username_2**: I wonder what's with the downvote, though. Was my answer that useless?  
> **username_0**: "I wonder what's with the downvote". I see both answers gained same votes (4). From that It seems to me more data is needed to make more definite conclusion. Thanks for your effort!  
> **username_2**: @Martian2020 You're welcome, but I am pretty convinced by Peter Erwin's arguments, meaning that, to the best of our knowledge, it is _not_ torus-shaped but more [aerobie](https://www.google.com/search?q=aerobie)-shaped. So I think you can safely mark his answer as "Accepted" (regarding my downvote, clicking between the up and down arrows reveals that it has 5 ups and 1 down, for a total of 4).  

---

