<p>Galaxies are intrinsically fuzzy objects that usually fade away fairly smoothly in density as you go out in radius. So there isn't really such a thing as <em>the</em> diameter or size.</p>
<p>Isophotal diameters are an attempt to come up with a standard (and simple) way of measuring galaxy sizes in ways that can be compared between different galaxies. It basically means the diameter or radius (along the galaxy's major axis if it's not circular) at which the integrated stellar brightness has declined down to some standard value. The most popular such value is 25 magnitudes per square arc second in the <em>B</em> band. Other definitions exist: for example, there's the once-popular &quot;Holmberg radius&quot; (26.5 mag arcsec<span class="math-container">$^{-2}$</span>, also in the <em>B</em> band), and the <em>Spitzer</em> Survey of Stellar Structure in Galaxies (<a href="http://adsabs.harvard.edu/abs/2010PASP..122.1397S" rel="nofollow noreferrer">Sheth et al. 2010</a>) uses both 25.5 and 26.5 mag arcsec<span class="math-container">$^{-2}$</span> in the <em>Spitzer</em> 3.6 micron band.</p>
<p>Here's an attempt to illustrate this for the S0 galaxy NGC 3412. These are two <span class="math-container">$r$</span>-band images I obtained back in 2004 with the Isaac Newton Telescope's Wide Field Camera (<a href="https://ui.adsabs.harvard.edu/abs/2008AJ....135...20E" rel="nofollow noreferrer">Erwin et al. 2008</a>). I've plotted their isophotes (after smoothing the images with a median filter) using steps of 0.5 mag arcsec<span class="math-container">$^{-2}$</span>. I've converted both to <em>B</em>-band assuming a constant color of <span class="math-container">$B - R = 1.46$</span>; since this is an S0 galaxy with no recent star formation, assuming one color for the entire galaxy is not too terrible an approximation.</p>
<p>The first image is a 30-second exposure; the signal-to-noise is so low that I stop plotting isophotes at the level of 26.5 mag arcsec<span class="math-container">$^{-2}$</span>.</p>
<p>The second image is a combination of several exposures totalling 2400 seconds (you can see that it's actually saturated in the center); the faintest plotted isophote is 28.5 mag arcsec<span class="math-container">$^{-2}$</span>.</p>
<p>In both plots, I've outlined the 25.0 mag arcsec<span class="math-container">$^{-2}$</span> isophote in blue. This is pretty elliptical (a bit ragged in the &quot;30s&quot; plot since the S/N is so low) and has a semi-major-axis of <span class="math-container">$\sim 98$</span> arc seconds, for a &quot;diameter&quot; of <span class="math-container">$\sim 200$</span> arc seconds. At the 11 Mpc distance of NGC 3412, this translates to a diameter of <span class="math-container">$\sim 10.7$</span> kpc, or about 35,000 light years.</p>
<p>(For comparison, the <a href="https://ned.ipac.caltech.edu/cgi-bin/datasearch?search_type=Cat_id&amp;objid=24416&amp;catname=RC3&amp;img_stamp=YES&amp;hconst=73.0&amp;omegam=0.27&amp;omegav=0.73&amp;corr_z=1&amp;of=table" rel="nofollow noreferrer">RC3</a> 25 mag arcsec<span class="math-container">$^{-2}$</span> diameter for this galaxy is 218 arc seconds, measured on photographic plates.)</p>
<p><a href="https://i.sstatic.net/65osjsiB.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/65osjsiB.png" alt="B-band isophote plot of S0 galaxy NGC 3412, based on 30-second exposure. Thick blue isophotes indicates 25th mag/arcsec^2 isophote level." /></a></p>
<p><a href="https://i.sstatic.net/QSN6CJVn.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QSN6CJVn.png" alt="B-band isophote plot of S0 galaxy NGC 3412, based on combined 2400s-second exposure. Thick blue isophotes indicates 25th mag/arcsec^2 isophote level." /></a></p>

CC BY-SA 4.0
2024-07-29T13:38:38.510Z
Perfect, thanks for clearing things up. An answer from an expert in the field is always the best one can look for.
jack_O'Dim
CC BY-SA 4.0
2024-07-29T13:16:32.210Z
2024-07-31T22:02:34.963Z
2024-07-31T22:02:34.963Z
Peter Erwin
<p>I heard that isophotal diameters are a common way to measure a galaxy diameter.
How do they work, and what relationship is there between the two values?</p>

CC BY-SA 4.0
2024-07-27T16:33:03.297Z
Yeah I mean, that is what I also think, but I surely have said that badly. With "real" I wanted to mean the part of a galaxy that we usually associate with the term, e.g. the disk for a spiral. Again, my fault, thanks for noticing
jack_O'Dim
CC BY-SA 4.0
2024-07-27T21:41:24.263Z
"I heard that..." if there's any way to add more context about where or how you herd it - especially if you can link to it, it will be helpful to potential answer authors.
uhoh
CC BY-SA 4.0
2024-07-27T23:08:26.227Z
@Miss_Understands "The outermost closed isophote loop is the diameter" -- no, the diameter is defined by the isophote corresponding to a particular value of the surface brightness. The most common definition is 25 magnitudes per square arc second in the *B* band, but there are other definitions using other values in other filters/wavelengths.
Peter Erwin
CC BY-SA 4.0
2024-07-28T17:40:16.233Z
@Miss_Understands Perhaps you could convert your comment (minus the aggressive tone) and add something along the lines that Peter Erwin mentioned.
StephenG - Help Ukraine
CC BY-SA 4.0
2024-08-06T10:49:06.350Z
I don't see why the site would be improved by preventing further answers (which is what closing will do at this point) voting to **leave open**.
uhoh
<galaxy><size><isophote>
What are isophotal diameters? And how do they relate to galaxies' size?
jack_O'Dim
# What are isophotal diameters? And how do they relate to galaxies' size?

(Asked by: username_0 on 2024-07-27)

I heard that isophotal diameters are a common way to measure a galaxy diameter.
How do they work, and what relationship is there between the two values?

**Tags:** galaxy, size, isophote
> **Comments:**
> **username_0**: Yeah I mean, that is what I also think, but I surely have said that badly. With "real" I wanted to mean the part of a galaxy that we usually associate with the term, e.g. the disk for a spiral. Again, my fault, thanks for noticing  
> **username_1**: "I heard that..." if there's any way to add more context about where or how you herd it - especially if you can link to it, it will be helpful to potential answer authors.  
> **username_2**: @Miss_Understands "The outermost closed isophote loop is the diameter" -- no, the diameter is defined by the isophote corresponding to a particular value of the surface brightness. The most common definition is 25 magnitudes per square arc second in the *B* band, but there are other definitions using other values in other filters/wavelengths.  
> **username_3**: @Miss_Understands Perhaps you could convert your comment (minus the aggressive tone) and add something along the lines that Peter Erwin mentioned.  
> **username_1**: I don't see why the site would be improved by preventing further answers (which is what closing will do at this point) voting to **leave open**.  

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## 1 answers

### Answer by username_2 on 2024-07-29. Score: 5 | Accepted answer

Galaxies are intrinsically fuzzy objects that usually fade away fairly smoothly in density as you go out in radius. So there isn't really such a thing as *the* diameter or size.

Isophotal diameters are an attempt to come up with a standard (and simple) way of measuring galaxy sizes in ways that can be compared between different galaxies. It basically means the diameter or radius (along the galaxy's major axis if it's not circular) at which the integrated stellar brightness has declined down to some standard value. The most popular such value is 25 magnitudes per square arc second in the *B* band. Other definitions exist: for example, there's the once-popular "Holmberg radius" (26.5 mag arcsec$^{-2}$, also in the *B* band), and the *Spitzer* Survey of Stellar Structure in Galaxies ([Sheth et al. 2010](http://adsabs.harvard.edu/abs/2010PASP..122.1397S)) uses both 25.5 and 26.5 mag arcsec$^{-2}$ in the *Spitzer* 3.6 micron band.

Here's an attempt to illustrate this for the S0 galaxy NGC 3412. These are two $r$-band images I obtained back in 2004 with the Isaac Newton Telescope's Wide Field Camera ([Erwin et al. 2008](https://ui.adsabs.harvard.edu/abs/2008AJ....135...20E)). I've plotted their isophotes (after smoothing the images with a median filter) using steps of 0.5 mag arcsec$^{-2}$. I've converted both to *B*-band assuming a constant color of $B - R = 1.46$; since this is an S0 galaxy with no recent star formation, assuming one color for the entire galaxy is not too terrible an approximation.

The first image is a 30-second exposure; the signal-to-noise is so low that I stop plotting isophotes at the level of 26.5 mag arcsec$^{-2}$.

The second image is a combination of several exposures totalling 2400 seconds (you can see that it's actually saturated in the center); the faintest plotted isophote is 28.5 mag arcsec$^{-2}$.

In both plots, I've outlined the 25.0 mag arcsec$^{-2}$ isophote in blue. This is pretty elliptical (a bit ragged in the "30s" plot since the S/N is so low) and has a semi-major-axis of $\sim 98$ arc seconds, for a "diameter" of $\sim 200$ arc seconds. At the 11 Mpc distance of NGC 3412, this translates to a diameter of $\sim 10.7$ kpc, or about 35,000 light years.

(For comparison, the [RC3](https://ned.ipac.caltech.edu/cgi-bin/datasearch?search_type=Cat_id&objid=24416&catname=RC3&img_stamp=YES&hconst=73.0&omegam=0.27&omegav=0.73&corr_z=1&of=table) 25 mag arcsec$^{-2}$ diameter for this galaxy is 218 arc seconds, measured on photographic plates.)

[![B-band isophote plot of S0 galaxy NGC 3412, based on 30-second exposure. Thick blue isophotes indicates 25th mag/arcsec^2 isophote level.](https://i.sstatic.net/65osjsiB.png)](https://i.sstatic.net/65osjsiB.png)

[![B-band isophote plot of S0 galaxy NGC 3412, based on combined 2400s-second exposure. Thick blue isophotes indicates 25th mag/arcsec^2 isophote level.](https://i.sstatic.net/QSN6CJVn.png)](https://i.sstatic.net/QSN6CJVn.png)

> **Comments:**
> **username_0**: Perfect, thanks for clearing things up. An answer from an expert in the field is always the best one can look for.  

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