<p><strong>Pogson's equation</strong><br/>
<a href="https://academic.oup.com/mnras/article/17/1/12/956950" rel="nofollow noreferrer">Pogson's equation</a> for magnitude <span class="math-container">$m$</span>, given in modern form in terms of the base-10 logarithm:
<span class="math-container">$$m = m_{ref} - 2.5\log_{10}\frac{I}{I_{ref}}$$</span>
where <span class="math-container">$m_{ref}$</span> is the magnitude of a reference object, <span class="math-container">$I$</span> is the intensity of the object in question, and <span class="math-container">$I_{ref}$</span> is the intensity (brightness) of the reference object.  Intensities are in units of power density (for example Watts per square meter). (see Wikipedia <a href="https://en.wikipedia.org/w/index.php?title=Magnitude_(astronomy)&amp;oldid=1064062972#Apparent_magnitude" rel="nofollow noreferrer">Apparent magnitude</a>)</p>
<p><strong>Visual Magnitude</strong><br/>
This of course leaves out the <a href="https://en.wikipedia.org/w/index.php?title=Photometric_system&amp;oldid=1033561215" rel="nofollow noreferrer">frequency band</a> over which the magnitude is measured resulting in numerous magnitudes depending on the region of the spectrum specified.  Previously there was a notion of <em>visual magnitude</em>, <span class="math-container">$m_{vis}$</span>, based on how objects appear to the eye.  As this is subjective and individuals vary, visual magnitude was supplanted by magnitude measured through a standard <a href="https://en.wikipedia.org/w/index.php?title=Photometric_system&amp;oldid=1033561215#Photometric_letters" rel="nofollow noreferrer">V-band</a> filter. The V-band filter response is typically centered at 551 nm and has a half-power width of 88 nm.  This is in contrast to <a href="https://en.wikipedia.org/w/index.php?title=Apparent_magnitude&amp;oldid=1068565762#Apparent_bolometric_magnitude" rel="nofollow noreferrer">bolometric magnitude</a>, <span class="math-container">$m_{bol}$</span>, where total received power over all wavelengths is considered.</p>
<p><a href="https://i.sstatic.net/tADJWm.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/tADJWm.png" alt="UBV-System" /></a><br/>
Image from <a href="https://en.wikipedia.org/w/index.php?title=UBV_photometric_system&amp;oldid=1064690389" rel="nofollow noreferrer">UBV photometric system</a> at Wikipedia.</p>
<p><strong>0 Magnitude</strong><br/>
To anchor the scale it is also necessary to select what constitutes 0 magnitude.  Traditionally, <a href="https://en.wikipedia.org/w/index.php?title=Vega&amp;oldid=1069891755" rel="nofollow noreferrer">Vega</a> has been selected as the 0 magnitude reference (see <a href="https://skyandtelescope.org/astronomy-resources/the-stellar-magnitude-system/" rel="nofollow noreferrer">The Stellar Magnitude System</a> from Sky &amp; Telescope).  A more precise definition is typically used such as the one found in <a href="https://ui.adsabs.harvard.edu/abs/1988PASP..100.1134B/abstract" rel="nofollow noreferrer">JHKLM Photometry: Standard Systems, Passbands, and Intrinsic Colors</a>.  In this system, Vega has a V magnitude of 0.03.</p>
<p><strong>Impact of Observer</strong><br/>
With the subjective version of visual magnitude the characteristics of an individual's eyes and observing experience with factor into magnitude estimations.  In addition, while the magnitude scale is linear, perceived brightness is not.  Quoting from the above referenced Sky &amp; Telescope link:</p>
<blockquote>
<p>The resulting stellar magnitude system is logarithmic, in neat
agreement with the 1850s belief that all human senses are logarithmic
in their response to stimuli. The decibel scale for rating loudness
was likewise made logarithmic. Alas, it's not quite so, not for
brightness, sound, or anything else. Our perceptions of the world
follow power-law curves, not logarithmic ones. Thus a star of
magnitude 3.0 does not in fact look exactly halfway in brightness
between 2.0 and 4.0. It looks a little fainter than that. The star
that looks halfway between 2.0 and 4.0 will be about magnitude 2.8.
The wider the magnitude gap, the greater this discrepancy.</p>
</blockquote>
<p><strong>Impact of Atmosphere</strong><br/>
The atmosphere can indeed reduce the apparent brightness of objects, but this is not included in the definition of magnitude as the reduction is a function of location, time, and elevation angle.  A correction can be applied to estimated magnitudes such as the one contained in <a href="http://www.icq.eps.harvard.edu/ICQExtinct.html" rel="nofollow noreferrer">Correcting for Atmospheric Extinction</a>.</p>

CC BY-SA 4.0
2022-02-06T16:17:30.197Z
2022-02-06T16:17:30.197Z
GrapefruitIsAwesome
<p>Well surely the eye has different sensitivity to various wavelengths... Will the Pogson's equation  not be affected if say I compare Betelgeuse and Rigel (both having a marked difference in spectral lines)...</p>
<p>Also, how can we ever draw a standard in stellar magnitude observed if both location and individual visual acuity?</p>
<p>Does atmosphere also have any effect??</p>
<p>Please refer some good resources for further study in this topic.</p>

CC BY-SA 3.0
2018-02-21T16:44:57.897Z
@barrycarter Please convert this into an answer.
called2voyage
<magnitude>
A question about stellar magnitude and the pogson's equation
sugatasen
# A question about stellar magnitude and the pogson's equation

(Asked by: username_0 on 2014-10-13)

Well surely the eye has different sensitivity to various wavelengths... Will the Pogson's equation not be affected if say I compare Betelgeuse and Rigel (both having a marked difference in spectral lines)...

Also, how can we ever draw a standard in stellar magnitude observed if both location and individual visual acuity?

Does atmosphere also have any effect??

Please refer some good resources for further study in this topic.

**Tags:** magnitude
> **Comments:**
> **username_1**: @barrycarter Please convert this into an answer.  

---

## 1 answers

### Answer by username_2 on 2022-02-06. Score: 1

**Pogson's equation**  
[Pogson's equation](https://academic.oup.com/mnras/article/17/1/12/956950) for magnitude $m$, given in modern form in terms of the base-10 logarithm:
$$m = m\_{ref} - 2.5\log\_{10}\frac{I}{I\_{ref}}$$
where $m\_{ref}$ is the magnitude of a reference object, $I$ is the intensity of the object in question, and $I\_{ref}$ is the intensity (brightness) of the reference object. Intensities are in units of power density (for example Watts per square meter). (see Wikipedia [Apparent magnitude](https://en.wikipedia.org/w/index.php?title=Magnitude_(astronomy)&oldid=1064062972#Apparent_magnitude))

**Visual Magnitude**  
This of course leaves out the [frequency band](https://en.wikipedia.org/w/index.php?title=Photometric_system&oldid=1033561215) over which the magnitude is measured resulting in numerous magnitudes depending on the region of the spectrum specified. Previously there was a notion of *visual magnitude*, $m\_{vis}$, based on how objects appear to the eye. As this is subjective and individuals vary, visual magnitude was supplanted by magnitude measured through a standard [V-band](https://en.wikipedia.org/w/index.php?title=Photometric_system&oldid=1033561215#Photometric_letters) filter. The V-band filter response is typically centered at 551 nm and has a half-power width of 88 nm. This is in contrast to [bolometric magnitude](https://en.wikipedia.org/w/index.php?title=Apparent_magnitude&oldid=1068565762#Apparent_bolometric_magnitude), $m\_{bol}$, where total received power over all wavelengths is considered.

[![UBV-System](https://i.sstatic.net/tADJWm.png)](https://i.sstatic.net/tADJWm.png)  
Image from [UBV photometric system](https://en.wikipedia.org/w/index.php?title=UBV_photometric_system&oldid=1064690389) at Wikipedia.

**0 Magnitude**  
To anchor the scale it is also necessary to select what constitutes 0 magnitude. Traditionally, [Vega](https://en.wikipedia.org/w/index.php?title=Vega&oldid=1069891755) has been selected as the 0 magnitude reference (see [The Stellar Magnitude System](https://skyandtelescope.org/astronomy-resources/the-stellar-magnitude-system/) from Sky & Telescope). A more precise definition is typically used such as the one found in [JHKLM Photometry: Standard Systems, Passbands, and Intrinsic Colors](https://ui.adsabs.harvard.edu/abs/1988PASP..100.1134B/abstract). In this system, Vega has a V magnitude of 0.03.

**Impact of Observer**  
With the subjective version of visual magnitude the characteristics of an individual's eyes and observing experience with factor into magnitude estimations. In addition, while the magnitude scale is linear, perceived brightness is not. Quoting from the above referenced Sky & Telescope link:

> The resulting stellar magnitude system is logarithmic, in neat
> agreement with the 1850s belief that all human senses are logarithmic
> in their response to stimuli. The decibel scale for rating loudness
> was likewise made logarithmic. Alas, it's not quite so, not for
> brightness, sound, or anything else. Our perceptions of the world
> follow power-law curves, not logarithmic ones. Thus a star of
> magnitude 3.0 does not in fact look exactly halfway in brightness
> between 2.0 and 4.0. It looks a little fainter than that. The star
> that looks halfway between 2.0 and 4.0 will be about magnitude 2.8.
> The wider the magnitude gap, the greater this discrepancy.

**Impact of Atmosphere**  
The atmosphere can indeed reduce the apparent brightness of objects, but this is not included in the definition of magnitude as the reduction is a function of location, time, and elevation angle. A correction can be applied to estimated magnitudes such as the one contained in [Correcting for Atmospheric Extinction](http://www.icq.eps.harvard.edu/ICQExtinct.html).


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