<p>The textbook I am working through, <em>Observational Astronomy</em> by Birney et al, cites the following relationship between the seeing disk and the air mass one is looking through:</p>
<p><span class="math-container">$$s=s_0X^\frac{3}{5}$$</span></p>
<p>(<span class="math-container">$s$</span>: true seeing disk; <span class="math-container">$s_0$</span>: zenith seeing disk; <span class="math-container">$X$</span>: air mass.)</p>
<p>This exponent, <span class="math-container">$\frac{3}{5}$</span>, is not elaborated upon in the text. But, it strikes me as a bit odd. The predominant factor lending itself to this exponent is turbulence, and turbulence is quite chaotic. That makes me curious about two things:</p>
<ol>
<li>Is this an approximation? To what degree is <span class="math-container">$\frac{3}{5}$</span> the accurate value of this exponent (even if the accuracy is limited by other restrictions)?</li>
<li>Regardless of the above, where does this number come from? Is it measured, or is there a theoretical justification for it?</li>
</ol>
<hr />
<p>My best guess is that it's related to the Kolmogorov power law for energy stored in turbulence, which predicts (for sufficiently large whirls) an energy spectrum of:</p>
<p><span class="math-container">$$E\propto x^\frac{-5}{3}$$</span></p>
<p>(<span class="math-container">$E$</span>: energy stored in a turbulent whirl; <span class="math-container">$x$</span>: the size of that whirl.)</p>
<p>Since the order of magnitude length scale seems to match possible cell sizes in the atmosphere, I'm wondering if this points to an underlying connection between these two laws. But this also seems like it might be a spurious correlation - the factors that feed into the seeing disk measurement are also optical, and I can't see a clear way to retain the 5/3 ratio on both sides.</p>

CC BY-SA 4.0
2021-11-03T11:33:19.653Z
Yes, that exponent has its origin in Kolmogorov turbulence, and is (very) approximately valid for seeings <0.9" and air masses <1.6, but with quite large variance I think. There's a derivation in the original article describing the most popular (I think) way of measuring seeing, the "DIMM": [Sarazin & Roddier (1990)](https://ui.adsabs.harvard.edu/abs/1990A%26A...227..294S).
pela
CC BY-SA 4.0
2021-11-03T12:43:52.433Z
You might enjoy :-) reading publications of Fried (former head of tOSC).  https://en.wikipedia.org/wiki/David_L._Fried
Carl Witthoft
CC BY-SA 4.0
2021-11-04T02:31:51.440Z
Could it perhaps have its derivations in a polytropic equation? The 3/5 comes up some times as an exponent in polytropic equations of states for ideal gases.
Justin T
CC BY-SA 4.0
2021-11-04T15:08:15.820Z
@pela Thank you for the reference, I will read it! Do you intend to write an answer using this paper? If not, I may write up what I learn for posterity.
Slate
CC BY-SA 4.0
2021-11-04T15:08:25.950Z
@CarlWitthoft Thanks for the recommendation, I will absolutely look into it :)
Slate
CC BY-SA 4.0
2021-11-04T20:23:27.520Z
No I don't intend to, because I'm unsure exactly how the exponent propagates, and I haven't read the paper in detail. I'd be happy to see your answer. You may also find the following useful: [Artamonov+ 10](https://ui.adsabs.harvard.edu/abs/2010ARep...54.1019A) show how much the seeing can vary around the $X^{3/5}$ relation, and [Ogane+ 21](https://ui.adsabs.harvard.edu/abs/2021MNRAS.503.5778O) and [Gilda+ 21](https://ui.adsabs.harvard.edu/abs/2021arXiv210700048G) are examples of newer literature discussing the DIMM (and also other methods, e.g. MASS).
pela
<turbulence>
Why does the seeing disk increase with a 3/5 exponent to air mass?
Slate
# Why does the seeing disk increase with a 3/5 exponent to air mass?

(Asked by: username_0 on 2021-11-03)

The textbook I am working through, *Observational Astronomy* by Birney et al, cites the following relationship between the seeing disk and the air mass one is looking through:

$$s=s\_0X^\frac{3}{5}$$

($s$: true seeing disk; $s\_0$: zenith seeing disk; $X$: air mass.)

This exponent, $\frac{3}{5}$, is not elaborated upon in the text. But, it strikes me as a bit odd. The predominant factor lending itself to this exponent is turbulence, and turbulence is quite chaotic. That makes me curious about two things:

1. Is this an approximation? To what degree is $\frac{3}{5}$ the accurate value of this exponent (even if the accuracy is limited by other restrictions)?
2. Regardless of the above, where does this number come from? Is it measured, or is there a theoretical justification for it?

---

My best guess is that it's related to the Kolmogorov power law for energy stored in turbulence, which predicts (for sufficiently large whirls) an energy spectrum of:

$$E\propto x^\frac{-5}{3}$$

($E$: energy stored in a turbulent whirl; $x$: the size of that whirl.)

Since the order of magnitude length scale seems to match possible cell sizes in the atmosphere, I'm wondering if this points to an underlying connection between these two laws. But this also seems like it might be a spurious correlation - the factors that feed into the seeing disk measurement are also optical, and I can't see a clear way to retain the 5/3 ratio on both sides.

**Tags:** turbulence
> **Comments:**
> **username_1**: Yes, that exponent has its origin in Kolmogorov turbulence, and is (very) approximately valid for seeings <0.9" and air masses <1.6, but with quite large variance I think. There's a derivation in the original article describing the most popular (I think) way of measuring seeing, the "DIMM": [Sarazin & Roddier (1990)](https://ui.adsabs.harvard.edu/abs/1990A%26A...227..294S).  
> **username_2**: You might enjoy :-) reading publications of Fried (former head of tOSC).  https://en.wikipedia.org/wiki/David_L._Fried  
> **username_3**: Could it perhaps have its derivations in a polytropic equation? The 3/5 comes up some times as an exponent in polytropic equations of states for ideal gases.  
> **username_0**: @pela Thank you for the reference, I will read it! Do you intend to write an answer using this paper? If not, I may write up what I learn for posterity.  
> **username_0**: @CarlWitthoft Thanks for the recommendation, I will absolutely look into it :)  
> **username_1**: No I don't intend to, because I'm unsure exactly how the exponent propagates, and I haven't read the paper in detail. I'd be happy to see your answer. You may also find the following useful: [Artamonov+ 10](https://ui.adsabs.harvard.edu/abs/2010ARep...54.1019A) show how much the seeing can vary around the $X^{3/5}$ relation, and [Ogane+ 21](https://ui.adsabs.harvard.edu/abs/2021MNRAS.503.5778O) and [Gilda+ 21](https://ui.adsabs.harvard.edu/abs/2021arXiv210700048G) are examples of newer literature discussing the DIMM (and also other methods, e.g. MASS).  

---

## 0 answers

