<p>The index of refraction for air <a href="https://en.wikipedia.org/wiki/Refractive_index#Density" rel="nofollow noreferrer">scales linearly with density</a>, and the density in an idea gas is proportional to pressure and inversely proportional to temperature. Under typical atmospheric conditions, dry air is reasonably ideal, so we can write:</p>
<p><span class="math-container">$$n_{1} = n - 1 = \frac{(n_0-1) T_0}{P_0}\frac{P}{T}$$</span></p>
<p>where <span class="math-container">$n_0$</span> is the index of refraction at some reference temperature <span class="math-container">$T_0$</span> and pressure <span class="math-container">$P_0$</span>.</p>
<p><a href="https://archive.org/details/jresv50n1p39/mode/2up" rel="nofollow noreferrer" title="'The Constants in the Equation for Atmospheric Refractive Index at Radio Frequencies', Journal of Research of the Notional Bureau of Standards Vol. 50, No.1, January 1953 Research Paper 2385">Smith and Weintraub</a> used a value of <span class="math-container">$n_0-1 = 288.04\times10^{-6}$</span> which was the average of 3 published values for visible, 9 GHz, and 24 GHz electromagnetic radiation measured at <span class="math-container">$T_0 = 273\,\textrm{K}$</span> and <span class="math-container">$P_0 = 1013.15\,\textrm{mb}$</span>, which gives</p>
<p><span class="math-container">$$n_{1} = n - 1 = 77.6\times 10^{-6} \frac{P}{T}$$</span></p>
<p>(I figured this out while wrestling with a similar commonly-used but obscurely-sourced equation relevant to my own question <a href="https://physics.stackexchange.com/questions/732028/how-distant-is-the-horizon-on-venus">&quot;How distant is the horizon on Venus?&quot;</a>.)</p>

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2022-10-17T03:34:27.697Z
2022-10-17T03:34:27.697Z
David Bailey
<p>In this equation, <span class="math-container">$n_{1} = n - 1 = \frac{77.6P}{T} \cdot 10^{-6}$</span> the variable P is the total air pressure and T is expressed in Kelvin. This equation describes the refractive index of moist air as EM waves pass through.</p>
<p>This formula appears in <a href="https://books.google.ca/books/about/Imaging_Through_Turbulence.html?id=nuIC-Mk0R4UC" rel="nofollow noreferrer" title="'Imaging through turbulence', Michael Roggemann and Byron Welsh, CRC Press 1996">Roggemann &amp; Welsh from 1996</a>, but a similar version existed in <a href="https://archive.org/details/jresv50n1p39/mode/2up" rel="nofollow noreferrer" title="'The Constants in the Equation for Atmospheric Refractive Index at Radio Frequencies', Journal of Research of the Notional Bureau of Standards Vol. 50, No.1, January 1953 Research Paper 2385">1953 (Smith, Weintraub)</a>. The current literature uses this formula extensively but its never been properly explained.</p>

CC BY-SA 4.0
2022-06-13T15:13:53.730Z
https://www.itu.int/dms_pubrec/itu-r/rec/p/R-REC-P.453-6-199705-S!!PDF-E.pdf suggests it's based on observation and curve fitting.
Barry Carter
<meteorology><atmosphere><atmosphere-modelling><atmospheric-radiation>
Where does this refractive index formula come from: $n_{1} = n - 1 = \frac{77.6P}{T} 10^{-6}$
Electrolyte
# Where does this refractive index formula come from: $n_{1} = n - 1 = \frac{77.6P}{T} 10^{-6}$

(Asked by: username_0 on 2022-06-13)

In this equation, $n\_{1} = n - 1 = \frac{77.6P}{T} \cdot 10^{-6}$ the variable P is the total air pressure and T is expressed in Kelvin. This equation describes the refractive index of moist air as EM waves pass through.

This formula appears in [Roggemann & Welsh from 1996](https://books.google.ca/books/about/Imaging_Through_Turbulence.html?id=nuIC-Mk0R4UC "'Imaging through turbulence', Michael Roggemann and Byron Welsh, CRC Press 1996"), but a similar version existed in [1953 (Smith, Weintraub)](https://archive.org/details/jresv50n1p39/mode/2up "'The Constants in the Equation for Atmospheric Refractive Index at Radio Frequencies', Journal of Research of the Notional Bureau of Standards Vol. 50, No.1, January 1953 Research Paper 2385"). The current literature uses this formula extensively but its never been properly explained.

**Tags:** meteorology, atmosphere, atmosphere-modelling, atmospheric-radiation

> **Comments:**
> **username_1**: https://www.itu.int/dms_pubrec/itu-r/rec/p/R-REC-P.453-6-199705-S!!PDF-E.pdf suggests it's based on observation and curve fitting.  

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## 1 answers

### Answer by username_2 on 2022-10-17. Score: 3

The index of refraction for air [scales linearly with density](https://en.wikipedia.org/wiki/Refractive_index#Density), and the density in an idea gas is proportional to pressure and inversely proportional to temperature. Under typical atmospheric conditions, dry air is reasonably ideal, so we can write:

$$n\_{1} = n - 1 = \frac{(n\_0-1) T\_0}{P\_0}\frac{P}{T}$$

where $n\_0$ is the index of refraction at some reference temperature $T\_0$ and pressure $P\_0$.

[Smith and Weintraub](https://archive.org/details/jresv50n1p39/mode/2up "'The Constants in the Equation for Atmospheric Refractive Index at Radio Frequencies', Journal of Research of the Notional Bureau of Standards Vol. 50, No.1, January 1953 Research Paper 2385") used a value of $n\_0-1 = 288.04\times10^{-6}$ which was the average of 3 published values for visible, 9 GHz, and 24 GHz electromagnetic radiation measured at $T\_0 = 273\,\textrm{K}$ and $P\_0 = 1013.15\,\textrm{mb}$, which gives

$$n\_{1} = n - 1 = 77.6\times 10^{-6} \frac{P}{T}$$

(I figured this out while wrestling with a similar commonly-used but obscurely-sourced equation relevant to my own question ["How distant is the horizon on Venus?"](https://physics.stackexchange.com/questions/732028/how-distant-is-the-horizon-on-venus).)


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