Optical depth

<h2 id="mathematical-definitions">Mathematical definitions</h2>
<h3>Optical depth</h3>
The optical depth of a material, denoted 
 
 
 
 τ
 
 
 {\textstyle \tau }
 
, is given by:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
 
 
 
 τ
 =
 ln
 
 
 (
 
 
 
 Φ
 
 
 e
 
 
 
 
 i
 
 
 
 
 Φ
 
 
 e
 
 
 
 
 t
 
 
 
 
 
 )
 
 =
 −
 ln
 ⁡
 T
 
 
 {\displaystyle \tau =\ln \!\left({\frac {\Phi _{\mathrm {e} }^{\mathrm {i} }}{\Phi _{\mathrm {e} }^{\mathrm {t} }}}\right)=-\ln T}
 
where

<ul><li>
 
 
 
 
 Φ
 
 
 e
 
 
 
 
 i
 
 
 
 
 
 {\textstyle \Phi _{\mathrm {e} }^{\mathrm {i} }}
 
 is the <a href="/facts/Radiant_flux/ak2d0btz">radiant flux</a> received by that material;</li>
<li>
 
 
 
 
 Φ
 
 
 e
 
 
 
 
 t
 
 
 
 
 
 {\textstyle \Phi _{\mathrm {e} }^{\mathrm {t} }}
 
 is the <a href="/facts/Radiant_flux/ak2d0btz">radiant flux</a> transmitted by that material;</li>
<li>
 
 
 
 T
 
 
 {\textstyle T}
 
 is the <a href="/facts/Transmittance/xlyCG7ks">transmittance</a> of that material.</li></ul>
The absorbance 
 
 
 
 A
 
 
 {\textstyle A}
 
 is related to optical depth by:
 
 
 
 τ
 =
 A
 ln
 ⁡
 
 10
 
 
 
 {\displaystyle \tau =A\ln {10}}

<h3>Spectral optical depth</h3>
The spectral optical depth in frequency (denoted 
 
 
 
 
 τ
 
 ν
 
 
 
 
 {\displaystyle \tau _{\nu }}
 
) or in wavelength (
 
 
 
 
 τ
 
 λ
 
 
 
 
 {\displaystyle \tau _{\lambda }}
 
) of a material is given by:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

τ
          
            ν
          
        
        =
        ln
        
        
          (
          
            
              
                Φ
                
                  
                    e
                  
                  ,
                  ν
                
                
                  
                    i
                  
                
              
              
                Φ
                
                  
                    e
                  
                  ,
                  ν
                
                
                  
                    t
                  
                
              
            
          
          )
        
        =
        −
        ln
        ⁡
        
          T
          
            ν
          
        
      
    
    {\displaystyle \tau _{\nu }=\ln \!\left({\frac {\Phi _{\mathrm {e} ,\nu }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\nu }^{\mathrm {t} }}}\right)=-\ln T_{\nu }}

τ
          
            λ
          
        
        =
        ln
        
        
          (
          
            
              
                Φ
                
                  
                    e
                  
                  ,
                  λ
                
                
                  
                    i
                  
                
              
              
                Φ
                
                  
                    e
                  
                  ,
                  λ
                
                
                  
                    t
                  
                
              
            
          
          )
        
        =
        −
        ln
        ⁡
        
          T
          
            λ
          
        
        ,
      
    
    {\displaystyle \tau _{\lambda }=\ln \!\left({\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}}\right)=-\ln T_{\lambda },}

where

<ul><li>
 
 
 
 
 Φ
 
 
 e
 
 ,
 ν
 
 
 
 t
 
 
 
 
 
 {\displaystyle \Phi _{\mathrm {e} ,\nu }^{\mathrm {t} }}
 
 is the <a href="/facts/Radiant_flux/ak2d0btz">spectral radiant flux in frequency</a> transmitted by that material;</li>
<li>
 
 
 
 
 Φ
 
 
 e
 
 ,
 ν
 
 
 
 i
 
 
 
 
 
 {\displaystyle \Phi _{\mathrm {e} ,\nu }^{\mathrm {i} }}
 
 is the spectral radiant flux in frequency received by that material;</li>
<li>
 
 
 
 
 T
 
 ν
 
 
 
 
 {\displaystyle T_{\nu }}
 
 is the <a href="/facts/Transmittance/xlyCG7ks">spectral transmittance in frequency</a> of that material;</li>
<li>
 
 
 
 
 Φ
 
 
 e
 
 ,
 λ
 
 
 
 t
 
 
 
 
 
 {\displaystyle \Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}
 
 is the <a href="/facts/Radiant_flux/ak2d0btz">spectral radiant flux in wavelength</a> transmitted by that material;</li>
<li>
 
 
 
 
 Φ
 
 
 e
 
 ,
 λ
 
 
 
 i
 
 
 
 
 
 {\displaystyle \Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}
 
 is the spectral radiant flux in wavelength received by that material;</li>
<li>
 
 
 
 
 T
 
 λ
 
 
 
 
 {\displaystyle T_{\lambda }}
 
 is the <a href="/facts/Transmittance/xlyCG7ks">spectral transmittance in wavelength</a> of that material.</li></ul>
Spectral absorbance is related to spectral optical depth by:

τ
          
            ν
          
        
        =
        
          A
          
            ν
          
        
        ln
        ⁡
        10
        ,
      
    
    {\displaystyle \tau _{\nu }=A_{\nu }\ln 10,}

τ
          
            λ
          
        
        =
        
          A
          
            λ
          
        
        ln
        ⁡
        10
        ,
      
    
    {\displaystyle \tau _{\lambda }=A_{\lambda }\ln 10,}

where

<ul><li>
 
 
 
 
 A
 
 ν
 
 
 
 
 {\displaystyle A_{\nu }}
 
 is the spectral absorbance in frequency;</li>
<li>
 
 
 
 
 A
 
 λ
 
 
 
 
 {\displaystyle A_{\lambda }}
 
 is the spectral absorbance in wavelength.</li></ul>
<h2 id="relationship-with-attenuation">Relationship with attenuation</h2>
<h3>Attenuation</h3>
Main article: <a href="/facts/Attenuation/XP4vX360">Attenuation</a>
Optical depth measures the attenuation of the transmitted radiant power in a material. Attenuation can be caused by absorption, but also reflection, scattering, and other physical processes. Optical depth of a material is approximately equal to its <a href="/facts/Attenuation/XP4vX360">attenuation</a> when both the absorbance is much less than 1 and the emittance of that material (not to be confused with <a href="/facts/Radiant_exitance/US1T54p8">radiant exitance</a> or <a href="/facts/Emissivity/LUxfMRRI">emissivity</a>) is much less than the optical depth:

Φ
          
            
              e
            
          
          
            
              t
            
          
        
        +
        
          Φ
          
            
              e
            
          
          
            
              a
              t
              t
            
          
        
        =
        
          Φ
          
            
              e
            
          
          
            
              i
            
          
        
        +
        
          Φ
          
            
              e
            
          
          
            
              e
            
          
        
        ,
      
    
    {\displaystyle \Phi _{\mathrm {e} }^{\mathrm {t} }+\Phi _{\mathrm {e} }^{\mathrm {att} }=\Phi _{\mathrm {e} }^{\mathrm {i} }+\Phi _{\mathrm {e} }^{\mathrm {e} },}

T
        +
        A
        T
        T
        =
        1
        +
        E
        ,
      
    
    {\displaystyle T+ATT=1+E,}

where

<ul><li>Φet is the radiant power transmitted by that material;</li>
<li>Φeatt is the radiant power attenuated by that material;</li>
<li>Φei is the radiant power received by that material;</li>
<li>Φee is the radiant power emitted by that material;</li>
<li>T = Φet/Φei is the transmittance of that material;</li>
<li>ATT = Φeatt/Φei is the attenuation of that material;</li>
<li>E = Φee/Φei is the emittance of that material,</li></ul>
and according to the <a href="/facts/Beer%25E2%2580%2593Lambert_law/nl64rI1C">Beer–Lambert law</a>,

T
        =
        
          e
          
            −
            τ
          
        
        ,
      
    
    {\displaystyle T=e^{-\tau },}
  
so:
  
    
      
        A
        T
        T
        =
        1
        −
        
          e
          
            −
            τ
          
        
        +
        E
        ≈
        τ
        +
        E
        ≈
        τ
        ,
        
        
          if
        
         
        τ
        ≪
        1
         
        
          and
        
         
        E
        ≪
        τ
        .
      
    
    {\displaystyle ATT=1-e^{-\tau }+E\approx \tau +E\approx \tau ,\quad {\text{if}}\ \tau \ll 1\ {\text{and}}\ E\ll \tau .}

<h3>Attenuation coefficient</h3>
Optical depth of a material is also related to its <a href="/facts/Attenuation_coefficient/ppCIsSGw">attenuation coefficient</a> by:
 
 
 
 τ
 =
 
 ∫
 
 0
 
 
 l
 
 
 α
 (
 z
 )
 
 
 d
 
 z
 ,
 
 
 {\displaystyle \tau =\int _{0}^{l}\alpha (z)\,\mathrm {d} z,}
 
where

<ul><li>l is the thickness of that material through which the light travels;</li>
<li>α(z) is the attenuation coefficient or Napierian attenuation coefficient of that material at z,</li></ul>
and if α(z) is uniform along the path, the attenuation is said to be a linear attenuation and the relation becomes: 
 
 
 
 τ
 =
 α
 l
 
 
 {\displaystyle \tau =\alpha l}

Sometimes the relation is given using the <a href="/facts/Cross_section_(physics)/lo014h2D">attenuation cross section</a> of the material, that is its attenuation coefficient divided by its <a href="/facts/Number_density/qtyDnLhz">number density</a>:
 
 
 
 τ
 =
 
 ∫
 
 0
 
 
 l
 
 
 σ
 n
 (
 z
 )
 
 
 d
 
 z
 ,
 
 
 {\displaystyle \tau =\int _{0}^{l}\sigma n(z)\,\mathrm {d} z,}
 
 where

<ul><li>σ is the attenuation cross section of that material;</li>
<li>n(z) is the number density of that material at z,</li></ul>
and if 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is uniform along the path, i.e., 
 
 
 
 n
 (
 z
 )
 ≡
 N
 
 
 {\displaystyle n(z)\equiv N}
 
, the relation becomes:
 
 
 
 τ
 =
 σ
 N
 l
 
 
 {\displaystyle \tau =\sigma Nl}

<h2 id="applications">Applications</h2>
<h3>Atomic physics</h3>
In <a href="/facts/Atomic_physics/7Jlm4I7k">atomic physics</a>, the spectral optical depth of a cloud of atoms can be calculated from the quantum-mechanical properties of the atoms. It is given by
 
 
 
 
 τ
 
 ν
 
 
 =
 
 
 
 
 d
 
 2
 
 
 n
 ν
 
 
 2
 
 c
 
 ℏ
 
 ε
 
 0
 
 
 σ
 γ
 
 
 
 
 
 {\displaystyle \tau _{\nu }={\frac {d^{2}n\nu }{2\mathrm {c} \hbar \varepsilon _{0}\sigma \gamma }}}
 
where

<ul><li>d is the <a href="/facts/Transition_dipole_moment/SN3ztKJ6">transition dipole moment</a>;</li>
<li>n is the number of atoms;</li>
<li>ν is the frequency of the beam;</li>
<li>c is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a>;</li>
<li>ħ is the <a href="/facts/Reduced_Planck_constant/qGHAe3qq">reduced Planck constant</a>;</li>
<li>ε0 is the <a href="/facts/Vacuum_permittivity/gWRUhPGY">vacuum permittivity</a>;</li>
<li>σ is the cross section of the beam;</li>
<li>γ is the <a href="/facts/Natural_linewidth/HuI4Bj5f">natural linewidth</a> of the transition.</li></ul>
<h3>Atmospheric sciences</h3>
See also: <a href="/facts/Beer%25E2%2580%2593Lambert_law/nl64rI1C">Beer–Lambert law</a>
In <a href="/facts/Atmospheric_sciences/HKjTA3F4">atmospheric sciences</a>, one often refers to the optical depth of the atmosphere as corresponding to the vertical path from Earth's surface to outer space; at other times the optical path is from the observer's altitude to outer space. The optical depth for a slant path is τ = mτ′, where τ′ refers to a vertical path, m is called the <a href="/facts/Airmass/EFw2A0sa">relative airmass</a>, and for a plane-parallel atmosphere it is determined as m = sec θ where θ is the <a href="/facts/Zenith_angle/tcm21e6t">zenith angle</a> corresponding to the given path. Therefore,
 
 
 
 T
 =
 
 e
 
 −
 τ
 
 
 =
 
 e
 
 −
 m
 
 τ
 ′
 
 
 
 
 
 {\displaystyle T=e^{-\tau }=e^{-m\tau '}}
 
The optical depth of the atmosphere can be divided into several components, ascribed to <a href="/facts/Rayleigh_scattering/o0MQtPMM">Rayleigh scattering</a>, <a href="/facts/Aerosols/X1jXFLp2">aerosols</a>, and gaseous <a href="/facts/Absorption_(electromagnetic_radiation)/xBYKt7yS">absorption</a>. The optical depth of the atmosphere can be measured with a <a href="/facts/Sun_photometer/GlqBrsGE">Sun photometer</a>.
The optical depth with respect to the height within the atmosphere is given by<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

τ
        (
        z
        )
        =
        
          k
          
            a
          
        
        
          w
          
            1
          
        
        
          ρ
          
            0
          
        
        H
        
          e
          
            −
            z
            
              /
            
            H
          
        
      
    
    {\displaystyle \tau (z)=k_{\text{a}}w_{1}\rho _{0}He^{-z/H}}

and it follows that the total atmospheric optical depth is given by<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

τ
        (
        0
        )
        =
        
          k
          
            a
          
        
        
          w
          
            1
          
        
        
          ρ
          
            0
          
        
        H
      
    
    {\displaystyle \tau (0)=k_{\text{a}}w_{1}\rho _{0}H}

In both equations:

<ul><li>ka is the absorption coefficient</li>
<li>w1 is the mixing ratio</li>
<li>ρ0 is the density of air at sea level</li>
<li>H is the <a href="/facts/Scale_height/vErWlen3">scale height</a> of the atmosphere</li>
<li>z is the height in question</li></ul>
The optical depth of a plane parallel cloud layer is given by<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
 
 
 
 τ
 =
 
 Q
 
 e
 
 
 
 
 [
 
 
 
 9
 π
 
 L
 
 2
 
 
 H
 N
 
 
 16
 
 ρ
 
 l
 
 
 2
 
 
 
 
 
 ]
 
 
 1
 
 /
 
 3
 
 
 
 
 {\displaystyle \tau =Q_{\text{e}}\left[{\frac {9\pi L^{2}HN}{16\rho _{l}^{2}}}\right]^{1/3}}
 
where:

<ul><li>Qe is the extinction efficiency</li>
<li>L is the <a href="/facts/Liquid_water_path/5UVglIJy">liquid water path</a></li>
<li>H is the geometrical thickness</li>
<li>N is the concentration of droplets</li>
<li>ρl is the density of liquid water</li></ul>
So, with a fixed depth and total liquid water path, 
 
 
 
 τ
 ∝
 
 N
 
 1
 
 /
 
 3
 
 
 
 
 {\textstyle \tau \propto N^{1/3}}
 
.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h3>Astronomy</h3>
Main article: <a href="/facts/Optical_depth_(astrophysics)/sP56mpQa">Optical depth (astrophysics)</a>
In <a href="/facts/Astronomy/0ArJVM1p">astronomy</a>, the <a href="/facts/Photosphere/32a139LB">photosphere</a> of a star is defined as the surface where its optical depth is 2/3. This means that each photon emitted at the photosphere suffers an average of less than one scattering before it reaches the observer. At the temperature at optical depth 2/3, the energy emitted by the star (the original derivation is for the Sun) matches the observed total energy emitted.
Note that the optical depth of a given medium will be different for different colors (<a href="/facts/Wavelength/kcJU0ymz">wavelengths</a>) of light.
For <a href="/facts/Planetary_rings/KNCgIIX7">planetary rings</a>, the optical depth is the (negative logarithm of the) proportion of light blocked by the ring when it lies between the source and the observer. This is usually obtained by observation of stellar occultations.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Air_mass_(astronomy)/EFw2A0sa">Air mass (astronomy)</a></li>
<li><a href="/facts/Absorptance/n6C9pnDR">Absorptance</a></li>
<li><a href="/facts/Actinometer/DVPn7hlm">Actinometer</a></li>
<li><a href="/facts/Aerosol/X1jXFLp2">Aerosol</a></li>
<li><a href="/facts/Angstrom_exponent/d7bqgCc9">Angstrom exponent</a></li>
<li><a href="/facts/Attenuation_coefficient/ppCIsSGw">Attenuation coefficient</a></li>
<li><a href="/facts/Beer%25E2%2580%2593Lambert_law/nl64rI1C">Beer–Lambert law</a></li>
<li><a href="/facts/Pyranometer/bcz4D6gD">Pyranometer</a></li>
<li><a href="/facts/Radiative_transfer/nYE4KojG">Radiative transfer</a></li>
<li><a href="/facts/Sun_photometer/GlqBrsGE">Sun photometer</a></li>
<li><a href="/facts/Transparency_and_translucency/PRryNNW6">Transparency and translucency</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://scienceworld.wolfram.com/physics/OpticalDepth.html">Optical depth equations</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Absorbance". doi:10.1351/goldbook.A00028 <a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" target="_blank">/wiki/International_Union_of_Pure_and_Applied_Chemistry</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Absorbance". doi:10.1351/goldbook.A00028 <a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" target="_blank">/wiki/International_Union_of_Pure_and_Applied_Chemistry</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Christopher Robert Kitchin (1987). Stars, Nebulae and the Interstellar Medium: Observational Physics and Astrophysics. CRC Press. <a href="/wiki/CRC_Press" target="_blank">/wiki/CRC_Press</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Absorbance". doi:10.1351/goldbook.A00028 <a href="/wiki/International_Union_of_Pure_and_Applied_Chemistry" target="_blank">/wiki/International_Union_of_Pure_and_Applied_Chemistry</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Petty, Grant W. (2006). A first course in atmospheric radiation. Sundog Pub. ISBN 9780972903318. OCLC 932561283. <a href="9780972903318" target="_blank">9780972903318</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Petty, Grant W. (2006). A first course in atmospheric radiation. Sundog Pub. ISBN 9780972903318. OCLC 932561283. <a href="9780972903318" target="_blank">9780972903318</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Petty, Grant W. (2006). A first course in atmospheric radiation. Sundog Pub. ISBN 9780972903318. OCLC 932561283. <a href="9780972903318" target="_blank">9780972903318</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Petty, Grant W. (2006). A first course in atmospheric radiation. Sundog Pub. ISBN 9780972903318. OCLC 932561283. <a href="9780972903318" target="_blank">9780972903318</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
</ol>

Optical depth open-in-new

Optical depth