Spectral index

<h2 id="spectral-index-of-thermal-emission">Spectral index of thermal emission</h2>
<p>At radio frequencies (i.e. in the low-frequency, long-wavelength limit), where the <a href="/facts/Rayleigh%E2%80%93Jeans_law/qlvgQ5jb">Rayleigh–Jeans law</a> is a good approximation to the spectrum of <a href="/facts/Thermal_radiation/1PD9jtCv">thermal radiation</a>, intensity is given by

B
          
            ν
          
        
        (
        T
        )
        ≃
        
          
            
              2
              
                ν
                
                  2
                
              
              k
              T
            
            
              c
              
                2
              
            
          
        
        .
      
    
    {\displaystyle B_{\nu }(T)\simeq {\frac {2\nu ^{2}kT}{c^{2}}}.}

Taking the logarithm of each side and taking the partial derivative with respect to 
  
    
      
        log
        
        ν
      
    
    {\displaystyle \log \,\nu }
  
 yields

∂
              log
              ⁡
              
                B
                
                  ν
                
              
              (
              T
              )
            
            
              ∂
              log
              ⁡
              ν
            
          
        
        ≃
        2.
      
    
    {\displaystyle {\frac {\partial \log B_{\nu }(T)}{\partial \log \nu }}\simeq 2.}

Using the positive sign convention, the spectral index of thermal radiation is thus 
  
    
      
        α
        ≃
        2
      
    
    {\displaystyle \alpha \simeq 2}
  
 in the Rayleigh–Jeans regime. The spectral index departs from this value at shorter wavelengths, for which the Rayleigh–Jeans law becomes an increasingly inaccurate approximation, tending towards zero as intensity reaches a peak at a frequency given by <a href="/facts/Wien%27s_displacement_law/nrMVYWTF">Wien's displacement law</a>. Because of the simple temperature-dependence of radiative flux in the Rayleigh–Jeans regime, the <i>radio spectral index</i> is defined implicitly by<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>

S
        ∝
        
          ν
          
            α
          
        
        T
        .
      
    
    {\displaystyle S\propto \nu ^{\alpha }T.}

</p>

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

<ol>
<li id="fn:1"><p>Burke, B.F., Graham-Smith, F. (2009). An Introduction to Radio Astronomy, 3rd Ed., Cambridge University Press, Cambridge, UK, ISBN 978-0-521-87808-1, page 132. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Radio Spectral Index". Wolfram Research. Retrieved 2011-01-19. <a href="http://scienceworld.wolfram.com/astronomy/RadioSpectralIndex.html" target="_blank">http://scienceworld.wolfram.com/astronomy/RadioSpectralIndex.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Spectral index open-in-new

Spectral index