Editing Heat index (section)

==Formula==
[[File:heat_index_graph.svg|thumb|link={{filepath:heat_index_graph.svg}}|Comparison of NWS heat index values (circles) with the formula approximation (curves). In [{{filepath:heat_index_graph.svg}} the SVG file,] hover over a graph to highlight it.]]
There are many formulas devised to approximate the original tables by Steadman. Anderson et al. (2013),<ref>{{Cite journal | doi=10.1289/ehp.1206273 | pmid=23934704| pmc=3801457 | title = Methods to Calculate the Heat Index as an Exposure Metric in Environmental Health Research | journal=Environmental Health Perspectives| volume=121| issue=10| pages=1111–1119|year = 2013|last1 = Anderson|first1 = G. Brooke| last2=Bell| first2=Michelle L.| last3=Peng| first3=Roger D.}}</ref> NWS (2011), Jonson and Long (2004), and Schoen (2005) have lesser residuals in this order. The former two are a set of polynomials, but the third one is by a single formula with exponential functions.

The formula below approximates the heat index in degrees Fahrenheit, to within ±{{cvt|1.3|F-change|1}}. It is the result of a multivariate fit (temperature equal to or greater than {{cvt|80|F}} and relative humidity equal to or greater than 40%) to a model of the human body.<ref name="SteadmanI" /><ref>Lans P. Rothfusz. "The Heat Index 'Equation' (or, More Than You Ever Wanted to Know About Heat Index)", Scientific Services Division (NWS Southern Region Headquarters), 1 July 1990 [https://www.weather.gov/media/ffc/ta_htindx.PDF]</ref> This equation reproduces the above NOAA National Weather Service table (except the values at {{cvt|90|F}} & 45%/70% relative humidity vary unrounded by less than ±1, respectively).

<math display="block">\mathrm{HI} = c_1 + c_2 T + c_3 R + c_4 T R + c_5 T^2 + c_6 R^2 + c_7 T^2R + c_8 T R^2 + c_9 T^2 R^2 </math>
where
* HI = heat index (in degrees Fahrenheit)
* ''T'' = ambient [[dry-bulb temperature]] (in degrees Fahrenheit)
* ''R'' = relative humidity (percentage value between 0 and 100)

<blockquote><math display="inline">\begin{align}
c_1 &= -42.379, & c_2 &= 2.049\,015\,23, & c_3 &= 10.143\,331\,27,\\
c_4 &= -0.224\,755\,41, & c_5 &= -6.837\,83 \times 10^{-3}, & c_6 &= -5.481\,717 \times 10^{-2},\\
c_7 &= 1.228\,74 \times 10^{-3}, & c_8 &= 8.5282 \times 10^{-4}, & c_9 &= -1.99 \times 10^{-6}.
\end{align}</math></blockquote>

The following coefficients can be used to determine the heat index when the temperature is given in degrees Celsius, where
* HI = heat index (in degrees Celsius)
* ''T'' = ambient [[dry-bulb temperature]] (in degrees Celsius)
* ''R'' = relative humidity (percentage value between 0 and 100)

<blockquote><math display="inline">\begin{align}
c_1 &= -8.784\,694\,755\,56, & c_2 &= 1.611\,394\,11, & c_3 &= 2.338\,548\,838\,89,\\
c_4 &= -0.146\,116\,05, & c_5 &= -0.012\,308\,094, & c_6 &= -0.016\,424\,827\,7778,\\
c_7 &= 2.211\,732 \times 10^{-3}, & c_8 &= 7.2546 \times 10^{-4}, & c_9 &= -3.582 \times 10^{-6}.
\end{align}</math></blockquote>

An alternative set of constants for this equation that is within ±{{cvt|3|F-change}} of the NWS master table for all humidities from 0 to 80% and all temperatures between {{cvt|70|and(-)|115|F}} and all heat indices below {{cvt|150|F}} is:

<math display="block">\begin{align}
c_1 &= 0.363\,445\,176, & c_2 &= 0.988\,622\,465, & c_3 &= 4.777\,114\,035,\\
c_4 &= -0.114\,037\,667, & c_5 &= -8.502\,08 \times 10^{-4}, & c_6 &= -2.071\,6198 \times 10^{-2},\\
c_7 &= 6.876\,78 \times 10^{-4}, & c_8 &= 2.749\,54 \times 10^{-4}, & c_9 &= 0.
\end{align}</math>

A further alternate is this:<ref>
{{cite book
 | last = Stull | first = Richard
 | title = Meteorology for Scientists and Engineers, Second Edition
 | url = https://books.google.com/books?id=QrYRAQAAIAAJ&q=5.37941
 | publisher = Brooks/Cole
 | year = 2000 | page = 60
 | isbn = 9780534372149 }}</ref>

<math display="block">\begin{align}
\mathrm{HI} &= c_1 + c_2 T + c_3 R + c_4 T R + c_5 T^2 + c_6 R^2 + c_7 T^2 R + c_8 T R^2 + c_9 T^2 R^2 + \\
&\quad {} + c_{10} T^3 + c_{11} R^3 + c_{12} T^3 R + c_{13} T R^3 + c_{14} T^3 R^2 + c_{15} T^2 R^3 + c_{16} T^3 R^3
\end{align}</math>
where
<math display="block">\begin{align}
c_1 &= 16.923, & c_2 &= 0.185\,212, & c_3 &= 5.379\,41, & c_4 &= -0.100\,254,\\
c_5 &= 9.416\,95 \times 10^{-3}, & c_6 &= 7.288\,98 \times 10^{-3}, & c_7 &= 3.453\,72\times 10^{-4}, & c_8 &= -8.149\,71 \times 10^{-4},\\
c_9 &= 1.021\,02 \times 10^{-5}, & c_{10} &= -3.8646 \times 10^{-5}, & c_{11} &= 2.915\,83 \times 10^{-5}, & c_{12} &= 1.427\,21 \times 10^{-6},\\
c_{13} &= 1.974\,83 \times 10^{-7}, & c_{14} &= -2.184\,29 \times 10^{-8}, & c_{15} &= 8.432\,96 \times 10^{-10}, & c_{16} &= -4.819\,75 \times 10^{-11}.
\end{align}</math>

For example, using this last formula, with temperature {{convert|90|F|C}} and relative humidity (RH) of 85%, the result would be: {{convert|{{heat index|90|85|disp=out}}|F|C}}.