It’s Not the Heat …
Humidity, Dew Point, and Heat Index
Copyright © 2016–2023 by Stan Brown, BrownMath.com
Copyright © 2016–2023 by Stan Brown, BrownMath.com
What are the dew point and heat index, and how are they calculated? Those are the questions that this article answers.
The equations are kind of tedious, so this page also provides an Excel workbook and a TI-83/84 program that you can download.
The dew point is an objective measure of the amount of moisture in the air. It’s different from the “humidity” (really relative humidity) that you hear about in every weather report. The American Heritage Dictionary defines dew point as “The temperature at which air becomes saturated and produces dew.”
Here’s the idea. The warmer air is, the more moisture it can hold; cooler air can hold less moisture. If you take some air, and cool it, without changing its pressure or moisture content, how cool does it have to get before the air can’t hold the original amount of moisture? That temperature is the dew point.
You’ll often hear about humidity in weather reports. That’s the amount of moisture in the air, considered as a percentage of the amount of moisture that air at the same temperature could hold. (Officially this is relative humidity, because it’s a measure relative to complete saturation.)
For example, if the temperature is 80 °F (just under 27 °C) and the relative humidity is 65%, then the air is holding 65% of the moisture that 80-degree air could hold. If the temperature is 90 °F (a bit over 32 °C) and the relative humidity is 65%, then the air is holding 65% of the moisture that 90-degree air could hold. Warmer air can hold more moisture than cooler air, so the moisture content of the second sample is greater, even though the humidity is the same. The dew points give you more information: they are 67.2 °F (19.6 °C) and 76.6 °F (24.8 °C), so the warmer air is quite a bit wetter, even though the humidities are the same. (I’ll explain just below how you can convert between dew point and relative humidity.)
Comparisons are easy for two temperatures at the same humidity, or two humidities at the same temperature, but what about the normal case, when the two samples have different temperature and different humidity? For example, suppose it’s 80 °F (27 °C) and 65% relative humidity inside your house, but 72 °F (22 °C) and 80% relative humidity outside. Obviously opening the windows will cool your house, but will you actually be bringing in moister air for that clammy feel? To answer that, you need to know the dew points inside and outside.
Paroscientific’s Web page gives fairly simple equations for computing the dew point from the Celsius temperature (C) and relative humidity (H). I’ve adapted them a bit:
(1)dew point = 237.1 r / (17.27 − r)
where
(2)r = 17.27 C / (237.1 + C) + ln(H/100)
ln is the natural logarithm function.
These equations are valid for temperature 0–60 °C (32–140 °F), humidity 1–100%, and dew point 0–50 °C (32–122 °F), and the dew point is correct to within ±0.4 °C (±0.7 °F).
If you’ve got temperatures in Fahrenheit, remember that C = (F−32)/1.8, and F = 1.8C+32.
Example: What’s the dew point when the temperature is 90 °F (27 °C) and the relative humidity is 65%?
Solution: Equation 1 needs r, so begin by computing r according to equation 2:
r = 17.27 C / (237.1 + C) + ln(H/100)
C = (90−32)/1.8 ≈ 32.22, and H = 65. Continuing:
r = 17.27 × 32.22 / (237.1 + 32.22) + ln(65/100)
r = 556.4394 / 269.32 − 0.4308
r = 1.6353
Now you can use that number in equation 1:
dew point = 237.1 r / (17.27 − r)
dew point = 237.1 × 1.6353 / (17.27 − 1.6353)
dew point = 387.7296 / 51.6347
dew point ≈ 24.8 °C, or 1.8×24.8+32 ≈ 76.6 °F
These computations aren’t difficult, but they’re tedious. You should get the same result with the Excel workbook provided below. I’ve also written a program that you can download to your TI-83+ or TI-84+ calculator.
Example: Should you open your windows in these conditions?
Inside the house: 80 °F (26.7 °C) and 65% relative humidity.
Outside: 72 °F (22.2 °C) and 80% relative humidity.
Solution: When it’s room temperature or higher, the dew point has a lot to do with your comfort level, so compute both dew points. Let’s start inside the house. From equation 2:
r = 17.27 C / (237.1 + C) + ln(H/100)
r = 17.27 × 26.7 / (237.1 + 26.7) + ln(65/100)
r ≈ 1.3172
Then apply equation 1:
dew point inside = 237.1 r / (17.27 − r)
dew point inside = 237.1 × 1.3172 / (17.27 − 1.3172)
dew point inside ≈ 19.6 °C or 1.8×19.6+32 ≈ 67.2 °F
Next, compute the outside dew point:
r = 17.27 C / (237.1 + C) + ln(H/100)
r = 17.27 × 22.2 / (237.1 + 22.2) + ln(80/100)
r ≈ 1.2554
dew point outside = 237.1 r / (17.27 − r)
dew point outside = 237.1 × 1.2554 / (17.27 − 1.2554)
dew point outside ≈ 18.6 °C or 1.8×18.6+32 ≈ 65.5 °F
The outside air is cooler and drier, so opening your windows is a win-win.
Suppose you know the current temperature and the dew point. Can you get the relative humidity from them? Absolutely!
Going forward, you had to do equation 2 and then equation 1. Therefore, to go backward you have to undo the equations in the opposite order. Starting from equation 1, and writing D for dew point:
237.1 r / (17.27 − r) = D
237.1 r = 17.27 D − D r
D r + 237.1 r = 17.27 D
(D+237.1) r = 17.27 D
(3)r = 17.27 D / (D+237.1)
Next, undo equation 2:
17.27 C / (237.1+C) + ln(H/100) = r
ln(H/100) = r − 17.27 C / (237.1+C)
H/100 = exp( r − 17.27 C / (237.1+C) )
(4)H = 100 exp( r − 17.27 C / (237.1+C) )
where exp(blah) means eblah, the inverse of the natural logarithm.
Example: The weather report tells you that the temperature is 86 °F and the dew point is 70 °F. What’s the relative humidity?
Solution: Start by converting to Celsius. The temperature is C = (86−32)/1.8 = 30°, and the dew point is D = (70−32)/1.8 ≈ 21.1°. Next, use equation 3 to find r:
r = 17.27 D / (D+237.1)
r = 17.27 × 21.1 / (21.1+237.1)
r ≈ 1.4113
You need that to find the relative humidity, using equation 4:
H = 100 exp( r − 17.27 C / (237.1+C) )
H = 100 exp( 1.4113 − 17.27 × 30 / (237.1+30) )
H ≈ 59%
This computation, too, is done by my TI-83+/84+ program and my Excel workbook below.
Your metabolism is constantly producing heat. Your body has to get rid of that excess heat, or you’d cook yourself! In cool weather, your body can does that just by radiating heat — indeed, when you feel cold you may put on a coat or sweater to slow down the rate of cooling.
But in warmer weather, your body sweats. To evaporate, perspiration needs extra heat. It takes that from your body, thus cooling you by evaporation. When the weather is not only warm, but humid, there’s more moisture in the air, so perspiration can’t evaporate as quickly. This why you can feel just fine in 80 °F (27 °C) at 30% humidity, but uncomfortable and sticky at the same temperature with 90% humidity.
The heat index is a measure of “how hot it feels”, also known as the apparent temperature. This sounds subjective, but there are standard computations based on models of how efficiently the body can get rid of heat at various combinations of temperature and humidity. The National Weather Service has a chart where you look up the heat index from the actual temperature and the relative humidity. Chris Robbins has a more comprehensive version of the same chart, one using temperature and dew point and one using temperature and relative humidity.
What is the heat index? labels a heat index of 90–103 °F (32–39 °C) with a call for Extreme Caution, 103–124 °F (39–51 °C) as Danger, and anything above that as Extreme Danger, with heat stroke highly likely.
Lans Rothfusz (1998) [full citation in “References”, below], who devised equation 5 (below) for the heat index, cautions that the heat index assumes you’re
Any changes can mean you feel hotter or cooler. If you’re heavier than 147 pounds, as most US men are, your body is more efficient at retaining heat, which is great in the winter but not so good in the summer, so your “personal heat index” will be higher. If you’re exerting yourself, or you’re in direct sunlight, or both, the National Weather Service says that you can add as much as 15 °F (8 °C) to the index in the chart.
90 °F (32 °C) with 90% humidity gives a heat index of 122 °F (50 °C) even if you’re resting in the shade. Add the 15 °F (8 °C) for exertion in the sun, and there’s a real danger you could keel over in many areas of the US on summer days. And that’s without even considering the increment for weighing more than the standard 147 lb (67 kg).
In The Heat Index Equation, the National Weather Service explains where the numbers in the chart come from. If H is the relative humidity (0–100) and F is the Fahrenheit temperature (80° or higher), then you can use the Rothfusz equation:
(5)heat index = −42.379 + 2.04901523 F + 10.14333127 H − 0.22475541 F H − 0.00683783 F² − 0.05481717 H² + 0.00122874 F² H + 0.00085282 F H² − 0.00000199 F² H² (when F ≥ 80)
If you’re outside the United States, convert temperature to C = (F−32) / 1.8, plug it into the equations, and then convert the heat index back via F = 1.8 C + 32.
There are two special cases:
(6)correction = (3.25 − 0.25 H) × √1 − ( |F−95| / 17 )
Theoretically, this correction could amount to 3.25 °F (1.8 °C), but the conditions would be extremely unusual: 0% humidity at 95 °F (35 °C).
Note: |F−95| is the difference between F and 95, as a positive number.
(7)correction = (0.1 H − 8.5) × (17.4 − 0.2 F)
This correction could amount to 2.1 °F (1.2 °C), but it would take 100% humidity at 80 °F (27 °C), which is very rare in the US, according to Chris Robbins./li>
What is the heat index? says that equation 5 is correct to within ±1.3 °F (±0.7 °C). Most sources, including that one, omit equation 6 and equation 7.
Example: The temperature is 90 °F (32 °C), and the humidity is 82%. It’s a sunny day, and your favorite jogging path is in open country. Is it safe to go jogging?
Solution: Recall that various danger levels are listed under Interpreting the Heat Index. The temperature and humidity aren’t in the ranges for equation 6 and equation 7, so use just equation 5 to compute the heat index:
heat index = −42.379 + 2.04901523 F + 10.14333127 H − 0.22475541 F H − 0.00683783 F² − 0.05481717 H² + 0.00122874 F² H + 0.00085282 F H² − 0.00000199 F² H²
heat index = −42.379 + 2.04901523×90 + 10.14333127×82 − 0.22475541×90×82 − 0.00683783×90² − 0.05481717×82² + 0.00122874×90²×82 + 0.00085282×90×82² − 0.00000199×90²×82²
heat index ≈ 115.0 °F or 46.1 °C
115 °F or 46 °C is right in the middle of the Danger zone. And that’s the heat index for a gentle walk in the shade. You’re going to be running in full sun, so add 10 °F (6 °C) or so, more if you weigh more than 147 lb (67 kg). Now you’re in the Extreme Danger zone. Better do your jogging today at an indoor track, with air conditioning.
Example: As I write this, it’s 117 °F (47.2 °C) in Death Valley, with 2% humidity. How does this compare to your usual track as a spot for running?
Solution: Use equation 5 again, and compute the heat index at 105.2 °F (40.7 °C). Death Valley conditions are actually less dangerous than your regular running track. Of course, “less dangerous” doesn’t mean safe — I wouldn’t recommend you go running in Death Valley either. But the heat index is 10 °F (6 °C) lower, because the extremely low humidity means your body can do a better job of getting rid of heat. I’ve been in Death Valley in similar conditions, and it was surprisingly bearable, as long as I didn’t try to move too fast.
There’s one more wrinkle to computing the heat index. If the temperature is under 80 °F (27 °C), equation 5 can give absurd results. The Heat Index Equation gives this simpler equation, and says that “in practice” you compute the simple one first. If it’s under 80 °F, it’s the heat index; if it’s over, then you apply equation 5 and the corrections.
heat index = 0.5[F + 61.0 + 1.2 (F − 68) + 0.094 H]
which I’ve simplified to
(8)heat index = 1.1 F − 10.3 + 0.047 H (when F < 80)
Example: To illustrate equation 8, let’s take a temperature of 75 °F (24 °C), in 90% humidity:
heat index = 1.1 F − 10.3 + 0.047 H
heat index = 1.1×75 − 10.3 + 0.047×90
heat index = 76.4 °F or 24.7 °C
When the temperature is below about 80 °F or 27 °C, even very high humidity doesn’t make it feel much hotter, as long as you’re in the shade and not exerting yourself. For this reason, many sites won’t even compute a heat index when the temperature is below that threshold. But don’t forget — when you’re out in the sun exercising, you’ll start to feel much hotter pretty quick.
Unfortunately, equation 5 is a lot messier than equation 1. You can still solve for the humidity, but it’s ugly. But for you, Gentle Reader, no sacrifice is too great. (Don’t forget that the Excel workbook can do all of this for you.)
HI (the heat index) and F are known constants in any given problem. That just leaves us with a quadratic in H, which can be solved by formula. To keep things simple, I’m ignoring the two special cases, equation 6 and equation 7 — they don’t come up all that often anyway. Here goes!
HI = −42.379 + 2.04901523 F + 10.14333127 H − 0.22475541 F H − 0.00683783 F² − 0.05481717 H² + 0.00122874 F² H + 0.00085282 F H² − 0.00000199 F² H²
Rearrange the nine terms for H², H, and all other:
HI = 0.00085282 F H² − 0.00000199 F² H² − 0.05481717 H² + 0.00122874 F² H − 0.22475541 F H + 10.14333127 H + 2.04901523 F − 0.00683783 F² − 42.379
Did you see what I did there? I made sure that the first term for H², the first term for H, and the first constant term all have plus signs, so that I don’t screw up the signs when I factor out H or H², like this:
HI =
(0.00085282 F −
0.00000199 F² − 0.05481717) H² +
(0.00122874 F² − 0.22475541 F +
10.14333127) H +
(2.04901523 F −
0.00683783 F² − 42.379)
And finally:
(0.00085282 F −
0.00000199 F² − 0.05481717) H² +
(0.00122874 F² − 0.22475541 F +
10.14333127) H +
(2.04901523 F −
0.00683783 F² − 42.379 − HI) = 0
This is a quadratic in the form AH²+BH+C = 0, and we all know how to solve quadratics. In this situation, you want only a plus sign for the ± sign in the quadratic formula; using a minus sign would give you negative humidity.
(9)H = [ −B +
√(B²−4AC) ] / (2A), with
A = 0.00085282 F −
0.00000199 F² − 0.05481717
B = 0.00122874 F² − 0.22475541 F +
10.14333127
C =
2.04901523 F −
0.00683783 F² − 42.379 − HI
Example: If the temperature is 90 °F (32.2 °C), how humid does it have to be for the heat index to reach the danger level of 103 °F (39 °C) in the shade? Solution, using equation 9:
A = 0.00085282×90 − 0.00000199×90² − 0.05481717 = .005818
B = 0.00122874×90² − 0.22475541×90 + 10.14333127 = −0.13186
C = 2.04901523×90 − 0.00683783×90² − 42.379 − 103 = −16.3541
H = [ −B + √(B²−4AC) ] / (2A)
H ≈ 66%
By the way, do you remember that simpler formula, equation 8, for when the temperature is below 80 °F? In that case, we have
I = 1.1 F − 10.3 + 0.047 H (when F < 80)
I − 1.1 F + 10.3 = 0.047 H
0.047 H = I − 1.1 F + 10.3
And finally
(10)H = (I − 1.1 F + 10.3) / 0.047
It’s probably easiest to compute the humidity first, from equation 9, then the dew point, from equation 3 and equation 4. That’s what the Excel workbook does, and of course it’s a lot easier to just use the workbook instead of grinding through the equations.
You could also use the chart provided by Chris Robbins.
Try the accompanying Excel workbook. In addition to the temperature, you give it any one of relative humidity, dew point, and heat index, and it finds the other two. Of course it works in both Fahrenheit and Celsius.
I’ve also written a TI-83+/84+ calculator program that you can download from Dew Point, Relative Humidity, and Heat Index on TI-83/84. It does most of the calculations in this article, but doesn’t compute humidity or dew point from heat index.
You can find many Web-based calculators by searching “heat index” calculate or “dew point” calculate on the Web.
Updates and new info: https://BrownMath.com/bsci/