# Why does Earth's atmospheric density have a big “knee” around 100 km? Is there a good analytical approximation?

I've used a quick very rough approximation of the drop of atmospheric density with altitude in this answer and in this answer by using a single exponential and scale height parameter, but that's not what happens in reality. I've typed in some numbers from the U.S. Standard Atmosphere 1976 for $$\rho(Z)/\rho_0$$ and plotted it below.

The Standard Atmosphere document contains a thorough introduction, and I will try to read it, but in the mean time I'd still like to know if there is a very simple way to understand what changes in the atmospheric physics in the neighborhood 100 km that makes the density fall off so much more slowly beyond this region? Why the giant knee?

Is there a way to write a nice, smooth, analytic function that better approximates the density that reproduces the general behavior of this knee?

note: the three thin lines are simple scale-height plots with $$h_{scale}$$ of 6.5, 7, and 7.5 km, bottom to top, just for reference.

Python for plot:

info = """80, 1.5068E-05, 85, 6.7099E-06, 90, 2.789E-06,
95, 1.137E-06, 100, 4.575E-07, 105, 1.898E-07,
110, 7.925E-08, 115, 3.501E-08, 120, 1.814E-08,
125, 1.054E-08, 130, 6.655E-09, 135, 4.461E-09,
140, 3.128E-09, 145, 2.270E-09, 150, 1.694E-09,
155, 1.294E-09, 160, 1.007E-09, 165, 7.959E-10,
170, 6.380E-10, 175, 5.174E-10, 180, 4.240E-10,
190, 2.924E-10, 200, 2.047E-10, 210, 1.507E-10,
220, 1.116E-10, 230, 8.402E-11, 240, 6.415E-11,
250, 4.957E-11, 260, 3.871E-11, 280, 2.425E-11,
300, 1.564E-11, 320, 1.032E-11, 340, 6.941E-12,
360, 4.739E-12, 380, 3.276E-12, 400, 2.288E-12,
420, 1.612E-12, 440, 1.144E-12, 460, 8.180E-13,
480, 5.844E-13, 500, 4.257E-13"""

import numpy as np
import matplotlib.pyplot as plt

alti, ratio = [info.strip('/n').split(',')[i::2] for i in range(2)]

alti, ratio = [np.array([float(x) for x in thing]) for thing in [alti, ratio]]

alts = alti[:15]

rats = [np.exp(-alts/hs) for hs in [6.5, 7, 7.5]]

# https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770009539.pdf
# U.S. Standard Atmosphere, 1976, NOAA, NASA, USAF
# Altitude (Z, Geometric Height) is in kilometers above sea level.
# Data is from Table 4.
# Here ratio is rho(Z) / rho_0, and rho_0 seems to be 1.225 kg/m^3

plt.figure()
plt.yscale('log')
for rat in rats:
plt.plot(alts, rat, linewidth=0.6)
plt.plot(alti, ratio, '-k', linewidth=2)
plt.show()

• Reference: en.wikipedia.org/wiki/Thermosphere – Russell Borogove May 23 '17 at 6:40
• @RussellBorogove I see, plug that into here en.wikipedia.org/wiki/Scale_height. Oh, so there is a chance of a nice smooth analytical function for $\rho(Z)/\rho_0$ if one adds an analytical function for height-dependent temperature. Nice! – uhoh May 23 '17 at 6:47
• – David Hammen May 23 '17 at 11:12
• @DavidHammen I should have just searched here for "knee" :) So it's only the part about the analytical approximation that keeps this from being a duplicate. Not that I'm advocating using them. – uhoh May 23 '17 at 11:16
• The link found in this comment seems helpful. – uhoh May 23 '17 at 11:38

Q: Why does Earth's atmospheric density have a big “knee” around 100 km?

A: The proportion of the lighter component of atomic oxygen O versus O2 and N2.

The log-linear plot in the question shows a roughly straight line behavior before, and again after the broad "knee" around 100 to 200 km. The scale height approximation for the behavior of an atmosphere's pressure versus height yields an exponential with a characteristic 1/e constant $$H$$ given by

$$H=\frac{k_B T}{mg}$$

where $$k_BT$$ is the average kinetic energy of an atmospheric particle and $$mg$$ is the gradient of gravitational potential energy $$dU/dh$$ of the particle, and the dependence of pressure on height in this simple approximation as

$$P(h-h_0) = P_0 e^{(h-h_0)/H}$$

The plot below shows the fraction of different atmospheric components with height. The 78%/21% ratio of N2/O2 start falling off abruptly above 100 km with monotomic oxygen dominating around 180 km. Being roughly half of the mass of N2 or O2, it's scale height is double, resulting in transition to a slope half as large in the log-linear plot.

Q: Is there a good analytical approximation?

A: Yes, this piecewise combination of three analytical approximations for example. Here is the image contained on that page:

Source

• The last link in your answer doesn't work for me. – Cornelisinspace Nov 6 '19 at 14:02
• @Conelisinspace that happens sometimes to me to for grc.nasa.gov sites. I've added a bit of the page to the answer for now, I'll add more tomorrow. – uhoh Nov 6 '19 at 16:50