What is the density of the earth's atmosphere at an altitude of four hundred kilometers?

What is the density of the earth's atmosphere at an altitude of four hundred kilometers? I want to use it to calculate the drag on something in orbit near the ISS.

The Jacchia Reference Atmosphere is a reference atmospheric model that defines values for atmospheric temperature, density, pressure and other properties at altitudes from 90 to 2500 km.

says Wikipedia. That sounds great, but how do I get access to it?

• I'm no pro in maths and orbital mechanics, but might this article help? Apr 19 at 8:15
• @compi That did help. Thanks. So does this, which it led to: space.stackexchange.com/questions/18223/… Apr 19 at 8:22
• Ah, the ancient problem with Wikipedia. Ask "What is the temperature of boiling water" you'll get the answer "It's the temperature at which water turns from liquid to gaseous phase."
– SF.
Apr 20 at 15:03

What is the density of the earth's atmosphere at an altitude of four hundred kilometers?

Variation with solar activity

It can vary by a factor of 50 depending on solar activity (and the model that you choose to believe).

Here are good references.

One or more of these gives pressure and temperature rather than density. You can assume that at 400 km (and somewhat below and above it) the atmosphere is primarily atomic oxygen (mass = 16 not 32) to get density using the ideal gas law.

Note that you can get quite a range of densities depending on the solar activity specified. This answer to What is the ISS drag? shows altitude loss rate for the ISS varying from 7 m/day to 400 m/day depending on solar activity.

Found in this answer, from https://en.wikipedia.org/wiki/Scale_height Wertz et al. SSC12-IV-6, 26th Annual AIAA/USU Conference on Small Satellites.

Diurnal (daily) variation

Hat-tip to @Peter-ReinstateMonica for pointing this out.

There is definitely a diurnal component to the temperature of the thermosphere and that will lead to changes in density, and this even drives diurnal tides, but it's complicated (see Figure 1 below).

However since orbit decay is a slow process, we primarily worry about some kind of average density experienced over a given orbit. An equatorial orbit in LEO will sample roughly 15 day/night cycles per day, a Sun-synchronous orbit might sample 15 dawn/dusk cycles per day.

From "Dynamics of the Thermosphere", Jeffrey M. FORBES, Journal of the Meteorological Society of Japan, Vol. 85B, pp. 193--213, 2007

Fig. 1. Energy input, conversion and transport processes relevant to the Ionosphere-Thermosphere (IT) system. Green indicates energy sources from above the thermosphere, blue indicates influences of lower atmospheric regions on the thermosphere, and red indicates energy conversion and transport processes within the thermosphere.

From The Structure of the Thermosphere and Its Variations Harris I., Priester W. (1968) The Structure of the Thermosphere and Its Variations. In: Quiroz R.S. (eds) Meteorological Investigations of the Upper Atmosphere. Meteorological Monographs, vol 9. American Meteorological Society, Boston, MA. https://doi.org/10.1007/978-1-935704-37-9_9:

FIG. 1. Densities at a height of 250 km above sea level derived from (the unfortunately named) Injun III by Jacchia and Slowey (1963) for the time interval from 15 December 1962 through June 1963. The data are plotted as a function of local time of the perigee. The corre- sponding dates are given at the top, also as MJD (Modified Julian Dates). For comparison, the geomagnetic indices A v and the solar 10.7-cm flux Fare presented. The numbers next to the plotted densities indicate the geographic latitudes of the perigee. The curves represent the Harris-Priester models, S=90 and S= 100.

• If the density varies a lot with the solar activity, should it then not vary a lot with the day/night side as well? (That is, there is no single overall answer to the question?) Apr 19 at 20:54
• @Peter-ReinstateMonica Yes, there is no overall answer! There is definitely a diurnal component to the temperature of the thermosphere and that will lead to changes in density, and this even drives diurnal tides, but it's complicated (see Figure 1). However since orbit decay is a slow process all that the OP should worry about is some kind of average density of a given orbit. An equatorial orbit in LEO will sample roughly 15 day/night cycles per day, a Sun-synchronous orbit might sample 15 dawn/dusk cycles per day.
– uhoh
Apr 19 at 22:13
• Good point with the average as far as orbits are concerned. Apr 19 at 22:17
$$\begin{matrix} Altitude, km& Low Solar Activity&&&& Mean Solar Activity&&&& Extremely High Solar Activity&&&\\ &Temp., K& Density, kg/m3& Pressure, Pa& Mol. Wt., g/mol& Temp.& Density& Pressure& Mol. Wt.& Temp.& Density& Pressure& Mol. Wt.\\ 0& 300.2511& 1.17E+00& 1.01E+05& 28.9502& 300.2511& 1.17E+00& 1.01E+05& 28.9502& 300.2511& 1.16E+00& 9.98E+04& 28.9502\\ 20& 206.2085& 9.48E-02& 5.62E+03& 28.9502& 206.2085& 9.49E-02& 5.62E+03& 28.9502& 206.2085& 9.41E-02& 5.57E+03& 28.9502\\ 40& 257.6979& 4.07E-03& 3.01E+02& 28.9502& 257.6979& 4.07E-03& 3.02E+02& 28.9502& 257.6979& 4.04E-03& 2.99E+02& 28.9502\\ 60& 244.1212& 3.31E-04& 2.32E+01& 28.9502& 244.1212& 3.31E-04& 2.32E+01& 28.9502& 244.1212& 3.28E-04& 2.30E+01& 28.9502\\ 80& 203.1065& 1.69E-05& 9.81E-01& 29.1353& 196.3636& 1.68E-05& 9.45E-01& 29.0175& 172.2146& 1.68E-05& 8.42E-01& 28.529\\ 100& 168.7219& 5.77E-07& 2.89E-02& 28.0036& 184.016& 5.08E-07& 2.81E-02& 27.7137& 297.3338& 2.78E-07& 2.63E-02& 26.1997\\ 120& 356.8669& 1.70E-08& 1.92E-03& 26.3948& 374.9715& 1.80E-08& 2.17E-03& 25.8745& 430.8385& 2.34E-08& 3.55E-03& 23.6456\\ 140& 545.8594& 2.96E-09& 5.37E-04& 25.0665& 635.5703& 3.26E-09& 7.03E-04& 24.5349& 875.9174& 4.93E-09& 1.61E-03& 22.3209\\ 160& 630.0652& 9.65E-10& 2.13E-04& 23.7884& 787.5532& 1.18E-09& 3.31E-04& 23.4225& 1,143.54 &2.23E-09& 9.90E-04& 21.4577\\ 180& 667.8662& 3.90E-10& 9.62E-05& 22.5037& 877.6729& 5.51E-10& 1.80E-04& 22.4106&1,314.34 &1.28E-09& 6.76E-04& 20.7706\\ 200& 684.9187& 1.75E-10& 4.70E-05& 21.2516& 931.2806& 2.91E-10& 1.05E-04& 21.4734& 1,423.65& 8.28E-10 & 4.86E-04& 20.1836\\ 220& 692.6487& 8.47E-11& 2.43E-05& 20.0935& 963.2701& 1.66E-10& 6.44E-05& 20.6108& 1,493.79& 5.69E-10& 3.60E-04& 19.6664\\ 240& 696.1697& 4.31E-11& 1.31E-05& 19.0789& 982.4191& 9.91E-11& 4.09E-05& 19.8292& 1,538.92& 4.08E-10& 2.72E-04& 19.2046\\ 260& 697.7811& 2.30E-11& 7.31E-06& 18.23& 993.9173& 6.16E-11& 2.66E-05& 19.1337& 1,568.03& 3.00E-10& 2.08E-04& 18.7901\\ 280& 698.522& 1.27E-11& 4.20E-06& 17.5402& 1,000.84& 3.94E-11& 1.77E-05& 18.5256& 1,586.86& 2.25E-10& 1.61E-04& 18.4178\\ 300& 698.8644& 7.22E-12& 2.47E-06& 16.983& 1,005.03& 2.58E-11& 1.20E-05& 18.0015& 1,599.07& 1.71E-10& 1.26E-04& 18.0839\\ 320& 699.0233& 4.21E-12& 1.48E-06& 16.5214& 1,007.56& 1.72E-11& 8.20E-06& 17.5537& 1,607.02& 1.32E-10& 9.93E-05& 17.7852\\ 340& 699.0973& 2.50E-12& 9.01E-07& 16.1147& 1,009.10& 1.16E-11& 5.69E-06& 17.1721& 1,612.19& 1.03E-10& 7.86E-05& 17.5186\\ 360& 699.132& 1.51E-12& 5.57E-07& 15.7219& 1,010.04& 7.99E-12& 3.98E-06& 16.8449& 1,615.58& 8.05E-11& 6.26E-05& 17.2812\\ 380& 699.1483& 9.20E-13& 3.50E-07& 15.3028& 1,010.62& 5.55E-12& 2.81E-06& 16.5597& 1,617.79& 6.35E-11& 5.01E-05& 17.0699\\ 400& 699.1561& 5.68E-13& 2.23E-07& 14.8185& 1,010.97& 3.89E-12& 2.01E-06& 16.3044& 1,619.25& 5.04E-11& 4.02E-05& 16.8818\\ 420& 699.1597& 3.54E-13& 1.45E-07& 14.2332& 1,011.19& 2.75E-12& 1.44E-06& 16.0669& 1,620.21& 4.02E-11& 3.25E-05& 16.7142\\ 440& 699.1615& 2.23E-13& 9.61E-08& 13.5181& 1,011.32& 1.96E-12& 1.04E-06& 15.836& 1,620.84& 3.23E-11& 2.63E-05& 16.5643\\ 460& 699.1623& 1.42E-13& 6.54E-08& 12.6581& 1,011.40& 1.40E-12& 7.55E-07& 15.6008& 1,621.26& 2.60E-11& 2.13E-05& 16.4297\\ 480& 699.1627& 9.20E-14& 4.59E-08& 11.6594& 1,011.45& 1.01E-12& 5.53E-07& 15.3508& 1,621.54& 2.10E-11& 1.73E-05& 16.3079\\ 500& 699.1629& 6.03E-14& 3.32E-08& 10.5547& 1,011.48& 7.30E-13& 4.07E-07& 15.076& 1,621.72& 1.70E-11& 1.42E-05& 16.1967\\ 520& 699.163& 4.03E-14& 2.49E-08& 9.4006& 1,011.50& 5.31E-13& 3.03E-07& 14.7669& 1,621.84& 1.38E-11& 1.16E-05& 16.094\\ 540& 699.163& 2.75E-14& 1.94E-08& 8.2657& 1,011.52& 3.88E-13& 2.27E-07& 14.4148& 1,621.93& 1.13E-11& 9.50E-06& 15.998\\ 560& 699.1631& 1.93E-14& 1.55E-08& 7.2141& 1,011.52& 2.85E-13& 1.71E-07& 14.0125& 1,621.98& 9.21E-12& 7.81E-06& 15.9067\\ 580& 699.1631& 1.39E-14& 1.28E-08& 6.2904& 1,011.53& 2.11E-13& 1.31E-07& 13.5547& 1,622.02& 7.55E-12& 6.44E-06& 15.8187\\ 600& 699.1631& 1.03E-14& 1.09E-08& 5.5149& 1,011.53& 1.56E-13& 1.01E-07& 13.0389& 1,622.04& 6.20E-12& 5.31E-06& 15.7321\\ 620& 699.1631& 7.90E-15& 9.40E-09& 4.8864& 1,011.53& 1.17E-13& 7.89E-08& 12.4665& 1,622.06& 5.10E-12& 4.40E-06& 15.6457\\ 640& 699.1631& 6.24E-15& 8.27E-09& 4.3891& 1,011.54& 8.79E-14& 6.24E-08& 11.8428& 1,622.07& 4.20E-12& 3.65E-06& 15.5578\\ 660& 699.1631& 5.06E-15& 7.36E-09& 4.0012& 1,011.54& 6.65E-14& 5.01E-08& 11.1779& 1,622.08& 3.47E-12& 3.03E-06& 15.4672\\ 680& 699.1631& 4.21E-15& 6.62E-09& 3.6999& 1,011.54& 5.08E-14& 4.07E-08& 10.4854& 1,622.08& 2.88E-12& 2.52E-06& 15.3725\\ 700& 699.1631& 3.58E-15& 6.00E-09& 3.4648& 1,011.54& 3.91E-14& 3.36E-08& 9.7818& 1,622.09& 2.38E-12& 2.11E-06& 15.2723\\ 720& 699.1631& 3.09E-15& 5.48E-09& 3.2789& 1,011.54& 3.04E-14& 2.82E-08& 9.0847& 1,622.09& 1.98E-12& 1.76E-06& 15.1653\\ 740& 699.1631& 2.70E-15& 5.02E-09& 3.1289& 1,011.54& 2.39E-14& 2.39E-08& 8.4111& 1,622.09& 1.65E-12& 1.48E-06& 15.0503\\ 760& 699.1631& 2.39E-15& 4.63E-09& 3.0049& 1,011.54& 1.90E-14& 2.06E-08& 7.7753& 1,622.09& 1.37E-12& 1.24E-06& 14.926\\ 780& 699.1631& 2.13E-15& 4.28E-09& 2.8996& 1,011.54& 1.53E-14& 1.79E-08& 7.1884& 1,622.09& 1.15E-12& 1.05E-06& 14.7912\\ 800& 699.1631& 1.91E-15& 3.96E-09& 2.8075& 1,011.54& 1.25E-14& 1.58E-08& 6.6572& 1,622.09& 9.59E-13& 8.84E-07& 14.6447\\ 820& 699.1631& 1.73E-15& 3.68E-09& 2.7249& 1,011.54& 1.03E-14& 1.40E-08& 6.1849& 1,622.09& 8.04E-13& 7.48E-07& 14.4854\\ 840& 699.1631& 1.56E-15& 3.43E-09& 2.6492& 1,011.54& 8.64E-15& 1.26E-08& 5.7711& 1,622.09& 6.74E-13& 6.36E-07& 14.3123\\ 860& 699.1631& 1.42E-15& 3.21E-09& 2.5784& 1,011.54& 7.32E-15& 1.14E-08& 5.4132& 1,622.09& 5.67E-13& 5.42E-07& 14.1244\\ 880& 699.1631& 1.30E-15& 3.00E-09& 2.5113& 1,011.54& 6.28E-15& 1.04E-08& 5.1066& 1,622.09& 4.77E-13& 4.63E-07& 13.921\\ 900& 699.1631& 1.18E-15& 2.81E-09& 2.447& 1,011.54& 5.46E-15& 9.47E-09& 4.846& 1,622.09& 4.03E-13& 3.97E-07& 13.7015\\ \end{matrix}$$
• Nice data! Wondering about the slight increase in average molecular weight per mole at 80 kilometers. Obviously at low altitudes it is a weighted average of those of nitrogen ($\approx 28$) and oxygen ($\approx 32$). Wonder what happens :-) Apr 21 at 6:54
• The data are merely what I quoted, and maybe they just assumed a constant MW at the lower compositions. Hauman et al. indicate that this is the mesosphere where ions form. I would guess these could include some triatomic species that are more stable as ions than as neutral molecules, such as $\text{NO}_2^+$. Apr 22 at 9:57