# Percentage of Atmosphere Beneath Observatories

Quote of the Day

Truth … is much too complicated to allow anything but approximation.

John von Neumann. It took me a long time to accept that all models are wrong at some level, but that you can use them to produce useful results.

## Introduction

Figure 1: Observatories are usually placed on
remote mountaintops. Here is a picture of the
Sphinx Observatory (Source).

I often see popular descriptions of observatories that say things like the observatory "is above 40% of the Earth's atmosphere". I had not thought much about this kind of statement until I saw the Wikipedia's list of the world's highest-altitude observatories, which surprised me as to the height and remoteness of the largest telescopes.  I cannot imagine trying to build on these locations (Figure 1 is an extreme example). In some respects, the construction challenges remind me of what builders must have gone through on some lighthouses.

In this post, I will look at the highest altitude observatories and compute the percentage of atmosphere that they are above.

## Background

The atmospheric pressure at a location is a measure of the weight of air above that location. We can determine the percentage of the air column below a given altitude by computing the ratio of p(h)/p(0), which represents the percentage of atmosphere above altitude h , and subtracting that ratio from 100%.

 Eq. 1 $\displaystyle \%\text{AtmoBelow(h)}=100\%-\frac{{p(h)}}{{p(0)}}$

where

• %AtmoBelow(h) is the percentage of atmosphere below the altitude h.
• p(h) is the atmospheric pressure at altitude h.
• p(0) is the atmospheric pressure at sea level, h= 0.

In this post, I will:

• Verify some general statements about the amount of atmosphere below certain altitudes.
• Determine the amount of atmosphere below the world's highest observatories.

## Analysis

### Atmospheric Pressure Curve Fit

To estimate the barometric pressure at different altitudes, I grabbed a table of pressures from the web and did a simple interpolation so that my function, p(h), is continuous (Figure 2).

Figure 2: Mathcad Interpolation of Barometric Pressure.

### Web References

Table 1 shows three examples of references to the percentage of atmosphere below that reference level. Note that the reference for Everest (marked in red) got their percentage number wrong by ~10%.

 Statement Altitude (m) Stated Air % Below My Air % Below Mauna Kea rises 9,750 meters from the ocean floor to an altitude of 4,205 meters above sea level, which places its summit above 40 percent of the Earth's atmosphere (Source). 4,205 40 40.8% 57.8 percent of the atmosphere is below the summit of Mount Everest (Source). 8,848 57.8 68.9 72 percent of the atmosphere is below the common cruising altitude of commercial airliners (about 10,000 m) (Source). 10,000 72 73.8

Figure 3 shows how the calculation was performed using Equation 1.

Figure 3: Example Calculations.

### World's Highest Observatories

The Wikipedia has a list of the highest observatories in the world. I used Equation 1 to find the percentage of atmosphere below each observatory in Table 2. The University of Tokyo Atacama Observatory is unbelievably high – that has to be a challenge for those who work there.

 Observatory Name Elev.(m) Air % Below Observatory Site Location University of Tokyo Atacama Observatory (TAO) 5,640 51.0% Cerro Chajnantor Atacama Desert, Chile Chacaltaya Astrophysical Observatory 5,230 48.3% Chacaltaya Andes, Bolivia James Ax Observatory 5,200 48.0% Cerro Toco Atacama Desert, Chile Atacama Cosmology Telescope 5,190 48.0% Cerro Toco Atacama Desert, Chile Llano de Chajnantor Observatory 5,104 47.4% Llano de Chajnantor Atacama Desert, Chile Shiquanhe Observatory (NAOC Ali Observatory) 5,100 47.4% Shiquanhe, Ngari Plateau Tibet Autonomous Region, China Llano de Chajnantor Observatory 4,800 45.2% Pampa La Bola Atacama Desert, Chile Large Millimeter Telescope Alfonso Serrano 4,580 43.6% Sierra Negra Puebla, Mexico Indian Astronomical Observatory 4,500 43.0% Hanle Ladakh, India Meyer-Womble Observatory 4,312 41.6% Mount Evans Colorado, United States Yangbajing International Cosmic Ray Observatory 4,300 41.5% Yangbajain Tibet Autonomous Region, China Mauna Kea Observatory 4,190 40.7% Mauna Kea Hawaii, United States High-Altitude Water Cherenkov (HAWC) Gamma-Ray Observatory 4,100 40.0% Sierra Negra Puebla, Mexico Barcroft Observatory 3,890 38.3% White Mountain Peak California, United States Very Long Baseline Array (VLBA), Mauna Kea Site 3,730 37.0% Mauna Kea Hawaii, United States Llano del Hato National Astronomical Observatory 3,600 36.0% Llano del Hato Andes, Venezuela Sphinx Observatory 3,571 35.7% Jungfraujoch Bernese Alps, Switzerland Mauna Loa Observatory 3,394 34.3% Mauna Loa Hawaii, United States Magdalena Ridge Observatory 3,230 32.8% South Baldy New Mexico, United States Mount Graham International Observatory 3,191 32.5% Mount Graham Arizona, United States Gornergrat Observatory 3,135 32.0% Gornergrat Pennine Alps, Switzerland European Extremely Large Telescope 3,060 31.4% Cerro Armazones Atacama Desert, Chile Haleakala Observatory 3,036 31.2% Haleakala Hawaii, United States

## Conclusion

Just a quick note to explain where some of these atmosphere percentage numbers come from. I became interested in this after watching the movie Everest, which did an excellent job showing the effect of high altitudes on people.

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### One Response to Percentage of Atmosphere Beneath Observatories

1. Ronan Mandra says:

Wiki has a formula for air pressure based on height and other atmospheric factors,
ref: https://en.wikipedia.org/wiki/Atmospheric_pressure
which works up to 44,331 meters