Back-of-the-Envelope Calculations: The Scale of the Atmosphere

Barbara Tewksbury
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Let's imagine a scale model of the Earth and use a basketball to represent the Earth. Now, let's get ourselves some packages of fruit roll-ups and start covering the basketball with layers of fruit roll-ups. How many layers would we have to cover the basketball with in order to make the stack of fruit roll-ups as thick as the Earth's atmosphere, to scale?

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Just one layer!! 99% of our atmosphere lies below the top of the stratosphere, which lies about 50 km above the Earth's surface (although our weather occurs in the troposphere, which is the lower 18 km of the atmosphere). The Earth has a radius of about 6,400 km. 50 km is roughly 0.8% of the radius of the Earth. A basketball is about 120 mm in radius. If we find 0.8% of 120 mm, we should have the scale distance of the top of the atmosphere above our basketball. If we multiply 120 mm x 0.008, we get just under 1 mm (that's about 4/100 of an inch!). A nice, chewy fruit roll-up is about a millimeter thick, maybe a bit less. Kind of a shock, isn't it? The atmosphere isn't very thick—at the scale of a basketball, all that's separating us from the vacuum of space is a single fruit roll-up! Makes you want to take good care of your air, doesn't it?

References and Resources

This SERC page describes the use of Back of the Envelope Calculations

A View from the Back of the Envelope (more info) : This site has a good number of easy simulations and visualizations of back of the envelope calculations.

The Back of the Envelope : This page outlines one of the essays in the book "Programming Pearls" (ISBN 0-201-65788-0). The book is written for computer science faculty and students, but this portion speaks very well to back of the envelope calculations in general.