# Earth Atmosphere Equation – Metric

## Atmosphere of Earth

The Earth’s atmosphere is an extremely thin sheet of air extending from the surface of the Earth to the edge of space. If the Earth were the size of a basketball, a tightly held pillowcase would represent the thickness of the atmosphere. Gravity holds the atmosphere to the Earth’s surface. Within the atmosphere, very complex chemical, thermodynamic, and fluid dynamics effects occur. The atmosphere is not uniform; fluid properties are constantly changing with time and place. We call this change the weather.

## Variations in Air Properties

Variations in air properties extend upward from the surface of the Earth. The sun heats the surface of the Earth, and some of this heat goes into warming the air near the surface. The heated air is then diffused or convected up through the atmosphere. Thus, the air temperature is highest near the surface and decreases as altitude increases. The speed of sound depends on the temperature and also decreases with increasing altitude. The pressure of the air can be related to the weight of the air over a given location. As we increase altitude through the atmosphere, there is some air below us and some air above us. But there is always less air above us than was present at a lower altitude. Therefore, air pressure decreases as we increase altitude. The air density depends on both the temperature and the pressure through the equation of state and also decreases with increasing altitude.

## Aerodynamic Force

Aerodynamic forces directly depend on the air density. To help rocket designers, it is useful to define a **standard atmosphere **model of the variation of properties through the atmosphere. There are actually several different models available–a standard or average day, a hot day, a cold day, and a tropical day. The models are updated every few years to include the latest atmospheric data. The model was developed from atmospheric measurements that were averaged and curve fit to produce the given equations. The model assumes that the pressure and temperature change only with altitude. The particular model shown here was developed in the early sixties, and the curve fits are given in Metric units. Curve fits are also available in English units.

## Model

The model has three zones with separate curve fits for the troposphere, the lower stratosphere, and the upper stratosphere. The troposphere runs from the surface of the Earth to 11,000 meters. In the troposphere, the temperature decreases linearly and the pressure decreases exponentially. The rate of temperature decrease is called the **lapse rate**. For the temperature **T** and the pressure **p**, the metric units curve fits for the troposphere are:

\(\LARGE T=15.04-.00649h\)

\(\LARGE p=101.29(\frac{T+273.1}{288.08})^{5.256}\)

where the temperature is given in Celsius degrees, the pressure in kilopascals, and **h** is the altitude in meters. The lower stratosphere runs from 11,000 meters to 25,000 meters. In the lower stratosphere the temperature is constant, and the pressure decreases exponentially. The metric units curve fits for the lower stratosphere are:

\(\LARGE T=-56.46\)

\(\LARGE p=22.65e^{1.73-.000157h}\)

## Upper Stratosphere Model

The upper stratosphere model is used for altitudes above 25,000 meters. In the upper stratosphere, the temperature increases slightly, and the pressure decreases exponentially. The metric units curve fits for the upper stratosphere are:

\(\LARGE T=-131.21+.00299h\)

\(\LARGE p=2.488(\frac{T+273.1}{216.6})^{-11.388}\)

In each zone the density **rho (**\(\bf\rho\)**)** is derived from the equation of state:

\(\LARGE \rho=\frac{p}{.2869(T+273.1)}\)

This is the atmosphere model used in the RocketModeler simulator. An interactive simulation for the atmosphere model is also available. With the applet, you can change altitude and see the effects on pressure and temperature.