Читать книгу Introduction to Flight Testing - James W. Gregory - Страница 32

As we proceed with the development of the standard atmosphere, we must consider how gravitational acceleration varies with altitude. From Newton's law of universal gravitation, we know that gravitational acceleration varies inversely with the square of the distance to the center of the earth. Thus, we have

(2.4)

where g is the local gravitational acceleration (varies with altitude), g₀ is the gravitational acceleration at sea level (9.806 65 m/s² or 32.174 ft/s²), h_A is the distance from the center of the earth (defined here as the absolute altitude²), and r_Earth is Earth's mean radius, which is 6356.766 km (NOAA et al. 1976).

Despite the fact that gravity varies with altitude, it is convenient to derive the standard atmosphere based on the assumption of constant gravitational acceleration. In order to do so, we must define a new altitude, the geopotential altitude, h, which we will use in the hydrostatic equation with the assumption of constant gravity. Referring to Eq. (2.3), we can also write the hydrostatic equation as a function of geopotential altitude and constant gravitational acceleration,

(2.5)

Taking the ratio of (2.5) and (2.3), we have

(2.6)

since the differential pressure and density terms cancel out for a given change of pressure. The small difference between g₀ and g then leads to a small difference between the geopotential and geometric altitudes. Combining Eqs. (2.4) and (2.6) produces

(2.7)

which can be integrated between sea level and an arbitrary altitude to find

(2.8)

This expression defines the relationship between geopotential altitude, h, and geometric altitude, h_G, which can also be solved for geometric altitude,

(2.9)

In our derivation of the standard atmosphere, we will use geopotential altitude, h, and assume constant g₀. Properties of the standard atmosphere such as temperature, pressure, and density, i.e., (T, p, ρ), will be found as a function of geopotential altitude, h, and then mapped back to geometric altitude, h_G, by Eq. (2.9). In this work, we are focused on the lower portions of the atmosphere where most aircraft fly (h ≤ 20 km or 65, 617 ft). At that upper altitude limit, Eq. (2.9) predicts a maximum difference of 0.31% between the geometric and geopotential altitude. Thus, in many cases related to flight testing, this difference between geopotential and geometric altitudes can be neglected.