Problem 104 - Thomas Calculus

Question

Mass of planet's atmosphere \(A\) spherical planet of radius \(R\) has an atmosphere whose density is \(\mu=\mu_{0} e^{-c h},\) where \(h\) is the altitude above the surface of the planet, \(\mu_{0}\) is the density at sea level, and \(c\) is a positive constant. Find the mass of the planet's atmosphere.

Step-by-Step Solution

Answer

Integrate the density function over sphere shells from base to infinite height to get the total mass of the atmosphere.

Step 1: Understand the Problem

We need to find the total mass of a planet's atmosphere given that the density decreases exponentially with altitude. The planet is spherical with a known radius \(R\), and the density at any height \(h\) above the surface is given by \(\mu = \mu_0 e^{-c h}\).

Step 2: Set up the Integral

The atmosphere can be thought of as a series of thin spherical shells, each with thickness \(dh\) and radius \(R + h\). The mass of each shell is density \(\times\) volume. Thus, the mass of a small shell at height \(h\) is \(dM = \mu(h) \cdot 4\pi(R + h)^2 dh\).

Step 3: Substitute the Density Function

Substitute the given density function \(\mu = \mu_0 e^{-c h}\) into the expression for \(dM\):\[dM = \mu_0 e^{-c h} \cdot 4\pi(R + h)^2 dh\]

Step 4: Integrate to Find Total Mass

Integrate \(dM\) from \(h=0\) to \(h=\infty\):\[M = \int_{0}^{\infty} 4\pi \mu_0 e^{-c h} (R+h)^2 \, dh\]This expression represents the total mass of the atmosphere.

Step 5: Evaluate the Integral

The integral is evaluated by substituting \(x = R + h\), which simplifies it as follows:\[M = 4\pi \mu_0 \int_{R}^{\infty} e^{-c (x - R)} x^2 \, dx\] This integral can then be solved using integration techniques like integration by parts or by looking up in integral tables as it represents a standard form after substitution.

Key Concepts

Exponential Density Function

The exponential density function is a mathematical expression that models how density decreases with altitude in this context. In the problem, the density of the atmosphere, \( \mu \), is given by \( \mu = \mu_0 e^{-ch} \), where \( \mu_0 \) is the density at sea level, \( c \) is a positive constant, and \( h \) is the altitude. This form is particularly useful because it captures the real-world phenomenon where atmospheric density decreases exponentially as we move away from the planet's surface.
Understanding this function is crucial because it helps predict how light or heavy the atmosphere will be at different heights without actually measuring it. This kind of mathematical model simplifies complex predictions in atmospheric science and engineering. It also provides the foundation for calculating the mass of the atmosphere, as we will see when integrating this function across different layers of the atmosphere.

Spherical Shells

A spherical shell is a three-dimensional shape resembling the hollow part of a sphere. To understand how atmospheric calculations use spherical shells, imagine concentric layers of air wrapping the planet. Each layer is like a thin shell with a slight thickness, \( dh \).
As the altitude increases, these shells have a radius of \( R + h \), where \( R \) is the radius of the planet and \( h \) is the height above the surface. The volume of each thin shell is given by the formula for the surface area of a sphere multiplied by its thickness: \( 4\pi (R + h)^2 dh \).
Calculating the mass of each shell is straightforward: multiply its volume by the density of the atmosphere at that height, \( \mu = \mu_0 e^{-ch} \). This calculation is what allows us to find the mass for each shell as \( dM = \mu(h) \cdot 4\pi (R + h)^2 dh \). Using spherical shells simplifies the understanding of the distribution and mass of the atmosphere.

Integration Techniques

Integration techniques are mathematical methods used to calculate the integral of a function, which in this case, helps us find the total mass of the atmosphere. To get the total mass, we sum up the mass of all individual spherical shells from the surface to infinity. This is represented by the integral \[ M = \int_{0}^{\infty} 4\pi \mu_0 e^{-c h} (R + h)^2 \, dh \].
A common way to evaluate this kind of integral is by changing variables or substituting, known as a substitution method. By letting \( x = R + h \), the limits of integration change, and the expression becomes easier to handle: \[ M = 4\pi \mu_0 \int_{R}^{\infty} e^{-c (x - R)} x^2 \, dx \].
At this point, advanced techniques like integration by parts or consulting integral tables may be used to solve the integral. These methods are invaluable as they allow for the computation of complex real-world problems like the mass of an atmosphere in a mathematically rigorous way.

Altitude-Dependent Density

Altitude-dependent density describes how the density of the atmosphere changes with altitude. In our problem, density is not constant; it decreases exponentially as the altitude increases. This characteristic is shown in the density function \( \mu = \mu_0 e^{-ch} \).
Altitude-dependent density is an essential consideration for atmospheric science, as it influences everything from weather patterns to aircraft design. As air density decreases with altitude, it affects parameters like air pressure and temperature, which are crucial for various calculations in physics and engineering.
When we calculate the mass of an atmosphere, we take this variation into account by integrating over all altitudes, considering how each thin shell's density changes. This approach ensures that our calculation of the atmospheric mass is both accurate and reflective of how the atmosphere behaves in reality.