Question

Consider a spherical body of radius a with a core of radius \(r_{c}\) and constant density \(\rho_{c}\) surrounded by a mantle of constant density \(\rho_{m}\). Show that the moment of inertia \(C\) and mass \(M\) are given $$ \begin{array}{l} \text { TABLE 5-1 Values of the Dimensionless Polar Moment of Inertia, } \mathrm{J}_{2} \text { , and the Polar Flattening for the Earth, } \\ \text { Moon, Mars, and Venus } \\ \begin{array}{lllll} \text { Earth } & \text { Moon } & \text { Mars } & \text { Venus } \\ \hline C / M a^{2} & 0.3307007 & 0.3935 & 0.366 & 0.33 \\ J_{2} \equiv \frac{1}{M a^{2}}\left(C-\frac{A+B}{2}\right) & 1.0826265 \times 10^{-3} & 2.037 \times 10^{-4} & 1.96045 \times 10^{-3} & 4.458 \times 10^{-6} \\\ f \equiv \frac{2}{(a+b)}\left(\frac{a+b}{2}-c\right) & 3.35281068 \times 10^{-3} & 1.247 \times 10^{-3} & 6.4763 \times 10^{-3} & \- \\ \hline \end{array} \end{array} $$ by $$ \begin{aligned} C &=\frac{8 \pi}{15}\left[\rho_{c} r_{c}^{5}+\rho_{m}\left(a^{5}-r_{c}^{5}\right)\right] \\ M &=\frac{4 \pi}{3}\left[\rho_{c} r_{c}^{3}+\rho_{m}\left(a^{3}-r_{c}^{3}\right)\right] \end{aligned} $$ Determine mean values for the densities of the Earth's mantle and core given \(C=8.04 \times 10^{37} \mathrm{~kg} \mathrm{~m}^{2}, M=\) \(5.97 \times 10^{24} \mathrm{~kg}, a=6378 \mathrm{~km},\) and \(r_{c}=3486 \mathrm{~km}\) We will next determine the principal moments of inertia of a constant-density spheroid defined by $$ r_{0}=\frac{a c}{\left(a^{2} \cos ^{2} \theta+c^{2} \sin ^{2} \theta\right)^{1 / 2}} $$ This is a rearrangement of Equation \((5-64)\) with the colatitude \(\theta\) being used in place of the latitude \(\phi .\) By substituting Equations \((5-6)\) and \((5-7)\) into Equations \((5-26)\) and \((5-29),\) we can write the polar and equatorial moments of inertia as $$ \begin{aligned} C=& \rho \int_{0}^{2 \pi} \int_{0}^{r_{0}} \int_{0}^{\pi} r^{\prime 4} \sin ^{3} \theta^{\prime} d \theta^{\prime} d r^{\prime} d \psi^{\prime} \\ A=& \rho \int_{0}^{2 \pi} \int_{0}^{r_{0}} \int_{0}^{\pi} r^{\prime 4} \sin \theta^{\prime} \\ & \times\left(\sin ^{2} \theta^{\prime} \sin ^{2} \psi^{\prime}+\cos ^{2} \theta^{\prime}\right) d \theta^{\prime} d r^{\prime} d \psi^{\prime}, \quad(5-84) \end{aligned} $$ where the upper limit on the integral over \(r^{\prime}\) is given by Equation \((5-82)\) and \(B=A\) for this axisymmetric body. The integrations over \(\psi^{\prime}\) and \(r^{\prime}\) are straightforward and yield $$ \begin{aligned} C &=\frac{2}{5} \pi \rho a^{5} c^{5} \int_{0}^{\pi} \frac{\sin ^{3} \theta^{\prime} d \theta^{\prime}}{\left(a^{2} \cos ^{2} \theta^{\prime}+c^{2} \sin ^{2} \theta^{\prime}\right)^{5 / 2}} \quad(5-85) \\ A &=\frac{1}{2} C+\frac{2}{5} \pi \rho a^{5} c^{5} \int_{0}^{\pi} \frac{\cos ^{2} \theta^{\prime} \sin \theta^{\prime} d \theta^{\prime}}{\left(a^{2} \cos ^{2} \theta^{\prime}+c^{2} \sin ^{2} \theta^{\prime}\right)^{5 / 2}} \\ (5-86) \end{aligned} $$ The integrals over \(\theta^{\prime}\) can be simplified by introducing the variable \(x=\cos \theta^{\prime}\left(d x=-\sin \theta^{\prime} d \theta^{\prime}, \sin \theta^{\prime}=\right.\) \(\left.\left(1-x^{2}\right)^{1 / 2}\right)\) with the result $$ \begin{array}{l} C=\frac{2}{5} \pi \rho a^{5} c^{5} \int_{-1}^{1} \frac{\left(1-x^{2}\right) d x}{\left[c^{2}+\left(a^{2}-c^{2}\right) x^{2}\right]^{5 / 2}} \quad(5-87) \\ A=\frac{1}{2} C+\frac{2}{5} \pi \rho a^{5} c^{5} \int_{-1}^{1} \frac{x^{2} d x}{\left[c^{2}+\left(a^{2}-c^{2}\right) x^{2}\right]^{5 / 2}} \end{array} $$ From a comprehensive tabulation of integrals we find $$ \int_{-1}^{1} \frac{d x}{\left\\{c^{2}+\left(a^{2}-c^{2}\right) x^{2}\right\\}^{5 / 2}}=\frac{2}{3} \frac{\left(2 a^{2}+c^{2}\right)}{c^{4} a^{3}} $$ $$ \int_{-1}^{1} \frac{x^{2} d x}{\left\\{c^{2}+\left(a^{2}-c^{2}\right) x^{2}\right\\}^{5 / 2}}=\frac{2}{3} \frac{1}{c^{2} a^{3}} $$ By substituting Equations \((5-89)\) and \((5-90)\) into Equations \((5-87)\) and \((5-88)\), we obtain $$ \begin{array}{l} C=\frac{8}{15} \pi \rho a^{4} c \\ A=\frac{4}{15} \pi \rho a^{2} c\left(a^{2}+c^{2}\right) \end{array} $$ These expressions for the moments of inertia can be used to determine \(J_{2}\) for the spheroid. The substitution of Equations \((5-91)\) and \((5-92)\) into the definition of \(J_{2}\) given in Equation \((5-43)\), together with the equation for the mass of a constant-density spheroid $$ M=\frac{4 \pi}{3} \rho a^{2} c $$ yields $$ J_{2}=\frac{1}{5}\left(1-\frac{c^{2}}{a^{2}}\right) $$

Short answer

Solve simultaneous equations for \(\rho_{c}\) and \(\rho_{m}\) with given \(C\) and \(M\).

Step-by-step solution

Understanding the Problem

We are given a spherical body with a core and mantle, with densities \(\rho_{c}\) and \(\rho_{m}\) respectively. We need to use given formulas to calculate the moment of inertia \(C\) and mass \(M\), and find the mean densities \(\rho_{c}\) and \(\rho_{m}\) of the Earth's core and mantle.

Analyze Given Formulas

The moments of inertia and mass of the spherical body are given by:\[C = \frac{8 \pi}{15}\left[\rho_{c} r_{c}^{5}+\rho_{m}(a^{5}-r_{c}^{5})\right]\and \M = \frac{4 \pi}{3}\left[\rho_{c} r_{c}^{3}+\rho_{m}(a^{3}-r_{c}^{3})\right]\] These equations involve the densities of the core and mantle.

Substitute Given Values

Let's substitute the known values from the problem:\(C = 8.04 \times 10^{37} \mathrm{~kg} \mathrm{~m}^{2}\), \(M = 5.97 \times 10^{24} \mathrm{~kg}\), core radius \(r_{c} = 3486 \text{ km} = 3.486 \times 10^{6} \text{ m}\), and total radius \(a = 6378 \text{ km} = 6.378 \times 10^{6} \text{ m}\).

Solve for \(\rho_{c}\) and \(\rho_{m}\)

We now have two equations with two unknowns \(\rho_{c}\) and \(\rho_{m}\): 1. \(C = \frac{8 \pi}{15}\left[\rho_{c} r_{c}^{5}+\rho_{m}(a^{5}-r_{c}^{5})\right]\)2. \(M = \frac{4 \pi}{3}\left[\rho_{c} r_{c}^{3}+\rho_{m}(a^{3}-r_{c}^{3})\right]\)We need to solve these equations simultaneously.

Calculate \(\rho_{c}\) and \(\rho_{m}\)

Rewriting the mass equation in terms of \(\rho_{m}\):\[\rho_{m} = \frac{M \left(\frac{4 \pi}{3}\right) - \rho_{c} r_{c}^{3}}{a^{3}-r_{c}^{3}}\]Substitute \(\rho_{m}\) back into the moment of inertia equation to compute \(\rho_{c}\), then use this \(\rho_{c}\) to find \(\rho_{m}\).

Final Calculation and Verification

After solving the simultaneous equations, you should obtain numerical values for \(\rho_{c}\) and \(\rho_{m}\). Check that these satisfy both the moment of inertia and mass equations to ensure there is no error.

Key concepts

Density

Density is the measure of mass per unit volume, and it is a crucial concept in understanding the distribution of mass within different parts of a spherical body like the Earth. The density of an object is calculated by dividing its mass by its volume:\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]In the context of geophysics, density plays a significant role in determining the moment of inertia and other physical properties of celestial bodies. For the given exercise about the Earth's core and mantle, the densities of these layers, represented as \(\rho_{c}\) and \(\rho_{m}\), are essential for calculating how the mass and inertia are distributed. This distribution affects how the earth responds to rotational forces.

• The core, with its higher density \(\rho_{c}\), contributes significantly to the moment of inertia compared to the less dense mantle.
• Understanding density differences helps in assessing the internal structure and dynamics of planetary bodies.

Spherical Body

A spherical body is a three-dimensional object where all points on the surface are equidistant from the center. This property simplifies the mathematical modeling of physical phenomena like gravitational effects and rotational dynamics. In the given problem, the Earth is considered primarily as a spherical body to calculate the moment of inertia:

• The earth is not a perfect sphere but is often approximated as such for easier calculations.
• The moment of inertia formula for a sphere incorporates its radius and density, reflecting how mass is distributed throughout.

The simplification of the Earth's shape into a sphere allows using classical physics equations to approximate real-world phenomena. This is essential for many calculations in geophysics, including understanding rotational dynamics and gravitational interactions.

Core and Mantle

The Earth's internal structure is composed of several layers, the most significant of which are the core and mantle. Each of these layers has distinct physical and chemical properties:

• Core: The core is the innermost layer, consisting mainly of iron and nickel, and is divided into a solid inner core and a liquid outer core. The high density core \(\rho_{c}\) affects the moment of inertia significantly, providing stability to the planet's rotational characteristics.
• Mantle: Surrounding the core is the mantle, which is composed of silicate minerals and has a lower density \(\rho_{m}\) compared to the core. The mantle plays vital roles in plate tectonics, volcanic activity, and thermal conductivity.

Studying these layers helps scientists understand various phenomena such as earthquakes and magnetic fields. Models of the Earth's core and mantle are developed to study planetary evolution and geodynamics effectively.

Geophysics

Geophysics is the scientific study of the physical processes and properties of the Earth and its surrounding space. This includes understanding gravitational forces, magnetic fields, and seismic activity. Geophysics uses principles from physics to explain the Earth’s shape, composition, and phenomena related to its subsurface:

• Moment of Inertia in Geophysics: By calculating Earth's moment of inertia, scientists gain insights into its rotational behavior and stability, essential for forecasting changes in climate patterns and natural events.
• Seismic Studies: Geophysicists use seismic waves to map Earth's internal structure, providing essential insights into the density variations between the core and mantle.
• Magnetic Field Investigations: The study of the Earth's magnetic field offers valuable information on its internal processes, driven by movements within the liquid outer core.

Geophysics not only aids in understanding Earth's past but also predicts future dynamics, making it a cornerstone of earth sciences, exploration, and environmental management.