Question

Consider a simple two-layer model of a planet consisting of a core of density \(\rho_{c}\) and radius \(b\) surrounded by a mantle of density \(\rho_{m}\) and thickness \(a-b\). Show that the gravitational acceleration as a function of radius is given by $$\begin{aligned} g(r) &=\frac{4}{3} \pi \rho_{c} G r \quad 0 \leq r \leq b \\ &=\frac{4}{3} \pi G\left[r \rho_{m}+b^{3}\left(\rho_{c}-\rho_{m}\right) / r^{2}\right] \quad b \leq r \leq a \end{aligned}$$ and that the pressure as a function of radius is given by $$\begin{aligned} p(r)=& \frac{4}{3} \pi \rho_{m} G b^{3}\left(\rho_{c}-\rho_{m}\right)\left(\frac{1}{r}-\frac{1}{a}\right) \\ &+\frac{2}{3} \pi G \rho_{m}^{2}\left(a^{2}-r^{2}\right) \quad b \leq r \leq a \\\ =& \frac{2}{3} \pi G \rho_{c}^{2}\left(b^{2}-r^{2}\right)+\frac{2}{3} \pi G \rho_{m}^{2}\left(a^{2}-b^{2}\right) \\ &+\frac{4}{3} \pi \rho_{m} G b^{3}\left(\rho_{c}-\rho_{m}\right)\left(\frac{1}{b}-\frac{1}{a}\right) \\ 0 \leq r \leq b \end{aligned}$$ Apply this model to the Earth. Assume \(\rho_{m}=\) \(4000 \mathrm{~kg} \mathrm{~m}^{-3}, b=3486 \mathrm{~km}, a=6371 \mathrm{~km}\) Calculate \(\rho_{c}\) given that the total mass of the Earth is \(5.97 \times 10^{24} \mathrm{~kg} .\) What are the pressures at the center of the Earth and at the core-mantle boundary? What is the acceleration of gravity at \(r=b ?\)

Short answer

Calculate core density \( \rho_c \), find pressures at \( r = 0 \) and \( r = b \), and gravitational acceleration at \( r = b \).

Step-by-step solution

Define Variables and Constants

Given the Earth model, we have the core density \( \rho_c \), mantle density \( \rho_m = 4000 \, \text{kg/m}^3 \), core radius \( b = 3486 \, \text{km} \) (convert to m by multiplying by 1000), and total radius \( a = 6371 \, \text{km} \). The total mass of the Earth \( M = 5.97 \times 10^{24} \, \text{kg} \). Also, the gravitational constant is \( G = 6.674 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2} \).

Calculate the Volume of Core and Mantle

The volume of the core is \( V_{core} = \frac{4}{3} \pi b^3 \) and the volume of the mantle is \( V_{mantle} = \frac{4}{3} \pi (a^3 - b^3) \). These will be used to calculate the mass of the core and mantle.

Express Total Mass of Earth

Using the total mass of the Earth, we can write \( M = \rho_c V_{core} + \rho_m V_{mantle} \). Substitute the expressions for the volumes from the previous step to find \( \rho_c \).

Solve for Core Density \( \rho_c \)

Rearranging the equation for total mass, express \( \rho_c = \frac{M - \rho_m V_{mantle}}{V_{core}} \). Substitute all known values to calculate \( \rho_c \).

Calculate Gravitational Acceleration at Core-Mantle Boundary (r = b)

Use the given formula \( g(r) = \frac{4}{3} \pi G \left( r \rho_m + b^{3}(\rho_c-\rho_m) / r^{2} \right) \) for \( b \leq r \leq a \). Substitute \( r = b \) and all known values to calculate \( g(b) \).

Calculate Pressure at the Core-Mantle Boundary and Center

For \( r = b \), use the formula given for pressure for both regions. Calculate \( p(b) \) using the formula for \( b \leq r \leq a \). To find \( p(0) \) at the center, take the expression for \( 0 \leq r \leq b \) and substitute \( r = 0 \).

Simplify Pressure Formula for Specific Ranges

For \( b \leq r \leq a \), simplify: \( p(b) = \frac{4}{3} \pi G \rho_m b^3 (\rho_c-\rho_m) (\frac{1}{b}-\frac{1}{a}) + \frac{2}{3} \pi G \rho_m^2 (a^2-b^2) \). For \( 0 \leq r \leq b \), simplify: \( p(0) = \frac{2}{3} \pi G \rho_c^2 b^2 + \frac{2}{3} \pi G \rho_m^2 (a^2-b^2) + \frac{4}{3} \pi \rho_m G b^3 (\rho_c-\rho_m)(\frac{1}{b}-\frac{1}{a}) \).

Calculate and Compare Pressures and Gravitational Acceleration

Substitute all known values into the simplified pressure equations to find \( p(b) \) and \( p(0) \). Ensure the calculated pressures and gravitational acceleration match expectations or boundaries for the Earth model.

Key concepts

Gravitational Acceleration

Gravitational acceleration gives us an idea of how much force gravity exerts on an object relative to the object's distance from the center of the planet. In our two-layer model of Earth, with a dense core and a less dense mantle, this property changes as we move from the core to the surface. For 0 \(\leq r \leq b\), where \(r\) is the radial distance and \(b\) is the core radius, gravitational acceleration at any point is determined by just the core's contribution:

• It is given by the formula: \(g(r) = \frac{4}{3} \pi \rho_c G r\). Here, \(\rho_c\) is the core's density and \(G\) is the gravitational constant.

As we move beyond the core into the mantle (\(b \leq r \leq a\), where \(a\) is Earth's total radius), the formula adds the mantle's gravitational contribution:

• This expression becomes \(g(r) = \frac{4}{3} \pi G\left[r \rho_m + b^{3}(\rho_c-\rho_m) / r^{2}\right]\), where \(\rho_m\) is the mantle's density.

At the core-mantle boundary (\(r = b\)), there is a transition where core and mantle contributions meet. Calculating gravitational acceleration here helps in understanding dynamical processes within the Earth.

Core Density Calculation

Understanding core density, \(\rho_c\), is crucial as it influences Earth's overall mass and gravitational behaviors. We start by considering the total mass of the Earth, which combines masses from both core and mantle parts:

• The Earth's total mass, \(M\), is given by: \(M = \rho_c V_{core} + \rho_m V_{mantle}\).

The core volume, \(V_{core}\), is calculated using the formula for a sphere's volume: \(V_{core} = \frac{4}{3} \pi b^3\), while the mantle volume is \(V_{mantle} = \frac{4}{3} \pi (a^3 - b^3)\).

• Rearranging the mass equation to solve for core density results in: \(\rho_c = \frac{M - \rho_m V_{mantle}}{V_{core}}\).

By substituting known values such as Earth’s total mass and densities, you can find \(\rho_c\). Ensuring the accuracy of this calculation helps in modeling Earth's internal structure correctly.

Earth's Pressure Distribution

Pressure within Earth varies significantly from the surface to the center. In our simple two-layer model, pressure is influenced by gravitational acceleration and material density. Deeper layers experience higher pressures due to the weight of overlying materials.

• For 0 \(\leq r \leq b\), the pressure is calculated with: \(p(r) = \frac{2}{3} \pi G \rho_c^2(b^2 - r^2) + \frac{2}{3} \pi G \rho_m^2 (a^2-b^2) + \frac{4}{3} \pi \rho_m Gb^3 (\rho_c-\rho_m)(\frac{1}{b}-\frac{1}{a})\).

Closer to the core-mantle boundary (\(b \leq r \leq a\)), the formula adjusts to:

• \(p(r) = \frac{4}{3} \pi \rho_m G b^3 (\rho_c-\rho_m)(\frac{1}{r}-\frac{1}{a}) + \frac{2}{3} \pi G \rho_m^2 (a^2-r^2)\).

Finding the pressure at the center, \(p(0)\), helps us understand the conditions the Earth’s core materials endure. Additionally, understanding pressure distribution is fundamental when studying geophysics and Earth's geological processes.

Two-Layer Planet Model

The two-layer planet model simplifies Earth into two main parts: a dense core and a less dense mantle. This model allows for greater ease in calculations related to mass, density, and pressure. While Earth is more complex, this model captures key physical behaviors.

• Core: It is characterized by the core density \(\rho_c\) and radius \(b\).
• Mantle: Defined by its own density \(\rho_m\) and the region from \(b\) to the overall radius \(a\).

This approach is beneficial for studying how different layers of a planet affect its gravitational field and pressure distribution. By using known values and gravitational formulas, scientists make predictions about internal Earth dynamics. Simplified models are valuable for initial studies and gaining insights into otherwise complex planetary structures.