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 $$ \begin{aligned} g(r) &=\frac{4}{3} \pi \rho_{c} G r \quad 0 \leq r \leq b \\ &=\frac{4}{3} \pi G\left[r \rho_{m}+b^{3}\left(\rho_{c}-\rho_{m}\right) / r^{2}\right] \quad b \leq r \leq a \end{aligned} $$

Short answer

Gravitational acceleration depends on enclosed mass and is derived using mass equations for core and mantle regions.

Step-by-step solution

Understand the Problem

We are given a model of a planet with two layers: a core with density \(\rho_c\) and radius \(b\), and a mantle with density \(\rho_m\) and thickness \(a-b\). We need to derive the expression for gravitational acceleration \(g(r)\) at any radius \(r\) within the planet. The solution is divided into two regions: \(0 \leq r \leq b\) and \(b \leq r \leq a\).

Gravitational Acceleration Inside the Core

For the region \(0 \leq r \leq b\), we use the formula for gravitational acceleration inside a sphere, which accounts only for the mass enclosed within radius \(r\). The mass \(M(r)\) enclosed within radius \(r\) is \(\frac{4}{3}\pi r^3 \rho_c\). Therefore, the gravitational acceleration is given by \(g(r) = \frac{GM(r)}{r^2} = \frac{G \cdot \frac{4}{3} \pi r^3 \rho_c}{r^2} = \frac{4}{3} \pi \rho_c G r\).

Gravitational Acceleration in the Mantle

For the region \(b \leq r \leq a\), the gravitational acceleration is affected by the mass of both the core and the mantle. The mass from the core is \(\frac{4}{3} \pi b^3 \rho_c\), and the mass from the mantle layer is the additional mass beyond \(b\) which contributes \((\frac{4}{3} \pi r^3 \rho_m - \frac{4}{3} \pi b^3 \rho_m)\). The gravitational acceleration is then composed of contributions from both the core and mantle: \(g(r) = \frac{G}{r^2}(\frac{4}{3}\pi b^3 \rho_c + \frac{4}{3}\pi (r^3 - b^3) \rho_m) = \frac{4}{3} \pi G [r \rho_m + b^3(\rho_c - \rho_m) / r^2]\).

Key concepts

Planetary Structure

When studying planets, it's useful to think about how they're constructed, just like layers of an onion. In our simple model, we focus on two main layers: the core and the mantle. The core is the central part of the planet, characterized by its density \( \rho_{c} \) and radius \( b \). Outside the core lies the mantle, which has its own distinct density \( \rho_{m} \) and stretches from radius \( b \) to \( a \).

These two layers combined make up the planet. This structured approach helps us to break down and analyze planetary properties like mass and gravitational forces, which can differ significantly between the different layers.

Density Distribution

Density is a critical factor in understanding the characteristics of planetary layers. In our model, the core and the mantle have distinct densities, \( \rho_{c} \) and \( \rho_{m} \). Density is defined as mass per unit volume, and it determines how much mass is packed into a given space.

The core's density \( \rho_{c} \) is often higher because it consists of heavier materials, while the mantle's density \( \rho_{m} \) is typically lower. Understanding these densities helps in calculating how mass is distributed across the planet, impacting various phenomena, including gravitational pull at different locations within the planet.

Two-Layer Model

A two-layer model simplifies the complexity of planetary structure by reducing it to just two distinct parts: the core and the mantle. This abstraction allows us to focus on key physical principles, such as how mass is distributed and gravitational forces behave across the planet.

In our model, the core is characterized by a uniform density and fixed radius \( b \), forming an inner sphere. The mantle, with its thickness \( a-b \), envelopes the core with its own density. This model is a useful simplification to illustrate fundamental concepts in planetary physics without the complexity of additional layers.

Mass Enclosure

Understanding how mass is enclosed at various radii within the planet is essential for calculating gravitational forces. The mass of any spherical section is based on how much material is inside that radius, often expressed in terms of density and volume.

For the core, the mass \( M(r) \) enclosed within a radius \( r \leq b \) is calculated as \( \frac{4}{3}\pi r^3 \rho_{c} \). In the mantle region \( b \leq r \leq a \), we must also include the core's total mass along with the mantle's contribution. Mass enclosure has practical applications like determining gravitational acceleration at different depths.

Gravitational Field Derivation

Calculating gravitational acceleration involves understanding how the mass within a planet's layers affects it. Inside the core (\( 0 \leq r \leq b \)), only the mass enclosed within \( r \) affects gravitational pull, expressed as \( g(r) = \frac{4}{3} \pi \rho_{c} G r \).

In the mantle (\( b \leq r \leq a \)), gravitational acceleration at any point \( r \) is influenced by both the core and the mantle. The derived formula \( g(r) = \frac{4}{3} \, \pi \, G [r \, \rho_{m} + \frac{b^{3} (\rho_{c} - \rho_{m})}{r^{2}}] \) shows this interaction, where \( G \) is the gravitational constant. Derivation like this is crucial in astrophysics for understanding planetary behaviors.