Problem 26 - An Introduction to Modern Astrophysics

Question

The \(r^{-2}\) dependence of Coulomb's electrostatic force law allows the construction of Gauss's law for electric fields, which has the form $$\oint \mathbf{E} \cdot d \mathbf{A}=\frac{Q_{\mathrm{in}}}{\epsilon_{0}},$$ where the integral is taken over a closed surface that bounds the enclosed charge, \(Q_{\text {in }}\). Because Newton's gravitational force law also varies as \(r^{-2},\) it is possible to derive a gravitational Gauss's law." The form of this gravitational version is $$\oint \mathbf{g} \cdot d \mathbf{A}=-4 \pi G M_{\mathrm{in}},$$ where the integral is over a closed surface that bounds the mass \(M_{\mathrm{in}},\) and \(\mathrm{g}\) is the local acceleration of gravity at the position of \(d \mathbf{A}\). The differential area vector \((d \mathbf{A})\) is assumed to be normal to the surface everywhere and is directed outward, away from the enclosed volume. Show that if a spherical gravitational Gaussian surface is employed that is centered on and surrounds a spherically symmetric mass distribution, Eq. ( 56 ) can be used to solve for g. The result is the usual gravitational acceleration vector around a spherically symmetric mass.

Step-by-Step Solution

Answer

Using a spherical Gaussian surface, \( g = -\frac{G M_{\mathrm{in}}}{r^2} \).

Step 1: Understanding the Given Gravitational Gauss's Law

The given equation for gravitational Gauss's law is: \( \oint \mathbf{g} \cdot d \mathbf{A} = -4 \pi G M_{\mathrm{in}} \). Here, \( \mathbf{g} \) is the gravitational field, \( d \mathbf{A} \) is an area element of the closed surface, \( M_{\mathrm{in}} \) is the mass enclosed by the surface, and \( G \) is the gravitational constant. The left side of the equation is the surface integral of the gravitational field vector over a closed surface.

Step 2: Choosing a Suitable Gaussian Surface

We consider a spherical Gaussian surface centered on the spherically symmetric mass distribution. Let the radius of this sphere be \( r \). This choice simplifies the problem since the gravitational field has the same magnitude at every point on the surface and is directed radially.

Step 3: Simplifying the Surface Integral

Since \( \mathbf{g} \) is uniform on the spherical surface and directed radially, the dot product simplifies to \( g \cdot dA = g \, dA \). The integral over the closed surface \( \oint \mathbf{g} \cdot d \mathbf{A} \) becomes \( g \oint dA = g \, (4 \pi r^2) \), where \( 4 \pi r^2 \) is the surface area of the sphere.

Step 4: Equating and Solving for Gravitational Field

From the simplified integral, we have \( g \, (4 \pi r^2) = -4 \pi G M_{\mathrm{in}} \). Solving for \( g \), we get: \[ g = -\frac{4 \pi G M_{\mathrm{in}}}{4 \pi r^2} = -\frac{G M_{\mathrm{in}}}{r^2}. \] The negative sign indicates that \( g \) is directed towards the center of the sphere, consistent with gravitational attraction.

Step 5: Conclusion: Form of Gravitational Acceleration

The derived equation \( g = -\frac{G M_{\mathrm{in}}}{r^2} \) shows that the gravitational acceleration around a spherically symmetric mass is directed towards the center and follows an inverse square law, similar to the familiar form of gravitational field. This result is consistent with the gravitational force law \( F = -\frac{G m_1 m_2}{r^2} \).

Key Concepts

Inverse Square Law

The Inverse Square Law is a fundamental principle often encountered in physics, describing how a force diminishes with increased distance from the source. Specifically, when a force or field emanates from a point source, its intensity decreases with the square of the distance from the source. This means if you double the distance from the source, the force becomes one-quarter as powerful.

Mathematically, this law is expressed as: \[ g = -\frac{GM_{\text{in}}}{r^2} \]where:

• \( g \) is the gravitational acceleration,
• \( G \) is the gravitational constant,
• \( M_{\text{in}} \) is the mass within the enclosing surface,
• \( r \) is the distance from the center of the mass.

This behavior is crucial in understanding both gravitational and electrostatic forces, as both follow this law, ensuring their forces weaken quickly as you move away from the source.

Gravitational Field

A gravitational field is an invisible force field surrounding a mass. It exerts a gravitational force on nearby objects, causing them to accelerate towards the mass's center. The concept of a field is used to help visualize and calculate these forces over a space.

In the equation:\[ g = -\frac{GM_{\text{in}}}{r^2} \]The gravitational field (\( g \)) determines how strong the gravitational pull is at a certain distance (\( r \)) from the mass center. The negative sign indicates that this force is attractive, pulling objects towards the mass creating the field.

• The stronger the field, the greater the pull on nearby objects.
• Closer objects experience stronger gravitational pull than those further away due to the inverse square relationship.

Understanding gravitational fields helps us predict planetary orbits, the behavior of celestial bodies, and other phenomena influenced by gravity.

Spherical Symmetry

Spherical symmetry refers to a system where conditions are uniform in all directions from a central point. In terms of gravitational fields, it simplifies calculations because the field's strength depends only on the distance from the center, not the direction.

Imagine the mass as perfectly distributed in a sphere. Due to spherical symmetry, we can select a spherical Gaussian surface with uniform properties. At any point on this surface, the gravitational field (\( g \)) has the same magnitude, making calculations straightforward.

• Spherical symmetry allows us to use simple models and equations that apply universally across the surface.
• It is especially useful in celestial mechanics and astrophysics for modeling planets and stars.

This symmetry means changes in gravitational force or field strength only happen with changes in distance, not direction.

Surface Integral

The concept of a surface integral involves integrating a field over a closed surface, capturing the total 'flux' through that surface. In our context, the flux refers to the flow of the gravitational field through a spherical surface.

Mathematically, the surface integral is written as:\[ \oint \mathbf{g} \cdot d\mathbf{A} = -4 \pi G M_{\text{in}} \]Where:

• \( \oint \) signifies integrating over a closed surface.
• \( \mathbf{g} \cdot d\mathbf{A} \) is the dot product denoting how the field interacts with the surface area.
• \( d\mathbf{A} \) is a differential area element pointing outward.

This equation signifies the total effect of the gravitational field interacting with the entire surface.

The surface integral is a crucial concept when applying Gauss's Law, allowing for the calculation of field behavior by considering a symmetrical closed surface, simplifying complex calculations into manageable form. Understanding surface integrals is key to leveraging Gauss's Law to solve problems in both electrostatics and gravitation.