4.3 Newtonian potential and gravitation

Newtonian potential and gravitation form the foundation of classical gravity. These concepts describe how mass distributions create gravitational fields, allowing us to calculate forces, energies, and orbits in various systems.

From Gauss's law to Poisson's equation, these principles explain gravitational phenomena on Earth and in space. Understanding multipole expansions, self-energy, and tidal forces helps us analyze complex gravitational interactions in astrophysics and beyond.

Newtonian potential

• Fundamental concept in classical gravity that describes the gravitational influence of a mass distribution
• Scalar field that determines the gravitational potential energy of a test mass
• Enables calculation of gravitational forces and fields through spatial derivatives

Definition of potential

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• Gravitational potential $V(\vec{r})$ at a point $\vec{r}$ due to a mass distribution $\rho(\vec{r}')$
• Given by the integral: $V(\vec{r}) = -G \int \frac{\rho(\vec{r}')}{|\vec{r}-\vec{r}'|} d^3\vec{r}'$
  • $G$ is the gravitational constant
  • Integration is performed over the volume of the mass distribution

Fundamental properties

• Gravitational potential is a scalar field, meaning it assigns a single value to each point in space
• Potential is always negative, with a minimum at the center of mass
• Potential approaches zero at infinity, setting the reference point for gravitational potential energy
• Potential is additive for multiple mass distributions (superposition principle)

Relation to gravitational field

• Gravitational field $\vec{g}(\vec{r})$ is the negative gradient of the potential: $\vec{g}(\vec{r}) = -\nabla V(\vec{r})$
• Field points in the direction of the steepest decrease in potential
• Magnitude of the field is proportional to the rate of change of potential
• Gravitational force on a test mass $m$ is given by $\vec{F} = m\vec{g}$

Gravitational potential energy

• Energy associated with the configuration of masses in a gravitational field
• Determines the work required to assemble a system of masses from infinity

Definition and formula

• Gravitational potential energy $U(\vec{r})$ of a test mass $m$ at position $\vec{r}$ in a potential $V(\vec{r})$
• Given by the formula: $U(\vec{r}) = mV(\vec{r})$
• Potential energy is always negative, indicating that work is required to separate masses

Relation to work

• Work done by the gravitational field on a mass $m$ moving from $\vec{r}_1$ to $\vec{r}_2$ is the negative change in potential energy
• $W = -\Delta U = -m(V(\vec{r}_2) - V(\vec{r}_1))$
• Positive work is done by the field when a mass moves to a region of lower potential

Conservative nature

• Gravitational field is conservative, meaning the work done by the field is independent of the path taken
• Consequence of the gravitational force being the negative gradient of the potential energy
• Allows the definition of a unique potential energy for each configuration of masses
• Enables the use of energy conservation in gravitational systems

Gauss's law for gravity

• Relates the gravitational flux through a closed surface to the enclosed mass
• Fundamental law in Newtonian gravity, analogous to Gauss's law in electrostatics

Statement of the law

• Gravitational flux $\Phi_g$ through a closed surface $S$ is proportional to the total mass $M$ enclosed by the surface
• Mathematically: $\oint_S \vec{g} \cdot d\vec{A} = -4\pi GM$
  • $\vec{g}$ is the gravitational field
  • $d\vec{A}$ is the vector area element pointing outward from the surface

Relation to Newtonian gravity

• Gauss's law is a consequence of the inverse-square nature of the gravitational force
• Can be derived from Newton's law of universal gravitation and the divergence theorem
• Provides an alternative formulation of Newtonian gravity in terms of gravitational flux

Applications and examples

• Calculating the gravitational field of spherically symmetric mass distributions (planets, stars)
• Determining the gravitational flux through a Gaussian surface enclosing a mass
• Proving the shell theorem: a uniform spherical shell exerts no net gravitational force on a mass inside it

Poisson's equation

• Differential equation relating the gravitational potential to the mass density
• Fundamental equation in Newtonian gravity, analogous to Poisson's equation in electrostatics

Derivation from Gauss's law

• Obtained by applying the divergence operator to both sides of Gauss's law
• $\nabla \cdot \vec{g} = -4\pi G\rho$
  • $\rho$ is the mass density
• Substituting $\vec{g} = -\nabla V$ yields Poisson's equation: $\nabla^2 V = 4\pi G\rho$

Relation to mass density

• Poisson's equation directly relates the gravitational potential $V$ to the mass density $\rho$
• The Laplacian operator $\nabla^2$ measures the curvature of the potential
• Regions with positive mass density have a positive Laplacian (concave potential)
• Regions with zero mass density have a zero Laplacian (harmonic potential)

Boundary conditions

• Solving Poisson's equation requires specifying boundary conditions on the potential
• Common boundary conditions include:
  • Dirichlet: specifying the value of the potential on a boundary surface
  • Neumann: specifying the normal derivative of the potential on a boundary surface
• Boundary conditions ensure the uniqueness of the solution to Poisson's equation

Multipole expansion

• Method for approximating the gravitational potential of an arbitrary mass distribution
• Expands the potential in terms of monopole, dipole, and higher-order moments of the mass distribution

Monopole term

• Lowest-order term in the multipole expansion, representing the total mass of the distribution
• Potential due to a monopole: $V_0(\vec{r}) = -\frac{GM}{r}$
  • $M$ is the total mass
  • $r$ is the distance from the center of mass

Dipole and higher-order terms

• Dipole term represents the asymmetry of the mass distribution, vanishing for symmetric distributions
• Higher-order terms (quadrupole, octupole, etc.) capture increasingly fine details of the mass distribution
• Contribution of higher-order terms decreases rapidly with distance, becoming negligible in the far-field

Far-field approximations

• At large distances from the mass distribution, the monopole term dominates the potential
• Far-field potential is well approximated by the monopole term: $V(\vec{r}) \approx V_0(\vec{r}) = -\frac{GM}{r}$
• Higher-order terms can be neglected, simplifying the calculation of the gravitational field and forces

Gravitational self-energy

• Energy associated with the gravitational interaction of a mass distribution with itself
• Arises from the non-linearity of the gravitational field equations

Definition and formula

• Gravitational self-energy $E_\text{self}$ of a mass distribution $\rho(\vec{r})$
• Given by the integral: $E_\text{self} = -\frac{1}{2}G \int \rho(\vec{r}) V(\vec{r}) d^3\vec{r}$
  • $V(\vec{r})$ is the gravitational potential due to the mass distribution itself
• Factor of $1/2$ accounts for double-counting of pairwise interactions

Importance in astrophysics

• Gravitational self-energy plays a crucial role in the stability and evolution of astrophysical systems
• Determines the binding energy of gravitationally bound systems (stars, galaxies, clusters)
• Affects the gravitational collapse and formation of compact objects (neutron stars, black holes)

Relation to binding energy

• Binding energy $E_\text{bind}$ is the energy required to disassemble a gravitationally bound system
• Related to the gravitational self-energy by: $E_\text{bind} = -E_\text{self}$
• Negative binding energy indicates a gravitationally bound system
• Positive binding energy implies an unbound system that will disperse without additional confinement

Orbits in a gravitational field

• Motion of a test mass in the gravitational field of a central mass
• Governed by the laws of conservation of energy and angular momentum

Kepler's laws

• Three empirical laws describing the motion of planets around the Sun, derived by Johannes Kepler
  1. Law of Ellipses: planets orbit the Sun in elliptical orbits with the Sun at one focus
  2. Law of Equal Areas: a line segment joining a planet and the Sun sweeps out equal areas in equal intervals of time
  3. Law of Periods: the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit

Circular and elliptical orbits

• Circular orbits: special case where the orbit has zero eccentricity and constant radius
  • Velocity is constant in magnitude and perpendicular to the radius vector
  • Centripetal acceleration is provided by the gravitational force
• Elliptical orbits: general case with non-zero eccentricity
  • Velocity varies in magnitude and direction along the orbit
  • Gravitational force provides both centripetal and tangential acceleration

Escape velocity

• Minimum velocity required for a test mass to escape the gravitational field of a central mass
• Obtained by setting the total energy (kinetic + potential) to zero at infinity
• Escape velocity from the surface of a spherical mass $M$ with radius $R$: $v_\text{esc} = \sqrt{\frac{2GM}{R}}$
• Objects launched with velocities greater than the escape velocity will follow hyperbolic trajectories and escape to infinity

Tidal forces

• Differential gravitational forces acting on extended bodies in a non-uniform gravitational field
• Arise from the variation of the gravitational field across the extent of the body

Origin of tidal forces

• Gravitational field of a central mass is not uniform, but decreases with distance
• Parts of an extended body closer to the central mass experience a stronger gravitational force than those farther away
• Differential force results in a stretching of the body along the line joining it to the central mass

Effect on extended bodies

• Tidal forces deform extended bodies, causing them to deviate from spherical symmetry
• Deformation is most pronounced along the line joining the body to the central mass
• Tidal bulges form on opposite sides of the body, aligned with the central mass
• Tidal forces can lead to tidal locking, where the body's rotation becomes synchronized with its orbital period

Roche limit

• Critical distance within which a celestial body held together by its own gravity will disintegrate due to tidal forces from a more massive body
• Occurs when the tidal forces exceed the self-gravitational forces holding the body together
• Roche limit for a fluid body orbiting a spherical mass $M$ with radius $R$: $d_\text{Roche} \approx 2.44R(\frac{\rho_M}{\rho})^{1/3}$
  • $\rho_M$ and $\rho$ are the densities of the central mass and the orbiting body, respectively
• Planetary rings and small moons can only exist outside the Roche limit of their parent body

Gravitational waves

• Ripples in the fabric of spacetime predicted by Einstein's theory of general relativity
• Generated by accelerating masses, analogous to electromagnetic waves produced by accelerating charges

Prediction from general relativity

• General relativity describes gravity as the curvature of spacetime caused by the presence of mass and energy
• Accelerating masses create disturbances in the spacetime curvature that propagate outward as gravitational waves
• Gravitational waves carry energy and momentum, causing a measurable effect on matter

Quadrupole formula

• Lowest-order approximation for the generation of gravitational waves by a mass distribution
• Gravitational wave amplitude is proportional to the second time derivative of the quadrupole moment of the mass distribution
• Quadrupole moment measures the deviation of the mass distribution from spherical symmetry
• Significant gravitational wave emission requires large, rapidly changing quadrupole moments (binary systems, asymmetric supernovae)

Detection methods

• Gravitational waves cause tiny distortions in the proper distance between test masses
• Laser interferometry: measure the differential change in the lengths of perpendicular arms using laser light
  • LIGO (Laser Interferometer Gravitational-Wave Observatory) and Virgo detectors
• Pulsar timing arrays: detect low-frequency gravitational waves through their effect on the arrival times of pulses from millisecond pulsars
• Space-based detectors: future missions (LISA, DECIGO) to detect gravitational waves in the mHz frequency range

Experimental tests

• Investigations designed to test the predictions of Newtonian gravity and its successors (e.g., general relativity)
• Provide crucial evidence for the validity and limitations of gravitational theories

Cavendish experiment

• Torsion balance experiment conducted by Henry Cavendish to measure the gravitational constant $G$
• Two small masses are attracted to two larger masses, causing a torsion beam to rotate
• Measuring the rotation angle and the torsion constant of the beam allows the determination of $G$
• First precise measurement of $G$, establishing the scale of the gravitational force

Gravitational redshift

• Shift of spectral lines towards longer wavelengths (redshift) in the presence of a gravitational field
• Predicted by the equivalence principle and general relativity
• Photons lose energy as they climb out of a gravitational potential well, resulting in a lower frequency (longer wavelength)
• Measured using high-precision spectroscopy of atomic transitions in the Sun's gravitational field and in the laboratory

Geodetic effect

• Precession of a gyroscope's spin axis in the presence of a gravitational field
• Consequence of the curvature of spacetime around a massive object
• Predicted by general relativity, with a magnitude of 6.6 arcseconds per year for a gyroscope in Earth orbit
• Measured by the Gravity Probe B experiment using superconducting gyroscopes in a polar Earth orbit
• Provides a direct test of the spacetime curvature predicted by general relativity