Introduction
Gravitational potential is potential energy per unit mass, a scalar field from which the vector gravitational field can be derived. Working with the scalar potential is often simpler than working directly with vector fields, and it provides a complete description of the gravitational field's conservative properties.
Definition of Gravitational Potential
Gravitational potential φ at a point is the potential energy per unit mass: φ = U/m. For a point mass M: φ = -GM/r. Units: J/kg or m2/s2. For multiple masses: φ = Σ(-GM_i/r_i). Potential is a scalar - easier to add than vector fields. The potential difference between two points equals work per unit mass needed to move between them.
Relation to Gravitational Field
The gravitational field is the negative gradient of potential: g = -∇φ = -(∂φ/∂x \(\hat{\imath}\) + ∂φ/∂y \(\hat{\jmath}\) + ∂φ/∂z \(\hat{k}\)). In spherical coordinates: g_r = -dφ/dr. Field points in direction of steepest decrease of potential. Equipotential surfaces (constant φ) are perpendicular to field lines. No work is required to move along equipotential surface.
\nIntroduction
Gravitational potential is potential energy per unit mass, a scalar field from which the vector gravitational field can be derived. Working with the scalar potential is often simpler than working directly with vector fields, and it provides a complete description of the gravitational field's conservative properties.
Definition of Gravitational Potential
Gravitational potential φ at a point is the potential energy per unit mass: φ = U/m. For a point mass M: φ = -GM/r. Units: J/kg or m2/s2. For multiple masses: φ = Σ(-GM_i/r_i). Potential is a scalar - easier to add than vector fields. The potential difference between two points equals work per unit mass needed to move between them.
Relation to Gravitational Field
The gravitational field is the negative gradient of potential: g = -∇φ = -(∂φ/∂x \(\hat{\imath}\) + ∂φ/∂y \(\hat{\jmath}\) + ∂φ/∂z \(\hat{k}\)). In spherical coordinates: g_r = -dφ/dr. Field points in direction of steepest decrease of potential. Equipotential surfaces (constant φ) are perpendicular to field lines. No work is required to move along equipotential surface.
\nEquipotential Surfaces
Equipotential surfaces are surfaces of constant gravitational potential. Properties: (1) Spherical around point masses; (2) Perpendicular to gravitational field lines; (3) No work done moving along them; (4) Closer spacing indicates stronger field. For Earth: approximately spherical surfaces (slightly oblate due to rotation). For uniform field: parallel planes.
Potential for Extended Distributions
For continuous mass distribution: φ = -∫(G dm/r). Examples: Uniform spherical shell - φ = -GM/R (constant) inside, φ = -GM/r outside. Uniform solid sphere - φ = -(GM/2R3)(3R2-r2) inside, φ = -GM/r outside. Potential is always continuous; its derivative (field) may be discontinuous at surfaces.
Applications
Applications include: calculating work done by gravity, analyzing fluid equilibrium (surfaces perpendicular to g), electrical analogies (similar mathematics for electrostatics), geodesy (Earth's gravitational field mapping), satellite orbit perturbation analysis. Potential formulation simplifies many problems that would be difficult with direct force calculations.
\nSolved Example: Work to Move Between Points
Calculate work needed to move 1000 kg satellite from Earth's surface (R_E = 6370 km) to geostationary orbit (r = 42,164 km from center). Solution: Gravitational potential at surface: φ1 = -GM_E/R_E. Potential at GEO: φ2 = -GM_E/r. Work per unit mass = φ2 - φ1 = -GM_E/r + GM_E/R_E = GM_E(1/R_E - 1/r). GM_E = gR_E2 = 9.8 × (6.37×10^6)2 = 3.98×10^14 m3/s2. Work/kg = 3.98×10^14 × (1/6.37×10^6 - 1/42,164×10^3) = 3.98×10^14 × (1.57×10^-7 - 2.37×10^-8) = 3.98×10^14 × 1.33×10^-7 = 5.29×10^7 J/kg. Total work = 1000 × 5.29×10^7 = 5.29×10^10 J ≈ 53 GJ. This is minimum energy needed (Hohmann transfer requires slightly more due to kinetic energy changes). Compare to surface escape energy: GM_Em/R_E = 6.25×10^10 J. Going to GEO requires about 85% of escape energy.
\n