The gravitational field and gravitational potential due to spherical bodies are fundamental concepts in classical mechanics and astrophysics, describing the attraction exerted by massive objects like planets or stars. In an inertial frame, these quantities are derived from Newton’s law of gravitation, assuming spherical symmetry simplifies calculations significantly. In a rotating frame with constant angular velocity, fictitious forces such as the centrifugal force and Coriolis force modify the dynamics, affecting the effective gravitational field and potential. This article explores the gravitational field and potential of spherical bodies in both inertial and rotating frames, their mathematical formulations, applications in astrophysics and engineering, and challenges in rotating frame analyses, with a focus on central force motion.
Table of Contents
Gravitational Field and Potential in an Inertial Frame
For a spherical body of mass \( M \), assumed to have spherical symmetry (e.g., a uniform or radially varying density), the gravitational field \(\mathbf{g}(\mathbf{r})\) at a point \(\mathbf{r}\) (distance \( r = |\mathbf{r}| \) from the center) is derived from Newton’s law of gravitation. Outside the body (\( r \geq R \), where \( R \) is the radius), the field is:
\mathbf{g}(\mathbf{r}) = -\frac{GM}{r^2} \hat{\mathbf{r}},
\]
where \( G \) is the gravitational constant, and \(\hat{\mathbf{r}}\) is the radial unit vector. This field is equivalent to that of a point mass \( M \) at the center, due to the shell theorem.
Inside a uniform spherical body (\( r < R \)), the gravitational field depends only on the mass enclosed within radius \( r \). For a uniform density \(\rho = \frac{M}{\frac{4}{3}\pi R^3}\), the enclosed mass is \( M_{\text{enc}} = \frac{4}{3}\pi r^3 \rho = M \frac{r^3}{R^3} \), and the field is:
\mathbf{g}(\mathbf{r}) = -\frac{G M r}{R^3} \hat{\mathbf{r}}.
\]
The gravitational potential \( V(\mathbf{r}) \) is defined such that \(\mathbf{g} = -\nabla V\). Outside the body (\( r \geq R \)):
V(r) = -\frac{GM}{r},
\]
with the potential set to zero at infinity. Inside a uniform sphere (\( r < R \)):
V(r) = -\frac{GM}{2R^3} (3R^2 – r^2),
\]
derived by integrating the field and ensuring continuity at \( r = R \). The potential is continuous and differentiable, reflecting the smooth transition of the gravitational field across the surface.
The field and potential describe a central force \(\mathbf{F} = m \mathbf{g}\), where \( m \) is the mass of a test particle, enabling the application of Kepler’s laws for orbital motion in the inertial frame.
Gravitational Field and Potential in a Rotating Frame
In a rotating frame with angular velocity \(\mathbf{\Omega}\), the equation of motion for a test particle of mass \( m \) includes fictitious forces:
m \frac{d^2 \mathbf{r}}{dt^2} = \mathbf{F} – m \mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r}) – 2m \mathbf{\Omega} \times \mathbf{v},
\]
where \(\mathbf{F} = m \mathbf{g} = -\frac{GMm}{r^2} \hat{\mathbf{r}}\) (for \( r \geq R \)) is the gravitational force, \(-\mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r})\) is the centrifugal acceleration, and \(-2 \mathbf{\Omega} \times \mathbf{v}\) is the Coriolis acceleration. The effective gravitational field in the rotating frame is:
\mathbf{g}_{\text{eff}} = \mathbf{g} – \mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r}) – \frac{2 \mathbf{\Omega} \times \mathbf{v}}{m},
\]
but the Coriolis term depends on velocity and does not contribute to the potential. The effective potential accounts for the gravitational and centrifugal contributions. Assuming the rotation is about the \( z \)-axis (\(\mathbf{\Omega} = \Omega \hat{\mathbf{z}}\)) and motion is in the \( xy \)-plane, the centrifugal acceleration is:
-\mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r}) = \Omega^2 r_\perp \hat{\mathbf{r}}_\perp,
\]
where \( r_\perp = \sqrt{x^2 + y^2} \) is the perpendicular distance from the rotation axis. The centrifugal potential is:
V_{\text{centrifugal}} = -\frac{1}{2} m \Omega^2 r_\perp^2.
\]
The effective potential for \( r \geq R \) is:
V_{\text{eff}}(r, r_\perp) = -\frac{GMm}{r} – \frac{1}{2} m \Omega^2 r_\perp^2.
\]
Inside the sphere (\( r < R \)), the gravitational potential becomes \( V(r) = -\frac{GM}{2R^3} (3R^2 – r^2) \), so:
V_{\text{eff}}(r, r_\perp) = -\frac{GMm}{2R^3} (3R^2 – r^2) – \frac{1}{2} m \Omega^2 r_\perp^2.
\]
The effective gravitational field is \(\mathbf{g}_{\text{eff}} = -\nabla V_{\text{eff}}\), excluding the Coriolis term, which affects motion but not the potential.
Conservation and Dynamics
In the inertial frame, the gravitational field of a spherical body leads to Keplerian orbits, with conserved angular momentum (\(\mathbf{L} = m \mathbf{r} \times \mathbf{v}\)) and energy. In the rotating frame, the Coriolis force introduces a torque, disrupting angular momentum conservation, but the Jacobi integral is conserved:
E_J = \frac{1}{2} m (v_r^2 + r^2 \dot{\theta}^2) + V_{\text{eff}}(r, r_\perp).
\]
This integral governs the dynamics, including equilibrium points where \(\nabla V_{\text{eff}} = 0\).
Applications
Astrophysics: Planetary and Binary Systems
For a planet or star, the gravitational potential in a rotating frame (e.g., co-rotating with a binary system) defines the Roche potential, combining gravitational and centrifugal terms:
V_{\text{eff}} = -\frac{GM_1}{|\mathbf{r} – \mathbf{r}_1|} – \frac{GM_2}{|\mathbf{r} – \mathbf{r}_2|} – \frac{1}{2} \Omega^2 r_\perp^2.
\]
This determines Lagrangian points for stable orbits or mass transfer in binary star systems. The Coriolis force affects particle trajectories, influencing accretion disk dynamics.
Geophysics: Earth’s Gravitational Field
In Earth’s rotating frame, the effective gravitational field includes the centrifugal force, causing the planet to be an oblate spheroid rather than a perfect sphere. The effective potential explains the shape of the geoid, critical for satellite navigation and geophysical modeling.
Engineering: Rotating Space Habitats
In a rotating space station designed to simulate gravity, the centrifugal force combines with the gravitational field of a central mass (if present). The effective potential determines the perceived “gravity” for occupants, guiding habitat design.
Fluid Dynamics: Rotating Stars or Planets
In rotating stars or planets, the gravitational potential and centrifugal potential shape the internal structure. The effective field influences convection and circulation patterns, affecting stellar evolution or planetary atmospheres.
Challenges and Considerations
The Coriolis force complicates dynamics by introducing velocity-dependent effects, requiring careful numerical modeling in applications like astrophysical simulations or climate models. The centrifugal force distorts the effective potential, creating new equilibrium points (e.g., Lagrangian points) that differ from the inertial frame. Transforming between inertial and rotating frames demands precise coordinate transformations to ensure consistency in energy and angular momentum. For non-uniform or non-spherical bodies, deviations from spherical symmetry require multipole expansions, complicating the potential.
Conclusion
The gravitational field and potential due to spherical bodies are modified in a rotating frame by the centrifugal force and Coriolis force, leading to an effective potential that governs motion. The Jacobi integral provides a conserved quantity, enabling analysis of orbits and equilibrium points. Applications include binary star systems, geophysical modeling, space habitats, and rotating fluids. By accounting for fictitious forces, physicists and engineers can accurately model gravitational effects in rotating systems, advancing our understanding of celestial and terrestrial dynamics.