Gravitation and Planetary Motion

Between 1609 and 1619, Johannes Kepler analyzed decades of precision astronomical observations of planetary positions compiled by Tycho Brahe, discovering three empirical kinematic laws that dismantled Ptolemaic and Copernican circular orbits.

1. Kepler's Three Laws of Planetary Motion

1. First Law (Law of Ellipses): The orbit of every planet is an ellipse with the Sun located at one of the two focal points: $$r(\theta) = \frac{p}{1 + e \cos\theta}$$ where $p = a(1 - e^2)$ is the semi-latus rectum, $a$ is the semi-major axis, and $e \in [0, 1)$ is the orbital eccentricity.
2. Second Law (Law of Equal Areas): A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time: $$\frac{dA}{dt} = \text{constant}$$
3. Third Law (Harmonic Law): The square of the orbital period $T$ of a planet is directly proportional to the cube of the semi-major axis $a$ of its orbit: $$T^2 \propto a^3 \implies \frac{T^2}{a^3} = \text{constant}$$

2. Mathematical Proof of Kepler's Second Law from Angular Momentum Conservation

Consider a planet of mass $m$ orbiting the Sun under an arbitrary central force $\mathbf{F}(\mathbf{r}) = F(r)\hat{\mathbf{r}}$. The net torque acting on the planet about the Sun is: $$\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}(\mathbf{r}) = \mathbf{r} \times [F(r)\hat{\mathbf{r}}] = \mathbf{0}$$ Since torque is the time rate of change of orbital angular momentum $\mathbf{L}$: $$\frac{d\mathbf{L}}{dt} = \boldsymbol{\tau} = \mathbf{0} \implies \mathbf{L} = \mathbf{r} \times m\mathbf{v} = \text{constant vector}$$ In polar coordinates $(r, \theta)$, in time $dt$, the radius vector rotates through angle $d\theta$. The infinitesimal triangular area swept out is: $$dA = \frac{1}{2} |\mathbf{r} \times d\mathbf{r}| = \frac{1}{2} r (r \, d\theta) = \frac{1}{2} r^2 \dot{\theta} \, dt$$ The areal velocity is therefore: $$\frac{dA}{dt} = \frac{1}{2} r^2 \dot{\theta} = \frac{L}{2m} = \text{constant}$$ Because angular momentum $L$ is strictly conserved for central forces, the areal velocity is constant! This proves Kepler's Second Law for any central force field.

In 1687, Sir Isaac Newton published the *Philosophiae Naturalis Principia Mathematica*, formulating the universal law of gravitation that unified celestial planetary mechanics with terrestrial falling bodies.

1. Formulation of the Universal Law

Every particle of mass $m_1$ in the universe attracts every other particle of mass $m_2$ with a force directly proportional to the product of their masses and inversely proportional to the square of the distance $r$ separating them: $$\mathbf{F}_{12} = -G \frac{m_1 m_2}{r^2} \hat{\mathbf{r}}_{12}$$ where $G = 6.67430 \times 10^{-11}\text{ N}\cdot\text{m}^2\text{/kg}^2$ is Newton's universal gravitational constant, and $\hat{\mathbf{r}}_{12}$ is the unit vector pointing from mass 1 to mass 2.

2. Deduction of the Inverse-Square Dependence from Kepler's Laws

Assume a planet of mass $m$ moves in a circular orbit of radius $r$ with speed $v = \frac{2\pi r}{T}$ around the Sun ($M$). The centripetal acceleration is provided exclusively by gravitational force: $$F = m \frac{v^2}{r} = m \frac{4\pi^2 r^2}{r T^2} = 4\pi^2 m \frac{r}{T^2}$$ Applying Kepler's Third Law ($T^2 = K r^3$): $$F = \frac{4\pi^2 m}{K r^2} \propto \frac{m}{r^2}$$ By Newton's Third Law of action-reaction, the force must also be proportional to the Sun's mass $M$, establishing $F \propto \frac{M m}{r^2}$.

3. The Superposition Principle

The net gravitational force exerted on mass $m_0$ by a collection of $N$ discrete point masses is the vector sum: $$\mathbf{F}_{\text{net}} = -G m_0 \sum_{i=1}^N \frac{m_i}{|\mathbf{r}_0 - \mathbf{r}_i|^3} (\mathbf{r}_0 - \mathbf{r}_i)$$ For a continuous mass distribution of density $\rho(\mathbf{r}')$: $$\mathbf{F}(\mathbf{r}) = -G m_0 \int_V \frac{\rho(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|^3} (\mathbf{r} - \mathbf{r}') \, d^3\mathbf{r}'$$

Newton delayed publishing the *Principia* for nearly two decades until he mathematically proved the **Shell Theorem**, justifying why planets and moons can be treated as idealized point masses located at their centers.

1. First Shell Theorem (External Field)

A uniform spherical shell of mass $M$ and radius $R$ attracts an external point mass $m$ located at distance $r > R$ from its center as if the entire mass of the shell were concentrated at its geometric center: $$\mathbf{g}(r) = -\frac{GM}{r^2} \hat{\mathbf{r}} \quad (r > R)$$

2. Second Shell Theorem (Internal Field)

A uniform spherical shell of mass $M$ exerts strictly zero net gravitational force on any particle located anywhere in its interior: $$\mathbf{g}(r) = \mathbf{0} \quad (r < R)$$

Proof: Divide the shell into pairs of opposing elemental surface areas $dA_1$ and $dA_2$ subtended by an infinitesimal double cone of solid angle $d\Omega$. The masses are $dm_1 = \sigma dA_1 = \sigma r_1^2 d\Omega$ and $dm_2 = \sigma dA_2 = \sigma r_2^2 d\Omega$. The opposing gravitational forces are: $$dF_1 = G \frac{m \, dm_1}{r_1^2} = G m \sigma d\Omega, \quad dF_2 = G \frac{m \, dm_2}{r_2^2} = G m \sigma d\Omega$$ Because $dF_1 = dF_2$, the opposing forces cancel identically for every direction throughout the $4\pi$ sphere!

3. Solid Sphere of Uniform Density ($ ho$)

For a solid planet of radius $R$ and total mass $M = \frac{4}{3}\pi R^3 \rho$:

• Inside the planet ($r \le R$): By the shell theorem, all outer spherical shells with radii $> r$ produce zero force. The force is due entirely to the inner sphere of radius $r$ containing mass $M(r) = M (r/R)^3$: $$g(r) = \frac{G M(r)}{r^2} = \frac{G M}{R^3} r = \frac{4}{3}\pi G \rho r$$ Gravity increases linearly from zero at the center to $g_0$ at the surface!
• Outside the planet ($r \ge R$): Gravity falls as the inverse square: $$g(r) = \frac{GM}{r^2}$$

Because gravity is the weakest fundamental force in nature, determining $G$ required exquisite experimental ingenuity. Henry Cavendish (1798) measured $G$ in his historic experiment commonly described as *"weighing the Earth"*.

1. The Torsion Balance Apparatus

Cavendish suspended a light horizontal wooden beam of length $L$ carrying two identical small lead spheres of mass $m$ from a thin, highly sensitive quartz/tungsten torsion wire with torsion constant $C$. Two large stationary lead spheres of mass $M$ were brought near the small spheres at center-to-center distance $d$.

2. Equilibrium Torque Balance

The gravitational attractive force between each pair of spheres is $F = G \frac{M m}{d^2}$. The deflecting gravitational couple is: $$\tau_{\text{grav}} = 2 F \left(\frac{L}{2}\\right) = F L = G \frac{M m L}{d^2}$$ In equilibrium, this torque is balanced by the restoring elastic torque of the twisted wire: $$\tau_{\text{elastic}} = C \theta \implies G \frac{M m L}{d^2} = C \theta$$ where $\theta$ is the angular deflection, measured using an optical optical lever (light beam reflected from a mirror on the wire onto a distant scale).

3. Determination of the Torsion Constant ($C$)

To eliminate the unknown wire stiffness $C$, Cavendish measured the period $T_0$ of free torsional oscillation of the suspended beam: $$T_0 = 2\pi \sqrt{\frac{I}{C}} \implies C = \frac{4\pi^2 I}{T_0^2}$$ where $I = 2 m (L/2)^2 = \frac{1}{2} m L^2$ is the moment of inertia of the small spheres. Substituting $C$ gives $G$ directly: $$G = \frac{2\pi^2 L d^2 \theta}{M T_0^2}$$

4. Mass and Density of the Earth

Once $G$ was known, Cavendish calculated Earth's mass from surface gravity $g = \frac{G M_E}{R_E^2}$: $$M_E = \frac{g R_E^2}{G} = \frac{(9.81)(6.371 \times 10^6)^2}{6.674 \times 10^{-11}} \approx 5.972 \times 10^{24}\text{ kg}$$ Earth's mean density is: $$\bar{\rho}_E = \frac{M_E}{\frac{4}{3}\pi R_E^3} = \frac{3 g}{4\pi G R_E} \approx 5,515\text{ kg/m}^3$$ Because surface rocks have density $\sim 2,700\text{ kg/m}^3$, this proved that Earth possesses a dense metallic core (iron/nickel)!

In physics, mass appears in two fundamentally distinct conceptual roles:

1. Two Definitions of Mass

1. Inertial Mass ($m_i$): The measure of a body's resistance to acceleration when acted upon by any force, defined through Newton's Second Law: $$\mathbf{F} = m_i \mathbf{a} \implies m_i = \frac{|\mathbf{F}|}{|\mathbf{a}|}$$
2. Gravitational Mass ($m_g$): The measure of a body's gravitational coupling strength (its gravitational 'charge'), defined through Newton's law of gravitation: $$\mathbf{F}_g = m_g \mathbf{g} = G \frac{M m_g}{r^2} \hat{\mathbf{r}}$$

2. The Weak Equivalence Principle

Setting inertial force equal to gravitational force for a freely falling object: $$m_i \mathbf{a} = m_g \mathbf{g} \implies \mathbf{a} = \left( \frac{m_g}{m_i} \\right) \mathbf{g}$$ If $m_g / m_i$ were different for different materials, a feather and a cannonball in a vacuum would accelerate at different rates. Galileo's Leaning Tower experiments, Newton's pendulum trials, and Loránd Eötvös's precision torsion balance experiments (1889) demonstrated that: $$\frac{m_g}{m_i} = 1.000000000000000$$ Modern satellite experiments (MICROSCOPE, 2022) confirm the equivalence to within 1 part in $10^{15}$.

3. Physical Significance: General Relativity

Albert Einstein recognized that the strict proportionality $m_i \equiv m_g$ is not a cosmic coincidence. It led directly to the **Einstein Equivalence Principle**:

The effects of a uniform gravitational field are physically indistinguishable from the effects of a uniformly accelerating frame of reference.

This principle forms the cornerstone of Einstein's General Theory of Relativity, where gravity is not a Newtonian force, but the curvature of four-dimensional spacetime.

The gravitational interaction is a conservative vector field described by a scalar potential.

1. Gravitational Field Intensity ($\mathbf{g}$)

The gravitational field $\mathbf{g}(\mathbf{r})$ at point $\mathbf{r}$ is the gravitational force experienced per unit test mass placed at that point: $$\mathbf{g}(\mathbf{r}) = \lim_{m_0 \to 0} \frac{\mathbf{F}_g}{m_0} = -\frac{GM}{r^2} \hat{\mathbf{r}}$$

2. Gravitational Potential ($V$)

Because gravity is a conservative force ($\boldsymbol{\nabla} \times \mathbf{g} = \mathbf{0}$), it can be expressed as the negative gradient of a scalar **gravitational potential** $V(\mathbf{r})$: $$\mathbf{g} = -\boldsymbol{\nabla}V \implies V(r) = -\int_\infty^r \mathbf{g} \cdot d\mathbf{r}' = -\int_\infty^r \left(-\frac{GM}{r'^2}\\right) dr' = -\frac{GM}{r}$$ taking $V(\infty) = 0$ as the reference zero-potential at infinite separation.

3. Gravitational Potential Energy ($U$)

The potential energy of a two-particle system with masses $M$ and $m$ is: $$U(r) = m V(r) = -\frac{GMm}{r}$$ Notice that $U(r)$ is strictly negative for all finite separations, reflecting an attractive bound state.

4. Gravitational Self-Energy of a Uniform Sphere

The work required to assemble a uniform solid sphere of mass $M$, radius $R$, and density $\rho$ by bringing infinitesimal mass shells $dm$ from infinity: $$U_{\text{self}} = -\int_0^R \frac{G M(r) \, dm}{r} = -\int_0^R \frac{G \left(\frac{4}{3}\pi \rho r^3\\right) \left(4\pi \rho r^2 dr\\right)}{r} = -\frac{16\pi^2 G \rho^2}{3} \int_0^R r^4 dr$$ $$U_{\text{self}} = -\frac{16\pi^2 G \rho^2 R^5}{15} = -\frac{3}{5} \frac{G M^2}{R}$$ This negative gravitational binding energy plays a critical role in stellar astrophysics and planetary core collapse.

Orbital dynamics governs the trajectories of artificial satellites, planets, and interstellar space probes.

1. Escape Velocity ($v_{\text{esc}}$)

The **escape velocity** is the minimum initial speed a projectile must have at the surface of a body of mass $M$ and radius $R$ to completely escape its gravitational well to infinity without further propulsion. By conservation of mechanical energy: $$E_i = \frac{1}{2} m v_{\text{esc}}^2 - \frac{GMm}{R} = E_f = 0 \implies v_{\text{esc}} = \sqrt{\frac{2GM}{R}} = \sqrt{2 g R}$$ For Earth ($R = 6,371\text{ km}, g = 9.81\text{ m/s}^2$): $$v_{\text{esc}} = \sqrt{2 \times 9.81 \times 6.371 \times 10^6} \approx 11,180\text{ m/s} \approx 11.2\text{ km/s}$$ For the Moon: $v_{\text{esc}} \approx 2.38\text{ km/s}$. For the Sun: $v_{\text{esc}} \approx 618\text{ km/s}$.

2. Orbital Velocity for a Circular Orbit

For a circular satellite orbit at altitude $h$ above Earth's surface ($r = R + h$): $$\frac{m v_{\text{orb}}^2}{r} = \frac{GMm}{r^2} \implies v_{\text{orb}} = \sqrt{\frac{GM}{r}} = \frac{v_{\text{esc}}}{\sqrt{2}}$$ At low Earth orbit ($r \approx R_E$): $v_{\text{orb}} \approx 7.91\text{ km/s}$ (First Cosmic Velocity).

3. The Vis-Viva Equation

For an arbitrary Keplerian orbit with semi-major axis $a$, the total mechanical energy is $E = -\frac{GMm}{2a}$. Equating kinetic plus potential energy to total energy: $$\frac{1}{2} m v^2 - \frac{GMm}{r} = -\frac{GMm}{2a} \implies v^2 = G M \left( \frac{2}{r} - \frac{1}{a} \\right)$$ This is the celebrated **Vis-Viva Equation**:

• Circular Orbit ($a = r$): $v = \sqrt{GM/r}$, $E < 0$.
• Elliptic Orbit ($a > 0$): Speed is maximum at periapsis ($r = r_{\min}$) and minimum at apoapsis ($r = r_{\max}$).
• Parabolic Orbit ($a \to \infty$): $v = \sqrt{2GM/r}$, total energy $E = 0$ (marginally unbound).
• Hyperbolic Orbit ($a < 0$): $E > 0$ (unbound interstellar trajectory).

The local acceleration due to gravity $g$ is not universally constant; it varies systematically with altitude, depth, latitude, and local geology.

1. Variation of $g$ with Altitude ($h$)

At an elevation $h$ above sea level ($r = R_E + h$): $$g(h) = \frac{G M_E}{(R_E + h)^2} = \frac{G M_E}{R_E^2 \left(1 + \frac{h}{R_E}\\right)^2} = g_0 \left(1 + \frac{h}{R_E}\\right)^{-2}$$ For altitudes small compared to Earth's radius ($h \ll R_E \approx 6371\text{ km}$), expanding via binomial series: $$g(h) \approx g_0 \left(1 - \frac{2h}{R_E}\\right)$$ Gravity decreases by approximately $0.03\%$ for every kilometer of altitude.

2. Variation of $g$ with Depth ($d$)

Inside a mine at depth $d$ below the surface ($r = R_E - d$), by the Shell Theorem: $$g(d) = \frac{G M(r)}{r^2} = \frac{4}{3}\pi G \rho (R_E - d) = g_0 \left(1 - \frac{d}{R_E}\\right)$$ Gravity decreases linearly to zero at Earth's center. Notice that gravity decreases twice as rapidly with altitude as it does with depth!

3. Variation with Latitude ($\phi$) due to Earth's Axial Rotation

Because Earth rotates with angular speed $\omega = 7.292 \times 10^{-5}\text{ rad/s}$, a body at latitude $\phi$ experiences an outward centrifugal acceleration: $$a_c = \omega^2 r_\perp = \omega^2 (R_E \cos\phi)$$ The effective gravity measured by a spring balance is the vector sum: $$g_{\text{eff}}(\phi) \approx g_0 - R_E \omega^2 \cos^2\phi$$

• At the **Equator** ($\phi = 0^{\circ}$): Centrifugal reduction is maximum: $$\Delta g = R_E \omega^2 = (6.378 \times 10^6)(7.292 \times 10^{-5})^2 \approx 0.0339\text{ m/s}^2$$ giving $g_{\text{equator}} \approx 9.780\text{ m/s}^2$.
• At the **Poles** ($\phi = 90^{\circ}$): Centrifugal effect vanishes: $g_{\text{pole}} \approx 9.832\text{ m/s}^2$.

Combined with Earth's equatorial bulge (flattening $f \approx 1/298$), objects weigh approximately $0.53\%$ more at the poles than at the equator!

Accurate geodetic measurements of $g$ require eliminating measurement errors caused by distributed pendulum mass.

1. The Compound (Physical) Pendulum

A compound pendulum is any rigid body free to oscillate in a vertical plane about a fixed horizontal axis. Let $m$ be the total mass, $d$ the distance from the pivot point $O$ to the center of gravity $G$, and $I$ the moment of inertia about $O$: $$I = I_G + m d^2 = m (k^2 + d^2)$$ where $k$ is the radius of gyration about $G$. The restoring torque for angular displacement $\theta$ is $\tau = -m g d \sin\theta \approx -m g d \theta$. The equation of motion is: $$I \ddot{\theta} + m g d \theta = 0 \implies T = 2\pi \sqrt{\frac{I}{m g d}} = 2\pi \sqrt{\frac{k^2 + d^2}{g d}} = 2\pi \sqrt{\frac{L_{\text{eff}}}{g}}$$ where $L_{\text{eff}} = \frac{k^2 + d^2}{d} = d + \frac{k^2}{d}$ is the length of the **equivalent simple pendulum**.

2. Center of Oscillation and Conjugate Points

Along the line passing through pivot $O$ and center of mass $G$, there exists a point $O'$ at distance $d' = k^2 / d$ on the opposite side of $G$. If the pendulum is inverted and suspended from $O'$, its period of oscillation is: $$T' = 2\pi \sqrt{\frac{k^2 + d'^2}{g d'}} = 2\pi \sqrt{\frac{k^2 + (k^2/d)^2}{g (k^2/d)}} = 2\pi \sqrt{\frac{d + k^2/d}{g}} = T$$ The points $O$ and $O'$ are **mutually conjugate**: the periods of oscillation about both points are identical! The distance between them is exactly $L_{\text{eff}} = d + d'$.

3. Kater's Reversible Pendulum

Invented by Captain Henry Kater (1818), this precision instrument consists of a rigid metal bar with two adjustable knife-edges facing inward and movable weights. By adjusting the weights until the period of oscillation about knife-edge 1 ($T_1$) equals the period about knife-edge 2 ($T_2 = T_1 = T$): $$g = \frac{4\pi^2 L}{T^2}$$ where $L$ is simply the physical distance between the two knife-edges, which can be measured with a traveling microscope to sub-millimeter precision! This completely eliminates the need to determine the unknown moment of inertia $I$, center of mass $G$, or radius of gyration $k$!