Gravity and Orbits

Section 6 of 6

Synthesis and Relativity

Part 6: Synthesis — The Unity of Celestial and Terrestrial Physics

What Newton Achieved

Before Newton, celestial and terrestrial physics were separate domains: the heavens were thought to obey different rules. Newton demolished this distinction. The same force that pulls an apple from a tree — gravity following an inverse-square law — keeps the Moon orbiting Earth and the planets orbiting the Sun. There is one physics, applying everywhere.

Kepler’s Laws	Newton’s Explanation
Orbits are ellipses	Solution to the equations of motion under inverse-square gravity
Equal areas in equal times	Conservation of angular momentum (central force → no torque)
$P^2 / a^3 = \text{const}$	Requires the inverse-square law; constant set by $G$ and the masses

This is the template for all of physics: observe patterns, propose mechanisms, derive the patterns as consequences, then use the mechanisms to predict new phenomena.

Forward Connections

Topic	Connection
Binary Stars	Orbital measurements → stellar masses via Newton’s Kepler III
Exoplanets	Transit timing, radial velocity → masses and orbits
Stellar Structure	Virial theorem (at equilibrium) → pressure-temperature balance
Compact Objects	Escape velocity → black-hole event horizons
Galaxy Dynamics	Rotation curves → dark-matter inference

Every time we infer mass from orbital motion, we’re using Newton’s framework.

Beyond Newton: A Glimpse of Einstein

Newton’s framework is extraordinarily powerful — it explains planetary motion, predicts eclipses, weighs distant stars, and guides spacecraft. For nearly all astrophysical situations in this course, it’s excellent. But it’s not the final word. In 1915, Einstein’s general relativity revealed that Newton’s inverse-square law is an approximation — an incredibly good one — to a deeper theory.

General relativity

Einstein’s 1915 theory of gravity as the curvature of spacetime: “mass tells spacetime how to curve; spacetime tells mass how to move.” Newtonian gravity is its weak-field, low-speed limit.

Newton’s View	Einstein’s View
Gravity is a force acting instantaneously across space	Gravity is the curvature of spacetime itself
Space and time are fixed, absolute backdrops	Space and time are dynamic, shaped by mass and energy
Objects feel a gravitational pull	Objects follow the straightest possible paths (geodesics) through curved spacetime

Grid representation of spacetime curved by a central mass, showing how the geometry of space itself is warped by the presence of matter. — Figure 10Gravity as Curved Spacetime: Einstein's view that mass curves spacetime, and objects follow the straightest possible paths (geodesics) through that curvature.cococubed.com

Newton’s framework needs corrections in three regimes: near extremely dense objects (neutron stars, black holes), at very high speeds (approaching $c$ ), and for precision measurements (GPS satellites need relativistic corrections or positioning drifts by ~10 km/day).

A teaser — black holes and the cosmic speed limit. Recall escape velocity, $v_{esc} = \sqrt{2GM/r}$ . What happens if you compress enough mass into a small enough region that escape velocity reaches the speed of light? Setting $v_{esc} = c$ and solving for $r$ :

This is the Schwarzschild radius — the boundary of a black hole’s event horizon. Inside it, not even light can escape.

The same $v_{esc}$ formula that describes rockets leaving Earth, pushed to its logical extreme, predicts the most exotic objects in the universe. Newton gave us the tools; Einstein showed us where they lead. (We’ll explore black holes in detail when we study compact objects later this semester.)

Quick Practice

Use these as a fast warm-up before the graded set.

Problem

Empirical vs. physical: Kepler observed $P^2 \propto a^3$ . Newton derived that the power of $r$ in gravity must be 2 to reproduce it. If gravity instead followed $F \propto 1/r^3$ , what relationship between $P$ and $a$ would we observe?

Problem

Orbital velocity scaling: Jupiter orbits at 5 AU, Earth at 1 AU. Without exact numbers, is Jupiter’s orbital velocity greater than, less than, or equal to Earth’s — and by what factor?

Problem

Escape velocity intuition: a neutron star has roughly the Sun’s mass but a radius of only 10 km $= 10^6$ cm. Compared to the Sun’s escape velocity (618 km/s), is the neutron star’s larger or smaller? Estimate the ratio.

Multiple choice

Energy and orbit type: a comet approaches the Sun with total energy $E = +3 \times 10^{42}$ erg (and doesn’t hit the Sun). Will it…

(a) stays bound(b) escapes(c) settles into a circular orbit

The 11 graded practice problems for this lecture live in the companion practice set.

Key Equations Summary

Quantity	Equation	CGS Units
Gravitational Force	$F_g = \dfrac{Gm_1m_2}{r^2}$	dyne
Centripetal Force	$F_c = \dfrac{mv^2}{r}$	dyne
Orbital Velocity	$v_{orb} = \sqrt{\dfrac{GM}{r}}$	cm/s
Escape Velocity	$v_{esc} = \sqrt{\dfrac{2GM}{r}} = \sqrt{2}\,v_{orb}$	cm/s
Kepler III (Newton)	$P^2 = \dfrac{4\pi^2}{G(M+m)}a^3$	$\mathrm{s^2}$
Gravitational PE	$U = -\dfrac{GMm}{r}$	erg
Total Energy	$E = \dfrac{1}{2}mv^2 - \dfrac{GMm}{r}$	erg
Bound Orbit Energy	$E = -\dfrac{GMm}{2a}$	erg
Angular Momentum	$L = mr^2\dot{\phi}$	$\mathrm{g\,cm^2\,s^{-1}}$
Areal Velocity	$\dfrac{dA}{dt} = \dfrac{L}{2m}$	$\mathrm{cm^2\,s^{-1}}$
Virial Theorem	$\langle K \rangle = -\tfrac{1}{2}\langle U \rangle$	(energy relation)

CGS constants: $G = 6.67 \times 10^{-8}\ \mathrm{cm^3\,g^{-1}\,s^{-2}}$ ; $M_\odot = 2 \times 10^{33}$ g; $R_\odot = 7 \times 10^{10}$ cm; $M_\oplus = 6 \times 10^{27}$ g; $R_\oplus = 6.4 \times 10^{8}$ cm; 1 AU $= 1.5 \times 10^{13}$ cm; 1 yr $= 3.16 \times 10^{7}$ s.

This reading emphasizes the conceptual journey from empirical patterns to physical explanations — one of the most important transitions in the history of science. As you work through the mathematics, keep asking: what does this equation tell me? What happens if I change one variable? And always check your units.

Glossary

Angular momentum: A conserved measure of orbiting/spinning motion, $L = mvr$ for circular motion. A central force exerts no torque, so $L$ stays constant — the deep reason behind Kepler’s Second Law.
Aphelion: The farthest point from the Sun, at distance $r_a = a(1+e)$ .
Areal velocity: The rate at which the planet-Sun line sweeps out area, $dA/dt = L/(2m)$ . Angular-momentum conservation makes it constant — which is Kepler’s Second Law.
Barycenter: The center of mass of a two-body system — the “balance point” about which both bodies orbit. The more massive body orbits closer to it.
Central force: A force directed along the line connecting two bodies (toward a center). Gravity is central — and any central force conserves angular momentum, which is why Kepler’s Second Law holds.
Centripetal acceleration: The center-directed acceleration of an object in circular motion, $a_c = v^2/r$ . It changes the direction of velocity, not its magnitude.
Centripetal force: The net inward force required to keep an object on a circular path, $F_c = mv^2/r$ . It is not a new kind of force — it’s a role played by whatever real force points to the center (gravity, tension, friction).
Eccentricity: A number $e$ measuring how “squashed” an ellipse is. $e = 0$ is a circle; as $e \to 1$ the ellipse becomes more elongated. Earth’s is about 0.017 — nearly circular.
Empirical law: A pattern extracted from data that describes what happens without explaining the underlying mechanism. Reliable within the range of the data, but silent on why.
Escape velocity: The minimum launch speed needed to coast from radius $r$ to infinity, $v_{esc} = \sqrt{2GM/r} = \sqrt{2}\,v_{orb}$ . The escaping object’s own mass cancels.
General relativity: Einstein’s 1915 theory of gravity as the curvature of spacetime: “mass tells spacetime how to curve; spacetime tells mass how to move.” Newtonian gravity is its weak-field, low-speed limit.
Gravitational potential energy: The energy stored in the gravitational field of two bodies, $U = -GMm/r$ with zero defined at infinity. It is negative because a bound object sits below the zero level — in a “well” it would need energy to climb out of.
Inertia: An object’s resistance to acceleration, set by its mass. For the same force, a more massive object accelerates less.
Orbital velocity: The speed required for a circular orbit at radius $r$ around mass $M$ , $v_{orb} = \sqrt{GM/r}$ . It depends only on $M$ and $r$ — not on the orbiting object’s mass.
Perihelion: The point of closest approach to the Sun, at distance $r_p = a(1-e)$ .
Physical law: A statement of mechanism, derived from first principles, that explains why a pattern holds — and predicts new situations beyond the original data.
Semi-major axis: Half the longest diameter of an ellipse — the average distance from the orbiting body to its host. We reserve the symbol $a$ for it throughout.
Torque: A twisting influence that changes angular momentum. A central force produces zero torque about the center, so it leaves $L$ unchanged.