Astronomy: Celestial Mechanics and Star Systems

This lecture introduces fundamental motion concepts needed for celestial mechanics: vector-based

velocity, velocity addition, and angular velocity. Using familiar examples and diagrams, the lecture

transitions from linear to angular motion, laying the foundation for orbital dynamics.

This lecture introduces linear and angular momentum—quantities that remain conserved even as

a system changes. These are essential for understanding how celestial systems evolve over time,

including planets, stars, and spacecraft in orbit.

This lecture introduces the concept of acceleration, both linear and angular, and how it connects to

motion, force, and momentum. Acceleration is at the heart of dynamics in celestial mechanics, and

we will discuss its various forms including centrifugal and gravitational acceleration, as well as its

relation to Newton’s laws.

This lecture introduces the moment of inertia, a rotational analogue to mass in linear motion. We

explore how the mass distribution of an object affects its resistance to rotational acceleration, and

how to calculate it for various simple solid bodies. Concepts such as the center of mass and symmetry,

Steiner’s theorem, and the connection between moment of inertia and angular momentum are all

covered. Finally, we examine torque and Newton’s second law for rotational motion in more depth.

This lecture introduces the concept of energy, a key physical quantity central to understanding

motion, forces, and celestial systems. We focus on kinetic and potential energy in multiple settings

— point masses, extended bodies, Earth-bound systems, and general gravitational setups. These

concepts will later be combined into total energy expressions and used in the next lecture on

conservation laws. Finally, we introduce the concepts of mechanical work and energy loss.

This lecture introduces the core conservation laws used in celestial mechanics. These include

conservation of momentum (linear and angular), energy, and magnetic flux. Each law will be

introduced, derived where appropriate, and supported with structured examples.

This lecture expands on the gravitational force and its potential, both foundational to celestial

mechanics. We introduce acceleration from gravity, the concept of first cosmic velocity, and integrate

examples involving conservation of energy, momentum, and angular momentum. More compelling

scenarios involving Earth, space missions, and Newton’s laws will illustrate these ideas.

This lecture begins our study of stellar orbits. We generalize the concept of orbit types from purely

circular to conic-section trajectories: ellipses, parabolas, and hyperbolas. We present the geometrical

basis using conical intersections, distinguish open and closed orbits, and relate them to the energy of

the system. This sets the foundation for discussing real astrophysical orbits of planets, comets, stars,

black holes and so on.

This lecture focuses on binary systems: two objects orbiting their common center of mass. We

examine how their orbits are influenced by mass, separation, and conservation laws. We explore

the concepts of reduced mass and gravitational binding energy and extend to practical examples

including neutron star binaries, white dwarf systems, and hierarchical triples.

This lecture introduces Kepler’s three laws of planetary motion. We explore each law with its

geometric and physical interpretation, provide the relevant equations, and walk through examples.

A derivation of the third law is given in general and simplified forms, including applications to real

astrophysical systems.

This lecture aims to give students practical experience using the orbital laws covered so far. The

focus is on worked examples and applications of Kepler’s laws, especially in non-trivial settings.

Emphasis is placed on interpretation and modeling over derivation. We consider only bound, closed

orbits, primarily through the lens of semi-major axis and orbital area.

In this lecture, we explore how orbital inclination affects the observed motion in binary and planetary

systems, focusing on the relation between observed and actual orbital motion. We then move on

to another type of inclination, that of the line of apsides, which in turn influences the observed

behavior of the system.

In this lecture, we explore the motion of artificial satellites and natural bodies under gravitational

attraction in elliptical orbits. We discuss launch trajectories, orbit classifications, energy considerations,

and introduce the so-called satellite paradox, along with its full derivation. Atmospheric

effects such as friction are also accounted for, leading into more realistic models of near-Earth

satellite behavior.

In this lecture, we formally define the parameters of elliptical orbits, including the semi-major

and semi-minor axes, eccentricity, semi-latus rectum, and focal distances. We discuss the physical

interpretation of these quantities, how they relate to known quantities like perihelion and aphelion

distances, and introduce essential geometrical concepts that enhance our understanding of Kepler’s

Second Law.

We also revisit previous examples (e.g., satellite launches) under this new framework and prepare

the groundwork for the full mathematical treatment of elliptical trajectories.

In this lecture, we explore the mathematical equations that describe elliptical orbits. While we will

not derive them from first principles (e.g., from Newton’s laws or central force potentials), we will

understand their form, how to interpret them geometrically and physically, and how the quantities

we previously defined appear naturally in them. We also briefly introduce ellipsoids and discuss

geometric approximations useful in astrophysics.

So far, we’ve discussed the geometry of elliptical orbits. In this lecture, we turn to dynamics:

what energy and velocity profiles arise when two bodies move in elliptical orbits under their mutual

gravity? We’ll derive a compact and surprisingly elegant formula for the total mechanical energy of

the system and explore how orbital speed varies along the ellipse. These results are foundational in

celestial mechanics and underlie models from binary stars to satellites around Earth.

In this lecture, we introduce a key element of celestial mechanics: Kepler’s Equation, which allows

us to connect time with position along an elliptical orbit. While we’ve previously dealt with spatial

geometry and energy in elliptical trajectories, we now seek to understand how orbital position evolves

in time. To achieve this, we define three angular parameters — the mean anomaly, eccentric

anomaly, and true anomaly — and relate them via geometric and trigonometric arguments. The

culmination is Kepler’s famous transcendental equation, which will be our principal tool to track a

planet’s location along an ellipse at any time t:

M = E − e sinE

In this lecture, we explore the orbital parameters of open trajectories—parabolic and hyperbolic

orbits. Building upon our understanding of elliptical orbits, we extend the concepts to these unbound

paths. We will discuss their defining equations, distances, angles, and conservation laws, providing

a comprehensive framework for analyzing open orbital motions.

In this lecture, we explore the concept of the second cosmic velocity, the minimum speed needed

to escape a body’s gravitational field. We derive this from fundamental principles of energy conservation

and relate it to the parabolic orbit. We also discuss the Schwarzschild radius — a key

concept from general relativity — and the event horizon it defines. Finally, we examine how escape

velocity affects a planet’s ability to retain an atmosphere, including a practical rule of thumb for

atmospheric loss.

In celestial mechanics, the shape of an orbit reveals not just where a body moves, but how gravity

governs its path. The so-called “optical properties” of conic sections — such as how rays reflect

through their foci — may seem geometric, but they have deep physical consequences. These reflection

behaviors are directly tied to orbital symmetries, conservation laws, and time-reversal in

Newtonian gravity.

This lecture focuses on interpreting these properties mechanically, showing how geometric symmetry

relates to gravitational interactions — from planetary returns to scattering trajectories.

Radial velocity curves are among the most important observational tools for understanding the

motion of celestial objects. From binary stars to exoplanets to interstellar objects, they offer insight

into orbits even when we cannot directly resolve the motion. These profiles encode not only

orbital speed but also the orientation of the orbit and the nature of the motion (bound or unbound).

In this lecture, we will explore how the orbital parameters (eccentricity, inclination, argument of

periapsis) shape the observed velocity curves, and how our own motion must be accounted for to

interpret them correctly.

In this lecture, we will analyze two key gravitational concepts that arise in multi-body systems —

the Hill Radius and the Roche Limit. Both define important boundary distances in celestial

mechanics:

• The Hill Radius defines the region around a body where it dominates the gravitational

influence over its satellites.

• The Roche Limit describes the distance within which a celestial body held together by

self-gravity will be torn apart by tidal forces.

These two concepts are crucial when studying satellites, rings, exoplanet systems, and more.

In previous lectures, we defined:

• The First Cosmic Velocity — the minimum horizontal speed to enter circular orbit above

the surface of a body (e.g., around Earth).

• The Second Cosmic Velocity — the escape velocity from a planet’s gravitational field.

In this lecture, we introduce the Third Cosmic Velocity, the speed needed to escape the solar

system from Earth (or another body), i.e., to overcome the Sun’s gravitational potential while

already being within the gravitational field of a planet. This connects two gravitational wells:

planetary and solar.

In previous lectures, we examined escape velocities and gravitational interactions in multi-body

systems. In this lecture, we introduce the concept of Lagrange points - special positions in a

rotating two-body system where a third object can remain stationary relative to the two primary

bodies. These points are central to orbital mechanics, satellite positioning, and long-term stability

analysis.

In this lecture, we complete our discussion of the Lagrange points by analyzing L3, L4, and L5 with

full derivations, similar in rigor to our treatment of L1 and L2. We also investigate the stability

of these points through a small perturbation analysis in the rotating frame.

In this lecture we analyze binary star systems in which only one of the two components is directly

observable. From radial velocity measurements and a few basic assumptions, we will derive an

important relation known as the mass function. This function allows us to place constraints on

the mass of the unseen companion — be it another star, a white dwarf, neutron star, or black hole.

Tidal forces arise from the differential gravitational pull of one body across another. They are

responsible for phenomena such as ocean tides, tidal locking, orbital evolution, and even axis reorientation.

They are the reason the Moon is slowly drifting away and Venus rotates differently.

Many astronomical systems feature multiple layers of orbital motion — satellites orbit planets,

which orbit stars, which themselves may orbit in binary or multiple star systems. Understanding

how these nested orbits interact is key to predicting long-term stability, resonance phenomena, and

the exchange of angular momentum between different components.

In previous lectures we developed the mathematical and physical tools to describe the orbital dynamics

of celestial bodies and artificial satellites. These principles allow us to understand the

motions of planets, moons, and stars, but they also form the foundation of space exploration. A

spacecraft is not simply placed into a final orbit: it must be carefully guided through sequences

of orbital changes, relying on precise energy and momentum transfers. Similarly, celestial bodies

themselves can undergo long-term orbital changes due to tidal forces, close encounters, or external

perturbations.

This lecture will focus on the mechanisms behind orbital changes, both natural and artificial. We

will examine the mathematics of orbit transfers (elliptic to circular, Hohmann transfers, bi-elliptic

transfers, and escape trajectories), gravitational assists or “slingshots,” and the conservation laws

governing these maneuvers. Real-world applications such as interplanetary probes and planetary

encounters will illustrate the generality of these principles. By the end, you should be able to

compute the energy and velocity requirements for various transfers, understand their efficiency, and

apply these ideas to both engineered and astrophysical scenarios.

In this lecture we investigate one of the most fundamental and elegant properties of planetary systems:

differential movement. Because gravitational attraction decreases with distance, planets

closer to the Sun orbit with higher angular and linear velocities than those further away. As a result,

alignments, oppositions, and repeating patterns emerge from the interplay of different orbital

periods.

Although our focus is on the Solar System, the same principles apply in extrasolar planetary systems,

satellite systems, and even in galaxies (with modifications). By working with simplified circular orbits

and uniform angular velocity, we gain powerful tools for predicting alignments and relative

motions.

We will introduce sidereal and synodic periods, derive their relation using angular velocities, and

discuss key phenomena such as lunar phases, retrograde motion, and resonances. The exact configurations

of planetary alignments, as well as the detailed treatment of lunar phases, will be covered

in a later lecture.

In the previous lecture, we studied differential movement and how different angular velocities

lead to synodic periods, retrograde motion, and resonances. We now take a step further to consider

the observable consequences of these relative motions: planetary configurations.

Planetary configurations describe the positions of planets relative to Earth and the Sun, such as

conjunctions, oppositions, elongations, and alignments. These configurations have been central to

both ancient and modern astronomy, providing the framework for calendar cycles, prediction of

eclipses, and planning of space missions.

Before addressing these geometrical relations, we begin with an empirical law of planetary spacing

— the Titius–Bode law — which historically offered the first approximate method to predict

orbital radii of planets.

In the previous two lectures, we explored differential motion and planetary configurations. These

ideas naturally lead to a deeper question: how do these relative motions produce the phases of

celestial bodies?

In this lecture, we begin with the phases of planets, which appear brighter or dimmer depending

on their position relative to Earth and the Sun. We then move to a detailed study of the Moon’s

phases — a fundamental cycle for both astronomy and human history. This discussion will include

the Moon’s sidereal and synodic periods, the geometric derivation of the visible illuminated fraction,

and applications to eclipses, libration, nutation, and tidal locking. Finally, we will connect these

astronomical facts to cultural terminology: blue moon, blood moon, and harvest moon.

This lecture consolidates key dynamical and geometrical principles, preparing us for deeper studies

of larger celestial structures.

So far, we have applied orbital mechanics to the Solar System and stellar systems. In this lecture,

we extend these principles to the scale of galaxies. The motion of stars and gas within galaxies

reveals striking departures from the predictions of Newtonian mechanics when applied only to visible

matter. This discrepancy is the foundation of modern dark matter theory. We will analyze galactic

mass distributions, expected vs. observed orbital speeds, and discuss theoretical explanations for

these phenomena.

In the previous lecture on Galactic Trends we inferred missing mass from rotation curves. To turn

kinematic data into masses in a model–independent way, we now develop the Virial Theorem.

We will derive it in the simplest possible way (center-of-mass frame), show how velocity dispersion

leads directly to mass estimates, and then build to the Jeans stability criterion (Jeans radius &

Jeans mass) that governs when clouds collapse to form stars. Throughout, we keep the emphasis

on transparent physical balances: kinetic vs. potential energy, pressure support vs. gravity.

This lecture discusses an International Physics Olympiad problem about the Evolution of the Earth-Moon System, combining many of the concepts discussed in the course.  You will be walked over the problem and solution in detail, and you will see how the previous topics come hand in hand to beautifully teach you about the history of our planet and its natural satellite, while being able to fully solve such an exercise from the highest level of competitions.

The 3-body problem asks a deceptively simple question: given three point masses that interact

only via Newtonian gravity, what are their motions for arbitrary initial positions and velocities?

Historically the problem was posed as soon as Newton gave his law of gravitation: while the two-body

problem is integrable and yields conic orbits, adding a third body makes the dynamics vastly

richer and far more difficult.

Important historical landmarks:

• Newton (late 17th c.): formulation of gravitational laws and recognition that more than

two bodies complicates motion enormously.

• Euler & Lagrange (18th c.): discovery of special exact solutions — the collinear (Euler)

and equilateral (Lagrange) families.

• Poincaré (late 19th c.): demonstrated the qualitative complexity of the problem, discovered

non-integrability and the seeds of chaos theory.

• Sundman (early 20th c.): showed a convergent (but impractical) series solution for the

3-body problem — existence but not usefulness for applied calculations.

• Modern era: numerical integration, regularization methods, discovery of many periodic and

“choreography” solutions (e.g. the figure-eight), and deep study of chaotic scattering and

stability.

This bonus lecture develops the physics and mathematics of the 3-body problem in a pedagogical

yet precise way: we present the equations, conserved quantities, special exact solutions, the circular

restricted approximation, notions of stability, the onset of chaos and its physical consequences, and

several conceptual worked examples that clarify the ideas.

We have glimpsed the Coriolis force whenever we moved to a rotating point of view in earlier

lectures (e.g. rotating frames for planetary motion, or the CR3BP). Here we finally tackle it head-on:

what it is, where it comes from, how to use it properly, and why it shapes flows from Earth’s

atmosphere to galactic disks. We start from the rotating-frame equations, keep the algebra minimal,

and then build physical intuition via terrestrial and astrophysical examples: falling bodies at the

equator, winds and hurricanes, the drain myth, and the behavior of gas and stars in rotating systems

such as accretion and galactic disks.