The Two Body Problem

“A rigorous, code-complete engineering textbook on two-body orbital mechanics — from Kepler’s Laws to a fully optimized Earth-to-Mars trajectory algorithm with working FORTRAN source code — written for practitioners who want to do the math, not just read about it.“

114 pages 
6x9 Trade book 
Available as a Paperback, PDF book, or ebook 
Read a 15-page preview at Blurb.com

About the Book

From Newton’s Law of Gravitation to a fully optimized Earth-to-Mars transfer trajectory — complete with working FORTRAN source code — The Two Body Problem & Applications: Mission: Mars is William H. Clark II’s most rigorously technical offering, and one of the most complete self-contained treatments of applied orbital mechanics available from an independent press.

Structured as both a systematic textbook and a practical engineering reference, the volume walks readers from foundational celestial mechanics — Kepler’s Laws, conic sections, orbital elements, coordinate transformations — through increasingly complex territory: perturbation theory, Lambert’s Problem, hyperbolic escape trajectories, planetary flyby geometry, and the patched conic method for multi-stage interplanetary missions. Every concept is grounded in the mathematics of mission planning, not abstract theory.

The book’s crown jewel is its final section: a detailed technical report presenting Clark’s original numerical algorithm for optimizing Earth-to-Mars trajectories without conventional nonlinear optimization tools, followed by the complete Mars Pathfinder FORTRAN source code. For aerospace engineers, astrodynamics students, and computational scientists serious about getting to Mars, this is the working manual.

Target Audience

  • Aerospace and mechanical engineering students at upper-division and graduate levels
  • Orbital mechanics practitioners and mission planning professionals
  • Computational scientists and software developers working on trajectory simulation
  • Readers of Clark’s Temporal Mechanics seeking the rigorous engineering foundation behind his theoretical claims
  • Technical library collections in aerospace, physics, and applied mathematics

Key Themes

  • Classical astrodynamics from first principles — every major result derived from Newton and Kepler, building a complete mathematical framework step by step
  • Coordinate systems and orbital elements — mastery of IJK frames, orbital element sets, and the transformations between them as the backbone of mission planning
  • Perturbation theory in practice — atmospheric drag, solar radiation pressure, the J2 equatorial bulge, and third-body effects treated as engineering realities, not footnotes
  • Computational trajectory optimization — Clark’s original numerical algorithm for Earth-to-Mars transfer planning, bypassing standard nonlinear solvers
  • The patched conic method — multi-stage interplanetary mission architecture (escape, heliocentric transfer, planetary capture) explained geometrically and analytically
  • Open-source engineering — full FORTRAN source code provided, reflecting Clark’s conviction that scientific work should be reproducible and practically executable

Detailed Table of Contents

  • Chapter 1 — Newton’s Law of Gravity: Derives the Two Body Equation from Newton’s Universal Law of Gravitation and establishes the fundamental constants of orbital motion.
  • Chapter 2 — Kepler’s First Law: Develops the geometry of elliptical orbits, defining eccentricity, true and eccentric anomalies, and the general conic section trajectory equation.
  • Chapter 3 — Time in Elliptical Motion: Introduces the eccentricity vector, proves Kepler’s Second and Third Laws, and derives Kepler’s Equation relating mean and eccentric anomalies.
  • Chapter 4 — Orbital Elements: Defines the IJK inertial coordinate system and the six classical orbital elements, with full transformation procedures between orbital planes and inertial reference frames.
  • Chapter 5 — Orbit Propagation: Presents methods for predicting spacecraft position over time using Kepler’s Equation via Newton-Raphson iteration and the f and g series functions.
  • Chapter 6 — Initial Orbit Determination: Covers the Laplace and Gauss methods for computing orbital state vectors from radar ranging and angular observation data.
  • Chapter 7 — Perturbations: Analyzes real-world forces that deviate from ideal two-body motion, including atmospheric drag, solar radiation pressure, third-body gravity, and the Earth’s J2 oblateness term.
  • Chapter 8 — Perturbed Orbit Determination: Explains how Two Line Element (TLE) sets and state transition matrices are used to refine orbital predictions under perturbation conditions.
  • Chapter 9 — Orbit Transfers: Details Hohmann Transfer maneuvers, rendezvous mechanics, and synodic period calculations for moving spacecraft between orbital regimes.
  • Chapter 10 — Lambert’s Problem: Examines the boundary value problem of connecting two position vectors with a trajectory, focusing on minimum-energy solutions and their mission planning applications.
  • Chapter 11 — Ballistic Trajectories: Analyzes thrust-free trajectories and identifies the optimal geometric conditions — chord passing through the empty focus — for maximum range.
  • Chapter 12 — Hyperbolic Orbits: Covers escape trajectories, hyperbolic flyby mechanics, and hyperbolic anomaly calculations essential for interplanetary mission design.
  • Chapter 13 — Velocity Diagrams: Provides graphical methods for analyzing relative velocities (v-infinity) during Earth departure and planetary arrival and capture maneuvers.
  • Chapter 14 — Planet Flyby Geometry: Explains the patched conic approximation for multi-stage interplanetary missions, including sphere-of-influence transitions and gravity assist geometry.
  • Chapter 15 — Gravity Field Force: Derives orbital acceleration from the gradient of the gravitational potential, introducing non-uniform mass distribution effects on spacecraft trajectories.
  • Chapter 16 — Gravitational Shock Wave: Presents Clark’s original theoretical proposal that the f and g functions represent distinct physical forces generating wave-fronts that influence interplanetary trajectory solutions.
  • Chapter 17 — Optimizing the Earth to Mars Trajectory: A full technical report presenting Clark’s numerical algorithm for Earth-to-Mars transfer optimization without conventional nonlinear optimization tools, including methodology and results.
  • Chapter 18 — Mars Pathfinder FORTRAN Code: Provides the complete, annotated FORTRAN source code implementing the optimization algorithm developed in Chapter 17, enabling direct computational reproduction.

Purchase the Book

Please visit the book’s landing page on Blurb.com for a 15-page preview and options to purchase the paperback or PDF.

Further Reading

Visit the author’s website 4 Theory.org for more extreme technology ideas + his latest tech papers in celestial mechanics.

Visit WH Clark.net for complete details of his dismissal from graduate school, where this book was a draft PhD dissertation.