Clark J. Calculate the Orbit of Mars!: An Observing Challenge and Historical Adventure

New York: Springer, 2021. — 319 p.This book shows readers how to calculate the orbit of Mars, based on their own observations and using observations made by the author. The historical, observational, and analytical aspects of the project to measure the orbit of Mars are all combined in this one book!
Determining the orbit of Mars is particularly important, as originally solving this problem required the founding of modern science. Clark discusses how people came to believe in the Newtonian model of the Solar System, works through the mathematical basis for the theory of gravity, and shows how Newton ruled out the possibility of alternative theories. Readers also learn how it became possible to accurately measure the positions of Mars from a moving, spinning platform―the Earth
This mid-level observational challenge is well within reach of most serious amateur astronomers. For the observations, only a telescope with auto-guiding capability and the ability to mount a digital single lens reflex (DSLR) camera is required. For the calculations, it is assumed that the reader has a science, engineering, or mathematics background and is familiar with calculus, vectors, and trigonometry.AcknowledgementsAbout the Author
In the Beginning
Ptolemy’s Model: Salient Features
The Copernican Revolution: The Pun That Keeps oving
Tycho Brahe, the Greatest Pre-telescope Observer
Tycho and Mathematics
Tycho’s Instruments
Galileo – Telescopes and Free Fall
Free Fall and Motion
The Telescope Bursts onto the Scene
Encounter with the Inquisition
Read My Ellipses
Kepler
Ellipses
The Behaviour of Two Bodies in Orbit
Christiaan Huyghens (1629–1695)
Sir Isaac Newton (1642–1727): The Great Synthesizer
Universal Gravitation
Gravity Due to Point Masses and Spheres
Angular Momentum and Its Implications
The Bit Where Conic Sections Come In
Time Evolution
A Brief Aside: the Word Anomaly
The Moon: A Challenge for Newtonian Physics
Canonical Units
Energy
The Orbit Tells Us What the Force Law Is
Least Squares Fit to Sets of Equations
Least Squares Fitting Subroutine
The Orbit of the Earth
Sunny Side Down, Please
From Solar Midday to Orbit
Sorting Out Some Definitions
Further Geometric Gymnastics with Ellipses
Tilting My Lance at Obliquity
Getting Around to Earth’s Orbit
Here’s Looking at You, Mars!
Equipment
Guiding
Photographic Technique
How I Made Measurements from Pictures Like Fig. 6.7
How to Obtain the Position of Mars from the Star Positios
Main Program
Angle Unit Changing Subroutines
Least Squares Fitting Subroutine LSQFIT Has to Be Added
Distance Unit Changing Subroutines
Results
The 2009–2010 Apparition of Mars
Later Observing Campaigns
Data from 2011 to 2012 (Table 6.5 and Fig. 6.31)
Data from 2020 (Table 6.6 and Fig. 6.32)
First Shot at the Orbit of Mars
Non-coplanar Orbits Facing in Random Directions
Rotations
What These Angles Are Used for
Herget’s Method
Lagrange’s f and g Functions
The Sector–Triangle Ratio
Introduction of Kepler’s Equation
Solution of Eqs. (7.89), (7.101) and (7.104)
Solving Eq. (7.1) with Known Magnitude but Unknown Direction of Sun–Planet Distance
Eureka! (Part One)
Obtaining the Orbital Elements from Two Radius Vectors
Software Listing
Main Program
Calculate Ascending Node
Convert Observation Times to Canonical Units
Use Escobal’s Methods to Calculate Various Quantities
Use Methods Given Above to Calculate the Longitude and Passage Time of Perihelion
This Is Basically the Subroutine Described in Chap. 4
Eureka! (Part Two)
Refining the Preliminary Orbit
Least-Squares Curve Fitting
Gambling on Monte Carlo
Monte Carlo Source Code
Manual Adjustment of Orbits
Using Knowledge of the 2020 Opposition
Conclusions Drawn
Da Capo al Fine: Can I Fit Tycho’s Data?
How Well Did I Do?
Appendix 1: Parabolas and Hyperbolas
Deriving the Equation of a Hyperbola in Polar Coordinates
Deriving the Equation of a Parabola in Polar Coordinates
Appendix 2: Multiplying Three Vectors
Appendix 3: Proof of Trigonometric Addition Formulae for All Angles