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

Description

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.   Preface Acknowledgements Contents About the Author Chapter 1: 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 Chapter 2: Read My Ellipses Kepler Ellipses Chapter 3: 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 Chapter 4: Least Squares Fit to Sets of Equations Least Squares Fitting Subroutine Chapter 5: 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 Chapter 6: 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) Chapter 7: 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) Chapter 8: 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 References Index