A Priori
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to view a Word Format File or a PDF Format File. Please read the preface
first to get a sense of how Orbits Explained evolved and what it intends to
convey.
are the seed of Orbits Explained click the
notebook photo links. These contain the
handwritten ideas and diagrams produced
from 2001 to 2004. Many of the ideas and
diagrams turned out to be false leads and
pitfalls. But finally a valid set of proofs
emerged.
Use your browser's zoom option to enlarge the
chapter pages if needed.
to disable the pop up blocking option of your web browser.
you might take a peek at the brief dedication page where I mention
those who have been patient with me.
R. Goodstein for writing the book, Feynman's Lost Lecture , The Motion of
Planets Around the Sun, published by Norton Press. In the preface to Orbits
Explained, I describe the quest to find an a priori proof for planetary laws and
how the Goodsteins and their book lured me to the cause.
help to familiarize you with the parts of the ellipse. The second chapter deals
solely with a derivation of the formula for the area of ellipses.
Chapter 3 describes vectors, which are arrows whose length and orientation
represent speed and direction. Chapter 4 discusses speed and velocity. If you
are well versed in vectors and their components you might want to skip these
chapters.
swept in equal times by a planet moving past a central Sun in the presence
of attractive force. We need this a priori proof as a starting point for a priori
proofs of all three of Kepler's Planetary Laws. So, Newton's proof is
explored in Chapter 5.
triangle. The base of the triangle is the distance to the Sun. The far side of the
triangle is the perpendicular distance traveled by the planet during the time it is
observed to move. In Chapter 6 we will see what this triangle looks like. We will
see that the velocity as measured perpendicularly to the radius to the Sun is called
tangential velocity. Finally, we will see that tangential velocity and distance to the
Sun must be inversely proportional to each other due to the law that equal areas
are swept in equal times.
the elliptical path and velocity of a planet orbiting the Sun. Hamilton's hodograph
construction is based on the premise that gravitational force is inversely related to the
square of the distance to the Sun. In Orbits Explained, our goal is to show that the
hodograph is valid without using the inverse square law of force and distance. But first
of course we must see Hamilton's hodograph and learn how it represents the position
and velocity of a planet. We will inspect the hodograph in Chapter 7.
task is to build or derive the hodograph using theory and logic without assuming an
elliptical orbit and without assuming the inverse square law of force. The first step in
accomplishing this is to create a mathematical device that can generate inverse
proportions. This device is the Inverse Proportion Machine or the hododyne. We will
need a special triangle for the proof of the hododyne. Chapter 8 introduces this
triangle. It is a right triangle whose hypotenuse has a perpendicular bisector.
also known as the hododyne. The hododyne is simply a straight line that is bent into
two segments. The segments spin about each other at the bend. Special moving
landmarks on the two segments delineate lengths that are inversely proportional to
each other. The proofs for the Inverse Proportional Machine are given in Chapter 9.
The proofs are given in their raw form exactly as they were first written on dinner
napkins and spare pieces of paper. I am sure they can be condensed mathematically
into a more concise proof. I invite you to find and submit a nicer derivation.
Hamilton's hodograph. Since the hodograph determines that the shape of the orbits
are elliptical and since we have used no assumptions about shape or force, we have
found an a priori proof that orbits are elliptical. In Chapter 10 we will pause for some
philosophy about the existence of orbits before we see the transition from hododyne
to hodograph in Chapter 11. It is in Chapter 11 where we see the a priori proof that
orbits are elliptical, Kepler's First Planetary Law.
CHAPTERS 12 to 24