A Priori
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hodograph to learn about how to interpret it. Since the Inverse Proportion Machine
creates segments that are inversely proportional to each other that lie on the same
line, we must make an adjustment. Tangential velocity and distance to the Sun
are our inversely proportional properties. By definition, they are perpendicular to
each other. Thus we must adjust by stating that the true direction of velocity is 90
degrees from where it is indicated to be on the hodograph. Chapters 12 and 13
will help us to visualize this.
velocity. But we are obviously interested in knowing the total velocity of the planet
anywhere along its orbit. In Chapter 14 we see how to find total velocity in the
hodograph diagram using our knowledge of components of vectors from Chapter 3.
planet points in the direction of the tangent to the ellipse? A few years ago I came
upon some beautiful logic in an article "The Pretrigonometry Proof of the
Reflective Property of the Tangent to the Ellipse" written by Zalman P. Usiskin. I
liked the logic and found that a similar approach could help to determine that the
velocity of a planet must indeed be along the tangent line to the ellipse. Chapter
15 describes this analysis of the total velocity direction at any instant.
as represented by the hodograph. In particular, we see how to measure the
angle from perihelion (the position where the planet is closest to the Sun) in the
orbit and in the hodograph.
components. They are radial velocity and tangential velocity. Chapter 17
defines them and shows us how to see them in the hodograph.
and the speed of a planet. In Chapter 19 we look at the angle of the planet as
measured from the Sun and how rapidly it changes with time. This is called angular
velocity. It is a property that will help us to prove Newton's Inverse Square Law of
Force and Distance in a priori fashion.
brilliant revelation that the change in velocity is directly proportional to the change in
angle to the Sun so that their ratio is a constant. This revelation is detailed in
Feynman's Lost Lecture, The Motion of Planets Around the Sun by David L. Goodstein
and Judith R Goodstein, Norton Press. We will need this relationship for the a priori
proof of Newton's Inverse Square Law of Force and Distance. Chapter 20 describes
this relationship between the change in velocity and the change in angle.
we use this knowledge and the relationship between the change in velocity and the
change in angle to prove that gravitational force varies inversely as the square of the
distance to the Sun in a priori fashion. This is Newton's Inverse Square Law of Force
and Distance.
planetary laws in a priori fashion. But here we depart from the a priori theme for
good reason. From this point onward in the proofs we will travel parallel roads, one
being a priori and the other being somewhat empirical. Let me explain what
somewhat empirical means and why we need to head in that direction just a little bit.
As we progress to a priori proofs of useful properties of orbits such as the energy of a
planet, it would be nice to be able to use experimentally measured constants in the
equations that we prove or derive. We benefit from this in two ways. Firstly, we can
use these derived formulae to make real life calculations of orbital positions, speeds,
and energies. Secondly, our formulae will look more recognizable in comparison to
the equations found in standard texts on celestial mechanics that are based upon
empirical methods. In any case, suppose we find in a priori fashion that force is
inversely proportional to the square of the distance to the Sun. Our somewhat
empirical deviation from a priori methods would be to experimentally calculate the
constant that regulates the proportion. Indeed, that has been done. The constant is
called the Gravitational Constant. Once we explain the Gravitational Constant in
Chapter 22, we can use it in many subsequent chapters to convert our a priori
formulae to practical equations. In effect, I am trying not to be stubborn. I see that it
is beneficial to allow empiricism to enter peripherally into the scheme of Orbits
Explained.
of planets in circular orbits. This will help us to prove Kepler's Third Law of
Planetary Motion, regarding the periods of the orbits, in a priori fashion in
Chapter 24.
Planetary Law. Later we will broaden the proof to include elliptical orbits.
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