Microworld: Calculus
in Action: Chapter 4, Section 1: The Geometry of Planetary Orbits
Click the Hyperlink
above to visit the Microworld.
Author:James
E. White
In
this Chapter, we will see Calculus in Action as it was first used -- to derive
John Kepler's three laws of planetary motion from Isaac Newton's single hypothesis
of Universal Gravitation.
Recall
that Kepler's laws assert:
1) The shape of each planet's orbit is an ellipse with the sun at one focus.
2) If an imaginary line is drawn from the sun to the planet, the line will sweep out equal areas in space in equal periods of time for all points in the orbit.
3) The ratio of the cube of the semimajor axis of the ellipse (i.e., the average distance of the planet from the sun) to the square of the planet's period (the time it needs to complete one revolution around the sun) is the same for all the planets.
In
one beautiful calculation, using his Calculus, Newton showed that all of these
would follow if the planets, in their motion around the Sun, obeyed his Universal
Law of Gravitation. And, of course, he developed a theory of Lunar motion
on the same principles. Thus, these laws about the motion of the planets,
derived from empirical observation, could now be deduced mathematically, with
the aid of the Calculus.
.This
first Section: The Geometry of Planetary Orbits will provide the necessary
background for a geometrical understanding of curves in space. In this section,
we will show that planetary orbits subject to Newtonian gravitation are actually
plane curves, and we will then be able to establish the existence of the first
conserved quantity, the angular momentum, which will now be viewed
as a vector quantity. While we could apply what we learned about plane curves
to establish Kepler's second law, we choose to follow a more geometric route,
now that we will have the tools to do it. So we will deduce that law in this
Section.
What
is that geometric route? We mentioned earlier that Gauss counselled us to
begin with the assumption of an underlying geometry before attempting to discover
the properties of plane curves or surfaces that are independent of the way
those curves or surfaces are presented (parameterized). There are properties
of curves in three-dimensional space that are also intrinsic properties of
those curves as geometrical objects. It may not be surprising for example
that space curves have osculating circles also, as long as they are parameterized
with non-vanishing velocity.
The
language for describing those geometric properties is called Linear Algebra.
This, together with notions of space velocity and space acceleration are key
elements of Newton's picture as he developed it in the Calculus. We will develop
the parts of this language that we will need to capture the essential geometry
of space curves in an especially simple and compact form.
In
order to set the stage in this Section for the use of linear algebra, we start
with a 3-Dimensional Model Solar System, in which you will see the
essential mysteries that faced Ptolemaic geocentric theories of planetary
motion, and for which Nicholas Copernicus had a very simple answer. With that
Solar System, you will be able to explore some of those mysteries for yourself,
and in particular, come to understand the 'retrograde motion' of the planet
Mars in the night sky. There is no calculus in this first part of the Section,
but there is a great deal to think about! The image below is a view from the
Earth along the plane of the ecliptic of the Sun, Mercury, Venus and Mars.
Obviously,
this is a daytime view (high noon) from the Earth, and so Mars in on the other
side of the Sun from us. Shortly, it will move to the same side of the Sun
as Earth, and the picture will automatically become a nighttime view showing
Mars alone in the sky, slowing down and changing direction,
as is scuttles backward across the sky for about 2 months.
The
same picture taken from the Solar System view shows the Earth also. It appears
at the top of the page.
The
remainder of this section is devoted to Geometry, with the final aim of establishing
that planetary orbits are plane curves, and discovering the angular momentum
invariant. We interpret this invariant as a (constant) approximation function
for the area swept out as the planet moves around the Sun, using time as a
parameter. Therefore, from the Fundamental Theorem of Calculus (See Chapter
3, Plane Curvature, Arc Length and the Fundamental Theorem of Calculus)
we will deduce Kepler's second law, that equal areas are swept out in equal
times.
The
following is the Table of Contents for this 6 page Microworld
- A Model Solar System
- The Algebra and Geometry of Plane Vectors
- Three Dimensional Vectors and Curves in Space
- Kepler's Second Law
- Symbolic Calculator
Each page
of the Microworld, including the Calculator page has the story for that page
under the 
icon.
Just click on this icon to read the story for the page. The Calculator is
quite versatile, and so we recommend you read through the instructions there
to become familiar with it.
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
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