🚀 go-pugleaf

On May 12, 2:16 pm, Yanick Toutain <yanicktout...@hotmail.com> wrote:
> Il suffit pour cela de remplacer vorb par la circonférence d'un cercle
> divisé par la période (pour la Lune, c'est quasiment le mois)
>
> Vorb = 2 * pi * L / mois
>
> 4 pi² * L² / mois ² = GM / L
>
> et on obtient
>
> mois²/ L^3 = 4 pi² / GM

I was just thinking about that a couple days ago, after PBS aired the
biography of Newton. In fact, this is none other than Kepler's third
law -- but now applied to the Earth and everything about it, rather
than to the sun!

The subject came up again for me, as I was reading through an edition
of the Encyclopedia of World History that I have. The mid 1970's
version of Langer's classic:

http://www.amazon.com/encyclopedia-history-medieval-chronologically-arranged/dp/B000CQCHUI/ref=sr_1_21/102-7610034-5457715?ie=UTF8&s=books&qid=1179362867&sr=8-21

Interestingly, there are details relating to the mid 1400's-1700's,
with respect to the science history, that are not well-known and
rarely mentioned.

For one, the first book on calculus was published when Newton was
around 13. This was all in a period of time when the knowledge-base of
Europe was being converted en mass over to book form, and this was
around the beginning of the new phase in history when new ideas and
publicasions began to come out. Its focus, of course, was not
systematized anywhere near as much as Newton or Leibniz's later
developments -- but both of them worked ultimately off of this
reference, which had the bare essentials of the theory of
infinitesimals and integrals.

It was later, Dr. Barrow (Newton's college professor) who introduced
(what we now call) the Fundamental Theorem of Calculus in his
lectures. Ultimately, that's what Newton was working off of and what
he expanded on.

Galileo did not come up with the Equivalence Principle. There was an
author published quite a bit earlier (in the 1500's) who first
hypothesized this principle.

Kepler was working off of a continually evolving set of observational
data, and various standards had been published and superseded by newer
standards over a couple generations of evolution before Kepler worked
on them. This had been going on for over a 100 years.

What's most interesting, is that Kepler originally only published the
first and second laws. The third law came a few years later in a
separate book.

But, from the first and second laws you can actually derive the third
law by a simple geometric argument (indeed, one that Kepler had
probably himself used, since the equatoin governing the motion around
an ellipse is now called Kepler's equation). This was the "geometric
derivation", also, that had apparently been alluded to in the PBS
biography on Newton -- the one that Newton sent Halley, when Halley
asked Newton what the solution of the problem was.

I came up with an entirely separate derivation long ago, while *I* was
working on the 2-body problem on my own. This one is completely
different from anything most people have seen before.

First, for rectilinear motion, consider escape velocity. For a
spherical body, assuming the field is symmetric, this velocity will be
only a function v(r) of the distance r from the center of the body. An
important feature is that if an object is going up at escape velocity
v(r) at position r, then it will STILL be going up at escape velocity
a short time, dt, later, but at a diminished speed v(r+dr) = v(r) -
(GM/r^2) dt, due to the effect of gravity. Meanwhile, its altitude
increases by dr = v(r) dt. Hence, the differential equation
   v'(r) dr = -(GM/r^2) dt; dr = v(r) dt
or, when combined,
   v'(r) v(r) = -GM/r^2.

Upon integration, one finds,
   v(r)^2 = A + 2GM/r
where A is a constant of integration. Since the v -> 0 as r ->
infinity, then it must be the case that A = 0. Thus, v^2 = 2GM/r.

This argument applies generally, for any radial motion. If the
velocity is less than escape velocity, then A = -2GM/a, where a is the
maximum altitude. Then
   v(r)^2 = 2GM/r - 2GM/a.
If the velocity is greater than escape velocity, then A = V^2, where V
is the asymptotic velocity. Hence,
   v(r)^2 = V^2 + 2GM/r.

Moreover, this argument generalizes to elliptical motion! Assume the
velocity is part of a field whose flow lines are ellipses with the
gravitational source at one of the ellipses' focus. Then, one has a
set of flow lines given in suitable coordinates by:
   x = a (cos E - e); y = b sin E
where a^2 = (ae)^2 + b^2 and 0 <= e < 1. The length of the radius
vector is then
   r^2 = x^2 + y^2 = (a(1 - e cos E))^2 --> r = a (1 - e cos E).
The set of ellipses are those given as a ranges from 0 to infinity.
The quantity e is fixed, and b is defined in terms of a and e.

This incorporates Kepler's first law. So, now the question is: how
does E vary with time? Or ... how does time vary with E, if you think
of E as being the independent measure of time (as our calendar
traditionally did).

Let W = dE/dt denote the rate of change of E. The quantity a is a
constant for the orbiting body, by assumption.

The velocity has components
   u = dx/dt = -a sin E W; v = dy/dt = b cos E W.
The acceleration has components
   du/dt = -a cos E W^2 - a sin E dW/dt; dv/dt = -b sin E W^2 + b cos
E dW/dt.
Over a short time, the radius vector will trace out an area given by
   H dt = (x dy - y dx)/2 = (x v - y u)/2 dt,
where H is the constant referred to by Kepler's second law.

Then
   2 H = xv - yu = a (cos E - e) b cos E W + b sin E a sin E W = ab W
(1 - e cos E) = r b W.
or
   2H/ab = (1 - e cos E) dE/dt = d/dt (E - e sin E).

Thus, the orbital motion is given by
   E - e sin E = 2H/ab (t - t_0)
and has a period (as E ranges from 0 to 2 pi) given by
   T = pi ab/H.

Since W = 2H/rb, and dr/dt = ae sin E W, then
   dW/dt = -2H/(b r^2) ae sin E W = -ae sin E/r (2H/rb)^2
so that the acceleration is given by

du/dt = -a cos E (2H/rb)^2 - a sin E (-ae sin E/r (2H/rb)^2) = -a (2H/
rb)^2 x/r
dv/dt = -b sin E (2H/rb)^2 + b cos E (-ae sin E/r (2H/rb)^2) = -a (2H/
rb)^2 y/r

Thus, the acceleration is inversely proportional to the square of the
distance from the center of the gravitating body, with a constant of
proportionality given by
   d^2 x/dt^2 = -K/r^2 (x/r); d^2 y/dt^2 = -K/r^2 (y/r); K = a (2H/
b)^2.

Substituting in for the orbital period, one obtains
   T^2 K = (pi ab/H)^2 (2H/b)^2 a = 4 pi^2 a^3
or
   T^2/a^3 = 4 pi^2/K.

Hence, Kepler's third law.

What Newton actually discovered is not so much this, but the relation
between K_{earth} and K_{sun}. This has the ratio

   K_{sun}/K_{earth} = (150000000 km/400000 km)^3 / (1 year/1 lunar
month)^2 = 375^3/13^2.

But since the sun and moon appear as about the same size, then the
respective diameters are in the ratio
   D_{sun}/D_{moon} = 375
and the respective volumes
   V_{sun}/V_{moon} = 375^3.

During lunar eclipses, one sees a shadow about 3 times larger than the
moon go across the moon. This shadow (as the Greeks knew) is the
Earth's shadow. Therefore,
   D_{earth}/D_{moon} = 3 and V_{earth}/V_{moon} = 3^3.
Thus,
   V_{sun}/V_{earth} = 375^3/27
and
   K_{sun}/K_{earth} = 27/169 V_{sun}/V_{earth}
or
   K_{sun}/V_{sun} ~~ 1/6 K_{earth}/V_{earth}.

This is consistent with the hypothesis that the constants are related
to the masses M_{sun} and M_{earth} by a universal constant
   K_{sun} = G M_{sun}; K_{earth} = G M_{earth}
provided one assumes the density of the sun's matter is about that of
water!

So, not only does one discover the law of gravity, but one also gets
the first hint as to what type of matter the sun is made out of.