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Newton’s Great...
Oversight
Galileo’s Falling Bodies and
Lagrange’s Trojan Asteroids
With Their Tadpole and Horseshoe Orbits
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2 NEWTON’S GRAVITY
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SECTIONS
2.1 Newton’s Laws
2.2 “Infinitesimals”
and Levels of Approximation
2.3 Simple Equations
2.4 A Simple 3-Body
Problem
2.5 A Quick Look
at the Separate Release Case… and Einstein’s “Relativity”
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2.1 Newton’s Laws
Newton’s laws of
gravity (see
equations) included as fundamental that:
-
all masses have a non-zero gravitational effect on all
other masses; each mass exerts a force on each other mass that is
proportional to each of the 2 masses and inversely proportional to the
square of the distance between them
-
all masses fall in/through space, e.g. toward each other
The first of these means that, in theory, no mass can
truly be classed as “infinitesimal”,
i.e. as having no gravitational attraction of other bodies (more, below).
The second means that the concept of “falling” must be other-body
relative, e.g. Earth-relative.
Using Newton’s laws, one can give simple
algebraic-trigonometric expressions for the initial instantaneous
accelerations of both the lighter and heavier falling bodies, as well as
that of the Earth, toward each other. To get the falling rate difference
of the lighter and heavier bodies we can, in modern terms, sum the vector
components of accelerations of the lighter body toward the Earth and of
the Earth toward the lighter body, likewise sum those of the heavier body
and the Earth, and take the difference. A variant of this would have been
simple enough even for Newton’s lesser contemporaries.
Here we deal only with the case that the lighter and
heavier bodies are equidistant from the 3rd, Earth-like body.
Newton’s laws make explicit that the distance between masses affects the
accelerations of bodies — i.e. the inverse square distance-force law — and
therefore affects their relative falling rates.
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SECTIONS
2.1 Newton’s Laws
2.2 “Infinitesimals”
and Levels of Approximation
2.3 Simple Equations
2.4 A Simple 3-Body
Problem
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2.2 “Infinitesimals”
and
Levels of Approximation
For review: it is a standard technique of
calculational convenience in physics to approximate some quantities, such
as e.g. relatively very small masses, as “infinitesimals”. An
“infinitesimal mass” is one which has “effectively zero” gravitational
effect on the other masses — in a technical but strict violation of
Newton’s Law of Gravity. So an infinitesimal mass would be “small enough”
to not attract any other masses, but would be “large enough” to
be attracted by other non-infinitesimal masses.
Using infinitesimals can make sub-classes of some
problems simple enough to be more conveniently solved, or even “solved” at
all. I.e. it can make mathematical analysis possible, or much easier (but
also potentially a much poorer approximation). It can make some
calculations much easier and/or faster since potentially very many
computations need not be performed. In fact this is how Galileo’s
experiment is usually implicitly analyzed. The 2 bodies that we gedanken-drop
from the Tower of Pisa have only a very small mass compared to the Earth,
and they are usually considered to be infinitesimal masses. Since “they do
not affect” the accelerations of other masses, we only get the
accelerations due to the gravity of the Earth that Eq. 1c describes, where
m2 in Eq. 1c would be the mass of the Earth.
But... (and this
needs to be emphasized):
When
we say that both the 2 falling bodies are “infinitesimal”, we are implicitly
assuming that their mass difference can be no greater than
“infinitesimal”; but when we say that 1 of the 2 falling bodies is
actually heavier than the other, we are explicitly assuming that their
mass difference is not “infinitesimal” (i.e. that it is
“effectively not zero”), rather that this mass difference is
infinitely greater than their assumedly “infinitesimal” masses.
These assumptions are inconsistent, and worse, they prejudice
the result so much that we altogether miss an extremely simple approach to
Trojan-Lagrangian points and bodies.
If we have 3 bodies, we can readily discern 4 levels
of abstraction and gravitational approximation:
0) none of
the bodies gravitationally affects any of the others
1) 1 of the
bodies gravitationally affects the others,
but is not affected by them
2) 2 of the
bodies gravitationally affect each other and the 3rd,
but are not affected by the 3rd
3) all 3
bodies affect each other gravitationally, as per Newton’s laws
The 0th level of approximation is not
without its uses since it is actually the kind used in e.g. thermodynamic
models of gas kinetics (usually with very many more bodies).
The 1st
level of approximation is the level at which Galileo is still
“scientifically correct”; it is the level which Kepler implicitly assumed
when he had the Sun, rather than the common center of mass, at the focus
of the elliptical orbits of the planets; it yields a good engineering
approximation for non-orbital falls (therefore of short duration) of
lighter and heavier bodies.
The 2nd level of approximation is
the level associated with Lagrange’s analysis of Trojan points; it does
not strictly hold here, though, since it also required (for “stability”)
that 1 of the 2 non-infinitesimal bodies be of an “intermediate infinitesimality”
(m2 < ~0.04 m1).
The
3rd level is the one we will examine here, and we will
make no assumptions about the relative masses. However, neither we will
analyze the “stability” of the “equilibrium” (i.e. when “small enough”
perturbing influences are acting throughout the not well-defined tadpole
or horseshoe regions).
At the 2nd and 3rd levels of
approximation where 2 or all 3 bodies respectively are considered
non-infinitesimal, lighter and heavier bodies exhibit a mostly
non-zero falling rate difference. Here it is important to note that we are
already accepting that the distance from the Earth or Earth-like body, the
distance from which we will release the lighter and heavier bodies, will
affect the falling rates of those bodies — already a theoretically
important deviation from precisely equal falling rates. As we noted
earlier, the falling rate difference is a function of their angle of
separation. Thus, we abstract to a situation where the lighter and heavier
bodies are point masses equidistant from the point mass of the Earth, and
thus that the 3 point masses form an isosceles triangle (2 sides and 2
angles the same).
When the falling rate
difference (due to the mass difference) is studied, one finds that:
-
it is proportional to the mass difference
-
it is inversely proportional to the square of the common distance
of the 2 masses from the point center of mass of e.g. the Earth
-
it is a function of the angular separation of the 2 bodies (with
e.g. Earth as the vertex)
-
it changes sign between 0 and 180 degrees, and, in fact,...
-
it zeroes at ±60 degrees
Now things start to get interesting because ± 60
degrees are the angles associated with the Trojan points predicted by
Lagrange, using his perturbation theory. The falling rate difference shows
itself during the prolonged fall of orbiting bodies, most visibly in the
dynamics of those mysterious and fascinating bodies today known as Trojan
asteroids.
Lagrange’s result concerning stability (which does
need at least something like his perturbation theory, and which is not
demonstrated here), however, assumes that there are 2 very large masses,
with 1 very much larger than the other, m2 < ~0.04 m1. He analyzed the planet Jupiter — our largest planet, about
0.1% the
mass of the Sun — in orbit around the Sun. (And it was a long time before
people started thinking about Trojan asteroids associated with other
planets — or moons. Jupiter seemed so massively unique.)
Lagrange studied what would happen to
infinitesimal bodies (see above) that found themselves near certain points in
relation to the 2 other very much larger masses. (Even large asteroids
were considered infinitesimal — in comparison to Jupiter — for Trojan
point analysis.) In particular, Lagrange studied what happens when the 3
bodies, subject to the above restriction on relative mass, form an
equilateral triangle and concludes that such a point in relation to the
others is a stable one for an infinitesimal body to occupy. I.e. asteroids
will orbit (in a fascinating “tadpole”
orbit; see
Figure 5) or “possibly” equilibrate at
one of the 2 equilateral triangle points
(the “Trojan points”; see
Figure 4) in the orbit of Jupiter, 1 leading Jupiter by 60 degrees, 1
trailing it, even when perturbed (within limits). It can take hundreds of
years to complete such an orbit.
But it is actually very simple to demonstrate the
result that, if unperturbed, any 3 arbitrary masses can
remain in an equilateral triangle — even one expanding and contracting —
in an equilibrium orbit around their common center of mass. Although here
we are still missing the critical aspect of stability, this result has
fascinating implications for both professional and amateur astronomy in
the 21st Century. We should remember that stability will always be
relative to the perturbing forces; i.e. if the perturbing force vector is
great enough and in the right direction, no perturbed body/associated
orbit will be “stable”. If
there is even a small potential energy well, but a smaller perturbing
force, any body/orbit can potentially be “stable”. Lagrange, too, missed that
lighter and heavier bodies fall at different rates, and that this gives
rise to Trojan points, so perhaps he also missed other interesting things in
his analysis. It is hoped that this last will inspire a new generation of
science and scientists.
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SECTIONS
2.1 Newton’s Laws
2.2 “Infinitesimals”
and Levels of Approximation
2.3 Simple Equations
2.4 A Simple 3-Body
Problem
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2.3 Simple Equations
Lagrange’s perturbation theory is notoriously complex
and difficult, even for professional physicists and astronomers. Also
notoriously complex and difficult for many is the mathematics of
differential equations, which is usually used to prove one of the results
that will be demonstrated here, i.e. that 3 arbitrary masses equidistant
from each other will theoretically remain equidistant from each other if
given the proper initial positions and velocities — given that they remain
free from perturbations, which never happens in the real world.
Unfortunately, the differential equation approach also misses the falling
rate difference.
Here, however, this unperturbed Trojan point result
is demonstrated using only Newton’s laws, high school algebra and trigonometry,
and of course the preliminary result that lighter and heavier bodies fall
at different rates except at the Trojan points. The usual differential
equation approach is not necessary. Not even calculus is needed, not
Newton’s, not even Leibniz’s. It is hoped that this will make the study of
Trojan points and bodies accessible even to high school and other
beginning physics and astronomy students. It is also hoped that the
derivation of these results from questioning the accepted “scientific law”
of equal falling rates will be an inspiration to all students and teachers, of
any level of experience or sophistication, of any subject.
The only mathematics and equations we need for a
fascinating reexamination of the falling bodies of Galileo, and a first
examination of Newton’s oversight, are quite simple and well known today, even
by high school standards. They are about as simple as mathematics and
equations can be and still do useful science. Any first year undergraduate
physics or astronomy major, even in an American university or college, is
expected to be able to handle such equations... or change majors. First we
have:
Newtonian gravity
infinite
speed of gravity (i.e. we ignore the finite speed of gravity)
mi are masses
F is Force
ai are accelerations
G is the
Gravitational constant
r is the
radial (linear) distance (between the masses)
Eq. 1a:
Then we have his equation
for his law of gravity:
Eq. 1b:
where the force on each mass
is directed toward the other mass. Since this last equation gives us the
force on each mass represented, combining them we get:
and thus
Eq. 1c:
Note
that, by Eq. 1c, the (initial, instantaneous, and, most
importantly, absolute) acceleration of a body in a gravitational
field does not depend on its own mass, but only on the mass of the other
body (or bodies), their mutual distance(s), and of course the
gravitational “constant”. (It is essential regarding the relevance to
relativity of this acceleration, at least of its equation, to note that it
is purely in terms of the absolute Newtonian reference frame, which
theoretically cannot
exist in relativity. And at least some astronomers and cosmologists
keep wondering from time to time whether it actually is constant; some are
now even including it as a parameter in calculations relating to “dark
matter”.)
It
is this last equation (Eq. 1c) that, taken by itself, has lead many
people to believe that Galileo was scientifically correct, and that
lighter and heavier bodies will fall at precisely the same rate (with
the implicit assumption that both those masses are “infinitesimals”;
see
comment). But in reality it is only one of many force components
that might act on each body, including the Earth. Since falling is e.g.
Earth relative, we must also take its acceleration into account in a
complete analysis.
Eq. 1d: 
Eq. 1e: 
Eq. 1f: 
(if  )
Eq. 1d shows the falling rate or
relative acceleration of 2 bodies in the reference frame of either of the
bodies, as opposed to the absolute acceleration of 1 of the bodies in the
absolute Newtonian frame of reference. Eq. 1e shows the difference in
falling rates, Earth relative, of 2 masses (released separately). Eq. 1f
shows the ratio of the falling rate difference to the absolute
acceleration due to Earth’s gravity (assumed to make up the vast majority
of the relative falling rate). Note that it doesn’t depend on either the
distance r or on the gravitational constant G, just on the
relative masses. Since the mass of the Earth is ~5.975·1024 kg,
the falling rate difference between a 2 kg mass and a 2 kg mass is ~1.67
parts in 1025 (except “relative” to an absolute Newtonian
reference frame). Newton should never have overlooked this.
Einstein should never have overlooked this. Eddington… well, it goes
without saying.
It is this last equation (Eq. 1c) that,
taken by itself, has lead many people to believe that Galileo was
scientifically correct, and that lighter and heavier bodies will fall at
precisely the same rate (with the implicit assumption that both those
masses are “infinitesimal”;
see
comment). But in reality it is only one of many force
components that might act on each body, including the Earth. (Since
falling is e.g. Earth relative, we must also take its acceleration
into account in a complete analysis.)
Here we will continue the usual policy and abstract
out viscosity, buoyancy, gravitational anomalies, the speed of gravity,
etc. We will consider only the Newtonian-gravitational forces of the 3 bodies taken as
point masses at their centers of mass. But, we still get 2 forces
acting on each body in this, our gedanken replay of Galileo’s Tower of Pisa experiment:
the gravitational forces from the other 2 of the 3 falling bodies. Most
people ignore the fact that the Earth falls, even though Newton’s theory
says it does. And in our case it falls/accelerates due to gravity at a
non-zero rate toward the 2 bodies. It falls slightly faster toward the
heavier one. And even more importantly, the 2 bodies whose falling rates
we are comparing fall toward each other.
Although we could “get away with it” and ignore these
terms as physicists do in the simpler levels of approximation, here we
will choose not to. In fact we need to take all the forces (of our
abstract case) into account to find the easy, low road to the Trojan
points — crediting Lagrange with taking the difficult, high road — the low
road that Newton, of all people, should not have missed... but did.
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SECTIONS
2.1 Newton’s Laws
2.2 “Infinitesimals”
and Levels of Approximation
2.3 Simple Equations
2.4 A Simple 3-Body
Problem
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2.4 A Simple 3-Body
Problem
Refer to
Figure 1. Of the 6 acceleration vector components there are 8
sub-components, Eqs. 2a through 2h, that are needed to compute the net
accelerations of the bodies toward each other at the instant of release —
we are not going to trace their trajectories, here — of both the lighter
body and the Earth toward each other, and the heavier body and the Earth
toward each other. Some of these will be a function of the angle between
the 2 bodies with the Earth point center of mass as the vertex. (Eqs. 2a
and 2b were already discussed above as Eq. 1c.)
3 point
masses, arranged in an isosceles triangle, masses constant
2 dimensions
will be sufficient for our purposes, even with 3 masses
 is
the mass of the Lighter body
 is
the mass of the Heavier body
 is
the mass of the Earth
r is the
distance from the Earth to either body (both are at the same distance in
this case)
q is the angle between the
lines between each body and the Earth (with the Earth as the vertex)
Eq. 2a:
 the
acceleration of
toward
due
to
Eq. 2b:
 the
acceleration of
toward
due
to
Eqs. 2a and 2b are both the standard “acceleration
due to gravity” (if r is the radius of the Earth) which are the
same for both the lighter and heavier bodies. Note that neither
acceleration is a function of the angle
q. Near Earth (i.e. in most
sub-orbital situations) they are adequate as an engineering approximation,
but we are interested here in a simple mathematical description of the
subtle differences between the standard approximation and reality that
will lead to the Trojan points.
The 6 following equations are the accelerations due
to gravity that are standardly ignored as “infinitesimal”:
Eq. 2c:
 the
acceleration of
toward
due
to
Eq. 2d:
 the
acceleration of
toward
due
to
We see that the Earth falls
faster toward the heavier body, and again, neither acceleration is a
function of the angle q.
This latter changes for the next 4 accelerations.
Eq. 2e:
 the
acceleration of
toward
due
to
Eq. 2f:
 the
acceleration of
toward
due
to
This is a good place to
remind ourselves that there is no necessity that the mass designated as
“Earth” must be the greatest mass, so this result will depend neither on
relative mass nor on the “infinitesimality” of any mass. If it is by far
the greater mass of the 3, then the terms from Eqs. 2e and 2f will of
course be much smaller than those from Eqs. 2g and 2h.
Eq. 2g:
 the
acceleration of
toward
due
to
Eq. 2h:
 the
acceleration of
toward
due
to
(NOTE that there is a
subtlety in the system of angles in this problem such that, if we switch
the positions of the 2 bodies, the angle that affects these last 2
equations (not actually the angle in
Figure 1 labeled q /2)
changes sign. The results are in fact symmetrical, and it is easiest to
take the absolute value to deal with the seeming asymmetry.)
When we combine all the terms to get the total
accelerations of each body and the Earth toward each other, we get:
Eq. 2i:
the total acceleration of
and
toward
each other
Eq. 2j:
the total acceleration of
 and
toward
each other
Combining these to get the
difference of falling rates of the heavier and lighter bodies we get:
NOTE that the 2 terms
representing the usual acceleration due to the gravity of Earth cancel.
NOTE WELL that these are the only terms that depend on
.
Also note that when performing a calculation on a computer it is best to
leave these terms out to help minimize floating point errors (accuracy).
Proceeding we get:
expanding which we get:
Combining terms we get:
The Falling
rate difference equation,
function of the mass difference and the
angle of separation:
Eq. 2k:
which is the simplified
expression for the difference of the falling rates of the heavier and
lighter bodies.
For a quick look at a plot
of a non-zero falling rate difference as a function of the angle between
them (sample masses), click on thumbnail, below. For a slightly more detailed
explanation, refer to
Figure 2.
NOTE that there are 2 factors that can be zero:
1) if the masses are equal then
their difference is zero, and the falling rate difference due to
their mass difference is zero, as it ought to be
2) and if the trigonometric
factor is zero, the falling rate difference will be zero. It is easy to
calculate that that factor will zero if and only if
q is ± 60° (± π/3 radians)
q is the angle between the
lines between each body and the Earth (with Earth as the vertex)
Eq. 3:
and we note that:
f (60°) = 1-cos(± 60°)-1/(4sin(|± 60°/2|))
= 1-1/2-1/(4×1/2)
= 1-1/2-1/2
= 0
It is an exercise in basic trigonometry to show that
± 60° are the only 2 roots of
 .
These are the angles — and the only angles — which place all 3 bodies at
the vertices of an equilateral triangle with regard to one another, i.e. 1
of them is in Lagrangian point L4 or L5. Refer to
Figure 2 to see the falling rate difference as a function of angle,
and a little more detailed explanation.
So, we have shown that Galileo was in fact
scientifically incorrect about lighter and heavier bodies
falling at (precisely) the same rate. If we consider just 3 point masses
and Newtonian gravity:
-
the heavier body and the Earth will fall together faster if
they are further apart than 60°
(with the center of mass of the Earth as the vertex)
-
only at precisely 60°
— i.e. the only place where Galileo is scientifically correct — will they
fall at precisely the same rate (approximately)
-
and paradoxically, the lighter body and the Earth will fall
together faster if they are closer together than 60°
Humorous remarks are
possible about Aristotle being 2/3 correct (between 60°
and 300°), Galileo being correct
only on a set of “measure zero” (between + 60°
and + 60°, and between - 60°
and - 60°), and, ironically,
neither being correct between + 60°
and - 60°, which last would have
pertained if Galileo had actually performed his apocryphal Tower of Pisa
experiment, but we will not indulge... at this time.
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2.1 Newton’s Laws
2.2 “Infinitesimals”
and Levels of Approximation
2.3 Simple Equations
2.4 A Simple 3-Body
Problem
2.5 A Quick Look at the Separate Release
Case… and Einstein’s “Relativity”
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2.5
A Quick Look at the Separate Release Case…
and Einstein’s
“Relativity”
Before moving on, we will take a quick look at the
separate release case, by looking at Eq. 2k. If we ignore the angle that
pertains to simultaneous release, this equation gives us the difference in the accelerations of the Earth toward the lighter and
heavier bodies, which difference is trivially non-zero. (The ratio of the
acceleration difference to the acceleration due to Earth’s gravity is
trivially the ratio of the mass difference of the lighter and heavier
bodies to the mass of the Earth. If we take the mass of the Earth as
5.975·1024 kg, a 1 kg mass difference will give rise to an
acceleration difference that is about 1.67 parts in 1025,
independent of the common distance of the bodies from the Earth.)
Only Eqs. 2a and 2b make it seem that the lighter and
heavier bodies experience “equal acceleration”. But they also make clear
the fact that this “equal acceleration” only occurs in the very first
instant (after which the differing masses have caused a different shift
of the other masses/energies), and is only relative to Newton’s absolute
space-time frame of reference, not what one can call a “truly
relativistic” frame of reference which would have to be be attached to
matter-energy based (non-inertial) frame.
This
last brings up an interesting possibility. Although it is pretty much purely theoretical, we can perform
a gedanken experiment such that if one can find a frame of reference in
which lighter and heavier bodies accelerate at the same rate (not counting
precisely 3 bodies in an equilateral triangle), then that frame of
reference must be an absolute Newtonian-style frame of reference. There is
no way in the real world to get an all-gravitational-conditions-equal
experimental framework for
the separate releases of 2 or more masses, but gedanken experiments have
no such limitations.
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