whitehorse456
(2007-08-07)
Is [Newtonian] gravity a theory or a law?
Everything becomes clear if you assign their proper meanings to words like "theory",
"law" etc.
In a scientific context, "theory" is not an insult
(as in the silly put-down "it's just a theory").
A theory is simply the most elaborate form
of consistent scientific knowledge
not yet disproved by experiment.
In experimental sciences, a theory can never be proved,
it can only be disproved by experiment.
This is precisely was makes a theory scientific.
A statement that cannot be disproved by experiment may still be highly respectable
but it's simply not part of any experimental science
(it could be mathematics, philosophy or religion, but it's not physics).
Now that we have the basic vocabulary straight, we may discuss gravity itself...
Gravity is a physical phenomenon which is obvious all around us.
As such, it's begging for a scientific theory to describe it accurately and consistently.
The rules within a theory are called laws
and the inverse square law of the Newtonian theory of gravitation
does describe gravity extremely well. Loosely stated:
Two things always attract in direct proportion to their masses and
in inverse proportion to the square of the distance between them.
However, the Newtonian laws are not the ultimate laws of gravity.
We do know that General Relativity (GR)
provides more accurate experimental predictions in extreme conditions
(e.g., a residual discrepancy in the motion of the perihelion of Mercury
is not explained by Newtonian theory but is accounted for by GR).
Does this mean Newtonian theory is wrong ?
Of course not.
Until we have a
theory of everything
(if such a thing exists) any
physical theory has its own range of applicability
where its predications are accurate
at a stated level of precision (stating the accuracy is very
important in Science; an experimental prediction is meaningless
if it does not come with a margin for error).
The Newtonian theory is darn good
at predicting the motion of planets within the Solar System to many decimal places...
That's all we ask of it.
Even General Relativity is certainly not
the ultimate theory of gravitation.
We know that much because GR is a classical theory,
as opposed to a quantum theory.
So, GR is not mathematically compatible with the quantum phenomena
which become so obvious at very small scales...
Science is mostly a succession of better and better approximations.
This is what makes it so nice and exciting.
If you were to insist at all times on "the whole truth and nothing but the truth"
in a scientific context, you'd never be able to make any meaningful statement
(unless accompanied by the relevant "margin for error").
As a consistent body of knowledge, each theory allows you to make such statements freely,
knowing simply that the validity of your discourse is only restricted by
the general conditions of applicability of a particular theory.
Without such a framework, scientific discourse would be crippled into utter uselessness.
(2008-08-22)
Comparing Gravity and Electrostatics
The inverse square law of Newtonian gravity is also valid for
electrostatics:
The force between two electric charges is proportional to the
charges and inversely proportional to the distance between them.
Consider two bodies with the same mass m carrying the
same electric charge q. If the following relation holds,
there won't be any force between them, as their gravitational attraction
is balanced by their electric repulsion, at
any distance:
G m 2 = q 2 /
4p e0
This happens when q/m =
Ö(4p e0 G)
=
8.617350(44) 10-11 C/kg.
In other words, two bodies carrying one elementary charge
(1.602 10-19 C)
have no net force between them if their mass is about
1.86 mg.
(2007-09-29)
Rigid Motion of a Rotating Triangle
A rigid motion of three equidistant gravitating bodies,
as they rotate around their common center of mass O.
The equilateral triangle at right tells the whole story:
If the bodies at A, B and C attract each other in direct proportion to their masses,
the so-called paralellogram law for
vector addition
does indicate that each body is subjected to a centripetal
acceleration toward O,
whose magnitude is proporttional to its distance to the common
center of mass O. (With a suitable scaling to
represent accelerations, the geometric construction of the center of mass
matches the parallelograms involved in vector addition, as depicted above.)
This means that the triangle ABC rotates rigidly
about its center of mass O.
Note that this much is true regardless of the dependence of forces on distance,
since the 3 bodies are at the same distance from each other.
Quantitatively, the square of angular velocity
w is the scaling factor of the above diagram:
To a distance R corresponds an acceleration
w2 R.
This remark allows the value of that scale to be obtained
geometrically in terms of Newton's
universal
gravitational constant (G) :
w
as a function of d = AB = AC = BC
|
w2 d 3
= G M
= G
( m A + m B + m C )
|
Proof :
In the diagram, we observe that the arrow extremities
divide each side (of length d) into three segments whose lengths are
proportional to the three masses (the coefficient
of proportionality being d/M).
Thus, an arrow toward B (from
A or C) translates (by scaling lengths into
accelerations) into the following component of
the acceleration, which is equated to its gravitational counterpart
(using Newton's inverse square law)
to yield the advertised relation.
w2
m B ( d / M )
= G m B / d 2
(This reduces to
Kepler's third law
when one body has negligible mass.)
(2007-10-08)
Lagrange points of two bodies in circular orbit
The 5 points where gravity balances the centrifugal force.
The above can be applied to the case of two bodies
in circular orbit around each other: A third body of
negligible mass would follow their rotation rigidly if it's
in the plane of rotation and forms an equilateral triangle with those two bodies.
There are two such points (called L4 and L5).
These are stable locations (in the sense that they seem
to attract nearby test masses)
provided the ratio of the larger mass to the smaller one exceeds 24.96
or, more precisely:
½
( 25 + 3 Ö69 )
= 24.959935794377112278876394117361238...
L4 (the "Greek" triangular point) leads the
smaller body in its orbit around the larger one, while
L5 (the "Trojan" or "trailing" triangular point) lags behind.
L4 and L5 are sometimes collectively known as the "Trojan points".
Several asteroids which reside there in the Sun-Jupiter system have been
named after legendary heroes of the Trojan war.
The leading triangular point L4 is home to the
Greek
camp led by
588 Achilles
(discovered in 1906 by Max Wolf)
with 659 Nestor,
911 Agamemnon,
1143 Odysseus,
1404 Ajax,
1583 Antilochus,
1437 Diomedes
and 1647 Menelaus.
The trailing Trojan point L5 marks the
Trojan
camp where
884 Priamus,
1172 Aeneas,
1173 Anchises
and 1208 Troilus
reside.
Early naming has left only two so-called "spies"
(both discovered in 1907 by August Kopff)...
617 Patroclus
is the lone Greek in the Trojan camp.
624 Hector
is the lone Trojan among the Greeks.
In addition,
there are three unstable Lagrangian points
(aligned with the two orbiting bodies) where the centrifugal force
exactly balances gravity.
L1 (the inner Lagrangian point) is located
between the two orbiting bodies. L2 is outside those two
bodies, on the side of the lighter one, while
L3 is on the side of the heavier one.
