Newtonian Gravity
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 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 has obvious manifestations 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 predictions are correct at a stated level of accuracy
(an experimental measurement 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
(which is roughly the mass of a dust mite).
(2011-03-15)
Binet's Formulas: Deriving Kepler's Laws
Motion of two isolated bodies gravitating around each other.
The two-body approximation discussed here applies with excellent
precision to the motion of two stars within a
binary system, because all other bodies are
either too small or too distant to influence their relative motion.
More importantly for the inhabitants of the
Solar system, this simple discussion also gives the
main motion of a planet around a star like the Sun.
As long as other planets do not come too close, their influences can be
either ignored or treated as small perturbations.
Historically, the two-body approximation
(with circular orbits) was first applied by
Isaac Newton to the
motion of the Earth and the Moon around each other.
Although the result is only a terse approximation of a very complicated
three-body motion (involving the Sun) the good numerical
agreement in this particular case
is what convinced Newton of the validity of his inverse-square law
of Universal Grabitation.
Because they are assumed to be isolated, the two gravitating bodies
revolve around their common center of mass
.../...
Jacques Binet |
|
Jacques Binet
(1786-1856; X1804)
is also known for another set of equations by the same name,
popular with recreational mathematicians and others:
Binet's formulas (1843)
give explicit expressions for the successive terms of the
Fibonacci sequence and other recursively defined sequences.
In practice, there's very little risk of confusion between the two sets of formulas.
Binet is also remembered for first describing the general rule
for matrix multiplication, in 1812.
He succeeded Poisson to the chair of
mechanics at Polytechnique in 1815.
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(2009-08-18)
Weighing the Earth (Airy's method, 1826 & 1854)
Pendulums at the top and bottom of a mine give the mass of the Earth.
In 1826, the idea occurred to
George
Biddell Airy (1801-1892) that the period
of a pendulum in a mine depends on the mass of the rock
above it and below it. The latter, which is essentially the whole Earth,
can be estimated from the former.
More precisely,
let's assume that the Earth, of radius R, has a mass distribution whose
density depends only on the distance r to its center.
That density, which varies with the depth
h = R-r, is r
near the surface and
its average is ro
ro =
3 M / (4p R 3 )
By Newton's theorem (the
theorem of Gauss applied to a spherically
symmetric distribution) the gravitational field g
at depth h
is the same as that which would be due to the total mass M located
at a greater depth if it was concentrated at the center of the Earth.
We'll obtain the variation of g by differentiation:
g = G M / r 2
dg = -2 G M / r 3 dr +
G / r 2 dM
For r slightly below R, we have
dM = -4pr 2 r dh (since dh = -dr) and:
dg/dh = 2 GM / r 3
- 4 G p r
=
2 GM / R 3
- 3 GM / R 3
(r/ro )
In the main, the gravitational field at a small depth h is thus:
|
g h = g 0 [ 1 +
( 2 - 3 r/ro )
h / R ]
|
This spherical model ought to be a good approximation of reality if we let
r be
the average density of the local
ground, since distant aspherical contribution to gravity
would yield nearly equal corrections at the top and bottom of the mine.
Airy had bad luck with two experimental attempts in 1826 and 1828 and he gave up
on the idea for a while (he went on with his life and became
Astronomer Royal in 1835).
In 1854 however, Sir Airy finally performed the
experiment conceived by his younger self, at the coal pit of
West Harton, near
South Shields.
Precision timing revealed that a pendulum placed at the bottom of the pit
was faster by about 2.24 s per day
(one part in 38572).
At a depth of 383 m gravity was thus
found to exceed surface gravity by one part in 19286.
Knowing that R = 6371000 m, the above equation
thus means that the average density of the Earth is
2.6374 times that of the rock at West Harton.
2 - 3 r/ro
= 6371000 / ( 19286 . 383)
so:
ro / r
= 2.63739747...
The value Airy published for the mean density of the Earth was
6.566 g/cc (the currently accepted value is around 5.5153).
This means that he estimated the density of the ground at West Harton
to be around 2.49, a value
reportedly provided to him by the mineralogist
William
H. Miller (1801-1880)
of Miller
index fame.
An estimate of 2.0912 would have given a perfect result.
The experimental details are delicate. Ideally, the pendulum should be located
near the center of a spherical cavity.
Local irregularities in the strata should be accounted for,
but this is impractical... Although
Robert von Sterneck
(1839-1910)
used better technology in 1882 and 1885,
he couldn't obtain consistent results from the similar experiments
that he conducted at the
St. Adalbert shaft of
Pribram
(Bohemia)
and the Abraham shaft of
Freiberg (Saxony).
In fact gravitational experiments involving large natural land masses are now mostly
used as evidence for the irregular distribution of superficial strata.
Apparently, similar experiments have never been carried out at sea...
Rigid pendulum (compound pendulum)
Wikipedia :
Pendulum
gravimeters
(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...
The Lagrange point L4 (the Greek
triangular point) leads the
smaller body in its orbit around the larger one, while
the Lagrange point 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 are
named after 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.
(2010-12-31)
Geosynchronous Orbit
How the braking effect of Earthly tides makes the Moon drift away.
Douglas G.
(2010-11-25)
Tidal forces from the Sun and the Moon
How the braking effect of Earthly tides makes the Moon drift away.
To investigate the causes and magnitudes of Earthly tides, we shall use
a simplified model where a perfectly spherical Moon moves along a circle
around the Earth which itself moves along a circular orbit around a
perfectly spherical Sun. We'll also make the drastic assumption that
the Earth is a solid sphere completely covered by a single ocean of
seawater (without any continents, islands or other irregularities
in the sea floor).
Consider two solid bodies (of masses m and M) with perfect spherical symmetry.
Their outer radii are respectively r and R. They are both gravitating around
their combined center of gravity (O) in a perfect circle.
The centers of the two spheres are at a distance D from each other:
- The distance from O to the center of m is MD / (M+m)
- The distance from O to the center of M is mD / (M+m)
- The whole system rotates rigidly with angular velocity
w
According to Newton's theorem
(i.e., Gauss's theorem
applied to Newtonian gravity)
the two spheres attract each other as would two masses concentrated at the center
of each sphere. The gravitational force between them is equal to:
F = G m M / D 2
= m w 2 MD / (M+m)
= M w 2 mD / (M+m)
Dividing the above by mM we obtain the following, in two different ways:
w 2 D 3
= G (m+M)
Let's consider a cartesian frame of reference rotating about O
at the angular speed w. The x-axis goes through
both of the centers of the spheres and the y-axis is parallel to the axis of rotation.
The z-axis is, of course, perpendicular to both of the above.
Let's study the apparent gravitational field at a distance R+z from the center of the sphere
of mass M (assuming that the sphere does not spin at all) at a point of latitude
q (with respect to the Ox axis) and longitude
j (where
j is defined to be zero for the half-meridian at the
surface of the M sphere in the xOy plane which is nearest to the m sphere.
It is the sum of three terms:
- The centrifugal field.
- The gravitational attraction of the sphere of mass M.
- The gravitational attraction of the sphere of mass m.
The vectorial sum of those three component, as computed below,
involves a main tidal term
(inversely proportional to the cube
of the distance D)
whose values on the Ox axis yield the size of the tidal
bulge of a liquid ocean coating the sphere of mass M.
(2010-12-31)
Asteroid 99942-Apophis and gravitational keyholes
Wake-up call: Detecting large asteroids on a collision course with Earth.
The asteroid dubbed 2004 MN4 was discovered in June 2004.
Initially, that asteroid
(now officially named 99942 Apophis)
was thought to have a 2.7% chance of impacting Earth on April 13, 2029.
Refined data shows that Apophis will come no closer
than 30000 km above Earth's surface on that date.
It will again come extremely close to the Earth in 2036 and 2068.
Near-Earth Object Program (NEO) at NASA
Techno-Science.net
(2005-11-03)
Refined
Path of Apophis toward Earth (2009-10-07)
analyzed by Steve Chesley & Paul Chodas (NASA / JPL)
Wikipedia :
99942 Apophis (2004 MN4 )
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