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Final Answers
© 2000-2018   Gérard P. Michon, Ph.D.    

     Newtonian Gravity

Nature and Nature's laws lay hid in night:
 God said "Let Newton be!" and all was light

Alexander Pope  (1688-1744)
 Isaac Newton 

Related articles on this site:

Related Links (Outside this Site)

The Universal Law of Gravitation  (Astronomy 161, University of Tennessee)
Lagrange Points (WMAP)  & mathematical complements  by  Neil J. Cornish.
Free-fall determination of G  (1997-1998)  with an  FG5 gravimeter.
On the importance of testing gravity at distances less than 1cm  by Silas R. Beane (1997)
Newton's Still Correct  (2001):  The Eöt-Wash Short-Range Experiment.
Wikipedia :   Lagrangian points   |   Trojan asteroids

Video :   MIT OpenCourseWare   Classical Mechanics by  Walter Lewin.
UC Berkeley  Physics for Future Presidents by  Richard A. Muller.
The Law of Falling Bodies (MU2) by  David L. Goodstein  (CalTech)  1 | 2 | 3


Newtonian Gravity

whitehorse456 (2007-08-07)   Is [Newtonian] gravity a theory or a law?

 Death mask of Isaac Newton 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).

 Coat-of-arms of 
Johannes Kepler (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

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.

 Come back later, we're
 still working on this one...

(2016-06-08)   Hodograph:  The curve traced by a  (celerity)  vector.
In a two-body Keplerian motion,  the  celerity hodograph  is a circle!

The term  hodograph  was coined in 1847 by  Hamilton  from the Greek words  'odoV  and  grajw,  with the intended meaning of  path written.

The  hodograph  of a vector is the trajectory of its extremity when its origin is fixed  (more pedantically, it's the locus of a point whose coordinates are the same as a varying vector's coordinates).

 Come back later, we're
 still working on this one...

"The Hodograph, or a New Method of Expressing in Symbolic Language the Newtonian Law of Attraction"
by  William R. HamiltonProceeding of the Royal Irish Academy3  (1847)  pp. 344-353.
That's Maths           Wikipedia :   Hodograph

(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

 Three equidistant masses 
 under mutual gravitation.
(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 from 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  wR.

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
w d 3   =   G M   =   G  ( m A +  m B +  m )

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       QED

(This reduces to Kepler's third law when one body has negligible mass.)

 Joseph-Louis Lagrange 
 1736-1813 (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.

Videos :   What are the Lagrange points?   by  Phil Plait  (TheBadAstronomer).
Lagrange points   by  Michael Merrifield  (in "Sixty Symbols" by Brady Haran).

(2010-12-31)   Geosynchronous Orbit
Satellites whose positions in the sky remain  nearly  fixed.

 Come back later, we're
 still working on this one...

(2013-03-27)   Gravitational self-energy of a uniform sphere
Negative, proportional to the mass squared and the inverse of the radius.

In a spherically-symmetric distribution of mass, the Newtonian gravitational field at any point is the same as what would be created at that point by whatever mass is located at a lesser distance from the center if that mass was concentrated at the center  (the field created by any homogeneous spherical shell at a greater distance is exactly zero).  This property of the inverse-square law is often called  Newton's theorem  (students of electrostatics may recognize it as a straight application of the theorem of Gauss).

In an homogeneous sphere of mass  M  and radius  R,  the mass within a distance  r  from the center varies as  r3  which makes the field directly proportional to  r.  All told, the radial field at distance  r  is:

- GM r / R3   when   r < R         and         - GM / r2   when   r > R

Multiplying this into an infinitesimal mass  dM  gives the gravitational force exerted on that mass at distance  r.  The  gravitational energy  dU  of  dM  within this mass distribution is obtained by integrating  that force from  r  to infinity  (making the convenient convention that attributes zero gravitational energy to very distant stuff, everything has negative gravitational energy):

dU    =   - G M dM   [   ò  R  x dx   +   ò  ¥  dx   ]
Vinculum Vinculum
r R3 R x2
=   - G M dM   [     R2-r2   +     1   ]
Vinculum Vinculum
2 R3 R

Now, the mass  dM  located in the shell between  r  and  r+dr  is equal to the shell's volume  4pr2dr  multiplied into the mass density  3M/(4pR).

dM   =   ( 3 M r2 / R3 ) dr

Therefore, the ball's total self-energy is given by the following integral:

U     =     ò  R   - G M   [     R2-r2   +     1   ]   ( 3 M r2 / R3 ) dr
Vinculum Vinculum
0 2 R3 R

Gravitational self-energy of an homogeneous sphere
of mass
  M  and radius  R :
U   =   - 6 G M2 / 5R   =   - 1.2 GM2/R

Something from nothing :

The above Newtonian expression of gravitational energy is sufficient to explain qualitatively how the entire Universe can have zero total energy.  Discarding the (interesting) effect of temperature, the total energy is:

M c2  -  6 GM2 / 5R

At face value, this would mean that a  (cold)  uniform ball of zero energy has a mass proportional to its radius.  In the presence of gravity, a ball of radius  R  created from  nothing  would thus have the following mass:

M   =   (5 c/ 6G)  R

Numerically, the constant   is   1.12224(14) 1027 kg/m.  This entails a density so large that Newtonian theory is no longer applicable; the above constant is 66.67% more than what's required to create black holes, according to General Relativity...

Thus, the problem is not so much to explain how gravity can create something from nothing, which turns out to be conceptually simple, but to realize that the mechanism is so powerful quantitatively that it cannot run  raw.  Somehow, the above calculation might qualitatively describe the creation of mass locked in particles of very small sizes  (essentially, tiny black holes whose ultimate Hawking evaporation is forbidden by the conservation of some quantized number, like electric charge).

Potential due to a uniform sphere  by  Richard Fitzpatrick,  UT Austin  (2011-03-31).
Deriving the gravitational potential energy   in "Dot Physics" by  Rhett Allain  (2012-03-27).
Gravitational potential energy   |   Sphere Gravitational Potential Energy  by  Eric Weisstein
Wikipedia :   Self-energy   |   Gravitational binding energy (for a uniform sphere)
Nordtvedt effet  (Dicke, 1964.  Nordtvedt, 1968)   |   Kenneth L. Nordtvedt (1939-)

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:

  1. The centrifugal field.
  2. The gravitational attraction of the sphere of mass M.
  3. 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.

 Come back later, we're
 still working on this one...

(2010-12-31)   Asteroid 99942-Apophis and gravitational keyholes
Wake-up call:  Detecting large asteroids on a collision course with Earth.

The  400 m  asteroid dubbed  2004 MN4  was discovered in June 2004.  Initially, that  Aten asteroid  (now called 99942 Apophis)  was given 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 Earth in 2036 and 2068.

On 2013-01-13, NASA officially ruled out the possibility of an impact in 2036  (based on fresh data)  as it also announced the record-setting flyby  of  2012DA14  (an asteroid about 40 m in size)  for Feb. 15, 2013.

Amazingly,  16 hours before  that actual event,  another  unrelated meteor  (about half the diameter of  2012DA14)  made front-page news as it exploded at an altitude of  23.3 km  in the atmosphere near the Russian town of Chelyabinsk, releasing more than 20 times the energy of the Hiroshima bomb!  Nobody had ever detected that one before it entered the atmosphere  (it's thought to have been an  Apollo asteroid).

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 )

(2013-01-21)   Dark Matter   (Oort, 1932.  Zwicky, 1933)
What makes galactical rims rotate so fast?

More than 99% of the mass of the  Solar system  is concentrated at its center  (within the Sun itself).  That's why orbital speeds around the Sun decrease fairly rapidly with distance.  (The orbital speed is the size of the orbit divided by the orbital period.  So, Kepler's third law implies that it varies inversely as the square root of the orbital radius.)

On the other hand, within a spherically symmetric distribution of mass of uniform density, the centripetal force of gravity is proportional to the radius  (this is a simple application of the theorem of Gauss, applicable to any inverse-square force field).  For the centripetal acceleration v2/R  to be proportional to R, the orbital speed (v) must be proportional to R as well.

The orbital speed  v  wouldn't depend on the radius  R  if the centripetal force was proportional to 1/R, which corresponds to a density of matter inversely proportional to the square of the radius  (by the theorem of Gauss).

In 1932, Jan Oort...

 Come back later, we're
 still working on this one...

The picture which might be emerging is that of a universe where the supermassive black holes formed first and pulled around them stuff which either gave birth to ordinary matter or didn't.

Dark Matter  with Meghan Gray, Ed Copeland & Michael Merrifield  (in "Sixty Symbols" by Brady Haran, 2010)
Wikipedia :   Galactical rotation curve   |   Jan Oort (1900-1992)   |   Fritz Zwicky (1898-1974)

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