F = G * m_{1} * m_{2} / r^{2},

where F is the gravitational force acting between two objects, the m's are the masses of the two objects, and r is the distance between the centers of the two objects.

With the exception of possible relativistic factors, as far as science knows, that equation is exact.

Newton used that equation to derive the equations of motion for two objects orbiting (due to gravitational force alone) each other, and he got:

T^{2} = 4 * PI^{2} * a^{3} / Mu,

where T is the (sidereal) orbital period, a is the semi-major axis
(essentially the average distance between the two in elliptic orbits),
and **Mu is the product of the Gravitational Constant and the total
mass in the system**.

It is easy to get **T** and **a** extremely accurately by careful
observation, and so this equation can give an extremely accurate value for Mu.
The problem is that **neither** the actual exact mass involved nor the
Gravitational Constant is known very accurately, and so **even with a
very precise value for Mu, no really precise value for either the
actual mass in the system or the Gravitational Constant has been
possible.**

We can use an example. The Moon and Earth mutually revolve around a common point in space, which we call a Barycenter. The Barycenter is always at the exact point of the center-of-mass of the Earth-Moon system, so the Earth-Moon system orbits the Sun in a nearly exact elliptic orbit. Therefore, all of the gravitation effects which occur in the Earth-Moon system always act as though it is all at the Barycenter (and NOT as it has always been assumed to be, at the exact center of the Earth). The Earth-Moon Barycenter happens to always be slightly over a thousand miles deep within the Earth (actually about 2902 miles or 4677 km away from the exact center of the Earth). The Earth and the Moon orbit around the Barycenter in their paths in a period of one sidereal month, or 27.3216610 days, or 2,360,591.5 seconds, T. This results in the Earth and Moon "wobbling" around the Barycenter in their two separate somewhat elliptic motions. The center of the Earth constantly wobbles around this (stationary) Barycenter at around 793 mph (or 1276 kph) due to our daily rotating and also approximately another 27 mph (or 44 kph) due to the Moon's orbiting. The center of the Earth (and us) therefore constantly "wobbles" around the Barycenter at around 793 ± 27 mph. This velocity varies depending on many variables such as the exact time, the latitude and longitude of the location on the Earth's surface, the celestial longitude and latitude of the Moon in the sky, and complexities in the Moon's orbit. We know that the Moon's orbit has a semi-major axis of 384,749,900 meters. The equation above then gives:

(2,360,591.5 sec)^{2} = 4 * 9.8696044 * 384,749,900^{3} / mu

or

5.5723922 * 10^{+12} = 2.2485124 * 10^{+27} / Mu

We can solve this for Mu and get:

**4.035 093 5 * 10 ^{+14}** (for the Earth-Moon gravitational
system)

**This number is probably accurate to its eight significant digits.**

Many scientists incorrectly neglect the mass of the Moon and the existence
of the Barycenter, and they publicize a value for Mu of **3.986 004 4
* 10 ^{+14}** (but that value is actually not really accurate, due
to that neglecting of the mass of the Moon and the existence of the
Barycenter inside the Earth.)

In the best laboratory experiments, done in a vacuum with the most
perfect equipment available, the value of G, the Gravitational
Constant has been determined as being 6.67 * 10^{-11}
and no more accurately, due to equipment limitations on such
ultra-sensitive experiments. (See a graph down below).

With this value for G, we can get the total mass of the Earth-Moon system to be Mu / G, or:

4.0350935 * 10^{+14} / 6.67 * 10^{-11}

or

6.05 * 10^{+24} kg.

The Moon is accurately known to be
1 / 81.270 of the mass of the Earth, so the Moon accounts for
0.08 * 10^{+24} kg which leaves

**5.97 * 10 ^{+24} kg**

as the best available estimate for the mass of the Earth! Only to three significant figures! As a Physicist, I am ashamed that after three hundred years of having the gravitational equations, and all the equipment that modern science has, that's the best we can do!

If you are a thoughtful person, you might think, Aha, I can simply drop a precisely known mass from a tall building and really accurately measure the speed (and therefore acceleration) it experiences while it falls. This is harder to do than it sounds, since air friction slows it down, but such an experiment can be done in a near vacuum, and extremely accurate values for the acceleration due to gravity are known. Since we know the mass of that object extremely well and also the acceleration, Newton's F = m * a means we can also know the exact force acting on it due to the Earth's gravitational attraction.

F = G * m_{1} * m_{2} / r^{2},

This is again Newton's universal gravitation equation, and we now know the left side extremely accurately, being m * a. So now this equation can be written:

m_{2} * a = G * m_{1} * m_{2} / r^{2}.

where m_{2} is the mass of our object, m_{1} is
the mass of the Earth, and r is the distance between the two, which
is the radius of the Earth. Continuing, we have:

a = G * m_{E} / r^{2},

Notice that again we have extremely accurate (measured) values
for a and for r, and so we can solve for an extremely accurate
value for the product G * m_{E}. That's Mu again!
Even though our experiments can determine Mu extremely accurately,
we still do not accurately know either G or the mass of the Earth.

Why don't we include a small object (1.000 000 kilogram, for example). After the spacecraft is a few million miles out from Earth, that object would be released, possibly on a temporary tether. Once it was at around a 10-meter distance to the main spacecraft, and it is given a slight velocity, it will orbit the spacecraft due to gravitation and the tether would be discarded.

If the spacecraft mass was 1,000 kilograms, then the equation above gives:

T^{2} = 4 * 9.8696044 * 10^{3} / Mu

where Mu is now 1001 * 6.67 * 10^{-11} or 6.67 * 10^{-8}

ycente T then is 769,000 seconds, or 8.90 days.

If the spacecraft had this satellite, it would be easy to determine the distance by radar ranging to many significant figures and a very accurate orbit could be determined, specifically the semi-major axis distance and the orbital period. This again gives an extremely accurate value for Mu, as before, which should be at least eight significant digits.

The mass of the spacecraft is rather accurately known, because we built it! As long as fuel load remaining is accurately known, and good practice is always having a very accurate fuel gauge, the mass of the spacecraft should be known to possibly the nearest gram. Out of a 1,000 kilogram spacecraft, that is one part in a million, which would then allow G to be known to an accuracy of one part in a million, six significant figures. That's a whole lot better than the three significant figures that three hundred years of science has gotten us so far, a thousand times more accurate.

**With G being known one thousand times more accurately, then the actual
mass of the Earth, Moon, Sun and everything else would also be known
one thousand times more accurately than now.** I would think that would be
tremendous incentive to include this very simple experiment on
one or more long distance spacecraft in the future.!

So far, I have not convinced either NASA or the ESA to add this simple and inexpensive addition to any of the spacecraft they have launched to the outer portions of the Solar System. Maybe some day, they will. (This article was first published on the Internet in February 2004.)

**Addendum of 2016:**

During the past forty years, about a dozen good attempts have been
made to do experiments to determine G (or actually mu) for the Earth's
gravitational field. **These dozen values have been amazingly different
from each other!** In 2015, J.D. Anderson, et al, created the following
graph. I am disappointed that Mr. Anderson did not include the statistical
error data for his graph, specifically an r^{2}, of at least 0.999,
but I trust that his graph had a good error factor.

**They ASSUME that all Gravitational Force exclusively arises from the
exact center of the Earth, which is wrong.** The Gravitational existence of
the Moon causes the existence the gravitational mass center of the Earth-Moon
system, which we call the Barycenter which is slightly over a thousand miles
deep in the somewhat liquid Lower Mantle layer of the Earth and which
constantly moves around inside the Earth. The center of the Earth
(and therefore the entire body of the Earth and us) constantly daily
"wobbles" around this Barycenter at around 793 mph (or 1276 kph) due to our
daily rotating and another 27 mph (or 44 kph) which is due to the Moon's
orbiting.

The complex resulting motion (daily and monthly) (velocities and accelerations) of the Earth around the Barycenter affects the detectable effects of all Gravitational effects within the Earth-Moon system, because all gravitational effects actually occur due to the gravitational attraction of the entire Earth-Moon mass (instantaneously) occurring from the location of the Barycenter.

A useful graph would therefore require the exact location and time of each experiment which is performed to be known, and also a calculation of the exact location of the Barycenter be known at that instant. Rather than the previously assumed Newtonian calculation being done for the exact center of the Earth, some Vector gravitational Force calculations are needed to be done for the space angle from the (instantaneous) Barycenter to that moment for that exact instantaneous experiment location. Only then could the correct data points be presented in such a graph. Such analysis and correction for each data point would be possible even now to make this graph to become useful.

Anderson et al make several astounding claims in the data in this graph.
The 13 data points are claimed to lie on an impressive sine-wave curve
which (amazingly) exactly resembles a totally different sine-wave curve,
which has to do with very tiny variations in the LOD (length of day). The
shown individual (red) error factors for each value seem to deny any validity for
such a really precise sine-wave curve. I also tried to duplicate the
r-squared data analysis which is critical to validity of any such
graph, where the common expectation is a r^{2} of at least 0.99.
When I tried to use their data points to generate an r^{2} error
value, I was unable to even get an r^{2} of 0.60.

They claim that the set of 13 measurements of G exhibit a 5.9-year periodic oscillation (solid curve) that closely resembles the 5.9-year oscillation in LOD measurements (dashed curve). The two outliers are a 2014 quantum measurement and a 1996 measurement.

(Phys.org) Newton's Gravitational constant, *G*,
has been measured about a dozen times over the last 40 years, but the
results have varied by much more than would be expected due to random
and systematic errors. Now scientists have found that the measured *G*
values oscillate over time like a sine wave with a period of 5.9 years.
It's not *G* itself that is varying by this much, they propose, but
more likely something else is affecting the measurements.

There are a variety of important aspects of this subject which few people seem to have ever explored. For starters, most "scientific constants" happen to be SCALAR values, but a few are VECTOR values. The peculiar behavior of G in these recent decades seems to bring up the possibility that G might be a Vector quantity, where math (Vector) formulas might be affected by other parameters.

I see that after Anderson et al published their paper in 2015, a variety of comments regarding it occasionally even seemed to be aware of my February 2004 publication of this paper, where I had first started trying to get NASA or the ESA to do a G experiment as part of the many rockets we have launched toward the outer Solar System, as described (and mathematically calculated) above. I notice that no one seemed willing to mention my work of eleven years or my name before their efforts.

I have a newer (Earth-based) experimental approach to offer now, which both attempts to establish if G is a Scalar quantity or a Vector quantity, and to possibly even determine the SPEED at which gravitational effects proceed, as well as far better experimental data to further explore just why the Earth's value of G seems so variable.

More Accurate Gravitational Constant. Another Important Gravitational Constant Experiment. (Feb 2017)

This presentation was first placed on the Internet in February 2004.

Conservation of Angular Momentum - An Exception or Violation (Sept 2006)

**Galaxy Spiral Arms** **Stability and Dynamics** A purely Newtonian gravitational explanation (Nov 1997, Aug 1998)

**Twins Paradox of Relativity Is Absolutely Wrong** (research 1997-2004, published Aug 2004)

Perturbation Theory. Gravitational Theory and Resonance (Aug 2001, Dec 2001)

Origin of the Earth. **Planetary Gravitational Resonances** (Dec 2001)

Rotation of the Sun (Jan 2000)

**Origin of the Universe**. Cosmogony - Cosmology (more logical than the Big Bang) (devised 1960, internet 1998)

**Time Passes Faster Here on Earth than on the Moon** (but only a fraction of a second per year!) (Jan 2009)

Globular Clusters. All Globulars Must Regularly Pass Through the cluttered Galaxy Plane, which would be very disruptive to their pristine form. (Nov 1997, Aug 1998)

Existence of Photons. A Hubble Experiment to Confirm the Existence of Individual Photons (experimental proof of quanta) (Feb 2000)

**Origin of the Moon - A New Theory** (June 2000)

**Planetary Rotation of Jupiter, Saturn, and the Earth** (Jupiter has a lot of gaseous turbulence which should have slowed down its rapid rotation over billions of years) (March 1998)

Cepheid Variable Stars. Velocity Graph Analysis (Feb 2003)

Compton Effect of Astrophysics. A Possible New Compton Effect (Mar 2003)

Olbers Paradox Regarding Neutrinos (Oct 2004)

Kepler and Newton. Calculations (2006)

Pulsars. Pulsars May Be Quite Different than we have Assumed (June 2008)

Sun and Stars - How the Sun Works - Nuclear Fusion in Creating Light and Heat (Aug 2006)

Stars - How They Work - Nuclear Fusion. Lives of Stars and You (Aug 2004)

Sundial Time Correction - Equation of Time. Sundial to Clock-Time Correction Factor (Jan 2009)

General Relativity - A Moon Experiment to Confirm It. Confirming General Relativity with a simple experiment. (Jan 2009)

General Relativity and Time Dilation. Does Time Dilation Result? (Jan 2009)

Geysers on Io. Source of Driving Energy (June 1998)

Mass Extinction, a New Explanation. A New Explanation for Apparent Periodicity of Mass Extinctions (May 1998, August 2001)

Precession of Gyroscopes and of the Earth. **Gyroscope Precession and Precession of the Earth's Equinoxes** (Apr 1998)

Ocean Tides - The Physics and Logic. Mathematical Explanation of Tides (Jan 2002)

Earth's Spinning - Perfect Energy Source (1990, Dec. 2009)

**Earth's Magnetic Field - Source and Logic**. Complex nature of the magnetic field and its source (March 1996)

**Earth Spinning Energy - Perfect Energy Source** From the Earth's Spinning (1990, Nov. 2002)

**Nuclear or Atomic Physics Related Subjects:**

**Nuclear Physics - Statistical Analysis of Isotope Masses** Nuclear Structure. (research 1996-2003, published Nov 2003)

Quantum Defect is NOT a Mathematical Defect- It Can Be Calculated **The Quantum Defect is a Physical Quantity and not a Fudge Factor**(July 2007)

Atomic Physics - NIST Atomic Ionization Data Patterns **Surprising Patterns in the NIST Data Regarding Atomic Ionization** (June 2007)

Nuclear Physics - Logical Inconsistencies (August 2007)

Neutrinos - Where Did they all Come From? (August 2004)

Neutrinos - Olbers Paradox Means Neutrinos from Everywhere (Oct 2004)

Quantum Nuclear Physics. A Possible Alternative (Aug 2001, Dec 2001, Jan 2004)

Quantum Physics - Quantum Dynamics. A Potential Improvement (2006)

Quantum Physics is Compatible with the Standard Model (2002, Sept 2006, Oct 2010)

Quantum Dynamics (March 2008)

Ionization Potential - NIST Data Patterns. Surprising patterns among different elements (March 2003)

Mass Defect Chart. (calculation, formula) (research 1996-2003, published Nov 2003)

**Assorted other Physics Subjects:**

Precession of Gyroscopes and of the Earth. **Gyroscope Precession and Precession of the Earth's Equinoxes** (Apr 1998)

**Earth's Magnetic Field - Source and Logic**. Complex nature of the magnetic field and its source (March 1996)

**Earth Spinning Energy - Perfect Energy Source** (1990, Nov. 2002)

Earth Energy Flow Rates due to Precessional Effects (63,000 MegaWatts) (Sept 2006)

Accurate Mass of the Earth. Gravitational Constant - An Important Gravitation Experiment. (Feb 2004)

Tornadoes - The Physics of How They Operate, including How they Form. **Solar Energy, an Immense Source of Energy, Far Greater than all Fossil Fuels** (Feb 2000, Feb 2006, May 2009)

Radiometric Age Dating - Carbon-14 Age Determination. Carbon-14, C-14 (Dec 1998)

Mass Extinction, an Old Explanation. An Old Explanation for Apparent Periodicity of Mass Extinctions (Aug 2003)

**Hurricanes, the Physics and Analysis** **A Credible Approach to Hurricane Reduction** (Feb 2001)

Sundial Time Correction - Equation of Time. Sundial to Clock-Time Correction Factor (Jan 2009)

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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago