Accurate Mass of the Earth

Precise Newtonian Gravitational Constant

An Important Gravitation Experiment

Modern science has extremely accurate values for most "constants" such as the speed of light, the value of pi, etc. However, the Gravitational Constant (G) is only known to around three significant digits! A relatively simple and inexpensive experiment could be done to greatly increase the accuracy of this important constant, by over a thousand times better precision.

Why it is not currently known better?

When Newton developed his gravitation theory, he arrived at a relatively simple equation,

F = G * m1 * m2 / r2,

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 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:

T2 = 4 * PI2 * a3 / 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.

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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 effect which occurs in the Earth-Moon system always acts 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 somewhat elliptic paths in a period of one sidereal month, or 27.3216610 days, or 2,360,591.5 seconds, T. This results the Earth and Moon "wobbling" around the Barycenter in two separate 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 "wobble" 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 * 384,749,9003 / mu


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


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!

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This example, and ALL other available experiments, can result in excellently accurate values for Mu (which is G times the total mass involved), but there has been no way of accurately determining either G or the total mass. Even though we know the product of those numbers really accurately, we only have a very poor idea of what either precisely is.

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 * m1 * m2 / r2,

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:

m2 * a = G * m1 * m2 / r2.

where m2 is the mass of our object, m1 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 * mE / r2,

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 * mE. 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.

A New Experiment

Whenever we send spacecraft to Mars or the other outer planets, once it has entirely escaped Earth's gravity (a few million miles out) it generally just coasts for maybe nine months (for Mars trips, longer for Jupiter and beyond) until a mid-course correction rocket burn adjusts the trajectory to arrive exactly where we want it to go. During those nine months, and also in the nine months after that rocket burn, virtually nothing happens.

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:

T2 = 4 * 9.8696044 * 103 / 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 r2, 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.

gravitational constant recent graph

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 r2 of at least 0.99. When I tried to use their data points to generate an r2 error value, I was unable to even get an r2 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.

( 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.

It certainly seems likely that this graph does not contain any information of value. The Anderson group seemed to totally neglect the importance of WHERE an experiment was performed, and exactly when, since the location of the gravitational Barycenter, the actual Locus of gravitational attraction in the Earth-Moon system, constantly moves around deep inside the Earth at around 793 mph (or 1276 kph) daily motion plus another 27 mph (or 44 km/hr) monthly motion at a distance about 2,902 miles (4677 km) from the exact center of the Earth. The DIRECTION and INTENSITY of any detected Gravitational Force in any experiment near the Earth's surface, which is actually due to the entire Earth-Moon gravitational mass attraction from the Barycenter, therefore constantly changes. Such information is critically important when trying to determine a Vector Force quantity in a very delicate experiment. This is especially true when both ends of such a force are moving at high speeds and moving in all three dimensions.

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)

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