Modern science has extremely accurate values for most 'constants'
such as the speed of light, the value of pi, etc, but the
Gravitational Constant is only known to around three significant
figures! There is a relatively simple and inexpensive experiment
that could be done to greatly increase the accuracy of this
important constant.
Why it is not currently known betterWhen Newton developed his gravitation theory, he arrived at a relatively simple equation,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 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 semimajor 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.


(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 for mu and get:
4.0350935 * 10^{+14}
This number is probably accurate to its eight significant digits.
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 ultrasensitive experiments. With this value for G, we can get the total mass of the EarthMoon 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 0.07 * 10^{+24 kg} which leaves
5.98 * 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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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 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, 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 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 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}
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 semimajor 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 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. I would think that would be tremendous incentive to include this very simple experiment on one or more long distance spacecraft in the future.!
Conservation of Angular Momentum A Violation of the Conservation of Angular Momentum(Sept 2006)
Galaxy Spiral Arms Stability and Dynamics A purely Newtonian gravitational explanation (Nov 1997, Aug 1998)
The Twins Paradox of Relativity Is Absolutely Wrong. (research 19972004, 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)
The Origin of the Moon (June 2000)
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 Variables Velocity Graph Analysis (Feb 2003)
Compton Effect. 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)
How the Sun Works in Creating Light and Heat (Aug 2006)
Nuclear Fusion in Stars. Lives of Stars and You (Aug 2004)
Equation of Time. Sundial to ClockTime Correction Factor (Jan 2009)
An Experiment on the Moon to Confirm General Relativity. 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)
A New Explanation for Mass Extinctions. A New Explanation for Apparent Periodicity of Mass Extinctions (May 1998, August 2001)
Precession of Gyroscopes and the Earth. Gyroscope Precession and Precession of the Earth's Equinoxes (Apr 1998)
The Physics of Ocean Tides. Mathematical Explanation of Tides (Jan 2002)
Perfect Energy Source From the Earth's Spinning (1990, Dec. 2009)
Source of the Earth's Magnetic Field. Complex nature of the magnetic field and its source (March 1996)
Perfect Energy Source From the Earth's Spinning (1990, Nov. 2002)
Nuclear or Atomic Physics related Subjects:
Statistical Analysis of SameAtomicWeight Isotopes Nuclear Structure. (research 19962003, published Nov 2003)
The Quantum Defect is NOT a Mathematical Defect The Quantum Defect is a Physical Quantity and not a Fudge Factor(July 2007)
NIST Atomic Ionization Data Patterns Surprising Patterns in the NIST Data Regarding Atomic Ionization (June 2007)
Logical Inconsistencies in Nuclear Physics (August 2007)
Where Did All the Neutrinos Come From? (August 2004)
Neutrinos from Everywhere (Oct 2004)
Quantum Nuclear Physics. A Possible Alternative (Aug 2001, Dec 2001, Jan 2004)
Quantum Physics. A Potential Improvement (2006)
Quantum Physics is Compatible with the Standard Model
Quantum Physics is Compatible with the Standard Model (2002, Sept 2006, Oct 2010)
Quantum Dynamics (March 2008)
Ionization Potential. Surprising patterns among different elements (March 2003)
The Mass Defect Chart. (calculation, formula) (research 19962003, published Nov 2003)
Assorted other Physics Subjects:
Precession of Gyroscopes and the Earth. Gyroscope Precession and Precession of the Earth's Equinoxes (Apr 1998)
Source of the Earth's Magnetic Field. Complex nature of the magnetic field and its source (March 1996)
Perfect Energy Source From the Earth's Spinning (1990, Nov. 2002)
Earth Energy Flow Rates due to Precessional Effects (63,000 MegaWatts) (Sept 2006)
Gravitational Constant. An Important Gravitation Experiment (Feb 2004)
The Physics of Tornadoes. The Physics of Tornadoes, including How they Form. Solar Energy, an Immense Source of Energy, Far Greater than all Fossil Fuels (Feb 2000, Feb 2006, May 2009)
Carbon14 Age Determination. Radiometric Age Dating, Carbon14, C14 (Dec 1998)
An Old Explanation for Mass Extinctions. An Old Explanation for Apparent Periodicity of Mass Extinctions (Aug 2003)
The Physics of Hurricanes A Credible Approach to Hurricane Reduction (Feb 2001)
Equation of Time. Sundial to ClockTime Correction Factor (Jan 2009)
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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago