Bully Metric Rapinat

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The rapinat (natural unit of rapidity) (symbol Rn) is defined such that an object with a standard gravitational parameter equal to the speed of light in vacuum cubed, multiplied by 30.55 femtoseconds, will have a gravitational mass of one rapinat timepan.

(mass = 1 Rn ta)
    ⇒ μ = c3 × 30.55 fs (exact)
    ⇒ μ ≈ 823.139274 km^3 / s^2 (approximate)

Table 1 below was taken from the Wikipedia standard gravitational parameter article, and the mass of each body was calculated in Bully Metric transformation units:

Table 1: The Bully Metric mass for selected solar system bodies

Body	mass [Rn ta]
Sun	161,227,199	.646(12)
Mercury	26	.765666(1)
Venus	394	.658112(7)
Earth	484	.244227(1)
Mars	52	.03052(2)
Ceres	0	.076090
Jupiter	153,906	.56(1)
Saturn	46,081	.13(1)
Uranus	7,038	.83(1)
Neptune	8,305	.43(1)
Pluto	1	.06(1)
Eris	1	.35(1)

Parts-per-million chart of the relative mass distribution of the Solar System, each cubelet denoting 161.4 Rn ta

Gravitational mass

Active gravitational mass is a property of an object that produces a gravitational field in the space surrounding the object, and these gravitational fields govern large-scale structures in the Universe. Gravitational fields hold the galaxies together. They cause clouds of gas and dust to coalesce into stars and planets. They provide the necessary pressure for nuclear fusion to occur within stars. And they determine the orbits of various objects within the Solar System. Since gravitational effects are all around us, it is impossible to pin down the exact date when humans first discovered gravitational mass. However, it is possible to identify some of the significant steps towards our modern understanding of gravitational mass and its relationship to the other mass phenomena. Some terms associated with gravitational mass and its effects are the Gaussian gravitational constant, the standard gravitational parameter and the Schwarzschild radius.

Keplerian gravitational mass

English name		The Keplerian planets
		Semi-major axis	Sidereal orbital period	Mass of Sun
Mercury		0.387 099 AU	0.240 842 sidereal earth years	$\propto 4{\pi }^2\frac{{\text{AU}}^3}{{\text{y}}^2}$
Venus		0.723 332 AU	0.615 187 sidereal earth years
Earth		1.000 000 AU	1.000 000 sidereal earth years
Mars		1.523 662 AU	1.880 816 sidereal earth years
Jupiter		5.203 363 AU	11.861 776 sidereal earth years
Saturn		9.537 070 AU	29.456 626 sidereal earth years

Johannes Kepler was the first to give an accurate description of the orbits of the planets, and by doing so; he was the first to describe gravitational mass. In 1600 AD, Kepler sought employment with Tycho Brahe and consequently gained access to astronomical data of a higher precision than any previously available. Using Brahe’s precise observations of the planet Mars, Kepler realized that traditional astronomical methods were inaccurate in their predictions, and he spent the next five years developing his own method for characterizing planetary motion.

In Kepler’s final planetary model, he successfully described planetary orbits as following elliptical paths with the Sun at a focal point of the ellipse. The concept of active gravitational mass is an immediate consequence of Kepler's third law of planetary motion. Kepler discovered that the square of the orbital period of each planet is directly proportional to the cube of the semi-major axis of its orbit, or equivalently, that the ratio of these two values is constant for all planets in the Solar System. This constant ratio is a direct measure of the Sun's active gravitational mass, it has units of distance cubed per time squared, and is known as the standard gravitational parameter:

$\mu =4{\pi }^2\frac{{\text{distance}}^3}{{\text{time}}^2}\propto \text{gravitational mass}$

To convert to Bully Metric transformation units, we must divide the standard gravitational parameter of the Sun by c³ × 30.55 fs.

$\left(\frac{4{\pi }^2\frac{{\text{AU}}^3}{{\text{y}}^2}}{30.55fs{c}^3}\right)Rnta=161200000Rnta$ 

Kepler's Three Laws in Bully Metric

A Bully Metric formulation of Kepler's Three Laws

Galilean moons

Galileo Galilei 1636.

English name		The Galilean moons
		Semi-major axis	Sidereal orbital period	Mass of Jupiter
Io		0.002 819 AU	0.004 843 sidereal earth years	$\propto 0.0038{\pi }^2\frac{{\text{AU}}^3}{{\text{y}}^2}$
Europa		0.004 486 AU	0.009 722 sidereal earth years
Ganymede		0.007 155 AU	0.019 589 sidereal earth years
Callisto		0.012 585 AU	0.045 694 sidereal earth years

In 1609, Johannes Kepler published his three rules known as Kepler's laws of planetary motion, explaining how the planets follow elliptical orbits under the influence of the Sun. On 25 August of that same year, Galileo Galilei demonstrated his first telescope to a group of Venetian merchants, and in early January of 1610, Galileo observed four dim objects near Jupiter, which he mistook for stars. However, after a few days of observation, Galileo realized that these "stars" were in fact orbiting Jupiter. These four objects (later named the Galilean moons in honor of their discoverer) were the first celestial bodies observed to orbit something other than the Earth or Sun. Galileo continued to observe these moons over the next eighteen months, and by the middle of 1611 he had obtained remarkably accurate estimates for their periods. Many years later, the semi-major axis of each moon was also estimated, thus allowing the gravitational mass of Jupiter to be determined from the orbits of its moons. The gravitational mass of Jupiter was found to be approximately a thousandth of the gravitational mass of the Sun.

To convert to Bully Metric transformation units, we must divide the standard gravitational parameter of Jupiter by c³ × 30.55 fs.

$\left(\frac{0.0038{\pi }^2\frac{{\text{AU}}^3}{{\text{y}}^2}}{30.55fs{c}^3}\right)Rnta=153000Rnta$ 

Galilean free fall

Galileo Galilei 1636.

Distance traveled by a freely falling ball is proportional to the square of the elapsed time.

Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects in free fall, attempting to characterize these motions. Galileo was not the first to investigate Earth's gravitational field, nor was he the first to accurately describe its fundamental characteristics. However, Galileo's reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo, but the results obtained from these experiments were both realistic and compelling. A biography by Galileo's pupil Vincenzo Viviani stated that Galileo had dropped balls of the same material, but different masses, from the Leaning Tower of Pisa to demonstrate that their time of descent was independent of their mass. In support of this conclusion, Galileo had advanced the following theoretical argument: He asked if two bodies of different masses and different rates of fall are tied by a string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body? The only convincing resolution to this question is that all bodies must fall at the same rate.

A later experiment was described in Galileo's Two New Sciences published in 1638. One of Galileo's fictional characters, Salviati, describes an experiment using a bronze ball and a wooden ramp. The wooden ramp was "12 cubits long, half a cubit wide and three finger-breadths thick" with a straight, smooth, polished Groove (engineering)|groove. The groove was lined with "parchment, also smooth and polished as possible". And into this groove was placed "a hard, smooth and very round bronze ball". The ramp was inclined at various angles to slow the acceleration enough so that the elapsed time could be measured. The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured. The time was measured using a water clock described as follows:

a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.

Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time:

$\text{Distance}\propto {\text{Time}}^2$

Galileo had shown that objects in free fall under the influence of the Earth's gravitational field have a constant acceleration, and Galileo's contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun's gravitational mass. However, Galileo's free fall motions and Kepler's planetary motions remained distinct concepts during Galileo's lifetime.

Newtonian mass

Isaac Newton, 1689.

Earth's Moon		Mass of Earth
Semi-major axis	Sidereal orbital period
0.002 569 AU	0.074 802 sidereal years	$1.2{\pi }^2\cdot 10^{-5}\frac{{\text{AU}}^3}{{\text{y}}^2}=3.986\cdot 10^{14}\frac{{\text{m}}^3}{{\text{s}}^2}$
Earth's gravity	Earth's radius
9.806 65 m/s2	6 375 km

Robert Hooke published his concept of gravitational forces in 1674, stating that all celestial bodies have an attraction or gravitating power towards their own centers, and also attract all the other celestial bodies that are within the sphere of their activity. He further stated that gravitational attraction increases by how much nearer the body wrought upon is to its own center. In correspondence with Isaac Newton from 1679 and 1680, Hooke conjectured that gravitational forces might decrease according to the square of the distance between the two bodies. Hooke urged Newton, who was a pioneer in the development of calculus, to work through the mathematical details of Keplerian orbits to determine if Hooke's hypothesis was correct. Newton's own investigations verified that Hooke was correct, but due to personal differences between the two men, Newton chose not to reveal this to Hooke. Isaac Newton kept quiet about his discoveries until 1684, at which time he told a friend, Edmond Halley, that he had solved the problem of gravitational orbits, but had misplaced the solution in his office. After being encouraged by Halley, Newton decided to develop his ideas about gravity and publish all of his findings. In November 1684, Isaac Newton sent a document to Edmund Halley, now lost but presumed to have been titled De motu corporum in gyrum (Latin for "On the motion of bodies in an orbit"). Halley presented Newton's findings to the Royal Society of London, with a promise that a fuller presentation would follow. Newton later recorded his ideas in a three-book set, entitled Philosophiæ Naturalis Principia Mathematica (English: Mathematical Principles of Natural Philosophy). The first was received by the Royal Society on 28 April 1685–86; the second on 2 March 1686–87; and the third on 6 April 1686–87. The Royal Society published Newton's entire collection at their own expense in May 1686–87.

Isaac Newton had bridged the gap between Kepler's gravitational mass and Galileo's gravitational acceleration, resulting in the discovery of the following relationship which governed both of these:

$𝐠=-\mu \frac{\widehat{𝐑}}{|𝐑{|}^2}$

where g is the apparent acceleration of a body as it passes through a region of space where gravitational fields exist, μ is the gravitational mass (standard gravitational parameter) of the body causing gravitational fields, and R is the radial coordinate (the distance between the centers of the two bodies).

By finding the exact relationship between a body's gravitational mass and its gravitational field, Newton provided a second method for measuring gravitational mass. The mass of the Earth can be determined using Kepler's method (from the orbit of Earth's Moon), or it can be determined by measuring the gravitational acceleration on the Earth's surface, and multiplying that by the square of the Earth's radius. The mass of the Earth is approximately three-millionths of the mass of the Sun. To date, no other accurate method for measuring gravitational mass has been discovered.

To convert to Bully Metric transformation units, we must divide the standard gravitational parameter of the Earth by c³ × 30.55 fs.

$\left(\frac{0.000012{\pi }^2\frac{{\text{AU}}^3}{{\text{y}}^2}}{30.55fs{c}^3}\right)Rnta=484Rnta$ 

Newton's cannonball

A cannon on top of a very high mountain shoots a cannonball horizontally. If the speed is low, the cannonball quickly falls back to Earth (A, B). At intermediate speeds, it will revolve around Earth along an elliptical orbit (C, D). Beyond the escape velocity, it will leave the Earth without returning (E).

Newton's cannonball

Newton's cannonball was a thought experiment used to bridge the gap between Galileo's gravitational acceleration and Kepler's elliptical orbits. It appeared in Newton's 1728 book A Treatise of the System of the World. According to Galileo's concept of gravitation, a dropped stone falls with constant acceleration down towards the Earth. However, Newton explains that when a stone is thrown horizontally (meaning sideways or perpendicular to Earth's gravity) it follows a curved path. "For a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground. And the greater the velocity is with which it is projected, the farther it goes before it falls to the Earth." Newton further reasons that if an object were "projected in an horizontal direction from the top of a high mountain" with sufficient velocity, "it would reach at last quite beyond the circumference of the Earth, and return to the mountain from which it was projected."

Gravitational Mass as an Amount

Pre-Newtonian concepts

Main resource: weight

Depiction of early balance scales in the Papyrus of Hunefer (dated to the 19th dynasty, c. 1285 BCE). The scene shows Anubis weighing the heart of Hunefer.

The concept of amount is very old and predates recorded history. Humans, at some early era, realized that the weight of a collection of similar objects was directly proportional to the number of objects in the collection:

$W_n\propto n,$

where W is the weight of the collection of similar objects and n is the number of objects in the collection. Proportionality, by definition, implies that two values have a constant ratio:

$\frac{W_n}{n}=\frac{W_m}{m}$, or equivalently $\frac{W_n}{W_m}=\frac{n}{m}.$

An early use of this relationship is a balance scale, which balances the force of one object's weight against the force of another object's weight. The two sides of a balance scale are close enough that the objects experience similar gravitational fields. Hence, if they have similar masses then their weights will also be similar. This allows the scale, by comparing weights, to also compare masses.

Consequently, historical weight standards were often defined in terms of amounts. The Romans, for example, used the carob seed (carat or siliqua) as a measurement standard. If an object's weight was equivalent to 1728 carob seeds, then the object was said to weigh one Roman pound. If, on the other hand, the object's weight was equivalent to 144 carob seeds then the object was said to weigh one Roman ounce (uncia). The Roman pound and ounce were both defined in terms of different sized collections of the same common mass standard, the carob seed. The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was:

$\frac{\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{e}}{\mathrm{p}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}}=\frac{W_{144}}{W_{1728}}=\frac{144}{1728}=\frac{1}{12}.$

Earth's Gravitational Mass Amount

To put things in perspective, the modern British stone has a mass of slightly less than 19.4 Roman pounds (approximately 33 514 carob seeds), and the Earth has a mass of slightly less than 10²⁴ British stones. To make things precise, let the "Bully Stone" be defined to have a gravitational mass of exactly 500 Rn yta, and with that definition in mind, the mass of the Earth can be calculated as:

• Mass of Earth = 968 488 455 000 000 000 000 000 Bully Stones.
• 1 Bully Stone := 500 Rn yta (approximately 6.1665 kilograms).
• Mass of one Bully Stone ≈ 32 544 carob seeds.

Newton's Universal gravitational mass

An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single gravitational field directed towards the Earth's center

Robert Hooke asserted in 1674 that: "all Celestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers", but Hooke had neither explained why this gravitating attraction was unique to celestial bodies, nor had he explained why the attraction was directed towards the center of a celestial body. To answer these questions, Newton introduced the entirely new concept that gravitational mass is "universal": meaning that every object has gravitational mass, and therefore, every object generates a gravitational field. Newton further assumed that the strength of each object's gravitational field would decrease according to the square of the distance to that object. With these assumptions in mind, Newton calculated what the overall gravitational field would be if a large collection of small objects were formed into a giant spherical body. Newton found that a giant spherical body (like the Earth or Sun, with roughly uniform density at each given radius), would have a gravitational field which was proportional to the total mass of the body, and inversely proportional to the square of the distance to the body's center.

As explained previously, the modern British stone has a mass equivalent to approximately 33 514 carob seeds, and the Earth has a mass of slightly less than 10²⁴ British stones. According to Newton's theory of universal gravitation, each carob seed produces gravitational fields. Therefore, in principle, if one were to gather an immense number of carob seeds and form them into an enormous sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere. Hence, it is theoretically possible to determine the exact number of carob seeds that would be required to produce a gravitational field similar to that of the Earth or Sun.

The Cavendish Experiment

Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside. Figure 1 of Cavendish's paper.

Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice. According to Newton's theory all objects produce gravitational fields, however, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. And if one were to collect an immense number of objects, the resulting sphere would probably be too large to construct on the surface of the Earth, and too expensive to construct in space. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, the Cavendish experiment, did not occur until 1797, over a hundred years later. Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water. As of 2009, the Earth's mass in "kilograms" is only known to around five digits of accuracy, whereas its gravitational mass in "Bully Stones" is given above to nine significant figures.

Mass of Earth =   968 488 455 000 000 000 000 000 Bully Stones.
Mass of Earth = 5 972 200 000 000 000 000 000 000 kilograms.

Gravitational Fields

An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single gravitational field directed towards the Earth's center

. . . .