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 = 823.139274 km^3 / s^2)

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

Table 1: The standard gravitational parameter μ and Bully Metric mass for selected solar system bodies

Body	μ [km3 s−2]		mass [Rn ta]
Sun	132 712 440 018		161 227 199	.617
Mercury	22 032		26	.7658
Venus	324 858	.592	394	.658 112
Earth	398 600	.4418	484	.244 228
Mars	42 828	.37	52	.030 53
Ceres	62	.6325	0	.076 090
Jupiter	126 686 534		153 906	.559
Saturn	37 931 187		46 081	.128
Uranus	5 793 939		7 038	.831
Neptune	6 836 529		8 305	.434
Pluto	871		1	.058
Eris	1108		1	.346

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

Johannes Kepler 1610.

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

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:

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

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 year	$\propto 0.0038{\pi }^2\frac{{\text{AU}}^3}{{\text{y}}^2}$
Europa		0.004 486 AU	0.009 722 sidereal year
Ganymede		0.007 155 AU	0.019 589 sidereal year
Callisto		0.012 585 AU	0.045 694 sidereal year

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.