Bully Metric

There are three variants of the apan defined within the Bully system of measurement units. The time apan (symbol ta) is by definition exactly 30.55 picoseconds. The length apan (or light apan) (symbol la) is by definition the distance light travels in vacuum in 30.55 picoseconds. The gravitational mass apan (symbol ma) is by definition the mass an object would have if its standard gravitational parameter were equal to the speed of light in vacuum cubed, multiplied by 30.55 picoseconds.

ta = 30.55 picoseconds (exact)

la = c × 30.55 picoseconds (exact)
   = 9.1586595919 millimeters (exact)

ma = c3 × 30.55 picoseconds / G (exact)
   ≈ 12.3330 ronnagrams (approximate)

The above definitions ensure normalization of the speed of light and Newton's gravitational constant when using Bully units:

$c=1.0\frac{la}{ta}$ (exact)

$G=1.0\frac{l{a}^3}{mat{a}^2}$ (exact)

Planck units and the Bully Metric

Table 1: Modern values for Planck's original choice of quantities

Name	Expression	Value (SI units)
Planck time	$t_{\text{P}}=\sqrt{\frac{\hslash G}{c^5}}$	5.391247(60)×10−44 s
Planck length	$l_{\text{P}}=\sqrt{\frac{\hslash G}{c^3}}$	1.616255(18)×10−35 m
Planck mass	$m_{\text{P}}=\sqrt{\frac{\hslash c}{G}}$	2.176434(24)×10-8 kg
Planck temperature	$T_{\text{P}}=\sqrt{\frac{\hslash c^5}{G{k}_{\text{B}}^2}}$	1.416784(16)×1032 K

Since c and G are both normalized in the Bully system, this ensures that Bully units have a simple relationship with Planck's units, as illustrated in table 2 below:

Table 1: Modern values for Planck's original choice of quantities

Name	Expression
Planck time	$t_{\text{P}}=\sqrt{\frac{\hslash G}{c^5}}=\sqrt{\frac{\hslash \frac{l{a}^3}{mat{a}^2}}{\frac{l{a}^5}{t{a}^5}}}=\sqrt{\frac{\hslash \frac{l{a}^3}{mat{a}^2}}{\frac{l{a}^5}{t{a}^5}}}$
Planck length	$l_{\text{P}}=\sqrt{\frac{\hslash G}{c^3}}=\sqrt{\frac{\hslash \frac{l{a}^3}{mat{a}^2}}{\frac{l{a}^3}{t{a}^3}}}$
Planck mass	$m_{\text{P}}=\sqrt{\frac{\hslash c}{G}}==\sqrt{\frac{\hslash \frac{la}{ta}}{\frac{l{a}^3}{mat{a}^2}}}$
Planck temperature	$T_{\text{P}}=\sqrt{\frac{\hslash c^5}{G{k}_{\text{B}}^2}}=\sqrt{\frac{\hslash \frac{l{a}^5}{t{a}^5}}{\frac{l{a}^3}{mat{a}^2}{k}_{\text{B}}^2}}$

, in that each Planck unit multiplied by 566.66 is approximately equal to the corresponding Bully unit multiplied by 10^-30. The relationship between Planck units and Bully units is illustrated in Table 1 below. The table uses the "quecto" (symbol "q") metric prefix with each Bully unit, where "q" means 10^-30. SI prefixes have the same meaning and conventions when used with apan variants as they have when used with standard SI units (see table in subsequent section for a complete list of prefixes).

Table 1: Modern values for Planck's original choice of quantities

Name	Expression	Value (SI units)	Value (Bully units)
Planck time	$t_{\text{P}}=\sqrt{\frac{\hslash G}{c^5}}$	5.391247(60)×10−44 s	$\frac{1.000001(10)}{566.66}qta$
Planck length	$l_{\text{P}}=\sqrt{\frac{\hslash G}{c^3}}$	1.616255(18)×10−35 m	$\frac{1.000001(10)}{566.66}qla$
Planck mass	$m_{\text{P}}=\sqrt{\frac{\hslash c}{G}}$	2.176434(24)×10-8 kg	$\frac{1.000001(10)}{566.66}qma$
Planck temperature	$T_{\text{P}}=\sqrt{\frac{\hslash c^5}{G{k}_{\text{B}}^2}}$	1.416784(16)×1032 K	TBD

The 'Bully Metric is an extremely efficient set of time and distance measurement units for representing earth's physical parameters. Bully units can efficiently represent the Earth's sidereal year and tropical year to eight digits; The Bully Metric can also efficiently represent four digit approximations for the Earth's radius (r ≈ 6371), Schwarzschild radius (R), standard gravitational parameter (μ = MG ≈ 3.984e14), and a typical gravitational acceleration on earth's surface (g ≈ 9.813 ).

$$1SiderealYear=1.033Eta=31,558,150s$$

$$1TropicalYear=(1.033-0.00004)Eta-2s=31,556,926s$$

$$r_{earth}\approx \frac{1Gla}{\sqrt{2\times 1.033}}=6371km$$

$$R_{earth}\approx \frac{la}{1.033}\approx 8.866mm$$

$${\mu }_{earth}\approx \frac{1}{2\times 1.033}\frac{l{a}^3}{t{a}^2}\approx 398,400,000,000,000\frac{m^3}{s^2}$$

$$g_{earth}\approx 1\frac{Ela}{Et{a}^2}\approx 9.813\frac{m}{s^2}$$

The Bully Constants

There are a surprising number of physical constants that can be approximated using various algebraic combinations of the following four numbers (click here to learn more):

1.033
30.55
2
0.00004

Sidereal year

The number of seconds in the Earth's sidereal year can be approximated as:

$$1SiderealYear=1.033\times 30.55Ms=31,558,150s$$

Tropical year

The number of seconds in the Earth's tropical year can be approximated as:

$$1TropicalYear=((1.033-0.00004)\times 30.55Ms)-2s=31,556,926s$$

Great year

The number of tropical years in a Great Year can be approximated as:

$$1GreatYear\approx (\frac{1.033}{0.00004})TropicalYears=25,825TropicalYears$$

Earth's radius (r)

An approximate relationship of the speed of light to the Earth's radius (r):

$$r_{earth}\approx \frac{c\times 30.55ms}{\sqrt{2\times 1.033}}=6371km$$

Earth's Schwarzschild radius (R)

An approximate relationship of the speed of light to the Earth's Schwarzschild radius (R):

$$R_{earth}=\frac{2\times G{M}_{earth}}{c^2}\approx \frac{c\times 30.55ps}{1.033}\approx 8.866mm$$

Earth's standard gravitational parameter (μ = MG)

An approximate relationship of the speed of light to the Earth's standard gravitational parameter (μ = MG):

$${\mu }_{earth}=G{M}_{earth}\approx \frac{c^3\times 30.55ps}{2\times 1.033}\approx 398,400,000,000,000\frac{m^3}{s^2}$$

Earth's gravity (g)

A typical gravitational acceleration on earth's surface can be approximated as:

$$g_{earth}=\frac{{\mu }_{earth}}{{r_{earth}}^2}\approx \frac{c}{30.55Ms}\approx 9.813\frac{m}{s^2}$$

Definition of the apan

The above approximations for earth's physical parameters can be further simplified by introducing new measurement units. The apan will be defined with two variants. The time apan (symbol ta) is by definition exactly 30.55 picoseconds. The length apan (or light apan) (symbol la) is by definition the distance light travels in vacuum in exactly 30.55 picoseconds.

SI prefixes will have the same meaning and conventions when used with the apan as the have when used with standard SI units (see table in subsequent section). For example:

One million ta = 1,000,000 ta = 1 mega time apan = 1 Mta = 30.55 μs
One million la = 1,000,000 la = 1 mega light apan = 1 Mla = 9.1586595919 km

Approximations for earth's physical parameters can be written in terms of the apan as follows:

Sidereal year

The number of seconds in the Earth's sidereal year can be approximated as:

$$1SiderealYear=1.033Eta=31,558,150s$$

Tropical year

The number of seconds in the Earth's tropical year can be approximated as:

$$1TropicalYear=(1.033-0.00004)Eta-2s=31,556,926s$$

Earth's radius (r)

An apan based approximation of Earth's radius (r):

$$r_{earth}\approx \frac{1Gla}{\sqrt{2\times 1.033}}=6371km$$

Earth's Schwarzschild radius (R)

An apan based approximation of the Earth's Schwarzschild radius (R):

$$R_{earth}\approx \frac{la}{1.033}\approx 8.866mm$$

Earth's standard gravitational parameter (μ = MG)

An apan based approximation of the Earth's standard gravitational parameter (μ = MG):

$${\mu }_{earth}\approx \frac{1\frac{l{a}^3}{t{a}^2}}{2\times 1.033}\approx 398,400,000,000,000\frac{m^3}{s^2}$$

Earth's gravity (g)

A typical gravitational acceleration on earth's surface can be approximated as:

$$g_{earth}=\frac{{\mu }_{earth}}{{r_{earth}}^2}\approx 1\frac{Ela}{Et{a}^2}\approx 9.813\frac{m}{s^2}$$

The apan prefix table

Prefix		Base 10	Bully Metric		SI Equivalent
Name	Symbol		Time	Length	Time	Length
quetta	Q	1030	Qta	Qla	30.55 Es	9.1586595919 Rm
ronna	R	1027	Rta	Rla	30.55 Ps	9.1586595919 Ym
yotta	Y	1024	Yta	Yla	30.55 Ts	9.1586595919 Zm
zetta	Z	1021	Zta	Zla	30.55 Gs	9.1586595919 Em
exa	E	1018	Eta	Ela	30.55 Ms	9.1586595919 Pm
peta	P	1015	Pta	Pla	30.55 ks	9.1586595919 Tm
tera	T	1012	Tta	Tla	30.55 s	9.1586595919 Gm
giga	G	109	Gta	Gla	30.55 ms	9.1586595919 Mm
mega	M	106	Mta	Mla	30.55 μs	9.1586595919 km
kilo	k	103	kta	kla	30.55 ns	9.1586595919 m
—	—	100	ta	la	30.55 ps	9.1586595919 mm
milli	m	10−3	mta	mla	30.55 fs	9.1586595919 μm
micro	μ	10−6	μta	μla	30.55 as	9.1586595919 nm
nano	n	10−9	nta	nla	30.55 zs	9.1586595919 pm
pico	p	10−12	pta	pla	30.55 ys	9.1586595919 fm
femto	f	10−15	fta	fla	30.55 rs	9.1586595919 am
atto	a	10−18	ata	ala	30.55 qs	9.1586595919 zm
zepto	z	10−21	zta	zla	30.55e-3 qs	9.1586595919 ym
yocto	y	10−24	yta	yla	30.55e-6 qs	9.1586595919 rm
ronto	r	10−27	rta	rla	30.55e-9 qs	9.1586595919 qm
quecto	q	10−30	qta	qla	30.55e-12 qs	9.1586595919e-3 qm