The Bully Mnemonic

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Bully Mnemonic Topics:

• The Bully Mnemonic
• The Bully Mnemonic & Calendar Variations

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The Bully Mnemonic is a technique for remembering the exact number of seconds that occur in Earth's sidereal year and tropical year; a good approximation of the Earth's Great Year; and a rough approximation of the Solar System's galactic year.

The following relationships are encoded in the Bully Mnemonic:

$${1\,Sidereal\,Year}={31,558,150\,Seconds}$$

$${1\,Tropical\,Year}={31,556,926\,Seconds}$$

$$1\,Great\,Year\approx 25,824\,Sidereal\,Years\approx 25,825\,Tropical\,Years$$

$${1\,Galactic\,Year}\approx 213,417,800\,Tropical\,Years$$

The following additional relationships are closely related to values used in the Bully Mnemonic

$${Earth's\,Standard\,Gravitational\,Parameter}={398,600,441,800,000\,{\frac {m^{3}}{s^{2}}}}$$

$${Earth's\,Schwarzschild\,Radius}={8.870\,056\,06\,mm}$$

Bully Mnemonic Steps

Initial Definitions

Step 1

The first step is to write down the first five digits:

$${\begin{matrix}1&2&3&4&5\end{matrix}}$$

Step 2

The second step is to select odd digits and intersperse them with zeros to form integers a) and b) as shown below: (important to remember that the first integer ends with 33 followed by a 0, whereas the second integer ends with 55 with no trailing 0)

$${\begin{matrix}{\color {Red}1}&\scriptstyle {\text{2}}&{\color {Red}3}&\scriptstyle {\text{4}}&{\color {Red}5}\end{matrix}}$$

$$a)\,{\color {Red}1}0{\color {Red}33}0$$

$$b)\,{\color {Red}3}0{\color {Red}55}$$

Step 3

The third step is to select even digits and define numbers c) and d) as shown below:

$${\begin{matrix}\scriptstyle {\text{1}}&{\color {Red}2}&\scriptstyle {\text{3}}&{\color {Red}4}&\scriptstyle {\text{5}}\end{matrix}}$$

$$c)\,{\color {Red}2}$$

$$d)\,0.{\color {Red}4}0$$

Sidereal & Tropical Years

Step 4

Multiply integers a) and b) from Step 2 to get the total number of seconds in a sidereal year.

$${\color {Red}1}0{\color {Red}33}0\times {\color {Red}3}0{\color {Red}55}=31558150={\frac {1\,Sidereal\,Year}{1\,Second}}$$

Using Long Multiplication:

       3055
×     10330
————————————
   3055
    0000
     9165
      9165
+      0000
————————————
   31558150

Step 5

The tropical year has a slightly shorter duration than the sidereal year. The approximate number of seconds in a tropical year is obtained by reducing integer a) by amount d), and then multiplying by b).

$$({\color {Red}1}0{\color {Red}33}0-0.{\color {Red}4}0)\times {\color {Red}3}0{\color {Red}55}=31556928\approx {\frac {1\,Tropical\,Year}{1\,Second}}$$

The exact number of seconds in a tropical year is obtained by reducing integer a) by amount d), multiplying by b), and then reducing by c).

$$(({\color {Red}1}0{\color {Red}33}0-0.{\color {Red}4}0)\times {\color {Red}3}0{\color {Red}55})-{\color {Red}2}=31556926={\frac {1\,Tropical\,Year}{1\,Second}}$$

Using the Distributive Property of Multiplication:

(10330 - 0.40) × 3055 = (10330 × 3055) - (0.40 × 3055)
                            =    31558150    -     1222
                            =    31556928

Great Years

Step 6

The Great Year is, by definition, a least common multiple of the sidereal year and the tropical year. From steps 4 and 5 above, we have that the ratio of tropical years to sidereal years is:

$${\frac {1\,Tropical\,Year}{1\,Sidereal\,Year}}\approx {\frac {(10330-0.40)\times 3055\,sec}{10330\times 3055\,sec}}$$

Divide top and bottom by amount d) and use the Distributive Property of Multiplication to obtain:

$${\frac {1\,Tropical\,Year}{1\,Sidereal\,Year}}\approx {\frac {({\frac {10330}{0.40}}-{\frac {0.40}{0.40}})\times 3055\,sec}{({\frac {10330}{0.40}})\times 3055\,sec}}$$

From whence:

$${\frac {1\,Tropical\,Year}{1\,Sidereal\,Year}}\approx {\frac {(25825-1)\times 3055\,sec}{(25825)\times 3055\,sec}}$$

Consequently:

$${\frac {25825\,Tropical\,Year}{25824\,Sidereal\,Year}}\approx {\frac {25825\times (25824)\times 3055\,sec}{25824\times (25825)\times 3055\,sec}}=1$$

Finally:

$$1\,Great\,Year\approx 25825\,Tropical\,Years\approx 25824\,Sidereal\,Years$$

In terms of Long Multiplication; 0.40, 25825, and 10330 are related as follows:

       0.40
×  25825
————————————
   080
    200
     320
      08.0
+      2.00
————————————
   10330.00

Galactic Years

Step 7

Multiply integer c) by the square of integer a) to get a rough approximate galactic year (the number of tropical years required for the Solar System to orbit once around the galactic center).

$${\color {Red}2}\times {{\color {Red}1}0{\color {Red}33}0}^{2}=213417800\approx {\frac {1\,Galactic\,Year}{1\,Tropical\,Year}}$$

Using Long Multiplication:

       10330
×      10330
——————————————
   10330
    00000
     30990
      30990
+      00000
——————————————
   106708900

And finally:

106708900 × 2 = 213417800

Additional Relationships

Step 8

Divide integer b) by the product of integer c) and integer a). The resulting value will be roughly ten orders of magnitude bigger than earth's standard gravitational parameter (μ) divided by the speed of light (c) cubed.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{{\color{Red} 3} 0 {\color{Red} 55}}{{\color{Red} 2} \times {{\color{Red} 1} 0 {\color{Red} 33} 0}} = 0.147\,870\,280\,736 \approx μ} Failed to parse (syntax error): {\displaystyle \frac{\{mu}}{{\color{Red} 2} \times {{\color{Red} 1} 0 {\color{Red} 33} 0}} = 0.147\,870\,280\,736 \approx μ}