Deeper Orbit Budget Scans

Radius-Square Field Budget Three-Row Scan

The same source value is divided by radius-square in three visible stages. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Radius 1 creates denominator 1

The source parameter stays 16. Radius 1 is squared first, making denominator 1. The field is then 16.

r2=1,g=161=16r^{2}=1,\quad g=\frac{16}{1}=16
Field budget row r=1The checked radius creates the checked field denominator.mu=16 m^3/s^2radius=1 mradiusSquared=1 m^2field=16 m/s^2

Radius 2 creates denominator 4

The source parameter stays 16. Radius 2 is squared first, making denominator 4. The field is then 4.

r2=4,g=164=4r^{2}=4,\quad g=\frac{16}{4}=4
Field budget row r=2The checked radius creates the checked field denominator.mu=16 m^3/s^2radius=2 mradiusSquared=4 m^2field=4 m/s^2

Radius 4 creates denominator 16

The source parameter stays 16. Radius 4 is squared first, making denominator 16. The field is then 1.

r2=16,g=1616=1r^{2}=16,\quad g=\frac{16}{16}=1
Field budget row r=4The checked radius creates the checked field denominator.mu=16 m^3/s^2radius=4 mradiusSquared=16 m^2field=1 m/s^2

The radius-square denominator carries the inverse-square pattern

The diagram fixes the source and moves the body outward. The table separates the three scan stages: radius, radius-square, then field.

rr⋅rμg11161624164416161\begin{array}{c|c|c|c}r&r\cdot r&\mu&g\\1&1&16&16\\2&4&16&4\\4&16&16&1\\\end{array}
Radius-square field budgetThe displayed row shows the largest denominator.r=4 -> r^2=16 -> g=1