Deeper Orbit Budget Scans

Circular Speed-Square Radius Budget Three-Row Scan

Field times radius gives the speed-square that closes a circular orbit. 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 4 needs speed-square 4

The field at this radius is 1. Multiplying field by radius gives speed-square 4. Dividing back by radius returns acceleration 1.

1⋅4=4,v2r=11\cdot4=4,\quad \frac{v^{2}}{r}=1
Circular closure row r=4The orbit accepts because acceleration equals the checked field.speedSquared=4 m^2/s^2centripetalAcceleration=1 m/s^2acceptedBit=1 bit

Radius 2 needs speed-square 8

The field at this radius is 4. Multiplying field by radius gives speed-square 8. Dividing back by radius returns acceleration 4.

4⋅2=8,v2r=44\cdot2=8,\quad \frac{v^{2}}{r}=4
Circular closure row r=2The orbit accepts because acceleration equals the checked field.speedSquared=8 m^2/s^2centripetalAcceleration=4 m/s^2acceptedBit=1 bit

Radius 1 needs speed-square 16

The field at this radius is 16. Multiplying field by radius gives speed-square 16. Dividing back by radius returns acceleration 16.

16⋅1=16,v2r=1616\cdot1=16,\quad \frac{v^{2}}{r}=16
Circular closure row r=1The orbit accepts because acceleration equals the checked field.speedSquared=16 m^2/s^2centripetalAcceleration=16 m/s^2acceptedBit=1 bit

Circular speed-square is a radius budget, not a separate guess

Each row starts from the checked field. The needed speed-square is field times radius, and the orbit check divides by the same radius to get back to the field.

rgg⋅rv⋅v/r414124841161616\begin{array}{c|c|c|c}r&g&g\cdot r&v\cdot v/r\\4&1&4&1\\2&4&8&4\\1&16&16&16\\\end{array}
Circular radius budgetThe middle row shows the multiply-then-divide closure.speedSquared=8 m^2/s^2centripetalAcceleration=4 m/s^2acceptedBit=1 bit4 * 2 = 8, 8 / 2 = 4