Relation Cross-Scans

Escape V-Square Double Circular Cross-Scan

Escape speed-square is twice circular speed-square at each checked radius. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Escape square doubles circular square at radius 4

Circular speed-square 4 doubles to escape speed-square 8.

2⋅4=8,ϵ=02\cdot4=8,\quad \epsilon=0
Escape row r=4The escape boundary cites the circular source.circularSpeedSquared=4 m^2/s^2escapeSpeedSquared=8 m^2/s^2kineticSpecific=4 J/kgpotentialSpecific=-4 J/kgtotalSpecificEnergy=0 J/kgacceptedBit=1 bit

Escape square doubles circular square at radius 2

Circular speed-square 8 doubles to escape speed-square 16.

2⋅8=16,ϵ=02\cdot8=16,\quad \epsilon=0
Escape row r=2The escape boundary cites the circular source.circularSpeedSquared=8 m^2/s^2escapeSpeedSquared=16 m^2/s^2kineticSpecific=8 J/kgpotentialSpecific=-8 J/kgtotalSpecificEnergy=0 J/kgacceptedBit=1 bit

Escape square doubles circular square at radius 1

Circular speed-square 16 doubles to escape speed-square 32.

2⋅16=32,ϵ=02\cdot16=32,\quad \epsilon=0
Escape row r=1The escape boundary cites the circular source.circularSpeedSquared=16 m^2/s^2escapeSpeedSquared=32 m^2/s^2kineticSpecific=16 J/kgpotentialSpecific=-16 J/kgtotalSpecificEnergy=0 J/kgacceptedBit=1 bit

Escape square stays twice circular square at every checked radius

Changing radius changes both circular and escape speed-square. The zero-energy boundary stays closed because escape speed-square is twice circular speed-square.

rvc⋅vcve⋅veϵ448028160116320\begin{array}{c|c|c|c}r&v_c\cdot v_c&v_e\cdot v_e&\epsilon\\4&4&8&0\\2&8&16&0\\1&16&32&0\\\end{array}
Escape double-square cross-scanThe middle radius row is displayed.circularSpeedSquared=8 m^2/s^2escapeSpeedSquared=16 m^2/s^2kineticSpecific=8 J/kgpotentialSpecific=-8 J/kgtotalSpecificEnergy=0 J/kgacceptedBit=1 bit