Stochastic Visibility Relation to Mean Chord Length and Opacity

Below is a practical guide on how to calculate the dimensionless quantity

$\Sigma \,\frac{4V}{A}$

where

• is the macroscopic collision/absorption rate (collisions per unit length) in a homogeneous medium.
• is the volume of the convex region .
• is the surface area of .

This product is often referred to as an optical thickness or opacity parameter in contexts such as radiative transfer, neutron transport, or acoustic scattering.

1. Determine (Collision Rate)

(sometimes called the “macroscopic cross section” in nuclear or radiative transport) is typically measured or estimated from physical properties of the medium:

1. From Microscopic Cross Section and Number Density
   $\Sigma = n\,\sigma,$ 

   where

   • is the number density of scattering/absorbing particles (number per unit volume),
   • is the microscopic cross section for each particle (units of area).
2. Experimental Measurement
   • Observe how beams or particles attenuate over a known distance .
   • Fit an exponential attenuation law .
   • Extract from the slope of .

Either way, has units of 1/length.

2. Compute (Volume of )

1. Analytical Formula (Simple Geometries)
   • Sphere of radius : .
   • Cuboid with sides (): .
   • Cylinder with base area and height : .
2. Numerical Integration / 3D Model
   • If is complex, use CAD software or voxel integration to estimate volume.
   • 3D laser scanning or CT data can approximate the shape and compute .

3. Compute (Surface Area of )

1. Analytical Formula (Simple Geometries)
   • Sphere of radius : .
   • Cuboid (): .
2. Numerical Methods
   • For irregular shapes, many computational geometry packages can approximate surface area from a polygon mesh or point cloud (e.g., using triangular surface meshes).

4. Put It All Together: Multiply

1. Compute the factor
   For a general shape: $$\langle \ell \rangle = \frac{4\,V}{A}.$$ 

   This is the mean chord length by Cauchy’s formula for a convex body.

2. Multiply by
   $\Sigma \,\frac{4V}{A}.$ 

   This final number is dimensionless—it tells you how “large” the collision rate is compared to the typical distance a chord travels in .

5. Example: Sphere of Radius

For a sphere:

1. Volume: .
2. Surface Area: .
3. Mean Chord Length:
   $$\frac{4V}{A} \;=\; \frac{4\bigl(\tfrac{4\pi}{3}r^3\bigr)}{4\pi r^2} \;=\; \frac{4 \cdot \tfrac{4\pi r^3}{3}}{4\pi r^2} \;=\; \frac{4r}{3}.$$ 

   
   

4. Multiply by :
   $$\Sigma \,\frac{4V}{A} \;=\; \Sigma \,\Bigl(\frac{4r}{3}\Bigr).$$ 

   
   

If , the sphere is “optically thick/opaque”; if , it is “optically thin/transparent.”

6. Physical Interpretation

• 
  • The collision/absorption rate is small compared to the typical distance .
  • High fraction of rays/particles traverse without interaction (nearly transparent).
• 
  • The collision rate is large relative to the size of .
  • Most rays/particles collide or are absorbed before exiting (opaque or optically thick).

7. Summary

1. Measure or estimate : from physical cross-section data or direct experiment.
2. Compute and : either by known geometry formulas or numerical methods.
3. Form the product .
4. Interpret:
   • → mostly free paths, high visibility/transparency.
   • → frequent collisions, high opacity, low visibility.

That’s the straightforward way to calculate and interpret for a convex region .