Re: Compute the Volume of a Sphere in Any Dimension

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An adaptation of the recent publications concerning detection of
precious metal isotopes is presented along with a block diagram
of equipment used (above). Remote detection of precious metals
in the asteroids would utilize a robotic flyby to assess the
qualitative nature of the surface of a NEA asteroid without
having to land the spacecraft to take extraneous samples of
regolith. Robotic landers are deemed unnecessary for purposes
of opening the window of opportunity for precious metal mining
of the asteroids. Two papers used for reference in this study
are at the bottom. It can be understood that from a feasibility
study, a measurement system from Monte-Carlo simulation can
be adapted for use in a robotic flyby of an NEA asteroid. Models
include total gamma quanta distribution and pulse height spectrum
of precious metal in asteroidal regolith, along with a Gamma Rate
Detector for remote observation.

The gamma quanta distribution in regolith takes the form:

G(x,y,E_i,E,cos)=Integral [P( E_i,x-x',y-y',h,E )rho(h)dx'dy'dh]

where P( E_i , x-x', y-y',h,E ) dx'dy'dh is the pointed source
function (known), with E_i being the initial gamma quantum energy,
E being the final gamma quantum energy, rho(h) is the concentra-
tion of precious metal solved for, knowing the gamma quanta
energy distribution for a specific flyby elevation, H_specific,
at any of the activation energies of either a gold or platinum
molecule.

In reference (1) step 2 lists using the EGS-4 Code from reference
(3), using a distribution N ( E, cos(theta)) N_1 quanta are
"generated uniformly on the bottom and collateral borders of
detector and their spectrum of deposited energy N_loss(E) is
determined. Then the pulse height spectrum is calculated due
to the formula

N_i(a_i , E) = integral [ N_loss(a_i , E)/sqroot[2(pi)(rho)(E')]
* exp[(E - E')2 / 2(rho)2E']dE'

where a_i is the parameter of isotopes distribution, rho(E) is
the function depending on the detector specific properties.

References

1. Calculation the concentrations of radioactive isotopes and
parameters of their distribution in soil for airborne gamma
ray spectrometer system - D. N. Matsukevich, A.I. Borodich,
A.A. Khrutchinsky and A.M. Belov, Institute of Nuclear Problems,
Belarussian State University, St. Bobruiskaya 11, Minsk, 220050,
Republic of Belarus

2. Electronic Dose Conversion Technique Using a NaI(Tl) Detector
for Assessment of Exposure Dose Rate from Environmental Radiation
- Gyuseong Cho and Ho Kyung Kim, Korea Advanced Institute of
Science and Technology, Taejon 305-701 Korea, and Heegon Woo
and Gilhwan Oh, Korea Electric Power Research Institute, Taejon
305-380 Korea, and Dal Kyu Ha, Samchand Enterprise Company,
Anyang 431-060 Korea

3. W.R. Nelson, H.Hirayama, D.W.Rogers. The EGS-4 Code System.
SLAC-Report-265, 1985. 383p. (SLAC = Stanford Linear Accelerator)

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