CHAPTER 4
RADIATION PASSIVE
SHIELD ANALYSIS AND DESIGNS FOR ORBITAL SPACE STATIONS
An overview of this chapter
1. Introduction
· Rational;
· State of the art
(current research status in literature);
· What we present;
· Our passive
shield solutions;
2. Source classification (based on source geometry)
· Pointwise sources
-
The Sun;
-
Magnetars;
-
Supernovae;
-
Hypernovae and gamma ray bursts;
· Circular
uniformly distributed source (approximation of the galactic cosmic radiation)
· General source case
(uniform distributed source and pointwise sources)
3. Statement of the general shield model
Particularizations of the general shield model and of the sources –
analysis cases:
4. Shield for pointwise sources
5. Shield for a single source (the Sun)
6. Semi-spherical, toroidal and cylindrical-shaped shields;
7. Determining the shield shape that guarantees a constant flux of radiation.
8. Shield for two pointwise sources
· Planar shield – 2 plates of variable lengths;
·
Parabola shaped shield (
parabolic-type function, 
circle generating
function);
· Simulation results.
9. Conclusions
10. ANNEX 1. Details upon types and sources of radiation in space
11. ANNEX 2. Calculus details
12. ANNEX 3. Axial and transversal sections
1. Introduction
Rational
We analyze passive shield designs and techniques for orbital space stations. The analysis is based on the shield geometry.
State of the art in literature
After extensive documentation, we have found that in literature there is a surprising lack of knowledge in designing of radiation passive shields. Lack of knowledge is found in determining outer space radiation sources’ position and their radiation flux output (e.g. at Magnetars). We also found a high degree of uncertainty in predicting bursts (e.g. gamma ray bursts or solar flares). This lack of knowledge combined with false hypotheses led to poor designs.
Due to lack of knowledge in the domain (like the position of hypernovae, Magnetars and even the distribution of the galactic cosmic radiation), we can only approximate the source distribution – by pointwise sources or circular distributed sources. Thus I considered as pointwise sources – the Sun, hypernovae and Magnetars and as a circularly distributed source the general galactic radiation. Of course, this is theoretically correct, yet it may be (from an engineering point of view) impractical.
General shield model conditions
The shield has to satisfy the following conditions:
1.
The
shield must protect in a specified area the people/equipment located in that
area.
2.
The
shield’s geometry must correspond to the variation of the flux of radiation
that enters the shield.
3.
The
shield has to ensure in the protected area a level of radiation below a given
value (infinity of possible shield geometries result).
4.
The
shield must have a minimal mass, therefore a minimal volume (minimization
problem).
What we present
The problems analyzed are: i) the efficiency of various shapes of shields; ii) the shield optimization, for several families of geometric shapes; iii) the distribution of radiation inside the shielded area (for one case). Notice that all the analysis is for primary radiation, not for the secondary generated radiation.
The later one will be analyzed in the future. Thus, recall that the secondary-radiation effect was not analyzed.
The following aspects have been treated:
1. A thorough
analysis and synthesis of the knowledge on sources of radiation in outer space.
2. A thorough
synthesis of the proposed passive shield designs.
3. A detailed
analysis of rational, new shielding solutions that are based on the knowledge
on the radiation source distribution in the outer space.
4. An optimization
analysis of the shields, aiming at the insuring of a safe level of radiation
while keeping the volume, hence the mass of the shield at the minimum.
Our passive shield designs:
·
Planar shield, formed of two interconnected plates;
·
Semi-spherical shield;
·
Discoid shield;
·
Parabolic shield;
·
Cylindrical/toroidal shield;
·
Optimized shield shapes.
Several cases are fully analyzed and come with simulation results. For the planar shield I have determined the best length ratio (for the two plates) such that the protected volume/shield volume is maximum. For the circularly distributed source case the solved problem was to determine the best-protected areas inside the station, if the station is a torus or a cylinder.
Each section or analysis case begins with a separate introduction briefly stating the problem and is ended with conclusions.
All simulations have been made for various coefficient values and were computed with Microsoft Excel and/or C programs.
Statement of the general problem
The shield has to satisfy the following conditions:
1. It has to protect in a given area the people/equipment located in that area.
2.
The shield has to correspond to the
irradiation geometry (so it has to correspond to the variation of the function 
).
3. The shield has to ensure in the protected area a level of radiation below a given value[1]. This condition gives us infinity of possible shield geometries.
4. The shield must have a minimal mass, therefore a minimal volume.
The second and third conditions give an expression through integral.
The fourth condition states a minimization problem.
Note: The secondary radiation produced by the decomposition of primary radiation when passing through the shield is neglected throughout the analysis. The shield solutions are analyzed only for primary radiation.
The variation of
the flux of radiation 
is considered known
and therefore, the radiation geometry. As an example, the radiation geometrical
distribution is considered as in figure 1. An example is given for a
corresponding shield shape to the given irradiation geometry.
Figure 1. An example of the incoming radiation geometrical
distribution. This is the variation of the flux of radiation with the angle 
formed by the rays
with the Ox axis. A polar angle-axis
coordinate system is considered.
For the general
case analysis, the variation of the flux of incoming radiation with the angle 
is considered known.
Therefore, it is possible to determine the total flux of radiation that enters
the shield:
(1)
The third condition states that:
, (2)
where
is the maximum
admitted total flux of radiation inside the protected area.
From the
attenuation law the expression of 
is derived:
, (3)
where the
following notations have been used:
·
the flux of radiation that passes through the shield and enters
the protected area for a given angle 
;
·
the flux of radiation that enters the shield for a given angle 
;
·
is the attenuation coefficient specific to a given material;[2]
·
is the thickness of the shield for a given angle 
(please see Figure
2);
Figure 2. Example of a representation (transversal section) of a possible shield shape for the example of irradiation geometrical distribution given in Figure 1.
In the precedent figure
a possible shield shape for a given irradiation geometrical distribution has
been shown. The representation is in a polar (angle-axis) coordinate system.
The contour of the transversal section of the shield is given by two functions,
(for the interior)
and 
(for the exterior).
The functions 
, 
and the function 
are defined below:
(4)
From (2), (3) and (4) we obtain:
(5)
Supposing that a
function 
was found –
satisfying all four conditions, then the volume of the shield will be in
dependence with[3]:
(6)
By applying the fourth condition, we will obtain an optimization problem regarding the mass of the shield. Therefore, the volume of the shield will have to be minimal.
The family of functions 
is infinite for this
general case. Unfortunately, mathematics at its present state of knowledge does
not permit us to optimize such a family of functions, given by (4), (5) and
(6). We cannot solve the problem in the general case.
The function 
has to be particularized
for different shield shapes and structures to partly solve this problem.
Starting from the statement of the general problem, first specific geometries
of the radiation sources are considered. Namely, we consider pointwise sources,
then a uniform circularly distributed source, and then mixed sources. The next
step is to particularize the functions 
and 
to determine 
. These cases determine specific shield shapes.
Pointwise sources
A. A single source (the Sun)
In this case, the mass shield is designed for protecting an orbital space station against a single pointwise source (the Sun). The shield is designed such that it offers a best mass versus attenuation ratio. We present first a solution given in many past designs – semi-spherical or cylindrical shields. The transversal section of the shield is in this case delimited by two circle-generating functions. This also covers the case of a toroidal-shaped shield (shield designed at protecting a toroidal space settlement; the toroidal shield rotates separately and covers the station).
1. Semispherical shield
The problem in
this case is to determine the attenuation function. For this, determining the
apparent thickness of the shield in different points is important. The shield’s
transversal section is considered as being delimited by two circle-generating
functions 
and 
defined below:
, (1)
where the following notations have been used:
·
the radius of the internal semicircle (the semicircle of radius 
is the internal
contour of the shield);
·
is the radius of the second semicircle (the semicircle of radius 
is the external
contour of the shield);
·
x is the point of reference; We consider for our analysis that 
Note: The numbering of the formulas is reiterated at the beginning of each section.
Figure 3. Transversal section of a cylindrical mass shield.
In Figure 3 the
transversal section of a semispherical shield is represented. The flux of radiation
that comes from the Sun is 
and is considered constant in this case. The
flux that comes out of the shield is variable with the position of the observer
in reference with the shield (the observer being in any point on the horizontal
axis between 
).
The apparent thickness of the shield is defined as:
(2)
I recall that the apparent thickness is the thickness that a ray from source S would experience passing through the shield.
For reference a
point 
on the Ox axis was considered. The flux of
radiation that passes through the shield in the 
point is 
. Notice that as the point of reference varies in the
interval 
, so does the apparent thickness of the shield in that point.
This implies that the attenuation varies with 
.
The flux of radiation that passes through the shield is determined from the attenuation law:
, (3)
where:
is the flux of radiation that enters the
shield;
is the flux of radiation that exits the
shield;
is the attenuation coefficient;
is the apparent shield thickness as defined
in (2).
The attenuation function is:
From relations (2) and (3) we determine the total flux of radiation that exits the shield:
(4)
The graph of the attenuation function is represented below:
Figure 4. Attenuation function graph. Notice that in this shield shape case a variable attenuation is obtained.
Figure 3 represents
also the variation of the function 
, as 
is just a scaling
factor. Please notice that the flux of radiation exiting the shield is not
uniform.
As shown in
Figure 3, a minimum attenuation will be obtained on the vertical axis that
contains the rotation center of the settlement (O) and a maximum attenuation on the vertical axis containing the
point 
. As the attenuation is minimal towards the center of
rotation of the settlement, this means that an excess of material is used for
this shielding solution.
The attenuation is not uniform due to the fact that the apparent thickness of the shield is variable. The non-uniform protection is an important disadvantage of this type of shield (people or equipment staying close to the symmetry axis of the cylindrical station will be the least protected).
Conclusions to this section
Due to the fact that a useless excess of material is used and that the shield does not ensure a uniform protection, this solution is not viable. This solution has been presented in different projects in the past. Any shield shape that has a transversal section as represented in Figure 2 – semi-sphere, or a semi-cylinder, or even the solution proposed in many past designs – a toroidal-shaped shield rotating over/covering a toroidal settlement – is not an economically viable solution.
We propose instead a different shield shape such that a uniform attenuation is obtained, thus no material excess:
2. Determining the shield shape that guarantees a constant flux of radiation
The problem in this case is to determine the shield shape that provides a uniform attenuation. This is equivalent to the fact that the flux of radiation that exits the shield is constant on its entire surface. The flux of radiation entering the shield is considered as a constant - the flux of radiation coming from the pointwise source is constant.
The same
notations are used as in Figure 2. The goal in this case is to determine the
function 
such that the flux of
radiation exiting the shield is constant (for any point on the [OA) segment (as
in Figure 2). The hypothesis is that 
, where 
is known.
From the
attenuation law (3) and considering 
constant, the
following expression is derived:
, (5)
where
denotes the apparent
thickness of the shield. In this case, 
is considered a constant.
The resulting shield shape is represented roughly in Figure 5.
Figure 5. Transversal section of a shield shape that provides a uniform apparent thickness and therefore a uniform attenuation
Please notice
that for this shield shape, the attenuation 
is a constant for any 
.
The apparent
thickness d is determined using the attenuation law (3) for which 
is considered known:
,
where
is the maximum
admitted radiation flux that exits the shield.
Conclusions to this section
This optimized
shield shape is designed for protecting a space station against a pointwise source
(e.g. the Sun). It ensures a uniform absorption and therefore a constant flux
of radiation exits the shield in any point 
. The flux of radiation that exits the shield in any point 
is constant (in the
hypothesis that 
is constant).
B. Two pointwise sources
In this analysis case we analyze which shield shape provides a maximum protected volume/shield volume ratio, in the hypothesis that the shield is designed to protect an orbital space station against two pointwise sources that provide a constant flux of radiation.
Two cases are
analyzed: planar shield composed from two interconnected plates and a
parabola-shaped shield (function 
is a parabolic-second
degree function and the function 
is a
circle-generating function[4]).
At the end of the “planar shield” case, numerical results and simulations are
shown. Simulations have been made to state the optimum ratio between the
lengths of the two interconnected plates that form the shield, such that the
protected volume/shield volume ratio is maximum.
1. Planar shield – 2 plates of variable lengths
The first
proposed solution is a planar shield. Two pointwise sources (
) are considered to form a known angle 
. In an oversimplified hypothesis, the flux of radiation from
each of the two sources is considered equal (
). Therefore, the thickness of the two plates that compose
the shield is equal, 
. The simplified hypothesis is not taken into account
in the “Simulations” section of this chapter.
The shield is
composed from two interconnected plates that form an angle 
. The problem is to determine the best ratio between the
lengths of the two plates such that the ratio (protected volume/shield volume)
is maximum.
The following notations have been used:
B - the point at which the two plates interconnect;
- the length of the first plate;
- the length of the second plate composing the shield;
- the height of the shield;
- the flux of radiation from the first source (
);
-the flux of radiation from the second source (
, e.g. a known magnetar or hypernovae);
- the protected area;
.
Figure 6. Transversal section of a mass shield consisting of two interconnected plates.
The function 
is defined as the
ratio between the protected volume and the shield’s volume, 
. I recall that 
states for the ratio
between the lengths of the two “plates” composing the shield, 
. 
is defined as
unitary.
The deriving
optimization problem is to obtain the value of 
for which 
is maximum. The mathematical problem is to determine the
maximum area of a quadrilateral if we know two of its segments (which are
consecutive) – in our case 
and 
and if we know all
its angles.
The volume of the shield is:
To calculate the protected volume we will determine the protected surface:
The following
notations have been used: 
; 
(1)
By changing in the expression (1) the formula of the sine of difference of angles the following expression is derived:
The expressions
of the angles 
and 
are shown below:
(2)
(3)
The values of the
segments 
are shown below:
(4)
(5)
The areas 
, 
are determined from
(4) and (5):
(6)
(7)
The total protected area is:
(8)
The total
protected volume is 
. The function 
is:
(9)
is considered unitary. This doesn’t change the problem – it is
just a scaling. From (9) the following expression of 
derives:
(10)
(11)
From (10) and (11) the final expression of 
is derived:
(12)
Method of obtaining the absolute maximum and minimum values of 
:
·
First determine all the local[5]
minimum and maximum points (for 
) from the following conditions:
(13)
·
Then determine the values of the
function 
in the points located
at the extremes of the domain of definition (in our case these points are 
and 
);
·
The final step is to compare the
resulted points of maximum and minimum to select the absolute minimum and the
absolute maximum of the function 
.
Determining the
values of the function at the extremes of the interval of definition is
important, because those points can also be points of maximum or minimum. For a
given function, there may be a number of local maximum/minimum points. All of
the maximum/minimum points of the function 
need to be compared
in order to determine the absolute maximum/minimum points. For each point in which the first derivative
is null, we must check if the second derivative is not null. If the second
derivative is null, then that point is not a local point of maximum/minimum,
but an inflection point.
Figure 7 presents a function with two local maximum/minimum points, out of which none are the absolute maximum/minimum points.
Figure 7. Representation of a function with two local extreme points out of which none are absolute extreme points.
In Figure 7, the absolute maximum and the absolute minimum are obtained for the extremes of the interval of definition of the function. This shows the importance to check the values of the function at the extremes of the interval of definition to determine the absolute extreme points.
The calculus of
the first derivative of the function 
is laborious. It is
presented in ANNEX 2. A program has been written to numerically solve the
system (13). The first and second derivative of 
had to be discretized
in order to solve the system numerically.
The definition of
states that:
As computers are
discrete machines, the limit has to be approximated in order to compute the derivative.
The limit is approximated by 
.
There are two
types of approximation: “left-side” or “right-side” approximation. Both
approximations are shown below (for 
and for 
):
“Left-side” approximation
(14)
(15)
“Right-side”
approximation
(14*)
(15*)
The program uses
a “left-side” approximation of the function 
in order to solve the
system (13). 
, 
are given as input.
The program returns the absolute maximum of the function for the given
interval.
2. Parabola shaped shield (
parabolic-type function, 
circle generating
function)
The shield’s
transversal section is considered as being delimited by a parabolic
second-degree function 
- to the exterior-
and by a circle generating function 
to the interior. This
shield shape is designed to provide radiation protection for an orbital space
station against two pointwise sources (
) that provide a constant flux of radiation. The shield’s
transversal section is presented in Figure 8.
Figure 8. Transversal section of a shield delimited by a
parabolic-type function at the exterior (
) and by a circle-generating
function to the interior (
).
The function 
is determined based
on the following criteria:
-The graph of the
function 
is a parabola
symmetrical to the vertical Oy axis
and has 2 intersections with the horizontal Ox
axis;
-The apex of the
parabola 
is a point of maximum
for the function;
The two
symmetrical intersections with the horizontal axis will be in the points 
and 
determined below:
Function 
is defined below:
,
Function 
is defined below:
The total area of the transversal section of the shield is:
, where:
(1)
(2)
is the primitive of 
:
From (1) and (2)
the expression of 
derives:
The total area of the transversal section is:
(3)
The function protected volume/shield volume is defined below:
(4)
In Figure 9 the protected area is shown, along with the two sources. The following notations have been used:
; 
; 
; 
; 
;
The protected area is derived as:
(5)
The angles 
are considered known.
have to be determined
from the following system of equations:
(6)
From the equation
system (6) and the expression (5) in this section, the total protected area can
be derived. Thus, the variation of function 
stating the ratio
protected volume/shield volume can be determined. The analysis is similar to
Case B.1. We will not insist on it.
Figure 9. Transversal section of shield and representation of the protected area.
From the equation
system (6) and the expression (5) in this section, the total protected area can
be derived. Thus, the variation of function 
stating the ratio
protected volume/shield volume can be determined. The analysis is similar to
Case B.1. We will not insist on it.
Simulation results
This section refers to the case B.1 for a planar shield composed of two interconnected
plates, designed at protecting a space station against two pointwise sources.
The equation (12)
and the system (13) have been solved numerically in Microsoft Excel and using a
program[6]. For different values of angle 
and for different
ratios 
has been computed. 
is computed in
variation with 
and graphs have been
plotted.
For the
simulations, the fluxes of radiation that enter the shield, 
and 
are considered different and therefore the thickness of the
two plates that form the shield are considered different -
-, case presented in Table 3 and in Figure 12.
Table 1.
Simulation results for 
, C represents the
sum of the plates’ lengths. For convenience, C is considered unitary. The last column shows the total protected
area, computed using the formulas (6), (7) and (8).
|
alpha=30 |
C=L1+L2=1 |
|
0.866025 |
|
0.5 |
|
|
|
L1 [*10m] |
L2 [*10m] |
lambda |
A1 |
A2 |
alfa_1 |
alfa_2 |
A1+A2 |
|
0 |
1 |
0 |
0 |
0.866025 |
0 |
0.523599 |
0.866025 |
|
0.1 |
0.9 |
0.111111 |
0.09866 |
0.791481 |
0.050636 |
0.472963 |
0.890141 |
|
0.2 |
0.8 |
0.25 |
0.19464 |
0.714257 |
0.102394 |
0.421204 |
0.908897 |
|
0.3 |
0.7 |
0.428571 |
0.28794 |
0.634355 |
0.155029 |
0.36857 |
0.922295 |
|
0.4 |
0.6 |
0.666667 |
0.37856 |
0.551773 |
0.208263 |
0.315336 |
0.930333 |
|
0.5 |
0.5 |
1 |
0.4665 |
0.466513 |
0.261803 |
0.261796 |
0.933013 |
|
0.6 |
0.4 |
1.5 |
0.55176 |
0.378573 |
0.315343 |
0.208256 |
0.930333 |
|
0.7 |
0.3 |
2.333333 |
0.63434 |
0.287955 |
0.368578 |
0.155021 |
0.922295 |
|
0.8 |
0.2 |
4 |
0.71424 |
0.194657 |
0.421213 |
0.102385 |
0.908897 |
|
0.9 |
0.1 |
9 |
0.79146 |
0.098681 |
0.472974 |
0.050625 |
0.890141 |
|
1 |
0 |
|
0.866 |
0 |
0.523611 |
-1.3E-05 |
0.866 |
A graph has been
plotted to show the variation of 
with 
for an angle 
:
Figure 10. Representation of the variation of the protected area
with lambda, for an angle 
. This graph was made considering that 
.
The following
representation of the variation of 
with 
was made for 
and for 
. Please notice that the maximum ratio is for 
:
Figure 11. Representation of the variation of the ratio protected
volume versus shield volume with 
. Please note that the best ratio is obtained for 
. The fluxes of radiation 
and 
are considered equal.
For another value
of angle 
different results are
obtained. The same hypothesis 
and 
has been used:
Table
2. Simulation results for 
. Notice that the protected area differs significantly for
small variations of 
- such as from 
to 
.
|
alpha=45 |
C=L1+l2=1 |
|
0.707107 |
|
0.707107 |
|
|
L1
|
L2 |
lambda |
A1 |
A2 |
alfa_1 |
alfa_2 |
A1+A2 |
|
0 |
1 |
0 |
0 |
0.5 |
0 |
0.785398 |
0.5 |
|
0.1 |
0.9 |
0.111111 |
0.09707 |
0.46863 |
0.072706 |
0.712693 |
0.5657 |
|
0.2 |
0.8 |
0.25 |
0.18828 |
0.43312 |
0.149087 |
0.636311 |
0.6214 |
|
0.3 |
0.7 |
0.428571 |
0.27363 |
0.39347 |
0.22848 |
0.556918 |
0.6671 |
|
0.4 |
0.6 |
0.666667 |
0.35312 |
0.34968 |
0.310015 |
0.475383 |
0.7028 |
|
0.5 |
0.5 |
1 |
0.42675 |
0.30175 |
0.392668 |
0.39273 |
0.7285 |
|
0.6 |
0.4 |
1.5 |
0.49452 |
0.24968 |
0.475323 |
0.310075 |
0.7442 |
|
0.7 |
0.3 |
2.333333 |
0.55643 |
0.19347 |
0.556867 |
0.228531 |
0.7499 |
|
0.8 |
0.2 |
4 |
0.61248 |
0.13312 |
0.636273 |
0.149125 |
0.7456 |
|
0.9 |
0.1 |
9 |
0.66267 |
0.06863 |
0.712672 |
0.072726 |
0.7313 |
|
1 |
0 |
|
0.707 |
#DIV/0! |
0.785398 |
0 |
|
In the following
simulations, the simplifying hypothesis that 
and 
are equal and thus 
is not taken
into account. The fluxes 
and 
are considered in the
ratios 1:1, 1:1.1 and 1:1.5 respectively. Computations have been made to
determine the optimal length ratio 
such that 
is maximum. The
computations are shown for a ratio 
in Table 3.
Table
3. Simulation results for a flux ratio of 
. Angle 
is considered of 
.
|
L1 |
L2 |
lambda |
A1 |
A2 |
alfa_1 |
alfa_2 |
A1+A2 |
(A1+A2)/(L1*d1+L2*d2) |
(A1+A2)/(L1*1+L2*1.2) |
|
0 |
1 |
0 |
0 |
0.866025 |
0 |
0.523599 |
0.866025 |
0.57735 |
0.787296 |
|
0.1 |
0.9 |
0.111111 |
0.09866 |
0.791481 |
0.050636 |
0.472963 |
0.890141 |
0.61389 |
0.816643 |
|
0.2 |
0.8 |
0.25 |
0.19464 |
0.714257 |
0.102394 |
0.421204 |
0.908897 |
0.649212 |
0.841572 |
|
0.3 |
0.7 |
0.428571 |
0.28794 |
0.634355 |
0.155029 |
0.36857 |
0.922295 |
0.683181 |
0.861958 |
|
0.4 |
0.6 |
0.666667 |
0.37856 |
0.551773 |
0.208263 |
0.315336 |
0.930333 |
0.715641 |
0.877673 |
|
0.5 |
0.5 |
1 |
0.4665 |
0.466513 |
0.261803 |
0.261796 |
0.933013 |
0.74641 |
0.888584 |
|
0.6 |
0.4 |
1.5 |
0.55176 |
0.378573 |
0.315343 |
0.208256 |
0.930333 |
0.775278 |
0.894551 |
|
0.7 |
0.3 |
2.333333 |
0.63434 |
0.287955 |
0.368578 |
0.155021 |
0.922295 |
0.801995 |
0.895432 |
|
0.8 |
0.2 |
4 |
0.71424 |
0.194657 |
0.421213 |
0.102385 |
0.908897 |
0.82627 |
0.891076 |
|
0.9 |
0.1 |
9 |
0.79146 |
0.098681 |
0.472974 |
0.050625 |
0.890141 |
0.847753 |
0.881328 |
|
1 |
0 |
|
0.866 |
0 |
0.523611 |
-1.3E-05 |
0.866 |
0.866 |
0.866 |
A graph has been
made to show the optimal length ratio 
for flux ratio 
of 1:1, 1:1.1 and
1:1.5, respectively. The graph shows the variation of the protected surface with
the length ratio 
of the two plates.
The vertical axis represents the ratio 
, while the horizontal axis represents the ratio 
. Notice that the results differ significantly for different
flux ratios 
.
Figure 12. Representation of the variation of the ratio protected
volume/shield volume with the length ratio 
.
The
representation in Figure 12 has been made to show the importance of the flux
ratios 
in determining the
optimal length ratio. For slightly different flux ratios – such as from 1:1 to
1:1.1 significantly different results are obtained.
Conclusions to this sub-section
Considering
that a mass shield is composed of two interconnected plates and it is designed
to protect a space station against two pointwise sources, an analysis has been
made to determine the optimal length ratio of the two plates such that the
ratio protected volume/shield volume is maximum. Simulations have been made for
different angles between the sources and for different flux ratios of the
sources - 
1:1, 1:1.1 and 1:1.5 respectively. The simulation revealed
that for small flux ratio variations – such as from 1:1 to 1:1.1- the results
differ significantly.
General conclusions to this chapter
In this chapter, I have re-stated the problem of the radiation shield design for space settlements, space colonies and other space habitable objects. The problem dealt with has been the shielding for primary radiation, with the aim of optimizing the shield effectiveness with respect to the mass of the shield. Several shield configurations and several radiation source geometries have been discussed. An essential problem discussed – which has never been discussed in the projects in this contest at least – is the distribution of the radiation inside the shielded volume. I have shown that the radiation distribution is far from being uniform. Therefore, care has to be observed when placing schools and other densely populated or critical habitable facilities inside the shield.
I hope this chapter is a useful contribution not only to the design of space settlements, but also to NASA’s work in general.
Bibliography
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