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

 

[1] “Radiation Hazards to Crews of Interplanetary Missions: Biological Issues and Research Strategies”, Task Group on the Biological Effects of Space Radiation Space Studies Board, Commission on Physical Sciences, Mathematics and Applications, National Research Council; National Academy Press, Washington D.C., 1996. http://www.nap.edu/openbook/0309056985/html/R1.html Accessed 11th September 2004

[2] M.J. Berger, J.H. Hubbell, S.M. Seltzer, J.S. Coursey, and D.S. Zucker, “X-COM: Photon Cross Sections Database”, 1998, http://physiscs.nist.gov/PhysRefData/Xcom/Text/XCOM.html, Accessed 13th December 2004

[3]“ASEN 5016 Lecture 17: Space Radiation”, http://www.colorado.edu/ASEN/asen5016/17-Rad.html, Accessed 13th December 2004

[4] Dr. Tony Phillips, “Solar Flares on Steroids”, 2004, http://science.nasa.gov/headlines/y2003/12sep_magnetars.htm, Accessed on 3rd November 2004

[5] A.A. Mikhailov, “Arrival directions and chemical composition of ultrahigh energy cosmic rays”, Proceedings of ICRC2001

[6] “Space environment (natural and artificial) – Model of radiation impact  by galactic cosmic rays”, ISO/DIS 15390, 2002

[7] W. Schoner, M. Hajek, M. Noll, R. Ebner, N. Vana, M. Fugger, Y. Akatov, V. Shurshakov, V. Arkhangelski, “Measurement of the depth dose - and LET distribution at the surface and inside of space station MIR”, 1999



[1]  The given value is the maximum flux of radiation admitted for people in the protected area.

[2] The material selection for the shield’s composition is based on its attenuation coefficient, on its availability and on the property of the material of generating less secondary radiation when hit by GCRs.

[3] The shield’s shape is considered a general cylinder with the transversal section given by the functions  and .  represents the height of the shield.

[4] Note: Here ,  have the same meanings as presented in the General Case: function  represents the inner contour of the transversal section of the shield, while function  represents the external contour of the transversal section of the shield.

[5] The local minimum and maximum points are the points located inside the domain of definition of the function, namely for values of  satisfying the condition:

[6] The program has been written in Borland C. It solves numerically the equation (12). It is presented in ANNEX 3.