Engineering tool for trapped proton flux anisotropy evaluation

Radiation Measurements, Vol. 26, No. 6, pp. 953-958, 1996

Pergamon PII: S1350-4487(96)00097-2

Copyright © 1996 ElsevierScienceLtd Printed in Great Britain. All fights reserved 1350-4487/96 $15.00+ 0.00

E N G I N E E R I N G TOOL F O R T R A P P E D P R O T O N F L U X ANISOTROPY EVALUATION M. KRUGLANSKI Belgian Institute for Space Aeronomy, Avenue Circulaire 3, B-1180 Brussels, Belgium Abstract--At low altitudes, the high-energy trapped proton fluxes are strongly anisotropic. The proton flux is controlled by the density distribution of the Earth's atmosphere that induces a steep pitch-angle distribution and an East-West effect, the latter caused by the finite size of the proton gyration radius. We have developed a software package, ANISO, to evaluate averaged energy spectra of trapped proton unidirectional fluxes for a given spacecraft orbit and attitude by deducing the angular-dependent proton flux spectra from the AP-8 omnidirectional flux. Included in the model are both the Armstrong et al. (1990) model and a model based on the Badhwar and Konradi (1990) pitch-angle distribution. Copyright © 1996 Elsevier Science Ltd

1. INTRODUCTION At low altitudes, the high-energy trapped proton fluxes are strongly anisotropic and may induce anisotropies in the radiation exposure of a spacecraft with stabilized attitude. The flux anisotropy is understood theoretically by the interaction of the trapped protons with the Earth's atmosphere and by the finite length of the proton gyroradius. The main purpose of modelling this flux anisotropy is to deduce angular-dependent proton flux spectra from standard omnidirectional flux databases which were, until recently, the only ones available. Such a model has been developed analytically by Watts et al. (1989) and has been applied to evaluating radiation shielding for manned spacecraft and to analyzing data from the Long Duration Exposure Facility (LDEF) satellite. In this article we briefly review the existing model of Armstrong et al. (1990). We describe a new model based on the Badhwar and Konradi (1990) pitchangle distribution. We then present the structure and a sample run of the program ANISO.

2. ANISOTROPY M O D E L S A trapped proton anisotropy model is intended to predict the trapped proton unidirectional flux at any point of observation as well as the unidirectional fluence obtained by the average of the flux along the trajectory of a satellite. The model comprises the following: • a model of the omnidirectional flux j0, such as the standard NASA magnetospheric proton models AP-8 MIN and AP-8 M A X (Sawyer and Vette, 1976); • a pitch-angle distribution caused by the atmospheric loss cone;

• an East-West asymmetry effect caused by the finite size of the gyration radius; • a normalisation factor which has to be determined so that the omnidirectional flux computed from the unidirectional flux remains equal to j0Heckman and Nakano (1969) assumed that the flux along a magnetic field line is inversely proportional to the atmospheric density and proposed a pitch-angle distribution based on a dipolar magnetic field and an exponential atmosphere. They obtained a pitch-angle distribution in the form of a gaussian function centred around the pitch angle :t = 90 ° with the square of the standard deviation given by 3 H (2 + cos2I), ~= ~

(1)

where I is the magnetic dip angle, H is the atmospheric scale height, and R is the distance from the Earth's centre. With similar assumptions, Lenchek and Singer (1962) proposed an azimuthal correction function for trapped particles at low altitude. This function considers that, for a given point of observation, the guiding centre of a proton coming from the West is above the guiding centre of a proton coming from the East. The function of Lenchek and Singer (1962) depends on the ratio between the gyroradius of the trapped particles and the atmospheric scale height at their mirror point. Watts et al. (1989) combined the Heckman and Nakano (1969) pitch-angle distribution with the Lenchek and Singer (1962) East-West asymmetry function to derive a two-dimentional distribution. They obtained the following anisotropy conversion 953

954

M. K R U G L A N S K I

factor to transform omnidirectional differential flux into unidirectional differential flux,

WVF= exp

[

(e -- /~/2)2 ] (r~cos I sin e sine ) ~ exp H ,

(2n),/2sineaerf(---~-~Io(rsc°slsine \a

/8/

\

(2)

j(e) = {K~=exp( - fl~=)0otherwise,When eL < • < It -- eL; (3)

/

H

where ct is the local pitch angle, 4) is the azimuthal angle measured to the East in the plane perpendicular to the magnetic field vector from the local vertical plane, and rg is the gyroradius of a locally mirroring particle (I0 is the modified Bessel function). Armstrong et al. (1990) used this anisotropy conversion factor with an atmospheric scale height H deduced from the Johnson and Smith (1985) atmospheric model. They applied it to the NASA models AP-8 MIN and AP-8 MAX to produce models of trapped proton unidirectional fluxes at low altitude. The models were called VF1MIN and VF1MAX, for solar minimum and solar maximum, respectively, by Colborn et al. (1990). Resulting from approximations, especially in the evaluation of the atmospheric scale height, Armstrong et al. (1990) restrict their model to the altitude range between 250 and 500 km. To extend the

1.0000

I

I

altitude range of validity, we developed a model based on a pitch-angle distribution proposed by Badhwar and Konradi (1990). The Badhwar and Konradi (1990) pitch-angle distribution has to be fitted using experimental data. It depends on a loss-cone-angle parameter ~L and a shape parameter fl and is given by

I

I

where K is a normalisation constant, ~= _ sine - sineL x/~ ,

(4)

and B is the local magnetic field intensity. We combined this pitch-angle distribution with the Lenchek and Singer (1962) East-West asymmetry function to deduce a new anisotropy conversion factor (Kruglansld, 1996). For eL < e < n-eL, the conversion factor is given by WBr = ~=exp( - fl~=)exp( r~coslsinesin4)H ) r,/2r [

4nJ. L/o[rec°slHine'7=,exp(- #~¢)dcose"] (5)

i

i

I

I

I

BK-MIN

.°..'""&'"",,

,,

..................

VF'I MIN

0.1000

.. :

;

?

0.0100 ?

0.0010 l

:. :. :. 0.0001

I

3O

I .:

45

I

I

60

75 90 05 Pitch ongle (degrees)

I"=

:, [

I

120

135

150

Fig. 1. Comparison of the pitch-angle distribution of the models V F I M I N and BK-MIN at 60°W, 35°S and altitude 450 kin. The conversion factor from omnidirectional differential flux to unidirectional flux is presented at E ffi 20 MeV and 4) = 0 ° for each model.

00001

0.0010

r,,,

,...

>

~n < >

0 "u

o,H

X >

nl

Z

0

o

>

0.0010 0.0001

©

©

©

,..]

C3

7.

7~

7~

00100

0.1000

1.0000

10.0000

BK-MIN

Fig. 2. Predicted anisotropy of trapped proton integral fluence (E > 100 MeV) obtained by the models VFIMIN and BK-MIN. The proton flux was averaged over Lwelve consecutive circular orbits (28.5 ° inclination, 450 km altitude). The polar angle is measured relative to the zenith direction and the azimuthal angle to geographic North direction in the horizontal plane.

•~

~'~ 0.0100

,~ 0.10001

1.00001

10.0000

VFIMIN

0.0100

0.10 O0

0000

a

0.0001

0.0010

0.0100

0.1000

1.0°00

100000

BK-MIN

Fig. 3. Predicted anisotropy of trapped proton integral fluence (E > 100 MeV) obtained by the models VFIMIN and BK-MIN. The proton flux was averaged over twelve consecutive circular orbits (28.5 ° inclination, 450 km altitude). The polar angle is measured relative to satellite velocity direction. The azimuthal angle is measured in the plane perpendicular to the velocity direction relatively to the trace of the orbital plane (looking away from Earth).

0.0001

-~ 0.0010

r,~,

I.

I 0"0000

VF1 MIN

,~coq5

70

5/3

C~ t" >.

E N G I N E E R I N G TOOL FOR TRAPPED PROTON FLUX ANISOTROPY EVALUATION outside this range it is 0. We combined Expression (5) with the NASA models AP-8 M I N and AP-8 MAX to produce new models called BK-MIN and BK-MAX, respectively. The parameters ~L and fl are functions of B and of the McIlwain's (1961) parameter L. These functions are obtained from a fit of the pitch-angle distribution to the AP-8 unidirectional proton fluxes at 20 MeV (Heynderickx and Lemaire, 1993). The scale height H is fixed at 100 km. In Fig. 1, we compare the conversion factor of the models VF1MIN and BK-MIN at 60°W, 35°S and altitude 450 km in the local vertical plane (~ = 0°). The proton kinetic energy E is set to 20 MeV. The (B,L) coordinates are computed with the Jensen and Cain (1962) geomagnetic model. The atmospheric scale heights of both models are quite similar (at 450 km, Hrnul~ ,~ 108 kln). The pitch-angle distribution used in the model BK-MIN is steeper than the one in the model VF1MIN. Within the model BK-MIN, the loss cone is defined explicitly by angle ~tL}. At the selected point, CtL= 79.5 °, the unidirectional flux is confined to a cone of 21 ° opening angle as seen clearly in Fig. 1. Within the model VF1MIN, the loss cone is defined implicitly through the parameter ~. In Fig. 1, cr = 10.3° induces a smoother anisotropy than the model BKMIN. Since the pitch-angle distribution used in the model BK-MIN constantly reproduces the flux of the NASA model AP-8 MIN, the results obtained with the new model BK-MIN are more consistent than those obtained with the model V F I M I N of Armstrong et al. (1990).

3. ANISO CODE Expressions (2) and (5) allow us to evaluate the trapped proton unidirectional differential fluxes at any position in space. These expressions are applied along a satellite orbit to deduce a unidirectional finence. The ANISO program has been implemented in the radiation analysis software package U N I R A D (Heynderickx et al., 1996). This package includes an orbit generator, the computation of the McIlwain (1961) parameter, empirical radiation environment flux models, and a convertor of energy spectra to radiation dose. The recently added program ANISO transforms trapped proton omnidirectional integral fluxes produced by the flux models into orbit-averaged unidirectional fluences. As output, it provides unidirectional finences for a set of directions specified by the user in a coordinate system (g, ~, ~) attached to the satellite. The program provides differential as well as integral fluences. To evaluate a unidirectional fluence, the program needs the attitude of the satellite along its orbit as input. In the code, the attitude of the satellite is defined by the transformation matrix between the coordinate system (~, fi, :D and the geographic

957

spherical coordinate system. Three common satellite orientations are predefmed in the package U N I R A D as follows: • The z-axis of the satellite points to the zenith. The x-axis and y-axis point in the horizontal plane to the geographic North and West directions, respectively. • The z-axis is parallel to the velocity vector of the satellite. The x-axis lies in the orbital plane pointing away from Earth. The y-axis is perpendicular to the orbital plane, pointing to the South direction. • The coordinate system (g, 35, ~) is parallel to the geographic equatorial inertial coordinate system. To satisfy a specific requirement, the user may easily adapt one of these attitudes according to the expected orientation of the satellite. As an example, we present in Fig. 2 the predicted anisotropy of the trapped proton integral flux ( E > 100 MeV) obtained by the models VF1MIN and BK-MIN for a satellite with a circular orbit of 28.5 ° inclination and 450 km altitude. The attitude of the satellite is set to the first predefined attitude where the satellite is stabilized along the gravity gradient. The omnidirectional proton spectra are taken from the AP-8 MIN model. The McIlwain (1961) parameter L and the magnetic field vector are computed using the Jensen and Cain (1962) geomagnetic field model. The fluxes presented in Fig. 2 correspond to an average over twelve consecutive circular orbits. The omnidirectional integral flux at E > 100 MeV, averaged over these orbits, is equal to 13.04 cm -2 s -1. The left panel is obtained with the model VF 1MIN while the right panel correspond to the prediction of the model BK-MIN. The unidirectional integral flux is given as a function of the look direction in the frame of the satellite. The azimuthal angles 0 °, 90 °, 180°, and 270 ° correspond, respectively, to particles coming from the geographic North, West, South, and East directions. The polar angles 0 ° and 180° correspond to the upward and downward directions while the polar angle 90° corresponds to the local horizontal plane. In each panel, the unidirectional integral flux varies significantly with polar and azimuthal angle. The angular distribution has the shape of two deep valleys that correspond to the mean directions of the upward and downward loss cone in the South Atlantic Anomaly region at 450 km altitude. The valleys are wider and steeper in the right panel so that the unidirectional flux anisotropy obtained by the model BK-MIN is higher than the anisotropy of the model VFIMIN. This feature is the result of a different description of the loss cone. In the model based on Badhwar and Konradi (1990), the trapped proton flux drops to zero in the loss cone directions while, in the model of Watts et al. (1989), the flux decreases more slowly.

958

M. K R U G L A N S K I

The East-West asymmetry obtained in both models is similar since the two models are based on the Lenchek and Singer (1962) function. The West/East ratios of the incident trapped-proton integral fluence (E > 100 MeV) are about 4.64 and 5.15 for the left and right panels, respectively. The slight difference is caused by the different value of the atmospheric scale heights ( H r n u m ~ 108 km and HBK-um = 100 km). In Fig. 3, we present the trapped proton integral flux (E > 100 MeV) observed by the same satellite of Fig. 2 when its attitude is set to the second predefined attitude, i.e. the satellite is stabilized along the velocity vector. The polar angles 0° and 180° now correspond to protons coming from the leading and the trailing sides of the satellite, respectively. When the polar angle is equal to 90 °, the azimuthal angles of 0 ° and 180° correspond to the upward and downward directions, respectively. The omnidirectional integral flux at E > 100 MeV remains the same as in Fig. 2 but differs in the angular distribution of the flux. The mean directions of the upward and downward loss cone appear now as two holes. As in Fig. 2, the unidirectional flux anisotropy obtained by the model BK-MIN is higher than the anisotropy of the model VFIMIN. As a result of the attitude characteristics, the trapped-proton fluences seen on the trailing side are, in both panels, substantially higher than on the leading side. This feature is caused by the East-West asymmetry. Figure 3 shows the feedback of the radiation environment modelling on the design of a spacecraft, which has an axis stabilized along its orbit trajectory. Sensitive tools should preferably be located in the leading part of the spacecraft; the trailing part of the spacecraft will play the role of shielding.

4. CONCLUSION We have developed two models, BK-MIN and BK-MAX, for the trapped proton anisotropy at low altitude. The new models as well as the models VF1MIN and VF1MAX are available in the program ANISO. This program can predict unidirectional trapped proton fluences for arbitrary orbits and attitudes. The unidirectional fluences may be used as an input to one-dimensional shielding model to deduce an anisotropic dose repartition inside a spacecraft. We showed that the models VF1MIN and BK-MIN produce different predictions. The discrepancies between these models must be solved to

obtain better description of the radiation suffered by a spacecraft. New theoretical developments as well as a comparison with the LDEF data may be helpful for this purpose. Acknowledgements The author thanks E.J. Daly and H. Evans for useful discussions. This work was supported by ESA/ESTEC/WMA TRP contract No. 10725/94/JG(SC).

REFERENCES Armstrong T. W., Colbom B. L. and Watts J. W. (1990) Characteristics of Trapped Proton Anisotropy at Space Station Freedom Altitudes. Science Applications International Corporation Report SAIC-90/ 1474. Badhwar G. D. and Konradi A. (1990) Conversion of Omnidirectional Proton Fluxes Into a Pitch Angle Distribution. d. Spacecraft and Roc. 27, 350. Colborn B. L., Watts J. W. and Armstrong T. W. (1990) Data Base Description and Retrieval Program for the Trapped Proton Vector Flux Data Bases VF1MAX and VFIMIN. Science Applications International Corporation Report SAIC-90/1475. Heckman H. H. and Nakano G. H. (1969) Low-Altitude Trapped Protons during Solar Minimum Period. J. Geophys. Res. 74, 3575-3590. Heynderickx D., Kruglanski M., Lcmaire J. and Daly E. J. (1996) The Trapped Radiation Software Package UNIRAD. Proc. of Workshop on Radiation Belts: Models and Standards, Brussels, October 17-20, AGU Monograph in press. Heynderickx D. and Lemaire J. (1993) Improvements to Trapped Radiation Software. Technical Note 1 of the TREND-2 Study, ESA/ESTEC Contract No. 9828/92/ NL/FM. Kruglanski M. (1996) Trapped Proton Anisotropy at Low Altitudes. Technical Note 6 of the TREND-3 Study, ESA/ESTEC Contract No. 10725/94/JG(SC). Jensen D. C. and Cain J. C. (1962) An Interim Geomagnetic Field. J. Geophys. Res. 67, 3568. Johnson D. L. and Smith R. E. (1985) The MSFC/J70 Orbital Atmosphere Model and the Data Bases for the MSFC Solar Activity Prediction Technique. NASA TM-86522. Lenchek A. M. and Singer S. F. (1962) Effects of Finite Gyroradii of Gcomagnetically Trapped Protons. J. Geophys. Res. 67, 4073--4075. McIlwain C. E. (1961) Coordinates for Mapping the Distribution of Magnetically Trapped Particles. J. Geophys. Res. 66, 3681-3691. Sawyer D. M. and Vette, J. I. (1976) AP-8 Trapped Proton Environment for Solar Maximum and Solar Minimum. NSSDCAVDC-A-R & $76-06. Watts J. W., Parnell T. A. and Heckman H. H. (1989) Approximate Angular Distribution and Spectra for Geomagnetically Trapped Protons in Low-Earth Orbit, Conf. on High-Energy Radiation in Background Space, (Eds A. C. Rester Jr. and J. I. Trombka), Santib¢l Island, FL 1987, Am. Inst. Phys. Conf. Proc, New York, pp. 75-85.