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Spacecraft shielding for a Mars mission
Advances in Space Research 36 (2005) 1731–1736 www.elsevier.com/locate/asr
Spacecraft shielding for a Mars mission Keran OÕBrien Department of Physics and Astronomy, Northern Arizona University, P.O. Box 6010, Flagstaff AZ 86011-6010, USA Received 25 September 2004; received in revised form 7 February 2005; accepted 8 February 2005
Abstract The stationary, source-free, one-dimensional solution to the cosmic-ray heliospheric transport equation has been applied to the treatment of cosmic-ray fluxes at solar minimum between the orbits of Earth and Mars. The effective dose rate from these radiations were calculated for cylindrical spacecraft hulls of various compositions both in space and on the surface of Mars. The spatial gradient of the flux between Earth and Mars is quite small so that the cosmic-ray dose-rate can be assumed spatially uniform in the interval between these orbits, between one and one and a half astronomical units. The resulting dose rates for a 2.5-year mission, assuming six months on the surface of Mars, were slightly over one sievert. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Radiation protection in space; Cosmic rays; Radiation transport in the heliosphere
1. Introduction The SunÕs corona is extremely hot, with a temperature of about a million K. As a result it expands supersonically to form the solar wind. The average physical properties of the solar wind can be understood in terms of spherically symmetric flow filling a roughly spherical volume called the heliosphere (Jokipii, 1991). The velocity of the solar wind is typically about 400 km/s. The heliosphere has two distinct boundaries, namely the solar wind termination shock at about 100 AU and the heliosheath beyond that. The current non-shock model cannot distinguish between these boundaries, but that is not important in the current context (Caballero-Lopez and Moraal, 2004) (CM). The solar wind is a highly-, if not totally-ionized, plasma which interacts with the galactic cosmic radiation incident on the solar system. The solar wind is not constant, and interacts with the incoming charged cosmic radiation in three ways. There are short-term statistical variations in the incident cosmic radiation called ‘‘quick decreases’’, longer-term
E-mail address: keran.o’[email protected].
variations, due to plasma clouds emitted from the Sun during coronal mass ejections, that shield the Earth from the lower energy cosmic rays, called Forbush decreases, and the eleven-year solar cycle, one half of the 22-year Hale cycle, due to the periodic reversal of the SunÕs magnetic field.
2. Cosmic-ray transport through the heliosphere The equation for the distribution of cosmic rays in the heliosphere is (CM): 1 o 2 _ oD=ot þ r S þ ðP hP iDÞ ¼ Q; ð1aÞ P 2 oP where rÞD S ¼ 4pP 2 ðC V D K
ð1bÞ
and D is the cosmic-ray phase-space density, P is the cosmic-ray rigidity, S is the differential current density, ¼ Kð r; P ; tÞ , the diffusion V the solar wind velocity, K tensor at some location r and time t in the heliosphere, hP_ i is the adiabatic rate of rigidity loss and C is the Compton–Getting coefficient, C = (1/3)o ln,D/o ln P,
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.02.014
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K. OÕBrien / Advances in Space Research 36 (2005) 1731–1736
or one third the spectral index in rigidity space. Q is a cosmic-ray source term in the heliosphere. Following CM, the stationary, source-free, onedimensional version of Eq.(1) is: oD 1 o 1 o 2 oD 2 oD rj rV P ¼0 ð2aÞ V 2 or r2 or or 3r or oP and /i ¼
Ai 2 P D: Zi
ð2bÞ
Here, r is the distance from the Sun, V is the solar wind, j = j(r)Pb is the radial diffusion coefficient, b = v/c, the ratio of the velocity of the cosmic-ray particle to the speed of light, /i is the cosmic-ray flux, Ai is the atomic weight and Zi is the atomic number, where i runs over all nuclei from hydrogen to nickel. Another stationary, source-free, one-dimensional result of interest is the heliocentric potential equation. As the validity of the similar diffusion–convection equation is limited to the outer heliosphere (Caballero-Lopez and Moraal, 2004), it shall not be discussed further. The heliocentric potential equation is 2 P ðT Þ /i ðr ¼ 1; EÞ ¼ /i ðrHP ; T Þ P ðEÞ ð3Þ Zi E ¼ T þ U: Ai Here, r is expressed in AU, the Earth–Sun distance (1 AU = 1.49597870660 · 1013 cm) (Groom and Dobbs, 2004) rHP is the distance from the Sun to the heliopause, which is taken to be 100 AU and /i (rHP, T) is the local interstellar spectrum. T is the particle energy in MeV of the local interstellar spectrum, and E is the energy of the modulated spectrum in MeV at 1 AU. The heliocentric potential equation is related to Eq. (2) by Z 1 rHP V dr: ð4Þ U¼ 3 1 jðrÞ The heliocentric potential has been widely used to treat solar modulation, and has been shown by CM to be valid in the inner heliosphere (r < 20 AU).
3. The 11-year solar cycle The 11-year solar cycle can be roughly approximated by a sine curve. In Fig. 1, the last four solar cycles, in terms of the heliocentric potential, have been approximated by 1 2pðt þ kÞ U¼ ðU max U min Þ sin þ ðU max þ U min Þ ; 2 K ð5Þ
Fig. 1. Heliocentric Potentials from 1958 to date. The Sine fit is from Eq. (5). The Integral fit is from Eq. (8). The tabulated data were compiled from five neutron monitor stations: Deep River, from 1958 to 1995; Thule Station, from 1995 to 2000; Goose Bay, from 1958 to 1961; Apatity, from 1995 to 2004; Oulu, from 1995 to 2004.
where Umax corresponds to the heliocentric potential at solar maximum, Umin corresponds to solar minimum, K is the period and k is the phase. Choosing Umax = 1300 MV, Umin = 466 MV and K = 11 years yields a fit to the data since 1958. While none of the last four cycles are fit especially well by this formula, no attempt is being made to fit any particular cycle. A reasonably good fit to any cycle could be made by judicious choices of solar maximum and minimum potentials, and the period. The tabular data shown in Fig. 1 were not used to represent the solar cycle. The monthly means that make it up have too coarse a time-structure and are quite irregular. The Runge–Kutta code used to transport cosmic rays through the heliosphere requires both the data and the differential of the data which are even more irregular (cf. OÕBrien, 2004, for instance). Serious numerical instabilities would result. Let s¼
AU rHP V
ð6Þ
the time it takes for a disturbance originating at the Sun to arrive at the heliopause, slightly over a year. Substituting in Eq. (5), yields 1 2 U¼ ðU max U min Þ sin ððrHP rÞs þ kÞ 2 K þðU max þ U min Þ : ð7Þ Substituting Eq. (7) in (4), and inserting numerical values for the period (11 years), the astronomical unit, the heliopause (100 AU), the solar wind, we get for U at 1 AU.
K. OÕBrien / Advances in Space Research 36 (2005) 1731–1736
U ¼ ð0:42706 sinð0:57120kÞ þ 0:45404 0:13269 cosð0:57120kÞÞU min 0:13269 cosð0:57120kÞ þ ð0:45404 þ 0:13269 cosð0:57120kÞ þ 0:42706 sinð0:57120kÞÞU max ð8Þ If both Umax and Umin are multiplied by 1.1, Eq. (8) yields results identical to Eq. (5) as a function of k as shown in Fig. 1. The reason for the renormalization of Umax and Umin is that as solar modulation approaches its maximum, the modulation disturbance proceeding out from the Sun is less than the modulation intensity at 1 AU. The converse applies at solar minimum. The incoming cosmic rays ‘‘look back’’ at modulation patterns that arose up to a year previously.
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4. Cosmic-ray fluxes and dose-rates between Earth and Mars Applying these modifications to a Runge–Kutta code made available by Caballero-Lopez and Moraal (2004), cosmic-ray proton- and complex nuclei-fluxes were calculated in the interplanetary space between Earth and Mars, that is, between 1 and 1.52 AU from the Sun for a recent solar minimum. The results are shown in Fig. 2(a) and (b). It is not clearly evident from those figures, but the gradient of the proton flux oup ðr; EÞ rup ðr; EÞ ¼ ; ð9Þ or
Fig. 2. (a) The proton flux between Earth and Mars as a function of energy and distance from the Sun at a recent solar minimum. (b) The flux of complex nuclei (nuclei heavier than hydrogen) between Earth and Mars as a function of energy and distance from the Sun at a recent solar minimum.
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K. OÕBrien / Advances in Space Research 36 (2005) 1731–1736 Table 1 Mass groups used in PLOTINUS
Fig. 3. The gradient of the proton flux between Earth and Mars as a function of energy and distance from the Sun.
where up(r,E) is the proton flux per (cm2 s sr MeV), r is the distance from the Sun in AU and E is the proton energy in MeV (calculated by taking first-differences of solutions to the Runge–Kutta code at intervals of 0.01 AU between 1 and 1.52 AU and dividing by Dr) and shown in Fig. 3 is quite small, on the order of 106 (or 104%) per AU. Since the rigidity of the complex nuclei is approximately twice the rigidity of the protons of the same energy, the gradient of the nuclear flux will be even less. Thus the dose rate resulting from exposure to these radiations can be taken to be constant in the interplanetary space between Earth and Mars. The cosmic-ray transport code PLOTINUS (OÕBrien, in press) is based on an analytical theory of the transport of high-energy radiation through the earthÕs atmosphere. The transport of primary and secondary particles is described by a solution of the Boltzmann transport equation. The transport of secondary particles is based on a solution of the Boltzmann equation separable into longitudinal and transverse components, applicable to high-energy hadron–nucleus collisions. The longitudinal component is based on work by Passow (1962) and reported by Alsmiller (1965), and the transverse component on the work of Elliott (1995) and Williams (1967). All secondary particles other than hadrons are mediated by meson production and decay. Cosmic rays consist of atomic nuclei ranging from hydrogen to nickel. PLOTINUS properly transports the primary nuclei. The breakup of primary nuclei as a result of collisions with atoms of air is treated by means of a generalized Rudstam (1966) formula. Thus the superposition approximation, which treats the incident nuclei as separate neutrons and protons, is not used in this code. The theory on which PLOTINUS is based has also been applied to the atmospheres of other solar system bodies (Molina-Cuberos et al., 2001).
Group
Z
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3–5 6–8 9–10 11–12 13–14 15–16 17–18 19–20 21–25 26–28
Calculations of nuclear transport with this code have been compared with the measurements of Webber and Ormes (1967). Webber and Ormes flew a Cerenkov-scintillator detector with an aperture of 50 sr cm2 on seven balloon flights between August 1964 and July 1965, at high latitudes. Webber and Ormes divided the nuclear groups into the traditional four, L (3 6 Z 6 5), M (6 6 Z 6 9), LH (10 6 Z 6 14) and H (20 6 Z 6 28). The source spectra, the local interstellar spectra used in PLOTINUS have been normalized to the data of Gaisser and Stanev (1998) and shown in Table 1. It can be seen that the L group corresponds to the group 3 used in PLOTINUS. It was assumed that the M group comprises all of group 4 and half of group 5, the LH group, half of group 5 and all of groups 6 and 7, and the H group corresponds to half of group 10 and all of groups 11 and 12. This approximation may lead to some error in the results, but it is difficult to say what would be the magnitude of the effect. In addition, there will be some uncertainty in the experimental partitioning of various nuclei among the four groups. The propagation of groups L, M, LH and H in a 70 g/ cm2 air barrier was then calculated and compared with experiment. Solar modulation was represented by the mean heliocentric potential over the period of the high-latitude flights, 550 MV. Zero geomagnetic cutoffs were used (zenith and azimuth); the data from the high-latitude data were intended to be directly comparable with data obtained by satellites. The results are shown in Fig. 4. The comparison in this case is absolute. The calculated results were not normalized to the measurements. There are some differences in the magnitude of the calculated and measured sources at the top of the atmosphere. Discrepancies between the experimental and measured data appear to be due entirely to these differences in the mass spectra, and not to errors in the transport calculations, as the calculated rates of attenuation closely match the experimental rate of attenuation.
K. OÕBrien / Advances in Space Research 36 (2005) 1731–1736
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using the Gaisser–Stanev metals flux distributions exceed the Webber–Ormes fluxes by only 6% at the top of the atmosphere.
5. Dose rates between Earth and Mars
The differences between the calculated and measured fluxes are less that 20% except for the L group. The measured L-group fluxes average about 40% higher than the calculated fluxes. That this is entirely due to the difference in the Gaisser and Stanev (1998) nuclear distribution and its associated adaption to the traditional group structure and the Webber and Ormes (1967) measurement can be seen in Fig. 5, where the calculated data have been normalized to the measured data at a depth of 2 g/cm2. At that small depth, transport effects are negligible. and the differences in the nuclear distributions are apparent. It is also evident that the normalized calculations reproduce the experimental data quite well, except for a small upward convexity in the L-group data at small depths. Should the Webber–Ormes data be more accurate than the Gaisser–Stanev data, the effect on the calculated dose rates would be quite small as the calculations
PLOTINUS was then adapted to cosmic-ray transport through aluminum and polystyrene, by changing appropriate cross-sections, densities, stopping powers and the like, and to cosmic-ray transport through the Martian atmosphere. The Martian atmosphere was assumed to be 90% CO2 and 10% noble gases. All calculations were carried out for a solar minimum of 470 MV. Sato et al.Õs (2003) transformation theory was applied to the effective proton dose calculations for isotropic incidence of Ferrari et al. (1997), using the definitions of effective dose in ICRP 92 (2003). The resulting transformation matrix was multiplied by the fluxes in each of the groups of Table 1 to give the effective dose rate for each of the groups. Effective dose rates for a recent solar minimum in cylindrical spacecraft hulls of large radii, one of aluminum and one of polyethylene are shown in Fig. 6 and compared with the dose equivalent to the blood-forming organs (BFO) calculated by Wilson et al. (1997). Though the radiation parameters are not identical, the largest disagreement is somewhat less than 20% in the region of 30 g/cm2 of polyethylene. The effective dose rate was then calculated for two compound spacecraft hulls of different composition, one of 2 g/cm2 of Al lined with 4 g/cm2 of polyethylene, and the other also of 2 g/cm2 of Al, but lined with 20 g/ cm2 of polyethylene, both in the Earth–Mars interplanetary space and on the surface of Mars. The Martian atmosphere was assumed to be 18 g/cm2 deep (MolinaCuberos et al., 2001). The major difference between
Fig. 5. Calculated and measured penetration of the cosmic-ray metals into the EarthÕs atmosphere. The measured data are by Webber and Ormes (1967). The calculated data are from this study. The calculated data have been normalized to the measured data at a depth of 2 g/cm2.
Fig. 6. The calculated effective dose rate from this study and the calculated equivalent dose rate to the blood-forming organs from Wilson et al. (1997). Solar activity level was a solar minimum of 470 MV. The solid lines are from this study (left axis). The points were calculated by Wilson et al. (1997) (right axis).
Fig. 4. Calculated and measured penetration of the cosmic-ray metals into the EarthÕs atmosphere. The measured data are by Webber and Ormes (1967). The calculated data are from this study.
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Table 2 Effective radiation exposure on a mission to Mars at a solar activity of 470 MV at 1 AU Spacecraft hull
Earth/Mars space (rem/year)
Martian surface (rem/year)
2.5 year missiona (rem)
2 g/cm2 Al, 4 g/cm2 CH2 2 g/cm2 Al, 20 g/cm2 CH2
63.75 55.02
17.17 11.51
136.1 115.8
a One year outbound, six months on the surface of Mars, and one year inbound.
the calculations on the Martian surface and the freespace calculations is one of geometry. The surface of the planet blocks radiation from one direction, whereas in free space, radiation is incident isotropically. The Martian calculations are, in addition, 3-medium calculations as there is the additional medium of the Martian atmosphere out the hull. The results are shown in Table 2. The effective dose received by a crew on a two and a half year mission, where the crew would be on the Martian surface for six months, would be a little over one sievert.
References Alsmiller, F.S. A general category of soluble nucleon-meson cascade equations, ORNL-3746, 1965. Caballero-Lopez, R.A., Moraal, H. Limitations of the force field equation to describe cosmic-ray modulation. J. Geophys. Res. 109, A01101, 2004. Elliott, J.P. MilneÕs problem with a point-source. Proc. Roy. Soc. A 228, 424, 1995 (London). Ferrari, A., Pelliccioni, M., Pillon, M. Fluence to effective dose conversion coefficients for protons from 5 Mev to 10 Tev. Radiat. Prot. Dosim. 71 (2), 85–91, 1997.
Gaisser, T.K., Stanev, T. Cosmic rays. Europ. Phys. J. 3, 132–137, 1998. Groom, D.E., Dobbs, M.A. Astrophysical constants and parameters. Phys. Lett. B 592, 92–93, 2004. International Commission on Radiation Protection. Publication 92, Relative biological effectiveness (RBE), Quality Factor (Q), and radiation weighting factor (wR), Annals of the ICRP 33, Pergamon Press, Oxford, 2003. Jokipii, J.R. Solar wind, in: Lerner, R.G., Trigg, G.L. (Eds.), Encyclopedia of Physics, 2nd ed. VCH Publishers, Inc., New York, Weinheim, Cambridge, Basel, 1991. Molina-Cuberos, G.J., Stumpner, W., Lammer, H., Ko¨mler, N.I., OÕBrien, K. Cosmic ray and UV radiation models on the ancient Martian surface. Icarus 154, 216–222, 2001. OÕBrien, K. The theory of cosmic-ray and high-energy solar-particle transport in the atmosphere, in: Proceedings of the Seventh International Symposium on the Natural Radiation Environment, Rhodes, Greece, in press. OÕBrien, K. The relative transit time of high-rigidity and low-rigidity cosmic rays through the solar system. Radiat. Prot. Dosimetry 109, 357–363, 2004. Passow, C. Phenomenologische Theorie zur Berechnung einer Kaskade aus schweren Teilchen (Nukleonenkaskade) in der Materie Phenomenological theory for the calculation of a cascade of heavy particles (nucleonic cascade) in matter. Notiz A 285, Deut. Elektron. Synchrotron, Hamburg, 1962. Rudstam, G. Zeitschrift fu¨r Naturforschung. Systematics of Spallation Yields 21a, 1027–1041, 1966. Sato, T., Tsuda, S., Sakamoto, Y., Yamaguchi, Y., Niita, K. Conversion coefficients from fluence to effective dose for heavy ions up to 3 GeV/A. Radiat. Prot. Dosim. 106, 137–144, 2003. Webber, W.R., Ormes, J.F. Cerenkov-scintillation counter measurements of nuclei heavier than helium in the primary cosmic radiation. J. Geophys. Res. 72, 5957–5976, 1967. Williams, M.M.R. Approximate solutions of the neutron transport equation in two and three dimensional systems. Nucleonik 9, 305– 312, 1967. Wilson, J.W., Cucinotta, A., Tai, H., Simonsen, L.C., Shinn, J. L., Thibault, A. Galactic and solar cosmic ray shielding in deep space, National Aeronautics and Space Administration, Langley Research Center, NASA Technical Paper 3682, 1997.
Spacecraft shielding for a Mars mission Keran OÕBrien Department of Physics and Astronomy, Northern Arizona University, P.O. Box 6010, Flagstaff AZ 86011-6010, USA Received 25 September 2004; received in revised form 7 February 2005; accepted 8 February 2005
Abstract The stationary, source-free, one-dimensional solution to the cosmic-ray heliospheric transport equation has been applied to the treatment of cosmic-ray fluxes at solar minimum between the orbits of Earth and Mars. The effective dose rate from these radiations were calculated for cylindrical spacecraft hulls of various compositions both in space and on the surface of Mars. The spatial gradient of the flux between Earth and Mars is quite small so that the cosmic-ray dose-rate can be assumed spatially uniform in the interval between these orbits, between one and one and a half astronomical units. The resulting dose rates for a 2.5-year mission, assuming six months on the surface of Mars, were slightly over one sievert. 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Radiation protection in space; Cosmic rays; Radiation transport in the heliosphere
1. Introduction The SunÕs corona is extremely hot, with a temperature of about a million K. As a result it expands supersonically to form the solar wind. The average physical properties of the solar wind can be understood in terms of spherically symmetric flow filling a roughly spherical volume called the heliosphere (Jokipii, 1991). The velocity of the solar wind is typically about 400 km/s. The heliosphere has two distinct boundaries, namely the solar wind termination shock at about 100 AU and the heliosheath beyond that. The current non-shock model cannot distinguish between these boundaries, but that is not important in the current context (Caballero-Lopez and Moraal, 2004) (CM). The solar wind is a highly-, if not totally-ionized, plasma which interacts with the galactic cosmic radiation incident on the solar system. The solar wind is not constant, and interacts with the incoming charged cosmic radiation in three ways. There are short-term statistical variations in the incident cosmic radiation called ‘‘quick decreases’’, longer-term
E-mail address: keran.o’[email protected].
variations, due to plasma clouds emitted from the Sun during coronal mass ejections, that shield the Earth from the lower energy cosmic rays, called Forbush decreases, and the eleven-year solar cycle, one half of the 22-year Hale cycle, due to the periodic reversal of the SunÕs magnetic field.
2. Cosmic-ray transport through the heliosphere The equation for the distribution of cosmic rays in the heliosphere is (CM): 1 o 2 _ oD=ot þ r S þ ðP hP iDÞ ¼ Q; ð1aÞ P 2 oP where rÞD S ¼ 4pP 2 ðC V D K
ð1bÞ
and D is the cosmic-ray phase-space density, P is the cosmic-ray rigidity, S is the differential current density, ¼ Kð r; P ; tÞ , the diffusion V the solar wind velocity, K tensor at some location r and time t in the heliosphere, hP_ i is the adiabatic rate of rigidity loss and C is the Compton–Getting coefficient, C = (1/3)o ln,D/o ln P,
0273-1177/$30 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.02.014
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K. OÕBrien / Advances in Space Research 36 (2005) 1731–1736
or one third the spectral index in rigidity space. Q is a cosmic-ray source term in the heliosphere. Following CM, the stationary, source-free, onedimensional version of Eq.(1) is: oD 1 o 1 o 2 oD 2 oD rj rV P ¼0 ð2aÞ V 2 or r2 or or 3r or oP and /i ¼
Ai 2 P D: Zi
ð2bÞ
Here, r is the distance from the Sun, V is the solar wind, j = j(r)Pb is the radial diffusion coefficient, b = v/c, the ratio of the velocity of the cosmic-ray particle to the speed of light, /i is the cosmic-ray flux, Ai is the atomic weight and Zi is the atomic number, where i runs over all nuclei from hydrogen to nickel. Another stationary, source-free, one-dimensional result of interest is the heliocentric potential equation. As the validity of the similar diffusion–convection equation is limited to the outer heliosphere (Caballero-Lopez and Moraal, 2004), it shall not be discussed further. The heliocentric potential equation is 2 P ðT Þ /i ðr ¼ 1; EÞ ¼ /i ðrHP ; T Þ P ðEÞ ð3Þ Zi E ¼ T þ U: Ai Here, r is expressed in AU, the Earth–Sun distance (1 AU = 1.49597870660 · 1013 cm) (Groom and Dobbs, 2004) rHP is the distance from the Sun to the heliopause, which is taken to be 100 AU and /i (rHP, T) is the local interstellar spectrum. T is the particle energy in MeV of the local interstellar spectrum, and E is the energy of the modulated spectrum in MeV at 1 AU. The heliocentric potential equation is related to Eq. (2) by Z 1 rHP V dr: ð4Þ U¼ 3 1 jðrÞ The heliocentric potential has been widely used to treat solar modulation, and has been shown by CM to be valid in the inner heliosphere (r < 20 AU).
3. The 11-year solar cycle The 11-year solar cycle can be roughly approximated by a sine curve. In Fig. 1, the last four solar cycles, in terms of the heliocentric potential, have been approximated by 1 2pðt þ kÞ U¼ ðU max U min Þ sin þ ðU max þ U min Þ ; 2 K ð5Þ
Fig. 1. Heliocentric Potentials from 1958 to date. The Sine fit is from Eq. (5). The Integral fit is from Eq. (8). The tabulated data were compiled from five neutron monitor stations: Deep River, from 1958 to 1995; Thule Station, from 1995 to 2000; Goose Bay, from 1958 to 1961; Apatity, from 1995 to 2004; Oulu, from 1995 to 2004.
where Umax corresponds to the heliocentric potential at solar maximum, Umin corresponds to solar minimum, K is the period and k is the phase. Choosing Umax = 1300 MV, Umin = 466 MV and K = 11 years yields a fit to the data since 1958. While none of the last four cycles are fit especially well by this formula, no attempt is being made to fit any particular cycle. A reasonably good fit to any cycle could be made by judicious choices of solar maximum and minimum potentials, and the period. The tabular data shown in Fig. 1 were not used to represent the solar cycle. The monthly means that make it up have too coarse a time-structure and are quite irregular. The Runge–Kutta code used to transport cosmic rays through the heliosphere requires both the data and the differential of the data which are even more irregular (cf. OÕBrien, 2004, for instance). Serious numerical instabilities would result. Let s¼
AU rHP V
ð6Þ
the time it takes for a disturbance originating at the Sun to arrive at the heliopause, slightly over a year. Substituting in Eq. (5), yields 1 2 U¼ ðU max U min Þ sin ððrHP rÞs þ kÞ 2 K þðU max þ U min Þ : ð7Þ Substituting Eq. (7) in (4), and inserting numerical values for the period (11 years), the astronomical unit, the heliopause (100 AU), the solar wind, we get for U at 1 AU.
K. OÕBrien / Advances in Space Research 36 (2005) 1731–1736
U ¼ ð0:42706 sinð0:57120kÞ þ 0:45404 0:13269 cosð0:57120kÞÞU min 0:13269 cosð0:57120kÞ þ ð0:45404 þ 0:13269 cosð0:57120kÞ þ 0:42706 sinð0:57120kÞÞU max ð8Þ If both Umax and Umin are multiplied by 1.1, Eq. (8) yields results identical to Eq. (5) as a function of k as shown in Fig. 1. The reason for the renormalization of Umax and Umin is that as solar modulation approaches its maximum, the modulation disturbance proceeding out from the Sun is less than the modulation intensity at 1 AU. The converse applies at solar minimum. The incoming cosmic rays ‘‘look back’’ at modulation patterns that arose up to a year previously.
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4. Cosmic-ray fluxes and dose-rates between Earth and Mars Applying these modifications to a Runge–Kutta code made available by Caballero-Lopez and Moraal (2004), cosmic-ray proton- and complex nuclei-fluxes were calculated in the interplanetary space between Earth and Mars, that is, between 1 and 1.52 AU from the Sun for a recent solar minimum. The results are shown in Fig. 2(a) and (b). It is not clearly evident from those figures, but the gradient of the proton flux oup ðr; EÞ rup ðr; EÞ ¼ ; ð9Þ or
Fig. 2. (a) The proton flux between Earth and Mars as a function of energy and distance from the Sun at a recent solar minimum. (b) The flux of complex nuclei (nuclei heavier than hydrogen) between Earth and Mars as a function of energy and distance from the Sun at a recent solar minimum.
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K. OÕBrien / Advances in Space Research 36 (2005) 1731–1736 Table 1 Mass groups used in PLOTINUS
Fig. 3. The gradient of the proton flux between Earth and Mars as a function of energy and distance from the Sun.
where up(r,E) is the proton flux per (cm2 s sr MeV), r is the distance from the Sun in AU and E is the proton energy in MeV (calculated by taking first-differences of solutions to the Runge–Kutta code at intervals of 0.01 AU between 1 and 1.52 AU and dividing by Dr) and shown in Fig. 3 is quite small, on the order of 106 (or 104%) per AU. Since the rigidity of the complex nuclei is approximately twice the rigidity of the protons of the same energy, the gradient of the nuclear flux will be even less. Thus the dose rate resulting from exposure to these radiations can be taken to be constant in the interplanetary space between Earth and Mars. The cosmic-ray transport code PLOTINUS (OÕBrien, in press) is based on an analytical theory of the transport of high-energy radiation through the earthÕs atmosphere. The transport of primary and secondary particles is described by a solution of the Boltzmann transport equation. The transport of secondary particles is based on a solution of the Boltzmann equation separable into longitudinal and transverse components, applicable to high-energy hadron–nucleus collisions. The longitudinal component is based on work by Passow (1962) and reported by Alsmiller (1965), and the transverse component on the work of Elliott (1995) and Williams (1967). All secondary particles other than hadrons are mediated by meson production and decay. Cosmic rays consist of atomic nuclei ranging from hydrogen to nickel. PLOTINUS properly transports the primary nuclei. The breakup of primary nuclei as a result of collisions with atoms of air is treated by means of a generalized Rudstam (1966) formula. Thus the superposition approximation, which treats the incident nuclei as separate neutrons and protons, is not used in this code. The theory on which PLOTINUS is based has also been applied to the atmospheres of other solar system bodies (Molina-Cuberos et al., 2001).
Group
Z
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3–5 6–8 9–10 11–12 13–14 15–16 17–18 19–20 21–25 26–28
Calculations of nuclear transport with this code have been compared with the measurements of Webber and Ormes (1967). Webber and Ormes flew a Cerenkov-scintillator detector with an aperture of 50 sr cm2 on seven balloon flights between August 1964 and July 1965, at high latitudes. Webber and Ormes divided the nuclear groups into the traditional four, L (3 6 Z 6 5), M (6 6 Z 6 9), LH (10 6 Z 6 14) and H (20 6 Z 6 28). The source spectra, the local interstellar spectra used in PLOTINUS have been normalized to the data of Gaisser and Stanev (1998) and shown in Table 1. It can be seen that the L group corresponds to the group 3 used in PLOTINUS. It was assumed that the M group comprises all of group 4 and half of group 5, the LH group, half of group 5 and all of groups 6 and 7, and the H group corresponds to half of group 10 and all of groups 11 and 12. This approximation may lead to some error in the results, but it is difficult to say what would be the magnitude of the effect. In addition, there will be some uncertainty in the experimental partitioning of various nuclei among the four groups. The propagation of groups L, M, LH and H in a 70 g/ cm2 air barrier was then calculated and compared with experiment. Solar modulation was represented by the mean heliocentric potential over the period of the high-latitude flights, 550 MV. Zero geomagnetic cutoffs were used (zenith and azimuth); the data from the high-latitude data were intended to be directly comparable with data obtained by satellites. The results are shown in Fig. 4. The comparison in this case is absolute. The calculated results were not normalized to the measurements. There are some differences in the magnitude of the calculated and measured sources at the top of the atmosphere. Discrepancies between the experimental and measured data appear to be due entirely to these differences in the mass spectra, and not to errors in the transport calculations, as the calculated rates of attenuation closely match the experimental rate of attenuation.
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using the Gaisser–Stanev metals flux distributions exceed the Webber–Ormes fluxes by only 6% at the top of the atmosphere.
5. Dose rates between Earth and Mars
The differences between the calculated and measured fluxes are less that 20% except for the L group. The measured L-group fluxes average about 40% higher than the calculated fluxes. That this is entirely due to the difference in the Gaisser and Stanev (1998) nuclear distribution and its associated adaption to the traditional group structure and the Webber and Ormes (1967) measurement can be seen in Fig. 5, where the calculated data have been normalized to the measured data at a depth of 2 g/cm2. At that small depth, transport effects are negligible. and the differences in the nuclear distributions are apparent. It is also evident that the normalized calculations reproduce the experimental data quite well, except for a small upward convexity in the L-group data at small depths. Should the Webber–Ormes data be more accurate than the Gaisser–Stanev data, the effect on the calculated dose rates would be quite small as the calculations
PLOTINUS was then adapted to cosmic-ray transport through aluminum and polystyrene, by changing appropriate cross-sections, densities, stopping powers and the like, and to cosmic-ray transport through the Martian atmosphere. The Martian atmosphere was assumed to be 90% CO2 and 10% noble gases. All calculations were carried out for a solar minimum of 470 MV. Sato et al.Õs (2003) transformation theory was applied to the effective proton dose calculations for isotropic incidence of Ferrari et al. (1997), using the definitions of effective dose in ICRP 92 (2003). The resulting transformation matrix was multiplied by the fluxes in each of the groups of Table 1 to give the effective dose rate for each of the groups. Effective dose rates for a recent solar minimum in cylindrical spacecraft hulls of large radii, one of aluminum and one of polyethylene are shown in Fig. 6 and compared with the dose equivalent to the blood-forming organs (BFO) calculated by Wilson et al. (1997). Though the radiation parameters are not identical, the largest disagreement is somewhat less than 20% in the region of 30 g/cm2 of polyethylene. The effective dose rate was then calculated for two compound spacecraft hulls of different composition, one of 2 g/cm2 of Al lined with 4 g/cm2 of polyethylene, and the other also of 2 g/cm2 of Al, but lined with 20 g/ cm2 of polyethylene, both in the Earth–Mars interplanetary space and on the surface of Mars. The Martian atmosphere was assumed to be 18 g/cm2 deep (MolinaCuberos et al., 2001). The major difference between
Fig. 5. Calculated and measured penetration of the cosmic-ray metals into the EarthÕs atmosphere. The measured data are by Webber and Ormes (1967). The calculated data are from this study. The calculated data have been normalized to the measured data at a depth of 2 g/cm2.
Fig. 6. The calculated effective dose rate from this study and the calculated equivalent dose rate to the blood-forming organs from Wilson et al. (1997). Solar activity level was a solar minimum of 470 MV. The solid lines are from this study (left axis). The points were calculated by Wilson et al. (1997) (right axis).
Fig. 4. Calculated and measured penetration of the cosmic-ray metals into the EarthÕs atmosphere. The measured data are by Webber and Ormes (1967). The calculated data are from this study.
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Table 2 Effective radiation exposure on a mission to Mars at a solar activity of 470 MV at 1 AU Spacecraft hull
Earth/Mars space (rem/year)
Martian surface (rem/year)
2.5 year missiona (rem)
2 g/cm2 Al, 4 g/cm2 CH2 2 g/cm2 Al, 20 g/cm2 CH2
63.75 55.02
17.17 11.51
136.1 115.8
a One year outbound, six months on the surface of Mars, and one year inbound.
the calculations on the Martian surface and the freespace calculations is one of geometry. The surface of the planet blocks radiation from one direction, whereas in free space, radiation is incident isotropically. The Martian calculations are, in addition, 3-medium calculations as there is the additional medium of the Martian atmosphere out the hull. The results are shown in Table 2. The effective dose received by a crew on a two and a half year mission, where the crew would be on the Martian surface for six months, would be a little over one sievert.
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