all been measured to an accuracy of a few percent between 600 MeV nucleon and 20 GeV nucleon~1 on the HEAO3

Transcription

1 THE ASTROPHYSICAL JOURNAL, 506:335È340, 1998 October 10 ( The American Astronomical Society. All rights reserved. Printed in U.S.A. A STUDY OF THE SURVIVING FRACTION OF THE COSMIC-RAY RADIOACTIVE DECAY ISOTOPES 10Be, 26Al, 36Cl, and 54Mn AS A FUNCTION OF ENERGY USING THE CHARGE RATIOS Be/B, Al/Mg, Cl/Ar, AND Mn/Fe MEASURED ON HEAO[3 W. R. WEBBER1 AND A. SOUTOUL Received 1997 November 6; accepted 1998 May 11 ABSTRACT We have used published HEAO3 data on the charge ratios Be/B, Al/Mg, Cl/Ar, and Mn/Fe to examine the decay of the radioactive cosmic-ray isotopes10be, 26Al, 36Cl, and 54Mn at energies from D600 MeV to 20 GeV nucleon~1. From these charge data we can obtain the surviving fractions of these isotopes at higher energy with a comparable accuracy to that obtained using directly measured lowenergy isotopic ratios. As a result of the relativistic time dilation of the lifetimes, these higher energy surviving fractions are a probe of propagation conditions outside the disk and into the halo of the Galaxy. The data are interpreted in terms of a simple leaky-box propagation model, then a more realistic di usion model, and Ðnally a still more realistic Monte Carlo di usion model for the propagation of cosmic rays. Interpreted in terms of a leaky-box model, the data for all four isotopes suggest to varying degrees that the cosmic-ray particles are propagating into regions of lower density away from the disk. The di usion model allows one to determine that the scale size of this extended propagation region in the Z-direction is 2È4 kpc. The data are consistent only with weak Galactic winds D20 km s~1 or less. Monte Carlo models require cosmic-ray lifetimes D20È30 ] 106yr and a halo size 2È3 kpc. The Monte Carlo calculations suggest that it is the scale size of the di usion coefficient in the Z-direction relative to the halo size that determines the surviving fractions of these radioactive isotopes at high energies rather than the details of the Z dependence of the matter density. Subject headings: acceleration of particles È cosmic rays 1. INTRODUCTION The four radioactive decay isotopes 10Be, 26Al, 36Cl, and 54Mn provide unique information on the propagation of cosmic rays in the galaxy. These secondary isotopes have half-lives di ering by a factor D5 from 3 ] 105 yr for 36Cl to 1.6 ] 106 yr for 10Be. If, as is commonly assumed, these particles di use through the magnetic Ðelds of the galaxy, then during a characteristic lifetime q for 10Be, for example, they will di use a distance l \ (Dq)1@2 which for a typical di usion coefficient D \ 2 ] 1028 cm2 s~1, gives l \ 1021 cm or D300 pc. For 36Cl this distance is D60 pc (see, for example, Ptuskin 1998). The measurement of the fraction of these isotopes that survive, f, compared with that expected if no decay occurs, can be used to derive an interstellar average matter density over a region of radius l corresponding to the radioactive lifetime. For example, the surviving fraction of 10Be at D100 MeV nucleon~1 is measured to be 0.20 ^ 0.04, which gives a matter density of 0.29 ^ 0.09 H atoms cm~3 using a leaky-box propagation model (e.g., Lukasiak et al. 1994a). This is less than the matter density D1.0 H atom cm~3 commonly assumed for the matter disk. Since this matter disk may be D100 pc thick, the cosmic-ray measurement suggests that these nuclei are di using beyond the disk into the lower density regions of the Galactic halo. To describe this motion requires more sophisticated di usion models. Measurements of the other radioactive nuclei at low energies give somewhat more imprecise values for f but seem to show a trend toward higher densities (as deduced in leakybox models) for shorter decay lifetimes, as would be expected if the matter density is decreasing with Z, the dis- 1 Astronomy Department, MSC 4500 New Mexico State University, Box 30001, Las Cruces, NM Service dïastrophysique, SAp CE-Saclay, Gif sur Yvette, CEDEX, France. tance from the center of the disk (Lukasiak et al. 1994b). For example, 26Al with a lifetime of 0.9 ] 106 yr and an average measured value of f \ 0.28 ^ 0.06 gives a matter density of 0.37 ^ 0.09 H atoms cm~3 (Lukasiak et al. 1994b; Simpson & Connell 1998). The present experimental situation at low energies for the di erent decay isotopes has been discussed within the context of a leaky-box Ñat halo model for di erent values of the di usion coefficient by Ptuskin (1998). He concludes that the di erences between a simple leaky-box and a halo model are not large and are difficult to observe using the presently available low-energy isotopic data alone. In fact, what is needed to examine this behavior in more detail is a range of radioactive lifetimes, particularly extending to longer q, corresponding to larger l. This can be accomplished by measuring the surviving fraction at higher energies from 1È10 GeV nucleon~1 where the relativistic time dilation increases the lifetime. Present-day experimental techniques cannot yet provide deðnitive resolution of the individual isotopes of at these higher energies, however; but there is an alternative that is, in fact, as good as any of the proposed isotopic experiments at higher energies and comparable in accuracy to the current measurements of these isotopes at lower energy. This is to use the ratios of the decaying charge to the decayed charge, e.g., Be/B, Al/Mg, Cl/Ar, and Mn/Fe as Ðrst discussed in the 1960s (e.g., Shapiro & Silberberg 1968). These charge ratios have now all been measured to an accuracy of a few percent between 600 MeV nucleon and 20 GeV nucleon~1 on the HEAO3 spacecraft (Engelmann et al. 1990). In what follows we shall give some illustrations to show that these charge ratio measurements are, in fact, comparable to the current low-energy isotopic measurements in terms of their ability to determine the surviving fraction, f. Consider 10Be. The total of three recent spacecraft mea- 335

2 336 WEBBER & SOUTOUL Vol. 506 surements, as summarized in Lukasiak et al. 1994) provide D50 well resolved 10Be nuclei. The statistical error alone on this measurement is thus D15%. At a few 100 MeV nucleon~1 the di erence between no decay and complete decay of 10Be changes the Be/B ratio from 0.34 to 0.26 or D25% as calculated using the leaky-box propagation model (see Fig. 1). The quoted errors on the individual HEAO measurements of the Be/B ratio are ^3%, which typically translate into an error of ^0.08 in f. This is a factor D2 times worse than the combined isotopic data at a single energy. However, the charge ratio is measured at 13 senergies between 0.6 and 20 GeV nucleon~1 and thus with time dilation, probes values of l up to 10 times larger than the isotopic measurement at low energies. For 26Al the average of the most recent Voyager and Ulysses isotopic data (Lukasiak et al. 1994b; Simpson & Connell 1998) gives a value of f \ 0.28 ^ 0.06, with an implied density of 0.37 ^ 0.09 H atoms cm~3 using the speciðc parameters and cross sections of the leaky-box propagation model described in this paper. The di erence between no decay and complete decay of 26Al changes the Al/Mg ratio from 0.20 to 0.15 at a few 100 MeV nucleon~1 or again about 25% (see Fig. 2). The quoted errors on the individual HEAO measurements of the Al/Mg ratio are about ^4%, which translate into an error of ^0.12 in f or about twice the error on the combined isotopic measurements at a single energy. For 36Cl the average of recent Voyager and Ulysses isotopic data is more uncertain, giving a value of f \ 0.16 ^ 0.06 (Webber 1998; Connell, Du Vernois, & Simpson 1997), with an implied density of 0.50 ^ 0.15 H atoms cm~3 using the speciðc parameters and cross sections of the leaky-box model described in this paper. The di erence between no decay and complete decay of 36Cl changes the Cl/Ar ratio from 0.81 to 0.44 or about 45% (see Fig. 3). The quoted errors on the individual HEAO measurements are ^7%, which translates into an error of ^0.14 in f. So overall the individual charge ratio measurements are competitive within a factor of 2 with the isotopic measurements for determining the value of f but have the advantage of up to 13 individual measurements at di erent energies compared with the isotopic measurements at essentially a FIG. 2.ÈSame as Figure 1 but for the Al/Mg ratio. Additional points labeled I and U at low energies come from the ISEE3 data (Leske and Weidenbeck 1993) and Ulysses data (DuVernois and Thayer 1996) respectively. single low energy for each decay isotope. At the same time, these charge ratio measurements probe the decay e ects at much longer lifetimes because of the relativistic time dilation and so probe Galactic propagation e ects at large distances from the disk. As a result, the data are more sensitive to the e ects of a galactic halo and possible convective e ects of a Galactic wind. In what follows we will interpret the HEAO data on the four ratios Be/B, Al/Mg, Cl/Ar, and Mn/Fe, Ðrst in terms of a standard leaky-box model (which is homogeneous and uniform) and then in terms of a di usion model which considers a Galactic halo and also the e ects of a Galactic wind. Finally, we will interpret the data in terms of a Monte Carlo di usion model which uses realistic astrophysical parameters. 2. DATA AND PREDICTIONS OF A STANDARD LEAKY-BOX MODEL Figures 1, 2, 3, and 4 show the currently available measurements of the Be/B, Al/Mg, Cl/Ar, and Mn/Fe ratios. This includes the HEAO3 data at energies above 600 MeV nucleon~1 and various other spacecraft data at lower energies, mostly in the range of 100È300 MeV nucleon~1. Asan illustration of how these charge fractions depend on the FIG. 1.ÈHEAO3 measurements of the Be/B ratio as a function of energy (Engelmann et al. 1990). Also shown is the Be/B ratio calculated for no decay and complete decay of 10Be and for a density of \0.30 H atoms cm~3 using a leaky-box propagation model. The Be/B ratio point at low energies labeled V ÏÏ comes from Lukasiak et al. (1994). FIG. 3.ÈSame as Fig. 2, but for the Cl/Ar ratio.

3 No. 1, 1998 COSMIC-RAY RADIOACTIVE DECAY ISOTOPES 337 FIG. 4.ÈSame as Fig. 2, but for the Mn/Fe ratio. fraction of the radioactive isotope that decays, we show the predictions for no decay and complete decay of the radioactive isotopes as well as for n \ 0.3, one of a family of curves for various average matter densities. As discussed below, these n \ 0.3 curves may not be appropriate for examining the decay of all of these isotopes and are shown here for illustration only. These speciðc calculations are described below and are for a modulation level \700 MV appropriate to the HEAO3 observations. This modulation level is used for all the calculations in Figures 1È8. For the Galactic propagation we Ðrst take the standard leaky-box model as described in Lukasiak et al. (1994a). This model uses an escape length j \ 31.6 br~0.6 [ 3.3 GV, esc j \ 12.8b at low rigidities, and a density n \ 0.3 cm~3. esc These and all other propagation and cross-section parameters are identical to those given in Lukasiak et al. (1994a). These parameters provide a very good Ðt to the B/C ratio at all energies and agree with the measured surviving fractions for 10Be and 26Al at D100 MeV nucleon~1. Since the isotopic data for 10Be at D100 MeV nucleon~1 imply an average density of about 0.30 H atoms cm~3 in the leaky-box model, we shall use this density as the base parameter for our calculations of the Be/B ratio at higher energies. In Figure 5 we show the predicted surviving fraction of 10Be as a function of energy for this density (calculated using a leaky box model), along with the HEAO data on the Be/B ratio as shown in Figure 1, now expressed in terms of the surviving fraction of 10Be as obtained from the leaky-box calculations. The HEAO data at all energies clearly lie below the prediction for n \ 0.3 cm~3 which Ðts the data at low energies. This would be expected if the particles were propagating into a region of lower density which was being probed further and further from the disk because of the increased 10Be lifetime at higher energies. For 26Al the the low-energy isotope data imply a density of D0.37 H atoms cm~3 in the leaky-box model. We therefore use this density as the base parameter for our calculations of the Al/Mg ratio as a function of energy. In Figure 6 we show the predicted surviving fraction of 26Al as a function of energy for this density as calculated using a leaky-box model. The HEAO data on the Al-Mg ratio as shown in Figure 2, expressed in terms of the surviving fraction of 26Al as obtained from the leaky-box calculations, are also shown in Figure 6. In this case there is a large scatter in the data points; however, all Ðve of the highest energy FIG. 5.ÈBe/B measurements in Fig. 1 converted into a surviving fraction for 10Be using a leaky-box propagation model. (Note that the lowenergy HEAO data for Be/B is not shown because of possible efficiency problems; see also Fig. 1.) Shown are the leaky-box calculations for matter densities of 0.3 and 0.1 H atoms cm~3. The dashed curve shows predictions of a di usion model with a halo thickness of 2 kpc. The solid line with letters X is the prediction for a Monte Carlo calculation with a cosmic-ray age of 25 ] 106 yr. The open circle at low energies is an average of lowenergy isotopic data on f as described in the text. points lie below this prediction for a constant density of 0.37 cm~3, presumably for the same reason as discussed above for the Be/B ratio. For 36Cl the low-energy isotope data implies a density of D0.50 H atoms cm~3 in the leaky-box model. We therefore begin by using this density as the base parameter for our calculations of the Cl/Ar ratio as a function of energy. In Figure 7 we show the predicted surviving fraction of 36Cl as a function of energy for this density as calculated using a leaky-box model. The HEAO data on the Cl/Ar ratio as shown in Figure 3, expressed in terms of the surviving fraction of 36Cl as obtained from the leaky-box calculations, are also shown in Figure 7. Again the six highest energy data points lie an average of 2.5 p each below the prediction for a constant density of 0.5 atoms cm~3, presumably for the same reasons as for the other two ratios. For 54Mn, because the lifetime is not known, the surviving fraction of 0.37 ^ 0.07 which can be measured quite FIG. 6.ÈSame as Figure 5, but for the surviving fraction of 26Al. Reference density of 0.37 H atoms cm~3 is based on the low-energy isotopic measurements.

4 338 WEBBER & SOUTOUL Vol. 506 FIG. 7.ÈSame as Fig. 5, but for the surviving fraction of 36Cl. Reference density of 0.50 H atoms cm~3 is based on the low-energy isotopic measurements. accurately (Lukasiak et al. 1997; DuVernois 1997) cannot immediately be translated into a density. However, if we assume a lifetime of 1.4 ] 10~6 yr, then this value of f gives a density of 0.30 H atoms cm~3, the same as for 10Be with approximately the same half-life. We therefore use this density as the base parameter for our calculation of Mn/Fe ratio as function of energy. In Figure 8 we show the predicted surviving fraction of 54Mn as a function of energy for this density as calculated using a leaky-box model. The HEAO data on the Mn/Fe ratio as shown in Figure 4, but expressed in terms of the surviving fraction of 54Mn as obtained from the leaky-box calculations, are also shown in Figure 8. Here the data at the highest energies also lie below the prediction for a constant density, but the e ect is less obvious than for the other isotopes. We should note that the actual reference density used in Figures 5È8 for each isotope is arbitrary. It can be adjusted up or down slightly depending on the density derived from the low energy isotope measurements. This will become apparent as one tries to Ðt the data at both high and low energies and, in fact, the high-energy data is useful for helping to deðne the low-energy reference density. FIG. 8.ÈSame as Fig. 5, but for the surviving fraction of 54Mn. Reference density of 0.30 H atoms cm~3 is based on the low-energy isotopic measurements assuming a lifetime of 1.4 ] 106 yr. In the preceding analysis we have carried the simple leaky-box propagation model about as far as it can go. This is a propagation picture in which the medium is homogeneous, and it does not utilize any further parameters of the interstellar medium other than an average density. As such it is useful, however, for examining trends in the data. It is evident from Figures 5È8 for all four decay isotopes that the high-energy charge ratio data from HEAO imply a progressively lower density with increasing energy interpreted in the leaky-box model. To put this in perspective we also show in Figures 5È8 the surviving fraction calculated using the leaky-box model for a density of 0.1 H atoms cm~3.itis clear that the higher energy data points above a few GeV nucleon~1 are more consistent with this lower density. This may be explained by the fact that the particles are propagating into a region the density of which decreases as a function of the distance from the sources which are assumed to be located in the disk. This e ect has also been discussed by Ptuskin (1998) on the basis of the low-energy isotopic data. His curves for the Be/B ratio (or the equivalent surviving fraction f ) calculated using a Ñat halo version of the leaky-box model are quite similar to those we show later using a di usion or Monte Carlo di usion model. To quantify these charge- and energy-dependent di erences we need a more sophisticated model that includes spatial variations of the important parameters, particularly in the Z-direction perpendicular to the plane of the disk. One such model is a di usion model. In what follows we shall examine the predictions of the di usion model presented by Webber, Lee, & Gupta (1992). This model includes the e ects of a less dense Galactic halo and also the e ects of possible convection perpendicular to the disk caused by a Galactic wind. The di usion models can only be solved analytically for certain limited Z dependences of the propagation parameters, however. Finally, to make the calculations still more realistic we consider a Monte Carlo propagation model that actually Ðts the Z dependence of the important parameters such as the di usion coefficient and the matter density to astrophysical measurements (Webber & Rockstroh 1997). 3. PREDICTIONS OF A DIFFUSION MODEL This model is described in Webber, Lee, & Gupta (1992, hereafter WLG). The matter is assumed to be concentrated in an inðnitely thin disk, and the particles are assumed to di use in a much broader, cylindrical halo region characterized by thicknesses in the range 1È10 kpc. The di usion coefficient is independent of Z throughout the di using volume but depends on rigidity. The sources, the interaction loss, and secondary production, are assumed to occur within the thin disk only. Radioactive decay occurs throughout the di using volume. For a di usion parameter K \ 6 ] 1027 cm2 s~1 and a total amount of matter traversed 0 necessary to reproduce the observed B/C ratio (D8 g cm~2 at energies of a few hundred MeV nucleon~1) the calculated surviving fractions for 10Be and 26Al are shown in Figures 12 and 13 of WLG. We superimpose the curves for a halo thickness L \ 2 kpc from Figures 12 and 13 on the 10Be decay data on Figure 5 and the 26Al decay data in Figure 6. This halo thickness is found by WLG to reproduce quite accurately the spectra of the primary nuclei such as He, O, Si, and Fe and the important secondary/primary ratios such as B/C and Z \ 21È23/ Fe as a function of energy. For 10Be the 2 kpc halo gives, on

5 No. 1, 1998 COSMIC-RAY RADIOACTIVE DECAY ISOTOPES 339 the average, somewhat larger values of f than the highenergy HEAO data, but it is a better Ðt to these data than a leaky-box with constant n \ 0.30 cm~3. For 26Al the 2 kpc halo Ðts all of the data quite well, better on average than a leaky box with n \ 0.37 cm~3. For both of these radioactive isotopes a value of the convective velocity as large as 80 km s~1 gives a very poor Ðt to the data (see Fig. 12 and 13 of WLG). The convective velocity has to be reduced to values D20 km s~1 or less, in which cases the curves with convection look much like those without, but for a slightly di erent halo size, before the predictions begin to Ðt the high-energy observations of f for 10Be and 26Al. This model utilizes a very simplistic two zone disk ] halo di using region, however, and it is worth examining a more realistic picture. 4. PREDICTIONS OF A MONTE CARLO MODEL The Monte Carlo model we use here has been described previously in the literature (Webber 1993; Webber & Rockstroh 1997). It is a one-dimensional (Z-dimension) adaption of the transport equation (two-dimensional if symmetry is assumed). A distribution of sources may be simulated by choosing the starting positions in Z for the injected particles. The description of the particlesï motion through physical space and energy space is as follows. In general, 104 particles are injected at each energy as part of an energy grid of 50 energies covering Ðve decades of energy from 10~2 to 103 GeV nucleon~1. The di usion mean free path or step size l is taken in units of the boundary distance L. Typical values of L /l may range from 102 to 104 steps. The distance L and the step size l are chosen to give a particular di usion coefficient. In our nominal case K \ 2 0 ]1028 cm2 s~1 at 1 GeV nucleon~1 so that the step size l \ 2 K /v \ 1.5 ] 1018 cm, and if L \ 6 ] 1021 cm (2 kpc) 0 then there are 4 ] 103 steps to the boundary. A particle is injected and allowed to random walk a certain member of steps (a big step), typically 0.25 times the number of steps to the boundary, in this case 103 steps. At that point, its location is evaluated and the probability of all the possible things that could have happened to the particle is determined. For nuclei, ionization loss and reacceleration gain are included as are nuclear interactions and radioactive decay. This may move the particle to new energy or a primary nucleus may produce one secondary nucleus in the interaction (e.g., C ] B or C] 10Be), and this secondary nucleus is followed throughout its lifetime including nuclear interactions and possible radioactive decay. Then the next particle is injected and all particles are followed to the next big step where the process is repeated. This process is continued until 104 particles have been injected at each energy. The total time for this overall injection of 104 particles is always taken to be greater than q \ L 2/2K, the e ective di usion lifetime of the cosmic rays. In this way the particle distribution reaches an equilibrium situation at all energies. The di usion coefficient (step size) is allowed to vary with both energy and distance from the plane. The various parameters such as the density are speciðed at Z \ 0, and their Z dependence is also speciðed. After all 104 particles have been injected at each energy, the number of particles of each species in each energy bin is evaluated and weighted by a chosen input spectrum which may be an energy or rigidity spectrum. The total number of particles in various Z bins and the time distribution of particles (in e ect the path length distribution) are also evaluated. In the model for these calculations we have taken a source distribution located at Z \ 0, and a spectral index in rigidity \[2.3 for all primary nuclei. The cross sections are taken from Webber, Kish, & Schrier (1990). The matter density is taken to be 1.2 H atoms cm~3 at Z \ 0 with a Z dependence Dexp ([Z/0.2Z ). The magnitude of the di usion coefficient and its rigidity B dependence and the distance to the boundary are varied and, in this calculation, are chosen to reproduce the observed B/C ratio as a function of energy (see Webber & Rockstroh 1997, Fig. 3). This corresponds to a value of K \ 2.2 ] 1028 cm2 s~1 with K D P0.6, a boundary distance 0 L \ 2 kpc, a lifetime q \ 25 ] 106 yr, and an integral of the matter density perpendicular to the disk, / ndl \ 1.4 ] 1021 cm~2. This later values agrees with the measured values of the sum of H1, H2, and H densities as summarized by Strong & YousseÐ (1995). The II total matter traversed at D1 GeV nucleon~1 is D10 g cm~2 which is about the same as required in the leaky-box models with j \ 31.6 br0.60 for esc R [ 3.3 GV. The results of these Monte Carlo calculations for the surviving fractions of the radioactive isotopes 10Be, 26Al, and 36Cl are shown as a series of xïs in Figures 5È7. First of all, we note that the values of f for 10Be and 26 Al from the Monte Carlo calculation agree quite closely with those from the di usion calculation for a very similar set of parameters. Both of these sets of calculations lie somewhat above the measurements for 10Be but agree with the measurements for 26Al as noted earlier. For 36Cl the Monte Carlo predictions agree quite well with the measurements of f from the HEAO data, perhaps lying slightly above them. For all three isotopes these Monte Carlo curves are better representation of the HEAO observations than a simple leaky box model with a constant density chosen to Ðt the low-energy isotopic data. 5. EFFECT OF PROPAGATION PARAMETERS ON THE PREDICTIONS OF f In the di usion model it is seen from Figures 12 and 13 of WLG that increasing the value of L, the halo size, reduces the predicted values of the surviving fraction at high energies. This is reasonable since particles will spend a longer time in a larger halo thus decaying further. The 10Be data, because of the lower measured values of f at higher energies, seems to Ðt a halo size D4 kpc better, whereas a halo size D2 kpc Ðts the 26 Al data better. For the Monte Carlo calculations the values obtained for f at high energies depend somewhat di erently on the parameters. We show in Figure 9 calculations for the surviving fraction of 10Be for di erent sets of parameters. Here also the predicted values of the surviving fraction at high energies are reduced from those of a leaky-box model of constant density. It is seen, however, that the value of the surviving fraction depends on the e ective lifetime chosen. Somewhat longer lifetimes than the reference lifetime of 25 ] 106 yr are therefore needed to Ðt the 10Be data. For 26Al the reference lifetime Ðts the data quite well, while for 36Cl a slightly longer lifetime intermediate between that for 26Al and 10Be is required to give the best Ðt to the data. We have also varied the characteristic value of the Z dependence of the matter density in the Monte Carlo model by a factor of 2, and it seems to have little e ect on the

6 340 WEBBER & SOUTOUL FIG. 9.ÈCalculated surviving fraction of 10Be in a Monte Carlo propagation model for cosmic-ray lifetimes of (1) \ 12 ] 106 yr, (2) \ 25 ] 106 yr, (3) \ 60 ] 106 yr, (4) \ 150 ] 106 yr. calculated values of f. This leads us to conclude that the most sensitive parameter in the Monte Carlo model is the value of the di usion coefficient and how it is related to the boundary size chosen. This in, turn, determines the lifetimes. This explains why in the simple di usion model the boundary distance (which is the only parameter available) is the important parameter. So, in e ect, we believe that the study of the surviving fraction at high energies is telling us more about the di usion coefficient and its relative value in the Z-direction with respect to the boundary distance (size of the halo) and less about how the matter density actually falls o in the Z-direction. 6. SUMMARY AND CONCLUSIONS We have used the HEAO3 results on the Be/B, Al/Mg, Cl/Ar, and Mn/Fe ratios to examine the decay of the radioactive isotopes 10Be, 26Al, 36Cl, and 54Mn at energies from D600 MeV to 20 GeV nucleon~1. From this data we can obtain the surviving fractions of these isotopes with a comparable accuracy to that obtained using directly measured low-energy isotopic ratios. These higher energy surviving fractions are important for exploring larger scale sizes of the cosmic-ray propagation volume as a result of the relativistic time dilation of the decay lifetime. This data is Ðrst intepreted in terms of a leaky-box propagation model, then a more realistic di usion model, and Ðnally a still more realistic Monte Carlo di usion model. Interpreted in terms of a leaky-box model, the data for all four isotopes from HEAO suggest to a greater or lesser degree that the density of the region in which the particles are propagating decreases with increasing energy. Assuming that this is due to the increasing lifetime of the decay nuclei as a result of time dilation, this suggests that cosmic rays are indeed propagating into regions beyond the matter disk where the density is much less than the commonly assumed disk value of 1.0 H atom cm~3. The di usion model calculations allow one to determine that the scale size of this extended halo-like propagating region (where the matter density is zero) in the Z-direction is in the range of 2È4 kpc. These di usion calculations can only Ðt the data for very weak Galactic winds 20 km s~1 or less. Monte Carlo models can Ðt the data with cosmic-ray lifetimes D20È30 ] 106 yr, halo sizes D2È3 kpc, and a matter density falling o as an exponential with a scale length D0.4 kpc. These Monte Carlo calculations show that it is the scale size of the di usion coefficient in the Z-direction relative to the halo size that is most important for Ðtting the data rather than the Z dependence chosen for the matter density. This work was prompted by discussions with Maurice Shapiro in New Hampshire in October 1996 regarding the very Ðrst measurements of 10Be decay 25 years ago which, in fact, used the Be/B ratio. These discussions led to the realization that we already have beautiful data on several radioactive decay charge ratios from the French-Danish HEAO3 experiment that can provide data on this radioactive decay at high energies, thus probing through the relativistic time-dilation e ect, conditions further from the plane of the Galaxy. REFERENCES Connell, J. J., DuVernois, M. A., & Simpson, J. A. 1997, Proc. 25th Inter- Lukasiak, A., McDonald, F. B., & Webber, W. R. 1997, ApJ, 488, 454 nat. Cosmic Ray Conf. (Rome), 2, 397 Ptuskin, V. S. 1998, Proc. ACE Symp. in press Simpson, J. A. & Connell 1998, ApJ, 497, L85 Shapiro, M. M., & Silberberg, R. 1968, Canadian J. Phys., 46, 5561 DuVernois, M. A. 1997, ApJ, 481, 241 Strong, A. W., & YousseÐ, 1995, Proc. 24th Internat. Cosmic Ray Conf. DuVernois, M. A., & Thayer, M. R. 1996, ApJ, 465, 982 (Rome) Engelmann, J. J., et al. 1990, A&A, 233, 96 Webber, W. R., 1993, ApJ, 402, 185 Leske, R. A., & Wiedenbeck, M. E., 1993, Proc. 23d Internat. Cosmic Ray ÈÈÈ. 1998, Proc. ACE Symp. in press Conf. (Calgary), 1, 571 Webber, W. R., Lee, M. A., & Gupta, M. 1992, ApJ, 390, 96 Lukasiak, A., Ferrando, P., MacDonald, F. B., & Webber, W. R. 1994a, Webber, W. R., Kish, J. C., & Schrier, D. A. 1990, Phys. Rev. C, 41, 566 ApJ, 423, 426 Webber, W. R., & Rockstroh, J. M. 1997, Adv. Space Res., 19, 817 Lukasiak, A., McDonald, F. B., & Webber, W. R. 1994b, ApJ, 430, L72