Studies report evidence for the one-time presence of SLRs, through decay product systematics, including 10Be, 26Al, 36Cl, 41Ca, and 53Mn, in CAIs in primitive carbonaceous meteorites at the nascence of the solar system  . The possible origins of these SLRs are widely varied and include stellar sources (AGB Stars, Wolf-Rayet stars, nova, and super nova) and energetic particle interaction, from either SEPs, or galactic cosmic rays (GCRs). Bricker & Caffee   proposed the solar wind implantation model (SWIM) for the incorporation of 10Be and 36Cl into CAIs early in primitive meteorites.
In the SWIM, the SLRs come into existence via SEP nuclear reactions in the proto-solar atmosphere of the young Sun, characterized by X-ray emissions orders of magnitude greater than main sequence stars. Studies of the Orion Nebulae indicate that pre-main sequence (PMS) stars exhibit X-ray luminosity, and hence SEP fluxes on the order of ~105 over contemporary SEP flux levels  . The irradiation produced SLRs are then trapped by magnetic field lines, and these solar wind SLRs eventually impregnate CAI precursors. This mode of production of SLRs, entrainment of SLRs in the solar wind, and implantation of SLR into solar system material is seen in the implantation of solar wind particles, e.g. 10Be   and 14C   , on the Moon.
10Be is produced via SEP spallation reactions, with oxygen serving as the chief target particle in the SWIM. Similar to 10Be, 7Be, half-life of 53 days  , is also primarily produced through SEP nuclear reactions with oxygen as the primary target particle, and 7Be has also recently been detected in stellar photospheres  In addition, the one-time presence of 7Be has been measured in CAIs in primitive meteorites (through the study of Li, the decay product of 7Be, systematics)   . Owing to the 53 day half-life, local irradiation is the only possible operation pathway for 7Be production. As such, the large difference in half-lives between 7Be and 10Be is of interest in terms of chronological processes associated with early solar system and CAI formation and evolution.
In this work, we consider the possible incorporation of 7Be into CAIs in primitive carbonaceous meteorites in the SWIM. Table 1 below characterizes berylliumisotopes found in CAIs.
2. Solar Wind Implantation Mode
In the SWIM, SLRs are produced in the solar nebula via SEP nuclear reactions on gaseous target material in the solar atmosphere ~4.6 Gyr, during the formation
Table 1. Beryllium isotopes found in CAIs.
Note: Radionuclide content in g−1 calculated from initial isotopic ratio and 9Be content in ppb. The 9Be content in CAIs is estimated 100 ppb   .
of the solar system. These newly produced nuclei are incorporated in the solar wind. The SLRs flow along magnetic field lines in the solar wind, and this particle flow intersects with materials which have fallen out of the main accretion flow, which was headed to hot-spots on the Sun. At the intersection of outflowing SLRs, and inflowing fallen CAI precursor material, the SLRs may become impregnated into the inflowing materials. The fundamental geometry for the implantation process described above and transportation of implanted CAIs to the asteroid zone can be gleaned from the X-wind model of Shu et al.    . Figure 1 below illustrates of the basic magnetic field geometry, 7Be production via SEP flaring activity, and subsequent implantation into CAI-precursor material from the main funnel flow onto the proto-Sun.
2.2. Refractory Mass Inflow Rate
The effective refractory mass inflow rate, S, i.e. the refractory mass that falls from the main funnel flow which was accreting onto the star at the X-region, is calculated from equation (1):
where is disk mass accretion rate, Xr is the cosmic mass fraction, and F is the fraction of material that enters the X-region from the main funnel flow  . For , we adopt 1 × 10−7 solar masses year−1. Disk mass accretion rates range from ~10−7 to ~10−10 solar masses year−1 for T Tauri stars from 1 - 3 Myr  , whereas embedded class 0 and class I PMS stars have mass accretion rates of ~10−5 to ~10−6 solar masses year−1  . Here we adopt for , a rate 1 × 10−7 solar masses year−1, corresponding to class II or III PMS stars. From Lee et al.  we utilize a cosmic mass fraction, Xr, and fraction of refractory material fraction F, of 4 × 10−3 and 0.01, respectively, in our model. Xr represents the fraction of refractory content in the inflowing material, and F represents the fraction of inflowing mass that does not accrete onto the proto-sun. The choice 0.01
Figure 1. SWIM magnetic field geometry for SLRs production via SEP nuclear reactions. The gray area represents the main accretion flow onto “hot spots” on the PMS star. SLRs produced close to the proto-solar surface are incorporated into CAI precursor material which has fallen from the accretion flow (figure after Shu et al.  ).
maximizes F, and corresponds to all the mass which comprises the planets falling from the accretion flow. F = 0.01 is the preferred value of Lee et al.  in their model. (See Lee et al.  for a detailed discussion of Xr and F) Employing Equation (1) and the parameters detailed above, we find the rate at which this refractory material reaches the x-region, called here the refractory mass inflow rate, S, is 2.5 × 1014 g s−1. In consideration of the extreme values of, S, S could be two orders of magnitude greater if the accretion rates of ~10−5 to ~10−6 solar masses year−1, or S could also be four orders of magnitude less if the mass accretion rate was ~10−8 to 10−10 solar masses year−1 and F ~0.0001.
2.3. Effective Ancient Production Rate
The effective ancient 7Be outflow rate, P in units of s−1, is given by:
where p is the ancient production rate and f is the fraction of the solar wind 7Be that enters the CAI-forming region; f = 0.1. (See Bricker & Caffee   for a discussion of factor f). The 7Be production rate is calculated assuming that SEPs are characterized by a power law relationship:
where r ranges from 2.5 to 4. For impulsive flares, i.e. r = 4, we use 3He/H = 0.1 and 3He/H = 0.3, and for gradual flares, i.e. r = 2.5, we use 3He/H = 0. For all spectral indices, we assume α/H = 0.1. Contemporary SEP flux rates at the Sun-Earth distance of 1 AU are ~100 protons cm−2×s−1 for E > 10 MeV  . We assume an increase in ancient particle fluxes over the current particle flux of ~4 × 105   , yielding an energetic particle flux rate of 3.7 × 1012 protons cm−2×s−1 for E > 10 MeV at the surface of the proto-Sun.
The production rates for cosmogenic nuclides can be calculated via:
where i represents the target elements for the production of the considered nuclide, Ni is the abundance of the target element (g×g−1), j indicates the energetic particles that cause the reaction, is the cross section for the production of the nuclide from the interaction of particle j with energy E from target i for
the considered reaction (cm2), and is the differential energetic particle
flux of particle j at energy E (cm−2×s−1)  . We assume gaseous oxygen target particles of solar composition  .
The cross-section we use to calculate 7Be production from protons and 4He pathways is from Sisterson et al.  , and the cross-section we use for production from 3He is from Gounelle et al.  . The Sisterson et al.  cross-section is experimental obtained, and the Gounelle et al.  cross-section is a combination of experimental data, fragmentation and Hauser-Feshbach codes. The uncertainty associated with model codes are at best a factor of two. Taking into account both target abundance and nuclear cross-sections, the reaction with oxygen as the target is the primary production pathway. Any other nuclear reaction would add little to the overall 7Be production rate. Table 2 shows the nuclear reactions considered in the calculations.
The content of 7Be found in refractory material, in atoms g−1, predicted by SWIM is given by:
where P is given atoms s−1 and S is given in g×s−1.
Using the refractory mass inflow rate, S, of 2.5 × 1014 g×s−1 from Equation (1), and calculations of P, the effective ancient 7Be outflow rate, from Equation (2) & Equation (4), we determine the content of 7Be in CAIs in atoms g−1 using Equation (5), and find the associated isotopic ratio for different flare parameters given in Table 3. Figure 2 depicts the 7Be isotopic ratio predicted by the SWIM from SEPs.
Similar to 10Be, the primary target for SEP production of 7Be is oxygen. As such, the SEP origin of 7Be and 10Be are uniquely intertwined. The estimated 7Be/10Be production ratio from MeV SEPs in the early solar system is estimated to be ~70  . Using the production rate from Equation (4) and the production rate for 10Be from Bricker & Caffee  from SEP interaction with oxygen targets, we obtain a production ratio of ~50, which is similar to Leya  . It would then be expected that the original ratio of 7Be/9Be found in CAIs would be ~50 times greater than the 10Be/9Be ratio, assuming the simple SWIM mechanism described above. Using 9.5 × 10−4  as the canonical 10Be/9Be ratio, the 7Be/9Be ratio would scale to 4.8 × 10−2. We find this ratio is reproducible within a factor of ~5, the uncertainty associated with SWIM, for spectral indices r > 3.2. The SWIM can account for the scaled up 7Be/9Be ratio. Figure 3 below details the ratio of 7Be/9Be from SWIM to 4.8 × 10−2.
Experimentally obtained measurements for the original 7Be/9Be ratio in CAIs are limited and a matter of considerable debate. Limited experimentally determined values for the ratio range from about 1.2 × 10−3  to 6.1 × 10−3  . The experimentally obtained ratios are at least a factor of 10 less than SWIM
Table 2. Nuclear reactions considered in this paper.
Table 3. Predicted 7Be content in CAIs.
Figure 2. Predicted 7Be content in CAIs from energetic protons as a function of solar flare parameter.
Figure 3. Ratio of 7Be/9Be found from SWIM.A ratio of one indicates exact match, a ratio greater than one indicates overproduction, and a ratio less than one indicates underproduction.
calculations, and also a factor of at least 10 less than the scaled up 7Be/9Be found from scaling the canonical 10Be/9Be ratio to 7Be and 10Be production rates. Figure 4 depicts the ratio of SWIM obtained ratio to the canonical 7Be/9Be ratio.
Clearly, some other mechanism is needed to explain the overproduction of the
Figure 4. Ratio of SWIM 9Be/10Be ratio to canonical 7Be/9Be ratio. A factor greater than one indicates overproduction relative to canonical.
Figure 5. Days to canonical ratio vs. spectral index.
7Be/9Be ratio, both in terms of SWIM calculations and the scaling of the 10Be/9Be to relative 7Be and 10Be production rates.
An assumption of SWIM is that radionuclides are produced via SEP interaction and then immediately incorporated into CAI precursor materials. With a half-life of 53 days, it is possible that some temporal evolution occurs before 7Be becomes implanted. Figure 5 shows days to canonical ratio for spectral index.
Figure 5 shows that with a delay on the order of ~100 days from the time of production of 7Be to implantation in to CAI precursor materials, the canonical ratio is replicated. Taking into account the time from production of the radionuclide to implantation into CAI precursors, i.e., two half-lives of 7Be, explains the deficit in 7Be/10Be measured ratio in comparison to the 7Be/10Be production ratio. It is possible and likely for nuclei to have some finite residence time in the photosphere. Calculations of this residence time have not been performed and are beyond the scope of this paper. Our ad hoc choice of two half-lives of residence time for 7Be was to explain the 7Be/10Be measured ratio in comparison to the 7Be/10Be production ratio.