As mentioned the implementation of β−-decay in the postprocessing of data is done with no regards to the uncertainty of the halife of 18775Re or the uncertainty of the stellar enhancement factor. A more accurate representation of the physical uncertainties of β−-decay would be relevant for the analysis of cosmochronology.
Appendix A
Calculation of cosmochronology
Following the analytical approach ofLuck et al. (1980)andShizuma et al. (2005), which again follows the approach of Clayton (1964)who estimate the chemical evolution of the 18775Re - 18776Os -system.
The solar values and evolution of 18775Re , 18776Os , 18776Os come, predominantly from three main sources; s-process contribution to 18676Os and 18776Os , r-process contribution to 18775Re , andβ−-decay from 18775Re to 18776Os . A simple exponential form is adopted for the r-process contribution to 18775Re .
186
76Os= 18676Oss
187
76Os= 18776Oss+18776Osβ d18776Osβ
dt =λβ18775Re d18775Re
dt =A(t)−λβ18775Re
=A0e−τ−1t−λβ18775Re
103
Solving for 18775Re :
d18775Re
dt +λβ18775Re =A0e−τ−1t General solution to homogenous equation: 187˙
75Reh+λβ18775Reh = 0
Where A0 and A1 are scaling factors for the proposed form of r-process contri-bution, andτ−1 is the decay constant of the proposed form of r-process contri-bution. λβ = Tln 2
1/2 is the decay constant of radioactive 18775Re , and T1/2 is the half-life of radioactive 18775Re .
Solving for 18776Os :
Where the constants are the same as 18775Re .
Calculating fraction of 18776Os / 18775Re (f187 from here on):
187
Where ∆t is the time between the formation of the galaxy and the formation of the Solar system. This is, according to the model, all the time available to produce r-process isotopes outside of the Solar system before collapse.
Adopting meteoritic abundances for the solar values at the formation of the solar system from Shizuma et al. (2005) (which adopts the values from Faestermann (1998)), and assuming the uncertainties of 18775Re and 18776Os are uncorrelated:
187
Since 18776Os and 18775Re are separated from the galaxy after the time of formation of the Solar system, the fraction 18776Os / 18775Re at that time can be caluclated from simple β−-decay of 18775Re . The time of solar system formation is denoted tf,sos, and the current time is denoted t0. Note that normalization constants need to be re-calculated when solving the equations for β−-decay .
187
normalization and integration constants: C = 18776Os(t =tf,sos) A = 18775Re(t =tf,sos)
This leads to an equation for calculating the Os-Re-fraction, f187(tf,sos), at the formation of the Solar system from physical parameters. The physical parameters are; the current Os-Re-fraction, f187(t0), the decay-rate of 18775Re , λβ, and the age of the solar system, t0−tf,sos.
f187(tf,sos) = [f187(t0) + 1]e−λβ(t0−tf,sos)−1 (A.3)
Estimates of the physical parameters:
λβ = ln(2)/T1/2
T1/2 = 41.577±0.12Gyr fromSnelling (2015)
t0−tf,sos = ∆t = 4,568.2+0.2−0.4M yr = 4.568+0.2×10−0.4×10−3−3Gyr fromBouvier & Wadhwa (2010)
f187(tnow) = 0.226±0.0579 from eq.A.2
(A.4) The average value for f187(tsos) from eq. A.3:
f187(tsos) = [f187(tnow) + 1]e−λβ∆t−1 = 0.136 (A.5) Error propagation off187(tsos)from eq. A.3:
δf(x, y, z)
Parameter Value Source T1/2 41.577±0.12[Gyr] Snelling (2015)
∆t 4.5682±0.4×10−3 [Gyr] Bouvier & Wadhwa (2010)
f187(tnow) 0.226±0.0579 eq.A.2 (from Shizuma et al. (2005)) f187(tsos) 0.136±0.0323 eq.A.4
Table A.1: Adopted and calculated values for the halife of 18775Re , age of Solar system, cosmic clock fraction f187 =18776Os/18775Reat current time, and at the time of formation of the Solar system.
Inserting values for (∆t, T1/2, f187(tnow)) with uncertainties from eq.A.4 (note that the maximum uncertainty of the age of the Solar system will be used).
∆t= 4.5682Gyr δ∆t= 0.4×10−3Gyr T1/2 = 41.577Gyr δT1/2 = 0.12Gyr
f187(tnow) = 0.226 δf187(tnow) = 57.9×10−3
= 0.0564 δf187(tsos) = √
0.0564f187(tsos) = 0.0323
Appendix B
Components of heavy nuclei in the Solar system
B.1 r-process standard deviation from Solar sys-tem abundances
Most elements heavier than iron in the solar system can be attributed to synthesis by the slow neutron capture process or rapid neutron capture process. Since the astrophysical sites of s-process nucleosynthesis is relatively well known the iso-topic distribution can be well approximated by simulations and nuclear reaction networks.
By measuring the isotopic content of the solar system (from photosphere mea-surements and meteorites that are similar to the solar distribution), the total contribution from s-process and r-process is observed. Scaling the well known s-process distribution to s-only isotopes1, gives the distribution of r-process iso-topesArnould et al. (2007).
1Isotopes which are shielded from r-process nucleosynthesis and produced solely through the s-process.
109
isotope standard min max σlower σupper Re-187 0.0318 0.027 0.0359 -0.1509 0.1289 Re-185 0.0151 0.011 0.0176 -0.2715 0.1656 Os-188 0.0707 0.0633 0.0781 -0.1047 0.1047 Os-189 0.103 0.0961 0.109 -0.067 0.0583 Os-190 0.152 0.137 0.168 -0.0987 0.1053 Os-192 0.273 0.252 0.289 -0.0769 0.0586 Eu-151 0.0452 0.0267 0.0482 -0.4093 0.0664 Eu-153 0.0495 0.046 0.0526 -0.0707 0.0626
Table B.1: Table taken from(Arnould et al., 2007, table 1)σlower,σupperare calculated by the relative fraction between standard value and min, max respectively. Upper standard deviation is the relative dierence between standard value and maximum value, and lower standard deviation is the relative dierence between standard value and minimum value.
The issue with table B.1 is in the interpretation of the values. If the uncer-tianty is assumed to be gaussian in nature (which is a dicault assumtion to argue for), the standard value is interpreted as the mean value of the distribu-tions, while the minimum and maximum values are the -1 σ and +1 σ values of the distributions. From the calculated values in column 4 and 5 in table B.1 the sigma values are not equal. There is not always a linear relationship between the minimum, standard, and maximum values. Which value to use? Below, in gure B.1, is a set of gaussian distributions using both the lower standard deviation, the upper standard deviation, and the average of the two. In order to see for which isotope the eect is greatest.
From gure B.1 the greatest dierence lies in 15163Eu and 18575Re . For both isotopes, the lower limit is the greatest uncertainty/standard deviation. By using the greatest standard deviations, more uncertainty in the observations are cov-ered, but it is uncertain if the uncertainties can be extended in both directions of the standard value.
(a) Upper, lower, deviation for 18876Os .
(f) Upper, lower, and
Figure B.1: Figures a-h show the two dierent standard deviations calculated from table B.1 and their average, plotted as gaussian distributions. 1.0 on the x-axis represent the standard value, and all distributions have a maximum of 1.0 on the y-axis. The plots are merely to visualize the values of table B.1 as gaussian probability distributions, and the axes are therefore intentionally unlabelled.
B.2 s-process standard deviation from Solar sys-temabundances
The quality of these data has continually improved and the most recent compi-lation by Anders and Grevesse [89A] lists 37 elements, determined in the photo-sphere of the Sun, with errors below 25%(Palme & Beer, 1993, p.197)
Estimated accuracy of elemental osmium is 5% from CI chondrites2(Palme &
Beer, 1993, table2, p.203).
The number abundance of elemental osmium (logN(N(H) ≡ 1012) is given in log value. The uncertainty of the meteorite abundances is then ' 1.45% and the uncertainty in solar photosphere composition is'6.90%. The dierence be-tween the two observations are20% with respect to the meteorite number abun-dance(Palme & Beer, 1993, table3, p.205), for other types of meteorites the dierence is greater.
The solar system abundances, the tted s-process distribution, and the re-sulting r-processes from (Palme & Beer, 1993, table3, p.205) are included in gure B.2.
2A type of carbonaceous meteorites with an near solar compositionhttps://en.wikipedia.
org/wiki/CI_chondrite.
(a) The observed solar system distribution of iso-topes, plotted in num-ber abundance (scaled to Si) against mass number.
Data taken from mete-orites and solar photo-sphere. Figure 7, page 211, in article.
(b) The calculated, and tted to Solar system-abundances, s-process distribution of isotopes.
Plotted as number abun-dance (scaled to Si) times neutron capture cross sec-tion against mass number of isotopes. Figure 4, page 220, in article.
(c) The r-process abun-dances, resulting from subtracting calculated s-process abundance from Solar systemabundance.
Figure 5, page 209, in article.
Figure B.2: Solar systemnumber abundances and derived isotopic distributions from Palme & Beer (1993).
List of Figures
2.1 Chart of Nuclides: low-mass excerpt . . . 15
2.2 Atomic structure . . . 17
2.3 Nuclear shells . . . 17
2.4 Binding energy per nucleon . . . 19
2.5 Chart of nuclides from (Clayton, 1964, g.1)around A=187 . . . 25
2.6 Hidden secrets of a massive star-formation region . . . 27
2.7 Hertzsprung-Russel diagram . . . 30
2.8 A simplied visualization of some of the common initial mass func-tions in the literature. (Cappellari et al., 2012, and references therein), Salpeter (1955), Kroupa (2001), Chabrier (2003), Miller & Scalo (1979). image-credit: By JohannesBuchner [CC BY-SA 4.0 (https:// creativecommons.org/licenses/by-sa/4.0)], from Wikimedia Commons. . . 31
2.9 Stellar enrichment diagram . . . 35
3.1 Diagram inow/outow one-zone galaxy model . . . 41
3.2 (Côté et al., 2016, g.1) . . . 42
3.3 How the input parameters were determined from multiple sources in the literature. Values and standard deviations averaged to a probaiblity distribution, and then tted to a single gaussian dis-tribution. Images from(Côté et al., 2016, gure 2 and table 7). . . 44
3.4 (Côté et al., 2016, g.6) . . . 46
3.5 (Côté et al., 2016, g.6) . . . 46
3.6 (Guedes et al., 2011, g.2) . . . 49
3.7 (Guedes et al., 2011, g.1) . . . 53
3.8 (Shen et al., 2015, g.4) . . . 56
3.9 (Shen et al., 2015, g.6) . . . 57
4.1 Star formation rate in Omega for direct insertion-tting . . . 61 115
4.2 Stellar mass in Omega for direct insertion-tting . . . 61
4.3 Total mass of Fiducial Omega-modelfor initial/inow mass-tting . 64 4.4 Stellar mass of Fiducial Omega-modelfor initial/inow mass-tting 64 4.5 Stellar mass of Fiducial Omega-modelfor with outow mass-tting 65 4.6 spectroscopic iron of Fiducial Omega-modelfor with outow mass-tting . . . 65
4.7 Fit of type 1a supernovae Fiducial Omega-model . . . 67
4.8 NSM-ejectamass . . . 69
4.9 mergerfraction spectroscopic plot . . . 69
4.10 nbnsm spectro plot . . . 70
4.11 nal spectro plot . . . 70
4.12 combo'spectro' plot . . . 70
4.13 nal rate plot . . . 70
4.15 eect of nucelar uncertainties . . . 76
5.1 Yields before β−-decay . . . 85
5.2 Yields with β−-decay and removing negative isotope yields . . . 87
5.3 Rate of nucleosynthetic events . . . 90
5.4 Analytical models for cosmochronology . . . 92
5.5 Yields+IMFslope afterβ−-decay and removing negative isotope yields . . . 95
5.6 Yields+IMFslope+NSM afterβ−-decay and removing negative isotope yields . . . 97
5.7 Rate of nucleosynthetic events Yields+IMFslope+NSM . . . . 98
B.1 Dierence in upper and lower deviation from Arnould et al. (2007) . 111 B.2 Solar systemnumber abundances and derived isotopic distributions from Palme & Beer (1993). . . 113
List of Tables
4.1 Omega-parameters directly adopted from Eris . . . 62 4.2 Mass data Eris . . . 63 4.3 Omega-parameters from tting mass-content to Eris . . . 64 4.4 Omega-parameters from tting stars and supernovae to Eris . . . 68 4.5 Omega-parameters from tting r-process events to Eris . . . 68 4.6 Values and uncertainties of observed r-process abundances in the
Solar system . . . 75 4.7 eect of nuclear uncertainties . . . 77 5.1 Mass-table at Solar system formation and now for Yields . . . . 88 5.2 Rates and numbers of nucleosynthetic events in Yields . . . 89 5.3 Analytical models for 18776Osc/ 18775Re . . . 93 5.4 Mass-table as Solar system formation and now for Yields+IMFslope 96 A.1 Summary of parameters at Solar system formation . . . 107 B.1 Observed r-process abundances in Solar system (Arnould et al. (2007))110
117
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