Summary of methods - Modelling uncertainty of the Rhenium-Osmium cosmic clock

Attempting to probe the uncertainties of nuclear observables is dicult, because both Eris and Omega both use the observed r-process abundance from the Solar system as the yield tables for all r-process events (neutron star mergers). This mean that all r-process events assume to generate the same distribution of heavy metals, and that distribution is the same as the component measured in the Solar system. In order to vary these values in Omega a single isotope is chosen, along with a factor that applies to the yield of that isotope. When using Omega, all yield tables will multiply with this factor for that specic isotope alone. E.g.

('C-14', 1.3) would mean that the yields for ¹⁴₆C will be multiplied with 1.3 in all yield tables. This method does not only probe the nuclear uncertianties, varying the yield-tables of a chemical evolution code nds the eect of the accumulated uncertainty of the stellar evolution applied to chemical evolution methods.

There are many parameters that aect r-process production alone. Ejecta distribution and size from neutron star mergerand type II supernovae, and the ratio of ejecta and frequency between the two. Even in relatively simple galactic chemical evolution models, these parameters can interact in unpredictable ways.

In order to get a complete picture of errors and uncertainty in numerical models, a common approach has been Monte Carlo style simulation . Each parameter is randomly drawn from a distribution correspondin g to their determined uncer-tainty (usually the chosen distributions are gaussian), and after many repetitions of randomly drawing, the result can be viewed statistically in order to determine uncertainty behavior. This thesis will such an experiment with and see how the uncertainties of the r-process aect the uncertainty of the Re-Os dating method.

Chapter 5 Results

There are three main experiments;

Yields The yields of isotopes are varied within their standard deviation 4.6 Yields+IMFslope Côté et al. (2016)found that the slope of the initial mass

func-tion aects the uncertainty of the chemical evolufunc-tion most greatly. This experiment varies the stellar yields and the high mass slope of the initial mass function. The stellar yields are given the same distribution as Yields, and the slope of the initial mass function, α, is given the uncertainty dis-tribution found in Côté et al. (2016); which is σ_α = 8.73% around the mean hαi= 2.29.

Yields+IMFslope+NSM Same as Yields+IMFslope, with a ten percent un-certainty in parameters related to neutron star mergers. These are the fraction of binary neutron stars that turn to neutron star mergers (scales the rate of neutron star mergers), the ejecta mass (scales the total mass ejected from a neutron star merger), and the slope of the delay-time distri-bution (distridistri-bution of time between neutron star formation and merger for a system that will merge).

It should be noted that the 10% uncertainty in Yields+IMFslope+NSMis rather arbitrary. It is dieicult to nd realistic realizations of the probability dis-tribution of the neutron star merger parameters in Yields+IMFslope+NSM, and the experiments is merely used to compare the eect of varying the astro-physical source. The experiments pick a randomly generated number (within the appropriate gaussian distribution) and calculates the chemical evolution with

Omega. The calculations are performed 1500 times with randomly generated de-viations to the parameters. The number is also chosen rather arbitrarily, it is more than twice the amount of calculations for multiple parameters inCôté et al.

(2016)and should be sucient for the purpose of this thesis.

The solar system is formed from a collapse of interstellar gas. The gas is assumed to have separated from the interstellar medium at the formation of the solar system. The formation of the solar system is estimated from meteorites to be'4.5 Gyrs agoBouvier & Wadhwa (2010). From the semianalytical model, Omega, the total mass of ¹⁸⁷₇₆Os and ¹⁸⁷₇₅Re in the interstellar medium is calculated. The cosmic clock fraction between the two isotopes, f₁₈₇ = ¹⁸⁷187⁷⁶^Os

75Re, is also calculated.

The fraction between the nuclei is relevant because it can be determined from meteorites, unlike the total mass of isotopes in the Solar system at the time of formation. In the Eris-simulation the galactic age is 14 Gyrs, which means the solar system formed at 9.5 Gyrs. The uncertainties of the mass of each isotope come from the uncertainty of the input parameters in each experiment, Yields, Yields+IMFslope, and Yields+IMFslope+NSM. The cosmic clock fraction,f₁₈₇, is determined from observations to be 0.136±0.0323, see table A.1 in appendix A.

5.1 Without β

⁻

-decay

The cosmic clock fraction,f₁₈₇ = ¹⁸⁷187⁷⁶^Os

75Re, is calculated by taking the ratio of total mass in the interstellar medium. In gures 5.1 the mean and distribution is plotted for the cosmic clock fraction and the mass of ¹⁸⁷₇₆Os , ¹⁸⁷₇₅Re against time.

The mean, deviation, and extrema are calculated from 1500 dierent calculations.

In Omega, β⁻-decay from radioactive nuclei (like ¹⁸⁷₇₅Re ) are not taken into account (for the version used in this thesis). The mass and ratio shown in gures 5.1 are only representative of the nuclei synthesized in stars, supernovae and neutron star mergers.

It is quite easy to see that the fraction in gure 5.1c is far below the observed value in table A.1.

Some irragularities in the distribution (non-gaussian tendencies) will be dis-cussed in section 5.2.

(a) Total mass of ¹⁸⁷₇₅Re in the inter-stellar medium of the galaxy modelled by Omega, as a function of time.

(b) Same as 5.1a, but for ¹⁸⁷₇₆Os .

Figure 5.1: The mass and mass fractions in the interstellar medium before β⁻ -decay is applied. Only nucleosynthesis/production from stellar sources is consid-ered.

The left panel, in all three gures, show the evolution of the total mass in the interstellar medium. The two right panels, in all three gures, show the distri-bution at 9.5 and 14 Gyr. The black vertical lines in the evolution-panel shows where the distribution in the two right panels have been taken from.

5.2 With β

⁻

-decay

The cosmic clock fraction, f₁₈₇ = ¹⁸⁷187⁷⁶^Os

75Re, is expected to increase with β⁻-decay , because amounts of ¹⁸⁷₇₅Re decays to ¹⁸⁷₇₆Os following the dierential equation in section 2.2.4. As outlined in the methods, aβ⁻-decay calculation was applied to the data in postprocessing. The resulting gures show a clear increase in cosmic clock fraction, as well as a signicant decrease in uncertainty. The decrease in uncertainty is discussed in chapter 6.

Due to the gaussian distribution of the input parameters and relatively large sample size (1500 model calculations) some isotope yields will be negative. This is unphysical, and all negative yields will be set to zero in the calculation. Forcing the stellar yield is the closest physical interpretation of a negative yield-value.

This eect leads to an overpopulation of randomly sampled fudge-factors being zero, and the non-gaussian shape of the distribution causes outliers in the distribution. Such outliers also greatly aect the standard deviation of the result-ing distribution. One possible solution to this is to take a gaussian distribution set it to zero below parameter-value zero and scale it to the integral (as is the norm for statistical distributions). Applying this form of distribution numerically is beyond the scope of this project. An alternate method is suggested. When the statistical distribution is found from the data, all models with a parameter-value of zero or lowerYˆA

ZX≤0, is ignored.

The resulting distributions are shown in gures 5.2, and the values for the distribution extracts (both with and without the eect of β⁻-decay ) are shown in table 5.1.

(a) Mass of ¹⁸⁷₇₅Re in the interstel-lar medium of the galaxy modelled by Omega.

(b) Mass of ¹⁸⁷₇₆Os in the interstel-lar medium of the galaxy modelled by Omega.

Figure 5.2: The plots are similar to gures 5.1, however β⁻-decay has been ap-plied and only data with positive, non-zero isotope-yields, are considered. The values for the mean and standard deviation (green line) in the distribution ex-tracts can be found in table 5.1. The green band represent meteordata calculated in appendix A.

Postprocessed t = 9.5Gyr

β⁻-decay applied 0.11(31)±0.00(07) 0.18(28)±0.00(08) Table 5.1: Mean and standard deviation of mass and mass fraction in Omega.

Distributions with and withoutβ⁻-decay were considered after calculations with negative yields were neglected. Values represent the distributions shown in gures 5.2, which are excerpts of the time-evolution graphs at the black vertical lines.

5.3 Rate of nucleosynthetic events

In Omega, the nuclides in question are synthesized in stars and ejected into the interstellar medium through a limited list of events. Asymptotic giant branch stars, massive stars that turn into type 2 supernvoae, type 1a supernovae, and binary neutron star mergers. In this section, the rate and cumulative number of neutron star mergers, type 2, and type 1a supernovae are shown, see gures 5.3. The data was taken from Yields experiment, meaning that there is no uncertainty applied to the number and rate of events. Right panels show the distribution at 9.5 Gyr and 14 Gyr, and the values at these times are shown in table 5.2

From table 5.2 the number of neutron star mergers in the Galaxy is'10⁸when the Solar system formed. Assuming that Eris is representative for Milky Way Galaxy, that the process material is well mixed, and that the Solar system r-process-abundance is representative of the Galactic average r-process abundance, and assuming that the Fiducial Omega-model model is represntative of Eris this is the number of neutron star mergers required to get the solar r-process abun-dance. Omega is a one-zone semianalytic model and therefor unable to reproduce the local variations that can occur in a galaxy, which particle-hydrodynamics simulations, like Eris have a better hope at resolving. This means that local variations challenges the assumption of r-process abundances in the Solar system

being representative of the Galactic average.

In Rosswog et al. (2017) an estimate at the amount of nuetron star mergers required in our neighbourhood is estimated (see (Rosswog et al., 2017, g.6)). A rough estimate of one to three neutron star mergers per year per (Gpc)³ for an ejecta mass of 0.03 M . Accounting for the dierence in ejecta mass, the Fiducial Omega-model requires an average of nearly one million mergers in 14 billion years in one single galaxy.

Binary neutron star mergers

time Gyr rate Myr⁻¹ ΣN

14 0.201 114×10³

9.49 3.75 101×10³

Type 1a supernovae

time Gyr rate Myr⁻¹ ΣN

14Gyr 1.1×10⁻⁴⁴M yr⁻¹ '0M yr⁻¹ 29.6×10⁶ 9.49Gyr 821M yr⁻¹ 27.2×10⁶

type 2 supernovae

time Gyr rate Myr⁻¹ ΣN

14Gyr 0M yr⁻¹ 258×10⁶

9.49Gyr 8.67×10³M yr⁻¹ 233×10⁶

Table 5.2: Rates and total number of nucleosynthetic events for neuron star mergers, type 1a and 2 supernovae in Omega. The time is taken at '9.5 Gyrs (the formation of the solar system, and 14 Gyrs (now). Plots of the time evolution of nucleosynthetic events are shown in gures 5.3.

(a) Type 1a supernovae. (b) Type 2 supernovae.

Figure 5.3: All plots show the rate of nucleosynthetic events (blue), and cu-mulative sum of events (red) afterβ⁻-decay applied and negative isotope yields have been removed. The nucleosynthetic events are type 1a (5.3a) and type 2 (5.3b) supernovae, and binary neuron star mergers (5.3c). The rate of each event follows the star formation rate (see gure 4.1) with a scale factor and delay time distribution.

5.4 Comparing models for

¹⁸⁷₇₆

Os /

¹⁸⁷₇₅

Re

With a numerical model for ¹⁸⁷₇₆Os / ¹⁸⁷₇₅Re , the data can be compared to other, analytical models. All the analytical models presented here are variations of the

Clayton (1964)model for cosmochemical evolution of ¹⁸⁷₇₆Os /¹⁸⁷₇₅Re , which assumes that the rate of events declines exponentially in time. As can be seen in appendix A the actual number of events and the amount of ¹⁸⁷₇₅Re ejected from each is insignicant when calculating the fraction¹⁸⁷76Os_c/¹⁸⁷₇₅Re . ¹⁸⁷76Os_cis the component of ¹⁸⁷₇₆Os from cosmoradiogenic decay from ¹⁸⁷₇₅Re .

The calculations in appendix A follow the analytical model of cosmochronol-ogy fromLuck et al. (1980)andShizuma et al. (2005), which again follow the analytical steps from Clayton (1964). A summary of models, parameters and references can be found in table 5.3.

From gure 5.4 we see that the shape of the analytical models matchg the data from Omega reasonably well. However, due to simplications, the analyti-cal models end up with deviating estimates of parameter values. The analytianalyti-cal models of Clayton (1964) and Luck et al. (1980) consider extreme paradigms for nu-cleosynthesis. Omega, with a more realistic model of nucleosynthesis, end up in between the two extremes.

Figure 5.4: A comparison of the data from Omega-experiment Yields(black), with the analytical models of Clayton (blue), Shizuma (green), and Luck (magenta) for comparison.

The model from Shizuma is tted to the data from Omega, with the curvet function in SciPy¹ (seen in red). The green band represent meteordata calculated in appendix A.

COMPARINGMODELSFOR 18776OS/ 18775RE93

Model ¹⁸⁷76Os_c/ ¹⁸⁷₇₅Re λRe λ_rncp Reference

Clayton ^Λ−λ_λ e^λt_1−e^1−e−(Λ−λ)t^−Λt −1 λ = ^{ln 2}_τ

Re Λ Clayton (1964)

Clayton

Sudden synthesis e^λt−1 τRe= 47±10Gyr Λ→ ∞ Clayton (1964)

Clayton

Uniform synthesis ^1−e^λt^−λt −1 " Λ→0 Clayton (1964)

Luck ^λ^Re^/β(1−e_e−βt^−βt−e^)−(1−e⁻λRet⁻^λRet⁾ λ_Re = 1.62±0.08

×10⁻¹¹yr⁻¹ β Luck et al. (1980)

Sudden synthesisLuck " " β = 10⁻⁶yr⁻¹ Luck et al. (1980)

Steady stateLuck " " β= 10⁻¹²yr⁻¹ Luck et al. (1980)

Shizuma ^(1−e

−λe β t

)−(1−e^−λt)λ^e_β /λ e^−λ^e^β ^t−e^−λt

λ^e_β = ^{1.2 ln 2}_τ

Re = 2.00×10⁻¹¹[yr⁻¹] λ∈[0,2]Gyr⁻¹ Shizuma et al. (2005)

SciPy curvet

to Fiducial Omega-model-data " 1.33×10⁻¹¹

±2.767×10⁻¹⁴ [yr⁻¹] 5.42×10⁻¹⁰

±5.79×10⁻¹² [yr⁻¹]

Table 5.3: Table with the analytical models in the literature, stemming from Clayton (1964). Notation of parameters is attempted to be consistent with the articles they are taken from, not with eachother in the table. λ¹⁸⁷

75Re is the decay constant of radioactive

18775Re . λ_rncpis the decay constant of the rate of events for rapid neutron capture processes. λ^e_β is the eective netβ⁻-decay constant for ¹⁸⁷₇₅Re after thermal consitions of astration have been taken into account, equalt to 1.2 times β⁻-decay -constant of neutral

18775Re . Shizuma does not give any uncertainty for the halife of ¹⁸⁷₇₅Re , and the boundaries of λ are only found to be in good agreement with a Galctic age of 11-15 yrs. The basic model for Shizuma, Luck and Clayton are identical, even though they are written dierently. Scipy curvet is the parameters, with standard deviation, after tting the model fromShizuma et al. (2005) to the data produced in Omega; Yields-experiment.

5.5 Consider high mass slope of initial mass func-tion

The initial mass function gives the distribution of masses for a stellar population.

Changing the high-mass slope of the mass function gives more stars with higher initial mass². High mass stars end their life as type 2 core collapse supernovae, with a signicantly dierentdistribution of nuclei. It was found inCôté et al. (2016)

that the slope of the initial mass function had the most signicant impact on uncertainties in nucleosynthesis.

In a dierent set of Omega-calculation, Yields+IMFslope, the slope of the initial mass function is varied as well, as described in the introduction to this chapter. Figures 5.5 show the mass and massratio of ¹⁸⁷₇₆Os and ¹⁸⁷₇₅Re , the distribution at the formation of the solar system and current time are imbedded as distribution extracts. The mean and standard deviation of the distribution extract can be found in table 5.4, with and without β⁻-decay applied to the data. From the distribution extracts in gures 5.5 the distribution are clearly non-gaussian in nature, so the mean and standard deviation can be misleading.

The uncertainty of the slope of the initial mass function dominates over the uncertainty of the stellar yields, as can be seen from the additional term in the uncertainty distribution of gure 5.5c. The degeneracy between parameters are not taken into account in the analysis of the yields and slope. It would be more meaningful to divide the distributions in two and calculate the mean and standard deviation for the two separate distributions individually (especially for f₁₈₇ in gure 5.5c).

2Note that more stars with more mass does not change the total mass of gas turned to stars, but gives more high mass stars at the expense of fewer small mass stars.

(a) Mass of ¹⁸⁷₇₅Re in the interstel-lar medium of the galaxy modelled by Omega.

(b) Mass of ¹⁸⁷₇₅Re in the interstel-lar medium of the galaxy modelled by Omega.

Figure 5.5: The mass and mass fractions in the interstellar medium after β⁻ -decay is applied and uncertainty in the high mass slope of the initial mass func-tion. Nucleosynthesis/production from stellar sources is considered as well as the radioactive decay from ¹⁸⁷₇₅Re to ¹⁸⁷₇₆Os . The amount of type II supernovae are also varied because the high mass slope of the initial mass function gives more massive stars, which in turn give more type II supernovae.

The far left plot of all subgures represent the timeevolution of the mass/mass-fraction in the interstellar medium, while the two right plots represent the uncer-tainty distribution at a given point in time. The points in time are 9.5 Gyrs (the formation of the solar system) and 14 Gyrs (current time). The points in time are also shown by black vertical lines in the far left plot.

Postprocessed t = 9.5Gyr Solar system formation

t= 14Gyr Now

18775Re No β⁻-decay 12.82±6.54 M 13.97±7.12 M β⁻-decay applied 11.57±5.91 M 11.87±6.06 M

18776Os No β⁻-decay 0.17±0.09 M 0.20±0.11 M β⁻-decay applied 1.41±0.64 M 2.29±1.08 M f₁₈₇ No β⁻-decay 0.01(80)±0.01(70) 0.01(98)±0.01(89)

β⁻-decay applied 0.13±0.02 0.20±0.02

Table 5.4: Mean and standard deviation of mass and mass fraction in Omega;

experiment Yields+IMFslope. Values represent the mean and standard devia-tions in the distribution extracts in gures 5.5.

5.6 Consider events of binary neutron star merg-ers

In the Yields+IMFslope+NSM-experiment, the yields of stellar ejecta is var-ied as well as the slope of the initial mass function and the physical parameters for neutron-star mergers. These parameters are number of events, mass ejected per event, and the delay-time distribution of events. Due to the lack of observational constraints it is hard to nd quantiable uncertainties for these parameters. In order to estimate how the uncertainties of the neutron-star merger parameters aect the outcome a 10% uncertianty is added to all parameters.

Figures 5.6 show the mass and mass-ratio of ⁸¹⁷₇₆Os and ¹⁸⁷₇₅Re , while g-ures 5.7 show the rate and cumulative numbers of nucleosynthetic events in the Yields+IMFslope+NSM-experiment.

(a) Mass of ¹⁸⁷₇₅Re in the interstel-lar medium of the galaxy modelled by Omega.

(b) Mass of ¹⁸⁷₇₅Re in the interstel-lar medium of the galaxy modelled by Omega.

Figure 5.6: The mass and mass fractions in the interstellar medium after β⁻ -decay is applied to experiment Yields+IMFslope+NSM. Nucleosynthesis/pro-duction from stellar sources is considered as well as the radioactive decay from

18775Re to ¹⁸⁷₇₆Os .

(a) Type 1a supernovae. (b) Type 2 supernovae.

Figure 5.7: All plots show rate of nucleosynthetic events (blue), and cumulative sum of events (red) for Yields+IMFslope+NSM-experiment after β⁻-decay applied and negative isotope yields have been removed. The nucleosynthetic events are type 1a (5.7a) and type 2 (5.7b) supernovae, and binary neuron star mergers (5.7c). The rate of each event follows the star formation rate (see gure

In document Modelling uncertainty of the Rhenium-Osmium cosmic clock (sider 80-0)