Polarized radiation and the Emergence of Biological Homochirality on Earth and Beyond
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16, 2021
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POLARIZED RADIATION AND THE EMERGENCE OF BIOLOGICAL HOMOCHIRALITY ON EARTH AND BEYOND N o ´ emie G lobus A natoli F edynitch and R oger D. B landford Center for Cosmology & Particle Physics, New-York University, New-York, NY10003, USA. E-mail: [email protected] Center for Computational Astrophysics, Flatiron Institute, Simons Foundation, New-York, NY10003, USA Institute for Cosmic Ray Research, the University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8582, Japan. E-mail: [email protected] Kavli Institute for Particle Astrophysics & Cosmology, Stanford University, Stanford, CA 94305, USA. E-mail: [email protected]
ABSTRACTIt has been proposed that spin-polarized cosmic radiation can induce asymmetric changes in helical biopoly-mers that may account for the emergence of biological homochirality. The parity violation in the weak inter-action has direct consequences on the transport of polarization in cosmic ray showers. In this paper, we showthat muons retain their polarization down to energies at which they can initiate enantioselective mutagenesis.Therefore, muons are most likely to succeed in establishing the connection between broken symmetries inthe standard model of particle physics and that found in living organisms. We calculate the radiation dosesdeposited by primary and secondary cosmic rays at various prime targets for the searches of life in the solarsystem: Mars, Venus, Titan, icy moons and planetesimals, and discuss the implications for the enantioselectivemutagenesis proposed as to be the driver of homochiralization. Earth is unusual in that spin-polarized muonsdominate the cosmic radiation at its surface.
Keywords: cosmic rays, astrobiology — INTRODUCTIONThe origin of life is still a puzzle. However, it is a chiral puzzle: living organisms use only one class of enantiomer(molecule that has a mirror image) in the construction of pro-teins and nucleic acids, a fundamental property of life knownas homochirality (see Avnir 2020, for a recent review). Ho-mochirality allows biopolymers to adopt stable helical struc-tures (Bada 1995; Nanda et al. 2007), a natural conformationfor biological macromolecules to adopt and one that plays akey role in fundamental biological processes (Totsingan et al.2012). While helical biopolymers have not been found, yet,in extraterrestrial environments, enantiomeric excesses (e.e.)of a few percent have been reported in the amino acids con-tent of some meteorites (Pizzarello 2006; Burton & Berger2018), but their relation to biological homochirality remainsa mystery.An important clue pertaining to the emergence of ho-mochirality comes from studies of low energy, spin-polarizedelectrons which can promote enantioselective chemistry(e.g., Dreiling & Gay 2014; Rosenberg 2019). In addition,the Chirality-Induced Spin Selectivity, CISS, e ff ect showsthat helical biopolymers, specifically double strand DNA and α -helical peptide, can act as spin filters for sub-eV electrons(Naaman & Waldeck 2012). This may be related to biolog-ical function and may play a role in the emergence of ho-mochirality.Cosmic rays, play a major role in generating mutation andpromoting evolution on Earth. They create pions in the upper atmosphere which decay into highly polarized muons whichcomprise the most prevalent high energy particles at groundlevel. Muons decay into electrons and positrons which arealso polarized. In Globus & Blandford (2020), henceforthGB20, it was proposed that this polarization could imposea small chiral preference on the development of one set ofbiological enantiomers over the other which could evolve tohomochirality over millions or even billions of generations.Some mechanisms for expressing this preference were dis-cussed, supplying a causal connection from a fundamentalphysics asymmetry to an equally fundamental asymmetry inbiology.There are three stages to this scheme: I. Understandingcosmic ray polarization in di ff erent environments, II. Esti-mating the chiral preference present in the mutation rate witha given irradiation and III. Translating this into an influenceon biological evolution. In GB20, the focus was on stage IIand some semi-classical models of the interaction betweencosmic rays and a chiral distribution of electrons in sim-ple biopolymers were discussed. A quantity called lodacity, L = ˆ µ · ˆ v , was introduced because it is the cosmic ray mag-netic moment, µ , not the spin, that mediates this interaction.The e ff ects were small; larger e ff ects may be exhibited in aquantum mechanical description. However it was also ar-gued that the e ff ects could have been su ffi cient to induce aenantioselective biological response and establish homochi-rality during stage III, especially if enantiomeric conflict ispresent. a r X i v : . [ a s t r o - ph . E P ] F e b In this paper, we focus on Stage I. We are most concernedwith cosmic rays on Earth today. However, there are manyquite di ff erent environments present within the solar systemand evidence for life could be found here too. Or not. Eitherway, it is of considerable interest to understand the evolu-tion of cosmic ray polarization under di ff erent circumstancesand so we also consider Mars, Titan, Venus, icy moons, andsmaller bodies as well as Earth.In Section 2, we explain the origin of the asymmetry incosmic radiation. In Section 3, we discuss the depolarizatione ff ects for muons and electrons in water, before presentingthe di ff erent environments we consider in Section 4, and thedoses in section 5. We discuss the biological implications inSection 6. ORIGIN OF POLARIZATION ASYMMETRY INSECONDARY COSMIC RADIATIONBefore discussing the results of detailed shower simula-tions, it is helpful to outline the development of the cosmicrays and their polarization from relativistic to the atomic en-ergies that may be of most interest biologically. The cos-mic rays observed in the solar system predominantly con-sist of hydrogen and ionized, typically stable nuclear iso-topes (see the Particle Data Group (2020) and Gaisser et al.(2016) for reviews). If the total flux of cosmic ray nuclei isconverted into a flux of nucleons, one finds that only 15%are neutrons and the rest protons. This charge asymmetryis conserved in the fluxes of secondary hadrons once pro-tons interact with matter, since on average more π + and K + mesons are produced in the projectile fragmentation region.Hence, particles that emerge from hadronic processes andreach a “fount” (which we define as a location where trans-biotic polymers can develop, to distinguish it from a “habi-tat” where living organisms are well-established) will carry anet positive charge whereas particles from electromagneticcascades have a charge ratio of one above the γ → e + e − threshold. At Earth, today, most particles at ground level aremuons, which are tertiary cosmic rays predominantly pro-duced through two-body decays of charged pions and kaons, e.g. π + → µ + ν µ . Since neutrinos are e ff ectively massless,for our purposes, and produced with left-handed helicity andin the pion’s rest frame its zero spin is conserved, the muonis produced fully polarized with the opposite helicity of theneutrino. These two factors, charge asymmetry and parity vi-olation of the weak interaction, imply that helicity state andcharge have to be treated separately, resulting in four muoncomponents µ + L , µ + R , µ − L and µ − R (Volkova & Zatsepin 1965;Dermer 1986; Lipari 1993), where L ( R ) label correspondto negative (positive) helicity. Following (Lipari 1993), theminimal energy fraction the muon receives in the pion’s restframe is r π = ( m µ / m π ) ≈ .
57, when it is emitted againstthe direction of movement, or 1 when it coincides with thepion’s direction. From this follows that the muon on averagereceives (cid:104) x (cid:105) = − r π / ≈ .
78 of the pion’s energy. The probability to for the helicity of muons when the pions arerelativistic is P R ( x ) = − r π (cid:18) − r π x (cid:19) , P L ( x ) = − r π (cid:18) r π x − r π (cid:19) , (1)which gives roughly 2 / (cid:104) x (cid:105) . For the heavier chargedkaons, 95% of muons are right-handed what can be verifiedby replacing m π with m K ± . Since we are only interested inthe magnetic moment polarization, ˆ µ · ˆ v , we denote by (cid:24)(cid:22) as the sum of doses deposited by muons with µ + L + µ − R (i.e.ˆ µ · ˆ v <
0) and (cid:24)(cid:23) as the sum of doses deposited by µ + R + µ − L (i.e. ˆ µ · ˆ v > L , whichwe define to be the average of ˆ µ · ˆ v and is always of the samesign (GB20). DEPOLARIZATION EFFECTS3.1.
Muons
Hadronic cascades produce muons which are subject toionization loss and decay into electrons and positrons. Witha vertical atmospheric grammage ∼ − and a scaleheight ∼
10 km, the surface muon particle flux is broad inenergy but peaks at a few GeV. Lower energy muons mostlylose energy and decay before reaching the ground. Fewerhigh energy muons are produced and they are subject to ex-tra radiative and pair losses.For illustration, we consider underwater fate of few GeVmuons and this elucidates what happens in other environ-ments. Muons lose energy in water though ionization losson a timescale t µ loss ≡ | dt / d ln T | ∼ T . ns; 0 . (cid:46) T G (cid:46) , ∼ T . ps; 0 . (cid:46) T M (cid:46) , ∼
60 fs; 0 . (cid:46) T k (cid:46) , (2)where the kinetic energy T G , M , k ≡ T / (1 GeV , , ∝ T and so they essentially stop onthe spot.We show in Appendix A that, for a particle that slowsthrough ionization loss, the muon scattering is relativelyunimportant. This implies that muons continue in the samedirection and that they remain polarized because their spinsare much less deflected in encounters than their veloci-ties. The muon lodacity will decrease according to L ∝ exp( − K λ ), where the “lethargy” λ = sech − ( v / c ) and the de-polarization factor K = .
037 for muons in water (see Eq. A4in Appendix A). This implies a lodacity reduction by a fac-tor of only ∼ .
83 as the muons cool to a kinetic energy Kinetic energy (GeV)10 E ( c m s ) Earth all ± polarized ± Kinetic energy (GeV)all e ± polarized e ± l o g ( h / m . w . e . ) Kinetic energy (GeV)10 E ( c m s ) Planetesimal all ± polarized ± Kinetic energy (GeV)all e ± polarized e ± l o g ( h / m . w . e . ) Figure 1 . Underground fluxes of muons and electrons on Earth (top) and on an icy moon without an atmosphere (bottom). The spectrum andcomposition of the primary cosmic rays are the one observed at Earth at the present epoch. In the left panels, the total µ + + µ − spectra atdi ff erent depths are figured by the solid lines; the polarized muons by dashed lines. The fraction of polarized muons is close to unity since theymostly come from pion decay. In the right panels, the total e + + e − spectra, i.e. those produced by all lepto-hadronic processes and also frombremsstrahlung pairs, are shown by solid lines at di ff erent depths. The polarized e + + e − from decay of muons are shown by dashed lines. T k ∼ Down to few keV, the depolarization is ignorable. Belowa few keV, there is a significant di ff erence between positiveand negative muons. Positive muons capture electrons withatomic cross sections (e.g. Kulhar 2004) when their speedsare comparable with those of valence electrons, T k ∼
10, toform muonium, Mu. This reduces the muon polarization byby a half (Nagamine 2003). There will be further scatteringsand exchanges, leading to further depolarization as the Muthermalizes.Negative muons, by contrast, continue to cool throughatomic interactions mostly through excitation as they are tooslow to ionize valence electrons. However, their deflectionsremain small and they retain their polarization until theyreach atomic energies, T k ∼ .
03. At this point, they formmuonic atoms and rapidly cascade into high energy ground The degree of depolarization due to ionization during slowing-down canalso be estimated using a formula given by Akylas & Vogel (1977); they findthat depolarization is negligible. states very close to the host nuclei before decaying. The cap-ture cross section is spin-dependent and may be sensitive tothe chirality M of the molecule (GB20). Modest depolar-ization will occur within the atom through spin-orbit interac-tion. For instance, negative muons stopped in carbon retainabout 15% of their initial polarization in the K-shell of spin-zero nuclei before they decay (Garwin et al. 1957). The de-excitation cascade to the ground state is accompanied by theemission of Auger electrons and circularly polarized X-rayswhich can also transfer some polarization.To summarize, muons retain almost complete polarizationuntil they form Mu or muonic atoms. They transport polar-ization e ffi ciently from high to low energies where they canhave a biological impact. This is not the case for the elec-trons, as we discuss in the next section.3.2. Electrons
In Fig. 1, we show the underground fluxes of muons andthe accompanying electrons as a function of depth (in meterwater equivalent), on Earth and on a moon without an atmo-sphere. While the muons are highly polarized, the electronpolarization is negligible. The basic reason is that electronsscatter faster than they cool (Appendix B). This implies thatpolarized electrons made in muon decays in the atmosphereare rapidly depolarized and do not reach the surface. Elec-trons produced in muon decays below the surface are alsounimportant.However, the muons cool by creating unpolarized, knock-on electrons, an unavoidable “entourage” of high energyelectrons that must accompany the muons. High-energymuons ( T >> GeV) that dominate at larger depths emit pho-tons through bremsstrahlung µ ± → µ ± + γ and initiate elec-tromagnetic sub-cascades through γ → e + + e − . These con-tribute to the “entourage” of unpolarized electrons, hence di-luting e ff ects induced by the high-energy muon polarization.A given biological molecule will evolve in the presence ofa spectrum of both particles. While we do not understandtheir separate e ff ects, a first step is to estimate their relativedensities. This is discussed more in Appendices B and C. FOUNTSNow we have explored the e ff ect of depolarization and di-lution, we will calculate the polarized radiation doses at vari-ous environments: Mars, Titan, Venus, and bodies without anatmosphere (moons, asteroids). We are interested in founts ,that we defined in the introduction as places where trans-biotic polymers can develop. The search for life in our so-lar system has been focused on finding water, one of the keyprerequisites for life as we know it, in the form of subsurfacelakes or oceans. In some environments, geologic and hydro-logic activity could have enabled physical processes neces-sary for the growth of the first helical biopolymers (see Cam-prub´ı et al. 2019, for a review).4.1. Worlds with a thin or negligible atmosphere
Mars is a frozen world punctuated by periods of hydro-logic activity (Cabrol 2018). Recently, a 20-km-wide lakeof liquid water has been detected underneath solid ice inthe Planum Australe region, at approximately 1.5 kilometerdepth (Orosei et al. 2018). Although there is still no evidencefor life on Mars, the red planet has once deployed environ-ments characteristics of wet-dry cycling, a process thoughtto have enabled the production of the first helical biopoly-mers (Ross & Deamer 2016); the Mars Exploration RoverSpirit reported 3.65-billion-year-old hot spring deposits, ap-proximatively of the same age as the Dresser hot springs onEarth that showed evidence for microbial life (Van Kranen-donk et al. 2017; Damer & Deamer 2020). If microbial lifestarted in similar hot springs, it is likely that after Mars’ ge-ological death and the loss of its atmosphere, microbial lifewould have only be able to survive underground, since Mar-tian subsurface seems to have the key ingredients to supportlife for hundreds of millions of years (Michalski et al. 2018;Checinska Siela ff & Smith 2019).Life on Mars could have originated when its atmosphere was much denser than today and comparable to the currentEarth’s atmosphere. In consequence, the radiation dose andspectrum penetrating to the surface are likely to be quite dif-ferent for Noachian and present-day Mars as discussed byLingam et al. (2018). Mars magnetic field could have beenlarger in the Noachian (Mittelholz et al. 2018) but this wouldnot a ff ect the muon polarization (GB20). As for the possible(intermittent) variations in the cosmic ray flux, we considerthem in Section 5.Evidence has accumulated that subsurface liquid regions(lakes or oceans) exist beneath the surface of the icy moonof Jupiter, Europa, and the icy moon of Saturn, Enceladus(Thomas et al. 2016). Mass spectrometers aboard the Cassinispacecraft found that the plumes of Enceladus contain water,salts, ammonia, carbon dioxide, and small and large organicmolecules, suggesting that Enceladus’ ocean may entertainan organic chemistry (Khawaja et al. 2019). In these environ-ments, the atmosphere is negligible or absent (as in asteroids,comets or meteroids). Like on Mars, we will see that in theseworlds where the atmosphere is absent, the spin-polarizationtransport is di ff erent than on Earth.4.2. Worlds with a very dense atmosphere
We also consider worlds with atmospheres denser than thepresent Earth. Titan, Enceladus and Europa are among themost geologically active worlds in our solar system but incontrast to Enceladus and Europa, Titan possess a very denseatmosphere. The essential chemical building blocks for lifeare present in Titan’s atmosphere (Yung et al. 1984). Titan isalso thought to have a subsurface ocean of water (Iess et al.2012) . For Venus, the habitable region lies in the upper at-mosphere (Morowitz & Sagan 1967). POLARIZED RADIATION DOSESThe current absorbed dose rates from cosmic radiation indi ff erent environments, are presented in Figure 2. It is aninteresting coincidence that for Earth, at ground, 85% of ion-izing cosmic radiation is due to the polarized muonic com-ponent that increases to almost 100% two meter water equiv-alent (m.w.e.) below the surface.In other bodies of the solar system, the presence or ab-sence of atmosphere leads to di ff erent results. For Mars(middle panel of Fig. 2), cosmic rays are only slightly at-tenuated by the thin Martian atmosphere and the polarizedmuons become the dominant source of cosmic radiation at ∼
10 m.w.e. with a dose of ∼ .
01 mGy / yr, much lowerthan the 0 . / yr on Earth. For planetesimals withoutan atmosphere or smaller bodies of the solar system (bot- It is possible that major impacts on a moon like Titan, Europa or Ence-ladus could expel large, icy fragments with liquid, life-containing interiors(or even surface oceans) into interplanetary space. Absent a heating source,the water would soon freeze, fossilizing its contents. It is surely of long-terminterest to locate, visit and examine such bodies (Dyson 1997). A b s o r b e d d o s e i n m G y / y r atmosphere X v = 1033.8 g/cm underground Earth P o l . f r a c . ground 10 Underground depth (m.w.e.)All charged particles Polarized muons Muons Muons 10 A b s o r b e d d o s e i n m G y / y r atmosphere X v = 18.4 g/cm underground Mars P o l . f r a c . ground 10 Underground depth (m.w.e.)10 A b s o r b e d d o s e i n m G y / y r Titan, X v = 12228.4 g/cm Venus, X v = 105490.7 g/cm A b s o r b e d d o s e i n m G y / y r No atmosphereprompt largesmall020406080 Altitude above ground (km)0.00.51.0 P o l . f r a c . Underground depth (m.w.e.)0.00.51.0 P o l . f r a c . Figure 2 . Top and Middle panels: Absorbed dose rate due to cosmic ray radiation at Earth and Mars. The black curves are obtained by summingthe ionization losses from all charged particles. Red curves ( (cid:24)(cid:22) ) are the sum of doses deposited by muons with µ + L + µ − R whereas the blue ( (cid:24)(cid:23) ) arethose with µ + R + µ − L . The purple curve is the total muon component. At Earth polarized muons constitute the dominant source of cosmic radiationat sea level and shallow underground depths. On Mars, polarized muons become the main source of cosmic radiation at ∼
10 meters waterequivalent (m.w.e). To estimate the underground fluxes in rock, the x-axis of the right-hand side panels can be multiplied with ρ rock / (g / cm ).Bottom left panel: Same calculation for the dense atmospheres of Titan and Venus. Bottom right panel: Same calculation for planetesimals,like Earth’s moon, or asteroids, protoplanets or comets, with a negligible atmosphere. “Large” signifies bodies with size much larger than thedepth for which up-going particles can be neglected. In “small” bodies, the radiation deposited is roughly isotropic and shown for the centerof a sphere with radius equal to the depth. The additional unpolarized, prompt muons, shown in orange, originate from three-body decaysof short-lived charmed and unflavored mesons and from electromagnetic pair production (see Figure 3 for more detail). Note that the dose isplotted logarithmically for clarity. A b s o r b e d d o s e i n m G y / y r E i n G e V Underground depth (m.w.e.)10 D o s e f r a c . All charged particlesUnpolarized & prompt ± e ± (incl. entourage)Polarized ± Figure 3 . Dilution of polarized radiation by electronic entourage atdeep undergrounds. This figure extends the case of a ”large” plan-etesimal from the bottom right panel of Figure 2. The depths ofseveral km.w.e. are more representative environments found in (forexample) Enceladus’ crust. At greater depths low energy muonshave stopped or decayed, and those which survive have increas-ingly higher (cid:104) E µ (cid:105) (cid:29) GeV, as illustrated in middle panel. High-energy muons lose energy predominantly by emitting high-energybremsstrahlung rather than through ionization. These gamma raysconvert into pairs that join the electronic “entourage”. tom right panel of Fig. 2), the polarized muons componentdominates at ∼
15 m.w.e. and with only one tenth of the in-tensity. In a dense, solid medium, the interaction length ismuch shorter than the decay length, leading predominantlyto (re-)interaction of unstable hadrons rather than their de-cay. This suppresses the production of high energy muonsand leads a swiftly decreasing polarized muon component asa function of depth.In Titan and Venus, polarized muons dominate the radia-tion at altitudes 40, 50 km. The surface irradiation, compara-ble to that below ∼
400 m of rock is negligible.Figure 3 demonstrates that the dose deposition in a largeicy body at km.w.e. depths is many orders of magnitude fee-bler compared to the scenarios that we study at a few m.w.e.The fraction of the polarised dose drops due to two e ff ects:1) kilometer depths are reached predominantly by the higher-energy, unpolarized, prompt muons (Fedynitch et al. 2019), i.e. from decays of unflavored mesons and heavy charmedhadrons, or from electromagnetic pair production; 2) the av-erage muon energy grows because low energy muons stopor decay at shallower depths. The stopping power of high- energy muons become increasingly radiative, which depositsenergy into the unpolarized electromagnetic entourage viabremsstrahlung.The terrestrial polarized muonic dose has a maximum of ∼ . / yr at the current epoch at ground level, but couldhave been larger during the Hadean period where a youngsun could have accelerated cosmic rays beyond ∼
10 GeV.If the cuto ff of stellar cosmic rays arises below that energy(Stellar proton events (SPEs) or a young T Tauri cosmic rayspectrum as modeled by Padovani et al. (2018), this wouldonly change the radiation dose by a factor ∼
2, as we showin Fig. 4. All environments harbor other natural sources ofradiation that may exceed the dose deposited by cosmic rays.About 3.5 billion years ago, life was likely exposed to a back-ground radiation field of ∼ / yr (Karam & Leslie 1999).However, more energetic transient events such a nearby su-pernova could increase the doses due to cosmic muons upto ∼
100 mGy / yr at ground level.We show in Fig. 4 the ab-sorbed dose at Earth 100yrs after the explosion of a nearbysupernova (Melott et al. 2017) compared to the actual doses.Since the muons retain their lodacity, higher radiation dosesfrom cosmic muons could have altered enantioselectively theevolutionary trajectories in shorter timescales due to highermutation rates. The closest supernova explosion over theage of the Earth was roughly ∼ −
10 pc away (e.g. El-lis & Schramm 1995). The prompt electromagnetic (UV, X-ray, γ -ray) radiation could have had serious biological e ff ectsbut would not be significantly chiral. By contrast, the pulseof freshly-accelerated cosmic rays is chiral and should havedelivered a fluence ∼ −
100 MJ m − over several thou-sand years, overwhelming the defenses presented by the solarwind, the magnetosphere and the atmosphere. For compari-son, this is ∼ − times the fluence associated with agiant, ”Carrington”-strength solar flare, albeit delivered overa much longer timescale. Activity associated with the nu-cleus of our Galaxy, including tidal disruption events (Pacettiet al. 2020), might also be biologically consequential. It istherefore possible, though far from assured, that life was ini-tiated, not terminated, by such cataclysmic events. DISCUSSIONSo far, scenarios for the emergence of biological homochi-rality involved two processes. The first one is the generationof an initial enantiomeric excess (e.e.) in biologically rele-vant but still simple chiral monomers, such as amino acidsor sugars. Various ways to achieve an initial e.e. have beenproposed: parity-violating energy di ff erences at the molec-ular level (Szab´o-Nagy & Keszthelyi 1999, and referencestherein), asymmetric destruction due to irradiation by betaparticles in radioactive decay (Vester et al. 1959), di ff erentialabsorption of circularly polarized light (D’hendecourt et al.2019), asymmetric adsorption on chiral mineral surfaces likeclays (Hazen et al. 2001). Because the resulting enantiomericexcesses are very small (only up to few percent, see review A b s o r b e d d o s e i n m G y / y r atmosphereall chargedpol. muons ground 10 Underground depth (m.w.e.)undergroundPresent flux at Earth (GSF model) T Tauri (Young Sun) Supernova @ 50pc after 100yr
Figure 4 . Absorbed dose at Earth for the presently observed cosmic ray flux according to the GSF model (Dembinski et al. 2017) as in Fig. 2, aphase in which the young Sun is modeled with a T Tauri star spectrum from Fig. 10 in Padovani et al. (2018), and for a nearby supernova 100years after explosion at 50pc (Melott et al. 2017). Despite the high proton densities assumed in the Taurosphere, the impact on the undergroundfluxes is negligible since most protons have energies below the particle production threshold and hence do not produce additional muons. In thenearby supernova scenario the upper edge represents case A from Melott et al. (2017) and the lower case B, respectively. Shown is the maximalfluxes expected 100 years after explosion that drop significantly within a few kyr. by Burton & Berger 2018), a second necessary process isthe amplification of the initial e.e. from low to high values.Enantioselective chemistry might be able to amplify the e.e.up to 99.5% (Soai et al. 2000) but the validity of such chem-ical processes in natural environments is still debated.In contrast, the scenario proposed by GB20 only involvesenantioselective mutagenesis, for which the timescale de-pends on the mutation rates of the first trans-biotic polymersand the magnitude of the bias. In order for the particularenantiomeric choice observed in living systems on Earth tobe causally related to the choice expressed by the weak inter-action and mediated by cosmic rays, we argued in GB20 thatthere needs to be a coupling between the lodacity, L , of thecosmic rays and the molecular chirality, M , of the molecularstructure. The mutation rate has to include a term propor-tional to the product of these two pseudoscalars. The pos-sibilities include an electromagnetic interaction in which thescalar product of the electric and magnetic dipoles or, equiv-alently E · B had a preferred sign. There is no good reasonfor this to occur in an isolated atom. However, it is just whatis expected in a segment of a helical molecule containing aconduction or a magnetization current. In this case < E · B > is negative (positive) for a right- (left-) handed helix, inde-pendent of the polarity.We conclude with two thoughts. Firstly, Earth is the onlybody in the solar system with an atmosphere just the rightdensity and scale height to ensure that muons dominate thecosmic radiation at ground level and for ∼
10 m . w . e . under-ground. It is thick enough to stop the other constituents, thinenough for the muons to survive. For worlds with thin or neg-ligible atmosphere, such as Mars, polarized muons are thedominant source of cosmic radiation at depths of 10 m.w.e.while for worlds with dense atmospheres, like Venus or Ti-tan, muons dominate below ∼
40 km altitude but the radiation doses are negligible at the surface. Now, muons are the oneconstituent of cosmic radiation that is created with and re-tains high polarization as it decelerates to speeds (cid:46) α c whenit can promote mutation of trans-biotic biopolymers. Mightthis be connected with the fact that Earth is the only bodyknown to harbor life? However, we know that Mars, Venus,and Earth have undergone significant evolution. Early in theSolar system’s history, Mars probably had an atmosphere of0.5-1 bar (Lingam et al. 2018) and Venus may have possessedan atmosphere of ∼ ff erent from those we experi-ence today. Therefore the special character of Earth’s atmo-sphere might be more relevant to life’s development than toits origin.The second thought is that the relative importance of themuons and their electronic entourage for mutation is not un-derstood. In GB20, ionization was taken as a proxy for mu-tation. If this is the case, then collisional ionization by elec-trons is probably more frequent than muon capture, dilut-ing the chiral e ff ect of the cosmic rays. Muons have ad-vantages of better depth-dose distributions. However, evenionization may be reparable and not so conducive to geneticalteration. There could be an appreciable chiral contributionto the Hamiltonian for the formation of a muonic atom orMu in a helical molecule. The fact that negative muons re-tain their polarization down to the energies where they canundergo a spin dependent capture, may also have an e ff ecton enantioselective mutagenesis. A single muonic atom de-cay will produce polarized X-rays and Michel electrons and Michel electrons have a well known energy distribution with a sharppeak at 52.83 MeV and an asymmetric angular distribution if the parent will surely fracture the molecule. Even the pickup of an elec-tron in Mu formation could be quite disruptive. Perhaps it ismuons, not electrons, that dominate mutation of trans-bioticbiopolymers, thereby increasing the chiral transfer and accel-erating evolution to homochirality.These matters and the prospect of performing experimentalinvestigations will be discussed in a future paper. ACKNOWLEDGEMENTSWe are thankful to David Deamer, Bruce Dunham,Akimichi Taketa, for helpful discussions and encourage-ments. The research of NG was supported by New YorkUniversity and the Simons Foundation. AF completed hiswork as JSPS International Research Fellow (JSPS KAK-ENHI Grant Number 19F19750).REFERENCES
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SCATTERING AND IONIZATION LOSSThe rate of loss of kinetic energy T by a cosmic ray in an environment where the molecular density is n , is dominated by theionization of electrons and can be approximated by dTdt = − π nm e v (cid:32) e π(cid:15) (cid:33) ln Λ (cid:88) i Z i , (A1)where v is the speed, the sum is over all the atomic numbers Z i of the nuclei in the molecule and Λ is the ratio of the maximumto the minimum impact parameter given roughly by 2 p / m e I , where p is the momentum and I is a mean ionization potential. Asthe energy loss is dominated by distant encounters, the energy fluctuations are small and the kinetic energy can be regarded asfunction of time.Nuclear recoils do not contribute to the stopping power but they dominate the scattering. Consider a cosmic ray with momentum p making an angle θ with the direction of its magnetic moment, ˆ µ which we can regard as fixed in space. The mean square rateof change of the transverse momentum by nuclear and electron collisions can be similarly expressed as (cid:42) d < ∆ p > dt (cid:43) = π nv (cid:32) e π(cid:15) (cid:33) ln Λ (cid:88) i Z i + (cid:88) i Z i . (A2)Now, the lodacity, L ≡ < cos θ > = < ˆ µ · p > / p , and if we average over azimuth in the scattering, we find that ∆ ln < L > = − < ∆ p > / p . Again the fluctuations will be small and can be ignored.It is helpful to introduce a quantity λ = m (cid:90) ∞ T dT (cid:48) p = sech − vc = ln (cid:18) pcT (cid:19) , (A3)which we call the “lethargy” for a cosmic ray with rest mass m . We can then combine these three equations to obtain d ln L d λ = − K = m e m + (cid:80) i Z i (cid:80) i Z i , (A4)and so, L = L exp( − K λ ) = L ( T / pc ) K , where K is the depolarization factor. For cosmic ray muons, electrons in water, K ∼ . , . B. KNOCK-ON ELECTRONS FROM COSMIC RAY MUONS UNDER WATERConsider, as a specific example, muons incident upon the surface of the ocean on Earth. We have argued that muons deceleratethrough ionization loss while retaining most of their initial lodacity. For the energies that dominate the particle flux at the surfaceof the earth, ∼ −
10 GeV, they stop at a depth of tens of meters before they decay. When they become non-relativistic, theirrange scales ∝ T . After they cool, they will be captured on atoms, ( µ − ), or capture an electron, ( µ + ). By contrast, electronsscatter before they decelerate. Polarized electrons created through muon decay are rapidly depolarized and are unimportantcompared with the knock-on electrons which will cool and thermalize on the spot by creating more knock-on electrons, formingan unpolarized “entourage”, accompanying the muons.The polarized muons and the unpolarized electrons will compete in creating mutations of trans-biotic molecules, diluting thechiral transfer. In order to quantify this e ff ect, we first consider a fixed number density high energy muons crossing a volumethat is large enough to allow the knock-on electrons to cool locally and small enough for the muons not to cool in traversingit. Introduce a conserved number current in energy space of high energy muons I µ , the number of particles cooling through akinetic energy T per unit volume and time. Let this be established at some moderately relativistic energy T CR ∼ S µ e ( T (cid:48) ; T ), at which a muon, with kinetic energy T , produces electrons with kinetic energy T (cid:48) < T lying in dT (cid:48) . Now, (cid:82) T dT (cid:48) T (cid:48) S µ e ( T (cid:48) ; T ) ∼ T / t µ loss ( T ) c.f. Eq. (1). Likewise we introduce the current of electrons in energyspace I e ( T ), with (cid:82) T dT (cid:48) T (cid:48) S e e ( T (cid:48) ; T ) ∼ T / t e loss ( T ). I e ( T ) satisfies dI e dT + (cid:90) T dT (cid:48) T (cid:48) [ S e e ( T ; T (cid:48) ) t e loss ( T (cid:48) ) I e ( T (cid:48) ) + S µ e ( T ; T (cid:48) ) t µ loss ( T (cid:48) ) I µ ] = . (B5)Next multiply Eq. B5 by T and integrate over dT . We find the approximate solution I e ( T ) T ∼ I µ T CR (B6)0In other words, the knock-on electron energy current in energy space at low energy is roughly the same as the energy currentfor the muons at high energy.In this paper we are most interested in the muon and electrons spectra which can then contribute to the biological evolutionof trans-biotic polymers and introduce a chiral bias. Many processes contribute to the full picture as we shall discuss elsewhere.For the moment, as an example, let us consider ∼
10 keV electrons which are known to induce mutation in DNA. These areproduced by muons with energy T (cid:38) ∼ ∼
10 keV as electrons, we can conclude that the number current of ∼
10 keV electrons is roughly3 GeV /
10 keV ∼ × times the conserved number current of muons. The implications of this for biological processes thendepend on combining these rates with the relevant cross-sections. C. COMPUTATION OF PARTICLE FLUXESWe compute di ff erential particle fluxes Φ ( E , X ), where E is the kinetic energy and X ( h ) = (cid:82) ∞ h d (cid:96) ρ ( (cid:96) ) is the slant depth in g / cm ,using the one-dimensional cascade equation solver MCE q (Fedynitch et al. 2015) that is optimized for calculations of inclusiveatmospheric lepton fluxes at high energies above a few GeV (Fedynitch et al. 2019). We have extend the kinetic energy rangedown to 10 MeV and for computations in di ff erent media, such as carbon-dioxide and water by using hadronic cross sectionsand particle production tables from the DPMJET-III-19.1 event generator (Roesler et al. 2001). Electromagnetic cross sectionshave been computed using the numerical routines from Meighen-Berger & Li (2019). Ionization losses (cid:104) d E / d X (cid:105) are based ontables from (Particle Data Group 2020; Berger et al. 2017) and are tracked for each charged particle whereas energy deposition bytertiary electromagnetic cascades from fast neutrons are neglected. MCE q adapts the notation and formulae for muon polarizationin the relativistic approximation from Lipari (1993) that we verified to be valid for E kin ,µ >
10 MeV.The flux of cosmic rays in the solar system is considered to be isotropic and represented by the Global Spline Fit (GSF)(Dembinski et al. 2017), which is an modern parameterization of cosmic ray fluxes at Earth between rigidity of a few GV and thehighest observed energies at Earth. The rigidity cuto ff due to the planetary magnetic field, the location within the solar systemand the Sun’s level of activity a ff ects the low energy cosmic rays below a few GV / nucleon. The omission of magnetic fields andsolar modulation is considered as a source of uncertainty of this calculation but is not expected to qualitatively change the result,in particular not at larger depths or underground.The absorbed dose rate D in Gy / s is calculated from the di ff erential fluxes Φ with the default units (GeV cm s sr) − using D ( X ) = π (cid:88) p (cid:90) θ max d cos θ (cid:90) d E p Φ p ( E p , X ) (cid:42) d E p d X (cid:43) ( E p ) , where θ max = max( π/ , π − arcsin ( R / ( R + h )), h the altitude above ground and R the radius of the planet. The index p iterates overparticle species. The expression for cos θ max takes into account that at higher altitudes, particle cascades can develop upwardsrelative to the horizon.The models of the planetary atmospheres are the Standard Atmosphere (1976) for Earth, derived from Huygens AtmosphericStructure Instrument measurements for Titan (Fulchignoni et al. 2005), from the Soviet VeGa-2 probe (Linkin et al. 1986; Lebon-nois & Schubert 2017) for Venus, and from a fit of piece-wise barometric formulae to Mars Global Surveyor data Undergroundfluxes are computed for a water target with the ionization loss computed computed with ESTAR and PSTAR (Berger et al. 2017).Within the precision of this calculation and depths < ff erence in stopping power compared to a dedicated calculationin standard rock is negligible for muons.In Figs. C1 and C2 MCE q is compared with FLUKA 4.0 (Battistoni et al. 2015) simulations in water for 1 TeV proton projec-tiles. The technical low energy cuto ff in MCE q is 10 MeV, which is su ffi cient to capture a good description for the total numberof muons (blue curve in Fig. C1 and right panel of Fig. C2). At tens of meters in depth, the di ff erences between codes becomemore pronounced since the particle cascade is increasingly a ff ected by multiple subsequent interactions that amplify di ff erencesin production and absorption cross sections between the models. To simulate water targets with MCE q it was necessary to includeionization losses for all charged particles and obtain inelastic cross sections for water from the hadronic interaction models. Nonethe less the agreement is remarkably good since MCE q has not yet been employed for simulations down to low energies andwater targets. For electrons, simulated using MCE q and electromagnetic cross sections from Meighen-Berger & Li (2019), theagreement is good except at around the cascade maximum. Since the majority of electromagnetic energy is of hadronic originthrough the π → γγ process, the hadronic model di ff erences between DPMJET-III-19.1 and PEANUT / FLUKA have impact. Anestimate for the fraction of electrons that are lost due to the 10 MeV cuto ff in MCE q can be made from the comparison between https://github.com/afedynitch/MCEq (Version 1.3) https://github.com/afedynitch/DPMJET https://fluka.cern Distance from impact in ice (m)10 N p a r t ( E > M e V ) Vertical 1 TeV proton shower in ice.
MCEq muonsMCEq electrons FLUKA muonsFLUKA electrons
Figure C1 . Particle number in a particle cascade initiated by a 1 TeV proton in ice. In MCEq the hadronic interaction model is DPMJET-III-19.1and PEANUT is used as default model in FLUKA 4.0. The full colored triangles represent results for a lower integration cuto ff at E kin , e ± | µ ± > ff value of 10 keV. The di ff erence between the faint and full markers can be used toestimate the fraction particles lost due to the lower grid boundary of 10 MeV in MCE q . Kinetic energy (GeV)10 E d N / d E e + + e Kinetic energy (GeV)10 E d N / d E + + markers: FLUKA 4.0, defaultcurves: MCEq 1.3, DPMJET-III-19.11 TeV proton in water 0.18 m0.46 m1.18 m4.80 m12.26 m31.29 m Figure C2 . Comparison of electron + positron (left) and muon ( µ + + µ − ) (right) spectra between MCE q and FLUKA using the same settingsas in Fig. C1. For muon fluxes, the origin of the di ff erences is mostly related to the hadronic interaction model DPMJET-III-19.1 in MCE q vs PEANUT in FLUKA. The hadronic model also a ff ects the electromagnetic cascades, but di ff erences at energies below 100 MeV may arisefrom incomplete partially the case For electromagnetic cascades the di ff erences are partly related to, For very low energy muons, MCEq tendsto predict lower fluxes than FLUKA. This is likely related to using DPMJET-III at energies close to particle production threshold, where it isknown to be incomplete. the pale and dark red triangles in Fig. C1. Since the gradient of the electromagnetic component is very steep after the maximum,the di ff erences between MCE qq