Eﬀects of Planetesimal-Scale Evaporation on Pb Isotopic Evolution and Timing of the Last U/Pb Fractionation

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These processes result in a series of elemental fractionation between U and Pb for their individual geochemical characteristic (Malavergne et al., 2007;Bouhifd et al., 2013;Albarède et al., 2015).The ratios of radiogenic Pb isotopes to the stable 204 Pb ( 206 Pb/ 204 Pb and 207 Pb/ 204 Pb) are elevated with the decay time and their accumulation rates vary as 238 U/ 204 Pb, termed as μ, changes (Connelly & Bizzarro, 2016).Notability, the Pb isotopic composition in the BSE samples generally plots to the right of the Earth isochron, which is referred to as "the first terrestrial Pb-isotope paradox" (Murphy et al., 2003;Hofmann, 2007;Connelly & Bizzarro, 2016).
The genesis of the Pb paradox is Pb loss from the Earth mantle and an increase of μ.The major part of Pb loss is attributed to the early Earth accretion (Connelly & Bizzarro, 2016).The lowest value of μ in the initial solar nebula is 0.27, according to the observation data of the solar photosphere (Anders & Grevesse, 1989).In the first few million years, the tiny gas and dust of the solar nebula condensed into small bodies; then they proliferated into small undifferentiated planetesimals (Norman & Mittlefehldt, 2002;Chambers, 2004).The formation of large planetesimals (<1000 km) and planet embryos (thousands of km) derives from high-energy collisions (Williams & Cieza, 2011;Elkins-Tanton, 2012).During these periods, Pb is constantly lost in the high-temperature evaporation events and μ increases, for Pb has a much lower 50% condensation temperature (50% T C ) than U (Lodders, 2003;Wood et al., 2019).Besides, Pb can enter the metal phase or dive into the metal core as the form of sulfide or iron alloy in the stage of core formation, for Pb is a chalcophile element and has iron affinity under high pressure (Hart & Gaetni, 2006;Wood & Halliday, 2010;Wood et al., 2010;Burton et al., 2012;Ballhaus et al., 2013).As a result, the μ value in the silicate part of terrestrial planets e.g., Earth increases episodically during the early accretion.
Some evidence suggests that volatility dominants the Pb loss rather than the iron affinity (Albarède et al., 2015;Connelly & Bizzarro, 2016).For example, the contents of chalcophile and moderately volatile elements (like Zn, Cu, Rb, Pb, Tl, etc.) are related to their bond energy strength (Albarède et al., 2015).Besides, the composition of siderophile elements strongly deviates from the volatility trend line compared with other chalcophile and moderately volatile elements, including Pb (Palme & O'Neill, 2014).The evaporation of the melting rocks caused by violent collisions occurs on planetesimals of a few to hundreds of kilometers size (Yo ung, 2017; Davies et al., 2020).It affects the composition of elements and their isotopes, e.g., for major elements (Mg and Si) (Hin et al., 2017;Yo ung et al., 2019) and chalcophile moderately volatile elements (Ge, Zn and In) (Young, 2017).
Two types of vapor loss are proposed.One is the impact-induced evaporation and the other is the direct outflow released from magma ocean or magma pool (Dauphas et al., 2015;Hin et al., 2017).
However, for large bodies, the high escape velocity needed makes it difficult for vapor to overcome gravity and diffuse into space.Therefore, vapor loss is only effective for objects with masses less than 0.2 M e (Earth mass) (Hin et al., 2017;Benedift et al., 2020).A cluster of planetesimals that experience different times of collisions and multiple thermal evaporation events could have diverse volatile composition.The terrestrial planets like Earth and Mars inherit the characteristic of planetesimals with varying degrees of volatile depletion during numerous fractionation processes (Sossi et al., 2019).
Pb is a moderately volatile element and it has a low 50% T C in the form of iron alloy and sulfide (PbS) (Lodders, 2003;Wood et al., 2019).Pb is easily volatilized as sulfides and lost with silicate vapor under high-temperature conditions (Wood & Wade, 2013;Sossi et al., 2019).The planetesimal-scale evaporation driven by impacts causes a huge amount of Pb loss (Norris & Wood, 2017).As a result, Pb is comparatively depleted in the bulk silicate Earth relative to CI chondrites.
The content of Pb in BSE is only 0.07 times of that in CI chondrites (Palme & O'Neill, 2014).The partition of Pb in the silicate vapor loss is decided by the temperature and oxygen fugacity of silicate melt and it could be extremely high under the formation condition of magma ocean (Elkins-Tanton, 2012;Norris & Wood, 2017).As a result, the planetesimal-scale evaporation of Pb is thought to be a critical factor for the increase of μ and the excess of radiogenic Pb isotope (Connelly & Bizzarro, 2016).
It is complex to obtain the cumulative Pb loss on the accreted planets.First, we need a model to simulate the collisional process between planetesimals.The melting degree produced by each collision is estimated with the collisional parameters.Subsequently, we need to calculate the fraction of Pb loss under different temperatures and redox states of the melting pools.The purpose of this study is to find out how the Pb loss caused by planetesimal-scale evaporation in the early accretion contributes to the variation of U/Pb ratio and Pb isotopic composition of the bulk silicate Earth.

Numerical Methods
The terrestrial planets form from the collisions and merge between planetesimals and planet embryos.This process can be restored by the N-body collisional accretion simulation (Chambers, 1999;Hansen, 2009;Fischer & Ciesla, 2014;Carter et al., 2015;Fang & Deng, 2020).We use the Mercury6 package (Chambers, 1999) and the same initial conditions as Hansen (2009), setting a uniform areal density within a narrow annulus from 0.7 to 1 AU.The annulus is sampled with 400 planetesimals of the equal mass, 0.005 M e (that is 0.005 times the Earth mass).Through N-body simulation, three or four large planet bodies can be formed in the inner planetary disk, which are prototypes of terrestrial planets (Fang & Deng, 2020).N-body simulations roughly restore the evolutional history of these analogues within the first 200 million years (Myr).The Earth analogue completes mass growth within 50-160 Myr from the initial planetesimal of 0.005 M e according to 100 groups of simulations.The first 20% mass accretion of the Earth, from 0.005 M e to 0.2 M e , is completed within 3 Myr.The output of each simulation gives the times of collisions, the masses of impactors and targets, and the velocities and angles of collisions.Based on these parameters, the melt volume and depth induced by a collision can be calculated, as well as the vapor loss (de Vries et al., 2016;Hin et al., 2017).
The collisions between planetesimals or collisions between planetesimal and planet embryo produce shock melting on the targets.The melting volume and depth induced by the high-energy impact are proportional to the shock energy of projectiles (Barr & Citron, 2011).The shock energy depends on the mass and velocity of the projectile, the impact angle, and the composition of the target which determines the internal energy generated under impact pressure (Barr & Citron, 2011;Abramov et al., 2012;de Vries et al., 2016).Based on the dimensional analysis model of Bjorkman and Holsapple (1987), the melt volume is expressed as V melt = 0.22E m −0.85 (ρ p /ρ t )D p 3 v 1.7 sin 1.3 θ, where ρ p and ρ t represent the density of the projectile and target, respectively.D p is the diameter of the projectile.E m represents the special energy of melting as the Rankine-Hugoniot state (Bjorkman & Holsapple, 1987) and it changes with temperature.The energy required for melting decreases when the temperature of the target increases.This formula is widely used in previous studies (Abramov et al., 2012;de Vries et al., 2016;Davies et al., 2020).The result of the N-body simulation provides the size of the projectile, impact velocity, and angle.Each simulation assumes an identical density for the projectile and target (3000 kg/m 3 ) so that the mass and volume of collisional melting can be calculated via the above formula.Assuming the initial melt is roughly spherical, the average melting radius is R m = (3V m /4π) 1/3 (Abramov et al., 2012).From the above formula of collisional melting, the generated melt volume has a positive correlation with the diameter of the projectile and collisional velocity (Abramov et al., 2012;de Vries et al., 2016).When the collisional angle is closer to vertical, more collisional energy is produced by the impact.The statistics of 100 sets of simulations indicate that a single collision generally produces silicate melting with less than 60% mass of the target during the first 20% accretion (Figure 1).The melting depth ranges from 500 km to 3000 km.In the early stage of the Earth accretion, frequent but small-scale collisions produce minor melt.While the melting intensifies with time and accreting mass (Figure 1).The increase of melting mass results in a huge amount of silicate vapor loss with volatiles escaping into space.

The Variation of the μ Value
The ratio of U/Pb is expressed as μ ( 238 U/ 204 Pb).The value of μ is controlled by the fractionation of U and Pb in the geological process.U and Pb have very different geochemical properties.U is a refractory lithophile element and it easily concentrates in the initial planetesimals.Pb is a moderately volatile and chalcophile element.It volatilizes during the accretion process or possibly enters the metal core as sulfides during the core formation.In these processes, the μ value of the silicate part continuously increases.The geological bodies with larger μ value further concentrate more radiogenic Pb isotopes.

The μ values in the planetary bodies
As shown in Table 1, the µ value of CI chondrites is around 0.2-0.22 (Palme & O'Neill, 2014;Carlson et a., 2015) and it is the minimum value among planetary bodies.Because the CI chondrites have a similar elemental distribution with the initial solar nebula, its µ value can primely represent the µ value of the earliest undifferentiated bodies (Scott, 2007;Albarède, 2009).The µ value of the solar photosphere is 0.27 close to the initial value (Anders & Grevesse, 1989).In the accreted systems, the µ value is higher than 0.22.For example, other carbonaceous chondrites show µ values in the range of 0.30-0.62(Newsom et al., 1995).The previous research measured the µ value of the L3 ordinary chondrite to be 1.8 (Connelly & Bizzarro, 2016).The protoplanets which experience multi-stage U/Pb fractionation show a much higher µ value than the undifferentiated bodies.By measuring different types of Martian meteorites, the range of the Martian mantle's µ value is from 1.5 to 5 with a medium value of 3 (Bouvier et al., 2005;Gaffney et al., 2007;Bouvier et al., 2009;Yoshizaki & McDonough, 2020).Martian mantle is less depleted in the volatiles and it shows a lower µ value than the Earth and Moon (Allègre et al., 1995;Premo et al., 1999).The measured µ value of modern bulk silicate Earth is in the range of 8-10 with an average value of 9 (Allègre et al., 1995;McDonough & Sun, 1995;Palme & O'Neill, 2014).Therefore, samples of modern BSE have generally enriched in radiogenic Pb isotopes and their μ value is much higher than that of the original material in the Solar System (Galimov, 2011(Galimov, , 2019)).
Table 1 The μ value in planetary reservoirs Planetary reservoirs µ( 238 U/ 204 Pb) References Solar Photosphere 0.27 (Anders & Grevesse, 1989) CI chondrites 0.22 (Carlson et al., 2015) CM chondrites 0.33 (Newsom et al., 1995) CO chondrites 0.30 (Newsom et al., 1995) CV chondrites 0.62 (Newsom et al., 1995) L3 ordinary chondrite 1.8 (Connelly & Bizzarro, 2016) Bulk silicate Mars (BSM) 1.5-5 (Gaffney et al., 2007) Bulk silicate Earth (BSE) 8-10 (McDonough & Sun, 1995) 3.2.The upper limit of the cumulative Pb loss The amount of Pb loss driven by collisional evaporation on planetesimals is decided by the collisional melting degree as well as the partition of Pb between the gas phase and melt phase.The former can be calculated from the results of the N-body simulations.The distribution of volatile elements in melt and vapor is controlled by temperature, oxygen fugacity, and volatile species (Wood & Wade, 2013;Norris & Wood, 2017).Pb is moderately volatile and easily evaporate as sulfides (the 50% T C of PbS is 495 K) (Wood et al., 2019).In the silicate melt, Pb can also volatilize in the form of PbO under high temperatures (Wood & Wade, 2013;Sossi et al., 2019).When the temperature rises above 1400℃ and the logarithm of the oxygen fugacity (logfO 2 ) is below -10, almost all Pb in silicate melt volatilizes as PbO or Pb (Sossi et al., 2019).
Pb is more concentrated in sulfides compared with silicate minerals (D sulfide/silicate mineral =5-2000 and D sulfide/peridotite =114.4)(Burton et al., 2012).The initial content of sulfur before accretion is around 5.35% according to CI chondrite, much higher than that of modern BSE (0.02%) (Palme & O'Neill, 2014).Therefore, the total Pb content in the sulfide is several times that in the silicate at the beginning of accretion and it decreases as the sulfur is lost.We assume that the sulfur content decreases linearly with accreting mass.
Assuming that all Pb in the melting pool volatilizes as PbS, PbO, and Pb and no Pb enter the core, we obtain the largest degree of Pb loss (T>1400℃ and logfO 2 <-10).Figure 2

Increase of μ in the accreting planetesimal
The degree of increase of μ is directly related to the amount of Pb loss with a constant U content in the multi-stage evaporation event.Under the extreme condition of the maximal Pb loss, the value of μ after the nth evaporation is expressed as When the magma ocean is relatively oxidized, the degree of Pb evaporation is limited by the temperature and oxygen fugacity (Wood & Wade, 2013;Norris & Wood, 2017).According to the Hertz-Knudsen-Langmuir (HKL) theory, researchers obtain the volatility factors of volatile species like Pb (Sossi et al., 2019).Pb directly evaporates to PbO(g) at high oxygen fugacity and decomposes to Pb(g) at low oxygen fugacity.The two reactions generally coexist.The partial pressure of PbO(g) and Pb(g) under different oxygen fugacity and temperature determines the residual fraction of PbO in the silicate melt: f PbO (melt) = exp(−(K * (PbO) + K * (Pb)/O 2 1/2 ) 3/ rρ (M/2πRT) 1/2 (t − t 0 )) (Sossi et al., 2019).The results of f PbO (melt) are available under different oxygen fugacity at 1300℃ and 1400℃ respectively from the experimental data in Sossi et al. (2019).
In this case, the value of μ after the nth evaporation can be expressed as µ n = µ n−1 /(R PbO * f PbO (melt) * F n + (1 − F n )).R PbO is the proportion of Pb in silicate and this value increases as sulfur content decreasing.
The planetesimal-scale Pb loss and μ increment happen in the first 3 Myr (the first 20% accretion).The earliest condensation in the solar nebula produces Moon-size small bodies within 1 Myr (Chamber, 2004;Johansen et al., 2007) and μ starts to grow in this stage from 0.22.The initial μ value of the undifferentiated planetesimals is larger than 0.22.Each collisional melting and evaporation are accompanied by a raising of μ value.We use an initial μ value in the range of 0.22-0.6 and find out that the μ value rises up to 1.5-5 in the first 20% of Earth accretion (Figure 3).
In Figure 3, the temperature and oxygen fugacity have effects on the evolution of μ value.When T>1400℃ and logfO 2 <-10, Pb in the melt is totally lost and the cumulative μ value could increase to the highest level (Sossi et al., 2019).A decline of temperature or an increase of oxygen fugacity can both bring down the Pb loss fraction in the silicate melt, resulting in a lower cumulative μ value (Figure 3).As a result, a high-T and reduced condition contribute to a high μ value in the silicate Earth.There is at least 93% Pb loss from the proto-Earth according to the Pb content of CI chondrites and the primitive mantle in Palme and O'Neill (2014).However, in the planetesimal-scale evaporation model, the average fraction of Pb loss is around 84% and it cannot reach 93% in most cases (Figure 4).Meanwhile, the early evaporation hardly raises the μ value to the value of modern BSE which is around 8-10 ( Palme & O'Neill, 2014).Other events are proposed to explain the subsequent Pb loss, for example, Pb entering into the core in the stage of core formation, Pb evaporation in the last giant impact, or Pb missing with the iron sulfide melts (Hart & Gaetni, 2006;Wood & Halliday;2010;Ballhaus et al., 2013;Savage et al., 2015;Connelly & Bizzarro, 2016;Maltese & Mezger, 2020).
Proto-Mars experiences rapid accretion and completes its core formation within the first 30 Myr (Elkins-Tanton, 2012).Mars is considered to be an original planet embryo that has not suffered a giant fractionation event (Elkins-Tanton, 2012).The μ value of the Martian mantle is in the range of 1.8-5 (Borg et al.2005;Gaffney et al., 2007), which is generally consistent with the post-evaporation μ value of the proto-Earth.This suggests that the subsequent μ increase is probably related to a late U/Pb fractionation event (Figure 5), e.g., the Moon-forming event (Canup & Asphaug, 2001;Deng et al., 2019).We estimate the effect of such a large-scale fractionation on the final μ value and Pb isotopic composition in the following sections.

The Necessity of a Late Fractionation from the Rb-Sr System
Rubidium (Rb) is a volatile lithophile element.The 50% T C of Rb as feldspar phase is about 750-800 K (Wood et al., 2019).Rb is easily lost in the high-T evaporation events and is extremely depleted in the Moon (Galimov, 2011(Galimov, , 2019)).The early evaporation driven by collisional melting results in Rb loss and Pb loss at the same time.Therefore, the 87 Rb/ 86 Sr ratio, expressed as κ, decreases as μ increases.A previous study shows a generally anti-correlation between the κ ( 87 Rb/ 86 Sr) and μ ( 238 U/ 204 Pb) from samples of Earth, Martian, and lunar mantle because of the strong volatility of Rb and Pb (Gaffney et al., 2007).(Galimov, 2011).The additional increase of μ is caused by a late U/Pb fractionation.
Rb is less volatile than Pb and the volatility factor of Rb in the silicate melt is less than that of Pb at the same temperature and oxygen fugacity (Sossi et al., 2019).The residual fractions in silicate melt for Pb and Rb satisfy (Sossi et al., 2019),f Rb (melt)/f Pb (melt) = exp(K * (Rb)/O 2 1/4 M Rb 1/2 )/ exp((K * (PbO) + K * (Pb)/O 2 1/2 )M Pb 1/2 ), where K * is the modified equilibrium constant for volatile species.We find out that the ratio of Rb and Pb residual fraction (f Rb (melt)/f Pb (melt)) becomes large with an increasing oxygen fugacity using parameters from Sossi et al., (2019).The assumed temperature is 1400 ℃ and logfO 2 is -8 for melting pools.Under this condition, the cumulative fraction of Pb loss within the first 20% accretion is above 80%, meanwhile, the fraction of Rb is around 74%.Based on the content of Rb and Pb in the silicate Earth relative to CI chondrites, the loss of Pb (93%) is higher than that of Rb (74%) according to the measurement in Palme and O'Neill (2014).
We further consider the evolution of κ and μ (Figure 5).The κ value of CI chondrites (~0.92) can be used as the initial value of the solar nebula (Palme & O'Neill, 2014) and the initial μ is set to be 0.22.The κ value decreases during the condensation of the nebular dust, because of the lower condensation temperature of Rb relative to Sr. Subsequently, due to the Rb loss driven by the planetesimal-scale evaporation, the κ value continues to decline to the current value.The estimated κ of the modern BSE is 0.092 (Galimov, 2011;Carlson et al., 2015).The pre-evaporation κ value is constrained to be 0.35 by 74% Rb loss in simulations assumed a constant Sr content.However, the increasing μ is inversely related to the decreasing κ.It increases to 4-5 from the pre-evaporation value of 0.65.The final μ is expected to reach 8-10 for 93.7% Pb loss after the last large-scale U/Pb fractionation (Figure 5).

The Pb isotopic composition in BSE
The two U-Pb decay systems are 235 U-207 Pb (λ 1 =9.8485×10 -10 year -1 ) and 238 U-206 Pb (λ 2 =1.55125×10 -10 year -1 ).The formulas of isochrons for two systems are expressed in terms of μ: shows that the slope of 207 Pb-206 Pb isochron is not associated with the μ value and it is positively correlated with time.
The Pb isotopic composition in the bulk silicate Earth mainly derives from Pb isotope data of oceanic basalts in the PetDb Database (www.earthchem.org/petdb).The proportion of MORB and OIB in the oceanic basalts is about 93.75% and 6.25% (Crisp, 1984;Gale et al., 2013).The data of global MORB is from Stracke et al. (2005) in the PetDb database (www.earthchem.org/petdb), and that of OIB is from Tuvalu, Hawaii, and Baffin (Finlayson et al., 2018;DeFelice et al., 2019;Willhite et al., 2019).Generally, the OIB values are more scattered and the average value is higher than the MORB value (Figure 6).isochron of the Earth (Figure 6), which is known as the first terrestrial Pb paradox.The 4.567 Ga is regarded as the age of the Solar System by U-Pb dating of the oldest solids, calcium aluminum inclusions (CAIs) (Amelin et al., 2010). is set to be 0.22 and increase to 1.51 within 3 Myr.In the first case, the μ value after the fractionation is about 9.26 and the proper time of fractionation is about 164 Myr.The evolved 206 Pb/ 204 Pb and composition has a slight deviation from the current value and its μ value increases to 9.04 at 114 Myr (Figure 6).Therefore, this model can reasonably explain the Pb paradox.

Constraints on the Time of the Last U/Pb Fractionation Event
We show in Section 3.4 that the planetesimal-scale Pb evaporation cannot satisfy the total Pb loss and μ value of the modern BSE.Further Pb loss is probably caused by the multi-stage core segregation or a late giant fractionation event.The two-stage model driven by a late U/Pb fractionation can effectively explain the deviation of Pb isotopic composition in the silicate Earth to the right of the Earth isochron (Savage et al., 2015;Connelly & Bizzarro, 2016), also seen in Figure 6.However, the multi-stage core segregation makes little contribution to this deviation.A late global U/Pb fractionation event could be a Moon-forming giant impact or segregation of the iron sulfide melting termed as "Hadean matte" during the late accretion of the Earth (Savage et al., 2015;Connelly & Bizzarro, 2016;Rubie et al., 2016).However, the oldest age of lunar samples from the U-Pb dating of zircon is 4.417Ga (Nemchin et al., 2009), which is a lower limit of the solidification timing of the lunar magma ocean.Therefore, the last U/Pb fractionation obtained in our model occurs after the Moon formation.The segregation of Hadean matte in the late accretion (later than the Moon-forming giant impact and before 3.8 Ga) drives the second extraction of the chalcophile elements, e.g., Pb into the metal core or to the core-mantle boundary (O'Neill et al., 1991).This causes the last global U/Pb fractionation, resulting in the final increase of μ.In our model, the timing of the Hadean matte event is limited to 165 Myr with a pre-fractionation μ of 1.51 or to 240 Myr with a pre-fractionation μ of 4.12.55%-84% of Pb is removed from the bulk silicate Earth in this process.

Conclusions
The planetesimal-scaled evaporation induced by the collisional melting has a critical effect on the content of the moderately volatile element, e.g., Pb.It is the main factor of the Pb loss in BSE rather than the hidden Pb reservoir and the separation of the core.The episodic increment of U/Pb ratio (μ) causes a multi-stage evolution of U-Pb isotopic composition within the first 3 million years.
Besides, a further two-stage model driven by a late large-scale U/Pb fractionation is required for the additional increase of μ and the excess of the radiogenic Pb isotopes.
1.The collisions between planetesimals produce magma pools or magma oceans on the targets.For targets smaller than 0.2 M e , volatiles e.g., Pb can easily dissipate into space at the surface of magma pools.The N-body simulations provide the collisional parameters to calculate the volume and depth of the melt and the melting fraction.As a result, the melting mass generally increases with the accreting mass of the target (within 0.2 M e ) as well as the Pb evaporation.
2. In the first 20% accretion of proto-Earth, the cumulative Pb loss is 80%-90% to the maximum.Pb mostly evaporates as sulfides (PbS) and evaporates from the silicate melt as PbO(g) and Pb(g).The fraction of Pb loss as sulfides decreases with sulfur loss.Pb loss fraction in the silicate melt is controlled by the temperature and oxygen fugacity.High-T and reduced condition promotes the huge amount of Pb evaporation.
3. 238 U/ 204 Pb referred to μ is inversely correlated with Pb loss.When the initial μ value of planetesimals is set to be 0.22-0.6, the μ value of the proto-Earth can increase to 1.5-4 in the multi-stage accretion within 3 Myr.This is not consistent with the measured value of modern BSE (8-10).Therefore, a subsequent Pb loss in a late U/Pb fractionation event is required.
4. The values of 87 Rb/ 86 Sr (κ) and 238 U/ 204 Pb (μ) in various geological bodies have a generally inverse correlation, which confirms the critical effect of the early Pb and Rb evaporation events.
Because Pb is more volatile than Rb and Pb shows sulfur affinity or iron affinity, the increment of the μ (0.22-9.26) is greater than the decrement of the κ (0.92-0.09).
5. For the U-Pb decay system, a late large-scale U/Pb fractionation or a huge increase of μ can produce an excess of the radiogenic Pb isotopes.In the traditional model without the early evaporation, a fractionation at 140 Myr resulting in an average μ value of 9.26 can effectively explain "the first Pb paradox".When considering the early Pb evaporation, the proper timing of fractionation is postponed to 165 Myr for μ from 0.22 and to 240 Myr for μ from 0.6.
6.Although the age range (165-240 Myr) of our model satisfies a group of young ages of Moon formation, it exceeds the lower limit for the solidification time of the lunar magma ocean (4.417 Ga).
Therefore, the last U/Pb fractionation possibly represents the "Hadean matte" event after the Moon-forming giant impact.In conclusion, global segregation of iron sulfide melt carrying 55%-84% Pb into the core at 165-240 Myr, which causes an excess of radiogenic Pb isotopes in BSE.The current Pb loss fraction is estimated to be 0.93 (Palme & O'Neill, 2014).The accreted body like Mars shows κ in the range of 0.12-0.26and μ in the range of 1.5-5 (Gaffney et al., 2007;Yoshizaki & McDonough, 2020).The final κ and μ values can approximately fall in the interval of the observed values of the present Earth mantle (κ≈0.09;μ=8-10) (Galimov, 2011).The additional increase of μ is caused by a late U/Pb fractionation.Tables Table 1 The μ value in planetary reservoirs

Figure 1 .
Figure 1.The average melting mass (M melt ) driven by multiple collisions evolves with the accreted Figure 2 decreases.

Figure 2 .
Figure 2. Pb loss fraction for each impact (bars) and the cumulative Pb loss fraction (curves) in one

Figure 3 .Figure 4 .
Figure 3.The accreted mass with time and the temporal evolution of the μ value during the first 20%

Figure 7 .
Figure 7. Constraints from the 238 U-206 Pb and 235 U-207 Pb systems on the μ value and the time of the

Figure 2 .
Figure 2. Pb loss fraction for each impact (bars) and the cumulative Pb loss fraction (curves) in one

Figure 3 .
Figure 3.The accreted mass with time and the tempal evolution of the μ value during the first 20%

Figure 4 .
Figure 4.The relation of the μ value before the last U/Pb fractionation to the cumulative Pb loss