Icarus 266 (2016) 189–203

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Excavation and melting of the Hadean continental crust by Late Heavy
Bombardment
Yuhito Shibaike a,⇑, Takanori Sasaki b, Shigeru Ida ca

Department of Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japanc
Earth-Life Science Institute, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japanba r t i c l e

i n f oArticle history:
Received 2 April 2015
Revised 26 October 2015
Accepted 27 October 2015
Available online 10 November 2015
Keywords:
Earth
Impact processes
Prebiotic environments
Asteroidsa b s t r a c t
No Hadean rocks have ever been found on Earth’s surface except for zircons—evidence of continental
crust, suggesting that Hadean continental crust existed but later disappeared. One hypothesis for the disappearance of the continental crust is excavation/melting by the Late Heavy Bombardment (LHB), a concentration of impacts in the last phase of the Hadean eon. In this paper, we calculate the effects of LHB on
Hadean continental crust in order to investigate this hypothesis. Approximating the size–frequency distribution of the impacts by a power-law scaling with an exponent a as a parameter, we have derived
semi-analytical expressions for the effects of LHB impacts. We calculated the total excavation/melting
volume and area affected by the LHB from two constraints of LHB on the Moon, the size of the largest
basin during LHB, and the density of craters larger than 20 km. We also investigated the effects of thevalue of a. Our results show that LHB does not excavate/melt all of Hadean continental crust directly,
but over 70% of the Earth’s surface area can be covered by subsequent melts in a broad range of a. If there
have been no overturns of the continental crust until today, LHB could be responsible for the absence of
Hadean rocks because most of Hadean continental crust is not be exposed on the Earth’s surface in thiscase.
Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction

1.1. Late Heavy Bombardment

Hadean rocks have not been found on Earth until today, and theage of the oldest rock is about 4.0 Ga. No continental crust may
have existed on the Hadean Earth. However, some zircons thatformed during the Hadean eon were found in Jack Hills sedimentary rocks (Wilde et al., 2001). Since zircons are formed by igneous
activity at the same time as granite, this discovery suggests thatthe Hadean continental crust existed but later disappeared. The
disappearance of the continental crust could be accounted for bygeological activity such as reworking or plate tectonics, anderosion of the crusts (e.g., Kawai et al., 2009). Another possibility
is excavation/melting of the continental crust by the Late Heavy
Bombardment (LHB), a concentration of impacts considered tohave existed in the last phase of the Hadean eon. In this paper,
we calculate the effects of LHB on Hadean continental crust inorder to investigate this hypothesis.

A classic scenario of LHB has been based on the fact that radiometric dates of lunar basins’ impact melts were concentrated at
3.9 Ga (Tera et al., 1974). In this model, about 15 lunar basins are
considered to be formed between 3.9 and 3.8 Ga (e.g., Ryder,2002). Cohen et al. (2005) argued from Ar-geochronology of lunar
meteorites that the concentration at 3.9 Ga is not required andHartmann (2003) claimed that the decrease of lunar crater density
is just the tail of the planetesimal accretion. The most recent model
proposed a sawtooth-like timeline of impact flux (Morbidelli et al.,2012). If LHB was caused by a disturbance in the ancient main belt
asteroids, the impact flux decreased exponentially after the onsetof LHB (Bottke et al., 2012). This exponential curve fits the lunar
crater density curve very well if the disturbance occurred at
4.1 Ga (Morbidelli et al., 2012). It is adjusted in order to explain
the accreted mass of highly siderophile elements (HSE) of theMoon after the Moon formation giant impact. Gomes et al.
(2005) argued that the cause of LHB was the migration of Jupiterand other giant planets predicted in the Nice model. The migrationof the giant planets moved some resonances and scattered themain belt asteroids and outer comets.

⇑ Corresponding author.
E-mail address: shibaike.y@geo.titech.ac.jp (Y. Shibaike).
http://dx.doi.org/10.1016/j.icarus.2015.10.023
0019-1035/Ó 2015 Elsevier Inc. All rights reserved.

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Y. Shibaike et al. / Icarus 266 (2016) 189–203

1.2. Previous works

Nsfd ¼

Z 1

Am0a dm

0m

The estimate for coverage of the Earth’s surface by impact craters depends on the impactors’ size–frequency distribution (SFD).
The estimates with unconstrained SFD predicted diverse results(e.g., Frey, 1977, 1980; Grieve, 1980; Ryder et al., 2000). Furthermore, stochastic one or two huge impacts could give considerable
effects (Zahnle and Sleep, 1997).
Recently, using a more constrained SFD corresponding to thecurrent main belt asteroids with the total LHB mass of 2  1023 g,
Abramov et al. (2013) showed that LHB may not have melted allof Hadean continental crust. They computed 3D temperature distributions of the crust using an analytical shock-heating modelwith effects of impact melt generation, uplift, and ejecta heating.
The result is that 1.5–2.5% of the upper 20 km of the crust wasmelted during LHB, and only 0.3–1.5% was melted through LHB
period. They also indicated that 5–10% of the Earth’s surface area
was covered by over 1 km depth of impact melt sheets, and theentire surface was covered by impact ejecta close to 1 km deep.
On the other hand, Marchi et al. (2014) suggested that Hadean
impacts could explain the absence of early terrestrial rocks basedon the sawtooth-like timeline, with the current main belt asteroids’ SFD. The key point of Marchi et al. (2014) is the ‘‘stochastic”
nature of LHB. They showed that the melting volume deeply
depended on whether very large impactors hit Earth or not.
Another key point is the effect of ‘‘impact-induced decompression”
and subsequent adiabatic melting of rising material in the mantle,
which increases the total melting volume. They claimed that theseflood melts from under the crust emerge when the impactors’
diameters were greater than 100 km. The melts flowed on the
Earth’s surface like spherical caps. The melt sheets’ diameters wereabout 20–30 times as large as that of the impactors, for an assumed
melt thickness of 3 km. The result is that 70–100% of the Earth’s
surface area was covered by the melt sheets since 4.15 Ga, and400–600% was covered during the period 4.5–4.15 Ga. This valueof the total melting area on the Earth’s surface is about ten times
as large as that of Abramov et al. (2013).
1.3. Aims of this work

¼

A

a1

m1a ;

ð2Þ

when a > 1 (Dohnanyi, 1969).
We estimate the effects of LHB in following way. In Section 2,
we derive the semi-analytical expressions for the effects of LHB
impacts. We estimate the total excavation/melting volume andarea by integrating the effects of individual impacts, assuming a
power-law SFD. In Section 3, we evaluate total effects with fixed
total masses of impactors and the two lunar constraints—the maximum size of basin formed during LHB and the small craters’ density. We also investigate the dependence of the power index a of
SFD. In Section 4, we discuss whether LHB can explain the absenceof Hadean rocks. Then, we compare our results with those of theprevious works for a nominal value of a ¼ 1:61. We also discuss
the validity of our assumptions and models. In Section 5, we summarize this paper. Appendix A is devoted to the calculation of thevalue of a which fulfills both the lunar constraints.
2. Basic methods
2.1. Effects of a single impact
We consider a single impact causes direct excavation/melting
and subsequent excavation/melting. Formation of transient craterand melting by shock heating cause ‘‘direct excavation/melting,”
while formation of final crater and uplifted or excavated molten
rock spreading on and beyond the final crater cause ‘‘subsequent
excavation/melting,” respectively. In this work, these effects aredenoted V xcav ; V melt ; Sxcav and Smelt for direct effects, and Sxcav;f andSmelt;f for subsequent effects (Fig. 2). The volume of evaporated
rocks would be sufficiently small so that we did not include it inthis study.
We assume the direct excavation volume, V xcav , is the volume of
the transient crater (Abramov et al., 2012),

 0:66  1
gL
qt
1:3
V xcav ¼ 0:146 2
sin h
v
m
1:3 0:66¼ 0:127q0:22
q1
g
sin
i
t v

1:3

Because these previous works were carried out based on respective model of the impactor’s SFD and the effects of impact, theresults were inconsistent with each other. This work is going toinvestigate the effects of the slope of the impactor’s SFD, whichis the main new contribution of this work.
The SFD of LHB impactors is still controversial. Some studies
claim that the lunar craters’ SFD is consistent with the current
main belt asteroids’ SFD (e.g., Strom et al., 2005). Fig. 1 shows thatthe current main belt asteroids’ SFD can be approximated as
a ¼ 1:71, where (minus) a is the power index of the mass–frequency distribution (Bottke et al., 2005). On the other hand,
analytical theory of the evolution of SFD in a collision cascade
predicted that a ¼ 1:83 (Tanaka et al., 1996).
Corresponding to these uncertainties in the LHB impactors’ SFD,
we here consider a wide variety of SFDs. In order to reveal intrinsic
physics more clearly, we approximate the SFD as a power-law
function with a broad range of the power index a, rather thanadopting a single more detailed SFD. The number of impactors
heavier than m; N sfd (>m), is defined as

dNsfd¼ Ama ;dm
where A is a proportional constant. Then,ð1Þ

hm0:78 ;ð3Þ

where qi and qt are the densities of the impactor and the crust, andv ; h; L; m, and g are the impact speed and angle, impactor’s diameterand mass, and gravity, respectively. Eq. (3) is estimated from impact
experiments (e.g., Schmidt and Housen, 1987) and is consistent
with recent computer simulations. This volume includes the crust
not excavated but displaced, and the depth of the transient crater
would be deeper than the thickness of Hadean continental crust.
So, the excavated crust volume would be overestimated in thiswork.
According to Abramov et al. (2012), the diameter of the transient crater Dt is 0:22  1=3
gL
qt
Dt ¼ 1:44 2v
m¼ 1:37qi

0:22=3q1=3
v 0:44 g 0:22 m0:26 :t

ð4Þ

So the direct excavation (circular) area on the Earth’s surface (i.e.
horizontal cross section of the direct excavation region) is 0:44  2=3
gL
qt
Sxcav ¼ 1:63 2v
m¼ 1:48qi

0:44=3q2=3v 0:88 g 0:44 m0:52 :t

The direct melting volume is

ð5Þ

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Y. Shibaike et al. / Icarus 266 (2016) 189–203

10

7

Bottke et al. (2005)
Power-law Approximation, α=1.71

106

Cumulative Number

10

5

104

103

102

101

10

0

1014

1016

1018

1020

1022

1024

1026

Mass (g)Fig. 1. Current main belt asteroids’ SFD and its power-law approximation. Red dots are derived from observations (Bottke et al., 2005). The bin width increases by a factor of
0.5. The black line is a power-law approximation of these plots and a ¼ 1:71. We transformed the SFD into the mass–frequency distribution as the asteroids’ densities areqi ¼ 2:6 g/cm3; the value of density is consistent with the total mass of the main belt asteroids (Krasinsky et al., 2002). (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

Fig. 2. Vertical cross sections of single impact effects. Excavation regions are framed by dashed curves, and melting regions are framed by solid curves. We defined four direct
effects, ‘‘direct excavation volume, V xcav ,” ‘‘direct melting volume, V melt ,” ‘‘direct excavation area, Sxcav ,” and ‘‘direct melting area, Smelt .” Sxcav and Smelt are the horizontal crosssections of the direct excavation/melting regions. When the depth of the melting region is deeper than the thickness of the crust, we cut the volume beneath the crust. We
also defined two subsequent effects, ‘‘excavation area including gravitational collapse, Sxcav;f ” and ‘‘melting area including the area covered by melts, Smelt;f .” Sxcav;f is the area ofthe final crater. The rim of the transient crater collapses with gravity and the final crater is formed. Smelt;f includes both the direct melting area and the area covered by melts.
The direct melting area is surrounded by the area covered by melts. When the diameter of the impactor is smaller than 100 km, Smelt;f is equal to Sxcav;f . When the diameter islarger than 100 km, Smelt;f is calculated by Eq. (12).

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Y. Shibaike et al. / Icarus 266 (2016) 189–203

V melt ¼ 0:42

 2 0:84  
v
m

m

qt

1:3

sin

h;

ð6Þ

where m is the specific internal energy of the target (Abramov
et al., 2012). The direct melting volume is proportional to theimpactors’ mass. When the depth of the melting region is deeper
than the thickness of the crust, h, we use the following equationin place of Eq. (6).

(
)
2=3
3
3V melth:
V melt;h ¼ p
h
2p
3

ð7Þ

The melting region’s shape is considered to be a hemisphere. From
Eq. (6), the direct melting area on the Earth’s surface (i.e. horizontal
cross section of the direct melting region) is
2=3 2 0:56  2=3
3V meltvm
2:6=3
Smelt ¼ p
¼ 1:08
sin
h:
2p
m
qt

ð8Þ

Eqs. (6)–(8) implicitly assume that the target crusts have nogeothermal gradient and a homogeneous initial temperature of 0  C.
Then, we consider the subsequent effects. As craters collapse
from transient craters to final craters due to gravity, their diameters become larger. There is a relationship between these diameters (Abramov et al., 2012),

Dt ¼ ðD0:15
D0:85
Þ=1:2;cf

ð9Þ

where Dt ; Df , and Dc are the transient crater’s diameter, final crater’s
diameter, and critical diameter between simple and complex craters, respectively. The final crater’s diameter and the excavation
area including gravitational collapse are

Df ¼ 1:79qi

0:22=2:55

Sxcav;f ¼ 2:52qq0:1=2:55
v 0:44=0:85 g 0:22=0:85 D0:15=0:85
m0:26=0:85 ; ð10Þtc
4=2:55 0:88=0:85 0:44=0:85 0:3=0:85 0:52=0:85
qt
v
g
Dc
m
:ð11Þ

0:44=2:55
i

Excavated or uplifting melts spread around the final crater(Abramov et al., 2013; Osinski et al., 2011). We assume the melting
area including the area covered by melts, Smelt;f , is equal to Sxcav;f . Whenthe impactor is larger than 100 km, the melting area including thearea covered by melts can be expressed as the following equation,

Smelt;f ¼

p 2 2
4

f L

¼ 1:21f2 2=3
m

qi

;

ð12Þ

where f is the proportion of the diameter of the melt region to thatof the impactor. The effects of decompression and adiabatic melting
are included, and f reaches 20–30 (Marchi et al., 2014).
2.2. Integrating the effects of a single impact

A

a1

a
m1
max

 1;

ð13Þ

then,
a1
A  ða  1Þmmax
;
1=ða1Þ
A
mmax 
;a1

Ae ¼ 23Am ;ð16Þ

where Ae and Am are the A values for Earth and the Moon,
respectively.
The effects of all LHB impacts are obtained by calculating the
integral from mmin to mmax of the effects of a single impact. We
describe all of the effects of a single impact (Eqs. (3), (5), (6), (8),(11), and (12)) as one general equation form, Ivs , in order to easily
understand their characteristics,

Ivs ¼ Bj mbj ;ð17Þ

where bj is the power-law index of each m, and Bj is a constant fixedby impact velocity and density (see Table 1). The general form of theeffects of all LHB impacts is

Ivs;T ¼

Z mmax
mmin

¼ Bj

dNsfd
Ivs dmdm

Ae
1þb a1þb aðmmaxj  mmin j Þ:
1 þ bj  að18Þ

When mmin is small enough,

8
>
>
>
< Bj

Ae
1þb ammaxj
1 þ bj  a
Ivs;T >Ae
1þb a>>m j: Bj
a  bj  1 min

ð0 < a < 1 þ bj Þ
ð19Þ

ð1 þ bj < aÞ:

The total effects of the impacts are dependent on the impactor’s
maximum mass mmax when 0 < a < 1 þ bj , and on the impactor’s
minimum mass mmin when 1 þ bj < a. We only use Eq. (19) tounderstand the dependence of the effects on the impactor’s mass,
and always use Eq. (18) to estimate the scales of the effects.
The total melting volume is dependent on the impactor’s maximum mass, mmax (when a < 2) and is proportional to the total
mass, MT (see Eq. (21)). The total direct melting area is proportional to the total cross section of the impactors, and the total crosssection is proportional to m5=3a . It has often been stressed that theeffects of large impacts are greater than those of small impacts(e.g., Abramov et al., 2013). However, we found that large impacts
and their effects are important only when a is small, and small
impacts are more important than large impacts when 1:52 < aand 5=3 < a because the number of such small impacts is verylarge. In this case, thin melt sheets formed by small impacts would
have covered the entire surface of the Earth, and it can explain theabsence of Hadean rocks (see Section 4.1).
Especially when a is large, the total melting volume and area
are strongly dependent on the minimum mass, mmin . In this work,
mmin is determined from the minimum mass le which can survive
a fall through the Earth’s atmosphere (see Section 4.3). Assuming
the current atmosphere, le ¼ 1011:5 g (Bland and Artemieva, 2003).
These estimates do not include overlapping of craters. A better
estimate including overlappings for the total excavation/melting
area is expressed in the following equation (Frey, 1980),

The definition of N sfd ð> mmax Þ is

Nsfd ð> mmax Þ ¼

The ratio of the collision probability with Earth and the Moon is23:1 (Zahnle and Sleep, 1997; Ito and Malhotra, 2006). The proportional constants A have the following relationship,

ð14Þ
ð15Þ

where mmax is the maximum mass of the impactors (Zahnle andSleep, 1997). This mmax just represents the maximum mass value
with the highest possibility, so it would have larger or smaller
values actually.

Table 1
Coefficients.
Effects

1 þ bjBj

V xcav

1.78

V melt2

1:3 0:66
0:127q0:22
q1
g
sin1:3 ht vi 0:84
1:3
1 v 2
0:42qt m
sin h

Sxcav

1.52

1:48qi0:44=3qt2=3 v 0:88 g 0:44

 2 0:56

Smelt

5/3

1:08qtSxcav;f ; Smelt;f

1.61

Smelt;f (L > 100 km)

5/3

2:52 i0:17 0:78t
2 2=3
1:21f i

2=3

q

q

q

v
m

v

sin2:6=3 h

1:04 0:52

g

Dc0:35

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Total Current Continental Crust’s Volume = 1

Y. Shibaike et al. / Icarus 266 (2016) 189–203

0.5

Ceres
(MT=5*1023g)

a
0.4

1.61

20

Vxcav,T, MT=5*1023g
Vxcav,T, MT=2*1023g
Vxcav,T, MT=1*10 g
23
Vmelt,T, MT=5*1023g
Vmelt,T, MT=2*10 g
Vmelt,T, MT=1*1023g

0.3

0.2

Hadean Continental Crust
0.1

0
1

1.2

1.4

1.6

1.8

2

1.8

2

Power Index α
1

Ceres
23
(MT=5*10 g)

Earth’s surface area = 1

b
0.8

1.61

Sxcav,T, MT=5*1023g
Sxcav,T, MT=2*1023g
Sxcav,T, MT=1*1023g
Smelt,T, MT=5*1023g
Smelt,T, MT=2*1023g
Smelt,T, MT=1*1023g

0.6

0.4

0.2

0
1

1.2

1.4

1.6

Power Index α
Fig. 3. Total Direct effects of LHB with fixed total impactors’ mass. Panels (a) and (b) show the total direct excavation/melting volumes and areas, respectively. Green, blue,
and red curves show the effects when MT ¼ 1, 2, and 5  1023 g, respectively. Solid curves show the total melting volumes and areas, dashed curves are those of excavation.
The black line in panel (a) shows the total Hadean continental crust’s volume. We cut off the SFD larger than the size of Ceres when a < 1:35 (MT ¼ 5  1023 g), left side of theaqua vertical line. In panel (a), the solid curves (i.e., melting volumes) include the correction of the thickness of the crust, h ¼ 30 km, but the dashed curves (i.e., excavation
volumes) do not. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)



ST
;
Sr ¼ 1  exp 
Se

ð20Þ

where ST is Sxcav;T ; Smelt;T ; Sxcav;f;T or Smelt;f;T , and Se is the total surface
area of Earth. We use this correction in this paper. On the other
hand, estimating the excavation/melting volumes including overlapping is difficult, so we do not use any corrections for them andthey are probably overestimated.
3. Effects of LHB impacts
3.1. Effects of LHB impacts with fixed total impactors’ mass
We estimate the effects of LHB and their dependence on a,where M T , the total mass of impactors, is fixed. We assume the ato be 1 < a < 2. From Eq. (1), MT is

MT ¼

Z mmax
mmin

¼

dNsfd
mdm
dm

A
a
ðm2a  m2
min Þ:
2  a max

ð21Þ

a
Am2
max
;
2a

ð22Þ

Then,

MT 

where mmin is small enough and a < 2 (Dohnanyi, 1969). This equation shows that the total mass depends on the maximum mass. The
following equations derived from Eqs. (14) and (15),

2a
M ;
a1 T
A ¼ ð2  aÞa1 ða  1Þ2a M Ta1 :mmax 

ð23Þ
ð24Þ

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Y. Shibaike et al. / Icarus 266 (2016) 189–203

1

Earth’s surface area = 1

Ceres

1.61
Excavation(direct)0.8

0.6

Melting (f=30)
0.4

Melting (f=20)
0.2

Excavation(collapse)

Melting (direct)

0
1

1.2

1.4

1.6

1.8

2

Power Index α
Fig. 4. Total excavation/melting areas with fixed total impactors’ mass. Dashed and solid green curve show the direct excavation and melting areas, respectively. Red and blue
curves show the total melting area including the area covered by melts when f ¼ 30 and 20, respectively. Dashed red curve shows the total excavation area including
gravitational collapse. The dashed and solid green curves are consistent with the dashed and solid blue curves in Fig. 3(b), respectively. Including the area covered by melts,
the total melting area increases dramatically. The total mass that hit Earth, MT , is fixed at 2  1023 g. We cut off the SFD larger than the size of Ceres when a < 1:18, left side of
the aqua vertical line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The dependence of the effects on M T is derived from Eqs. (18), (23)and (24),

ð2  aÞa1 ða  1Þ2a
Ivs;T ¼ Bj
1 þ bj  a()1þbj a2a1þbj a leM
MTa1 :
a1 T
ð25Þ
23

The most typical value of M T is about 1–5 10 g (e.g., Gomes
et al., 2005; Jørgensen et al., 2009; Levison et al., 2001). Abramov
et al. (2013) also used the value MT ¼ 2  1023 g. In this paper,
we use the value MT ¼ 1, 2 and 5  1023 g to calculate the effectsof LHB.
Fig. 3(a) shows the estimated total excavation and melting volumes relative to the total Hadean continental crust’s volume. We
considered the thickness of the crust (30 km) for the estimates ofthe melting volume (see Section 2.1). We also consider the cut-offof the heavier side of the SFD to avoid the mass becoming unrealistically large. We choose the cut-off value as mceres ¼ 9:4  1023 g,
the mass of Ceres. Ceres is the largest object among the main belt
asteroids. For example, when M T ¼ 5  1023 g, we have to cut off
the SFD where a < 1:35. In this a region, me;max ¼ mceres . The estimated total excavation/melting volumes are normalized with thetotal current continental crust’s volume, 7:18  109 km3 (Cogley,1984). According to geochemical constrains of the mantle, we
assumed that the continental crust was formed at 4.5 Ga for the
first time and there was about 12% of the total current continental
crust’s volume in the last phase of the Hadean (McCulloch andBennett, 1994). We assume qi ¼ 2:6 g/cm3, qt ¼ 2:7 g/cm3,

v ¼ 21 km/s and m ¼ 5:2 MJ/kg. In the case of MT ¼ 2  1023 g in

Fig. 3(a), the total melting volume is smaller than the total volumeof Hadean continental crust in all a ranges. LHB impacts does not
melt all of Hadean continental crust in this case. On the other hand,
the total excavation volume exceeds that of the continental crustwhen a is larger than about 1.8, though we note that excavation
volume of the continental crust would be overestimated (see Section 2.1). LHB cannot excavate/melt all of the continental crust bydirect effects when a = 1.61.

Fig. 3(b) shows the direct excavation/melting area using Table 1,
and Eqs. (20), (24), and (25). This figure shows that LHB does not
excavate/melt all of the Earth’s surface area by direct effects whena = 1.61. If there were many small impacts (i.e., when a was large),
the excavation area would be able to expand the whole surface of
the Earth.
Fig. 4 shows the total excavation/melting area including gravitational collapse and the area covered by melts. In particular, thetotal melting area including the area covered by melts (red andblue solid curves) is dramatically increased compared to the total
direct melting area and approaches 60–70% of the Earth’s surface
area where a ¼ 1:61 and f = 20–30 (Marchi et al., 2014). As abecomes larger, the number of small impactors dramaticallyincreases. For small impactors, the area covered by melts is comparable to the excavated area on the Earth’s surface including gravitational collapse (Table 1). Therefore, the total melting area
including the area covered by melts (the red and blue solid curves)approaches the total excavation area including gravitational collapse (the red dashed curve) when a is large enough.
Fig. 5 shows the total melting area for each impactor’s size.
Fig. 5(a) shows the total direct melting area. It is mainly dependent
on the maximum mass when a > 5=3 and on the minimum mass
when a < 5=3 (also see Smelt in Table 1). In other words, as a
becomes larger, the number of small impactors dramaticallyincreases, and as a becomes smaller, the maximum mass to hit
Earth increases. This trend is the same as that of the total
impactors’ cross section. Also, even if we estimate the total direct
melting area from the size of the maximum LHB basin on the Moonor the lunar crater density, their dependence on m is not changed,
so that this trend of the total melting area is not changed (seeSection 3.2).
When the area covered by melts is included, this trend is changed. Fig. 5(b) suggests that the trend of the contributions can bedivided into three ranges. Small impactors mainly melt the Earth’s
surface when a is larger than about 1.7. Impactors of about 100 km
in size mainly melt the surface when a is about 1.5–1.7. Impactorsof over 500 km in size mainly melt the surface when a is less thanabout 1.3. Considering the ‘‘stochastic” view, an impactor of such

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Y. Shibaike et al. / Icarus 266 (2016) 189–203

0

10

5/3

Earth’s surface area = 1

a< 1 km
1-10 km

10-1

10-100 km
10

-2

10

-3

Melting (all sizes)> 100 km

1

1.2

1.4

1.6

1.8

2

Power Index α
1

1.61

Earth’s surface area = 1

b0.8< 1 kmMelting (all sizes)

0.6

0.4

> 100 km> 300 km0.2

> 500 km
0
1

1.2

1.4

1.6

1.8

2

Power Index α
Fig. 5. Total melting areas for each impactors’ size with fixed total impactor’s mass. Panel (a) shows the total direct melting area; panel (b) shows the total melting area
where f ¼ 30. Black curves in panels (a) and (b) show the total melting areas of all size impactors, consistent with the green and red solid curves in Fig. 4, respectively. In
panel (a), red, blue, green and violet curves show the total direct melting areas by impactors of <1, 1–10, 10–100 and >100 km in size, respectively. In panel (b), red, gray andyellow curves show the total melting areas by impactors larger than 100, 300 and 500 km, respectively. Dashed violet curve shows the total direct melting area by impactors
smaller than 1 km. MT is fixed at 2  1023 g. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 2
How to calculate each value.
Constraints
Total massAe
ð2  aÞ

me;maxa12aða  1Þa1MTImbrium sizea123ða  1Þmm;max

Crater densitya1 N23ða  1Þmm;20
m;20

2a
a1 M T
1=ða1Þ

23mm;max
1=ða1Þ
a1 N
23mm;20m;20size is the largest one in most cases. When a ¼ 1:61, the effects ofimpactors of about 100 km in size are dominant.

3.2. Estimate from lunar constraints
While constraints of LHB on Earth have been erased, theyremain on the Moon. We estimate the value of Am from lunar con-straints when 1 < a < 2. First, we estimate it based on the largest
basin formed by LHB impacts. The following equation is derived
from Eq. (14), where mm;max is the maximum mass to hit the Moon,a1
Am ¼ ða  1Þmm;max:

ð26Þ

Because the impactor formed the Imbrium basin is at the edge of
the SFD where the statistical fluctuation is the largest, this estimate
has uncertainty.
We assumed the largest impactor (mm;max ) formed the Imbriumbasin at 3.85 Ga. Although the South Pole-Aitken is the largest
lunar basin, it was considered to be formed before the onset of
LHB. The impactor’s mass m is estimated from Dt (Eq. (10)),m ¼ 0:149q0:11=0:35
q1=0:78v 0:22=0:13 g 0:11=0:13 D0:15=0:26
D0:85=0:26
:
t
c
i
f

ð27Þ

Using this equation, the mass of the impactor formed the Imbrium
is mm;max ¼ 8:02  1020 g, where qi ¼ 2:6 g/cm3, qt ¼ 2:9 g/cm3,

Y. Shibaike et al. / Icarus 266 (2016) 189–203

Total Current Continental Crust’s Volume = 1

196

0.5

Ceres

a

1.61

0.4

0.3

Excavation(exclude h)

0.2

Hadean Continental Crust
0.1

Melting
0
1

1.2

1.4

1.6

1.8

2

Power Index α
1

Ceres

Earth’s surface area = 1

b

1.61

Melting (f=30)

0.8

0.6

Melting (f=20)

0.4

Excavation(collapse)

0.2

Excavation(direct)

Melting (direct)

0
1

1.2

1.4

1.6

1.8

2

Power Index α
Fig. 6. Total effects of LHB estimated from the size of the Imbrium basin. Panels (a) and (b) show the total excavation/melting volumes and areas, respectively. In panel (a),red and blue curves show the total excavation and melting volumes, respectively. The black line shows the total Hadean continental crust’s volume. The melting volume
includes the correction of the thickness of the crust, h ¼ 30 km, but the excavation volume does not. In panel (b), dashed and solid green curves show the total direct
excavation and melting areas, respectively. Dashed red curve shows the total excavation areas including gravitational collapse. Red and blue solid curves show the total
melting areas including the area covered by melts where f ¼ 30 and 20, respectively. The Imbrium mass is mm;max ¼ 8:02  1020 g. We cut off the SFD larger than the size ofCeres when a < 1:44. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)v ¼ 18 km/s, Dc ¼ 18 km on the Moon, and the Imbrium basin’s
diameter, Df , is 1160 km (Spudis, 1993).
Then, we estimate the value of Am from the crater density on theMoon. According to Morbidelli et al. (2012), the following differential equation represents the number of impacts to hit the Moon:
dN20¼ 2:7  1016 expð6:93tÞ þ 5:9  107 ;
dt

ð28Þ

where N20 is the lunar crater density whose diameters are larger
than 20 km, and t is the age. The units are [/km2] and [Ga (Gyrago)]. In this expression, LHB started at 4.1 Ga (a sawtooth-liketimeline).
The number of impacts to hit area S between the age t and t þ dt
forming craters larger than D isnt;tþdt ð> DÞ ¼

dND
S:
dt

ð29Þ

Then, the proportional constant A is derived from Eq. (2),At;tþdt ¼ ða  1ÞSmDa1

dND
;
dt

ð30Þ

where mD is the impactor’s mass that forms a crater whose diameter is D. The total number of impacts which hit the Moon to form acrater larger than 20 km during LHB is

Nm;20 ¼ Sm N20
Z t0
dN20
¼ Sm
dt
dt
tf
¼ 3:79  107  8:76  105
¼ 3:32  103 ;ð31Þ

where Sm is the moon’s surface area, t 0 ¼ 4:5 and t f ¼ 0. This estimate of N 20 ¼ 8:76  105 km2 is exactly consistent with the real

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0

10

5/3

Earth’s surface area = 1

a< 1 km10-1

1-10 km
Melting (all sizes)
10-100 km

10-2> 100 km
-3

10

1

1.2

1.4

1.6

1.8

2

Power Index α
1

1.61

Earth’s surface area = 1bMelting(all sizes)

0.8< 1 km0.6

0.4> 100 km0.2

> 500 km> 300 km
0
1

1.2

1.4

1.6

1.8

2

Power Index α
Fig. 7. Total melting areas for each impactor’s size estimated from the size of the Imbrium basin. Panel (a) represents the direct total melting area; panel (b) shows the total
melting area including the area covered by melts where f ¼ 30. Black curves in panels (a) and (b) show the total melting areas of all size impactors, consistent with the green
and red solid curves in Fig. 6(b), respectively. In panel (a), red, blue, green and violet curves show the total direct melting areas by impactors of <1, 1–10, 10–100 and >100 km
in size, respectively. In panel (b), red, gray, and yellow curves show the total melting areas by impactors larger than 100, 300, and 500 km, respectively. Dashed violet curveshows the total direct melting area by impactors smaller than 1 km. Imbrium mass is mm;max ¼ 8:02  1020 g. (For interpretation of the references to color in this figure legend,
the reader is referred to the web version of this article.)

lunar crater density in the Nectaris basin (Marchi et al., 2012). The
key point is that we use the timeline only for calculating the lunar
crater density N20 and so this estimate does not depend on a specific
LHB model. Therefore, Am can be directly estimated only by observing the lunar crater density. Then,a1
Am ¼ ða  1Þmm;20
Nm;20 ;

ð32Þ

where mm;20 is the impactor’s mass which can make a 20-kmdiameter
crater
on
the
Moon.
Using
Eq.
(27),
mm;20 ¼ 1:38  1015 g. We summarize how to estimate Ae andme;max from lunar constraints in Table 2.
Figs. 6 and 7 show the total excavation/melting volume and area
estimated from the size of the Imbrium. Fig. 7(b) includes the subsequent effects, while Fig. 7(a) does not. On the other hand, Figs. 8
and 9 show those estimated from the crater density on the Moon.
Fig. 9(b) includes the subsequent effects, while Fig. 9(a) does not.

Figs. 6(a) and 8(a) show that LHB does not excavate/melt all of
Hadean continental crust in almost all a ranges. This is the same
result as that of the excavation/melting volume estimated fromthe total mass to hit the Earth (Fig. 3(a)). Figs. 6(b) and 8(b) show
that, although LHB does not excavate/melt all of the Earth’s surface
directly, most of the surface is covered by the melts from the subsequent effect in almost all a ranges. Figs. 7(a) and 9(a) show thetotal direct melting areas for each impactor’s size. They show thatthe total melting areas are mainly dependent on the maximum
masses when a > 5=3 and on the minimum masses when a < 5=3(also see Smelt in Table 1). Figs. 7(b) and 9(b) suggest that the trend
in contributions to covering the surface with melts are divided into
three a ranges like the contribution derived from the total mass
(Fig. 5(b)). The three a ranges are about 1.0–1.5, 1.5–1.7, and 1.7–
2.0 for both the estimates derived from the size of the Imbriumbasin and the lunar crater density. When a ¼ 1:61, the effects ofimpacts of about 100 km in size are dominant.

Y. Shibaike et al. / Icarus 266 (2016) 189–203

Total Current Continental Crust’s Volume = 1

198

0.5

Ceres

a

1.61

0.4

Melting

0.3

Excavation(exclude h)

0.2

Hadean Continental Crust
0.1

0
1

1.2

1.4

1.6

1.8

2

Power Index α
1

Ceres
1.61

Earth’s surface area = 1bMelting (f=20)
Melting (f=30)

0.8

Excavation(collapse)
0.6

0.4

Excavation(direct)

0.2

Melting (direct)
0
1

1.2

1.4

1.6

1.8

2

Power Index α
Fig. 8. Total effects of LHB estimated from the lunar crater density. Panels (a) and (b) show the total excavation/melting volumes and areas, respectively. In panel (a), red andblue curves show the total excavation and melting volumes, respectively. The black line shows the total Hadean continental crust’s volume. The melting volume includes thecorrection of the thickness of the crust, h ¼ 30 km, but the excavation volume does not. In panel (b), dashed and solid green curves show the total direct excavation andmelting areas, respectively. Dashed red curve shows the total excavation area including gravitational collapse. Red and blue solid curves show the total melting areas
including the area covered by melts where f ¼ 30 and 20, respectively. The density of lunar craters larger than 20 km is N 20 ¼ 8:76  105 km2. We cut off the SFD larger
than the size of Ceres when a < 1:55. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

We summarize the above results in Fig. 10. The red and blue
arrows represent the a ranges of LHB estimated from the size of
the Imbrium basin and the lunar crater density that can excavate/
melt 70% of Hadean continental crust’s volume or the Earth’s surface
area. Considering only the direct excavation/melting of the continental crust, it is difficult for LHB impacts for most a ranges to excavate/melt all of the continental crust. According to the estimate
from the size of the Imbrium basin, there is enough excavation/
melting only for limited conditions, when a > 1:7 (excavation)
and a > 1:9 (melting), the a ranges where the total effects areenough for both the volume and area. According to the estimate
from the lunar crater density, there is enough excavation and melting only for a < 1:3. When a ¼ 1:61, all estimated direct effects of
LHB impacts are not enough. On the other hand, when we consider
the subsequent effects, the a ranges in which LHB can excavate/melt
the continental crust expand. In particular, the subsequent melting

can cover over 70% of the Earth’s surface in all a ranges (estimatedfrom the size of the Imbrium) and when about a < 1:6 and a > 1:9(estimated from the lunar crater density). When a ¼ 1:61, the subsequent melting covers over 70% of the surface in both the estimates. In conclusion, our results show that LHB would not
excavate/melt all of the Hadean continental crust directly, but mostof the Earth’s surface area could be covered by melts.
4. Discussion
4.1. LHB and the absence of Hadean rocks
Can LHB explain the absence of Hadean rocks? When a crust is
excavated by impacts, it is broken into small pieces and scattered.
This helps Hadean continental crust to subduct with oceanic plates
and disappear from the Earth’s surface. When the crusts melt, their

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Y. Shibaike et al. / Icarus 266 (2016) 189–203

0

10

5/3

Earth’s surface area = 1

a> 100 km10-1

Melting(all sizes)

10-100 km
10-2

1-10 km< 1 km

-3

10

1

1.2

1.4

1.6

1.8

2

Power Index α
1

1.61

Earth’s surface area = 1bMelting(all sizes)

0.8

0.6

< 1 km> 300 km0.4

> 500 km0.2

> 100 km
0
1

1.2

1.4

1.6

1.8

2

Power Index α
Fig. 9. Total melting ares for each impactor’s size estimated from the lunar crater density. Panel (a) represents the total direct melting area; panel (b) shows the total melting
area including the area covered by melts where f ¼ 30. Black curves in panels (a) and (b) show the total melting areas of all size impactors, consistent with the green and redsolid curves in Fig. 8(b), respectively. In panel (a), red, blue, green and violet curves show the total direct melting areas by impactors of <1, 1–10, 10–100 and >100 km in size,
respectively. In panel (b), red, gray, and yellow curves show the total melting areas by impactors larger than 100, 300, and 500 km, respectively. Dashed violet curve shows
the direct melting rate by impactors smaller than 1 km. The density of lunar craters larger than 20 km is N 20 ¼ 8:76  105 km2. (For interpretation of the references to color
in this figure legend, the reader is referred to the web version of this article.)radiometric ages of zircons are reset (Abramov et al., 2013). It
erased the record of Hadean rocks. If the melts covered the entire
Earth’s surface area, and the stratigraphic succession has been preserved until today, the absence of Hadean rocks on the surface canbe explained. Our result that LHB is not able to excavate/melt all of
Hadean continental crust directly but can cover most of the Earth’s
surface with 3-km thick melts suggests that if there has not been alarge-scale folding or overturn of the crust until today, LHB could
explain why we have never found any Hadean rocks except theJack Hills zircons because most of Hadean continental crust is
not be exposed on the Earth’s surface in this case.
4.2. Comparison of results to previous works
We compare our results derived from the lunar constraints withprevious works with a ¼ 1:61. In Table 3, we summarize SFDs andthe excavation/melting volumes and areas of our calculations,
Abramov et al. (2013) and Marchi et al. (2014). Without the subsequent effects, both our calculation and Abramov et al. (2013) showthat a few crusts are excavated/melted by LHB. Not considering
crater overlapping, the total melting area including the area covered by melts, Smelt;f;T , is over 90%. This estimate is consistent withthat of Marchi et al. (2014). However, if we take into account crater
overlapping, the coverage is reduced to 60–74%.
Note that while we analytically estimated the values of LHB
effects at the case of highest probability, Marchi et al. (2014)
adopted Monte Carlo simulations so that their results have a‘‘stochastic” nature. To roughly take into account the stochastic
nature in our calculations, we have the maximum mass fluctuate
by a value of N sfd ð> me;max Þ from 1=e to e because the fluctuationof maximum mass impact has a large contribution to results(Eq. (13)). Fig. 11 shows the total melting area including the area

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Y. Shibaike et al. / Icarus 266 (2016) 189–203

aVxcav,T
Imbrium

Sxcav,T
Sxcav,f,T
Vxcav,T

Crater Density

Sxcav,T
Sxcav,f,T
1

1.2

1.4

1.6

1.8

2

Power Index αbVmelt,T
Imbrium

Smelt,T
Smelt,f,T
Vmelt,T

Crater Density

Smelt,T
Smelt,f,T
1

1.2

1.4

1.6

1.8

2

Power Index α
Fig. 10. The a ranges where LHB excavates (a) and melts (b) 70% of total Hadean continental crust. Red arrows show the a ranges of LHB estimated from the size of Imbriumbasin that can excavate/melt 70% of the total Hadean continental crust’s volume (i.e., 8.4% of the total current continental crust’s volume) or the Earth’s surface area. Bluearrows show the a ranges estimated from the lunar crater density. Black solid lines show a ¼ 1:61. In the estimate of Smelt;f;T , we assumed f ¼ 30. When we consider thesubsequent effects, the a ranges in which LHB can excavate/melt most of the continental crust expand. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article.)covered by melts for M T ¼ 2  1023 g. When a is small, the fluctuation affects the melting rate significantly because the contribution
from the maximum mass is larger for smaller a.

Table 3Comparison between our work and previous works.

SFDv (km/s)qi (g/cm3)qt (g/cm3)
le (g)

4.3. Validity of assumptions and models
Here we discuss the validity of the assumptions we adopt. First,
we did not consider the geothermal heat. We obtained V melt , that is
consistent with the analytical shock-heating model used in
Abramov et al. (2013), where the target has no geothermal gradient and the initial temperature is homogenized to be 0 °C. Because
geothermal heat increases the melting volume, V melt that we haveobtained could be an underestimate. If the Hadean Earth had ahigher geothermal gradient than that of the present Earth, themelting volume could increase by a factor of two or three at most(Abramov et al., 2013). On the other hand, the subsequent melting
includes the gradient’s effect. However, Marchi et al. (2014)claimed that if the Hadean mantle potential temperature washigher or the lithosphere was thinner than today, the melting volume may increase by 50–75%.
Second, our estimates of the minimum impactor mass may beoverestimated because the Hadean atmosphere may have hadhigher pressure than the current atmosphere. The minimum diameter Lmin can be shown analytically as

T surf (°C)
dT/dz (°C/km)h (km)MT (g)
V xcav;T (%)dV melt;T
Sxcav;T (%)g
Smelt;T
Sxcav;f;T
Smelt;f;T
Smelt;f;T ha

Our work

Abramov et al. (2013)

Marchi et al. (2014)

a = 1.61
21
2.6
2.7

MBAa
20
2.7
2.7

MBAa
25
3.314
‘‘granite”

1011:50030

1:41  1015 b
20
12, 70–

2:2  1023

2  1023

5:86  1018 c
20
11.25
30
–

3.6
2.7
20
2.2
51
60–74
92–1307e
2.1–3.6f
25
5–10
–
–
–

–
–
–
–
–
–
70–100Main belt asteroids’ SFD.
L = 1 km.
L = 15 km.
dPercent of the total current continental crust’s volume.
e‘‘Impact ejecta covered the entire surface of the LHB-era Earth to a depth close
to 1 km.”f‘‘1.5–2.5 vol.% of the upper 20 km of Earth’s crust was melted in the LHB.”g
Percent of the Earth’s surface area.
h
Not considering crater ovarlapping.
b
c

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Y. Shibaike et al. / Icarus 266 (2016) 189–203

1

Earth’s surface area = 1

1.61
0.8

0.6

0.4

Nsfd(>me,max)=1
0.2

Nsfd(>me,max)=1/e
Nsfd(>me,max)=e

0
1

1.2

1.4

1.6

1.8

2

Power Index α
Fig. 11. Total melting areas in a stochastic scenario with fixed total impactors’ mass. Blue and green dashed curves show the total melting areas including the area covered bymelts when the value of N sfd ð> mmax Þ are 1=e and e, respectively (Eq. (13)). Red solid curve shows our analytical estimate of the case with highest possibility. We do notconsider the cut off of the SFD by the mass of Ceres. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Lmin ¼ 0:15

Psurf

qi g surf sin h

;

ð33Þ

where P surf and g surf are, respectively, the atmospheric pressure at
the surface and the acceleration of gravity (Melosh, 1989). In thiscase, our estimates of the particular excavation/melting areas
decrease when a is large (Fig. 12).
Third, our estimate assumed impactors’ density, qi , of 2.6 g/cm3
and velocity, v, of 21 km/s during LHB, and these values impliedthat the impactors were asteroids. However, some previous works
claimed that the source of LHB impacts were comets or icy planetesimals (e.g., Levison et al., 2001; Jørgensen et al., 2009). In thesecases, the impactor’s density should be changed to qi  1 g/cm3
and the velocity to v  30 km/s. However, this change would notaffect the results significantly. According to Table 1, V xcav /

1

q0:22
v 1:3 m0:78 ; Sxcav / q0:44=3
v 0:88 m0:52 ; Sxcav;f / q0:17
v 1:0 m0:61 ; V melt /iiiv 1:68 m and Smelt / v 1:12 m2=3 , so these effects should increaseby only ð1:0=2:6Þ0:22þ0:78  ð30=21Þ1:3 ¼ 0:61; ð1:0=2:6Þ0:44=3þ0:52 ð30=21Þ0:88 ¼ 0:72; ð1:0=2:6Þ0:17þ0:61  ð30=21Þ1:0 ¼ 0:68; ð30=21Þ1:68 
ð1:0=2:6Þ ¼ 0:70 and ð30=21Þ1:12  ð1:0=2:6Þ2=3 ¼ 0:79 times,
respectively. Smelt;f does not depend on qi and v. If the effect isregulated by the impactor’s kinetic energy, f should be multiplied
ð1=3Þ

¼ 0:92 times, and the minimum size
ð1:0=2:6  ð30=21Þ2 Þ
impactor that can form the flood melt should be multipliedð1=3Þ

¼ 1:1 times.
ð1:0=2:6  ð30=21Þ2 Þ
Finally, we discuss the effects of pre-LHB impacts. In the sections above, we only estimated the effects of LHB. However, several
impacts occurred during the middle of the Hadean eon and much

10

SxcavT, 10 g
Earth’s surface area = 1

1011.5g
0.8

13

10 g
10
SmeltT, 10 g
11.5

0.6

10
g
13
10 g

0.4

0.2

0
1

1.2

1.4

1.6

1.8

2

Power Index α
Fig. 12. Dependence of the total direct excavation/melting areas on le estimated from the lunar crater density. Red, blue and green curves show the total direct excavation/
melting areas when le ¼ 1010 ; 1011:5 and 1013 g, respectively. Dashed and solid curves show the total direct excavation and melting areas, and blue ones are consistent withthe dashed and solid green curves in Fig. 8(b), respectively. These estimates are very dependent on the minimum size of the impactor, especially when a is large. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)202

Y. Shibaike et al. / Icarus 266 (2016) 189–203

more impacts occurred at the beginning of it. In the first few hundred million years of the Hadean eon, impacts melted (or excavated) Hadean continental crust right after they formed, and thecontinental crust must not have had time to grow sufficiently.

where

N m;20 ¼ 3:32  103 ; mm;max ¼ 8:02  1020 g,

andmm;20 ¼

1:38  1015 g (see Section 3.2).

References
5. Summary
We have investigated by analytical arguments the possibility
for LHB impacts to excavate/melt Hadean continental crust. In
order to reveal intrinsic physics, we adopt simple power-law
impactors’ SFD with various exponents a, rather than a single
detailed SFD. We divided the effects of impactors into two phases,
and derived general formulas of excavation/melting volume andarea as functions of a and the impactor’s mass multiplied by a factor determined by impact velocity, planetary gravity, bulk densityof impactors and the target planet. We estimated the total LHB
effects from the total mass of LHB impacts and two types of constraints on the Moon, the size of the largest basin during LHB
and the small crater density.
With the fixed total mass of LHB impactors, the total direct melting area on the Earth’s surface is generally regulated by small(large) impacts, for large (small) a. The estimates from the lunar
constraints suggest that LHB can excavate/melt almost all of
Hadean continental crust in narrow a ranges. Estimating from thesize of the Imbrium basin, LHB can remove the continental crust
only at a > 1:7 (excavation) and a > 1:9 (melting), while estimatingfrom the lunar crater density, only at a < 1:3 (excavation and melt).
In contrast, the subsequent melts which spread on and beyond thefinal craters can cover over 70% of the Earth’s surface in all a ranges(estimated from the size of the Imbrium basin) or when abouta < 1:6 (estimated from the lunar crater density). However, themost likely value of a is 1.6–1.7. We conclude that LHB impacts
would not excavate/melt all of Hadean continental crust directly,
but most of the Earth’s surface could be covered by melts of subsequent impact effects. It suggests the absence of Hadean rocks could
be explained by LHB if the stratigraphic succession has been preserved until today because most of Hadean continental crust is
not be exposed on the Earth’s surface in this case.
Acknowledgments
We thank S. Marchi and O. Abramov for very useful comments
as reviewers. Their comments helped us improve the paper a lot.
We thank S. Maruyama, K. Kurosawa, and H. Sawada for valuable
discussions. T. S. was supported by a Grant-in-Aid for Young Scientists (B), JSPS KAKENHI Grant No. 24740120, and Grant-in-Aid for
Scientific Research on Innovative Areas Number 2605, MEXT. This
research was supported by a grant for JSPS (23103005) Grant-inaid for Scientific Research on Innovative Areas.
Appendix A. Value of a that fulfills both lunar constraints
We calculated the a value that fulfills both lunar constraints ofthe size of the Imbrium basin and the lunar crater density. From
Table 2,a1a123ða  1Þmm;max¼ 23ða  1Þmm;20
Nm;20 ;

ðA1Þ

thus

a¼ 1þ

ln Nm;20lnðmm;max =mm;20 Þ¼ 1:61;

ðA2Þ

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