Supplementary Methods
--- Estimation of the mass of the gravitationally-attracted solar atmosphere --We consider a proto-atmosphere on a Mars-sized protoplanet. The hydrostatic
equation of a proto-atmosphere is given by
GM
1 dp
(SP1)
  2 pl ,
 dr
r
where  is the density of a proto-atmosphere, p is the pressure of a proto-atmosphere, r
is the radius from the center of the planet, G is the gravitational constant, and Mpl is the
planetary mass. From the assumptions of ideal gas and isothermal structure of a
proto-atmosphere, we can derive the density distribution of a proto-atmosphere as
 GM pl m  1
 
 
 ,
kT
r
r
Hill



   Hill exp 
(SP2)
where m is the molecular mass, k is the Boltzmann constant, T is the temperature, and
Hill is the density at the Hill radius (rHill). The Hill radius is defined by
 M pl
rHill  
 3M SUN
1/ 3

 a ,

(SP3)
where MSUN is the mass of the sun, and a is the distance from the sun.
In order to calculate the density distribution of a proto-atmosphere from Eq. (SP2),
Hill and T are required. These values are given by a model of the nebular disk. We
consider the minimum mass solar disk model [Hayashi, 1981]. According to Hayashi
[1981], the distributions of nebular temperature (Tn) and the surface gas density (g) are
respectively given by
 a 

Tn  280  
1
AU


1 2
 a 

 g  1.7 10  
 1 AU 
4
14
 L 


L
 SUN 
K,
(SP4)
kg/m2,
(SP5)
3 2
where LSUN is the solar luminosity at the present, and L is the solar luminosity at stages
considered. Since the solar luminosity at the stage of the planetary formation is
considered to be about 0.7  LSUN, the nebular temperature is 256 and 301 K at the
present Earth’s and Venusian orbits, respectively. We adopt these temperatures as the
atmospheric temperature (T). From isothermal structure of nebula in the direction of
disk’s depth (so-called z direction of disk), the density distribution at the central plane
of the disk is written as
 a 

 n  1.4  10  
 1 AU 
6
11 4
 L 


 LSUN 
1 / 8
kg/m3.
(SP6)
From Eq. (SP6), the nebular density is 1.46  10-6 and 3.57  10-6 kg/m3 at the present
Earth’s and Venusian orbits, respectively. We adopt these densities as the density at the
Hill radius (Hill).
We assume that the mass of a proto-atmosphere (Matm) is that within the Bondi
radius (rBondi), which is defined by
rBondi 
GM pl m
.
(SP7)
kT
At the Bondi radius, the thermal energy of a proto-atmosphere equals to the potential
energy of the protoplanet. Then the mass of the atmosphere is given by
M atm  
rBondi
rpl
4 r 2  dr ,
(SP8)
where rpl is the planetary radius. When we consider the Mars-sized planet (i.e., Mpl = 6.4
 1023 kg, rpl = 3.4  106 m) and solar composition gas with molecular weight 2.34
g/mol (i.e., m = 3.89  10-27 kg), we can numerically obtain Matm from Eq. (SP8) as 7.8
 1019 and 3.4  1019 kg at the present Earth’s and Venusian orbits, respectively. At the
stage of giant impacts, surrounding nebula gas is already lost. However, the atmosphere
within the Bondi radius is trapped by the gravity of the protoplanet. Although Matm
depends on the planetary orbit (a), we have written down 4  1019 kg on the manuscript
as the typical value of Matm.
References
Hayashi, C. Structure of the solar nebula, growth and decay of magnetic fields and
effects of magnetic and turbulent viscosities on the nebula. Prog. Theor. Phys.
Suppl. 70, 35–53 (1981).