Ph. D. Thesis
Numerical Study of Multi-Component
Hydrodynamic Escape:Application to the Early Venusian AtmosphereTakanori Sasaki
Department of Earth and Planetary Science,
The University of Tokyo
2008

Abstract
Venus is free of molecular oxygen but it is thought to have contained a primordial
water ocean. Present lack of water is considered to be due to photodissociation of
H2 O followed by thermal escape of H at early stage. Then, the lack of oxygen on
present Venus is puzzling. Meanwhile, Venus may have lost some Ne, since it isenriched about 10% in 22 Ne/20 Ne with respect to solar abundances. On the other
hand, Venus appears to have lost no Ar, as indicated by near solar 36 Ar/84 Kr and38

Ar/36 Ar ratios. Then, the fractionation of Ne and Ar is puzzling. A key to solvethese problems would be a hydrodynamic escape of oxygen or noble gases from theearly Venusian atmosphere.
Hydrodynamic escape is one of a few mechanisms that can change the compositionof a planetary atmosphere irreversibly. So far, several numerical models have beendeveloped to solve the hydrodynamic escape problem. Previous models, however, areunsatisfactory to apply actual atmospheric evolutions quantitatively because theycannot solve transonic solutions or multi component solutions. Their quantitative
uncertainties cause their conclusions of the Venusian problems to be all different eachother. These problems should be solved by use of the numerical method to derive theprecise quantitative solutions. Here I present the first study on the multi-component
time-dependent hydrodynamic equations for transonic neutral gas escape from theearly Venusian atmosphere.
The numerical model consists of the one-dimensional time-dependent non-viscous
Euler equations with thermal conduction in spherical geometry. The effects of collisions with other species are included in the momentum equation to solve the multicomponent hydrodynamic escape. I develop the numerical code of hydrodynamic
escape using CIP method, which is well suited to solve the transonic flow. A twodimensional energy deposition calculation method is adapted to provide an accurate
approximation of the complicated radiative transfer processes in an extended sphericalatmosphere. I also use the power-law fits of solar fluxes against solar age to estimate
the energy deposition by solar EUV heating in numerical calculations; the emission of
the younger Sun in EUV wavelength intervals is orders of magnitude larger than thecurrent solar flux. The solution technique is validated in an isothermal atmosphere,
polytropic atmosphere, and adiabatic atmosphere. I apply the developed numericalmodel to the escape of hypothetical early Venusian atmosphere to consider (1) water
loss from early Venus and (2) fractionation of noble gases on Venus.
(1) Water loss from early Venus: According to the results, water loss from early
Venus can be achieved if the homopause number densities of H2 and O are maintained
around 9.0 × 1015 [molecules/m3 ] for a long time; in such a case, timescale of water

loss as amount of terrestrial ocean is about 1.5 × 108 years. Except in that case(i.e., when the homopause number densities vary with time), the hydrogen molecules

escape more than oxygen molecules from the water-rich Venusian atmosphere, andthen the relative abundance of oxygen to hydrogen inevitably increases with time.
Moreover, the larger the relative abundance of oxygen to hydrogen is, the harder theoxygen escapes from the Venusian atmosphere, so that the oxygen relative abundance
would increase at an accelerating pace.
Eventually, the results indicate that the complete water loss from early Venus byhydrodynamic escape of oxygen as well as hydrogen would be very difficult but not
be impossible. Under special circumstances, hydrodynamic escape of H2 and O byitself can solve the water loss problem for Venus. This is an important consequenceof this study. However, in the most cases, the oxygen would be left behind while thehydrogen would escape from the Venus, so that the oxygen consequently concentrates
in the Venusian atmosphere. How the oxygen concentrate and how the concentrated
oxygen was consumed and/or escaped are also discussed.
(2) Fractionation of noble gases on Venus: I show the fractionation of Ne and Ar
would not be explained simply by mass-dependent hydrodynamic escape of primordial
Venusian atmosphere. So, I consider the CO rotational cooling to realize the finetuning of hydrogen escape flux, which makes Ne escape and Ar unescape. Whencarbon monoxide is the same amount as hydrogen at the homopause in the early
Venusian atmosphere, the Ne/Ar fractionation by hydrodynamic escape with CO
cooling efficiently occurs. However, even in this case, quantitatively appropriate
Ne/Ar fractionation cannot be achieved by hydrogen hydrodynamic escape. The
results indicate that the present Venus’ Ne/Ar abundance ratio cannot be explainedby hydrodynamic escape of early Venusian atmosphere with solar-composition.
Present Venusian Ne/Ar abundance ratio could not be achieved by hydrodynamic
escape even if an atmospheric temperature controlled by CO cooling is taken into
consideration. However, other cooling mechanism such as H2 O cooling and othernoble gases’ sources such as solar implanted materials and/or comets might make it
2

possible to achieve the proper Ne/Ar fractionation. Other possibilities of achieving
noble gas patterns of present Venusian atmosphere are also discussed.
The results imply that simple explanations for these problems in previous studies
are actually not enough to account for the present Venusian atmosphere. Thus, theresults also imply that more detailed studies are required to understand the Venusian
atmospheric evolution history.

3Contents
1 General Introduction

11

I

13

Development of Numerical Model

2 Introduction to Hydrodynamic Escape
2.1

Hydrodynamic Escape . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.2

Applicability of Hydrodynamic Escape . . . . . . . . . . . . . . . . .

16

3 Previous Numerical Studies of Hydrodynamic Escape

19

3.1

Analytical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

3.2

Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

4 Single-Component Numerical Model

23

4.1

Basic Equations for Single-Constituent Atmosphere . . . . . . . . . .

23

4.2

Energy Deposition of Solar EUV Radiation . . . . . . . . . . . . . . .

25

4.3

Evolution of the Solar Activity . . . . . . . . . . . . . . . . . . . . . .

28

5 Validation of Numerical Model

33

5.1

Validation in an Isothermal Atmosphere . . . . . . . . . . . . . . . .

33

5.2

Validation in a Polytropic/Adiabatic Atmosphere . . . . . . . . . . .

34

6 Multi-Component Numerical Model

II

15

39

6.1

Extension of Hydrodynamic Escape Code . . . . . . . . . . . . . . . .

39

6.2

Crossover Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Application to the Early Venusian Atmosphere

7 Introduction to the Problems
7.1

43
45

Introduction to Water Loss from Early Venus . . . . . . . . . . . . .
5

45

7.2

Introduction to Fractionation of Noble Gases on Venus . . . . . . . .

8 Water Loss from Early Venus

46
51

8.1

Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

8.2

Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

8.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

8.4

Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

8.4.1

Oxygen Concentration . . . . . . . . . . . . . . . . . . . . . .

63

8.4.2

Oxidation of Surface Minerals . . . . . . . . . . . . . . . . . .

64

8.4.3

Nonthermal Escape . . . . . . . . . . . . . . . . . . . . . . . .

65

8.4.4

Contribution of Water from Comet . . . . . . . . . . . . . . .

66

8.4.5

D/H Fractionation . . . . . . . . . . . . . . . . . . . . . . . .

67

9 Fractionation of Noble Gases on Venus
9.1

69

Previous Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

9.1.1

Proposed Ne/Ar Fractionation Mechanism . . . . . . . . . . .

71

Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

9.2.1

CO Rotational Cooling . . . . . . . . . . . . . . . . . . . . . .

73

9.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

9.4

Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

9.4.1

H2 O Rotational Cooling . . . . . . . . . . . . . . . . . . . . .

82

9.4.2

Additional Origins of Noble Gases . . . . . . . . . . . . . . . .

84

9.2

10 Conclusions

87

A CIP Method

89

A.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

A.2 Nonadvection phase . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

A.3 Advection phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

B Supporting Materials for Numerical Model95

B.1 Data for Solar EUV Spectrum . . . . . . . . . . . . . . . . . . . . . .

95

B.2 Data for Cross-Sections of Photoabsorption for Molecular Hydrogen .

95

B.3 Two-Dimensional Energy Deposition . . . . . . . . . . . . . . . . . .

95

B.4 Data for Binary Diffusion Coefficient . . . . . . . . . . . . . . . . . .

98

6

C Analytical Solutions

101

C.1 Parker Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
C.2 Quasi-Analytical Solution for Polytropic Atmosphere . . . . . . . . . 101
Acknowledgments

105

Bibliography

107

7

List of Figures
4.1

Escape Flux Rates vs. the Position of the Heating Layer . . . . . . .

26

4.2

Flux Densities at Venus Orbit . . . . . . . . . . . . . . . . . . . . . .

30

5.1

Comparison Between Numerical Solution and Analytical Solution . .

34

5.2

Difference Between Numerical Solution and Analytical Solution . . .

35

5.3

Convergence to Steady State . . . . . . . . . . . . . . . . . . . . . . .

36

5.4

Difference Between Numerical Solution and Quasi-Analytical Solution

37

5.5

Difference Between Polytropic Solution and Adiabatic Solution . . . .

38

6.1

Schematic of 2-Component Hydrodynamic Escape . . . . . . . . . . .

40

7.1

Relative Abundances of Noble Gases in Meteorites and Terrestrial
Planets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

8.1

Hydrogen Escape Flux Rate for Some Solar Ages . . . . . . . . . . .

55

8.2

Number Density of H2 and O atmosphere . . . . . . . . . . . . . . . .

56

8.3

Gas Flow Velocity of H2 and O atmosphere . . . . . . . . . . . . . . .

57

8.4

Hydrogen temperature of H2 and O atmosphere . . . . . . . . . . . .

58

8.5

Escape Conditions for Some O Mixing Ratio . . . . . . . . . . . . . .

61

8.6

Oxygen Escape Flux Rate for Some O2 Mixing Ratio . . . . . . . . .

62

9.1

Required Hydrogen Flux to Uplift Heavy Elements . . . . . . . . . .

71

9.2

Relative Abundances of Noble Gases in Meteorites and Venus. . . . .

73

9.3

Hydrogen temperature of H2 -Ne-Ar-CO atmosphere . . . . . . . . . .

79

9.4

Escape Flux Rate of H2 , Ne, and Ar. . . . . . . . . . . . . . . . . .

80

9.5

Cooling Efficiency Ratio between CO and H2 O . . . . . . . . . . . . .

84

B.1 Correlation between body size and metabolic rate. . . . . . . . . . . .

98

8

List of Tables
4.1

Integrated Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

4.2

Parameters of the Power-Law Fits to the Measured Integrated Fluxes

31

7.1

Elemental Abundances in Solar, Meteorites, Comets and Terrestrial
Planet Atmospheres and Known Crustal Reservoirs . . . . . . . . . .

47

7.2

Elemental Abundances Normalized by Averaged Solar Abundances

.

48

8.1

Definition of Numerical Parameters . . . . . . . . . . . . . . . . . . .

53

9.1

Employed Free Parameters . . . . . . . . . . . . . . . . . . . . . . . .

73

9.2

Molecular line data for the CO model molecule . . . . . . . . . . . . .

75

9.3

Calculated Escape Flux Ratio (FAr /FNe ) . . . . . . . . . . . . . . . .

78

B.1 Reference Spectrum for Solar EUV . . . . . . . . . . . . . . . . . . .

96

B.2 Photoabsorption Cross-Sections for Molecular Hydrogen. . . . . . .

97

B.3 Binary Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . .

99

9Chapter 1
General Introduction
Understanding how primordial atmospheres evolved to their present compositions is
closely connected to the challenge of understanding the evolution of the planets themselves, in particular their geophysical histories of differentiation and outgassing, andemergences of coupled atmosphere-interior systems. Standard theories for the planetary formation predict that the following four processes should mainly determine theplanetary atmospheres (Zahnle, 1998): capture of high-reduced gas from the primordial solar nebula, degassing from protoplanets and planetesimals, atmospheric loss
by giant impacts, and atmospheric escape by hydrodynamic escape. When a planet
grows up to be a lunar-size in the nebula gas, the planet starts to gravitationallycapture the nebula gas. According to the modern planetary formation theory (e.g.,
Kokubo & Ida, 2000), terrestrial planets may capture the primordial solar nebulain the middle of planetary formation because the timescale of planetary formation
would be smaller than that of dissipation of nebula gas (Strom et al., 1993). On theother hand, high velocity impacts of planetesimals into the growing planets have been
considered to occur during accretion, which results in degassing of volatile materials
contained in the planet forming planetesimals (Abe & Matsui, 1986). Therefore, theprimordial planetary atmosphere should be mixture of atmospheres originated fromnebula gas and degassing volatile materials (Abe et al., 2000). And a primordial planetary atmosphere should be reducing chemical composition initially, irrespective thesource of atmosphere (Hashimoto et al., 2007). After the stage of planetary formationand mixed atmosphere formation, the atmosphere may have evolved to the present
atmosphere following extensive loss and escape of the primordial atmosphere. Particularly, it has been implicated that the process of atmospheric escape could mainly
bring the atmospheric and surficial differences of terrestrial planets (Earth, Venus,
and Mars). Atmospheric loss mechanism by giant impacts have been studied in detail1112

by Genda & Abe (2003, 2005), and they have estimated the effect of atmospheric loss
process following giant impacts. So far, however, we have no quantitatively reliable
study for hydrodynamic escape of primordial terrestrial atmospheres yet, although
there exists many qualitative studies of that (e.g., Zahnle & Kasting, 1986).
Hydrodynamic escape of the primordial atmospheres on terrestrial planets is key
to account for the characteristics of planetary noble gas abundances as follows: With
Venus, it is the lack of water and the related, extremely high D/H ratio - the latteran obvious remnant of large scale hydrogen escape. For Earth, it is the presence of
strange isotopic anomalies in noble gases, especially neon and xenon. For Mars, it
is all of the above, plus other isotopic anomalies (argon, nitrogen), and most tellingof all, a general 100-fold depletion of nonradiogenic atmophiles when compared to
Earth.
In this paper, I develop a numerical code of multi-component hydrodynamic escape to study the atmospheric escape process quantitatively, and also apply the numerical code to early Venusian atmosphere to give some constraint in the Venusian
atmospheric evolution. The significance of this study is not exclusive to solve someremaining issues of Venusian atmospheres. It is noteworthy to develop a general numerical code of hydrodynamic escape because many atmospheric evolution problems
can be solved using the code. The numerical code could be applicable to wide variety of planetary atmospheres including extrasolar planets, since early atmosphericevolution following hydrodynamic escape would be an inevitable evolutionary process
for general planets in the universe. Advances of planetary formation and continued
discovery of extrasolar planets have revealed large variety of planets and planetaryatmospheres (Ida & Lin, 2004). It will be possible to illustrate the variety of evolutional history for each planet by use of the developed numerical code of hydrodynamic
escape in this study. Therefore, I believe that this study has large possibility andexpansibility to lead to more general study of planetary science including extrasolar
planets, and of course astrobiology in the future; this huge potential is the greatest
feature of this study.

Part I
Development of Numerical Model

13

Chapter 2
Introduction to Hydrodynamic
Escape2.1

Hydrodynamic EscapeThe solar and polar winds are generally considered to be fluid-like or hydrodynamic
and can be described by considering the equations of hydrodynamics rather than theparticle picture of the collisionless exosphere. The theory of hydrodynamic escapeof an atmosphere, as it applies to the solar wind, is described in standard references (e.g., Parker, 1964). Similar hydrodynamic models have been used to describe
the loss of a planetary atmosphere for which the hydrodynamic escape parameter(HEP; see section 5.1) is small and the flow speed is large (e.g., Watson et al., 1981).
Such large flow speeds may have existed during early planetary atmospheric formation. Hydrodynamic escape occurs when hydrogen-rich primordial atmospheres arestrongly heated at high altitudes by intense extreme-ultraviolet (EUV) radiation fromthe young sun (e.g., Pepin, 1991). The likeliest energy source driving escape would
have been solar EUV radiation (λ < 100 nm), i.e., radiation of short enough wavelength that it can be directly absorbed by hydrogen (Sekiya et al., 1980; Watson etal., 1981). Outflows from a hydrogen-rich atmosphere, consistent with hydrodynamic
escape driven by atmospheric heating, have been directly observed during transits ofan extrasolar giant planet in close orbit around a solar-type star (Vidal-Madjar etal., 2003).
The heated hydrogen escapes to space, and in the process its outward flow exerts
upward drag forces on heavier atmospheric constituents. With powerful energy input, escape fluxes can be large enough to lift species as massive as noble gases outof terrestrial planets atmospheres (Hunten et al., 1987). Lighter species are more
readily entrained and lost with the escaping hydrogen, leading to mass fractionation
15

2.2. APPLICABILITY OF HYDRODYNAMIC ESCAPE

16of the residual atmosphere. Hydrogen escape fluxes high enough to sweep out andfractionate species as heavy as noble gases from terrestrial planets require energy
inputs 2 to 3 orders of magnitude greater than presently supplied by solar EUV radiation (Pepin, 1991). The early Sun likely maintained a high EUV luminosity for
only several hundred million years (Ribas et al., 2005). Thus, conditions favoringhydrodynamic escape were restricted to early epochs in planetary history. Therefore,
the hydrodynamic escape of primordial atmospheres exposed to early strong EUV
radiation should be an important process for terrestrial planets’ atmosphere as large
fractionations of atmospheric components, especially noble gases.
Hydrodynamic escape is one of a few mechanisms that can change the compositionof a planetary atmosphere irreversibly (Hunten, 1990). The isotopic fractionation
argues for hydrodynamic escape, which is fractionating, while the general deficiencies
argue for erosion by impacts as a relatively violent process that does not produce
much fractionation (Genda & Abe, 2005). So far, several numerical models have beendeveloped to solve the hydrodynamic escape problem (Watson et al., 1981; Kasting& Pollack, 1983; Tian et al., 2005). Actually, hydrodynamic escape of hydrogen-richprimary atmospheres and outgassed volatile elements from the terrestrial planets,
operating in an astrophysical environment for the early solar system inferred fromobservation of young star-forming regions in the galaxy, can qualitatively account formost of the known details of noble gas distributions in their present day atmospheres(Pepin 1991).
Previous models, however, is unsatisfactory to apply actual atmospheric evolution quantitatively because they cannot solve transonic solutions or multi component
solutions. Here I present the first study on the multi component time-dependent hydrodynamic equations for transonic neutral gas escape from planetary atmospheres.

2.2

Applicability of Hydrodynamic EscapeIn this study, I apply the developed numerical model to the escape of hypothetical
early Venusian atmosphere. This numerical code of course can be applied to other terrestrial planets’ atmospheres as well as the Venusian atmosphere. Mass fractionationin hydrodynamic escape may have contributed to explain the fractionation pattern of
the isotopes of xenon on Earth (Hunten et al., 1987; Sasaki & Nakazawa, 1988; Pepin,2006) and Mars (Pepin, 1991), the high Martian 38 Ar/36 Ar ratio (Pepin, 2006), andisotopic differences between atmospheric and mantle neon (Sasaki & Nakazawa, 1988;
Pepin, 2006). Mass fractionation by hydrodynamic escape could also account for theCHAPTER 2. INTRODUCTION TO HYDRODYNAMIC ESCAPE

17

observed modest elemental fractionation of the Martian noble gases with respect toEarth (Hunten et al., 1987). However, it cannot easily explain both the elemental
and the isotopic fractionations, because the relatively strong isotopic fractionations
are incompatible with the relatively weak elemental fractionation, and vice versa. A
more precise quantitative study by use of the developed numerical code would givesome clear explanations for these elemental and isotopic fractionations.

Chapter 3
Previous Numerical Studies ofHydrodynamic Escape
Several analytical and numerical models have been developed previously to solve
the hydrodynamic escape problem (e.g., Sekiya et al., 1980; Watson et al., 1981;
Kasting & Pollack, 1983; Zahnle & Kasting, 1986; Sasaki & Nakazawa, 1988; Chassefiére, 1996; Tian et al., 2005). However, because a singularity point (where thegas flow velocity equals the sound speed) exists in the transonic time-independenthydrodynamic equations, a self-consistent transonic solution of the time-independentequations cannot be obtained easily.

3.1

Analytical Studies

Several models have been developed before to study the hydrodynamic escape fromhydrogen-rich planetary atmospheres by solving the time-independent hydrodynamic
equations directly. Here, some typical previous studies of hydrodynamic escape by
analytical methods are introduced.
Watson et al. (1981) used a trial-and-error method to solve the problem. For acertain value of escape flux, this method calculates the required density and velocity
distribution in the steady state. A 10% variation of the mass flux at the criticalpoint can cause a change of density and velocity at the lower boundary by a factorof 10 or more (sometimes more than 4 orders of magnitude). Therefore, it is difficult
to find solutions for a pure hydrogen atmosphere using this method as pointed by
Kasting & Pollack (1983). Watson et al. (1981) assumed that heating by solar EUV
radiation occurred in a single layer. However, because the optical depth depends on
the absorption cross-section, the altitude of unity optical depth is not the same for19

3.2. NUMERICAL STUDIES

20

solar radiation at different wavelengths. In fact, the heating rate distribution has abroad structure (Fox & Dalgarno, 1979; Chassefiére, 1996). Their model and method
are too simple to discuss hydrodynamic escape precisely.
Kasting & Pollack (1983) included photochemistry as well as infrared cooling by
H2 O and CO2 in a study of the hydrodynamic escape from the atmosphere of early
Venus. They developed a numerical method to solve the momentum and the energy equations iteratively. Their model included a self-consistent temperature profile.
However, note that their method was limited to subsonic solutions. For most casesof interest, however, the actual outflow would have been transonic. The procedureused by their model is not valid under such circumstances because they neglected
nonlinear terms in the momentum equation of the heavier constituent and because
the boundary conditions imposed at large distances from the planet are inappropriate
for transonic flow.
Chassefiére (1996a) matched the hydrodynamic escape flux to a modified Jeansescape flux at the exobase level and considered the interaction between the solar wind
and the early Venusian atmosphere. This model forbids transonic solutions, because
the exobase level is set to be below the transonic level. The single-layer heating
assumption and the subsonic approximation limit the possibility of a self-consistent
study of the hydrodynamic escape of neutral gases from planetary atmospheres.
In either case, developed models are unsatisfactory to apply actual atmosphericevolution of terrestrial planets because they cannot solve transonic solutions exactly.
This is the fatal flaw of analytical studies.

3.2

Numerical Studies

Tian et al. (2005) developed a numerical model to study the hydrodynamic escape ofneutral gas from planetary atmospheres by solving the time-dependent hydrodynamic
equations. A robust numerical scheme capable of solving the transonic hydrodynamic
escape problem with a distributed energy source is also developed. They presented
the first study on the time-dependent hydrodynamic equations for single-component
transonic neutral gas escape from planetary atmospheres. However, the hydrodynamic escape model in Tian et al. (2005) contains only one component - hydrogenmolecules. As a result, the effects of the existence of other heavier components on
the hydrogen hydrodynamic escape process (drag force on the hydrogen escape flowby heavier components, extra heating/cooling by the heavier components, etc.) are

CHAPTER 3. PREVIOUS NUMERICAL STUDIES OF HYDRODYNAMICESCAPE21

not clear in their studies. Including other heavier gases into the model because of
the downward drag force exerted on hydrogen by other gases can further decrease
the expected escape rate. One should also include minor species, some of which may
be radiatively active, because photochemical simulations to date suggest radiatively
active species would be present in the primordial atmospheres of terrestrial planets.
Therefore, the model in Tian et al. (2005) cannot solve multi-component hydrodynamic escape precisely. The multi-component numerical model of hydrodynamic
escape is absolutely essential for considering the fractionation of noble gases on terrestrial planets as well as the hydrogen escape itself.
From the following chapter, first, I describe the development of numerical codefor single-constituent hydrodynamic escape. Then next, I show some validation of
the numerical code. Finally, I extend the code to multi-constituent case as a quantitative, self-consistent hydrodynamic escape model that includes the effects of mutual
collisions and atmospheric heating/cooling by some constituents.

Chapter 4
Single-Component Numerical
Model
4.1

Basic Equations for Single-Constituent Atmosphere

The singular behavior in the time-independent hydrodynamic equations no longer
exists in the time-dependent model, thus allowing the transonic solution to be obtained through time marching to a steady state. My hydrodynamic model is onedimensional, since a spherically symmetric atmosphere is assumed. The model solves
the one-dimensional time-dependent non-viscous Euler equations for a single-constituent
atmosphere with thermal conduction in spherical geometry:∂∂(ρr 2 ) + (ρur 2 ) = 0
(4.1)
∂t
∂r
∂
∂
∂p
(ρur 2 ) + (ρu2 r 2 ) = −ρGM − r 2
(4.2)
∂t
∂r
∂r
!#
"
∂
∂T
∂
∂u
∂
κr 2
(4.3)
(pr 2 ) + (pur 2 ) = (γ − 1) −pr 2− 2pur + qr 2 +
∂t
∂r
∂r
∂r
∂r
Here, ρ is the gas density, r is the distance from the planet center, u is the gas flow
velocity, p is the pressure, G is the universal gravitational constant, M is the massof the planet, γ is the adiabatic constant, q is the volume heating rate (more detailsof heating rate are given in section 4.3), κ is the thermal conductivity, and T is thetemperature, where T = p/ρR and R is the molar gas constant. ρur 2 is a measureof the globally integrated escape rate and must be a constant in space in the steadystate. κ is given by a function of temperature (Watson et al., 1981)κ = κ0



T
T0

23

s

(4.4)

4.1. BASIC EQUATIONS FOR SINGLE-CONSTITUENT ATMOSPHERE

24

where κ0 is 4.45 × 104 [ergs/cm·s·K] (Hanley et al., 1970), T0 is temperature at thebottom of atmosphere, and s is about 0.7 for a primordial atmosphere (Banks &Kockarts, 1973). In practice, these equations are reduced to the following set of
equations to be solved by numerical method:∂n∂n∂u 2nu
+u
= −n
−
∂t
∂r
∂r
r
∂u
1 ∂p GM
∂u
+u
=−
− 2
∂t
∂r
nm ∂r
r
!
∂p
∂p
∂u 2γpu
γ −1 ∂
2 ∂T
κr
+u
= −γp
−
+ (γ − 1)q +
∂t
∂r
∂r
r
r 2 ∂r
∂r

(4.5)
(4.6)
(4.7)

where, n is the number density, m is the mass of the molecule.
If the gas flow velocity is greater than the sound speed, the flow is supersonic. In
the case of hydrodynamic escape, the flow is slow at the lower boundary (subsonic)
and fast at the upper boundary (usually supersonic). Tian et al. (2005) used LaxFriedrichs scheme (Toro, 1999; De Sterck et al., 2001) to solve the system. The firstorder Lax-Friedrichs scheme was used to solve the hydrodynamic equations. To solve
the transonic flow, however, the Lax-Friedrichs scheme is not the optimum scheme. I
use CIP method (Yabe and Aoki, 1991; Yabe et al., 2001) alternatively which is well
suited to solve the transonic flow. Details of the CIP method are given in Appendix
A.

When using an explicit time-marching method, the maximum time step is constrained by the CFL condition, which is dependent on the size of the grid cells. The
distribution of grid cells in the model is not uniform. The grid size decreases exponentially with decreasing r to compensate for the nearly exponential increase ofdensity near the lower boundary (Keppens & Goedbloed, 1999). A global time stepis then determined as the minimum of the local time steps, which can be definedby applying the CFL condition to each grid cell separately. In the numerical model,
two ghost cells each are located below the lower and above the upper boundaries.
Boundary conditions are imposed in the ghost cells, or the variables are extrapolated
linearly to the ghost cells if there is no boundary condition to be imposed. The initialconditions in the numerical model are set so that density is proportional to r 2 , andvelocity is set to be a constant for all grid cells. It is note that the steady state results
are insensitive to the initial conditions. Convergence to a steady state is measured

CHAPTER 4. SINGLE-COMPONENT NUMERICAL MODEL

25

by the normalized difference of all variables between time steps,
v
u
u
u
∆=u
t

N
var
X



Ngrid

1
 1

Nvar i=1 Ngrid grid=1
Xδin+1 − δin(δin )2

2 



(4.8)

Here n represents the time level, Nvar is the number of variables in the system, andNgrid is the number of grid cells in the system. Sensitivity tests show that if thesteady state tolerance is set to a value that is not small enough, the steady statesolution may not be approached sufficiently (Tian et al., 2005). In the model thesteady state tolerance is set at ∆ = 10−5 .

4.2Energy Deposition of Solar EUV Radiation

Assumption that heating due to solar EUV radiation occurs only in a single layer is
not necessarily a good approximation, because the absorption cross-section depends
on wavelength. The unity optical depth level occurs at a different altitude for eachwavelength. As a result, the energy absorption altitude distribution may have acomplicated structure. In the case of Earth’s atmosphere, most of the energy isabsorbed at altitudes above 1000 km, and the double-peaked feature between 103 and104 km is caused by the wavelength dependence of the energy deposition (Chassefiére,1996). The energy deposition levels off and then starts to increase at around 105
km because of the geometric effect (Chassefiére, 1996) of the increasing size of theextended planetary atmosphere. The hydrogen molecule escape rates corresponding
to different heating layer positions and energy deposition rates are calculated for theatmosphere of a Earth-like planets, with the lower boundary hydrogen density is set
to be 1020 molecules/m3 (Fig. 4.1). These nonlinear relations between escape rates
and the position of the heating layer under the single-layer heating are also shownby Tian et al. (2005). Thus, the escape rates depend on the position of the heating
layer as well as on the absolute amount of energy input. It is difficult for a modelwith a single heating layer to represent this dependence accurately.
For the application to the primordial atmospheres, I assume that the dominant
gas in the upper atmosphere is hydrogen and the only energy source to drive thehydrodynamic escape is the solar EUV radiation below 105 nm. The solar EUV
spectrum is taken from Woods & Rottman (2002). The spectrum of solar EUV
radiation entering the top of the atmosphere is based on a 1994 rocket measurementfrom 0 to 105 nm (Woods & Rottman, 2002). The absorption cross-section data for4.2. ENERGY DEPOSITION OF SOLAR EUV RADIATION

26

Figure 4.1: Nonlinear relations between escape flux rates and the position of the heating layer under the single-layer heating assumption such as Watson et al. (1981). The
escape flux rates are normalized to the value at the bottom of atmosphere (100km).
Input EUV energy is double for the upper green line. The nonlinear relation also
depends on the amount of input EUV energy.

CHAPTER 4. SINGLE-COMPONENT NUMERICAL MODEL27

H2 gas are from Avakyan et al. (1998). In Appendix B.1 and B.2, the data arepresented on the spectrum of solar EUV radiation (Woods & Rottman, 2002) andcross-sections of photoabsorption for molecular hydrogen (Avakyan et al., 1998) at
0.1 to 105 nm. The interval numbers are given according to the practice accepted in
the aeronomic calculations (Richards et al., 1994). Insignificant Rayleigh scattering
is neglected. The same solar spectra and absorption cross-section data are used forall simulations in this paper.
All energy deposition calculations in the literature assume that solar radiationenters the top of the atmosphere isotropically. To account for the angular distributionof solar radiation, the mass path is usually calculated for an average solar zenith angleof about 60◦ (Kasting & Pollack 1983; Chassefiére, 1996). Although this approach isa good approximation for a thin, plane-parallel atmosphere, it is not appropriate fora dense, extended atmosphere, in which radiative transfer in spherical geometry isneeded. Because of the lack of spherical symmetry, the problem of radiative transfer inan extended spherical atmosphere is complicated, and easy analytical expressions are
not available (Sobolev, 1975). Then, a two-dimensional energy deposition calculation
method (Tian et al., 2005) is adapted to provide a more accurate approximation of
the complicated radiative transfer processes in an extended spherical atmosphere.
Details of the geometrical consideration are described in Appendix B.3.
Not all energy absorbed by hydrogen molecules will be used to drive hydrodynamic
escape flow. Some will be lost to space through thermal emission. Although thethermal emission process is not included in this paper, a heating efficiency (η) is
defined as the percentage of energy absorbed by molecules that is effectively converted
into the kinetic energy of the molecules. The heating efficiency was fixed at 15% inWatson et al. (1981) and Tian et al. (2005). I use the same value in this study
as the first approximation, where I discuss transonic hydrodynamic escape from theatmosphere of the terrestrial planet. We should discuss in more detail the importanceof different heating efficiency values for the simulation results in future work.
The hydrodynamic escape model used in this paper is a simple model, so the temperature is fixed to a constant at the lower boundary. A variation of solar EUV input
will probably have an influence on the lower boundary temperature, and thus will
have a more complicated impact on the hydrogen escape rate than we have considered. Intuitively a higher temperature will result in a stronger escape flux. Because
the temperature, as well as the density at the lower boundary, is influenced by lower
atmosphere dynamics and radiative transfer processes, other models working in the4.3. EVOLUTION OF THE SOLAR ACTIVITY

28

lower part of the atmosphere are required to accurately determine these parameters
for the primordial atmospheres. It will be an important future study to include a
detailed photochemical model solving the energy absorption, radiation, and net deposition.

4.3

Evolution of the Solar Activity

Because of ever accelerating nuclear reactions in its core, the Sun is a slowly evolving
variable star that has undergone about 40% increase in luminosity over the last 4.5
Gyr, as predicted by the standard solar evolution model (e.g., Girardi et al., 2000).
The fundamental question of whether the Sun has always been a relatively inactive
star or, in contrast, has experienced some periods of stronger magnetic activity has astrong impact on the evolution of the solar system. Compelling observational evidence(Güdel et al., 1997) shows that zero-age main-sequence solar-type stars rotate over
10 times faster than today’s Sun. As a consequence of this, young solar-type stars,
including the young Sun, have vigorous magnetic dynamos and correspondingly strong
high-energy emissions. Simon et al. (1985), and others showed that the Sun loses
angular momentum with time via magnetized winds (magnetic braking), thus leading
to a secular increase of its rotation period (Durney, 1972). In response to slower
rotation, the solar dynamo strength diminishes with time, causing the Sun’s highenergy emissions also to undergo significant decreases.
“The Sun in Time” program (Dorren & Guinan, 1994) was established some 20
years ago to study a sample of accurately selected solar proxies (G0-G5 V stars)
with different ages across the electromagnetic spectrum. The Sun in Time sample
contains nearby single (or widely separated binary) G0-G5 stars that have known
rotation periods and well-determined temperatures, luminosities, and metallicities.
In addition, we have been able to estimate the stellar ages using their memberships inclusters and moving groups, rotation period-age relationships, and, for the older stars,
fits to stellar evolution models. Data from the following space missions have been usedin the present study: the Advanced Satellite for Cosmology and Astrophysics (ASCA),
the Rontgen Satellite (ROSAT), the Extreme Ultraviolet Explorer (EUVE), the Far
Ultraviolet Spectroscopic Explorer (FUSE), the Hubble Space Telescope (HST), andthe International Ultraviolet Explorer (IUE). Ribas et al. (2005) present the resultsof the investigation of the long-term magnetic history of the Sun focusing on the highenergy emissions using the data obtained with the ASCA, ROSAT, EUVE, FUSE and
IUE satellites from a homogeneous sample of single nearby G0 V main-sequence stars

CHAPTER 4. SINGLE-COMPONENT NUMERICAL MODEL

29

Table 4.1: Integrated Fluxes
λ Interval (Å)
1-20
20-100100-360
360-920920-1180

0.10Gyr 0.30Gyr 0.65Gy 1.6Gyr 4.56Gyr 6.7Gyr
180.2
21.5
7.76
0.976
0.15
0.048
82.4
14.8
10.7
2.80
0.70
0.321
187.2
69.4
22.7
7.7
2.05
1.37
45.6
15.2
7.0
2.85
1.00
0.68
18.1
8.38
2.90
1.70
0.74
0.50

Note. Units of fluxes are [erg/cm2s]. The data are normalized to a distance of 1AU and to the radiusof a 1 M⊙ star, and scaled (using a simple radius-squared relationship) to the expected radius of
the Sun at the star’s age (Ribas et al., 2005). The intervals considered are 1-20 Å(ASCA), 20-100
Å(ROSAT), 100-360 Å(EUVE), and 920-1180 Å(FUSE). The fluxes in 360-920 Åare calculated fromthe observed solar flux.

which have known rotation periods and well-determined physical properties, including
temperatures, luminosities, metal abundances and ages. The resulting stellar fluxes
are presented in Table 4.1.
From the data from these space missions, we can plot the total integrated fluxes
and a power-law fit that yields an excellent fit to the data (Ribas et al., 2005). In
the complete wavelength interval, which is frequently used in aeronomy calculations,the flux (F [ergs/cm2 s]) as a function of stellar age (τ ) is accurately reproduced by
the expression
F = ατ β(4.9)

where the stellar age τ is in gigayears. Parameters of the power-law, α and β, arepresented in Table 4.2. Also illustrative is the plot in Figure 4.2, which represents thestellar fluxes transformed into the values at Venus orbit normalized to the current
solar values as a function of age. The steeper decrease of the higher energy emissions
is evident in this plot. Also note that the emissions of the youngest stars in all
wavelength intervals are orders of magnitude larger than the current solar flux. I
use the power-law fits of solar fluxes to estimate the energy deposition by solar EUV
heating in numerical calculations.

4.3. EVOLUTION OF THE SOLAR ACTIVITY

30

Figure 4.2: (top) Smoothed fits to the integrated irradiances in Table 4.1 for different
wavelength intervals. The flux density is transformed into the values at Venus orbit.
The parameters of the resulting power-law best-fitting relationships are given in Table
4.2. (bottom) Present solar-normalized fluxes vs. age for different stages of theevolution of solar-type stars for different wavelength intervals.

CHAPTER 4. SINGLE-COMPONENT NUMERICAL MODEL

31

Table 4.2: Parameters of the Power-Law Fits to the Measured Integrated Fluxes
λ Interval (Å)
1-20
20-100100-360
360-920920-1180

α
2.40
4.45
13.5
4.56
2.53

β
-1.92
-1.27
-1.20
-1.00
-0.85

Chapter 5
Validation of Numerical Model
5.1

Validation in an Isothermal Atmosphere

The solution technique is validated in an isothermal atmosphere, polytropic atmosphere, and adiabatic atmosphere. Here I validate my solution technique against
analytical solutions for an isothermal atmosphere. Similar validation test can befound in Keppens & Goedbloed (1999) and Tian et al. (2005). In an isothermal
atmosphere, Eq. (4.5)-(4.7) can be reduced to the density and the momentum equations. An analytical solution with a nonzero velocity distribution is available (Parker,1964). Details of the analytical solution are described in Appendix C.1. Figure 5.1
shows that the resulting velocity of numerical solution is a good approximation tothat of the analytical Parker solution with a few percent uncertainty (The numbersof grid cell are set to 1000).
In the numerical model, the density is imposed, and the mass flux is extrapolated
at the lower boundary. The ratio of the gravitational potential energy and the molecular kinetic energy is important to identify the dynamic characteristics of the escape.
The traditional Jeans escape parameter is defined by parameters at the exobase level.
In an atmosphere in which hydrodynamic escape occurs, the location of the exobase
is unknown prior to the simulations, so we define the hydrodynamic escape parameter (HEP) using the parameters at the lower boundary level: HEP = GMm/r0 kT0 ,where k is Boltzmann’s constant, and r0 and T0 are the values of r and T in Eq.
(4.5)-(4.7) at the lower boundary, respectively. When the kinetic energy of molecules
is comparable to their gravitational potential energy, the HEP value is small and theescape flow is strong. When the gravitational energy is much larger than the kinetic
energy, the HEP is large and the escape flow is weak.
Figure 5.2 shows that for different HEP values, the analytical solutions are approached increasingly well by the numerical solutions when the number of grid cells
33

5.2. VALIDATION IN A POLYTROPIC/ADIABATIC ATMOSPHERE

34

Figure 5.1: Comparison between the numerical solution and the analytical solutionof velocity for the HEP = 5 case in an isothermal atmosphere.

is increased. In the case of hydrogen-dominated atmospheres, the simulation process
is concluded if the escape rate in the steady state does not change by more than 10%
when the number of grid cells is doubled. In this study, the number of grid cell is set
to 3000, which presents enough resolution to consider the Venus’ problem (Chapter
8 and 9).
Figure 5.3 shows that the normalized difference of all variables between time steps(Eq. (4.8)) converge a steady value in a few months. This means that it takes only
a few months from the initial state to converge a steady value.
These simulations of an isothermal atmosphere produce a good approximation of
the steady state hydrodynamic escape rate and large-scale atmospheric structure ina short timescale.

5.2

Validation in a Polytropic/Adiabatic Atmosphere

Next, I validate my solution technique against analytical solutions for polytropic
atmosphere and adiabatic atmosphere. In a polytropic atmosphere, Eq. (4.7) can be

CHAPTER 5. VALIDATION OF NUMERICAL MODEL

35

Figure 5.2: Comparison between the numerical solution and the analytical solutionof flux. Analytical solutions of hydrodynamic escape from an isothermal atmosphere
are approached when the number of grid cells is increased. The vertical axis is therelative deviation of the escape flux in the numerical solution compared with that in
the analytical solutions for different HEP values.

5.2. VALIDATION IN A POLYTROPIC/ADIABATIC ATMOSPHERE

36

Figure 5.3: Normalized difference of all variables between time steps for the HEP =5 case in an isothermal atmosphere.

replaced by the polytropic relation
T = T0



n
n0

α−1

(5.1)

where α is the polytropic index. In the special case of adiabatic outflow of an idealgas, α = cp /cv . With the polytropic assumption, Euler equations can be solved quasianalytically. Details of the quasi-analytical solution are described in Appendix C.2.
Figure 5.4 shows that the resulting density, velocity, and temperature of numericalsolution are good approximations to those of the quasi-analytical solution.
In an adiabatic atmosphere, the volume heating rate q in Eq. (4.7) is zero. Figure
5.5 shows that the resulting density, velocity, and temperature of numerical solution
for adiabatic atmosphere are very good much with those for polytropic atmospherein the case of α = cp /cv .
These validation results show that the numerical method is capable of accurately
solving equation (4.1)-(4.3) for transonic flows.

CHAPTER 5. VALIDATION OF NUMERICAL MODEL

37

Figure 5.4: Comparison between the numerical solutions and the quasi-analytical solutions of polytropic atmosphere for different polytropic index. Normalized hydrogen
number density, normalized velocity, and temperature are shown from above to bottom. Polytropic indices are 1.1 (red, purple), 1.2 (blue, cyan), 1.3 (yellow, brown),
and 1.4 (green, black).

5.2. VALIDATION IN A POLYTROPIC/ADIABATIC ATMOSPHERE

38

Figure 5.5: Comparison between the numerical solutions of polytropic atmosphereand adiabatic atmosphere. Normalized hydrogen number density, normalized velocity,and temperature are shown from above to bottom. Polytropic index is 1.1. Adiabatic
solutions are shown by broad red lines for convenience of comparison of two lines.

Chapter 6
Multi-Component Numerical
Model
Hydrodynamic escape of hydrogen from a planetary atmosphere can remove heavier
gases, which can be dragged along with the escaping hydrogen (Hunten et al., 1987).
The heated hydrogen escapes to space, and in the process its outward flow exerts
upward drag forces on heavier atmospheric constituents. Figure 6.1 illustrates themechanism of that for an atmosphere heated at high altitude by intense EUV radiation. Collisions of outward bound H2 molecules with other atmospheric constituentsexert net upward drag forces on them. Equilibrium between this upward drag andgravity leads to an increased average height for these species. However, if the H2
escape flux is large enough, drag wins for Ne and it is entrained with the escaping H2
and swept out of the atmosphere; progressively higher fluxes also remove Ar, Kr, and
Xe. Under a constant H2 escape flux, lighter species are more readily entrained andlost with the escaping hydrogen, leading to mass fractionation of the residual atmosphere. The importance of hydrodynamic escape, in the context of observed noble gas
isotope distribution, is that it can implement loss and fractionation of species more
massive than Ne which could not escape so much on their own by classical processes
such as Jeans escape (Chamberlain & Hunten, 1987).

6.1

Extension of Hydrodynamic Escape CodeThe one-dimensional time-dependent nonviscous hydrodynamic equations for eachcomponent in a multi-constituent atmosphere with thermal conduction in sphericalgeometry are deviated from Eq. (4.5)-(4.7):∂n∂u 2nu
∂n
+u
= −n
−
∂t
∂r
∂r
r

(6.1)
39

6.1. EXTENSION OF HYDRODYNAMIC ESCAPE CODE

40

Figure 6.1: Hydrodynamic escape in a hydrogen-rich primordial planetary atmosphere, illustrated for input of external energy at high altitudes. Exospheric H2 isheated to escape velocity by adsorption of strong solar EUV radiation. Escapinghydrogen is replenished by rising hydrogen wind. Collisions of outward bound H2
molecules with other atmospheric constituents (illustrated here for Ne and Xe) exert
net upward drag forces on them. This figure is after Fig. 4 in Pepin (2006).

CHAPTER 6. MULTI-COMPONENT NUMERICAL MODEL
"

nx ∂T
∂u
1 ∂p GM X
αT k
∂u
+u
=−
− 2 +
(ux − u)nx ki −
∂t
∂r
nm ∂r
r
m nx + n ∂r
x
!
∂p
∂T
∂p
∂u 2γpu
γ−1 ∂
κr 2
+u
= −γp
−
+ (γ − 1)q + 2
∂t
∂r
∂r
r
r ∂r
∂r

41
!#

(6.2)
(6.3)

where, physical quantities labeled with subscript x means the corresponding quantities for the other constituents besides itself. In the momentum equation, the effectsof collisions with other species are included in the last two terms (Zahnle & Kasting,1986). The first of these accounts for the transfer of momentum from one mean flow
to the other; the parameter ki stands for the momentum transfer collision rate. The
last term accounts for thermal diffusion; αT is the thermal diffusion factor, and k
is Boltzmann’s constant. In Table B.3, the relevant binary diffusion coefficients arecollected. The binary diffusion coefficient bi is identified with the quantity kT /mki .
Thermal diffusion concentrates the heavier constituent at lower temperatures. Thus itaids escape if temperature drops with altitude; it hinders escape for positive temperature gradients. The thermal diffusion factor αT is given by some references (Mason
et al., 1966; Chapman & Cowling, 1970). However, the inclusion of thermal diffusion
is not expected to have a large impact on calculation actually. On the other hand,the fact that bi is itself temperature dependent can have a much larger effect.

6.2

Crossover Mass

Here, it is convenient to define the crossover mass mc between hydrogen and component x as follows (Hunten et al., 1987):
mc = mH2 +kT FH2bi gXH2(6.4)

where, mH2 is the mass of hydrogen, k is the Boltzmann constant, T is the temperature, FH2 is the flux of hydrogen, bi is the binary diffusion coefficient betweenhydrogen and component x, g is the acceleration due to gravity, and XH2 is the mole
fraction of hydrogen. For mx = mc (mx is the mass of component x), the composition
is still independent of height; the drag terms are just sufficient to equalize the density scale heights, but not to lift constituent x out of the atmosphere (Hunten, 1973,1985). If mx is greater than mc , then the flux of component x remains zero, but thedensity scale heights are no longer the same. For masses smaller than the crossover
mass, constituent x is swept out of the atmosphere with a flux proportional to themole fraction and linearly varying with mass. This behavior is easily understood interms of the competition between the downward weight of an atom or molecule and6.2. CROSSOVER MASS

42

the upward drag force, which has been taken to be independent of mass by the assumption of a universal value for bi . At the crossover mass, the two forces are equal;
lighter masses feel a net upward force, and heavier ones a downward force.
It is note that the crossover mass is the approximate solution. A more complete
treatment of the problem would replace the solution of hydrodynamic equation. However, Zahnle & Kasting (1986) considered a more general case, and verify it is a good
approximation for escape of a trace constituent provided thatmH2
mc − mx>
mc − mH2
mx(6.5)

which shows that the expressions of crossover mass are nearly correct when the difference in the masses of the two constituents is large. A similar argument is derived
by Sasaki & Nakazawa (1988). For noble gases escaping in a hydrogen-dominatedatmosphere, Eq. (6.4) is good for Ne, Ar, Kr, and Xe; it is less good for He because He is only weakly bound in terrestrial planet atmospheres and escape readily
by hydrodynamic escape itself or by other processes. So I use the crossover mass for
relatively heavy elements to roughly estimate the minimum required flux of hydrogen
for Venusian atmosphere (chapter 8 and 9).

Part II
Application to the Early Venusian
Atmosphere

43Chapter 7
Introduction to the Problems
7.1

Introduction to Water Loss from Early Venus

Venus is free of oxygen but it is thought to have contained a primordial water ocean,giving rise to a dense steam atmosphere and a subsequent strong greenhouse effect.
On the basis of the analysis of the Pioneer Venus large-probe neutral mass spectrometer data it was found that Venus’ atmosphere is enriched in D over H relative to
Earth by a factor of about 120±40 (de Bergh et al., 1991). They report a D/H ratio
in the Venus clouds of (1.6 ±0.2)×10−2 , or about 100 times higher than the terrestrial
value of 1.56 × 10−4 . If Venus and Earth had similar D/H ratios initially, this implies

that Venus must have started out with at least 100 times as much water as is now
present.
Geiss & Reeves (1981) have suggested that this isotopic fractionation is a resultof low-temperature equilibrium reactions between gaseous and condensed phases ininterstellar space prior to the formation of the solar nebula. Thus, the simplest solution is to assume that the initial D/H ratio on Venus was equal to the present value.
However, theories of planetary formation indicate that any large spatial gradient ofisotopic fractionation should have been smoothed out due to mixing during the accretion phase (Wetherill, 1981). And H2 O could have been present also on early Venus
in approximately terrestrial amounts. So, lacking a plausible chemical separation
mechanism during the formation phase of the planetary atmospheres which would
favor a water-rich early Earth and very dry Venus, one can hypothesize that early
Venus may have had a comparable H2 O inventory as the Earth.
So we should consider a hypothesis that the present lack of water is supposed tobe due to photodissociation of H2 O followed by thermal escape of H at early stage.
Matsui & Abe (1986) and Abe & Matsui (1986) showed that due to thermal blanketing
during the first 108 years of a planet’s life, temperatures as high as 1500 K might45

7.2. INTRODUCTION TO FRACTIONATION OF NOBLE GASES ON VENUS46

be reached in the steam atmosphere heated by impacts. A part of H2 O is dissolvedin a hypothesized magma ocean. In the following cooling phase, H2 O condenses
and a water ocean is formed. The climate models of Kasting (1988) indicated thatthe runaway greenhouse effect occurs under these conditions nearly independentlyof the amount of CO2 in the atmosphere. At high temperature, water vapor may
be present in large amounts in the thermosphere and hydrodynamic escape of H,
produced by photodissociation of H2 O, quite efficient. Ingersol (1969) has found that
if the H2 O mixing ratio at the mesopause exceeds a value of about 0.1, water vapor
would have remained a major atmospheric species up to altitudes where it could be
photodissociated and atomic hydrogen became the main species which could escapehydrodynamically. The atmosphere at this time was therefore rich in water vapor,yielding H and H2 by photodissociation.
Part of the oxygen can be dragged off to space along with the escaping hydrogen,but if not, or if O escape is only partial, an O2 -rich atmosphere can accumulate during
the early history of the planet. On the other hand, we know that there are negligible
quantities of O2 in the present Venus atmosphere. The lack of oxygen on Venus ispuzzling. So, in the first instance, to pursue the idea if all oxygen can be escaped by
H2 drag in a quantitative fashion, one must examine in detail the processes by whichhydrogen has escaped to space.
In chapter 8, I apply the developed numerical model to the escape of hypothetical
early Venusian atmosphere to consider the water loss from early Venus.

7.2

Introduction to Fractionation of Noble Gases
on Venus

How and from what sources the atmospheres of Earth, Mars and Venus formed, andhow they evolved to their present states, are classic problems in the planetary sciences.
Carbon dioxide is the principal species on Mars and Venus; on Earth, it’s nitrogen.
One might expect that these “sister” planets, not very different in mass or distancefrom the sun, would have acquired similar primordial atmospheres in their youth andfollowed similar evolutionary paths to the present. These primordial atmospheres
were probably acquired in the same way from the same reservoir, not surprising
for two planets with comparable masses and heliocentric distances. However, thedifference between Venus and the other two sister planets is clearly demonstrated.
Table 7.1 lists the elemental abundances of noble gases, hydrogen, and carbon
in solar, meteorites, comets, and atmospheres of the terrestrial planets and known

1

a

Solar
Meteorites
CI chondritesb
E chondritesc
Cometsd
Planets
EartheVenusfMarsg

H
7.58 (-01)
6.68±0.83 (-03)
6.8±1.5 (-02)
3.10 (-05)

12C
3.90 (-03)

20Ne
1.56 (-03)

36Ar
8.97 (-05)

84

130

Kr
5.84 (-08)

Xe
8.07 (-10)

3.7±0.70 (-02) 2.89±0.77 (-10) 1.25±0.10 (-09) 3.57±0.15 (-11)
4.2±1.6 (-03)
2.0±2.0 (-11)
3.3±2.8 (-10)
3.0±2.7 (-12)
1.7±0.3 (-01)< 6.2 (-07)
3.4 (-05)
2.6 (-06)

7.0±1.9 (-12)
4.1±2.9 (-13)
9.5 (-09)

1.0+0.7−0.6 (-05) 1.00±0.01 (-12) 3.45±0.01 (-11) 1.66±0.02 (-12) 1.40±0.02 (-14)
2.60±0.04 (-05)
2.9±1.3 (-10) 2.51±0.97 (-09)
4.7+0.6
8.9+2.5−0.4 (-12)−6.8 (-14)
1.11±0.20 (-08) 4.38±0.34 (-14) 2.16±0.43 (-13) 1.76±0.10 (-14) 2.08±0.27 (-16)

Note. Powers of ten multipliers in parentheses.
aAverage of data from Cameron (1982) and Anders & Ebihara (1982).
b
Kerridge (1985) for hydrogen and carbon; Mazor et al. (1970) for noble gases.
cGrady et al. (1986) for carbon; Crabb & Anders (1981) for noble gases.
dDelsemme (1988) and Dauphas & Marty (2002) for hydrogen and carbon in comets; Owen et al. (1992), Bar-Nun & Owen (1998), and Dauphas (2003)
for measurements of noble gases trapped in amorphous water ice (55K).
e
Rubey (1951) for hydrogen (from H2 O); Walker (1977) for carbon (uncertainties include the lower estimate of Rubey (1951) and the higher estimateof McElroy et al. (1977)).
f
von Zahn et al. (1983) for carbon, neon, and argon; Donahue & Pollack (1983) for krypton; Donahue (1986) for xenon.
g
Owen et al. (1977) Viking data for carbon; average of data from Wiens et al. (1986) and Swindle et al. (1986) for neon; Wiens et al. (1986) for argon;
average of data from Becker & Pepin (1984) and Swindle et al. (1986) for krypton and xenon.

CHAPTER 7. INTRODUCTION TO THE PROBLEMS

Table 7.1: Elemental Abundances in Solar, Meteorites, Comets and Terrestrial Planet Atmospheres and Known Crustal Reservoirs

477.2. INTRODUCTION TO FRACTIONATION OF NOBLE GASES ON VENUS48

Table 7.2: Elemental Abundances Normalized by Averaged Solar Abundances
1

Solar
Meteorites
CI chondrites
E chondrites
CometsPlanets
Earth
Venus
Mars

H
1

8.81±1.09 (-03)
9.0±2.0 (-02)
4.09 (-05)

12

C
1

20

Ne
1

36

Ar
1

84

Kr
1

9.5±1.8 (0) 1.85±0.49 (-07) 1.39±0.11 (-05) 6.11±0.26 (-04)
1.1±0.4 (0)1.3±1.3 (-08)
3.7±3.1 (-06)
5.1±4.6 (-05)
4.4±0.8 (+01)< 4.0 (-04)
3.8 (-01)
4.5 (+01)

130

Xe
1

8.7±2.4 (-03)
5.1±3.6 (-04)
1.2 (+01)

2.6+1.8−1.5 (-03) 6.41±0.06 (-09) 3.85±0.01 (-07) 2.84±0.03 (-05) 1.73±0.02 (-05)
6.67±0.10 (-03)
1.9±0.8 (-07) 2.80±1.08 (-05)
8.0+1.0
1.1+0.3−0.7 (-05)−0.8 (-04)
2.85±0.51 (-06) 2.81±0.22 (-11) 2.41±0.48 (-09) 3.01±017 (-07) 2.58±0.33 (-07)

CHAPTER 7. INTRODUCTION TO THE PROBLEMS

49

Figure 7.1: Relative abundances of noble gases in meteorites and the terrestrial planet
atmospheres with respect to solar abundances. All the data used in this figure arelisted in Table 7.2

crustal reservoirs (Pepin (1991) and references therein). The solar abundance represents the nebula one. The hydrogen and carbon contents on the atmospheres of
the terrestrial planets are important as well as noble gases, because they are major
components of the atmospheres of the terrestrial planets in the forms of CO, CO2 ,H2 O, and so on. Table 7.2 lists the elemental abundances normalized by averaged
solar abundance, and Figure 7.1 is plotted from the noble gases’ values listed in Table
7.2. On Venus, not only is the heavy noble gas abundance pattern different from thaton Earth and Mars, but also the abundances of both argon and neon are remarkably
high (Fig. 7.1).
Free from the entanglements of chemical interactions, compositions of noble gasses
are clues to the characteristics of their source reservoirs, and are ideal recorders ofmass fractionations imprinted by physical processing such as escape from planetarygravitational fields. Information of noble gasses abundances for Venus comes fromatmospheric measurements by mass spectrometers on the Venera and Pioneer Venus
spacecraft (Wieler, 2002). Venus may have lost some Ne, since it is enriched about
10% in 22 Ne/20 Ne with respect to solar abundances. On the other hand, Venus
appears to have lost no Ar, as indicated by near solar 36 Ar/84 Kr and 38 Ar/36 Ar ratios(Pepin, 1991). Nonradiogenic argon (36 Ar, 38 Ar) is extremely abundant on Venus

7.2. INTRODUCTION TO FRACTIONATION OF NOBLE GASES ON VENUS50
compared to Earth. On a gram per gram basis, 36 Ar is more abundant on Venus thanin all but a few of the most gas-rich meteorites (Pepin, 1991).
In chapter 9, I apply the developed numerical model to the escape of hypothetical
early Venusian atmosphere to consider the fractionation of noble gases on Venus.
Note that the assumed atmospheres in chapters 8 and 9 are different ones. The
assumed atmosphere in chapter 8 consists of only dissociated H2 O, i.e., the abundance
ratio of hydrogen and oxygen is about 1:1. The case that hydrogen, which would beoriginated from solar nebula and/or degassing atmosphere, is much more abundant
than oxygen is not considered in this study. On the other hand, the assumed atmosphere in chapter 9 consists of hydrogen as a major component and noble gases as
minor components. And CO is treated as a parameter (minor component to major
component). Therefore, these two problems are treated independently in this study.

Chapter 8
Water Loss from Early Venus
8.1

Previous Studies

Watson et al. (1981) have investigated hydrodynamic escape from Venus under theassumption that the upper atmosphere consisted of pure molecular hydrogen. They
found that XUV-driven energy-limited hydrodynamic conditions would have resulted
in the upper limit of the hydrogen escape flux of about 1012 /cm2 s. Such an escape flux
would evacuate the amount of a terrestrial water ocean in about 280My. However,Kasting & Pollack (1983) has pointed out that their calculation had several weak
points. The escape flux calculated from their model depends sensitively upon thenumber density of hydrogen, which is treated essentially as a free parameter. Clearly,
if one wishes to calculate the actual escape rate instead of simply an upper boundone must construct a model, which calculates thermospheric hydrogen densities selfconsistently.
The hydrodynamic escape of hydrogen from Venus has been studied by Kasting& Pollack (1983) through a detailed one-dimensional photochemical-dynamic modelof the high atmosphere. Their escape flux is approximately four times lower than theenergy-limited flux calculated by Watson et al. (1981), because it is source limited at
the base of the expansion. Kasting & Pollack (1983) showed that if water vapor wasa major constituent of Venus’ upper atmosphere, an amount of hydrogen comparable
to that contained in Earth’s oceans could have escaped over a time span of no morethan a few hundred million years. They indicated, however, some of the oxygen left
behind from the escape of hydrogen, and they would be consumed by oxidation of
surface minerals. And then, by considering that H primitive escape is limited byavailable solar EUV energy, Chassefiére (1996) have confirmed the finding by Kasting& Pollack (1983), showing that planets like Venus or Mars are unlikely to have lostan important fraction of their oxygen, swept away together with hydrogen. However,51

8.2. NUMERICAL MODEL

52

their models are not valid for a transonic flow; and the actual outflow would have beentransonic. On the other hand, Zahnle & Kasting (1986) addressed the problem witha hydrodynamic escape model similar to that developed by Kasting & Pollack (1983)
but eliminating that model’s most obvious deficiencies, especially the assumption of
subsonic flow. They concluded that substantial quantities of water may have been
lost without the need to oxidize large amounts of the crust, which is contrast to
Kasting & Pollack (1983).
As described above, many studies have been done to examine the problem “Water
loss from early Venus” so far. However, their solutions to this problem are to abandon the goal of finding the exact critical solution and to simply approximate it as
closely as possible by following subsonic paths. And their quantitative uncertainties
cause their conclusions of this problem to be all different each other. This problem
should be solved by use of the numerical method to derive the precise quantitative
solution. In Part I, I succeeded in developing the numerical code of multi-componenthydrodynamic escape for the first time. So, in this chapter, I apply the numericalcode to consider the water loss problem, and determine whether the complete water
loss from early Venus can be achieved or not.

8.2

Numerical Model

The model and the energy calculation method are applied to the atmosphere of a
Venus-like planet in which the thermosphere is composed only of hydrogen and oxygen molecules. The lower boundary is fixed at approximately the altitude of thehomopause (100km), and the lower boundary temperature is fixed at 330 K. The
upper boundary is at an altitude of roughly 50000 km. The solar EUV radiation flux
is set to four cases (0.05, 0.1, 0.5, 1.0 [Gyr]). I assume that hydrogen is the only
energy reservoir of solar EUV radiation for hydrodynamic escape. The hydrogen andoxygen number densities at the lower boundary are treated as free parameters. The
numerical conditions are summarized in Table 8.1.
In this study, constituent 1 is H2 and constituent 2 is O. The question of whether
constituent 2 could be H2 O or O2 may be raised in view of previous study (Kasting &Pollack, 1983). By considering their “case D” (Fig. 10 in their paper), the O mixing
ratio at the base of the expansion (=200 km altitude) is small due to the rapid
recombination of OH, produced by photodissociation of H2 O, with O, produced byphotodissociation of CO2 , which yields O2 and H. H2 O is nevertheless more abundant

CHAPTER 8. WATER LOSS FROM EARLY VENUS

53

Table 8.1: Definition of Numerical Parameters
Symbol
R
M
L
r0
rL
n0
T0
αT
η
τ

Meaning
Planetary radiusPlanetary mass
The number of numerical grid cells
Homopause altitude
Upper boundary altitude
Homopause hydrogen number density
Homopause temperature
Thermal diffusion factor
Heating efficiency
Age of the central star

Values
1RVenus = 6052 km
1MVenus = 4.869×1024 kg
3000 grid
RVenus +100 km = 6152 km
8RVenus = 48416 km
3.0 × 1014 ∼ 2.0 × 1016 /m3
330 K
0.3
0.15
0.05, 0.1, 0.5, 1.0 Gyr

than O2 by a factor of 3 at 200 km. The ratio n(H2 O)/n(H) is about 1/2 at 200 km,
and there is therefore 1 O atom for 2 H atoms at the base of the expansion. H2 O,
which is not much heavier than O, can be levitated by hydrodynamically escaping
H but, for a flow velocity of about 3 cm/s, it may be roughly estimated that H2 O
molecules will be photodissociated in the first 100 km above the base of the expanding
flow, O becoming the final form of escaping oxygen. Released OH radicals recombine
quickly with O, over less than 10 km height, to give O2 . The fate of O2 molecules in
the high atmosphere is to be photodissociated in about 1 day, providing additional O
atoms. Since solar EUV flux consumed in escape is deposited over a height of at least
1000 km, no major change is expected concerning O escape with respect to the caseof dominant O at the base of the expansion (“case A” of Kasting & Pollack (1983);
Fig. 9 in their paper). In “case A”, O is dominant at 200 km and is still in a ratio of
about 1/2 with respect to hydrogen. It may be therefore assumed that constituent 2
is oxygen. The conclusion is that escaping oxygen must be finally under the form of
atomic oxygen.

8.3

Results

Because solar radiation and particle fluxes play a major role in all atmospheric processes, the evolution of the Venusian atmosphere can only be understood within thecontext of the evolving solar energy and particle fluxes. Figure 8.1 shows the changeof hydrogen escape flux rate with varying solar age. I also calculate the requiredhydrogen escape flux rate to uplift oxygen, which is estimated roughly by use ofcrossover mass (Eq. 6.4). Under the lower limit of hydrogen flux, oxygen cannot

8.3. RESULTS

54

escape, so that the parameter of hydrogen homopause number density is selected toachieve enough hydrogen flux to be over the lower limit. The higher hydrogen escapeflux rate is obtained in the case of younger solar age and the case of higher hydrogenhomopause number density. The result is reasonable in view of physics. (This would
provide assurance that our numerical code is correct.)
Assuming that the primitive Venus had H2 as amount of terrestrial ocean, Fig.
8.1 indicates that timescales of water loss are 5.0 × 106 − 5.0 × 107 (0.05Gy), 1.5 ×

107 −1.5×108 (0.1Gy), 5.0×107 −5.0×108 (0.5Gy), and 1.5×108 −2.0×109 (1.0Gy).

In considering the water loss from Venus, it is reasonable to select the solar age as
several hundred million years because the runaway greenhouse effect occurs after abrief interval from the completion of Venus formation, and also because the hydrogendissociated from H2 O would continue to escape about a few hundred million years(Kasting & Pollack, 1983). In the following, I discuss the detail features of water loss
mechanism adapting the solar age as 1.0 billion years, which may correspond to thelast stage of water loss from Venus.
Figures 8.2, 8.3, and 8.4 show the steady state distributions of number density, gas
flow velocity, and temperature for the different homopause hydrogen number densities
in the case that the solar age is 1.0 Gyr.
The density distributions in the steady state of hydrodynamic escape from ahydrogen-oxygen-dominated atmosphere of a Venus-like planet are shown in Fig. 8.2.
The calculated number density distributions deviate severely from those in hydrostatic equilibrium (Tian et al. (2005)). Therefore, the atmospheric structure cannot
be described properly by the concept of the scale height (the altitude within which
the density drops by a factor of e). Thus, the traditional definition of the exobase (thealtitude at which the mean free path of particles is comparable to the scale height of
the atmosphere) is not appropriate, and an equivalent definition of the exobase is thealtitude at which the probability of collisions with the particles above the altitude isunity.
Near the upper boundary, the gas flow velocities approach and become super sonic(Fig. 8.3). Because of the decline of gravity with distance, near the upper boundary
the escape velocity approaches to the gas flow velocity. Because the escape velocity
will continue to decrease and the gas flow velocity remains super sonic beyond theupper boundary, it can be lost from the planet. In the case that n0 = 3.0 × 1014

[molecules/m3 ] (blue lines in Fig. 8.3), the column number density of hydrogen in
the model is so small that the hydrogen escape flux rate is not enough to accelerate

CHAPTER 8. WATER LOSS FROM EARLY VENUS

55

Figure 8.1: Hydrogen escape flux rate in the steady state of hydrodynamic escapeof Venusian atmosphere. Abundance of H2 and O is 1:1. Four solar age cases (0.05,
0.1, 0.5, 1.0 [Gyr]) are calculated. Hydrogen number densityies at the bottom of
the atmosphere are 3.0×1014 (blue), 1.0×1015 (green), 1.0×1016 (red), and 2.0×1016(purple) [molecules/m3 ]. Orange dashed line shows the roughly estimated lower limitof required hydrogen escape flux rate to uplift oxygen in the Venusian gravitational
field, which is calculated using crossover mass (see section 6.2).

8.3. RESULTS

56

H₂ = 2.0×10¹⁶
1.0×10¹⁶
1.0×10¹⁵
3.0×10¹⁴

Figure 8.2: Number density distributions in the steady state of hydrodynamic escapeof Venusian atmosphere. Solar age is 1.0 Gyr. Abundance of H2 and O is 1:1.
Hydrogen number densities at the bottom of the atmosphere are 3.0×1014 (blue),
1.0×1015 (green), 1.0×1016 (red), and 2.0×1016 (purple) [molecules/m3 ]. The thinnerlines correspond H2 and the thicker ones correspond O.

CHAPTER 8. WATER LOSS FROM EARLY VENUS

57

H₂ = 2.0×10¹⁶
1.0×10¹⁶
1.0×10¹⁵
3.0×10¹⁴

Figure 8.3: Gas flow velocity distribution in the steady state of hydrodynamic escapeof Venusian atmosphere. Solar age is 1.0 Gyr. Abundance of H2 and O is 1:1.
Hydrogen number densities at the bottom of the atmosphere are 3.0×1014 (blue),
1.0×1015 (green), 1.0×1016 (red), and 2.0×1016 (purple) [molecules/m3 ]. The thinnerlines correspond H2 and the thicker ones correspond O.

8.3. RESULTS

58

H₂ = 2.0×10¹⁶
1.0×10¹⁶
1.0×10¹⁵
3.0×10¹⁴

Figure 8.4: Temperature distribution in the steady state of hydrodynamic escape of
Venusian atmosphere. Solar age is 1.0 Gyr. Abundances of H2 and O is 1:1. Hydrogen
number density at the bottom of the atmosphere is 3.0×1014 (blue), 1.0×1015 (green),
1.0×1016 (red), and 2.0×1016 (purple) [molecules/m3 ].

CHAPTER 8. WATER LOSS FROM EARLY VENUS

59

the oxygen. Then next case (n0 = 1.0 × 1015 [molecules/m3 ], green lines in Fig.
8.3), the oxygen is accelerated sufficiently, which velocity is comparable to that ofhydrogen, with increasing the hydrogen number density (i.e., with increasing thehydrogen escape flux rate). Meanwhile, the gas flow velocity of hydrogen decreases
with increasing the drag effect by surrounding oxygen. Then next case (n0 = 1.0×1016[molecules/m3 ], red lines in Fig. 8.3), because the acceleration of hydrogen by EUV
heating is greater than the deceleration of hydrogen by surrounding oxygen’s drag,the gas flow velocities of hydrogen and oxygen increase together. Finally, In thecase that n0 = 2.0 × 1016 [molecules/m3 ] (purple lines in Fig. 8.3), with increasing

oxygen atoms in the atmosphere, downward acceleration of oxygen due to gravity
overcomes the upward lifting of oxygen by hydrogen hydrodynamic escape. Then,the power relationships between the acceleration and the deceleration are reversed
so that the gas flow velocity of oxygen decreases dramatically. On the other hand,since the acceleration of hydrogen by EUV heating is increasing, the gas flow velocity
of hydrogen continues to increase. These trends are also appeared in the density
distributions (Fig. 8.2).
The temperature (Fig. 8.4) varies with escape flow because the escape flow carries
out energy and produces adiabatic cooling. In addition to the hydrogen homopause
number density, there are two effects related to the energy deposition and distribution that affect the escape rate. First, the total amount of EUV radiation absorbed
is smaller in a low homopause density atmosphere than in a high homopause density atmosphere. Second, most of the energy absorption occurs at lower altitudes
when the homopause hydrogen density is low, which causes a large temperature gradient near the lower boundary. With a large temperature gradient, energy is quickly
lost through thermal conduction out of the lower boundary, which contributes to a
weak escape flow. These nonlinear effects combined together intricately produce theresulted escape flux rate of atmosphere.
Altough the temperature is fixed to a constant at the lower boundary in the model,
variation of the lower boundary temperature will have effects on the calculated hydrogen escape rate. Intuitively a higher temperature will result in a stronger escapeflux. Moreover, the calculated value of the hydrodynamic escape flux depends upon
the hydrogen number density in the lower thermosphere that, in turn, is controlled
by a combination of chemical, diffusive, and advective processes. Heating and coolingrates, heat capacities, and thermal conductivities are also functions of atmosphericcomposition. These complicated effects and relationships should be built in the numerical code to study the early Venusian atmosphere more precisely.

8.3. RESULTS

60

Figures (8.2) to (8.4) show the atmospheric structures of hydrogen-oxygen mixing
atmospheres. Then next, I discuss the results of atmospheric escape flux, which is
directly concerned with the water loss from early Venus. The hydrogen and oxygen
molecule escape flux rates corresponding to different abundances of H2 and O arecalculated for the Venusian atmosphere, with the lower boundary hydrogen densities
set to be parameter (Fig. 8.5). And Fig. 8.6 shows only oxygen molecule escape flux
rates corresponding to different abundances of H2 and O in one figure. Lines-points
in Fig. 8.5 show the range that oxygen can obtain the upward velocity accelerated
by hydrogen’s uplift effect as well as the escape flux rate of each molecule. In otherwords, the x-axis range where lines-points are not plotted means that oxygen has
the downward velocity (i.e., the oxygen molecule cannot escape from the Venusian
atmosphere) in this x-axis range. Out of this shown range, only hydrogen can escapefrom the Venusian atmosphere.
Figure 8.5 indicates the following three results. (1) Range of homopause hydrogen number density where oxygen can escape becomes narrower with increasing therelative abundance of oxygen to hydrogen in the Venusian atmosphere. (2) Absoluteamount of escaping hydrogen and oxygen becomes smaller with increasing the relative
abundance of oxygen to hydrogen in the Venusian atmosphere. This decline trend isshown obviously in Fig. 8.6. (3) Escape flux rate of oxygen is always smaller thanthat of hydrogen. Although there are narrow ranges where the escape flux rate ofoxygen and hydrogen are approximately same amount (homopause number density
is around 9.0 × 1015 [molecules/m3 ]), hydrogen is easy to escape in most hydrogen

number density ranges including the ranges where oxygen cannot escape.

According to the results, water loss from early Venus can be achieved if the homopause number densities of H2 and O are maintained around 9.0×1015 [molecules/m3 ]for a long time; in such a case, timescale of water loss as amount of terrestrial ocean
is about 1.5 × 108 years. Except in that case (i.e., when the homopause number den-sities vary with time), the hydrogen molecules escape more than oxygen moleculesfrom the water-rich Venusian atmosphere, and then the relative abundance of oxygento hydrogen inevitably increases with time. Moreover, the larger the relative abundance of oxygen to hydrogen is, the harder the oxygen escapes from the Venusian
atmosphere, so that the oxygen relative abundance would increase at an accelerating
pace, which is often called “positive feedback.”
Eventually, the results indicate that the complete water loss from early Venus by

CHAPTER 8. WATER LOSS FROM EARLY VENUS
Figure 8.5: Hydrogen (red) and Oxygen (green) escape flux rate in the steady state of hydrodynamic escape of Venusian atmosphere. Solar age is 1.0 Gyr. The horizontal axis is hydrogen number density at the bottom of the atmosphere. Abundances ofH2 and O is 1:1 (upper left), 1:1.1 (upper right), 1:1.5 (lower left), and 1:2 (lower right).
61

8.3. RESULTS

62

Figure 8.6: Oxygen escape flux rate in the steady state of hydrodynamic escape of
Venusian atmosphere. Solar age is 1.0 Gyr. The horizontal axis is hydrogen number
density at the bottom of the atmosphere. Abundances of H2 and O is 1:1 (red), 1:1.1(green), 1:1.5 (blue), and 1:2 (purple).

CHAPTER 8. WATER LOSS FROM EARLY VENUS

63hydrodynamic escape of oxygen as well as hydrogen would be very difficult but not
be impossible. Under special circumstances, hydrodynamic escape of H2 and O byitself can solve the water loss problem for Venus. This is an important consequence of
this study. However, in the most ranges of the parameter as the homopause number
densities, the oxygen would be left behind while the hydrogen would escape from theVenus, so that the oxygen consequently concentrates in the Venusian atmosphere.
Hence, we should next consider how the concentrated oxygen was consumed and/or
escaped afterward.
Finally, I note that the assumed atmosphere in this chapter consists of only dissociated H2 O, i.e., the abundance ratio of hydrogen and oxygen is about 1:1. Under thecase that hydrogen, which would be originated from solar nebula and/or degassing
atmosphere, is much more abundant than oxygen, it would be easier to lose the complete water from early Venus by hydrodynamic escape. This case study should be an
important future work.

8.4

Discussions

In this section, some possible mechanisms to consume/escape oxygen in the early
Venusian atmosphere are reviewed according to some previous studies.

8.4.1

Oxygen Concentration

The topic of abiotic O2 concentrations has been studied extensively in relation to the
problem of the origin and early evolution of life on Earth. Although work on thisproblem dates back to the early 1960’s, the problem was first formulated correctly byWalker (1977), who realized that the hydrogen released by volcanism and by water
vapor photolysis must be balanced by escape of hydrogen to space. The amount of
O2 that could be produced abiotically can be estimated using a one-dimensional photochemical model (Kasting et al., 1984). An up-to-date discussion of this problem isgiven by Kasting (1993). There, the problem of balancing the atmospheric hydrogenbudget is reviewed, and it is pointed out that rainout of reduced and oxidized gases
should be taken into account. Lightning-induced decomposition of CO2 is also a potentially important source of O2 at low altitudes in such atmospheres. The calculated
O2 mixing ratio peaks at about 50 km at a value of just under 0.1% by volume. This
high-altitude peak is produced by photolysis of CO2 followed by recombination of O
atoms with each other and with OH to form O2 . The O2 concentration decreases
rapidly towards the surface as a consequence of reaction with reduced volcanic gases,

8.4. DISCUSSIONS

64

catalyzed by radical species produced by H2 O photolysis. At early stage of the planetary formation, the conclusion that atmospheric O2 concentrations should be low isrobust, because the primordial planets would have much reduced gases such as H2
and CO. Higher solar UV fluxes in the past could have enhanced the O2 concentrationin the upper atmosphere, but they should have had little effect on ground-level O2
concentrations (Canuto et al. 1982, 1983; Kasting et al. 1984; Kasting 1985).
On the other hand, Kasting (1996) point out that abiotically-produced O2 concentrations could be much higher on planets for which the preceding criteria do not
apply; for example, a Venus-like planet could build up a transient O2 -rich atmosphereas a consequence of rapid water loss and hydrogen escape. They implied that O2 partial pressures of tens or hundreds of bars are theoretically possible in this case. (Thepartial pressure of a fully vaporized terrestrial ocean is 270 bars, 240 of which comefrom oxygen.) However, as shown in the previous section, the escape flux of hydrogenby hydrodynamic escape dramatically decreases under the high-concentration of oxygen in the primordial atmosphere. So, rapid water loss and hydrogen escape, which
would bring about the transient high O2 concentration in the primordial Venusian
atmosphere, could not be achieved easily. In other words, it is inversely possible thatthe H2 O is formed from the dissociated H2 and O by backward reaction. In the case ofthat, complete loss of water on the Venus cannot be achieved. Unfortunately, we does
not have valid criterion in deciding which atmospheric circumstances would bring the
backward reaction yet. Therefore, the research of conditions of the H2 O backward
reaction must be critical to study the mechanism of water loss on the Venus.
If the backward reaction of H2 O does not occur and the dissociated H2 escapefrom the Venusian atmosphere completely, remaining oxygen could be consumed byoxidation of surface minerals or nonthermal escape of atmospheric oxygen. A problem
is the question of how much the oxygen presumed to have been left behind after thehydrogen had escaped could be disposed.

8.4.2

Oxidation of Surface Minerals

Lewis and Prinn (1984) have pointed out that the loss of 110 bars of O2 (a fullterrestrial ocean contains close to 240 bar of oxygen) would entail the oxidation of
about 80 km of Earth-like crust. Acting over 4.5 Gyr this would require an averagerate of tectonic recycling approximately 15 times greater than is seen presently on
Earth. And, if we suppose that the rate of resurfacing of crustal material by tectonic

CHAPTER 8. WATER LOSS FROM EARLY VENUS

65

activity during the first gigayear of Venus’ existence was the same as on Earth, atypical quantity of about one-third amount of terrestrial ocean has been possibly
absorbed by Fe oxidation, carbonate formation, and oxidation of the atmosphere(Chassefiére, 1996). However, since Venus today appears to be less tectonically active
than Earth, the problem of disposing of this oxygen is potentially severe.

8.4.3

Nonthermal EscapeThe upper parts of an atmosphere are partially ionized, by the agency of solar photons
with wavelength generally below 100 nm. These ions are often suprathermal, and theirinteractions can produce suprathermal neutral atoms as well. These suprathermal
processes are limited to modest quantities of gas, nevertheless, they can be importantfor minor constituents, such as N and O at present Mars and H at present Venus andMars. (Hunten, 1993)
Another process involving ions pertains to Venus and Mars, which lack a significant intrinsic magnetic field (Luhmann and Kozyra, 1991). These planets posses aboundary, the ionopause, that separates the bound ionosphere from an outer regionin which the solar wind is diverted and flows around and past the planet. This region
still contains some neutral gas, and if such atoms are ionized by solar photons orelectron impact, they are swept up in the flow. Some of these ions can re-impact
the upper atmosphere with enough energy to remove additional atoms. Study of thisfield is still in its infancy, but there are suggestions that the process is significant for
loss of O from Venus (McElroy et al., 1982).
Venus could have lost its magnetic field very early in its history. If a planetaryatmosphere like that of Venus is not protected by a strong intrinsic magnetic field, theplanetary neutral gas in the exosphere can be ionized due to charge exchange with thedeflecting solar wind plasma flow, by electron impact and by the XUV radiation, andpicked up by the solar wind plasma flow around the planet. The principal ionizing
radiations responsible for the heating of the upper planetary atmospheres and theformation of planetary ionospheres from stellar or solar sources are ionizing X-rays
and EUV radiation (XUV). It has been also suggested that the solar wind may havebeen much more massive in the distant past (Wood et al., 2002). Compelling observational evidence shows that zero-age-main-sequence solar-type stars rotate morethan 10 times faster compared to the present Sun (Güdel et al., 1997). As a consequence, young solar-type GV stars, including the young Sun, have vigorous magnetic
dynamos and correspondingly strong high-energy emissions. With enhanced high-8.4. DISCUSSIONS

66

energy emissions and frequent flares young solar-type stars are also expected to havemore powerful stellar winds. Without the protection of magnetic field, a strong solar
wind of the young Sun could easily remove substantial amounts of oxygen that could
have been left from a hydrodynamically escaping ocean.
Kulikov et al. (2006) studied the early Venusian thermosphere-exosphere region
and the efficiency of the O+ ion loss by the pick up process due to the evolving solar
radiation and particle environment over the planet’s history. They used minimum
and maximum values of the solar wind mass flux inferred from stellar wind particle
collisions with the interstellar medium which surrounds the solar proxies with different
ages (Wood et al., 2002; 2005). If the incoming solar wind plasma penetrates a scaleheight deeper into the thermosphere, which is not protected by a strong intrinsic
magnetic field, they obtain a total integrated oxygen loss over the planet’s historyof about 180-280 bar (= 70-110% Terrestrial Ocean) for the maximum solar windvalues, about 40-60 bar (= 15-25% TO) for the average solar wind, and about 10-15
bar (= 4-6% TO) for the minimum solar wind.
We should keep in mind that their study took into account only the ion pickup process and no other nonthermal loss processes like atmospheric sputtering andatmospheric erosion by plasma instabilities. Additional loss processes like detachedplasma clouds and sputtering will enhance the oxygen loss from an unprotected early
Venus atmosphere to larger amounts, so that their calculations may represent only
lower limits for nonthermal escape of oxygen. Therefore, It is expected that the
Venusian oxygen, which might have remained in the atmosphere after the evaporationof an early water ocean could have also been lost to space by ion pick up for solar
wind fluxes, if the flux exceed the average solar wind’s value, and if the early Venusian
atmosphere was not protected by an intrinsic Earth-like magnetic field.

8.4.4

Contribution of Water from Comet

Chyba (1990) has argued that the heavy bombardment by cometary material could
have provided a full terrestrial ocean to early Earth, with a continuous supply ofwater during this period. If it is the case, water was brought at a relatively slow rate.
Venus and the Earth are expected to have accreted about the same water amountthrough cometary impacts during the heavy bombardment phase (Chyba 1990), andmy calculations show no significant O escape could occur on Venus over such a long
period (over 108 years). If Chyba (1990) is right, that is, if 0.2-0.7 TO was accreted

CHAPTER 8. WATER LOSS FROM EARLY VENUS

67

over 1 Gyr from impacting comets, the large amount of oxygen that could not be lostby hydrodynamic escape left behind on the early Venus. The contribution of watersupply to early Venus from cometary heavy bombardment would make oxygen escapefrom Venusian atmosphere even harder.

8.4.5

D/H Fractionation

The hydrogen escape flux rate would have declined at lower H2 O density, and theflux may have been smaller than found here if more O had accumulated in the Venus
atmosphere. The last portion of the water would have been much more difficult to
get rid of, because of the drastic decline in the H escape rate at low H2 O density andpossible oxygen accumulation after hydrodynamic escape of substantial amount of H2
in the Venusian atmosphere. During the last stage of the hydrodynamic escape process the escape flux would have decreased rapidly and significant D/H fractionation
should have occurred (Kasting & Pollack, 1983).
However, we should note that it is difficult to estimate the initial H2 O reservoir onearly Venus from the present D/H ratio, because all estimations depend on the nonsteady or steady state Venus’ water content and possible unknown D/H fractionation
processes which could have had different rates over the planet’s past compared to thepresent time. Although the amount of water on early Venus is implicitly assumed tobe that of terrestrial ocean in this section, actually the amount of water is unknown.
The question may be answered someday by performing chemical analyses to determine
the oxidation state of Venusian crust in detail.

Chapter 9
Fractionation of Noble Gases on
Venus
9.1

Previous Studies

Hydrodynamic escape driven by EUV heating has been envisioned for fractionating
noble gases on terrestrial planets (Zahnle & Kasting, 1986; Sasaki & Nakazawa, 1988;
Pepin, 1991; Pepin, 2006). The rapid escape of a light gas would have exerted an
aerodynamic drag on heavier gases. The balance between the upward aerodynamic
drag and the downward gravitational attraction resulted in a mass-dependent loss ofnoble gases. Mechanisms operating today on the terrestrial planets cannot remove
species as massive as the heavy noble gases, and yet there is clear evidence that massdependent loss signatures are imprinted on atmospheric Ar, Kr, and Xe isotopes. So,
the hydrodynamic escape process would have occurred on earlier epochs of planetaryhistories, when now-vanished energy sources such as strong EUV heating of the young
Sun (Pepin, 2006) could have driven intense outflows of atmospheric gases. Actually,on the early stage of planetary formation, a massive hydrogen atmosphere could havebeen acquired by capture of nebular gases or could have resulted from the reduction
or photodissociation of water (Kasting & Pollack, 1983), and the appropriate energy
could have been deposited in the atmosphere as extreme ultraviolet radiation fromthe young evolving Sun (Ribas et al., 2005).
The noble gas depletion patterns in terrestrial planets’ atmospheres are simply
considered to be created from initially solar compositions via thermal escape of theatmospheres (Fig. 7.1). However, we should note that the atmosphere consisted ofsolar composition would be lacking in major components of present terrestrial planets’ atmospheres such as H, C, and N. Anyway, the fractionation process by thehydrodynamic escape of primordial atmospheres would have a possibility of account69

9.1. PREVIOUS STUDIES

70

ing for at least the noble gas patterns as the following observations: The Ar/Kr ratio
on Venus is nearly solar while the Ar/Kr ratio on Earth and Mars is nearly meteorites, the absolute abundance of Ar on Venus is similar to or larger than that of themost argon-rich meteorites, the Ne/Ar elemental ratio is roughly the same on Earth,Mars, Venus and meteorites, and neon on planets is isotopically between solar windand meteorites neon, and argon on Mars is isotopically heavier than either. So, itshould be key to understand the atmospheric evolutional history of terrestrial planetsthat studying in detail the hydrodynamic escape process of a primordial planetaryatmosphere, which is energize by EUV irradiation of the young Sun.
Recently, Pepin (2006) presented an outline of how atmospheric evolution may
have proceeded on the terrestrial planets. Their objective was to account in detail forcontemporary elemental and isotopic compositions. Although they indicated a possible evolution scenario of terrestrial planetary atmospheres, the fundamental questionof weather nature actually shaped the atmospheres is still unanswered. Many mechanisms could have contributed to gain or loss of primordial planetary gases. Their
model, however, has many uncertainties such as the origins, compositions, and lifetimes of primary reservoirs, the mechanisms responsible for transporting and fractionating volatile elements, and the environments in which such processes operated.
Regretfully, we have no model-independent ways to assess the relative importanceof possible primordial sources and evolutionary processes in shaping present atmospheres. So, we should not take complicated evolution scenario simply for the sake ofaccounting for the present atmospheric elementary patterns. We should consider thefundamental question of whether nature actually shaped the atmospheres in this way.
Therefore, in contradiction to Pepin (2006), I take the simplest evolution scenario andconsider which process would determine the atmospheric profile in this chapter.
In the first approximation, noble gas depletion patterns were assumed to be created from initially solar compositions by gravitational escape. Gravitational escapeof atmospheric constituents is intrinsically mass-dependent, no matter the specificloss mechanism, and so both elements and isotopes in the residual atmospheres are
fractionated with respect to initial compositions. High abundance of argon on Venus
carry powerful evidence that their primordial parents had solar compositions, at leasta certain amount, and that gravitational escape played a major role in their evolution. Analytical approximations to mass fractionation during hydrodynamic escapeindicated that these fractionation could be achieved in some instances for the caseof a heavy constituent being dragged along by a much lighter one (Zahnle & Kasting, 1986). In this study, the developed numerical model applied to the escape ofCHAPTER 9. FRACTIONATION OF NOBLE GASES ON VENUS

71

Figure 9.1: The hydrogen escape flux rate at which the various heavy elements would
begin to escape. Note that required hydrogen flux to uplift Ne is like CO and slightly
smaller than Ar.

noble gases in hydrogen escaping flow from the early Venus to examine whether thefractionation can be achieved or not.

9.1.1

Proposed Ne/Ar Fractionation Mechanism

Hydrodynamic escape of noble gases is a mass-dependent escape process, i.e., thelighter an element is, the more it escapes. So, is it possible to transform Ne/Ar
ratio of Venusian atmosphere from solar nebula abundance (Ne/Ar ∼ 1) into present

Venusian abundance (Ne/Ar ∼ 0.01) by mass-dependent hydrodynamic escape? In

Fig. 9.1, I have indicated the points at which noble gases would have begun to escapecalculated using crossover mass (Eq. (6.4)). Figure 9.1 shows that the difference ofthe required hydrogen escape flux of Ne and Ar is very slight. Hence, a fine-tuning ofhydrogen escape flux is required to achieve a proper Ne/Ar fractionation. However,the long-term fine-tuning of hydrogen escape flux (i.e., the long-term fine-tuning of
input solar EUV energy) would not be realistic. Thus, the fractionation of Ne andAr, would not be explained simply by mass-dependent hydrodynamic escape.
Here, I consider a photochemical reaction to realize the fine-tuning of hydrogenescape flux, which make neon escape and argon unescape. I take notice of carbon

9.2. NUMERICAL MODEL

72

monoxide. Any wind capable of lifting Ne is also capable of lifting CO (Fig. 9.1). At
the top of the atmosphere, infrared cooling by CO might then have radiated away
EUV energy that would otherwise speed the flow. In other words, the wind may
only have become strong enough to reach the threshold of removing CO, at which
point any additional EUV energy input would be radiated away. On the other hand,
Ar might not be removed by hydrogen hydrodynamic escape at the threshold speed,
because the required hydrogen escape flux is slightly larger than Ne and CO. Thus,
the infrared cooling by CO might control the atmospheric temperature (that is in
proportion to the hydrogen escape flux) by self-sustained way to maintain the finetuning of hydrogen escape flux. It would therefore be expected that some loss of Ne
occurred, however that no loss of the heavier Ar took place. The fractionation of theVenus’ noble gases might be explained by this mechanism (Zahnle & Kasting, 1986).
Transient reducing atmospheres produced by mantle reactions with metallic ironhave been studied by Kuramoto & Matsui (1996). The possible presence of free ironin the crust, during the earlier stages of accretion, suggests that the atmosphere could
be reducing, with H2 rather than H2 O, and CO (and/or CH4 ) rather than CO2 . And
also, it is likely that the early atmosphere of Venus is also reducing similarly to thatof Earth since the atmospheres on both planets would be generated by degassing ofplanetesimals (Hashimoto et al., 2007). Carbon monoxide is, like N2 , an inevitableproduct of impact shock chemistry (e.g., Zahnle, 1990; Hashimoto et al., 2007). As
one cannot build a planet or satellite without impacts, CO figures to have been amajor constituent of the earliest atmospheres. Therefore, it is likeliest to have beenpresent as a component of transient reducing atmospheres, which contain substantial
amount of carbon monoxide.

9.2

Numerical Model

The model and the energy calculation method are applied to the atmosphere of a
Venus-like planet in which the thermosphere is composed only of hydrogen, noble
gases, and carbon monoxide molecules. The lower boundary is fixed at approximately
the altitude of the homopause (100km), and the lower boundary temperature is fixedat 330 K. The upper boundary is at an altitude of roughly 50000 km. The H2 , Ne, Ar,
and CO number densities at the lower boundary is treated as a free parameter. The
parameters are chosen in reasonable ranges obtaining from the possible atmosphericsources shown in Fig. 9.2. The employed free parameters are summarized in Table
9.1. The effect of CO cooling in the atmosphere is taken into account using line-by-line

CHAPTER 9. FRACTIONATION OF NOBLE GASES ON VENUS

73

Figure 9.2: Relative abundances of noble gases in solar, meteorites, and the Venusian
atmosphere normalized to each Ar abundance. All the data used in this figure arelisted in Table 7.2

method as described below.

9.2.1

CO Rotational CoolingTemperature change of the atmosphere by CO cooling is calculated by the following
equationρCvdT= −ΛCOdt(9.1)

where Cv is the specific heat of the atmosphere. Term in the right hand side represent
the cooling term of CO. The radiative cooling rate ΛCO due to a transition from level
Table 9.1: Employed Free Parameters
Symbol
nH2
nNe
nAr
nCO
τ

Meaning
homopause hydrogen number density
homopause neon number density
homopause argon number density
homopause carbon monoxide number density
age of the central star

Values1013 ∼ 1018 /m3
10−7 ∼ 10−3 × nH2= nNe
10−4 ∼ 1.0 × nH2
0.01, 0.05, 0.1 Gyr9.2. NUMERICAL MODEL

74

i to level j of CO is given byΛCO (νij ) = nCO (i)Aij hνij βesc (τij )SCO (νij ) − P (νij )SCO (νij )(9.2)

where nCO (i) is the population density in level i of CO, Aij is the spontaneous transition probability, hνij is the energy difference between level i and level j, βesc (τij )is the photon escape probability at the optical depth τij of the line (Hollenbach &McKee, 1979; Tielens & Hollenbach, 1985a). The source function SCO (νij ) is givenby2hν 3
SCO (νij ) = 2 ij
c

gi nj
−1
gj ni

!−1

(9.3)

where gi and gj are the statistical weight of the levels and c is the speed of light.
P (νij ) is the intensity of the background radiation. In calculating cooling rates dueto the rotational lines of CO, the effects of the background radiation field are assumed
to be ignored. So the cooling rates ΛCO (νij ) is simplified from Eq. (9.2) to be thefollowing equation
ΛCO (νij ) = nCO (i)Aij hνij βesc (τij )(9.4)

The optical depth averaged over the line is given bynj (z ′ )gidz ′Aij c3 Z z
′
n
(z
)
−
1
τij (z) =i
8πνij3 0
ni (z ′ )gj
δνd
"

#

(9.5)

where δνd is the turbulent Doppler line width (= 2.7 km/s; Tielens & Hollenbach,1985b), and the integration is carried out from the top of the atmosphere (z = 0) to
depth z. Following de Jong & Boland (1980), the probability that a photon escapes
through the nearest boundary is approximated byβesc (τ ) =1 − exp(−2.34τ )
,
4.68τ


"

τ
= 4τ ln √
π

!#0.5

τ <7
−1


,

(9.6)
τ ≥7

(9.7)

In these formulae it is implicitly assumed that half of the photons, those that are
emitted to the planet’s ground, do not escape. Because all opacity is assumed to
be local in the escape probability formalism, these inwardly directed photons do notcontribute to the cooling.
It is assumed that cooling rates due to the levels in non-LTE (Local Thermodynamic Equilibrium) are ignored. From the assumption, cooling efficiency of theCHAPTER 9. FRACTIONATION OF NOBLE GASES ON VENUS

75

Table 9.2: Molecular line data for the CO model molecule
J
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

E[cm−1 ] λ[mm]
3.85
2.601
11.54
1.300
23.07
0.867
38.45
0.650
57.67
0.520
80.74
0.434
107.640.372
138.390.325
172.980.289
211.400.260
253.670.237
299.770.217
349.700.200
403.460.186
461.050.174
522.480.163
587.720.153
656.790.145
729.680.137
806.380.130
886.900.124
971.230.119
1059.37
0.113
1151.320.109
1247.06
0.104

Aij [s−1 ]7.17 (-8)
6.87 (-7)
2.48 (-6)
6.09 (-6)
1.22 (-5)
2.13 (-5)
3.40 (-5)
5.11 (-5)
7.29 (-5)
1.00 (-4)
1.33 (-4)
1.73 (-4)
2.19 (-4)
2.72 (-4)
3.33 (-4)
4.02 (-4)
4.80 (-4)
5.66 (-4)
6.60 (-4)
7.64 (-4)
8.77 (-4)
9.98 (-4)
1.13 (-3)
1.27 (-3)
1.42 (-3)

ncr [cm−3 ]5.0×T −0.66 (3)1.9×T −0.45 (4)4.6×T −0.35 (4)8.3×T −0.28 (4)1.3×T −0.22 (5)2.0×T −0.18 (5)2.9×T −0.16 (5)4.1×T −0.14 (5)5.5×T −0.12 (5)7.2×T −0.11 (5)9.2×T −0.10 (5)1.1×T −0.09 (6)1.4×T −0.08 (6)1.7×T −0.075 (6)2.0×T −0.07 (6)2.4×T −0.065 (6)2.8×T −0.06 (6)3.2×T −0.055 (6)3.7×T −0.05 (6)4.2×T −0.045 (6)4.7×T −0.04 (6)5.3×T −0.035 (6)5.9×T −0.03 (6)6.5×T −0.025 (6)6.8×T −0.020 (6)Note. Powers of ten multipliers in parentheses. From left to right: upper level of the rotational lines
J → J − 1, energy of the upper level, wavelength of the lines, Einstein transition probabilities for
spontaneous emission, and fits to the critical densities. The data are given according to Kamp &van Zadelhoff (2001).

9.3. RESULTS

76

transition from level i is taken into account at the r if n(r) > ncr , where ncr is thecritical density. Concerning the levels above J = 26, the critical densities exceed
1017 molecules/m3 . So, I use the number density of CO that is always less than 1017
molecules/m3 in the calculations. It consists of 26 levels up to J = 25 and has 351
CO-H2 collisional rate coefficients (Schinke et al., 1985). The pumping of CO levels
by infrared radiation is included. The molecular line data is summarized in Table
9.2. The data in Table 9.2 is applicable to wide temperature range (McKee et al.,1982).
Therefore, a formula for the CO cooling is
ΛCO =

25
X

hνJJ ′ nJ (CO)AJJ ′ βesc (τJJ ′ )

(9.8)

J=1

where J ′ = J − 1. The level population nJ (CO) is calculated in the assumption ofLTE

nJ (CO) =n(CO)gJ exp − kEBJT
Z



(9.9)

where n(CO) is total number density of CO, gJ = 2J + 1 is the statistical weight, Z
is the partition function.

9.3

Results

Although I use a broad parameter range (Table 9.1), the calculated results can beclassified into three categories in a simple way: Typical results are shown in Table
9.3. (1) Both neon and argon escape in almost same escape flux rate by uplifting ofhydrogen hydrodynamic escape (△ in Table 9.3). (2) Neon obtain more escape flux
rate than argon by well-balanced force condition between an acceleration by upliftingof hydrogen hydrodynamic escape and deceleration by cooling of carbon monoxide’s
emission (escape flux ratios (FAr /FNe ) are roughly shown in Table 9.3). (3) Neither
neon nor argon escape, because the cooling of carbon monoxide’s emission excessively
weakens uplifting force of hydrogen hydrodynamic escape or because the downward
drag force of carbon monoxide overcomes uplifting force of hydrogen (▽ in Table 9.3).
Typically, the result changes from category (1) to (3) along with increasing thehomopause carbon monoxide number density. In other words, along with increasing
the cooling effect of carbon monoxide’s emission, the temperature of Venusian atmosphere drops and the hydrogen escape flux rate decreases, so that the noble gases
cannot get enough upward force from surrounding hydrogen.

CHAPTER 9. FRACTIONATION OF NOBLE GASES ON VENUS

77

Table 9.3 also indicates that neon/argon fractionation before the 50 Myr could
not occur because the solar EUV irradiation from the central star would be too highto fractionate neon and argon. In this case, even in the most extreme case (i.e., nNe,Ar
is 10−4 × nH2 and nCO is 1.0 × nH2 ), CO cooling does not decrease the atmospherictemperature (and consequently the hydrogen escape flux rate) enough to fractionateof neon and argon. On the other hand, at solar age of 100 Myr, the neon/argon
fractionation occurs only in the limited cases such as nNe,Ar is 10−4 × nH2 and nCO

is 1.0 × nH2 . However, it is note that, if the homopause number densities of H2 andCO are more than 1.0 × 1017 , downward drag force of CO to H2 becomes too large sothat noble gases cannot escape by hydrogen hydrodynamic escape regardless of their
number densities at homopause.

Figure 9.3 shows calculated temperature of Venusian atmosphere under the various homopause CO number densities. The homopause hydrogen number density set
to 1.0×1015 [molecules/m3 ]. With increasing the homopause CO number density,
the atmospheric temperature is decreasing especially in the case of CO = 1.0×1015[molecules/m3 ]. The result means that the CO cooling is actually effective in decreasing the atmospheric temperature.
Figure 9.4 shows calculated escape flux rates under the various homopause CO
number densities for the typical case that Ne/Ar fractionation occurs. The homopause hydrogen number density are 1.0×1015 , and both Ne and Ar is 1.0×1010[molecules/m3 ]. Hydrogen escape flux rate is decreasing with increasing homopause
CO number density, and along with that, the escape flux rate of neon and argon
are also decreasing (Fig. 9.4). Then when carbon monoxide is the same amount ashydrogen at the homopause, the Ne/Ar fractionation (FAr /FNe = 0.1) occurs. In thiscase, the fractionation of neon and argon due to the CO cooling actually occurs as
expected in section 9.1.1 or Zahnle & Kasting (1986).
So, can we get the observed Ne/Ar ratio of present Venusian atmosphere via
hydrodynamic escape of proto atmosphere with gravitationally attracted solar nebula
components? Unfortunately, the answer would be “No.” Here, let us consider thefollowing three extreme cases for compositions of primordial Venusian atmosphere.
(1) The case that the Venusian primordial atmosphere consists entirely of solar
nebula compositions: According to Genda & Abe (2003, 2005), without an ocean
at the surface of a planet, the atmospheric mass after the stage of giant impacts is9.3. RESULTS

78

Table 9.3: Calculated Escape Flux Ratio (FAr /FNe )
τ
0.01
0.01
0.01
0.05
0.05
0.05
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1

nH2
1.0e141.0e151.0e161.0e141.0e151.0e161.0e141.0e141.0e141.0e141.0e151.0e151.0e151.0e151.0e151.0e151.0e151.0e151.0e161.0e161.0e161.0e161.0e161.0e161.0e17

nNe,Ar
1.0e101.0e111.0e121.0e101.0e111.0e12
1.0e9
1.0e101.0e101.0e10
1.0e9
1.0e101.0e101.0e101.0e101.0e111.0e111.0e111.0e111.0e111.0e111.0e121.0e121.0e121.0e13nCO
1.0e141.0e151.0e161.0e141.0e151.0e161.0e141.0e101.0e121.0e141.0e111.0e111.0e131.0e151.0e151.0e111.0e131.0e151.0e121.0e141.0e161.0e121.0e141.0e161.0e17

FAr /FNe
△
△
△
△
△
△
△
△
△
∼ 0.1
△
△
△
∼ 0.1
△
△
△
▽
△
△
∼ 0.9
△
△
▽
▽

Note. A small portion of the results are shown here.
△: Both neon and argon escape in almost same flux along with hydrogen.
▽: Neither neon nor argon escape along with hydrogen.

CHAPTER 9. FRACTIONATION OF NOBLE GASES ON VENUS

79

Figure 9.3: Temperature distribution in the steady state of hydrodynamic escapeof Venusian atmosphere. Solar age is 0.1 Gyr. Abundance of H2 and CO is parameter. Hydrogen number density at the bottom of the atmosphere is 1.0×1015[molecules/m3 ]. CO number densities at the bottom of the atmosphere are 1.0×1012(red), 1.0×1013 (green), 1.0×1014 (blue), and 1.0×1015 (purple) [molecules/m3 ].

9.3. RESULTS

80

Figure 9.4: Hydrogen, Neon, and Argon escape flux rates in the steady state ofhydrodynamic escape of Venusian atmosphere. Solar age is 0.1 Gyr. The H2 number
densities at the bottom of the atmosphere are 1.0×1015 , Ne is 1.0×1010 , and Ar is
1.0×1010 [molecules/m3 ]. The horizontal axis is CO number density at the bottomof the atmosphere. The escape flux rates of Ne (red) and Ar (green) refer to the left
vertical axis, and that of H2 (blue) refers to the right.

CHAPTER 9. FRACTIONATION OF NOBLE GASES ON VENUS

81

approximately a few times the atmospheric mass of a protoplanet. Hence, because
the early Venus is considered not to have an ocean at the surface, the amount of a
protoplanet’s atmosphere approximately gives the one of the Venusian atmosphere
after the giant impacts stage. The mass of the nebula gas which a protoplanet gravitationally attracts in the nebula gas is 8.4 × 1021 kg, which is the total atmosphericmass within the Hill radius (Hayashi, 1979). This mass would be upper limit of primordial atmosphere on the protoplanet because the decrease of the density at Hill
radius results in the decrease of total atmospheric mass (Mizuno et al., 1982). The
atmosphere of 8.4×1021 kg with the solar abundance contains krypton of 4.9×1014 kg(Table 7.1). On the other hand, the present Venusian atmosphere contains kryptonof 2.3 × 1014 kg. Therefore, the pre giant impact atmosphere is enough to contain thepresent Venusian krypton content; The same is true for argon and xenon in Venu-

sian atmosphere. Because the krypton would be too heavy to escape and retain theinitial amount to the present, the mass of the pre giant impact atmosphere can be
estimated inversely to be order of 1020 kg to satisfy the present Venusian krypton
contents (Genda & Abe, 2005).
The atmosphere of order of 1020 kg with the solar abundance contains argon oforder of 1016 kg, which is approximately the same amount of the present Venusianargon of 1.2 × 1016 kg. In the meanwhile, the amount of neon of the Venusian post

giant impact atmosphere is 1.6 × 1017 kg, which is about two order more than thepresent Venusian neon contents of 1.4×1015 kg. This overabundant neon must escapeby hydrogen hydrodynamic escape, which does not take off Venusian argon so much.
However, CO abundance in the solar nebula would be much smaller than hydrogen
abundance (about two order smaller). Hence, CO cooling is ineffective in fractionating
the Ne/Ar escape flux rate (Table 9.3 and Fig. 9.4). Therefore, we cannot get theobserved Ne/Ar ratio of present Venusian atmosphere via hydrodynamic escape ofsolar abundance atmosphere.
(2) The case that the Venusian primordial atmosphere consists entirely of degassing compositions: Unlike with the case (1), CO abundance in meteorites at the
maximum would be same as hydrogen abundance (Table 7.1). So, there is a possibility that effective Ne/Ar fractionation due to the CO cooling would occur in this case
(Table 9.3 and Fig. 9.4). However, this degassing dominant atmosphere has a fatal
flaw in the first place, i.e., even the most volatile-rich meteorites have less abundantargon than present Venusian atmosphere. In other words, the atmosphere consisted
entirely of degassing compositions from meteorites is absolutely lacking in the amount9.4. DISCUSSIONS

82

of argon as compared to present Venusian atmosphere. Therefore, we cannot get theobserved Ne/Ar ratio of present Venusian atmosphere via hydrodynamic escape ofdegassing atmosphere.
(3) The case that the Venusian primordial atmosphere consists of mixture of solar
nebula and degassing compositions: The mixing ratio of solar nebula and degassing
compositions is unknown. So, let us consider the most optimistic case, i.e., noble gases
are supplied from solar nebula (that is, enough to contain the present Venusian argon
content) and carbon monoxide is supplied from most volatile meteorites’ degassing(that is, the same amount as hydrogen). In this case, it seems possible to achieve
present Venusian Ne/Ar ratio by hydrodynamic escape of neon and argon under theeffective CO cooling circumstance in the primordial Venusian atmosphere.
However, in practice, the maximum fractionation ratio as FAr /FNe = 0.1 (Fig.
9.4) does not account for the present Ne/Ar ratio of the Venusian atmosphere. The
atmosphere of 1020 kg with the solar abundance contains neon of 1.6 × 1017 kg (Table

7.1), and meanwhile the present Venusian atmosphere contains neon of 1.4 × 1015 kg.
So, neon should be reduced to over two order lower amount via escape process. Then,

if the argon escape rate is tenth part of the neon escape rate, argon in the primordial
Venusian atmosphere almost vanish because the atmosphere of 1020 kg with the solar
abundance contains argon of only 9.0 × 1015 kg, which is over one order less thanneon in the atmosphere. Therefore, even in the most optimistic case, we cannot get

the observed Ne/Ar ratio of present Venusian atmosphere via hydrodynamic escapeof mixture atmosphere of solar nebula and degassing compositions.

9.4

Discussions

As shown in the previous section, even in the most optimistic case, appropriate Ne/Ar
fractionation cannot be achieved by hydrogen hydrodynamic escape of primordial
Venusian atmosphere even if the temperature control by CO cooling is consideredin the numerical model. In this chapter, other possibilities of achieving noble gas
patterns of present Venusian atmosphere are discussed.

9.4.1

H2 O Rotational CoolingAt high H2 O levels, cooling in the water vapor rotation band becomes important
as well as CO rotational cooling. However, it is very difficult to calculate the H2 O
cooling rate precisely because H2 O has broad continuous rotational absorption lines

CHAPTER 9. FRACTIONATION OF NOBLE GASES ON VENUS

83

so that line-by-line method cannot be applied. So, approximate expressions are usedhere to roughly estimate the effect of H2 O cooling compared with CO cooling. H2 O
molecules are excited to higher rotational states by collisions with other molecules,
mostly H atoms produced from H2 O photolysis. Infrared cooling of a mixture of
H and H2 O has been studied in connection with interstellar shocks (Hollenbach and
McKee, 1979) and I have simply adopted their parameterization as Kasting & Pollack(1983). The cooling is approximated by:
ΛH2 O = n(H)n(H2 O)LH2 O(9.10)

where n(H) and n(H2 O) are the number densities of atomic H and H2 O vapor and
LH2 O is a rotational line cooling coefficient given by:
2
2(kT )2 A0
2 + ym + 0.6ym
LH2 O =
nE0
1 + Cτ + nncr + 1.5( nncr )1/2"#

(9.11)The parameters in Eq. (9.11) are:
ym
Cτ
τT
ncr

"

Cτ
= ln 1 +1 + 10 nncr"#

(9.12)
 #!1/2

τT 2= τT 2π ln 2.13 +e
4N(H2 O)A0=
ηJT (1.18 × 107 )(∆V )5 (E0 /k)2
4kT A0=E0 σH VT

VT =

8kT
πm0

!1/2



(9.13)(9.14)(9.15)(9.16)

where n = n(H); m0 is mass of H atom; A0 = 2.3×102 sec−1 ; e = 2.718; E0 /k = 17 K;
σH = 3 ×1016 cm2 ; ηJT = 7.7(T /1000)1/2; (∆V )5 = 1 is Doppler width of H2 O lines inunits of 105 cm/s; and N(H2 O) is column density of H2 O out to inf. The derivationof Eq. (9.11) and a discussion of the individual terms is given in Hollenbach and
McKee (1979). They estimate that this formulation is accurate to within a factor of
2 at all optical depths.
Figure 9.5 shows the cooling efficiency ratio between CO and H2 O (ΛH2 O /ΛCO )in the case that both CO and H2 O have the same homopause number density (=1.0 × 1015 molecules/m3 ) as hydrogen. Though the cooling efficiency ratio differs foreach altitude, H2 O cooling is always much more effective than CO cooling (Fig. 9.5).

9.4. DISCUSSIONS

84

Figure 9.5: Cooling efficiency ratio between CO and H2 O (ΛH2 O /ΛCO ). The homopause number density of both CO and H2 O is 1.0×1015 and that of H2 is 1.0×1016[molecules/m3 ].

Therefore, with substantial amount of H2 O component in the primordial Venusian
atmosphere, it could be possible to drop the atmospheric temperature enough to
fractionate neon and argon effectively (i.e., the difference of escape flux rate between
neon and argon could be larger than 10 : 1). However, because H2 O is much easier to
uplift than neon and argon (Fig. 9.1), the self-sustaining temperature control cannot
occur in Ne-H2 O-Ar atmosphere unlike in Ne-CO-Ar atmosphere. So, how H2 O
cooling affects the fractionation of neon and argon is not a simple question. It will
be an important future work to discuss the effect of H2 O cooling to the primordial
Venusian atmosphere quantitatively. Additionally, wether a “cold trap” could beformed in the lower part of the atmosphere should be considered. The cold trap isdefined as the altitude at which the saturation H2 O mixing ratio reaches its minimum
value. If H2 O can go through the cold trap properly, the H2 cooling would be effective
to decrease the temperature of upper part of the atmosphere.

9.4.2

Additional Origins of Noble Gases

Although hydrodynamic escape of primordial Venusian atmosphere would play a rolein atmospheric evolution, particularly affecting the fractionation of neon, VenusianCHAPTER 9. FRACTIONATION OF NOBLE GASES ON VENUS

85argon might be escaped too much by this mechanism. A way to overcome this problem
would be addition of other atmospheric sources to the Venusian atmosphere after acompletion of hydrodynamic escape of primordial Venusian atmosphere. Possible
additional origins of noble gases for Venusian atmosphere have been proposed suchas solar wind implanted materials (Whetherill, 1981; Sasaki, 1991; Ballentine, 2005)
or extremely low-temperature condensates in comets which quantitatively trap argon(neon freezes only if hydrogen freezes, Owen et al., 1999).
Implantation of solar wind is considered to be a noble gas source for lunar soils(Eberhardt et al., 1970), gas-rich meteorites (Suess et al., 1964; Goswani et al.,
1984; Caffee et al., 1988) and Venus (Wetherill, 1981; McElroy & Prather, 1981).
Anisotropy of high-energy particle tracks in meteorite grains suggests that noble
gases in gas-rich meteorites were implanted at the surfaces of parent bodies (Goswani
et al., 1984). However, there is also evidence that implies earlier irradiation by highenergy particles before accretion (Heymann and Palma, 1986; Caffee et al., 1987).
The dust and small fragments are envisioned as constantly being formed by collisions
and re-accreted into planetesimals. Most of these materials are eventually incorporated into planets. Because much of the planetary material becomes exposed to thesolar wind, noble gases accrete early and more or less homogeneously. Enrichmentof Venus in implanted gas would be a consequence of the irradiated material being
initially confined to the innermost edge of the radially opaque circumsolar planetesimal disk. Wetherill (1981) speculated that the high noble gas abundances on Venus,
comparable to those in gas-rich meteorites, could be the result of accretion of planetesimals or their fragments that have trapped significant amounts of ancient solar
wind. And Sasaki (1989, 1991) implied that an appreciable amount of solar windspecies is trapped and the high abundances of solar-type noble gases in gas-rich meteorites and Venus can be explained. However, although the solar-wind implantation
can account for nearly solar Ar/Kr in Venusian atmosphere, it fails to account for
low Ne/Ar.
On the other hand, comets would deliver an already fractionated mixture of gas.
Fractionation in icy planetesimals occurs for temperatures above 30 K because thediffering sizes and polarizabilities of the atoms lead to competition for the available
sites in the ice (Notesco et al., 2003). However, neon is not trapped in amorphous
ice except at temperatures below 20 K (Laufer et al., 1987; Bar-Nun et al., 1988). A
relatively late impact by a large comet (or group of comets) formed at a temperaturenear or above 30 K would can account for the fractionation of noble gases on Venus by9.4. DISCUSSIONS

86

bringing in a nearly solar mixture of the noble gases except neon after the completionof hydrodynamic escape of primordial atmosphere (Wetherill, 1990).
Unfortunately, we have never acquired reliable observed values of nobles gases inany comets nor solar-implanted materials. So, how these additional origins of atmosphere quantitatively affect the Venusian noble gas patterns has been unknown yet.
Future observations of noble gases’ abundance in comets and researches of primordialmaterials such as solar-implanted materials are important future work to explanationof noble gas patterns of Venusian atmosphere.

Chapter 10
Conclusions
In this paper, a robust numerical model is developed to study multi-component hydrodynamic escape from planetary atmospheres. The numerical code is solved using
CIP method, which is well suited to solve the transonic flow. A two-dimensional
numerical model is also developed to calculate the deposition of solar EUV radiationin an extended atmosphere.
The numerical model is applied to the early Venusian atmosphere to consider“water loss from early Venus” and “fractionation of noble gases on Venus” problems
quantitatively. The former application (water loss from early Venus) indicates that
Venus was able to lost entire water, that is assumed to be terrestrial ocean amount,under the special circumstances such that the homopause number densities of H2 andO are maintained around 9.0×1015 [molecules/m3 ] for about 1.5×108 years. However,except in that special case, Venus could not have lost much oxygen, which dissociatedfrom water on early Venus. This means that a substantial amount of oxygen left on
Venus after hydrodynamic escape of hydrogen. This result implies that other ways oftaking the oxygen away from Venusian atmosphere should be discussed more precisely
in the future. On the other hand, the latter application (fractionation of noble gases
on Venus) indicates that present Venusian Ne/Ar abundance ratio cannot be achievedby hydrodynamic escape even if an atmospheric temperature controlled by CO coolingis taken into consideration. However, other cooling mechanism such as H2 O coolingand other noble gases’ sources such as solar implanted materials and/or comets mightmake it possible to achieve the proper Ne/Ar fractionation. This result implies that
more detailed studies of atmospheric cooling processes and other noble gases’ sources
are required to understand the fractionation of Venusian atmospheric noble gases.
Moreover, time-evolutional numerical code is required for the future study to solve
the Venusian atmospheric evolution more precisely. Because both the solar EUV flux87

88

and the hydrogen supply are expected to decline with time, the lighter elements are
more depleted and relatively more fractionated than the heavier elements by realhydrodynamic escape.

Appendix A
CIP Method
A.1

Basic Concepts

Although nature operates in a continuous world, a discretization process is unavoidable for implementing numerical simulations. The CIP method tries to constructa solution inside the grid cell that is close enough to the real solution of the given
equation, with some constraints (Takewaki & Yabe, 1987; Yabe & Aoki, 1991; Yabeet al., 2001). Yabe et al. proposed an approximation of the profile as shown below.
If we differentiate Eq. (A.1) with spatial variable x , we get Eq. (A.2)
∂f
∂f
+u
=0
(A.1)
∂t
∂x
∂g
∂g
∂u
+u
=− g
(A.2)
∂t
∂x
∂x
Equation (A.2) gives a simple translational motion of a wave with velocity u, andg stands for the spatial derivative of f , i.e., ∂f /∂x. In the simplest case wherethe velocity u is constant, Eq. (A.2) coincides with Eq. (A.1) and represents the
propagation of a spatial derivative with velocity u. Using these equations, we cantrace the time evolution of f and g. If g is predicted after propagation, the profile after
one step is limited to a specific profile. It is easy to imagine that by this constraint,
the solution becomes much closer to the initial profile that is the real solution. Most
importantly, the solution thus created gives a profile consistent with Eq. (A.1) eveninside the grid cell. If two values of f and g are given at two grid points, the profile
between these points can be interpolated by the cubic polynomial
F (x) = ax3 + bx2 + cx + d(A.3)

Thus, the profile at the n + 1 step can be obtained by shifting the profile by u∆t so
that
f n+1 = F (x − u∆t)
89

(A.4)

A.1. BASIC CONCEPTS

90

g n+1 =dF (x − u∆t)
dx

(A.5)

Then, f and g values at ith grid cell are given by
fin+1 = ai ξ 3 + bi ξ 2 + gin ξ + fin(A.6)

gin+1 = 3ai ξ 2 + 2bi ξ + gin
gi + giup 2(fi − fiup )
+
ai =
D2
D3
3(fiup − fi ) 2gi + giup
bi =
−
D2
D

(A.7)
(A.8)
(A.9)

where we define ξ = −u∆t. Here, D = −∆x, iup = i − 1 for u ≥ 0 and D = ∆x,

iup = i + 1 for u < 0. The CIP advection scheme can be sorted out as a kind of semi-

Lagrangian method in the sense that the CIP advection scheme employs a Lagrangianinvariant solution. Semi-Lagrangian methods that trace back along the characteristics
in time depend on an interpolation of the initial profile to determine the value atthe upstream departure point (which may not coincide with the computational grid
point). To solve nonlinear problems, it is expedient to separate the solution procedure
into several fractional steps.
Let us consider a hyperbolic equation,
∂f u
∂f
+
=g
∂t
∂x

(A.10)

where the source term g can be a function of f . The equation can be split into twophases; the nonadvection phase
∂f
=G
∂t

(A.11)

∂f
∂f
+u
=0
∂t
∂x

(A.12)and the advection phase

Here I have moved the compression term f ∂u
on the left-hand side of Eq. (A.10) to∂x
the right-hand side and have introduced,
G=g−f

∂u
∂x

(A.13)The CIP solvers that were given in Eq. (A.6)-(A.9) are applied only to the advection
phase after the nonadvection phase is solved with a finite-difference approach. For
the spatial derivative of Eq. (A.10), we have
∂f ′
∂f ′
∂u
+u
= G′ − f ′
∂t
∂x
∂x

(A.14)

APPENDIX A. CIP METHOD

91

This equation is also split into two phases:∂f ′∂u= G′ − f ′
∂t
∂x
∂f ′
∂f ′
+u
=0
∂t
∂x

(A.15)(A.16)

Actual method to solve the Euler equations (Eq. (4.5)-(4.7)) separately for the nonadvection and the advection phases is described in the following subsections.

A.2

Nonadvection phase

The equation for f , ∂f /∂t = G, is simply solved using finite differencing,fi∗ = fin + Gi ∆t,(A.17)

where ∗ on f means the time after one time step in the nonadvection phase. Even if
G includes the diffusion term and must be solved implicitly, the implicit solver can beapplied to this phase. We need not convert Eq. (A.14) for f ′ into a finite-difference
form. As has been discussed previously (Takewaki & Yabe, 1987), a simple treatment
is possible. The time evolution of f ′ associated with G′ can be estimated from Eq.
(A.17). Thus, we approximate the time evolution of f ′ with the finite difference ofthe time evolution of f to be∗nnf ∗ − fi−1
− fi+1+ fi−1fi′∗ − fi′n
= i+1
∆t
2∆x∆t

(A.18)

Another term f ′ ∂u/∂x in Eq. (A.15) can simply be solved with finite differencing;thus we have∗nnf ∗ − fi−1
− fi+1+ fi−1fi′∗ − fi′n
ui+1 − ui−1
= i+1
− fi′n
∆t
2∆x∆t
2∆x

(A.19)

Here we adapted an approach that uses staggered meshes to solve these equations.
First, we describe a solution in a staggered mesh system; that is, ρ and p are definedat the grid point xi and u is defined at xi+1/2 . The finite-difference forms in the
nonadvection phase of Eq. (4.5)-(4.7) can be written asun− uni−1 2nni uni+1/2n∗i − nni
n i+1/2
= −ni
−
∆t
∆x
xi
n
n
u∗i+1/2 − uni+1/2
pi+1 − pni + νi+1
− νin GM
2
=− n
− 2
∆t
ρi+1 + ρni∆xxi(A.20)(A.21)

A.3. ADVECTION PHASE

92

uni+1/2 − uni−1 2γpni uni+1/2p∗i − pni
+ (γ − 1)qi
= −γpni
−
∆t
∆x
xi
"

#

γ−1
2 Ti − Ti−1
2 Ti+1 − Ti
+ 2
/∆x
− κxi
κxi
xi
∆x
∆x
i+1
i

(A.22)(A.23)

We adopted the numerical viscosity ν, which has the same dimension as pressure,used by Nishiguchi and Yabe (1983). Large number ranges and steep gradients posea difficult challenge to numerical approximation techniques, and it can generally be
expected that in regions of large gradients numerical accuracy may decrease. So, inthe non-advection phase of conservation of momentum equation, an explicit diffusion
term is required. This diffusion acts to damp sharp gradients on the grid, providing
numerical stability. The numerical viscosity ν is given by the equation
νi = −cν γρi (uni+1/2 − uni−1/2 )

"s

pni
γ+1 n
(ui+1/2 − uni−1/2 )
−
n
ρi
2

#

(A.24)

if uni+1/2 − uni−1/2 < 0. If uni+1/2 − uni−1/2 ≥ 0, νi = 0. Here cν is the numerical viscosity

coefficient, and we adapt cν = 0.75 in this study.

Because of the large density curvature near the lower boundary (especially for thelarge-HEP case), the diffusion term is not negligible there. In the steady state, thetime derivative term in the continuity equation goes to zero. If the explicit diffusion
term is small, a nearly constant momentum can be obtained. However, near the lower
boundary for large HEP the momentum must adjust itself to meet the large density
curvature in the diffusion term, and thus it deviates from the analytical solution. As aresult, in the numerical results the momentum and the flow velocity are not calculated
as accurately in the region of the domain close to the lower boundary as in the rest of
the domain. However, it is interesting to note that the deviation in momentum close
to the lower boundary at low resolution does not influence the calculated value of themass flux at the outflow boundary significantly (Tian et al., 2005). This is because
the diffusion term becomes very small in most of the atmosphere since the gradientsare low.

A.3

Advection phase

After f and f ′ are advanced in the nonadvection phase, the CIP solver is applied
to the advection phase. The interpolated profile is determined from Eq. (A.6)-(A.9)
using f ∗ and f ′∗ on the right-hand side.

APPENDIX A. CIP METHOD

93

These two phases complete one time step of the calculation; then after replacingfn+1

and f ′n+1 by f n and f ′n , we return to the nonadvection phase again. These

procedures are iterated until achieving the steady state solution.

Appendix B
Supporting Materials for
Numerical Model
B.1

Data for Solar EUV Spectrum

For aeronomic studies, reference solar irradiance spectra representative of conditions
during solar cycle 22 (1986-1996) are presented here. These reference spectra areprimarily established from actual observations; however, results from selected solar
irradiance proxy models are used in creating and validating the reference solar spectrain the XUV and EUV range. These reference spectra are given in 1 nm intervals on
0.5 nm centers. A reference spectrum for solar cycle 22 minimum conditions is listed
in Table B.1.

B.2

Data for Cross-Sections of Photoabsorption
for Molecular Hydrogen

In Table B.2 the data are presented on cross-sections for photoabsorption. The accuracy of the recommended absolute values of cross-sections determined by averagingof the available experimental data is better than ±20%.

B.3

Two-Dimensional Energy Deposition

As shown in Fig. B.1, a solar radiation beam is absorbed (Beer’s Law) in a layer withdepth ∆s. The energy deposition in a given layer along each path is then multipliedwith the area of the ring, 2πD∆D (Fig. B.1), to obtain the total energy deposited into
the shell segment. Here D is the distance from the planet center to the solar radiation
beam. Note that for one shell with radius r, shell segments may have different D95

B.3. TWO-DIMENSIONAL ENERGY DEPOSITION

96

Table B.1: Reference Spectrum for Solar EUV
λ (nm)
0.5
5.5
10.5
15.5
20.5
25.5
30.5
35.5
40.5
45.5
50.5
55.5
60.5
65.5
70.5
75.5
80.5
85.5
90.5
95.5
100.5

@λ+0nm @λ+1nm @λ+2nm @λ+3nm @λ+4nm
3.700
6.000
7.000
7.455
7.834
8.282
8.288
8.409
8.247
8.395
8.000
7.902
7.739
7.755
8.218
8.206
8.542
9.192
8.827
8.782
8.422
8.285
8.645
8.392
8.778
8.986
8.381
8.723
8.590
8.393
9.873
9.045
8.799
8.807
8.849
8.750
8.892
8.381
8.025
7.980
8.215
8.003
7.976
8.268
7.957
7.923
8.471
8.099
8.337
8.628
8.532
8.061
8.093
8.192
8.048
8.700
8.172
8.156
8.903
8.286
8.734
8.403
9.024
8.410
7.823
7.691
7.682
7.688
8.224
8.077
8.541
8.174
8.012
8.133
8.229
8.443
8.865
8.735
8.964
8.663
8.641
8.702
8.876
9.150
8.981
9.032
9.063
9.144
9.207
9.296
9.375
9.269
8.821
8.777
8.632
8.438
8.460
9.699
8.775
8.968
8.582
8.649
9.568
9.589
8.802

Note. A reference spectrum for solar cycle 22 minimum condition is listed in 1 nm intervals on 0.5
nm centers and at a solar distance of 1 AU (Woods and Rottman, 2002). The values of the log(base-10) of the irradiance are the irradiances in units of [photons/cm2 s].

APPENDIX B. SUPPORTING MATERIALS FOR NUMERICAL MODEL

Table B.2: Photoabsorption Cross-Sections for Molecular Hydrogenλ (nm)
0.1-0.3
0.3-0.44
0.44-0.8
0.8-1.8
1.8-2.3
2.3-3.1
3.1-4.4
4.4-5.0
5.0-6.15
6.15-10
10-15
15-20
20-25
25-30
30-35
35-40
40-45
45-50
50-55
55-60
60-65
65-70
70-75
75-80
80-85
85-90
90-95
95-100100-105

σ (10−18 cm2 )1.5e-6
1.8e-6
4.1e-6
4.1e-51.2e-43.1e-41.0e-32.0e-33.4e-31.2e-24.8e-2
0.14
0.40
0.76
1.34
2.1
2.9
3.8
5.0
6.3
7.8
8.9
11.1
12.8
14.3
16.0
26.3
32.9
25.2

Note. The intervals of wavelength are given according to Avakyan et al. (1998).

97B.4. DATA FOR BINARY DIFFUSION COEFFICIENT

98

Figure B.1: Schematic of the two-dimensional energy deposition calculation method.
(left) Solar beam is in the plane of the figure. (right) Solar beam is perpendicular tothe surface of the paper. This figure is modified after Fig. 6 in Tian et al. (2005)

values (from D = 0 to D = r). When all shell segments belonging to the same shell
are included, the total energy deposited into the shell is obtained. The total energy
Etotal divided by the total volume of the shell Vshell gives the volume heating rate q
in the shell:q(r) =

EtotalVshell

(B.1)

Z r

(B.2)

Etotal =0∆S(r) · E(r) · σ · n(r) · 2πD∆D

4
4
Vshell = πr 3 − π(r − ∆r)3
3
3

(B.3)

where, E(r) is the solar EUV energy entering the shell (see section B.1); σ is the
cross-section of photoabsorption for molecular hydrogen (see section B.2); n(r) is thenumber density of molecular hydrogen in the shell. ∆S(r) is described by
s


s


∆r 2
∆r
− D2 −
r+r−∆S(r) =
2 q
2
√
2
2
2
2
r − D − (r − ∆r) − D
=


2

− D2

(B.4)
(B.5)

And E(r) is given byE(r) = E(r + 1) · (1 − n(r + 1) · σ · ∆S(r + 1))

B.4

(B.6)

Data for Binary Diffusion Coefficient

In Table B.3, the relevant binary diffusion coefficients are collected. Also shown
are relative values of Φi , normalized so that Φi = 1 for 36 Ar in H2 at T = 300 K.

APPENDIX B. SUPPORTING MATERIALS FOR NUMERICAL MODEL

99

Table B.3: Binary Diffusion CoefficientSpecies
4
He
16
O
18
H2 O
20
Ne
28
CO
36
Ar
84
Kr
130
Xe

Φi
4.83
2.10
2.7
1.29
1.36
1.00
0.494
0.358

bi in H2 [1017 /cm·s]
5.23×T 0.75
3.0×T 0.75
5.23×T 0.75
4.37×T 0.731
2.65×T 0.75
2.81×T 0.75
2.3×T 0.76
2.7×T 0.712

Note. The data are given according to Zahnle & Kastig (1986) and references therein.

This parameter Φi is a dimensionless ratio of the relative importance of upward dragto gravity on the minor constituent (Zahnle & Kasting, 1986). These provide anindication of the relative ease with which different constituents may escape (Zahnle& Kasting, 1986). It is worth noting that from the perspective of fractionation duringhydrodynamic escape, Ne and CO are effectively equivalent. It is also apparent that
O and H2 O are lost much more easily than any noble gas save He.

Appendix C
Analytical Solutions
C.1

Parker Solution

For the isothermal atmosphere, the exact solution (Parker, 1964) is
v2
v2
v02
v02
2w 2
1
−
ln
=
−
ln
+
4
ln
ξ
−
1−
2
2
2
2
2
c0
c0
c0
c0
c0
ξ

!

(C.1)

where, v is the velocity, v0 is the velocity at the bottom of atmosphere, c0 is thesonic velocity at the bottom of atmosphere, ξ = r/a is the dimensionless variable, ais the radius of planet, and w 2 = GM/a is the gravitational potential at the bottom
of atmosphere. The critical point is
ξc =

w2
,
2c20

v(ξc ) = c0

(C.2)

so that v0 is given by
v02
v02
w2
2w 2
−
ln
=
−3
−
4
ln
+
c20
c20
2c20
c20

(C.3)

v2
v2
2c20 ξ 2w 2
−
ln
=
−3
−
4
ln
+ 2
c20
c20
w2
c0 ξ

(C.4)

and v(ξ) is given by

According to Eq. (C.4), we can get the exact solution of velocity normalized sonicvelocity v/c0 for the isothermal atmosphere.

C.2

Quasi-Analytical Solution for Polytropic Atmosphere

For the polytropic atmosphere, the following relations are assumed.
p
p0

! γ−1
γ

=

ρ
ρ0

!γ−1

101

=

T
T0

(C.5)

C.2. QUASI-ANALYTICAL SOLUTION FOR POLYTROPIC ATMOSPHERE102

Equation (C.5) is reduced to
1 dp
1 dρ
=γ
p dr
ρ dr

(C.6)

The terms dp/dr and dρ/dr are eliminated from Eq. (4.1), (4.2), and Eq. (C.5), we
can getkTru2 − γ
u
m

!

du
kT
GM
= 2γ
−
dr
m
r

(C.7)

Then, dp/dr and dρ/dr are determined using du/dr
du GM
dp= −ρ u + 2
dr
dr
r
!
dρ
1 du 2= −ρ+dr
u dr r

!

(C.8)
(C.9)

If the critical point (= sonic point) does in fact exist, the following relations are
required at this pointkTc
u2c − γ
=0

s

kTc
m
m
kTc GM
1 GMm
2γ= 0 ⇔ rc =−m
rcγ 2kTc⇔ uc =(C.10)

γ(C.11)

where, subscript c means the value at the critical point. Then, the slope (du/dr)r at
this critical point may be evaluated from Eq. (C.7) by use of L’Hôpital’s rule:
du
dr

!

=

c

(df /dr)c(dg/dr)c(C.12)

Here, f and g are!

kTGM
f = u 2γ
−
m
r
!
kT
2
g = r u −γm

(C.13)(C.14)

Equation (C.13) and (C.14) are differentiated respectively,
df
kT
du
GM
2γ
=
−
dr
dr
m
r
⇔

df
dr

!

c

"

!

k
= uc 2γ
m
=

k dT
GM
+ u 2γ
+ 2
m dr
r
dT
dr

−2(γ − 1)u2c

!

GM
+ 2
rc
c

du
dr

!

!

(C.15)#u3c− 2(2γ − 3)rcc(C.16)

APPENDIX C. ANALYTICAL SOLUTIONS
kTdg= u2 − γ
dr
m

!

!

"

⇔

dg
dr

c

103

du
k dT
+ r 2u − γdrm dr= rc 2uc

dudr= (γ + 1)rc uc

!

k
−γ
m
c
du
dr

!

c

dT
dr

!

(C.17)! #c+ 2(γ − 1)u2c(C.18)

Eq. (C.12), (C.16), and (C.18) are reduced to the following quadratic equation(γ + 1)ξ 2 + 4(γ − 1)ξ + 2(2γ − 3) = 0
where, ξ = urcc



(C.19)

du, and Eq. (C.19) can be solved using the quadratic formula. The
dr ccorrect critical point slope is obtained by choosing the smaller root of the resulting quadratic equation in (du/dr)c. The larger root corresponds to the unphysicaltransonic solution where u is subsonic at large distances. For the polytropic winds
considered here all the partial derivatives in Eq. (C.12) may be evaluated analytically, so that the slope (du/dr)r is determined exactly. The solution to Eq. (C.7)
is then obtained by numerically integrating u inward and outward from the criticalpoint, using the usual Runge-Kutta method. In the same way, p and ρ are obtainedby numerically integrating Eq. (C.8) and (C.9).

Acknowledgments
I would like to express my sincere appreciation to my supervisor Dr. Y. Abe. His
advice, encouragement, and comment guided this study properly. And many discussions with him improve not only this thesis but also my philosophy of planetaryscience.
I am deeply grateful to Dr. H. Genda for his helpful advises and comments
throughout this work. His constructive comment on numerical study is essential todeveloping the numerical code.
I would like to express my gratitude to Dr. E. Tajika, Prof. K. Kuramoto, Prof.
S. Sasaki, and Dr. N. Iwagami for their meaningful reviews and comments, which
greatly improve this thesis.
I would like to thank Prof. M. Ozima, Prof. H. Nagahara, Dr. S. Sugita, Dr. S.
Tachibana, and other members in Earth and Planetary System Science Colloquium
and Seminar for their fruitful suggestions and comments.
I would like to thank Dr. R. Machida, K. Hamano, K. Morizono, A. Sakamoto, K.
Watanabe, K. Tetsuzaki, D. Tsuihiji, K. Abuku, and J. Miki for fruitful discussions
and comments. I would also like to thank S. Watanabe, T. Ishii, A. Suzuki, H.
Ichikawa, H. Sato, K. Ishibashi, K. Kitazato, Y. Saito for friendly competitions andencouragements each other.
This thesis was financially supported by JSPS Research Fellowship.
Finally, I am grateful to my parents and grandparents for their enduring encouragements and supports.

105

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