Download Modeling of terrestrial extrasolar planetary

Modeling of terrestrial extrasolar
planetary atmospheres in view of habitability
vorgelegt von
Diplom-Physikerin
Barbara Stracke
aus Pointe Claire (Kanada)
Von der Fakultät II - Mathematik und Naturwissenschaften
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
Dr. rer. nat
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. rer. nat. Mario Dähne
Berichterin/Gutachterin: Prof. Dr. rer. nat. Heike Rauer
Berichter/Gutachter: Prof. Dr. rer. nat. Erwin Sedlmayr
Tag der wissenschaftlichen Aussprache: 10. Juli 2012
Berlin 2012
D83
Diese Arbeit wurde im Institut für Planetenforschung am Deutschen Zentrum für
Luft und Raumfahrt e.V. in Berlin-Adlershof in der Abteilung ’Extrasolare Planeten
und Atmosphären’ unter Betreuung von Prof. Dr. rer. nat. H. Rauer angefertigt.
Abstract
The Habitable Zone (HZ) is generally defined as the orbital region around a star, in
which life supporting (habitable) planets can exist. Taking into account that liquid
water is a fundamental requirement for the development of life as we know it, the
HZ is mainly determined by the stellar insolation, which is sufficient to maintain
liquid water at the planetary surface.
The aim of this thesis was to address two key scientific questions about the inner
boundary of the HZ: Firstly, where is the inner boundary of the HZ located in the
Solar System and in other stellar systems, and secondly, is the runaway greenhouse
effect important for the determination of the inner boundary of the HZ?
To investigate the physical processes relevant for the determination of the inner
boundary of the HZ a one-dimensional radiative-convective atmospheric model was
improved, validated, and tested to be able to answer the addressed questions. The
feedback processes for increased solar insolations between the surface temperature
and the greenhouse effect of water vapor on the boundary of the inner HZ are
calculated self-consistently.
With this atmospheric model the inner limit of the HZ is determined self-consistently
for the Solar System. Criteria for the inner boundary of the HZ are the critical point
of water and the loss of water vapor due to atmospheric escape. The influence of
specific parameters on the inner boundaries of the HZ is investigated like e.g. the
surface albedo, relative humidity, and the size of the water reservoir. The inner
boundary of the HZ is also determined for different central stars and compared to
HZ scalings.
The occurrence of a runaway greenhouse for increased stellar insolation and surface
temperatures is determined by the application of radiation limits of the outgoing
infrared flux in previous studies. If the stellar flux exceeds this radiation limit, the
surface temperature of the planetary atmosphere is assumed to increase until all the
liquid water available on the surface is evaporated. It is shown in this study that the
so-called ’tropospheric radiation limit’, where the planetary outgoing infrared flux
approaches a constant value with increasing solar insolation, is not a good criterion
for the occurrence of a runaway greenhouse effect. For the investigated model scenarios no runaway greenhouse is approached, although a tropospheric radiation limit
of the outgoing infrared flux occurs. This is caused by enhanced Rayleigh scattering
of water for water dominated atmospheres, which leads to a constant net shortwave
flux for increasing insolation. This constant net shortwave radiation flux is able to
balance the constant net infrared radiation flux and thus radiative equilibrium is
possible at the top of the atmosphere.
Zusammenfassung
Die Habitable Zone (HZ) ist allgemein definiert als der Bereich um einen Stern, in
der lebensfreundliche (habitable) Planeten existieren können. Mit flüssigem Wasser
als Grundvoraussetztung für Leben wie wir es kennen, kann die HZ um einen Stern
bestimmt werden durch die Einstrahlung des Sterns, die ausreichend stark sein muss,
damit Wasser auf der Planetenoberfläche flüssig sein kann.
Ziel dieser Arbeit ist es, zwei Schlüsselfragen bezüglich der inneren Grenze der HZ
zu beantworten: Zum einen wo befindet sich die innere Grenze der HZ im unserem Sonnensystem und wo um andere Sterne und zum anderen inwieweit ist der
selbstverstärkende Treibhauseffect (runaway greenhouse effect) wichtig ist für die
Bestimmung der inneren Grenze der HZ.
Um die physikalischen Prozesse, die für die Bestimmung der inneren Grenze der
HZ relevant sind, zu untersuchen, wurden Verbesserungen an einem eindimensional
radiativ-konvektiven Atmosphärenmodell vorgenommen. Die Rückkopplungsprozesse
für erhöhte Einstrahlung zwischen Oberflächentemperature und dem Treibhauseffekt
von Wasser werden nun selbstkonsistent berechnet.
Mit diesem Atmosphärenmodell wurde die inner Grenze der HZ bestimmt, indem
die stellare Einstrahlung erhöht und verschiedene Modellszenarien selbstkonsistent
berechnet wurden. Kriterien für die Bestimmung der inneren Grenze der HZ sind,
dass die Oberflächentemperature den kritischen Punkt von Wasser erreicht oder
dass Wasserdampf durch atmosphärische Verlustprozesse verloren wird. Die innere
Grenze der HZ wurde für beide Kriterien bestimmt. Des weiteren wurde der Einfluss
von bestimmten Parametern auf die innere Grenze untersucht, wie zum Beispiel
die Oberflächenalbedo, die relative Feuchte und die Größe des Wasserreservoirs.
Zudem wurde die innere Grenze auch für andere Zentralsterne bestimmt und mit
HZ-Skalierungen verglichen.
Das Auftreten eines selbsverstärkenden Treibhauseeffektes für erhöhte Einstrahlung
und Oberflächentemperaturen wird in vielen früheren Studien durch die Anwendung von Grenzwerten für den infraroten Strahlungsfluss bestimmt. Wenn die
Einstrahlung des Sterns höher ist als dieser Grenzwert, wird angenommen, dass
die Oberflächentemperatur so stark ansteigt, dass das gesamte Wasserreservoir verdampft wird. In dieser Arbeit wird gezeigt, dass dieser Grenzwert für die infrarote
Strahlung kein gutes Kriterium ist, um zu bestimmen, ob ein selbstverstärkender
Treibhauseffekt auftritt. Der Grund hierfür ist, dass bei den untersuchten Modelszenarien für erhöhte stellare Einstrahlung kein selbstverstärkender Treibhauseffekt erreicht wird, obwohl ein konstanter Grenzwert für den ausgehenden infraroten
Strahlungsfluss auftritt. Dies ist bedingt durch den Effekt der Rayleigh-Streuung
von Wasserdampf für wasserdominierte Atmosphären, was zu einem konstanten
kurzwelligen Nettostrahlungsfluss führt. Dieser konstante kurzwellige Strahlungsfluss kann den konstanten ausgehenden infrarot Strahlungsfluss an der oberen Grenze
der Atmosphäre ausgleichen, und damit ist globales Strahlungsgleichgewicht möglich.
Contents
1 Introduction
1
1.1
Aim of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.2
Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2 Habitability of terrestrial planets and habitable zones
2.1
2.2
7
Habitability of terrestrial planets . . . . . . . . . . . . . . . . . . . .
7
2.1.1
Energy sources . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.2
Carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.3
Liquid water . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.1.4
Additional habitability requirements . . . . . . . . . . . . . .
10
Habitable zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2.1
Different Definitions of the HZ . . . . . . . . . . . . . . . . .
11
2.2.2
Extensions and restrictions to the classical HZ concept . . . .
16
3 Basic atmospheric physics
21
3.1
Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.2
Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . .
22
3.3
Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.4
Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.4.1
The global energy balance . . . . . . . . . . . . . . . . . . . .
23
3.4.2
Global energy balance with greenhouse effect . . . . . . . . .
26
Energy transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.5.1
29
3.5
Radiative transfer . . . . . . . . . . . . . . . . . . . . . . . .
CONTENTS
ii
3.6
3.5.2
Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.5.3
Schwarzschild criterion . . . . . . . . . . . . . . . . . . . . . .
34
Water in the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . .
35
3.6.1
Characteristics of water vapor in the atmosphere . . . . . . .
35
3.6.2
The hydrological cycle . . . . . . . . . . . . . . . . . . . . . .
36
3.6.3
Clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.6.4
Optical properties of water vapor . . . . . . . . . . . . . . . .
37
4 Physical processes determining the inner HZ boundary
4.1
4.2
4.3
General conditions for instability of liquid water on the surface of a
planet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Runaway greenhouse effect
40
. . . . . . . . . . . . . . . . . . . . . . .
4.2.1
Stratospheric radiation limit
. . . . . . . . . . . . . . . . . .
41
4.2.2
Tropospheric radiation limit . . . . . . . . . . . . . . . . . . .
44
Loss of water from the atmosphere . . . . . . . . . . . . . . . . . . .
46
4.3.1
Thermal escape . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.3.2
Hydrodynamic processes . . . . . . . . . . . . . . . . . . . . .
48
4.3.3
Non-thermal escape . . . . . . . . . . . . . . . . . . . . . . .
48
4.3.4
Diffusion-limited escape flux . . . . . . . . . . . . . . . . . . .
48
5 Previous studies related to the inner HZ boundary
5.1
5.2
39
51
Runaway Greenhouse Limit . . . . . . . . . . . . . . . . . . . . . . .
51
5.1.1
Ingersoll (1969) . . . . . . . . . . . . . . . . . . . . . . . . . .
51
5.1.2
Komabayasi (1967) . . . . . . . . . . . . . . . . . . . . . . . .
52
5.1.3
Pollack (1971) . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.1.4
Kasting (1988) and Kasting et al. (1993) . . . . . . . . . . . .
55
5.1.5
Nakajima et al. (1992) . . . . . . . . . . . . . . . . . . . . . .
58
5.1.6
Rennó et al. (1994)
. . . . . . . . . . . . . . . . . . . . . . .
60
5.1.7
Pujol and North (2002) . . . . . . . . . . . . . . . . . . . . .
61
5.1.8
Ishiwatari et al. (2002) . . . . . . . . . . . . . . . . . . . . . .
62
5.1.9
Sugiyama et al. (2005) . . . . . . . . . . . . . . . . . . . . . .
63
5.1.10 Selsis et al. (2007) . . . . . . . . . . . . . . . . . . . . . . . .
64
5.1.11 Goldblatt and Watson (2012) . . . . . . . . . . . . . . . . . .
64
5.1.12 Advantages and disadvantages of the runaway greenhouse studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
Water Loss Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
5.2.1
67
Kasting (1988) and Kasting et al. (1993) . . . . . . . . . . . .
CONTENTS
iii
6 Model description
71
6.1
Model requirements . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
6.2
Model history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
6.3
Model improvements . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
6.4
Model assumptions and simplifications . . . . . . . . . . . . . . . . .
74
6.5
Schematic overview of the model . . . . . . . . . . . . . . . . . . . .
75
6.6
Basic model description . . . . . . . . . . . . . . . . . . . . . . . . .
77
6.7
Radiative transfer
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
6.7.1
Thermal radiation . . . . . . . . . . . . . . . . . . . . . . . .
78
6.7.2
Shortwave radiation . . . . . . . . . . . . . . . . . . . . . . .
82
Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
6.8.1
Adiabatic lapse rate of water . . . . . . . . . . . . . . . . . .
85
6.8.2
Adiabatic lapse rate of carbon dioxide . . . . . . . . . . . . .
87
Calculation of the temperature profile . . . . . . . . . . . . . . . . .
87
6.10 Calculation of the water profile . . . . . . . . . . . . . . . . . . . . .
92
6.8
6.9
7 Test and validations of the model
95
7.1
Influence of infrared radiative transfer codes . . . . . . . . . . . . . .
95
7.2
Conditions for convection . . . . . . . . . . . . . . . . . . . . . . . .
97
7.3
Calculation of water mixing ratios . . . . . . . . . . . . . . . . . . .
98
7.4
Summary of further tests and validations . . . . . . . . . . . . . . . 100
7.4.1
Validation of MRAC . . . . . . . . . . . . . . . . . . . . . . . 100
7.4.2
Test of the numerical scheme . . . . . . . . . . . . . . . . . . 101
8 Results
103
8.1
Input and boundary parameters . . . . . . . . . . . . . . . . . . . . . 104
8.2
Inner boundary of the HZ determined by the critical point of water . 105
8.2.1
Discussion of the runaway greenhouse effect . . . . . . . . . . 110
8.2.2
Influence of Rayleigh scattering of water on the determination
of the inner boundary of the HZ . . . . . . . . . . . . . . . . 119
8.3
Inner boundary of the HZ determined by the water loss limit . . . . 123
8.4
Influence of surface albedo of the planet . . . . . . . . . . . . . . . . 125
8.5
8.4.1
Critical point of water . . . . . . . . . . . . . . . . . . . . . . 125
8.4.2
Water loss limit . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Impact of the relative humidity . . . . . . . . . . . . . . . . . . . . . 127
8.5.1
Critical point of water . . . . . . . . . . . . . . . . . . . . . . 129
8.5.2
Water loss limit . . . . . . . . . . . . . . . . . . . . . . . . . . 130
CONTENTS
iv
8.6
Role of the water reservoir . . . . . . . . . . . . . . . . . . . . . . . . 131
8.7
Case study: Kepler 22b-like planet . . . . . . . . . . . . . . . . . . . 133
8.8
Influence of different stellar types on the inner boundary of the HZ . 136
8.8.1 HZ Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.9
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9 Summary and Outlook
9.1
147
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9.1.1 Model improvements . . . . . . . . . . . . . . . . . . . . . . . 147
9.1.2
Where is the inner boundary of the HZ located in the Solar
System and in other stellar systems? . . . . . . . . . . . . . . 148
9.1.3
9.2
Is the runaway greenhouse important for the determination of
the inner boundary of the HZ? . . . . . . . . . . . . . . . . . 149
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.2.1
Model improvements . . . . . . . . . . . . . . . . . . . . . . . 150
9.2.2
Model scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 151
List of Figures
2.1
HZ for different stars (Kasting et al. (1993)) . . . . . . . . . . . . . .
14
3.1
Diagram of the energy fluxes for a planet without an atmosphere . .
25
3.2
Diagram of the energy fluxes for a planet with an atmosphere. . . .
27
3.3
Absorption spectra for the Earth’s atmosphere . . . . . . . . . . . .
31
4.1
Phase diagram of water . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.2
Relationship between the optical depth and the temperature at the
tropopause (Nakajima et al. (1992)) . . . . . . . . . . . . . . . . . .
44
Solar insolation versus the surface temperature for pure water vapor
models with a 50% cloud cover (Pollack (1971)) . . . . . . . . . . . .
54
Outgoing net infrared flux and net solar flux versus the surface temperature (Kasting et al. (1993)) . . . . . . . . . . . . . . . . . . . . .
55
Effective solar insolation versus the surface temperature (Kasting
(1988)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Outgoing infrared flux versus the surface temperature (Nakajima et
al. (1992)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
Optical depth versus the surface temperature and the mole fraction
of H2 O (Nakajima et al. (1992)) . . . . . . . . . . . . . . . . . . . .
60
5.6
Surface temperature versus the solar forcing (Rennó et al. (1994)) .
61
5.7
Outgoing infrared flux versus the surface temperature (Pujol and
North (2002)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
Meridional distribution of the zonal mean outgoing infrared (longwave) radiation (OLR) (Ishiwatari et al. 2002) . . . . . . . . . . . .
63
5.1
5.2
5.3
5.4
5.5
5.8
LIST OF FIGURES
vi
5.9
Vertical profile of H2 O mixing ratio for selected surface temperatures
(Kasting et al. (1993)) . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.10 Variation of stratospheric H2 O with effective solar flux (Kasting et
al. (1993)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
6.1
Model scheme of the radiative-convective model . . . . . . . . . . . .
76
6.2
Input temperature profile to the radiative-convective model . . . . .
88
6.3
Calculation of the temperature profile taking radiative equilibrium
into account. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
Calculation of the temperature one layer above the surface taking
radiative equilibrium into account. . . . . . . . . . . . . . . . . . . .
90
Calculation of the surface temperature from one layer above with
convective adjustment. . . . . . . . . . . . . . . . . . . . . . . . . . .
91
Calculation temperatures from one layer above the surface up to the
tropopause with convective adjustment. . . . . . . . . . . . . . . . .
91
Relative humidity profile for the Earth of Manabe and Wetherald
(1967). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
Temperature profiles over altitude and pressure for different infrared
schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
Temperature and water profiles over altitude for different convection
conditions for a RH of Manabe and Wetherald (1967). . . . . . . . .
97
Temperature and water profiles over altitude for different convection
conditions for RH=100%. . . . . . . . . . . . . . . . . . . . . . . . .
98
Temperature and water profiles over altitude for different water calculation approaches for a RH of Manabe and Wetherald (1967).. . .
99
6.4
6.5
6.6
6.7
7.1
7.2
7.3
7.4
7.5
Temperature and water profiles over altitude for different water calculation approaches for RH=100%. . . . . . . . . . . . . . . . . . . . 100
8.1
Temperature and water profiles over altitude for increased solar insolations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.2
Total radiative flux at TOA for increased solar insolations. . . . . . . 107
8.3
Infrared fluxes over altitude for increased solar insolations. . . . . . . 108
8.4
Shortwave fluxes over altitude for increased solar insolations. . . . . 108
8.5
Surface temperatures for increased solar insolation . . . . . . . . . . 109
8.6
Surface temperatures versus the distance of the planet to the star. . 110
8.7
Net infrared fluxes for increased solar insolation. . . . . . . . . . . . 111
8.8
Net infrared fluxes for increased surface temperature. . . . . . . . . . 112
8.9
Infrared fluxes over wavelength for increased solar insolations . . . . 113
LIST OF FIGURES
vii
8.10 Mean optical depth versus wavelength for different altitudes for different solar insolations. . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.11 Mean optical depth over altitude and atmospheric temperatures over
mean optical depths for spectral band 11 (3.64-4.17μm) for increased
solar insolations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.12 Mean optical depth over altitude and atmospheric temperatures over
mean optical depths for spectral band 16 (5.41-7.41μm) for increased
solar insolations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.13 Mean optical depth over altitude and atmospheric temperatures over
mean optical depths for spectral band 18 (9.01-10.00μm) for increased
solar insolations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.14 Temperature profiles over pressure for increased solar constants.
. . 117
8.15 Net infrared fluxes and net solar fluxes for increased solar insolation. 119
8.16 Temperature profiles over altitude for increased solar insolations with
and without Rayleigh scattering of H2 O. . . . . . . . . . . . . . . . . 120
8.17 Surface temperatures for increased solar insolations with and without
Rayleigh scattering of H2 O. . . . . . . . . . . . . . . . . . . . . . . . 121
8.18 Net infrared fluxes and shortwave fluxes for increased solar insolations
with and without Rayleigh scattering of H2 O. . . . . . . . . . . . . . 122
8.19 Planetary albedos for increased solar insolations with and without
Rayleigh scattering of H2 O. . . . . . . . . . . . . . . . . . . . . . . . 123
8.20 Temperature and water profiles over altitude for increased solar insolations with an other water calculation approach. . . . . . . . . . . . 124
8.21 Stratospheric water mixing ratios for a series of runs versus increased
solar insolations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.22 Planetary albedos for increased solar insolations for different surface
albedos Asurf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.23 Temperature and water profiles over altitude for different relative humidity profiles for S = 1.00S0 . . . . . . . . . . . . . . . . . . . . . . 128
8.24 Temperature and water profiles over altitude for different relative humidity profiles for S = 1.41S0 . . . . . . . . . . . . . . . . . . . . . . 129
8.25 Surface temperatures for increased solar insolation for RH of Manabe
and Wetherald (1967). . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.26 Stratospheric water mixing ratios for increased solar insolations for
RH of Manabe and Wetherald (1967). . . . . . . . . . . . . . . . . . 131
8.27 Position of the inner boundary of the HZ for model scenarios with
different water reservoirs. . . . . . . . . . . . . . . . . . . . . . . . . 133
8.28 Temperature and water profiles over altitude of Kepler 22b-like planet
for different relative humidity profiles. . . . . . . . . . . . . . . . . . 135
8.29 Stellar Input spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
viii
LIST OF FIGURES
8.30 Temperature and water profiles over altitude for different star for
S = 1.00S0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.31 Surface temperatures for increased stellar insolations for three different host stars of the planet. . . . . . . . . . . . . . . . . . . . . . . . 139
8.32 Stratospheric water mixing ratios versus increased stellar insolations
for three different host stars of the planet. . . . . . . . . . . . . . . . 140
8.33 Planetary Albedo over increased stellar insolations for planets around
three different host stars. . . . . . . . . . . . . . . . . . . . . . . . . 141
8.34 Distances of the inner boundaries of the HZ around different stars. . 142
8.35 Relationship between the solar insolation and the effective temperature of stars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.36 Relationship between the distance and the effective temperature of
stars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
CHAPTER
1
Introduction
The search for life is not only restricted to the Earth, although it is the only planet,
where life has been found so far. In the Solar System searching for life is performed
via remote sensing and also via in-situ methods. Mars is of especial interest, since
the detection of fluvial features and hydrated minerals suggests that liquid water
existed on the surface of early Mars (e.g. Masson et al. (2001)) and liquid water is
a fundamental requirement for life as we know it (see subsection 2.1.3).
Even beyond the boundaries of the Solar System the search for life has become
scientifically justifiable and not only science fiction, with the first detection of an
extrasolar planet around a main sequence star in 1995 (Mayor and Queloz, 1995).
Many of the extrasolar planets detected so far are gas giants close to their central
star (’hot jupiters’). Since for the most successful detection methods, namely the
radial velocity method and the transit method, these planets are easier to find. The
detection sensitivity has been improved down to smaller planets e.g. via space-based
extrasolar planet surveys like CoRoT and Kepler. Furthermore, longer observing
times enhance the probability of the detection of planets farther away from their
central star.
With over 50 low mass planets detected today (so-called Super-Earths, planets with
masses from one to ten Earth masses (ME )) the question as to determine whether
such planets could host life becomes more important. A central indicator to assess,
if a planet could be habitable (life supporting) (see section 2.1), is its surface temperature. The surface temperature should be appropriate for liquid water a basic
requirement for life (see subsection 2.1.3).
Usually from detection measurements of extrasolar planets only the following planetary characteristics are available:
Introduction
2
• Distance of the planet to its central star, from the orbital period
• The mass of the planet (for radial velocity method only the minimum mass)
• The radius of the planet (only by transit method)
Taking the above known characteristics of detected extrasolar planets and information about the central stars into account, an effective temperature of the planet (see
section 3.4.1) can be estimated for an assumed planetary albedo. The planetary
albedo describes how much of the incoming stellar radiation is reflected back to
space and depends on the characteristics of the planetary atmosphere.
The atmosphere plays an even more important role for the planetary surface temperature due to the greenhouse effect (see section 3.4.2). For example, for the Earth the
atmospheric greenhouse effect due to radiatively active gases produces an increase in
surface temperature of 33 K. This leads on the Earth to a mean surface temperature
of 288 K, which allow liquid water on the planetary surface. Without its atmosphere
the Earth would have a surface temperature below the freezing point of water and
the temperature difference between the day and night side would increase. Thus,
without an atmosphere the Earth would not be habitable.
Information on the atmospheres of extrasolar planets such as for example the temperature and chemistry can be gained by characterizing these planets spectroscopically.
Characterization of atmospheres with spectra has so far only been possible for a few
tens of extrasolar planets, mainly ’hot jupiters’ and ’hot neptunes’ (e.g. Charbonneau et al. (2002); Vidal-Madjar et al. (2004); Stevenson et al. (2010); Madhusudhan
and Seager (2011)).
Due to the lack of information about the atmospheres of smaller extrasolar planets,
modeling of planetary atmospheres can give insights into the habitability of such
objects. By performing atmospheric modeling of different atmospheric compositions and masses it can be assessed whether a planet could have habitable surface
temperatures for a specific, assumed atmospheric composition.
However, it is not only interesting to determine whether an existing specific planet
could be habitable, but also to determine boundaries for habitable conditions. The
habitable zone (HZ) (see section 2.2) is defined as the orbital region around a star,
in which life-supporting (habitable) planets can exist. Taking into account that
liquid water is a fundamental requirement for the development of life as we know it,
the HZ around a star is strongly influenced by the stellar insolation, which should
be sufficient enough to maintain liquid water on the surface of the planet. The
boundaries of the HZ are therefore controlled by processes that can lead to too high
or too low surface temperatures for liquid water on the planetary surface.
The boundaries of the HZ for the Sun and also other stars were investigated in studies, which are reviewed in section 2.2. In the seminal study by Kasting et al. (1993)
the HZ boundaries for different planetary characteristics and different stars using
a one-dimensional radiative-convective model of the atmosphere were determined.
The results of the Kasting et al. (1993) study are used for scalings in recent studies
3
which aim at determining whether detected extrasolar planets could be habitable
(e.g., Selsis et al. (2007), Underwood et al. (2003)) (see subsection 2.2.2).
However, a disadvantage of the study by Kasting et al. (1993) is that the atmospheric
temperature, water profiles and radiative fluxes were not calculated self-consistently
(see subsection 5.1.4). For example, to determine the inner boundary of the HZ
Kasting et al. (1993) prescribed temperature profiles by fixing the surface temperature and the stratospheric temperature. With these temperatures and resulting
water profiles, infrared and shortwave fluxes were calculated. The (effective) solar
insolation and thus the corresponding orbital distance required to maintain a given
surface temperature was determined by assuming global energy balance of the atmosphere. Thus, for the determination of the inner boundary, Kasting et al. (1993)
did not include feedback processes between the solar insolation, the surface temperature and the greenhouse effect of water. Furthermore, they did not calculate an
atmosphere in global energy balance.
Processes determining the inner boundary of the HZ in general are thought to be
relevant for the explanation of the history of Venus’ atmosphere, which might have
lost an Earth-like ocean in its early history. Also with regard to the future of
the water budget on Earth, the processes which determine the inner HZ limit, are
interesting to investigate, considering the projected increase in luminosity of the Sun
with time.
Reasons for inhabitability due to too high surface temperature and thus no liquid
water on the surface of the planet at the inner side of the HZ are (see chapter 4)
• Too much energy input from the star
• Too large greenhouse effect
• Or the coupling of both of the above effects
The runaway greenhouse effect (see section 4.2) and its effect on the surface temperature is discussed for terrestrial planets with an assumed water reservoir (comparable
to the Earth). Briefly, the runaway greenhouse refers to a coupling of a high stellar
insolation and the greenhouse effect of water vapor, which leads to such high surface
temperatures that the complete water reservoir would be evaporated. Atmospheric
modeling calculations determining the runaway greenhouse focus on radiation limits
of the outgoing thermal radiation (see section 4.2.1 and 4.2.2). It is usually assumed
that if the stellar flux exceeds this radiation limit, the surface temperature increases
until the complete surface water reservoir of the planet is evaporated.
The inner boundary of the HZ can also be determined considering the loss of the
planetary water reservoir within the lifetime of the planet for wet planetary atmospheres assuming photo-dissociation of water vapor and subsequent escape of
hydrogen to space, discussed in section 5.2.
Introduction
4
1.1
Aim of this Thesis
Focusing on the determination of the inner boundary of the HZ, in the most cited
study investigating this boundary by Kasting et al. (1993), the temperatures, water
concentrations and radiative fluxes of the planetary atmospheres were not calculated
self-consistently for increased solar insolations.
To investigate the effect of feedback processes for increased solar insolation between
the surface temperature and the greenhouse effect of water vapor on the boundary
of the inner HZ, a self-consistent atmospheric model has to be applied. One aim
of this Thesis is to develop and apply an atmospheric model which would allow
the investigation of the inner HZ more consistently than in the previous study by
Kasting et al. (1993).
This model is designed to be able to calculate the feedback of temperature, water
vapor concentration, and wavelengths dependent radiative fluxes self-consistently
taking the basic physical processes acting in planetary atmospheres into account.
Temperature and water vapor concentrations in the model have to respond to increased solar insolations and also to increased insolations of different central stars.
A goal of this Thesis is to investigate the physical processes relevant for the determination of the inner boundary of the HZ and to address the following scientific
questions applying this self-consistent atmospheric model:
Where is the inner boundary of the HZ located in the Solar System and
in other stellar systems?
To answer this question processes important to determine the inner HZ are investigated. With the self-consistent atmospheric model the inner limit of the HZ is
determined for the Solar System. Conditions for the inner boundary of the HZ are
the critical point of water and the loss of water vapor due to atmospheric escape.
The influence of specific parameters on the inner boundaries of the HZ is investigated
like e.g. the surface albedo, relative humidity, and the size of the water reservoir.
The inner boundary of the HZ is also determined for different central stars.
Is the runaway greenhouse important for the determination of the inner
boundary of the HZ?
The occurrence of the runaway greenhouse for increased stellar insolation and surface
temperatures is determined by the application of radiation limits of the outgoing
infrared flux. If the stellar flux exceeds this radiation limit, the surface temperature
of the planetary atmosphere is assumed to increase until all the liquid water available
on the surface is evaporated. The conditions leading to such radiation limits will be
investigated and discussed for the results of the inner boundary of the HZ determined
by the critical point of water. It is tested whether these conditions imply a runaway
greenhouse effect.
1.2 Outline of the Thesis
1.2
Outline of the Thesis
This Thesis is organized as follows:
In chapter 2 habitability and the habitable zone concept are discussed. The basic
atmospheric physics are reviewed in chapter 3. Chapter 4 deals with the most
important physical processes for the determination of the inner boundary of the HZ,
for which a literature overview is given in chapter 5. The one dimensional radiativeconvective model used and adapted to calculate the inner boundary of the HZ is
described in chapter 6 and validated in chapter 7. Chapter 8 presents the results of
the atmospheric model to determine the inner boundary of the habitable zone. The
Thesis ends with summary and an outlook in chapter 9.
5
6
Introduction
CHAPTER
2
Habitability of terrestrial planets and habitable zones
In order to describe the Habitable Zone (HZ) a general definition of habitability is
given for terrestrial planets. Although a clear definition for life as we know it on the
Earth is lacking, the most important requirements for life on Earth are known and
presented in detail. Furthermore, an overview of the different definitions of the HZ
is given based on the habitability requirements. Extensions and restrictions to the
classical HZ concept are presented at the end of this chapter.
2.1
Habitability of terrestrial planets
Habitability is broadly defined as the potential of an environment (past or present)
to support life of any kind (Steele et al., 2006). Thereby, two kinds of habitability for
planets can be distinguished, namely indigenous habitability, which is the potential
to support life that originated on the planet and exogenous habitability, which is the
potential to support life that originated somewhere else and was then carried to the
planet. Planetary habitability relates to the existence of the essential requirements
for life on the planet and not that life actually exists there. Therefore, not only
habitable planets are possible which are inhabited but also habitable planets which
are uninhabited.
The basic requirements for life on Earth, which is the only kind of life we know so
far, are energy, carbon and liquid water (McKay, 2007), which are explained in more
detail.
Habitability of terrestrial planets and habitable zones
8
2.1.1
Energy sources
Energy as an external source is needed as a precondition for living systems, because
they require energy to maintain a low state of entropy, to sustain their metabolism
and to perform work. Additionally, energy allows the existence of liquid water.
Life on the Earth utilizes only two kinds of energy, namely energy in the form of
solar radiation and chemical energy (methanogenic microbial ecosystems). Energy
is stored by the cells of life in the form of high energy phosphate compounds such
as ATP (adenosine triphosphate), which is used by all living organisms on Earth for
the biosynthesis of cellular components and for cell functions (Javaux and Dehant,
2010).
Microorganisms, which obtain energy from organic chemicals are termed chemoorganotrophs and those, which obtain energy from inorganic chemicals chemolithotrophs.
A disadvantage for life which depends on chemical energy is the strong dependence
on chemical resources.
The advantage of the use of electromagnetic radiation (visible light) emitted from
the Sun is that it provides a continuous source of energy. To capture this energy
phototrophic organisms, which use light as an energy source, utilize photosynthesis. Photosynthesis refers to the metabolic process of building organic carbon from
carbon dioxide by harvesting the energy of sunlight and water. Oxygenic photosynthesis is the process where water is split using the energy of sunlight, the extracted
hydrogen is used to reduce carbon dioxide to organic carbon and oxygen is produced
as a consequence (Catling and Kasting, 2007).
However, the total solar luminosity is not constant. In addition to variation on small
time scales, such as the 11-year solar activity cycle, on long time scales, the luminosity has increased since the ZAMS (Zero Age Main Sequence) by about 30% (Gough,
1981). Therefore, for the Early Earth also other energy sources might have been
important as summarized in Sephton (2003): For example, the decay of radioactive forms of uranium and potassium could have provided heat from the interior of
the Earth. Primordial heat could have been generated as the Earth’s accretion released gravitational energy and volcanism. Meteors and meteorites passed through
the atmosphere could also have contributed to the energy available to synthesize
molecules.
2.1.2
Carbon
The basic building blocks of life as we know it (e.g. lipids, carbohydrates, proteins
and nucleic acids) are composed of organic compounds, which are based on carbon.
Carbon, which is also one of the most abundant higher mass elements in the universe,
acts as the backbone molecule of biochemistry (McKay, 2007).
Carbon atoms form four bonds, and the single bond joining two carbon atoms is
strong and remain joined for a long time for moderate temperatures (∼288 K) (Ricardo and Benner, 2007). The bonds between carbon atoms are stronger than for
2.1 Habitability of terrestrial planets
other elements, which form chains (e.g. silicon). The ability of carbon to form chemical bonds with itself and other atoms (e.g. hydrogen, oxygen, or nitrogen) allows
chemical complexity and versatility required to conduct the reactions of biological
metabolism and propagation (Pace, 2001). Due to this ability to form long, stable
chains with itself and functional groups (composed of hydrogen, oxygen, nitrogen,
phosphorus, sulfur, and a host of metals, such as iron, magnesium, and zinc) carbon
is able to construct molecules with a high information content.
Furthermore, carbon in its oxidized form carbon dioxide (CO2 ) acts in the atmosphere as an important greenhouse gas. For the Earth the resulting greenhouse
effect helps to maintain surface temperatures above the triple point of water (273.15
K) and thus allows water to be liquid on the surface. Besides, the exchange of the
atmospheric carbon compound CO2 throughout the Carbonate-Silicate cycle helps
to stabilize the planetary climate, which may be crucial for the habitability of a
planet (Walker et al., 1981; Kasting et al., 1993; Gaidos et al., 2005), as explained
in section 2.2.1.
2.1.3
Liquid water
Liquid water (H2 O) is made from two of the most abundant elements in the universe
(hydrogen and oxygen) and is an necessary ingredient for life on Earth.
Liquid water can act as a dissolving medium for molecules of living systems. It is
a polar solvent, since the hydrogen atoms are positively charged, while the oxygen
atoms are negatively charged, which allows water molecules to form hydrogen bonds
with themselves and with organic molecules. This allows life on Earth to form
independent, stable cellular structures (Javaux and Dehant, 2010). Liquid water
contributes to biochemical reactions during metabolism and biosynthesis and can
also be a product of metabolic reactions (Baross et al., 2007).
Furthermore, liquid water is necessary for large scale processes on the planet, which
are able to support (the development of) life. The presence of water is crucial in the
generation of plate tectonics on the Earth, which might be essential for supporting
Earth-like life, since plate tectonics is required to provide minerals and nutrients
for life (Javaux and Dehant, 2010). Additionally, water is central to weathering of
rocks, which leads to the transport of nutrients into the ocean. Weathering is also
a part of the Carbonate-Silicate cycle (see Section 2.2.1), which is a fundamental
mechanism to stabilize the climate over long time scales on the Earth and for which
also plate tectonics is necessary. This cycle may have had a non-negligible influence
on the development of life.
Water is liquid in the temperature range from 0o C to 100o C for normal Earth conditions, where the surface pressure is 1 bar. Note that also organic molecules are
in this temperature range sufficiently stable and reactive. This range increases with
e.g. higher salinity, which leads to the reduction of the freezing point and the elevation of the boiling point. With increased pressure, pure water boils at higher
temperatures up to the critical point of water at 647 K. More details about the
9
Habitability of terrestrial planets and habitable zones
10
properties of water are summarized in section 3.6.
In the Solar System sunlight and the elements required for life are quite common on
the terrestrial planets. However, liquid water appears to be the limiting factor for
life.
2.1.4
Additional habitability requirements
Brownlee and Kress (2007) and Lammer et al. (2009) discuss habitability requirements in a broader context.
Brownlee and Kress (2007) proposed three minimum conditions which should be met
in order that life as we know it from the Earth could originate and evolve to more
complex life forms. These conditions are (a) a solid planet present in the HZ, (b) the
existence of the essential materials for life (liquid water and carbon compounds), and
(c) the planets should provide suitable environments for the origin and long-term
support of life.
Lammer et al. (2009) gave a summary of the basic habitability requirements suggesting: (a) a certain time span over which a celestial body can accumulate enough
building blocks necessary for the origin of life, (b) liquid water which is in contact
with these building blocks, and (c) external and internal environmental conditions
which allow liquid water to exist on a celestial body over a time span necessary for
life to evolve
The additional most important contribution from Brownlee and Kress (2007) and
Lammer et al. (2009) however is the inclusion of the time as a parameter, which
should be long enough to allow life to evolve and to support itself thereafter, and
the environment i.e. standing liquid water bodies, rocky planets.
2.2 Habitable zones
2.2
Habitable zones
An overview is given of the most important definitions and models of the HZ, which is
based mainly on the habitability requirement of liquid water on the surface of a rocky
planet. The reviewed limits of the HZs are summarized in Table 2.1. Furthermore,
proposed extensions and restrictions to this classical definition of the HZ based on
surface habitability, and a classification of habitable bodies are presented.
2.2.1
Different Definitions of the HZ
Huang (1959a,b)
The term ’Habitable Zone’ was introduced by Huang (1959a). He defined the HZ as
the region around a star within which planetary temperatures are neither too high
nor too low for life. Huang assumed this zone to be a function of the amount of energy
received per unit time and unit area facing the star. Huang also suggested that the
distance of the HZ from the star changes with stellar type. Thus, the distance of
the HZ from the star increases with increased stellar luminosity, and decreases for
stars with lower luminosity (Huang, 1959b). This was discussed qualitatively rather
than demonstrated explicitly.
Dole (1964)
Dole (1964) defined the HZ, which he termed ’ecosphere’, as region in space, in the
vicinity of a star, in which suitable planets can have surface conditions compatible
with the origin, evolution to complex life forms, and continuous existence of land life
and in particular surface conditions suitable for humans (Dole and Asimov, 1964).
Therefore, the mean surface temperature should vary between 0o C and 30o C, with
extremes not exceeding -10 or 40o C over 10% of the whole surface.
Dole used empirical methods to determine planetary surface temperatures. Therefore, an Earth-like planet was assumed with a thin, transparent atmosphere and
a cloud cover of approximately 45%. Theoretical temperature were calculated for
a simplified model of the Earth, represented by a rapidly rotating, non-conducting
black sphere, which was half illuminated by a distant point source. Dole estimated a
HZ that ranges from 0.725 Astronomical Units (AU) to 1.24 AU, which is wide range
compared to the rather narrow temperature range (0o C and 30o C, with extremes
not exceeding -10 or 40o C over 10% of the whole surface).
Kasting et al. (1993) discussed the reasons for this wide range of the HZ of Dole:
Firstly, the assumed optically thin atmosphere and secondly, the fixed planetary
albedo of the planets. An optically thin atmosphere is in contrast to conditions on
Earth, where the atmosphere is optically thick over most infrared (IR) wavelengths,
which leads to a greenhouse effect, which is not considered in the study of Dole
(1964). Furthermore, the albedo of the Earth fluctuates caused by changes in the
11
Habitability of terrestrial planets and habitable zones
12
distribution of snow, ice and clouds. The greenhouse and the albedo effect could
lead to positive feedbacks, which could destabilize planetary climate.
Hart (1978,1979)
Hart did not only take the surface temperature of the planet into account, but also
feedback processes which could lead to these temperatures. Hart used the runaway
greenhouse effect as the inner boundary of the HZ. Runaway glaciation is the cause
for the outer limit of the HZ, which Hart defined by ice covering of the oceans raising
the surface albedo and thus leading to further decrease of temperature. Hart (1978)
was the first to take into account that the boundaries of the HZ around stars are
not constant with time.
Hart (1978) calculated surface temperatures over geological timescales taking into
account the evolution of the Earth’s atmosphere. The model includes the changes
in the solar luminosity, the variation in Earth’s albedo, the greenhouse effect using
a grey atmosphere approximation, variations in the biomass, and geochemical processes. As a starting point for the simulations Hart assumed that the Earth had no
atmosphere 4.5 billion years ago and an albedo of 0.15 at that time. At the end of
each time step (2.5 · 106 years) the model calculated the mass of the oceans, the mass
and composition of the atmosphere, the quantities of dissolved gases, the albedo, the
effective temperature, the surface temperature and various related quantities. For
the optimum run which resulted in the best fit to the observed data the assumed
composition of the juvenile volatiles results in an atmospheric mixture of 84% H2 O,
14% CO2 , 1% CH4 and 0.2% N2 . The parameters of this optimum run were used
to determine the inner and outer boundary of the HZ by varying the Earth-Sun
distance. Furthermore taking into account that main sequence stars slowly increase
in luminosity with age and their HZs gradually move outward. The Continuously
Habitable Zone (CHZ) is defined as the region of continuous overlap of all the previous HZs of a star. Hart’s calculated CHZ ranges from 0.95 to 1.01 AU for the Solar
System.
The extension of the CHZ concept to other main sequence stars was made in Hart
(1979). He concluded that the CHZs for main sequence stars less massive than the
Sun are not only closer but also generally narrower compared to the CHZ of our Sun.
Furthermore, for stars later than K0 stars, Hart determined that there is probably
no CHZ possible.
Kasting et al. (1993)
As a reason for the rather narrow HZ range of Hart (1978), especially for the outer
boundary of the HZ, Kasting et al. (1993) announced the negligence of an important
negative feedback between atmospheric CO2 partial pressure and mean global surface temperature, the Carbonate-Silicate cycle. Since Hart ruled out the existence of
a HZ around stars later than K0 stars, Kasting et al. (1993) repeated the estimation
2.2 Habitable zones
of the HZ by including the Carbonate-Silicate cycle.
Carbonate-Silicate Cycle
The Carbonate-Silicate cycle was proposed by Walker et al. (1981). They suggested that the partial pressure of CO2 in the atmosphere is buffered over geological
time scales (about 5·105 years) by a negative feedback mechanism, in which the
weathering of silicate minerals depends on the surface temperature, and the surface
temperature in turn, depends on CO2 partial pressure through the greenhouse effect. Atmospheric CO2 dissolves in rainwater, forming carbonic acid, H2 CO3 . The
acidic rainwater erodes silicate rocks by dissolving silicate minerals. The calcium
and bicarbonate ions, which result, are transported via rivers, ground water, etc. to
the oceans. There, organisms that live in the surface region use them to make shells
out of calcium carbonate (CaCO3 ). After the organisms die, a carbonate sediment is
formed on the sea floor. Over thousands of years these sediments are transported via
plate tectonics to subduction zones. There pressures and temperatures are so high
that the calcium carbonate reacts with silicon dioxide, a mechanism called carbonate
metamorphosis, which results in the reformation of silicates minerals and gaseous
CO2 . This gaseous CO2 can be released back into the atmosphere by volcanism.
Considering a warming of the atmosphere can lead to faster weathering, which remove CO2 from the atmosphere, which counteracts the warming and thus stabilizes
the climate. Considering a cooling of the atmosphere can leads to a freezing of
the oceans and a decrease of weathering, but the outgassing of CO2 via volcanism
continues and thus also stabilizes the climate.
Temperature, for which water can be liquid on the planetary surface, are the condition that a planet is placed in the HZ of Kasting et al. (1993).
Kasting et al. (1993) took for the outer boundary two modeling limits into account.
At 1.67 AU the outermost limit occurs where a maximum greenhouse effect of CO2
(for a partial pressure of CO2 of 8 bars) fails to keep the surface of the planet above
the freezing point. Another outer limit is located at 1.37 AU and corresponds to
the distance from the star where CO2 starts to condense, which would enhance the
cooling due to the albedo effect.
The inner boundary of the HZ was also calculated in two ways by Kasting et al.
(1993), which were introduced in Kasting (1988). One limit is based on the ’moist
greenhouse’, which results in the loss of an Earth ocean within the age of the Earth.
This limit is located at a distance from the star of 0.95 AU. The other limit is at
0.84 AU, where the surface temperature is 647K, critical point of water, above which
water cannot sustain liquid on the planetary surface. This limit is termed in Kasting
et al. (1993) the runaway greenhouse limit. Both of these inner limits are explained
in more detail in chapter 5.
For the HZ calculations Kasting et al. (1993) applied a one-dimensional radiativeconvective climate model for the Sun and two other main sequence stars to determine
the limits of the HZ and the CHZ. They took CO2 and H2 O as radiative gases and
assumes N2 as the background gas. The calculations differ from previous studies of
13
14
Habitability of terrestrial planets and habitable zones
Figure 2.1: Stellar mass versus the widths of the HZ. The dashed line marks the probable
terrestrial planet accretion zone. The dotted line shows the distance for which an Earth-like
planet would be locked into synchronous rotation. Taken from Kasting et al. (1993), their
Figure 16.
the HZ by a more accurate treatment of energy transport in the atmosphere and
by taking into account the Carbonate-Silicate cycle for the calculations of the outer
boundary.
Nevertheless, the atmospheric temperature, water profiles and radiative fluxes were
not calculated self-consistently (see subsection 5.1.4). Kasting et al. (1993) did not
include feedback processes between the solar insolation, the surface temperature and
the greenhouse effect of water. Temperature profiles were prescribed and the (effective) solar insolation required to support a given surface temperature is determined
assuming global energy balance of the atmosphere.
Kasting et al. (1993) estimated the CHZ over 4.6 billion years (Ga). To determine
the outer boundary of the CHZ, the value for the (effective) solar insolation at the
outer boundary of the HZ was divided by 0.7, which is derived from the initial solar
luminosity of 70% of the Sun’s present value (Gough, 1981). The inner limit of
the CHZ is assumed to be the same as for the current HZ. For the extreme limits,
runaway greenhouse and maximum greenhouse, the CHZ ranges from 0.84 to 1.39
AU. The range of the calculated CHZ by Kasting et al. is much wider than that of
Hart (1978) due to the inclusion of the climate stabilizing Carbonate-Silicate cycle.
Kasting et al. also investigated the CHZs around two other main sequence stars.
Based on their climate calculations the CHZs of F-type stars should be narrower
because they evolve more rapidly than G stars and the CHZs for M-type stars
2.2 Habitable zones
HZin [AU]
0.725
15
HZout [AU]
1.24
0.95
1.01
0.84
0.95
0.84
1.37
1.67
1.39
1.2
Boundary conditions
Surface temperature of 30o C
Surface temperature of 0o C
CHZ for 4.5 Ga
(Runaway greenhouse)
CHZ for 4.5 Ga
(Runaway glaciation)
Runaway greenhouse
Water loss limit
1st CO2 condensation
Maximum greenhouse
CHZ for 4.6 Ga (extreme limits)
Surface temperature of 0o C
and CO2 >10ppm
Reference
Dole (1964)
Dole (1964)
Hart (1978)
Hart (1978)
Kasting et al. (1993)
Kasting et al. (1993)
Kasting et al. (1993)
Kasting et al. (1993)
Kasting et al. (1993)
Franck et al. (2000)
Table 2.1: Summary of the HZ boundaries
should be wider, because these stars evolve more slowly. Figure 2.2.1 shows the
habitable zone for different stars as a function of the stellar mass.
The effect of water clouds on the inner boundary of the HZ, determined by the
critical point of water was discussed in Kasting (1988). This study presents that
clouds tend to lower the surface temperature on a warm and moist planet. For
even a single-layer cloud at a few tenth of bar the inner boundary of the HZ would
be located between 0.46 AU (100%) and 0.67 AU (50%) dependent on the cloud
coverage.
In a further study Williams and Kasting (1997) suggested that it is unlikely for a
planet to remain habitable for the case of the maximum greenhouse to the outer
boundary (1.67 AU), because the planet would be covered by CO2 clouds. If these
clouds extended down towards the surface, as they probably would near the maximum greenhouse limit, their radiative effect would almost certainly be to cool the
planet. Nevertheless, the effect of CO2 clouds is uncertain. In more recent studies
by Forget and Pierrehumbert (1997) it is usually assumed that CO2 clouds warm.
The two-dimensional model by Williams and Kasting (1997) demonstrated that the
clouds first appear at the poles around 1.30 AU, then become widespread between
1.40 and 1.45 AU. Thus, 1.45 AU would be a more conservative choice for the outer
edge of the Sun’s present HZ.
Franck et al. (2000)
Franck et al. (2000) defined the HZ by surface temperature boundaries from 0o C
and 100o C. Additionally, for the outer limit a CO2 partial pressure above 10−5
bar is required to ensure that conditions are suitable for biological productivity via
photosynthesis. This sets the outer boundary of the HZ for the present Solar System
Habitability of terrestrial planets and habitable zones
16
at 1.2 AU
The model of Franck et al. (2000) (based on Caldeira and Kasting (1992)) couples
increasing solar luminosity, silicate-rock weathering rate, and global energy balance
to estimate the partial pressure of atmospheric carbon dioxide, the mean global
surface temperature, and the biological productivity, as a function of time, in the
geological past and future. Their outer boundary of the HZ for the present Solar
System is located at 1.2 AU
All of the reviewed HZ boundaries of the different studies are summarized in Table
(2.1).
2.2.2
Extensions and restrictions to the classical HZ concept
The classical concept of the HZ is defined via the surface temperature of the planet,
which has to be such that liquid water can exist on the planetary surface. The
results of Kasting et al. (1993) for the width of the HZ for different stars were used
to determine an HZ scaling, which is applied to determine the HZ of other stars and
to assess whether detected extrasolar planet would be habitable.
The following extensions and restriction of the classical HZ concept take into account
the advantages and disadvantages of UV radiation, the existence of liquid water
outside the classical HZ in the subsurface of planets, and the extreme temperature
limits for life as we know it on the Earth. Furthermore, the classes of planetary
habitats of Lammer et al. (2009) are summarized.
HZ scaling
Based on the results of the HZ boundary for three different stars by Kasting et al.
(1993) (F-type (Tef f =7200 K), G-type (Tef f =5700 K), and M-type star (Tef f =3700
K)), a HZ scaling was introduced for other central stars by Underwood et al. (2003)
and Selsis et al. (2007). These HZ scalings are also applied in recent studies to
determine the boundaries of the HZ for detected extrasolar systems (e.g. Kaltenegger
and Sasselov (2011) and Kane and Gelino (2012)) and to decide if detected extrasolar
planets would be habitable.
Underwood et al. (2003) uses for the HZ scaling the relationship between the solar
insolations needed to determine the boundaries of the HZ and the effective temperature Tef f of the stars. The HZ scaling is presented for the inner boundary determined
by the runaway greenhouse and the water loss and for the outer boundary for the
first condensation and the maximum greenhouse:
Inner boundaries of the HZ:
Runaway greenhouse:
2
−4
S = 4.190 · 10−8 Tef
f − 2.139 · 10 Tef f + 1.268 (2.1)
2.2 Habitable zones
17
2
−5
S = 1.429 · 10−8 Tef
f − 8.429 · 10 Tef f + 1.116 (2.2)
Water loss:
Outer boundaries of the HZ:
First CO2 condensation:
Maximum Greenhouse:
2
−5
S = 5.238·10−9 Tef
f −1.424·10 Tef f +0.4410 (2.3)
2
−5
S = 6.190 · 10−9 Tef
f − 1.319 · 10 Tef f + 0.2341 (2.4)
Selsis et al. (2007) based their HZ scaling on the relationship between the orbital
distances d and the effective temperature Tef f and luminosity L of the star:
Inner boundary of the HZ:
Runaway greenhouse: d = (0.84AU − 2.7619 · 10
−5
T∗ − 3.8095 · 10
−9
T∗2 )
L
1/2
LSun
(2.5)
Water loss:
d = (0.95AU − 2.7619 · 10−5 T∗ − 3.8095 · 10−9 T∗2 )
1/2
L
(2.6)
LSun
Outer boundaries of the HZ:
Maximum Greenhouse: d = (1.67AU − 1.3786 · 10
−4
T∗ − 1.4286 · 10
−9
T∗2 )
L
1/2
LSun
(2.7)
where T∗ = Tef f − 5700 K and LSun the luminosity of the Sun.
Note, this scaling depends on the assumption that are made in the study of Kasting
et al. (1993), where an Earth-like planet with a 1bar N2 and CO2 atmosphere was
considered taking into account that the atmospheric water vapor mixing ratio depends on the surface temperature and the water reservoir, which assumed to have
the size of the Earth’s oceans.
The ultraviolet HZ
Buccino et al. (2006) defined their HZ limits based on surface ultraviolet (UV)
radiation. They analyzed the evolution of this UV HZ during the main sequence
stage of solar-type and M-type stars. The disadvantages of UV radiation are that
18
Habitability of terrestrial planets and habitable zones
it inhibits photosynthesis, induces DNA destruction, and causes damage to a wide
variety of proteins and lipids. In particular, UV radiation between 200 and 300 nm is
very damaging to most terrestrial biological systems (Lindberg and Horneck, 1991).
The advantages of UV radiation are that it is one of the most important energy
sources on the surface of the primitive Earth for the synthesis of many biochemical
compounds and, therefore, essential for several biogenesis processes.
The inner limit of the UV HZ is determined by the levels of UV damaging radiation
tolerable by DNA and the outer limit is characterized by the minimum UV radiation
needed in the biogenic process. The UV criterion for the HZ analyzes the biological
conditions needed for the origin and the development of life once the liquid water
scenario is already satisfied. The UV criterion is more restrictive. Buccino et al.
(2006) took the attenuation effect of the atmosphere on UV radiation only into account by a factor, which is the ratio between the radiation received on the planetary
surface and the incident radiation on top of the atmosphere.
The implications of UV surface radiation for the habitability of planets around Ftype and K-type stars was also investigated by Kasting et al. (1997). They concluded
that the UV radiation does not have a great effect on the habitability of planets
orbiting F-type and K-type stars, because they receive less UV radiation at their
surface than Earth. This effect results from the assumption of a 1bar, 21% O2
atmosphere for their calculations and the interaction with the stellar radiation.
The cold circumstellar HZ (CCHZ)
A terrestrial planet or satellite that is found beyond the HZ of its star could still harbor life, which does not use starlight as a source of energy, living below the surface.
Biological activity could for example be possible in the subsurface of Mars, Jupiter’s
satellite Europa or/and Saturn’s satellite Enceladus. Should future research demonstrate the presence of life in a subsurface ocean of Europa or Enceladus, it would
imply the existence of a zone outside the classical HZ, which could also support habitable planetary bodies. This extension to the HZ was termed by Penã-Cabrera and
Durand-Manterola (2004) the cryo-ecosphere or cold circumstellar habitable zone
(CCHZ). In general the width of the CCHZ is larger than the width of the classical
HZ. This leads to a higher probability of finding planets in the cryo-ecosphere than
in the classical HZ. Nevertheless, for extrasolar planets and also in the solar system,
probing such a CCHZ is difficult, since the influence of the possible biology upon
the surface and the atmosphere is probably negligible.
The possibility of the development of life in icy planetary bodies in the cryoecosphere is affected by the internal energy of the planet, required to provide liquid
water. For the Jovian system the provided heat results from gravitational interaction between Europa (and Callisto) on the one hand, and Jupiter and the other
Galilean moons on the other hand.
2.2 Habitable zones
The HZ for extremophiles
Penã-Cabrera and Durand-Manterola (2004) discussed a zone, which they termed
’circumstellar ecosphere’, which is determined by radiative equilibrium and the maximum and minimum temperatures for extremophiles on the Earth. Extremophiles
are organisms that are able to live in extreme environments determined by e.g.
temperature, salinity, pH value, UV radiation. Maximum temperatures of 386 K
are tolerable for the thermophilic microorganism Pyrolobus fumarii (Rothschild and
Mancinelli, 2001). Minimum temperature of 255K are tolerable for the psychrotolerant blue-green algae Phormidium sp. (Kohshima, 1984). However, Penã-Cabrera
and Durand-Manterola (2004) neglected the planetary atmosphere, which has a
strong influence on the surface temperature.
Selsis et al. (2007) pointed out that the temperature, which are tolerable for thermophilic microorganisms is close to the surface temperature of the inner boundary
of the HZ determined by water loss (Kasting et al., 1993).
HZ classifications
Lammer et al. (2009) distinguish four different classes of planetary habitats based
on the assumption that liquid water is the basic requirement for life. The first two
classes are based on the classical HZ definition taking surface water into account,
whereas the last two classes are based on the requirement of liquid water on/in the
planet.
• Class I: bodies on which stellar and geophysical conditions allow Earth-analog
planets to evolve so that complex multi-cellular life forms may originate
• Class II: bodies on which life may evolve but due to stellar and geophysical
conditions the planet rather evolve towards Venus- or Mars-type worlds where
complex life-forms may not develop
• Class III: bodies where subsurface water oceans exist which interact directly
with a silicate-rich core
• Class IV: bodies which have liquid water layers between two ice layers, or
liquids above ice
19
20
Habitability of terrestrial planets and habitable zones
CHAPTER
3
Basic atmospheric physics
The existence of an atmosphere plays an important role to determine if a planet
is habitable due to its influence on the surface temperature. A planetary atmosphere can be described by pressure p, hydrodynamic velocity v, mass density ρ,
and temperature T . These values are related and can be determined with the help
of the basic equations like the equation of state, and the conservation of momentum,
mass and energy, which are reviewed in the following chapter. In the context of the
conservation of energy, the global energy balance of a planet, introducing the greenhouse effect, and the transport processes of energy in an atmosphere are explained.
This chapter ends with an overview of the effects of water in the atmosphere.
3.1
Equation of state
The equation of state describes the relationship between pressure p, volume V , and
absolute temperature T :
p = f (V, T )
(3.1)
The gases in the atmosphere can be assumed to be ideal gases, since a real gas can
be approximated by the ideal gas law when intermolecular forces are weak, which
is the case for low enough pressures or temperatures high enough for the gas to
be sufficiently diluted (Jacobson, 2005). Local thermodynamical equilibrium (LTE)
in the atmosphere is assumed, where thermodynamic quantities are defined at any
point in the atmosphere and the atmosphere consists of particles interacting to be
in equilibrium with each other.
The equation of state for an ideal gas is given by the ideal gas law:
Basic atmospheric physics
22
p=
nRT
= N kB T
V
(3.2)
where R is the universal gas constant (8.31 J mol−1 K−1 ), N = nA/V is the number
concentration of gas particles, A the Avogadro number (6.022·1023 mol−1 ), which is
the number of particles in one mole, n the amount of substance of the gas in mole,
and kB = R/A is the Boltzmann constant (1.28·10−23 J K−1 ).
3.2
Conservation of momentum
In general the Navier-Stokes equation describes the conservation of momentum,
which arise from applying Newton’s second law to fluid motion taking into account
viscosity. The Navier-Stokes equation includes the impact of the pressure force
(−1/ρ∇p, where ρ is the mass density), gravitational force (∇ϕ, where ϕ is the
gravitational potential), friction force (Ffriction ) and Coriolis force (2Ω × v, where
Ω is the angular velocity and v is the flow velocity of the gas fluid) such that
1
∂v
+ (v · ∇)v = − ∇p + ∇ϕ + Ffriction + 2Ω × v
∂t
ρ
(3.3)
where t is the time. The solution of this Navier-Stokes equation can be approximated
evaluating the different forces.
Investigating the vertical direction of the Earth’s atmosphere friction and dynamical
processes can be neglected and thus the pressure force approximately balances the
gravity force. The gravity of the atmosphere can be neglected due to the small mass
of the atmosphere compared to the planetary mass. Furthermore, the change of
gravity with height is assumed to be negligible since the atmospheres of terrestrial
planets is small compared to the radius of the planetary body. Assuming furthermore
a plane-parallel atmosphere, leads to the hydrostatic equation:
dp(z)
= −ρ(z)g
dz
(3.4)
where p is the pressure, z the vertical height, and g the acceleration of gravity. For
the horizontal direction on the Earth the pressure force is balanced by the Coriolis
force in a first approximation.
3.3
Conservation of mass
The conservation of mass is described by a continuity equation for the mass density
ρ. The atmospheric gas can be approximated as a fluid. The continuity equation
states that the local rate of increase in mass density ρ is equivalent to the fluid’s
convergence
3.4 Conservation of energy
23
∂ρ
= −∇ · (ρv)
∂t
(3.5)
The continuity equation is important for the description of the chemistry and dynamics taking place in the atmosphere.
3.4
Conservation of energy
Based on the conservation of energy the temperature of the atmosphere can be
determined by taking into account that the atmosphere is assumed to be in local
thermal equilibrium (LTE).
The equation of energy conservation results from the first law of thermodynamics
and can be written as
dQ = Cv dT + pdV
(3.6)
where dQ is the amount of heat exchanged, Cv is the specific heat capacity for a
constant volume, Cv dT is the inner energy and pdV the energy of the work for a
constant volume.
The conservation of energy in the atmosphere is closely related to the transport
processes of energy in the atmosphere. Energy transport processes in the atmosphere
are radiation, convection and conduction. Conduction is neglected here because it
is not relevant for the lower atmospheres of terrestrial planets.
The energy budget of the atmosphere is described assuming global energy balance.
For the terrestrial planets in the Solar System the effective emission temperature
based on the global energy balance deviates from the observed planetary surface
temperature. This implies that the surface temperature not only depends on the
global energy balance, but also on atmospheric properties. These atmospheric properties are responsible for the greenhouse effect. For the terrestrial planets in the
Solar System the most important greenhouse gases are water vapor (H2 O) and carbon dioxide (CO2 ), which absorb and emit at infrared wavelengths, in which most
terrestrial planetary surfaces in the Solar System primarily radiate.
3.4.1
The global energy balance
The energy balance is formally stated by the first law of thermodynamics, which
declares that the energy in a closed system is conserved. To achieve global energy
balance in the atmosphere of a planet, the energy returned to space by the planet’s
radiative emission has to balance the incoming energy from the star.
Basic atmospheric physics
24
The incoming stellar radiation flux
The stellar luminosity L is the total rate at which energy is released by central stars.
The central star is usually the dominant energy source for planets, e.g the Sun for
the terrestrial planets in the Solar System. The star’s energy is produced by nuclear
fusion in its core, where lighter elements are converted into heavier ones, releasing
energy during this process.
Assuming that the star radiates like a blackbody, the luminosity of a star L is related
to the effective stellar temperature Tef f,star and the radius Rstar of the star via the
Stefan-Boltzmann law:
2
4
σTef
L = 4πRstar
f,star
(3.7)
where σ is the Stefan-Boltzmann constant (5.67 · 10−8 Wm−2 K−4 ). This luminosity
L is equal to the amount of energy passing outwards through a sphere with the star
at its center neglecting sources and sinks between star and this sphere. The amount
of energy per time at a planetary orbital distance d is
L = 4πd2 S
(3.8)
where S is the flux of stellar energy at the planet’s orbit, the stellar insolation.
For the Earth, the solar insolation, S, at the Earth orbit is defined as the solar
constant S0 = 1367 Wm−2 for the mean luminosity of the Sun L0 = 3.9 · 1026 W
and the mean distance between the Earth and the Sun d = 1.5 · 1011 m, which is
1 Astronomical Unit (AU). Correspondingly, the solar insolation at the Venus orbit
(d = 0.72 AU) is S = 1.93 S0 and at the orbit of Mars (d = 1.52 AU) S = 0.43 S0 .
The stellar power incident on a planet is defined as the stellar insolation S at the
mean orbital distance times the cross-sectional area of the planet (πrP2 , where rP is
the radius of the planet) intercepting the stellar energy flux. Taking the planetary
albedo AP to be the ratio of reflected to incident stellar energy, the stellar energy
flux that is absorbed by the planet is equal to
absorbed stellar radiation = (1 − AP )SπrP2
(3.9)
For the Earth with present mean conditions of cloud, snow and ice cover, the planetary albedo AP 0.30. For Venus the value of the planetary albedo is 0.77 and for
Mars 0.24.
The thermal infrared flux
As an approximation the surfaces of terrestrial planets exhibit blackbody radiation
with the effective emitting temperature Te . The emitted planetary radiation can be
determined by the Stefan-Boltzmann law
3.4 Conservation of energy
25
Figure 3.1: Diagram of the energy fluxes per unit area for a planet without an atmosphere
.
emitted planetary radiation = 4πrP2 σTe4 (3.10)
where is the emissivity, which is for simplicity reasons assumed to be unity.
The effective temperature
Assuming global energy balance of a planet neglecting its atmosphere, the absorbed
stellar flux is set to be equal to the emitted planetary flux, see Fig 3.1
(1 − AP )
S
= σTe4
4
(3.11)
Thus,
S(1 − AP )
Te =
4σ
1
4
(3.12)
The effective temperature Te for the Earth is 255 K, with the planetary albedo AP =
0.3. Note that the planetary albedo is normally used in Equation (3.12), although
this includes implicitly an atmosphere, which is not taken into account. For Venus
Te = 227K, which is lower than for the Earth due to a higher planetary albedo of
Venus (AP = 0.77). For Mars Te = 211 K (with AP = 0.22), which is also lower than
Basic atmospheric physics
26
the value for the Earth due to the increased orbital distance. These temperatures
derived from the global energy balance deviate from the observed global mean surface
temperature TS of these planets, which are for the Earth TS = 288 K, for Venus
TS = 735 K and for Mars TS = 218 K. The reason for this difference arises from the
greenhouse effect.
3.4.2
Global energy balance with greenhouse effect
Potential differences in the effective emission temperature and the surface temperature for hypothetical planets at the same distance from their star with the same
surface albedo and size arise from the presence of an atmosphere. A planet without
an atmosphere would have a surface temperature TS equal to the effective temperature Te . The energy balance of a planet with an atmosphere can be affected by
greenhouse gases, which absorb and emit in the infrared radiation, i.e., where terrestrial planetary surfaces radiate primarily, and scattering. To determine the contribution of an atmospheric species to the greenhouse effect its absorption features in
the infrared wavelength region are important. Since re-emission of absorbed energy
is homogeneous and isotropic, a part of the emission is radiated back to the surface.
Thus, the surface receives not only stellar radiation, but also infrared radiation emitted by the greenhouse gases. This additional heating in the lower atmosphere and
of the planetary surface is termed the greenhouse effect. Scattering of radiation on
atmospheric molecules is also an important process in the atmosphere to determine
the global energy balance. Rayleigh scattering is mostly important for shortwave
radiation because of the λ−4 -dependence.
Simple model of the greenhouse effect
In order to calculate a more realistic surface temperature of a planet, it is necessary
to take into account its atmosphere and thus the resulting greenhouse effect. As a
first approach a simple model of the atmosphere is considered. Regarding energy
transport, only radiative transport is considered and global energy balance is imposed. The atmosphere is furthermore represented by a single layer. This layer
acts as a blackbody. The emitted radiation depends on the temperature of the
atmospheric layer. The following assumptions are made for the atmospheric layer:
• It is transparent to wavelengths corresponding to the incoming stellar energy
(shortwave radiation) (τvis = 0).
• It is opaque to wavelengths at which the planet emits most of its thermal
energy (longwave radiation) (τIR > 1).
With these assumptions the incoming solar radiation can pass through the atmospheric layer without energy loss and is then absorbed or reflected by the surface.
4
). This emitted radiation by
The surface emits thermal infrared radiation (σTsurf
3.4 Conservation of energy
27
Figure 3.2: Diagram of the energy fluxes per unit area for a planet with a single layer
atmosphere , that is transparent for solar radiation and opaque to planetary radiation
the ground is assumed to be fully absorbed by the atmospheric layer and heats the
atmosphere to a temperature TA . The atmospheric layer then radiates as a blackbody with the temperature TA (σTA4 ). These processes are represented in Figure
3.2. At the top of the atmosphere based on the global energy balance in this model
the average net incoming stellar flux per unit area, must be equal to the outgoing
radiation there. This outgoing thermal radiation depends on the temperature TA at
which the atmospheric layer emits. Therefore,
(1 − AP )
S
= σTA4
4
(3.13)
has to be fulfilled, i.e. Te = TA .
To determine the energy balance at the surface the energy flux radiated by the atmospheric layer, which is emitted in all directions has to be taken into account. Thus,
for the surface there is (in addition to the incoming stellar radiation (1 − AP )S/4)
the energy flux in the downward direction that is emitted by the atmospheric layer
which causes additional heating of the surface. The two fluxes in the downward
direction are then equal to the thermal flux radiated by the planetary surface due
to energy conservation
4
σTsurf
= (1 − AP )
S
+ σTe4
4
(3.14)
Basic atmospheric physics
28
With these two equations 3.14 and 3.13 the surface temperature is obtained to
Tsurf
(1 − AP )S
=
2σ
1
4
1
= 2 4 Te
(3.15)
This warming effect of the surface temperature caused by the existence of an atmosphere is called the greenhouse effect. Applying equation (3.15) for the Earth,
by using the effective emitting temperature Te = 255 K, results in a surface temperature of Tsurf = 303 K. This is a greenhouse effect of 48 K, that is larger than
the observed 33 K for the Earth. This difference comes from the simplicity of the
assumed atmospheric model, where the atmospheric layer is assumed to be completely optically thick, which is not the case for the Earth (see section 3.5.1) and
thus over-estimates the effect. Applying equation (3.15) for Venus results to 270
K, which is 490 K smaller than the observed Tsurf , thus the greenhouse effect is
under-estimated because the atmosphere of Venus is so thick and CO2 -rich that
greenhouse enhancement should be taken into account for many atmospheric layers.
On the other hand for Mars with equation (3.15) the calculated temperature with
the simple model is 251 K, which is 20 K hotter than the observed Tsurf , thus the
greenhouse effect is over-estimated as for Earth.
Generally the greenhouse effect in an atmosphere warms the surface, because the
atmosphere is (usually) much more transparent for incoming shortwave radiation,
than for outgoing longwave radiation.
Comments on the greenhouse effect in more realistic atmospheres
For the Earth the surface temperature calculated with the simple model is overestimated caused by the fact that the atmosphere of Earth is not completely opaque
for thermal radiation. In fact, the Earth’s atmosphere has an infrared window, where
the relevant greenhouse gases do not absorb the emitted thermal infrared radiation
of the Earth’s surface. Thus, the Earth’s surface can radiate directly to space in the
atmospheric window region from 8-12 μm (see Figure 3.5.1 section 3.5.1). This is a
cooling effect and results in a smaller greenhouse effect and therefore lower surface
temperature than obtained with the simple model.
A better way to describe an atmosphere more realistically is to apply an atmospheric
model which consists of a larger number of layers which absorb, emit and scatter
radiation. The temperature in an atmosphere (including the greenhouse effect) can
be calculated by solving the radiative transfer equations in each of the atmospheric
layers for the shortwave radiation as well as the thermal infrared radiation emitted
by the planetary surface. Therefore, it is important to take the relevant absorption
features of the greenhouse gases in the atmosphere into account.
Furthermore, for the simple model of the atmosphere it was assumed that the energy
transport is purely radiative. In reality, however, for dense atmospheric regions,
which are opaque to most infrared wavelength, the transfer of energy is dominated
by convection and not by radiation (subsection 3.5.2).
3.5 Energy transport
3.5
29
Energy transport
This section reviews energy transport via radiation and convection in the atmospheres of terrestrial planets.
3.5.1
Radiative transfer
The radiative transfer characterizes the path of stellar and thermal radiation through
the atmosphere of a planet including changes in the intensity due to absorption,
emission, or scattering.
Considering a plane-parallel atmosphere the radiative transfer is described with the
radiative transfer equation
μ
dIν
= I ν − Sν
dτν
(3.16)
• Iν is the spectral intensity and describes the energy which is radiated into a
solid angle per frequency, time and unit area.
• τν is the optical depth and is here used as the vertical coordinate, since the
absorption coefficients kν and density of the absorbing species ρ are functions
only of height due to the plane-parallel approach. The optical depth τν is the
radiative path from the top of the atmosphere to the height z and is defined
as
∞
τν =
kν ρdz
(3.17)
z
• μ is the cosine of the zenith angle θ.
• Sν is the source function which is defined as
Sν =
ην,emission +
sν,scattering (k → k)Iν (k )d2 k kν + sν,scattering
(3.18)
where ην,emission is the emission coefficient, kν is the absorption coefficient
and sν,scattering is the scattering coefficient and sν,scattering (k → k)Iν (k )d2 k describes the part of the radiation that is scattered into the line-of-sight, where
k is the direction of the photon.
To solve the radiative transfer equation angular integration of Iν and frequency
integration of the resulting Fν have to be performed in order to obtain the net
fluxes Fnet .
With the angular integration of Iν the spectral flux Fν can be calculated:
Basic atmospheric physics
30
1
Fν =
2π
μ
−1
Iν (μ, ϕ)dϕdμ
(3.19)
0
where ϕ is the azimuthal angle.
The flux density Fnet can be obtained by integrating the monochromatic irradiance
Fν over the relevant frequency interval
ν2
Fnet =
Fν dν
(3.20)
ν1
The atmospheric temperature can be derived from angular and wavelength integrated fluxes Fnet applying the relationship between temperature and energy
(E = T V ρCp ) and the hydrostatic equation (see equation 3.4):
dT
g dFnet
=−
dt
Cp dp
(3.21)
Absorption properties
Absorption, emission, and scattering of photons in the atmosphere depends on
gaseous composition. Each atmospheric gas has its own characteristic absorbing
features. These characteristics depend on the configuration of the molecule. For
example, a greenhouse gas, which absorbs infrared radiation, must usually be a
molecule with a permanent dipole.
Due to the emission or absorption of a photon an energy transition of the molecule
can occur. The absorption of radiation from a molecule leads to the transfer of energy
from the photon to the molecule that absorbed it. In general, internal energy of a
molecule is stored in electronic, vibrational, rotational or translational states. Thus,
a molecule in the atmosphere can only absorb radiation if the energy of the photon
corresponds to the difference between the energy of two allowable excitation states
of the molecule. These transitions between energy levels determine the wavelength
dependence of the absorption properties (Hartmann, 1994).
Electronic transitions in molecules can take place for photons with wavelengths
around 1 μm.
The rotational energy for molecules is much lower (102 -10−4 μm). Bent triatomic
molecules, like water, with a permanent dipole moment have pure rotational bands.
Transitions between energy levels of the molecule, which lead to rotational bands,
correspond to photons with wavelength from the far-infrared to the microwave region
(Goody and Yung, 1989).
The vibrational energy of a molecule is stored in the vibrations of the atoms of the
molecule about a stable point, where the forces of attraction and repulsion of these
3.5 Energy transport
Figure 3.3: Absorption spectra of important atmospheric gases for the Earth and the absorption spectrum of the Earth’s atmosphere. The so-called window region from 8-12μm,
describing a wavelength region where only little absorption of infrared radiation by water
vapor takes place can be identified in the viewgraph. Taken from Goody and Yung (1989),
their Figure 3.1
31
32
Basic atmospheric physics
atoms of the molecule are in balance at the appropriate distance. For a triatomic
molecule, like water, there are three independent modes of vibration: a symmetric
stretching mode, an antisymmetric stretching mode, and a bending mode. During vibrational transitions triatomic molecules develop temporary dipole moments,
which leads to a rotational transition in addition to the vibrational transition. The
combination of these two transitions allows the molecule to absorb and emit radiation at a large number of closely spaced wavelengths, which leads to absorption
bands (Hartmann, 1994). Transitions between levels lead to vibrational bands in
the spectrum of the molecule extending from the near- to far-infrared (1-100 μm)
(Goody and Yung, 1989). As a consequence, the vibrational and rotational energy
transitions are mainly responsible for the absorption and emission of the thermal
emission of a terrestrial planet like the Earth (Hartmann, 1994).
These energy transitions of an atmospheric species take place at discrete frequencies,
which produce absorption lines. The line widths of these absorption lines depend on
broadening processes. Pressure broadening results from collisions between molecules,
which can supply or remove amounts of energy during the radiative transitions, allowing photons of a broader range of wavelengths to produce a particular transition
of a molecule. It causes a characteristic shape and width of the absorption lines,
a Lorentz line shape. Away from the line center the probability of absorption decreases but it remains significant in some wavelength interval so-called wings of the
absorption line, which can be very broad in the case of high pressure (Hartmann,
1994). Doppler broadening occurs due to the movement of molecules relative to a
photon, which causes a Doppler shift in the frequency of radiation. This allows a
broader range of wavelength to cause a particular energy transition. Due to this
broadening processes, the absorption spectrum may have substantial absorption between the line centers. However, the pressure broadening is important in the lower
atmosphere, whereas the Doppler broadening is relevant in the stratosphere
An overview of the absorption spectra of the most important species in the Earth’s
atmosphere is presented in Figure (3.5.1) and it includes also the complete absorption
spectrum of the Earth’s atmosphere.
3.5 Energy transport
3.5.2
33
Convection
Convection refers to the transport of thermal energy between different places in the
atmosphere due to the transport of particles. Convection is a dynamical, threedimensional phenomenon.
For the Earth, this process can be envisaged as follows: Air is heated at the surface.
An air parcel rises, because due to the higher temperature of the air parcel its
density is lower than the density of the surrounding air. This happens without
exchanging energy due to radiative transfer or heat conduction. Due to the decrease
of air pressure with height the air parcel is expanding and performs work against
the actual air pressure. Since this process is adiabatic, the same energy has to be
subtracted from the inner energy of the gas. Thus, the air cools and obtains a
density that is either equal to its surrounding than convective ascend stops or it is
even more dense than its surrounding and sinks.
These considerations are for particular air parcels. In reality such air parcels are
heated over and over at the surface and rise up, which results in a convective motion.
The atmosphere in this region will have an asymptotic temperature distribution,
which is consistent with the adiabatic temperature gradient.
To determine the temperature in a convective atmosphere the first law of thermodynamics is used here, which describes the heat exchange between the air parcel and
its surroundings during the vertical motion (see equation 3.22)
dQ = Cv dT + pdV
(3.22)
where dQ is the amount of heat exchanged, Cv is the specific heat per mole of air
at a constant volume and Cv dT is the inner energy of the air parcel, and pdV the
work used by the air parcel.
Conduction is very low and the parcel of air absorbs roughly the same amount of
radiation as it emits. Hence, for dense atmospheres nearly no energy is exchanged
dQ ≈ 0, it can be treated as an adiabatic change and equation (3.22) can be written
as
pdV = −Cv dT
(3.23)
The ideal gas law (equation 3.2) in its differential form is
d(pV ) = pdV + V dp = RdT = (Cp − Cv )dT
(3.24)
Multiplying the volume V to the hydrostatic equation (dp = −ρgdz) gives
V dp = −V ρgdz = −M gdz
Including the last two equations in equation (3.23) leads to:
(3.25)
Basic atmospheric physics
34
(Cp − Cv )dT + M gdz = −Cv dT
(3.26)
dT
Mg
g
=−
= − = −Γdry
dz
Cp
cp
(3.27)
which can be formed to
where Γdry is termed the adiabatic lapse rate.
For example, for the Earth the lapse rate for a dry atmosphere is Γdry ∼
=10 K/km.
Moist convection
For an atmosphere that contains a condensable gas such as water vapor or CO2 ,
the lapse rate, which describes the convective energy transport, has to be modified.
Because if an air parcel, which contains water vapor, rises up and cools it might reach
the saturation limit and thus the water vapor can condense. When condensation
of water occurs, latent heat L, stored by the evaporation process of liquid water, is
re-released. This released latent heat contributes to inner energy of the air parcel,
which depends on the amount of water vapor within it. For the moist air parcel the
inner heat is not zero as for the dry air parcel, but dQ = −Ldρsat , which has to be
taken into account in the first law of thermodynamics for a moist air parcel, where
ρsat is the density of the saturated water vapor
−Ldρsat = Cv dT + pdV
(3.28)
With the conversions as in subsection 3.5.2 the moist adiabatic lapse rate, Γwet , for
an atmosphere containing water vapor, which describes the convection, results in
dT
1
g
= −Γwet
=−
dz
cp (1 + L dρsat )
(3.29)
Cp dT
Taking water vapor into account leads to a weakening in the lapse rate, due to latent
heat released. For example for the Earth, the lapse rate for a moist atmosphere is
Γwet ∼
=6.5 K/km.
3.5.3
Schwarzschild criterion
Whether or not the atmosphere is stable against convection can be determined
via the Schwarzschild criterion. To distinguish between the two fundamental energy transport processes, radiative transfer and convection in the atmosphere the
Schwarzschild criterion is introduced. This indicates that convection can take place
if the temperature gradient for an atmosphere in radiative equilibrium exceeds the
adiabatic temperature gradient, because the heated air parcels can posses a lower
density than its surrounding when they are expanding adiabatically during ascending
3.6 Water in the Atmosphere
35
dT
dT
>
dz
dz
rad
conv
(3.30)
For the Earth convection determines energy transport in the troposphere, where the
atmosphere is dense, whereas in the stratosphere the important process for transport
of energy is radiative transfer.
3.6
Water in the Atmosphere
Since liquid water is an important requirement for habitability, this section reviews
the characteristics of water vapor in the atmosphere. Basic physical processes are
presented, which are important to determine the water content and its distribution
in a terrestrial atmosphere.
3.6.1
Characteristics of water vapor in the atmosphere
A common way to describe the water vapor content (and also the content of other
chemical components) in the atmosphere is its mass and volume mixing ratio. The
mass mixing ratio of water vapor cH2 O,m is the ratio of the mass of water vapor to
the mass of air in a volume with the total pressure p. The total pressure p is the
sum of the partial pressures of water vapor pH2 O and of dry air pdry , respectively:
p = pdry + pH2 O . Taking into account the ratio of the molecular weight of water
MH2 O and of air M , the mass mixing ratio of water vapor cH2 O,m can be written
taking the ideal gas law into account as
cH2 O,m =
MH2 O
0.622pH2 O
=
M
p
(3.31)
The volume mixing ratio of water cH2 O,v is the ratio of the number of water molecules
nH2 O per volume to the total number of molecules of air n per volume
cH2 O,v =
nH 2 O
pH 2 O
=
n
p
(3.32)
Saturation
Water vapor is saturated in the atmosphere, when phase state changes of evaporation
and condensation are in equilibrium. The saturation vapor pressure, which gives
the water vapor content at this equilibrium, can be calculated with the ClausiusClapeyron equation, which assumes water vapor as an ideal gas
Lpsat,H2 O
dpsat,H2 O
=
dT
RT 2
(3.33)
Basic atmospheric physics
36
where L is the latent heat. The latent heat L is the stored energy, which is released,
by condensation, freezing, and solid deposition or absorbed during evaporation, melting, and sublimation. It is the energy required to change from one phase state to
another at a constant temperature.
Relative humidity
In general, atmospheric humidity is the amount of water vapor carried in the air.
The relative humidity RH relates the amount of water in the air to the equilibrium
value at saturation and is expressed in percentage. Thus, RH is the ratio of the
actual partial pressure of water pH2 O to the saturation vapor pressure psat,H2 O , hence
demonstrates how far the atmosphere is from the thermodynamic equilibrium state
in respect to water vapor:
RH(z) =
pH2 O (T (z))
· 100%
psat,H2 O (T (z))
(3.34)
If the air is saturated with water vapor the relative humidity will be 100%.
3.6.2
The hydrological cycle
On the Earth water can exist in all three aggregation states on the surface and in the
atmosphere. The water content in the atmosphere depends on the water reservoir
available on the planet and thus, on the temperature.
The hydrological cycle generally describes the movement and phase changes of water
in the atmosphere and on the surface of a planet. With the assumption of a liquid
(and solid) water reservoir on the surface of a terrestrial planet, hence a habitable
planet, a hydrological cycle will emerge like on the Earth. The hydrological cycle
starts with the evaporation and sublimation of water above oceans, land and ice
surfaces. The energy needed for the evaporation, which comes mainly from solar radiation, is stored in the evaporated water, as latent heat. The main part of the water
vapor in the atmosphere is provided by the surface oceans, but also by lakes, rivers,
moist soil and moist vegetation. In the atmosphere water vapor is transported by
dynamical processes until condensation occurs. The requirement for the formation
of clouds is a minimum supersaturation of the water vapor. The supersaturation is
reached by the cooling of air with height, whereby water vapor reaches its dew point.
Condensation of water vapor mostly occurs on small atmospheric particles, which
act as condensation nuclei. The stored latent heat is re-released to the atmosphere
during the condensation process, which is important for the atmospheric energy balance system. The result of this latent heat release is that clouds are warmer than
their surrounding air, which leads to turbulent effects. The subsequent precipitation
of water to the surface due to the influence of planetary gravity closes the hydrological cycle. With the evaporation of the liquid water and the sublimation of ice on
the planetary surface the cycle can start again.
3.6 Water in the Atmosphere
3.6.3
Clouds
For habitable planets, which have a liquid water reservoir on the surface, clouds
are forming due to the hydrological cycle. The presence and nature of clouds in
the atmosphere is important for the climate of a planet. An overview about the
general influence of clouds is given in Kitzmann et al. (2010). Clouds affect the
climate by enhancing the greenhouse effect and by scattering of the incoming stellar
radiation. For the Earth high-level ice clouds are responsible for the heating of the
surface by increasing the greenhouse effect. Low-level water clouds, on the other
hand, have a cooling effect on the surface of the Earth due to scattering of stellar
radiation. For mid-level water clouds both climatic effects are canceling each other.
The greenhouse effect and the albedo effect due to scattering are directly correlated
with the wavelength-dependent optical properties of the cloud particles. Thus, to
determine the surface temperature the wavelength dependence of the incident stellar
spectra and of the optical properties together with the particle sizes of clouds need
to be known.
3.6.4
Optical properties of water vapor
Water vapor is an important greenhouse gas due to its absorption features in the
thermal wavelength region. For the Earth it is the most important one (IPCC,
2007). An additional reason that atmospheric water vapor is such an important
greenhouse gas is the large reservoir of liquid (or solid) water on the surface of the
Earth. For example for the Earth an increase in surface temperatures causes liquid
water from the reservoir to evaporate. This evaporation leads to more water vapor
in the atmosphere produces, which an enhanced greenhouse effect.
Water (H2 O) is a molecule consisting of two hydrogen atoms and one oxygen atom.
The bond length between the hydrogen and oxygen molecule is 95.8 pm and the bond
angle is 104.45◦ (Goody and Yung, 1989). Thus, water vapor is a triatomic and also
bent molecule. Due to the bent structure water has a permanent dipole moment,
which is responsible for pure rotation bands in addition to vibration-rotation bands.
Water vapor has many rotational absorption lines at closely spaced wavelengths,
which form a rotation band that absorbs at wavelength between 12 and 200 μm.
Furthermore, an important vibration-rotation band of water vapor is located near
6.3 μm. For the Earth, these bands are important for the absorption of the thermal
infrared radiation. Between these two absorption features of water vapor there is
relatively weak absorption by water vapor. Hence, since thermal infrared radiation
can only pass through the atmosphere in this wavelengths region (8-12μm) it is called
the water vapor window. An additional vibrational-rotational band is produced at
2.7μm. Electronic bands in the near-infrared with six distinguishable groups are
located between 0.9 and 2.2μm and a weak, visible band in the wavelength range
from 0.56 to 0.91μm (Goody and Yung, 1989). Hence, in the Earth’s atmosphere
the weak near-infrared bands absorb a large amount of the incoming solar radiation.
The absorption features of water are shown in Figure (3.5.1).
37
38
Basic atmospheric physics
The continuum absorption due to water vapor has posed a complex problem. Nevertheless, it is important for the wavelength regions of the atmospheric spectrum in
the microwave and the infrared regime with the greatest transparency, where the
window regions are located (Clough et al., 1989). One explanation for the continuum absorption could be the far wings of the absorption lines (Clough et al., 1989).
Another reason for the continuum might be dimers of water molecules in the vapor phase, which are two water molecules loosely bound by a hydrogen bond, as a
possible component of the water self-continuum absorption (Varanasi, 1988).
CHAPTER
4
Physical processes determining the inner HZ boundary
Liquid water on the surface of a terrestrial planet is an important requirement for
life (see subsection 2.1.3). This condition is also used to determine the HZ (e.g Hart
(1978); Kasting et al. (1993) in section 2.2).
The inner boundary of the HZ, which is in the focus of this Thesis, is determined
by the distance of a planet to the central star, where liquid water cannot exist on
the planetary surface. For example the runaway greenhouse effect, a dynamical
and self-enhancing process is assumed to lead to the total evaporation of oceans.
Furthermore, escape processes can cause the loss of the complete water reservoir
of a planet via the loss to space of hydrogen produced by the photo-dissociation of
water. These are crucial processes for the determination of the inner boundary of
the HZ.
4.1
General conditions for instability of liquid water on
the surface of a planet
The condition for occurrence of the inner boundary of the HZ can be:
1. Thermodynamics: Surface temperature Tsurf is above the critical temperature
for water Tcrit which is 647 K (illustrated in the phase diagram of water shown
in Figure 4.1). Above this temperature liquid water cannot exist thermodynamically on the surface, which is the absolute strict upper limit independent
of the mass of the water reservoir.
Tsurf > Tcrit
Physical processes determining the inner HZ boundary
40
Figure 4.1: Phase diagram of water. The critical point of water is at Tcrit =647K and
pcrit =22 MPa (pcrit =220 bar) Taken from [http://www.lsbu.ac.uk/water-/phase.html]
2. Water reservoir : Tsurf is too high for liquid water being stable on the surface
of a planet. With the saturation vapor pressure curve of water the temperature can be determined above which an ocean, which would result in a pressure
pOcean when completely evaporated, could no longer be stable. This temperature depends on the mass of the ocean.
Psat (Tsurf ) > POcean
3. H2 O cycle: Tsurf and the atmospheric temperatures are so high, that the water
cycle breaks down, because water exists only in the gaseous phase and cannot
condense or freeze in the atmosphere.
PH2 O < Psat (Tz ) for all z
4.2
Runaway greenhouse effect
The runaway greenhouse effect is assumed to cause an increase in surface temperature that can lead to a complete evaporation of the water reservoir of the planet.
This process results from the coupling of a high energy flux from the star with
the enhanced greenhouse effect of a condensable greenhouse gas, which is assumed
here to be water vapor. It is a dynamical process usually thought to be forced by
the incoming stellar flux, which increases Tsurf and leads to more water vapor in
the atmosphere via evaporation. This enhances the optical depth τ and thus the
greenhouse effect by increasing the downwelling infrared flux Fir↓ , which heats again
4.2 Runaway greenhouse effect
the surface and Tsurf increases. The amount of the greenhouse gas water vapor is
temperature dependent, since water vapor in the atmosphere is controlled by the
hydrological cycles involving evaporation and precipitation (see section 3.6.2). It
leads to a runaway greenhouse effect if the optical depth τ increases faster than the
outgoing infrared flux Fir↑ can balance the incoming stellar flux. This feedback is
assumed to continue until the water reservoir of the planet is completely evaporated.
If the complete water reservoir is evaporated and resides in the atmosphere and the
surface temperature increases further until a new radiative equilibrium is possible.
The runaway greenhouse effect is a dynamical and time-dependent process. In the
literature the runaway greenhouse (e.g., Ingersoll (1969); Nakajima et al. (1992);
Sugiyama et al. (2005)) is defined mostly by the existence of an upper limit of
the outgoing infrared flux Fir↑ , which results from the fact that the amount of the
greenhouse gas water vapor is temperature dependent and makes the atmosphere
optically thick. Two mechanisms, which can lead to such an upper limit on the
outgoing IR radiation are distinguished and explained in the next subsections: one
limit is based on stratospheric conditions (stratospheric radiation limit) and the
other limit is based on tropospheric conditions (tropospheric radiation limit). If the
incident stellar flux exceeds the smaller of the two limits, it is assumed that the
temperature would continue to increase until all the liquid water available on the
surface is evaporated.
4.2.1
Stratospheric radiation limit
Komabayasi (1967) and Ingersoll (1969) analytically derived conditions for the stratospheric radiation limit of Fir↑ by utilizing radiative equilibrium models. These studies
are summarized in Nakajima et al. (1992) and Satoh (2005), for example.
For these calculations the following assumptions have been made:
• Only the stratosphere assumed to be in radiative equilibrium is taken into
account.
• An expected underlying saturated troposphere is represented by a saturated
tropopause only, which implies that the liquid water reservoir on the surface
and the water vapor in the atmosphere are in equilibrium.
• Water vapor is the only absorbing gas for the infrared radiation.
• The absorption coefficient of water vapor of the infrared radiation is independent of wavelength (gray approach).
• No scattering in the atmosphere is considered.
• The atmosphere is assumed to be transparent to shortwave radiation.
• The atmosphere is considered as plane-parallel.
41
Physical processes determining the inner HZ boundary
42
• The atmosphere is considered to be in local thermal equilibrium (LTE).
The radiative transfer equation (see equation 3.16) for an atmosphere, that does not
scatter any radiation and that is in LTE, is given by
μ
dIν (τν , μ)
= Iν (τν , μ) − Bν (T (τν ))
dτ
(4.1)
where Iν is the spectral intensity of the radiation, τ the optical depth, μ the cosine
of zenith angle, Bν the Planck function, and the subscript ν denotes the frequency
↑
for the atmosphere with a gray
of the radiation. The upward infrared flux Fir,ν
approach, where absorption coefficients and subsequently the optical depth τ are
independent of ν, can be calculated by multiplying μ to equation 4.1 and integration
over μ
d
2π
dτ
1
↑
Iν (τ, μ)μ2 dμ = Fir,ν
(τ ) − πB(T (τ ))
(4.2)
0
Taking into account the assumption that Iν (τ, μ) is isotropic and independent of
↑
μ in each hemisphere: Fir,ν
(τ ) = πIν (τ, μ = 1) (Eddington approximation), the
integration of the left hand site of equation 4.2 can be written as
1
2π
Iν (τ, μ)μ2 dμ =
0
2π
2 ↑
Iν (τ, μ = 1) = Fir,ν
(τ )
3
3
(4.3)
By integrating equation 4.2 over the frequency ν the outgoing infrared flux, Fir↑ can
be determined to
2 dFir↑ (τ )
= Fir↑ (τ ) − πB(T (τ ))
3 dτ
(4.4)
With some conversions (see Satoh (2005) and Nakajima et al. (1992)) the following
relation can be obtained
F ↑ (τ )
πB(τ ) = ir
2
3
τ + 1 = σT 4 (τ )
2
(4.5)
The optical depth τ derived from this radiative equilibrium assumption can be written
4
τ=
3
σT 4 (τ )
1
−
↑
2
Fir (τ )
(4.6)
4.2 Runaway greenhouse effect
43
This approach is a first order approximation for the stratosphere where energy is
dominated by radiative transfer.
In general, the optical depth τ of the atmosphere from equation 3.17 can be written
by applying the hydrostatic equation as
z
τ=
p
kρdz =
∞
k
0
dp
p
=k
g
g
(4.7)
where k, ρ and p are the absorption coefficient, density and partial pressure of water
vapor and z the vertical coordinate. Assuming furthermore an atmosphere, which
is saturated in water vapor and therefore a water vapor partial pressure is equal to
the saturation vapor pressure psat , the optical depth τ can be written
τ =k
psat (T )
g
(4.8)
Thus, the optical depth at the tropopause τtp can be defined according to two methods (i.e. equation 4.6 and equation 4.8). The first method assumes radiative equilibrium at the tropopause
4
τtp =
3
4
σTtp
Fir↑
1
−
2
(4.9)
and the second method assumes hydrostatic gas-liquid equilibrium at the tropopause
τtp = k
psat (Ttp )
g
(4.10)
The assumption for the tropopause that it is in radiative equilibrium and also saturated, is only achieved, if a solution for both τtp equations is possible. Figure 4.2.1
shows the relationship between the optical depth τtp and the tropopause temperature Ttp from equation 4.9, given in thick lines as a function of different outgoing
infrared fluxes Fir↑ . The thin line in this Figure represents τtp from equation 4.10 as a
function of the saturation vapor pressure, which depends on the temperature of the
tropopause. In Figure 4.2.1 the curves for equation 4.10 and equation 4.9 intersect
but only for values below a certain value of outgoing infrared fluxes Fir↑ (Fir↑ ∼ 385
Wm−2 ). Above this value of Fir↑ the curves do not cross at all. Therefore, no τtp
value can be found where both equations are fulfilled.
This limited value of Fir↑ is in some studies called the stratospheric or KomabayasiIngersoll limit. In Nakajima et al. (1992) the reasons for the appearance of this value
are explained as follows: The amount of absorbing matter in equation 4.9 required for
preserving the radiative equilibrium and the amount of absorbing matter in equation
4.10 required for preserving the gas-liquid equilibrium lead to a contradiction. In
order to emit outgoing radiation above this limit, the atmospheric temperature
Physical processes determining the inner HZ boundary
44
Figure 4.2: The relationship between the optical depth τtp and the temperature Ttp at the
tropopause. The thick lines represent Equation (4.9) with F as a parameter. The thin line
represents Equation (4.10). Taken from Nakajima et al. (1992), their Figure 1.
must be sufficiently high, but the absorbing matter obtained from the gas-liquid
equilibrium at that temperature causes the optical thickness to become too high to
allow for the required radiation, which depends on the shape of the saturated vapor
pressure curve.
The stratospheric limit for Fir↑ can be derived only in the case of a gray atmosphere,
where the simple relationship as used in this approach between the absorbing matter
and temperature can be applied (Nakajima et al., 1992).
4.2.2
Tropospheric radiation limit
In studies investigating the runaway greenhouse effect with radiative-convective
models (reviewed in section 5.1) and taking into account the troposphere and surface conditions additionally to the stratosphere, an asymptotic limit of the Fir↑ can
also be found. Nakajima et al. (1992) suggested that this limit operates in a different way than the stratospheric radiation limit introduced by Ingersoll (1969) and
Komabayasi (1967). Furthermore, Nakajima et al. (1992) reproduced the results
for the asymptotic limit for Fir↑ which can be seen in Kasting (1988). Note, that
this asymptotic limit is not discussed as the reason for a runaway greenhouse effect
by Kasting et al. (1993). The ’runaway greenhouse limit’ by Kasting (1988) and
Kasting et al. (1993) is defined when the surface temperature reaches the critical
point of water.
4.2 Runaway greenhouse effect
45
Nakajima et al. (1992) utilized a simple one-dimensional radiative-convective model,
with gray infrared absorption coefficients, a stratosphere in radiative equilibrium and
a moist adiabatic troposphere. They also identified the physical mechanism causing
this limit of Fir↑ . Sugiyama et al. (2005) derived analytically the conditions for the
tropospheric radiation limit, which are presented in this section.
For their calculations the following assumptions are made:
• Radiative-convective equilibrium is considered that would develop in response
to a prescribed surface temperature.
• The surface radiates like a blackbody with an emissivity of 1.
• Scattering in the atmosphere is neglected.
• The atmospheric temperature is a function of any vertical coordinate and the
surface temperature.
The radiative transfer equation 4.1 in Eddington approximation can be written for
↑
the outgoing monochromatic infrared radiation flux Fir,ν
μ
↑
dFir,ν
dτν
↑
= Fir,ν
− πBν
(4.11)
The solution for this radiative transfer equation 4.11 at the top of the atmosphere
(toa) is
τν,surf (Tsurf )
↑
Fir,ν,toa
= πBν (Tsurf ) exp −
μ
τν,surf
(Tsurf )
πBν [T (τν , Tsurf )] exp
+
0
τ
− ν
μ
dτν
μ
(4.12)
Differentiating equation 4.12 and applying the Leibniz theorem leads to
dπBν (Tsurf )
τν,surf (Tsurf )
=
exp −
dTsurf
dTsurf
μ
πBν (Tsurf ) − Bν [T (τν (Tsurf ), Tsurf )]
−
μ
τν,surf (Tsurf ) dτν,surf (Tsurf )
· exp −
μ
Tsurf
↑
dFir,ν,toa
τν,surf (Tsurf )
+
0
∂{Bν [T (τν , Tsurf )]}
τ
dτν
exp − ν
∂Tsurf
μ
μ
(4.13)
Physical processes determining the inner HZ boundary
46
↑
From this equation 4.13 a radiations limit can be derived, if dFir,ν,toa
/dTsurf → 0
for all frequencies ν, which is fulfilled for the following conditions:
1.
2.
τν,s (Tsurf )
dπBν (Tsurf )
exp −
→0
dTsurf
μ
(4.14)
∂{Bν [T (τν , Tsurf )]}
→ 0 for all ν
∂Tsurf
(4.15)
The first condition is achieved when the total optical depth of the atmosphere becomes large at high surface temperatures, such that even in the window region of
water vapor (8-12μm), where the absorption is weak but not zero, the thermal emission from the surface cannot reach space. This is the case due to a rapid increase in
τν,surf , which enhances the greenhouse effect, because τν,surf roughly scales linearly
with saturation vapor pressure.
The second condition is achieved when the temperature in the atmosphere is independent of the surface temperature, which is caused by the water mixing ratio
becoming unity. The pseudoadiabatic lapse rate, which determines the temperature
profile in the troposphere, approaches the saturation vapor pressure curve because
as Tsurf increases, more water vapor gets into the atmosphere and the mixing ratio
of water becomes unity.
The reasons that a limiting flux exists based on tropospheric conditions are summarized following Sugiyama et al. (2005):
1. The total opacity of the atmosphere is large enough to prevent thermal infrared
emission from the surface from reaching space. This condition is satisfied at
high temperatures, because the optical depth of the entire atmosphere scales
roughly as saturation vapor pressure under the assumptions made.
2. Atmospheric temperature becomes a function of the optical depth only at
each frequency, thereby trapping the atmospheric emission regardless of Tsurf .
When water vapor dominates the atmosphere at high temperatures this condition is attained causing the temperature profile to approach to the saturation
vapor pressure curve.
4.3
Loss of water from the atmosphere
Determining the inner boundary of the HZ, it is clearly important to investigate
processes, which lead to the loss of the significant parts of water reservoir of a
planet. This section focuses on escape processes in an atmosphere to space that lead
to changes in the composition of a planetary atmosphere. Different mechanisms are
presented categorized into thermal and non-thermal loss processes. Considering the
4.3 Loss of water from the atmosphere
47
loss of a light atmospheric constituent via thermal escape for the Earth, the limiting
factor is in this case not the escape rate itself but instead the rate at which this
light constituent is transported from the lower to the upper atmosphere, where the
escape takes place.
4.3.1
Thermal escape
Atmospheric loss via thermal escape depends on exospheric temperatures (Bauer and
Lammer, 2004). If atmospheric species in an atmosphere have such high velocities
corresponding to the high energetic part of the Maxwell distribution that they exceed
the escape velocity vesc , Jeans escape will occur.
The escape velocity is then determined by
vesc =
2g(R)R =
2GMp
.
R
(4.16)
whereR is the planetocentric distance and g(R) GMp /R2 the gravity acceleration
of the planet, where G is the gravity constant and Mp the mass of the planet.
Atmospheric escape takes place at the boundary between the atmosphere and space,
the exobase, which is the height for which the mean free path is larger than the scale
height, and the particles move on ballistic paths.
The escape flux at the exobase is given by the Jeans escape flux φesc (Lecavelier des
Etangs et al., 2004)
v0
2
nexo mH √ e−X (1 + X).
φesc = 4 · πrexo
2 π
(4.17)
where nexo is the density of the hydrogen at the height rexo of the exobase and
mH the mass of hydrogen. Hydrogen (a product of water photo-dissociation (see
subsection 4.3.4)) is the lightest gas in the atmosphere and easiest to be lost to
space. The most probable velocity v0 distribution of the Maxwell distribution of
hydrogen is given by
v0 = (2kb Texo /mH )1/2
(4.18)
where mH is the mass of a hydrogen atom and Texo is the exospheric temperature
of the planet.
The escape parameter X is defined as
X=
vesc
v0
2
.
(4.19)
exo
where X is an important characteristic of planetary atmospheres. The lower this
parameter the greater is the escape flux.
Physical processes determining the inner HZ boundary
48
4.3.2
Hydrodynamic processes
For very high temperatures the mean kinetic energy of the hydrogen particles approaches the potential energy of the gravity field of the planet. The escape parameter
X attains small values due to the approach of the mean velocity of hydrogen atoms
to the escape velocity (v0 → vesc ). Therefore, the hydrogen particles can escape with
high mass loss rates. This hydrodynamic flux can carry away more massive particles
with it (Hunten, 1992; Vidal-Madjar et al., 2004), which is termed ’blow-off’.
4.3.3
Non-thermal escape
Non-thermal processes do not depend on the intrinsic thermal energy from the
Maxwell distribution but instead rely on specific processes involving e.g. interaction
with the stellar wind, meteor impacts, etc.
4.3.4
Diffusion-limited escape flux
Jeans escape is negligible for most atmospheric constituents of terrestrial atmospheres, but can be significant for the lightest gases of atmospheres such as hydrogen
and helium. Therefore, hydrogen can escape to space when it reaches the exosphere,
from which escape is possible. The loss rate of hydrogen is limited to the rate at
which hydrogen and its compounds are transported upwards from the lower to the
upper levels of the atmosphere.
This transport is described by the flux of constituent i through an ambient atmosphere consisting of constituent a and it can be derived from the combined eddy
diffusion and molecular diffusion equation, which is presented in e.g. Hunten (1973)
and summarized in Walker (1997). The difference in the vertical velocities wi of constituent i and wa of constituent a are described by molecular diffusion and eddy mixing. With some re-arranging this diffusion equation for the vertical flux Φi = wi ni
of constituent i can be written as
dci
(4.20)
dz
where K is the eddy mixing coefficient, Di the diffusion coefficient, na the number
density of constituent a, and ci the mixing ratio (defined as ni /na ). Φl in Equation
(4.20) is the termed the limiting flux and described as
Φi = Φl − (K − Di )na
Φl
bi c i
(ma − mi g) αi dT
=
−
(1 − ci )
kT
T dz
bi ci
1 1
=
(1 + ci )
Ha Hi
(4.21)
(4.22)
where bi is the binary diffusion parameter, αi the thermal diffusion factor and Ha
and Hi are the scale heights of constituent a and constituent i, respectively.
4.3 Loss of water from the atmosphere
If the mixing ratio ci is independent of height the vertical flux Φi is equal to the limiting flux φl . In this case the flux is determined by molecular diffusion because eddy
mixing leads to transport only when the mixing ratio varies. Taking into account
that the constituent i is a light gas which is embedded by a heavier background gas,
it can be derived from Equation (4.21) that the limiting flux Φl is positive and thus
directed upwards.
For the Earth, Walker (1997) presented that the rate of loss of hydrogen is limited
by the diffusion limited transport and not by Jeans escape. This can be shown
by comparing the mean velocities for Jeans escape and diffusion-limited transport,
where the smaller of both velocities will identify the limiting process. The limiting
diffusive velocity can be derived from equation 4.21 with the assumptions that ci 1
and mi ma to Φl /ni ≈ bi /na Ha . With a binary diffusion parameter of 2.73·1019
cm−1 s−1 for hydrogen in air at a temperature of 208 K, a scale height of 6.41 km and
the total number density of the background gas component na of 9.35·1012 cm−2 , the
limiting diffusive velocity is about 4cm s−1 . For Jeans escape the mean expansion
velocity for hydrogen at an exospheric temperature Texo =1500 K is vesc =7.26·103
cm s−1 . Thus, the limiting diffusive velocity is the bottleneck and therefore the
escape flux of hydrogen is limited by the diffusion-limited transport and is equal to
the limiting flux. To calculate the escape rate for hydrogen, not the Jeans escape
has to be calculated, but the limiting flux at levels where the hydrogen mixing ratio
is known.
For hydrogen it is important to take the following conditions into account (Walker,
1997). The flux of hydrogen in all forms is carried by eddy mixing and does not
depend on the compounds in which hydrogen occurs. The limiting flux of hydrogen
in the diffusion region is insensitive to whether the hydrogen is atomic or molecular
since molecular hydrogen and indeed hydrogen-containing molecules in general are
rapidly converted into atomic hydrogen above the base of the thermosphere.
Important sources of atomic hydrogen are e.g. water vapor, methane and molecular
hydrogen from the region around 30 km (Hunten and Strobel, 1974), which diffuse
and mix upward. From there up to a height of about 100 km these compounds of
hydrogen are converted into H2 and H, whereas the total mixing ratio of hydrogen
atoms remains nearly constant. Above this height the molecular hydrogen flows
upwards at the diffusion-limited rate (see equation 4.21). The molecular hydrogen
is converted to atomic hydrogen due to reactions with atomic oxygen. Hence, the
escape flux is proportional to the mixing ratio of all hydrogen atoms containing
compounds above a height of 30 km in the stratosphere. Hunten and Strobel (1974)
showed that the details of the chemistry are not important to determine the escape
flux. Only the overall effect of the photochemistry has to be taken into account which
can be represented as a change from mainly H2 O in the stratosphere to mainly H2
at about 100 km and then to atomic hydrogen in the heights above.
To summarize, the rate of escape of hydrogen is proportional to the concentration
of hydrogen compounds in the stratosphere and is insensitive to atmospheric temperature, eddy mixing rates, or the details of photochemistry. The most abundant
49
50
Physical processes determining the inner HZ boundary
hydrogen compound in the stratosphere is water vapor. Therefore, Walker (1997)
pointed out that one possible source of change in the escape flux is a change in
the temperature of the tropopause. Higher tropopause temperatures would permit
more water vapor to enter the stratosphere and lead to larger hydrogen escape fluxes,
which is important for the determination of the inner boundary of the HZ.
CHAPTER
5
Previous studies related to the inner HZ boundary
This chapter provides a literature review of the processes that are important to
determine the inner boundary of the HZ. It is focused on the runaway greenhouse
effect and loss processes of water, which were introduced in chapter 4.
Studies using different approaches to determine the runaway greenhouse are summarized. Furthermore, the so-called water loss limit of the HZ by Kasting et al. (1993)
is introduced, which is defined by the loss of the complete water reservoir of a planet
to space within the lifetime of a planet due to atmospheric escape processes.
5.1
Runaway Greenhouse Limit
Understanding the runaway greenhouse effect was relevant in the context of studies
on the history of water on Venus (e.g. Ingersoll (1969); Pollack (1971); Kasting
(1988)). In some studies about the HZ the runaway greenhouse effect was used
to determine the inner boundary of the HZ (e.g. Kasting et al. (1993)), which
should result in a climate state where no liquid water is existing on the surface of
an Earth-like planet. In this section an overview is given about the determination
of the runaway greenhouse effect in the most relevant studies. Also, the applied
models, assumptions, and approaches are presented. The basic physical principles
to describe the runaway greenhouse effect are presented in section 4.2.
5.1.1
Ingersoll (1969)
The investigation of the history of water on Venus was the motivation for introducing
the classical runaway greenhouse in the study of Ingersoll (1969). The main aim of
Previous studies related to the inner HZ boundary
52
this study was to establish that the runaway greenhouse effect exists in principle
and not to develop a detailed model of the early atmosphere of Venus.
Ingersoll (1969) proposed the existence of a critical value of the solar insolation.
Above this critical value liquid water cannot exist at the planetary surface and the
atmosphere is then considered to be in a runaway greenhouse state. This may occur
in the atmosphere of a planet if the amount of a greenhouse gas, in this case water
vapor, which absorbs infrared (IR) radiation, is determined by the vapor pressure
of the gas. The IR opacity is determined by water vapor, whose concentration in
the atmosphere is controlled by equilibrium with the solid or liquid phases at the
surface. Due to this reason a positive feedback loop arises between the IR opacity
of the atmosphere and the surface temperature, which is caused by the well-known
greenhouse effect of H2 O molecules. If the solar insolation is greater than a certain
critical value water exists only as vapor in the atmosphere, which is then called to
be in a runaway greenhouse state.
The model of Ingersoll (1969) introduced in subsection 4.2.1 takes only the stratosphere and the tropopause in radiative equilibrium into account to determine this
critical value for the incoming radiative flux. The tropopause is also assumed to be
close to saturation (due to assumed dynamics and convection in the troposphere).
This condition, i.e., that the base of the stratosphere is saturated results from the
presence of solid or liquid water on the surface of the planet. The relative humidity
of the complete atmosphere is set to 100%. Other properties of the troposphere
are not considered explicitly. For radiative transfer only IR radiation is taken into
account in the stratosphere. The IR absorption coefficients are independent of wavelength (gray approach). Therefore, also the optical depth of the atmosphere does
not depend on wavelength. To determine the temperature profile of the stratosphere
from radiative equilibrium in such a gray model atmosphere the Eddington approximation is applied. The atmosphere consists of a non-condensable and a condensable
component, which is water vapor. Both gases are assumed to behave like an ideal
gas and the mass fraction of the absorbing gas is considered to be constant.
The method of Ingersoll (1969) to derive the critical incoming solar flux value causing the runaway greenhouse state is explained in detail in subsection 4.2.1 of this
Thesis. If the incoming solar insolation exceeds a certain upper limit of the outgoing
↑,toa
at the top of the atmosphere, the atmosphere cannot remain in
infrared flux FIR
an equilibrium state with the ocean. The ocean will evaporate regardless of its mass.
↑,toa
Ingersoll obtained values for the critical flux are FIR
=321-655 Wm−2 and corresponding tropopause temperatures Ttp = 239 - 289 K, depending on the absorption
coefficients of water vapor, which were varied from k = 0.01 m2 kg−1 to k = 0.002
m2 kg−1 .
5.1.2
Komabayasi (1967)
Independently, Komabayasi (1967) studied with a comparable approach as Ingersoll
(1969) a hypothetical planetary scenario with an atmosphere and hydrosphere com-
5.1 Runaway Greenhouse Limit
posed of gaseous and liquid water. In his study Komabayasi considered qualitatively
the thermal equilibrium states of an ocean and a vapor atmosphere.
The atmospheric model used takes only the stratosphere into account, which is
assumed to be in radiative equilibrium. The radiative transfer in the infrared wavelength region is treated in gray approximation and the ocean is assumed to emit
radiation as a blackbody. As for the Ingersoll study, the Eddington approximation
is used to determine the temperature in radiative equilibrium. The atmosphere consists of only water vapor. The stratosphere is assumed to be water vapor saturated,
i.e. the relative humidity is 100%. The approach of Komabayasi (1967) to determine
an upper limit of the outgoing flux also explained in detail in subsection 4.2.1.
The main result of the study by Komabayasi (1967) is that, if the surface temperature, which in this study is the sea water temperature, exceeds a certain value,
the greenhouse effect increases the surface temperature Tsurf so strongly that the
ocean will boil off. No explicit critical solar insolation is given, at which this scenario will occur, but it is shown that in general a critical flux could be determined
taking into account a gray atmosphere and that the ocean and the atmosphere are
in equilibrium below this value.
It was shown in both studies that for a gray atmosphere a critical solar flux exists
above which the runaway greenhouse effect occurs. This limit does not depend on
the tropospheric structure, hence the surface temperature. The conditions for this
limit to exist are a stratosphere in radiative equilibrium and a saturated tropopause
and are termed by Sugiyama et al. (2005) as the stratospheric radiation limits.
For the studies of Komabayasi (1967) and Ingersoll (1969) the critical flux is derived
by a relatively simple relationship between the absorbent matter and temperature in
the atmosphere, which is caused by the independence of wavelength. For a realistic
treatment the wavelength dependence of the relevant absorption coefficients should
to be taken into account, which makes the determination of the critical flux to
determine the stratospheric radiation limit much more difficult.
5.1.3
Pollack (1971)
Pollack (1971) described the runaway greenhouse in the context of explaining the
history of Venus investigating the dependence of the surface temperature for a planet
with an assumed water reservoir upon the incident solar flux. The molecular opacities used in for the radiative transfer are wavelengths dependent, which is an advantage compared to the studies by Ingersoll (1969) and Komabayasi (1967), which used
a gray approach for the atmosphere. Furthermore, additionally to the stratosphere
also the troposphere is taken into account
The runaway greenhouse effect is assumed to occur in this study if the surface
temperature reaches the temperature of the critical point of water Tcrit = 647 K.
Above this temperature liquid water can not exist on the surface of the planet and
thus, all water will be in the gaseous state in the atmosphere. Furthermore, runaway
temperatures are defined by Pollack (1971) to be the temperatures for which smaller
53
54
Previous studies related to the inner HZ boundary
Figure 5.1: Solar insolation versus the surface temperature for pure water vapor models
with a 50% cloud cover. Taken from Pollack (1971), their Figure 6
water reservoirs than that of the Earth are not able to exist on the surface dependent
on the saturation vapor pressure curve of water.
The model calculates wavelength dependent thermal fluxes and planetary albedos.
The radiative active gas for infrared absorption is water. The continuum absorption
for H2 O is taken into account for 9 to 13 μm. Clouds could be included in the
model, which are assumed to radiate as blackbodies at their local temperature and
are optically thick. The clouds are assumed to be located at a height, where the
temperature is 30 K cooler than the surface temperature. Furthermore, Rayleigh
scattering is included.
The temperature profiles are prescribed. Starting from a surface temperature, the
temperature is monotonically decreasing with altitude in the troposphere until 215
K is reached, which assumed to be constant temperature of the stratosphere. The
lapse rate in the troposphere for water dominated atmosphere is dependent on the
saturation pressure. For increased surface temperature the emitted thermal flux,
the fraction of solar energy absorbed (=1-planetary albedo) and the incident solar
flux are investigated. The incident solar flux is obtained from the assumed balance
of the thermal emission to space against the amount of solar energy absorbed.
Pollack (1971) concluded that the solar insolation for present Venus is sufficiently
high, to reach the temperature of the critical point of water for a cloud cover of 50%
and a relative humidity of 25% and 100% for a pure H2 O atmosphere, which is shown
in Figure 5.1. For the 100% cloud cover the runaway greenhouse could not occur
5.1 Runaway Greenhouse Limit
caused by the rapid increase of the albedo with increasing surface temperature.
5.1.4
Kasting (1988) and Kasting et al. (1993)
Kasting (1988) investigated the runaway greenhouse effect in the context of the
evolution of the atmospheres of Earth and Venus. In a further study of Kasting
et al. (1993) these calculation where used to determine the inner boundaries of the
HZ around the Sun as well as around other main sequence stars.
Figure 5.2: Outgoing net infrared flux FIR and net solar flux FS at the top of the atmosphere (TOA) versus the surface temperature. Taken from Kasting et al. (1993), their
Figure 1(a)
The runaway greenhouse in both studies, terms the state of the atmosphere and
planet when the oceans are entirely evaporated. The effective solar flux (SEF F =
FIR (T OA)/FS (T OA)) at which this runaway greenhouse should occur, is calculated
for a surface temperature of 647 K, the critical point of water as in Pollack (1971).
Above this critical point liquid water cannot exist thermodynamically on the surface
of the planet but is rather in a supercritical state (see section 4.1).
Kasting (1988) and Kasting et al. (1993) used a one-dimensional radiative-convective
model of the atmosphere. The atmosphere is assumed to be fully saturated by
water vapor and cloud-free. The radiative transfer in the stratosphere is contrary to
the earlier gray calculations of Ingersoll (1969) and Komabayasi (1967) wavelength
dependent. To calculate the absorption and scattering of the shortwave radiation a
δ two-stream scattering formulation is used for a wavelength range from 0.2 to 4.4
μm (in 38 spectral bands) and includes Rayleigh scattering of H2 O. The absorption
of the outgoing IR radiation was determined by using band models. The wavelength
55
56
Previous studies related to the inner HZ boundary
range of the infrared model reaches from 0.39 to 500 μm (in 62 spectral intervals).
The stratosphere was taken to be isothermal with a temperature of 200 K. For the
troposphere the temperature is determined using the moist pseudo-adiabatic lapse
rate, where condensed water leaves the system as soon as it appears, for temperature
between 273 K and 647 K, and a dry adiabatic lapse rate above the temperature
of the critical point of water. Radiative species accounted for are water and carbon
dioxide. The assumed background atmosphere is Earth-like with of 1 bar N2 and 300
parts per million (ppm) CO2 (and an Earth-like amount of O2 for Kasting (1988)).
The relative humidity was assumed to be equal to 100%. The surface albedo is set
to 0.22, for which the model reproduces Earth’s mean surface temperature of 288K.
The planet parameters are that of the Earth meaning the same size and mass as
well as a water reservoir in the mass of an Earth ocean (1.4 ·1021 l).
With this model Kasting (1988) and Kasting et al. (1993) determined the temperature profiles by fixing the surface temperature and then calculated the atmospheric
temperature upwards in the troposphere with a moist pseudo-adiabatic or dry adiabatic lapse rate depending on the surface temperature. This is done until the
calculated tropospheric temperature reaches the fixed value for the stratosphere of
200 K. Above this layer the temperature in the stratosphere is kept at this value.
With this prescribed temperature profile the water profile is calculated. The resulting temperature and water profile are taken to calculate the radiative fluxes.
This method, since atmospheric temperatures, water concentrations, and radiative
fluxes are determined without iteration, ignore the feedback of the water vapor concentration on the greenhouse effect and therefore an enhancement of the surface
temperature. The resulting effective solar insolation SEF F = FIR (T OA)/FS (T OA)
is determined (FS = net incoming solar radiation and FIR = net outgoing IR radiation) corresponding to a given surface temperature Tsurf . Thus, SEF F (normalized
to the solar constant S0 ) is required to support a given surface temperature taking into account that the net incoming solar radiation should be equal to the net
outgoing IR radiation at the top of the atmosphere to reach radiative equilibrium.
With the explained approach the surface temperature for various runs was raised
incrementally from 220 to 2000K. The net solar flux FS and the net IR flux FIR
at the top of the atmosphere for increased surface temperatures for the runs are
shown in Figure 5.2, where a constant FIR of about 310 Wm−2 can be seen for surface temperatures above 400 K. But this occurrence was not discussed by Kasting
et al. (1993) in relation to a tropospheric radiation limit as condition for a runaway
greenhouse effect (see section 4.2.2).
Both studies (Kasting (1988) and Kasting et al. (1993)) obtained an effective solar flux SEF F , where the runaway greenhouse effect starts, here defined where the
surface temperature reaches the critical point of water (647 K), of SEF F =1.4 S0 ,
which corresponds to an orbital distance of d =0.84 AU shown in Figure 5.3.
In contrast to these prescribed temperature profile approach of these studies, Kasting
et al. (1984a) calculated temperature profiles self-consistently for increased solar
insolation. Temperatures were obtained up to values of solar insolation of S =
5.1 Runaway Greenhouse Limit
Figure 5.3: Effective solar insolation Sef f versus the surface temperature. Taken from
Kasting (1988), their Figure 7.
1.45S0 , but the resulting surface temperature was only 384.2 K, since this study did
not take finite absorption coefficients for water in every spectral band (≥ 0.63μm)
into account.
Furthermore, in Kasting et al. (1993) the partial pressures of the background gas
CO2 and also N2 were varied, but this was found to have not a large influence on the
calculated runaway greenhouse limit. The variation of the planetary mass leads for a
Mars-sized planet (1/10 of the mass of the Earth ME ) to a larger orbital distance of
d = 0.88 AU due to the larger column depth, which increases the greenhouse effect
to reach 647K. The runaway greenhouse (647 K) will occur for a bigger planet (10
·ME ) at a smaller distance of d = 0.81 AU. For planets around other main sequence
stars the distances where the runaway greenhouse starts are for an M0 star d =0.24
AU with SEF F = 1.05S0 and for an F0 star d =1.50 AU with SEF F = 1.90S0 .
Compared with the previous studies by Ingersoll (1969) and Komabayasi (1967) the
studies by Kasting (1988) and Kasting et al. (1993) have the advantage that they
did not use a gray approach for the radiative transfer. In the studies a detailed
wavelength dependent IR radiative transfer was used as well as a wavelength dependent radiative transfer to calculate the shortwave flux in the atmosphere, for which
also Rayleigh scattering was included. Furthermore, not only the stratosphere was
taken into account, but also the troposphere.
A disadvantage of the studies by Kasting (1988) and Kasting et al. (1993) is that
the atmospheric temperature is not calculated self-consistently but prescribed. As
a result of their model approach, in which the temperature profile is not calculated
self-consistently, but is fixed based on the assumed Tsurf at the beginning of the
calculation and thus, the radiative fluxes are calculated once, radiative equilibrium
57
Previous studies related to the inner HZ boundary
58
in the stratosphere is not reached. No iterative feedback between the temperature
and the water profile and the radiative fluxes is considered, which is important to
take the greenhouse effect into account. Furthermore, the radiative fluxes and the
temperature profile should be converged when radiative equilibrium is reached in
the stratosphere in a iterative self-consistent way.
5.1.5
Nakajima et al. (1992)
The motivation of the study by Nakajima et al. (1992) was to investigate the possible
evaporation of an ocean during the evolution of the atmosphere of a terrestrial planet.
Therefore, the study focused on the relationship between the surface temperature
and the emission of infrared radiation at the top of the atmosphere. Nakajima
et al. (1992) compared and analyzed in this context the studies of Ingersoll (1969),
Komabayasi (1967) and Kasting (1988).
An atmosphere, where the surface temperature Tsurf is so high that the ocean of
this planet cannot exist in an equilibrium state, is understood to be in runaway
greenhouse state. For the runaway greenhouse as defined by Nakajima et al. (1992)
the critical point of water and the mass of the ocean plays no role, only the infrared
absorption of water vapor is assumed to be relevant for the evaporation of the ocean.
To determine the start of a runaway greenhouse Nakajima et al. (1992) found that
↑top
the occurrence of singularities in the behavior of the outgoing infrared flux FIR
,
such as an asymptotic value (see Kasting (1988)) or an upper limit (see Ingersoll
(1969)) seems to be necessary. An atmosphere is then assumed to be in a runaway
greenhouse state when it receives more incoming solar radiation than the radiation
limit.
Nakajima et al. (1992) applied a radiative-convective model for the troposphere
and stratosphere. In the stratosphere radiative equilibrium is assumed. The model
atmosphere is assumed to be transparent to solar radiation and the absorption coefficient of the IR radiation is assumed to be constant and independent of wavelength
(gray approach). For integration of the radiative transfer equations the Eddington
approximation is used. A moist pseudo-adiabatic temperature lapse rate is assumed
in the troposphere. Furthermore, the troposphere is assumed to be saturated with
respect to water vapor and with a relative humidity of 100%. The height where
the net radiation flux convergence becomes positive in the upper levels of the atmosphere, where the temperature is sufficiently low, is defined to be the position of the
tropopause. The atmosphere consists of a non-condensable gas and a condensable
component in its gas phase. Both are assumed to be ideal gases. The molecular
weight of the non-condensable component is assumed to be the same as that of the
condensable component, so that the average molecular weight becomes independent
of the mixing ratio of the non-condensable component.
Nakajima et al. (1992) examined the stratospheric radiation limit, which they termed
Komabayasi-Ingersoll limit in this study, and the tropospheric radiation limit, which
was termed the asymptotic limit.
5.1 Runaway Greenhouse Limit
↑,top
Figure 5.4: Outgoing infrared flux FIR
versus the surface temperature TS . Taken from
Nakajima et al. (1992), their Figure 3.
For the calculations of the stratospheric radiation limit only the stratosphere and the
condensable component are taken into account. The critical flux for the stratospheric
↑top
=385 Wm−2 and the corresponding tropopause temperature
radiation limit is FIR
Ttp =255 K (see Figure 4.2.1 in subsection 4.2.1).
To model the tropospheric radiation limit the troposphere was included and a background pressure of the non-condensable component of 1 bar was assumed. The
↑top
calculated. As could be
surface temperatures were increased and the resulting FIR
↑top
does not reach
seen in Figure (5.4) the maximum of the outgoing infrared flux FIR
↑top
the Komabayasi-Ingersoll limit (stratospheric radiation limit). The value of FIR
tends to be constant for surface temperatures above 400 K. This asymptotic value
↑top
= 293 Wm−2 .
is the tropospheric radiation limit of about FIR
Nakajima et al. (1992) identified the physical mechanism, which explains the tropospheric radiation limit. For high surface temperature Tsurf the temperature structure approaches the saturation vapor pressure curve. Correspondingly, the mixing
ratio of the water vapor approaches unity. By examining the temperature structure
according to the optical depth τ , it is shown that the temperature structure with
regard to τ approaches a limiting curve as the surface temperature increases. As
a result, the temperature structure in the vicinity of τ = 1, which determines the
↑top
, does not depend on the surface temperature Tsurf anymore. The
value of FIR
temperature structure is fixed with respect to τ . This is explained to be the reason
for the tropospheric radiation limit, shown in Figure 5.5.
59
Previous studies related to the inner HZ boundary
60
Figure 5.5: Optical depth τ versus the surface temperature TS and the mole fraction of
H2 O. Taken from Nakajima et al. (1992), their Figure 5.
Once the incoming solar flux exceeds the smaller of the stratospheric or the tropospheric radiation limit, the surface temperature is assumed to continue to increase
until all the liquid water evaporates.
5.1.6
Rennó et al. (1994)
Rennó et al. (1994) investigated the influence of changes in solar forcing with a
radiative-convective model, including explicitly a hydrological cycle in the model.
The rapid increase of surface temperature when the solar forcing approaches a critical
value is indicated for the runaway greenhouse effect to occur for surface temperatures
up to 330 K (see Figure 5.6).
Rennó et al. (1994) used a one-dimensional radiative-convective equilibrium model.
To calculate the radiative fluxes a broadband radiation parameterization (Chou,
1992) is used. As radiative gases water vapor, carbon dioxide, and ozone are taken
into account. Furthermore, clouds are included and considered to be black to infrared radiation, except for high clouds which are assumed to be half black. Clouds
are characterized by the vertical distribution and the optical properties. The hydrological cycle is described by the moisture profile, which is interactively calculated by
a cumulus convection scheme. Instead of the moist pseudoadiabatic lapse rate as in
e.g., Nakajima et al. (1992) or Kasting et al. (1993), the convection is parameterized
by six different schemes, where some of these schemes also include mass transport.
The critical value of the solar insolation for the runaway greenhouse lies for clear
sky conditions of the radiative fluxes between 1.1 and 1.41 S0 , and for fixed cloud
conditions between about 1.22 and 1.49 S0 dependent on the used cumulus convection scheme. For cumulus convection schemes including mass flux schemes the
5.1 Runaway Greenhouse Limit
Figure 5.6: Surface temperature Tsurf versus the the solar forcing for various cumulus
convection schemes. Fixed cloud covers are assumed in the radiative modeling. Taken from
Rennó et al. (1994), their Figure 10.
runaway greenhouse occurs very rapidly compared to cumulus convection schemes
with convective adjustment schemes. In Figure 5.6 the surface temperatures for increased solar flux obtained with various cumulus convection schemes are shown for
fixed cloud cover.
An additional advantage to the inclusion of cumulus convection schemes to describe
the hydrological cycle and the inclusion of clouds, is the investigation of the influence
of increased solar insolation on the temperatures of the atmosphere in Rennó et al.
(1994). This is contrary to the studies of Nakajima et al. (1992) and Kasting et al.
(1993), which only increase the surface temperatures.
Disadvantages of this study are that the radiative transfer is only valid for temperatures between 160 and 330K, scattering is neglected, the cloud-radiation feedback
is not well known, and uncertainties occur in water vapor continuum outside the
window region.
5.1.7
Pujol and North (2002)
Pujol and North (2002) investigated the runaway greenhouse with a semi-gray approach instead of the gray approach by Nakajima et al. (1992). A semi-gray atmosphere is in their study represented by a single window region in the infrared
spectrum, where the absorption coefficient is set to zero or small values, whereas
for the rest of the spectrum a higher constant absorption coefficient is used (gray
approach). The runaway greenhouse in the study of Pujol and North (2002) is based
on an upper boundary on the outgoing infrared flux (see tropospheric radiation limit
of Nakajima et al. (1992)).
Pujol and North (2002) used a one-dimensional radiative-convective model, based
61
Previous studies related to the inner HZ boundary
62
Figure 5.7: Outgoing infrared flux F versus the surface temperature with variation of the
effective width β of the atmospheric window from completely transparent (β=1) to infrared
radiation (without window β=0). Taken from Pujol and North (2002), their Figure 7.
on the model of Nakajima et al. (1992), but including in the infrared an atmospheric
window. The stratosphere is assumed in radiative equilibrium and the troposphere
is assumed to be completely saturated with water vapor, which is the semi-gray gas.
Contrary to the previous study by Nakajima et al. (1992) for the semi-gray atmosphere relative limits of the outgoing infrared flux are found. For an atmosphere
with zero absorption in the window region an upper limit of the outgoing infrared
flux is not found. Whereas, a relative radiation limit can be found for atmospheres
with narrow windows and furthermore relative and absolute radiation limits with low
absorption within the atmospheric window. In Figure 5.7 the outgoing infrared flux
F is shown as a function of temperature, for effective width β of the atmospheric
window, which is varied from completely transparent (β=1) to infrared radiation
(without window β=0).
5.1.8
Ishiwatari et al. (2002)
In the study of Ishiwatari et al. (2002) a general circulation model (GCM) was used
to investigate the runaway greenhouse. The runaway greenhouse is defined as in
studies before (e.g., Nakajima et al. (1992); Pujol and North (2002)), which results
from a upper limit of the solar insolation for the atmosphere to reach equilibrium.
The model of Ishiwatari et al. (2002) is based on the one-dimensional radiativeconvective model of Nakajima et al. (1992) but includes three-dimensional atmospheric motion. The model includes a gray radiation scheme and neglect clouds.
The solar insolation was increased from 1200 to 1800 Wm−2 for eight different runs
with the initial condition of an isothermal atmosphere (280K) with constant specific
5.1 Runaway Greenhouse Limit
humidity (10−3 ). On the surface of the planet a ’swamp ocean’ is assumed, which
is is completely wet, has no heat capacity, and is always in heat balance.
Figure 5.8: Meridional distribution of the zonal mean outgoing infrared (longwave) radiation (OLR). Taken from Ishiwatari et al. (2002), their Figure 4a.
Even if the effect of atmospheric motion is included in the model, there exist an
upper limit of the outgoing infrared flux (=outgoing longwave radiation (OLR)).
This upper limit in the study of Ishiwatari et al. (2002) is about 400 Wm−2 and
corresponds to a critical value of the solar constant exceed 1600 W−2 for the runaway greenhouse. This upper limit is determined by the vertical structure of the
atmosphere in the equatorial region, since the solar insolation increases, the temperature distribution at the levels of optical depth around unity becomes uniform in the
latitudinal direction, and the values of outgoing infrared radiation at all latitudes
approaches the equatorial limit. The meridional distribution of the zonal mean outgoing infrared radiation (OLR) is shown in Figure 5.8. The upper limit calculated
with the GCM in the study of Ishiwatari et al. (2002) corresponds to the value of
Nakajima et al. (1992) with a relative humidity of 60%, which is a typically value
obtained by the GCM.
The conditions for a runaway greenhouse are more specified in a three-dimensional
model as in a one-dimensional model, since the global mean value of incident flux,
neither the maximum nor the minimum of the latitudinal distribution of incident
flux, exceeds the upper limit.
5.1.9
Sugiyama et al. (2005)
The study of Sugiyama et al. (2005) analyzes on the tropospheric radiation limit
for a non-gray atmosphere. In the study of Sugiyama et al. (2005) the tropospheric
radiation limit was analytically derived, which is explained in detail in this thesis in
subsection 4.2.2.
The atmosphere becomes sufficiently optically thick, which results from the high
63
Previous studies related to the inner HZ boundary
64
temperatures because the optical depth scales with the saturation vapor pressure.
And furthermore, also for high temperatures the pseudoadiabatic temperature structure approaches the saturation vapor pressure curve and thus leads to atmospheric
temperature, which are only a function of optical depth at each frequency. This results in a limiting outgoing infrared flux. As in Nakajima et al. (1992), it is assumed
to occur if the atmosphere would receive more incoming stellar radiation then the
radiation limit of the outgoing infrared flux.
Furthermore, the influence of interactive relative humidity is investigated in the
study of Sugiyama et al. (2005). For allowed changes of the relative humidity with
surface temperature, then there can occur a new kind of radiation limit or multiple
equilibria.
5.1.10
Selsis et al. (2007)
In the study of Selsis et al. (2007) based on investigation about the habitability of
the planets Gliese 581c and Gliese 581 d, the conditions for the inner habitable zone
are summarized. The inner limits of the HZ are taken from Kasting et al. (1993).
An additional necessary condition was introduced to assess whether a planet is
habitable or not based on the effective temperature Tef f of the planet. It is proposed
that if Tef f is lower than 270 K a planet could be habitable. For this Tef f the thermal
emission derived from the Stefan-Boltzmann law is ∼300 Wm−2 . This value could
be derived from the studies of Kasting (1988) and Kasting et al. (1993), where a
limiting infrared flux of FIR = 310 Wm−2 for surface temperatures above 400 K can
be seen (see Figure 5.2).
The existence of such an asymptotic limit depends on the model, the model approach, and the model assumptions. For example for the study of Pujol and North
(2002) multiple equilibria are possible and thus no limiting infrared flux can be determined. Furthermore, in the study of Ishiwatari et al. (2002), which takes into
account a general circulation model, an asymptotic behavior for the outgoing infrared flux appears for ∼400Wm−2 , which results in an equilibrium temperature of
290K.
5.1.11
Goldblatt and Watson (2012)
Goldblatt and Watson (2012) reviewed the physical basics of the runaway greenhouse. They defined the runaway greenhouse also by radiation limits of the outgoing thermal radiation for warm and wet atmospheres and distinguished between
the stratospheric and tropospheric radiation limit. Goldblatt and Watson (2012)
followed the approach by Nakajima et al. (1992).
They discussed qualitatively the influence of Rayleigh scattering for thick atmospheres and assumed that the runaway greenhouse would be initiated later as solar
absorption decreases as the infrared emission gets constant.
5.1 Runaway Greenhouse Limit
5.1.12
65
Advantages and disadvantages of the runaway greenhouse
studies
In the following Tables the main advantages and disadvantages of most of the reviewed studies are summarized.
Ingersoll (1969)
Advantages
Analytical model
Nakajima et al. (1992)
Includes troposphere
Kasting (1988) Kasting
et al. (1993)
Wavelength dependent absorption, includes troposphere, includes infrared and shortwave radiation, includes Rayleigh
scattering
Rennó et al. (1994)
Wavelength dependent absorption, advanced description of
the convection, increased solar insolation, includes clouds
Pujol and North (2002)
Gray radiative transfer but with atmospheric window regions, includes troposphere
Ishiwatari et al. (2002)
Includes troposphere, GCM: includes motion, increased solar flux, time dependent, adjustment of surface pressure
Table 5.1: Advantages of the runaway greenhouse studies
Previous studies related to the inner HZ boundary
66
Ingersoll (1969)
Disadvantages
Gray radiative transfer, only infrared, only stratosphere, neglects troposphere and surface conditions
Nakajima et al. (1992)
Gray radiative transfer, variation of Tsurf , only infrared, no
self-consistent approach, Neglecting scattering
Kasting (1988) Kasting
et al. (1993)
Variation of Tsurf , no self-consistent approach, no global radiative equilibrium, only isothermal stratosphere, H2 O continuum only in the window region
Rennó et al. (1994)
Cloud-radiation feedback not well known, no scattering included, uncertainties in water vapor continuum outside the
window region, radiative transfer only valid for 160-330K,
multiple equilibria possible
Pujol and North (2002)
Not really wavelength dependent, variation of Tsurf , no selfconsistent approach, neglecting the feedback of the greenhouse effect upon Tsurf , no scattering included
Ishiwatari et al. (2002)
Gray radiative transfer, neglecting scattering of solar flux
Table 5.2: Advantages of the runaway greenhouse studies
5.2 Water Loss Limit
5.2
5.2.1
67
Water Loss Limit
Kasting (1988) and Kasting et al. (1993)
Kasting et al. (1993) determined a second inner limit of the habitable zone (additionally to the runaway greenhouse limit) termed the water loss limit of a terrestrial
planet.
This water loss limit is based on the diffusion-limited flux, which is described by
the bottleneck process for the loss of hydrogen (see subsection 4.3.4). Kasting et al.
(1993) investigated the rate Φesc , at which water can be lost for an Earth-like planet.
They implied that water vapor in the stratosphere is entirely photo-dissociated and
the hydrogen produced can be lost to space.
The critical escape rate of water Φesc , which leads to the loss of an ocean within the
lifetime of the planet, can in general be described as
Φesc =
water content of the ocean
age of the planet
(5.1)
To calculate this escape rate of water for the Earth, an ocean content of 1.4·1021 l, as
well as the age of the Earth i.e. 4.5 billion years, have to be taken into account. The
water content of the ocean has to be converted in the number of hydrogen atoms
of the ocean per area in cm2 (1.87 · 1028 H atoms of the ocean per cm2 ). With
the rounded value of 2 · 1028 , which in Kasting et al. (1993) was applied for the
hydrogen atoms per cm2 , the critical escape rate Φesc needed for the Earth to lose
an Earth-ocean over its lifetime can be calculated to
2 · 1028 H atoms cm−2
4.5Ga
11
= 1.32 · 10 H atoms cm−2 s−1
Φesc =
(5.2)
(5.3)
On the other hand, hydrogen escape on Earth is limited by diffusion (Hunten, 1973)
as described in subsection 4.3.4. Thus, the diffusion-limited escape rate for hydrogen
as written in Kasting et al. (1993) is
Φesc (H) 2 · 1013 cH,tot cm−2 s−1
(5.4)
here, cH,tot is the total hydrogen mixing ratio in the stratosphere. This escape
rate Φesc is the diffusion limited escape rate Φesc = Φl , which can be written for
light minor constituent as Φl ≈ bi ci /Ha . In Kasting et al. (1993) cH,tot corresponds
to ci and the numerical factor (2 · 1013 ) results from the scale height Ha and the
binary diffusion parameter bi , which depends on the atmospheric composition. As
discussed in subsection 4.3.4, the mixing ratio of the total stratospheric hydrogen
compounds cH,tot is determined by the water vapor mixing ratio in the stratosphere
68
Previous studies related to the inner HZ boundary
Figure 5.9: Vertical profile of H2 O mixing ratio for selected surface temperatures. Taken
from Kasting et al. (1993), their Figure 2(b).
cH2 O,stratosphere . The diffusion-limited escape rate Φesc is then proportional to the
stratospheric water mixing ratio cH2 O,stratosphere (Hunten, 1973).
Substituting equation 5.2 in equation 5.4 gives a critical value for the stratospheric
mixing ratio of water for the Earth of
cH2 O,crit 3 · 10−3
(5.5)
The temperature and water profile and the resulting shortwave and infrared radiative fluxes are calculated following the approach and using the same assumptions
as for the runaway greenhouse limit (647 K) by Kasting et al. (1993) explained in
subsection 5.1.4. Figure 5.10 shows the water profiles for increased surface temperatures. The stratospheric mixing ratio of water cH2 O,stratosphere , calculated for
different surface temperatures versus the effective solar insolation SEF F = FIR /FS
are presented in Figure 5.10. The effective solar flux SEF F , for which the critical
value of the stratospheric water vapor cH2 O,strat = 3 · 10−3 is reached, is obtained at
an effective solar flux of SEF F =1.1S0 . This corresponds to a distance of d =0.95
AU from the Sun. This distance is defined by Kasting et al. (1993) to be the distance
of the water loss limit of the habitable zone. Thus, for the distance of the water loss
limit the water vapor mixing ratio in the stratosphere is so high that a water reservoir in the size of the Earth oceans can be lost via diffusion-limited escape within
the lifetime of the Earth.
Note that Kasting et al. (1993) did not calculate an escape flux or include pho-
5.2 Water Loss Limit
Figure 5.10: Variation of stratospheric H2 O with effective solar flux. Taken from Kasting
et al. (1993), their Figure 3.
todissociation processes. This limit only depends on the value of the critical mixing
ratio of water in the the stratospheric cH2 O,crit . Neither are the temperature profiles
calculated self-consistently depending on radiative fluxes.
69
70
Previous studies related to the inner HZ boundary
CHAPTER
6
Model description
Based on the overview about the studies considering processes relevant to determine
the inner boundary of the HZ, an improved atmospheric model is needed that is
able to include most of the advantages of the previously reviewed studies. Thus, the
model should be able to determine the inner boundary of the HZ considering the
feedback processes for increased stellar insolation between the surface temperature,
water mixing ratio, and the radiative fluxes self-consistently.
The therefore needed requirements of the atmospheric model to answer the addressed
scientific questions are summarized. A short overview of the history of the used onedimensional radiative-convective model is given. The improvements of this model
are introduced which were implemented in the context of this Thesis. Assumptions
and simplifications of the model and physical limitations when calculating the inner
HZ are discussed. A description of the treatment of the main energy transport
processes, radiation and convection in the model is given. A detailed description of
the calculations of temperature and water profiles is presented.
6.1
Model requirements
The aim of this Thesis is to answer the addressed scientific questions (see section
1.1):
• Where is the inner boundary of the HZ located in the Solar System and in
other stellar systems?
• Is the runaway greenhouse important for the determination of the inner boundary of the HZ?
Model description
72
Therefore, an atmospheric model is needed which fulfills the following conditions:
• Basic physical processes acting in planetary atmospheres, e.g. energy transport
by radiation and convection have to be taken into account in the model (see
chapter 3).
• The model has to be able to calculate for a variety of increased solar/stellar
insolations temperature and water profiles self-consistently until temperature
profiles are converged and radiative equilibrium is reached in the stratosphere.
Starting from ’normal’ Earth conditions (temperature and water profiles) with
the increased solar/stellar insolation a new corresponding steady-state solution
has to be found.
• The radiative transfer as the dominant energy transport process in the stratosphere has to include shortwave radiation resulting from the stellar insolation
and the longwave radiation resulting from the planetary radiation, which has
to be treated wavelength dependent.
• For the shortwave radiation scattering effects of all species has to be taken into
account additionally to the gaseous absorption.
• For the longwave radiation, which has to be able to treat the expected high
temperature, pressure and water concentration, additionally to the gaseous
absorption also the continuum absorption of water and CO2 has to be taken
into account.
• As energy transfer process in the troposphere convection has to be considered.
• The surface pressure has to increase with increasing water vapor in the atmosphere.
• The model should include the important atmospheric species of the atmospheres of the terrestrial planet in the Solar System (N2 , CO2 and H2 O)
• The model has to be able to calculate temperatures up to the critical point of
water.
6.2
Model history
The one-dimensional radiative-convective model described in this chapter fulfills
these conditions and helps to answer the scientific questions. It calculates temperature, pressure, water profiles and radiative fluxes of an atmosphere self-consistently
taking radiative equilibrium in the stratosphere into account.
This atmospheric model is based on a one-dimensional radiative-convective model
by Kasting et al. (1984a) and Kasting et al. (1984b) and was further developed
by Kasting (1988), Kasting (1991), Kasting et al. (1993), Mischna et al. (2000) and
6.3 Model improvements
Pavlov et al. (2000). Furthermore, Segura et al. (2003) updated the thermal radiative
transfer code using RRTM (Rapid Radiative Transfer Model) by Mlawer et al. (1997)
and coupled this climate model with a photochemical model. This coupled model
(from web site: vpl.ipac.caltech.edu/sci/AntiModels-/models/frontpage.html) was
the basis for further modifications by e.g. Grenfell et al. (2007) and Rauer et al.
(2011).
Taking only the climate model of this coupled model into account, von Paris et al.
(2008) adapted the model that it can be applied for CO2 -dominated atmospheres.
An infrared radiative transfer code was developed to handle high CO2 concentrations. This code is termed MRAC (Modified RRTM for Application to CO2 dominated atmospheres) and based on RRTM. RRTM is only applicable for atmospheres similar to that of the Earth, because it is only valid for a range of temperatures, pressures, and concentrations that do not deviate too much from the
standard Earth-profile of temperature (e.g. ± 30 K from the US Standard Atmosphere (1976)).
6.3
Model improvements
In this thesis further modifications to this atmospheric climate column model are
performed to be able to answer the addressed questions by being able calculate
temperature, pressure, water profiles and radiative fluxes of an atmosphere selfconsistently.
Therefore, the infrared radiative transfer code MRAC was improved that it can not
only be applied to CO2 -dominated atmospheres but also to warm (up to 647 K)
and water-rich atmospheres. Compared to the previous MRAC version of von Paris
et al. (2008) the following improvements have been implemented (see section 6.7):
• Absorption coefficients for H2 O and CO2 are accounted for all spectral bands
(except CO2 in spectral band 25, see section 6.7.1)
• The k-distributions are calculated for a wider temperature and pressure range:
temperature range from 100 to 700K and pressure range from 10−5 to 103 bar
• The binary species parameter needed to calculate the k-distributions is determined independently from a reference atmosphere
• The continuum absorption of H2 O and CO2 is updated and taken into account
for the complete wavelength spectrum
The improvements on the infrared radiative transfer scheme were done in close
collaboration with Philip von Paris. These improvements where presented in a
collaborative study on the first potentially habitable planet Gliese 581d (von Paris
et al. (2010)) and in von Paris (2010) (Thesis).
73
Model description
74
A further improvement was introduced concerning the condition to distinguish between the convective and radiative energy transport (see section 6.9). This model is
appropriate for warm and water-rich atmospheres, where convection is assumed to
be the dominant energy transport mechanism up to high altitudes. The reason for
this is the increased optical depths for enhanced water vapor concentrations in the
atmosphere.
6.4
Model assumptions and simplifications
The one-dimensional radiative-convective model gives as results global mean conditions for the temperature, pressure, and water abundance in the atmosphere. Thus,
longitudinal and latitudinal dynamical effects are neglected.
This is reasonable for fast-rotating planets (fast compared to the thermal response
of their atmospheres), where a mean insolation can be announced. On such planets,
day-night and meridional contrasts are small, hence one-dimensional global averages
give reasonable approximations.
Hydrostatic equilibrium (see section 3.2) gives the vertical pressure profile and is used
in the model to determine the height of the model from initial pressure conditions at
the surface and the top of the atmosphere. This is appropriate since the atmosphere
modeled is assumed to be small compared to the planetary mass (g is constant),
the scale height of the atmosphere is small compared to the planetary radius, the
vertical forces are dominated by the barometric pressure gradient, and the planetary
radius is large enough that a plane-parallel atmosphere can be assumed.
Energy transport is performed only via radiative transfer and convection, which is
reasonable for the troposphere and stratosphere because the model lid at the top
of the atmosphere is set at a height, where heat conduction as a further process for
energy transport is negligible.
The radiative transfer in the model atmosphere is separated into the treatment of
the shortwave radiation from star and the thermal radiation of the planet. This is
possible for the Earth and the Sun because the overlap of the spectra of the stellar
incoming flux and the thermal outgoing flux emitted by the planet is negligible.
Transport of mass is not taken into account in this model explicitly. It is implicitly
included for water due to the description of the convection and the subsequent
calculation of the water concentration in the atmosphere. Convection in general
refers to the transport of thermal energy between different places in the atmosphere
due to the transport of particles, but in this model convection is described by the
adiabatic temperature gradient (see subsection 3.5.2).
In the one-dimensional radiative-convective model only H2 O and CO2 as greenhouse
gases and N2 as background gas are taken into account. H2 O and CO2 are the most
important greenhouse gases for the Earth and N2 is found in significants amounts
in the terrestrial atmospheres of the Solar System like Venus, Earth, Mars and
Titan. Gases like O2 , O3 , CH4 , N2 O, and SO2 , which are present in the atmosphere
6.5 Schematic overview of the model
of terrestrial planets of the Solar System are neglected, because they are extremely
dependent on the biosphere, volcanism, outgassing, and formation of a planet, which
is until now not possible to determine for extrasolar planets.
The hydrological cycle in this model is represented by a relative humidity profile.
Thus, processes like for example evaporation, condensation, or precipitation are not
included explicitly in the model.
Clouds are not explicitly incorporated in the model version used for the investigations
of this Thesis, although they are important in the atmospheres of terrestrial planets
in the Solar System. Their effect could be implicitly included by adjusting the
surface albedo, which is an input parameter of the model. The adjusted albedo
is able to compensate for the missing reflections by clouds and reproduce, e.g.,
the mean surface temperature for the Earth (288 K). The effect of clouds in the
investigated atmospheres is uncertain, thus the cloud-free model is a first approach
to investigate the influence of the basic physical processes like radiative transfer and
convection in a model atmosphere.
The surface albedo is assumed to be independent of wavelength, which is a simplification.
Furthermore neglected are photochemical reactions and chemistry in the atmosphere. In this model the atmospheric molecules taken into account (H2 O, CO2
and N2 ) are only involved in radiative and/or convective processes.
Despite these simplifications and assumptions the one-dimensional radiative-convective
model is able to treat the basic physical processes of an atmosphere such as energy
transport.
6.5
Schematic overview of the model
Figure 6.1 summarizes how the model works in schematic way as described in the
next sections.
75
76
Model description
Figure 6.1: Model scheme of the radiative-convective model
6.6 Basic model description
6.6
77
Basic model description
The model atmosphere calculated with the one-dimensional radiative-convective
model is divided into 52 layers. With the two boundary conditions for the pressure at the top of the atmosphere p0 and at the surface psurf = pback + pH2 O a
logarithmic pressure grid is determined by applying the hydrostatic equilibrium.
As initial condition a temperature profile (US Standard Atmosphere 1976) and the
concentration profiles for H2 O (US Standard Atmosphere 1976) and CO2 (modern
Earth value from 1992: 355ppm) are used. CO2 is assumed to be well-mixed in the
atmosphere as it is for the Earth.
Radiative transfer for the shortwave radiation including molecular absorption and
Rayleigh scattering is taken into account as well as thermal radiation of the planet
and atmosphere, where in addition to the molecular absorption the continuum absorption is also treated. The convection is described by adiabatic lapse rates.
The species taken into account in the model for the investigations of this Thesis are
N2 , H2 O and CO2 as stated above. The mixing ratio of N2 cN2 is calculated via
cN2 = 1 − cH2 O − cCO2 . Water and carbon dioxide are important radiative gases
taken into account for the shortwave radiation as well as for the thermal radiation.
All three species contribute to the heat capacity and additionally to the shortwave
radiative transfer via Rayleigh scattering. The water and carbon dioxide content
in the atmosphere are linked to the surface temperature due to the evaporation or
condensation of these gases, which is described with the adiabatic lapse rate of these
two molecules.
The surface pressure psurf = pback + pH2 O and the mixing ratios of H2 O and CO2
are temperature-dependent and re-calculated in each iteration step.
Further parameters are important for the model and specified by the user, e.g., the
gravity acceleration, the surface albedo of the planet, and the stellar zenith angle.
To determine the water profile, a relative humidity profile must be chosen. One
possible input profile is derived by Earth’s observation (Manabe and Wetherald,
1967), another possible input profile is an iso-profile set to 100%, which is reasonable for a troposphere saturated with water, and two further profiles, which are a
combination of both dependent on the surface temperature or the water content of
the atmosphere. Steam tables for H2 O and CO2 are input to the model, which are
needed for the determination of the convective lapse rates in the troposphere and
the saturation vapor pressure.
The heat capacity Cp of the atmosphere is needed for the calculation of the radiative
equilibrium temperature (see equation 3.21) and the adiabatic lapse rate (e.g., see
equation 3.29). It is calculated as the sum of the heat capacities of the considered
species (N2 , H2 O, and CO2 ):
Cp (T, z) = cN2 (z)Cp,N2 (T ) + cCO2 (z)Cp,CO2 (T ) + cH2 O(z)Cp,H2 O (T )
where Cp,x are the heat capacities and cx the mixing ratios of species x.
(6.1)
Model description
78
For the different species different temperature dependencies are parameterized:
• N2 :
• CO2 :
Cp,N2 (T ) = 6.76 + 6.06 · 10−4 · T + 1.3 · 10−7 · T 2
(6.2)
Cp,CO2 (T ) = 7.7 + 5.3 · 10−3 · T − 8.3 · 10−7 · T 2
(6.3)
• H2 O:
−2
2
3
Cp,H2 O (T1000 ) = 30.092 + 6.832 · T1000 + 6.793 · T1000
− 2.534 · T1000
+ 0.082 · T1000
(6.4)
For H2 O the above equation represents the Shomate equation (Parks and Shomate, 1940), where T1000 is T /1000 and the coefficients are taken from Chase
(1998).
6.7
Radiative transfer
For the stratospheric region in the atmosphere, radiative transfer is the dominant
energy transport process, which determines the temperature profile (see equation
3.21 and section 3.5.1 for the basic physics of the radiative transfer).
In the model atmosphere the thermal fluxes from the planetary radiation and the
shortwave flux from the star are calculated separately. Two numerical schemes
are utilized to solve the monochromatic radiative transfer equation for the relevant
wavelength region (see equation 3.16) for the spectral intensity. With the angular
integration of the monochromatic intensity (equation 3.19) and the frequency integration of the resulting monochromatic flux (equation 3.20) the net radiative fluxes
are calculated. These net radiative fluxes of the shortwave FS,net and thermal infrared radiation FIR,net are summed in order to obtain the total radiative flux Ftot
in the atmosphere.
6.7.1
Thermal radiation
The numerical scheme to calculate the thermal infrared radiation is termed MRAC,
which is described by von Paris et al. (2008) and based on RRTM (Mlawer et al.,
1997). MRAC is improved in the context of this Thesis to be applicable to warm,
H2 O-dominated atmospheres, work which was done in close collaboration with Philip
von Paris.
The thermal infrared model to calculate the infrared fluxes FIR has a spectral range
from 1 to 500 μm, which is subdivided into 25 spectral intervals. In the model
only H2 O and CO2 are considered as radiative species. The spectral bands and the
species taken into account in a spectral interval (marked with an x) are summarized
6.7 Radiative transfer
79
in Table 6.1. The infrared radiative model was improved to consider the relevant
processes for water-dominated atmosphere.
Absorption by H2 O and CO2 is included in all wavelengths intervals, except for CO2
in spectral band 25, where no absorption lines are shown in the Hitemp database
(Rothman et al., 1995).
Spectral interval
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
Spectral interval [μm]
1.00 - 1.34
1.34 - 1.43
1.43 - 1.67
1.67 - 1.87
1.87 - 2.17
2.17 - 2.44
2.44 - 2.67
2.67 - 2.95
2.95 - 3.28
3.28 - 3.64
3.64 - 4.17
4.17 - 4.44
4.44 - 4.65
4.65 - 5.00
5.00 - 5.41
5.41 - 7.41
7.41 - 9.09
9.09 - 10.00
10.00 - 11.05
11.05 - 12.20
12.20 - 13.70
13.70 - 16.67
16.67 - 19.05
19.05 - 21.74
21.74 - 500.0
H2 O
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
CO2
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
-
Table 6.1: Spectral intervals for MRAC and species taken into account in these intervals for
the gaseous absorption. x for species considered in this interval, - for species not considered
in this interval
Thermal radiative transfer takes into account emission by the surface and the atmosphere and absorption of radiative species. The thermal radiation in this model
considers not only molecular gas absorption but also continuum absorption. The
upwelling thermal flux at the surface is determined as a boundary condition from
the Stefan-Boltzmann law assuming a blackbody temperature with a thermal surface emissivity of unity. As a second boundary condition for the radiative fluxes, the
downwelling thermal flux at the top of the atmosphere is assumed to be zero, which
Model description
80
implies that no atmospheric effects are taken into account above the model lid.
The radiative transfer equation for a particular layer and interval in the model is
the same as in RRTM:
R = R0 + (Bef f − R0 ) · (1 − T )
(6.5)
where R and R0 are the outgoing and incoming radiation, Bef f the effective Planck
function, and T the transmission (T = e−τ ) of the layer (Mlawer et al., 1997). The
effective Planck function Bef f is described by
Bef f = Blay + (Blev − Blay ) 1 − 2 ·
T
1
−
τ
1 − T
(6.6)
where Blev is the Planck function for the temperature in the layer, Blay the Planck
function between two adjacent layers, and τ the optical depth.
The spectral fluxes of the thermal radiation are obtained from the spectral intensity
via angular integration approximated by a first-order Gaussian quadrature. For the
hemispheric integration over μ = cos θ, where θ is the zenith angle in the radiative
transfer equation (3.19) the diffusivity approximation is applied, which only takes a
single quadrature point into account. The best value, which is approximately correct
for averages over spectral intervals and for all heights (Goody and Yung, 1989) is
1
, which corresponds to a zenith angle θ = 52.95◦ (Elsasser, 1942).
μ = 1.66
The frequency integration of the radiative transfer equation (see equation 3.20) in
the thermal wavelength region is performed using the correlated-k method (e.g.,
Mlawer et al. (1997), Goody and Yung (1989)). This method reorders the spectral
absorption coefficients to an ascending order, which is done by mapping the absorption cross section kν from spectral space to a probability space. The cumulative
probability function f (k) is the fraction of the absorption coefficients in the interval
smaller than the absorption coefficient k and is termed the k distribution (Mlawer
et al., 1997). Because this function is strictly monotonic the number of radiative
transfer operation can be reduced. Each of the spectral bands is subdivided in 16
Gaussian subintervals, where each subinterval has a characteristic absorption coefficient derived by the arithmetic mean. The extension of this approach from a
homogeneous to an inhomogeneous atmosphere is the correlated-k method (Mlawer
et al., 1997).
The absorption cross sections were calculated with the line-by-line radiative transfer
model SQuIRRL (Schreier and Böttger, 2003) using the Hitemp database for spectral
input data. With these absorption coefficients the k distributions are determined.
These k distributions were calculated for a fixed pressure and temperature grid and
depend on the relative concentrations of H2 O and CO2 , taken into account via the
binary species parameter. These calculated k distributions were included as lookup tables in the MRAC code. For a specific temperature, pressure, and relative
concentration of H2 O and CO2 conditions, the particular values required for the
radiative transfer in the model are derived from linear interpolation from the stored
6.7 Radiative transfer
81
k distributions. These interpolated absorption coefficients are then multiplied by
the appropriate column density to yield the optical depths τgaseous .
The calculations of these k distribution were a main part of this Thesis, because
the range of the pressure, temperature and relative concentration have to be wide
enough to be appropriate for the determination of the inner boundary of the HZ.
• The temperatures, for which the k distributions were calculated, vary from
100 to 700K, where 50K-steps from 100 to 400K and 100K-steps from 400 to
700K are taken into account.
• For the pressure grid pressures from 10−5 to 103 bars in g equidistant logarithmic steps are utilized.
• Furthermore, the binary species parameter η is considered for the calculation of
the optical depths (except band 25, where only H2 O is considered). The binary
species parameters η reflects the relative abundances of these two radiative
species
η = log
WH 2 O
WCO2
(6.7)
where WH2 O and WCO2 are the column densities of H2 O and CO2 , respectively.
The sum of these both column densities is the effective column density Wef f
Wef f = WH2 O + WCO2
(6.8)
The optical depths is then calculated with
τ = κef f · Wef f
(6.9)
where κef f is the effective cross section for a gas mixture containing H2 O
and CO2 in the specified amounts. These cross sections are calculated for
18 different values of the binary species parameter η. The range for binary
species parameter η varies from -9 to 6 with one point per order of magnitude
in relative concentration. The interpolation in η is performed linearly.
Continuum absorption
Additionally to the gaseous absorption of the thermal radiation also the continuum
absorption is included in MRAC. For H2 O and CO2 both self and foreign continuum
are taken into account. The contribution of these continua to the overall absorption
coefficient is added to the optical depth.
The description of the self and foreign continuum of H2 O is based on the approach of
Clough et al. (1989) (CKD continuum). The implementation of the CKD continuum
formulation is adopted from the line-by-line model SQuIRRL (Schreier and Böttger,
Model description
82
2003). The formulation of these continua takes into account the contributions of the
far wings of collisionally-broadened spectral lines, which are treated semi-empirically
to be comparable with experimental results for the continuum absorption and the
temperature-dependence. This continuum is applied for the the complete thermal
wavelength range of MRAC, which is needed for dense, H2 O-dominated atmospheres.
The foreign continuum of CO2 is also based on the approach of the CKD continuum.
The derived optical depths of these continuum absorption τself −cont,H2 O , τf oreign−cont,H2 O ,
and τf oreign−cont,CO2 are then added to the optical depths determined by the gaseous
absorption τgaseous .
In contrast the CO2 self continuum formulation relies on the approach of Kasting
et al. (1984b), which result from weak, pressure-induced transitions near 7μm and
beyond 20μm of the CO2 molecule. The optical depth τself −cont,CO2 is determined
to be
τself −cont,CO2 = Xi W · pE
T0
T
t i
(6.10)
where Xi is the pressure induced CO2 absorption, W is the column amount of CO2 ,
pE is the effective broadening pressure pE = (1 + 0.3 · cCO2 ) · p, T0 =300K and ti
incorporates the temperature dependence.
The optical depth of the CO2 self continuum is considered constant over a spectral
interval, thus a constant term is added to the total optical depth in each of the
16 Gaussian subinterval of each spectral band. Since the spectral intervals of the
approximation and the radiative transfer scheme do not coincide, there will be an
adjustment procedure, which calculates the mean value of the optical depth over the
spectral interval.
6.7.2
Shortwave radiation
The spectral range of the shortwave radiation code is subdivided into 38 spectral
layers from 0.2376 to 4.545 μm to calculate the shortwave fluxes FS . The numerical
scheme is based on Kasting et al. (1984b) and Kasting (1988) and was improved in
von Paris (2010)a and von Paris et al. (2010)b. Spectra of different kinds of stars
can be used as input. These spectra are normalized in terms of net radiative energy
input to the incoming solar insolation at the top of the atmosphere of the Earth.
In addition to the gaseous absorption of shortwave radiation, Rayleigh scattering of
shortwave radiation on atmospheric molecules is also taken into account. H2 O, CO2 ,
CH4 , O3 , and O2 are able to contribute in the model to the gaseous absorption. For
Rayleigh scattering CO2 , N2 , H2 O, and O2 are included in the radiative transfer
model for the shortwave region.
For the calculation performed in this Thesis to determine the inner boundary of
the HZ, only H2 O, CO2 and N2 are considered, because the others species (O2 , O3 ,
and CH4 ) are influenced by life. Therefore, in the following description only the
6.7 Radiative transfer
83
contributions of H2 O, CO2 and N2 to gaseous absorption and Rayleigh scattering
are described. The spectral intervals and the species taken into account for the
gaseous absorption of radiative transfer (marked by an x) of the shortwave radiation
are summarized in Table 6.2.
As a boundary condition the downwelling shortwave radiative flux at the top of the
atmosphere is determined by the stellar spectrum and the stellar insolation (in S0 )
divided by four to take into account the uniform distribution of the energy over the
planetary sphere. Both are input parameters. The upwelling shortwave radiative
flux at the surface is given by the product of the downwelling stellar flux at the
surface and the surface albedo.
The angular integration of the intensity (see equation 3.19) is performed with a
quadrature δ-2-stream approximation (Toon et al., 1989). The frequency integration of the radiative transfer equation (see equation 3.20) in each spectral band
is parameterized by a 4-term correlated-k exponential sum in each interval (e.g.,
Wiscombe and Evans (1977)). The absorption cross-sections σabs for the gaseous
absorption of the shortwave radiation are taken from Pavlov et al. (2000) using the
HITRAN 1992 database (Rothman et al., 1992). To obtain the net shortwave radiative flux in each layer the upward and downward stellar fluxes are added. To
take into account the diurnal variation, the shortwave radiative fluxes are multiplied
with the factor 0.5. Global, daily-mean results for the one-dimensional model are
represented with a cosine of the stellar zenith angle of 60◦ .
Rayleigh scattering
The cross section σray,i (λ) for the Rayleigh scattering of a specific species i (N2 ,
CO2 , and H2 O) for the complete spectral range are described by the approach of
Vardavas and Carver (1984)
−21
σray,i (λ) = 4.577 · 10
6 + 3 · Di
6 − 7 · Di
Pi2
λ4
(6.11)
where the λ is the wavelength in μm, the conversion factor 4.577·10−21 is taken from
Allen (1973), Di is the depolarization factor, and P = (10−5 ·Ai (1+10−3 ·Bi /λ2 ))2 for
CO2 and N2 , where Ai and Bi are material parameters for the the specific molecule
i. The values for Di , Ai , and Bi for N2 and CO2 are taken from Vardavas and
Carver (1984) and Allen (1973).
For H2 O in equation 6.11 the depolarization factor DH2 O is 0.17 (Marshall and
Smith, 1990), the factor PH2 O is described by P = r2 , where r is the refractivity.
The refractivity r of water is determined by r = 0.85 · rdry (Edlén, 1966) with the
refractivity of dry air rdry approximated by Bucholtz (1995):
rdry = 10
−8
5.7918 · 106
1.679 · 105
+
2.38 · 102 − λ−2 57.362 − λ−2
(6.12)
Model description
84
Spectral interval
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
26
27
28
29
30
31
32
33
34
35
36
37
38
Spectral interval [nm]
237.6 - 275.0
275.0 - 285.0
285.0 - 307.1
307.1 - 329.2
329.2 - 341.2
341.2 - 390.0
390.0 - 450.0
450.0 - 540.0
540.0 - 549.5
549.5 - 566.6
566.6 - 605.0
605.0 - 625.0
625.0 - 666.7
666.7 - 691.0
691.0 - 752.0
752.0 - 784.0
784.0 - 842.0
842.0 - 891.0
891.0 - 962.0
962.0 - 1036.0
1036.0 - 1070.0
1070.0 - 1130.0
1130.0 - 1203.0
1203.0 - 1307.0
1307.0 - 1431.0
1431.0 - 1565.0
1565.0- 1688.0
1688.0 - 1862.0
1862.0 - 2020.0
2020.0 - 2203.0
2203.0 - 2481.0
2481.0 - 2660.0
2660.0 - 2920.0
2920.0 - 3239.0
3239.0 - 3577.0
3577.0 - 4010.0
4010.0 - 4172.0
4172.0 - 4545.0
H2 O
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
CO2
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Table 6.2: Spectral intervals for the shortwave spectrum and atmospheric species taken into
account in the column model for the wavelength interval shown for gaseous absorption. x for
species considered in this interval, - for species not considered in this interval
6.8 Convection
6.8
85
Convection
Convection is the dominant energy transport mechanism in the lower atmosphere.
The temperature profile determined via convection is described in the model with
the adiabatic lapse rates.
For the stratosphere the convection is described with a standard dry adiabatic lapse
rate
Cp
d ln P
=
d ln T
R
(6.13)
where Cp is the heat capacity and R the universal gas constant.
For the troposphere wet adiabatic lapse rates are applied to calculate the temperature profile. The considered condensing species are H2 O and CO2 . In the convective
scheme of the model only one of these species is considered at once.
For the investigations of this Thesis the lapse rate of water is the most important of
the two, because for the investigated cases with high surface temperatures to determine the inner boundary of the HZ high water vapor concentrations are expected.
Carbon dioxide is present for these studies nearly exclusively in the gaseous form
due to the high temperatures considered.
6.8.1
Adiabatic lapse rate of water
The wet lapse rate assumed is pseudoadiabatic, which means that the condensed
phase leaves the system as soon as it appears (Ingersoll, 1969) and thus αc = ρc /ρ =
0, the relation between the mass density of H2 O in the condensed phase ρc to the
mass density of the atmosphere ρ. For the wet pseudoadiabatic lapse rate different
temperature regimes are distinguished and described in Kasting (1988):
1. T < 273 K: The lapse rate is
⎞
⎛
d ln Psat,H2 O ⎝
1
d ln P
⎠ d ln αv
=
−
MA
d ln T
d ln T
d ln T
1 + αv M
(6.14)
H2 O
where P is the atmospheric pressure, T the temperature, Psat,H2 O the saturation vapor pressure, αv = ρv /ρ the mass density of H2 O of the vapor phase ρv
relative to the mass density of the atmosphere ρ, MA the molecular weight of
the atmosphere and MH2 O the molecular weight of H2 O.
For the temperature regime considered, water vapor behaves like an ideal gas
and the Clausius-Clapeyron equation is valid for the first term on the right
hand side of equation 6.14:
d ln Psat,H2 O
MH2 O L
=
d ln T
RT
(6.15)
Model description
86
where L is the latent heat per mass of evaporation or sublimation and R the
universal gas constant.
The factor of the last term of equation 6.14 is:
d ln αv
=
d ln T
MH2 O L
−γ
MA T
·
1
L
T
+
(6.16)
R
αv MA
dL
where γ = Cpv + αv (Cpc − TL + dT
) with Cpv and Cpc the specific heats at
constant pressure for the non-condensable gas and the condensed phase. The
dL
is assumed to be zero (Kasting,
change of latent heat with temperature dT
1988), because no evaporation or condensation is taken into account for the
considered temperature range.
2. 273 K > T > 647 K:
The lapse rate for this temperature region follows the approach of Ingersoll
(1969):
Psat,H2 O
d ln P
=
d ln T
P
d ln Psat,H2 O
d ln ρv
βMH2 O
d ln αv
1+
+
+
(6.17)
d ln T
αv MA
d ln T
d ln T
where β = ρv RT /Psat,H2 O mv is the inverse compressibility, which describes
the non-ideality of the gas. The last term can be written as
d ln αv
=
d ln T
R d ln ρv
MA d ln T
dsc
v
− Cvn − αv dds
ln T − αc d ln T
αv (sv − sc ) + R/MA
(6.18)
where Cvm is the specific heat at constant volume for the non-condensable
species and sv and sc are the specific entropies of the vapor and the condensed
phase. In this equations, the terms for Pv , ρv , d ln Pv /d ln T , d ln ρv /d ln T ,
d ln sv /d ln T , and (sv − sc ) are interpolated from stored values in the water
steam tables available at 5 K intervals included in the model.
3. T > 647 K:
Above the critical temperature of water the dry adiabatic lapse rate is used
d ln Psat,H2 O
Cp
d ln P
= cH 2 O
+ (1 − cH2 O )
d ln T
d ln T
R
(6.19)
The wet lapse rate of water in the model is used, if CO2 does not condense, which
is the case for the investigated scenarios in this Thesis.
6.9 Calculation of the temperature profile
6.8.2
87
Adiabatic lapse rate of carbon dioxide
CO2 condensation takes place in cold, CO2 -dominated atmospheres. It is assumed to
occur for atmospheres which are supersaturated in CO2 . Supersaturation in respect
to CO2 occurs, if the supersaturation ratio Ssat is higher than unity:
PCO2 (z)
= Ssat ≥ 1
Psat,CO2 (z)
(6.20)
where pCO2 is the partial pressure and psat,CO2 the saturation vapor pressure of CO2 .
The value of Ssat in the model taken from Glandorf et al. (2002) Ssat = 1.34 is based
on measurements.
The description of the wet adiabatic lapse rate of CO2 follows the approach by
Kasting (1991) and Kasting et al. (1993)
⎞
⎛
d ln Psat,CO2 ⎝
d ln P
1
⎠ d ln αv
=
−
M
α
v
A
d ln T
d ln T
d ln T
1+ β M
(6.21)
CO2
where MCO2 is the molecular weight of CO2 and the terms d ln αv /d ln T and β are
interpolated from values stored in the steam tables available in the model (Kasting,
1991).
For the saturation vapor pressure of CO2 Psat,CO2 two temperature regimes are
distinguished below and above the triple point of CO2 by Ambrose (1956) :
• T < 216.6 K:
d ln Psat,CO2
= 2.303 · T
d ln T
867.2124
+ 18.65612 · 10−3
T2
−2 · 72.4882 · 10
−6
· T + 3 · 93 · 10
−9
T
(6.22)
2
• T ≤ 216.6 K:
d ln Psat,CO2
= 2.303 · T
d ln T
6.9
1284.07
+ 1.256 · 10−4
(T − 4.718)2
(6.23)
Calculation of the temperature profile
The important processes which determine the temperature structure have been introduced in the previous sections. This section explains how the temperature profile
is calculated in detail in each iteration step of the model.
88
Model description
1. As an input parameter the start temperature profile together with the input
profile of the water mixing ratios (US Standard Atmosphere (1976)) are used
in the first iteration step, shown in Figure 6.2. For the following iteration steps
the temperature profile from the previous timestep is taken into account for
calculations.
Figure 6.2: Input temperature profile to the radiative-convective model
2. With this temperature profile and the gas profiles of H2 O, CO2 and N2 , the
radiative fluxes are calculated by the infrared and shortwave radiative transfer
codes (as explained in section 6.7). With the down- and upwelling parts of
these radiative fluxes the total radiative flux for every layer in the atmosphere
↓
↑
− FIR
.
can be calculated via Ftot = FS↓ − FS↑ + FIR
Taking further the heat capacity Cp into account, which considers H2 O, CO2
and N2 , the temperature profile can be calculated from the fluxes of the radiative transfer for layer j =1 at the top of the atmosphere to layer j =(ND-1),
which is the layer above the surface via the central equation 3.21
Ftot,i (j + 1, Ti−1 (j)) − Ftot,i (j, Ti−1 (j))
Ti (j) − Ti−1 (j)
g
=−
(ti − ti−1 )
Cp (j, Ti−1(j) )
pi (j + 1) − pi (j)
(6.24)
where T is the temperature, t the timestep, g the gravity acceleration, Cp the
heat capacity, p the pressure, subscript j relates to the layer, which can vary
from 1 to ND-1, and subscript i indicates the iteration step, see Figure 6.3.
6.9 Calculation of the temperature profile
89
Figure 6.3: Calculation of the temperature profile taking radiative equilibrium into account.
3. Based on the approach of Pavlov et al. (2000) the temperature in the layer
immediately above the surface Ti (N D − 1) is then re-calculated depending on
the total flux Ftot,i (1, Ti−1 (1)) at the top of the atmosphere, which is presented
in Figure 6.4:
Ftot,i (1, Ti−1 (1))
Ti (N D − 1) − Ti−1 (N D − 1)
g
=−
(ti − ti−1 )
Cp (N D − 1, Ti−1(N D−1) ) pi (N D) − pi (N D − 1)
(6.25)
4. With the temperature from equation 6.25 the surface temperature is then
determined via convective adjustment, which is explained in section 6.8 and
shown in Figure 6.5 by applying the adiabatic lapse rate. The lowest layer of
the atmosphere is thus always determined via convection.
5. Starting with this temperature one layer above the surface j =(ND-1)(from
equation 6.25) the temperatures above are determined by continually comparing the temperature calculated via convective adjustment Tconv (z) with the
temperature Trad (z) calculated via radiative transfer with equation 3.21. The
condition for the calculation of the temperature in a layer j via convection is
Tconv (j) > Trad (j)
(6.26)
and then the layers belong to the troposphere. If this condition is not fulfilled
for a layer the radiative temperature Trad from equation 6.24 is taken for this
90
Model description
Figure 6.4: Calculation of the temperature one layer above the surface taking radiative
equilibrium into account.
and the above layers and this model region of the atmosphere is the stratosphere, which is depicted in Figure 6.6. The highest layer of the troposphere
is defined as the tropopause (JCONV).
An additional condition to this simple condition was introduced by von Paris
(2010). It takes into account that if radiative equilibrium is not reached in
a layer and also not calculated by convective adjustment, the temperature of
the layer is forced to be calculated with the adiabatic lapse rate and thus
dominated by convection.
Another condition, introduced in this Thesis, can also be applied additionally
to the simple condition. It is appropriate for Earth-like and warmer atmospheres. If the simple condition is not fulfilled, but the top of the convective
region (JCONV) in the investigated timestep i is smaller than in the timestep
before (i − 1), then convection is also forced to take place and the temperature
is determined with the adiabatic lapse rate. This additional condition was
implemented because for warm temperatures (above 340K) the temperature
profiles in the lower atmosphere determined by radiative transfer (see equation
3.21) and by convective adjustment become very similar. This approach is appropriate because for warm and wet atmospheres the atmosphere is optically
thick up to high levels
6.9 Calculation of the temperature profile
Figure 6.5: Calculation of the surface temperature from one layer above with convective
adjustment.
Figure 6.6: Calculation temperatures from one layer above the surface up to the tropopause
with convective adjustment.
91
Model description
92
6. At the end of each iteration step a smoothing of the temperature profile is
applied for the layers in the stratosphere:
T (j) = 0.5 · T (j) + 0.25 · [T (j − 1) + T (j + 1)]
6.10
(6.27)
Calculation of the water profile
The water profile in each iteration step is calculated with the temperature profile
before the temperature smoothing in the stratosphere. In the lower atmosphere the
mixing ratio of water vapor cH2 O is calculated based on the saturation mixing ratio
csat,H2 O , which is the ratio of the saturation vapor pressure of water psat,H2 O and
the total pressure p in the layer and the relative humidity RH:
cH2 O (z) = csat,H2 O (z) · RH(z) =
psat,H2 O (T )
· RH(z)
p(z)
(6.28)
To determine the water profiles in the upper atmospheric layers, three different
approaches can be used in this model.
1. The water mixing ratio cH2 O from equation 6.28 determines the water profile
in the entire model atmosphere.
2. Below the cold trap (JCOLD) like 1. Above the cold trap the mixing ratio of
water is set to the value at cold trap. The cold trap is located at the height
where csat,H2 O starts to increase. This approach is used for example in the
following studies e.g. Grenfell et al. (2007); von Paris et al. (2008); von Paris
et al. (2010).
3. Below the tropopause (JCONV) like 1. Above tropopause the mixing ratio of
water is set to the value of tropopause.
In the model different approaches can be used to describe the relative humidity.
• For atmospheres with a high water content a relative humidity of unity (100%)
can be applied. This over-estimates the amount of water for Earth-like conditions, but for water-rich atmospheres it is a more appropriate approach, since
these atmospheres must be saturated (hydrostatic equilibrium).
• To account for an Earth-like atmosphere the relative humidity profile of Manabe and Wetherald (1967) is appropriate, which is shown in Figure 6.7 and is
based on Earth observations and uses a surface relative humidity RHsurf of
77%:
RHmw (z) = RHsurf
p(z)
psurf
− 0.02
0.98
(6.29)
6.10 Calculation of the water profile
93
Figure 6.7: Relative humidity profile for the Earth of Manabe and Wetherald (1967).
• The relative humidity profile of Cess (1976) takes into account the increase of
the relative humidity with increasing surface temperature:
RH(z) = (RHmw )Ω
(6.30)
Ω = 1 − 0.03(Tsurf − 288K)
(6.31)
where
For temperatures close to the Earth’s mean surface temperature (288K) the
profile of Manabe and Wetherald (1967) is considered because Ω is about unity
and for high surface temperature and water concentration Ω is about 0 and the
relative humidity profiles becomes an isoprofile with the value of the relative
humidity profile at the surface RHsurf . For surface temperature Tsurf ≥ 321K
the relative humidity is set to 100%.
• The relative humidity profile of Kasting and Ackerman (1986) is a modification
of the relative humidity profile of Cess (1976) and takes instead of the surface
temperature the saturation mixing ratio csat,H2 O into account:
Model description
94
RH(z) = (RHmw )Ω
(6.32)
where
Ω=1−
csat,H2 O − cp
0.1 − cp
(6.33)
where csat,H2 O is the saturation mixing ratio of water at the surface and cp is
the value for csat,H2 O for the present Earth atmosphere (cp =0.0166). Thus, for
low water concentrations like on Earth the profile of Manabe and Wetherald
(1967) is considered because Ω is about unity and for high surface temperature and water concentration Ω is about 0 and the relative humidity profiles
becomes an isoprofile with the value of the relative humidity profile at the
surface RHsurf . For surface temperature Tsurf ≥ 321K the relative humidity
is set to 100%.
For the calculation of the saturation vapor pressure of water psat,H2 O three different
temperature regimes are distinguished:
• T < 273K:
Here, sublimation is taken into account to describe the saturation vapor pressure
psat,H2 O = p0 · e
−
MH O L
MH O L
2
2
− RT
RT
0
(6.34)
with T0 =273.15 K, and p0 = 0.0061 bar
• 273 < T < 647 K:
The vapor pressure of water psat,H2 O is interpolated from the steam tables of
water vapor included in the model.
• T > 647K:
The value of the water vapor pressure psat,H2 O is set to an arbitrarily high
value (about 1030 bar).
CHAPTER
7
Test and validations of the model
In this chapter tests and validations of the one-dimensional radiative-convective
model introduced in the previous chapter are presented. Temperature profiles determined with different infrared radiative transfer codes are analyzed. The influence of different conditions to assess whether an atmospheric layer is dominated by
convection or radiative transfer is tested. Furthermore, the different approaches to
calculate the water mixing ratio in the upper atmosphere of the model are compared.
Additional, detailed tests and validations were performed in close collaboration with
Philip von Paris. The results of these test and validation, which were described in
von Paris (2010) are shortly summarized at the end of this section.
7.1
Influence of infrared radiative transfer codes
The temperature profiles calculated with the one-dimensional radiative-convective
model using MRAC for the infrared radiative transfer, which was introduced in subsection 6.7.1, is compared to temperature profiles using an other infrared radiative
transfer code, RRTM Mlawer et al. (1997), which is also implemented in the model.
The radiative species taken into account by MRAC are CO2 and H2 O. The other
thermal radiative transfer code RRTM Mlawer et al. (1997) can take more atmospheric species into account, which are relevant for Earth conditions, H2 O, CO2 , O3 ,
CH4 , and N2 O. Therefore, model runs using RRTM including all radiative species
relevant for the determination of an Earth-like temperature profile are performed as
well as runs with N2 , CO2 , and H2 O only for comparison with MRAC.
Figure 7.1 shows the temperature profiles over altitude and pressure using the different infrared radiative transfer codes for a solar insolation of the Earth (S = 1S0 ),
96
Test and validations of the model
Figure 7.1: Temperature profiles over altitude (left) and pressure (right) for different infrared codes with Asurf =0.21 and a relative humidity profile of Manabe and Wetherald
(1967). The the boundaries of the temperature validation range of RRTM are shown in
grey dashed lines.
a surface albedo Asurf = 0.21 and the relative humidity profiles of Manabe and
Wetherald (1967).
The surface and tropospheric temperatures determined via convective adjustment
are in good agreement for the investigated cases, whereas the temperatures in the
stratosphere, determined by radiative equilibrium deviate. MRAC and reduced
RRTM include only CO2 , H2 O and RRTM include CO2 , H2 O, O3 , CH4 , and N2 O
as radiative species. The inclusion of additional atmospheric species (O3 , CH4 , and
N2 O) in the atmospheric model leads to a higher altitude of the model atmosphere
and an increase of the stratospheric temperature due to the absorption of shortwave
radiation by O3 . This stratospheric temperature increase does not occur for the
temperature profiles including only CO2 , H2 O and N2 as atmospheric species.
The temperature profile using the reduced RRTM taking only CO2 , H2 O and N2
into account, is comparable to the temperature profile, where MRAC is applied for
the radiative transfer. Only some temperature deviation occur in the stratosphere.
Note, that RRTM is only valid for temperature range ±30 K to the temperature
profiles of the US 1976 standard atmosphere. The temperature boundaries of the
validated region of RRTM are shown in 7.1 in grey dashed lines. The temperature
profiles determined with the reduced RRTM are outside the validation range of
RRTM resulting, which results in negative optical depths, which is not physical.
This was the reason to develop the infrared radiative transfer code MRAC, which
is capable to calculate temperature profiles including only N2 , CO2 , and H2 which
are the basic molecules in the atmospheres of the terrestrial planets of the Solar
System. The temperatures T , pressures p and relative concentration r of H2 O and
CO2 for which MRAC is valid are much wider than for RRTM (100 < T < 700 K,
10−5 < p < 102 bar, 10−9 < r < 106 ).
7.2 Conditions for convection
7.2
Conditions for convection
Figure 7.2: Temperature (left) and water profiles (right) for different convection condition
with Asurf =0.21 and a relative humidity profile of Manabe and Wetherald (1967).
Different conditions to decide if an atmospheric layer is dominated by convection or
radiative transfer can be used in the model.
The simple condition used for example in Grenfell et al. (2007) compares the temperature derived from the radiation fluxes Trad and the temperature determined with
the adiabatic lapse rate Tconv (see section 6.9). If Tconv > Trad , then the temperature in this layer is dominated by convection and Tconv is used. This temperature
comparison is performed for the layers starting from one layer above the surface up
to the layer, where the following condition if fulfilled: Tconv < Trad . If this condition
is fulfilled, then the temperature in this layer is dominated by radiation and Trad is
used. The uppermost layer of the convective region is termed the tropopause and
all the convective layers below are the troposphere. The radiative dominated layers
above are the stratosphere of the model atmosphere.
An additional condition to this simple condition was introduced by von Paris (2010).
It takes into account that if radiative equilibrium is not reached in a layer and also
not calculated by convective adjustment, the temperature of the layer is forced to
be calculated with the adiabatic lapse rate and thus dominated by convection.
Another condition, introduced in this Thesis, can also be applied additionally to the
simple condition. It is appropriate for Earth-like and warmer atmospheres. If the
simple condition is not fulfilled, but the top of the convective region (JCOLD) in the
investigated timestep i is smaller than in the timestep before (i − 1), then convection
is also forced to take place and the temperature is determined with the adiabatic
lapse rate. This additional condition was implemented because for warm temperatures (above 340K) the temperature profiles in the lower atmosphere determined
by radiative transfer (see Equation 3.20) and by convective adjustment become very
similar. This approach is appropriate because for warm and wet atmospheres the
97
Test and validations of the model
98
atmosphere is optically thick up to high levels (see Figure 8.10).
Figure 7.2 shows the temperature (left panel) and water profiles (right panel) for a
surface albedo Asurf = 0.21 and with the relative humidity profile of Manabe and
Wetherald (1967) using the different conditions for convection for a solar insolation
of the Earth (S = 1S0 ). The different convection condition have no influence on the
temperature and water profiles.
Figure 7.3: Temperature (left) and water profiles (right) for different convection criterion
with Asurf =0.21 and RH=100%.
The same behavior can be seen in Figure 7.3, where all the parameters are the same
but the relative humidity profile was changed and assumed to be 100%. This results
in higher temperatures and water mixing ratios in the entire model atmosphere, but
this has also no influence on the agreement of the temperature and water profiles
using the different convection conditions.
7.3
Calculation of water mixing ratios
Different approaches were used for the calculation of the water mixing ratio cH2 O in
the upper atmosphere. These approaches were introduced in section 6.10.
1. cH2 O (z) = csat,H2 O (z) · RH(z) =
atmosphere.
psat,H2 O (T )
p(z)
· RH(z) for the complete model
2. Below the cold trap (JCOLD) like 1. Above the cold trap the mixing ratio of
water is set to the value at cold trap. The cold trap is located at the height
where csat,H2 O starts to increase.
3. Below the tropopause (JCONV) like 1. Above tropopause the mixing ratio of
water is set to the value of tropopause.
7.3 Calculation of water mixing ratios
Figure 7.4: Temperature (left) and water profiles (right) for different water calculation approaches with Asurf =0.21 and a relative humidity profile of Manabe and Wetherald (1967).
The grey dash dotted line is the pressure (p(z)) and grey dash long dashed line is the saturation pressure of water (psat,H2 O (z)).
Figure 7.4 shows the temperature (left panel) and water profiles (right panel) for
the three water profiles for a solar insolation of the Earth (S = 1S0 ), a surface
albedo Asurf = 0.21 and the relative humidity profile of Manabe and Wetherald
(1967). Additionally, for better understanding also the pressure profile p (grey dotted
line) and the saturation pressure profiles psat (grey dashed line) are shown, since
p
(T )
2O
· RH(z).
cH2 O (z) = sat,H
p(z)
The water profiles determined by approach 1 and approach 2 agree very well in the
lower atmosphere and deviate only above the cold trap (JCOLD). Although there
is the difference above the cold trap in the water mixing ratios, the temperature
profiles of the two approaches are comparable (difference of 2 K for Tsurf ).
For the troposphere the temperatures and water mixing ratios for all three water
calculation approaches are comparable. Above the tropopause (JCONV) the mixing
ratio of water is higher. This results compared to the water calculation by approach
1 due to cooling by H2 O in lower temperatures in the stratosphere (about 6-10 K
lower).
Figure 7.5 shows again the temperature and water profiles for the same surface
albedo, but with a relative humidity profiles of 100%. The atmospheric temperatures
and water profiles are higher than using the relative humidity profiles of Manabe
and Wetherald (1967). The overall behavior of the water profiles is the same for the
different water calculation approaches but the tropopause (JCONV) and the cold
trap (JCOLD) are located at higher altitudes.
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Test and validations of the model
100
Figure 7.5: Temperature (left) and water profiles (right) for different water calculation
approaches with Asurf =0.21 and a relative humidity profile unity.
7.4
Summary of further tests and validations
Additional tests and validations of the one-dimensional radiative-convective model
(see chapter 6) were performed in close collaboration with Philip von Paris. A short
summary is given of the results of these tests and validations (von Paris, 2010). The
infrared radiation transfer scheme MRAC was validated and the influence of the
numerical scheme of the model on the calculated atmospheric profile was tested.
7.4.1
Validation of MRAC
MRAC, the infrared radiative scheme, improved in this Thesis, was tested against
line-by-line (lbl) calculation with SQuIRRL (Schreier and Böttger, 2003). In the troposphere (5km) the net fluxes are in agreement within 0.2%. However, the deviation
of four near-infrared bands, which do not contribute to the overall energy budget, is
higher than 10%. At 50 km the net fluxes are in agreement within 3.5%. For eight
near-infrared bands the deviation is higher than 10% up to 76% in the 4.3μm band,
which results from the differences in the compared temperature profiles (MRAC:
H2 O and CO2 , SQuIRRL: H2 O, CO2 , CH2 , O3 and N2 O). The infrared flux emitted
in this band is small, hence the also the influence of the temperature profile is small.
The H2 O continuum included in MRAC is in good agreement with lbl calculation
with SQuIRRL, which also uses the CKD continuum formulation (Clough et al.,
1989). Only in the far-infrared of the foreign continuum contribution of the lbl
calculations is higher than with MRAC. The comparison of the self- and foreign
continuum of water with the self-continuum formulation of water applied in Kasting
et al. (1993) agrees well in the window region (8-12 μm), but for warmer atmospheres
the difference in these two formulations increases. Which shows that the contribution
of the self-and foreign continuum for the complete thermal wavelength range becomes
more important for warm and water rich atmospheres and should not be neglected.
7.4 Summary of further tests and validations
7.4.2
Test of the numerical scheme
It was shown that pressure variations at the top of the atmosphere, a boundary condition of the model, has no significant influence on the lower atmospheric conditions
and surface temperature.
The choice of the initial temperature profile is presented to have nearly no influence
on the resulting temperature profiles of the atmosphere.
The choice of the maximum time step is crucial. For a too large maximum time step
temperature fluctuations between consecutive iterations become too large and thus
the model does not converge. For a too small maximum time step the smoothing
dominates the temperature profile and temperature changes produced by radiative
transfer and convection can not be seen.
The change in vertical grid spacing and the number of the vertical grid levels have
only a small influence on the surface temperature.
Different data bases (Hitemp, Hitran 1986, 1992, 1996, 2004) for the calculation
of the absorption coefficients used for the k-distribution in MRAC produce surface
temperatures which vary by about 1 K.
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102
Test and validations of the model
CHAPTER
8
Results
The aims of this thesis are the determination of the inner boundary of the HZ
and the investigation of the runaway greenhouse effect. To determine the inner
boundary of the HZ different criteria are considered (summarized in chapter 4). The
one-dimensional radiative-convective model described in chapter 6 is applied with
increased stellar insolations to determine these inner HZ boundaries. The influence
of increased stellar insolation on temperature and water profiles is calculated selfconsistently until temperature profiles are converged and radiative equilibrium is
reached in the stratosphere. The fulfillment of one of the criteria for the nonexistence of liquid water on the planetary surface for a specific solar insolation,
determines the inner boundary of the HZ for this criterion.
After reviewing the parameters taken for the model scenarios, the results for the
inner boundary of the HZ are presented. Two criteria for determining the inner
boundary of the HZ are applied. Firstly, the surface temperature Tsurf reaches
the temperature of the critical point of water (Tcrit = 647 K). It is discussed as
to whether the way in which the inner boundary is determined by reaching the
temperature of the critical point of water corresponds to a runaway greenhouse
effect (introduced in section 4.2). Secondly, the water loss limit is applied, which
takes into account processes that lead to the escape of the complete water reservoir
of the planet within the lifetime of the planet. The basic principles that lead to
the water loss limit are presented in section 4.3. An overview of the influence of
changing parameters like e.g. the surface albedo, the relative humidity, and the
water reservoir on the results of the model is given. The model is applied to a planet
like the detected extrasolar planet Kepler 22b, to investigate if such a planet is in
or outside the HZ for a particular assumed atmospheric composition. Furthermore,
the influence of different host stars on the inner boundary of the HZ is investigated
Results
104
and the results are compared to HZ scalings.
8.1
Input and boundary parameters
A modern solar spectrum is used as input parameter. The solar insolation S =
1.00S0 is given by the solar constant S0 of 1366 Wm−2 , which is the amount of
solar radiation per unit area at 1 AU. The input stellar spectrum determines the
downwelling shortwave flux FS↓ in the model atmosphere. To determine the inner
boundaries of the HZ the solar insolation is increased.
The assumed planet is characterized by the mass and radius of the Earth, described
by the gravity acceleration of g =9.8 ms−2 . The background atmospheric pressure
pback is also comparable to that of the Earth and assumed to be 1 bar at the surface.
At the upper boundary of the model the pressure is set to 6.6·10−5 bar, which
corresponds to atmospheric pressures in the Earth’s mid-mesosphere.
The atmospheric constituents are assumed to be comparable to the constituents of
the atmospheres of the terrestrial planets in the Solar System, whereas atmospheric
components produced directly or indirectly by life are neglected. Therefore, as
atmospheric constituents N2 and CO2 are considered, which are constant background
gases and H2 O, which responds to changes in atmospheric temperatures. For CO2
the modern-day atmospheric value of 355 parts per million (ppm) is used, the input
mixing ratio of water vapor at the surface is 1.4 · 10−2 and decreases to values of 3.5 ·
10−7 at the upper model lid (US Standard atmosphere 1976), and the rest of the 1 bar
background atmosphere is filled with N2 . Absorbing gases for the radiative transfer
in the infrared and shortwave length range are H2 O and CO2 . Continuum absorption
of H2 O and CO2 is taken into account for the thermal radiative transfer. H2 O, CO2 ,
and N2 are included for the Rayleigh scattering of the shortwave radiation.
A reservoir of water on the surface of the planet is assumed to have the same
size as the oceans of the Earth (1.4·1021 l). The relative humidity in the entire
model atmosphere is set to 100%. In such cases, a completely saturated atmosphere
results. This overestimates the amount of water in the atmosphere for Earth-like
moderate distances of the planet to the star, but it is a reasonable assumption for
an atmosphere close to the inner edge of the HZ, where a lot of water vapor is
expected to be in the atmosphere. Different surface albedos Asurf are applied as
input parameters: Asurf =0.13 (Kitzmann et al. (2010) and Rossow and Schiffer
(1999)) is the mean surface albedo for the Earth; higher albedos are also taken into
account to mimic the effect of clouds, which are not included in the model used:
Asurf =0.21 is the surface albedo required to reach 288K with a relative humidity
profile of the Earth (Manabe and Wetherald, 1967); Asurf =0.29 is the surface albedo
required to reach 288K with a relative humidity profile of 100%; and Asurf =0.22 is
chosen for comparisons because it is used in the studies of Kasting et al. (1993).
The parameters and initial conditions, which were used for the runs that are shown
in the next section, are summarized in Table 8.1. Where other parameters used for
8.2 Inner boundary of the HZ determined by the critical point of water
particular runs this is stated explicitly.
Property
Star
Solar constant (S0 ) [Wm− 2]
Gravity acceleration [ms−2 ]
Background surface pressure (pback ) [bar]
Atmospheric pressure at TOA (pT OA ) [bar]
Atmospheric composition
Absorbing gases
Water ocean [l]
Relative humidity (RH)
Surface albedo (Asurf )
Value
Sun
1366
9.8
1
6.6·10−5
N2 , CO2 and H2 O
H2 O and CO2
1.4·1021
100%, Manabe and Wetherald (1967)
0.13, 0.21, 0.22, 0.29
Table 8.1: Model parameters used for the determination of the inner boundary of the HZ
8.2
Inner boundary of the HZ determined by the critical
point of water
To determine the inner boundary of the HZ with the criterion that the surface
temperature reaches the temperature of the critical point water, model runs are
performed with increased solar insolation S. For different solar insolations S, temperature, water and radiative flux profiles are calculated self-consistently until temperature profiles are converged and radiative equilibrium is reached in the stratosphere. In this context the runaway greenhouse effect is discussed and the effect of
Rayleigh scattering of H2 O on it.
The results in this section are presented for a surface albedo of Asurf =0.22, a
relative humidity RH of 100% and for water profiles determined by the first approach
(cH2 O = csat,H2 O · RH for the entire atmosphere, see section 6.10). The influence of
different surface albedos Asurf and different relative humidity profiles are discussed
in section 8.4 and 8.5.
Figure 8.1 shows a selection of temperature (left panel) and water profiles (right
panel) over altitude for increased solar insolations S. The corresponding surface
temperatures Tsurf are summarized in Table 8.2. The model is designed to calculate
temperatures up to the temperature of the critical point of water (647K). The highest
value of solar insolation for the model runs presented in this section before reaching
the the critical point of water is S = 1.41S0 .
Note that the temperature profile for the solar insolation S = 1.00S0 does not
correspond to the Earth’s temperature profile, since the relative humidity RH is
assumed to be 100%, which over-estimates the H2 O mixing ratio and produces higher
surface temperatures, and important radiative species of the Earth’s atmosphere are
105
Results
106
Figure 8.1: Temperature (left) and water profiles (right) over altitude for model runs with
increased solar insolations S of 1.00, 1.10, 1.20, 1.30, and 1.41 S0 with Asurf =0.22 and
RH=100%.
S [S0 ]
Tsurf [K]
1.00
305.16
1.10
364.98
1.20
436.14
1.30
490.01
1.41
626.66
Table 8.2: Surface temperatures Tsurf for increased solar insolations S
neglected, like e.g. O3 , which leads to lower stratospheric temperatures.
Figure 8.1 and Table 8.2 show that the surface temperatures increase with increasing solar insolation. With increasing solar insolation S the altitude of the
modeled atmosphere increases, which is caused by the increasing surface pressure
(psurf = pback + pH2 O ) due to enhanced water vapor in the atmosphere (see Figure
8.14). The higher water content is also the reason for the convective zone to increase
until convection is the main energy transport mechanism in the model atmosphere,
except for the uppermost layer which is in radiative equilibrium. This can be seen
in the temperature profiles for solar insolation S ≥ 1.10S0 in Figure 8.1, which are
determined by the wet adiabatic lapse rate for nearly the entire model atmosphere.
This is appropriate because the atmosphere is optically thick even at high altitudes.
This results from the increased water vapor content in the atmosphere. Furthermore, the optical thickness in the upper layers increases caused by the fact that the
height of a model layer is larger for increased solar insolation due to the increased
atmosphere height of the model. Additionally, the lapse rate from S = 1.00S0 to
S = 1.10S0 gets smaller due to the influence of latent heat with increased solar
insolation.
The water profiles become an isoprofile for increased solar insolations, which results
from the fact that if water vapor is the major atmospheric constituent, the atmospheric pressure p is determined by the water in the atmosphere, hence the saturation
vapor pressure psat (T ). Thus, the mixing ratio of water vapor (cH2 O (T ) = psat (T )/p)
8.2 Inner boundary of the HZ determined by the critical point of water
approaches an isoprofile.
Figure 8.2: Total radiative flux at the top of the atmosphere for increased solar insolations
S with Asurf =0.22 and RH=100%.
Figure 8.2 shows the total radiative flux Ftot at the top of the atmosphere for in↓
↑
− FIR
has to
creased solar insolation The total radiative flux Ftot = FS↓ − FS↑ + FIR
be zero to reach radiative equilibrium. Radiative equilibrium is reached at the top
of the model atmosphere for all presented solar insolations. For the S = 1.00S0 case
radiative equilibrium is reached for layers above around 18km. The layers below
are determined by the moist adiabatic lapse rate. For solar insolations S ≥ 1.09S0
radiative equilibrium is only reached in the uppermost layer of the model atmosphere. All the layers below are dominated by convection and calculated via the
moist adiabatic lapse rate.
Figure 8.3 shows the profiles of the up- and downwelling infrared fluxes for model
runs with increased solar insolations S. The upwelling infrared radiative fluxes
↑
FIR
(dotted lines) at the surface are enhanced for increased solar insolations. This
results from the increased surface temperature Tsurf (see the left panel in Figure
8.1), which determines the upwelling infrared flux at the surface via the Stefan↓
Boltzmann law. The downwelling infrared radiative flux FIR
at the top of the model
atmosphere is determined by the boundary condition, which sets the value to zero.
The greenhouse effect increases with increasing solar insolations S. This is indicated
by a downwelling infrared flux (solid line) of the same value as the upwelling infrared
flux (dotted line). For the solar insolation S = 1.00S0 the greenhouse effect is
significant for altitudes up to about 5 km, whereas for the solar insolation of S =
1.41S0 it is significant up to altitudes of about 180 km. The reason for this can be
107
108
Results
Figure 8.3: Infrared fluxes FIR increased solar insolations S of 1.00, 1.10, 1.20, 1.30, and
1.41 S0 with Asurf =0.22 and RH=100%. Downwelling fluxes in solid lines and upwelling
fluxes in dotted lines.
seen in the water profiles (see the right panel in Figure 8.1), where the water mixing
ratio is high (isoprofile of almost 1) for all atmospheric layers and this increases the
greenhouse effect significantly.
Figure 8.4: Shortwave fluxes FS over altitude for increased solar insolations S of 1.00,
1.10, 1.20, 1.30, and 1.41 S0 with Asurf =0.22 and RH=100%. Downwelling fluxes in solid
lines and upwelling fluxes in dotted lines.
As a result of the increased solar insolation the downwelling shortwave radiative
flux FS↓ (solid line) in Figure 8.4 at the top of the model atmosphere increases. The
upwelling shortwave radiative flux FS↑ (dotted line) increases at the upper atmo-
8.2 Inner boundary of the HZ determined by the critical point of water
sphere for S ≥ 1.10S0 due to the influence of Rayleigh scattering of water, which is
discussed in detail in section 8.2.1.
Figure 8.5: Surface temperatures for increased solar insolation S with Asurf =0.22 and
RH=100%.
The development of the surface temperature with increased solar insolation is shown
in Figures 8.5. Each surface temperature point results from a converged run, where
the temperature profile is converged and radiative equilibrium is reached in the
stratosphere, i.e., the steady-state solution. The surface temperature increases with
increased solar insolation and reaches the temperature of the critical point of water
for a solar constant S = 1.42S0 . The corresponding model scenario is not shown in
Figure 8.5 because the run breaks up due to the fact that the model is not designed
to calculate temperature above the temperature of critical point of water.
As discussed in section 4.1 reaching the temperature of the critical point of water
is an absolute strict upper limit for liquid water on the surface of a terrestrial
planet. Thus, the last run, where the surface temperature is below the critical point
corresponds to a solar insolation S = 1.41S0 , is the last habitable run performed.
This value of the solar insolation S corresponds to a distance d from the star of 0.84
AU, using d = 1AU/S 1/2 . In Figure 8.6 the surface temperatures are plotted versus
the distance from the star d. The inner boundary of the HZ, determined by reaching
the temperature of the critical point of water at the surface, is located at 0.84 AU.
This result is independent from the water mixing ratio calculation approach (see
section 6.10) and from the surface albedo Asurf , which is discussed in section 8.4.1.
This self-consistently determined inner boundary of the HZ (d = 0.84 AU) is in
agreement with the inner boundary of the HZ of Kasting et al. (1993) determined
using the same criterion the critical point of water. Kasting et al. (1993) termed this
109
Results
110
Figure 8.6: Surface temperatures versus the distance d of the planet to the star with
Asurf =0.22 and RH=100%.
limit of the HZ the runaway greenhouse limit. Kasting et al. (1993) did not calculate
the inner boundary self-consistently for increased solar insolation but determined the
inner boundary of the HZ by increasing the surface temperature (see section 5.1.4).
8.2.1
Discussion of the runaway greenhouse effect
For many studies reviewed in section 5.1, for an increased solar insolation or increased surface temperatures a runaway greenhouse effect occurs. Kasting et al.
(1993) also termed their inner boundary of the HZ, which is located where the temperature of the critical point of water (647K) is reached, the runaway greenhouse
limit.
The following section evaluates if a runaway greenhouse based on the radiation
limit is approached for the investigated model scenarios. Since the troposphere and
the stratosphere are taken into account, the reason for a runaway greenhouse to
approach is assumed to be the tropospheric radiation limit (see subsection 4.2.2).
The conditions for the tropospheric radiation limit are briefly summarized here:
↑
(TOA) is constant.
1. The outgoing infrared flux FIR
2. The atmosphere is optically thick.
3. The moist adiabatic lapse rate approaches the water vapor saturation pressure
curve.
8.2 Inner boundary of the HZ determined by the critical point of water
This section investigates if these conditions are fulfilled for the investigated model
scenarios. The results are shown for surface albedos of Asurf =0.22 and relative
humidity profiles of 100%, but the main conclusions hold also for the model runs
with the other surface albedos.
↑
Constant outgoing infrared flux FIR
(TOA)
It is investigated in this section if the outgoing infrared flux is constant for increased
solar insolation .
Figure 8.7: Net infrared fluxes Fnet,IR (TOA) for increased solar insolation S with
Asurf =0.22 and RH=100%.
Figure 8.7 shows the net infrared flux at the top of the atmosphere for increased
solar insolations S. The net infrared flux at the top of the atmosphere Fnet,IR (TOA)
↑
is equivalent to the upwelling infrared flux FIR
(TOA), because the downwelling
↓
infrared flux at the top of the atmosphere FIR (TOA) is set to zero in the model as
a boundary condition. The net infrared flux Fnet,IR (TOA) increases with increasing
↑
(TOA) due to the
solar insolation S ≤ 1.10S0 , which results from the increase in FIR
Stefan-Boltzmann law for increasing atmospheric temperatures. If the atmosphere
becomes dominated by water vapor, the outgoing infrared flux becomes constant for
increased solar insolation S until the surface temperature Tsurf reaches temperature
of the critical point of water. The outgoing infrared flux is again shown in Figure
8.8, this time versus surface temperature. This plot shows as expected the same
behavior for the outgoing infrared flux as in Figure 8.7 and is displayed for better
comparison with the runaway greenhouse studies such as e.g. Nakajima et al. (1992),
111
112
Results
which increases the surface temperature Tsurf instead of the solar insolation S and
did not include the shortwave radiation in their calculations.
The behavior of the outgoing infrared flux for increased solar insolation S or surface
temperature Tsurf confirms the first condition for the tropospheric radiation limit
i.e. the outgoing infrared flux is constant.
Figure 8.8: Net infrared fluxes Fnet,IR (TOA) for increased surface temperature with
Asurf =0.22 and RH=100%.
Furthermore Figure 8.9 shows the upwelling infrared fluxes at the top of the atmo↑
(TOA) for the different spectral bands for increased solar insolations S,
sphere FIR
which correspond to the net infrared flux at the top of the atmosphere Fnet,IR (TOA).
For all spectral bands the Fnet,IR (TOA) approaches a constant flux for increased solar insolation S.
With increasing solar insolation S, Fnet,IR (TOA) increases for every spectral band
via Planck’s law due to increasing atmospheric temperatures Tsurf , except the spectral bands in the wavelength region between 8-12 μm. For this region the gaseous
absorption of infrared radiation is small, the so-called window region. But with
the increase in water vapor the continuum absorption of water vapor in this region
↑
is absorbed. Thus, for the specis increased and the upwelling infrared flux FIR
tral bands in the window region the net infrared flux at the top of the atmosphere
Fnet,IR (TOA) decreases with increasing solar insolation S. However, also for the
window region a constant infrared flux is approached for increased solar insolation
S. The first condition for the tropospheric radiation limit is also fulfilled in every
spectral band.
8.2 Inner boundary of the HZ determined by the critical point of water
Figure 8.9: Infrared fluxes Fir (T OA) for every spectral band over wavelength for increased
solar insolations S with Asurf =0.22 and RH=100%.
Atmospheric optical thickness
The behavior of the optical thickness for increased solar insolation is investigated.
Figure 8.10: Mean optical depth τ versus wavelength for the surface (left panel) and one
layer below the top of the atmosphere (right panel) for solar insolations S of 1.00, 1.10,
1.20, 1.30, and 1.41 S0 with Asurf =0.22 and RH=100%.
Figure 8.10 shows the mean optical depths (over all subintervals) for each spectral
interval of the infrared radiative transfer code for different solar insolations S at
the surface (left panel) and for one layer below the top of the atmosphere (right
113
Results
114
panel). The optical thickness in every spectral band at the surface increases with
increasing solar insolation S due to the increase in water vapor (see right panel in
Figure 8.1). At the surface for a solar insolation S = 1.00S0 (see black line in the
left panel of Figure 8.10) the mean optical depths are higher than unity (marked
with the black dotted line) in all band, except for band 11 (3.64-4.17 μm), and
thus optically thick for infrared radiation. This is in contrast to normal Earth’s
conditions (Tsurf = 288K), where in the window region (8-12 μm) the atmosphere
is optically thin for infrared radiation at the surface, which leads to cooling. The
cause for this difference is the assumed relative humidity profile of 100%, which
overestimates the amount of water in these atmospheres and leads to higher surface
temperatures (Tsurf = 305.16 K).
For one layer below the top of the atmosphere the optical depths are lower than
at the surface as expected. For a solar insolation of S = 1.00S0 the atmosphere
is completely optically thin for infrared radiation. With increasing solar insolation
the optical depth in each spectral band increases. However, the rate of increase of
the optical depths decreases with increasing solar insolation. The largest increase
of optical thickness occurs between solar insolations S = 1.00S0 and S = 1.10S0 .
This results from the huge increase in water vapor in the atmosphere between these
two values (see right panel of Figure 8.1). For the solar insolation of S = 1.41S0 ,
in some spectral bands (2, 4, 5, 7, 8, 15, 16, 17, 24, 25) the top of the atmosphere
is optically thick (τ ≥1) for infrared radiation. A few of these spectral bands (7, 8,
16, 25) are already optically thick at the top of the atmosphere for a solar insolation
S = 1.10S0 .
Figures 8.11 to 8.13 show the vertical mean optical depth profiles for three spectral
bands for increased solar insolation S. These bands have different characteristics at
the surface for a solar insolation of S = 1.00S0 , which are summarized in Table 8.3.
Spectral band
11
16
18
Wavelength region [μm]
3.64-4.17
5.41-7.41
9.01-10.00
Characteristic
optically thin
optically thick
located in window region
Table 8.3: Spectral bands, wavelength regions and characteristics of the chose wavelength
bands with its characteristics at the surface for a solar insolation of S = 1.00S0 .
Spectral band 11 (3.64-4.17μm)
The mean optical depths for spectral band 11 are below unity for a insolation S =
1.00S0 for all atmospheric layers (see left panel in Figure 8.11). With increasing
solar insolation the optical depth of the atmosphere increases. The vertical region
where the model atmosphere is optically thick (τ ≥1) increases from about 10 km
altitude for a solar insolation S = 1.10S0 to about 90 km for a solar insolation
S = 1.41S0 .
8.2 Inner boundary of the HZ determined by the critical point of water
Figure 8.11: Mean optical depths over altitude (left) and atmospheric temperatures over
mean optical depths (right) for spectral band 11 (3.64-4.17μm) for increased solar insolations
with Asurf =0.22 and RH=100%. Dotted black line marks where the mean optical depth is
unity.
The atmospheric temperatures for increased solar insolation become fixed in regard
to the optical depth for spectral band 11 see left panel in Figure 8.11.
Spectral band 16 (5.41-7.41μm)
Figure 8.12: Mean optical depths over altitude (left) and atmospheric temperatures over
mean optical depths (right) for spectral band 16 (5.41-7.41μm) for increased solar insolations
with Asurf =0.22 and RH=100%.
Figure 8.12 is as for Figure 8.11 but for spectral band 16. For a solar insolation
S = 1.00S0 the model atmosphere is optically thick up to an altitude of about 11
km. For solar insolations S ≥ 1.10S0 the model atmospheres are completely optically
thick from the surface to the top of the atmosphere.
For band 16 the atmospheric temperatures for increased solar insolation are fixed in
regard to the optical depth (see left panel in Figure 8.12).
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Results
116
Spectral band 18 (9.01-10.00μm)
Figure 8.13: Mean optical depths over altitude (left) and atmospheric temperatures over
mean optical depths (right) for spectral band 18 (9.01-10.00μm) for increased solar insolations with Asurf =0.22 and RH=100%.
Spectral band 18 in Figure 8.13 is located in the window region behaves similar to
spectral band 11. However, the lowest atmospheric layers of the atmosphere are
optically thick for a solar insolation S = 1.00S0 . The optically thick region increases
up to altitudes of about 183 km with increasing solar insolation.
Also for band 18 the atmospheric temperatures for increased solar insolation become
fixed in regard to the optical depth (see left panel in Figure 8.13).
To conclude with increasing solar insolation S the atmosphere is always optically
thick at the surface and the altitude where the atmosphere is optical thick (reaches
τ = 1) increases up to heights of about 100 km. Furthermore, the atmospheric
temperatures become fixed in regard to the mean optical for each spectral band.
Note that this fulfills the second condition for the tropospheric radiation limit.
Approaching of the moist adiabatic lapse rate to the water vapor saturation pressure curve
The behavior of the temperature profiles for increased solar insolation is investigated.
Figure 8.14 shows the temperature profiles over pressure. As explained before for
increased solar insolation the temperature profiles are almost exclusively dominated
by convection as the dominant energy transport process. The moist adiabatic lapse
rate approaches the saturation vapor pressure curve of water (plotted as a grey
dotted line) for solar insolations S > 1.10S0 , which is related to the water mixing
ratio becoming an isoprofile of unity (see right panel in Figure 8.1).
Furthermore, the influence of the evaporated water in the model atmosphere can be
seen in Figure 8.14 resulting in an increase of surface pressure (psurf = pback + pH2 O )
for increased solar insolation. For example for a solar insolation of S = 1.41S0 the
8.2 Inner boundary of the HZ determined by the critical point of water
Figure 8.14: Temperature profiles over pressure for increased solar constants with
Asurf =0.22 and RH=100%. Saturation vapor pressure curve of water plotted as gray dotted
line.
surface pressure results to psurf = 173.51 bar.
The approaching of the moist adiabatic lapse rate to the saturation vapor pressure
curve of water for high solar insolations fulfills the third condition for the tropospheric radiation limit.
Conclusion for the tropospheric radiation limit
The results of the model scenarios for increased solar insolations S confirm that the
conditions for the tropospheric radiation limit are fulfilled.
The studies investigating the runaway greenhouse by reaching a tropospheric radiation limit of the outgoing infrared flux (reviewed in section 5.1) implied that the
constant outgoing infrared flux is not able to balance the increasing incoming solar
insolation and radiative equilibrium is not possible at the top of the atmosphere.
They assumed that a solar insolation higher than the constant outgoing infrared
flux would lead to a such strong increase in surface temperature and in greenhouse
effect that the water reservoir would evaporate completely. Note, that the influence
of the solar insolation on the shortwave radiation in the model atmosphere was not
calculated in many of these studies (e.g. Nakajima et al. (1992); Pujol and North
(2002)).
However, for the results of the model runs with increased solar insolation, where
the treatment of shortwave radiation is calculated self-consistently in the applied
atmospheric model, a strong increase in surface temperature for optical thick atmo-
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Results
spheres, which would describe a runaway greenhouse, is not approached, although
the conditions for the tropospheric radiation limit are fulfilled. Reason for this are
discussed in the next subsection.
8.2 Inner boundary of the HZ determined by the critical point of water
8.2.2
Influence of Rayleigh scattering of water on the determination of the inner boundary of the HZ
Figure 8.15: Net infrared fluxes Fir (red x) and net solar fluxes Fs (green x) for increased
solar insolations with Asurf =0.22 and RH=100%.
However, the one-dimensional radiative-convective model used to calculate the atmospheric temperatures includes the incoming solar insolation and a consistent treatment of the shortwave radiative flux. Figure 8.15 shows that analogous to the
constant net infrared radiation flux at the top of the atmosphere, the increase of
the solar insolation leads also to a constant net shortwave flux at the top of the
atmosphere. The net infrared and the net shortwave flux can balance each other
at the top of the atmosphere up to solar insolation S = 1.41S0 . Thus, at the top
of the model atmosphere radiative equilibrium can still be maintained although the
tropospheric radiation limit occurs. As a result a runaway greenhouse, which would
lead to a large increase in surface temperature due to the not balanced incoming
solar insolation is not approached for the investigated model scenarios.
The reason that the net shortwave flux FS,net at the top of the atmosphere is constant, is the upwelling shortwave radiative flux FS↑ . Because an increase in solar
insolation S increases directly the downwelling shortwave radiative flux FS↓ and does
not lead to a constant flux (see Figure 8.4). The upwelling shortwave flux at the top
of the atmosphere depends at the surface on the surface albedo Asurf and in the
atmosphere on Rayleigh scattering.
The influence of Rayleigh scattering of H2 O on the results is shown in Figures 8.16
to 8.19, which show the comparison of results with and without Rayleigh scattering
of water. Rayleigh scattering of N2 and CO2 is included in all model scenarios.
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Results
Note that the runs with the highest solar insolations for both cases, including and
excluding Rayleigh scattering of H2 O, breaks up, since the model is only designed
to calculate temperature up to critical point of water. But for these model runs the
temperatures were calculated consistently up to the critical point. Thus, the output
from the last iteration step before reaching the critical point of water for S = 1.42S0
with and S = 1.08S0 without Rayleigh scattering of water are shown in Figures 8.16
to 8.19.
Figure 8.16: Temperature profiles over altitude for increased solar insolations with (left)
and without (right) Rayleigh scattering of H2 O with Asurf =0.22 and RH=100%.
Figure 8.16 shows a selection of temperature profiles for increased solar insolation
over altitude. Without Rayleigh scattering of H2 O the critical point of water is
reached at a much smaller value of the solar insolation S at 1.08S0 compared to
1.42S0 with Rayleigh scattering of H2 O. Furthermore, by increasing the solar insolation from S = 1.07S0 to S = 1.08S0 for the temperature profiles without Rayleigh
scattering of H2 O a huge increase for the surface temperature Tsurf from about 340
K to 646 K can be seen. This large increase in surface temperature (over 300K) for
a solar insolation increase of ΔS = 0.01S0 does not occur for runs, which include
the Rayleigh scattering of water.
The surface temperatures Tsurf for increased solar insolations S are shown in Figure
8.17. For these scenarios including Rayleigh scattering of H2 O the surface temperature increases moderately until the surface temperature reaches the temperature of
the critical point of water and then the model breaks up. Without H2 O Rayleigh
scattering the surface temperature increases also moderately with increasing solar
insolation up to 1.07S0 , but for a solar insolation of 1.08S0 the surface temperature strongly increases until the surface temperature reaches the temperature of the
critical point of water and crashes.
This effect is also apparent for the net shortwave and infrared radiative fluxes at
the top of the atmosphere, which are plotted in Figure 8.18. The shortwave and
the infrared fluxes at the top of the atmosphere for the model runs with Rayleigh
8.2 Inner boundary of the HZ determined by the critical point of water
Figure 8.17: Surface temperatures for increased solar insolations S with (blue) and without
(red) Rayleigh scattering of H2 O with Asurf =0.22 and RH=100%.
scattering of H2 O are the same and thus radiative equilibrium in the stratosphere
is reached even for the 1.42S0 run, which is not converged. This is in contrast to
the model runs, which do not include Rayleigh scattering of H2 O, and for which
radiative equilibrium is maintained for values of the solar insolation S of 1.07S0 ,
but for the run with the solar insolation S = 1.08S0 , which is not converged, the
net shortwave flux FS is higher than the net infrared flux FIR at the top of the
atmosphere. This might lead to a further increase in surface temperature Tsurf ,
because the infrared flux is not able to balance the shortwave flux.
It is revealing to investigate the planetary albedo Ap for increased solar insolations S (see Figure 8.19 ), which is calculated from the amount of the incoming
shortwave radiation, which is scattered back to space at the top of the atmosphere
(Ap = FS↑ (T OA)/FS↓ (T OA)). With increasing solar insolation S the planetary
albedo Ap decreases due to the absorption of shortwave radiation in the near-infrared
wavelength region by water vapor. This behavior changes when water vapor becomes
the major constituent of the atmosphere. For water dominated atmospheres the
planetary albedo increases if Rayleigh scattering of H2 O is included. This behavior
is not the case when Rayleigh scattering of H2 O is neglected.
However, Kasting (1988) and Kasting et al. (1993) also experience a tropospheric
radiation limit (FIR = 310 Wm−2 ). The tropospheric radiation limit of Kasting
(1988) was discussed in Nakajima et al. (1992). The condition for the runaway
greenhouse is defined in Kasting et al. (1993) by reaching the critical point of water
(d = 0.84 AU). This limit by Kasting et al. (1993) is in agreement with the inner
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Results
Figure 8.18: Net infrared fluxes Fir and shortwave fluxes FS at the top of the atmosphere for increased solar insolations S with and without Rayleigh scattering of H2 O with
Asurf =0.22 and RH=100%.
boundary of the HZ, calculated in this section taking also the critical point of water
into account (d = 0.84 AU).
Kasting (1988) and Kasting et al. (1993) included in their investigations also the
shortwave radiation including Rayleigh scattering. But they did not calculate the
radiative equilibrium in the upper atmosphere, they assumed that it should occur
to support a given surface temperature (for further information about their model
calculation approach see subsection 5.1.4).
For the model scenarios in this Thesis the temperature and water profiles are calculated self-consistently for a given solar insolation S until a new radiative equilibrium
is reached in the upper atmosphere. Rayleigh scatting of H2 O is the cause that
no runaway greenhouse (strong temperature increase due to the optical thick atmosphere) is approached for the model scenarios of this Thesis for increased solar
insolation, although the conditions for the tropospheric radiation limit are fulfilled.
Rayleigh scattering for H2 O dominated atmospheres increases the planetary albedo
so strong, that the net shortwave flux is constant too at the top of the atmosphere
and thus able to balance the constant infrared flux and radiative equilibrium is possible at the top of the atmosphere for increased solar insolations. This was also
qualitatively assumed by Goldblatt and Watson (2012), but not calculated.
8.3 Inner boundary of the HZ determined by the water loss limit
Figure 8.19: Planetary albedos for increased solar insolations S with (blue) and without
(red) Rayleigh scattering of H2 O with Asurf =0.22 and RH=100%.
8.3
Inner boundary of the HZ determined by the water
loss limit
The inner HZ can also be determined by another criterion than the previously described, i.e. instead of temperature reaching the critical point of water, the solar insolation S can be determined, for which the complete water reservoir (size
of the Earth’s oceans) of a planet can be lost during the lifetime of the planet
(age of the Earth). This criterion is adopted in Kasting (1988) and Kasting et al.
(1993) by defining the critical mixing ratio of stratospheric water vapor to be
cH2 O,crit (Stratosphere) = 3·10−3 described in section 5.2. If this critical mixing
ratio is reached in the stratosphere an Earth ocean can be lost within the lifetime
of the Earth due to diffusion-limited escape of hydrogen. The diffusion-limited flux
is the bottleneck process for the loss of hydrogen (see subsection 4.3.4). Mass transport, photo-dissociation, and the loss processes are not calculated explicitly.
Those studies assumed an isoprofile of water for the stratosphere with the value of
the tropopause. To compare with those studies, the same approach was incorporated
for the model scenarios presented in the following: the water profile is calculated by
approach 3 described in section 6.10 using cH2 O = psat /p for the troposphere and in
the stratosphere the value of the water mixing ratio at the tropopause, the uppermost
layer of the convective region, is taken throughout the complete stratosphere, hence
a constant water vapor mixing ratio.
Figure 8.20 shows a selection of temperature and water profiles with the new ap-
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Results
Figure 8.20: Temperature (left panel) and water profiles (right panel), with isoprofiles in the
stratosphere over altitude for increased solar insolations with Asurf =0.22 and RH=100%.
proach for the water calculation (approach 3). The new approach has little influence
on the surface temperatures compared to resulting surface temperatures using the
water calculation approach 1 (cH2 O = psat /p for the entire atmosphere). It has
only a little influence on the overall structure of the temperature profile, if water is
not dominant and nearly no influence if water is dominant (see left panel of Figure
8.1). However, the water profiles in the stratosphere appear different from the one
presented in the right panel of Figure 8.1 if water is not dominant.
To determine the water loss limit Figure 8.21 shows the water mixing ratios in the
stratosphere plotted for increasing solar insolations S. When the water mixing ratio
reaches the critical value of water vapor in the stratosphere of cH2 O,crit (Stratosphere) =
3 · 10−3 the complete water reservoir (= Earth’s water reservoir) can be lost during
the lifetime of the planet (= age of the Earth) assuming diffusion-limited escape and
an inner limit of the HZ is reached.
The stratospheric water mixing ratio is higher than the critical stratospheric mixing
ratio for a solar insolation S = 1.09S0 . This solar insolation corresponds to a distance
of d = 0.96 AU of the planet to the star, which is the inner boundary of the HZ
determined by the water loss limit. This inner boundary of the HZ is independent
from the water mixing ratio calculation approach (see section 6.10) but it depends
on the surface albedo Asurf , which is discussed in subsection 8.4.2.
The size of the ocean reservoir, the age of the planet and also the planetary radius can
influence the critical value of water vapor in the stratosphere of cH2 O,crit (Stratosphere).
But these changes of cH2 O,crit (Stratosphere) might not be important for the determination of the water loss limit, because for a small increase in solar insolation
ΔS = 0.02S0 from S = 1.08S0 to S = 1.10S0 the mixing ratio of water in the
stratosphere changes over five orders of magnitude.
8.4 Influence of surface albedo of the planet
125
Figure 8.21: Stratospheric water mixing ratios for increased solar insolations with
Asurf =0.22 and RH=100%. The dotted line marks the critical stratospheric mixing ratio that leads to the escape of the water reservoir of the planet within the lifetime of the
planet.
8.4
Influence of surface albedo of the planet
So far all the results were shown for a surface albedo Asurf = 0.22. The influence of
changes in the surface albedo on the two criteria to determine the inner boundary
of the HZ are investigated.
8.4.1
Critical point of water
The influence of the different surface albedos for solar insolations S of 1.00S0 and
1.41S0 are shown in Table 8.4.
Asurf
Tsurf [K] for S=1.00S0
Tsurf [K] for S=1.41S0
0.13
326.81
627.29
0.21
308.19
626.74
0.22
305.16
626.66
0.29
287.92
626.59
Table 8.4: Surface temperature Tsurf for different values of the solar insolation S and for
four surface albedos Asurf
The higher surface temperatures for a solar insolation S of 1.00S0 than 288K are a
result of the assumed relative humidity of 100%, which due to resulting enhanced
H2 O greenhouse effect leads to higher surface temperatures. The surface temper-
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Results
ature Tsurf decreases with increasing surface albedo Asurf , since a higher Asurf ,
leads to more reflection of incoming stellar radiation at the surface and thus to less
absorption and therefore smaller Tsurf . The differences in Tsurf are about 40 K for
an increase from Asurf =0.13 to Asurf =0.29. For a solar insolation of 1.41 S0 , which
is close to the temperature of the critical point of water, the increase in Asurf from
0.13 to 0.29 leads to a decrease in Tsurf of only less than 1 K. The model breaks
up due to reaching the critical point of water for the model scenarios for a solar
insolation of 1.42S0 , independent of the surface albedo Asurf .
Figure 8.22: Surface temperatures for increased solar insolations for surface albedos Asurf
of 0.13 (upper left), 0.21 (upper right), 0.22 (lower left), and 0.29 (lower right) with RH =
100%.
The reason for this behavior is shown in Figures 8.22, which shows for the different
surface albedos the planetary albedo Ap , which is calculated from the amount of
the incoming shortwave radiation, which is scattered back to space at the top of
the atmosphere (Ap = FS↑ (T OA)/FS↓ (T OA)). With increasing solar insolation the
planetary albedo Ap decreases due to the increased absorption of shortwave radiation by atmospheric H2 O. This behavior changes when water vapor becomes the
major constituent of the atmosphere. This appears when the water profile reaches
an isoprofile of mixing ratios for Asurf =0.13 at about S=1.05S0 , for Asurf =0.21 at
8.5 Impact of the relative humidity
127
about S=1.09S0 , for Asurf =022 at about S=1.10S0 , and for Asurf =0.29 at about
S=1.13S0 . For higher solar insolations water is the major constituent in the atmosphere and its influence is to increase the upwelling shortwave flux increases due to
the Rayleigh scattering of H2 O, hence leads to an increase in the planetary albedo
Ap . This is the reason why the behavior of the planetary albedo Ap is the same for
higher solar insolations S independent of the surface albedo.
Therefore, the scattering properties of water dominated atmospheres lead to the
result, that the surface temperature Tsurf reaches the critical temperature of water,
for a solar insolations S of 1.41S0 independent of the applied surface albedo. For
water dominated atmospheres the scattering properties of the atmosphere dominate
over the scattering (reflection) of the surface.
8.4.2
Water loss limit
For the water loss limit the different solar insolations S, for which the stratospheric
water mixing ratio is higher than the critical stratospheric mixing ratio, and the
corresponding distances d of the planet to the star, are summarized in Table 8.5 for
different surface albedos Asurf .
Asurf
S [S0 ]
d [AU]
0.13
1.05
0.98
0.21
1.09
0.96
0.22
1.09
0.96
0.29
1.13
0.94
Table 8.5: Water loss limit as a function of solar insolations S and distances d from the
planet to the star for four surface albedos Asurf
The inner limit of the HZ determined by the water loss depends on the surface
albedos Asurf , see Figure 8.5. For increasing surface albedo from Asurf = 0.13 to
Asurf = 0.29 the inner limit is shifted inwards from d =0.98 AU to d =0.94 AU.
It is located at a distance where the solar insolation is high enough that the water
profile reaches the isoprofile. As can be seen in Figure 8.22 the effect of Rayleigh
scattering becomes then important for higher solar insolations. This is in contrast
to the inner limit of the HZ determined by reaching the temperature of the critical
point of water, where the result does not depend on different surface albedos (see
subsection 8.4.1).
8.5
Impact of the relative humidity
This section investigates the influence of different relative humidity profiles. Results
presented so far have assumed a constant relative humidity profile of 100%. This is
an appropriate approach for atmospheres which are water vapor dominated, which
is assumed to be the case when the surface temperature reaches the critical point
Results
128
of water. For the determination of the water loss limit however such an approach
could overestimate the amount of water in the atmosphere and could lead to water
loss limits, which are unrealistically distant from the star.
This section uses different relative humidity profiles, which were introduced in section
6.10.
Relative humidity (RH)
100%
Manabe and Wetherald (1967)
Cess (1976)
Kasting and Ackerman (1986)
Surface Temperature (Tsurf ) [K]
305.16
286.24
285.95
286.15
Table 8.6: Surface temperatures Tsurf for different relative humidity profiles for a solar
insolation S = 1.00S0 and a surface albedos Asurf = 0.22.
Figure 8.23: Temperature (left panel) and water profiles (right panel) for different relative
humidity profiles for a solar insolation S = 1.00S0 with Asurf =0.22. Blue, green and orange
lines all overlie.
The effect of the different relative humidity profiles on the surface temperature Tsurf
for a surface albedo Asurf = 0.22 and a solar insolation S = 1.00S0 is summarized
in Table 8.6. The surface temperatures Tsurf using the relative humidity profiles of
Manabe and Wetherald (1967), Cess (1976), and Kasting and Ackerman (1986) are
comparable (around 286K), but the surface temperature is about 20 K lower than
for the relative humidity profile of 100%. Figure 8.23 shows temperature and water
profiles for the different relative humidity profiles for a solar insolation S = 1.00S0 .
As expected the temperature and water profiles computed with the relative humidity
profiles of Cess (1976) and Kasting and Ackerman (1986) are equal to the profiles
computed with the relative humidity profile of Manabe and Wetherald (1967) for a
solar insolation of S = 1.00S0 .
On the other hand for Earth-like conditions, the constant relative humidity profiles of
8.5 Impact of the relative humidity
100% for the model atmosphere overestimates the amount of water in the atmosphere
for a solar insolation S = 1.00S0 (see right panel of Figure 8.23), which leads due to
the greenhouse effect of water to the higher atmospheric temperatures.
8.5.1
Critical point of water
Figure 8.24: Temperature (left panel) and water profiles (right panel) for different relative
humidity profiles for a solar insolation S = 1.41S0 with Asurf =0.22. Red, green and orange
lines all overlie.
Figure 8.24 shows temperature and water profiles for a solar insolation of S = 1.41S0
for the different relative humidity profiles. The temperature and water profiles calculated with the relative humidities of Cess (1976) and Kasting and Ackerman (1986)
are now similar to the profiles with a relative humidity of 100%. Thus, the inner
boundary of the HZ determined by reaching the temperature of the critical point
of water is similar for all three relative humidity profiles. The surface temperature reaches the temperature of the critical point of water for these three relative
humidity profiles for a solar insolation S = 1.41S0 , which corresponds to an inner
boundary of the HZ at 0.84 AU.
However, the temperature and water profiles computed with the relative humidity
profile of Manabe and Wetherald (1967) deviate from the results with the other
relative humidities for a solar insolation of S = 1.41S0 (see Figure 8.24). The atmospheric temperatures modeled with the relative humidity of Manabe and Wetherald
(1967) are around 300 K smaller than for the other relative humidity approaches.
The water profile is also much smaller around an order of magnitude for the surface
and around 8 to 9 orders of magnitude for the top of the atmosphere. This is due
to the fact that the relative humidity profile of Manabe and Wetherald (1967) keeps
the stratospheric artificially dry.
To determine the inner boundary for the model scenarios using the relative humidity
profile by Manabe and Wetherald (1967), Figure 8.25 shows the surface temperatures
for increased solar insolations S. The surface temperatures Tsurf increase more
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Results
130
Figure 8.25: Surface temperatures for increased solar insolation with Asurf =0.22 and RH
of Manabe and Wetherald (1967).
slowly with increasing solar insolation S compared with the results using a relative
humidity profile of 100% (see Figure 8.6). The critical point of water is reached for a
solar insolation of S = 2.57S0 , which would correspond to an inner boundary of the
HZ at 0.63AU using the relative humidity profile of Manabe and Wetherald (1967).
But using the relative humidity profile of Manabe and Wetherald (1967) for water
dominated atmospheres is not physical. The relative humidity could not have been
much smaller than 100% for a water dominated atmosphere without violating the
barometric law (Kasting, 1988).
Thus, the assumption of a relative humidity of 100% seems to be appropriate for the
determination of the inner boundary of the HZ, determined by reaching the critical
point of water.
8.5.2
Water loss limit
Figure 8.26 shows different water mixing ratios in the stratosphere for increasing
solar insolations S calculated with the relative humidity profile of Manabe and
Wetherald (1967). The water profile is calculated with the same approach as to
determine the water loss limit (see section 8.3). Thus, the water mixing ratio in the
stratosphere is fixed to the value at the tropopause (approach 3).
When the water mixing ratio reaches the critical value of water vapor in the stratosphere of cH2 O,crit (Stratosphere) = 3 · 10−3 the complete water reservoir (= Earth’s
water reservoir) can be lost during the lifetime of the planet (= age of the Earth)
8.6 Role of the water reservoir
Figure 8.26: Stratospheric water mixing ratios for a series of runs (x) versus increased
solar insolations with Asurf =0.22 and RH of Manabe and Wetherald (1967). The dotted
line marks the critical stratospheric mixing ratio that leads to the escape of the water reservoir
within the lifetime of the planet.
and an inner limit of the HZ is reached. For a solar insolation of S = 1.7S0 the
water loss limit occurs with a relative humidity profile of Manabe and Wetherald
(1967). The corresponding inner boundary of the HZ determined by the water loss
limit is located at 0.77 AU using the relative humidity of Manabe and Wetherald
(1967). This limit is even located closer to the star than the inner boundary of the
HZ determined by reaching the critical point of water (647 K) for a relative humidity
of 100%.
8.6
Role of the water reservoir
The inner HZ boundary, as determined by the critical point of water (Tcrit,H2 O ) (see
section 8.2) arises since it is impossible for liquid water to exist on the surface of a
planet above this temperature (see section 4.1). In this study, the size of the water
reservoir of the planet is assumed to be of the size of the Earth’s oceans (1.4·1021 l),
since most of the water on the Earth’s surface is stored in the oceans. If this water
reservoir would be evaporated completely, it would produce an atmospheric pressure
at the surface of 270 bar. This pressure is 50 bar above the pressure at the critical
point of water (pcrit,H2 O = 220 bar). Thus, for all water reservoirs, which would
produce a pressure of above 220 bar when vaporized completely, liquid water would
be impossible on the planetary surface if the temperature of the critical point is
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Results
132
reached.
For smaller water reservoirs (below 220 bar) liquid water cannot exist on a planetary
surface for lower surface temperatures (Tsurf < Tcrit,H2 O ). A similar approach was
introduced by Pollack (1971) to determine temperatures (termed runaway temperatures) at which different amounts of water cannot exist on the planetary surface.
To determine the pressure pocean , which is produced by the complete evaporation
of the water reservoir, it is assumed that the oceans are composed of pure water
(ρH2 O =1kg/l). This is a simplification which neglects the salt content of the oceans,
which could slightly change the density compared to pure water. With this assumption the mass of the ocean is mocean =1.4 · 1021 kg. Therefore, the resulting pocean can
be calculated by using
pocean =
F
mocean · g
=
A
A
(8.1)
where A = 5.1 · 1014 m2 is the surface area of the Earth, and g = 9.81ms−2 the
acceleration of gravity of the Earth. With this pressure pocean the temperature for
the surface Tsat , above which liquid water is not stable, can be calculated from the
saturation vapor pressure curve of water (see Section 4.1). The complete evaporation
temperatures Tsat of the oceans for smaller water reservoirs are summarized in Table
8.7.
Earth ocean mass
100%
50%
10%
5%
1%
0.5%
0.1%
0.05%
mocean [kg]
1.4 · 1021
7 · 1020
1.4 · 1020
7 · 1019
1.4 · 1019
7 · 1018
1.4 · 1018
7 · 1017
pocean [bar]
270
135
27
13.5
2.7
1.35
0.27
0.135
Tsat [K]
647 (Tcrit,H2 O )
607.0
501.2
466.5
403.0
381.3
339.8
324.9
S[S0 ]
1.41
1.40
1.31
1.26
1.14
1.11
1.07
1.05
d [AU]
0.84
0.85
0.87
0.79
0.94
0.95
0.97
0.98
Table 8.7: Summary of the pressures pocean , temperatures Tsat , the solar insolations S and
the distances to the star d of the inner boundary of the HZ for water reservoirs with different
masses mocean
The surface temperature Tsat necessary for the complete evaporation decreases with
decreasing mass of the ocean mocean . The temperature Tsat for the evaporation of
water reservoirs that could produce pressures pocean > 220 bar, if the temperature
reaches the critical point of water, Tcrit,H2 O .
The solar insolations S and distances d to the star to reach these temperatures
are given in Table 8.7. The solar insolations S are derived from Figure 8.5 for a
surface albedo Asurf = 0.22 and a relative humidity of 100%. The distances d of
the inner boundary related to the mass of the water reservoir of the planet is shown
8.7 Case study: Kepler 22b-like planet
in Figure 8.27. The distances d of the inner boundary of the HZ decreases with
increasing mass of the planetary water reservoir. The inner habitable region (from
the distance where S = 1.00S0 to the distance of the considered inner boundary of
the HZ) becomes wider with a greater water reservoirs on the planet. Note that the
inner boundary of the HZ determined by the water vapor saturation pressure curve
for a 1% Earth ocean (d = 0.94 AU) is nearly at the same distance as the water loss
limit (d = 0.95 AU).
Figure 8.27: Position of the inner boundary of the HZ for model scenarios with different
water reservoirs (ocean masses) with Asurf = 0.22 and RH=100%.
The result that the inner boundary moves away from the star for smaller ocean
masses is in contrast to the study by Abe et al. (2011). They proposed that for dry
planets (a desert world with limited surface water) the inner boundary of the HZ
would be be closer to the star. An important difference to this study is that for the
results presented here a saturated atmosphere is assumed, in contrast to Abe et al.
(2011), who assumed unsaturated air.
8.7
Case study: Kepler 22b-like planet
Additionally to the determination of the inner boundary of the HZ in general, the
atmospheric model can also be used to determine if detected extrasolar planets could
be located inside the HZ. In this section a planet that is comparable to the extrasolar
planet Kepler 22b is investigated. Therefore, the parameters that are known from
the detection are introduced and the assumptions for model scenarios are discussed.
Due to the lack of information about the atmospheric composition of these planet,
133
Results
134
an atmospheric composition and mass has to be assumed for the model scenarios.
For the investigation of a Kepler 22b-like planet the stellar and planetary parameters
are taken from Borucki et al. (2012) and summarized in Table 8.8 and Table 8.9 .
Stellar parameters
Stellar Type
Luminosity [LSun ]
Effective Temperature [K]
Value
G5
0.79
5518
Table 8.8: Stellar parameters of Kepler 22
Planetary Parameters
Semi Major Axis [AU]
Radius [REarth ]
Upper mass limits [MEarth ] for circular orbit
Upper mass limits [MEarth ] for eccentric orbit
Value
0.849
2.38
27
36
Table 8.9: Planetary parameters of Kepler 22b
To determine the surface temperature Tsurf of Kepler 22b Borucki et al. (2012)
added the Earth’s greenhouse effect of 33 K to the effective temperature Te of the
planet, which they calculated to be 262 K by assuming a planetary albedo of 0.3.
Thus, their surface temperature Tsurf is 295 K and therefore it is assumed that
Kepler 22b would be habitable.
The atmospheric model described in chapter 6 is applied to give more in-depth
information on the surface temperature including radiative transfer and convection
as energy transport processes.
The following assumptions and simplifications were made for the Kepler 22b-like
planet:
• Atmospheric species H2 O, CO2 and N2 are considered. Assuming an Earth
profile as start value for H2 O, an isoprofile of 355 ppm for CO2 , and N2 as
background gas.
• The water profile is calculated with the same approach as for the water loss
limit (see Section 8.3), where the water mixing ratio in the stratosphere is set
to the value of the water mixing ratio at the tropopause.
• The atmospheric background pressure is assumed to be 1 bar.
• Since only upper mass limits are determined in Borucki et al. (2012), the mass
of a Kepler 22b-like planet is assumed in this study to be 10MEarth , which
is around the maximum mass for a Super-Earth. This yields a gravitational
acceleration of g = 1.77gEarth with the radius of 2.38REarth
8.7 Case study: Kepler 22b-like planet
• Since Kepler 22 is a G-type star it is assumed that the spectral distribution is
similar to that of the modern Sun (Gueymard, 2004) and the solar spectrum
scales with a solar insolation of S = 1.10S0 , which considers the smaller luminosity of the star of 0.79LSun and the smaller orbital distance of the planet of
d = 0.849AU.
Figure 8.28: Temperature (left panel) and water profiles (right panel) of Kepler 22b-like
planet over altitude for a relative humidity profiles of 100% (solid line) and the relative
humidity profile of Manabe and Wetherald (1967) (dashed line) with Asurf = 0.22
Figure 8.28 shows resulting temperature (left panel) and the water (right panel)
profiles for the Kepler 22b-like planet for a surface albedo of Asurf = 0.22 for
different relative humidity profiles. Only the extreme cases for relative humidity
RH are shown; solid lines show the relative humidity profile of 100% whereas the
dashed line the relative humidity profile of Manabe and Wetherald (1967). The top
of the atmosphere is in general smaller for the Kepler 22b-like planet than for the
Earth-like planets investigated before, due to the influence of the higher gravity.
Changing the relative humidity profile from Manabe and Wetherald (1967) to 100%
results due to higher temperatures and water mixing ratios in an increase of the
lid of the atmospheric model from about 29 to 34 km. Also the convective zone
increases, which can be clearly seen in the water mixing ratios (see left panel of
Figure 8.28) where the tropopause, above which the mixing ratio is an isoprofile,
increases from 7 to 19km.
For different surface albedos Asurf the surface temperatures Tsurf and water mixing
ratios in the stratosphere cH2 O (stratosphere) for the two relative humidity profiles
(i.e. 100% and the relative humidity profile of Manabe and Wetherald (1967)) are
shown in Table 8.10. The surface temperatures Tsurf decreases for both relative
humidities RH for increasing surface albedo Asurf . The surface temperature Tsurf
for a relative humidity profiles RH of 100% decrease about 66 K from 365.97 K to
299.93 K for surface albedos of 0.13 and 0.29, whereas the surface temperature Tsurf
for the profile by Manabe and Wetherald (1967) decrease about 15 K from 303.32
135
Results
136
K to 288.38 K. The higher surface temperatures for the RH =100% result from the
higher water mixing ratios and the stronger greenhouse effect.
Asurf
RH=100%:
Tsurf
cH2 O (Stratosphere)
RH from
Manabe and Wetherald (1967):
Tsurf
cH2 O (Stratosphere)
0.13
0.21
0.22
0.29
365.97
1.75 · 10−1
324.62
5.53 · 10−6
321.67
3.23 · 10−5
299.93
6.75 · 10−6
303.32
1.04 · 10−5
296.15
1.06 · 10−5
294.90
6.79 · 10−6
288.38
1.52 · 10−5
Table 8.10: Surface temperatures Tsurf and stratospheric water mixing ratios
cH2 O (Stratosphere) for a Kepler 22b-like planet for two different relative humidity
profiles
The surface temperatures for the model scenarios with different surface albedos
Asurf and relative humidities RH indicate habitable surface conditions. All surface
temperatures Tsurf are below the temperature of the critical point of water.
To determine the habitability of the Kepler 22b-like planet in view of the water loss
limit, the stratospheric water mixing ratios were investigated. To conclude for all,
but one scenario for the Kepler 22b-like planet with a surface albedo Asurf = 0.13
and RH=100%, the mixing ratios are below the critical value for the stratospheric
water mixing ratio cH2 O,crit (strat) and therefore also within the HZ.
Overall, compared to the estimated surface temperature of Kepler 22b in Borucki
et al. (2012), which took only the Earth’ greenhouse effect into account, the results
of this more detailed study are in good agreement with relative humidity profiles of
Manabe and Wetherald (1967). The surface temperature with a relative humidity
profile of 100% are larger than those calculated by Borucki et al. (2012) depending
on the surface albedo from ΔTsurf =3 K for Asurf =0.29 to ΔTsurf =70 K for
Asurf =0.13.
Although the surface temperatures suggest that the Kepler 22b-like planet is habitable it is not clear if this planet is a rocky planet. For simplicity in this work a
planetary mass from ten Earth masses was assumed for Kepler 22b, but the observations estimated upper limits of the masses from 27 to 36 MEarth .
8.8
Influence of different stellar types on the inner boundary of the HZ
This section investigates the influence of different stellar types on the inner boundary
of the HZ. The planet is placed such that the total stellar energy flux of these stars
is scaled to the present total stellar insolation S0 at the top of the atmosphere for
8.8 Influence of different stellar types on the inner boundary of the HZ
the Earth (S0 =1366 Wm−2 ). The input spectra for the F-type and K-type star and
the Sun are obtained from high resolution stellar spectra. For a detailed description
of the determination of the stellar spectra for these stars see von Paris (2010).
The input spectra of these stars binned to the 38 spectral bands of the shortwave
radiation code are shown in Figure 8.29. In contrast to this, Kasting et al. (1993)
approximated the emission from the central star with blackbody radiation for their
investigations of HZ around an M0 star and an F0 star. The details for the stars
investigated in this Thesis and the position, where the planet receives the same
incoming stellar flux at the top of the atmosphere as the Earth from the Sun S0 , are
given in Table 8.11.
Figure 8.29: Binned Stellar input spectra for the three stars.
Stellar type
F2V
G2V
K2V
Star name
σ Bootis
Sun
Eridani
Tef f [K]
6733
5780
5000
Position [AU]
1.89
1
0.605
Table 8.11: Characteristics of the investigated stars, taken from von Paris (2010)
Figure 8.30 shows the temperature and water profiles for the different stars for
a stellar insolation of S = 1.00S0 and a relative humidity profile of 100%. The
temperature profiles are different to the temperature profiles presented in Segura
et al. (2003) and Grenfell et al. (2007) for the same stars due the following reasons:
Segura et al. (2003) and Grenfell et al. (2007) included more atmospheric species,
especially O3 is important which produces the large increasing temperature in the
stratosphere for the Sun and the F-type star; the relative humidity profile of Segura
137
Results
138
Figure 8.30: Temperature (left panel) and water profiles (right panel) for different stars
for a stellar insolation S = 1.00S0 with Asurf =0.22 and a relative humidity of 100%.
et al. (2003) and Grenfell et al. (2007) is taken to be that of the Earth (Manabe and
Wetherald, 1967), which results in lower atmospheric temperature; the definition of
the planetary orbit of the star in Segura et al. (2003) is located where the stellar
fluxes are accordingly to reach a surface temperature of Tsurf = 288K; also chemistry
is included.
The water mixing ratio is determined with approach 2 of section 6.10. Although
the total energy flux of the stars is the same for all three investigated cases, the
temperature profiles deviate quite significantly. This is a result of the different
spectral distribution of the stellar spectra. The surface temperatures Tsurf for the
extreme cases for the relative humidity profiles are summarized in Table 8.12. The
difference between the surface temperatures Tsurf of the F-type and the K-type star
for a relative humidity of 100% is about 40 K, whereas the temperature difference
for the relative humidity profile by Manabe and Wetherald (1967) is only about 12
K.
Stellar type
RH=100%:
Tsurf
RH from
Manabe and Wetherald (1967):
Tsurf
F2V
Sun
K2V
287.23
305.16
328.64
279.22
286.24
291.55
Table 8.12: Surface temperatures Tsurf for different host stars and different relative humidity profiles
Figure 8.31 shows the surface temperatures Tsurf for increasing stellar insolations S
for the different stars. The solar insolation needed to reach the temperature of the
critical point of water is highest for the the F-type star (S = 1.73S0 ) and lowest for
8.8 Influence of different stellar types on the inner boundary of the HZ
139
the K-type star (S = 1.18S0 ).
The stratospheric water mixing ratios cH2 O,crit (Stratosphere) are shown in Figure
8.8 for the different stars. For the K-type star the stellar insolation S needed to reach
a stratospheric water mixing ratio above the critical value cH2 O,crit (Stratosphere)
= 3·10−3 determining the water loss limit, is the smallest (S = 1.04S0 ), whereas it
is largest for the F-type star (S = 1.19S0 ).
Figure 8.31: Surface temperatures for increased stellar insolations S for three different host
stars of the planet: F2V star (blue), Sun (green), and K2V star (red) with Asurf = 0.22
and RH=100%
Stellar type
S [S0 ] for HZ boundary: Tcrit,H2 O :
d [AU] for HZ boundary: Tcrit,H2 O :
S [S0 ] for HZ boundary: cH2 O,crit (Stratosphere)
d [AU] for HZ boundary: cH2 O,crit (Stratosphere)
F2V
1.73
1.44
1.19
1.73
G2V
1.41
0.84
1.09
0.96
K2V
1.18
0.55
1.04
0.59
Table 8.13: Stellar insolations S for different stellar types to determine the inner boundary of the HZ by reaching the temperature of the critical point of water Tcrit,H2 O and
for the water loss limit by reaching the critical mixing ratio of water in the stratosphere
cH2 O,crit (Stratosphere), respectively with Asurf =0.22 and RH=100%
In Table 8.13 the stellar insolations S are summarized for which the inner boundary
of the HZ is reached determined by the surface temperature reaching the critical
point of water and by reaching the water loss limit. Although the total energy flux
for a stellar insolation of S = 1.00S0 is the same for all three stars, the stellar
140
Results
Figure 8.32: Stratospheric water mixing ratios versus increased stellar insolations S for
three different host stars: F2V star (blue), Sun (green), and K2V star (red) with Asurf =0.22
and RH=100%. The dotted line marks the critical stratospheric mixing ratio that leads to
the escape of the water reservoir within the lifetime of the planet.
insolation needed to determine the two inner boundaries of the HZ vary. Hence, not
only the total amount of energy but also the stellar flux distribution is crucial for the
determination of the inner boundaries of the HZ. Furthermore, the inner boundary
does not depend on the approach used to determine the water profile.
The results for the inner boundary of the HZ for the F2V star (S = 1.73S0 ) with
an effective temperature of Tef f = 6733 K are quite similar to the results of the
F0 star with an effective temperature Tef f = 7200 K as calculated in Kasting et al.
(1993). They determined the inner boundary of the HZ (their so-called ’runaway
greenhouse limit’), where the critical point of water is reached for a stellar insolation
S = 1.90S0 (d = 1.50 AU) and the water loss limit for a solar insolation S = 1.25S0
(d = 1.85 AU). The higher stellar insolations of Kasting et al. (1993) result from
the higher effective temperature of their F-type star. Kasting et al. (1993) did not
calculate boundaries of the HZ for a K-type star, but for an M-type star.
The planetary albedos Ap for the different stars are shown in Figure 8.33 for increased stellar insolation S. The overall behavior of the planetary albedo is the
same for all three stars. With increasing stellar insolation the planetary albedo decreases due to increased absorption of near-infrared radiation by water vapor. For
surface temperature Tsurf > 340K the water vapor becomes the major constituent
of the atmosphere, which leads to the increased Rayleigh scattering of H2 O and this
increases the planetary albedo Ap .
8.8 Influence of different stellar types on the inner boundary of the HZ
Figure 8.33: Planetary Albedo over increased stellar insolations for planets around three
different host stars: F2V star (blue), Sun (green), and K2V star (red) with Asurf =0.22 and
RH=100%.
The planetary albedo is the highest for the F-type star, and the lowest for the K-type
star. The spectrum of the F-type is shifted towards lower wavelengths (see Figure
8.29), thus more flux is available in the ultraviolet and visible wavelengths range.
This leads to an increase of Rayleigh scattering, which is proportional to λ−4 and
thus to higher planetary albedos Ap .
The spectrum of the K-type star lead to a higher flux in the near-infrared and less
in the ultraviolet and visible wavelength region compared to the Sun and the Ftype star. Therefore, the absorption of CO2 and H2 O is stronger than the Rayleigh
scattering and the amount of shortwave radiation flux absorbed by the atmosphere
increases, which leads to a lower planetary albedo (Kasting et al., 1993).
141
Results
142
Figure 8.34: Distances of the inner boundaries of the HZ around different stars. The
grey diamonds mark the positions of the planet, where the energy input at the top of the
atmosphere is S = S0 = 1366Wm−2 . The red crosses mark the position of the inner limit of
the HZ determined by the water loss limit. The orange stars mark the position of the inner
boundary of the HZ determined by reaching the critical point of water.
Figure 8.34 shows the positions of the inner boundaries of the HZ for the three stars.
Changing the stellar type from F-type to K-type star results in a decrease of orbital
distance where the inner boundaries of the HZ are situated. The decrease is caused
by the overall decreasing luminosity from F-type to K-type stars.
The inner habitable region is defined as the region between the position of the planet,
where the stellar insolation S is scaled to the solar constant S = 1.00S0 and the
position of the inner boundary of the HZ determined by reaching the critical point
of water. This inner habitable region decreases with decreasing effective temperature
Tef f of the star. It is the widest for the F-type star due to the large influence of
Rayleigh scattering, hence the high planetary albedo (see Figure 8.33).
8.8.1
HZ Scaling
The relationship between the stellar insolations S needed to determine the innermost
boundary of the HZ and the effective temperature Tef f of the stars of the last section
is compared to the HZ scalings introduced in section 2.2.2. Underwood et al. (2003)
used for the HZ scaling the results of the HZ boundary for the three different stars
by Kasting et al. (1993) (F-type (Tef f =7200 K), G-type (Tef f =5700 K), and M-type
star (Tef f =3700 K)).
They applied the following scaling to describe the relationship between the solar
8.8 Influence of different stellar types on the inner boundary of the HZ
insolation S needed to reach the critical point of water (runaway greenhouse limit
by Kasting et al. (1993)) and the effective temperature Tef f of the stars:
Runaway greenhouse:
2
−4
S = 4.190 · 10−8 Tef
f − 2.139 · 10 Tef f + 1.268 (8.2)
Figure 8.35: Relationship between the solar insolation S needed for reaching the critical
point of water and the effective temperature of stars.
Figure 8.35 shows comparison of the solar insolations S needed to reach the critical
point of water for different stars as results of this study (red stars) and of Kasting
et al. (1993) (black crosses). Furthermore, the HZ scaling of Underwood et al.
(2003) connects the results of Kasting et al. (1993) (see equation 8.2). The results
of this thesis are in good agreement for the F2V star with an effective temperature
of Tef f =6733 K. However, for cooler stars it is indicated that the solar insolation
S needed to reach the critical point of water for this study are lower than the solar
insolations derived from HZ scaling of Underwood et al. (2003). This might result in
a smaller inner habitable region (region between S = 1.00S0 and S for reaching the
critical point of water) for lower effective temperature for the results of this study
compared to the HZ scaling of Underwood et al. (2003).
For comparison also the results are shown for the scaling of Selsis et al. (2007). They
used the relationship between effective temperature and luminosity of the star and
the distance of the inner boundary of the HZ determined by the critical point of
water
143
Results
144
Runaway greenhouse: d = (0.84AU − 2.7619 · 10
−5
T∗ − 3.8095 · 10
−9
T∗2 )
L
1/2
LSun
(8.3)
where T∗ = Tef f − 5700 K and LSun the luminosity of the Sun. The effective
temperatures, luminosities of the stars and distances of the inner boundary are also
taken from Kasting et al. (1993) for the M0, G2, and F0 star:
Due to the reason that it is not clear which effective temperature and luminosities
for other stars than for the M0, G2, and F0 star from Kasting et al. (1993) were
assumed, only the scaling results of Selsis et al. (2007) for these stars are shown in
Figure 8.36.
Note, although the scaling is assumed to represent the values of Kasting et al. (1993)
there are some deviations for the F0 star.
Figure 8.36: Relationship between the distance of the inner HZ for reaching the critical
point of water and the effective temperature of stars.
8.9
Conclusions
In this chapter the inner boundary of the HZ was determined in two ways: Firstly,
via reaching the critical point of water (647 K), where water is then supercritical
and thus thermodynamically not able to exit in the liquid phase (extreme limit of
the inner boundary of the HZ). Secondly, via the water loss limit defined by Kasting
(1988), which could lead to the loss of an Earth’s ocean within the lifetime of a
planet.
8.9 Conclusions
For a solar insolation of S = 1.41S0 , which corresponds to an orbital distance of
d = 0.84 AU the first condition for the inner boundary of the HZ is fulfilled, reaching
the critical point of water. This limit is independent of the surface albedo due to
the effect of increased Rayleigh scattering on the shortwave radiation flux for water
vapor dominated atmospheres.
A detailed discussion was presented on the runaway greenhouse effect for the limit
determined by reaching the critical point of water. The tropospheric radiation limit
by e.g. Nakajima et al. (1992), which gives the conditions for a runaway greenhouse
effect, is reached for the model scenarios of this thesis with increasing solar insolation.
However, for the results of the model scenarios with increased solar insolation, where
the treatment of shortwave radiation is included in the applied atmospheric model,
a strong increase in surface temperature for optical thick atmospheres, which would
describe a runaway greenhouse, is not approached, although the conditions for the
tropospheric radiation limit for the outgoing infrared radiation flux are fulfilled. The
reason is the inclusion of Rayleigh scattering of H2 O for the shortwave radiation,
which increases with increasing water vapor content in the atmosphere, which leads
to a constant net shortwave radiation flux for increased solar insolation. Therefore,
radiative equilibrium is reached at the top of the atmosphere. The results are in
agreement with the result of the inner boundary of the HZ (647 K) determined by
Kasting et al. (1993), which also included Rayleigh scattering but did not calculate
atmospheres in radiative equilibrium. The advantage of the investigated model
scenarios of this thesis is, that temperatures, water concentrations, and radiative
fluxes are calculated self-consistently to investigate the physical feedback processes.
The water loss limit is reached with a surface albedo of Asurf = 0.22 for a solar
insolation of S = 1.09S0 (d = 0.96 AU). This limit depends on the choice of the
surface albedo, which varies for Asurf = 0.13 from a solar insolation of S = 1.05S0
(d = 0.98 AU) to a solar insolation of S = 1.13S0 (d = 0.94 AU) for Asurf = 0.29.
Taking the relative humidity profile of the Earth (Manabe and Wetherald, 1967) into
account, results in a higher solar insolation to reach the inner boundaries of the HZ.
Reaching the critical point of water would need a solar insolation of S = 2.57S0 and
a corresponding orbital distance of d = 0.63 AU. However, the relative humidity of
Manabe and Wetherald (1967) keeps the stratosphere artificially dry and is thus not
appropriate to determine this inner limit. The inner boundary of the HZ determined
by the water loss limit is located at 0.77 AU with a corresponding solar insolation
of S = 1.7S0 , which would be even higher than the solar insolation (S = 1.41S0 ) to
reach the critical point of water with a relative humidity profile of 100%.
A smaller water reservoir is not stable on the surface of a planet for temperatures
smaller than the temperature of the critical point of water (temperature taken from
the saturation vapor pressure curve of water). Therefore, the solar insolation decreases with decreasing mass of the water reservoir to reach the temperature, where
the water reservoir is no longer stable assuming a water saturated atmosphere.
The model was utilized to perform a case-study to investigate a planet like the
extrasolar planet Kepler 22b. It was assumed that the mass of this planet is that of
145
146
Results
the maximum mass assumed for a Super-Earth (∼10ME ) and the atmosphere was
assumed to consist of N2 , CO2 and H2 O. The surface temperature for the Kepler
22b-like planet are calculated to be between 288 and 366 K (below 647 K) depending
on the assumed surface albedo and relative humidity profile. For all, but one scenario
(for a surface albedo Asurf = 0.13 and RH=100%), the mixing ratios are below the
critical value for the stratospheric water mixing ratio cH2 O,crit (strat) and therefore
also within the HZ taking into account the water loss limit as the inner boundary
of the HZ.
Both inner limits were also calculated for two other solar-like main sequence stars.
The orbital distance of the inner HZ decreases from an F-type to a K-type star caused
by the decreasing luminosity and also the inner habitable region (from distance with
S = 1.00S0 to distance for reaching the critical point of water) decreases.
CHAPTER
9
Summary and Outlook
This chapter summarizes the most important results of this Thesis. An outlook
is given on suggested additional improvements to the one-dimensional radiativeconvective model and on further model scenarios investigating the inner boundary
of the HZ.
9.1
Summary
A goal of this Thesis was to investigate the physical processes relevant for the determination of the inner boundary of the HZ. An atmospheric model is applied to
investigate the effect of feedback processes between the surface temperature and
the greenhouse effect of water vapor for increased solar insolations on the boundary
of the inner HZ. The applied one-dimensional radiative-convective model is able to
calculate the inner boundary of the HZ more consistently than in previous studies
(see e.g. Kasting et al. (1993)). This model is able to calculate the feedback of
temperature, water concentration, and wavelengths dependent radiative fluxes for
increased stellar insolations self-consistently.
The self-consistent model is applied to investigate the physical processes relevant for
the determination of the inner boundary of the HZ to answer the addressed scientific
questions (see section 1.1).
9.1.1
Model improvements
The improved one-dimensional radiative-convective model presented in chapter 6
allows the investigation of warm (up to 647 K) and water-rich atmospheres needed
Summary and Outlook
148
for the determination of the inner boundary of the HZ. Improvements were implemented into this model in order to answer the addressed questions by calculating
temperature, pressure, water profiles and wavelength dependent radiative fluxes of
an atmosphere self-consistently for increased stellar insolations.
The infrared radiative transfer code MRAC was improved to be applicable to warm
and water-rich atmospheres. Compared to the previous MRAC version of von Paris
et al. (2008) the following improvements have been implemented (see section 6.7):
• Absorption coefficients for H2 O and CO2 are accounted for all spectral bands
(except CO2 in spectral band 25).
• The k-distributions are calculated for a wider temperature and pressure range:
temperature range from 100 to 700K and pressure range from 10−5 to 103 bar.
• The binary species parameter needed to calculate the k-distributions is determined independently from a reference atmosphere.
• The continuum absorption of H2 O and CO2 is updated and taken into account
for the complete wavelength spectrum.
Furthermore, the condition to distinguish between the convective and radiative energy transport (see section 6.9) was improved for warm (up to 647 K) and water-rich
atmosphere. For these atmospheres convection is assumed to be the dominant energy transport mechanism up to high altitudes due to increased optical depths for
enhanced water vapor concentrations.
9.1.2
Where is the inner boundary of the HZ located in the Solar
System and in other stellar systems?
With this one-dimensional radiative-convective model the inner limit of the HZ
is determined self-consistently by increasing the stellar insolation and taking the
feedback processes between the surface temperature and the greenhouse effect of
water vapor into account until a new steady-state solution is reached.
Two conditions to determine the inner boundary of the HZ were considered: Firstly,
reaching the critical point of water, where water can thermodynamically not exist
in the liquid phase and secondly, the water loss limit (e.g. Kasting (1988)), which
could lead to the loss of an Earth’s ocean within the lifetime of the Earth.
The critical point of water as a condition for the inner boundary of the HZ is reached
for a solar insolation of S = 1.41S0 , which corresponds to an orbital distance of the
inner boundary of the HZ of d = 0.84 AU. This limit is nearly independent of
the surface albedo due to the effect of increased Rayleigh scattering of H2 O on the
shortwave radiation flux for water dominated atmospheres.
The water loss limit is reached with a surface albedo of Asurf = 0.22 for a solar
insolation of S = 1.09S0 (d = 0.95 AU). This limit depends on the choice of the
9.1 Summary
surface albedo, which varies from a solar insolation of S = 1.05A0 (d = 0.98 AU)
for Asurf = 0.13 to a solar insolation of S = 1.13A0 (d = 0.94 AU) for Asurf = 0.29,
because the atmosphere is not completely dominated by water.
A water reservoir smaller than the water reservoir of the Earth’s is not stable on
the surface of a planet for temperatures smaller than the temperature of the critical
point of water (saturation vapor pressure curve of water) and thus determines also
an inner boundary of the HZ. The solar insolation (orbital distance) for this inner
boundary of the HZ decreases (increases) with decreasing mass of the water reservoir
to reach the temperature, where the water reservoir is no longer stable, assuming
that the model atmosphere is saturated.
For two other solar-like main sequence stars (F-type and K-type star) both inner
limits were also calculated self-consistently. The orbital distance of the inner HZ
decreases from an F-type (d = 1.44 AU) to a K-type star (d = 0.55 AU) caused by
the decreasing luminosity. Furthermore, the inner habitable region (the distance,
where S = 1.00S0 to the distance, where the critical point of water is reached)
decreases due to the effect of the spectral distribution and its influence on Rayleigh
scattering and near-infrared absorption of H2 O.
The inner boundaries of the HZ determined by reaching the critical point of water
and the water loss limit confirm the results of Kasting et al. (1993), which also
included Rayleigh scattering of water vapor. The advantage of this Thesis compared
to the study by Kasting et al. (1993) is that the physical processes leading to the
inner boundary of the HZ are calculated self-consistently until radiative equilibrium
is reached with the atmospheric column model by taking into account feedback
processes triggered by increased solar insolation, between the surface temperature
and the water vapor content in the model atmosphere.
9.1.3
Is the runaway greenhouse important for the determination
of the inner boundary of the HZ?
The occurrence of the runaway greenhouse for increased stellar insolation and surface temperatures is determined in many previous studies (see section 5.1) by the
application of radiation limits of the outgoing infrared flux. If the stellar flux exceeds this radiation limit, the surface temperature of the planetary atmosphere is
assumed to increase until all the liquid water available on the surface is evaporated.
The conditions leading to such radiation limits were investigated in this Thesis .
It is shown in this study that the tropospheric radiation limit of the outgoing infrared
flux is not a good criterion for the occurrence of a runaway greenhouse effect. This
was because, for the investigated model scenarios up to reaching the critical point of
water for increased solar insolation (S = 1.41S0 ) and stellar insolation (F-type star:
S = 1.73S0 and K-type star: S = 1.18S0 ) no runaway greenhouse is approached,
although a tropospheric radiation limit of the net infrared flux occurs.
The above result is related to the effect of H2 O Rayleigh scattering on the upwelling
shortwave flux for water dominated atmosphere, which contributes to the net short-
149
Summary and Outlook
150
wave flux. This leads for water dominated atmospheres to a constant net shortwave
flux of the same value as the constant net infrared flux (radiation limit of the outgoing infrared flux) and thus radiative equilibrium at the top of the atmosphere.
The contribution of the upwelling shortwave radiation flux was neglected in previous
studies investigating the tropospheric radiation limit as condition for the occurrence
of a runaway greenhouse effect (e.g. Nakajima et al. (1992), Sugiyama et al. (2005))
or only briefly discussed but not calculated (Goldblatt and Watson, 2012).
9.2
9.2.1
Outlook
Model improvements
The one-dimensional radiative-convective model was improved in this Thesis to be
able to calculate the inner boundary of the HZ determined by reaching the critical
point of water and by the water loss limit. Additional improvements could be done
to this atmospheric model to answer further scientific questions about the inner
boundary of the HZ.
Inner boundary of the HZ including water clouds
The effect of clouds is interesting to investigate for the determination of the inner
HZ, because it might influence the surface temperature. The influence of clouds
on the surface temperature on the Earth depends on the height of the clouds. For
the Earth for example, high-level ice clouds are responsible for the heating of the
surface by increasing the greenhouse effect. Low-level water clouds, on the other
hand, have a cooling effect on the surface of the Earth due to scattering of stellar
radiation (Kitzmann et al., 2010).
Inner boundary of the HZ including loss processes
The effect of atmospheric escape is important for the determination of the inner
boundary of the HZ. The one-dimensional radiative-convective model could be coupled to a photochemical model with a ’reduced’ chemistry (e.g. H2 O, CO2 , and
N2 and related products). An outgoing escape flux of hydrogen, as a result of the
photolysis of water could be introduced in this photochemical model.
Schwarzschild Criterion
The climate model could be improved to be able to calculate the temperature profile more consistently applying the Schwarzschild criterion, to determine where the
energy transport changes from convectively dominated to radiatively dominated regions. Therefore, the atmospheric model should be able to calculate temperature
9.2 Outlook
profiles in radiative equilibrium only, to be able to apply the Schwarzschild criterion
correctly (see section 3.5.3).
9.2.2
Model scenarios
With this existing one-dimensional radiative-convective model a variety of model
scenarios have been performed to determine the inner boundary of the HZ. The
model can additionally be applied to further model scenarios to investigate the inner
boundary of the HZ more extensively.
Inner boundary of the HZ for M-type stars
The inner boundary of the HZ could be determined for M-type stars. For the inner
boundary of the HZ around an M-type star the effect of the Rayleigh scattering
of H2 O on the planetary albedo is assumed to be important. The planetary albedo
would be even lower than for the K-type star (see Figure 8.33) because the spectrum
of the star is shifted to even longer wavelengths.
Inner boundary of the HZ for ’non-Earth-like’ planets
For the model scenarios shown, an Earth-like planet was assumed for surface gravity
(g = 9.8 ms−2 ) and background atmospheric composition (CO2 and N2 ). Additionally, to the changes of the surface albedo (see section 8.4), the relative humidity
(see section 8.5) and the water reservoir of the planet (see section 8.6), for the
case study of the Kepler 22b-like planet also the planetary gravity was changed
(g = 1.77gEarth )(see section 8.7).
Future model scenarios could calculate the inner boundary of the HZ with different
atmospheric compositions, e.g. taking more CO2 into account which enhances the
greenhouse effect. Furthermore, the influence of changing the gravity of the planet
on the inner boundary of the HZ can be investigated.
151
152
Summary and Outlook
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Danksagung
Zunächst einmal möchte ich Frau Prof. Dr. Heike Rauer für die Betreuung dieser
Arbeit danken. Besonderer Dank gilt Frau Prof. Dr. Heike Rauer dafür, dass sie sich
die Zeit genommen hat meinen Arbeitsprozess kontinuierlich zu begleiten und die
Arbeitsergebnisse immer wieder kritisch zu diskutieren. Ich bedanke mich, dass sie
mir ermöglicht hat, diese Arbeit abzuschließen und nicht zuletzt für die Begutachtung dieser Dissertationsschrift.
Den weiteren Mitgliedern der Prüfungskommission Prof. Dr. Erwin Sedlmayer und
Prof. Dr. Mario Dähne möchte ich dafür danken, dass sie sich Zeit für die Begutachtung dieser Arbeit nehmen.
Philip von Paris möchte ich für die sehr gute Zusammenarbeit an dem dem Atmosphärenmodell danken, vor allem an der infrarot Strahlungstransportroutine und für
intensive Diskussion über das Model, Buffy und Gott und die Welt.
Bei Lee Grenfell, Mareike Godolt, Philip von Paris und Anders Erikson möchte
ich für das gründliche Korrekturlesen dieser Arbeit bedanken und für wertvolle
Ratschläge zur Verbesserung dieser Arbeit. Besonders Lee Grenfell und Mareike
Godolt seien hier hervorgehoben, die mich großartig unterstützt haben.
Mareike Godolt möchte ich an dieser Stelle nochmal ganz besonders herzlich danken,
vor allem für unsere montäglichen Treffen und auch sonstigen Diskussionen. Diese
Treffen haben mich sehr motiviert und mir Begeisterung an der Arbeit zurückgegeben.
Vor allem bin ich sehr dankbar für die wertvolle Freundschaft, die daraus entstanden
ist.
Bei Beate Patzer möchte ich mich für ihre Diskussionsbereitschaft bedanken, die
dazu führte mich noch intensiver mit dem Phänomen ’Runaway Greenhouse’ zu
beschäftigen.
Prof. Dr. Tilman Spohn möchte ich danken, dass er mir ermöglicht hat auch im Rahmen meiner Arbeit als Graduate Training Manager der Helmholtz Allianz ’Planetary
Evolution and Life’ weiterhin für meine Doktorarbeit zu forschen und zudem auch
für motivierende Diskussionen über diese Arbeit.
Ich danke allen Mitarbeitern der Abteilung ’Extrasolare Planeten und Atmosphären’
des DLR für die nette Atmosphäre und das gute Arbeitsklima. Insbesondere möchte
ich der Atmos-Gruppe danken für die wichtigen und konstruktiven Diskussionen.
Alexander Hölscher, Daniel Kitzmann und Joachim Stock möchte ich danken, die
mich in Programmierfragen zu IDL und Fortran beraten haben.
Bedanken möchte ich mich auch bei Beate Richter, die mir geholfen hat strukturierter an dieser Arbeit zu arbeiten.
Von ganzem Herzen danke ich meiner Familie, vor allem meinen Eltern und meinem
Bruder, dafür, dass sie mich in jeglicher Hinsicht unterstützt und aufgebaut haben.
Zudem danke ich auch meinen Freunden, die für mich da waren, wenn ich sie
brauchte und mir starken Rückhalt gegeben haben. Besonderer Dank geht an dieser
Stelle an Ivana.