Atmospheres
●
Terrestrial planet atmospheres
96% CO2 4% N2
96% CO2 4% N2
78% N2 21% O2
“Atmospheres”
●
Jovian worlds
“Atmospheres”
●
Jovian worlds
“Atmospheres”
●
Jovian worlds
Atmospheres
90+% N2 + CH4
Detecting Pluto's Atmosphere via Stellar Occultation
Consider a single, completely elastic, gas atom...
●
Estimate velocity vs. kT...
1
3
2
m v = kT
2
2
●
√
3kT
v=
m
Consider a particle launched from the surface at this speed,
how high does it get?
v initial
t=
g
2
initial
1 2 v
x = gt =
2
2g
On Earth, presuming a nitrogen molecule, this particle bounces up to an
altitude of about 10 kilometers – pretty consistent with the thickness of
the lower atmosphere.
Atmospheric Retention and Escape Velocity
●
Reference the previous equation to consider escape to infinity.
Find T such that
v escape
3kT
=
α
m
√
v escape =
√
2GM planet
R
As opposed to the
“average” velocity,
particles that can
escape are in the
high-velocity “tail” of
the
Maxwell-Boltzmann
distribution (α= 5-6)
Atmospheric Retention and Escape Velocity
●
The retention condition then is that the mean thermal velocity
be several times (α) lower than the escape velocity.
–
Different molecules/atoms have different escape conditions.
–
Heavy molecules can stick around. Hydrogen is most likely to
escape.
–
T, of course, is a function of distance from the Sun. More distant
objects have a better chance of holding on to an atmosphere.
T =
2 G M planet mmolecule
2
3 α k R planet
Atmospheric Retention and Escape Velocity
●
●
●
Molecules escape from the “exosphere” - the region of the
atmosphere where particle trajectories are unlikely to collide.
For Earth the temperature of this layer is about 1000K.
One can calculate an escape criterion based on the mean
molecular weight of the species, µ.
m molecule = μ m H
T ex
M
μ>7.1
1000K M earth
(
●
)(
−1
R
) (R )
earth
The Earth readily holds on to N2 and O2. Hydrogen atoms
escape easily.
T =
2 G M planet m molecule
2
3 α k R planet
Escape from the Exosphere
●
The Exosphere/Thermosphere is the region of the Earth's
atmosphere that is so tenuous that gas atoms are on ballistic
trajectories (few collisions). Gas escapes to space from this
environment, which, energetically, is much warmer than the
surface.
The Earth's Geocorona
●
●
The Lyman-alpha transition of hydrogen atoms scatters
sunlight to make visible the escaping hydrogen in Earth's
atmosphere.
The outer boundary (edge of the exosphere) is established
where solar radiation pressure exceeds the gravitational
influence of the Earth.
The Lunar “Atmosphere”
●
At a density of 10-14 that of Earth's, the Moon has a transient
atmosphere maintained by radioactive decay (helium, argon)
and sputtered by the solar wind and cosmic rays (potassium,
sodium).
http://nightsky.jpl.nasa.gov/download-view.cfm?Doc_ID=483
http://www.nasa.gov/mission_pages/LADEE/news/lunar-atmosphere.html#.UoomszyJAjA
The Solar Wind
●
●
●
At the solar photospheric temperature of
6000K hydrogen is secure from escape.
However, through a process that is not yet
well understood, solar magnetic fields heat
the upper atmosphere substantially.
The escape velocity from the surface of the
Sun is 620 km/s and the tenuous outer solar
atmosphere escapes.
v proton
T
= 160 km/ s 6
10 K
(
)
1
2
The Solar Wind
●
●
The wind is tenuous – about 107 atoms/m3 or
10-21 kg/m3 measured 1AU from the Sun.
The mean velocity is about 400 km/s –
consistent with the thermal energy of the
escaping gas – and is constant with
increasing distance from the Sun since it is
well above escape velocity.
–
●
Given the above two facts the density must
fall off as 1/r2
The mass flux is about 108 kg/s
2
Ṁ =4 π r v ρ
v proton
T
= 160 km/ s
6
10 K
(
)
1
2
The Heliopause
●
The mass flux of the solar
wind represents momentum.
–
As long as this momentum
dominates the
momentum/inertia of the
surrounding (interstellar)
medium the solar wind
flows unimpeded.
–
At some point the solar
wind's density falls to the
point that the interstellar
medium begins to
dominate – the heliopause.
–
In reality things are more
complicated due to shocks,
magnetic fields, and other
complexities
The Heliopause
●
The mass flux of the solar
wind represents momentum.
–
As long as this momentum
dominates the
momentum/inertia of the
surrounding (interstellar)
medium the solar wind
flows unimpeded.
–
At some point the solar
wind's density falls to the
point that the interstellar
medium begins to
dominate – the heliopause.
–
In reality things are more
complicated due to shocks,
magnetic fields, and other
complexities
Solar Wind Interaction with Earth's Magnetic Field
●
●
●
Unimpeded, the solar wind would interact directly with Earth's
outer atmosphere and erode it.
The low pressure of the martian atmosphere is likely a
consequence of this effect (or maybe not).
The Earth's magnetic field deflect (charged) solar wind
protons.
Martian Atmospheric Loss
●
●
All evidence for liquid flowing water on the surface of Mars
points to an era when Mars had a substantially thicker
atmosphere.
Simply accounting for the Maxwell velocity distribution Mars
should be able to hold on to a thick CO2 atmosphere, however
solar wind stripping may play a significant role in planets with
marginally bound atmospheres and weak magnetic fields.
The Mars Atmosphere and Volatile
EvolutionN experiment (launching
today) aims to characterize Mars'
atmospheric composition.
It will orbit in a 6000 x 125 km ellipse
dipping into the Martian exosphere
each orbit with instruments that can
directly sample atmospheric
composition. It will correlate
atmospheric composition and structure
with solar activity.
Hydrostatic Equilibrium and Scale Height
●
Each layer of atmosphere must support the force/weight of the
atmosphere above. The differential force (that is pressure)
across any layer (dr) of the atmosphere is:
G
M
(r
)ρ
dP = −ρ g dr = −
dr
2
r
●
The ideal gas law relates P and ρ
P=
ρk T
μ m proton
g μ mp
dP
∫ P =− k T ∫ dr
for constant g near a planetary surface, i.e. a
simple plane-parallel atmosphere
Hydrostatic Equilibrium and Scale Height
●
Each layer of atmosphere must support the force/weight of the
atmosphere above. The differential force (that is pressure)
across any layer (dr) of the atmosphere is:
G
M
(r
)ρ
dP = −ρ g dr = −
dr
2
r
●
The ideal gas law relates P and ρ
P=
ρk T
μ m proton
−Δ r
P (r)=P o exp
H
(
)
kT
H=
g μ m proton
Atmospheric “scale height”
Conditions in the Solar Core
●
Integrate the same equation, but accounting for the change in
gravitational acceleration (and density) with radius.
G M (r )ρ
−dP = ρ g dr =
dr =
2
r
●
●
4 3
G π r ρ(r ) ρ(r )
3
(
)
r
2
dr
Uh oh, this could get messy (they'll tell you all about it in
graduate school)
In the meantime..
2
dP ≃ P central = g ρ = G M sun M sun
2
dr
R sun
R sun 4 π R 3
sun
3
( )
P central ≃
3 G M sun
4π R
4
sun
Temperature in Solar Core
●
Appeal to the ideal gas law (not a bad approximation since
temperatures are so high we are dealing with individual
elementary particles)
dP ≃ P central = g ρ = G M ρ
2
dr
R
R
¿
¿
¿
P = k T ≃ G M sun
ρ
m particle
R
G M sun m particle
Tc =
R sun k
P=
ρk T
μ m proton
Temperature in Solar Core
●
Appeal to the ideal gas law (not a bad approximation since
temperatures are so high we are dealing with individual
elementary particles)
dP ≃ P central = g ρ = G M ρ
2
dr
R
R
¿
¿
P=
ρk T
μ m proton
¿
P = k T ≃ G M sun
ρ
m particle
R
G M sun m particle
Tc =
R sun k
P.E.
= 12 million K =
k
skin crawling physics moment... the
central temperature of the Sun is simply
related to the gravitational potential
energy of a single proton at the Sun's
surface
The Virial Theorm
●
In a self-gravitating system in equilibrium kinetic and
gravitational energy are equi-partitioned such that
2∗K.E. = − P.E.
Why isn't Jupiter a Star
●
Jupiter is 1/1000th the mass of the Sun and 1/10th the radius,
so it's central temperature should be roughly 1/100th that of
the Sun or about 100,000K – far too low for nuclear fusion.
G M sun m particle
M
Tc =
α
R sun k
R
●
But the real question is why can't it become a star...
–
As it radiates away thermal energy it should be able to contract,
giving up gravitational potential energy and since M is constant
and R is getting smaller – get hotter. In fact getting hot enough
to light its nuclear furnace stabilizing the configuration.
●
●
The description above is exactly how a star “ignites” and achieves
long term stability.
Any self-gravitating sphere of hydrogen should be able to get hot
enough eventually to achieve hydrogen fusion, however....
Degeneracy Pressure
●
●
...the scenario on the previous slide would be true if simple
gas pressure were the only pressure in play.
At high densities, that is small volume per individual particle,
quantum mechanical degeneracy pressure (the desire of an
electron not to be spatially confined) begins to dominate gas
pressure and stops contraction.
ΔxΔ p ≃ ℏ
●
●
Gas pressure supports the Sun. Degeneracy pressure
intervened long ago to halt the collapse of Jupiter and limit the
core temperature.
You'll learn about white dwarfs and neutron stars (related
phenomena) in detail next semester.
Brown Dwarfs
●
So, degeneracy pressure intervened to prevent Jupiter from
becoming a star, but such was not the case for the Sun.
–
●
●
●
Somewhere in between these two masses (0.001Msun and
1.0 Msun) lies the transition between brown dwarfs and stars.
Brown dwarfs are star-like self gravitating balls of hydrogen
where degeneracy pressure has thwarted achieving the
required internal temperature necessary for thermonuclear
fusion (a few million degrees)
Object with masses lower than 0.08Msun (80 times the mass
of Jupiter) are brown dwarfs – which cool steadily after an
initially hot formation.
Mind stretcher – the luminosity of a star has to do with it's
configuration – not with the fact that nuclear burning is
happening inside. The energy release from nuclear burning
maintains the configuration...
Back Down to Earth
●
The temperature structure of the atmosphere is dictated by
heating, cooling and bulk motion.
Convection in the Troposphere
●
Heating vs. cooling dictates the temperature profile at any
layer of the atmosphere.
–
●
Heating vs. cooling translates to absorption vs. ease of
radiation.
Sunlight heating the surface dumps the most energy into the
atmosphere.
–
The atmosphere itself radiates poorly so bulk motion of the
atmosphere is required to get the energy out → convection.
Convection in the Troposphere
●
●
●
If a parcel of atmosphere is displaced upward it will find itself
in a region of different pressure, temperature, and, in
particular, density.
In moving to a region of different pressure the parcel expands
doing work against its environment and cooling – adiabatic
expansion.
The parcel is the equivalent of a hot air balloon, if it's new
density is less than the local density it will be buoyant and
continue to rise.
Cloud Decks
●
●
Clouds condense at the altitude where the temperature, set by
adiabatic convection hits the condensation (dew) point.
Ever notice that the cloud base for cumulus is lower when it is
more humid?
Temperature Profile in the Troposphere
●
●
●
●
Adiabatic cooling thus sets the temperature profile in the
troposphere due to convection.
Temperature profiles can get complicated at night as
temperature inversion turns off convection and the lower
atmosphere can stratify.
The vertical temperature profile is set by the “adiabatic lapse
rate” which depends on the amount of water in the
atmosphere.
–
Dry adiabatic lapse rate = 9.8C / kilometer – 5o F/thousand feet
–
Wet adiabatic lapse rate = 5C / kilometer – 3oF/thousand feet
Clouds form at the level where the cooling crosses the dew
point.
More about Convection

Convection occurs when the vertical temperature profile is
less steep than the adiabatic lapse rate.
−


A rising parcel of air finds itself less dense than its surroundings
and is thus buoyant.
Convection is an extremely efficient conveyor of energy.
−
Once turned on, convection can easily carry the excess energy
from the “hot” side of the fluid to the “cold”.
−
Convective atmospheres have their temperature profile set
exactly to the adiabatic profile. No matter how much energy you
put in at the bottom, convection can deal with the excess.
Jupiter and low-mass (less massive than about 0.3Mo) are
fully convective on the interior.
−
Their internal temperature profiles are adiabatic.