Escape, temperature and atmosphere relationship

What is the critical escape speed needed to hold atmosphere, and how
will it depend on the temperature?

Let us take the major Solar System examples, from outside in:

Pluto: 1,18 km per second, approaches to 30 a. u.. Appreciable nitrogen
atmosphere restricted by freezing.
Triton: 1,45 km per second, orbits at 30 a. u.. Appreciable nitrogen
atmosphere restricted by freezing.
Titan: 2,64 km per second, orbits at 10 a. u.. A nonfreezing nitrogen
atmosphere, the inventory being 160 000 Pa.
Io: 2,57 km per second, orbits at 5,2 a. u.. Not much atmosphere.
Europa: 2,03 km per second, orbits at 5,2 a. u.. Not much atmosphere.
Ganymede: 2,73 km per second, orbits at 5,2 a. u.. Not much atmosphere.
Callisto: 2,44 km per second, orbits at 5,2 a. u.. Not much atmosphere.
Mars: 5,03 km per second, orbits at 1.52 a. u.. A freezing carbon
dioxide atmosphere, contains nonfreezing nitrogen inventory of 20 Pa.
Moon: 2,38 km per second, orbits at 1 a. u.. Not much atmosphere.
Earth: 11,2 km per second, orbits at 1 a. u.. Nitrogen atmosphere of
approximately 80 000 Pa.
Venus: 10,4 km per second, orbits at 0,72 a. u.. Carbon dioxide
atmosphere with nitrogen inventory of about 300 000 Pa.
Mercury: 4,44 km per second, orbits at 0,39 a. u.. Not much atmosphere.

I wonder if there is any rule between the nitrogen contents of the
three dense nitrogen atmospheres?

wrote:

What is the critical escape speed needed to hold
atmosphere, and how will it depend on the temperature?

Well a very very simple constraint is that the average speed of a
gas molecule should be below escape velocity. If a gas molecule at the
top of the atmosphere is moving faster than escape velocity, it's not
going to remain in the atmosphere (this is often termed "thermal
escape" or "Jeans escape").
Again very roughly, compare the RMS thermal velocity of a particular
type of molecule at the top of the atmosphere with the escape velocity:
v_esc / v_rms = 3 -- half-life of days
v_esc / v_rms = 4 -- half-life of decades
v_esc / v_rms = 5 -- half-life of 10's of millions of years
v_esc / v_rms = 6 -- half-life of billions of years

The real problem with this is that the loss mechanism is almost
never purely thermal, and even then the temperature of interest is that
of the exobase (where the mean free path of a molecule is on the order
of an atmospheric scale height), which is hard to simply predict. For
instance, Earth's exobase temperature is roughly 1000 K, while Venus
has an exobase temperature closer to 300 K, even though it's closer to
the Sun.

The nitrogen inventory question is more interesting - try searching
for information about secondary atmospheres, nitrogen, and stable
isotopes as tracers.

--
Brian Davis

wrote:
What is the critical escape speed needed to hold atmosphere, and how
will it depend on the temperature?

Let us take the major Solar System examples, from outside in:

Pluto: 1,18 km per second, approaches to 30 a. u.. Appreciable nitrogen
atmosphere restricted by freezing.
Triton: 1,45 km per second, orbits at 30 a. u.. Appreciable nitrogen
atmosphere restricted by freezing.
Titan: 2,64 km per second, orbits at 10 a. u.. A nonfreezing nitrogen
atmosphere, the inventory being 160 000 Pa.
Io: 2,57 km per second, orbits at 5,2 a. u.. Not much atmosphere.
Europa: 2,03 km per second, orbits at 5,2 a. u.. Not much atmosphere.
Ganymede: 2,73 km per second, orbits at 5,2 a. u.. Not much atmosphere.
Callisto: 2,44 km per second, orbits at 5,2 a. u.. Not much atmosphere.
Mars: 5,03 km per second, orbits at 1.52 a. u.. A freezing carbon
dioxide atmosphere, contains nonfreezing nitrogen inventory of 20 Pa.
Moon: 2,38 km per second, orbits at 1 a. u.. Not much atmosphere.
Earth: 11,2 km per second, orbits at 1 a. u.. Nitrogen atmosphere of
approximately 80 000 Pa.
Venus: 10,4 km per second, orbits at 0,72 a. u.. Carbon dioxide
atmosphere with nitrogen inventory of about 300 000 Pa.
Mercury: 4,44 km per second, orbits at 0,39 a. u.. Not much atmosphere.

I wonder if there is any rule between the nitrogen contents of the
three dense nitrogen atmospheres?

Probably not, but the very nature of atmospheric leakage means that
lighter molecules will leak at a greater rate than heavier ones.
Leakage occurs when the average speed of a molecule is about that of
escape speed at the exosphere interface. Lighter molecules with the
same energy have a greater speed, and so the lighter molecules will tend
to escape at a greater rate than heavier ones.

So it's not surprising that hydrogen and helium are nowhere to be found
in these terrestrial atmospheres. (For jovian planets like Jupiter,
that's changed because their enormous gravity now means that the escape
speed at the exosphere is very high, and so they can retain much lighter
molecules all the way down to H2.)

So it's not surprising that the atmospheres we see on the list are
nitrogen, carbon dioxide, and oxygen -- with only the Earth having
significant quantities of O2, since that's kept up by life. If there's
any significant to N2 being listed over CO2 in this list, I'd expect
it's a chemical favoritism in terms of carbon getting absorbed and
interacted with more readily than nitrogen by the stuff that tends to
sit around on the surfaces of terrestrial worlds.

--
Erik Max Francis && && http://www.alcyone.com/max/
San Jose, CA, USA && 37 20 N 121 53 W && AIM erikmaxfrancis
If the sky should fall, hold up your hands.
-- (a Spanish proverb)

Erik Max Francis wrote:
wrote:
What is the critical escape speed needed to hold atmosphere, and
how
will it depend on the temperature?

Let us take the major Solar System examples, from outside in:

Pluto: 1,18 km per second, approaches to 30 a. u.. Appreciable
nitrogen
atmosphere restricted by freezing.
Triton: 1,45 km per second, orbits at 30 a. u.. Appreciable
nitrogen
atmosphere restricted by freezing.
Titan: 2,64 km per second, orbits at 10 a. u.. A nonfreezing
nitrogen
atmosphere, the inventory being 160 000 Pa.
Io: 2,57 km per second, orbits at 5,2 a. u.. Not much atmosphere.
Europa: 2,03 km per second, orbits at 5,2 a. u.. Not much
atmosphere.
Ganymede: 2,73 km per second, orbits at 5,2 a. u.. Not much
atmosphere.
Callisto: 2,44 km per second, orbits at 5,2 a. u.. Not much
atmosphere.
Mars: 5,03 km per second, orbits at 1.52 a. u.. A freezing carbon
dioxide atmosphere, contains nonfreezing nitrogen inventory of 20
Pa.
Moon: 2,38 km per second, orbits at 1 a. u.. Not much atmosphere.
Earth: 11,2 km per second, orbits at 1 a. u.. Nitrogen atmosphere
of
approximately 80 000 Pa.
Venus: 10,4 km per second, orbits at 0,72 a. u.. Carbon dioxide
atmosphere with nitrogen inventory of about 300 000 Pa.
Mercury: 4,44 km per second, orbits at 0,39 a. u.. Not much
atmosphere.

I wonder if there is any rule between the nitrogen contents of the
three dense nitrogen atmospheres?

Probably not, but the very nature of atmospheric leakage means that
lighter molecules will leak at a greater rate than heavier ones.
Leakage occurs when the average speed of a molecule is about that of
escape speed at the exosphere interface. Lighter molecules with the
same energy have a greater speed, and so the lighter molecules will
tend
to escape at a greater rate than heavier ones.

So it's not surprising that hydrogen and helium are nowhere to be
found
in these terrestrial atmospheres. (For jovian planets like Jupiter,
that's changed because their enormous gravity now means that the
escape
speed at the exosphere is very high, and so they can retain much
lighter
molecules all the way down to H2.)

So it's not surprising that the atmospheres we see on the list are
nitrogen, carbon dioxide, and oxygen -- with only the Earth having
significant quantities of O2, since that's kept up by life. If
there's
any significant to N2 being listed over CO2 in this list, I'd expect
it's a chemical favoritism in terms of carbon getting absorbed and
interacted with more readily than nitrogen by the stuff that tends to

sit around on the surfaces of terrestrial worlds.


Exactly.

However: given the observed inertness of nitrogen, why does Earth have
4 times less of it than Venus and twice less than Titan?

Also: the first principles computations would show that the temperature
falls with the square root of distance, and the velocity of a molecule
of given mass falls with the fourth root of the distance.

Pluto approaches to 30 a. u. and keeps appreciable amounts of nitrogen
though the escape speed is 1,18 km/s. Jupiter orbits at one sixth the
distance, therefore the temperature would be about 2,5 times higher
than on Pluto and the speed of molecules about 160 % that on Pluto. All
Galileian satellites have escape speeds of over 2 km/s, but for some
reason lack nitrogen atmospheres.

Titan orbits at about 9,5 a. u. and has a dense nitrogen atmosphere of
160 000 Pa. Mars is about 6 times closer and ought to have about 250 %
the temperature of Titan, molecules having about 160 % the speed. Yet
though the escape speed from Mars is 190 % that of Titan, Mars only has
20 Pa of nitrogen.

On 7 May 2005 wrote:

[snip]
However: given the observed inertness of nitrogen, why does Earth have
4 times less of it than Venus and twice less than Titan?

Also: the first principles computations would show that the temperature
falls with the square root of distance, and the velocity of a molecule
of given mass falls with the fourth root of the distance.
[snip]


This whole conversation has completely neglected the role of interference
from other bodies.

The Earth has a large moon to skim it's upper exosphere aggressively.
The Moon's "exophere" is completely disrupted by the Earth.

The remarkable thing is that any of the gas giant moons have
an atmosphere at all!

The fact that they do is probably a function of renewal and "gas torus"
mechanics. The gravity well around Jupiter and Saturn is a lot "steeper"
than the solar gradient.

It also wouldn't surprise me to discover that the composition of the
atmospheres of satellites is linked to off-gassing from their parent
(gas-giant) planets.

I firmly believe that Earth is going to turn out to be a fluke... due to
the presence of our large moon. (Which pushes the boundaries of calling
Earth/Moon a binary planet rather than a proper planet and satellite.)

Gene P.

--
Alcore Nilth - The Mad Alchemist of Gevbeck

chornedsnork wrote:

given the observed inertness of nitrogen, why does
Earth have 4 times less of it than Venus and twice
less than Titan?

Partially because it's not inert - nitrogen can be fixed (even by
abiotic processes) in an atmosphere containing oxygen, so it might be
better to say that part of Earth's nitrogen inventory is tied up in
reservoirs other than the atmosphere. The actual ratio of, say, CO2 to
N2 for Earth & Venus are very close - what's not is the form it's in.
Just because it's a "volatile" doesn't mean it's in the atmosphere
(most of Earth once-free O2 isn't there now, for instance, there's more
"free oxygen" floating around in the oceans as sulfates than in the
atmosphere even now).

the first principles computations would show that the
temperature falls with the square root of distance, and
the velocity of a molecule of given mass falls with the
fourth root of the distance.

Which sounds great, but doesn't work out very well at all. Venus is
too hot by that standard, and yet it's temperature at the exobase is
actually *lower* than Earth's. First principles are great... but if a
calculation based on first principles doesn't work (or often even if it
does), then you have to get beyond them.

Also, if you are comparing atmospheres, pressures can be misleading,
as that's a product of how much gas you have as well as the surface
gravity (i.e. - if Mars & Earth had "sweated out" the same amount of
atmosphere per unit mass, they still wouldn't have remotely the same
surface pressure). Mass of the atmosphere (or particular gas) as a
fraction of planetary mass might be better, or even for some
calculations column density (the pressure divided by the surface
gravity).

--
Brian Davis

Gene P. wrote:

The Earth has a large moon to skim it's upper exosphere aggressively.
=A0
The Moon's "exophere" is completely disrupted by the Earth. =A0

Hmm. Seeing as how the exosphere for the Earth is around 500 km, I
suspect the Moon is doing absolutely nothing with respect to
atmospheric escape from the Earth-moon system. After all, if a molecule
or atom leaves the exobasse at escape velocity, it... leaves. The Moon
will not gravitationally confine it to the system, and it's chances of
hitting any molecule leaving the exobase is about 1 in 200,000.

The remarkable thing is that any of the gas giant moons
have an atmosphere at all!

Why? They are shielding from solar wind stripping by large external
magnetic fields, are very cold so thermalJeans escape mechanism are
low, and as you point out, at least for very thin atmospheres, the
presence of a gas torus slows loss.

I firmly believe that Earth is going to turn out to be a fluke...

Fair enough. I'm waiting for evidence. The best argument I've seen
to that effect is the "Earth's obliquity is stabilized by our large
Moon", which is true. But it's only destabilizede in the first place by
Jupiter. If Jupiter had a lower mass, or a different location in the
solar system, then the obliquity of the Earth without the Moon would be
stable. Lasker's paper on this topic is fairly clear (OK, that's not
true; but I think the above is fairly clearly stated).

--=20
Brian Davis

wrote:
chornedsnork wrote:

given the observed inertness of nitrogen, why does
Earth have 4 times less of it than Venus and twice
less than Titan?

Partially because it's not inert - nitrogen can be fixed (even by
abiotic processes) in an atmosphere containing oxygen,

Which means nitrate. But the ocean on Earth holds no large amounts of
nitrates.

so it might be
better to say that part of Earth's nitrogen inventory is tied up in
reservoirs other than the atmosphere. The actual ratio of, say, CO2
to
N2 for Earth & Venus are very close - what's not is the form it's in.
Just because it's a "volatile" doesn't mean it's in the atmosphere
(most of Earth once-free O2 isn't there now, for instance, there's
more
"free oxygen" floating around in the oceans as sulfates than in the
atmosphere even now).

the first principles computations would show that the
temperature falls with the square root of distance, and
the velocity of a molecule of given mass falls with the
fourth root of the distance.

Which sounds great, but doesn't work out very well at all. Venus
is
too hot by that standard, and yet it's temperature at the exobase is
actually *lower* than Earth's. First principles are great... but if a
calculation based on first principles doesn't work (or often even if
it
does), then you have to get beyond them.

Also, if you are comparing atmospheres, pressures can be
misleading,
as that's a product of how much gas you have as well as the surface
gravity (i.e. - if Mars & Earth had "sweated out" the same amount of
atmosphere per unit mass, they still wouldn't have remotely the same
surface pressure). Mass of the atmosphere (or particular gas) as a
fraction of planetary mass might be better, or even for some
calculations column density (the pressure divided by the surface
gravity).

Very well.
For a given partial pressure, the column density on Venus is about 9%
bigger, atmospheric mass is 0,2 % smaller and mass ratio is 22,4 %
bigger. Thus Venus has 5 times more nitrogen than Earth.

On Mars, the column density is 266%, mass is 75,5 % and mass ratio is
706 %. Thus Mars has still over 500 times less nitrogen.

On Titan, column density is about 7 times that on Earth, total mass is
about 120 % and the mass ratio is about 55 times bigger. Thus Titan has
100 times the nitrogen inventory of Earth.

--
Brian Davis

chornedsnork wrote:

[a bunch of good stuff]

Dang, you ask some good question, sir! Hmm...

Which means nitrate. But the ocean on Earth holds no
large amounts of nitrates.

You're right. Doing some research ("Biogeochemistry", Schlesinger)
there's an estimated 3.9e+18 kg N in the atmosphere, 3.5e+12 kg in
biomasss, & around 120e+12 kg in organic material. The same reference,
however, does mention when comparing planets (Venus, Earth, & Mars)
that Mars is light enough & warm enough to have lost some N via thermal
process (and there's some evidence for this, in the stable isotope
ratios; Mars has the light isotope significantly enriched over
terrestrial values). As to the difference between Earth & Venus, it's
mentioned that it's very difficult to determine the C/N ratio
accurately for most planets, so it is the *books* contention that both
Earth & Venus have fairly identical *ratios* of the main materials.

Look at it this way: assume the atmospheres of the terrestrial
planets are secondary, outgassed from the bulk planet. Earth has a pN2
of 79,118 Pa, or a column density of 8,065 Kg/m^2, or a total
atmospheric inventory of 4.122e+18 kg N2, which I'm assuming outgassed
from a planet massing 5.98e+24 kg. That's an inventory of 6.9e-7 kg N2
per kg of planet. For Venus, that's a pN2 of 304,000 Pa, a column
density of 34,150 kg/m^2, which is a total inventory of 1.57e+19 kg and
relative to planet mass an inventory of 3.2e-6 kg of N2 per kg of
planet, or about 4.7 times Earth's. But given the number of assumptions
and unknowns, I'd say these are not (at this point) completely out of
line. Actually, to me, the surprise is that the inventories for Earth &
Venus are so close
Another reference ("Moons & Planets, 4th ed.", Hartmann) lists
identical inventories (per unit planet mass) for CO2 & N2 for both
Earth & Venus (Table 11-2, p320; sources listed therein). These figures
are off mine by almost a factor of 10 in some cases, and none of them
are quoted to more than at most two significant figures (no error
estimates), so i don't think the figures are well known in any event.

On Titan, column density is about 7 times that on Earth,
total mass is about 120 % and the mass ratio is about 55
times bigger. Thus Titan has 100 times the nitrogen inventory of
Earth.

pN2 of 136,830 Pa, column density 97,740 kg/m^2, total mass 8.1e+12
kg N2, or as a ratio 6.1e-5 kg N2 per kg "planet" mass, about 88 times
Earth's. But I'd expect titan's inital inventory of potential N source
would be much larger, due to the materials it would have formed from
(NH3 or NH3-H2O eutetic ices would have potentially been a large
contributor to such an outer solar system object; they would be very
very rare in planetismals around 0.7 to 3.0 AU, where the terrestrial
planets formed).

In other words, relative to body masses, Earth & Venus have very
similar inventories of N2, while Titan has a lot more (as would be
expected, due to potential for incorperating low-temperature ices in
the accretion phase), and Mars has a lot less (due to thermal escape,
which would be theoretically possible, and which would explain the
stable isotope data).

Does that makes sense? Good questions!

--
Brian Davis

wrote:
Erik Max Francis wrote:
wrote:
What is the critical escape speed needed to hold atmosphere, and
how
will it depend on the temperature?

Let us take the major Solar System examples, from outside in:

Pluto: 1,18 km per second, approaches to 30 a. u.. Appreciable
nitrogen
atmosphere restricted by freezing.
Triton: 1,45 km per second, orbits at 30 a. u.. Appreciable
nitrogen
atmosphere restricted by freezing.
Titan: 2,64 km per second, orbits at 10 a. u.. A nonfreezing
nitrogen
atmosphere, the inventory being 160 000 Pa.
Io: 2,57 km per second, orbits at 5,2 a. u.. Not much atmosphere.
Europa: 2,03 km per second, orbits at 5,2 a. u.. Not much
atmosphere.
Ganymede: 2,73 km per second, orbits at 5,2 a. u.. Not much
atmosphere.
Callisto: 2,44 km per second, orbits at 5,2 a. u.. Not much
atmosphere.
Mars: 5,03 km per second, orbits at 1.52 a. u.. A freezing carbon
dioxide atmosphere, contains nonfreezing nitrogen inventory of 20
Pa.
Moon: 2,38 km per second, orbits at 1 a. u.. Not much atmosphere.
Earth: 11,2 km per second, orbits at 1 a. u.. Nitrogen atmosphere
of
approximately 80 000 Pa.
Venus: 10,4 km per second, orbits at 0,72 a. u.. Carbon dioxide
atmosphere with nitrogen inventory of about 300 000 Pa.
Mercury: 4,44 km per second, orbits at 0,39 a. u.. Not much
atmosphere.

I wonder if there is any rule between the nitrogen contents of
the
three dense nitrogen atmospheres?

Probably not, but the very nature of atmospheric leakage means that
lighter molecules will leak at a greater rate than heavier ones.
Leakage occurs when the average speed of a molecule is about that
of
escape speed at the exosphere interface. Lighter molecules with
the
same energy have a greater speed, and so the lighter molecules will
tend
to escape at a greater rate than heavier ones.

So it's not surprising that hydrogen and helium are nowhere to be
found
in these terrestrial atmospheres. (For jovian planets like
Jupiter,
that's changed because their enormous gravity now means that the
escape
speed at the exosphere is very high, and so they can retain much
lighter
molecules all the way down to H2.)

So it's not surprising that the atmospheres we see on the list are
nitrogen, carbon dioxide, and oxygen -- with only the Earth having
significant quantities of O2, since that's kept up by life. If
there's
any significant to N2 being listed over CO2 in this list, I'd
expect
it's a chemical favoritism in terms of carbon getting absorbed and
interacted with more readily than nitrogen by the stuff that tends
to

sit around on the surfaces of terrestrial worlds.


Exactly.

However: given the observed inertness of nitrogen, why does Earth
have
4 times less of it than Venus and twice less than Titan?

Also: the first principles computations would show that the
temperature
falls with the square root of distance, and the velocity of a
molecule
of given mass falls with the fourth root of the distance.

Since luminosity is inversely proportional to the square of the
distance, and luminosity is proprotional to the 4th power of the
temperature, temperature falls off with the square of the distance, not
the square root- A planet 4 times as far from the sun will have about
half the absolute temerature- A. McIntire