Download Sublimation • In a vapour in thermal equilibrium, the molecules of

Sublimation
• A sample of ice in a vacuum will sublimate, and the vapour will fill the
⎛ HL HL ⎞
⎟⎟ ,
−
kT
kT
0
⎝
⎠
chamber until it reaches the vapour pressure, pv = p0 exp⎜⎜
where HL is the latent heat of vapourization (the amount of energy
required to remove one molecule). T0, p0 are constants
Gas
p0
(Pa)
T0
(K)
HL
(10-20 J/molecule)
N2
133
47
1.1
CH4
133
67
1.5
CO2
133
139
3.8
NH3
133
164
5.0
H2O
133
256
7.6
• In a vapour in thermal equilibrium, the molecules of mass m
are moving randomly, with velocities v, and the kinetic
energy is equal to the thermal energy:
µm H v 2
2
=
3
kT
2
• The number of molecules per unit volume striking a surface,
per unit time, is just
⎛ 1 ⎞⎛ 1 ⎞
Z = ⎜ ⎟⎜ ⎟nv ,
⎝ 2 ⎠⎝ 3 ⎠
where the factor 1/6 is due
to the fact that we only count particles moving in the (say)
+x direction
• Using the ideal gas law to relate n and vapour pressure pV,
we get an expression for the outflow rate (molecules per
unit time per unit area sublimating):
⎛1⎞ p
Z =⎜ ⎟ v
⎝ 6 ⎠ kT
3kT
pv
=
µmH
12kTµmH
• Comet activity at >3 AU indicates the presence of volatile
ices such as NH3 (ammonia), CH4 (methane), CO2, and N2.
1. We see the sublimated gas because of the photons the
atoms absorb from solar radiation and then reemit.
Comet Orbits
• Long-period comets have:
¾ Highly eccentric orbits
¾ Large semi-major axes and, therefore, aphelion
distances
¾ Originate from all directions
¾ Postulated to originate from the Oort cloud, a
spherical shell surrounding the SS at great
distance (at least ~104 AU).
ƒ Short-period comets:
¾ Represent about 20% of all known comets
¾ Eccentricities typically 0.6-0.8, lower than for
long-period comets
¾ Most orbital inclinations within ~30 degrees of
ecliptic
¾ Likely originated in the Kuiper belt.
¾ Prograde orbits
¾ Halley’s comet is an exception, with a high
inclination, retrograde orbit, e=0.97, but P=76 yr.
Halley and comets like it probably originated in the
Oort cloud but had their orbit perturbed.
ƒ Orbits can change because of
¾ Gravitational effects
¾ Mass loss (leads to jets and evaporation)
Origin and Evolution
• How many passes can a comet makes?
¾ Halley’s comet loses 0.1% of its total mass on
each passage through the inner solar system
¾ Therefore it can only survive ~1000 orbits
¾ With a period of ~100 years, it indicates a
typical lifetime of 105 years.
• Only the very longest period comets (P>millions of
years) could conceivably survive for the lifetime of
the solar system
¾ Must be only occaisionally ejected from the
Oort cloud or Kuiper belt.
¾ Ejected from Oort cloud by gravitational
perturbations from passing, nearby stars, or
tidal action of the Milky Way disk.
¾ To account for the number of comets seen,
Oort cloud must contain about 1012 comet
nuclei!
• Comets will experience strongest outgassing on
their first passage. Afterwards, they probably
develop a rocky crust that prevents escape of gas.
¾ Therefore many comets that visit the inner
solar system will either disappear or become
much smaller (fainter) on subsequent
approaches.
Coma
• The large, bright spherical head of a comet is
called the coma.
¾ Spectra of comets tell us the gases present in
the coma and in approximately what proportion.
¾ There is also a much larger hydrogen coma, that
emits at UV wavelengths
•
•
•
•
Tails
Tails are extremely elongated features which can
extend millions of km from the coma and as much
as a third of the way across the sky.
The tail always points in a direction away from the
Sun.
Usually two distinct tails can be seen:
¾ a broad, curved, yellowish dust tail and
¾ a narrow, straighter, bluish ion tail.
The dust tail is pushed away from the Sun by
radiation pressure, while the ion tail is dragged out
by the solar wind (which drags magnetic field
lines). Radiation pressure is insignificant for ions,
which only absorb specific frequencies of light.
Comet Nuclei
• Difficult to observe because when the comet is
close, the coma is bright
• Size can be estimated from brightness when comet
is at large distances (>4 AU). Generally smaller than
10 km
• From appearance of coma and tails, we know the
nucleus is primarily dirty water ice.
• Images from visiting spacecraft show nuclei are:
¾ Irregular in shape
¾ Heavily pitted
¾ Low albedo (3-5%)
¾ Close to solar abundance
• Are they solid masses of ice (icebergs), or loosely
bound pebbles with icy mantles?
• Several lines of evidence point to the iceberg theory:
¾ Comets can revisit the inner solar system many
times and still show bright coma/ tail.
• Comets can survive close passages by the Sun, within
the Roche limit.
Halley’s comet
• Within 4600 km of nucleus, there is no solar
wind or solar magnetic field. Gas is pure
outflowing cometary gas, at about 300 K
• Gas near the nucleus is 80% water, 10% CO,
3% CO2, 2% methane, <1.5% NH3 and 0.1%
HCN.
• Dust contains rocky material but also some
organic molecules
• Nuclear material is closer to solar abundances
than even carbonaceous chondrites. Thus
comets may be the most primordial objects in
the SS.
• Relatively large nucleus, 12km x 8 km
¾ Implies a low density of only 300 kg m-3.
¾ Nucleus is honeycombed with voids and
tunnels.
¾ Due to evaporation of ices? Or loose
binding of planetesimals?
• Losing gas at a rate of 20 tonnes per second.
Losing dust at 3-10 tonnes per second.
Fate of Comets
1. Orbit can be perturbed by giant planets, and
ejected from the solar system
2. Weaker tidal effects could alter orbits of the
longest-period comets such that perihelion no
longer brings it into the inner solar system.
3. Could run out of volatiles to sublime, and become
an asteroid.
4. Collisions with planets or the Sun are rare, but do
occur.
Exercises
1.
Calculate the vapour pressure of water at 273
K and at 177 K.
The vapour pressure of a given substance at temperature
T is given by :
where HL is the latent heat of
⎛H
H ⎞
pv = p0 exp⎜⎜ L − L ⎟⎟ vaporization, and p0 is the vapour
⎝ kT0 kT ⎠ pressure at some temperature T0.
⎛ 7.6 × 10 −20
pv = 133Pa exp⎜⎜
− 23
⎝ 1.38 × 10
1 ⎤⎞
⎡ 1
−
⎢⎣ 256 273 ⎥⎦ ⎟⎟ = 507Pa
⎠
⎛ 7.6 × 10 −20
At 177 K we get pv = 133Pa exp⎜⎜
− 23
×
1
.
38
10
⎝
1 ⎤⎞
⎡ 1
⎢⎣ 256 − 177 ⎥⎦ ⎟⎟ = 0.009Pa
⎠
Note the pressure due to the solar wind is about 10-15 Pa,
so water ice will sublimate if T>90 K.
2.
Write the energy balance equation for an
object a distance r from the Sun, with radius R,
and including sublimation.
The heating term (visible light absorbed from the Sun)
must be equal to the cooling due to thermal radiation in
the infrared and sublimation. These terms have been
previously defined, so we have:
LSun
(1 − AV ) 2 πR 2 = (1 − AIR )4πσR 2T 4 + 4πR 2 ZH L
4πr
The πR2 term cancels out, so we have:
LSun
4
(1 − AV )
(
)
−
Z
(
T
)
H
=
1
−
A
σ
T
L
IR
16πr 2
For a given material, for which we know the heat of
vapourization and the visible/IR albedos, we can solve for
the equilibrium temperature as a function of distance r.
3. Calculate the change in period caused by a small
change in velocity as a comet approaches the Sun.
The change in period is related to a
change in semimajor axis, a:
4π 2 3
P =
a
GM
4π 2 2
2 PdP = 3
a da
GM
dP 3 da
=
2 a
P
2
Using the vis-viva equation, we can
relate da/a to a change in velocity, dv (at fixed r):
⎡1 1 ⎤
v 2 = 2GM ⎢ −
⎥
⎣ r 2a ⎦
GM
2vdv = 2 da
a
dv GM da
=
v2
v
2a a
Assuming a is large, we can
approximate v2~2GM/r:
2GM dv GM da
=
2a a
r
v
da
⎛ a ⎞ dv
= 4⎜ ⎟
a
⎝r⎠ v
dP 3 da
⎛ a ⎞ dv
=
= 6⎜ ⎟
2 a
P
⎝r⎠ v
For a=40 AU, a 1% change in velocity at r=1 AU results in
a factor 2.4 change in Period.
4.
Consider a hypothetical comet, with a
pure water-ice nucleus 1km in radius. If the
sublimation rate is Z~1022 molecules/m2/s,
how many passages will the comet be able to
make through the inner solar system?
For a nucleus of pure water, the mean molecular mass is
18mH. So the rate of mass loss due to sublimation is:
4πR 2 Zµm H = 4π (1km ) 10 22 (18)(1.67 × 10 −27 kg ) = 3800 kg / s
2
Sublimation of water begins when the comet is within
r~2AU of the Sun. If we assume that the semimajor
axis a is much larger than
2GM
2AU, then the velocity is
v≈
≈ 3 × 10 4 m / s
r
approximately
To guess how long the comet spends near the Sun,
assume it travels at that velocity over a distance 4AU
(directly to the Sun and back), so t~4AU/v~200 days.
So during each passage within r=2AU, the comet loses
a mass m=6.6x1010 kg. The total mass, assuming a
density 900 kg/m3, is M=4/3πR3ρ=3.8x1013 kg. So it
will survive ~57 passages (~11,000 years if P=200 y).