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Transcript
Lecture 10. Roche Limit / Comets
Thursday, December 20, 12
Lecture 10. Roche Limit / Comets
Thursday, December 20, 12
Outline
• Roche limit
• One word on Pluto
• Comets
- why they are interesting?
- latest results from sample return mission
-
2
Thursday, December 20, 12
Roche limit
• Roche limit is the closest distance an object can
come to another object without being pulled
apart by tidal forces.
• Inside the Roche limit, orbiting material will tend
to disperse and form rings, while outside the
limit, material will tend to coalesce.
• Notable examples :
- Saturn rings
- Shoemaker - Levy 9 comet breakup
• Notable exceptions
- Mars’ moon Phobos
- Jupiter’s moon Metis
- held together by their internal strength.
3
Thursday, December 20, 12
Roche limit
• Roche limit is the closest distance an object can
come to another object without being pulled
apart by tidal forces.
• Inside the Roche limit, orbiting material will tend
to disperse and form rings, while outside the
limit, material will tend to coalesce.
• Notable examples :
- Saturn rings
- Shoemaker - Levy 9 comet breakup
• Notable exceptions
- Mars’ moon Phobos
- Jupiter’s moon Metis
- held together by their internal strength.
3
Thursday, December 20, 12
Roche limit
• Roche limit is the closest distance an object can
come to another object without being pulled
apart by tidal forces.
• Inside the Roche limit, orbiting material will tend
to disperse and form rings, while outside the
limit, material will tend to coalesce.
• Notable examples :
- Saturn rings
- Shoemaker - Levy 9 comet breakup
• Notable exceptions
- Mars’ moon Phobos
- Jupiter’s moon Metis
- held together by their internal strength.
3
Thursday, December 20, 12
Roche limit
• Roche limit is the closest distance an object can
come to another object without being pulled
apart by tidal forces.
• Inside the Roche limit, orbiting material will tend
to disperse and form rings, while outside the
limit, material will tend to coalesce.
• Notable examples :
- Saturn rings
- Shoemaker - Levy 9 comet breakup
• Notable exceptions
- Mars’ moon Phobos
- Jupiter’s moon Metis
- held together by their internal strength.
3
Thursday, December 20, 12
Roche limit
• Roche limit is the closest distance an object can
come to another object without being pulled
apart by tidal forces.
• Inside the Roche limit, orbiting material will tend
to disperse and form rings, while outside the
limit, material will tend to coalesce.
• Notable examples :
- Saturn rings
- Shoemaker - Levy 9 comet breakup
• Notable exceptions
- Mars’ moon Phobos
- Jupiter’s moon Metis
- held together by their internal strength.
3
Thursday, December 20, 12
Roche limit
• Roche limit is the closest distance an object can
come to another object without being pulled
apart by tidal forces.
• Inside the Roche limit, orbiting material will tend
to disperse and form rings, while outside the
limit, material will tend to coalesce.
• Notable examples :
- Saturn rings
- Shoemaker - Levy 9 comet breakup
• Notable exceptions
- Mars’ moon Phobos
- Jupiter’s moon Metis
- held together by their internal strength.
3
Thursday, December 20, 12
Tidal force (repeat)
ˆr - is a unit vector from M to m
G - gravitational constant
M, m - masses of two bodies
R - distance between the two
Consider a small particle at a distance of Δr
use series to expand acceleration
first term is gravity at the center of mass
on Earth tides are to pulls by both Sun
(0.54x10-7g) and the Moon (1.1x10-7g)
4
Thursday, December 20, 12
Tidal force (repeat)
ˆr - is a unit vector from M to m
G - gravitational constant
M, m - masses of two bodies
R - distance between the two
Consider a small particle at a distance of Δr
use series to expand acceleration
first term is gravity at the center of mass
on Earth tides are to pulls by both Sun
(0.54x10-7g) and the Moon (1.1x10-7g)
4
Thursday, December 20, 12
Roche limit (2)
• Roche limit
- gravitational and tidal forces
2!"#!
!! =
!
!!
!"#
!! = ! ! !
!
- gravitational force equals tidal force
!! = !! !
- simple derivation and expression in terms of diameter
! 2!!
= ! !
!!
!
!"# 2!"#!
! ! =
!
!
!!
2!
!=!
!
!
!
!
4!!! !!
!
where ! =
3
2!!
!=!
!!
!
!
!
5
Thursday, December 20, 12
Roche limit (3)
• Rigid satellites
2!!
!=!
!!
!
!
!
- where R - radius of the primary body, ρM - density of the primary body and ρm - density of the satellite
• Fluid satellites (more accurate!)
⎛ ρ M ⎞
d ≈ 2.44R ⎜
⎝ ρ ⎟⎠
m
1
3
Body
Satellite
Roche limit (rigid)
Distance
(km)
Earth
Moon
Earth
average
Comet
Sun
R
Roche limit (fluid)
Distance
(km)
R
9,496
1.49
18,261
2.86
17,880
2.80
34,390
5.39
Earth
554,400
0.80
1,066,300
1.53
Sun
Jupiter
890,700
1.28
1,713,000
2.46
Sun
Moon
655,300
0.94
1,260,300
1.81
Sun
average
Comet
1,234,000
1.78
2,374,000
3.42
6
Thursday, December 20, 12
Another explanation of Saturn’s ring formation
NATURE | LETTER
Origin of Saturn’s rings and inner moons by mass removal from a lost Titan-sized satellite
Robin M. Canup
Nature (2010)
doi:10.1038/nature09661 Received 24 May 2010
Accepted 05 November 2010
online 12 December 2010
Published
The origin of Saturn’s rings has not been adequately explained. The current rings are more than 90 to 95 per
cent water ice1, which implies that initially they were almost pure ice because they are continually polluted by
rocky meteoroids2. In contrast, a half-rock, half-ice mixture (similar to the composition of many of the satellites
in the outer Solar System) would generally be expected. Previous ring origin theories invoke the collisional
disruption of a small moon3, 4, or the tidal disruption of a comet during a close passage by Saturn5. These
models are improbable and/or struggle to account for basic properties of the rings, including their icy
composition. Saturn has only one large satellite, Titan, whereas Jupiter has four large satellites; additional large
satellites probably existed originally but were lost as they spiralled into Saturn6. Here I report numerical
simulations of the tidal removal of mass from a differentiated, Titan-sized satellite as it migrates inward towards
Saturn. Planetary tidal forces preferentially strip material from the satellite’s outer icy layers, while its rocky
core remains intact and is lost to collision with the planet. The result is a pure ice ring much more massive than
Saturn’s current rings. As the ring evolves, its mass decreases and icy moons are spawned from its outer edge7
with estimated masses consistent with Saturn’s ice-rich moons interior to and including Tethys.
7
Thursday, December 20, 12
Why Pluto is now a dwarf planet.
• Was considered a planet from 1930 to 2006
- it’s existence was predicted from perturbations to the Neptune’s orbit
• Why not a planet any more.
- Percival Lowell looked for this planet (Planet X), but with advances in our understanding of Neptune’s
orbit the need for Planet X disappeared.
- In the last decade many other similar objects were discovered
‣ 2060 Chiron
‣ Eris (MEris ≈1.28 x Mpluto)
- Highly inclined orbit crosses Neptune’s
orbit
• Exploration
- New Horizons mission
- closest approach with Pluto in 2015
Original plates from Clyde Tombaugh's discovery of Pluto. (Apparent magnitude +15.1)
8
Credit: Lowell Observatory Archives, from [1].
Thursday, December 20, 12
Comets
• Comet is a small solar system body which has a body and a tail
• In general comets consist of
- Nucleus
‣ mostly ice and gas with some dust and other matter relatively solid and stable
- Coma
‣ carbon dioxide and other gases from the nucleus in a dense cloud of water
- Dust Tail
‣ the most-visible part of a comet smoke-sized dust particles up to 10 million km long driven off the
nucleus by escaping gases
- Ion Tail
‣ plasma laced with rays and streamers up to 100 million km long caused by interactions with the solar
wind
- Hydrogen Cloud
‣ huge sparse envelope of neutral hydrogen millions of km in diameter
• Where do comets come from
- short period comets originate from Kuiper Belt (beyond orbit of Neptune)
- long period comets come from Oort Cloud (far beyond Kuiper Belt)
9
Thursday, December 20, 12
Comets
• Comet is a small solar system body which has a body and a tail
• In general comets consist of
- Nucleus
‣ mostly ice and gas with some dust and other matter relatively solid and stable
- Coma
‣ carbon dioxide and other gases from the nucleus in a dense cloud of water
- Dust Tail
‣ the most-visible part of a comet smoke-sized dust particles up to 10 million km long driven off the
nucleus by escaping gases
- Ion Tail
‣ plasma laced with rays and streamers up to 100 million km long caused by interactions with the solar
wind
- Hydrogen Cloud
‣ huge sparse envelope of neutral hydrogen millions of km in diameter
• Where do comets come from
- short period comets originate from Kuiper Belt (beyond orbit of Neptune)
- long period comets come from Oort Cloud (far beyond Kuiper Belt)
9
Thursday, December 20, 12
Comets
• Comet is a small solar system body which has a body and a tail
• In general comets consist of
- Nucleus
‣ mostly ice and gas with some dust and other matter relatively solid and stable
- Coma
‣ carbon dioxide and other gases from the nucleus in a dense cloud of water
- Dust Tail
‣ the most-visible part of a comet smoke-sized dust particles up to 10 million km long driven off the
nucleus by escaping gases
- Ion Tail
‣ plasma laced with rays and streamers up to 100 million km long caused by interactions with the solar
wind
- Hydrogen Cloud
‣ huge sparse envelope of neutral hydrogen millions of km in diameter
• Where do comets come from
- short period comets originate from Kuiper Belt (beyond orbit of Neptune)
- long period comets come from Oort Cloud (far beyond Kuiper Belt)
9
Thursday, December 20, 12
Comets in History
Woodcut showing destructive influence of a fourth century comet from Stanilaus
Lubienietski's Theatrum Cometicum (Amsterdam, 1668). Click on image for larger view.
Image credit: NASA/JPL
The Mawangdui silk, a 'textbook' of
cometary forms and the various
disasters associated with them, was
compiled sometime around 300 B.C.
Click on image for larger view. Image
credit: NASA/JPL
Types of cometary forms, illustrations from
Johannes Hevelius' Cometographia (Danzig, 1668)
Click on image for larger view. Image credit: NASA/
JPL
10
Thursday, December 20, 12
Why study comets
• May be the oldest, most primitive bodies in the solar system preserving the
earliest record of material from the nebula which formed the sun and the
planets.
• Bring volatile light elements to the planets, playing a role in forming oceans
and atmospheres.
• Are the most organic-rich bodies in the solar system providing ready-formed
molecules possibly involved in the origin of life on Earth.
• Impact the Earth and other planets at hypervelocities, causing major changes
in climate and dramatically affecting the ecological balance, possibly
including the extinction of the Dinosaurs.
• Are the building blocks of planetary systems around other stars.
11
Thursday, December 20, 12
Exploration
Deep Impact – NASA Flyby (2005–2006) - Tempel I
Stardust – NASA Sample Return (1999–2006) - Wild 2
Deep Space 1 (1998–2001) - Borelly
Galileo – NASA Shoemaker-Levy 1994 flyby (1995–2003)
Giotto – ESA flyby (1985–1992)
Sakigake – ISAS flyby (1985–1998)
Suisei – ISAS flyby (1985–1998)
Vega 1– (Venera-Halley 1) Soviet Comet Halley flyby (1984–1986)
Vega 2 (Venera-Halley 2) Soviet Comet Halley flyby (1984–1986)
Source: Lunar and Planetary Institute
12
Thursday, December 20, 12
Shoemaker Levy 9 comet and impact
Source: University of Washington
Thursday, December 20, 12
13
SL9 impact into Jupiter
First (far left) and
consequent impacts of
SL9 fragments into
Jupiter. All signs have
of breakup have soon
disappeared.
An enigmatic crater
chain on Ganymede,
which may have formed
due to a comet breakup
when it came near
Jupiter
14
Thursday, December 20, 12
Recently visited comets
Cometary nuclei visited in the last decade: (left) Tempel 1 (NASA, University of Maryland, and Deep
Impact Team), (middle) Borrelly [NASA, Jet Propulsion Laboratory (JPL), and Deep Space 1 Team], and
(right) Wild 2 (NASA, JPL, and Stardust Team). The longest dimensions are 8 km for Borrelly, 6 km for
Tempel 1, and 5.5 km for Wild 2. Note the differences in overall shape, even though all are comparable
in size. The smooth areas on Tempel 1 are low, whereas the only smooth area on Borrelly is
topographically high. On Wild 2, the smooth areas are at the bottoms of circular depressions. The circular
features on Wild 2 have a morphology that is very different from those on Tempel . Source: A’Hearn,
Whence Comets?, Science, Dec 2006.
15
Thursday, December 20, 12
Comets to scale
Images at the same scale for all
cometary nuclei observed by
spacecraft. Differences in overall
shape are dramatic, as are the
differences in contrast between
the nuclei and their associated
jets, which are brighter than the
nucleus at Halley and Hartley 2
and much fainter than the nucleus
at other comets. Credit: Science/
AAAS.
16
Thursday, December 20, 12
Halley comet
Tapestry of Bayeux (Normandy) with
Halley's comet. Text reads ISTI MIRANT
STELLA: "These (people) are looking in
wonder at the star."
Venera - Halley spacecraft (Венера Галлей). Vega 1 made its closest
approach on March 6, around 8,890
km from the nucleus, and Vega 2
made its closest approach on March
9 at 8,030 km. The data intensive
examination of the comet covered
only the three hours around closest
approach.
The nucleus of Halley's Comet,
imaged by the Giotto probe in 1986.
The dark colouration of the nucleus
can be observed, as well as the jets
of dust and gas erupting from its
surface.
17
Thursday, December 20, 12
Stardust sample return
• Sample return from comet Wild/2
- sample collected in an aerogel
This image and diagram show the comet Wild 2, which
NASA's Stardust spacecraft flew by on Jan. 2, 2004.
The picture on the left is the closest short exposure of
the comet. The listed names on the right are those used
by the Stardust team to identify features. "Basin" does
not imply an impact origin. Image credit: NASA/JPLCaltech.
Thursday, December 20, 12
This is an artist's concept depicting a view of
comet Wild 2 as seen from NASA's Stardust
spacecraft during its flyby on Jan. 2, 2004.
Image credit: NASA/JPL-Caltech.
18
Stardust
• Deep Impact
- Tempel 1 in July 2005
- launched a probe to look at impact results
• Schedule
- encounter with Wild 2 : Jan 2 2004
- Capsule returns to Earth, Jan 19, 2006
- 2007 - retargeting Tempel 1 as Stardust Next
- Encounter - February 14, 2011
‣ distance 178km
‣ best resolution : 12 m/ pixel
• Key conclusions
- surface albedo ~ 4%, no exposed ice
- globally layered
- small outbursts occur
19
Thursday, December 20, 12
Comet surface features
Key to examples of surface features on
Tempel 1.
(A) Composite of ITS frames with examples
of types of features
(B) Topographic regions of Tempel 1. Unit
names from (A) with generalized physical
descriptions.
(C) Portion of MRI frame MV173728497
showing smooth areas near terminator and
associated complex topography.
(D) Definition of “facets.”
(E) Strongly stretched portion of the ITS
composite image; arrows denote scarp
bounding smooth unit.
From Thomas et. al (2007), Icarus, Deep Impact Mission
to Comet 9P/Tempel 1
20
Thursday, December 20, 12
EPOXI flyby of Hartley 2
• Flyby - November 4, 2010
• Discovered that not only
water is emitted by the
comet but CO2 jets as well.
• Emits snow-ball size chunks
of ice.
• The comet acts differently
than others we know.
21
Thursday, December 20, 12
EPOXI flyby of Hartley 2
• Flyby - November 4, 2010
• Discovered that not only
water is emitted by the
comet but CO2 jets as well.
• Emits snow-ball size chunks
of ice.
• The comet acts differently
than others we know.
21
Thursday, December 20, 12
EPOXI flyby of Hartley 2
• Flyby - November 4, 2010
• Discovered that not only
water is emitted by the
comet but CO2 jets as well.
• Emits snow-ball size chunks
of ice.
• The comet acts differently
than others we know.
21
Thursday, December 20, 12
Stardust results
• First, they found tiny flecks of once-molten minerals--material very different from the raw, primordial dust they
expected to see. Such unaltered, so-called presolar material was the prime ingredient of the rocky planets and
was thought to abound in icy comets.
• Before Stardust's return, cosmochemists thought of comets as vaults where the primitive ingredients of the
planetary recipe had been locked up.
- Their best look at the likely ingredients list came from the study of certain meteoritic particles
collected in Earth's stratosphere by retired spy planes. Because of their exotic isotopic composition,
these particular interplanetary dust particles (IDPs) looked as though they might be comet dust.
- Presumably, such primitive dust fell into the cold, outer reaches of the nebula that gave rise to the
planets and combined with nebular ices to form comets, in which the dust has been preserved ever
since.
• Wild 2 is more like an asteroid than a primitive comet
- Rather than preserving the original ingredients of planets, comets--or at least Wild 2--seem to be loaded
with materials first altered by the great heat near the young sun,
- no one is at all sure where the solar system's lingering primitive materials might reside.
22
Thursday, December 20, 12
Stardust latest results
• Found tiny flecks of once-molten minerals - material very different from the
raw, primordial dust they expected to see.
- Such unaltered, so-called presolar material was the prime ingredient of the rocky planets and was
thought to abound in icy comets.
• Before Stardust's return, cosmochemists thought of comets as vaults where
the primitive ingredients of the planetary recipe had been locked up.
- Because of their exotic isotopic composition, these particular interplanetary dust particles (IDPs)
looked as though they might be comet dust. Presumably, such primitive dust fell into the cold, outer
reaches of the nebula that gave rise to the planets and combined with nebular ices to form comets, in
which the dust has been preserved ever since.
• Wild 2 is more like an asteroid than a primitive comet
- Rather than preserving the original ingredients of plants, comets--or at least Wild 2--seem to be
loaded with materials first altered by the great heat near the young sun,
- It is not clear where the solar system's lingering primitive materials might reside.
• EPOXI - role of CO2 has to be clarified
Source:Kerr, Where Has All the Stardust Gone? Science 25 January 2008
23
Thursday, December 20, 12
Linear regression (1)
• Taken from Numerical Recipes, Third Edition, http://www.nr.com
- pages 788 through 783, Fitting data to a Straight Line
• We are trying to model our data using simple linear regression
! ! = ! + !" !
• We use chi-square merit function, which is
- if the measurement is errors are normally distributed then this merit function will give maximum
likelihood parameter estimations of a and b
!!!
! ! !, ! =
!!!
!! − ! − !!!
!!
!
!
• Chi-square merit function is minimized w.r.t to a and b
0=
!! !
!"
!!!
= −2
!!!
!! − ! − !!!
!! !
!
0=
!! !
!"
!!!
= −2
!!!
!! − ! − !!!
!!
!! !
!
24
Thursday, December 20, 12
Linear regression (3)
• Define Sums
!!!
!≡
!!!
1
!!!!!!!! ≡
!! !
!!!
!!!
!!
!!!!!! ≡
!! !
!!!
!!!
!!
!!
!! !
!!!
!!!! ≡
!!!
!!!
!!!!!!" ≡
!! !
!!!
!!!
!! !!
!
!! !
• Then solution of equations are
∆≡ !!!!! − !!
!!
!!! !! − !! !!"
!=
!!
∆
! !!! − !! !!
!=
!!
∆
• In case where measurements are unweighted (our case, σi=1)
define
1
!!
!! =
! −
!!!!
!! ! !
!
1
!
!
!!! =
1+
!
!!!!!
!!!
and
!!! !
!!! ≡
!!!
!!!
1
= !
!!!
!
where σa and σb are error
estimates
for our regression
coefficients
25
Thursday, December 20, 12