LECTURE 2 : COMETS OORT CLOUD DYNAMICS - PHYSICS …on.br › cce › 2016 › en › arq › L2.pdfCiclo de Cursos Especiais - Lecture 2 12. The ux of Oort cloud comets per unit of

LECTURE 2 : COMETS

OORT CLOUD DYNAMICS - PHYSICS AND

CHEMISTRY OF THE NUCLEUS

∗ External perturbers on Oort cloud comets.∗ Injection rate of comets into the planetary region.∗ Comet showers.∗ The comet nucleus: the “dirty snowball” and the “rubble pile” models.∗ Collisional or primordial rubble piles?∗ Outbursts and splittings of comet nuclei.∗ Chemical composition.∗ Formation of radicals and ions in the coma.∗ Formation of the comet tails.

Ciclo de Cursos Especiais - Lecture 2 1

Perturbations by nearby stars

Geometry of a stellar encounter.

Let us consider a star of mass M passing at a high relative velocity V with respect tothe Sun, such that V >> vc where vc is the orbital velocity of the comet.


We can use the impulse approximation

∆vc =

∫ +∞

−∞F∗ × dt =

∫ +∞

−∞

GM

(D2 + x2)1/2

dx

V=

2GM

VD

where x is the distance along the star’s straight path to the point Kc of closest approachto the comet, and D is the distance of closest approach of the star to the comet. Thevelocity of the star can be expressed as : V = dx/dt.

Since the sum of the impulses along the trajectory cancel out, the only nonzero componentof the perturbing force exerted by the star on the comet will follow the direction ~D. Thevector ~∆vc will have the same direction as ~D.

The star will also impart an impulse to the Sun that can be calculated in the samemanner. Let K�, ~D�, and D� be the point of closest approach to the Sun, the vector ofclosest approach of the star to the Sun and its magnitude. The impulsive change in thecomet’s velocity with respect to the Sun is

~∆v = ~∆vc − ~∆v� =2GM

VD

~D

D− 2GM

VD�

~D�D�


If the star gets much closer to the Sun than to the comet, namely if D� << D, theprevious equation can be approximated by

|∆v| ' |∆v�| =2GM

VD�

It is also possible that occurs the opposite situation, namely D << D�. The change|∆v| will be the same with the subtitution of D� by D.

For distant stellar encounters in which r << D� and r << D, the vectors ~D y ~D� willbecome quasiparallel and the impulse equation reduces to

|∆v| ' 2GMr cosβ

V D2�

where β is the angle between the vectors ~r and ~D�.

During an orbital revolution of period P the comet will be perturbed by many stars. Inorder to compute the overall perturbation by passing stars, we have to know the stellarflux Φ∗ in the Sun’s vicinity.


Stellar flux in the Sun’s vicinity

Stellar populations in the Sun’ vicinity(1)

Class ρ∗,i (M� pc−3) n∗,i (pc−3) σ∗,i (km s−1)giants 0.0006 0.0005 17.0

MS(2) MV < 2.5 0.0031 0.0013 7.5MS 2.5 < MV < 3.0 0.0015 0.0010 10.5MS 3.0 < MV < 4.0 0.0020 0.0015 14.0MS 4.0 < MV < 5.0 0.0024 0.0021 19.5MS 5.0 < MV < 8.0 0.0074 0.0090 20.0

MS MV > 8.0 0.014 0.05 20.0white dwarfs 0.005 0.008 20.0brown dwarfs 0.008 0.12 20.0

(1) Source: Holmberg & Flynn (2000)

(2) MS: main sequence stars with absolute visual magnitudes MV in the indicated range.

By combining all the spectral types of the table (except the brown dwarfs), we obtain astellar density in the Sun’s neighborhood n∗ = 0.073 pc−3 and a mass density ρ∗ = 0.036M� pc−3, that gives an average mass per star ∼ 0.5 M�.


Galactic tidal force

∗ The tidal force of the Galactic disc dominates over that of the Galactic bulge.

The Galactic disc can be approximately modelled as a homogeneous disc of density ρdiskin the mean plane of the Galaxy, such that its potential can be expressed as (Heisler &Tremaine 1986, Morris & Muller 1986, Torbett 1986)


U = Uo + 2πGρdiskz2

where Uo is a constant and z is the distance to the Galactic mean plane.

Mass density in the mean plane of the Galaxy at the distance of the Sun to the Galacticcenter: ρdisk = 0.10 M� pc−3.

The Sun will experience oscillations in the vertical direction to the Galactic plane describedby the equation of motion

d2z/dt2 = −4πGρdiskz

With a half-period τz =√π/Gρdisk ∼ 42 Myr, reaching a maximum altitude of about

70 pc.

From the potential U , we obtain the tidal force of the Galactic disc acting on a comet ata Galactic latitude φ

~Fdisk = [(dU/dz)c − (dU/dz)�]z = 4πGρdiskr sinφz


r: distance Sun-comet, r sinφ = zc − z� is the difference between the distances of thecomet and the Sun to the mean Galactic plane, and z is a unit vector perpendicular tothe Galactic plane.

The tidal force of the Galactic disc acts changing the comet’s perihelion distance by theamount (per revolution)

q1/2f = q

1/2i + 4.5

√2π2M−1

� ρdiska7/2 cosα sin 2φ

qi, qf : initial and final perihelion distances, α is the angle between the orbital plane andthe plane perpendicular to the Galactic disc containing the Sun-comet radius.


Change in the perihelion distance by external perturbers

Relative change in q per orbital revolution (Fernandez 2005).

∗ The effects of the external perturbers (passing stars and the tidal force of the Galacticdisc) start to be significant for semimajor axes a >

∼ 104 au.


From the Oort cloud to the inner planetary region

Fictitious comet starts at the star and ends at the square (Kaib & Quinn 2009)

∗ An inner Oort cloud comet (a <∼ 2 × 104 au) must require many revolutions to change appreciably its

perihelion distance. Once it approaches the Jupiter-Saturn barrier, planet perturbations are strong enough

to raise its semimajor axis to values where ∆q/q ∼ 1 in a single revolution =⇒ The comet can then

overshoot the Jupiter-Saturn barrier to reach the terrestrial planet zone.


Dependence of the tidal force of the Galactic disc on the Galacticlatitude: Observational evidence

Distribution of Galactic latitudes de 151 new and “young” comets (aorig > 500 au). Thehistograms were fitted to normalized sinusoidal curves (Fernandez 2005).


Very close stellar encounters: Definition of the loss cone

The diffusion of the perihelion distances of Oort cloud comets with small q by externalperturbers leads to an uniform q-distribution. The fraction of thermalized comets withperihelion distances < q is

Fq ∼ 2q/a

valid for q << a. The number of comets with perihelion distances in the range (q, q+dq)is

fq(q)dq =

(dFqdq

)dq =

2

adq

∗ Loss cone: Oort cloud comets with semimajor axis a are thermalized, namely thevelocity vectors are isotropically distributed. However, vectors very close to the solardirection will be quickly removed by planetary perturbations generating a loss cone. Thefraction of comets with perihelion distances q < qL (where qL ∼ 15 au) is FL ∼ 2qL/a

⇒ the solid angle of the loss cone is 4πFL and its radius 2F1/2L .


The flux of Oort cloud comets per unit of q that reach the Earth’s vicinity is

nEarth =1

qL

∫ +∞

afill

FL1

PΓ(a)da

where FL = 2qL/a, P = a3/2, and Γ(a)da is the number of Oort cloud comets withsemimajor axes in the range (a, a + da). Numerical simulations lead to the followingempirical relation

Γ(a)da ∝ a−γda

where γ ' 2− 4 (Bailey 1983, Fernandez and Ip 1987).


A very close stellar encounter can suddenly fill the loss cones of those comets close tothe star’s path causing a sudden increase of the comet flux in the Earth’s vicinity.


Anomalies in the distribution of the comet aphelia

The anomalous aphelion clusterings shown in the map could uncover very close starpassages in the recet past (Fernandez & Ip 1991).


Comet showers

(Heisler 1990)

Large perturbations cause the fillingof loss cones, thus provoking a substantialincrease in the comet flux in the Earth’sneighborhood. This phenomenon is calleda comet shower.Could the Earth have been exposedto comet showers during its lifetime??Could comet showers have been responsiblefor any of the mass extinctions thatpunctuated the development of life on Earth?


Fred Whipple’s “dirty snowball model”

Artist conception of the comet Fred Whipple shows his model ofnucleus according to Whipple’s dirty snowball consistent in amodel in vogue in the 1970s. snowball of 250 kg covered

with dirt.


Comet nuclei observed from spacecraft

Images of comets Halley, Borrelly, Tempel 1 and Wild 2 taken from spacecrafts at shortdistance (left). The “rubble-pile” model (e.g. Weissman 1986) (right).


Space missions to comets

The Giotto mission of the European Space Agency (ESA) was the first flyby to a comet(Halley) in 1986.


The Deep Impact mission

The Deep Impact mission of NASA had as a goal the study of the material of comet9P/Tempel 1. The spacecraft arrived to the comet on July 4, 2005 releasing an impactorthat collided with the comet nucleus, releasing a dust cloud that was monitored.


The Rosetta mission

The Rosetta mission of ESA was launched in 2004 in a long 10-yr journey targeted tocomet 67P/Churyumov-Gerasimenko.


Images of comet Churyumov-Gerasimenko

The Rosetta spacecraft is now orbiting the comet nucleus at about 30 km of its surface.Spectacular images of the nucleus have been taken that show 2 lobes united by a neck.The comet is about 4 km length. There is currently a discussion on whether this bodyis primordial, a result of a gentle collision of two planetesimals, or if it is a product ofevolution.


Rosetta and Philae

The Rosetta spacecraft carried a lander called Philae (an old Egiptian city). Unfortunately,the landing was not as good as expected and Philae finally laid down in a shaded place,being unable to receive enough solar energy.


Physical aspects

∗ The equation of thermal equilibrium

(1−Av)F�e

−τ

r2AU

πR2N = 4πR2

N(1−AIR)σT 4 +QLSNA

+ 4πR2Nκ(T )

∣∣∣∣∂T∂z∣∣∣∣z=0


Outbursts and splittings of comet nuclei

Spectacular brightness increase (∼ 14 magnitudes) of the Jupiter family comet17P/Holmes (October 2007).

Splitting process of comet C/1975 V1 (West) in 4 main pieces between 8-18/March/1976(New Mexico State University, Las Cruces).


Estimate of the mass released in the outbursts of comets41P/Tuttle-Giacobini-Kresak and 17P/Holmes

Approximated calculation of the dust content in the coma:

Id =N(S)πs2p(λ)φ(α)

πr2∆2F�

Id : Intensity in the continuum (measured in the visible)N(S) : Number of dust particles of typical radius s within the circular aperture of radiusSp(λ) : geometric albedo of the dust particlesφ(α) : phase function of phase angle αr : heliocentric distance (expressed in au)∆ : geocentric distance

Total mass of dust:

Md = N(S)× 4

3πs3ρd

ρd : density of the dust grains


Cont.

Brightness of the comet nucleus:

IN =F�r2

R2Npvφ

′(α)

∆2

Ratio of the dust coma to the nucleus brightness:

IdIN

=N(S)s2p(λ)φ(α)

R2Npvφ

′(α)

Converting into magnitudes:

mN −md = 2.5 log

[N(S)s2p(λ)φ(α)

R2Npvφ

′(α)

]For the calculations we adopt a typical dust grain size: s = 1 µm (= 10−4 cm)


41P/Tuttle-Giacobini-Kresak

Absolute nuclear magnitude: HN = 18.4, nucleus radius: RN = 0.7 km

1st Outburst : 25 May 1973mmax(1UA) ' 4.0, previous magnitude ∼ 13, r ∼ q = 1.152 au2nd Outburst : 5 July 1973mmax(1UA) ' 4.0, previous magnitude ∼ 13, r = 1.25 au

Released mass of dust : Md = 1.3× 1013 g

Mass of the comet nucleus : MN = 7.2× 1014 g

⇒Md

MN= 0.018


17P/Holmes

Absolute nuclear magnitude: HN = 16.6, nucleus radius: RN = 1.6 km

Outburst : 23 October 2007mmax(1UA) ' 1.4, previous magnitude ∼ 15.4, r = 2.435 au

Released mass of dust : Md = 1.42× 1014 g

Mass of the comet nucleus : MN = 8.6× 1015 g

⇒Md

MN= 0.016

⇒ The mass released in splittings and outbursts can be greater than the continuousrelease by sublimation


Frequency of splittings

JFCs ∼ 1 splitting / comet every ∼ 77 revolutions

LPCs ∼ 1 splitting / comet every ∼ 25 revolutions (Fernandez 2005)

Is there a correlation with the heliocentric distance?

0 1 2 3 4 5

perihelion distance (AU)

0

0.02

0.04

0.06

spli

ttin

g r

ate

(co

me

t−1 r

evo

lutio

n−

1)

Number of observed splittings as a function of the perihelion distance.


Possible causes of splittings and outbursts

∗ Tidal force of the Sun, Jupiter or a terrestrial planet.∗ Phase transition of amorphous ice to crystalline ice.∗ Sublimation of pockets of a very volatile material (e.g. CO, CO2).∗ Thermal stresses that cause fractures.∗ High rotation speed of the comet nucleus leading to rotational instability.∗ Solar activity.∗ Collisions with meteoroids.∗ Collisions with fragments of the nucleus itself.


Generation of internal pressure and fragmentation

Fragmentation of a “rubble pile” nucleus due to the increase of the pressure of gasesthat cannot escape freely through the regolith mantle (right-hand side). If there werenot regolith mantle, the gases would escape thus preventing the increase of the pressure(left-hand side) (Samarasinha 2001).


Cometary spectra

Spectrogram of comet C/1975 N1 (Kobayashi-Berger-Milon) showing the main molecularbands in the visible and near UV. The narrow dark strip in the middle of the spectrumalong the λ-axis is the continuum produced by the scattering of sunlight by the dustparticles in the inner coma (radius ∼ 104 km) (Wyckoff 1982).


UV spectrum

UV spectrum of comet Hale-Bopp takenwith the Faint Object Spectrographof the Hubble Space Telescope. The spectrumshows the Cameron bands of CO, a bandof CS and the conspicuos band of OH(Weaver et al. 1997).


Chemical composition

Relative abundances of molecular species in cometsMolecule Mass fraction

H2O ∼ 100

CO ∼ 7-8

CO2 ∼ 3

H2CO ∼ 0-5 (formaldehide)

NH3 ∼ 1-2

HCN ∼ < 0.02-0.1

CH3OH ∼ 1-5 (methanol)

Other detected molecules of biological interest :

HNCO, HC3N, OCS, H2CS,NH2CHO, HCOOH,

HCOOCH3, CH3CHO, HNC, C2H2, C2H6


Sublimation of volatiles

Sublimation rate of different volatile molecules as a function of the temperature (Mukaiet al. 2001).


The distribution of parent molecules and radicals in the coma

The observed radicals in the cometary spectrum cannot exist in free state. They areproduced when the sublimated parent molecules are photodissociated and ionized underthe Sun’s UV radiation.

Let us take the example of the most important parent molecule: H2O. Its photodissociationleads to the following products:

H2O + hν −→

H + OHH2 + O2H + OH2O+

H + OH+

where hν represents the photon energy. A percentage of 85.5% of the water moleculesare dissociated in H and OH. The conclusion is that a comet must be surrounded by ahuge corona of hydrogen atoms.


Hydrogen corona around comet Hale-Bopp when it approached the Sun in 1997. Theimage in UV light was obtained by the SWAN instrument onboard of the SOHO spacecraft.The corona occupies an extension of about 108 km.


Another interesting example is:

CO2 + hν → CO + O(1D)

where the oxygen is left in the excited 1D state from which it is de-excited emittingfluorescent radiation in the red line at 6300 A.

Formation of the ion tail

The decomposition of the parent moleculesleads to the production of many differentions, in particular CO+, N+

2 ,CH+, CO+

2 , and H2O+. The ionsare trapped by the magnetic fieldsassociated to the solar wind anddragged in the opposite direction to theSun forming the ion tail. Its bluecolor is due to the fluorescentradiation emitted by the ion CO+.


Formation of the dust tail

The motion of the dust particles will be ruled by 2 opposite forces: the gravitationalattraction of the Sun FG and the force associated to the solar radiation pressure FR. Fora dust particle of radius a and density ρ at a heliocentric distance r, these 2 forces canbe expressed as

FG =GM�r2

(4

3πa3ρ

)

FR =Qprc

(L�

4πr2

)πa2

where c is the light speed and Qpr is the efficiency factor for the radiation pressure. Sinceboth forces FG y FR are radial, opposite, and vary as r−2, a dust particle will follow aKeplerian trajectory that corresponds to an “effective” gravitational field reduced by thefactor (1− β)FG where

β =FRFG

= 0.585× 10−4Qprρa

(ρ and a are expressed in cgs units).


Dynamics of the dust particles

The dust particles released from the nucleus at different times t1, t2, t3 ... will followdiverging trajectories from the nucleus in such a way that appear collectively as a curvedtail opposite to the Sun.


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LECTURE 2 : COMETS OORT CLOUD DYNAMICS - PHYSICS …on.br › cce › 2016 › en › arq › L2.pdfCiclo de Cursos Especiais - Lecture 2 12. The ux of Oort cloud comets per unit of