Migration of small bodies and
dust in the Solar System
Sergei Ipatov
Alsubai Establishment for Scientific Studies,
Doha, Qatar.
• Papers and slides from presentations on the considered items are
presented on the website http://faculty.cua.edu/ipatov/ (e.g.,
http://faculty.cua.edu/ipatov/list-publications.htm
and
http://faculty.cua.edu/ipatov/present.htm).
• Slides
from
this
presentation
can
be
found
on
http://faculty.cua.edu/ipatov/st-andrews2011.ppt)
1
Main problems to be considered in the presentation:
• Migration of small bodies, including migration of trans-Neptunian
objects and comets to near-Earth space.
• Migration of planetesimals and planet embryos.
• Migration of dust.
• Sources of the zodiacal dust cloud.
• Probabilities of collisions of migrating bodies and dust particles with
planets.
• Angular momenta of rarefied preplanetesimals and formation of
small-body binaries.
• Cavities as a source of outbursts from comets.
• Recognition of cosmic ray signatures on Deep Impact images.
• Spectra of a model exo-Earth.
• I will tell shortly about several obtained results. Those who want
to understand some problems in more detail can read the file
http://faculty.cua.edu/ipatov/st-andrews2011.ppt with the slides
from this presentation or read my papers devoted to these
problems (via http://faculty.cua.edu/ipatov/list-publications.htm ).
2
Migration of small bodies. Initial data.
The orbital evolution of ~30,000 Jupiter-crossing objects (JCOs) with
initial period P<20 yr was considered under the gravitational influence
of 7 planets (Venus-Neptune).
 First series (n1): Orbital elements were the same as those of 20 real comets (with numbers 7,






9, 10, 11, 14, 16, 17, 19, 22, 26 30, 44, 47, 51, 57, 61, 65, 71, 73, 75) with 5<P<9 yr, but
different values of the mean anomaly were considered.
Second series (n2): Orbital elements were the same as those of 10 real comets (with numbers
77, 81, 82, 88, 90, 94, 96, 97, 110, 113) with 5<P<15 yr, but different values of the mean
anomaly were considered.
In each of other series initial orbits were close to the orbit of one comet (2P, 9P, 10P, 22P,
28P, 39P, or 44P) or to test long-period or Halley-type comets. For 2P we considered also
Mercury.
Bodies initially located at the 3:1 and 5:2 resonances with Jupiter.
Methods of integration. We used the SWIFT package by Levison and Duncan (Icarus, 1994,
v. 108, 18-36). Evolution of N orbits was calculated using the Bulirsh-Stoer method
(BULSTO) with the error per integration step less than =10-9, or =10-8, or some value
between these two values. Also =10-12 and =10-13 were used. We also used a symplectic
method with an integration step 3≤ds≤10, or ds=30 days (RMVS3).
The considered time interval in one run usually was equal to the largest dynamical lifetime of
bodies in the run (until all bodies reached 2000 AU or collided with the Sun). Sometimes it
reached several hundreds Myr.
Earth, Moon, and Planets, v. 92, 89-98 (2003); Advances in Space Research, v. 33, 15241533 (2004). Annals of the New York Acad. of Sci., v. 1017, 46-65 (2004); Proc. IAU Symp.
236 “Near-Earth Objects, Our Celestial Neighbors: Opportunity and Risk”, pp. 55-64 3(2007).
Motion of JCOs in NEO orbits for a long time
• The motion of former JCOs (Jupiter-crossing objects) inside Jupiter’s orbit for a
long time was obtained both for integration with the use of BULSTO and with a
symplectic method.
• A few considered JCOs got orbits located inside Jupiter’s orbit
and moved in such orbits for millions or even hundreds of
millions of years. The probability of a collision of such object with
a terrestrial planet can be greater than the total probability of
thousands of other JCOs.
• For one object (from 10P runs with BULSTO) its probability of collisions with
Earth and Venus was 0.3 and 0.7, respectively.
• For another object (from 2P runs) during its lifetime (352 Myr) its probability of
collisions with Earth, Venus and Mars was 0.172, 0.224, and 0.065, respectively.
For 12,000 other objects with BULSTO such probability was 0.2, 0.18, and 0.04,
respectively.
• For series 2P, at ds=3 days (i.e., for a symplectic method), a lifetime of one object
was 400 Myr, and it moved on Inner-Earth, Aten, Apollo, and Amor orbits during
2.5, 2.2, 44.9, and 80.8 Myr, respectively. At t=6.5 Myr this object got an orbit with
e=0.03 and a=1.3 AU, and then until 370 Myr the eccentricity was less than 0.4 and
often was even less than 0.2. The probability of a collision of this object with the
Earth was about 1, and it was greater than that for all other 99 objects in that run by
two orders of magnitude, i.e. greater by four orders of magnitude than the mean
4
probability for one 2P object.
Time variations in semi-major axis (left), eccentricity and sini (right)
10P (a, d), 2P (b), resonance 3:1 ©. BULSTO (a-c), RMVS (d)
•
5
Distribution of all migrating objects in runs with BULSTO
(time in Myr during which a was in interval with a width of 0.005 AU (a-b) or 0.1
AU (c-d)). Though only a small fraction of considered objects had a<2 AU, the
mean time spent by former JCOs in orbits with a<2 AU was comparable with
that at a=3 AU.
6
Comparison of the results obtained by symplectic and BULSTO integrators.
n1
n1
n1
2P
2P
2P
2P
n2
n2
Method
BULSTO
RMVS3
RMVS3
BULSTO
RMVS3
BULSTO
BULSTO
BULSTO
RMVS3
•
10-9
ds10 days
ds10 days
10-9
ds=10 days
=10-12
=10-13
10-9
ds10 days
V
N
Pr
1900 2.42
1200 25.4
1199 7.88
501
141
251
262
50
>434
50
58
4000 9.9
10000 14.7
V
E
T
Pr
4.23
4.51
13.8
40.1
9.70
4.76
345
110
307
344
>388 >213
373
40
24.4 11.6
24.8 14.9
E
T
7.96
24.0
12.6
397
348
>433
432
35.6
36.1
M
Pr
6.15
2.48
0.76
10.5
28
>17.5
2.7
2.12
2.88
M
T
30.
35.2
16.8
430
364
>450
448
56.3
56.1
There is no principal difference in the values of Pr=1,000,000P and T, obtained by
different methods and with different values of  and ds (at ds10 days), or if we
delete objects with perihelion distance q<2rSun (exclusive for some runs with
Comet 96P in series n2). Such difference is about the difference obtained
for different N for the same method and integration step. One additional
object can increase the values of P and T by an order of magnitude. In this table
the collision probabilities with the Earth for a symplectic method and for 10-12
are not smaller than those for 10-910-8.
7
Migration of trans-Neptunian objects to the near-Earth space
• The number of TNOs migrating to the inner regions of the Solar System can be
evaluated on the basis of simple formulas and the results of numerical integration.
Let NJ=PNpJNNTNO be the number of former TNOs with d>D reaching Jupiter's
orbit for the given time span TSS, where NTNO is the number of TNOs with d>D; PN
is the fraction of TNOs leaving the trans-Neptunian belt and migrating to Neptune's
orbit during TSS; and pJN is the fraction of Neptune-crossing objects which reach
Jupiter's orbit for their lifetimes.
• The current number of Jupiter-crossers that originated in the zone with 30<a<50
AU equals NJn=NJtJ/TSS, where tJ is the average time during which the object
crosses Jupiter's orbit. According to [1], PN0.1-0.2 at TSS=4 Gyr and
pJN=0.34. As mutual gravitational influence of TNOs also takes place, we take
PN=0.2.
• Hence, at tJ=0.13 Myr and NTNO=5109 (d>1 km), we have NJn=104.
• The number of former TNOs now moving in Earth-crossing
orbits equals NE=NJnT/tJ. For T=0.0065 Myr (this value of T, the mean time
spent in Earth-crossing orbits, is less than that for series n1 and n2) and tJ=0.13
Myr, we have NE=500. It is about 2/3 of the estimated number of
Earth-crossers with d>1 km (750). The latter number doesn’t include
•
NEOs with large values of e and i. It is also probable that the number of 1-km TNOs
is smaller by a factor of several than 5109, and PN can also be smaller. We do not
consider destruction of bodies at these estimates.
8
[1] M.J. Duncan, H.F. Levison, and S.M. Budd, Astron. J., 110, 3073-3081 (1995).
Conclusions on migration JCOs and TNOs to the Earth
• Results of our runs testify in favor of at least one of these conclusions:
•1) the fraction of 1-km former trans-Neptunian objects (TNOs) among nearEarth objects (NEOs) can exceed several tens of percents,
•2) the number of TNOs migrating inside solar system could be smaller by a factor of
several than it was earlier considered,
•3) it is more probable that most of 1-km former TNOs that had got
NEO orbits disintegrated into mini-comets and dust during a smaller
part of their dynamical lifetimes if these lifetimes are not small.
•The mean collision probabilities of Jupiter-crossing objects (from initial orbits close
to the orbit of a comet) with the terrestrial planets can differ by more than two orders
of magnitude for different comets.
•For initial orbital elements close to those of some comets (e.g. 2P and 10P), about
0.1% of objects got Earth-crossing orbits with semi-major axes a<2 AU and moved in
such orbits for more than a Myr (up to tens or even hundreds of Myrs).
•A relatively small fraction (~0.0001-0.001) of JCOs which transit to orbits
with aphelia inside Jupiter’s orbit (Q<4.2 AU) and reside such orbits
during more than 1 Myr may contribute significantly in the NEO
supply. A probability of a collision of one object with a terrestrial
planet can be larger than that of 10,000 other objects.
•Former JCOs can get any Amor, Apollo, Aten and inner-Earth objects’ orbits and
9
move in such orbits for millions years.
Formation of scattered disk objects (SDOs)
• In 1987 (Ipatov S.I., Earth, Moon, and Planets, v. 39, 101-128, 1987), five years before
the first TNO was discovered in 1992, based on my runs of the formation of the
giant planets, I supposed that there were two groups of TNOs and, besides
TNOs formed beyond 30 AU and moving in low eccentric orbits,
there were former planetesimals from the zone of the giant
planets now moving in highly eccentric orbits beyond Neptune. A
•
very small fraction of such planetesimals could left in eccentrical orbits beyond Neptune and
became so called ''scattered disk objects'' (SDOs). Later on similar model of the formation of
SDOs were considered by several authors in more detail. The end of the bombardment of
terrestrial planets could be caused mainly by the planetesimals that had got highly eccentric
and inclined orbits located mainly beyond Neptune.
The total mass of planetesimals in the feeding zone of Uranus and Neptune could exceed
100m. Most of these planetesimals could still move in this zone when Jupiter and Saturn
had accreted the bulk of their masses. My computer runs (published in 1991) showed
that the embryos of Uranus and Neptune could increase their
semimajor axes from 10 AU to their present values, moving
permanently in orbits with small eccentricities, due to gravitational interactions with the
planetesimals that migrated from beyond 10 AU to Jupiter, which ejected most of them into
hyperbolic orbits. In these old runs, the mutual gravitational influence was taken
into account by the method of spheres. Later on, similar results were obtained
by other scientists by numerical integrations using much faster computers.
•
•
•
Ipatov S. I. (1987) Earth, Moon, and Planets, 39, 101-128.
Ipatov S.I. (1991) Sov. Astron. Letters, 17, 113-119.
Ipatov S.I. (1991) LPSC, 607-608 ; Ipatov S. I. (1993) Solar Syst. Res., 27, 65-79.
10
Migration of embryos of Uranus and Neptune
Distribution of planetesimals over their semimajor axes at several stages of disk
evolution. (Sov. Astron. Letters, v. 17, 113-119, 1991; LPSC, 607-608, 1991; Solar
System Res. v. 27, 65-79, 1993. Presented at conf. “Planetesimal dynamics" in Sept.
1992 in Santa-Barbara and later). The total mass of a disk of planetesimals equaled to
135Me. Arrows show positions of planets. Initial semi-major axes of the giant planets: 5.5,
6,5, 8, and 10 AU.
11
Variations of semi-major axes and eccentricities of embryos of
Uranus and Neptune (vs. the number of planetesimals in a disk)
Results published in 1991-1993. Initial eccentricities of embryos of Uranus and
Neptune (with masses equal to 10 Mearth): 0.75 and 0.82 (left), 0.02 (right). Results
obtained by the method of spheres of action are closed to those obtained about ten years
later by E.W. Thommes, M.J. Duncan, and H.F. Levison (Nature, 1999, v. 402, 635638; Astron. J., 2002, v. 123, 2862-2883) by numerical integrations.
12
Uranus and Neptune were not formed if their massive embryos
had not been considered (a figure from Solar System Res., 27, 65-79, 1993) .
13
The Nice model. The outer edge of the disk is assumed at ∼34 AU. It was found
that in order to most accurately reproduce the characteristics of the outer planetary
system the total mass of the disk must have been ∼35 M⊕.
• Giant planet evolution in the Nice
model. (A) Each planet is represented
by a pair of curves—the top and bottom
curves are the aphelion and perihelion
distances, respectively. In this
simulation Jupiter and Saturn cross
their 1:2 mean-motion resonance at
878 My, which is indicated by the
dotted line (from Gomes et al., 2005).
The resonance caused the jumps in
variations of semi-major axes of
Uranus and Neptune.
• (B) The eccentricity of Neptune.
14
Mixing of planetesimals in the zone of the terrestrial planets.
Initial planetesimals belonged to four groups characterized by different semimajor axes. (A figure from Solar System Research, v. 27, 65-79, 1993) .
15
Mixing of planetesimals in the zone of the terrestrial planets
Composition of the largest planets formed at different distances was similar for
different planets and was close to that of the initial disk.
16
The choice of the pairs of contacting bodies
•
•
•
•
•
•
•
The mutual gravitational influence of bodies in disks was taken into account by the action
spheres method, that is two two-body problems were considered.
Initially I used the "probability" algorithm of choosing pairs of encountering objects
during the evolution of discrete systems with binary interactions. The pairs of bodies
encountering to the distance equal to the radius rs of action sphere were chosen in the
proportion to the probability of their encounter.
Later on the efficient "deterministic" algorithm was developed, for which for a pair of
encountering bodies i and j the moment tij of the first isolated encounter to rs is
minimum. In our algorithm we don't store the matrix {tij}. The required memory (exclusive
for the memory for parameters of the bodies) is equal to 3N, where N is the number of bodies,
but the result of evolution is the same as for the case when we store all this matrix {tij}.
If the number of bodies in a disk is not small, then for the deterministic algorithm the
time needed for the growth of embryos is ten times smaller than that for the
"probability" algorithm.
When the evolution of the disks corresponding to the terrestrial feeding zone was modeled by
the deterministic method, it was found that the time to form 80% of the mass of the largest
planet (Earth) did not exceed 10 Myr (i.e., was shorter than analytic estimates), and the total
evolution times of the disks were of the order of 100 Myr.
Ipatov, S.I., Computer modelling of the process of solar system formation, Proc. Intern. IMACS
Conference on Mathematical modelling and applied mathematics (June 18-23, 1990, Moscow).
Ed. by A.A. Samarskii and M.P. Sapagovas. Elsevier. Amsterdam, pp. 245-252 (1992).
Ipatov, S.I., Methods of choosing the pairs of contacting bodies for investigations of the evolution of
discrete systems with binary interactions, Matematicheskoe Modelirovanie (Mathematical Modeling), v.
5, N 1, pp. 35-59 (1993), in Russian.
17
Gravitational interaction of two planetesimals
•
•
•
Regions of initial semimajor axis and an initial angle between the directions to bodies from
the Sun vertex were studied for the following types of orbital evolution: motion about
triangular libration points on tadpole and horseshoe synodic orbits (types M and N), the case
of close approaches of the objects (type A), and chaotic variations of the orbital elements
during which close approaches of the bodies are impossible (type C).
There must be 1.6 mu1/2 [not 1.6 mu1/3] to the right.
Ipatov S.I. Gravitational interaction of two planetesimals moving in close orbits, Solar System
Research, v. 28, 494-512 (1994).
18
Mutual gravitational influence of transNeptunian objects and planetesimals
• Runs (Preprint KIAM, 1980): Gravitational interaction of three or four
TNOs with initially circular orbits and masses close to that of Pluto.
For the planar model: Growth of emax to 0.04, 0.1, and 0.2 in 100
Myr, 1 Gyr, and 3 Gyr, respectively (at 50 AU).
• Estimate: For iav=10 deg and N=100, the increase of emax up to 0.2
in 4 Gyr. Sol. Syst. Res. 1995, 29, 9-20.
• Runs of N=50 identical objects (eo=0.0001, io=0.00005 rad). For
1000 km objects (density 2 g/cm3), growth of eav to 0.04, 0.08, and
0.11 (iav to 0.7, 1.5, and 2 deg) was during 106, 107, and 108
revolutions around the Sun, respectively. For fifty 100 km objects,
growth of eav to 0.01 was during 108 revolutions, but increase of eav
to 0.005 in 103 revolutions. LPSC 2007, #1260.
• eav and iav are proportional to sqrt{N encounters}. Estimate: eav
reach 0.01 during 3×105 revolutions of 100 km objects with a total
mass m=Mearth.
19
Orbital evolution of asteroids at resonances
• This plot illustrates that the region of initial values of a and e for which orbits
become Mars-crossing in 0.1 Myr coincides with the 5:2 Kirkwood gap.
•
•
•
Regions of initial data corresponding to different types of interrelations of between orbital
elements were studied. Limits of variations in e are different for different types.
Ipatov: Sov. Astron. Lett. 1989, 15, 324-328; Icarus 1992, 95, 100-114; Solar System
Research 1992, 26, 520-541.
The probability of a collision of an asteroid (eo=0.15, io=10 deg) during its lifetime with the
Earth is 5×10-4 for the 3/1 resonance and 10-4 for the 5/2 resonance with Jupiter.
20
Delivery of volatiles by dust
• Although the role of dust in the accretion mechanism was probably relatively
small, it could have played an important role in the evolution of the terrestrial
planets. Specifically, the dust could be more efficient than larger bodies
in delivery of organic (prebiogenic) or even biogenic matter to the
planets.
• This is explained by the fact that dust particles are not subjected to intense
heating (usually higher than 100-150 C) when they enter the atmosphere, because
they have a high total surface-to-mass ratio, which allows the excess heat to be
radiated effectively. Therefore, they can decelerate in the upper atmosphere for
several seconds or tens of seconds (depending on the entry angle) without any
change in their internal structure and composition and then softly precipitate onto
the surface, delivering in this way the primitive interplanetary or even interstellar
matter.
• Dust can be considered as a potential carrier of biogenic material from outer space,
without considering the efficiency of the material's transportation between the inner
planets. From this point of view, life forms radically different from the terrestrial
variants are unlikely to be found, for example, on Mars (if they exist there now
or existed before).
• The main sources of interplanetary dust are comets, asteroids, and
trans-Neptunian bodies. A considerable amount of dust is produced in the process
of the sublimation of the dust-ice material of the nuclei of comets during their orbits
to perihelion and in numerous collisions of small bodies and their fragmentation.
21
Migration of dust.
Initial data and methods of integration
• Migration of 20,000 dust particles under the gravitational influence
of all planets (Pluto was considered only for trans-Neptunian particles),
radiation pressure, Poynting-Robertson drag, and solar wind drag was integrated
using the Bulirsh-Stoer method (BULSTO) with the error per
integration step less than =10-8. We used the SWIFT package by Levison and
Duncan (Icarus, 1994, v. 108, 18).
• A wide range of particle sizes (from 1 micron to several
millimeters) was studied. The calculations were made for values of β
(the ratio between the radiation pressure force and the gravitational
force) equal to 0.0001, 0.0002, 0.0004, 0.001, 0.002, 0.004, 0.005, 0.01, 0.05, 0.1,
0.2, 0.25, and 0.4. For silicates at density of 2.5 g/cm3, such β values
correspond to particle diameters d of about 4700, 2400, 1200, 470, 240, 120,
47, 9.4, 4.7, 2.4, 1.9, and 1.2 microns, respectively. For water ice d is greater
by a factor of 2.5 than that for silicate particles.
• In each run we took N250, because for N≥500 the computer time per
calculation for one particle was greater by a factor of several than that for
N=250.
•
•
Ipatov, S.I., Mather, J.C., and Taylor, P.A., Annals of the New York Academy of Sciences, v.
1017, pp. 66-80 (2004).
22
Ipatov, S.I. and Mather, J.C., Advances in Space Research, v. 37, N 1, 126-137 (2006)
Initial coordinates and velocities
• The initial positions and velocities of the asteroidal particles
considered were the same as those of the first numbered main-belt
asteroids (JDT 2452500.5), i.e., dust particles are assumed to leave
the asteroids with zero relative velocity.
• The initial positions and velocities of the trans-Neptunian particles
were the same as those of the first TNOs (JDT 2452600.5). Our initial
data for dust particles were different from those in previous papers.
• The initial positions and velocities of cometary particles were the
close to those of Comet 2P Encke (a≈2.2 AU, e≈0.85, i≈12 deg), or
Comet 10P/Tempel 2 (a≈3.1 AU, e≈0.526, i≈12 deg), or Comet
39P/Oterma (a≈7.25 AU, e≈0.246, i≈2 deg), or test long-period
comets (e=0.995, q=a(1-e)=0.9 AU or q=0.1 AU, i was distributed
between 0 and 180 deg, particles started at perihelion), or test Halleytype comets (e=0.975, q=0.5 AU).
• We considered Encke particles starting near perihelion (runs denoted
as Δto=0), near aphelion (Δto=0.5), and when the comet had orbited
for Pa/4 after perihelion passage, where Pa is the period of the comet
(such runs are denoted as Δto=0.25).
23
Distribution of migrating asteroidal dust particles with their
semimajor axes. The time in orbits with a<3 AU was greater for smaller beta.
Runs with (right) or without (left) planets.
24
Migration of asteroidal dust particles.
Resonances n/(n+1) with the terrestrial planets [e.g., 6:7, 5:6, 3:4, and 2:3
resonances with Earth]. The peaks in the distribution of particles over a at
resonances with the Earth are better seen for smaller β (for greater
sizes of particles).
25
Distribution of dust particles in semimajor axis and eccentricity
(a) asteroidal particles at beta=0.01, (b) asteroidal particles at beta=0.4, © transNeptunian particles at beta=0.1
26
Distribution of dust particles with distance from the Sun and
height above the initial plane of the orbit of the Earth
(a) asteroidal particles at beta=0.01, (b) asteroidal particles at beta=0.4,
© trans-Neptunian particles at beta=0.1
27
Sources of the zodiacal dust cloud
• Our studies of sources of the zodiacal dust were
based on comparison of some observations with
our models based on results of integration of the
motion of >20,000 asteroidal, cometary, and transNeptunian dust particles under the gravitational
influence of all planets (Pluto was considered only for transNeptunian particles), radiation pressure, PoyntingRobertson drag and solar wind drag. The motion of particles was
integrated using the Bulirsh-Stoer method (BULSTO) with the error per integration
step less than =10-8. We used the SWIFT package by Levison and Duncan (Icarus,
1994, v. 108, 18).
28
Number density of different migrating dust particles vs. a
for particles produced by different small bodies.
Observations showed that number density is constant at 3-18 AU from the Sun
[Humes D.H., J. Geophys. Res., 1980, 85, 5841]. Therefore, the obtained plots show that
asteroidal dust doesn't dominate at a>3 AU, trans-Neptunian dust does not
dominate at 3-7 AU, and a lot of dust particles at 3-7 AU were produced by comets.
29
Our model of calculations of changes of the solar
spectrum after the light was scattered by dust particles
• Based on positions and velocities of migrating particles taken from
one our run with a fixed β, we studied the variations in solar
spectrum after the light was scattered by dust particles and
reached the Earth. For each such stored position, we calculated many (~102-104
•
•
•
•
•
•
•
•
depending on a run) different positions of a particle and the Earth during the period Pr of
revolution of the particle around the Sun. All positions of particles during their dynamical
life-times were considered.
Three different scattering functions were considered:
(1) the scattering function depended on a scattering angle θ in such a way: 1/θ for θ<c, 1+(θ-c)2 for θ>c,
where θ is in radians and c=2π/3 radian.
(2) we added the same dependence on elongation ε (considered eastward from the Sun).
(3) the scattering function didn't depend on these angles at all.
ε is the angle with a vertex in the Earth between directions to the Sun and a particle,
θ is the angle with a vertex in a particle between direction to the Earth and the direction from the Sun to a
particle.
The intensity of light that reaches the Earth was proportional to λ2(R×r)-2, where r is the distance between
a particle and the Earth, R is the distance between the particle and the Sun, and λ is a wavelength of light.
For each considered positions of particles, we calculated velocities of a dust particle relative to the Sun
and the Earth and used these velocities and the scattering function for construction of the solar spectrum
received at the Earth after the light was scattered by different particles located at some beam (line of
sight) from the Earth. The direction of the beam is characterized by ε and inclination i. Particles in the
cone of 2.5 deg around this direction were considered.
30
The shape of the Fraunhofer line produced by our model and at the observations of
the zodiacal light spectrum near the solar Mg I λ5184 absorption line by Reynolds,
Madsen and Moseley (ApJ, 2004, 612, 1206) (ast1 and tn1 – for function 1, ast2 and
tn2 – for function 2, ast and tn – for function 3; ast – asteroidal dust, tn – transNeptunian dust, com – dust particles started from comet 2P at perihelion)
31
Calculation of ‘velocity-elongation’ plots
• Based on our plots of the intensity of the scattered light obtained at
the Earth vs. ∆λ (λ is the length of the wave near the solar Mg I
λ5184 absorption line and ∆λ=λ-λo, where λo corresponds to the minimum of solar
spectrum near this line) we calculated the shift ∆λs of the plot, which is
based on our model distribution of dust particles, relative to the
plot of the solar spectrum. Considering that v/c=∆λs/λ (where v is a
characteristic velocity of particles and c is the velocity of light), we calculated
the characteristic velocity of particles at different elongations.
• The plots of this velocity vs. elongation (the angle with a vertex in the
Earth between directions to the Sun and a dust particle) were compared
with those obtained by Reynolds et al. (2004) at observations.
•
•
Velocity-elongation plots for different scattering functions are denoted as c1 and m1 for
function 1, as c2 and m2 for function 2, and c and m for function 3.
For lines marked by ‘c’, we considered a shift of a centroid (the ‘center of mass’ of the region
located upper than a plot of intensity vs. ∆λ and restricted by the maximum value of the
intensity) for scattered light obtained at the Earth relative to the centroid for the solar light.
For lines marked by ‘m’, we considered a shift of the minimum of the plot of intensity of a
scattered light from the minimum for the solar spectrum.
• Velocity-elongation plots obtained for different scattering functions and for
‘centroid’ and ‘minimum’ models are close to each other, exclusive for the cases when
ε is close to 0.
32
Comparison of plots of velocities of Mg I line
versus elongation with the observations
• Ipatov, S.I., Kutyrev, A., Madsen, G.J., Mather, J.C., Moseley, S.H., Reynolds, R.J.
(e.g., 37th LPSC, #1471, 2006; Icarus, v. 194, N. 2, 769-788, 2008)
compared the plots of velocities of Mg I line (at zero inclination) versus elongation ε
(measured eastward from the Sun), with the observational plots obtained by
Reynolds et al. (ApJ, 2004, 612, 1206-1213).
• The velocity-elongation plots obtained for different considered scattering functions
were close to each other for 30<ε<330 deg, the difference was greater for more
close direction to the Sun.
• The difference between different plots for several sources of dust was
maximum at ε between 90 and 120 deg. For future observations of velocities of
the zodiacal light, it is important to pay particular attention to this interval of ε.
• In our opinion, the main conclusion of the comparison of such curves is that
asteroidal dust doesn't dominate in the zodiacal light and a lot of
zodiacal dust particles were produced by comets.
• This conclusion is also supported by the comparison of a spatial density of different
migrating dust particles with the observational result that a spatial density is constant
at 3-18 AU from the Sun.
• Significant contribution of cometary dust to the zodiacal dust was considered by
several other authors (e.g., Zook, 2001; Grogan et al. 2001, Nesvorny et al. ApJ,
2010, 713, 816-836).
33
Velocity-elongation plots for asteroidal, trans-Neptunian, and
cometary zodiacal dust. For asteroidal dust, the velocity-elongation
curves are below the observational curve at elongation ε<240 deg. For
trans-Neptunian dust, the observational curve is mainly inside the region covered by curves
for different β, but at 180<ε<270 deg it is mainly above the trans-Neptunian curves. For
particles produced by long-period and Halley-type comets, the plot was above the
observational curve at ε<180 deg and it was below at ε>180 deg ..
34
Velocity-elongation curves for particles originated from comets 10P
and 39P. 10P curves are below the observational curve, but the
difference is smaller for larger particles. For some particles produced by
comets with eccentricities ~0.25-0.5, the model curves can be close to the
observational curve.
35
Velocity-elongation plots for particles originating from comet 2P
at perihelion (per), aphelion (ap), and T/4 (T is the period of revolution) after perihelion
passage (m). The curves for 2P particles originating at perihelion are relatively close to
each other for different β, and they are close to the observational curve.
36
Mean eccentricity of particles at different distances from the Sun at
several values of β (see the last number in the legend) for particles originated from
comets 2P, 10P, 39P, and trans-Neptunian objects. At eccentricities ~0.3-0.4,
velocity-elongation plots better fit the observations.
37
Distribution of dust particles with semimajor axis and eccentricity
at several values of β for particles originating from Comet 2P at the
middle of the orbit (2P 0.25t) and at aphelion (2P 0.5t). Right plots are
for smaller beta.
38
Mean (for elongation between 30 and 330 deg.) width (FWHM = Full-Width HalfMaximum) of MgI line at several values of β. The values of the width were based on
spectra obtained for particles produced by different bodies. The mean observed value
was equal to 76.6 km/s. Asteroidal or 2P particles alone do not satisfy to the
observations.
39
Values of α in the proportionality of number of particles n(R)~R-α obtained by
comparison of the values of the number density n(R) at distance R to the Sun equal
to 0.3 and 1 AU (a), at R=0.8 and R=1.2 AU (b), and at R equal to 1 and 3 AU (c). The
values are presented at several values of β for particles produced by Comets 10P and 39P (10P
and 39P), from trans-Neptunian objects (tno), and from long-period comets (lp) at e=0.995, q=0.9
AU and i between 0 and 180 deg. Observational values are 1.3 for (a) and 1.5 for (c).
Comparison of model distributions of particles over R with observations testifies
against a considerable zodiacal contribution of particles produced by comets
moving in very high eccentric orbits (such as Comet 2P).
40
Sources of zodiacal dust particles
• There is no principal difference in the ratio of fractions of
asteroidal/cometary/trans-Neptunian dust for our
spectroscopic studies at different ε, as our studies showed that
the main contribution to the spectrum is from particles
on a distance less than 1 AU from the earth and similar
vertical shifts of the `velocity-elongation' curves (different
for different sources of dust) from the observational curve
were obtained for many values of ε. The principal difference
in the fractions will be if we consider particles in different
parts of the solar system, separated by several AU.
• Besides, our calculations showed that the difference
between characteristic velocities of Mg I line for different
sizes of particles was usually less than the difference for
different sources of particles. It would mean that for our
conclusions about the fraction of asteroidal dust we need not
know the exact distribution of zodiacal particles in 41
masses.
•
Sources
of
zodiacal
dust
particles
Comparison of the WHAM observations (both of velocity and width
of Mg I line) and the observations of the number density with the
results based on our models for particles originating from different
small bodies testify in favour of a considerable fraction of particles
originating from comets, including those originating beyond Jupiter’s
orbit, and trans-Neptunian objects, but it did not contradict to 30% of
asteroidal dust needed for explanation of formation of dust bands.
• Fractions of asteroidal particles, cometary particles
originated inside of Jupiter's orbit, and particles
originated beyond Jupiter's orbit can be about 1/3 each,
with a possible deviation from 1/3 up to 0.1-0.2.
• The velocity amplitudes of the Mg I line (in plots of the velocity vs.
elongation) are greater for greater mean eccentricities and inclinations,
but they depend also on distributions of particles over their orbital
elements. The mean eccentricities of zodiacal particles
located at 1-2 AU from the Sun that better fit the WHAM
observations are between 0.2 and 0.5, with a more probable value
of about 0.3.
42
Sources of zodiacal dust particles
• The differences between the curves of the characteristic velocity of
the scattered Mg I line in the zodiacal light vs. the solar elongation for
several sources of dust reached its maximum at elongation between 90
deg and 120 deg. For future observations of velocity shifts in the
zodiacal spectrum, it will be important to pay particular attention to
these elongations.
• The estimated contribution of particles produced by long-period
and Halley-type comets to zodiacal dust does not exceed 0.1-0.15.
The same conclusion can be made for particles originating from
Encke-type comets (with e~0.8-0.9).
• The velocity amplitudes of the Mg I line (in plots of the velocity vs.
elongation) are greater for greater mean eccentricities and inclinations,
but they depend also on distributions of particles over their orbital
elements.
43
Calculations of the characteristic time elapsed up to the
encounter of two objects to radius of sphere rs
• T2=6.28·kp·Ts ·R ·kv/(rs·kfi) - planar model,
• T3=T2·Δi·R/rs - spatial motel;
• R is the distance of encounter from the Sun, kfi is the sum of angles
(in radians) with apices in the Sun, within which the distance
between orbits is less than rs, Ts is the synodic period of revolution,
kp=P2/P1, where P2>P1, Pi is a period of revolution of the i-th body
around the Sun.
• In order to take into account that velocity at distance R from the Sun
differs from the mean velocity, we used coefficient kv=sqrt{2a/R-1}.
• In contrast to the approach used by Opik (1951) and Arnold
(1965), T depends on orbital orientations and on a synodic
period.
• Ipatov, S.I., Evolution times for disks of planetesimals, Soviet
Astronomy, v. 32 (65), 560-566 (1988).
• Ipatov, S.I. and Mather, J.C., Comet and asteroid hazard to the
terrestrial planets, Advances in Space Research, v. 33, 1524-1533
(2004).
44
Probabilities P=10-6Pr of collisions of bodies with the terrestrial
planets. The probability of a collision of JFCs with the Earth was greater than 4×10-6.
N
V
Pr
V
Tr
E
Pr
E
Tr
E
Tc
M
Pr
M
T
n1 10-9
1900
2.4
4.2
7.9
1760
6.1
30.0 0.7 20
n1 10d
1200
25.4
13.8
24.0
600
2.48
35.2 3.0 25.7
n1 10d
1199*
7.88
9.70
12.6
2650
0.76
16.8 2.8 10.3
9.9
24.4
35.6
3060
2.12
56.3 2.7 7.7
14.7
141
1.3
28.3
1.44
1.7
1.06
2.58
1286
101
24.8
345
1.8
41.3
2.98
21.8
1.72
15.8
1886
173
4.5
40.1
4.76
11.6
14.9
36.1
397
4.1
71
4.87
34.7
3.03
24.9
2747
371
2420
3610
1100
1970
2770
18260
2550
6210
1450
1160
2.88
10.5
0.7
10.3
0.74
0.44
0.31
0.75
488
209
56.1 3.1
430 18.
9.7 1.2
16.4 1.6
11.0 1.6
68.9 1.9
6.82 1.6
46.3 2.0
4173 2.7
1455 0.5
, ds
n2
10-9 4000
n2 10d
2P 10-9
9P 10-9
10P 10-9
22P 10-9
28P 10-9
39P 10-9
44P 10-9
3:1 10-9
5:2 10-9
10000
501*
800
2149*
1000
750
750
500
288
288
110
3.7
35.6
1.76
1.9
1.19
4.01
1889
318
r
Td
90.1
249
2.6
107
1.5
0.1
2.7
8.6
5169
1634
Tr (the mean time in a planet-crossing orbit) and Td are in Kyr, Tc=Tr/Pr in Myr, Pr =106P =106P/N for
Venus=V, Earth=E, Mars=M. Td is the mean time spent in orbits with Q=a(1+e)<4.2 AU. r is the ratio
of times spent in Apollo and Amor orbits. For one object (from 10P runs), its probability of collisions
with Earth and Venus was 0.3 and 0.7, respectively. For another object (from 2P runs) during its lifetime
(352 Myr), its probability of collisions with Earth, Venus and Mars was 0.172, 0.224, and 0.065,
respectively. For 12,000 other objects with BULSTO such probability was 0.2, 0.18, and 0.04,
respectively. Results with the BULSTO code at 10-910-8 are marked as 10-9, those at 10-12 are
marked as 10-12, and those with the RMVS3 code are at integration step ds. For the lines which do not
include all bodies in a series of runs, the number of objects N is marked by *.
45
Delivery of water to the terrestrial planets
during the formation of the giant planets
• The total mass of water delivered to the Earth during the
formation of the giant planets is Mw=MJPJEki, where MJ is the total mass of
planetesimals from the feeding zones of these planets that got Jupiter-crossing orbits during
evolution, PJE is the probability P of a collision of a JCO with the Earth during
its lifetime, and ki is the portion of water ices in planetesimals.
• For MJ=100m (where m is the mass of the Earth), ki=0.5, and PJE=410-6
(this value is even less than those in series n1 and n2), we have Mw=210-4m. This
value is about the mass of the Earth’s oceans.
•
The larger value of P for Earth we have calculated compared to those argued by Morbidelli et
al. 2000 (P  (1-3)10-6) and Levison et al. 2001 (P = 410-7 ) is caused by the fact that in our
runs we considered a larger number of Jupiter-crossing objects and the main portion of the
probability of collisions was caused by a small fraction (0.01-0.001) of bodies, each of
which moved at least several hundreds of thousands years in the Earth-crossing orbits with
Q<4.7 AU. We also considered that the mass supply from the Uranus-Neptune region is about
100m , in contrast to (20-30)m estimated by other authors, which increase the volatiles
delivery.
The total mass of water delivered to Venus can be of the same
order of magnitude and that delivered to Mars can be less by a
factor of 3 or 4 than that for Earth. Ancient Venus and Mars
could have large oceans.
•
Proc. of the IAU Symposium No. 236 “Near-Earth Objects, Our Celestial Neighbors:
46
Opportunity and Risk”, pp. 55-64 (2007). Also several previous papers.
Probabilities of collisions of migrating particles with the Earth
Fig. 1. The
probability P of collisions of dust particles and bodies (during their
dynamical lifetimes) with the Earth versus β (the ratio between the radiation
pressure force and the gravitational force) for particles launched from asteroids (ast),
trans-Neptunian objects (tno), Comet 2P/Encke at perihelion (2P per), Comet 2P/Encke at aphelion (2P
aph), Comet 2P/Encke in the middle between perihelion and aphelion (2P m), Comet 10P/Tempel 2
(10P), Comet 39P/Oterma (39P), long-period comets (lp) at e=0.995 and q=0.9 AU, and Halley-type
comets (ht) at e=0.975 and q=0.5 AU (for lp and ht runs, initial inclinations were from 0 to 180o). If
there are two points for the same β, then a plot is drawn via their mean value. Probabilities presented at
β~10-5 are for small bodies (β=0). Probabilities presented only for bodies were calculated for initial
orbits close to orbits of Comets 9P/Tempel 1 (9P), 22P/Kopff (22P), 28P/Neujmin (28P), 44P/Reinmuth
2 (44P), and test asteroids from resonances 3:1 and 5:2 with Jupiter at e=0.15 and i=10o (‘ast 3:1’
and
47
‘ast 5:2’). For ‘n1’ and ‘n2’, initial orbits of bodies were close to 10-20 different Jupiter-family comets.
Probabilities of collisions of migrating particles
with Venus and Mars
The ratios of probabilities of collisions of JFCs with Venus, Mars, and Mercury to the mass of
a planet usually were not smaller than those for Earth. Probabilities of collisions of considered
particles with Venus were of the same order as those for Earth, and those for Mars were about
an order of magnitude smaller.
• Fig. 2. The probability P of collisions of dust particles and bodies (during their
dynamical lifetimes) with Venus (left) and Mars (right) versus β (the ratio between
the radiation pressure force and the gravitational force). Designations are the48same
as those for Fig. 1.
Probabilities of collisions of migrating particles
with Mercury
• Fig. 3. The probability P of collisions of dust particles and bodies (during their
dynamical lifetimes) with Mercury versus β (the ratio between the radiation
pressure force and the gravitational force). Designations are the same as those
49 for
Fig. 1.
Probabilities of Collisions of Migrating Bodies and
Dust Particles with the Earth
• The probability of a collision of Comet 10P with the Earth during the dynamical
lifetime of the comet was PE≈1.4∙10-4, but 80% of this mean probability was due
only to one body among 2600 considered bodies with initial orbits close to that of
Comet 10P. For runs for Comet 2P, PE≈(1-5)∙10-4. For most other considered JFCs,
10-6<PE<10-5. For Comets 22P/Kopff and 39P/Oterma, PE≈(1-2)∙10-6; and for
Comets 9P/Tempel 1, 28P/Neujmin 1 and 44P, PE≈(2-5)∙10-6.
• For all considered JFCs, PE>4∙10-6 even if we exclude a few bodies
for which the probability of a collision of one body with the Earth
could be greater than the sum of probabilities for thousands of other
bodies. The Bulirsh-Stoer method of integration and a symplectic
method gave similar results.
• For dust particles produced by comets and asteroids, PE was found
to have a maximum (~0.001-0.005) at 0.002≤β≤0.01, i.e., at d~100
μm (this value of d is in accordance with observational data). These
maximum values of PE were usually (exclusive for Comet 2P) greater at least by an
order of magnitude than the values for parent comets.
50
Probabilities of collisions of migrating particles with
Jupiter and Saturn
Probabilities of collisions of considered particles and bodies with Jupiter during their dynamical
lifetimes are smaller than 0.1. They can reach 0.01-0.1 for bodies and particles initially
moved beyond Jupiter’s orbit. For bodies and particles initially moved inside Jupiter’s orbit,
the probabilities are usually smaller than the above range and can be equal to zero.
Fig. 4. The probability P of collisions of dust particles and bodies (during their
dynamical lifetimes) with Jupiter (left) and Saturn (right) versus β (the ratio
between the radiation pressure force and the gravitational force). Designations51are the
same as those for Fig. 1.
Probabilities of collisions of migrating particles
with Uranus and Neptune
Probabilities of collisions of migrating particles (exclusive for trans-Neptunian
particles) with other giant planets were usually smaller than those with Jupiter. The
total probability of collisions of any typical considered body or particle with all
planets didn’t exceed 0.2.
Fig. 5. The probability P of collisions of dust particles and bodies (during their dynamical
lifetimes) with Uranus (left) and Neptune (right) versus β (the ratio between the radiation
pressure force and the gravitational force). Designations are the same as those for Fig. 1.
52
References to ‘migration’ studies
•[1] Ipatov, S.I. (1987) Accumulation and migration of the bodies from the zones of giant planets,
Earth, Moon, and Planets, v. 39, N 2, pp. 101-128.
•[2] Ipatov, S.I. (1993) Migration of bodies in the accretion of planets, Solar System Research
(translated from Astronomicheskii Vestnik), v. 27, N 1. pp. 65-79.
•[3] Ipatov, S.I. (2001) Comet hazard to the Earth, Adv. in Space Research, v. 28, N 8, pp. 1107-1116.
•[4] Ipatov, S.I. and Mather, J.C. (2003) Migration of trans-Neptunian objects to the terrestrial
planets, Earth, Moon, and Planets, v. 92, 89-98.
•[5] Ipatov, S.I. and Mather, J.C. (2004) Migration of Jupiter-family comets and resonant asteroids to
near-Earth space, Annals of the New York Academy of Sciences, v. 1017, pp. 46-65.
•[6] Ipatov, S.I. and Mather, J.C. (2004) Comet and asteroid hazard to the terrestrial planets, Adv. in
Space Research, v. 33, N 9, 1524-1533.
•[7] Ipatov, S.I. and Mather, J.C. (2006) Migration of small bodies and dust to near-Earth space, Adv.
in Space Research, v. 37, N 1, 126-137.
•[8] Ipatov, S.I. and Mather, J.C. (2007) Migration of comets to the terrestrial planets, Proc. of the
IAU Symposium No. 236 “Near-Earth Objects, Our Celestial Neighbors: Opportunity and Risk” (14-18
August 2006, Prague, Czech Republic), ed. by A. Milani, G.B. Valsecchi & D. Vokrouhlický, pp. 5564.
•[9] Ipatov, S.I., Kutyrev, A., Madsen, G.J., Mather, J.C., Moseley, S.H., Reynolds, R.J. (2008)
Dynamical zodiacal cloud models constrained by high resolution spectroscopy of the zodiacal light,
Icarus, v. 194, N. 2, 769-788
•[10] Marov, M. Ya. and Ipatov, S.I. (2005) Migration of dust particles and volatiles delivery to the
terrestrial planets, Solar System Research, v. 39, N 5, 374-380
•[11] Ipatov, S.I., (2010) Collision probabilities of migrating small bodies and dust particles with
planets, Proceedings of the IAU Symposium 263 "Icy bodies in the Solar System" (Rio de Janeiro,
Brazil, 3-7 August, 2009), ed. by D. Lazzaro, D. Prialnik, R. Schulz, J.A. Fernandez, pp. 41-44.
http://arxiv.org/abs/0910.3017
53
•Files with the above papers are available on http://faculty.cua.edu/ipatov/list-publications.htm
Angular momenta of rarefied preplanetesimals and
formation of small-body binaries. Introduction.
• The binary fractions in the minor planet population are about 2% for main-belt
asteroids, 22% for cold classical TNOs, and 5.5% for all other TNOs (Noll 2006).
http://www.johnstonsarchive.net/astro/asteroidmoons.html
• There are several hypotheses of the formation of binaries for a model of solid
objects. For example, Goldreich et al. (2002) considered the capture of a secondary component inside Hill
sphere due to dynamical friction from surrounding small bodies, or through the gravitational scattering of a
third large body. Weidenschilling (2002) studied collision of two planetesimals within the sphere of influence
of a third body. Funato et al. (2004) considered a model for which the low mass secondary component is
ejected and replaced by the third body in a wide but eccentric orbit. Studies by Astakhov et al. (2005) were
based on four-body simulations and included solar tidal effects. Gorkavyi (2008) proposed multi-impact
model. Ćuk, M. (2007), Pravec et al. (2007) and Walsh et al. (2008) concluded that the main mechanism of
formation of binaries with a small primary (such as near-Earth objects) could be rotational breakup of ‘rubble
piles’. More references can be found in the papers by Richardson and Walsh (2006), Petit et al. (2008), and
Scheeres (2009).
• In recent years, new arguments in favor of the model of rarefied
preplanetesimals - clumps have been found (e.g. Makalkin and Ziglina 2004, Johansen et
al. 2007, Cuzzi et al. 2008, Lyra et al. 2008). These clumps could include meter sized boulders in
contrast to dust condensations earlier considered. Sizes of preplanetesimals could be up to their
Hill radii.
• My studies of the formation of binaries at a stage of rarefied preplanetesimals
presented below testify in favor of existence of rarefied preplanetesimals and can
allow one to estimate their sizes at the times of collisions of preplanetesimals.
• Ipatov, S.I., The angular momentum of two collided rarefied preplanetesimals and the
54
formation of binaries, Mon. Not. R. Astron. Soc., v. 403, 405-414 (2010)
http://faculty.cua.edu/ipatov/mnras-binaries.pdf, http://arxiv.org/abs/0904.3529
Scaling the size ladder from dust to planets
•1. Particle-particle sticking. (micron -> mm)
•2. Vertical settling into thin sublayers. (micron -> cm)
•3. Gravitational instability assisted by gas. (mm – tens of km)
•4. Streaming instabilities clump particles. (cm – tens of m)
•5. Gravitations collapse triggered by turbulent clumping
can produce preplanetesimals with masses of solid bodies with radii of many km.
(cm – many km)
•6. Pairwise, gravitationally focused collisions. (greater km)
•Dust can settle vertically into a dense sublayer (e.g., Chiang 2008,
Barranco 2009); pile up as it drifts radially (Youdin & Shu 2002, Youdin
& Chiang 2004); remain behind as stellar ultraviolet radiation
photoevaporates gas (e.g., Throop & Bally 2005); and be concentrated
on small scales by passively responding to turbulent fluctuations
and by actively driving drag instabilities with gas (e.g., Goodman &
55
Pindor 2000, Youdin & Goodman 2005, Johansen & Youdin 2007).
•
Boulders can undergo efficient gravitational collapse in locally overdense regions in the
midplane of the disk. The boulders concentrate initially in transient high pressure regions in
the turbulent gas, and these concentrations are augmented a further order of magnitude by a
streaming instability driven by the relative flow of gas and solids. Johansen et al. (2007)
found that gravitationally bound clusters form with masses comparable to
dwarf planets and containing a distribution of boulder sizes. After only seven orbits, peak
densities in these clusters approach 10,000 that of gas or a million times the average
boulder density in the disk. Figs from: A. Johansen, J. S. Oishi, M.-M. Mac Low, H. Klahr,
T. Henning, & A. Youdin, Rapid planetesimal formation in turbulent circumstellar disks,
Nature, 2007, 448, 1022-1025
56
Particle column density, showing the formation of seven gravitationally bound
clumps in a 3D, vertically stratified, shearing box simulation of unmagnetized gas and
superparticles (Johansen et al. 2009). The box-averaged hpi/hgi=0.02. This snapshot is
taken 5 orbits after self-gravity was turned on. The bound fragments contain ∼20%
of the total mass in solids; each has a mass comparable to a compact
planetesimal having a size 100–200 km. Fig. from: Johansen A, Youdin A,
MacLow M.M. 2009. Particle Clumping in Protoplanetary Disks Depends Strongly on
Metallicity. ArXiv e-prints.
57
Formation of trans-Neptunian objects
•In the models of accumulation of trans-Neptunian objects (TNOs)
from smaller solid planetesimals, accumulation took place at small
(0.001) eccentricities and a massive belt. In my opinion, published
e.g. in 2001, it is probable that, due to the gravitational influence of the
forming giant planets, TNOs, and migrating planetesimals, small
eccentricities of TNOs could not exist during all the time needed for
the solid-body accumulation of TNOs with diameter d>100 km.
•Therefore TNOs with d100 km moving now in not very eccentric
orbits could be formed directly by the compression of large rarefied
dust condensations (with a>30 AU), but not by the accretion of
smaller solid planetesimals.
• Probably, some planetesimals with d100-1000 km in the feeding zone of the giant planets and
even some large main-belt asteroids also could be formed directly by the compression of rarefied
dust condensations. Some smaller objects (TNOs, planetesimals, asteroids) could be debris of
larger objects, and other such objects could be formed directly by compression of condensations.
• As in the case of accumulation of planetesimals, there could be a “run-away” accretion of
condensations. It may be possible that, during the time needed for compression of condensations
into planetesimals, some largest final condensations could reach such masses that they formed
planetesimals with diameter equal to several hundreds kilometers.
• Ipatov S.I., (2001) LPSC, (#1165). http://www.lpi.usra.edu/meetings/lpsc2001/pdf/1165.pdf
• Ipatov S.I. (2004) Proc. of the 14th Annual Astroph. Conf. in Maryland “The Search for Other
Worlds”, ed. by S.S. Holt and D. Deming, American Inst. of Physics, 277-280.
• Ipatov, S.I. (2004) http://www.astro.umd.edu/~ipatov/ipatov-darwin-conf-p045.doc 58
• Ipatov, S.I. (2005) http://www.lpi.usra.edu/meetings/ppv2005/pdf/8054.pdf
Scenarios of formation of binaries at the stage
of rarefied preplanetesimals
• Application of previous solid-body scenarios to preplanetesimals. The models of
binary formation due to the gravitational interactions or collisions of future binary
components with an object (or objects) that were inside their Hill sphere, which
were considered by several authors for solid objects, could be more effective for
rarefied preplanetesimals. For example, due to almost circular heliocentric orbits,
duration of the motion of preplanetesimals inside the Hill sphere could be longer and
the minimum distance between centers of masses of preplanetesimals could be
smaller than for solid bodies, which usually moved in more eccentric orbits.
• Two centers of contraction. Some collided rarefied preplanetesimals had a greater
density at distances closer to their centers, and sometimes there could be two
centers of contraction inside the preplanetesimal formed as a
result of a collision of two rarefied preplanetesimals.
• For such model, binaries with close masses separated by a large
distance (up to a radius of a Hill sphere) and with any value of the eccentricity
of the orbit of the secondary component relative to the primary component could
be formed. The observed separation distance can characterize sizes of
encountered preplanetesimals. Most of rarefied preasteroids could contract into
solid asteroids before they collided with other preasteroids.
59
Scenarios of formation of binaries at the stage
of rarefied preplanetesimals
• Excessive angular momentum. Formation of some binaries could be caused by
that the angular momentum that they obtained at the stage of
rarefied preplanetesimals was greater than that could exist for
solid bodies. During contraction of a rotating rarefied preplanetesimal, some
material could form a cloud (that transformed into a disk) of material moved
around the primary. One or several satellites of the primary could be
formed from this cloud.
• Our studies showed that the angular momentum of any discovered transNeptunian binary is smaller than the typical angular momentum of two
identical rarefied preplanetesimals having the same total mass and
encountering up to the Hill sphere from circular heliocentric orbits.
• Hybrid scenario. Both above scenarios could work at the same time. In this case, it
is possible that besides massive primary and secondary components, there could
be smaller satellites moving around the primary (and/or the secondary) at smaller
distances. For binaries formed in such a way, separation distance between main
components can be different (e.g. large or small).
60
Data
presented
in
the
Table
• For six binaries, the angular momentum K of the present primary and
scm
secondary components (with diameters dp and ds and masses mp and ms), the
momentum
Ks06ps=KsΘ=vτ∙(rp+rs)∙mp∙ms/(mp+ms)=kΘ∙(G∙MSun)1/2∙(rp+rs)2∙mp∙ms∙(mp+ms)-1∙a-3/2 (vτ is
the tangential component of velocity vcol of collision) of two collided Hill
spheres -preplanetesimals with masses mp and ms moved in circular
heliocentric orbits at kΘ≈(1-1.5∙Θ2)=0.6 (this value of |kΘ | characterizes the mean
momentum; the difference in semimajor axes equaled to Θ∙(rp+rs), rp+rs was the sum of
radii of the spheres), and the momentum Ks06eq of two identical collided
preplanetesimals with masses equal to a half of the total mass of the binary components
(i.e. to 0.5mps, where mps=mp+ms) at kΘ=0.6 are presented in the Table. All these
three momenta are considered relative to the center of mass of the system. The
resulting momentum of two colliding spheres is positive at
0<Θ<(2/3)1/2≈0.8165 and is negative at 0.8165<Θ<1. Formulas and other
details of calculation of momenta can be found in Ipatov (2010a).
• Kspin=0.2π∙χ∙mp∙dp2∙Tsp-1 is the spin momentum of the primary (χ=1 for a homogeneous
sphere; Tsp is the period of spin rotation of the primary). L is the distance between the
primary and the secondary. In this Table we also present the values of 2L/dp and L/rHtm,
where rHtm is the radius of the Hill sphere for the total mass mps of the binary.
• Three velocities are presented in the last lines of the Table, where vτpr06 is the
tangential velocity vτ of encounter of Hill spheres at present masses of components
of the binary, vτeq06 is the value of vτ for encounter of Hill spheres at masses equal to
0.5mps each, and vesc-pr is the escape velocity on the edge of the Hill sphere of 61
the
primary.
Table. Angular momenta of several small-body binaries
(90842)
(87)
(90)
Orcus 2000 CF105 2001 QW322 Sylvia
Antiope
-----------------------------------------------------------------------------------------------------binary
Pluto
a, AU
dp, km
ds, km
mp, kg
ms, kg
39.48
39.3
43.8
43.94
2340
950
170
108?
1212
260
120
108?
22
20
18
1.3×10
7.5×10
2.6×10 ? 6.5×1017 ?
1.52×1021 1.4×1019
9×1019 ? 6.5×1017 ?
for ρ=1.5 g cm-3
L, km
19,570
8700
23,000
120,000
L/rHtm
0.0025
0.0029
0.04
0.3
2L/dp
16.9
18.3
271
2200
Tsp, h
153.3
10
Kscm, kg∙km2∙s-1 6×1024 9×1021
5×1019
3.3×1019
Kspin, kg∙km2∙s- 1023
1022
1.6×1018
2×1017
at Ts=8h
Ks06ps, kg∙km2∙s-1 8.4×1025 9×1022 1.5×1020
5.2×1019
Ks06eq, kg∙km2∙s-1 2.8×1026 2×1024 2.7×1020
5.2×1019
3.49
286
18
1.478×1019
3×1015
for ρ=1 g cm-3
1356
0.019
9.5
5.18
1017
4×1019
at Ts=8h
3×1017
8×1020
3.156
88
84
4.5×1017
3.8×1017
171
0.007
3.9
16.5
6.4×1017
3.6×1016
6.6×1018
6.6×1018
(Kscm+Kspin)/Ks06eq 0.02
0.01
0.2
0.63
0.05
0.1
vτeq06, m∙s-1
2.2
0.36
0.26
2.0
0.82
0.3
0.8
0.26
0.53
1.3
5.3
0.82
1.7
vτpr06, m∙s-1
vesc-pr, m∙s-1
6.1
5.5 1.8
15.0 5.8
62
Comparison of angular momenta of present binaries
with model angular momenta
•For the binaries presented in the Table, the ratio rK=(Kscm+Kspin)/Ks06eq (i.e.,
the ratio of the angular momentum of the present binary to the typical
angular momentum of two colliding preplanetesimals – Hill spheres moving
in circular heliocentric orbits) does not exceed 1, i.e. the angular
momentum of any discovered trans-Neptunian binary is smaller than
the typical angular momentum of two identical rarefied
preplanetesimals having the same total mass and encountering up to
the Hill sphere from circular heliocentric orbits. For most of observed
binaries, this ratio is smaller than for the binaries considered in the Table.
Small values of rK for most discovered binaries and the separation distances,
which are usually much smaller than radii of Hill spheres, can be due to that
preplanetesimals had already been partly compressed at the moment of
collision (could be smaller than their Hill spheres and/or could be denser
for distances closer to the center of a preplanetesimal).
•Petit et al. (2008) noted that most other models of formation of binaries
cannot explain the formation of the trans-Neptunian binary 2001 QW322.
For this binary we obtained that the equality KsΘ=Kscm is fulfilled at kΘ≈0.4
and vτ≈0.16 m/s. Therefore in our approach this binary can be explained
even for circular heliocentric orbits of two collided preplanetesimals.
•The angular momentum obtained at collisions of two preplanetesimals
was of the same order same as that used by D. Nesvorny et al. (AJ, 2010,
785-793) in their model of gravitational collapse that caused formation of
binaries. They supposed the momentum was acquired at a stage of formation of
63
condensations, but in this case the momentum must be only positive, though
there are observed binaries with negative momentum.
Formation of axial rotation of Pluto and
inclined mutual orbits of components
Pluto has three satellites, but the contribution of two satellites (other than
Charon) to the total angular momentum of the system is small. To
explain Pluto’s tilt of 120o and inclined mutual orbit of 2001 QW322
components (124o to ecliptic), we need to consider that thickness of a
disk of preplanetesimals was at least of the order of sizes of
preplanetesimals that formed these systems.
Inclined mutual orbits of many trans-Neptunian binaries
testify in favor of that momenta of such binaries were
acquired mainly at single collisions of rarefied
preplanetesimals, but not due to accretion of much
smaller objects (else primordial inclinations of mutual
orbits relative to the ecliptic would be small).
It is not possible to obtain reverse rotation if the angular momentum was
caused by a great number of collisions of small objects with a larger
preplanetesimal (for such model, the angular momentum Ks and
period Ts of axial rotation of the formed preplanetesimal were studied
by Ipatov 1981a-b, 2000).
64
Discussion
• In the considered model, sizes of preplanetesimals comparable
with their Hill spheres are needed only for formation of
binaries at a separation distance L close to the radius rHtm of
the Hill sphere (such as 2001 QW322). For other binaries
presented in the Table (and for most discovered binaries), the ratio
L/rHtm does not exceed 0.04. To form such binaries, sizes of
preplanetesimals much smaller (at least by an order of
magnitude) than the Hill radius rHtm are enough. The
observed separation distance L can characterize the sizes of
contracted preplanetesimals.
• Density of rarefied preplanetesimals was very low, but relative
velocities vrel of their encounters up to Hill spheres were
also very small, and they were smaller than escape velocities
on the edge of the Hill sphere of the primary (see Table). It
is not needed that all encounters up to the Hill sphere resulted in
collision of preplanetesimals. It is enough that there were such
encounters only once during lifetimes of some preplanetesimals.
65
Discussion
• For a primary of mass mp and a much smaller object, both in
circular heliocentric orbits,
• vτ/vesc-pr=kΘ∙3-1/6∙(MSun/mp)1/3∙a-1 (designations are presented on
page 7 in “Data presented in the Table”). This ratio is smaller for
greater a and mp. Therefore, the capture was easier for more
massive preplanetesimals and for preplanetesimals in
the trans-Neptunian region than in the asteroid belt.
• The ratio of the time needed for contraction of preplanetesimals
to the period of rotation around the Sun, and/or the total mass of
preplanetesimals could be greater for the trans-Neptunian region
than for the initial asteroid belt. It may be one of the reasons of a
larger fraction of trans-Neptunian binaries than of binaries in
the main asteroid belt.
• At greater eccentricities of heliocentric orbits, the probability of
that the encountering objects form a new object is smaller (as
collision velocity and the minimum distance between centers of
mass are greater and the time of motion inside the Hill sphere is
smaller) and the typical angular momentum of encounter up to the
Hill sphere is greater.
66
N bodies simulation of a rotating spherical condensation using the PGKRAV code.
Initial rotation. David Nesvorn´y, Andrew N. Youdin, and Derek C. Richardson,
FORMATION OF KUIPER BELT BINARIES BY GRAVITATIONAL COLLAPSE,
Astron. J., 140, 785–793, 2010. Simulation of the evolution of a rotating spherical
condensation, initially consisted of a few thousands objects that coagulate at
collisions. Nesvorny et al. supposed that axial rotation was acquired during
formation of condensations, but in this case it is not possible to explain reverse
rotation of some observed binaries.
67
References
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Gorkavyi N. N., 2008, abstract, Asteroids, Comets, Meteors, #8357
Ipatov S. I., 1981a, AS USSR Inst. Applied Math. Preprint N 101, Moscow, in Russian
Ipatov S. I., 1981b, AS USSR Inst. Applied Math. Preprint N 102, Moscow, in Russian
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68
Cavities as a source of outbursts from comets
• Introduction. On July 4, 2005 370 kg the impactor module collided with Comet
9P/Tempel 1 at velocity of 10.3 km/s (A’Hearn et al. 2005). It was an oblique impact,
and the angle above the horizon was about 20-35o. Evolution of the cloud of ejected
material was observed by Deep Impact (DI) cameras, by space telescopes (e.g. Rosetta,
Hubble Space Telescope, Chandra, Spitzer), and by over 80 observatories on the Earth.
• The values of the projection of the velocity vle of the leading edge of the dust cloud
of ejected material onto the plane perpendicular to the line of sight were about 100-200
m/s for observations made 1-40 h after impact.
• Our studies of ejection of material from this comet were based on
analysis of the images made by Deep Impact (DI) cameras during the
first 13 minutes.
• Jorda et al. (2007) concluded that particles with d<2.8 μm represent more than 80%
of the cross-section of the observed dust cloud. The velocities discussed in our
presentation correspond mainly to particles with d<3 μm, which probably
constitute not more than 7% of the total ejected material.
• The total mass of ejected dust particles with diameter d less than 2, 2.8, 20, and 200
μm was estimated to be about 7.3×104–4.4×105, 1.5×105–1.6×105, 5.6×105-8.5×105,
and 106-1.4×107 kg, respectively.
•The main aims of our studies: Time variations in velocities and relative amount of
material ejected from Comet 9P/Tempel 1. The role of a triggered outburst in the
ejection. Excessive ejection in a few directions (rays of ejected material). Cavities in
comets as sources of outbursts.
Ipatov, S.I. and A’Hearn, M.F., The outburst triggered by the Deep Impact collision
69
with Comet Tempel 1, Mon. Not. R. Astron. Soc., v. 414, 76-107 (2011).
http://faculty.cua.edu/ipatov/di-mnras.pdf , http://arxiv.org/abs/0810.1294.
We analyzed several series of DI images. In each series, the
integration time and the size of image were the same. For series Ma, Ha,
and Hc, we analyzed the differences in brightness between a current image and that
before the impact. These series are marked by “(dif)”. For other series, we analyzed the
brightness in current images. Observed brightness of the cloud of ejected material
was mainly due to particles with diameters d<3 micron, and we discuss mainly
ejection of such particles.
Series
Instru- INTTIME,
ment
seconds
Size, pixels
EXPID
IMPACTM,
seconds
min, max
min, max
Ma (dif)
MRI
0.0514
64×64
9000910, 9000910
0.001, 5.720
Mb
MRI
0.3
1024×1024
9000942, 9001067
77.651, 802.871
Ha (dif)
HRI
0.1
512×512
9000910, 9000945
0.215, 109.141
Hb
HRI
0.6
1024×1024
9000931, 9001002
39.274, 664.993
Hc (dif)
HRI
0.6
512×512
9000927, 9000942
27.664, 86.368
Hd
HRI
0.1
1024×1024
9000934, 9000961
50.715, 251.525
He
HRI
0.5
1024×1024
9001017, 9001036
70
719.805, 771.95
Contours corresponding to CPSB (calibrated physical surface
brightness) equal to 1, 0.3, 0.1, and 0.03, for series Mb. Our studies
were based mainly on analysis of distances from the place of ejection
to the contours of fixed brightness. The bumps on the contours
correspond to rays of excessive ejection.
71
Rays
of
ejected
material
Considerable excessive ejection to a few directions (some rays)
•
began approximately at the same time te~10 s when the direction
from the place of ejection to the brightest pixel changed, the peak
brightness began to increase, and there was a local peak of the rate
of ejection. These features could be caused by the outburst triggered
by the impact.
• The excess ejection of material to a few directions (rays of ejected
material) was considerable during the first 100 s, took place during
several minutes, and was still observed in images at t~500-770 s even
close to the place of ejection. It shows that the outburst could
continue up to ~10 min.
• The sharpest rays were caused by material ejected at te~20 s.
• The upper-right excessive ejection (perpendicular to the direction of
impact) began mainly at te~15 s (though there was some ejection at te~2
s), could reach maximum at te~25-50 s, could still be considerable at
te~100 s, but then could decrease, though it still could be seen at te~400 s.
The value of te~15 s is correlated with the changes of the direction to the
brightest pixel at t~12-13 s.
• The upper bump of the outer contours is clearly seen at 66<t<665 s,
especially at t~200-350 s. The direction from the place of impact to72 this
bump is not far from the direction opposite to the impact direction.
Time variations in sizes L (in km) of regions inside contours of CPSB=const. The number after a
designation of the series in the figure legend shows the value of brightness of the considered contour. The
curves have local minima and maxima that were used for estimates of velocities at several moments in
time. Based on the supposition that the same particles correspond to different local maxima (or minima) of L
(e.g., to values L1 and L2 on images made at t1 and t2), we calculated the characteristic velocities v=(L1L2)/(t1-t2) at te=t1-L1/v. For series Ma, we considered L as the distance from the place of impact to the contour
down in y-direction. For other series, we considered the difference between maximum and minimum values
of x for the contour.
73
Typical projections vmodel of velocities (in
km/s) on the plane perpendicular to the line of
sight at time te of ejection for the model for
which velocities vmodel at te are the same as
velocities vexpt=c×(t/0.26)-α of particles at the
edge of observed bright region at time t.
The values of velocities of ejection
marked by vyobs and vxobs are based
on our analysis of local minima and maxima
of plots on the previous slide.
The distance from the place of ejection to
the edge was used to find the dependence of t
on te.
As the first approximation, the
characteristic velocity at te>1 s can be
considered to be proportional to te-0.75 or
te-0.7 (i.e. α~0.7-0.75; 0.71 corresponds to sand;
0.75, to the ejection mainly governed by
momentum).
The values of vymin and vxmin show the
minimum velocities of particles needed to
reach the edge of the bright region (in an image
considered at time t) from the place of ejection
moving in y or x-direction, respectively.
74
Calculations of the relative rate of ejection
Considering that the time needed for particles to travel a distance Lr to
the edge of the bright region is equal to dt=Lr /vexpt (where
vexpt(t)=c×(t/0.26)-α ), we find the time te=t−dt of ejection of material
of the contour of the bright region considered at time t.
The volume Vol of a spherical shell of radius Lr and width h is
proportional to Lr2h, and the number of particles per unit of volume is
proportional to rte∙(Vol∙v)-1, where v is the velocity of the material.
Here rte corresponds to the material that was ejected at te and reached
the shell with Lr at time t. The number of particles on a line of sight,
and so the brightness Br, are approximately proportional to the
number of particles per unit of volume multiplied by the length of the
segment of the line of sight inside the DI cloud, which is proportional
to Lr. Actually, the line of sight crosses many shells characterized by
different rte, but as a first approximation we supposed that Br is
proportional to rte(v∙Lr)-1. For the edge of the bright region,
Br≈const. Considering v=vexpt, we calculated the relative rate of
ejection as rte=c∙Lr∙t-α.
75
Relative rate of ejection at different times te of ejection for the model
in which characteristic velocities of particles constituting the edge of the observed
bright region at time t are equal to vexpt=c×(t/0.26)-α. At 1<te<3 and 8<te<60 s, the plot
of time variation in the estimated rate rte of ejection of observed material was
essentially greater than the exponential line connecting the values of rte at 1 and 300
s. (The exponential monotonic decrease of the rate is predicted by theoretical
models.) There was a local maximum of the rate at te~10 s with typical
projections of velocities vp~100 m/s, and there was a sharp decrease
of rte at te~60 s. Such variations in rte can be mainly due to the triggered
outburst. Our studies do not contradict to a continuous ejection of material
during at least 10 minutes after the collision.
76
Relative amount fev of observed material ejected with velocities
greater than v vs. v for the model VExp in which characteristic velocities of
particles at the edge of the observed bright region in an image at time t are equal to
v=vexpt=c×(t/0.26)-α for five pairs of α and c. fev=1 for material ejected before the time
te803 of ejection of particles located at the edge of the bright region in an image made at
t=803 s. At velocities of several tens of meters per second, the model
amount is greater than for theoretical estimates. For theoretical models,
exponents of the velocity dependence of the relative volume fev of material ejected with velocities
greater than v, equal to -1, -1.23, -1.66, and -2, correspond to α equal to 0.75, 0.71, 0.644, and
0.6, respectively (α=0.71 is for sand).
77
‘FAST’ and ‘SLOW’ OUTBURSTS
• The rates and velocities of material ejected after the DI impact
were different from those for experiments and theoretical models.
Holsapple and Housen (2007, Icarus 187, 345-356) concluded that the difference
was caused by vaporization of ice in the plume and fast moving gas.
• In our opinion, the greater role in the difference for particles
located at a few km from the comet could be played by the outburst
triggered by the impact (by the increase of ejection of bright particles).
• The ‘fast’ outburst could be caused by ejection of material from
cavities with dust and gas under gas pressure. ‘Slow’ outburst
ejection could be similar to the ejection from a ‘fresh’ surface of a
comet and could be noticeable at 30-60 min after the formation of the crater.
• The jump of the contribution of the DI outburst to the brightness of the DI cloud
began at te~8 s. The ejection rate at te=10 s was greater by a factor of 1.4 than that
obtained at the supposition that the previous decrease of the rate was prolonged with the
same exponent. This factor may characterize the increase of the role of the outburst in
the ejection of small particles if at that time veo~ve.
•There could be a sharp decrease of the outburst at te~60 s.
•The contribution of the outburst to the brightness of the cloud could be considerable,
but its contribution to the total ejected mass could be relatively small because typical
sizes of particles ejected at the outburst probably were smaller than those ejected at the
78
normal ejection.
Velocities of ejected particles
• Projections of velocities of most of observed material ejected at te~0.2
s were about 7 km/s. Analysis of DI observations that used different
approaches showed that at 1<te<100 s the time variations in the
projections vp of characteristic velocity onto the plane perpendicular to
the line of sight can be considered to be approximately proportional to
te-α with α~0.7-0.75.
• For the VExp model with vp proportional to te-α at any te>1 s, the
fractions of observed material ejected (at te≤6 and te≤15 s) with
vp≥200 and vp≥100 m/s were estimated to be about 0.1-0.15 and 0.20.25, respectively, if we consider only material observed during the
first 13 min.
• The ‘fast’ outburst with velocities ~100 m/s probably could last
for at least several tens of seconds, and it could significantly
increase the fraction of particles ejected with velocities ~100 m/s,
compared with the estimates for the VExp model and for the normal
ejection.
79
Velocities and acceleration by gas
Our estimates of velocities of particles are in accordance
with the estimates (100-200 m/s) of projection of velocity of
the leading edge of the DI dust cloud that were based on
various ground-based observations and observations made by
space telescopes.
Destruction, sublimation, and acceleration of particles did
not affect much on our estimates of velocities because we
considered the motion of particles along a distance of a few
km during not more than a few minutes.
During the considered motion of particles (at a distance
not more than a few km) with initial velocities vp≥20 m/s,
the increase of their velocities due to the acceleration by
gas did not exceed a few m/s.
80
Cavities with material under gas pressure
Analysis of observations of the DI cloud and of outbursts
from different comets testifies in favor of that there can be
large cavities with material under gas pressure below a
considerable fraction of a comet’s surface.
Internal gas pressure (e.g. due to crystallization of
amorphous ice and/or sublimation of the CO ice) and the
material in the cavities can produce natural and triggered
outbursts and can cause splitting of comets.
The outburst ejection of material from a cavity could be
greater for a specific direction. Therefore, its role in the
direction from the place of ejection to the brightest pixel
could be greater than in the total ejection rate.
81
Location of the upper border of the main
excavated cavity
Our estimates testify in favor of the location of the upper border of
the main excavated cavity (with dust and gas under pressure) at a
depth dcav~5-10 meters. For example, at time teb=8 s, the depth of a crater
dcr=df×dh/(Te/teb)γ=12.5 m for df=100 m, dh=0.25, Te=80 s, and (Te/teb)γ=10γ=2; for the
same data and Te=400 s, dcr≈12.5/1.62≈8 m. For theoretical models (Holsapple &
Housen 2007), radius of a crater is proportional to teγ, where γ is about 0.25-0.4. For
small cavities excavated at te=1 s, the value of dcr (~4-5 m) was smaller by a factor of 8γ
(i.e. by about a factor of 2) than at te=8 s.
The distance dcav between the pre-impact surface of the comet’s nucleus and the upper
border of the cavity could be smaller than dcr because the excavated cavity could be
located at some distance from the center of the crater (not below the center). On the
other hand, due to cracks caused by the impact, the outburst from the cavity could begin
before excavation of the upper border.
The distances from the upper borders of large cavities to the surface of a comet of
about 5-10 m, and sizes of particles inside the cavities of a few microns are in a good
agreement with the results obtained by Kossacki and Szutowicz (Icarus, 2011)
82
Conclusions
At time of ejection te~10 s, there was a local maximum of the rate of
ejection of observed particles (mainly with diameter d<3 μm) with
typical projections of velocities vp~100 m/s. At 1<te<3 s and 8<te<50
s the estimated rate of ejection of observed material was essentially
greater than that for theoretical monotonic exponential decrease. Such
difference was caused by that the impact was a trigger of an outburst.
At te~55-72 s, the ejection rate sharply decreased and the direction
from the place of ejection to the brightest pixel quickly returned to the
direction that was before 10 s. It could be caused by a sharp decrease of
the outburst that began at te~10 s.
Analysis of observations of the DI cloud and of outbursts from
different comets testifies in favor of that there can be large cavities
with material under gas pressure below a considerable fraction of
comet’s surface. The upper boarder of relatively large cavities can be
located at about 5-10 meters below the surface of the comet.
83
Results of our studies of ejection of material after the DI impact are
presented in the below papers:
[1] Ipatov S.I. and A’Hearn M.F., The outburst triggered by the Deep
Impact collision with Comet Tempel 1, Mon. Not. R. Astron. Soc.,
2011, Mon. Not. R. Astron. Soc., v. 414, N 1, 76-107.
http://faculty.cua.edu/ipatov/di-mnras.pdf ,
http://arxiv.org/abs/0810.1294.
[2] Ipatov S.I., Cavities as a source of outbursts from comets, In
“Comets: Characteristics, Composition and Orbits”, Nova Science
Publishers, in press, http://faculty.cua.edu/ipatov/cavities.pdf ,
http://arxiv.org/abs/1103.0330 .
[3] Ipatov S.I. and A'Hearn M.F., Deep Impact ejection from Comet
9P/Tempel 1 as a triggered outburst, Proc. IAU Symp. S263 "Icy
bodies in the Solar System“, Cambridge University Press, 2010, pp.
317-321. http:/faculty.cua.edu/ipatov/di-iau.pdf ,
http://arxiv.org/abs/1011.5541 .
A full list of Ipatov’s publications (and copies of most of them)84can
be found on http://faculty.cua.edu/ipatov/list-publications.htm .
Recognition of cosmic ray signatures on Deep Impact CCDs
Ipatov, S.I., A’Hearn, M.F., and Klaasen, K.P., Automatic removal of cosmic ray
signatures on Deep Impact images, Advances in Space Research, v. 40, 160-172,
(2007). http://www.astro.umd.edu/~ipatov/adsr-2007.pdf
The flight module of the DI spacecraft has two telescopes: MRI
(medium resolution instrument) and HRI (high resolution instrument).
Dark raw images
•CR events are most easily detected in dark images. We analyzed
dark raw images to derive statistics for CR events.
•Only pixels with digital numbers DN>370 for MRI and DN>390 for
HRI were considered candidates for being CR events (i.e., signals more
than ~15 DN above the bias level).
•On the left side of the below slide, the observed number of CR events
per cm2 per second is plotted vs. the number of pixels affected by the
event for dark images taken during both typical solar activity and during
a flare.
•On the right side of the slide, the mean value of DN (including the bias
level) within the CR signature is plotted vs. the number of pixels in the
event.
•This slide is based on analysis of several MRI images with exposure
85
time 9≤t≤30 s.
The number of CR events per second per square centimeter
(left) and mean values of Digital Number (right) vs. the number of
pixels at a peak of solar activity (crosses) and for out-of peak
activity (stars) on dark MRI images with 9≤t≤30 s.
86
MRI and HRI images at small t
For small t, the greatest number of detected CRs in a series is usually for the first
image in the series.
In Ken’s opinion, for this case the CR integration time is not the same as the image
integration time. It must also include the shutter close and on average 1/2 the readout
time. So a 4-ms exposure in Mode 1 really collects CRs for more like 0.82 s on
average. The CR integration period is about 0.82 s longer on average than the total
image exposure time.
For most HRI and MRI visual images made during low solar activity at t>4 s (t
= exposure time), the number of CR events per second per square
centimeter of CCD was about 2-4 (typically ~3), and generally there were
no events consisting of more than 2t pixels.
Most CR events in an image consist of not more than 4 pixels.
The largest CR signatures have a linear form in contrast to the more circular form for
star images.
At high solar activity, the CR event rate can increase by a factor of
5 compared to that at low activity, and long CRs due to grazing paths of the
cosmic ray through the detector membrane can exceed 8.5t.
At t=30 s the maximum number of pixels in one long streak exceeded 200, while no
CR events consisted of more than 45 pixels at t=30 s for images outside the period
87 of
solar flares.
MRI and HRI images
The ratio of the number of CR events consisting of
n pixels obtained at high solar activity to that at
low solar activity was greater for greater n.
For example, this ratio was greater at high solar
activity to that for out-of-peak activity by a factor of
about 1.5, 2, 3, 3.5, 7 for rays consisted of 1, 2, 3, 4,
and 5 pixels, respectively.
This suggests that events caused by energetic
particles from the Sun tend to produce larger
signatures than do interstellar CRs.
88
MRI and HRI images
• Based on comparison of CR signatures on dark and sky images,
we can make two main conclusions that can be used for recognition
of CRs:
• (1) Even for out-of peak of solar activity, most CR events consist of a
small number of pixels, while well-exposed star images are typically
larger (especially for the out-of-focus HRI). At exposure time t≥4 s,
almost all (>80%) 1-4 pixel charge clusters in typical sky images are
CRs (excluding images of dense conglomerations of stars).
• (2) Large CR events have a linear form in contrast to the more
circular form of star images. We calculated the ratio kp=npix/(dxx 2
+dyy 2) for different clusters, where dxx and dyy are the maximum
differences of coordinates x and y in a charge cluster each increased
by 1, and npix is the total number of pixels in the cluster. At npix>30
we found that kp<0.17 for all CRs (dark images) and kp>0.17 for all
stars. However, when charge clusters consist of ≤10 pixels, it is
difficult to distinguish between CRs and stars based on kp.
89
•
MRI
and
HRI
images
The most reliable way to recognize CR events in astronomical images is to compare
different images of the same region of the sky, but it is not always possible to do
this. We analyzed the performance of four codes (imgclean, crfind,
di_crrej, and rmcr) written by E. Deutsch, R. White, D. Lindler, and S. Ipatov,
respectively, that seek to recognize CRs in a single image.
• First three codes run well in many cases (e.g., for typical calibrated
sky images, for analysis of which they were created), but usually
they do not work well with raw images.
• Some of the codes have problems with long (oblique entry) rays, and they delete
pixels near the edge of a comet nucleus or from its coma.
• Crfind and di_crrej only find pixels corresponding to CRs, but do not replace these
pixels.
• The rmcr code was written to work on both raw and calibrated images and to replace
detected CRs with values of their neighboring pixels.
• Below we briefly discuss the performance of these codes, giving main attention to
those images for which the above codes do not work well.
90
Description of rmcr code
• In my rmcr code, which was adapted expressly for Deep Impact, only those
pixels for which raw DNs or calibrated radiances are greater than some limit lim are
considered to be possible CRs. This limit can be an input parameter (e.g., for raw
images) or it can be calculated as lim=limit0*klim (e.g., klim=3), where limit0 is the
median value of all pixels on an image. For calibrated images, one may not know lim
in advance, so it is better to use the latter calculation of lim.
• This code deletes all ‘long’ CRs – objects with kp<0.17.
• Clusters consisting of not more than nlimit pixels are considered
as CRs.
• Depending on a considered image and problem, the input parameter nlimit can be
chosen to take any value (e.g., 10). In one variation of the code, the objects
(exclusive for ‘long’ objects) that are closer than dss (e.g., dss=10 pixels) to a defined
rectangle that includes the comet and its coma are not considered as CRs.
• For small values of nlimit, it may be useful to run rmcr for calibrated images two
times - first with a greater lim, and then with a smaller lim.
• The code runs slowly when there are a lot of pixels in all objects (e.g., a comet
occupies a considerable part of an image) because the code analyzes the entire image
at once, rather than by small portions of the image at a time as do the other codes.
91
Recognition of CRs by different codes
• The below three slides illustrate clusters recognized as CRs by the
different codes on three images. The expected number of CRs is
about 15, 4, and 20, respectively.
• On all three images, crfind and di_crrej defined too many pixels as
CRs that are not CRs.
• Imgclean does not recognize the whole long CR on the below slide,
but it deletes fewer pixels from the coma than rmcr (which does
recognize the whole long CR). So imgclean seems to be the best
technique to use for this slide.
• Pixels erroneously classified as CRs by di_crrej and imgclean at the
edges of an image are actually due to bright pixels at the boundary
between the imaging array and the overclocked pixels. Rmcr
excludes these pixels from its search for CRs.
92
Calibrated MRI image 6002420 (256x256 pixels). From left
to right: initial image, pixels recognized as CRs by di_crrej,
crfind, imgclean, rmcr (at klim=3 and lim=10).
93
Recognition of CRs by different codes on an image with Comet Tempel 1
• In the case of the below slide, the choice of the best code depends on the purpose
for which the frame is being analyzed:
• di_crrej did not delete pixels near the edge of the comet, but it considered about 80
small clusters (not well seen on the figure) as CRs instead of expected number of
~4;
• imgclean worked normally far from the comet, but it deleted many pixels around
the comet;
• it is possible to find parameters for suitable operation of rmcr, but they are not
those which are usually used;
• crfind did run to normal completion, so it gave the worst results (but for some other
images it can give the best results).
• In the case of the below slide, imgclean replaced the brightness of some pixels
recognized as CRs near the coma by a wrong brightness and after such replacement
the image looked like as some material was ejected from the comet. In such cases it
may be better to use images for which CRs are not deleted or try to find input
parameters for which rmcr gives suitable results.
94
Central part (256x256) of calibrated MRI image 9000907 (1024x1024). From left to
right: an initial image with Comet Tempel 1, pixels recognized as CRs by di_crrej,
crfind, imgclean, rmcr (at klim=15 and lim=10), image after imgclean.
95
Calibrated HRI image with a large star. From left to right: initial image, pixels
recognized as CRs by di_crrej, crfind, imgclean, rmcr (at nlimit=0.0005 and lim=30).
On the below slide, imgclean recognized as CRs pixels that belong to the star, but did
not recognize as CRs the objects near the star, which must be deleted for the considered
problem. So for this slide, rmcr gave better results.
96
Images with maximum radiance of ~0.0001
• For the calibrated images considered (dark images
and images of conglomerations of stars) with
maximum radiance of ~0.0001, di_crrej and crfind
did not work normally if we used the same default
parameters for which these codes worked normally
with calibrated images with maximum radiance of
~1.
• For the small radiance case, they deleted a lot of
pixels of background.
• We have not found parameter settings that work well
at small radiances for these codes.
97
Conclusions on a choice of the code
• For a calibrated dark image at a peak of solar activity, imgclean
removed most of the CRs, but a few tens of them were left on an
image. Even several long rays were left. Imgclean did not recognize
well long CRs on some other images. For a calibrated image of
conglomerations of stars, imgclean left most of stars, but sometimes it
deleted many small stars. It recognizes as CRs a lot of pixels near the
comet which are not CRs.
• It is a fast code and is easy to use.
• Imgclean is a more reliable code if one needs to remove
CRs automatically from a large number of images,
• but depending on a specific image and a specific
problem, other codes can work better (e.g., sometimes crfind
is the best when there is a large image of a comet).
• Imgclean was updated in order not to delete clusters from the edges of
DI images (the version is named imgcleane)
98
Raw ITS images
• The telescope on the impact DI module was named as ITS
(impactor targeting sensor). For raw ITS images, only rmcr
works.
• The number of detected CRs on dark ITS images per second per
quadrant was obtained to be about the same as that for MRI (medium
resolution instrument) and HRI (high resolution instrument) if we
consider only pixels with DN greater by about 50 than the median
value of DN for a quadrant. The difference between the median
values for different quadrants of ITS dark images could exceed 40.
• If we consider pixels with DN greater by 20 than the median value of
DN for a quadrant, then the number of clusters recognized by rmcr as
CRs is greater by a factor of 5-10 than that for MRI and HRI.
• The fraction of clusters detected as CRs and common for a pair of
images is about 5-20% of clusters detected as CRs on one image.
99
Calibrated ITS images
• CR detection has proved more difficult in ITS images than
in either MRI or HRI images. The problem may be
because the values of the background DNs in an ITS raw
dark image vary more than those for MRI and HRI images
(by factors of 2 or more) due to the higher operating
temperature and increased dark current of the ITS CCD.
• None of the codes considered worked well with
calibrated dark ITS images. The number of charge
clusters consisting of ≤4 pixels deleted by imgclean and
rmcr was greater by a factor of several than even the
expected number of CRs at the peak of solar activity, so
most of deleted objects were not real CRs.
• Crfind and di_ccrej designated even more pixels as
cosmic rays.
100
Calibrated IR images
•
•
•
•
Imgclean was run on two successive dark
frames and the number of charge clusters that
were different between the two runs was
about 5-10 per cm2 per second (the time was
considered as a sum of exposure times for two
images).
This number does not include pixels which
coincide with bad pixels, but only 1.5-2 % of
pixels which were detected as CRs and were
different between the runs coincide with pixels
in tables of bad pixels.
Most of these CR clusters consist of only 1-5
pixels (see Figure).
Caption to Figure. The number of clusters
detected as CR events per second per square
centimeter vs. the number of pixels for out-of
peak activity in a typical dark IR frame
(1024x512) with 7.7≤t≤9.3 s. Crests
correspond to left quadrant and stars
correspond to the right quadrant.
101
Calibrated IR images
• Long CRs were seen less often on IR images than on MRI and HRI
images (at the same exposure times).
• The ratio of the number of CR events per unit area tends to be less in
the middle of an image than for the left and right edges of the image.
For example, we divided dark 1024x512-pixel IR images into four
parts (256x512); the total number of objects in the left and right parts
was greater than that in two central parts by a factor of about 1.3-1.4.
• The number of clusters at the left quadrant was greater by a
factor of 4/3 than that at the right quadrant.
• Probably not all of the clusters presented in the above Figure (the
number of clusters vs number of pixels) are caused by CRs because
the numbers presented in this figure are greater than those for visual
images for out-of peak activity and the number of CRs can not be
different at different parts of a frame. Some of these clusters are
caused by bright background pixels which have different brightness
on different images but are not included in the table of bad pixels.
102
Clusters detected as CRs by imgclean on a dark IR
image 1024x512 (integration time was 7.7 s)
103
Pixels different for two dark IR images 1024x512 with
clusters detected as CRs (total integration time was 15.4 s)
104
Calibrated IR images
• The number of clusters classified as CRs by imgclean run on a
single calibrated IR image is greater by a factor of about 6 than the
number of the clusters on this image that are not present on a
neighboring image (we take only half of the number of clusters that
are different for two images as these differences include CRs from
both images).
• Only about 1/6 of pixels in the above clusters coincide with bad
pixels as defined in the bad pixel maps. Most of the clusters
deleted by imgclean from a calibrated IR (infrared) image can
also be found in the same locations on other images and are not
true CRs. They are most likely pixels with variable brightness in
the detector, most of which are not included in the bad pixel maps.
• Coordinates of such ‘common’ clusters can be stored in computer
memory and can be compared with the coordinates of clusters that
are recognized by imgclean as CRs after a run for one image.
Clusters with such coordinates are not CRs.
105
Calibrated IR images
• In our test runs with IR images, only imgclean
worked. Crfind and di_crrej had problems running to
completion with all parameter settings we tried. Rmcr
needs relatively constant background. Imgclean
overestimates (by about a factor of 6 or more) the
number of CRs on calibrated IR images.
• We do not think it worthwhile generally to use codes
that correct CRs on IR images. Sometimes such codes
may be useful for removing only long cosmic rays if they
are present on IR images, but in this case one must be
careful not to remove any long lines which belong to the
spectrum or to stars (i.e., remove only oblique CRs, but
not vertical or horizontal lines).
106
Triggering presolar cloud collapse and injection of short-lived radioisotopes
(such as 26Al and 60Fe). In collaboration with Alan Boss
(ApJ, 708, 1268-1280, 2010; ApJL, 686, L119-123, 2008)..
FLASH code was used to calculate the evolution of a
presolar cloud (initially the Bonner sphere) caused
by a moving shock front. 2D (cylindrical
coordinates) and 3D (Cartesian coordinates) models
were considered. Max density of the sphere equaled to
6.2e-19 g/cm3; R_init=1.79e17 cm, T_ambient=10 K,
front: max dens=3.6e-20, other: 3.6e-22.
We obtained that shocks with isothermal
thermodynamics and speeds in the range of 5 to 30
km/s are able to both trigger collapse and inject
shock wave material. For 2D and 5 levels, log density
equal to -12 was reached in 250, 180, and 160 Kyr for
speed of 5, 10, and 20 km/s, respectively. At v=30 km/s,
log dens=-12.3 was reached. More material of the
front was left near the sphere at v=5 km/s than
for greater velocities. There was no collapse to log
dens=-15 at v≤3.5 and v≥35 km/s. For 3 levels (3D or
2D), max increase of density was by a factor of ~20-30
(in 120-130 Kyr). There were also dense regions outside
the main dense region. For a pure sphere without a front,
the sphere slowly expended outside.
Upper figure: Initial data for 2D model.
Lower figure: 3D model, v=20 km/s, 60 Kyr, xy plane.
Only material outside the sphere is considered in the
lower figure, and “fingers” of this material are seen.
107
SYNTHETIC SPECTRA OF A MODEL EXO-EARTH
With James Cho. Presentation on this item can be found on
http://www.dtm.ciw.edu/~ipatov/lpsc2008atm.ppt
• Several Earth-like planets outside the solar system have recently been detected. Many more are expected to
be detected in the near future.
• How does physical properties of an “exo-Earth”, such as the rotation period, affect the observed
spectra? How the spectra change if we look at the exo-Earth from different directions?
• Based on a series of 3-D general circulation model simulations for atmospheres of Earth
and “exo-Earth” (i.e., Earth with a 100-day rotation period, initial data were identical for both
planets, and runs were made for 2 yr), we generated synthetic spectra of planets using a full 1-D
radiative transfer code SBDART (LPSC, # 2554, 2008).
• For both planets, the maximum flux decreases from equator to poles (more sharply to the south
pole). For all surface of a planet, the plots of flux vs. wavelength λ are very close for Earth,
exo-Earth, and initial state.
• There is essentially no difference in the spectra near the equator for an exo-Earth and Earth. There can be
a distinguishable difference for latitudes closer to the poles (especially to the South Pole), mainly at ~5-10
and ~13-16 µm (see above figures). These bands would be the best wavelength ranges for observations.
• When integrated over the planetary disk, the differences in spectral signal between Earth and an exo-Earth
are small (especially if one observes the whole disk from different directions close to the equator) but can
still be seen. If a whole disk is seen from the equator or exactly from a pole, variations in spectra with time
can be less than for other viewing angles. Based on spectra, we can discuss a viewing angle.
The local minimum at wavelength about 9.4-10 µm was absent in plots of total upward
flux
108
at 11 km for ozone density equal to 0. As ozone is important for life, the interval of wavelengths about 9.4-10 µm may be important for future observations of earth-like planets.
SYNTHETIC SPECTRA FROM A GCM SIMULATION
OF A MODEL EXO-EARTH. With James Cho.
• Several Earth-like planets outside the solar system have recently been detected.
Estimated 35% of stars harbor Earth-like planets. Hence, many more are expected
to be detected in the near future. To assess the detectability of biosignatures that
may be present on these planets, we have performed a series of general circulation
model (GCM) simulations of putative “extrasolar Earths”. These planets are
identical in all respects to the model Earth, except for different rotation
periods. In this work, we use the outputs from the GCM simulations to
compute model spectra. Such synthetic spectra can be useful for guiding and
interpreting observations.
• The GCM used is CCM3, a global atmospheric model. It is a spectral model that
solves the primitive equations. The model includes parameterizations of various
important physical processes and boundary conditions such as shortwave and
longwave radiation, moist convection, cloud fraction, and land surface types.
Temperature-pressure profiles at each grid point over the globe, along with
distributions of radiatively active species, are used to generate the spectra off-line.
• For the spectra calculation, we use SBDART, a code that computes planeparallel radiative transfer in the atmosphere in clear and cloudy conditions
(Ricchiazzi et al. BAMS, 1998, 79, 2101-2114). The code solves the radiative
transfer equation and is based on a collection of highly developed, reliable109
physical
models.
Model
We have checked the SBDART code using climatological temperature and species
distributions of the Earth (which is also the initial conditions of our GCM simulations)
as inputs. We have also checked the code against results described by Ricchiazzi et al.
(1998). We analyze the total upward flux at 1 km and 11 km altitudes in the
wavelength range between 1 and 18 µm and also between 0.3 and 1 µm. The GCM
simulation resolution is T85L19, which corresponds to 128 longitude (0-357 deg) and
64 latitude (from -88 deg to 88 deg) points and 18 layers (pressure levels) over the full
globe.
SBDART is used to compute the mean spectrum for different regions over the model
planet. For all regions, we compare the spectra for the initial atmosphere with
atmospheres after 2 years of model runs for planets with rotation period P from 0.167
to 100 days. Here we present results from P = 1 and P = 100 cases – “Earth” and “exoEarth”, respectively. We considered spectra from the common initial condition, Earth,
and exo-Earth. SBDART makes calculations of spectrum for one point on the Earth
(for a pair of values of longitude and latitude), but it analyses the atmosphere at
different heights. Using SBDART as a subroutine, we calculated the mean spectrum
for some region on a planet.
110
Model
•
Two models were considered. All calculations of spectra were made for initial state, Earth, and exoEarth.
•
Model 1. Model for which mean radiative flux depends on the area considered, but not
on the angle at which we see different parts of this area.
•
For model 1 for calculation of the mean radiative flux for some region, the weighting coefficient for one
SBDART run is proportional to the area corresponding to the pair (lon, lat) of longitude and latitude.
This area is proportional to cos(lat), but did not depend on lon. Model 1 corresponds to the abstract
model when one sees all points of considered surface at the same angle. The size of the area
corresponding to each pair (lon, lat) is 2.8 deg × 2.8 deg, and this pair shows the center of the region.
Model 1a. For series of runs 1a, we considered a fixed latitude and all values of longitude (0-360 deg).
Different values of latitude (-88 deg [the South Pole], -82 deg, -77 deg, -66 deg, -43 deg, 1 deg
[equator], 46 deg, 77 deg, and 88 deg [the North Pole]) were considered.
Model 1b. For series of runs 1b, we calculated mean values of flux based on SBDART runs for all pairs
(lon, lat), i.e., we considered all surface of a planet.
Model 1c. For model 1c, mean values of flux were based on SBDART runs for the southern hemisphere
(-90 deg ≤lat≤0, 0≤lon≤360 deg).
•
•
•
•
Model 2. Model for which mean radiative flux depends on the angle at which one sees
different parts of this area.
•
For model 2, the mean flux is calculated for the case when a half of a planet is observed from some
point located far from the planet. In this model we take into account that we see different parts of
surface at different angles. If one looks from the equator at lon=lon0, then the total weighting
coefficient k for one SBDART run equals to abs(cos(lat)×cos(lat)×cos(lon-lon0)). One cos(lat) is due to
that the area corresponding to a single SBDART run is smaller for latitudes close to poles (same as for
model 1), and another cos(lat) is due to that one sees areas at different latitudes at different angles. If
one looks from a pole, then k=abs(cos(lat)×sin(lat)).
111
CLOUD AND TEMPERATURE MAPS
• Below we consider cloud and temperature maps on planets. Cloud coverage was
calculated as 1-[(1-CLD1)×(1-CLD2)×...(1-CLD18)], where CLD1,…,CLD18 are
values of cloud coverage at 18 different values of height.
• GCM simulations show significant differences in the distribution of fields important
for spectra, due to different rotation periods. The differences in the distribution are
not simple. For example, the distribution of total cloud (Fig. 1) is fairly zonal for
both the Earth and the exo-Earth, but less “banded” on the exo-Earth: three bands in
the meridional direction, from the North Pole to the South Pole, are present on the
Earth but only one (near the equator) on the exo-Earth with P = 100 days. In the
latter planet, the southern polar region is nearly devoid of clouds, and cloud
coverage is close to 1 near the equator. For Earth and initial maps, there are a lot of
clouds near both poles; for the Earth map there are even less clouds near the equator
than at the poles.
• Near the equator the temperature maps (Fig. 2) do not differ much for P=1 day and
P=100 days. For exo-Earth, the temperatures T are more uniform than for Earth. For
example, for exo-Earth, the region of T>290 K is smaller than that for Earth. For all
surface of exo-Earth T>230 K, but there are some regions with T<230 K for Earth.
The main difference in T-maps for Earth and exo-Earth is near the South Pole. At
the South Pole, the temperature is warmer for exo-Earth than for Earth and initial
state. Near the South Pole, the line T=240 K is long for the Earth and is absent for
exo-Earth. For initial planet, there was even line T=220 K near the South Pole. The
difference between T-maps for exo-Earth and Earth was less than the differences
between these maps and the initial T-map. The temperature at the North Pole (265
112
K) was greater than that at the South Pole.
• Fig. 1. Cloud
maps.
• (a) Cloud map
for the model
Earth case (P=1
day) ;
• (b) Cloud map
for the model
exo-Earth
(P=100 days).
113
• Fig. 2. Surface
temperature maps.
• (a) Surface
temperature (K) for
the model Earth case
(P=1 day).
• (b) Surface
temperature (K) for
the model exo-Earth
(P=100 days).
114
•
Spectra
at
wavelength
1-18
μm
Longitudinal averages of outgoing long-wave flux at fixed latitudes (not adjusted
for surface element orientation) show that the flux is greatest at the equator and
decreases toward the poles on both the Earth and the exo-Earth (Table 1, Figs. 3-4).
For model 1a for initial planet, the maximum value Fmax of flux (usually reached
at wavelength λ≈10 μm) at the South and North Poles was respectively by a factor
of 4 and 1.7 smaller than at the equator. The values of Fmax for initial planet (or
Earth, or exo-Earth) almost didn’t depend on lat at lat≤-77deg (Table 1). For the
same planet, the values of flux at λ~10-12 μm at lat=-77 deg were greater by a
factor of 1.1 than that at the South Pole, and were greater by a factor of 1.05 at
lat=77 deg than at the North Pole. For the South Pole, such values were greater for
Earth and exo-Earth than for initial state by a factor of 1.3 and 1.7, respectively.
For the North Pole, maximum values of flux were close for Earth, exo-Earth, and
initial state. For the flux at 1 km, the spectra plots were close for the three planets at
-43 deg≤lat≤88 deg.
• For both planets, spectra near the equator (at 1 and 11 km altitudes) at the end of
the GCM run duration are essentially the same. Moreover, the spectra are not
different from that of the initial state. In contrast, the fluxes of the two planets
differ significantly near the poles in the ~5-10 μm and ~13-16 μm bands. The
difference is well seen also at latitude lat≤-43 deg and lat≥77 deg. The differences
between the fluxes at 11 km and 1 km are smaller if we compare Earth and initial
state, than if we compare exo-Earth and initial state. For all plots and h=11 km,
there was a local minimum of flux at λ~14-16 μm, and a smaller local minimum at
9.5 μm. All of the above features are primarily due to the different cloud coverage
115
on the exo-Earth compared to that on the model Earth.
• Fig. 3. Spectra (outgoing LW flux) at the equator (model 1a) for two different
altitudes, 1 km (upper line) and 11 km (lower line). The spectra is nearly identical
for the model Earth case (P=1 day) and the model extrasolar planet (P = 100
days). In the model Earth case, the spectrum is almost same as in the initial state.
• Fig. 4a. Longitudinally averaged spectra at latitude = -77 deg (model 1a) for initial
state. The lines are as in Fig. 3.
116
• Fig. 4b. Longitudinally averaged spectra at latitude = -77 deg (model 1a) for model
Earth (P=1 day). The lines are as in Fig. 3. Compare with the spectra at the equator
in Fig. 3: the flux is significantly reduced at all wavelengths.
• Fig. 4c. Longitudinally averaged spectra at latitude = -77 deg (model 1a) for model
exo-Earth (P=100 days). The lines are as in Fig. 3. Compare with the spectra at the
equator in Fig. 4b: absorption in the ~5-10 µm range is significantly reduced,
117
although not in the ~13-16 µm range at 11 km altitude.
• Fig. 5a. Disk-integrated spectra including the weighted contribution of the surface
element orientation for Earth (model 2). The planet is seen from the South Pole.
• Fig. 5b. Spectra as in Fig 5a for the exo-Earth.
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Maximum flux at different latitudes
• Table 1. Maximum flux (in W/m^2/micron) at wavelength about 10-12 micron for
several values of latitude (model 1a)
• latitude, deg -88 -82 -77 -66 -43 1 46 77 88
• initial
6.5 8 7.5 13 23 30 25 18 17.5
• earth
9 10 9.5 13.5 23 30 25 18 17.5
• exo-Earth
12 12.5 12.5 14 22 29.5 25 18 17
• Table 2. Maximum flux (in W/m^2/micron) from half of planet surface at
wavelength about 10.0-10.4 micron at altitude equal 1 km (maximum values at 11
km are usually smaller by a factor of 1.01-1.02 than those at 1 km) for several
directions of view on a planet (model 2)
•
South Pole North Pole Equator
Equator
•
lon=90 deg lon=270 deg
• initial
6.61
8.09
8.27
8.49
• earth
6.63
7.97
8.05
8.31
• exo-Earth
6.61
7.77
8.25
8.09
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Spectra at wavelength 1-18 μm
• For all surface of a planet (model 1b), the plots of flux vs. λ were
very close for Earth, exo-Earth, and initial state. The value of Fmax
for all surface of the southern hemisphere (model 1c) is about 15 W m^{-2}
micron^{-1} and is smaller than that (16 W m^{-2} micron^{-1}) for a whole
surface (model 1b) by about 7% (these values are about twice less than those for
the equator). It means that there is a difference between plots for the southern
and northern hemispheres (e.g., it may be caused by Antarctica). For models 1b
and 1c, the differences between values of Fmax for Earth, exo-Earth, and initial
planet are smaller (<2%) than that for two hemispheres. Therefore the
influence of a line of sight on observed flux can be greater than
that of a period of rotation (see also results for model 2). For the
southern hemisphere at wavelength about 14-16 μm, the flux at
11 km for exo-Earth was greater by a factor of 1.2 than that for
Earth and initial state.
• The maximum value Fmax of flux for spectrum plots is greater for hotter Tregions. Cloud coverage was maximum in a wide region near the equator and it
was minimum near the South Pole of exo-Earth. Such peculiarities of cloud
coverage may help to understand why differences between the fluxes at 1 and 11
km at the South Pole differ from those at the equator.
120
Spectra at wavelength 1-18 μm
• When the influence of viewing geometry is taken into account
(model 2), the total flux when the planet is viewed is reduced by a
factor of 2 (compared to model 1) since the line of sight is not
perpendicular to the surface elements.
• However, the general behavior is not different than that described
above, when the flux for each surface element is simply integrated
unadjusted for variation in the orientation. This is consistent with
the similarity of the spectra in the equatorial region on both planets:
areas from the higher latitudes do not contribute significantly.
• The peak values of the spectra (see Table 2) also differ by no more
than 5% when the planet is viewed centered at lon = 90 deg or
270 deg with lat = 0 – or from the North Pole (with full longitude
range). For the view centered on the South Pole, the peak is
smaller by a factor of 1.2 compared to that for the view
centered on the North Pole, again showing the asymmetry of the
two poles. In the South Pole view, the difference between the
fluxes at 11 and 1 km at ~6-8 μm band for the exo-Earth is smaller
than that for the Earth (Fig. 5).
121
Spectra at wavelength less than 1 μm
• The previous discussion was for analysis of plots with wavelength between 1 and
18 microns. In Fig. 6 we present spectra for wavelength between 0.3 and 0.8
micron for total upward flux at 11 and 1 km. The plot is presented for the North
Pole of exo-Earth, but plots for other latitudes (model 1a), Earth, and the initial
state look similar to Fig. 6 (e.g., all plots have a local minimum at 0.76 micron).
The form of spectrum at 11 km is the same as that at 1 km, but the values are
greater by about a factor of 5. Note that at wavelength between 3 and 18 microns,
the flux at 11 km always doesn’t exceed that at 1 km.
• At wavelengths less than 1 micron, there are no practically differences for
plots for Earth, exo-Earth, and initial state (therefore such wavelengths are not
recommended for studies of differences in period of planet rotation). At the
equator and the North Pole, the Earth plot could be higher than the exo-Earth plot
by not more than 7 %; at the South Pole the difference is smaller (in contrast to
larger wavelengths). The difference “Earth – exo-Earth” is greater than the
difference “Earth – initial”. For the South Pole, the fluxes could be a few percent
smaller than those for the North Pole or for the equator, and plots for the equator
could be a little lower than those for the North Pole, the relative difference is
greater for 1 km than for 11 km.
122
•
Fig. 6. Spectra (outgoing flux) at the North Pole (model 1a) for two different altitudes, 1 km
(lower line) and 11 km (upper line). The plot is for the model of extrasolar planet (P = 100
days), but spectra are nearly identical for the model Earth case (P=1 day) and in the initial
state. For such wavelengths, spectra are also similar for different latitudes.
•
Fig. 7. Spectra at the North Pole (model 1a) for the model Earth case (P = 1 day) if ozone is
absent. The local minimum at wavelength about 9.4-10 micron was absent on plots of total
upward flux at 11 km for ozone density equal to 0 though such minimum was on plots
for two
123
other models that include ozone (there was no difference in plots for the latter two models).
Spectra at different ozone distributions
• Spectrum plots were considered for a model with some distribution of ozone with
height and also for a model for which ozone density equaled to 0.00005 g/m3 at all
heights and at all points on surface. The plots for these two models (at
wavelengths between 0.3 and 18 microns) practically coincided, and it was not
possible to see any difference. It means that the model with a fixed ozone density
equaled to 0.00005 g/m3 can be considered if we do not know exact distribution of
ozone.
• We also considered the third model with zero ozone density. For this model, the
difference with two other models was only at two wavelength intervals. The
local minimum at wavelength about 9.4-10 micron was absent in
plots of total upward flux at 11 km for ozone density equal to 0
(Fig. 7) though such minimum was in plots for two other models (for flux at 1 km
there was no such difference). At wavelengths less than about 0.35 micron
(SBDART does not consider wavelength less than 0.25 micron), there were some
differences in plots (both for fluxes at 1 and 11 km) obtained for the model
without ozone from the plots for two other models. As ozone is important
for life, the interval of wavelengths about 9.4-10 micron may be
important for future observations of earth-like planets.
124
Conclusions
• In this work, we observe the following common features in the synthetic spectra: (1)
both planets (Earth and exo-Earth with rotation period equal to 1 and 100 days,
respectively) have a broad CO_2 absorption band centered around 14 μm; (2) clouds
tend to muffle long-wave spectral signatures; (3) there is essentially no difference in
the spectra near the equator for exo-Earth with 100-day rotation period (compared
with Earth at P = 1 day); however, in some regions (e.g., near the South Pole), there
can be a distinguishable difference, indicating that viewing angle matters; (4) when
integrated over the planetary disk, the differences in spectral signal observed from
different directions are reduced; however, even when integrated signal, differences
between Earth and an exo-Earth can still be seen (e.g., spectra at 11 km altitude
viewed from directions of the South and North Poles are not the same), but the
differences are very small if one observes the whole disk from different directions
close to the equator.
• In summary, one-dimensional modeling, which cannot explicitly incorporate
circulation, is clearly limited because of anisotropy. Our results here suggest that
spectral signal cannot be used to infer rotation rate since viewing
geometry is in general not known a priori. For observations, our calculations also
suggest that ~5-10 and ~13-16 µm bands would be the best wavelength ranges to
consider, at least for a model exo-planet which is Earth-like in all respects except
for the rotation period. Analyzing spectra at wavelength of about 9.5-10 micron,
one can make a conclusion whether an extra-solar planet has ozone or not.
125