Anisotropy of Long-period Comets
Explained by Their Formation Process
Arika Higuchi
1,21 University of Occupational and Environmental Health, Japan (2020.7.1-) 2 RISE Project, National Astronomical Observatory of Japan (-2020.6.30)
Published 2020.8.26 in AJ, available at https://doi.org/10.3847/1538-3881/aba94d
Abstract
Long-period comets coming from the Oort cloud are
thought to be planetesimals formed in the planetary region on the ecliptic plane. We have investigated the orbital
evolution of these bodies due to the Galactic tide.
We extended our previous work and derived analytical solutions to the Galactic longitude and latitude of the direction of aphelion. Using the analytical solutions, we predict that Oort cloud comets returning to the planetary region are concentrated on the ecliptic plane and a second plane, which we call the “empty ecliptic.”
Sun
Earth Galactic plane
The Oort cloud
Introduction
(1) Outward transportation by planetary scattering
(2) Evolution into the spherical structure by perturbations from the Galaxy
• q outside the planetary region.
• i changed.
• qi in the planetary region and ei~1. • on the ecliptic (inclination ii ~ 60 deg.)
The Oort cloud comets are planetesimals formed in the protoplanetary disk and initially on the ecliptic plane with the perihelion distances q in the planetary region.
z: the position of the comet in the Galactic coordinates
(3) Observed as long-period comets when q come back to the planetary region.
Aim of this paper
n
The Galactic longitude and latitude of the direction of aphelion,
L
and
B
(or those of perihelion), are
used to evaluate the distribution of observed long-period comets
(e.g., Biermann et al. (1983), Luest (1984), Delsemme (1986, 1987), Matese & Whitmire (1996), Matese et al. (1999)).n
Based on the standard formation scenario, the Oort cloud comets are planetesimals formed in the
protoplanetary disk and initially on the ecliptic plane with the perihelion distances near the giant
planets
(e.g., Dones et al. 2004).n
As long as the Oort cloud is not completely destroyed by close stellar encounters, the memory of the
initial distribution can be found as anisotropies in the present distribution.
n
We investigate the evolution of the aphelia of comets
initially on the ecliptic plane
under the
axisymmetric approximation of the Galactic tide with the same procedure as in
Higuchi et al. (2007).
n
Using the analytical solutions, we predict the distribution of long-period comets on the L−B plane.
• Biermann, L., Huebner, W. F., and Lust, R., “Aphelion Clustering of ``New'' Comets: Star Tracks through Oort's Cloud”, Proceedings of the National Academy of Science, vol. 80, no. 16. pp. 5151–5155, 1983. doi: 10.1073/pnas.80.16.5151.
• Luest, R., “The distribution of the aphelion directions of long-period comets”, Astronomy and Astrophysics, vol. 141, no. 1. pp. 94–100, 1984. • Delsemme, A. H., “Cometary evidence for a solar companion?”, in The Galaxy and the Solar System, 1986, pp. 173–203.
• Delsemme, A. H., “Galactic Tides Affect the Oort Cloud - an Observational Confirmation”, Astronomy and Astrophysics, vol. 187. p. 913, 1987.
• Matese, J. J., Whitman, P. G., and Whitmire, D. P., “Cometary Evidence of a Massive Body in the Outer Oort Clouds”, Icarus, vol. 141, no. 2. pp. 354–366, 1999. doi: 10.1006/icar.1999.6177. • Dones, L., Weissman, P. R., Levison, H. F., and Duncan, M. J., “Oort cloud formation and dynamics”, in Comets II, 2004, p. 153.
• Higuchi, A., Kokubo, E., Kinoshita, H., and Mukai, T., “Orbital Evolution of Planetesimals due to the Galactic Tide: Formation of the Comet Cloud”, The Astronomical Journal, vol. 134, no. 4. pp. 1693– 1706, 2007. doi: 10.1086/521815.
Analytical solutions
Anisotropy of Long-period Comets 3
Using the orbital elements, the unit vector of the direction of aphelion in the Galactic coordinates rQ is written as
rQ = 0 B @ Qx Qy Qz 1 C A = 0 B @
cos ! cos ⌦ + sin ! sin ⌦ cos i cos ! sin ⌦ sin ! cos ⌦ cos i sin ! sin i
1 C
A (1)
Then L and sin B are written as
L = atan ✓ Qy Qx ◆ = ⌦ + ✓, (2)
sin B = Qz = sin ! sin i, (3)
where
✓ = (
atan sin ! cos icos ! for cos ! < 0
⇡ + atan sin ! cos icos ! for cos ! > 0. (4)
2.2. Conserved quantities
Assume that the Galactic tide is much smaller than the solar gravity. The time-averaged Hamiltonian of a body moving under the approximated Galactic potential is given as
hHi = GM
2a R, (5)
where G is the gravitational constant, M is the solar mass, a is the semimajor axis of the body, and R is the disturbing function R = ⌫ 2 0 4 a 2 sin2 i 1 e2 + 5e2 sin2 ! , (6)
where ⌫0 = p4⇡G⇢ is the vertical frequency and ⇢ is the total density in the solar neighborhood (e.g., Heisler &
Tremaine 1986). From Equation (6) and Lagrange’s planetary equation for da/dt, we know a is constant. Then we introduce a new simplified Hamiltonian:
c = sin2 i 1 e2 + 5e2 sin2 ! . (7)
The simplified z-component of the angular momentum, which is a conserved quantity under the axisymmetric approx-imation of the potential, is written as
j = p1 e2 cos i. (8)
Substituting Equation (8) into Equation (7), one can draw equi-Hamiltonian curves on the ! e plane for given c and j using Equation (7). From the Hamiltonian curves, we can learn the overall behavior of ! and e without solving the equation of motion. For some cases, equi-Hamiltonian curves circulate with ! and for other cases they librate around ! = 90 or 270 . This libration is essentially the von Zeipel-Lidov-Kozai mechanism (von Zeipel 1910; Kozai 1962; Lidov 1962; Ito & Ohtsuka 2019). The condition for circulation is to have a solution to e for ! = 0, i.e.,
c + j2< 1 (circulation)
c + j2> 1 (libration) (9)
and c + j2 = 1 gives the separatrix (e.g., Higuchi et al. 2007). This leads to the necessary condition on i for libration, sin i > p1/5.
Substituting Equations (3) and (8) into Equation (7), the Hamiltonian is given with B instead of i and !,
c = 1 e2 j2 + 5e2 sin2 B. (10)
The separatrix with Equation (10) is written as
c + j2 = 1 e2 1 5 sin2 B = 1. (11)
For e2 > 0, the sufficient condition on B for libration is given as sin|B| >
r 1
5. (12)
Matese & Whitman (1989) defined the value B = asinp1/5 ' ±26.6 as a barrier that the latitude of perihelion cannot migrate across.
Anisotropy of Long-period Comets 3
Using the orbital elements, the unit vector of the direction of aphelion in the Galactic coordinates rQ is written as
rQ = 0 B @ Qx Qy Qz 1 C A = 0 B @
cos ! cos ⌦ + sin ! sin ⌦ cos i cos ! sin ⌦ sin ! cos ⌦ cos i sin ! sin i
1 C
A (1)
Then L and sin B are written as
L = atan ✓ Qy Qx ◆ = ⌦ + ✓, (2)
sin B = Qz = sin ! sin i, (3)
where
✓ = (
atan sin ! cos icos ! for cos ! < 0
⇡ + atan sin ! cos icos ! for cos ! > 0. (4)
2.2. Conserved quantities
Assume that the Galactic tide is much smaller than the solar gravity. The time-averaged Hamiltonian of a body moving under the approximated Galactic potential is given as
hHi = GM2a R, (5)
where G is the gravitational constant, M is the solar mass, a is the semimajor axis of the body, and R is the disturbing function R = ⌫ 2 0 4 a 2 sin2 i 1 e2 + 5e2 sin2 ! , (6)
where ⌫0 = p4⇡G⇢ is the vertical frequency and ⇢ is the total density in the solar neighborhood (e.g., Heisler &
Tremaine 1986). From Equation (6) and Lagrange’s planetary equation for da/dt, we know a is constant. Then we introduce a new simplified Hamiltonian:
c = sin2 i 1 e2 + 5e2 sin2 ! . (7)
The simplified z-component of the angular momentum, which is a conserved quantity under the axisymmetric approx-imation of the potential, is written as
j = p1 e2 cos i. (8)
Substituting Equation (8) into Equation (7), one can draw equi-Hamiltonian curves on the ! e plane for given c and j using Equation (7). From the Hamiltonian curves, we can learn the overall behavior of ! and e without solving the equation of motion. For some cases, equi-Hamiltonian curves circulate with ! and for other cases they librate around ! = 90 or 270 . This libration is essentially the von Zeipel-Lidov-Kozai mechanism (von Zeipel 1910; Kozai 1962; Lidov 1962; Ito & Ohtsuka 2019). The condition for circulation is to have a solution to e for ! = 0, i.e.,
c + j2 < 1 (circulation)
c + j2 > 1 (libration) (9)
and c + j2 = 1 gives the separatrix (e.g., Higuchi et al. 2007). This leads to the necessary condition on i for libration, sin i > p1/5.
Substituting Equations (3) and (8) into Equation (7), the Hamiltonian is given with B instead of i and !,
c = 1 e2 j2 + 5e2 sin2 B. (10)
The separatrix with Equation (10) is written as
c + j2 = 1 e2 1 5 sin2 B = 1. (11)
For e2 > 0, the sufficient condition on B for libration is given as sin|B| >
r 1
5. (12)
Matese & Whitman (1989) defined the value B = asinp1/5 ' ±26.6 as a barrier that the latitude of perihelion cannot migrate across.
Anisotropy of Long-period Comets 3
Using the orbital elements, the unit vector of the direction of aphelion in the Galactic coordinates rQ is written as
rQ = 0 B @ Qx Qy Qz 1 C A = 0 B @
cos ! cos ⌦ + sin ! sin ⌦ cos i cos ! sin ⌦ sin ! cos ⌦ cos i sin ! sin i
1 C
A (1)
Then L and sin B are written as
L = atan ✓ Qy Qx ◆ = ⌦ + ✓, (2)
sin B = Qz = sin ! sin i, (3)
where
✓ = (
atan sin ! cos icos ! for cos ! < 0
⇡ + atan sin ! cos icos ! for cos ! > 0. (4)
2.2. Conserved quantities
Assume that the Galactic tide is much smaller than the solar gravity. The time-averaged Hamiltonian of a body moving under the approximated Galactic potential is given as
hHi = GM
2a R, (5)
where G is the gravitational constant, M is the solar mass, a is the semimajor axis of the body, and R is the disturbing function R = ⌫ 2 0 4 a 2 sin2 i 1 e2 + 5e2 sin2 ! , (6)
where ⌫0 = p4⇡G⇢ is the vertical frequency and ⇢ is the total density in the solar neighborhood (e.g., Heisler &
Tremaine 1986). From Equation (6) and Lagrange’s planetary equation for da/dt, we know a is constant. Then we introduce a new simplified Hamiltonian:
c = sin2 i 1 e2 + 5e2 sin2 ! . (7)
The simplified z-component of the angular momentum, which is a conserved quantity under the axisymmetric approx-imation of the potential, is written as
j = p1 e2 cos i. (8)
Substituting Equation (8) into Equation (7), one can draw equi-Hamiltonian curves on the ! e plane for given c and j using Equation (7). From the Hamiltonian curves, we can learn the overall behavior of ! and e without solving the equation of motion. For some cases, equi-Hamiltonian curves circulate with ! and for other cases they librate around ! = 90 or 270 . This libration is essentially the von Zeipel-Lidov-Kozai mechanism (von Zeipel 1910; Kozai 1962; Lidov 1962; Ito & Ohtsuka 2019). The condition for circulation is to have a solution to e for ! = 0, i.e.,
c + j2< 1 (circulation)
c + j2> 1 (libration) (9)
and c + j2 = 1 gives the separatrix (e.g., Higuchi et al. 2007). This leads to the necessary condition on i for libration, sin i > p1/5.
Substituting Equations (3) and (8) into Equation (7), the Hamiltonian is given with B instead of i and !,
c = 1 e2 j2 + 5e2 sin2 B. (10)
The separatrix with Equation (10) is written as
c + j2 = 1 e2 1 5 sin2 B = 1. (11)
For e2 > 0, the sufficient condition on B for libration is given as sin|B| >
r 1
5. (12)
Matese & Whitman (1989) defined the value B = asinp1/5 ' ±26.6 as a barrier that the latitude of perihelion cannot migrate across.
Anisotropy of Long-period Comets 3
Using the orbital elements, the unit vector of the direction of aphelion in the Galactic coordinates rQ is written as
rQ = 0 B @ Qx Qy Qz 1 C A = 0 B @
cos ! cos ⌦ + sin ! sin ⌦ cos i cos ! sin ⌦ sin ! cos ⌦ cos i sin ! sin i
1 C
A (1)
Then L and sin B are written as
L = atan ✓ Qy Qx ◆ = ⌦ + ✓, (2)
sin B = Qz = sin ! sin i, (3)
where
✓ = (
atan sin ! cos icos ! for cos ! < 0
⇡ + atan sin ! cos icos ! for cos ! > 0. (4) 2.2. Conserved quantities
Assume that the Galactic tide is much smaller than the solar gravity. The time-averaged Hamiltonian of a body moving under the approximated Galactic potential is given as
hHi = GM2a R, (5)
where G is the gravitational constant, M is the solar mass, a is the semimajor axis of the body, and R is the disturbing function R = ⌫ 2 0 4 a 2 sin2 i 1 e2 + 5e2 sin2 ! , (6)
where ⌫0 = p4⇡G⇢ is the vertical frequency and ⇢ is the total density in the solar neighborhood (e.g., Heisler &
Tremaine 1986). From Equation (6) and Lagrange’s planetary equation for da/dt, we know a is constant. Then we
introduce a new simplified Hamiltonian:
c = sin2 i 1 e2 + 5e2 sin2 ! . (7)
The simplified z-component of the angular momentum, which is a conserved quantity under the axisymmetric approx-imation of the potential, is written as
j = p1 e2 cos i. (8)
Substituting Equation (8) into Equation (7), one can draw equi-Hamiltonian curves on the ! e plane for given c and j using Equation (7). From the Hamiltonian curves, we can learn the overall behavior of ! and e without solving the equation of motion. For some cases, equi-Hamiltonian curves circulate with ! and for other cases they librate around ! = 90 or 270 . This libration is essentially the von Zeipel-Lidov-Kozai mechanism (von Zeipel 1910; Kozai
1962; Lidov 1962; Ito & Ohtsuka 2019). The condition for circulation is to have a solution to e for ! = 0, i.e.,
c + j2 < 1 (circulation)
c + j2 > 1 (libration) (9)
and c + j2 = 1 gives the separatrix (e.g., Higuchi et al. 2007). This leads to the necessary condition on i for libration, sin i > p1/5.
Substituting Equations (3) and (8) into Equation (7), the Hamiltonian is given with B instead of i and !,
c = 1 e2 j2 + 5e2 sin2 B. (10)
The separatrix with Equation (10) is written as
c + j2 = 1 e2 1 5 sin2 B = 1. (11)
For e2 > 0, the sufficient condition on B for libration is given as sin|B| >
r 1
5. (12)
Matese & Whitman (1989) defined the value B = asinp1/5 ' ±26.6 as a barrier that the latitude of perihelion
cannot migrate across.
Anisotropy of Long-period Comets 3
Using the orbital elements, the unit vector of the direction of aphelion in the Galactic coordinates rQ is written as
rQ = 0 B @ Qx Qy Qz 1 C A = 0 B @
cos ! cos ⌦ + sin ! sin ⌦ cos i cos ! sin ⌦ sin ! cos ⌦ cos i sin ! sin i
1 C
A (1)
Then L and sin B are written as
L = atan ✓ Qy Qx ◆ = ⌦ + ✓, (2)
sin B = Qz = sin ! sin i, (3)
where
✓ = (
atan sin ! cos icos ! for cos ! < 0
⇡ + atan sin ! cos icos ! for cos ! > 0. (4) 2.2. Conserved quantities
Assume that the Galactic tide is much smaller than the solar gravity. The time-averaged Hamiltonian of a body moving under the approximated Galactic potential is given as
hHi = GM2a R, (5)
where G is the gravitational constant, M is the solar mass, a is the semimajor axis of the body, and R is the disturbing function R = ⌫ 2 0 4 a 2 sin2 i 1 e2 + 5e2 sin2 ! , (6)
where ⌫0 = p4⇡G⇢ is the vertical frequency and ⇢ is the total density in the solar neighborhood (e.g., Heisler &
Tremaine 1986). From Equation (6) and Lagrange’s planetary equation for da/dt, we know a is constant. Then we
introduce a new simplified Hamiltonian:
c = sin2 i 1 e2 + 5e2 sin2 ! . (7)
The simplified z-component of the angular momentum, which is a conserved quantity under the axisymmetric approx-imation of the potential, is written as
j = p1 e2 cos i. (8)
Substituting Equation (8) into Equation (7), one can draw equi-Hamiltonian curves on the ! e plane for given c and j using Equation (7). From the Hamiltonian curves, we can learn the overall behavior of ! and e without solving the equation of motion. For some cases, equi-Hamiltonian curves circulate with ! and for other cases they librate around ! = 90 or 270 . This libration is essentially the von Zeipel-Lidov-Kozai mechanism (von Zeipel 1910; Kozai
1962; Lidov 1962; Ito & Ohtsuka 2019). The condition for circulation is to have a solution to e for ! = 0, i.e.,
c + j2 < 1 (circulation)
c + j2 > 1 (libration) (9)
and c + j2 = 1 gives the separatrix (e.g., Higuchi et al. 2007). This leads to the necessary condition on i for libration, sin i > p1/5.
Substituting Equations (3) and (8) into Equation (7), the Hamiltonian is given with B instead of i and !,
c = 1 e2 j2 + 5e2 sin2 B. (10)
The separatrix with Equation (10) is written as
c + j2 = 1 e2 1 5 sin2 B = 1. (11)
For e2 > 0, the sufficient condition on B for libration is given as sin|B| >
r 1
5. (12)
Matese & Whitman (1989) defined the value B = asinp1/5 ' ±26.6 as a barrier that the latitude of perihelion
cannot migrate across.
Derivations are here -> https://doi.org/10.3847/1538-3881/aba94d
•
The Galactic longitude and latitude of the direction of the aphelion, L and B:
i: ω: Ω:
inclination
argument of perihelion
longitude of ascending node *all in the Galactic coordinates
Anisotropy of Long-period Comets 3
Using the orbital elements, the unit vector of the direction of aphelion in the Galactic coordinates rQ is written as
rQ = 0 B @ Qx Qy Qz 1 C A = 0 B @
cos ! cos ⌦ + sin ! sin ⌦ cos i cos ! sin ⌦ sin ! cos ⌦ cos i sin ! sin i
1 C
A (1)
Then L and sin B are written as
L = atan ✓ Qy Qx ◆ = ⌦ + ✓, (2)
sin B = Qz = sin ! sin i, (3)
where
✓ = (
atan sin ! cos icos ! for cos ! < 0
⇡ + atan sin ! cos icos ! for cos ! > 0. (4)
2.2. Conserved quantities
Assume that the Galactic tide is much smaller than the solar gravity. The time-averaged Hamiltonian of a body moving under the approximated Galactic potential is given as
hHi = GM
2a R, (5)
where G is the gravitational constant, M is the solar mass, a is the semimajor axis of the body, and R is the disturbing function R = ⌫ 2 0 4 a 2 sin2 i 1 e2 + 5e2 sin2 ! , (6)
where ⌫0 = p4⇡G⇢ is the vertical frequency and ⇢ is the total density in the solar neighborhood (e.g., Heisler &
Tremaine 1986). From Equation (6) and Lagrange’s planetary equation for da/dt, we know a is constant. Then we introduce a new simplified Hamiltonian:
c = sin2 i 1 e2 + 5e2 sin2 ! . (7)
The simplified z-component of the angular momentum, which is a conserved quantity under the axisymmetric approx-imation of the potential, is written as
j = p1 e2 cos i. (8)
Substituting Equation (8) into Equation (7), one can draw equi-Hamiltonian curves on the ! e plane for given c and j using Equation (7). From the Hamiltonian curves, we can learn the overall behavior of ! and e without solving the equation of motion. For some cases, equi-Hamiltonian curves circulate with ! and for other cases they librate around ! = 90 or 270 . This libration is essentially the von Zeipel-Lidov-Kozai mechanism (von Zeipel 1910; Kozai 1962; Lidov 1962; Ito & Ohtsuka 2019). The condition for circulation is to have a solution to e for ! = 0, i.e.,
c + j2< 1 (circulation)
c + j2> 1 (libration) (9)
and c + j2 = 1 gives the separatrix (e.g., Higuchi et al. 2007). This leads to the necessary condition on i for libration, sin i > p1/5.
Substituting Equations (3) and (8) into Equation (7), the Hamiltonian is given with B instead of i and !,
c = 1 e2 j2 + 5e2 sin2 B. (10)
The separatrix with Equation (10) is written as
c + j2 = 1 e2 1 5 sin2 B = 1. (11)
For e2 > 0, the sufficient condition on B for libration is given as sin|B| >
r 1
5. (12)
Matese & Whitman (1989) defined the value B = asinp1/5 ' ±26.6 as a barrier that the latitude of perihelion cannot migrate across.
•
The time-averaged disturbing function that arises from the vertical component of the Galactic tide
a: semimajor axis e: eccentricity•
Two Patterns of evolution of L and B and the ratio of their periods:
P
L/P
B〜
2
Pattern 1: L circulates, B oscillates around 0 (and ω librates)
P
L/P
B〜
∞
Pattern 2: L changes little, B oscillates (and ω circulates)
* In both case, the period of the oscillation of q is the same as PB. * These relations are established only for ei〜1.
1 10 100 1000 10000 0 1 2 3 4 4.5 q [au] time [109 yr] -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 4.5 sin B time [109 yr] 0 60 120 180 240 300 360 0 1 2 3 4 4.5 L [deg] time [109 yr] -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 (4) ai=2x10 4 au sin B time [109 yr] qi=10 au, i=310 deg qi=30 au, i=210 deg q [a u] sin B L [de g] time [Gyr] 1 10 100 1000 10000 0 1 2 3 4 4.5 q [au] time [109 yr] -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 4.5 sin B time [109 yr] 0 60 120 180 240 300 360 0 1 2 3 4 4.5 L [deg] time [109 yr] -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 (4) ai=2x104 au sin B time [109 yr] qi=10 au, i=310 deg qi=30 au, i=210 deg 1 10 100 1000 10000 0 1 2 3 4 4.5 q [au] time [109 yr] -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 4.5 sin B time [109 yr] 0 60 120 180 240 300 360 0 1 2 3 4 4.5 L [deg] time [109 yr] -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 (4) ai=2x10 4 au sin B time [109 yr] qi=10 au, i=310 deg qi=30 au, i=210 deg 1 10 100 1000 10000 0 1 2 3 4 4.5 q [au] time [109 yr] -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 4.5 sin B time [109 yr] 0 60 120 180 240 300 360 0 1 2 3 4 4.5 L [deg] time [109 yr] -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 (4) ai=2x10 4 au sin B time [109 yr] qi=10 au, i=310 deg qi=30 au, i=210 deg
Result
: Relation of B and L in the planetary region
Number of returns to
the planetary region k k = even(including initial condition) k = odd
pattern 1
B
1L
1-B
1L
1pattern 2
B
2L
2B
2L
2-π
relation satisfied
eq.
(1)
eq. (2)
18
Higuchi
the prediction of i
Ein the observable region using the analytical solution is as difficult as much as that for i (see Figure
4) and as sensitive to q as much as that for ⌦. Panels (5) and (6) in Figure
6
are the same as panels (1) and (2) but
for a
i= 5
⇥ 10
4au. The drift of the curves is seen more obviously than in panel (1) since bodies with a
i= 5
⇥ 10
4au
make more oscillations in 4.5 Gyr.
Panels (3) and (4) in Figure
6
show the behavior of L and sin B for a
i= 2
⇥ 10
4au, respectively, and panels (7) and
(8) in Figure
6
are the same as panels (3) and (4), respectively, but for a
i= 5
⇥ 10
4au. Both L and sin B are almost
independent of q for q < 50 au. The agreement of the results of numerical calculations and the analytical solutions is
good for a
i= 2
⇥ 10
4au. For a
i= 5
⇥ 10
4au, the disagreement is larger than
⇠ 30 for some bodies beyond 4.5 Gyr
but still much better than that in ⌦ or i
E. Therefore, we conclude that the relation among q, L, and sin B in Table
1, for any m, is safely satisfied for q in the observable region.
4.4. The Empty Ecliptic
Based on the standard scenario of the formation of the Oort cloud, the initial orbital elements of the Oort cloud
comets are restricted as follows; q
i. 30 au to be near a giant planet, i
i' i = 60 and ⌦
i' ⌦ = 186 to be on the
ecliptic plane. Also !
iis uniformly distributed for 0
!
i< 360 , L
iis given by Equation (2), and sin B
i, in order to
be on the ecliptic plane, is given by
tan B
i= sin(L
i⌦ ) tan i .
(100)
The relation between L and sin B in Table
1
defines two planes in the Galactic coordinates. As L
iand B
iare assumed
to be on the ecliptic plane the points that satisfy 0
L < 360 and sin B given by Equation (
100) for L draw a curve
on the L
sin B plane, by definition. Another set of points for an odd m that satisfy 0
L < 360 and
sin B draw
a curve that defines a second plane, which is formed by a rotation of the ecliptic around the Galactic pole by 180 :
tan B =
sin(L
⌦ ) tan i .
(101)
We call this plane as “the empty ecliptic,” since it is not initially populated and this plane and the ecliptic are
symmetrical about the plane perpendicular to the Galactic plane through the intersection of the ecliptic plane and
the Galactic plane (just like the focus and the empty focus of an ellipse). If L and B of long-period comets in the
observable region are concentrated on these two planes, it would constitute observational evidence that the comets
were on the ecliptic plane at t = 0. Comets with relatively small values such as a
⇠ 10
4au, which satisfy P =4.5 Gyr
(i.e., m = 1 for present), are predicted to be on the empty ecliptic for their first return to the planetary region.
5.
OBSERVATIONAL DATA
Figure
7
shows solar system bodies with q > 1 au and a > 10
3au or e > 1 taken from the JPL Small Body Database
Search Engine on 2020 June 5
1on the L
sin B plane. 277 bodies with e
1 and 296 bodies with e > 1 are indicated
with open and filled circles, respectively. The bodies are divided into three groups with L as indicated by colors.
Two interstellar objects, 1I/2017 U1 (’Oumuamua) and 2I/2019 Q4 (Borisov), are additionally shown with squares for
reference. We used the osculating orbital elements to calculate L and B using Equations (2) and (3). To calculate L
and B for bodies with e > 1, we replace ! in Equations (2) and (3) as
!
! ! + (⇡
f
1),
(102)
where
cos f
1=
1
e
.
(103)
To evaluate the concentrations on the two planes, we define the new angle " as
tan " =
tan B
sin(L
⌦ )
.
(104)
The angle " is interpreted as a longitude around the intersection of the ecliptic and the Galactic plane. For the ecliptic
and empty ecliptic planes, " = i
= 60 and " =
i
=
60 , respectively. The solid and dashed curves in Figure
1
https://ssd.jpl.nasa.gov/ query.cgi
18
Higuchi
the prediction of i
Ein the observable region using the analytical solution is as difficult as much as that for i (see Figure
4
) and as sensitive to q as much as that for ⌦. Panels (5) and (6) in Figure
6
are the same as panels (1) and (2) but
for a
i= 5
⇥ 10
4au. The drift of the curves is seen more obviously than in panel (1) since bodies with a
i= 5
⇥ 10
4au
make more oscillations in 4.5 Gyr.
Panels (3) and (4) in Figure
6
show the behavior of L and sin B for a
i= 2
⇥ 10
4au, respectively, and panels (7) and
(8) in Figure
6
are the same as panels (3) and (4), respectively, but for a
i= 5
⇥ 10
4au. Both L and sin B are almost
independent of q for q < 50 au. The agreement of the results of numerical calculations and the analytical solutions is
good for ai
= 2
⇥ 10
4au. For ai
= 5
⇥ 10
4au, the disagreement is larger than
⇠ 30 for some bodies beyond 4.5 Gyr
but still much better than that in ⌦ or i
E. Therefore, we conclude that the relation among q, L, and sin B in Table
1
, for any m, is safely satisfied for q in the observable region.
4.4. The Empty Ecliptic
Based on the standard scenario of the formation of the Oort cloud, the initial orbital elements of the Oort cloud
comets are restricted as follows; q
i. 30 au to be near a giant planet, i
i' i = 60 and ⌦
i' ⌦ = 186 to be on the
ecliptic plane. Also !
iis uniformly distributed for 0
!
i< 360 , L
iis given by Equation (
2
), and sin B
i, in order to
be on the ecliptic plane, is given by
tan B
i= sin(L
i⌦ ) tan i .
(100)
The relation between L and sin B in Table
1
defines two planes in the Galactic coordinates. As L
iand B
iare assumed
to be on the ecliptic plane the points that satisfy 0
L < 360 and sin B given by Equation (
100
) for L draw a curve
on the L
sin B plane, by definition. Another set of points for an odd m that satisfy 0
L < 360 and
sin B draw
a curve that defines a second plane, which is formed by a rotation of the ecliptic around the Galactic pole by 180 :
tan B =
sin(L
⌦ ) tan i .
(101)
We call this plane as “the empty ecliptic,” since it is not initially populated and this plane and the ecliptic are
symmetrical about the plane perpendicular to the Galactic plane through the intersection of the ecliptic plane and
the Galactic plane (just like the focus and the empty focus of an ellipse). If L and B of long-period comets in the
observable region are concentrated on these two planes, it would constitute observational evidence that the comets
were on the ecliptic plane at t = 0. Comets with relatively small values such as a
⇠ 10
4au, which satisfy P =4.5 Gyr
(i.e., m = 1 for present), are predicted to be on the empty ecliptic for their first return to the planetary region.
5.
OBSERVATIONAL DATA
Figure
7
shows solar system bodies with q > 1 au and a > 10
3au or e > 1 taken from the JPL Small Body Database
Search Engine on 2020 June 5
1on the L
sin B plane. 277 bodies with e
1 and 296 bodies with e > 1 are indicated
with open and filled circles, respectively. The bodies are divided into three groups with L as indicated by colors.
Two interstellar objects, 1I/2017 U1 (’Oumuamua) and 2I/2019 Q4 (Borisov), are additionally shown with squares for
reference. We used the osculating orbital elements to calculate L and B using Equations (
2
) and (
3
). To calculate L
and B for bodies with e > 1, we replace ! in Equations (
2
) and (
3
) as
!
! ! + (⇡
f
1),
(102)
where
cos f
1=
1
e
.
(103)
To evaluate the concentrations on the two planes, we define the new angle " as
tan " =
tan B
sin(L
⌦ )
.
(104)
The angle " is interpreted as a longitude around the intersection of the ecliptic and the Galactic plane. For the ecliptic
and empty ecliptic planes, " = i
= 60 and " =
i
=
60 , respectively. The solid and dashed curves in Figure
1 https://ssd.jpl.nasa.gov/ query.cgi
The initial condition (= The ecliptic)
The empty ecliptic
Anisotropy of Long-period Comets 11
we confirmed that these opposite evolutions seen in ⌦ and L are due to the radial component of the Galactic tide that breaks the conservation of j. In panels (6) and (8), the disagreement that arises from the shifts of the periods is quite significant; however, the amplitudes of the oscillations show good agreement even after 4.5 Gyr.
From the above comparisons, we conclude that the analytical solutions are basically useful for describing the orbital evolution, except for i and ⌦ of comets in the observable region (i.e., q . 102 au). For bodies with a = 5 ⇥ 104 au, the small di↵erences between the periods given by the analytical solutions and the ones obtained from the numerical integrations pile up and are quite large at t ⇠ a few Gyr. This could be understood simply as the results of the shifts of oscillation/circulation of orbital evolution. Therefore, the time evolution normalized by the periods obtained by numerical integrations is well reproduced by the analytical solutions.
4. QUASI-RECTILINEAR APPROXIMATION AND APPLICATION TO LONG-PERIOD COMETS
In this section, we apply the analytical solutions derived in Section 2 to fictional observable long-period comets entering the planetary region from the Oort cloud. We assume that the comets initially have very elongated orbits given by planetary scattering on the ecliptic plane. For these comets, setting ei ' 1 is a good approximation and it
makes the solutions simple. We call this approximation as the quasi-rectilinear approximation. Since the planetary scattering does not give high inclinations (see Appendix), we assume ii = i = 60 and ⌦i = ⌦ = 186 to be on the
ecliptic plane. Using this approximation, we compare the periods of , ⌦, and L and investigate the relation among them especially for comets in the observable region.
4.1. Preparation
In this section, we first derive the explicit expression for ↵0, ↵1, and ↵2 (eq.(70)) and their relation, k2, w21, and w22
(eqs. (32), (37), and (51)) under the quasi-rectilinear approximation, which gives 0 < j2 ⌧ 1. Using the expressions, we calculate the values of ⇧ that appear in solutions to ⌦ and L.
4.1.1. solutions and parameters
Substituting e = ei, B = Bi, and 1 e2i = j2/ cos2i = 4j2 into Equation (10), we obtain
c = 3j2 + 5(1 4j2) sin2Bi. (63)
Using i Bi i , we have the minimum and maximum values of c as
cmin = 3j2 ' 0 for Bi = 0
cmax= 3j2 + 154 (1 4j2) ' 154 for Bi = i
. (64)
where j2 ⌧ 1 is used.
From Equation (27), the explicit expressions of the solutions are approximated as
⇤ 1' 5j2 5 + 4j2 c + 100j4 (5 + 4j2 c)3 +O(j 6) ⌧ 1 (65) ⇤ 2' 5 + 4j2 c 4 5j2 5 + 4j2 c + O(j 4) (66)
where j2 ⌧ 1 is used but the term O(j4) in Equation (65) is left for the comparison with j2.
To evaluate the relation between j2 and ⇤1, we substitute c = cmin since the di↵erence between j2 and ⇤1 becomes
minimum for c = cmin. It is calculated as ⇤ 1 j2' 5j2 5 + 4j2 c min + 100j 4 (5 + 4j2 c min)3 j2 = j 4(75 10j2 j4) (5 + j2)3 > 0. (67)
Therefore, the relation between j2 and ⇤1 is always as j2 < ⇤1.
Anisotropy of Long-period Comets 11
we confirmed that these opposite evolutions seen in ⌦ and L are due to the radial component of the Galactic tide that breaks the conservation of j. In panels (6) and (8), the disagreement that arises from the shifts of the periods is quite significant; however, the amplitudes of the oscillations show good agreement even after 4.5 Gyr.
From the above comparisons, we conclude that the analytical solutions are basically useful for describing the orbital evolution, except for i and ⌦ of comets in the observable region (i.e., q . 102 au). For bodies with a = 5 ⇥ 104 au, the small di↵erences between the periods given by the analytical solutions and the ones obtained from the numerical integrations pile up and are quite large at t ⇠ a few Gyr. This could be understood simply as the results of the shifts of oscillation/circulation of orbital evolution. Therefore, the time evolution normalized by the periods obtained by numerical integrations is well reproduced by the analytical solutions.
4. QUASI-RECTILINEAR APPROXIMATION AND APPLICATION TO LONG-PERIOD COMETS
In this section, we apply the analytical solutions derived in Section 2 to fictional observable long-period comets entering the planetary region from the Oort cloud. We assume that the comets initially have very elongated orbits given by planetary scattering on the ecliptic plane. For these comets, setting ei ' 1 is a good approximation and it
makes the solutions simple. We call this approximation as the quasi-rectilinear approximation. Since the planetary scattering does not give high inclinations (see Appendix), we assume ii = i = 60 and ⌦i = ⌦ = 186 to be on the
ecliptic plane. Using this approximation, we compare the periods of , ⌦, and L and investigate the relation among them especially for comets in the observable region.
4.1. Preparation
In this section, we first derive the explicit expression for ↵0, ↵1, and ↵2 (eq.(70)) and their relation, k2, w12, and w22
(eqs. (32), (37), and (51)) under the quasi-rectilinear approximation, which gives 0 < j2 ⌧ 1. Using the expressions, we calculate the values of ⇧ that appear in solutions to ⌦ and L.
4.1.1. solutions and parameters
Substituting e = ei, B = Bi, and 1 e2i = j2/ cos2 i = 4j2 into Equation (10), we obtain
c = 3j2 + 5(1 4j2) sin2Bi. (63)
Using i Bi i , we have the minimum and maximum values of c as
cmin = 3j2 ' 0 for Bi = 0
cmax= 3j2 + 154 (1 4j2) ' 154 for Bi = i
. (64)
where j2 ⌧ 1 is used.
From Equation (27), the explicit expressions of the solutions are approximated as
⇤ 1' 5j2 5 + 4j2 c + 100j4 (5 + 4j2 c)3 + O(j 6) ⌧ 1 (65) ⇤ 2' 5 + 4j2 c 4 5j2 5 + 4j2 c +O(j 4) (66)
where j2 ⌧ 1 is used but the term O(j4) in Equation (65) is left for the comparison with j2.
To evaluate the relation between j2 and ⇤1, we substitute c = cmin since the di↵erence between j2 and ⇤1 becomes
minimum for c = cmin. It is calculated as ⇤ 1 j2' 5j2 5 + 4j2 c min + 100j 4 (5 + 4j2 c min)3 j2 = j 4(75 10j2 j4) (5 + j2)3 > 0. (67)
Therefore, the relation between j2 and ⇤1 is always as j2 < ⇤1.
………… (1)
………… (2)
In the planetary region, L and B satisfy equation (1) or (2).
22 Higuchi
To evaluate the concentrations on the two planes, we define a new angle " as tan " = tan B
sin(L ⌦ ). (104)
For the ecliptic and empty ecliptic planes, " = i = 60 and " = i = 60 , respectively. The solid and dashed curves in Figure 7 show the ecliptic and empty ecliptic planes, respectively. Curves for " = 0, ±30 , and ±80 are also shown in thin dashed curves in Figure 7.
Figure 8 shows the distribution of ". There are two sharp peaks not exactly at the ecliptic or empty ecliptic plane but near them. For an isotropic distribution, it would be flat in ".
0 20 40 60 80 -90 -60 -30 0 30 60 90 E E’ ε [deg]
Figure 8. Distribution of " defined by equation (104) for all bodies in Figure 7 except U1. Filled bars show bodies with e > 1.
Panel (1) in Figure 9 shows the distribution of sin B. The depletions around b = 0 and b = ±90 found by Luest
(1984) and Delsemme (1987) is seen (note that b = B), however, the shape of the distribution depends on the regions of L0 = L ⌦ . Panels (2)-(4) in Figure 9 show the distribution of sin B for regions blue ( 30 < L0 < 30 , 150 < L0 < 210 ), orange (30 < L0 < 60 , 120 < L0 < 150 , 210 < L0 < 240 , 300 < L0 < 330 ), and dark orange (60 < L0 < 120 , 240 < L0 < 300 ), respectively. The non-double, rather a broad peak at sin B = 0 in panel (2) and the sharpest double peaks at | sin B| > 0.5 in panel (4) are explained as a consequence of the double peaks in the distribution of ". If the depletions are the result of the strength of the Galactic tide as Delsemme (1987) explained, the distributions are expected to be the same in any regions since the strength of the Galactic tide is independent of L. Therefore, we conclude that the concentration of comets on the ecliptic and empty ecliptic planes is a better explanation than that by Delsemme (1987).
Discussion
: Distribution of observed long-period comets
sin
B
L [deg]
• On the L - sin B plane: • The distribution of the longitude ε defined by
ε = 60 deg: Ecliptic
ε = -60 deg: Empty Ecliptic
-1 -0.5 0 0.5 1 0 60 120 180 240 300 360
Empty Ecliptic Ecliptic
Galactic plane, =0 =-80 =-60 =-30 =80 =60 =30 ’Oumuamua Borisov sin B L [deg]
Two sharp peaks near the ecliptic or empty ecliptic plane. *An isotropic distribution would be flat in ε.
from JPL/NASA, comets with a > 1000 au || e ≧ 1 and q > 1 au
