Estimations of the energy quasi-integral of the restricted three-body problem
We consider the restricted three-body problem for a dust particle in the vicinity of a spherical cometary nucleus in an eccentric orbit about the Sun. The differential equations of the particle’s spatial motion are integrated both analytically and numerically to obtain and estimate the energy quasi-...
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| Дата: | 2005 |
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Головна астрономічна обсерваторія НАН України
2005
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| Цитувати: | Estimations of the energy quasi-integral of the restricted three-body problem / G.F. Chorny // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 500-503. — Бібліогр.: 11 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859784969894756352 |
|---|---|
| author | Chorny, G.F. |
| author_facet | Chorny, G.F. |
| citation_txt | Estimations of the energy quasi-integral of the restricted three-body problem / G.F. Chorny // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 500-503. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Кинематика и физика небесных тел |
| description | We consider the restricted three-body problem for a dust particle in the vicinity of a spherical cometary nucleus in an eccentric orbit about the Sun. The differential equations of the particle’s spatial motion are integrated both analytically and numerically to obtain and estimate the energy quasi-integral.
|
| first_indexed | 2025-12-02T10:12:10Z |
| format | Article |
| fulltext |
ESTIMATIONS OF THE ENERGY QUASI-INTEGRAL
OF THE RESTRICTED THREE-BODY PROBLEM
G. F. Chörny
Main Astronomical Observatory, NAS of Ukraine
27 Akademika Zabolotnoho Str., 03680 Kyiv, Ukraine
e-mail: chorny@mao.kiev.ua
We consider the restricted three-body problem for a dust particle in the vicinity of a spherical
cometary nucleus in an eccentric orbit about the Sun. The differential equations of the particle’s
spatial motion are integrated both analytically and numerically to obtain and estimate the energy
quasi-integral.
INTRODUCTION
The three-dimensional generalization of a dust motion in the cometary orbital plane is a problem of natural
interest, because it enables one to study macroscopic volume formations in a cometary atmosphere [3, 4, 7–10].
Dust particles coming out of the nuclear region are being acted upon by the forces of radiation pressure and
gravitation. The resulting action depends upon the ratio of these two forces, which is generally denoted as β
[1, 5, 11]. For the particles with β = 1, one may term these dust particles resonant. In the equation of motion
of a resonant particle, gravitational effect of the cometary nucleus will be remained as well beyond a region of
the nuclear influence on the non-resonant dust. Because of this, for the resonant particle, one has the three-body
problem, that under known conditions can be reduced to the restricted three-body problem, in which the orbit
of comet is conic section of arbitrary eccentricity and the trajectory of the dust particle is a spatial curve [6].
In this paper we consider some consequences of the specified eccentric restricted problem in its general case.
DIFFERENTIAL EQUATIONS OF MOTION
The differential equations of motion of a separate dust particle in a non-inertial cometocentric reference system
(CRS) are as follows:
ẍ′
1 = −μs(1 − β)
r + x′
1
y3
− μc
x′
1
x3
+ θ̈x′
2 + θ̇2x′
1 + 2θ̇ẋ′
2 +
μs
r2
,
ẍ′
2 = −μs(1 − β)
x′
2
y3
− μc
x′
2
x3
− θ̈x′
1 + θ̇2x′
2 − 2θ̇ẋ′
1, (1)
ẍ′
3 = −μs(1 − β)
x′
3
y3
− μc
x′
3
x3
,
where x′
n (n = 1, 2, 3) are components of the position vector x of the particle in the CRS, μs = Gms is the Sun
gravitational parameter, μc = Gmc is the gravitational parameter of the comet; r, θ̇, θ̈ are comet’s heliocentric
distance, the angular rate, and the angular acceleration about the Sun, respectively; x = (x′
1
2 + x′
2
2 + x′
3
2)1/2
and y = ((r + x′
1)
2 + x′
2
2 + x′
3
2)1/2.
AN ENERGY QUASI-INTEGRAL
Complex-analysis tools [2] permits one to derive an energy quasi-integral from Eqs. (1).
Denote by v the orbital velocity of the particle at the heliocentric distance y. Then the expression for
the particle’s energy quasi-integral assumes the form
v2
2
= (1 − β)
μs
y
+
μc
x
− μc
∫ t
t0
(x′
1ṙ + θ̇x′
2r)
dt′
x3
+ H0, (2)
c© G. F. Chörny, 2004
500
where the constant
H0 =
v2
0
2
− (1 − β)
μs
y0
− μc
x0
(3)
is the energy integral at the initial time t0.
It is well known that
v2 = |vex + v′|2 = v2
ex + v′ 2 + 2vex · v′, (4)
where vex denotes the reference frame velocity of a point fixed in the CRS and v′ is the dust particle velocity
relative to the CRS. So, if the velocity v′ of the dust particle relative to the nucleus is equal to zero, then v2 in
the left side of the Eq. (2) can be replaced by v2
ex and we obtain the so-called equation of the surfaces of zero
relative velocity of the particle in the CRS:
2μc
x
+ 2(1 − β)
μs
y
− 2μc
∫ t
t0
(x′
1ṙ + θ̇x′
2r)
dt′
x(t′)3
− v2
ex = C, (5)
where C = −2H0.
ESTIMATIONS AND DISCUSSION
It is useful to estimate terms in Eq. (2). For convenience, let us consider the minimal and maximal hypothetical
comets. Take the nucleus radius of the minimal comet to be 0.4 km, as Sugano–Saiguse–Fujikawa Comet has,
the radius of the maximal comet to be 20 km, as in the case of P/Schwassmann–Wachmann 1 Comet. Let us
assume that the mean density of their nuclei equals 1 · 103 kg m−3. Then masses of the nuclei are 2.681 · 1011 kg
and 3.351 · 1016 kg, respectively. This leads to inequalities:
17.890 m3 s−2 ≤ μc ≤ 22.361 · 105 m3 s−2. (6)
At the same time μs = 13.273 · 1019 m3 s−2.
In order to estimate the constant H0, it is necessary to select a boundary in the coma out of which the dust
particles are mainly decoupled from the outflowing gas – reach their terminal velocity. Early coma models set it
at a few tens of the nucleus radius [3]. Also, temporary captured particles of submillimeter size and larger can
be found in the circumnuclear volume of a mean radius of 100 the nucleus radii [4, 8]. In comparison, the data
obtained in [10] indicate that within jets the boundary may be removed to a distance of a few thousand nucleus
radius. Use for our purposes the distance x0 = 100 the nucleus radii. Constant x0 defines the distance to
the starting position of the particle in Eq. (3). Let the starting particle have the position vector y0 along
the Sun – comet axis. Then constant y0 in Eq. (3) equals y0 = r0 ± x0 = r0 ± |x′
1|0, where r0 is the heliocentric
distance of the cometary nucleus at the moment of the particle start. Because the value H0 is being looked for
at the single point (r0 + x′
1,0, 0, 0), an acceptable approximation of v2
0/2 can be obtained by putting
v2
0
2
=
v2
c,0
2
= μs
(
e − 1
2q
+
1
r0
)
, (7)
where vc,0 is the orbital velocity of the comet in an orbit with the eccentricity e and the perihelion distance q
at the moment of the particle start. As to the value of β, for majority of materials 0 ≤ β < 1. Only for iron,
graphite or magnetite particles, and also for the fluffy ones, there is β ≥ 1 [1, 11]. Using above numerical values
shows that to first order it is sufficient to approximate H0 by the expression:
H0 = μs
(
e − 1
2q
+
β
r0
)
. (8)
It is seen that H0 < 0 for β < (1 − e)r0/2q, that is only for comets in an elliptic orbit. Otherwise H0 ≥ 0.
On the whole, when β < (1 + e)/2, there is a region of r0 on an elliptic orbit where H0 ≤ 0.
To estimate the integral in Eq. (2), assume that a particle is started from a surface x = x0 around a comet at
t = t0 and arrived at a point (x′
1∗, x
′
2∗, x
′
3∗) at t = t∗ with the relative speed v′ = 0. Let at this point a net zero
acceleration act on the particle in the rotating frame CRS. Thus, the particle will remain in this equilibrium
point of the CRS at least by moment t > t∗. Then the integral from Eq. (2) may be broken up into the sum:
I ≡
∫ t
t0
(ṙx′
1 + rθ̇x′
2)
dt′
x3
=
∫ t∗
t0
(ṙx′
1 + rθ̇x′
2)
dt′
x(t′)3
+
∫ t
t∗
(ṙx′
1∗ + rθ̇x′
2∗)
dt′
x3∗
. (9)
501
Table 1. Results of Estimations of Terms in Equation (2)
Comet: 46P/Wirtanen C/1995 O1(Hale–Bopp) 29P/Schwassmann–
Wachmann 1
v2
ex/2 4.211 5.614 0.761
μs/y 5.614 5.617 1.490
μc/x 2.301·10−15 3.952·10−12 6.139·10−12
−μcI 1.899·10−9 2.475·10−7 2.732·10−6
H0(β) 6.719β−1.389 7.184β+0.046 1.517β−0.743
Note. All the quantities are expressed in terms of 108 m2 s−2 unit. The quantities in the right-hand side of Eq. (2) are
calculated at the point (r(θ = 80o)+x′
10, x′
10, x′
10), where x′
10 ≡ (x′
1)0 = −100 nuclear radii. The fourth order Runge–Kutta
method was used to solve the system of the differential equations (1) at the calculation of the integral I in Eq. (9). The value
of H0 is estimated at the point (r(θ0 = 60o) + x′
10, x′
10, x′
10), with the relative starting velocity v′0 = (ẋ′
10, ẋ′
20, ẋ′
30) =
(140,−14, 14) m s−1. The orbit parameters e, q are taken from [http: //cfa-www.harvard.edu/iau/Ephemerides/Comets/].
The epoch of osculation of the orbital elements is 2003 June 10.0 TT. It is supposed that the mean density of the comet
nuclei is equal to 1 · 103 kgm−3, the mass of Comet Hale–Bopp is 2.1 · 1015 kg, the mean radius of Comet 46P/Wirtanen is
690 m, and that of Comet Schwassmann–Wachmann 1 is 2 · 104 m.
To evaluate the first integral in the right-hand side of this relation, differential Eqs. (1) must be solved. But
the second integral may be written in the form
I∗ ≡ x′
1∗
x3∗
∫ t
t∗
ṙdt′ +
x′
2∗
x3∗
∫ t
t∗
rθ̇dt′ =
x′
1∗
x3∗
[r(t) − r(t∗)] + p
x′
2∗
x3∗
∫ θ
θ∗
dϑ
1 + e cosϑ
, (10)
where p is the semilatus rectum of the comet’s orbit, and θ∗ = θ(t∗) is the true anomaly of the comet at t = t∗.
Performing the quadrature yields the formulas
∫ θ
θ∗
dϑ
1 + e cosϑ
=
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
2e−1
∗ (π∗ + arctanA), (e < 1, e∗ =
√|1 − e2|),
e−1
∗ ln[(1 + A)/(1 − A)], (e > 1, e∗ =
√|1 − e2|),
2A, (e = 1, e∗ = 1),
(10∗)
where
π∗ =
{
0, if tan(θ/2) tan(θ∗/2) > (e + 1)/(e − 1),
sign(tan(θ/2)) · π, if tan(θ/2) tan(θ∗/2) < (e + 1)/(e − 1),
and
A =
e∗ · sin((θ − θ∗)/2)
cos((θ − θ∗)/2) + e · cos((θ + θ∗)/2)
.
Table 1 lists illustrative results of the estimations for some of the representative comets.
STANDARD TREATMENT OF ENERGY QUASI-INTEGRAL
Another interpretation of Eq. (2) arises after introducing some habitual quantities.
Consider the energy quasi-integral (2). Since |v| ≡ v ≥ 0, the left member of Eq. (2) is greater than or
equals zero. Therefore, for the right-hand side of the equation, the following condition is satisfied:
(1 − β)
μs
y
+
μc
x
− μc
∫ t
t0
(x′
1ṙ + θ̇x′
2r)
dt′
x3
+ H0 ≥ 0. (11)
Define
E0 = mpH0, T0 = mpv
2
0/2, (12)
U = −mp
[
(1 − β)
μs
y
+
μc
x
− μc
∫ t
t0
(x′
1ṙ + θ̇x′
2r)
dt′
x3
]
, (13)
502
where mp is the mass of the dust particle. Then the condition (11) becomes
E0 − U ≥ 0, E0 = T0 + U0, (14)
where U0 = U |t=t0 . According to the standard treatments of the introduced quantities, mechanical motion of
the particle is possible in a space region where U ≤ E0, that is where inequality (14) is fulfilled. If at once
the initial energy of the particle E0 < 0, the motion is limited to the condition U = E0. This equality defines
the surfaces of zero relative velocity v′ of the dust particle motion by means of Eq. (5). If under condition (14)
E0 > 0 (that is H0 > 0) then the motion of the particle is unrestricted in space.
As can be seen from Table 1, any value of β leads to H0 > 0 for Comet Hale–Bopp when a particle is started
under the conditions listed in the Note.
Acknowledgements. Dr. Yu. V. Babenko is gratefully acknowledged for eagerness to help in a numerical
integration of the system of the differential equations of a dust particle motion. Unfortunately, author picked
out another calculus software.
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| id | nasplib_isofts_kiev_ua-123456789-79709 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0233-7665 |
| language | English |
| last_indexed | 2025-12-02T10:12:10Z |
| publishDate | 2005 |
| publisher | Головна астрономічна обсерваторія НАН України |
| record_format | dspace |
| spelling | Chorny, G.F. 2015-04-03T19:58:49Z 2015-04-03T19:58:49Z 2005 Estimations of the energy quasi-integral of the restricted three-body problem / G.F. Chorny // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 500-503. — Бібліогр.: 11 назв. — англ. 0233-7665 https://nasplib.isofts.kiev.ua/handle/123456789/79709 We consider the restricted three-body problem for a dust particle in the vicinity of a spherical cometary nucleus in an eccentric orbit about the Sun. The differential equations of the particle’s spatial motion are integrated both analytically and numerically to obtain and estimate the energy quasi-integral. Dr. Yu. V. Babenko is gratefully acknowledged for eagerness to help in a numerical integration of the system of the differential equations of a dust particle motion. Unfortunately, author picked out another calculus software. en Головна астрономічна обсерваторія НАН України Кинематика и физика небесных тел MS5: Dynamics and Physics of Solar System Bodies Estimations of the energy quasi-integral of the restricted three-body problem Article published earlier |
| spellingShingle | Estimations of the energy quasi-integral of the restricted three-body problem Chorny, G.F. MS5: Dynamics and Physics of Solar System Bodies |
| title | Estimations of the energy quasi-integral of the restricted three-body problem |
| title_full | Estimations of the energy quasi-integral of the restricted three-body problem |
| title_fullStr | Estimations of the energy quasi-integral of the restricted three-body problem |
| title_full_unstemmed | Estimations of the energy quasi-integral of the restricted three-body problem |
| title_short | Estimations of the energy quasi-integral of the restricted three-body problem |
| title_sort | estimations of the energy quasi-integral of the restricted three-body problem |
| topic | MS5: Dynamics and Physics of Solar System Bodies |
| topic_facet | MS5: Dynamics and Physics of Solar System Bodies |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79709 |
| work_keys_str_mv | AT chornygf estimationsoftheenergyquasiintegraloftherestrictedthreebodyproblem |