Determination of minimum distance between orbits of celestial bodies
A method is suggested to determine of minimum distance between orbits of two celestial bodies. The method is based on finding of the singular points of one variable function. The orbits of celestial bodies can be elliptic, parabolic or hyperbolic. The orbits of celestial bodies can also be described...
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| Zitieren: | Determination of minimum distance between orbits of celestial bodies / Yu. Babenko // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 423-426. — Бібліогр.: 2 назв. — англ. |
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Babenko, Yu. 2015-04-03T19:27:27Z 2015-04-03T19:27:27Z 2005 Determination of minimum distance between orbits of celestial bodies / Yu. Babenko // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 423-426. — Бібліогр.: 2 назв. — англ. 0233-7665 https://nasplib.isofts.kiev.ua/handle/123456789/79692 A method is suggested to determine of minimum distance between orbits of two celestial bodies. The method is based on finding of the singular points of one variable function. The orbits of celestial bodies can be elliptic, parabolic or hyperbolic. The orbits of celestial bodies can also be described universal variable. On the basis of the given method the program is developed, using which the accounts of minimal distance between orbits of comets and orbits of planets are executed. The given method can also be used for finding of the minimal distances both between orbits of asteroids and orbits of planets, and between asteroids orbits. en Головна астрономічна обсерваторія НАН України Кинематика и физика небесных тел MS4: Positional Astronomy and Global Geodynamics Determination of minimum distance between orbits of celestial bodies Article published earlier |
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Determination of minimum distance between orbits of celestial bodies |
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Determination of minimum distance between orbits of celestial bodies Babenko, Yu. MS4: Positional Astronomy and Global Geodynamics |
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Determination of minimum distance between orbits of celestial bodies |
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Determination of minimum distance between orbits of celestial bodies |
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Determination of minimum distance between orbits of celestial bodies |
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Determination of minimum distance between orbits of celestial bodies |
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determination of minimum distance between orbits of celestial bodies |
| author |
Babenko, Yu. |
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Babenko, Yu. |
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MS4: Positional Astronomy and Global Geodynamics |
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MS4: Positional Astronomy and Global Geodynamics |
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2005 |
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English |
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Кинематика и физика небесных тел |
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Головна астрономічна обсерваторія НАН України |
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A method is suggested to determine of minimum distance between orbits of two celestial bodies. The method is based on finding of the singular points of one variable function. The orbits of celestial bodies can be elliptic, parabolic or hyperbolic. The orbits of celestial bodies can also be described universal variable. On the basis of the given method the program is developed, using which the accounts of minimal distance between orbits of comets and orbits of planets are executed. The given method can also be used for finding of the minimal distances both between orbits of asteroids and orbits of planets, and between asteroids orbits.
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0233-7665 |
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https://nasplib.isofts.kiev.ua/handle/123456789/79692 |
| citation_txt |
Determination of minimum distance between orbits of celestial bodies / Yu. Babenko // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 423-426. — Бібліогр.: 2 назв. — англ. |
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AT babenkoyu determinationofminimumdistancebetweenorbitsofcelestialbodies |
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2025-11-24T21:07:39Z |
| last_indexed |
2025-11-24T21:07:39Z |
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1850494680635015168 |
| fulltext |
DETERMINATION OF MINIMUM DISTANCE
BETWEEN ORBITS OF CELESTIAL BODIES
Yu. Babenko
Astronomical Observatory, National Taras Shevchenko University of Kyiv
3 Observatorna Str., 04053 Kyiv, Ukraine
e-mail: babenko@observ.univ.kiev.ua
A method is suggested to determine of minimum distance between orbits of two celestial bodies.
The method is based on finding of the singular points of one variable function. The orbits of
celestial bodies can be elliptic, parabolic or hyperbolic. The orbits of celestial bodies can also be
described universal variable. On the basis of the given method the program is developed, using
which the accounts of minimal distance between orbits of comets and orbits of planets are executed.
The given method can also be used for finding of the minimal distances both between orbits of
asteroids and orbits of planets, and between asteroids orbits.
INTRODUCTION
The minimal distances between orbits of celestial bodies are heavy weighed. If they are small enough that
probably close approach celestial bodies. It is especially important in case of comet, when the small value of
this minimal distance can serve indication that the close approach of the comet and planet could take place, as
a result of which there could be an essential change of elements of an orbit of a comet. There is one more case,
when it is important to know this minimal distance is a case of orbits asteroids and orbit of the Earth, when
small distance between their orbits can be indication on an opportunity of collision this asteroid with the Earth.
DETERMINATION OF MINIMUM DISTANCE BETWEEN ORBITS
In [2] the case of elliptic orbits of two bodies is considered and is shown that the given task can be reduce
to finding of zero function by one variable. By the author [1] is shown that the finding of minimal distance
between an elliptic orbit both hyperbolic and parabolic orbits can also be reduce to a finding of zero function by
one variable. In connection with this it represents the great interest, including practical, whether a finding of
minimal distance between orbits of any type, including in a case, when for the description of an orbit are used
universal variable, is possible, to reduce a finding of zero function by one variable, to that the present work is
devoted.
Let us consider a task of a finding of minimal distance between orbits of two celestial bodies in the most
general case.
The system of coordinates, where the XY plane coincides with a celestial body orbital plane, and the X axis
is directed in an orbit perihelion may be written as
{
x = rx(a, e, φ),
y = ry(a, e, φ), (1)
where rx and ry are also functions dependent on major axis, eccentricity and variable φ, which will be concretized
below. It is clear that the radius-vector r =
√
x2 + y2 will be function of the same variable.
As it is known, in ecliptic system of coordinates the radius-vector r can be expressed by rp
x, rp
y , and also by
argument perihelion ω, longitude of the ascending node Ω and inclination of an orbit i:
r = Prx + Qry, (2)
where P and Q are the orbital vectors of the constants, which are expressed by ω, Ω, and i:
Px = cosω cosΩ − sin ω sinΩ cos i, Qx = − sinω cosΩ + cosω sinΩ cos i,
Py = cosω sin Ω + sin ω cosΩ cos i, Qy = − sinω sin Ω − cosω cosΩ cos i,
Pz = sin ω sin i, Qz = cosω sin i.
(3)
c© Yu. Babenko, 2004
423
Marking everything, concerns to the first body by 1, and to the second one by 2 it is possible to write
the following formula for a square of distance between bodies ρ :
ρ2 = (r2 − r1, r2 − r1) = r22 − 2(r2, r1) − r21, (4)
where ρ is a function of two variables φ1 and φ2. It will depend on parameters of orbits of two celestial bodies.
As it is known, for a finding of its minimum it is necessary to equate with zero its derivative of variable and
to solve concerning them the turned out equations system. If such decisions will be a little bit that to find
distances for all values of the decisions and minimal of the found distances will be required minimal distance
between orbits of these two celestial bodies. From above-stated it is received the following equations system:
⎧⎪⎪⎨
⎪⎪⎩
∂ρ2
∂φ1
= 2r1
dr1
dφ1
− 2
(
r2,
dr1
dφ1
)
= 0,
∂ρ2
∂φ2
= 2r2
dr2
dφ2
− 2
(
dr2
dφ2
, r1
)
= 0.
The obtained system can also be rewrote as follows:
⎧⎪⎪⎨
⎪⎪⎩
r1
dr1
dφ1
−
(
r2,
dr1
dφ1
)
= 0,
r2
dr2
dφ2
−
(
dr2
dφ2
, r1
)
= 0.
(5)
With (1) and (2) the equations system (5) is possible to rewrite in the following kind:
⎧⎪⎨
⎪⎩
r1
dr1
dφ1
− S1
r1x
dφ1
− S3
r1y
dφ1
= 0,
r2
dr2
dφ2
− S2r1x − S4r1y = 0,
(6)
where S1 = P1P2r2x + P1Q2r2y , S2 = P1P2
dr2x
dφ2
+ P1Q2
dr2y
dφ2
, S3 = Q1P2r2x + Q1Q2r2y, S4 = Q1P2
dr2x
dφ2
+
Q1Q2
dr2y
dφ2
, and where by P1P2, Q1P2, P1Q2, and Q1Q2 defined scalar products (P1,P2), (Q1,P2), (P1,Q2),
(Q1,Q2).
The further consideration we shall take at once for all types of orbits, including for a case of use for
the description of an orbital universal variable. In Table 1 for all these cases the values of rx, ry, r, drx/dφ,
dry/dφ, dr/dφ are given without their expressions in order to save place for our paper. For an elliptic orbit
the eccentric anomaly E is chosen as independent variable, for a parabolic orbit σ = q tan
f
2
is chosen, where q
is the perihelion distance, f is the true anomaly, for a hyperbolic orbit H is chosen, which is as follows expressed
by true anomaly: tanh
H
2
=
√
e + 1
e − 1
tan
f
2
, for a case of use for the description of an orbital universal variable,
i.e., generalized anomaly s, determined by expression μ
dt
ds
= r, where μ is the gravitational constant, p is
an orbital parameter. The functions Cn(z) are defined by the formulas:
Cn(z) =
k=∞∑
k=0
(−1)k zk
(2k + n)!
; n = 1, 2, 3, ... . (7)
Let us note that there are 10 variants of the equations system (6): E − E, E − P , E − H , E − U , P − P ,
P −H , P −U , H −H , H−U , U −U , where for orbits types the following designations are entered: E is elliptic,
P is parabolic, H is hyperbolic, U is used for the description of an orbital universal variable. We shall consider
only four cases, when the first orbit is consistently chosen elliptic, parabolic, hyperbolic and written down in
universal constant while for the second orbit we shall keep recording in a general view.
Elliptic orbit. The equations system (6) with the accounting Table 1 accepts the following form:
{
M sin E1 + N cosE1 = K sin E1 cosE1,
A sin E1 + B cosE1 = 0,
(8)
424
Table 1. Expression for rx, ry, r, drx/dφ, dry/dφ, dr/dφ for various types of orbits
Expression Elliptic orbit Parabolic orbit Hyperbolic orbit Universal variable
rx a(cos E − e) q(1 − σ2) | a | (e − cosh H) q − s2C2(αs2)
rx a
√
1 − e2 sin E 2q σ | a | √e2 − 1 sinhH)
√
qsC1(αs2)
r a(1 − e cos E) q(1 + σ) | a | (e cosh H + 1) q C0(α s2)s2C2(αs2)
drx
dφ
−a sinE −2qσ − | a | sinh H sC1(αs2)
√
μ
r
dry
dφ
a
√
1 − e2 cos E 2q | a | √e2 − 1 cosh H C0(αs2)
√
p μ
r
dr
dφ
ae sinE 2q σ | a | e sinh H (1 − αq)sC1(αs2)
where M = S1 − a1e1, N = −√
1 − e2
1S3, K = −a1e1, A = a1
√
1 − e2
1S4, B = a1S2, C = r2
dr2
dφ2
. It follows
from (8) that A, B, C, M , N , and K depend only on φ2 and do not depend on E1. Thus, to reduce the decision
of the given equations system to a finding of zero function by one variable it is necessary to find sinE1 and
cosE1 from the second equation and to substitute the received expressions in the first equation.
By solving the second equation from the equations system (8) we obtain the following expressions for sin E1
and cosE1:
sin E1 =
AC ± B
√
A2 + B2 − C2
A2 + B2
,
cosE1 =
BC ∓ A
√
A2 + B2 − C2
A2 + B2
.
(9)
As follows from (9) the decision of the equations system (8) only exists, when the condition: A2+B2−C ≥ 0
is satisfied.
Substituting the obtained expressions for sinE1 and cosE1 in the first equation of the system (8) after
algebraic transformations one can get:
F = [CẼ(MA + NB) + KAB(G − C2)] − G[KC(B2 − A2) − E(MB − NA)]2 = 0, (10)
where Ẽ = A2 + B2, G = Ẽ − C2.
Parabolic orbit. The equations system (6) with the accounting Table 1 accepts the following form:
{
Mσ3
1 + Nσ1 = K,
Aσ2
1 + 2Bσ1 = C,
(11)
where M = q1, N = q1 − S1, K = S3, A = q1S2, B = q1S4, C = r2
dr2
dφ2
− q1S2. Solving the second equation of
the system (11) the following equation for σ may be written:
σ1,2 =
−B ±√
B2 − AC
A
. (12)
As follows from (12) the decision of the equations system (11) only exists, when the condition: B2 −AC ≥ 0 is
satisfied.
In this case substituting the obtained expression for σ1,2 in the second equation of the system (11) similarly
previous one the following function F by one variable is obtained:
F = (3MB2 + B2 − AC + A2N)(B2 − AC) − (A3K + MB3 + 3B2 − 3BAC + A2NB) = 0. (13)
Thus, and in this case the searching for a minimal distance between a parabolic orbit and other orbit form was
reduced to a finding of zero function by one variable.
Hiperbolic orbit. The equations system (6) with the accounting Table 1 may be written by the equations:
{
M sinh H1 + N coshH1 = K sinh H1 coshH1,
A sinh H1 + B coshH1 = 0,
(14)
425
where M =| a | −S1, N = −√
e2
1 − 1S3, K =| a1 | e1, A =| a1 | √
e2
1 − 1S4, B = − | a1 | S2, C = r2
dr2
dφ2
S2.
From the second equation in (14) we find:
sinh H1 =
−AC ± B
√
A2 + C2 − B2
B2 − C2
,
coshH1 =
BC ∓ A
√
A2 + C2 − B2
B2 − C2
.
(15)
As follows from (15) the decision of the equations system (14) only exists, when the condition: A2 +C2−B2 ≥ 0
is satisfied. The values sinh H1, coshH1 in the first equation of the system (14) were substituted for these values
from (15) and the function F was obtained:
F = [CẼ(AM − NB) + KAB(C2 + G)]2 − G[Ẽ(MB − NA) + KC(A2 + B2)]2 = 0, (16)
where Ẽ = A2 − B2, G = Ẽ + C2.
The case of use for the description of an orbital universal variable. We can write the equations
system (6) with the accounting Table 1 as follows:{
Ms1C1(α1s
2
1) + Ns2
1c2(α1s
2
1) − L = Ks1C1(α1s
2
1)s
2
1C2(α1s
2
1),
A(s1C1(α1s
2
1))2 + 2Bs1C1(α1s
2
1) + G = 0,
(17)
where M =
μS1
r1
− q1(1 − α1e1), N = −α1
√
μp1
r1
, L =
√
μp1S3, K = 1 − α1q1, A = α1 + (α1q1S4)2,
B =
√
q1S4
(
α1r2
dr2x
dφ2
− (1 − α1q1)S2
)
, C =
(
α1r2
dr2x
dφ2
− (1 − α1q1)S2
)2
− 1.
Let us note that at reception of the second equation of the system (17) the following expression is used:
αs2C2(αs2) = 1 −
√
1 − α(sC1(αs2))2.
A solution of the second equation may be presented as:
(s C1(αs2))1,2 =
−B ±√
B2 − AG
A
. (18)
As follows from (18) the decision of the equations system (17) only exists, when the condition: B2 − AQ ≥ 0
is satisfied. The following expression for function F is derived after substitution of the obtained decision in
the first equation of the system (17) taking into account (18):
F = U2 − E2K2W
A2
, (19)
where U = (N − L)2 − 2(N − L)
(
M−K
A
)
B +
(
M−K
A
)
B2 + W
(
M−K
A
)2 − R
A2
(
V 2 + WK2
A
)
− 4BWV
A2 , E =
2B
(
M−K
A
)2− 2RV
A2 − 2B
A2
(
V 2 + K2W
A
)
− 2R
A2
(
KB
A
)
, V = KB
A +N, W = B2−AG, R = A2 +B2−AG−α1B
2.
Thus, one see in this case the searching for a minimal distance between orbits of two bodies was reduced to
a finding of zero function by one variable as well as in the three above-mentioned cases too.
It is necessary to note that our research was carried out without a concrete definition of the second orbit.
However, from stated above clearly that the transition to the concrete kind of the second orbit is reduced to
replacement in the received formulas of general expressions by the concrete expressions according to Table 1.
In summary we shall note that from our consideration is followed that the definition of minimal distance
between orbits of two celestial bodies can be executed as follows. To find a dependence on types of orbits or
form of their description it is necessary to find zero function by one variable. Then, for these values variable
the second variable is founded, and the distances between celestial bodies orbits are founded for the obtained
pairs of variables. Minimum of these distances will be minimal distance between the given orbits of celestial
bodies.
[1] Babenko Yu. G. // Astrometry and Astrophys.–1983.–49.–P. 22–26.
[2] Vasiljev N. N. // Bull. Inst. Theor. Astron.–1978.–14.–P. 266–268.
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