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Attitude drift toward collinearity of a wayward planet
Mechanics ResearchCommunications, Vol. 28. No. 6, pp. 601-6C9,2001
Pergamon
Copyright 8 2001 Elsevier Science Ltd Printed in the USA. All rights reserved 0093.6413/01/$-see front matter
PII: Soo93-6413(02)00211-2
ATTITUDE DRIFT TOWARD COLLINEARITY OF A WAYWARD PLANET
F.P.J. Rimrott and F.A. Salustri Department of Mechanical and Industrial Engineering University of Toronto, Toronto, Ontario, Canada M5S 3G8
A model planetary system is studied, where all planets are on circular orbits in one and the same plane, except one planet whose orbital plane is inclined at an angle. By means of the collinearity principle [l] and a Ward spiral [2] it is shown how the wayward planet moves towards the common plane to achieve eventually collinearity of all angular momenta. Our own solar system, consisting of the Sun and of, say, 9 planets is a good example [3]. It is essentially characterized by collinearity, i.e. all angular momenta are (almost) parallel and are pointing in the same direction, with one exception, the planet Pluto, whose angular momentum deviates some 17.2” from the global angular momentum. A further distinction, one that we have ignored in our model, is that Pluto moves on an elliptic orbit, while all other orbits are (almost) circular. There is obviously some gravitational interaction between Pluto and neighbouring planets, a circumstance that led to the discovery of Pluto in the first place.
t
Figure 1. Planetary System with Wayward Planet 601
F. P. J. RIMROTT and F. A. SALUSTRI
602
The Collinearitv Princiule The collinearity principle [l] describes the behaviour of torque free gyroscopic systems in time. As far as planetary systems are concerned it stipulates that 1. an independent planetary system, such as our own solar system, will tend to acquire unidirectional collinearity of the individual angular momenta with respect to the master mass in the course of time. 2. the associated orbital energy of the planetary system will tend to acquire a minimum value in the course of time. Our own solar system is thus a good example of a system that is close to its eventual final state. It almost satisfies that all orbits are unidirectional in one and the same plane. All orbits are also almost circular ,thus they are close to a minimum of orbital energy [4].
Ward Soiral For a point satellite of mass m in the field p of a point master subjected to a constant aerodynamic drag force D on a nearly circular orbit of initial radius r. Dr. Charles A. Ward has shown that such a satellite’s orbit (Figure 3) is spiral [2] and given by ‘0
r =
(l+ce)2
(1)
where c is a characteristic constant with c = 0 for a circular orbit 0 < c
c=-
D KO
(2)
where (3) A small value of c then implies that D << K,
(4)
COLLINEARITY
OF WAYWARD
PLANET
or, in words, the drag force must be much smaller than the Kepler force.
Angular Momenta For the angular momenta (Figure 1) we have H, where
= H,t
H
(5)
H g = (constant) global angular momentum HS = angular momentum H = angular momentum
of system without wayward planet of wayward planet
For our model system we let (H 1 -cc (H g ( At the end of the collinearization
(6)
process we have (Figure 2). = Hs, t
%
Hf
(7)
Since the angular momenta are now all collinear we may also write
Hsf + HI
%
=
H4
= HsiCOSP
(8)
with
= Ho cosa
Hf
(9) (10)
and since Ho sina:
=
H,isinP
(11)
we get =
Uff
Ho!!!!!!?
During the drifting process toward collinearity
H=
Hf cos(cr
(12)
tan0
-
y)
[l] the wayward planet’s angular momentum
= Ho
is
cosa cos(a-y)
(13)
604
F.P.J.RIMROTT
and F. A. SALUSTRI
Ni = H 0. With y = a we obtain the final
With y = 0 we obtain the initial momentum momentum HI
= HocosIX
(14)
E
sf
I ..:: ..::i:; ::j:::: ‘::$:; :... ‘_..
5:; .:.:.> ;:::::::: :::I$::::
‘::;:i:;:; .‘..... ‘.....
H
x
H ‘...
g
I
I \\ Initial Attitude Hsisin i3= H,sincr
Attitude DriZt Hf = Hcos(a-y)
Figure 2. Angular Momenta
7 Ef a
Final Attitude
Hf = H,cosa
COLLINEARITY
OF WAYWARD
PLANET
605
Apart from the primary attraction by the point master, there is also some secondary gravitational attraction by the other planets, indicated by the force F in Figure 3, which produces a torque about the point master, which causes the angular momentum of the wayward planet to describe a cone, as shown in Figure 3, with a base radius of initially bo
=
Ho sinew
(15)
and then, in the course of the drifting process, by virtue of equation (13) b
=
H sin ((x - y) = Ho cosa tan(a-y)
(16)
Snirallinq Let us now stipulate that there is some drag (solar wind, gravitational interaction between planets, meteorite strikes) such that the orbit of the wayward planet is continuously losing in size as given by a Ward spiral i 1) ‘0
r = where c is a very small coefficient, circular.
(17)
(ltce)2
small enough to ensure that the planet orbit remains near-
Let p =GM denote the gravitational parameter of the central body, the product of the universal gravitational constant and the mass of the central attracting body, where G = 6.67 x 10-1’m3kg-‘s-2 represents the universal gravitational constant. By Newton’s law, for a particle of mass m moving in a circular orbit under the gravitational force due to M, we have
v2
m
-
r
Pm
=
v =
7
r I
r
(18)
Accordingly, since in a circular motion the instantaneous velocity vector is always perpendicular to the position vector, and taking into account equation (17), the magnitude of the angular momentum of the moving particle m and the radius vector are related by
H
q
mrv
HO =
m&
(19)
and because of equation (13) we have 1 -= 1tce
cosa cos(a
- y)
(20)
606
F. P. .I. RIMROTT and F. A. SALUSTRI
Figure 3. Initial -4ngular Momentum Cone
COLLrNEARITY OF WAYWARD PLANET
607
When y = 0 we have initial conditions with 8 = 0. When y = a we have collinearity and final conditions, i.e. 1
-
= cosa
It&f
(21)
From equations (13) and (15) we conclude that the orbit radius is r
‘0
=
cos2(a-y)
cos'a
(22)
giving an initial radius, when y = 0 , of ri
=
(23)
‘0
and a final radius, for y = a of ‘f
= ro cos201
(24)
The polar angle 8 of planet motion is related to the cone angle of the angular momentum cone by equation (20), specifically by e
_
-
L COW-y) _ c i
cosa
1
1
(25)
Initially when y = 0 we have 8 = 0. Finally when y = a, assuming that c remains constant during the whole collinearization process, we have
Of =
:(A -1)
(26)
Since 0, is obviously extremely large, the coefficient c must be extremely small.
Using Pluto Data Pluto’s orbit [3] is elliptical, with semi-major axis a = 5946 (106) km and eccentricity E = 0.253. Its equivalent circular orbit, which Pluto can attain by means of an orbital energy reduction at constant angular momentum about the Sun is, according to reference [4], TO
q
a(l-a2)
= 5946 (106) (1 - 0.2532)
q
5565 (106) km
(27)
We can now use the equations of the present paper to study Pluto’s attitude drift towards collinearity with the rest of the solar system. Pluto’s orbital plane is presently inclined some
608
F. P. .I. RJMROTT and F. A. SALUSTRI
17.2’ from the ecliptic. When collinearity is attained the final radius of Pluto’s orbit about the sun is given by equation (24) and obtained as = ro cos2a
‘f Pluto’s (including
Charon’s)
=
5565(106) ~0~~17.2”
mass is m =
5078 (106) km
q
15( 102’) kg and its original
(28)
angular
momentum
about the Sun is H,
= m&
=
15(10z1)
while the global angular momentum constant
(with H,
collinearity
132.7(109) 5565(106)
= 407.6 (J03’) kg km Is
(29)
of all planets and the Sun itself about the Sun centre remains
= 31 500 000 (103’) kg km2 /s).
The angular
momentum
of Pluto, when
has been attained, is = Ho cosa
Hf
= 407.6(1030) cos 17.2” = 389.4 (103’) kgkm2 /s
(30)
Pluto’s orbital energy is initially
Eo = -
132.7 (109) 15(10z1)
Pm
-= zr
=
- 178.83(1027) J
2(5565)106
0
(31)
and finally
Ef
=
-2”1= Colhnearization
132.7 (109) 15(10*‘)
Pm
2(5078)106
-
then has been accompanied
(32)
by an energy change of
AE = Ef - E, The energy change is negative.
- 195.97(10z7) J q
Consequently
=
- 17.14(1027) J
(33)
it represents a loss,
Pluto’s orbital period about the sun changes from
r.
= 2#
= 2n\/z
= 7.16(109) s = 227 y
(34)
COLLINEARITY
r/
= 2nE
q
OF WAYWARD
2z/%
PLANET
GO0
= 198 y
= 6.24 (109)s
(35)
The most interesting question, however, as to how long it will take for Pluto to achieve collinearity with the rest of the solar system cannot be answered because it is not known how much drag-like resistance Pluto is subject to. Let us, for instance, say that it would take 100 million change lo, from 17.2“ to 16.2”. That means it takes 100 (106)
n =
=
lOO(106) 227
TO
years for Pluto’s orbital inclination
= 440 528 rev
to
(36)
or 8 = 2701 = 2rr(440 528)
= 2 767 923 rad
(37)
From equation (25)
c
=
-$s - 1)=
1.897 (10m9) rad-’
(38)
and 5565 (106)
‘0
r =
(ltc8)2
=
(1+1.897(10+)
= 5507 (106) km
(39)
2 767 923)’
By means of a model planetary system and by making use of the collinearity principle, it has been shown how a wayward planet would strive towards collinearity with all the other angular momenta.
1.
Rimrott, F.P.J.: Das Kollinearitatsprinzip,
Technische Mechanik,
2.
Rimrott, F.P.J. and Salustri, CANCAN (2001), 305-306.
3.
Rimrott, F.P.J.: Introductory
4.
Rimrott, F.P.J. and Cleghom, W.L.: The Collinearity Orbits, Technische Mechanik, 20,4, (2000), 305-3 10.
18, 1 (1998), 57-68.
F.A.: The Ward Sprial in Orbit Dynamics,
Orbit Dynamics, Vieweg, Wiesbaden, Principle
Proceedings,
(1989), 193 p. and Minimum
Energy
Pergamon
Copyright 8 2001 Elsevier Science Ltd Printed in the USA. All rights reserved 0093.6413/01/$-see front matter
PII: Soo93-6413(02)00211-2
ATTITUDE DRIFT TOWARD COLLINEARITY OF A WAYWARD PLANET
F.P.J. Rimrott and F.A. Salustri Department of Mechanical and Industrial Engineering University of Toronto, Toronto, Ontario, Canada M5S 3G8
A model planetary system is studied, where all planets are on circular orbits in one and the same plane, except one planet whose orbital plane is inclined at an angle. By means of the collinearity principle [l] and a Ward spiral [2] it is shown how the wayward planet moves towards the common plane to achieve eventually collinearity of all angular momenta. Our own solar system, consisting of the Sun and of, say, 9 planets is a good example [3]. It is essentially characterized by collinearity, i.e. all angular momenta are (almost) parallel and are pointing in the same direction, with one exception, the planet Pluto, whose angular momentum deviates some 17.2” from the global angular momentum. A further distinction, one that we have ignored in our model, is that Pluto moves on an elliptic orbit, while all other orbits are (almost) circular. There is obviously some gravitational interaction between Pluto and neighbouring planets, a circumstance that led to the discovery of Pluto in the first place.
t
Figure 1. Planetary System with Wayward Planet 601
F. P. J. RIMROTT and F. A. SALUSTRI
602
The Collinearitv Princiule The collinearity principle [l] describes the behaviour of torque free gyroscopic systems in time. As far as planetary systems are concerned it stipulates that 1. an independent planetary system, such as our own solar system, will tend to acquire unidirectional collinearity of the individual angular momenta with respect to the master mass in the course of time. 2. the associated orbital energy of the planetary system will tend to acquire a minimum value in the course of time. Our own solar system is thus a good example of a system that is close to its eventual final state. It almost satisfies that all orbits are unidirectional in one and the same plane. All orbits are also almost circular ,thus they are close to a minimum of orbital energy [4].
Ward Soiral For a point satellite of mass m in the field p of a point master subjected to a constant aerodynamic drag force D on a nearly circular orbit of initial radius r. Dr. Charles A. Ward has shown that such a satellite’s orbit (Figure 3) is spiral [2] and given by ‘0
r =
(l+ce)2
(1)
where c is a characteristic constant with c = 0 for a circular orbit 0 < c
c=-
D KO
(2)
where (3) A small value of c then implies that D << K,
(4)
COLLINEARITY
OF WAYWARD
PLANET
or, in words, the drag force must be much smaller than the Kepler force.
Angular Momenta For the angular momenta (Figure 1) we have H, where
= H,t
H
(5)
H g = (constant) global angular momentum HS = angular momentum H = angular momentum
of system without wayward planet of wayward planet
For our model system we let (H 1 -cc (H g ( At the end of the collinearization
(6)
process we have (Figure 2). = Hs, t
%
Hf
(7)
Since the angular momenta are now all collinear we may also write
Hsf + HI
%
=
H4
= HsiCOSP
(8)
with
= Ho cosa
Hf
(9) (10)
and since Ho sina:
=
H,isinP
(11)
we get =
Uff
Ho!!!!!!?
During the drifting process toward collinearity
H=
Hf cos(cr
(12)
tan0
-
y)
[l] the wayward planet’s angular momentum
= Ho
is
cosa cos(a-y)
(13)
604
F.P.J.RIMROTT
and F. A. SALUSTRI
Ni = H 0. With y = a we obtain the final
With y = 0 we obtain the initial momentum momentum HI
= HocosIX
(14)
E
sf
I ..:: ..::i:; ::j:::: ‘::$:; :... ‘_..
5:; .:.:.> ;:::::::: :::I$::::
‘::;:i:;:; .‘..... ‘.....
H
x
H ‘...
g
I
I \\ Initial Attitude Hsisin i3= H,sincr
Attitude DriZt Hf = Hcos(a-y)
Figure 2. Angular Momenta
7 Ef a
Final Attitude
Hf = H,cosa
COLLINEARITY
OF WAYWARD
PLANET
605
Apart from the primary attraction by the point master, there is also some secondary gravitational attraction by the other planets, indicated by the force F in Figure 3, which produces a torque about the point master, which causes the angular momentum of the wayward planet to describe a cone, as shown in Figure 3, with a base radius of initially bo
=
Ho sinew
(15)
and then, in the course of the drifting process, by virtue of equation (13) b
=
H sin ((x - y) = Ho cosa tan(a-y)
(16)
Snirallinq Let us now stipulate that there is some drag (solar wind, gravitational interaction between planets, meteorite strikes) such that the orbit of the wayward planet is continuously losing in size as given by a Ward spiral i 1) ‘0
r = where c is a very small coefficient, circular.
(17)
(ltce)2
small enough to ensure that the planet orbit remains near-
Let p =GM denote the gravitational parameter of the central body, the product of the universal gravitational constant and the mass of the central attracting body, where G = 6.67 x 10-1’m3kg-‘s-2 represents the universal gravitational constant. By Newton’s law, for a particle of mass m moving in a circular orbit under the gravitational force due to M, we have
v2
m
-
r
Pm
=
v =
7
r I
r
(18)
Accordingly, since in a circular motion the instantaneous velocity vector is always perpendicular to the position vector, and taking into account equation (17), the magnitude of the angular momentum of the moving particle m and the radius vector are related by
H
q
mrv
HO =
m&
(19)
and because of equation (13) we have 1 -= 1tce
cosa cos(a
- y)
(20)
606
F. P. .I. RIMROTT and F. A. SALUSTRI
Figure 3. Initial -4ngular Momentum Cone
COLLrNEARITY OF WAYWARD PLANET
607
When y = 0 we have initial conditions with 8 = 0. When y = a we have collinearity and final conditions, i.e. 1
-
= cosa
It&f
(21)
From equations (13) and (15) we conclude that the orbit radius is r
‘0
=
cos2(a-y)
cos'a
(22)
giving an initial radius, when y = 0 , of ri
=
(23)
‘0
and a final radius, for y = a of ‘f
= ro cos201
(24)
The polar angle 8 of planet motion is related to the cone angle of the angular momentum cone by equation (20), specifically by e
_
-
L COW-y) _ c i
cosa
1
1
(25)
Initially when y = 0 we have 8 = 0. Finally when y = a, assuming that c remains constant during the whole collinearization process, we have
Of =
:(A -1)
(26)
Since 0, is obviously extremely large, the coefficient c must be extremely small.
Using Pluto Data Pluto’s orbit [3] is elliptical, with semi-major axis a = 5946 (106) km and eccentricity E = 0.253. Its equivalent circular orbit, which Pluto can attain by means of an orbital energy reduction at constant angular momentum about the Sun is, according to reference [4], TO
q
a(l-a2)
= 5946 (106) (1 - 0.2532)
q
5565 (106) km
(27)
We can now use the equations of the present paper to study Pluto’s attitude drift towards collinearity with the rest of the solar system. Pluto’s orbital plane is presently inclined some
608
F. P. .I. RJMROTT and F. A. SALUSTRI
17.2’ from the ecliptic. When collinearity is attained the final radius of Pluto’s orbit about the sun is given by equation (24) and obtained as = ro cos2a
‘f Pluto’s (including
Charon’s)
=
5565(106) ~0~~17.2”
mass is m =
5078 (106) km
q
15( 102’) kg and its original
(28)
angular
momentum
about the Sun is H,
= m&
=
15(10z1)
while the global angular momentum constant
(with H,
collinearity
132.7(109) 5565(106)
= 407.6 (J03’) kg km Is
(29)
of all planets and the Sun itself about the Sun centre remains
= 31 500 000 (103’) kg km2 /s).
The angular
momentum
of Pluto, when
has been attained, is = Ho cosa
Hf
= 407.6(1030) cos 17.2” = 389.4 (103’) kgkm2 /s
(30)
Pluto’s orbital energy is initially
Eo = -
132.7 (109) 15(10z1)
Pm
-= zr
=
- 178.83(1027) J
2(5565)106
0
(31)
and finally
Ef
=
-2”1= Colhnearization
132.7 (109) 15(10*‘)
Pm
2(5078)106
-
then has been accompanied
(32)
by an energy change of
AE = Ef - E, The energy change is negative.
- 195.97(10z7) J q
Consequently
=
- 17.14(1027) J
(33)
it represents a loss,
Pluto’s orbital period about the sun changes from
r.
= 2#
= 2n\/z
= 7.16(109) s = 227 y
(34)
COLLINEARITY
r/
= 2nE
q
OF WAYWARD
2z/%
PLANET
GO0
= 198 y
= 6.24 (109)s
(35)
The most interesting question, however, as to how long it will take for Pluto to achieve collinearity with the rest of the solar system cannot be answered because it is not known how much drag-like resistance Pluto is subject to. Let us, for instance, say that it would take 100 million change lo, from 17.2“ to 16.2”. That means it takes 100 (106)
n =
=
lOO(106) 227
TO
years for Pluto’s orbital inclination
= 440 528 rev
to
(36)
or 8 = 2701 = 2rr(440 528)
= 2 767 923 rad
(37)
From equation (25)
c
=
-$s - 1)=
1.897 (10m9) rad-’
(38)
and 5565 (106)
‘0
r =
(ltc8)2
=
(1+1.897(10+)
= 5507 (106) km
(39)
2 767 923)’
By means of a model planetary system and by making use of the collinearity principle, it has been shown how a wayward planet would strive towards collinearity with all the other angular momenta.
1.
Rimrott, F.P.J.: Das Kollinearitatsprinzip,
Technische Mechanik,
2.
Rimrott, F.P.J. and Salustri, CANCAN (2001), 305-306.
3.
Rimrott, F.P.J.: Introductory
4.
Rimrott, F.P.J. and Cleghom, W.L.: The Collinearity Orbits, Technische Mechanik, 20,4, (2000), 305-3 10.
18, 1 (1998), 57-68.
F.A.: The Ward Sprial in Orbit Dynamics,
Orbit Dynamics, Vieweg, Wiesbaden, Principle
Proceedings,
(1989), 193 p. and Minimum
Energy