Reins Model for the Restricted [600423]

Rein's Model for the Restricted
Elliptic Three-Body Problem
ith dwith drag
Mihai Barbosu(1)Ciprian Chiruta(2)Tiberiu Oproiu(3)
(1) RIT (Rochester Institute of Technology), USA
(2) The University of Agricultural Sciences and Veterinary Medicine Ia și
(3)Astronomical Observatory, Cluj
The Eighth Congress of Romanian Mathematicians
June 26 – July 1, 2015, Iasi1

Consider theplanar elliptic three body problem :studying the1) The problem
Consider theplanar elliptic three -body problem :studying the
motion of a body P under the gravitational action of two bodies P1,
P2, with the following restrictions:
(i) The body P has infinitesimal mass (it does not influencethe motion of the bodies P1 and P2) which have finite masses m
1
d ti l andm2, respec tively
(ii) The motions of P1and P2are given as solution of a two-body
bl i l / lli ti bit f ith problem: circular/ellipticorbits – common focus inthem a s s
centre O, the eccentricity e, the semi-major axes a1and a2, and
the mean motion n. The motion of P takes place in the orbital
plane of P1and P2
iii)Solutions: no anal ytic solutions, but fo rthecircular restricted ) y
three-body problem – one first integral – Jacobi’s
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2) Coordinates
Focus on the elliptic case:
Uniforml yrotatin gframe ,with the constant an gularvelocit yn.
y g , g y
In the rotating frame, the bodies P1and P2describe closed
curves around the”mean ”points P1(−a1,0)and P2(a2,0), curves around the mean points P1(a1,0)and P2(a2,0),
respectively. The positions of Pi,i=1 ,2 ,a r eg i v e ni nt h i sf r a m e
by (Rein, 1940):
where vstands for the true anomaly.
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3) Equations of motion of P
The planar motion of the infinitesimal mass point P with
respect to the frame is described by the differentials
ystem of equations:
y
The force function becomes an explicit function of t
Remark : The differential system of eq. of motion (3) does
notadmit a first integral analogous to Jacobi’s integral which
appears in the case of the circula rrestricted three-body
problem.
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4) Replace U with a time-averaged function
We shall use an averaging method: Rein’s ”semi-averaging”
scheme:
A)Express thetime dependence ofthecoordinates of A)Express thetime-dependence ofthecoordinates of
Pi, i = 1, 2 by means of infinite power series of the eccentricity
and then obtain finite expressions (depending on t):
B)Instead oftheforce function Uwetake itstime B)Instead oftheforce function U,wetake itstime-
averaged value:
ith T 2 / with T = 2 π/n
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5) Use of elliptic coordinates
U can be written as:
with
while K( χi) represent the elliptic integrals of first kind:
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Furthermore, the relationship between elliptic coordinates and
new Cartesian coordinates leads to:
Eventually the differential system of equations of the relative
ti fPiRi’ i dh hthf motion ofPinRein’ss e m i-average dscheme hastheform:
Equations (15) have a first integral of motion simila rto the
Jacobi’s integral from the circular restricted three-body problem:
with h, the constant of integration. 7

6) Effect of perturbing drag
The equations oftheplanar motion ofthethird body perturbing The equations oftheplanar motion ofthethird body perturbing
by an arbitrary external force Fare:
From eq. (18) we have
If we denote we have:
If we denote we have:
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6) Effect of perturbing drag
We conside rdrag force proportional to the velocity of the
particle in the rotating frame ,
Th ti fth l ti fthbd P
The equa tion oftheplanarmotion ofthebodyP,a r e :
From (20) we have:
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7) Application to the Earth-Moon System
We consider Earth-Moon system, let P be the spacecraft, P1
the Earth and P2the Moon. The eccentricity and
the Moon to Earth ratio
The equilibrium points of the system (18)
ib
are g iven by:
The system (23) has three collinear points
and two quasi -equilateral points The
and two quasi equilateral points The
coordinates for the collinear points are:
and for quasi -equilateral points are:
and for quasi equilateral points are:
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8) Numerical results
For all three collinear points we choose K= 0 001 For all three collinear points we choose K=-0.001
Fig. 1. Results for L1 Fig. 2. Results for L3
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8) Numerical results
For quasi equilateral points we choose different values for K For quasi -equilateral points we choose different values for K
Drag
constant
K=00 2
K=-0.02
Fi 3 R lt f L Fi 4 R lt f L
Drag constant
K=0 00625Fig. 3. Results for L4 Fig. 4. Results for L5
K=-0.00625
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CONCLUSIONS
™In this model a spacecraft motion around the equilibrium points
isconducted onunstable orbits isconducted onunstable orbits
™Rein’s model for the restricted three-body problem presents a
real interest because the expression ofthe force function real interest because the expression ofthe force function
doesn’t depend on time explicitly
™F lll fth ti i t thRi’ dli ™Forsmallvalues o fthe eccen tricitytheRein’sm o delisv e r y
good approximation for the elliptic restricted three body problem
™IAt d i fthEt h M t thi dli ™InAstrodynam ics,fortheEarth-Moon sys tem, thism o delis
more accurate than the classical restricted circular three bodyproblem, since the Moon has an elliptical orbit
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THANK YOU!
THANK YOU!
Thank you !
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