Canonical Treatments for the Two Bodies Problem with Varying Mass Taking into Consideration the Periastron Effect ()
1. Introduction
The problem of the two bodies with varying mass has roots going back in the history since the middle of the 19th century. Since F. Mestschersky’s initial study a great many researchers have dedicated much of their time to this problem, telling it became one of the classics of celestial mechanics, which has become known as the Gylden-Mestschersky.
An ample bibliography can be found in the published works of E. N. Polyakova [1] and C. Prieto [2] . The specific case which results in a slow isotropic mass loss has also been the focus of exhaustive studies carried out by researchers, for instance, J. D. Hadjidemetriou [3] [4] , to name but a few. The vast majority of these, in search of the stellar application, have taken the so-called Eddington-Jeans law [5] [6] , as a law of the variation of mass,
(1)
where a and n are real numbers, the first of them is positively proximate to zero and the second is varying between 1.4 and 4.4.
Prieto [7] [8] intended to present an approximate analytic solution of the two-body problem with slowly decreasing mass which is obtained through the integration of the Hamilton equations using Deprit’s method of perturbations. The solution, obtained through the Eddington-Jeans law, is put into practice in a specific case and compared with F. Mestschersky’s exact equation; n = 2; and with that which results from numerically integrating the equations.
In the framework of celestial mechanics, this problem has been exhaustively addressed by Docobo [9] , Andrade [10] , among others.
In 2002, M. Andrade [11] analyzed the dynamics of binary systems with time-dependent mass loss and periastron effect i.e., a supposed enhanced mass loss during periastron passage by means of analytical and numerical techniques.
M. Andrade and J. A. Docobo [12] studied the dynamics of binary systems with small parameter perturbation model, the time-dependence of the whole set of orbital elements, concluded, could be calculated over long timescales and even for high eccentricities. In these models, they studied the following timeand distance-dependent mass-loss law:
(2)
where the first term represent time-dependent mass loss, and the last one introduces the periastron effect, where “r” is the distance between the two components, 
is the total angular momentum and β is another small parameter close to zero.
W. A. Rahoma et al. [13] were introduction paper concerned with the two-body problem with varying mass in case of isotropic mass loss from both components of the binary systems. The law of mass variation used gives rise to a perturbed Keplerian problem depending on two small parameters. The problem is treated analytically in the Hamiltonian frame-work and the equations of motion are integrated using the Lie series developed and applied, separately by Delva [14] and Hanslmeier [15] . A second order theory of the two bodies eject mass was also constructed, returning the terms of the rate of change of mass up to second order in the small parameters of the problem.
W. A. Rahoma et al. [16] studied the two-body problem with varying mass in case of isotropic mass loss. In that work, the problem was treated analytically in the Hamiltonian frame-work and the equations of motion ware integrated using the Lie operator and Lie series.
Our work is aimed to introduce an analytical solution for the problem of two bodies with varying mass, taking into consideration the periastron effects, using the canonical perturbation technique derived by Hori and developed by Kamel.
2. The Hamiltonian of the Problem
The Hamiltonian is constructed in terms of Delaunay’s variables as described in Deprit [17] :
(3)
The first term in the Hamiltonian function, described by Equation (3), is the Hamiltonian in the Keplerian case (with constant mass) 
while the second term represents the varying mass effects. In terms of the Delaunay variable, we can develop the Hamiltonian as:
(4)
That is meant that the Hamiltonian is function of
, 
and 
as:
The variation of 
may be retained from one of the two masses 
or 
and this is the case of one body eject mass. Otherwise the case of the two bodies ejects masses. We will concern with the first case. Then;
(5)
The Hamiltonian 
represented by Equation (4) is implicit depending on time through the variable mass 
and its time derivative
. By modifying Docobo’s law for the rate of change of mass assigned by Equation (2) and use Jeans law described by Equation (1) we get:
(6)
By applying Equation (6) in the Hamiltonian function (4) which can be written as:
(7)
with n positive number varying between 1.4 and 4.4.
Since the variable mass can be expressed as a Taylor series expansion as:
that will lead to a Hamiltonian with explicit dependence of time. So we must extend the phase space by introducing a new pair of variable
. The first is “
” assigned as the variable mass and the second its conjugate momentum “
”.
The new systems of canonical equations of motion are:
(8)
where K is the new Hamiltonian in the extended phase space which can be expressed as:
(9)
The first term in Equation (9) is contribution of the two body with constant mass with 
is total mass of the system at specific time
. The small parameter 
is close to zero as well as the small parameter
. In this stage, we can handle the problem as two bodies with two small parameters. But, if we assume that the two small parameters 
and 
has the same order of magnitude the problem will turn into one small parameter problem.
The Hamiltonian (9) can be written in the form:
, (10)
where,
(10.1)
with,
(10.2)
and,
The canonical equations of motion for the Hamiltonian K are:
(11)
Applying the last Equation for the Hamiltonian function represented by Equation (10) yields:
(11.1)
(11.2)
(11.3)
(11.4)
(11.5)
(11.6)
(11.7)
(11.8)
3. Short Period Solution Using Hori Method’s
The Hameltonian function K is function of 
and
. The integrable part of the Hameltonian, K0, is function of
. So the variable 
can be consedered as the fast variable.
In the next stage we will use Hori’s method [18] , developed by Kamel [19] to elemeinate the short period termes from the Hameltonian. And the transformed Hameltonian, 
, can be written, up to second orders, as follows:-
3.1. Zero Order
By applying the Hori’s Perturbation technique developed by Kamel yields:
(12)
3.2. First Order
The first order transformed Hamiltonian derived by Horis’ can be outlined as:
(13.1)
where;
. (13.2)
And by choosing;
. (13.3)
then
(13.4)
So the first order generating function, S1, is therefore obtained by integral:
. (13.5)
By averaging the Hameltonian, 
, over the mean anomaly 
we can determine the transformed first order Hamiltonian, 
, as:-
After calculating the required integrals needed in the last equation we get:
. (14)
By applying Equation (13.4) to calculate the periodic part of the Hamiltonian as:
(15)
To calculate the first order generating function, 
, we must use Equation (13.5) (with the help of Equations (12) and (15)) which lead, after calculate the required derevativs and products, to the results:-
(16)
where B’s are function of 
and given by:
(16.1)
(16.2)
(16.3)
3.3. Second Order
The seconed order transformed Hamiltonian derived by Horis’ can be outlined as:-
(17)
where;
(17.1)
Choosing;
(17.2)
and;
(17.3)
Hence S2 therefore can be obtained by;
(17.4)
To construct the function
, we need, first, to calculate the Poisson bracket
. By the help of Equations (10.2), (14) and (16) with 
yeilds:
(18)
where the coeffecients
, 
and 
are function of 
defined as:-
Then, applying Equation (17.2) by averaging over the mean anomaly 
for the last Equation (18), we can determine the transformed second order Hamiltonian, 
, as:-
(19)
Substituting from Equation (18) in the last Equation yeilds:-
(20)
A useful paper for Ahmed (1994) [20] , was used to calculate the required integral. While the last integral was solved except for the case of hyperbolic orbits.
where 
and 
are coefficients, ware derived during the calculation of the Poisson brackets
, defined as:
Finally, the required transformed second order Hamiltonian can be written as:
(21)
where Y is function of L1 and L2 and defined by:
The last, second orders Hamiltonian, is valid in all cases of orbits except in the case of hyperbolic orbits (e > 1).
3.4. The Transformed Hamiltonian
The second order transformed Hamiltonian; 
can be constructed in the form:
Which can be written by using Equations (12), (14) and (21) as:
(22)
The equations of motion in the transformed augmented phase space can be derived as follows:
(23-1)
(23-2)
(23-3)
(23-4)
(23-5)
(23-6)
(23-7)
(23-8)
where 
is the partial differentiation for the function Ψ with respect to the momentum L2 and given by:
The Hamiltonian and the system of differential equation, described by Equations (23), has one degree of freedom since the variable 
is explicitly appears in them.
3.5. The Element of Short Period Transformation and Its Inverse
The elements of short period transformation can be calculated using equations:
(24.1)
(24.2)
The inverse transformation equations can be calculated as follows:
(25.1)
(25.2)
where
, 
, 
and 
can be calculated using Equation (13.5).
The prime over the variables means that variables in the transformed phase space which will be omitted for the sake of simplicity.
The variable mass 
is implicitly function of time through the relation (7’), so we will complete the solution using the method which was introduced by Delva [14] and Hanslmeier [15] , separately.
4. The Final Solution
A special linear differential operator, the Lie operator, produces a Lie series to construct an approximate solution. The convergence of the Lie series is similar to that of Taylor series, since the Lie is only another analytic form of the Taylor series. The algorithm for the method was outlined in Delva [14] and Hanslmeier [15] .
The Lie operator, D, can be defined as:
(26.1)
and the second order “D” operator, which can be noted as D2, constructed as:
(26.2)
The coefficients of the operator D, 
and
, can be replaced by the canonical equations of motion represented by Equations (23). Then, Lie operator for our case can be written as:
(27)
Now, we can calculate the solution by applying the D operator, represented by Equation (27), up to second orders, to the variables 
and the results are as follows:
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
where the quantities
, 
, 
, 
, 
, 
, 
and 
are the orbital elements at time 
Using the initial values for the orbital elements, with the help of Equations (28) - (35), we can calculate the new perturbed orbital elements due to the varying mass.
5. Discussion and Conclusions
The last group of equations is a second order solution for the two body problem with m1 varying mass. In those equations, the following results are included.
1. The varying mass gives rise to the change of mean anomaly 
and the argument of preside
. That rises influenced by both the varying mass and the periastron effect.
2. A new varying, second order, mass formula was derived and represented by Equation (31).
3. The varying mass, as well as the periastron effects, does not affect the argument of ascending node 
and the momenta
, 
and 
in the extended transformed phase space.
4. A second order equation for calculation of the momentum associated to the varying mass, 
, as well as the periastron effect, was derived and represented by Equation (35).
Future work: We plane to apply these equations for deferent systems as well as comparing the results with others and with observations.