Third Order Hamiltonian for a Binary System with Varying Masses Including Preastron Effect

This work concerns of the effects of the variation in the masses for two attracting bodies on the orbiter orbital elements. The formulation of the problem was done in the frame of Hamiltonian mechanics. Moreover, constructing the Hamiltonian function of the varying masses of a binary system including, periastron effect, in canonical form in the extended phase space, up to third order of the small parameter αi, to be able to solve using canonical perturbation techniques. The Hamiltonian is explicit function of time through the variable masses, so we will extend the phase space by introducing two canonical variables l4 and l5 represents the change of masses while L4 and L5 are represent their conjugate momentum. Finaly we will drive the new Hamiltonian in the extended phase space.


Introduction
The problem of the two bodies with varying mass has roots going back in the history since the middle of the 19th century. The comprehensive work done by many scientists in this problem was using the Newtonian frame of work. Dufour (1866) he is the first one who examine the astronomical phenomenon of variable mass by relating the secular variation of lunar acceleration with the increase of the Earth mass due to the impact of meteorites. After that, Gylden (1884) set out the solution to the system of differential equations which describes the two-body motion when the masses are subject to variations. Rahoma et al. (2009) was introduced paper concerned with the two-body problem with varying mass in case of isotropic mass loss from both components of the binary system. The law if mass variation used gives rise to a perturbed Keplerian problem depending on two small parameters. The problem is treated analytically in the Hamiltonian framework and the equations of motion are integrated using the Lie series developed and applied separately by Delva (1984) and Hanslmeier (1984). 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.
M.I.EL-Saftawy, Amirah R. AL-Gethami(2014), in their work, the Hamiltonian of the two body problem with varying mass was developed in the extended phase space taking into consideration the periastron effect. The short period solution was obtained through constructing a second order canonical transformation using "Hori's" method (Hori, 1966) developed by "Kamel" (Kamel, 1969). The element s of the transformation as well as the invers transformation were obtained too. The final solution of the problem was derived using "Delva-Hanslmeier" method.
M.I.EL-Saftawy, F.A.Abd EL-Salam,(2017), the model of varying mass function, including periastron effect in terms of Delaunay variables was expanded. The Hamiltonian of the problem is developed in the extended phase space by introducing a new canonical pair of variables (q 4 , Q 4 ). The first "q 4 "is defined as explicit function of time and the initial mass of the system. The conjugate momenta "Q 4 " is assigned as the momenta raises from the varying mass. The short-period analytical solution through a second-order canonical transforming using "Hori's" method developed by "Kamel" is obtained. The variation equation for the orbital elements are obtained too. The result of the effect of the varying mass and the periastron effect in the case of n = 2 are analyzed.

2.
Formulation of the problem.
The Hamiltonian for the two-body problem expressed in term Delaunay variables, which derived firstly by Deprit, A. (1983), is: where the usual Delaunay Variable defined by: Substitute frome the second equation into the first one we get: where, ̇=̇1 +̇2.
With the help of Jeans law (Jeans 1924(Jeans ,1925 ̇= − , (k=1,2), which yields: We can expand the function about its value at time 0 , up to 3 ed. order, to be: Where the required derivatives are, Now, calculate the quantity ̇ for the second term of the Hamiltonian.
Let us expand μμ as function of t around t = t0 , as: Finaly, if we assume 1 , 2 , have the same order of magnitude, then we will write the Hamiltonian, up to order 2, in summation form: Where (ℓ i , i ) is arbitrary function of the old variables and momenta. We can choose it to be the old Hamiltonian.
The new Hameltonian, K, in the extended phase space, up to second order, is given by: