_{1}

^{*}

The axi-symmetric satellite problem including radiation pressure and drag is treated. The equations of motion of the satellite are derived. An energy-like is given for a general drag force function of the polar angle
*θ*, and then it is used to find a relation for the orbit equation of the satellite with initial conditions satisfying the vanishing of arbitrarily choosing higher derivatives of the velocity.

The classical two body problem is one of the most important topics in the field of celestial mechanics, specially the applications of the theory of artificial satellites. Since Brouwer and Hori [

Marvaganis [

In this work, an attempt is made to get a solution for the problem of an axi-symmetric satellite under drag and radiation pressures, which all the effects are included in the equation of motion from the beginning by using energy like integral. A relation for the orbit equation is derived first for a general air drag function and then for the case of Danby’s drag. Finally, the solution of an almost constant speed satellite has been given.

The equation of motion of an axi-symmetric satellite under the gravitational force of a spherical body with an additional force due to the resistance force and radiation pressure can be modeled such as Mavraganis and Michalakis (1994), and El-Shaboury and Mostafa (2014).

The air resistance is taken as a general function R of the polar coordinates

Now, vector product Equation (1) with r, and remembering that

which gives immediately,

This expression admits a first vector which is the constant direction

Let the resistance be a general function of the polar angel

We get,

where, h is the constant angular momentum in the absence of the drag force, and

Now, let

The vector equation of motion is thus resolved to

where

and using the substitution

Equation (5) becomes,

This gives,

where

Taking R in the form

which implies by using Equation (4)

In order to integrate the required integration, we expand V in Taylor series of the polar angle

where

We write,

where,

Thus,

where,

Therefore, we have the integration,

Thus, using Equations (12), (14a), and (15) we get a relation for the orbit equation in the form,

where

Equation (16) describes a relation for the orbit equation of an axi-symmetric satellite with oblateness coeffi-

cient k under radiation pressure of coefficient

The convergence of the involved series is guaranteed for initial velocity satisfying the vanishing of

for all n > N, where N can be chosen arbitrary.

In the special case of Danby’s drag (Dabny, 1962),

Substituting,

we get the integration (18) in the form

for n when it is even or odd,

When n is even, we write

and then we use the expansion

thus we get

And when n is odd, we write

then we use the expansion

thus we get

We can collect the two cases together in one case to get,

Substituting from Equations (17 - 20) into Equation (16), we get after simplification

Equation (21) gives a relation for the orbit equation of an axi-symmetric satellite under the gravitational effect of a radiating body and air resistance described by Danby’s drag.

If the satellite is of almost constant speed, then we assume that the first derivative is of small value, and all the higher derivatives to be zero (e.g. Mavraganis, 1991), we get the solution

Equation (22) is a special case of Equation (21) when the satellite is of almost constant speed.

In this paper, the motion of an axi-symmetric satellite under the effect of a radiating body in the presence of air drag is studied. An energy-like integral for the problem has been evaluated using a Taylor expansion for the velocity around the initial value of the polar angel. The convergence of the integral is guaranteed by the assump-

tion that the derivative

The energy-like integral has been used to get a relation for the orbit equation of the satellite. The relation is derived first for a general air drag function and then for the case of Danby’s drag. Finally, the solution of an almost constant speed satellite has been given.

AhmedMostafa, (2015) Use of an Energy-Like Integral to Study the Motion of an Axi-Symmetric Satellite under Drag and Radiation Pressure. International Journal of Astronomy and Astrophysics,05,148-154. doi: 10.4236/ijaa.2015.53019