### 3.2 Surface integral approach and body zone

One way to evaluate the gravitational force acting on a star is the volume integral approach. In the
Newtonian case, the gravitational force on star becomes
The integral region covers star but does not cover the star (the companion star). To evaluate
the above integral we must know the internal structure, which is generally a difficult task even in the
Newtonian dynamics, needless to say in general relativity. By means of the surface integral approach we can
put off dealing with the internal structure problem until tidal effects affect the orbital motion. In the
Newtonian case, using the Poisson equation and Gauss’s law, we can rewrite the above volume integral into
a surface integral,
Note that the sphere has no intersection, neither with star nor star . Thus, we can evaluate
the gravitational force acting on star without knowledge about the internal structure of star
.
The surface integral approach was used by Einstein, Infeld, and Hoffmann in general relativity [73].
They used the vacuum Einstein equations only, and their method can be applied to any object including a
black hole. We will take the surface integral approach in this article. But in our formalism, we shall treat
only regular objects like a neutron star.

Now let us introduce the body zone to provide the surfaces of the surface integral approach.
The scalings of and motivate us to define the body zone of star () as
and the body zone coordinates of the star as .
is a representative point of the star , e.g. the center of the mass of the star . , called
the body zone radius, is an arbitrary length scale (much smaller than the orbital separation and not identical
to the radius of the star) and constant (i.e. . With the body zone coordinates, the
star does not shrink when , while the boundary of the body zone goes to infinity (see
Figure 1).

Then it is appropriate to define the star’s characteristic quantities such as the mass, the
spin, and so on with the body zone coordinates. On the other hand the body zone serves us
with a surface , through which gravitational energy momentum flux flows, and in turn
it amounts to the gravitational force acting on star (see Figure 2). Since the body zone
boundary is far away from the surface of star , we can evaluate the gravitational
energy momentum flux over with the post-Newtonian gravitational field. In fact we shall
express our equations of motion in terms of integrals over and be able to evaluate them
explicitly.