Surface integral approach and body zone

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 $F i 1$ on star $1$ becomes

The integral region $B1$ covers star $1$ but does not cover the star $2$ (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,

&#x222E; F i= &#x2212; dS tij, &#x0028;43 &#x0029; 1 &#x2202;B1 j &#x0028; ij &#x0029; tij &#x2261; -1- &#x2202;-&#x03A6;-&#x2202;&#x03A6;--&#x2212; &#x03B4;---&#x2202;&#x03A6;--&#x2202;&#x03A6;- . &#x0028;44 &#x0029; 4&#x03C0; &#x2202;xi &#x2202;xj 2 &#x2202;xk &#x2202;xk

Note that the sphere $∂B1$ has no intersection, neither with star $1$ nor star $2$ . Thus, we can evaluate the gravitational force acting on star $1$ without knowledge about the internal structure of star $1$ .

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 $R$ and $m$ motivate us to define the body zone of star $A$ ( $A = 1, 2$ ) as $BA ≡ {xi||⃗x − ⃗zA(τ)| < εRA }$ and the body zone coordinates of the star $A$ as $i αA ≡ ε−2(xi − ziA(τ))$ . $ziA (τ )$ is a representative point of the star $A$ , e.g. the center of the mass of the star $A$ . $RA$ , 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. $dRA ∕dτ = 0 )$ . With the body zone coordinates, the star does not shrink when $ε → 0$ , while the boundary of the body zone goes to infinity (see Figure 1 ).

Figure 1:

Body zone coordinates and near zone coordinates. In the near zone coordinates $(τ,xi)$ , both the body zone and the star shrink as $ε$ (thin dotted arrow) and $ε2$ (thick dotted arrow) respectively. Both the thin and thick arrows point inside. In the body zone coordinates $(τ,αi-) A$ , the star does not shrink while the body zone boundary goes to infinity as $−1 ε$ (thin dotted line pointing outside).

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 $∂BA$ , through which gravitational energy momentum flux flows, and in turn it amounts to the gravitational force acting on star $A$ (see Figure 2 ). Since the body zone boundary $∂BA$ is far away from the surface of star $A$ , we can evaluate the gravitational energy momentum flux over $∂BA$ with the post-Newtonian gravitational field. In fact we shall express our equations of motion in terms of integrals over $∂BA$ and be able to evaluate them explicitly.

Figure 2:

Gravitational energy momentum flux through the body zone boundary. The meshed two circles represent stars $1$ and $2$ . Each star is surrounded by the body zone represented here by a striped area. The arrows around star $1$ represent the gravitational energy momentum flux flowing through the body zone boundary.