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### 5.3.4 Mass object in gravitationald field of another object

#### 5.3.4.1 Interface area

If a mass object orbits around cosmic object, or anyhow is situated in its gravitational field, it has its own open gravitational field, if its specific mass is high enough, to create the gravitational space density on its surface higher than the gravitational space density of a respective position in a gravitational field of a dominant gravitational object. If its specific mass is not high enough, it is without its own open gravitational field, despite its individual parts, like atoms and nucleons, have always their own open gravitational fields (because of their very high specific mass).
Note : To explain the fact that an object does not have its own open gravitational field, we must remind that the gravitational effects of such object are manifested by means of the hidden gravitational field (that is, the gravitational field of the respective object in a space of the density defined by the dominant object). The matter will be dealt with later.
The boundary between gravitational field of the object Ml and the superior gravitational field of the object M is defined as the interface area. See Fig18.
Fig18

Fig18 shows that the open part of the gravitational field, if any, of the mass object in a gravitational field of a dominant gravitational object is enclosed in an egg-shaped interface area, defined by:
Dl
lower interface limit
Du
upper interface limit
Da
central interface limit
Applying equations (107) and (170), we can derive conditions of the equality of a gravitational space density:
for lower interface limit
for upper interface limit
for central interface limit
where
M (M1)
stands for mass of a superior (inferior) object
rxm (rxm1)
stands for radius according to equation (169) of an object M (M1)
Rorl
stands for radius rlmx of an orbit around object M, at which object M1 is located.
Substituting for rx and vm/x from equations (169) and (170) into the equation (114), we can easily derive :
(501)

It means, the expression vzx/m represents the Zct (contemporary) speed of the spacetime structure, when the relativistic effect is ignored.Therefore, except of case when the mass of the gravitational object approaches zero, or infinity, we may write :
(502)

Applying the equation (502) and the derived conditions of equality of the gravitational space density, we can determine the interface limits :
(503)

(504)

(505)

#### 5.3.4.2 Gravitational acceleration due to exterior objects.

The gravitational interaction originated by the exterior object M1 (M2) has to be added to the gravitational interaction of the examinated object M2 (M1), to calculate the resultant gravitational interaction in the gravitational field of the object M2 (M1).

In case, the object M2 is situated inside the gravitational field of the dominant object M1, we can find out the contribution of the gravitational field of the object M1 to the acceleration caused by the gravitational object M2, according to equations derived in chapter 5.3.2.2. However this calculation is based on condition that the space density of the respective point in the gravitational field M2 is defined by gravitational field of the object M1 (not M2). Since the space density due to gravitational field of the object M2 becomes higher now than the space density that is originated by the object M1, the gravitational field of the object M1 becomes hidden in this zone. Therefore the acceleration due to gravitational field of the object M1 must be re - calculated on space density of the gravitational field of the object M2.

Let us imagine the two following events :
1. An object or a spacetime structure itself moves in a background of the spacetime density (consequently, in a time density ) at the frame intrinsic speed v1.
2. The same object or a spacetime structure itself passes from the background of the space density to the space density , without being affected by gravitational (or any other) interaction.
Under these circumstances we must admit that if an object, or a spacetime structure itself in a background of the spacetime density passes distance l1 frame intrinsic meters within t1 frame intrinsic seconds, it will pass

in the background of the space density .

Therefore the inertial passage from the background of the space density to the background of the space density brings the change of the frame intrinsic speed v1 to v2:
(506)

Examining gravitational effects of the dominant object M1 (of the mass M1) in point P, situated in a gravitational field of the inferior object M2 (of the mass M2) (see Fig19), we can derive :
Fig 19

(506a)

(506b)

Applying the equation (169) for rx1 and rx2, we receive :
(506c)

and, substituting from (506c) to (242), we have:
(507)

Examining the speed of the spacetime structure in point P, it must be :
(507a)

where
vx/m1
stands for the frame intrinsic speed, in direction P-M1, of the spacetime structure of the gravitational field of the object M1 ,in case, if the point P was situated in the space density g1. This speed is defined by equation (170).
vx/m1-2
stands for the frame intrinsic speed, in direction P-M1, of the spacetime structure of the gravitational field of the object M2 (due to gravitational field of the object M1), that would have to be added (in vector superposition) to speed vm/x2, (representing the frame intrinsic speed in direction P-M2, caused by gravitational field of the object M2), if we wanted to know the total frame intrinsic speed of the spacetime structure in point P,
rlmx1
stands for the distance P-M1 in length units of the chosen frame, and,
rlmx2
stands for the distance P-M2 in length units of the chosen frame.
Thus, respecting the equation (108), it must be :
(508)

where vzx/m1-2

We may write, respecting also the equations (169), (170), and (242a) :

(509)
The acceleration of the space flow, manifested by the speed vzx/m1-2 in gravitational field of the object M2 can be expressed as
(509b)

Since
substituting from equations (510) and (511) to equation (509), we receive :
(512)

The gravitational acceleration gzx/m1-2 represents the Zct gravitational acceleration in gravitational field of the object M2, due to gravitational field of the object M1, that has to be added to the Zct gravitational acceleration in gravitational field M2, in vector superposition, to calculate the total Zct acceleration. The negative sign in equation (512) define the direction P-M1 of the acceleration.

In case when rlmx2 / rx2 = rlmx1 / rx1, occuring in all points of the interface area between gravitational fields M1 andM2, the gravitational acceleration reaches
(513)
According to equations (170) and (166) :

(514)

Applying rx1/rx2=vm/x1/vm/x2= rlmx1/rlmx2, we can derive for the ratio of the gravitational acceleration induced from object M1 (in direction P - M1) to gravitational acceleration caused by object M2 (in direction P - M2) in points of the interface area (like the ratio of the components of the gravitational acceleration on from Sun and Earth on the interface area around Earth):
(514a)
The equation (251) is identical with the equation (182), confirming that the Zct acceleration gzx/m1-2 becomes identical with acceleration gzx/m1 of the gravitational field M1 in all points of interface area.

In case when rlmx2 = rx2, occurring, for instance on Earth surface ( if the reference frame on Earth surface was taken as the chosen one), the equation (512) becomes
(515)

and, applying the equation (138), we can derive for the ratio of the gravitational acceleration induced from object M1 (in direction P - M1) to gravitational acceleration caused by object M2 (in direction P - M2) in the chosen reference frame (like the the ratio of the components of the gravitational acceleration from Sun and Earth on the Earth surface):
(515a)
This result is in compliance with the Newton's theory of universal gravitation, confirming that the gravitational effects of the Sun's gravitational field on Earth surface, calculated by spaceflow theory, are in correspondence with experiment. The contribution of the gravitational acceleration of the Sun to the gravitational acceleration of the Earth according to the equation (249) however, differs from the acceleration expected by Newton's theory along with the height above Earth.

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