5.3.3.4 The example of calculation of orbital trajectories in solar system
Basic parameters of the sun's gravitational field
The basic parameters of the Sun's gravitational field, namely its frame intrinsic radius and space-flow speed, not influenced by the relativistic relationships, should be calculated fro the equations (169) and (170). Using these equations we have :
(253)
(254)
where
r '_{x}
stands for the frame intrinsic radius of the Sun's gravitational field, not influenced by relativistic relationships,
v '_{m/x}
stands for the frame intrinsic space-flow speed of the Sun's gravitational field, not influenced by relativistic relationships,
stands for gravitational constant = 6,673x10-11m3kg^{-1} s^{-2 },
T '_{g}
stands for the time constant of the Sun's gravitational field, not influenced by relativistic relationships = 570,48 seconds and,
m
stands for the mass of the Sun = 1,989*10^{30} kilograms
The relativistic behaviour of the space-flow mechanism, creating gravitational field, results in a slight modification of the basic parameters, that could be neglected in common calculations, but not in the precise calculations. This "relativistic modification" is defined by means of the factor p, by the equations (237), (244) and (246). Since the factor p varies with the radius, it must be determined for each orbital zone (e.g. for the mean radius of the planet's trajectory) separately. See Tab13b-1. Tab13b-1
The initial radius at which the planet Mercury reaches the upper culminating point (Aphelion)
r_{k}=7,1740*10^{10}m
The initial Z_{cr} vertical speed (Z_{cr} ... in units of the chosen frame)
v_{ol}=0
The initial Z_{cr} orbital speed (Z_{cr} ... in units of the chosen frame)
v_{ook}=3,37220*10^{4}m/sec
Calculated parameters
Substituting for r_{x}, v_{m/x}, r_{k} and v_{ook} to equation (227), we can calculate the initial Z_{ct} orbital speed (Z_{ct} ... length units in the frame at the radius at which Mercury occurs, time units in the chosen reference frame)
v_{zok}= 3,806988*10^{4}
Substituting for r_{x}, v_{m/x}, r_{k} and v_{zok} to equation (216), we can calculate the final radius r_{f} at which planet Mercury reaches the lower culminating point (Perihelion)
(255)
Since r_{k} > r_{f}, the radius r_{k} represents aphelion (determining the greatest distance from the Sun) and, the radius r_{f} represents perihelion (determining the minimal distance from the Sun).
Applying equation (216a) we receive for the excentricity of the Mercury's trajectory :
(256)
The apparent space density.
Applying the equation (224-1) we can calculate the apparent space density in the range of radia between r_{k} and r_{f}. See Fig16.
Since the gravitational space density r_{x}/r_{lmx} in range of radia r_{k} - r_{f} reaches values 6,1680*10^{-3} - 9,5758*10^{-3} (r_{x}/r_{k} - r_{x}/r_{f}), that is smaller than the apparent space density, the condition (224b) is fulfilled. Therefore the apparent density as shown on Fig16 stays valid all over the Mercury's trajectory.
The orbital Z_{ct} speed (the speed expressed in length units of the frame of the radius at which Mercury takes place and in time units of the chosen frame)
Applying equation (210) we can calculate the Z_{ct} orbital speed v_{zo} for the range of radia r_{k} - r_{f}. See Fig16a.
The space deformation density.
Applying the equation (248), we can determine the space deformation density in range of radia r_{k} - r_{f}. See Fig16b.
The Z_{cr} orbital speed (that is, orbital speed in length and time units of a chosen frame) in a deformed spacetime structure.
The Z_{cr} orbital speed in a deformed spacetime structure is defined by the equation (249). The curve of Z_{cr}, orbital speed (v_{ood}) vs radius, in the range of radia r_{k} - r_{f}, is shown on Fig16a.
The angle speed in a deformed spacetime structure is defined by the equation (250a). The curve angle speed vs radius, in the range of radia r_{k} - r_{f}, is shown on Fig16c.
The vertical Z_{ct}v_{zl} speed (that is, the speed expressed in length units of the frame of the radius at which Mercury takes place and in time units of the chosen frame), on condition that the initial speed v_{zlk}=0, according to the equation (213), is shown on Fig16d in the range of radia r_{k} - r_{f}.
The vertical Z_{cr} speed v_{ol}
The vertical Z_{cr} speed v_{ol} (that is, orbital speed in length and time units of a chosen frame) according to the equation (229), is shown on Fig16d in the range of radia r_{k} - r_{f}.
Time interval T_{od} of motion from the aphelion r_{k} to the perihelion r_{f}
Since the integral
is very complicated, it was calculated by the numerical method : T_{od}=3,80125*10^{6} seconds
The angle _{}, the radius Sun - Mercury carries out during motion from the aphelion r_{k} to the perihelion r_{f}.
Since the integral
is very complicated, it was calculated by the numerical method :
Calculation of the parameters of Mercuru's trajectory for the motion from perihelion to aphelion
For the upward motion of the planet Mercury, from the perihelion to the aphelion, the initial Z_{cr} orbital speed v_{ook} in non deformed spacetime structure shall be calculated from the parameters of downward motion, as the speed v_{oo} according to the equation (226) at the perihelion :
The radius r_{f} from the calculation of the downward motion shall be now declared as the initial radius - r_{k}.
All other parameters as calculated fot the downward motion will become identical with those fot the upward motion, including:
- The time interval T_{ou} of the motion from the perihelion to the aphelion, T_{ou}=3,801025*10^{6} sec
- The angle _{} , the radius Sun - Mercury carries out during motion from the perihelion r_{k} to the aphelion r_{f},
Siderical period of Mercury (period of rotation)
(257)
The advance of culminating points (perihelion or aphelion advance) within one revolution
(258)
The advance of culminating points (perihelion or aphelion advance) within 100 years