Abstract
Motions in the two-body problem against the background of the cosmic vacuum are considered. The potential function of the problem is taken in the form U = μ/r + H2r2/2, where μ is the gravitational parameter, H is the effective Hubble constant, which represents cosmic vacuum and r is a distance from attracting centre. Precise analytical solution of the equations of motion is obtained. The formulae for calculating a moving particle's coordinates for any instance of time with given initial conditions are derived. The chain of the formulae contains a transcendent equation analogous to the Kepler's equation in the two-body problem. The formulae are expressed using incomplete elliptic integrals of the first and third kind. It is shown that finite motions take place only within a circle with a radius |$\overline{r}=(\mu /H^2)^{1/3}$| centred in an attracting body. In this case, the minimum distance cannot exceed the value of r★ = (μ/4H2)1/3. Circular motions can take place in the area of finite motions. The circular motions with the radii in the range |$(r^*,\overline{r})$| are orbitally unstable. This means that the slightest variations in the initial conditions turn the circular motion either into finite motion with a great amplitude of distance's variation or into infinite motion. The latter can exist for any value of minimum distance. Numerical integration of the equations of motion has been carried out for a number of instances. It is shown that, if the equations are solved for the rectangular coordinates, numerical integration can give erroneous solutions if the trajectory approaches the sphere with the radius |$\overline{r}$|. This is a result of the accuracy's loss in the case when two close values are subtracted. A change of variables in the equations of motion is found which gives no loss of accuracy so that numerical integration can be carried out with the same accuracy for any trajectory of motions. Comparison of analytical solution with the numerical integration is made for a number of sets of initial conditions. It gives complete coincidence of solutions at the time intervals covering several dozens of revolutions of the particle around the attracting centre.
Highlights
Nowadays, it is an obvious fact that all the bodies of the Universe are subjected to the mechanical influence of some additional force not related to the action of other bodies
Later, we shall examine the way in which the parameters r2 and V2 can be obtained on the basis of arbitrary initial conditions, when the particle, at initial moment of time, has arbitrary values of distance and velocity
We have examined the properties of motions in the two-body problem against the cosmic vacuum background
Summary
It is an obvious fact that all the bodies of the Universe are subjected to the mechanical influence of some additional force not related to the action of other bodies. The step in modelling the motions of the bodies of the Universe within the frames of the simplified model of cosmic vacuum is studying the motion of two bodies influenced by both their gravitational attraction and cosmic vacuum Such an approach has already been applied by Nandra, Lasenby & Hobson (2012) where some features of the motions were obtained using both Newtonian mechanics and general relativity theory. It is necessary to obtain the general solution of the equations of motion, i.e. the coordinates x, y of the moving particle, as explicit functions of time and arbitrary constants These functions can be given by a chain of formulae where transcendent equations are permissible analogous to the Kepler equation in the classical two-body problem. The first one is to obtain the general analytical solution depending on four arbitrary constants This method provides formulae for direct calculations of a moving particle’s coordinates for any instance of time if initial conditions are given.
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