I work out the Newtonian and general relativistic effects due to an
isotropic mass loss

In this paper I investigate the orbital effects induced by an isotropic variation

Many treatments of the mass loss-driven orbital dynamics in the framework of the Newtonian mechanics, based on different approaches and laws of variation of the central body's mass, can be found in literature; see, for example, [

The plan of the paper is as follows. Section

By defining

By limiting ourselves to realistic astronomical scenarios like our solar system, it is quite realistic to assume that

The unperturbed Keplerian ellipse at epoch

The Gauss equation for the variation of the semimajor axis

The Gauss equation for the variation of the eccentricity is [

As a consequence of the found variations of the osculating semimajor axis and the eccentricity, the osculating orbital angular momentum per unit mass, defined by

The osculating total energy

Moreover, the osculating Keplerian period

The Gauss equation for the variation of the pericentre

The Gauss equation for the mean anomaly

At first sight, the results obtained here may be rather confusing: if the gravitational attraction of the Sun reduces in time because of its mass loss, the orbits of the planets should expand (see the trajectory plotted in Figure

Black continuous line represents the true trajectory obtained by numerically integrating with MATHEMATICA the perturbed equations of motion in Cartesian coordinates over 2 years; the disturbing acceleration (

In fact, there is no contradiction, and my analytical results do yield us realistic information on the true evolution of the planetary motion. Indeed,

Now, if I compute the radial change

Difference

Since Figure

Radial and transverse perturbations

Let us now analytically compute it. According to [

From a vectorial point of view, the radial and transverse perturbations to the Keplerian radius vector

Since the approximate calculations of other researchers often refer to circular orbits, and in view of the fact that when a Sun-like star evolves into a giant tidal interactions circularize the orbit of a planet [

Black continuous liner represents true trajectory obtained by numerically integrating with MATHEMATICA the perturbed equations of motion in Cartesian coordinates over 2 years; the disturbing acceleration (

Difference

The field equations of general relativity are nonlinear, but in the slow-motion

Black continuous line represents true trajectory obtained by numerically integrating over 3 years the equations of motion perturbed by (

Here I will briefly review some of the results obtained by others by comparing with ours.

Hadjidemetriou in [

Black continuous line represents true trajectory obtained by numerically integrating with MATHEMATICA the perturbed equations of motion in Cartesian coordinates over 2 years; the disturbing acceleration (

Schröder and Smith in [

Noerdlinger in [

Krasinski and Brumberg in [

Concerning the observationally determined increase of the Astronomical Unit, more recent estimates from processing of huge planetary data sets by Pitjeva point towards a rate of the order of

I started in the framework of the two-body Newtonian dynamics by using a radial perturbing acceleration linear in time and straightforwardly treated it with the standard Gaussian scheme. I found that the osculating semimajor axis

As a suggestion to other researchers, it would be very important to complement my analytical two-body calculation by performing simultaneous long-term numerical integrations of the equations of motion of all the major bodies of the solar system by including a mass-loss term in the dynamical force models as well to see if the N-body interactions in presence of such an effect may substantially change the picture outlined here. It would be important especially in the RGB phase in which the inner regions of the solar system should dramatically change.

The authors thanks Professor K.V. Kholshevnikov, St. Petersburg State University, for useful comments and references.