Essential Relativistic Celestial Mechanics

Victor Brumberg

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351449687
• Edition: 1
• Topic: Naturwissenschaften
• Subtopic: Astronomie & Astrophysik

Essential Relativistic Celestial Mechanics presents a systematic exposition of the essential questions of relativistic celestial mechanics and their relation to relativistic astrometry. The book focuses on the comparison of calculated and measurable quantities that is of paramount importance in using general relativity as a necessary framework in the discussion of high-precision observations and for the construction of accurate dynamical ephemerides. It discusses the results of the general relativistic theory of motion of celestial bodies and describes the relativistic theory of astronomical reference frames, time scales, and the reduction of observations.

1

Mathematical Tools

1.1 ELEMENTS OF NEWTONIAN CELESTIAL MECHANICS

1.1.1 Two-body problem

Newtonian celestial mechanics is based on Newton’s law of universal gravitation (theory of the Newtonian potential) and the laws of Newtonian mechanics (theory of motion). The simplest problem of Newtonian celestial mechanics is the problem of two bodies (point masses). The differential equations of motion of this problem in some fixed rectangular reference system x, y, z have the form

$\begin{array}{l}{{\displaystyle \ddot{r}}}_1=-G{M}_2\frac{r_1-r_2}{r^3}\hfill \\ {{\displaystyle \ddot{r}}}_2=-G{M}_1\frac{r_2-r_1}{r^3}\hfill \end{array}$

(1.1.1)

where M_i are the masses, r_i = (x_i,y_i,z_i) are the position vectors of the two bodies, r = |r₂ – r₁| is the mutual distance between the bodies and G is the gravitational constant. Replacing r₁ and r₂ by the position vector of the Newtonian centre of mass

$\begin{array}{cc}{r}_0=\frac{1}{M}(M_1{r}_1+M_2{r}_2)& M=M_1+M_2\end{array}$

and the relative position vector of the second body with respect to the first one

$r=r_2-r_1$

system (1.1.1) is split into two systems:

${{\displaystyle \ddot{r}}}_0=0$

(1.1.2)

and

${\displaystyle \ddot{r}}=-GM\frac{r}{r^3}.$

(1.1.3)

From (1.1.2) it follows that the centre of mass of the two-body system is in uniform rectilinear motion (a reference system is called barycentric provided that the centre of mass is at rest at the origin of this system). Equations (1.1.3) determine the relative motion of M₂ with respect to M₁.

The trajectories of the two-body problem are conic sections (straight lines in the degenerate case). In what follows only the cas...