## I1 The Precession of the Perihelion of Mercury

Should a single, isolated planet orbit the sun then it must do so according to Kepler and Newton in an exact ellipse. Newton already recognized that this is not the case in the solar system because the planets affect each other gravitationally. An exact solution to the ‘three-body problem’ evaded even great people like Poincaré (whose attempt at a solution deeply penetrated into the territory known today as ‘chaos theory’). Today, iterative numeric methods can calculate the orbits of all the planets with high precision for a long time into the future. The apsis, the straight line through the aphelion (furthest point from sun) and perihelion (closest point to sun) of the orbit precedes very slowly under the influence of the outer planets, in the same direction in which the planets orbit. This results in a rosette-like path, where the effect as shown in the diagram is greatly exaggerated. By the way, these numerical simulations have also shown that the solar system will remain stable even over very long periods [36-315ff]. |

There is, however, a small difference between the calculated values for the precession of the perihelion within Newtonian physics and those measured observationally. The following table shows the numerical values in the units ‘arc seconds per century’. The fuzziness of the values can be read from the column ‘difference’:

Planet | Newtonian Value | Observed Value | Difference | Prediction GTR |

Mercury mmmm | mmmm 532.08 mmmm | mmmm 575.19 mmmm | mm 43.11 ± 0.45 mm | mmmm 43.03 mmmm |

Venus | 13.2 | 21.6 | 8.4 ± 4.8 | 8.6 |

Earth | 1165 | 1170 | 5 ± 1.2 | 3.8 |

The difference between the calculated and the measured value, especially in the case of Mercury was so big that it demanded an explanation. The French astronomer Urbain Le Verrier, who predicted in 1845 the existence and location of the new planet Neptune based on the interference of the planet Uranus, postulated in 1859 the existence of another planet Vulcan, whose orbit was closer to the sun than Mercury’s.

The GTR explains precisely this difference between the Newtonian theory and observation. Einstein was overjoyed when he calculated near the end of 1915 that his new theory predicted an addition of just 43 arc seconds per century to the precession of the perihelion of Mercury! He derived the following formula:

where Δφ is the extra rotation per orbit in radians; R_{S} is the Schwarzschild radius of the sun; a is the length of the semi-major axis of the orbit; and ε is the eccentricity of the ellipse.

The effect decreases with increasing distance from the sun and is also greater with highly elliptical orbits than with circular orbits. Therefore Mercury was an ideal candidate. The small eccentricity of the orbit of Venus not only weakens the effect but also makes it difficult to observe the precession. The values in the last column of the table can be obtained from Einstein's formula, if the result is multiplied by the number of revolutions in 100 years and then converted from radians to arc seconds (See problem 1).

That this effect must occur is made nearly self-evident by Epstein’s ‘barrel’ region [15-166]:

In the first drawing space is flat and the planet moves in its ellipse, however, for Epstein in an unconventional direction (one always looks from the north to the ecliptic). That is the situation according to Newton.

Now we cut the plane along the apsis. We make the cut from the aphelion to the sun.

As discussed in section **H6** a cone should now arise with its tip in the sun. To this end we must push the edges on both sides of the incision over each other (thus crafts one a cone!). This forces an advance (precession) of the aphelion in the direction of the planet’s orbit!

Amazingly, it is even possible to determine the magnitude of the phenomenon from Epstein’s paper model. Almost with no computation we come surprisingly close to the results of the formula, whose derivation forced Einstein to tussle with elliptic integrals.

The red curve shows the cross-section of Epstein's ‘space bump’ (problem 5 in **I10** deals with the analysis of this function). The central mass sits at the origin, and with increasing distance x from the origin the spatial curvature decreases. If our planet has an average distance a from the central mass, we can then approximate the space bump with a local cone. The appropriate angle of inclination φ between the surface line of the cone and the plane through the center of the central mass can be easily determined for the planet at point a. It is cos (φ) = Δx(r, ∞) / Δx(r, r) = 1 - α / a according to **G4**. If the cone has a surface line of length 1 then the base circle measures a radius of (1 - α / a). We now cut the cone along a surface line and press it flat:

How big is the angle β of the missing sector?

β / (2π) is equal to the ratio of the 'missing' arc length to the circumference, i.e.,

β / (2π) = [2π - 2π • (1 - α / a)] / (2π) = 1 - (1 - α / a) = α / a

We thus obtain the expression for β

β= (2π) • α / a = π • 2 • α / a = π • R_{S} / a

Reform the cone and you will find either a circle or an ellipse offset by about the size of this angle β, that is, β specifies the amount of precession per orbit of the apsis.

We only obtain about one third of the correct value (compare with the formula above in this section). This should not concern us since we have only taken the influence of space curvature into account, and that also using very modest means. We are, in any case, within a correct order of magnitude.