Advancement of Mercury's perihelion

[Alan Grayson]

According to Wiki, the phenomenon is largely due to the other planets. See,  https://en.wikipedia.org/wiki/Tests_of_general_relativity . My question is this; is the advancement of the perihelion accompanied by raising the orbit of Mercury? AG

[Lawrence Crowell]

The periapsis or perihelion advance of Mercury is largely a result of classical perturbation theory in classical mechanics. About 10% of the perihelion advance could not be accounted for by perturbation methods in classical mechanics. 

This has to be admired in some ways. Finding the ephemeris of Mercury is tough, for the planet makes brief appearances near the sun in mornings and evenings. Finding an orbital path from its course across the sky is not easy. The second issue is that perturbation methods in classical mechanics are difficult. These were developed arduously in the 19th century and Le Verrier worked on this to find the planet Neptune from the perturbed motion of Uranus in 1848. These methods were worked on through the 19th century. The later work of von Zeipel and Poincare were used to compute the periapsis advance of Mercury, but there was this persistent 43arc-sec/year that resisted these efforts.

It was general relativity that predicted this anomaly in ways that are far simpler than the classical perturbation methods. This post-diction of GR was an initial success in the theory, followed up shortly by the Eddington expedition that found the optical effects of GR in a solar eclipse in 1919.

LC

[Alan Grayson]

On Sunday, August 2, 2020 at 1:51:34 PM UTC-6, Lawrence Crowell wrote:

The periapsis or perihelion advance of Mercury is largely a result of classical perturbation theory in classical mechanics. About 10% of the perihelion advance could not be accounted for by perturbation methods in classical mechanics. 

This has to be admired in some ways. Finding the ephemeris of Mercury is tough, for the planet makes brief appearances near the sun in mornings and evenings. Finding an orbital path from its course across the sky is not easy. The second issue is that perturbation methods in classical mechanics are difficult. These were developed arduously in the 19th century and Le Verrier worked on this to find the planet Neptune from the perturbed motion of Uranus in 1848. These methods were worked on through the 19th century. The later work of von Zeipel and Poincare were used to compute the periapsis advance of Mercury, but there was this persistent 43arc-sec/year that resisted these effort.

You mean 43 arc-sec/CENTURY. My question is this; why don't the perturbations due to other bodies in our solar system ALSO cause radial increases in Mercury's orbital energy, to produce an outward expansion of its orbit, rather than just rotations of the ellipse characterizing its orbit?

 TIA, AG

[Lawrence Crowell]

Yes it is per century. 

The Einstein correction will not adjust the major axis. There are changes, and I have argued with with respect to Earth and the Earth orbit has changed from .87AU to 1AU over the course of the 3.5 billion years. However, these perturbations are in a power series in the gravitational coupling G = 6.67x10^{-11}N-m^2/kg^2, which is fairly small It is also in a power series in the ratio of the orbital radius of the satellite and the larger body, say Earth or Mercury and Jupiter.v It is also to the 4th power. That is also a small number. Don't quote me on this and I would have to look it up. However, this is quite a small perturbation.

LC

[smitra]

On 03-08-2020 00:35, Alan Grayson wrote:
> On Sunday, August 2, 2020 at 1:51:34 PM UTC-6, Lawrence Crowell wrote:
>
>> The periapsis or perihelion advance of Mercury is largely a result
>> of classical perturbation theory in classical mechanics. About 10%
>> of the perihelion advance could not be accounted for by perturbation
>> methods in classical mechanics.
>>
>> This has to be admired in some ways. Finding the ephemeris of
>> Mercury is tough, for the planet makes brief appearances near the
>> sun in mornings and evenings. Finding an orbital path from its
>> course across the sky is not easy. The second issue is that
>> perturbation methods in classical mechanics are difficult. These
>> were developed arduously in the 19th century and Le Verrier worked
>> on this to find the planet Neptune from the perturbed motion of
>> Uranus in 1848. These methods were worked on through the 19th
>> century. The later work of von Zeipel and Poincare were used to
>> compute the periapsis advance of Mercury, but there was this
>> persistent 43arc-sec/year that resisted these effort.
>
> You mean 43 arc-sec/CENTURY. My question is this; why don't the
> perturbations due to other bodies in our solar system ALSO cause
> radial increases in Mercury's orbital energy, to produce an outward
> expansion of its orbit, rather than just rotations of the ellipse
> characterizing its orbit?
> TIA, AG
>

There are oscillations in the orbital parameters and if resonances occur
this can lead to instabilities. Mercury is close to getting onto a
resonance with Jupiter and due to the chaotic nature of the orbital
motion of the planets, this means that there is a chance that the inner
solar system can get destabilized:

https://www.nature.com/articles/nature08096

"Here we report numerical simulations of the evolution of the Solar
System over 5 Gyr, including contributions from the Moon and general
relativity. In a set of 2,501 orbits with initial conditions that are in
agreement with our present knowledge of the parameters of the Solar
System, we found, as in previous studies, that one per cent of the
solutions lead to a large increase in Mercury’s eccentricity—an increase
large enough to allow collisions with Venus or the Sun. More
surprisingly, in one of these high-eccentricity solutions, a subsequent
decrease in Mercury’s eccentricity induces a transfer of angular
momentum from the giant planets that destabilizes all the terrestrial
planets ∼3.34 Gyr from now, with possible collisions of Mercury, Mars or
Venus with the Earth."


"The most surprising collision is the one of Venus with the Earth,
which occurs in S468 in a five-stage process (Figs 2 and 3). The first
step is the increase in the eccentricity of Mercury, obtained through
perihelion resonance with Jupiter at 3.137 Gyr. This step is essential,
as it allows a transfer of non-circular angular momentum from the outer
planets to the terrestrial planets. The eccentricity increase
of Venus, the Earth and Mars, is then obtained through secular
resonances among the inner planets while the eccentricity of Mercury
decreases between 3.305 and 3.325 Gyr. Once Mars and the Earth
acquire large eccentricities, close encounters occur and collisions
become possible, as in S468 (Fig. 3c). In S468, the collision with
Mars does not occur, but several close encounters (Fig. 3c) lead to the
diffusion of Mars’s semi-major axis (Fig. 3b) until secular resonances
produce a decrease in the eccentricity of Mercury together with an
additional increase in the eccentricity of Venus and the Earth at about
3.347.3 Gyr (Fig. 3c). At this point, close encounters between Venus
and the Earth occur, with several exchanges of the planets’ orbits
(Fig. 3b) before a final collision at 3.352891 Gyr (Fig. 3c)."

Saibal

[Lawrence Crowell]

It is the case that planetary orbits change in time. I think the Earth is being nudged outward by Jupiter over time. This might give added time for life to continue as the sun heats up over the next billion years and more. 

LC