Someday I should continue and finish this series. I never got too far into the promised land of 4d polytopes.

]]>• J. Laskar, A. Fienga1, M. Gastinea, and H. Manche, La2010: A new orbital solution for the long-term motion of the Earth, *Astronomy and Astrophysics* **532** (2011), A89.

We present here a new solution for the astronomical computation of the orbital motion of the Earth spanning from 0 to −250 Myr. The main improvement with respect to our previous numerical solution La2004 is an improved adjustment of the parameters and initial conditions through a fit over 1 Myr to a special version of the highly accurate numerical ephemeris INPOP08 (Intégration Numérique Planétaire de l’Observatoire de Paris). The precession equations have also been entirely revised and are no longer averaged over the orbital motion of the Earth and Moon. This new orbital solution is now valid over more than 50 Myr in the past or into the future with proper phases of the eccentricity variations. Owing to the chaotic behavior, the precision of the solution decreases rapidly beyond this time span, and we discuss the behavior of various solutions beyond 50 Myr. For paleoclimate calibrations, we provide several different solutions that are all compatible with the most precise planetary ephemeris. We have thus reached the time where geological data are now required to discriminate between planetary orbital solutions beyond 50 Myr.

Luckily, 50 million years is enough to study the onset of the glacial cycles we’re experiencing now.

]]>where the variables are defined at http://casa.colorado.edu/~ajsh/schwp.html. The reason for this change is to have a metric that fully adheres to GR’s equivalence principle (EP) and is otherwise preferred by Occam’s razor.

Be in free fall, initially straddling the horizon of a black hole (massive enough that you don’t detect a tidal force), beside a freely falling stone that’s above the horizon and escaping to infinity. GR predicts that the stone is passing you in the outward direction, away from the black hole. If GR adhered to its EP I’d be able to add the condition “Let the stone be passing you in the inward direction”, or even “Let the stone completely pass you in the inward direction at a speed close to c as you measure”, for in some other locally inertial frame that freedom of relative movement is possible. Which is to say, the laws of physics in your local frame differ from those in some other locally inertial frame. Note I didn’t specify that you’re falling through the horizon.

In the river model of black holes (http://jila.colorado.edu/~ajsh/insidebh/waterfall.html) an analogy is made that a fish is inexorably swept to the bottom of a waterfall when the water falls faster than the fish can swim through it. The problem with GR working analogously is that there’s no EP for rivers. The page says “[In] general relativity, space itself can do whatever it likes.” A theory of gravity that postulates the EP is thereby constrained on what space itself can do.

A solution is to limit escape velocity to less than c everywhere. The escape velocity embedded in the metric above is:

No test that I know of rules out the equations above. For example the new metric still predicts 42.98 arcseconds per century for Mercury’s anomalous perihelion shift. The problems around black holes vanish, including incompatibility between GR and QM regarding singularities, which for the new metric can be a limiting case.

If you allow it, later I’ll suggest how the new metric can solve the mystery behind dark matter, without making new assumptions. I apologize if my comments are out of line here.

]]>You can have constant energy E or constant T but not both (as fas as I can see)

… I thought it was worth trying to set the record straight on the same page.

If you fix the masses of two bodies orbiting under their mutual gravitational attraction, then the total energy (kinetic plus potential) , the semi-major axis of their orbit, , and the period of their orbit, , are all determined by specifying just one of these three quantities. For example, in terms of , we have:

where and are the total mass and reduced mass of the two bodies, and for something like the Earth-Sun system are very close to being just the mass of the Sun and the mass of the Earth.

Contrary to the assumptions in the question, you can change the angular momentum of the system while holding the energy, semi-major axis and period of the orbit constant. We have:

where is the eccentricity.

So changes in angular momentum in an orbit of fixed energy (and hence fixed period and fixed semi-major axis) just correspond to changes in eccentricity. The period *doesn’t* need to change in order for the angular momentum to change; rather, the change in angular momentum arises from the change in the *shape* of the orbit.