Here’s a list of Earth’s perihelia and aphelia through 2100. As you can see, the time these happen moves around within about a 4-day period, sometimes changing 2 days from year to year. This is not the kind of variation that the solstices and equinoxes go through. So what is it? It has to be something that can be calculated but is apparently irregular. I’m having trouble thinking of what that could be.
Gravity is the issue; as your linked article says:
BTW a difference of 29,000 km is down in the noise. You might also look up Milankovitch Cycles for a longer-term perspective.
Note that a there are ~365.4 days in an Earth orbit. So the timing of such events gets ~6 hours later each year until you hit a leap year which takes off a day by itself.
This can best be seen by the solstice/equinox chart from here.
But that doesn’t include all the effects on perihelion/aphelion:
The Earth and Moon do a little dance and the difference in orbital distance is subtle, esp. at those points. So depending on where the Earth is in regards to it’s mutual orbit with the Moon, this can effect the timings of those as well.
A small point but you’ve hit on one of the basic problems: what, exactly, is a year?
Strictly speaking, anything we can calculate a hundred years into the future is not “irregular” … just saying …
Irregular is not the same as random, which is not the same as unpredictable.
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I agree irregular doesn’t always mean random, but I disagree about something irregular can be predicted … I give you an irregular hexagon with one side measuring 1 meter, can we predict the lengths of the other five sides? … certainly not with any accuracy …
How can we predict the exact shape of the Mariod without measuring it?
Irregular means “not conforming to rules or expectations” … if there are no rules or expectations for predicting the time of the apsides, then how can we publish table listing these times so far into the future? … clockworks look confusing and complicated, but they still operates like clockworks …
In this case, “irregular” means not periodic. It is still predictable, via numerical integration and other analytic techniques. The “irregular” motion includes some regular components like apsidal precession, but the shorter-term effects are harder to describe.
The classical irregular orbit is the Moon’s orbit itself. No one is arguing it is unpredictable or chaotic, and indeed positions are tabulated in ephemerides for use by sailors and astronomers, but anyone who thinks it’s so easy is invited to reproduce those calculations.
The moon’s position is the biggest effect causing the variation described in the OP; it won’t be in the same place on the same date each year.
Consider the digits of pi. These are irregular (no pattern) but completely predictable.
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Well, according to the idjit you quoted it’s 365.4 days long. :smack: (Thought “and a quarter” and typed a “4”.)
Yeah, anomalistic year applies to the OP’s question. But it’s close enough to sidereal and tropical (esp. the former) years to account for the majority of effects the OP noticed.
A general order of the largest effects here, from most to least as far as impact.
tidal perturbations => General Relativity => rotational perturbations => rotational * => tidal *
Ignoring GR concepts, people tend to get to get tides wrong, so here is the only description that I personally know of that is accessible and mostly correct.
It is important to realize that Floating Point is finite math, and has representation errors. Because floating point units have lagged the increase in precision that integer units have gained for most of the previous 30 years the costs of even JPL increasing precision is a limiting factor.
JPL moved to double precision via arbitrary precision library when they upgraded from a Univac to the Alpha based systems for some portions and then actually had to reduce precision when moving to the UltraSparc due to issues with accumulated errors.
While not perfect, given enough budget for compute time we can model the interactions to a much higher degree than most people believe.
DE430 is a high precision release that covers planetary and lunar ephemeris from Dec 21, 1549 to Jan 25, 2650
DE431 is slightly less precise Aug 15, -13200 to March 15, 17191.
While FPU’s are actually tending towards reduced precision for speed for machine learning, it may be possible to reduce the costs of the higher precision models with CPUs like Power 9 that have 128 bit decimal hardware acceleration that removes the rounding and representation errors we run into with binary FPUs. Machine epsilon, and rounding block the full use of the entire range of FP numbers BTW. Or in other terms, for these problems you have to stick to the range where FPUs don’t count by twos.
While there aren’t analytical solutions yet for the problem, the numerical models are accurate for quite a long while if you can afford the compute time. The above mentioned FPU’s are currently the limiting factor on needs like calculating problems like the Sun–Jupiter Lagrangian points. The FP rounding errors impact the calculations of the relativistic terms so much that there is no point using PN values and one just deals with the lack of precision of the pure Newtonian model as an example.
I should also add in, that some of this oddness is related to treating gravity and the Coriolis term as non-fictional forces (forces that arise to to a reference frame). While there is no analytical solution in GR, if you take the individual terms and solve for geodesics in a model that doesn’t require those fictional forces one at a time they become more predictable.
Unfortunately this is hard to capture in words and I am unable to visualize 4D, but if you want to take on the process of learning this is the book you want to eventually work through.
You still have to simplify but the geometry of spacetime is clearly the primary cause.
Solar oblateness, the interaction other planets, gravitoelectric effects, and the fact that rotating planets actually warp spacetime by spinning (frame dragging) are quite beautiful and while the math is quite scary symbolic math tools make it far more accessible then in the past.
Because the math is terrifying, non-linear and to be honest unnecessary for practical needs tools like the Post-Newtonian expansions are added as corrective terms to the classic theories.
Noting I am a fan of PBS’s Space Time’s intent to be accurate yet accessible; and with the reality that most people will not take years to learn the math; This play list is a good resource to start to understand the implications of GR and curved spacetime.
As the PDF for DE410 is not in typical text form and thus unsearchable and this is GQ, here is a cite showing that the move to the Sun UltraSPARC caused issues. This demonstrates how just changing platforms can induce errors due to the finite nature of floating point mathematics.
ftp://ssd.jpl.nasa.gov/pub/eph/planets/ioms/de410.iom.pdf
When they moved to quad precision for the Newtonian part, yet staying with double on the GR part it took 16 minutes per century. As no shipping UltraSPARC had quad precision in hardware (but it is in the spec) the impact is large. As demonstrated where they call out that a pure double precision run takes 35 seconds per century.
ftp://ssd.jpl.nasa.gov/pub/eph/planets/ioms/de414.iom.pdf
The ~30 times longer run time cost is due to arbitrary precision arithmetic.
The GR numbers tend to be small, and matrix math and complex numbers are harder to implement. So most likely the costs of doing those in quad precision was judged too expensive for the added precision.
As an add-on question, for example the distance Earth to Sun on 02 Jan 2005 is 0.9832968 AU. What are the two points this distance is between? I assume one is the center of Earth but where is the other?
For any natural “Body” E.G. the sun, planet, satellite, comet, asteroid; The center of a body-fixed frame is the center of the body.
The distance will be related to the distance measured from the inertial frame which is always the solar system barycenter.
Note that the center of the sun, and the center of the solar system barycenter are not the same thing, although it is possible for them to be the same location.
Here is a usable overview of the ICRF Inertial Frame
I did specify the strictest case …
I believe this is periodic, each of the parameters themselves are periodic so the sum will also be periodic … with a very very long period such that this 100 years is a tiny chunk … it “looks” irregular when in fact it is not …
This page shows a graph of the sum of the biorhythm functions and makes this comment
Oscillations within oscillations within oscillations … if each is regular, then the sum will be regular … the same applies to orbits …
If you aren’t scared away by tensors, the following document covers a lot of the effects.
ftp://ssd.jpl.nasa.gov/pub/eph/planets/ioms/ExplSupplChap8.pdf
Note that zonal spherical harmonics are basically Fourier transforms for a sphere.
Here is the first six terms of the Fourier series of a square wave such as in sound; an ideal square wave can be thought of as the sum of all odd-integer harmonic frequency sine waves for a fundamental frequency.
And from the first reference you will see that for the moon they consider:
While not directly related to the perturbations, this short video will show just how interesting these harmonics can be. But the take away is that the previous linked graphs are more opaque partly due to the reduced dimensions of the graph.
Here is another video, using sound and an Oscilloscope to draw shapes which may help with the concept.
Jerobeam Fenderson demonstrates how adding just a few sine waves results in complex shapes.
Not everything is periodic; there are secular effects as well.
I think a reasonable answer to the OP is: 1. The date of apsis increases by about 1.7 days per century. 2. In the short term, the apsides of the Earth-Moon barycentre are pretty regular. 3. The remaining variation principally depends on the phase of the Moon at the time the barycentre reaches perihelion/aphelion; e.g. a first quarter will make the perihelion happen at a later date, and a last quarter will make perihelion happen at an earlier date.
I think it is still an open question, but aren’t most of the changes generally attributed the secular acceleration of the Moon and secular slowing down of the Earth’s rotation to tidal friction?
The Milankovitch cycles mentioned earlier and the precession of the equinoxes seems to be mostly related to the precession of the Earth’s axis over time.
But past that and the influences of Mercury to Neptune gets into the Post-Keplerian parameters which is past my current understanding by a mile and fortunately small enough to ignore.
VSOP87 claims that the Earth-Moon barycenter max drift should be below 1" for the next ~3980.
JPL ephemerides are better, but .612 arc-seconds for the next ~2980 years seems pretty good.
As VSOP87C has about a thousand terms in each series, so I am sure I am missing something. If you can help feed my most likely unhealthy interest in ephemerides I would appropriate any tips on where to look. Obviously this isn’t an analytical solution and to avoid wasting others time I am not claiming this is a solved problem. I am just curious about secular effects I have missed.