which mode would it be?
Has Sun ever been observed to go pear-shaped?
If no, what is the lowest mode of Sun to have ever been confirmed to occur?
which mode would it be?
Has Sun ever been observed to go pear-shaped?
If no, what is the lowest mode of Sun to have ever been confirmed to occur?
do you mean shaped like a pear?('pear shaped' in the UK can mean going wrong)
I don't see how it could become shaped like a pear.
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The Earth's shaped like a pear, to a small extent - that is, it has a non-zero J3 gravitational zonal harmonic. I presume chornedsnorkack is asking about a vibration mode that would make the sun bulge alternately in its northern and southern hemispheres.
Grant Hutchison
Established Member
Yes, thanks. So the designation is J3. Bulge out at the stalk pole, bulge in on the stalk hemisphere, bulge out on the hemisphere opposite to stalk, bulge in again at the pole opposite to stalk. Literally going pear shaped.
Note how egg shaped is also pear shaped. Not actually concave bulge in around sharp end, but bulge out at the stalk pole forms the sharp end, etc....
That's interesting! Pear-shaped is actually one step more complex
than egg-shaped. Is there a continuum of shapes from spherical to
ellipsoidal to egg-shaped to pear-shaped to something one step more
complex than pear-shaped, and then to one more step beyond that,
and so forth? Do the number of different possible shapes increase
with each step of increase in complexity? An ellipsoid can be either
prolate or oblate. Is apple-shaped the same degree of complexity
as pear-shaped? Does apple shape have a conventionalized
description or definition like pear shape does? Are they actually
the same shape? Or is apple shape more complex because it is
indented at both poles? Are there other possible shapes at that
level, fruity or otherwise?
What branch of mathematics would describe this progression?
A branch of analytical geometry?
-- Jeff, in Minneapolis
But what kills this is:
What shape would you call a pear that was shaped exactly like the Earth?
You would call it a near perfect sphere. Because the human eye could never distinguish the misshapenness in an object 3.5 inches across.
Time wasted having fun is not time wasted - Lennon
(John, not the other one.)
I'm not sure that kills it. The Earth is pear-shaped in a detectable way, because we can pick up the effect of the "pear shaped" component of the gravitational field on orbiting satellites.
Grant Hutchison
For some reason I'm imagining the sun, or the earth, as wobbling in and out of this pear shape. Could the gravitation field fluctuations seen by earths satellites be attributed to earths wobble rather than an actual shape change?
I know that I know nothing, so I question everything. - Socrates/Descartes
is this question related to the news I heard recently that our model of the sun might be wrong, missing mass at the core or a bunch of dark matter? It was in this week's New Scientist. There were questions raised about whether our standard candle is not what we thought it was.
sicut vis videre esto
When we realize that patterns don't exist in the universe, they are a template that we hold to the universe to make sense of it, it all makes a lot more sense.
Originally Posted by Ken G
True!
The pear-shape is a degree 3 order 0 spherical harmonic. Spherical harmonics are used to model three-dimensional surfaces in a way similar to how Fourier series model functions. Just as each set of two Fourier functions of a given degree can be combined into a single function, with an offset (or phase shift), so can spherical harmonics. There are six other degree 3 harmonics (2n+1), which were not detectable in the early sixties when the pear-shapedness was first announced. A few of them are greater than the pear-shape! If you combine all of them, they almost completely resolve into a single degree 3 order 2 shape that is tilted with respect to the earth's axis. That shape would be described as a rounded off tetrahedron.
ETA: AGU Fall 1992 https://forum.cosmoquest.org/showthr...024#post699024
Last edited by grapes; 2017-Oct-24 at 02:39 PM. Reason: Reference
I would say, just "smooth" out the indentation of the neck of the pear. That makes less variation in the surface--but when it comes to describing it in spherical harmonics, it probably means that it has infinite components.
Kinda like the simple saw-tooth function, easy to describe, but it has an infinite series of fourier functions to represent it--and of course you get the furious ringing at the nodes if you truncate the series.
What I would say is that an egg is a sum of a lemon and a pear.
My bold. The next one my dad mentioned back in the 1960s and early '70s was the slight elongation of the Earth's cross section shape at the equator. That was just the start. He needed something like 17 such terms to get good ephemerides empirically for U.S. Navy beacon satellites. He was doing this as a mathematical adjunct to his primary work on the electronics aboard the satellites and in the surveillance radar stations on the ground. I would venture a guess that GPS needs even more to get the current hairsplitting accuracy that was beyond anything being done 40 to 50 years ago.
The geoid is generally modeled with higher and higher degrees of spherical harmonics.
https://en.m.wikipedia.org/wiki/Geoi...representation
That article there says that the current best model is EGM96 (Earth Gravity Model 1996) with 130317 components. That's (360+1)^{2}-4. The "missing" four are the spherical harmonics that represent the radius of the earth (degree 0), and the three components of degree 1 which represent an offset from the origin and are identically zero if the origin is the center of figure (which is the center of mass for an equipotential figure).
With degree 4, there would be (4+1)^{2}-4 such components, or 21. The two degree 2 order 1 components are dynamic and almost zero, they result in the Chandler wobble. That would leave 19 components, the elongation of the equator are represented by the two degree 2 order 2 terms. Maybe that's the point your dad was at.
You mean the equatorial bulge which makes Earth approximate an
oblate spheroid? (Or oblate ellipsoid.) Calling it an "elongation" sounds
like you mean a prolate spheroid (or prolate ellipsoid), which I doubt is
what you intended. Earth's equatorial diameter is about 43 km greater
than its polar diameter, while the pear-shaped bulges are only on the
order of ten or twenty meters.
I suppose that the Earth must also be very slightly prolate in the
directions of the Moon and the Sun, roughly across the equator.
I wonder if the GPS uses a numerical description of the Earth rather
than an analytical description.
One thing I have never understood: Earth's south pole is supposed
to be depressed as part of the pear-shapedness, yet Antarctica is
said to be a high plateau. How can both be true? The sticking up
of the continental landmass would be much larger than the 10 or
20 meter depression of pear-shapedness.
-- Jeff, in Minneapolis
My bold. That is not what I meant. In a perfect oblate ellipsoid or in one with only the pear-shaped deviation, the cross section in the plane of the equator would be perfectly circular. It actually is out of round by some tens of meters, far more than the tidal stretching, and is pretty much frozen like the pear shape. If I remember correctly, this term is roughly triangular, and it creates sweet spots in longitude for geostationary satellites. They need less station-keeping fuel than they would at other locations.
The depression in the south polar region is that of the geoid, which is the calculated surface at which sea level would be in the absence of the polar ice cap. Of course the continents stick up by amounts that are large compared with the departures of the geoid from a perfect oblate ellipsoid, and this complicates the calculation of the geoid beyond my mathematical capability. If we could drill a deep well at the South Pole and connect it to the ocean with a suitable aqueduct, the water level in the well would settle down at the depressed geoid elevation.
I'm pretty sure that what the elongation refers to the difference in diameters along the equator itself. There is a net "pinching" of the equator, which results in a tendency of geostationary satellites to drift towards two diametrically opposite points. See ETA below.
Continental topography is compensated by its roots (isostasy). The geoid represents the shape of the earth's gravity field, it is an equipotential surface that cuts through the continental land masses--it is more-or-less "sea level". The high plateau is relative to that depressed part of the pear-shape (although, as I said, there is no pear shape per se).Earth's equatorial diameter is about 43 km greater
than its polar diameter, while the pear-shaped bulges are only on the
order of ten or twenty meters.
I suppose that the Earth must also be very slightly prolate in the
directions of the Moon and the Sun, roughly across the equator.
I wonder if the GPS uses a numerical description of the Earth rather
than an analytical description.
One thing I have never understood: Earth's south pole is supposed
to be depressed as part of the pear-shapedness, yet Antarctica is
said to be a high plateau. How can both be true? The sticking up
of the continental landmass would be much larger than the 10 or
20 meter depression of pear-shapedness.
ETA: https://en.wikipedia.org/wiki/Geosta...ital_stability
A second effect to be taken into account is the longitude drift, caused by the asymmetry of the Earth – the equator is slightly elliptical. There are two stable (at 75.3°E and 252°E) and two unstable (at 165.3°E and 14.7°W) equilibrium points. Any geostationary object placed between the equilibrium points would (without any action) be slowly accelerated towards the stable equilibrium position, causing a periodic longitude variation.[8] The correction of this effect requires station-keeping maneuvers with a maximal delta-v of about 2 m/s per year, depending on the desired longitude.
Last edited by grapes; 2017-Oct-26 at 12:31 PM. Reason: ETA
Perhaps I was mistaken in attributing a triangular component to the equator. I may have confused it with something else.
I envision the continents as chunks of lower density rock floating in the mantle sort of like icebergs floating in sea water. If we replace them with equal masses of denser material, with the tops a couple of miles below sea level, the geoid will be nearly unchanged, if I am not mistaken.
It may not be exact...I just noticed that the listed unstable points are 180 degrees apart, but the stable points are 176.7 degrees apart.
The degree 2 order 2 field do have points 180 degrees apart, but those could be modified by higher degree components.
ETA: The degree 3 order 3 field would add a "triangular" component to the shape of the equator, but the effect of the fields at the satellite decreases as the inverse nth power of radius, so the geosynchronous satellites would be even less affected than satellites in low earth orbit.
Last edited by grapes; 2017-Oct-26 at 03:06 PM. Reason: higher components
Here is what purports to be a listing of the EGM96 coefficients: ftp://cddis.gsfc.nasa.gov/pub/egm96/...96_to360.ascii
The second one of each of the degree-n, order-0 coefficients is identically zero.
Notice that the degree 2 order 0 coefficient is a few orders of magnitude greater than any other, that's the equatorial centrifictional bulge (although it does show up a little in the degree 4 order 0 coefficient). The degree 2 order 1 coefficients (Chandler wobble) are much smaller, by several magnitudes, than any of the other low degree coefficients, essentially zero.