Why Is There More Mass But Less Gravity At The Equator?

Do you even know Newton's equation of gravitation?

 

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No, it isn't. If you scale the Earth larger you also increase its total mass, so you would expect surface gravity to go up. But the deformation of the Earth into an oblate spheroid does not increase ts total mass, it only changes its shape. Since the force of gravity you feel is due to the sum of all parts of the Earth pulling on you (not just that part between you and the center, and sitting at the equator puts you further from the main bulk of the Earth that provides that pull, you feel a reduction in gravity.

 

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This has been a trollable question since at least 2005.

http://www.physicsforums.com/showthread.php?t=59484
http://www.physicsforums.com/showthread.php?t=131329

1) The Earth is not homogeneous, as anyone who owns a gold mine will tell you.
2) The Earth is not spherically symmetric. It's lumpy. Not as lumpy as the moon.
3) Therefore the exact treatment of the gravitational force of the Earth as that of a point mass is not expected to hold. Therefore, for satellite applications the Earth's gravity is approximated by spherical harmonics and for local applications where exactness is required, locally measured values are used. A one-pound weight may not weigh one pound due to local conditions but is sure to weigh the same as another one-pound weight at the same location, because it is actually a mass standard in disguise. (The SI has no such problem since formally we would say our weight is a 0.4536 kg mass and measure a weight of 4.45 newtons in one place and 4.44 newtons in another. But when you use only balance scales the distinction between mass and weight is harder to see.)

4) The surface of the Earth rotates in almost rigid fashion (albeit slower than it did billions of years ago).
5) Therefore any measurement of acceleration or weight done to a reference to the surface of the Earth. is going to conflate gravity and centrifugal acceleration. This is Newton's version of the equivalence principle.

6) The surface of the Earth is largely the surface of a fluid (oceans) and where it is solid, that solid (continents) floats on top of fluid. Indeed, plumb bobs lean away from mountain ranges since continental rock is lighter than the fluid which it displaces.
7) Therefore the surface of the Earth will closely approximate an equipotential surface (sea level) which is perpendicular to the direction of acceleration (which is gravity + centrifugal acceleration).

8) Because the Earth rotates below a speed needed to tear it apart the Earth is approximately spherical.
9) Because the Earth rotates only about 100 times slower than this top speed, the dominant shape of the Earth is an oblate spheroid. This has the poles nearer the center than the equator.
10) Since the force of gravity is proportional to the gradient of the potential, and the potential is constant in the center no matter which way you approach it, and the equator is further from the center, then the potential must be changing slower near the equator than at the poles, and therefore the measured acceleration (weight) must therefore be lower at the equator, which is what is measured.

 

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Sorry for nitpicking, but I was under the impression that there is no such thing as a centrifugal "force" - it's just an effect of acceleration.

 

In that sense, there is no such thing as a gravitational "force", either. Gravitation, like centrifugal acceleration, is just an artifact of a non-inertial observer. In general relativity, an inertial frame is a reference frame that is in free-fall. Centrifugal force and gravitational force, along with the Coriolis force and the frame acceleration (aka third-body effects) are all pseudo forces.

There is no device that can be constructed that can directly measure any pseudo force. In particular, just as no device can be constructed to directly measure centrifugal acceleration, no device can be constructed to directly measure gravitational acceleration.

 

You're all full of sh*t. The only way there could be less gravity at the equator is if there was less density. But there is no real reason why this should be so. The centrifugal force argument is an obvious red herring for anyone with a modicum of intelligence. I give up on this site. You're all thick as sh*t.

(I've had a fw ciders, and I'm off on holiday tomorrow, so I don't care)

 

Density does not play part in gravitation.

Don't leave, you haven't showed us all your maths yet.

 

Oooook get going.

yeah that nitpicking for no reasons: we all know it not really a force.

 

Nothing wrong here (or anywhere in your post) but I just wanted to point out why APL/JHU was paid By US Navy to determine the coefficients of the spherical harmonics (also called the Tesserial harmonics, I think, except for spelling errors) far out to very minor time variable terms, I think. At least twice as many as are public (not secrete, but even exactly how many are known is secrete or was.)

During the MAD era US's most secure nuclear deterrent was the Polaris sub. It needed to know exactly where it was, which way is gravitational up there and during powered flight, where the target of its ICBM is, and the details of the bumpy gravitational field it would fly thru ballistically getting there. A friend I car pooled with ran this more than decade long program. (He died just a few months ago), but he could not tell me anything detailed about it. APL solved all of these problems for the US Navy.

Ironically, the Sputnik was a big help with the sub's "Where am I exactly?" problem: Two APLers recorded its transmitted tone, which is Doppler shifted. Fitting that shift exactly allowed them to determine the orbit, which was not disclosed initially, more accurately than any other measurement method at the time. They realized that the math could be inverted. I.e. if you already knew the orbit precisely you could find where you were. So we make first the TRIAD satellites and then the DISCOS satellites.

Triad got told by uplink the exact parameters of its orbit every few days (They changed by various forces including the then unknown lumpiness of Earth's gravity, solar winds, earth's magnetic field acting on satellite currents, variable atmospheric drag, and some more I think but have forgotten)

In the insane logic of the era this need for an up link every few days was not good enough. -If the bad guys struck first and everyone was dead in the US the sub might not get to firing position, while the satellite still knew its orbit well and not be able to kill all the bad guys too, weeks later. DISCOS solved that problem as it knew where it was for (not sure I can say how long, but long enough) without any up links.

DISCOS had a free floating sphere about 1cm in diameter that was at the mass center of the satellite, called the "proof mass" (hereafter PM). The PM was an alloy of gold and platinum (I think) that was neither dia- nor para- magnetic. The PM was the true satellite the main body shield it from all the disturbances other than gravity and fired tiny thruster to follow the PM's orbit exactly.

DISCOS was expensive to make, as every resistor etc was weighed and precisely located so the CoM could be accurately set to the exact center of the PM's tiny chamber. If the CoM were ahead of the PM, the gravitational attraction of the satellite would constantly be accelerating the PM not only ONLY the Earth's gravity field. (BTW the higher order term of the harmonics "wiggle" via mass movements such as high atmospheric pressure air masses, the tides, winter snows, etc.) Eventually DISCO was retired and now we have GPS to tell the sub where it is. I suspect that DISCOS was more accurate, but as the bombs got bigger, it did not matter so much what the size of the error circle was.

I played a very minor role in all this, mainly designing a less expense version of the DISCOS system. The PM became a hollow cylinder with AC current along rigid wire on the axis. I.e. eddy currents in the PM interacted to support the PM against radial gravitational forces of the satellite It really only need to be "free floating" along the orbit to do its job adequately. You and Janus58 probably know better than me why those "in orbit trajectory direction" accelerations had to be due to only the Earth's gravity and small transverse acceleration could have other forces acting.

I did the physical modeling and set up the equations / calculations to show that the eddy current damping would be large enough to make sure the radial oscillation never could become large enough for the cylinder PM to actually hit the wire, and made an experimental test ON EARTH in tall vacuum chamber with PM as the "bob" of fine, long-wire pendulum to confirm my analysis. When the AC was on, the pendulum oscillations were damped more, just as I had predicted. - I did need some help with some of the integrals. It was the toughest applied math I had ever seen.

 

The primary driver for developing high-fidelity gravity models is that there is no way to measure the acceleration due to gravity. All inertial navigation systems must instead estimate the acceleration due to gravity. Since the vast majority of the flight of a ballistic missile is ballistic (hence the name), any errors in the estimation of the gravitational acceleration will lead to errors in the projected impact point.

The current best model of the Earth's gravity field is EGM2008. A link: http://earth-info.nga.mil/GandG/wgs84/gravitymod/egm2008/index.html. As an aside, WTF is the "National Geospatial-Intelligence Agency"?

Nuclear subs have an incredibly bad idea of where they are, at least when viewed from the perspective of launching an SLBM and making it accurately hit a target thousands of miles away. The Polaris and Poseidon missiles were accurate enough to kill Russian women and children. In other words, such a missile targeted at the center of Moscow would probably have missed by tens of miles. The addition of star trackers made the Trident I missile able to kill Russian train stations and manufacturing plants: "A miss is as good as a mile" takes on a different meaning when a near miss is enhanced by a nuclear explosion.

This lack of accuracy was the primary driver in the development of the NAVSTAR system. NAVSTAR is now know as GPS. With GPS, Trident II missiles are capable of killing Russian silos. Taking out a hardened silos requires accuracy on the order of a hundred yards or better. Not bad when the initial error is on the order of a mile or so.

 

Yes they were almost "lost" with weeks of inertial guidance. To be secure they needed to stay submerged and just trail a near surface antenna radars could not see. I may be biased, but think the DISCOS system was much more accurate than GPS if no up dates are available from the ground to tell the satellite where it is (so it can tell the ground user)

GPS is really a "time of flight" system. From three known points in space where three satelites are "now" - The user is at the point where three spheres centered on those known point intersect. If there is error accumulating in the satellite's "idea" as to its location (or it clock) then that error transfers, larger I think, to the ground observer's computed idea of his position.

DISCOS was just not cost effective when mid-course corrections of the IBM via stellar observations became possible but it was very accurate and only required one satellite to survive a war in space, not three. (It still used APL invented Dopper curve fit to the over fly data, not the intesection of three spheres.)
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DH: "WTF is the "National Geospatial-Intelligence Agency"?"
BT" Have no idea, but your link seems to be providing a "geogrid of gravity" not the coefficients of the expansion; however one clever with math should be able to back the many terms of the spherical harmonic expansion out of that data. I doubt if the results would be as accurate as they are know, but still secrete. I did not skim deep enough to know where their data came from, but it may be of interest to oil companies etc. Probably not of much use if you want to know details of the "wiggles" in a ballistic missile’s flight, I would guess.

 

From the site:

Introduction The official Earth Gravitational Model EGM2008 has been publicly released by the U.S. National Geospatial-Intelligence Agency (NGA) EGM Development Team. This gravitational model is complete to spherical harmonic degree and order 2159, and contains additional coefficients extending to degree 2190 and order 2159.​

The 2190x2159 tide-free spherical harmonics coefficients are the primary elements product of the EGM2008 model. The link to those coefficients is right there, on the cited Web site; no secrets. They do not provide a "geogrid of gravity" per se; they do, however, provide tools to compute such a thing.

 

Thanks for the correction. My footnote was wrong. I skimmed to quickly and went to some sub-links to see if APL got any credit for its contribution - could not find any.