地球内部的引力耦合与内核的耦合转动徐速
地球内部的引力耦合 与内核的耦合转动 徐 速
Newton Theorem.The gravitationalp=constforce shouldvanisheverywhere insideanellipsoidal homoeoidf2Newton,1.,1686a-cPrincipia MathematicaaBookI,Prop.XClCor. 3.fi=f2
Newton Theorem •The gravitational force should vanish everywhere inside an ellipsoidal homoeoid. Newton, I., 1686, Principia Mathematica, Book I, Prop. XCI, Cor. 3
Xu and Szeto Theorem.For a homogeneous ellipsoidal shell whose inner and outersurface flattening values are f1and f2 respectively,thegravitational potential in any internal point of the shell isexactly@ = A+ Br? P (COSO)wheretantanA=-2元pGee.tantartantanB=2元pGe2f-f2(1-f)
Xu and Szeto Theorem • For a homogeneous ellipsoidal shell whose inner and outer surface flattening values are f1 and f2 respectively, the gravitational potential in any internal point of the shell is exactly ( ) 2 = + A Br P COS 2 where 1 1 2 2 1 2 1 2 1 2 tan tan 2 e e A G a a e e − − = − − 1 1 1 1 1 2 1 2 2 2 3 3 1 2 1 2 1 2 1 1 tan tan tan tan 2 e e e e B G e e e e e e − − − − = − − − − − ( ) 2 2 2 2 1 f f e f − = −
Inferences (1). Whenf1 =f2.Newton's theorem is recovered as aspecial case.: Newton's theorem predicts a constant potential inside acomposite body consisting of concentric,alignedhomoeoidalshells of varying density.If thesehomoeoidallayers are replaced by axially symmetricellipsoidal shells of varying flattening,Newton's theoremis incapable of predicting the potential inside sucha bodyThe newtheorem extends to this situation. We find adegree 2harmonic potential whosemagnitude dependsupon the radial gradient of flattening
Inferences (1) • When f1 = f2, Newton’s theorem is recovered as a special case. • Newton’s theorem predicts a constant potential inside a composite body consisting of concentric, aligned homoeoidalshells of varying density. If these homoeoidallayers are replaced by axially symmetric ellipsoidal shells of varying flattening, Newton’s theorem is incapable of predicting the potential inside such a body. The new theorem extends to this situation. We find a degree 2 harmonic potential whose magnitude depends upon the radial gradient of flattening
Inferences (2)Parameter B,and hence the internal attraction,depends neither on the thickness of the shell northe locations of the inner and outer boundariesThis is consistent with Newton's theorem, since ahomoeoidof arbitrary thickness can be added tothe outside of a shell, followed by the removalfromtheinside out'ofanotherhomoeoidwithoutaffecting B ortheinternal attraction
Inferences (2) Parameter B, and hence the internal attraction, depends neither on the thickness of the shell nor the locations of the inner and outer boundaries. This is consistent with Newton’s theorem, since a homoeoidof arbitrary thickness can be added to the outside of a shell, followed by the removal ‘from the inside out’of another homoeoidwithout affecting B or the internal attraction