SIO 229 Gravity and Geomagnetism. Lecture 6. J 2 for Earth. J 2 in the solar system. A first look at the geoid.

SIO 229 Gavity and Geomagnetism Lectue 6. J 2 fo Eath. J 2 in the sola system. A fist look at the geoid.

The Thee Big Themes of the Gavity Lectues 1.) An ellipsoidal otating Eath Refeence body (mass + shape) fo descibing Eath s geopotential, gavity, and topogaphy. 2.) Deviations fom the ellipsoidal Eath Global deviations of geopotential, gavity and topogaphy fom that of the efeence body, integating to ZERO ove the suface of Eath. These ae vey small elative to the efeence. 3.) Local plana appoximations The way most investigations ae actually done.

Multipole Expansion of Eath s Gavitational Potential But by definition 9 R 2 = - s 2 = 2 + s 2-2 s (5) Diffeentiating (6): o = i i + s i s i - 2 i s i. (6) 2R R s i = 2s i - 2 i (7) R s i = s i - i R. (8) Now we substitute (8) into (4) and s i 1 R = i - s i R 3. (9) Looking back at (2) we need to evaluate (9) at s i = 0, whee R = : æ 1 ö = i è s i Rø 0. (10) 3 We spae the student the details and simply asset that in a simila way we can show: æ 2 1 ö ç = 3 i j è s i s j R ø 5 0 - d ij 3. (11) We gathe togethe (10), (11) and (3) into the fomula fo potential ( 39 ): - V G = ò d3 s (s) é 1 ê B ë + s i i + s is j 3 2 æ 3 i j è 5 We eaange the last tem a bit to sepaate tems in s j and j : æ 3 i j s i s j è 5 - d ij 3 ö ø +... ù ú û. (12) - d ij 3 ö ø = 3 i j s i s j - 2 d ij s i s j 5 (13) = 3 i j s i s j - s 2 2 5 (14) whee m B = ò d3 s (s) B c i = ò d3 s (s) s i /m B B Q ij = ½ ò d3 s (s)(3s i s j - s 2 d ij ) B 10 (17a) and m B is the mass of B, c is a vecto fom O to the cente of mass of B and Q ij is called the quadupole tenso of the mass distibution. In fact this is an expansion in invese powes of, which wecan see moe clealy if we wite things in tems of the unit vecto ˆ: - V G = m B + ˆ cm B 2 + ˆ i ˆ j Q ij 3 +... (18) This equation gives the fist thee tems of a multipole expansion of the potential of the body B: the fist tem is the monopole, the second the dipole tem, the thid the quadupole, etc in Geek. To obtain the infamous MacCullagh s fomula we estate the tem in Q ij in tems of the inetia tenso. Q ij = ½ ò d3 s (s)(3s i s j - s 2 d ij ). (19) B We now tun to (1.19) and (1.20) to compute the tems unde the integal: Thus Q ij = ½[3(½M kk d ij - M ij ) - (½M kk )d ij ] (20) = ½M kk d ij - 3 2 M ij. (21) ˆ i ˆ j Q ij = ½M kk - 3 2 ˆ i ˆ j M ij (22) = ½( A + B + C) - 3 I(ˆ). (23) 2 This is MacCullagh s fomula in its geneal fom: - V G = m B + cm B 3 + 1 3 [½( A + B + C) - 3 2 I(ˆ)] +... (24) = i j 5 (3s is j - s 2 d ij ). (15) So, putting (15) into (12) and moving the integals aound: - V () G = 1 B ò d3 s (s) + i ò 3 d3 s (s)s i + i j ò 5 d3 s (s)( 3 2 s is j - ½s 2 d ij ) +... (16) B B = m B + i c i m B 3 + i j 5 Q ij +... (17)

Gavitational Potential and Moments of Inetia fo an Ellipsodial Eath MacCullagh s This is MacCullagh s Fomula fomula in geneal in its Catesian geneal fom: fom (Eq 3.24) - V G = m B + cm B 3 + 1 3 [½( A + B + C) - 3 2 I(ˆ)] +... (24) monopole dipole quadupole + + + Mass as if it wee all at oigin of coodinate system Location of cente of mass along the coodinate axes + Mass distibution off each axis, aanged symmetically on both sides of oigin

Gavitational Potential and Moments of Inetia fo an Ellipsodial Eath MacCullagh s This is MacCullagh s Fomula fomula in geneal in its Catesian geneal fom: fom (Eq 3.24) - V G = m B + cm B 3 + 1 3 [½( A + B + C) - 3 2 I(ˆ)] +... (24) monopole dipole quadupole MacCullagh s Fomula adapted to Eath, Catesian coodinates (Eq 4.4) - V G = m B monopole + C - A 1-3 ˆ 2 3 3 2 quadupole +... (4) MacCullagh s Fomula adapted to Eath, Spheical Coodinates (Eq 4.5) - V G = m E é ê1 - J 2 P 2 (cos q ) a2 ë +... ù 2 ú (5) û

Gavitational Potential and Moments of Inetia fo an Ellipsodial Eath MacCullagh s This is MacCullagh s Fomula fomula in geneal in its Catesian geneal fom: fom (Eq 3.24) - V G = m B + cm B 3 + 1 3 [½( A + B + C) - 3 2 I(ˆ)] +... (24) V((x,y,z)) monopole dipole quadupole MacCullagh s Fomula adapted to Eath, Catesian coodinates (Eq 4.4) - V G = m B monopole + C - A 1-3 ˆ 2 3 3 2 quadupole +... (4) V((x,y,z)) MacCullagh s Fomula adapted to Eath, Spheical Coodinates (Eq 4.5) - V G = m E é ê1 - J 2 P 2 (cos q ) a2 ë 2 +... ù ú û (5) V(, θ, φ)

Gavitational Potential and Moments of Inetia fo an Ellipsodial Eath MacCullagh s This is MacCullagh s Fomula fomula in geneal in its Catesian geneal fom: fom (Eq 3.24) - V G = m B + cm B 3 + 1 3 [½( A + B + C) - 3 2 I(ˆ)] +... (24) V((x,y,z)) monopole dipole quadupole MacCullagh s Fomula adapted to Eath, Catesian coodinates (Eq 4.4) - V G = m B + C - A 1-3 ˆ 2 3 3 2 +... (4) V((x,y,z)) MacCullagh s Fomula adapted to Eath, Spheical Coodinates (Eq 4.5) - V G = m E é ê1 - J 2 P 2 (cos q ) a2 ë 2 +... ù ú û (5) V(, θ, φ) J 2 = C - A m E a 2 = = Legende polynomial of degee 2

J 2 fo Eath

A C

Change in Eath s J 2 (1976~2011) C A

How to detemine J2

J 2 13 fo this ae completely mysteious. Why should we be inteested in these quantities? Oiginally the main inteest came fom the constaints they place on the adial density stuctue of the Eath. If we assume the main vaiation in density is with s the distance fom the Eath s cente (Yes, now O is at the cente of mass!) (1.19) shows us that: a A + B + C = 8p ò (s) s4 ds (8) and of couse we have 0 a m E 4p = ò (s) s2 ds. (9) 0 Thus we have two integals of the density pofile to which we can fit models. Fo most planets in the sola system these ae the only two constaints available! Anothe eason, which we come to late is that the main shape of the Eath and the moment of inetia ae connected and simple models of the Eath s flattening due to otation can pedict J 2 (and they get it wong). Fist let s look at estimation of J 2,defined in (4.13). Hee we give an ovesimplified teatment; the eal thing needs spheical hamonics and a lot of compute modeling. We look at the motion of an atificial satellite. If the Eath had the gavitational potential of only the fist tem in (3.31), and we ignoe attactions of the sun, moon and othe planets, the obit of a nea-eath satellite would lie in a plane passing though the Eath s cente of mass. But the equatoial bulges exet a toque on the satellite tending to pull the plane towads the equato, which has the effect of making the plane of the obit pecess in the opposite diection to that of the satellite, called etogade pecession. This motion is called egession of the nodes, the nodes being the points whee the obit intesects the equatoial plane. To a fist appoximation, the ate of pecession depends on J 2. Afte a tedious calculation (see Section A in the Annex) in which one computes the aveage toque in one evolution due to the J 2 tem, it can be shown that T 1,the peiod of pecession is given by T 1 = 2 2 1 T 0 3 a 2 J 2 cos Q whee T 0 is the peiod of the satellite in its obit, is the mean adius of the obit, Q is the angle between the instantaneous plane of the obit and û C,the Eath s otation axis. Clealy the fastest ate of pecession is when Q = 0(when the plane of the obit lies in the Eath s equatoial plane and so the pecession becomes invisible); since J 2» 0. 001 we can compute that T 1 ³ 38 days. Obviously having J 2 is not enough to detemine C and A sepaately (We assume that to a good enough appoximation A = B). The second piece of infomation comes fom the ate of pecession of the Eath s otation axis because of tidal toques fom the sun and moon acting on the equatoial bulges. The pecession makes itself known by the movement of the fixed point in the night sky elative to the stas. It is called the pecession of the equinoxes because the time of the equinox (when day and night have equal length) occus when the spin axis of the Eath is (10)

!" # Eath

$% & # Satellite! ' Eath

"# $! Satellite ' Δ' ) % Δ' (fom equatoial mass) Eath

"# $!! Satellite ' Δ' ) % Δ' Eath Regession of the nodes... whee node = longitude whee satellite obit cosses the equatoial plane

Finding J2

J 2 fo the planets

Gavitational Potential and Moments of Inetia fo an Ellipsodial Eath This is MacCullagh s fomula in its geneal fom: 13 - V G = m B + cm B 3 + 1 3 [½( A + B + C) - 3 2 I(ˆ)] +... (24) 4. Detemination of the Moments of Inetia of the Eath MacCullagh s fomula, equation (3.24), can be witten moe simply if we make some obvious choices. Fist, choose O to be at the cente of mass of B. This emoves the dipole tem fom the expansion (3.24) because then c = 0. Second, let the axes of the coodinates be the pincipal axes of B. Then in the thid tem we use (2.12) ˆ i ˆ j Q ij = ½( A + B + C) - 3 2 ( A ˆ2 1 + B ˆ 2 2 + C ˆ 2 3) (1) Finally, let us look at a degeneate case with two equal eigenvalues: A = B < C as it is faily accuately fo the Eath. ˆ i ˆ j Q ij = ½(2 A + C) - 3 2 [ A( ˆ2 1 + ˆ 2 2) + C ˆ 2 3]. (2) = ½(C - A)(1-3 ˆ 2 3) (3) because ˆ 2 = ˆ 2 1 + ˆ 2 2 + ˆ 2 3 = 1. With all this in effect (3.24) becomes - V G = m B + C - A 1-3 ˆ 2 3 +... (4) 3 2 When we analyze the Eath s gavity field this is often witten as follows fo easons soon to become appaent - V G = m E é ê1 - J 2 P 2 (cos q ) a2 ë +... ù 2 ú (5) û whee m E is the Eath s mass, a is its equatoial adius, J 2 = C - A m E a 2 (6) and cos q = ˆ 3 = û C ˆ whee q is the geocentic colatitude and P 2 is the Legende polynomial of degee 2: P 2 (µ) = ½(3µ 2-1). (7) Equation (5) is an example of a vey simple spheical hamonic expansion of the potential V. The quantity J 2 is a dimensionless numbe, known fom the analysis of satellite obits to high accuacy fo the Eath: J 2 =.0010826265. You should emembe it is about 10-3. Now at last we ae beginning to discuss the Eath. The constants of geodesy that I will use ae all taken fom one souce: Global Eath Physics: A Handbook of Physical Constants, Edito: T. J. Ahens, 1995. Chaptes 1 and 2 ae a athe dy compilation of the cuent best estimates of geodetic popeties of the Eath and the othe planets. Actually because geodesists and astonomes love to give as many significant figues as they can, values fo vaious numbes ae given to extavagant pecision and don t always agee with each othe in diffeent pats of the book! The accuacy of ou measuements of J 2 means that we can actually see it vaying in time! The expected diection is a decease in J 2 as the Eath ecoves fom glacial loading in the pola egions the Eath is becoming less oblate. But vey ecent wok (C. M. Cox and B. F.Chao, Science, 297, pp 831-3, 2002) shows that the annual decline of about 3 10-11 suddenly evesed in about 1997. The easons fo this ae completely mysteious. Why should we be inteested in these quantities? Oiginally the main inteest came fom the constaints they place on the adial density stuctue of the Eath. If we assume the main vaiation in density is with s the distance fom the Eath s cente (Yes, now O is at the cente of mass!) (1.19) shows us that: a A + B + C = 8p ò (s) s4 ds (8) and of couse we have (and they get it wong). T 1 2 0 a m E 4p = ò (s) s2 ds. (9) 0 Thus we have two integals of the density pofile to which we can fit models. Fo most planets in the sola system these ae the only two constaints available! Anothe eason, which we come to late is that the main shape of the Eath and the moment of inetia ae connected and simple models of the Eath s flattening due to otation can pedict J 2 (and they get it wong). Fist let s look at estimation of J 2,defined in (4.13). Hee we give an ovesimplified teatment; the eal thing needs spheical hamonics and a lot of compute modeling. Welook at the motion of an atificial satellite. Ifthe Eath had the gavitational potential of only the fist tem in (3.31), and we ignoe attactions of the sun, moon and othe planets, the obit of a nea-eath satellite would lie in a plane passing though the Eath s cente of mass. But the equatoial bulges exet a toque on the satellite tending to pull the plane towads the equato, which has the effect of making the plane of the obit pecess in the opposite diection to that of the satellite, called etogade pecession. This motion is called egession of the nodes, the nodes being the points whee the obit intesects the equatoial plane. Toafist appoximation, the ate of pecession depends on J 2. Afte a tedious calculation (see Section A in the Annex) in which one computes the aveage toque in one evolution due to the J 2 tem, it can be shown that T 1,the peiod of pecession is given by simple models of the Eath s flattening due to otation can pedict J2 = 2 1 T 0 3 a 2 J 2 cos Q whee T 0 is the peiod of the satellite in its obit, is the mean adius of the obit, Q is the angle between the instantaneous plane of the obit and û C,the Eath s otation axis. Clealy the fastest ate of pecession is when Q = 0(when the plane of the obit lies in the Eath s equatoial plane and so the pecession becomes invisible); since J 2» 0. 001 we can compute that T 1 ³ 38 days. Obviously having J 2 is not enough to detemine C and A sepaately (We assume that to a good enough appoximation A = B). The second piece of infomation comes fom the ate of pecession of the Eath s otation axis because of tidal toques fom the sun and moon acting on the equatoial bulges. The pecession makes itself known by the movement of the fixed point in the night sky elative to the stas. It is called the pecession of the equinoxes because the time of the equinox (when day and night have equal length) occus when the spin axis of the Eath is (10)

J 2 vs. Spin Rate in the Sola System 0.020 Satun 0.015 Jupite J2 0.010 0.005 Uanus Neptune Moon Mas Eath 0.000 0.0 0.5 1.0 1.5 2.0 2.5 Spin Rate (evolutions/day)

J 2 vs. Spin Rate in the Sola System 0.020 0.1000 Satun 0.015 Jupite 0.0100 J2 0.010 0.0010 0.005 Uanus Neptune Moon Mas Eath 0.000 0.0001 0.0 0.5 1.0 1.5 2.0 2.5 Spin Rate (evolutions/day)

Mass vs. Spin Rate 1000.00 J 2 vs. Spin Rate in the Sola System Jupite 0.020 0.1000 100.00 Satun J2 0.015 0.010 Satun Jupite 0.0100 mass (% eath) 10.00 1.00 Uanus Eath Neptune 0.0010 0.005 Uanus Neptune Mas Eath Moon 0.000 0.0001 0.0 0.5 1.0 1.5 2.0 2.5 Spin Rate (evolutions/day) 0.10 Moon Mas 0.01 0.0 0.5 1.0 1.5 2.0 2.5 Spin Rate (evolutions/day)

Mass vs. Spin Rate 1000.00 J 2 vs. Spin Rate in the Sola System Jupite 0.020 0.1000 100.00 Satun J2 0.015 0.010 Satun Jupite 0.0100 0.0010 mass (% eath) 10.00 1.00 Venus Uanus Eath Neptune 0.005 Uanus Neptune Mas Eath Moon 0.000 0.0001 0.0 0.5 1.0 1.5 2.0 2.5 Spin Rate (evolutions/day) 0.10 Moon Mas 0.01 0.0 0.5 1.0 1.5 2.0 2.5 Spin Rate (evolutions/day)

Why do we cae about J 2, o inetia tenso eigenvalues (A, B, C)?