420:
260:. How the gravitational force and the centrifugal force add up to a force orthogonal to the geoid is illustrated in the figure (not to scale). At latitude 50 deg the off-set between the gravitational force (red line in the figure) and the local vertical (green line in the figure) is in fact 0.098 deg. For a mass point (atmosphere) in motion the centrifugal force no more matches the gravitational and the vector sum is not exactly orthogonal to the Earth surface. This is the cause of the
268:
25:
1002:
869:
686:
1418:
is the vector pointing to the point considered straight from the Earth's rotational axis. It can be shown that this pseudo-force field, in a reference frame co-rotating with the Earth, has a potential associated with it in terms of Earth's rotation rate, ω:
2551:
representation). In this context the geopotential is taken as the potential of the gravitational field of the Earth, that is, leaving out the centrifugal potential. Solving for geopotential in the simple case of a nonrotating sphere, in units of or :
2153:, and captures the detailed, complex variations of the true gravity field of the actually existing Earth from point-to-point, as distinguished from the overall global trend captured by the smooth mathematical ellipsoid of the normal potential.
1881:
539:
1413:
2004:
787:
1495:
2437:
1069:
1598:
1349:
2350:
997:{\displaystyle \mathbf {g} =\nabla W=\mathrm {grad} \ W={\frac {\partial W}{\partial X}}\mathbf {i} +{\frac {\partial W}{\partial Y}}\mathbf {j} +{\frac {\partial W}{\partial Z}}\mathbf {k} }
576:
2607:
2694:
2241:
2502:
2033:
2766:
1668:
1213:
1518:
2268:
2136:
1715:
1691:
1284:
1101:
397:
359:
332:
2186:
2093:
2069:
1260:
1236:
1161:
1141:
1121:
1025:
847:
305:
218:
35:
1738:
427:
465:
1360:
1941:
1912:, or Geodetic Reference System 1980, which the Global Positioning system uses as its reference. Its geometric parameters are: semi-major axis
2889:
715:
275:
The geoid is a gently undulating surface due to the irregular mass distribution inside the Earth; it may be approximated however by an
2998:
2952:
1425:
2367:
2555:
1925:
1721:
2611:
1030:
1540:
1304:
2191:
185:
50:
93:
65:
2936:
2071:
has been constructed matching the known GRS80 reference ellipsoid with an equipotential surface (we call such a field a
2288:
681:{\displaystyle \mathbf {\bar {F}} =-Gm_{1}\int \limits _{V}{\frac {\rho _{2}}{r^{2}}}\mathbf {\hat {r}} \,dx\,dy\,dz}
72:
2701:
1902:
404:
149:
of the geopotential, without the negation. In addition to the actual potential (the geopotential), a theoretical
79:
2466:
2009:
Obviously, this value depends on the assumption that the potential goes asymptotically to zero at infinity (
705:
276:
2012:
1164:
549:
61:
287:), approximates the geoid to within a little over ±100 m. One can construct a simple model geopotential
307:
that has as one of its equipotential surfaces this reference ellipsoid, with the same model potential
419:
3020:
2843:
2544:
2040:
2035:), as is common in physics. For practical purposes it makes more sense to choose the zero point of
1725:
1641:
1613:
280:
267:
197:
189:
283:. The currently most widely used reference ellipsoid, that of the Geodetic Reference System 1980 (
2848:
1524:
1185:
852:
2994:
2948:
2885:
2540:
1528:
1176:
1503:
2879:
2858:
2548:
1729:
1628:
363:
193:
134:
2246:
2109:
1700:
1676:
1269:
1074:
370:
337:
310:
2982:
1891:
565:
261:
181:
86:
2853:
2833:
2171:
2078:
2054:
2036:
1245:
1221:
1146:
1126:
1106:
1010:
832:
826:
452:
290:
203:
142:
3014:
2356:
2278:
In the special case of a sphere with a spherically symmetric mass density then ρ = ρ(
126:
2911:
2944:
1924:
is equal to the known mass of the Earth (including atmosphere), as involved in the
1612:. So, it is accurate to approximate the geopotential by a field that has the Earth
822:
403:. Many observable quantities of the gravity field, such as gravity anomalies and
1876:{\displaystyle \Psi \approx (GM/r)(1-(a/r)^{2}J_{2}((3/2)\cos ^{2}(\phi )-1/2))}
560:. For a non-pointlike object of continuous mass distribution, each mass element
557:
177:
173:
24:
435:
408:
221:
534:{\displaystyle \mathbf {F} =-G{\frac {m_{1}m_{2}}{r^{2}}}\mathbf {\hat {r}} }
2456:
2162:
1609:
1408:{\displaystyle \mathbf {p} =X\mathbf {i} +Y\mathbf {j} +0\cdot \mathbf {k} }
856:
251:
118:
2990:
1999:{\displaystyle U_{0}=62636860.850\ {\textrm {m}}^{2}\,{\textrm {s}}^{-2}}
146:
271:
Balance between gravitational and centrifugal force on the Earth surface
1298:
The centrifugal force—per unit of mass, i.e., acceleration—is given by
782:{\displaystyle V=-G\int \limits _{V}{\frac {\rho _{2}}{r}}\,dx\,dy\,dz}
2543:, the geopotential is typically described by a series expansion into
1732:
in terms of spherical coordinates; truncating the series results in:
1071:
is the orthonormal set of base vectors in space, pointing along the
1920: = 1/298.257222101. If we also require that the enclosed mass
1007:
which expresses the gravity acceleration vector as the gradient of
2838:
1909:
418:
284:
266:
256:
166:
122:
42:
138:
1175:
Both gravity and its potential contain a contribution from the
564:
can be treated as mass distributed over a small volume, so the
2881:
Geometrical
Geodesy: Using Information and Computer Technology
829:
and of the direction from the volume element to point mass 1.
18:
1490:{\displaystyle \Phi ={\frac {1}{2}}\omega ^{2}(X^{2}+Y^{2}).}
2432:{\displaystyle {\bar {F}}=-{\frac {GMm}{R^{2}}}\ {\hat {r}}}
2165:, an alternative version of the geopotential is used called
1523:
The centrifugal potential can also be expressed in terms of
2355:
These integrals can be evaluated analytically. This is the
1064:{\displaystyle \{\mathbf {i} ,\mathbf {j} ,\mathbf {k} \}}
1593:{\displaystyle \Phi =0.5\,\omega ^{2}r^{2}\sin ^{2}\phi }
1344:{\displaystyle \mathbf {g} _{c}=\omega ^{2}\mathbf {p} ,}
2075:) we can subtract it from the true (measured) potential
46:
2704:
2614:
2558:
2469:
2370:
2291:
2282:); i.e., density depends only on the radial distance
2249:
2194:
2174:
2112:
2081:
2057:
2015:
1944:
1741:
1703:
1679:
1644:
1543:
1506:
1428:
1363:
1307:
1272:
1248:
1224:
1188:
1149:
1129:
1109:
1077:
1033:
1013:
872:
849:
is the gravitational potential energy per unit mass.
835:
718:
579:
468:
373:
340:
313:
293:
206:
2043:, and refer the potentials of other points to this.
2760:
2688:
2601:
2496:
2431:
2344:
2262:
2235:
2180:
2130:
2087:
2063:
2027:
1998:
1875:
1720:A closed-form exact expression exists in terms of
1709:
1685:
1662:
1592:
1512:
1489:
1407:
1343:
1278:
1254:
1230:
1207:
1155:
1135:
1115:
1095:
1063:
1019:
996:
841:
781:
680:
533:
411:), can be expressed in this disturbing potential.
391:
353:
326:
299:
212:
651:
586:
525:
2345:{\displaystyle s={\sqrt {x^{2}+y^{2}+z^{2}}}\,.}
165:"Geop" redirects here. For the unit prefix, see
1908:The most recent Earth reference ellipsoid is
1500:This can be verified by taking the gradient (
1027:, the potential of gravity. The vector triad
129:. For convenience it is often defined as the
8:
2188:, which are reckoned from the geoid upward:
2095:of the real Earth. The result is defined as
1058:
1034:
423:Diagram of two masses attracting one another
254:is close to one equigeopotential called the
220:. The surfaces of constant geopotential or
176:applications, gravity is distinguished from
51:introducing citations to additional sources
2602:{\displaystyle \Psi (h)=\int _{0}^{h}g\,dz}
1932:= 3986005 × 10 m·s, we obtain for the
2689:{\displaystyle \Psi =\int _{0}^{z}\leftdz}
1179:due to the Earth's rotation. We can write
2884:. Springer Berlin Heidelberg. p. 9.
2735:
2722:
2703:
2667:
2640:
2630:
2625:
2613:
2592:
2583:
2578:
2557:
2479:
2468:
2418:
2417:
2406:
2389:
2372:
2371:
2369:
2338:
2330:
2317:
2304:
2298:
2290:
2274:Simple case: nonrotating symmetric sphere
2254:
2248:
2219:
2193:
2173:
2145:is numerically a great deal smaller than
2111:
2080:
2056:
2014:
1987:
1981:
1980:
1978:
1972:
1966:
1965:
1949:
1943:
1916: = 6378137.0 m, and flattening
1859:
1835:
1820:
1805:
1795:
1783:
1757:
1740:
1702:
1678:
1643:
1578:
1568:
1558:
1553:
1542:
1505:
1475:
1462:
1449:
1435:
1427:
1400:
1386:
1375:
1364:
1362:
1333:
1327:
1314:
1309:
1306:
1271:
1247:
1223:
1204:
1187:
1148:
1128:
1108:
1076:
1053:
1045:
1037:
1032:
1012:
989:
969:
961:
941:
933:
913:
890:
873:
871:
834:
772:
765:
758:
747:
741:
735:
717:
671:
664:
657:
646:
645:
637:
627:
621:
615:
605:
581:
580:
578:
520:
519:
511:
500:
490:
483:
469:
467:
372:
345:
339:
318:
312:
292:
205:
2236:{\displaystyle C=-\left(W-W_{0}\right),}
2051:Once a clean, smooth geopotential field
41:Relevant discussion may be found on the
2968:
2966:
2964:
2870:
1616:as one of its equipotential surfaces.
2987:An Introduction to Dynamic Meteorology
2497:{\displaystyle \Psi =-{\frac {GM}{r}}}
2161:In practical terrestrial work, e.g.,
1934:potential at the reference ellipsoid:
428:Newton's law of universal gravitation
361:of the geoid; this model is called a
7:
2460:
2361:
2028:{\displaystyle R\rightarrow \infty }
1635:) is constructed as a two-part sum:
855:field can be derived from a gravity
709:
570:
430:states that the gravitational force
2817:is the average radius of the earth,
1619:Like the actual geopotential field
568:over the extent of object 2 gives:
2705:
2615:
2559:
2470:
2270:is the geopotential of the geoid.
2022:
1742:
1704:
1680:
1657:
1651:
1544:
1507:
1429:
1273:
1201:
980:
972:
952:
944:
924:
916:
900:
897:
894:
891:
881:
14:
2823:is the geometric height in meters
2536:is the total mass of the sphere.
1728:). It can also be expressed as a
16:Energy related to Earth's gravity
1926:standard gravitational parameter
1722:ellipsoidal-harmonic coordinates
1401:
1387:
1376:
1365:
1334:
1310:
1054:
1046:
1038:
990:
962:
934:
874:
648:
583:
522:
470:
34:relies largely or entirely on a
23:
1520:) operator of this expression.
224:of the geopotential are called
2785:is the gravitational constant,
2664:
2651:
2568:
2562:
2423:
2377:
2019:
1870:
1867:
1850:
1844:
1828:
1814:
1811:
1792:
1777:
1768:
1765:
1748:
1717:is the centrifugal potential.
1695:normal gravitational potential
1608:The Earth is approximately an
1481:
1455:
1:
2972:Torge, Geodesy. 3rd ed. 2001.
2937:Heiskanen, Weikko Aleksanteri
2761:{\displaystyle \Psi =Gm\left}
2539:For the purpose of satellite
1663:{\displaystyle U=\Psi +\Phi }
1627:(not to be confused with the
2989:(4th ed.). Burlington:
180:. Gravity is defined as the
1208:{\displaystyle W=V+\Phi \,}
405:deflections of the vertical
238:geopotential level surfaces
192:. Likewise, the respective
3037:
2359:saying that in this case:
1903:second dynamic form factor
230:(sometimes abbreviated as
164:
153:and their difference, the
2939:; Moritz, Helmut (1967).
2801:is the mass of the earth,
2141:The disturbing potential
1724:(not to be confused with
227:equigeopotential surfaces
200:called the geopotential,
1238:is the potential of the
1177:centrifugal pseudo-force
264:for atmospheric motion.
196:can be added to form an
1513:{\displaystyle \nabla }
1103:coordinate axes. Here,
706:gravitational potential
277:ellipsoid of revolution
252:Global mean sea surface
184:of gravitation and the
157:, can also be defined.
2878:Hooijberg, M. (2007).
2762:
2690:
2603:
2498:
2433:
2346:
2264:
2237:
2182:
2132:
2089:
2065:
2029:
2000:
1877:
1711:
1687:
1664:
1594:
1514:
1491:
1409:
1345:
1280:
1256:
1232:
1209:
1165:geocentric coordinates
1157:
1137:
1117:
1097:
1065:
1021:
998:
843:
783:
682:
550:gravitational constant
535:
424:
393:
355:
334:as the true potential
328:
301:
272:
242:equipotential surfaces
214:
2763:
2691:
2604:
2499:
2434:
2347:
2265:
2263:{\displaystyle W_{0}}
2238:
2183:
2133:
2131:{\displaystyle T=W-U}
2090:
2066:
2030:
2001:
1878:
1712:
1710:{\displaystyle \Phi }
1688:
1686:{\displaystyle \Psi }
1665:
1595:
1515:
1492:
1410:
1346:
1294:Centrifugal potential
1281:
1279:{\displaystyle \Phi }
1257:
1233:
1210:
1158:
1138:
1118:
1098:
1096:{\displaystyle X,Y,Z}
1066:
1022:
999:
844:
784:
683:
536:
422:
394:
392:{\displaystyle T=W-U}
356:
354:{\displaystyle W_{0}}
329:
327:{\displaystyle U_{0}}
302:
270:
215:
2702:
2612:
2556:
2467:
2368:
2289:
2247:
2192:
2172:
2157:Geopotential number
2110:
2101:disturbing potential
2079:
2055:
2047:Disturbing potential
2013:
1942:
1739:
1726:geodetic coordinates
1701:
1677:
1642:
1541:
1504:
1426:
1361:
1305:
1270:
1246:
1222:
1186:
1147:
1127:
1107:
1075:
1031:
1011:
870:
863:) field as follows:
833:
716:
577:
466:
401:disturbing potential
371:
338:
311:
291:
204:
155:disturbing potential
47:improve this article
2844:Geopotential height
2635:
2588:
2545:spherical harmonics
2455:with corresponding
2167:geopotential number
2041:reference ellipsoid
1623:, the normal field
1614:reference ellipsoid
1240:gravitational field
704:with corresponding
434:acting between two
281:reference ellipsoid
198:effective potential
145:is obtained as the
2849:Geopotential model
2758:
2686:
2621:
2599:
2574:
2494:
2429:
2342:
2260:
2233:
2178:
2128:
2085:
2061:
2039:to be that of the
2025:
1996:
1873:
1707:
1683:
1660:
1590:
1525:spherical latitude
1510:
1487:
1405:
1341:
1276:
1252:
1228:
1205:
1153:
1133:
1113:
1093:
1061:
1017:
994:
839:
779:
740:
678:
620:
531:
425:
389:
351:
324:
297:
273:
210:
2891:978-3-540-68225-7
2751:
2730:
2698:Integrate to get
2674:
2541:orbital mechanics
2518:
2517:
2492:
2453:
2452:
2426:
2416:
2412:
2380:
2336:
2181:{\displaystyle C}
2088:{\displaystyle W}
2064:{\displaystyle U}
1984:
1969:
1963:
1529:geocentric radius
1443:
1288:centrifugal field
1255:{\displaystyle W}
1231:{\displaystyle V}
1156:{\displaystyle Z}
1136:{\displaystyle Y}
1116:{\displaystyle X}
1020:{\displaystyle W}
987:
959:
931:
906:
842:{\displaystyle u}
803:
802:
756:
731:
702:
701:
654:
643:
611:
589:
528:
517:
367:. The difference
300:{\displaystyle U}
236:), also known as
213:{\displaystyle W}
194:scalar potentials
186:centrifugal force
112:
111:
97:
3028:
3005:
3004:
2983:Holton, James R.
2979:
2973:
2970:
2959:
2958:
2941:Physical Geodesy
2933:
2927:
2926:
2924:
2922:
2908:
2902:
2901:
2899:
2898:
2875:
2859:Physical geodesy
2822:
2816:
2815:
2813:
2800:
2799:
2797:
2784:
2783:
2781:
2767:
2765:
2764:
2759:
2757:
2753:
2752:
2750:
2736:
2731:
2723:
2695:
2693:
2692:
2687:
2679:
2675:
2673:
2672:
2671:
2649:
2641:
2634:
2629:
2608:
2606:
2605:
2600:
2587:
2582:
2512:
2503:
2501:
2500:
2495:
2493:
2488:
2480:
2461:
2447:
2438:
2436:
2435:
2430:
2428:
2427:
2419:
2414:
2413:
2411:
2410:
2401:
2390:
2382:
2381:
2373:
2362:
2351:
2349:
2348:
2343:
2337:
2335:
2334:
2322:
2321:
2309:
2308:
2299:
2269:
2267:
2266:
2261:
2259:
2258:
2242:
2240:
2239:
2234:
2229:
2225:
2224:
2223:
2187:
2185:
2184:
2179:
2137:
2135:
2134:
2129:
2094:
2092:
2091:
2086:
2073:normal potential
2070:
2068:
2067:
2062:
2034:
2032:
2031:
2026:
2005:
2003:
2002:
1997:
1995:
1994:
1986:
1985:
1982:
1977:
1976:
1971:
1970:
1967:
1961:
1954:
1953:
1882:
1880:
1879:
1874:
1863:
1840:
1839:
1824:
1810:
1809:
1800:
1799:
1787:
1761:
1730:series expansion
1716:
1714:
1713:
1708:
1692:
1690:
1689:
1684:
1669:
1667:
1666:
1661:
1629:potential energy
1604:Normal potential
1599:
1597:
1596:
1591:
1583:
1582:
1573:
1572:
1563:
1562:
1519:
1517:
1516:
1511:
1496:
1494:
1493:
1488:
1480:
1479:
1467:
1466:
1454:
1453:
1444:
1436:
1414:
1412:
1411:
1406:
1404:
1390:
1379:
1368:
1350:
1348:
1347:
1342:
1337:
1332:
1331:
1319:
1318:
1313:
1285:
1283:
1282:
1277:
1261:
1259:
1258:
1253:
1237:
1235:
1234:
1229:
1214:
1212:
1211:
1206:
1162:
1160:
1159:
1154:
1142:
1140:
1139:
1134:
1122:
1120:
1119:
1114:
1102:
1100:
1099:
1094:
1070:
1068:
1067:
1062:
1057:
1049:
1041:
1026:
1024:
1023:
1018:
1003:
1001:
1000:
995:
993:
988:
986:
978:
970:
965:
960:
958:
950:
942:
937:
932:
930:
922:
914:
904:
903:
877:
848:
846:
845:
840:
797:
788:
786:
785:
780:
757:
752:
751:
742:
739:
710:
696:
687:
685:
684:
679:
656:
655:
647:
644:
642:
641:
632:
631:
622:
619:
610:
609:
591:
590:
582:
571:
540:
538:
537:
532:
530:
529:
521:
518:
516:
515:
506:
505:
504:
495:
494:
484:
473:
398:
396:
395:
390:
364:normal potential
360:
358:
357:
352:
350:
349:
333:
331:
330:
325:
323:
322:
306:
304:
303:
298:
219:
217:
216:
211:
190:Earth's rotation
151:normal potential
135:potential energy
107:
104:
98:
96:
55:
27:
19:
3036:
3035:
3031:
3030:
3029:
3027:
3026:
3025:
3011:
3010:
3009:
3008:
3001:
2981:
2980:
2976:
2971:
2962:
2955:
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2700:
2699:
2663:
2650:
2642:
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2610:
2609:
2554:
2553:
2527:
2510:
2481:
2465:
2464:
2445:
2402:
2391:
2366:
2365:
2326:
2313:
2300:
2287:
2286:
2276:
2250:
2245:
2244:
2215:
2208:
2204:
2190:
2189:
2170:
2169:
2159:
2108:
2107:
2077:
2076:
2053:
2052:
2049:
2011:
2010:
1979:
1964:
1945:
1940:
1939:
1900:
1892:semi-major axis
1831:
1801:
1791:
1737:
1736:
1699:
1698:
1675:
1674:
1640:
1639:
1606:
1574:
1564:
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1539:
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1502:
1501:
1471:
1458:
1445:
1424:
1423:
1359:
1358:
1323:
1308:
1303:
1302:
1296:
1268:
1267:
1244:
1243:
1220:
1219:
1184:
1183:
1173:
1145:
1144:
1125:
1124:
1105:
1104:
1073:
1072:
1029:
1028:
1009:
1008:
979:
971:
951:
943:
923:
915:
868:
867:
853:Earth's gravity
831:
830:
808:
795:
743:
714:
713:
694:
633:
623:
601:
575:
574:
566:volume integral
507:
496:
486:
485:
464:
463:
450:
443:
417:
369:
368:
341:
336:
335:
314:
309:
308:
289:
288:
262:coriolis effect
202:
201:
182:resultant force
170:
163:
108:
102:
99:
56:
54:
40:
28:
17:
12:
11:
5:
3034:
3032:
3024:
3023:
3013:
3012:
3007:
3006:
2999:
2974:
2960:
2953:
2928:
2912:"Geopotential"
2903:
2890:
2869:
2868:
2866:
2863:
2862:
2861:
2856:
2854:Normal gravity
2851:
2846:
2841:
2836:
2834:Dynamic height
2829:
2826:
2825:
2824:
2818:
2802:
2786:
2756:
2749:
2746:
2743:
2739:
2734:
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2716:
2713:
2710:
2707:
2685:
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2639:
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2624:
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2617:
2598:
2595:
2591:
2586:
2581:
2577:
2573:
2570:
2567:
2564:
2561:
2525:
2516:
2515:
2506:
2504:
2491:
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2451:
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2409:
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2400:
2397:
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2341:
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2329:
2325:
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2303:
2297:
2294:
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2257:
2253:
2232:
2228:
2222:
2218:
2214:
2211:
2207:
2203:
2200:
2197:
2177:
2158:
2155:
2139:
2138:
2127:
2124:
2121:
2118:
2115:
2084:
2060:
2048:
2045:
2037:normal gravity
2024:
2021:
2018:
2007:
2006:
1993:
1990:
1975:
1960:
1957:
1952:
1948:
1898:
1884:
1883:
1872:
1869:
1866:
1862:
1858:
1855:
1852:
1849:
1846:
1843:
1838:
1834:
1830:
1827:
1823:
1819:
1816:
1813:
1808:
1804:
1798:
1794:
1790:
1786:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1760:
1756:
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1750:
1747:
1744:
1706:
1682:
1671:
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1659:
1656:
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1647:
1605:
1602:
1601:
1600:
1589:
1586:
1581:
1577:
1571:
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1561:
1557:
1552:
1549:
1546:
1509:
1498:
1497:
1486:
1483:
1478:
1474:
1470:
1465:
1461:
1457:
1452:
1448:
1442:
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1434:
1431:
1416:
1415:
1403:
1399:
1396:
1393:
1389:
1385:
1382:
1378:
1374:
1371:
1367:
1352:
1351:
1340:
1336:
1330:
1326:
1322:
1317:
1312:
1295:
1292:
1275:
1251:
1227:
1216:
1215:
1203:
1200:
1197:
1194:
1191:
1172:
1169:
1152:
1132:
1112:
1092:
1089:
1086:
1083:
1080:
1060:
1056:
1052:
1048:
1044:
1040:
1036:
1016:
1005:
1004:
992:
985:
982:
977:
974:
968:
964:
957:
954:
949:
946:
940:
936:
929:
926:
921:
918:
912:
909:
902:
899:
896:
893:
889:
886:
883:
880:
876:
838:
827:volume element
806:
801:
800:
791:
789:
778:
775:
771:
768:
764:
761:
755:
750:
746:
738:
734:
730:
727:
724:
721:
700:
699:
690:
688:
677:
674:
670:
667:
663:
660:
653:
650:
640:
636:
630:
626:
618:
614:
608:
604:
600:
597:
594:
588:
585:
556:is the radial
542:
541:
527:
524:
514:
510:
503:
499:
493:
489:
482:
479:
476:
472:
453:centre of mass
448:
441:
416:
413:
399:is called the
388:
385:
382:
379:
376:
348:
344:
321:
317:
296:
246:level surfaces
209:
188:caused by the
162:
159:
143:gravity vector
141:, so that the
110:
109:
62:"Geopotential"
45:. Please help
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
3033:
3022:
3019:
3018:
3016:
3002:
3000:0-12-354015-1
2996:
2992:
2988:
2984:
2978:
2975:
2969:
2967:
2965:
2961:
2956:
2954:0-7167-0233-9
2950:
2946:
2942:
2938:
2932:
2929:
2917:
2913:
2907:
2904:
2893:
2887:
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2864:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2831:
2827:
2819:
2807:
2803:
2791:
2787:
2782:10 Nm/kg
2775:
2771:
2770:
2769:
2754:
2747:
2744:
2741:
2737:
2732:
2727:
2724:
2718:
2714:
2711:
2708:
2696:
2683:
2680:
2676:
2668:
2660:
2657:
2654:
2646:
2643:
2637:
2631:
2626:
2622:
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2596:
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2575:
2571:
2565:
2550:
2546:
2542:
2537:
2535:
2531:
2523:
2514:
2507:
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2462:
2459:
2458:
2449:
2442:
2440:
2420:
2407:
2403:
2398:
2395:
2392:
2386:
2383:
2374:
2364:
2363:
2360:
2358:
2357:shell theorem
2339:
2331:
2327:
2323:
2318:
2314:
2310:
2305:
2301:
2295:
2292:
2285:
2284:
2283:
2281:
2273:
2271:
2255:
2251:
2230:
2226:
2220:
2216:
2212:
2209:
2205:
2201:
2198:
2195:
2175:
2168:
2164:
2156:
2154:
2152:
2148:
2144:
2125:
2122:
2119:
2116:
2113:
2106:
2105:
2104:
2102:
2098:
2082:
2074:
2058:
2046:
2044:
2042:
2038:
2016:
1991:
1988:
1973:
1958:
1955:
1950:
1946:
1938:
1937:
1936:
1935:
1931:
1927:
1923:
1919:
1915:
1911:
1906:
1904:
1897:
1893:
1889:
1864:
1860:
1856:
1853:
1847:
1841:
1836:
1832:
1825:
1821:
1817:
1806:
1802:
1796:
1788:
1784:
1780:
1774:
1771:
1762:
1758:
1754:
1751:
1745:
1735:
1734:
1733:
1731:
1727:
1723:
1718:
1696:
1654:
1648:
1645:
1638:
1637:
1636:
1634:
1630:
1626:
1622:
1617:
1615:
1611:
1603:
1587:
1584:
1579:
1575:
1569:
1565:
1559:
1555:
1550:
1547:
1537:
1536:
1535:
1533:
1530:
1526:
1521:
1484:
1476:
1472:
1468:
1463:
1459:
1450:
1446:
1440:
1437:
1432:
1422:
1421:
1420:
1397:
1394:
1391:
1383:
1380:
1372:
1369:
1357:
1356:
1355:
1338:
1328:
1324:
1320:
1315:
1301:
1300:
1299:
1293:
1291:
1289:
1265:
1264:gravity field
1249:
1241:
1225:
1198:
1195:
1192:
1189:
1182:
1181:
1180:
1178:
1170:
1168:
1166:
1150:
1130:
1110:
1090:
1087:
1084:
1081:
1078:
1050:
1042:
1014:
983:
975:
966:
955:
947:
938:
927:
919:
910:
907:
887:
884:
878:
866:
865:
864:
862:
858:
854:
850:
836:
828:
824:
820:
816:
812:
799:
792:
790:
776:
773:
769:
766:
762:
759:
753:
748:
744:
736:
732:
728:
725:
722:
719:
712:
711:
708:
707:
698:
691:
689:
675:
672:
668:
665:
661:
658:
638:
634:
628:
624:
616:
612:
606:
602:
598:
595:
592:
573:
572:
569:
567:
563:
559:
555:
551:
547:
512:
508:
501:
497:
491:
487:
480:
477:
474:
462:
461:
460:
458:
454:
447:
440:
437:
433:
429:
421:
414:
412:
410:
406:
402:
386:
383:
380:
377:
374:
366:
365:
346:
342:
319:
315:
294:
286:
282:
278:
269:
265:
263:
259:
258:
253:
249:
247:
243:
239:
235:
234:
229:
228:
223:
207:
199:
195:
191:
187:
183:
179:
175:
168:
160:
158:
156:
152:
148:
144:
140:
136:
132:
128:
127:gravity field
124:
120:
116:
106:
95:
92:
88:
85:
81:
78:
74:
71:
67:
64: –
63:
59:
58:Find sources:
52:
48:
44:
38:
37:
36:single source
32:This article
30:
26:
21:
20:
2986:
2977:
2945:W.H. Freeman
2940:
2931:
2919:. Retrieved
2915:
2906:
2895:. Retrieved
2880:
2873:
2805:
2789:
2773:
2697:
2538:
2533:
2529:
2521:
2519:
2508:
2454:
2443:
2354:
2279:
2277:
2166:
2160:
2150:
2146:
2142:
2140:
2100:
2096:
2072:
2050:
2008:
1959:62636860.850
1933:
1929:
1921:
1917:
1913:
1907:
1895:
1887:
1885:
1719:
1694:
1672:
1632:
1624:
1620:
1618:
1607:
1531:
1522:
1499:
1417:
1353:
1297:
1287:
1286:that of the
1263:
1262:that of the
1239:
1217:
1174:
1006:
861:geopotential
860:
851:
823:mass density
818:
814:
810:
804:
793:
703:
692:
561:
553:
545:
543:
459:is given by
456:
445:
438:
436:point masses
431:
426:
400:
362:
274:
255:
250:
245:
244:, or simply
241:
237:
232:
231:
226:
225:
171:
154:
150:
130:
115:Geopotential
114:
113:
100:
90:
83:
76:
69:
57:
33:
2916:ametsoc.com
1171:Formulation
558:unit vector
455:separation
279:called the
222:isosurfaces
178:gravitation
174:geophysical
3021:Gravimetry
2897:2023-09-11
2865:References
2798:10 kg
415:Background
409:plumb-line
73:newspapers
2814:10 m
2733:−
2706:Ψ
2623:∫
2616:Ψ
2576:∫
2560:Ψ
2477:−
2471:Ψ
2457:potential
2424:^
2387:−
2378:¯
2213:−
2202:−
2163:levelling
2123:−
2023:∞
2020:→
1989:−
1854:−
1848:ϕ
1842:
1775:−
1746:≈
1743:Ψ
1705:Φ
1681:Ψ
1658:Φ
1652:Ψ
1610:ellipsoid
1588:ϕ
1585:
1556:ω
1545:Φ
1508:∇
1447:ω
1430:Φ
1398:⋅
1325:ω
1274:Φ
1202:Φ
981:∂
973:∂
953:∂
945:∂
925:∂
917:∂
882:∇
857:potential
821:) is the
745:ρ
733:∫
726:−
652:^
625:ρ
613:∫
596:−
587:¯
526:^
478:−
384:−
137:per unit
119:potential
103:July 2014
43:talk page
3015:Category
2991:Elsevier
2985:(2004).
2921:14 April
2828:See also
2549:spectral
161:Concepts
147:gradient
131:negative
2768:where:
1901:is the
1693:is the
1631:, also
825:at the
805:where ρ
548:is the
133:of the
121:of the
117:is the
87:scholar
2997:
2951:
2888:
2534:dxdydz
2520:where
2415:
2243:where
2099:, the
1962:
1886:where
1673:where
1527:φ and
1354:where
1266:, and
1218:where
905:
544:where
89:
82:
75:
68:
60:
2839:Geoid
2810:6.378
2794:5.975
2778:6.673
1910:GRS80
451:with
285:GRS80
257:geoid
167:geop-
123:Earth
94:JSTOR
80:books
2995:ISBN
2949:ISBN
2923:2023
2886:ISBN
1894:and
1697:and
1163:are
1143:and
809:= ρ(
552:and
444:and
233:geop
172:For
139:mass
66:news
2524:= ∫
2149:or
1890:is
1833:cos
1576:sin
1551:0.5
125:'s
49:by
3017::
2993:.
2963:^
2947:.
2943:.
2914:.
2808:=
2792:=
2776:=
2528:ρ(
2103::
1930:GM
1928:,
1905:.
1534::
1290:.
1242:,
1167:.
1123:,
817:,
813:,
562:dm
554:r̂
248:.
240:,
3003:.
2957:.
2925:.
2900:.
2821:z
2812:×
2806:a
2796:×
2790:m
2780:×
2774:G
2755:]
2748:z
2745:+
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2728:a
2725:1
2719:[
2715:m
2712:G
2709:=
2684:z
2681:d
2677:]
2669:2
2665:)
2661:z
2658:+
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2652:(
2647:m
2644:G
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2572:=
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2566:h
2563:(
2547:(
2532:)
2530:s
2526:V
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2511:4
2509:(
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2421:r
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2404:R
2399:m
2396:M
2393:G
2384:=
2375:F
2340:.
2332:2
2328:z
2324:+
2319:2
2315:y
2311:+
2306:2
2302:x
2296:=
2293:s
2280:s
2256:0
2252:W
2231:,
2227:)
2221:0
2217:W
2210:W
2206:(
2199:=
2196:C
2176:C
2151:W
2147:U
2143:T
2126:U
2120:W
2117:=
2114:T
2097:T
2083:W
2059:U
2017:R
1992:2
1983:s
1974:2
1968:m
1956:=
1951:0
1947:U
1922:M
1918:f
1914:a
1899:2
1896:J
1888:a
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1868:)
1865:2
1861:/
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1778:(
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1769:(
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1749:(
1655:+
1649:=
1646:U
1633:U
1625:U
1621:W
1580:2
1570:2
1566:r
1560:2
1548:=
1532:r
1485:.
1482:)
1477:2
1473:Y
1469:+
1464:2
1460:X
1456:(
1451:2
1441:2
1438:1
1433:=
1402:k
1395:0
1392:+
1388:j
1384:Y
1381:+
1377:i
1373:X
1370:=
1366:p
1339:,
1335:p
1329:2
1321:=
1316:c
1311:g
1250:W
1226:V
1199:+
1196:V
1193:=
1190:W
1151:Z
1131:Y
1111:X
1091:Z
1088:,
1085:Y
1082:,
1079:X
1059:}
1055:k
1051:,
1047:j
1043:,
1039:i
1035:{
1015:W
991:k
984:Z
976:W
967:+
963:j
956:Y
948:W
939:+
935:i
928:X
920:W
911:=
908:W
901:d
898:a
895:r
892:g
888:=
885:W
879:=
875:g
859:(
837:u
819:z
815:y
811:x
807:2
798:)
796:2
794:(
777:z
774:d
770:y
767:d
763:x
760:d
754:r
749:2
737:V
729:G
723:=
720:V
697:)
695:1
693:(
676:z
673:d
669:y
666:d
662:x
659:d
649:r
639:2
635:r
629:2
617:V
607:1
603:m
599:G
593:=
584:F
546:G
523:r
513:2
509:r
502:2
498:m
492:1
488:m
481:G
475:=
471:F
457:r
449:2
446:m
442:1
439:m
432:F
407:(
387:U
381:W
378:=
375:T
347:0
343:W
320:0
316:U
295:U
208:W
169:.
105:)
101:(
91:·
84:·
77:·
70:·
53:.
39:.
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