Gravity An Introduction

Transcription

1 Gravity A Itroductio Reier Rummel Istitute of Advaced Study (IAS) Techische Uiversität Müche Lecture Oe 5th ESA Earth Observatio Summer School 2-13 August 2010, ESA-ESRIN, Frascati / Italy

2 gravitatio ad space sciece micro-g eviromet: e.g. boilig water fudametal physics: e.g. equivalece priciple earth scieces e.g. gravitatioal field of earth moo ad plaets

3 global map of gravity aomalies

4 the pla Lecture Oe: Theoretical basics of gravitatio as applied to the earth [gravitatioal law, properties, mathematical represetatio] The laguage for the two other lectures Lecture Two: The role of the earth s gravitatioal field i earth scieces [gravity aomalies, geoid as a referece, temporal variatios] Lecture Three: Priciples of satellite gravimetry; i their logic derived from free fall tests i a laboratory o earth [the orbit, priciples of GRACE, ESA s missio GOCE ad satellite gradiometry]

5 itroductio to gravitatio Newto s law of gravitatio: mm mm F G e G x x A B A B A 2 AB 3 A B AB x x A B Newto s secod law: F A m I A a A F A A gravitatioal acceleratio: m m aa G e m A I A B 2 AB AB AB e AB B

6 itroductio to gravitatio from sigle mass, to may masses, to a cotiuum a G e d B A 2 AB B AB Fudametal properties of Newto s law of gravitatio: cetral force actio = reactio iverse square distace superpositio of all partial forces istataeous

7 itroductio to gravitatio a G e d B A 2 AB B AB a A is a vector field i space with the followig properties: a 0 curl free a V A A A i.e. there exists a gravitatioal potetial V A ad i outer space, we get: a source free V 2 0 0

8 itroductio to gravitatio a G e d B A 2 AB B AB a A is a vector field with the followig properties: a 0 curl free a V A A A i.e. there exists a gravitatioal potetial V A ad i outer space, we get: a source free V B VA G d AB B V V V x y z

9 example: satellite orbit 2 V 2 x a ice applicatio: a satellite orbit x A aa AV " perturbatios " ad iitial coditios : x ; x 0 0

10 example: satellite orbit a ice applicatio: a satellite orbit sphere flatteed sphere real earth Kelpleria ellipse precessig ellipse precessig ellipse plus gravitatioal code

11 example: satellite orbit What about the attractio of su, moo ad plaets? Aswer: They determie the earth s orbit about the su Tides are acceleratio relative to the earth s ceter of mass Marshak, 2005

12 gravitatio ad gravity o the surface of the rotatig earth oe measures: gravity= gravitatio + cetrifugal acceleratio g a z ad W V Z z a g

13 size of gravity sigals gravity (i laboratory at TU Müche) m/s 2 statioary variable 10 0 spherical Earth 10-3 flatteig & cetrifugal acceleratio 10-4 moutais, valleys, ocea ridges, subductio 10-5 desity variatios i crust ad matle 10-6 salt domes, sedimet basis, ores 10-7 tides, atmospheric pressure 10-8 temporal variatios: oceas, hydrology 10-9 ocea topography, polar motio geeral relativity

14 geometry of the earth s gravity field gravity vector plumb lie geoid W W cost 0. level surface

15 gravity related quatities ) <1m to 2m

16 gravity related quatities map with geoid heights relative to the GRS80 ellipsoid

17 series represetatio of gravitatioal field V V V x y z Laplace equatio (PDE) solutio i Cartesia coordiates (after determiatio):,, exp k exp VA x y z V k z c i kx y with z 0 k solutio i spherical coordiates (after determiatio): 1 R VA,, r V0 Pm Cm cosm Sm sim 0 r m0 1 1 R R V0 C P cos m S P si m m m m m m0 m r m r almost a Fourier series

18 series represetatio of gravitatioal field log 500km 333km 250km 200km 20000km [ km] max spectral represetatio of the earth s gravitatioal field: triagular plot of the spherical harmoic coefficiets C ; S m m

19 series represetatio of gravitatioal field C 00 A 00 C 11 S 11 C 10 S 22 S 21 C 20 C 21 C 22 S 33 S 32 S 31 C 30 C 31 C 32 C 33 S 44 S 43 S 42 S 41 C 40 C 41 C 42 C 43 C 44 S 55 S 54 S 53 S 52 S 51 C 50 C 51 C 52 C 53 C 54 C 55 sectorial tesseral zoal tesseral sectorial surface spherical harmoic fuctios: Ym, Pm cos m sim

20 series represetatio of gravitatioal field Degree C 00 S 11 C 10 C 11 S 22 S 21 C 21 scale origi orietatio S m Order m C m

21 series represetatio of gravitatioal field sigal degree variace 2 2 ( m m) m0 c C S ave c 21 i aalogy to sigal processig: characteristics of sigal ad oise Here: degree variaces correspod to power spectral desity

22 series represetatio of gravitatioal field power law by WM Kaula error c / sigal ad error degree variaces (dimesioless)

23 series represetatio of temporal variatios of gravitatioal field Degree Stadard Deviatio i Geoid Height [m] Atmosphere daily ECMWF Ocea daily MIT Hydrology mothly Europe GRACE Error Predictio Degree rapid time variable geoid sigals (RMS) [ to be divided by the earth radius i order to arrive at dimesioless uits]

24 series represetatio of temporal variatios of gravitatioal field Degree Stadard Deviatio i Geoid Height [m] Ocea semi-aual Ocea aual GRACE Error Predictio Atmosphere aual Degree slow geoid time variable sigals (RMS) [ to be divided by the earth radius i order to arrive at dimesioless uits]

25 three levels of gravity quatities o earth ad i space 1 R VA,, r V0 Pm Cm cosm Sm sim 0 r m0 1 1 R R V C P 0 cos m S P si m m m m m m0 m r m r r 1 r 2 δv δ r V δ rr V disturbace potetial or geoid gravity disturbaces or gravity aomalies gravity gradiets or torsio balace

26 various gravity quatities o earth ad i space Gravity model EGM08, D/O1000

27 h = 0km δv δv r δv rr Gravity model EGM08, D/O1000

28 satellite altitude: r earth s Surface: R δv δ r V δ rr V 1 R 2 R δv δ r V δ rr V 1 r 2 r 1 R r 2 R r 3 R r three levels of gravity quatities o earth ad i space

29 various gravity quatities o earth ad i space Gravity model EGM08, D/O1000

30 δv r h = 400km h = 250km h = 0km Gravity model EGM08, D/O1000

31 T r T rr T 1 R 2 R T r T rr T 1 r 2 r 1 R r 2 R r 3 R r Gravity model EGM08, D/O1000

32 various gravity quatities o earth ad i space x y z x y V ik [E] z

33 satellite-to-satellite trackig low-low satellite-to-satellite trackig high-low satellite gradiometry satellite altitude r r 1 r 2 δv δ r V δ rr V R r 1 R r 2 R r 3 earth s surface R R 1 R 2 δv δ r V δ rr V disturbace potetial or geoid gravity disturbaces or gravity aomalies gravity gradiets or torsio balace

34 summary of lecture Oe 1. Newto s law of gravitatio describes all its relevat properties such as iverse square distace, priciple of superpositio, its statioary part beig vorticity free, ad source free outside the earth (Laplace equatio) 2. Gravity is the sum of gravitatio ad the cetrifugal part 3. Satellite orbits are essetially described by gravitatio 4. Tides are a accelertatio (a force) relative to the earth s ceter of mass 5. The global gravitatioal field is represeted as a series of spherical harmoics, beig a solutio of Laplace partial differetial equatio (Dirichlet) 6. Spherical harmoics o a sphere are aalogous to a Fourier series i a plae 7. Therefore there exists a closed theory of sigal ad oise processig 8. With icreasig distace from the earth sphere the series coefficiets are dampeig out per degree like (R/R+h) With each radial derivative of the gravitatioal potetial the series coefficiets are amplified per degree like (+1) 10. The strategy of satellite missios GRACE ad GOCE rests o the priciple of compesatig the damperig effect by amplificatio (see 8. ad 9.)