Gravimetry of the Planets - J.P. Barriot

Transcription

1 Gravimetry of the Planets - J.P. Barriot

2 Programme of the training Thursday 18 November 9h-12h30 (with coffee break at 10h30) "Current tectonic and geomorphology of Mars", Ernst Hauber 14h30-18h (with coffee break at 16h) "Gravimetry and geodesy of the planets", J.P Barriot and Veronique Dehant Evening conference ( 20h): "Life and habitability on Mars", Frances Westall (English spoken) Friday 19 November 9h-12h30 (with coffee break at 10h30) "Formation and evolution of Mars, early Mars", Doris Breuer 14h30-18h (with coffee break at 16h) "In-situ geophysical exploration of the planets", Philippe Lognonné Evening conference(20h): "Water on mars", F. Costard (French spoken) Outreach program Saturday morning 20 November (9h30-13h) Instrument mock-up of the Martian exploration Recent pictures of Mars Diaporama Discussions with scientists Gravimetry of the Planets - J.P. Barriot

3 Training course : "Gravimetry and geodesy of the planets", J.P Barriot and Veronique Dehant Part I : Gravimetry by JP Barriot A The Orbit as a resonator to be tuned B- Determination of the thicknesses of the crusts of the planets from an isostatic hypothesis Gravimetry of the Planets - J.P. Barriot

4 Introduction The motion of a space probe around a planet is governed by the wellknown equation of Newtonian dynamics r m r Γ = r F (1) where F is the sum of all the forces acting on the spacecraft, and resulting acceleration, m being its mass. r Γ is its This equation must be referred to a Galilean reference frame, that is to say a referential with no net "acceleration", i.e. a referential "fixed" with respect to the stars, located at the barycentre of our Solar System (this definition is somewhat messy, and a consistent physical background can be r only set up by considering General Relativity). By a suitable modification of F, the origin of this frame can be translated from the barycentre of the Solar system to the barycentre of the planet, but with an orientation still fixed with respect to the stars. As the gravitational forces coming from the planet the principal forces, Equation (1) is separated as follows r r Γ = r X = r U( X r ) + f r /m (2), the forces f being "second order" forces in amplitude (attraction from other planets, radiation pressure, atmospheric drag...). The potential U is generated by the repartition of masses inside the planet. As the planet is rotating with respect to the stars, this potential is a function of the (Newtonian) time t. It is then better to compute it with respect to a reference frame linked to the planet, where these masses are at rest (at least up to the first order, as the planet cannot be considered completely as a rigid r X = M r body (tides, tectonic motions,..)). Then we have x, where M is the rotation matrix linking the two frames, with M 1 = M T. Equation (2) can then be rewritten as r X = M r U(M r T X ) + f r /m (3) Gravimetry of the Planets - J.P. Barriot

5 whereu is now time "independent", with M = M(t, orientation parameters op), U = U(gravity field parameters gp) r and f = r f (t, small forces parameters fp). Mathematically, the orientation parameters can be the Euler's angles, or any set of parameters characterizing a rotation in space (three parameters). A more complex set of parameters is used in practice, as the underlying physical phenomena (precession, nutations, polar motion) have very different time scales. The gravity field parameters are, almost exclusively, the so-called global harmonic coefficients, from the splitting of the Laplace equation in vacuum in spherical coordinates. The observed quantities linked with Equation (3) are the most often quantities derived from the position or velocity of the spacecraft with respect to a direction of observation r ν (t) (and with respect to the own motion of the observer), i.e. r X r r ν or X r ν. quantities q linearly derived from The goal of geodesy is to derive the orientation parameters op, the gravity field parameters gp and the small forces parameters fp from the observed quantities q. There is often an a priori knowledge about the looked-for orientation parameters op, the gravity field parameters gp and the small forces parameters fp. If there is none, a zero value is assumed. Let us call these a priori estimates op O, gp O and fp O. Then one can estimate r r X O = M O U O (M r T O X O ) + f r O /m... (4) from M O = M(t, orientation parameters op O ), U O = U(gravity field parameters gp O ) r f O = f r (t, small forces parameters fp O ) and Gravimetry of the Planets - J.P. Barriot

6 and so writes r X r X O = [M r U(M r T X )] (gp gp O, op op 0 ) (op,gp) op= op O gp= gp O + r f ( fp fp O ) /m ( fp) fp= fpo...(5) This process is called "linearization" and is as the base of all data processing in space geodesy. The set of Equation (4) and (5) then permit to compute estimated observations q O, to form the differences q q O and to write finally a linear system of equations called "equation of observations" linking the observed q q O differences to the unknown (gp gp O, op op 0, fp fp O ) differences, system solved numerically by least squares methods. Instead of continuing to speak about generalities, I have taken the option to present, with sufficient depth, a very important application of Equation (5), where the rotation matrix is reduced to the diurnal rotation of the planet (moreover with a constant rate) and where the a priori gravity parameters considered are the central term of the potential and the flattening coefficient J2. This study was first done by Kaula (1960), and show that the orbit of a space probe acts as a resonator excited by the gravity field of the planet. Secondly, I give an example of the joint interpretation of the topography and gravity in terms of the so-called "compensation depth", in a global scale. This study can be generalized in term of regional, elastic or viscoelastic, equilibrium. JPB, Dec. 9 th, Gravimetry of the Planets - J.P. Barriot

7 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

8 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

9 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

10 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

11 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

12 A- The Orbit as a Resonator to be Tuned ORBITAL PARAMETERS From

13 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

14 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

15 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

16 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

17 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

18 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

19 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

20 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

21 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

22 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

23 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

24 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

25 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

26 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

27 A- The Orbit as a Resonator to be Tuned With the kind permission of A. Cazenave

28 A- The Orbit as a Resonator to be Tuned From Theory of Satellite Geodesy, W. Kaula

29 Observation (continue) de la trajectoire Analyse des perturbations ==> géopotentiel A- The Orbit as a Resonator to be Tuned With the kind permission of G. Balmino 1. Observations SST-hl GOCE sat. GPS(i) La force de la dynamique

30 A- The Orbit as a Resonator to be Tuned With the kind permission of G. Balmino Harmoniques sphériques : P lm (sinϕ) cos mλ, P lm (sinϕ) sin mλ DEGRE L ϕ : latitude λ : longitude Résolution d'un modèle = π R (planète) / L = km / L, pour la Terre 0 ORDRE DEGRE L

31 Perturbations gravitationnelles : une orbite filtre et résonne log (pert. 3-D) > 3.0 < -2.0 A- The Orbit as a Resonator to be Tuned With the kind permission of G. Balmino Perturbations gravitationnelles en position (mètre), par degré et ordre des harmoniques l [R/(R+h)] l e -2π h/ λ (λ= 2π R/ l). k n orb m θ Terre m 0 l ORDRE

32 A- The Orbit as a Resonator to be Tuned MARS EXPRESS PERIASTRE 250 KM

33 A- The Orbit as a Resonator to be Tuned MARS ORBITER 400 KM

34 A- The Orbit as a Resonator to be Tuned MARS ORBITER 400 KM MARS EXPRESS PERIASTRE 250 KM (inclinaison 0!)

35 A- The Orbit as a Resonator to be Tuned MARS ORBITER 400 KM MARS EXPRESS PERIASTRE 250 KM (inclinaison 45!)

36 A- The Orbit as a Resonator to be Tuned MARS ORBITER 400 KM MARS EXPRESS PERIASTRE 250 KM (inclinaison 90)

37 B - Determination of the Thickness of the Crust of a Planet

38 B - Determination of the Thickness of the Crust of a Planet

39 B - Determination of the Thickness of the Crust of a Planet

40 B - Determination of the Thickness of the Crust of a Planet

41 B - Determination of the Thickness of the Crust of a Planet

42 B - Determination of the Thickness of the Crust of a Planet

43 B - Determination of the Thickness of the Crust of a Planet

44 B - Determination of the Thickness of the Crust of a Planet

45 B - Determination of the Thickness of the Crust of a Planet

46 B - Determination of the Thickness of the Crust of a Planet

47 B - Determination of the Thickness of the Crust of a Planet

48 B - Determination of the Thickness of the Crust of a Planet

49 B- Determination of the Thickness of the Crust of a Planet From Venus Book II, University of Arizona Press

50 B- Determination of the Thickness of the Crust of a Planet From Venus Book II, University of Arizona Press

51 B- Determination of the Thickness of the Crust of a Planet Mars Topography and gravity anomalies: From Science, Vol 287., 2000

52 2. (Visco -) Elastic Flexure d 0 d 1 B- Determination of the Thickness of the Crust of a Planet With the kind permission of G. Balmino Detailed properties of a planet/satellite Gravity + Topography Study of properties of the lithosphere and upper mantle Modeling Examples 1. Isostasy ex : Airy model density d 0 d > d 0 d > d 0 Venus : the "pancakes"

53 CROÛTE : 40 KM CROÛTE : 70 KM B- Determination of the Thickness of the Crust of a Planet From COUPE DE LA CROÛTE MARTIENNE DU NORD (GAUCHE) AU SUD (DROITE), LONGITUDE 0

54 B- Determination of the Thickness of the Crust of a Planet From JGR, Vol 109, E08002, 2004

55 B- Determination of the Thickness of the Crust of a Planet MOON / CLEMENTINE DATA From Science, Vol. 266, 1994

56 References Theory of Satellite Geodesy, W. Kaula, Dover Publication, 2000, ISBN Crustal Structure of Mars from Gravity and Topography, G.A. Neumann et al., Journal of Geophysical Research, Vol. 109, E08002, Internal Structure and Early Thermal Evolution of Mars from Mars Global Surveyor Topography and Gravity, Zuber et al., Science, Vol 287, The Shape and Internal Structure of the Moon from the Clementine Mission, Zuber et al., Science, Vol. 266, 1994 Venus II, The Venus Gravity Field and other geodetic parameters, Sjogren et al., University of Arizona Press, ISBN , Gravimetry of the Planets - J.P. Barriot