gravity - an introduction · pdf fileprinciples of satellite gravimetry; in their logic...

GravityAn Introduction

Reiner RummelInstitute of Advanced

Study

(IAS)

Technische Universität München

Lecture

One

5th ESA Earth Observation Summer School

2-13 August 2010, ESA-ESRIN, Frascati

/ Italy

micro-g

environment:

e.g. boiling

water

fundamental physics:

e.g. equivalence

principle

earth

sciencese.g. gravitational

field

of earth

moon

and planets

gravitation

and space

science

global map

of gravity

anomalies

Lecture

One:Theoretical

basics

of gravitation

as applied

to the

earth[gravitational

law, properties, mathematical

representation]The

„language“

for

the

two

other

lecturesLecture

Two:The

role

of the

earth‘s

gravitational

field

in earth

sciences[gravity

anomalies, geoid

as a reference, temporal variations]Lecture

Three:Principles

of satellite

gravimetry; in their

logic

derived

from

free

fall testsin a laboratory

on earth[the

orbit, principles

of GRACE, ESA‘s

mission

GOCE and satellite

gradiometry]

the

plan

Newton‘s

law

of gravitation:

Newton‘s

second law:

gravitational

acceleration:AB

A

B

AF

ABe

introduction

to gravitation

2 3A B A B

A AB A BAB A B

m m m mF G e G x xx x

IAA AmF a

2B

A ABAB

AIA

mm

ma G e

Fundamental properties

of Newton‘s

law

of gravitation:• central

force• action

= reaction• inverse

square

distance• superposition

of all partial forces• instantaneous

introduction

to gravitation

from

single

mass, to many

masses, to a continuum

2B

A AB BAB

a G e d

Aa

is

a vector

field

in spacewith

the

following

properties:

i.e. there

exists

a gravitational

potential Vand in outer

space, we

get:

introduction

to gravitation

2B

A AB BAB

a G e d

0A A Aa curl free a V

20 0Aa source free V

Aa

is

a vector

fieldwith

the

following

properties:

i.e. there

exists

a gravitational

potential Vand in outer

space, we

get:

introduction

to gravitation

2B

A AB BAB

a G e d

0A A Aa curl free a V

20 0Aa source free V

2 2 2

2 2 2 0V V Vx y z

BA B

AB

V G d

2

2

Vx

example: satellite

orbit

a nice

application: a satellite

orbit

0 0

" "

: ;

A A Ax a V perturbations

and initial conditions x x

sphere flattened

sphere real earth

a nice

application: a satellite

orbit

Kelplerian

ellipse precessing

ellipse precessing

ellipseplus „gravitational

code“

example: satellite

orbit

example: satellite

orbit

What

about

the

attraction

of sun, moon

and planets?

Answer: They

determine

the

earth‘s

orbit

about

the

sun

Tides

are

acceleration

relative

to the

earth‘s

center

of mass

Marshak, 2005

aa

zz

gg

on the

surface

of the

rotating

earth

one

measures:gravity= gravitation

+ centrifugal

acceleration

gravitation

and gravity

g a zandW V Z

10 0 spherical

Earth10-3 flattening

& centrifugal

acceleration

10-4 mountains, valleys, ocean

ridges, subduction10-5 density

variations

in crust

and mantle

10-6 salt

domes, sediment

basins, ores10-7 tides, atmospheric

pressure

10-8 temporal variations: oceans, hydrology10-9 ocean

topography, polar motion

10-10 general

relativity

gravity

(in laboratory

at TU München)9.807 246 72 m/s2

stat

iona

ryva

riabl

esize

of gravity

signals

level

surfacegeoid

plumb

line

gravity

vector

0 . W W const

geometry

of the

earth‘s

gravity

field

<1m to 2m

)

gravity

related

quantities

-150.00 -100.00 -50.00 0.00 50.00 100.00 150.00

-50.00

0.00

50.00

-100.00

-80.00

-60.00

-40.00

-20.00

0.00

20.00

40.00

60.00

80.00

map

with

geoid

heightsrelative to the

GRS80 ellipsoid

gravity

related

quantities

…almost a Fourier series

Laplace

equation

(PDE)

solution

in Cartesian

coordinates

(after

determination):

solution

in spherical

coordinates

(after

determination):

series

representation

of gravitational

field

2 2 2

2 2 2 0V V Vx y z

2 20, , exp exp

0

A kk

V x y z V k z c i kx y

with z

1

00 0

1 1

00

, , cos sin

cos sin

n n

A nm nm nmn m

n n

nm nm nm nmm n m n m

RV r V P C m S mr

R RV C P m S P mr r

20000[ ]max

kmkmn

series

representation

of gravitational

field

500km

333km

250km

200km

spectral

representation

of the

earth‘s

gravitational

field:triangular

plot

of the

spherical

harmonic

coefficients ;nm nmC S

log

C00

A00 C11S11 C10

C20

C30

C40

S22 S21

S31S33 S32

S41S42S44 S43

C21 C22

C31

C41

C32 C33

C42 C43 C44

C50S51S52S54 S53 C51 C52 C53 C54 C55S55

zonalsectorial sectorialtesseral tesseral

series

representation

of gravitational

field

surface

spherical

harmonic

functions: cos,

sinnm nm

mY P

m

scale

origin

orientation

Degree n

Order m

CC00

SS21

SS11

SS22 CC21

CC11CC10

nm nmS C

series

representation

of gravitational

field

in analogy

to signal

processing: characteristics

of signal

and noiseHere: degree

variances

correspond

to power spectral

density

signaldegree

variance

2 2

0

( )

n

n nm nmm

c C S2 1

nn

caven

series

representation

of gravitational

field

signal

and error

degree

variances

(dimensionless)

10 31.6 10 /nc n

2n

error

series

representation

of gravitational

field

power „law“

by

WM Kaula

Degree

Deg

ree

Sta

ndar

dD

evia

tion

inG

eoid

Hei

ght[

m]

10

10

20

20

30

30

40

40

50

50

10-6 10-6

10-5 10-5

10-4 10-4

10-3 10-3

Atmosphere dailyECMWFOceandailyMITHydrologymonthlyEuropeGRACE ErrorPrediction

rapid time variable geoid

signals (RMS)

[ to be divided by the earth radius in order to arrive at dimensionless units]

series

representation

of temporal variations

of gravitational

field

Degree

Deg

ree

Sta

ndar

dD

evia

tion

inG

eoid

Hei

ght[

m]

10

10

20

20

30

30

40

40

50

50

10-6 10-6

10-5 10-5

10-4 10-4

10-3 10-3

Ocean semi-annualOcean annualGRACE Error PredictionAtmosphere annual

“slow”

geoid

time variable signals (RMS)[ to be divided by the earth radius in order to arrive at dimensionless units]

series

representation

of temporal variations

of gravitational

field

δV δr

V δrr

V1r

n2r

n

disturbancepotential orgeoid

gravity

disturbancesorgravity

anomalies

gravity

gradientsortorsion

balance

three

levels

of gravity

quantities

on earth

and in space

1

00 0

1 1

00

, , cos sin

cos sin

n n

A nm nm nmn m

n n

nm nm nm nmm n m n m

RV r V P C m S mr

R RV C P m S P mr r

various

gravity

quantities

on earth

and in space

Gravity

model

EGM08, D/O1000

h = 0km

δV

δVr

δVrr

Gravity

model

EGM08, D/O1000

satellitealtitude: r

earth‘sSurface: R δV δr

V δrr

V1R

n2R

n

δV δr

V δrr

V1r

n2r

n

1nRr

2nRr

3nRr

three

levels

of gravity

quantities

on earth

and in space

various

gravity

quantities

on earth

and in space

Gravity

model

EGM08, D/O1000

h = 0km

h = 250km

h = 400km

δVr

Gravity

model

EGM08, D/O1000

T rT rrT1R

n2R

n

T rT rrT1r

n2r

n

1nRr

2nRr

3nRr

Gravity

model

EGM08, D/O1000

x y

x

z

y

z

Vik

[E]

−0.5 0 0.5

various

gravity

quantities

on earth

and in space

satellitealtitude

r

earth‘ssurface

R

satellitegradiometry

satellite-to-satellitetrackinghigh-low

satellite-to-satellitetrackinglow-low

disturbancepotential orgeoid

gravity

disturbancesorgravity

anomalies

gravity

gradientsortorsion

balance

δV δr

V δrr

V1R

n2R

n

δV δr

V δrr

V1r

n2r

n

1nRr

2nRr

3nRr

summary

of lecture

One1.

Newton‘s

law

of gravitation

describes

all its

relevant properties

such asinverse

square

distance, principle

of superposition, its

stationary

partbeing

vorticity

free, and source

free

outside

the

earth

(Laplace

equation)2. Gravity

is

the

sum

of gravitation

and the

centrifugal

part3. Satellite

orbits

are

essentially

described

by

gravitation4. Tides

are

an accelertation

(a force) relative to the

earth‘s

center

of mass5. The

global gravitational

field

is

represented

as a series

of spherical

harmonics,being

a solution

of Laplace

partial differential equation

(Dirichlet) 6. Spherical

harmonics

on a sphere

are

analogous

to a Fourier

series

in a plane7. Therefore

there

exists

a closed

theory

of „signal

and noise

processing“8. With

increasing

distance from

the

earth

sphere

the

series

coefficients

aredampening

out per degree

n like

(R/R+h)n+1

9. With

each

radial derivative

of the

gravitational

potential the

series

coefficientsare

amplified

per degree

n like

(n+1)10. The

strategy

of satellite

missions

GRACE and GOCE rests

on the

principleof compensating

the

dampering

effect

by

amplification

(see

8. and 9.)