1

Introduction to Astrometry

Toshio FUKUSHIMANational Astronomical Observatory of Japan

2006

2

Index 0. Summary 1. Observation 2. Time 3. Space 4. Coordinate System 5. Motion of Celestial

Bodies 6. Rotation

7. Earth Rotation 8. Keplerian Motion 9. Signal Propagation 10. Least Squares

Method 11. Crush Course of

General Relativistic Effects

12. References

3

0. Summary

What is Astrometry? General Principles Basic Elements of Astrometry

References: Time, Space, Units Motion: Linear, Orbital, Rotational Signal Propagation: 1-way, Round-trip

Mathematical Tools

4

Astrometry is … Quest for Universe through

Position/Motion of Celestial Objects Also called: Fundamental Astronomy Astronomy in “Astronomy & Astrophysics”

Related with Celestial Mechanics Geodesy Special/General Theories of Relativity

5

General Principles

4-dim. Continuous Spacetime Law of Causality Time Arrow Definiteness Deterministic Principle Existence of Inertial Frame Principles of Relativity

6

Reference Systems RS=Coordinate System + Unit System Time Coordinate System

Astronomical, Physical, Broadcasting Space Coordinate System

Horizontal, Equatorial, Ecliptic Solar System Barycentric, Geocentric,

Terrestrial(=Earth-Crust-Fixed) Unit System: International, Astronomical

7

Motion Cosmic Expansion Quasi-Linear Motion: Far Objects

Stars, Galaxies, Quasars Orbital Motion

Quasi-Keplerian: Binary, Comet, Asteroid Complicated: Planet, Satellite, Space Vehicle

Rotation Earth, Moon, Planet, Satellite, Asteroid, etc.

8

Signal Propagation Electro-Magnetic Wave

Visible, IR, Radio, UV, X, Gamma Geometric Optics Approx.: Photon Path Relativistic Treatments

Cosmic Ray = High Energy Particle Gravitational Wave

9

Mathematical Tools Vector Analysis Linear Algebra Solution of Non-Linear Equation Method of Least Squares Fourier Analysis Numerical Integration of Ordinary

Differential Equations

10

1. Observation Global Quantities: Non-Measurable

Coordinates, Finite Length

Local Quantities: Measurable Clock Reading, Angle, Frequency, etc.

Measuring Methods Passive, Semi-Passive, Active

New Observing Facilities

11

Observables

Clock Reading Epoch: Arrival Time, Emission Time Time Interval = Duration Time

Angle: Difference in Incoming Vectors Others

Frequency = Energy Pattern, Code Embedded Artificially

12

Passive Observation Astro-Camera: 2D Angles

CCD Array, Video, Photographic Plate

Theodolite, Meridian Circle: 1D Angle Interferometer: Precise 1D Angle

VLBI=Very Long Baseline Interferometer Radio, Optical, IR, X-ray, …

Ground-based VS in-the-Space

13

Passive Observation (2) Detector: Arrival Time, Energy

PMT (Photo Multiplier Tube), Photo Diode CCD (Charge Coupled Device), Bubble Chamber

Clock Reading Event Time: Arrival, Eclipse, Occultation, etc.

Time Series: Light Curve, Decay Pattern Doppler Shift: Radial Velocity

Spectrometer, Emission/Absorption Lines

14

Semi-Passive Obs.

Doppler Shift Up/Down Link with Artificial Satellite or Space

Vehicle

Integrated Doppler Shits ~ Range Difference NNSS, DORIS/PRARE

Semi-Passive VLBI: ALSEP, RISE Difference Time Obs.: GPS, GLONASS,

GALILEO

15

Global Positioning System

US DoD Flying Atomic Clocks

16

Active Observation RADAR Bombing

Inner Planets, Near-Earth Asteroids Range and Range-Rate (R&RR)

Artificial Satellite, Space Vehicle Radio Transponding

Artificial Satellite, Space Vehicle LASER Ranging

Artificial Satellite (SLR), Moon (LLR)

17

LASER Ranging Satellite LR Lunar LR

3 Apollo + 2 Luna

18

RADAR Bombing Haystack, MIT Arecibo

19

New Facilities Optical/IR Interferometer

NPOI, PRIMA/VLTI, SIM, TPF-I Orbital Telescope

HIPPARCOS, JASMINE, GAIA VLBI

VLBA, VSOP, VERA, e-VLBI

20

NPOI US Navy Prototype Optical Interferometer Flagstaff, Arizona, USA

21

PRIMA/VLTI Phase-

Referenced Imaging and MicroarcsecondAstrometry

ESO, Chile VLT Outrigger

22

SIM Space Interferometer Mission, NASA

23

TPF-I Terrestrial Planet Finder-Interferometer

24

HIPPARCOS

First Satellite dedicated to Astrometry

ESA Great

Achievements

25

JASMINE Japanese Astrometry Satellite Mission

26

GAIA Post-HIPPARCOS ESA Will be Launched

in Summer 2011

27

VLBA VLBI Array 10 Stations

in USA NRAO, USA

28

VSOP First Space

VLBI Mission ISAS/NAOJ,

Japan

29

VERA Japanese VLBI Array 4 Stationsin Japan

Two Beam Differenced Observation

NAOJ

30

e-VLBI Online VLBI via High Speed Internet

31

2. Time Basic Concepts Ideal Time Systems

Integrated, Dynamical, Broadcasting Practical Time Scales

Atomic Time, Universal Time Solar System Barycentric/Coordinate Time

Units and Expression Julian Date

32

Concepts of Time Newtonian Viewpoint Absolute Time Time Transformation: 1 to 1 Ordering: Chronology Precision VS Accuracy

Essential Question on Repeatability

)(tft =

33

Integrated Time System

Assumption: Constant Duration of A Certain Phenomenon

Time = Number of Phenomena Example

Astronomical: Day, Month, Year Mechanical: Pendulum, Spring Physical: Quartz, Molecule, Atom

34

Dynamical Time System

Time Argument in Equation of Motion Epoch Determined Inversely from

Observation Example

Mean Longitude of the Sun L(T)= Epehemeris Time: ET=T(L)

2089113129602769044841279 T". T". ".' ++

35

Broadcasting Time System

Time Signals in the Air: JJY, TV, NTT NTP: TS on Computer Network GPS Time: TS from GPS Satellites

Standard Time Time Zone: 15 degree = 1 Hour

Japan Standard Time: JST JST = UTC + 9 h

36

Atomic Time Definition of SI Second: CGPM (1967)

9192631770 Periods Specific Radiation from Cesium 133

International Atomic Time: TAI Steered by BIPM (Paris) Hundreds of Cesium Atomic Clocks+ Several Hydrogen Maser Clocks

Relative Precision: 15-16 Digits

37

Cesium Atomic Clock HP/Agilent

5071A

38

Atomic Fountain Clock

39

Hydrogen Maser Clock

40

Universal Time Dynamical TS based on Earth Rotation

UT = GMT (Greenwich Mean Solar Time) 3 Variations: UT0, UT1, UT2 Monitored by IERS

UTC (Coordinated Universal Time) Leap Second

Secular Deceleration of Earth Rotation

41

Solar System Dynamical Time

Official TS of IAU (1984-1991) General Relativistic Effects

Considered TDB: SS Barycentric Dynamical Time TDT: Terrestrial Dynamical Time Unit Adjustment: <TDB> = <TDT>

TDT = TAI+32.184s

42

Solar System Coordinate Time

Official TS of IAU (1991-) No Unit Adjustment TCB: SS Barycentric Coordinate Time TCG: Geocentric Coordinate Time TT: Terrestrial Time TT = TDT = TAI+32.184s

TCB-TCG: Time Ephemeris Harada and Fukushima (2003)

43

Time Units 1 day=24 hours=1440 min.=86400 s Julian Century: jc, Julian Year: jy

1 jc = 100 jy = 36525 days

Besselian Year = Mean Solar Year = 365.2421897… days

ms, s, ns, ps, fs, … Speed of Light: c = 299792458 m/s

44

Time Expression Year, Month, Day, Hour, Minute, Second

Day of Week, Day of Year

Julian Date: JD J2000.0 = 12 O’clock, Jan. 1st, 2000= JD2451545.0

Modified Julian Date: MJD MJD = JD – 2400000.5

45

Julian Date From (Y, M, D, h, m, s) to JD

L=int((M-14)/12); I=1461*(Y+4800+L); J=367*(M-2-12*L); K=int((Y+4900+L)/100); N=int(I/4)+int(J/12)-int((3*K)/4)

46

Julian Date (2) JD0=N+D-32075; JD1=JD0-0.5; JD2=h/24.0+m/1440.0+s/86400.0; JD=JD1+JD2 or JD = (JD1,JD2)

47

Julian Date (3) From JD to (Y, M, D, h, m, s)

JD0=int(JD-0.5); JD1=JD0-0.5; L=JD0+68569; N=int((4*L)/146097); K=L-int((146097*N+3)/4); I=int(4000*(K+1))/1461001); P=K-int((1461*I)/4)+31;

48

Julian Date (4) J=int((80*P)/2447); D=P-int((2447*J)/80); Q=int(J/11); M=J+2-12*Q; Y=100*(N-49)+I+Q; JD2=JD-JD1; h=int(JD2*24) m=int(JD2*1440-h*60); s=JD2*86400-h*3600-m*60;

49

Day of Week

I=JD0-7*int((JD0+1)/7)+2; I: 1,2,3,4,5,6,7 I=1: Sunday

50

3. Space Space Coordinate and Unit Spacial Coordinate Transformation

Rectangular, Spherical, Spheroidal Inertial Coordinate System

Parallel Transport of Coordinate Origin, Rotation around Origin

Velocity and Acceleration

51

Spatial Coordinates Rectangular

Spherical

Spheroidal

),,( zyx

),,( ),,( lflq rr

),,( hlj

52

Spherical Coordinate Horizontal

Ecliptic Equatorial Galactic

)Az,El();Az,Alt();,();,( , AaAzr

dap ,,, ,r b l

,, bp

53

Horizontal Coordinate Radius: r, Zenith Distance: z Altitude (Angle)

a = Alt = El = 90 deg – z

Azimuth(al Angle)：A = Az, Left-Handed

÷÷÷

ø

ö

ççç

è

æ-=

÷÷÷

ø

ö

ççç

è

æ-=

÷÷÷

ø

ö

ççç

è

æ

aAa

Aar

zAz

Azr

zyx

sinsincos

coscos

cossinsin

cossin

54

Ecliptic Coordinate Ecliptic ~ Mean Earth Orbit For Solar System Objects Obliquity of Ecliptic:

Radius: r Longitude: Latitude:

Ecliptic

Equator

Vernal Equinox

55

Equatorial Coordinate Basic Representation

Right Ascension (R.A.) = Declination (Decl.) = (Annual) Parallax:

1 AUsinr

p - æ ö= ç ÷è ø

AU

r

S

E

P

56

Angle Units Radian: rad

180 deg = rad

Degree: deg = ° Minute of Arc: min = arc minute = ' Second of Arc: second = arc second = "

= arcsec = as

57

Angle Units (2) 1 deg = 60 arcmin = 3600 arcsec 180 deg = rad 1 arcsec ~ 4.848 rad

20 arcsec ~ 0.1 mrad: Aberration 0.001 arcsec = milli-arcsec: mas 0.000001 arcsec = micro-arcsec: as

58

Length Units SI meter: Defined via SI Second

Speed of Light: c = 299792458 m/s

Astronomical Unit (of Length): AU Rough: Mean Radius of Earth Orbit Rigorous: AU = c, = 499.00478353… s

Parsec (pc), Light Year (ly) 1 pc = AU/sin 1” ~ 30.9 Pm ~ 3.26 ly 1 ly = c x 1 jy ~ 9.5 Pm

59

Spheroidal Coordinate Geographic Latitude: Longitude: Height from Reference Ellipsoid: h

cos coscos sin

sin

N

N

Z

xyz

r j lr j l

r j

æ ö æ öç ÷ ç ÷=ç ÷ ç ÷ç ÷ ç ÷è ø è ø

60

Geographic Latitude Geocentric Latitude: Geographic/Geodetic Latitude:

Equator

PoleP

r

Geocenter H

Zenith

Nadir

Horizon

61

Spheriodal Coord. (2) Ellipsoid Normal: N

=Radius of Curvature ACROSS Meridian

( )j

rr

22

2

sin1 ,

1 ,

eddaN

hNehN ZN

-==

+-=+=

62

Ellipse Semi-Major Axis: a Semi-Minor Axis: b

12

2

2

2

2

2

=++bz

ay

ax

a

b

63

Flattening Factor Flattening Factor: f Eccentricity: e, Complimentary Ecc.: ec

2

2 22 2

2

, 1 1

2

ca b bf e e f

a aa be f f

a

-º º = - = -

-º = -

64

Spherical to Rectangular

sin cos cos cossin sin cos sin

cos sin

xy r rz

q l f lq l f lq f

æ ö æ ö æ öç ÷ ç ÷ ç ÷= =ç ÷ ç ÷ ç ÷ç ÷ ç ÷ ç ÷è ø è ø è ø

65

Rectangular to Spherical

),atan2

),,atan2sin

),,atan2cos

,

1

1

22222

x(y

p(zrz

z(prz

yxpzyxr

=

=÷øö

çèæ=

=÷øö

çèæ=

+=++=

-

-

l

f

q

66

Spheroidal to Rectangular

cos coscos sin

sin

N

N

Z

xyz

r j lr j l

r j

æ ö æ öç ÷ ç ÷=ç ÷ ç ÷ç ÷ ç ÷è ø è ø

( )j

rr

22

2

sin1 ,

1 ,

eddaN

hNehN ZN

-==

+-=+=

67

Rectangular to Spheroidal

Difficult Inverse Problem Easy: Longitude Eliminating Longitude

-> Latitude Equation

),atan2 x(y=l

( )( )( )

2 2

2

cos

1 sin

N h p x y

N e h z

j

j

ì + = º +ïí

- + =ïî

68

Latitude Equation After Elimination of h

2 2

2

sin cossin cos1 sin

where

Cp ze

C ae

j jj jj

- =-

=

69

Latitude Equation (2) Variable Transformation Transformed Equation Derivation and Solution

cott j=

2

2

( ) 0

where 1

Ctf t zt pg t

g e

º + - =+

º -

70

Derivation of Lat. Eq.

( )

2 2

2 2 2 22

2

2 2

1sin ,cos1 1

111 1 1 11

1

1

tt t

p zt C tt t t te

tCtp zte t

j j= =+ +

\ - =+ + + +-

+

- =- +

71

Solution of Lat. Eq.

(0) 0, ( ) 00

f p f zt Ct= - £ +¥ » + ³

\ £ £ +¥

Localization (Northern Hemisphere) Variable Domain after Localization

Newton Method Initial Guess 0 /

ptz C g

=+

z£0

72

Newton Method Effective to Solve Nonlinear Eq.

Essence Linearization

Newton Iteration

0)( =xf

)(')()(*

xfxfxxf -º)(* xfx ®

y=f(x)

xx0 x1x

y

73

Newton Method (2) Quadratic Convergence

Doubling Effective Digits Fast but Unstable Slow when Multiple Roots Key Points

Bracketing to Assure Uniqueness Selecting Stable Starters

74

Stable Starter Bracketing

Assumption 1

Assumption 2

Stable Starter: Upper Bound of Solution

RL xxx <<

( ) ( )RL xfxf << 0

0)('',0)(' >>®<< xfxfxxx RL

75

Application to Lat. Eq. Preparation

( )

( )

( )

2

32

52

( )

( 0 ) 0

'( ) 0

3''( ) 0

C tf t z t pg t

f p fC gf t z

g t

C g tf tg t

º + -+

= - £ £ + ¥

= + >+

-= <

+

76

Application (2) Newton Iteration

Stable Starter: Lower Bound of Solution*

0 0 (0)/

pt fz C g

= ® =+

( )( )

32 3

*3

2

( )( )'( )

p g t Ctf tf t tf t z g t Cg

+ -º - =

+ +

77

Velocity & Acceleration Velocity = Variation of Position Acceleration = Variation of Velocity Jerk = Variation of Acceleration

2 3

2 3

d d d, , dt dt dt

= = =x x xv a j

78

Velocity in Spherical CS

d d d ddt dt dt dt

r r

rr

v v vf f l l

f lf l

æ ö¶ ¶ ¶æ ö æ ö= = + +ç ÷ ç ÷ç ÷¶ ¶ ¶è ø è øè ø= + +

x x x xv

e e e

d d d, , cosdt dt dtrrv v r v rf l

f lf= = =

Vector Representation

Component Representation

79

Coordinate Triad in Spherical CS

÷÷÷

ø

ö

ççç

è

æ=÷

øö

çè涶

ºf

lflf

sinsincoscoscos

rrxe

÷÷÷

ø

ö

ççç

è

æ--

=÷÷ø

öççè

涶

ºf

lflf

ff

cossinsincossin

1 xer

÷÷÷

ø

ö

ççç

è

æ-=÷

øö

çè涶

º0

cossin

cos1 l

l

lflxe

r

80

Velocity in SpheroidalCS

d d ddt dt dt h hh v v v

h j j l lj l

j læ ö¶ ¶ ¶æ ö æ ö= + + = + +ç ÷ ç ÷ç ÷¶ ¶ ¶è ø è øè ø

x x xv e e e

( )( )

2

32 2

d d d, , cosdt dt dt

1,

1 sin

h M N

M

hv v v

a eM h M

e

j lj lr r j

rj

= = =

-= + =

-

Vector Representation

Component Representation

81

Coordinate Triad in Spheroidal CS

÷÷÷

ø

ö

ççç

è

æ=÷

øö

çè涶

ºj

ljlj

sinsincoscoscos

hhxe

÷÷÷

ø

ö

ççç

è

æ--

=÷÷ø

öççè

涶

ºj

ljlj

jrj

cossinsincossin

1 xeM

÷÷÷

ø

ö

ççç

è

æ-=÷

øö

çè涶

º0

cossin

cos1 l

l

ljrlxe

N

82

Radius of Curvature in Spheroidal CS

RC Across Meridian: East-West Direction

RC In Meridian: North-South Direction( )

( )322

2

sin1

1

je

eaM-

-=

j22 sin1 eaN

-=

( ) ( )d cos d sinsin , cos

d dN Z

M M

r j r jr j r j

j j= =

( ) hNehN ZN +-=+= 21 , rr

hMM +=r

83

5. Coordinate System 4-Dim. Coordinate System (CS)= Time System + Spatial CS

Inertial CS Accelerated CS and Inertial Force

Rotating CS: Coriolis F, Centrifugal F Coordinate Transformation

Galilean CS, Rigid-Body Rotation

84

Inertial Coordinate System

CS where Law of Inertia Holds Newton’s Law of Inertia

No Force -> Linear Motion

Galilei’s Principle of Relativity Law of Physics is Invariant at Any ICS

Parallel Transport of Coordinate Origin ICS to ICS

85

Parallel Transport of Coordinate Origin

Galactic Center in Quasar-Rest Frame Cosmic Expansion

Local Standard of Rest in Galactic CS Local Standard of Rest (LSR) = Solar

System Barycenter Feature of Local Motion: Oort’s Constant

86

Parallel Transport of Coordinate Origin (2)

Geocenter in Solar System Barycentric CS Planetary Ephemeris

Averaged Crust in Geocentric CS Earth Rotation

Observer in Terrestrial CS Fixed to Earth Surface (= Averaged Crust) Surface Motion (Aircraft, Ship, Car, etc)

87

Ephemeris and Almanac Numerical Table on Complicated Motion

Orbit: Planets, Satellites, Asteroids Rotation: Planets, Satellites

Astronomical Almanac (US+UK) Japanese Ephemeris NASA/JPL DE series, DE413/408

Most Precise, Machine Callable

88

Spatial Coordinate Transformation

General Transf.

Taylor Expansion w.r.t. New Coordinates

( )txXXx kjjk ,=¬

( ) ( )

( ) ( )

3

1

3

1

0, 0,

jj j k

k k

j jk kk

XX X t t x

x

A t B t x

=

=

¶= + +

¶

= + +

å

å

89

Linear Transformation

( ) ( )xAX tt B+= General Affine Transformation

Static: 12-Parameter Transformation

xAX B+=

90

Coefficient MatrixQ++= SDB

Scaling: Diagonal Component

Shear: Non-Diagonal, Symmetric

Infinitesimal Rotation: Asymmetric

kjjk ¹= if 0D

( )S S S 0 if jk kj jk j k= = =

kjjk Q-=Q

91

7-Parameter Transformation

CT between Similar Two CS Isotropic Scaling Origin Shift Rotation

Ex.: Transf. among Geocentric CSs World Geod. System (ITRFnn, GRS80) Tokyo Datum and JGD 2000

( )xXX Q++= Is0

92

6. Motion of Celestial Bodies

Rest: Quasar Linear: Most of Stars Rotation: Earth, the Moon, Satellite Kepler: Binaries Quasi-Kepler: Asteroid, Satellite Complicated: Planet, Space Vehicle

93

Resting Body Quasar: Practically Being Rest Position Expression

Epoch Mean Place at Epoch Parallax at Epoch

Quasar Catalogs: IAU, ICRFnn

0t( )00 ,da

0p

94

Linear Motion

Different Treatment for Radial Comp. Proper Motion = Linear Motion on

Celestial Sphere

( ) ( )000 ttt -+= vxx

÷÷÷

ø

ö

ççç

è

æ=

dadad

sinsincoscoscos

rx ( )0

0

0

0

ttVrr R

-÷÷÷

ø

ö

ççç

è

æ+÷÷÷

ø

ö

ççç

è

æ@

÷÷÷

ø

ö

ççç

è

æ

d

a

mm

da

da

95

Star Catalog Epoch, Mean Place, and Parallax Propor Motion Radial Velocity Astrophysical Information

Luminosity, Color, Variability, etc. Astrometric Star Catalogs

HIPPARCOS, FKn, PPM, AGKn

( )da mm ,

RV

96

7. Rotation Rotation = Orthogonal Transformation

Infinitesimal Rotation: Vector Product Finite Rotation: Orthogonal Matrix

Euler’s Theorem Fundamental Rotation Angular Velocity

97

OrthogonalTransformation

Distance Invariant in Euclidean Space

Rotation: A Linear Transformation

Orthogonality

( ) ( )22 xX D=D

xX D=D R

( ) ( )TT

T

RRIRRRRR==\

D=DD=D1

2T2

or -

xxxx

98

Finite Rotation

Expression: Matrix, Spinor, Quarternion Rotational Operation: Matrix Product Rotation Matrix = Coordinate Triad= Trio of Orthonormal Basis

( )TX Y Z= e e eRX

YZ

99

Euler’s Theorem

Any Finite Rotation = Triple Product of Fundamental Rotation Matrices

Euler Angles = 3 Fundamental Rotation Angles

( ) )()()(,, abggba ijkijk RRRRR º=

( )( ) ( )abggba ---=- ,,,, 1kjiijk RR

100

Fundamental Rotation Operation

Rotate around z-axis by the angle

)()(3 qq zRR =

X

Y

xy P

101

Fundamental Rotation Operation (2)

Rotation Around j-th Axis by

Reverse Rotation

)(qjR

( )( ) ( )qq -=-jj RR 1

102

Fundamental Rotation Matrix

Ex.: Ecliptic-Equatorial Transf. Obliquity of Ecliptic

÷÷÷

ø

ö

ççç

è

æ-=

1000cossin0sincos

)(3 qqqq

qR

( )e1Re

103

Fundamental Rotation Matrix (2)

Small Angle Approximation

( )

( ) ´÷÷ø

öççè

æ-@\

´-=÷÷÷

ø

ö

ççç

è

æ-+@

åÕj

jjj

jj e

e

qq

qqq

q

IR

IIR 33 0000000

104

Euler Rotation Combinations of Euler Angles: 3x2x2=12 3-1-3 (=X) Convention

Most Popular, the Euler Angles Used in Classic Rotational Dynamics

( ) ( ) ( ) ( )yqffqy 313313 ,, RRRR =

105

3-1-3 Rotation Matrix

( )÷÷÷

ø

ö

ççç

è

æ

-+---+-

=

qfqfq

qyfqyfyfqyfy

qyfqyfyfqyfy

fqyCCSSS

SCCCCSSSCCCSSSCCSSCSCSCC

,,313R

÷÷÷

ø

ö

ççç

è

æ

-+---+-

=qfqfqqyfqyfyfqyfyqyfqyfyfqyfy

coscossinsinsinsincoscoscoscossinsinsincoscoscossinsinsincoscossinsincossincossincoscos

106

3-1-3 Euler Angles

X

Z

YN

P

107

Weak Point of 3-1-3 Convention

( ) ´÷÷÷

ø

ö

ççç

è

æ

+-@

yf

qfqy 0,,313 IR

Degeneracy in Small Angles

Recipe Use 3-2-1 and Other Convention with

All Different Indices

108

3-2-3 Convention Alias: Y-Convention, Ex.: Precession

Screw: Rotation Around Fixed Direction

÷÷÷

ø

ö

ççç

è

æ=

jljlj

cossinsincossin

n

( ) ( ) ( )323 , , I+ sin 1 cosl j c c c= ´+ - ´ ´n n nR

( )AAA z--= ,,323 qzRP

109

Other Conventions 1-3-1: Nutation

2-1-3: Polar Motion + Sidereal Rotation

1-2-3: Aerodynamics, Attitude Control One of Most Desirable Conventions

( )( )eeye D+-D-= AA ,,131RN

( )pp xy --Q= ,,312RWS

110

Rotation and Velocity Transformation

= Þ

= +

= -W= - ´

X x

V v x

v xv ω x

RdRRdt

RR

111

Angular Velocity

( )

dddt dt

j j j jjj

jj

j

q q

q

æ ö= @ - ´ç ÷

è øé ùæ ö

= - ´ = - ´ = -Wê úç ÷è øë û

åÕ

å

e

e ω

R R I

R

ddt

jj

j

q=åω e

112

Infinitesimal Rotation 3D Anti-Symmetric Matrix ~ Axial Vector

True Meaning of Vector Product

xθx D´=QD

÷÷÷

ø

ö

ççç

è

æ

--

-=Q

00

0

xy

xz

yz

qqqqqq

÷÷÷

ø

ö

ççç

è

æ=

z

y

x

qqq

θ

113

Small Angle Rotation( )

´÷÷÷

ø

ö

ççç

è

æ-@

÷÷÷

ø

ö

ççç

è

æ

-++--+-+

=

=

gba

abggba

ababb

agabgagabgbg

agabgagabgbg

I

RRRR

CCSCSSCCSSCCSSSCSSSCSCCSSSCCC

)()()(,, 123123

114

8. Earth Rotation Base of Coordinate Transformation between

Geocentric and Terrestrial Coordinate System Sidereal Rotation (S) … Rotation Angle UT1 Motion of Figure Axis

Quasi-Diurnal: Polar Motion = Wobble (W) Others: Precession (P) + Nutation (N)

Matrix Representation

WSNPR =

115

Precession + Nutation Figure Axis Motion (Other than Wobble) 2 Components in Ecliptic CS

Longitude, Obliquity Precession=Very Long Periodic Motion

50 arcsec/y, ~26000y Period Nutation=Other Periodic Motion

18.6y, 0.5y, 9.3y, etc New Model Soon Appears

Ecliptic

Ecliptic Pole

z

116

Precession Discovery: Hipparchus (~150BC) Old Model: IAU1976

Lieske et al. (1976, A&A) Dynamics: Newcomb’s Theory Correction of Planetary Masses Adding Geodesic Precession

Theory: in Ecliptic CS Formula: in Equatorial CS

117

Precession (2)

3 Precession Angles in Equatorial CS

Unit: 1 arcsec T =(JD-2451545.0)/36525

( )AAA z--= ,,323 qzRP

32

018203.0041833.0

017998.0

09468.142665.0

30188.0

2181.23063109.20042181.2306

TTTzA

A

A

÷÷÷

ø

ö

ççç

è

æ-+

÷÷÷

ø

ö

ççç

è

æ-+

÷÷÷

ø

ö

ççç

è

æ=

÷÷÷

ø

ö

ççç

è

æqz

118

Precession (3) Approximation of Precession Matrix

Correction in R.A. and Decl.

÷÷÷

ø

ö

ççç

è

æ --@

1001

1

A

A

AA

qf

qfP AAA z+ºzf

tan sin , cosP A A P Aa f q d a d q aD = + D =

119

Precession (4) Approximation of Precession Angle

Precession (Speed) in R.A. and Decl.

Approximate Correction Formula

TnTm PAPA @@ qf ,

/jy"3109".2004 ,/jy"4362.4612 == PP nm

( )( )Tn

Tnm

PP

PPP

adada

cos ,sintan

@D+@D

120

Nutation Discovery: Bradley (1747) Old Model: IAU1980

Seidelmann et al. (1981, CM) Rigid Earth: Kinoshita (1977, CM) Non-Rigidity: Wahr (1981, GJRAS)

Mean Obliquity (Lieske et al., 1976)

32 001813".000059".0 8150".46448".21'2623

TTTA

+-

-°=e

121

Nutation (2)

( )( )eeye D+-D-= AA ,,131RN Matrix Representation

Nutation in Longitude Nutation in Obliquity Analytic Expression

åå=

=÷÷ø

öççè

æ=÷÷

ø

öççè

æDD 5

1 ,

cossin

jjjk

k kk

kk ΩnAAA

ey

ey

122

Delauney Angles Main 5 Angles in Nutation Theory

Mean Anomaly of Moon Mean Anomaly of Sun Mean Argument of Latitude of Moon Mean Elongation Mean Longitude of Ascending Node of Moon

Details: Seidelmann et al. (1981)

'

F'LLD -º

Ω

123

Rough Approx. of Nutation Precision: 0.1 arcsec Unit: 1 arcsec

+÷÷ø

öççè

æ+÷÷ø

öççè

æ+÷÷ø

öççè

æ-+

÷÷ø

öççè

æ-

+÷÷ø

öççè

æ-+÷÷ø

öççè

æ-=÷÷

ø

öççè

æDD

0sin1.0

0'sin1.0

2cos1.02sin2.0

2cos1.02sin2.0

'2cos6.0'2sin3.1

cos2.9sin2.17

LL

ΩΩ

LL

ΩΩ

ey

124

Approx. Nutation Approximation of Nutation Matrix

Nutation in R.A. and Decl.

÷÷÷

ø

ö

ççç

è

æ

DDD-DD-D-

@1

11

enemnm

N

AA eyneym sin,cos D=DD=D

125

Approx. Nutation (2) Correction in R.A. and Decl.

( )aeand

aeandmasincos

,cossintanD+D=D

D-D+D=D

N

N

126

Sidereal Rotation

( )3= QS R

Almost Uniform Quasi-Diurnal Rotation 0 = 7.2921150(1) x 10-5 radian/s

Angular Rotation = 360 degree/Sidereal Day ~ 365.2422.../366.2422... Rot./Day

Greenwich Apparent Sidereal Time (GAST)

127

Deviation from Uniform Rotation

UTC → UT1 → GMST → GAST DUT1 = UT1-UTC: Unpredictable GMST = GMST0 + r UT1 + ... Ratio of Sidereal/Universal Time: r r ~ 1.0027379... GAST = GMST + cos + ...

Length of Day (LOD) = 2/

128

Polar Motion = Wobble

( ) ( )2 1p px y= - -W R R

Slow Motion of Pole Viewed on Earth Symbol: (xp, yp), Size: 0.1 arcsec ~ 30m Periods: Annual, Chandler (~14 month)

Unpredictable = To be Monitored

129

EOP Earth Orientation Parameters

DUT1, LOD, xp, yp,, Pole Offsets Old Terms: Earth Rotation Parameters (ERP)

Pole Offset = Error in Prec./Nut. Theory International Earth Rotation Service (IERS)

Since 1984, Joint Service of IAU and IUGG Homepage: http://www.iers.org/

130

9. Keplerian Motion Solution of Two-Body Problem Gravitational Constant

Orbital Element = 6 Constants Shape of Orbit Orientation of Orbital Plane Location in Orbit

xx32

2

rdtd m-

=( )mMG +=m

ea,w,, IΩ

T

131

Unit of Mass SI Unit of Mass: kg Astronomical Unit: Solar Mass Universal Constant of Gravitation: G Observable = GM = Gravitational

Constant of Central Body Heliocentric GC = Sun’s GM Geocentric GC = Earth’s GM

SGM

EGM

SM

132

Orbital Elements Semi-Major Axis: a Orbital Eccentricity: e Longitude of Ascending Node: Orbital Inclination: I Argument of Pericenter: Time of Pericenter Passage: T

133

Ellipse Semi-Major Axis: a Semi-Minor Axis: b

12

2

2

2

=+by

ax

a

b

134

Orbital Eccentricity Eccentricity: e, Complimentary Ecc.: e’

22

22

1' , eabe

abae -=º

-º

ae

F

135

Orbital Plane 3-1-3 Euler Angles of Orbital Plane

3 Important Direction Vectors Origin of Longitude: X-axis Ascending Node: N Pericenter: P

( ) ( ) ( ) ( )W=W 313313 ,, RRRR II ww

136

Z

P

g

N

I

Orbital Plane (2)

137

Keplerian Orbit Elliptical: e < 1

Planet, Satellite, Binary Parabola: e = 1

Good Approximation of Comet Orbit Quasi-Parabola: e ~ 1

Comet, Peculiar Asteroids Hyperbolic: e > 1

Space Vehicle, Close Encounter

138

Element to Position and Velocity (Elliptic)

Solve (Elliptic) Kepler’s Equation

Speed of Ecc. Anomaly E PV in Orbital Plane

( )îíì

=-=Eb

eEasin

cosh

x

( )TtnEeE -=- sin

EenEcos1-

=

îíì

=-=

EEbEEa

cossin

hx

139

Element to Position and Velocity (Parabolic)

Solve Barker’s Eq. = Parabolic Kepler’s Eq. Speed of PV in Orbital Plane

( )21

2

q

q

x t

h t

ì = -ïí

=ïî

( )3

33 2t T

qt mt + = -

2 3

11 2q

mtt

=+

22

qq

x tth t

ì = -í

=î

140

Element to Position and Velocity (Hyperbolic)

Solve (Hyperbolic) Kepler’s Equation

Speed of F PV in Orbital Plane

( )coshsinh

a e Fb F

xh

ì = -í

=î

( )sinhe F F n t T- = -

cosh 1nF

e F=

-

sinhcosh

aF FbF F

xh

ì = -í

=î

141

Element to PV (2) Reverse Euler Rotation

( ) ( )÷÷÷

ø

ö

ççç

è

æ

---=00

,, hx

hx

w

ΩI, 313Rvx

142

Kepler’s Equation First Transcendenal Equation in History Elliptic

Parabolic

Hyperbolic

MEeE =- sin

PM=+3

3tt

HMFFe =-sinh

143

Elliptic Kepler’s Eq. Eccentric Anomaly: E Mean Anomaly: M Kepler’s 3rd Law True Anomaly: f

MEeE =- sin

( )îíì

===-=

frEbfreEa

sinsincoscos

hx

32an=m

( )M n t T= -

144

Solution of Kepler’s Eq.

Domain Reduction

Newton Method( )

( ) ( )( )

( )

*

* cos sin' 1 cos

E f E

f E M e E E Ef E E

f E e E

®

- -º - =

-

( ) 0sin =--º MEeEEf

0 0M M Ep p-¥ < < ¥Þ £ < Þ £ <

145

Stable Starter of Newton Merthod

Stability Theory ofNewton Method

Upper Bound as Stable Starter Examples

( ) ( )( ) ( ) 0'',0'

,00>>

<£EfEf

ff p

( ) ( )

÷øö

çèæ

++

+-

=

÷÷ø

öççè

æ÷øö

çèæ=

eeMeM

eM

fffE

1 ,,

1min

,2

,0min ***0

p

pp

( )0* Ef

146

Perturbed Keplerian Orbit

Element = Slow Function of Time

Perturbation Theory Polynomials + Fourier Series

( ) ( )tΛTΩIeaΛ =º ,,,,, w

( )å ++

+++=

kkkkk tStC

tΛtΛΛΛnn sincos

2210

147

Complicated Orbit Eq. of Motion Solution

Numerical: Numerical Integration Analytical: Perturbation Theory

Parameters Estimation by Fitting Solution to Obs. Data

Result: Astronomical Ephemeris

+-= xx

32

2

rdtd m

148

Astronomical Ephemeris Numerical: DE (NASA/JPL, USA) Analytical: VSOP/ELP (BdL, France) DE: Available through NAOJ/ADAC

Software (Fortran/C) + Binary Files DE408: BC10000-AD10000, UNIX/Win/Mac PV of Sun, Moon, and 9 Major Planets

Whole Solar System Bodies: HORIZONS http://ssd.jpl.nasa.gov/

149

10. Signal Propagation Geometric Optics Approximation Basic: One-way Propagation Application: Multi-way Prop. Light Direction: Aberration & Parallax Doppler Shift Propagation Delay

150

One-Way Propagation Photon: Linear Motion

Constant Speed of Light Special Theory of Relativity

( ) ( )000 ttt -+= VXX

c=0V

Source

Observer

t = t0

t = t1

Photon

151

Passive Observables Arrival Epoch

Incoming Direction

Observed Wavelength

1t

1d

1l

152

Eq. of Light Time Within Solar System Departure Epoch Arrival Epoch Light Time = Duration

Equation to Solve LT ( )10c Rt t=

S

O

01 tt -ºt

0t1t

153

Eq. of Light Time (2) Diff. in Departure/Arrival Position

( )1 0 1 0t t- = -x x V

Evaluate Magnitude of Diff. Vector

Assume that Source/Observer Motions are Known

10R Vt=

( ) ( )tt OS xx ,

154

Eq. of Light Time (3)

V c=

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

0 0 1 1 10 10

10 1 0 1 1

, , ,S O

O S S

t t R

t t t

t t

t t

= = º

= - = - -

x x x x R

R x x x x

Use Constant Speed of Light

Final: Equation of Light Time

( )10c Rt t=

155

Eq. of Light Time (4)

Newton Method

Correction Formula

( ) ( ) 0f c Rt t tº - =

( )*ft t®

( ) ( )( )

( ) ( )( )

* '' '

f R Rf

f c Rt t t t

t tt t

-¢ º - =-

156

Eq. of Light Time (5) Initial Guess: Infinite Speed of Light One Newton Correction

Next Stage: General Relativity Needed

( ) ( )1 * 0 SO

SO

Rfc V

t º =-

( ) ( )( ) ( )( )1 1 1 11 1 , S S

SO S SOSO

t tR t V

R- × -

= - =v v x x

x x

157

Light Direction

Aberration: Effect of Observer’s Velocity Parallax: Effect of Observer’s Position Periods: Annual, Diurnal, Monthly, etc. Correction for Light Time: MUST within

Solar System

101

1 10R--

= =RVd

V

158

Aberration Bradley (1727) Finiteness of Speed of Light

Ex.: Raindrops Trails on Side Window Vector Expression of Aberration

( ) ( )1 1

1 1

' cc c

- - - ×+= = » +

- +1 1 1

1

V v v d v dd vd dV v d v

159

Annual Aberration Effect of Orbital Motion of Earth (Annual) Aberration Constant

Angle Expression

"2010km/s 103

km/s 30 45 »=

´»º -

cvEk

' sinq q k q@ -

S

E0

’

E1

vE

160

Annual Aberration (2) Ecliptic Coordinate System is Useful Approximation Formula

Mean Longitude of Sun: L Aberration Ellipse

( )( ) ( )îíì

--»D-»D

lklblbkb

LL

A

A

coscossinsin

( ) 1sin

cos22

=÷÷ø

öççè

æ D+÷

øö

çèæ D

bkb

klb AA

161

Diurnal Aberration Effect of Earth Rotation Equatorial CS is Useful Diurnal Aberration Constant

Approx.Formula Sidereal Rotation Angle: Geoc. Lat.:

( )( ) ( )îíì

-Q-»D-Q»D

afkadadfkd

coscos''cossinsincos''

A

A

"3.0106.1m/s103

m/s480' 68 »´=

´»º -

cR EEwk

162

Parallax Bessel (1838): 81 Cygni Deviation of Observer’s Position from

its Mean Value Ex.: Direction Difference between

Right/Left Eye’s View

Vector Expression of Parallax( )1 0 1 00 1 0 1 0

00 1 0 1 0 0

rR r r

- ×- -= = = » -

- -x d x dx x d xRd d

x x d x

163

Annual Parallax Effect of Earth Orbital Motion Alternative Distance Measure

Angle Expression0

AU 1r

»p

00 sinqpqq +»Sun E

S

0

164

Annual Parallax (2) Approximation Formula in Ecliptic CS

Note: 90 deg Phase Diff. from Aberration Parallactic Ellipse

( )( ) ( )îíì

-»D-»D

00

00

sincoscossin

lplblbpb

p

p

LL

( ) 1sin

cos2

0

20 =÷÷

ø

öççè

æ D+÷

øö

çèæ D

bpb

plb pp

165

Diurnal Parallax Effect of Earth Radius: Moon, Artificial Sat. Approximation Formula in Equatorial CS

Yet Another Distance Measure: Horizontal Parallax

( )( ) ( )îíì

-Q»D-Q»Dafpadadfpd

p

p

sincos''coscossincos''

ppp 51 104AU1

sin' -- ´»÷øö

çèæ»÷

øö

çèæº EE R

rR

166

Doppler Shift Classic (= Non-Relativistic)

Approximation

Outgoing Object = Red Shift Incoming Object = Blue Shift

( )c

z dvv ×-=

-º 10

0

01

lll

167

Doppler Shift (2) Similar to Aberration

Again Aberration Constant

Annual Doppler Shift

Diurnal Doppler Shift

( )lbk -»D Lz sincos

( )adfk -»D Θz sincoscos''

168

Propagation Delay Vacuum Delay: General Relativity

Color Independent

Medium Delay Eminent in Longer Wavelength (Radio, etc.) Inter-Galactic/Stellar Matter, Solar Corona Ionosphere, Troposphere Atomosphere

169

Wavelength-Dependent Delay

Elimination by Multiple Wave Observation Geodetic VLBI: S-band + X-band GPS: L1-band + L2 band Space Vehicle: Up-Link + Down-Link

Use of Empirical Model: Not-Good Solar Corona, Ionosphere, Troposphere

( ) +++=D 2fC

fBAft

170

Delay Model Solar Corona

Muhleman and Anderson (1981)

Troposphere (Chao 1970): Zenith Angle, z

ò=D dsNcf e2CORONA

3.40t += 6rANe

045.0cot0014.0cos

ns7TROP

++

=D

zz

t

171

Refraction Variation in Incidental Zenith Angle

Saastamoinen (1972)

P: Atmosph. Pressure (Unit: hP)PW: Water Vapor Pr. (Unit: hP) T: Absolute Temperature (Unit: K)

++=D zbzaz 3tantan

z÷øö

çèæ -

=T

PPa W156.0271".16

172

Multi-Way Propagation Appl. of One-Way Prop. Series of Eq. of Light Time

Ex.: Three-Way (t3, t2, t1, t0 )

Delay in Relay Optical: 0 Radio: Constant

Specific to Transponder

Source

Observer

Relay 1

Relay 2

t0

t1

t2

t3

173

Round-Trip Propagation Typical Active Observation Observable

Emission/Reception Epochs Useful even when Target

Motion is Unknown Sum of One-way Prop. Cancellation at 1st Order Observer

Target

t2

t0

t1

174

Round-Trip Light Time Approximation of Reflection Epoch

Approximation of Distance at Reflection

20 2 2 0

1 2 2t t t tvt O

cæ ö+ -æ öæ ö= + ç ÷ç ÷ ç ÷ç ÷è ø è øè ø

( ) ( ) ( )2

2 01 11 ,

2SO SO S O

c t t vR O R t tc

é ù- æ ö= + = -ê úç ÷è øê úë û

x x

175

Quasi-Simultaneous Propagation

t2

Almost Same Arrival Pair of Eq. of Light Time Difference in Arrival Epoch

t1

t0

Observer 1

Observer 2

Source

b

k12 tt -=t

176

Interference Observation Equation

( )( )

2 1

0 1 2

0 1 2

/ 2/ 2

= -

- +=

- +

b x xx x x

kx x x

Difference in Eq. of Light Time Alias: VLBI Observation Eq.

Baseline Vector b Midpoint Direction k

ct = - ×b k

177

Quasi-Periodic Propagation

Arrivals with Similar Interval Series of Eq. of Light Time Initial Arrival Epoch

Assumption Const. Interval at Source

0N Nt t tD = -

T0 ,X0

Observer

t0 , x0

…

TN ,XN

tN , xN

Source

N-th

0N NT T T N TD = - = D

0th

178

Arrival-Time Observation Equation

( ) ( )0 0

0 00

0 0

N N N= - - -

-=

-

B x x X XX xKX x

Diff. from Initial Eq. of Light Time Pulsar Arrival-Time Observation Eq.

Baseline Vector B Initial Direction K

00

NN N

Bc t cN T OR

æ öD = D - × + ç ÷

è øB K

179

11. Least Square Method (LSM)

Gauss (1801): Ceres Orbit Determination Typical Optimization Problem Objective Function: ()

Optimization = 0 PD of Objective Function= Set of Linear Equation (=Normal Eq.)

( ) ( ) 2,j j

jf t gl lé ùF = -ë ûå

180

Application of LSM Data Analysis by Model Fitting

Linear Motion: Mean Place/Proper Motion Kepler Ellipse: Binary Orbit Determination Kepler Parabola: Comet Orbit Determination Offset: Correction to Existing Model Model Parameters: Geopotential Coefficients Initial Conditions: Numerical Ephemeris Proper Elements: Analytical Orbit Theory

181

Zero Partial Derivative Optimization = Zero PD Taylor Expansion

Usage of Newton Method Normal Equation H blD = -

( )2

0 0

jji i i j

l ll l l l

æ öæ ö æ ö¶F ¶F ¶ F= + D +ç ÷ç ÷ ç ÷ ç ÷¶ ¶ ¶ ¶è ø è ø è ø

å

0il

¶F=

¶

182

Normal Equation Hessian Matrix: Positive Def., Symmetric Standard: Modified Cholesky Method Caution!: Rank Deficiency, Degeneracy Recipe

General Inverse: Popular in Geodesy Orthogonal Basis Expansion Check Correlation Among Variables Good Initial Guess

183

Extension of LSM Weighted LSM

Chi-Square Fitting Non-Linear LSM

Gaussian Approx., Quasi-Newton Method

LSM Associated with Dynamical System Integration of Variational Eq. of Motion

184

Error Estimation Variance-Covariance Matrix:

Correlation among Parameters Diagonalization of Hessian Matrix

Determine Error Ellipsoid Minimum of Obj. F.

No Meaning if Non-Diagonalized Practical Estimate: Very Difficult

02j

jjHs F

=

185

12. Crush Course of General Rel. Effects

Theories and Principles Galilean Approximation Newtonian Approximation Post-Galilean Approximation Post-Newtonian Approximation Dragging of Inertial Frame

186

Relativistic Theory

Special Theory of Relativity Einstein’s General Theory of Relativity Other Gravitational Theories

Brans-Dicke, Nordvegt, Ng, … Scalar-Vector, Scalar-Tensor, … Parametrized Post-Newtonian (PPN)

Approximation

187

Principles Special Theory of Relativity (STR)

Principle of Special Relativity Principle of Constancy of Light Speed Principle of Limit of STR

General Theory of Relativity (GTR) Principle of General Relativity Principle of Equivalence Principle of Limit of GTR

188

Principle of Limit Unspoken but Important Special Theory of Relativity

Limit of Infinite c = Newton Mechanics General Theory of Relativity

Limit of Infinite c = Newton Mechanics + Law of Universal Attraction

Limit of Zero Gravity = STR

189

4-Dim. Spacetime 3+1 Dimension Expression

Metric Tensor

( )0,1,2,3 =mmx

( )3

2

, 0

d d ds g x xm nmn

m n =

= å

ctx =0

190

Proper Time

( ) ( )2 22 d dc st = - Definition

Reading of a Clock Moving with Observer

4-Velocity ddxum

m

t=

191

Galilean Metric

÷÷÷÷÷

ø

ö

ççççç

è

æ-

º@

1000010000100001

mnmn hg

÷÷ø

öççè

æ-=@

I00T

HG1

192

Lorentz Transformation

( )( ) ( )

cosh sinh sinh cosh

T

Ly y

y yæ ö

= ç ÷ç ÷Äè ø

nn n n

Basic Formula (1-D Space)

General Formula (3-D Space)

ˆ cosh sinhˆ sinh cosh

c tc txx

y yy y

Dæ öD æ öæ ö=ç ÷ ç ÷ç ÷DD è øè øè ø

1tanh vc

y -=

vvn =

193

Poincare Transformation Natural Extension of Lorentz Transf.

= Parallel Shift of Origin + Lorentz Transf. + Rotation

( ) mam

ama xPxxx Oˆˆˆ +=

÷÷ø

öççè

æ=

R001

RP LR=

194

Newtonian Metric

÷÷

ø

ö

çç

è

æ +-@I0

0T

cG 2

21 f

Gravitational Force Function Note Signature: > 0

195

Time Dilatation Newtonian Approximation

Lorentzian TD: Moving Clock Delays Gravitational TD: Delay Under Grav. F. Meaning of Effective Grav. Potential

2eff

2 2

d 11 1dt 2

vc c

ft fæ ö

» - + = -ç ÷è ø

196

Wavelength Shift Phase = Gauge Invariant

Independent on Choice of CS

2nd Order Doppler Shift Gravitational Red Shift

tt

wwq D-

=D

=D

Þ=Dff0

197

Post-Galilean Metric

÷÷÷÷

ø

ö

çççç

è

æ

÷øö

çèæ +

+-@

I1 2

2

2

21

c

cG

T

gf

f

0

0

198

PPN Formalism

C.F. Will (1981) Parametrized Post-Newtonian (PPN) F. PPN Parameters: =1, , , … =1

Principle of Equivalence One of Principles of Limit (GTR)

199

PPN Parameter GTR: = = 1、他は0 Nonlinearity of Grav. F.: Spatial Curvature: All Experiments Support GTR

Planetary Motion: = 1.00 Radio Bending by Sun: = 1.000

200

Geodesic Extension of Straight Line

Extended Law of Inertia in GTR

Timelike Geodesic: World Line (WL) of Massive Particle

Null Geodesic: WL of Particle with Zero Rest Mass (Photon, etc.)

Spacelike Geodesic: Spatial Coord. Axis

201

Eq. of Geodesic Principle of Equivalence

Gravity is Not A Force

Path of Free-Fall Particle = Geodesic Equation of Timelike Geodesic

d 0dua Γ u um

m m n rnrt

= + =

202

Christoffel’s Symbol

÷÷ø

öççè

æ¶

¶-

¶

¶+

¶

¶= r

mnnmr

mrnlrl

mn xg

xg

xg

gΓ21

Not A Tensor = Depends on CS Can Be 0 at One Point by Coord. Transf.

Extension of Gravitational Acceleration

Inverse Metric Tensor nl

mnlm d=gg

203

Eq. of Motion of Photon Path of Photon = Null Geodesic

Newtonian Gravitational Acceleration: a Solution by Successive Approximation

0dk Γ k kd

mm n rnrl

+ =

( )2 2

d 1 dt c c

g ×é ù+æ öÞ = + -ê úç ÷è ø ë û

a v vv 0 a

204

Gravitational Lensing Grav. Field = Convex Lense Deflection Angle

Large Defl.: 2~4 Images, Ring Microlensing = Light Amplification

Detection of MACHO

( )2

1tan

2S

SEc rg m yq

+D = S

E

P

205

Gravitational Delay Shapiro Effect (Shapiro 1964)

Radar Bombing of Planets Pulsar Arrival Time Observation

Solar System: Sun, Jupiter, Earth Binary Pulsar: Companion

S

P

E

( )3

1logS SE SP PE

SE SP PE

r r rc r r rg m

t+ æ ö+ +

D = ç ÷+ -è ø

206

Post-Newtonian Metric

÷÷÷÷

ø

ö

çççç

è

æ

÷øö

çèæ +

++-@

I1 23

342

2

221

cc

ccΦ

cG

T

gf

f

g

g

Nonlinear Scalar Potential Vector (Gravito-Magnetic) Potential g

2Φ bf= +

207

4-Acceleration 4-Dim. Acceleration

Absolute Derivative, D Proper (=Rest) Mass, m 4-Force

D dd du ua Γ u um m

m m n rnrt t

º = +

mm maf =

208

PN Eq. of Motion EIH（Einstein, Infeld, Hoffmann）Eq. of Motion

( )2 2

ddt

1 3 4

KK

J JK JK JK JKJ

J K JK JK

A Bc r r

m g¹

=

æ ö é ù++ + +ç ÷ ê ú

è ø ë ûå

v a

r v a

3 , J JKK JK J K

J K JKrm

¹

= = -å ra r r r

209

PN Eq. of Motion (2)

( ) ( )

( ) ( )

( ) ( )

2

22

,

2 2 1

3 1 2 1 ,2 2

2 2 1 2

JK J K

L LJK K

L K L JKL JL

JK J JK JJ J K

JK

JK JK K J

Ar r

r

B

m mb g b g

g g

g g

¹ ¹

= -

= - + - - +

æ ö× ×+ + - + × - +ç ÷

è ø= × + - +é ùë û

å å

v v v

v

r v r av v v

r v v

210

Dragging of Inertial Frame

Locally Parallel Shift of Origin Global Non-Rotation No Coriolis Force Rest w.r.t. Quasar

Fermi Transportation GR Extension of Parallel Shift of Origin

Proper CS = Fermi-Transported CS

211

Dragging of Inertial Frame (2)

Rotation Velocity of Proper CS STR: Thomas Precession GTR

Geodesic Precession ~1.92 arcsec/jc De Sitter (1917)

Lense-Thirring Effect Gravito-Magnetic Effect

3cav´

( )3

1c

g f+ ´Ñv

3cÑ´g

212

12. References Kovalevsky et al. (eds); 1989, Reference

Frames, Kluwer Acad. Publ. Seidelmann (ed.); 1992, Expl. Suppl. To

Astr. Almanac, Univ. Sci. Books. Soffel; 1989, Relativity in Astrometry,

Cele. Mech. & Geodesy, Springer-Verlag. Woolard and Clemence; 1966, Spherical

Astronomy, Acad. Press.

213

References (2) Kovalevsky and Seidelmann; 2004,

Fundamentals of Astronomy, Cambridge Univ. Press.

McCarthy and Petit (eds); 2004, IERS Convention 2003, IERS Tech. Note 32.

Smart; 1956, Spherical Astronomy, Cambridge Univ. Press.

214

Author Toshio FUKUSHIMA,Prof. Dr.

National AstronomicalObservatory of Japan (NAOJ)

2-21-1, Ohsawa, MitakaTokyo 181-8588, Japan

Toshio.Fukushima@nao.ac.jp http://chiron.mtk.nao.ac.jp/~toshio/