Introduction to Astrometry
Transcript of Introduction to Astrometry
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Introduction to Astrometry
Toshio FUKUSHIMANational Astronomical Observatory of Japan
2006
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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
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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
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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
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General Principles
4-dim. Continuous Spacetime Law of Causality Time Arrow Definiteness Deterministic Principle Existence of Inertial Frame Principles of Relativity
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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
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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.
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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
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Mathematical Tools Vector Analysis Linear Algebra Solution of Non-Linear Equation Method of Least Squares Fourier Analysis Numerical Integration of Ordinary
Differential Equations
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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
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Observables
Clock Reading Epoch: Arrival Time, Emission Time Time Interval = Duration Time
Angle: Difference in Incoming Vectors Others
Frequency = Energy Pattern, Code Embedded Artificially
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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
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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
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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
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Global Positioning System
US DoD Flying Atomic Clocks
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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)
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LASER Ranging Satellite LR Lunar LR
3 Apollo + 2 Luna
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RADAR Bombing Haystack, MIT Arecibo
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New Facilities Optical/IR Interferometer
NPOI, PRIMA/VLTI, SIM, TPF-I Orbital Telescope
HIPPARCOS, JASMINE, GAIA VLBI
VLBA, VSOP, VERA, e-VLBI
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NPOI US Navy Prototype Optical Interferometer Flagstaff, Arizona, USA
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PRIMA/VLTI Phase-
Referenced Imaging and MicroarcsecondAstrometry
ESO, Chile VLT Outrigger
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SIM Space Interferometer Mission, NASA
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TPF-I Terrestrial Planet Finder-Interferometer
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HIPPARCOS
First Satellite dedicated to Astrometry
ESA Great
Achievements
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JASMINE Japanese Astrometry Satellite Mission
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GAIA Post-HIPPARCOS ESA Will be Launched
in Summer 2011
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VLBA VLBI Array 10 Stations
in USA NRAO, USA
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VSOP First Space
VLBI Mission ISAS/NAOJ,
Japan
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VERA Japanese VLBI Array 4 Stationsin Japan
Two Beam Differenced Observation
NAOJ
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e-VLBI Online VLBI via High Speed Internet
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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
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Concepts of Time Newtonian Viewpoint Absolute Time Time Transformation: 1 to 1 Ordering: Chronology Precision VS Accuracy
Essential Question on Repeatability
)(tft =
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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
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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". ".' ++
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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
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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
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Cesium Atomic Clock HP/Agilent
5071A
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Atomic Fountain Clock
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Hydrogen Maser Clock
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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
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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
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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)
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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
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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
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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)
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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)
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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;
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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;
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Day of Week
I=JD0-7*int((JD0+1)/7)+2; I: 1,2,3,4,5,6,7 I=1: Sunday
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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
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Spatial Coordinates Rectangular
Spherical
Spheroidal
),,( zyx
),,( ),,( lflq rr
),,( hlj
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Spherical Coordinate Horizontal
Ecliptic Equatorial Galactic
)Az,El();Az,Alt();,();,( , AaAzr
dap ,,, ,r b l
,, bp
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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
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Ecliptic Coordinate Ecliptic ~ Mean Earth Orbit For Solar System Objects Obliquity of Ecliptic:
Radius: r Longitude: Latitude:
Ecliptic
Equator
Vernal Equinox
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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
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Angle Units Radian: rad
180 deg = rad
Degree: deg = ° Minute of Arc: min = arc minute = ' Second of Arc: second = arc second = "
= arcsec = as
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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
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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
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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
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Geographic Latitude Geocentric Latitude: Geographic/Geodetic Latitude:
Equator
PoleP
r
Geocenter H
Zenith
Nadir
Horizon
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Spheriodal Coord. (2) Ellipsoid Normal: N
=Radius of Curvature ACROSS Meridian
( )j
rr
22
2
sin1 ,
1 ,
eddaN
hNehN ZN
-==
+-=+=
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Ellipse Semi-Major Axis: a Semi-Minor Axis: b
12
2
2
2
2
2
=++bz
ay
ax
a
b
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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 = -
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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
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Rectangular to Spherical
),atan2
),,atan2sin
),,atan2cos
,
1
1
22222
x(y
p(zrz
z(prz
yxpzyxr
=
=÷øö
çèæ=
=÷øö
çèæ=
+=++=
-
-
l
f
q
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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
-==
+-=+=
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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
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Latitude Equation After Elimination of h
2 2
2
sin cossin cos1 sin
where
Cp ze
C ae
j jj jj
- =-
=
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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 -
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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= =+ +
\ - =+ + + +-
+
- =- +
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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
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Newton Method Effective to Solve Nonlinear Eq.
Essence Linearization
Newton Iteration
0)( =xf
)(')()(*
xfxfxxf -º)(* xfx ®
y=f(x)
xx0 x1x
y
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Newton Method (2) Quadratic Convergence
Doubling Effective Digits Fast but Unstable Slow when Multiple Roots Key Points
Bracketing to Assure Uniqueness Selecting Stable Starters
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Stable Starter Bracketing
Assumption 1
Assumption 2
Stable Starter: Upper Bound of Solution
RL xxx <<
( ) ( )RL xfxf << 0
0)('',0)(' >>®<< xfxfxxx RL
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Application to Lat. Eq. Preparation
( )
( )
( )
2
32
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( )
( 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 + -+
= - £ £ + ¥
= + >+
-= <
+
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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 - =
+ +
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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Resting Body Quasar: Practically Being Rest Position Expression
Epoch Mean Place at Epoch Parallax at Epoch
Quasar Catalogs: IAU, ICRFnn
0t( )00 ,da
0p
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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
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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
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7. Rotation Rotation = Orthogonal Transformation
Infinitesimal Rotation: Vector Product Finite Rotation: Orthogonal Matrix
Euler’s Theorem Fundamental Rotation Angular Velocity
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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
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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
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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
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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
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
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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
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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
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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
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
[email protected] http://chiron.mtk.nao.ac.jp/~toshio/