Signal Propagation Electro-Magnetic Signal Geometric Approximation ~ Fast Particle Approximation ...
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Transcript of Signal Propagation Electro-Magnetic Signal Geometric Approximation ~ Fast Particle Approximation ...
Signal Propagation Electro-Magnetic Signal Geometric Approximation
~ Fast Particle Approximation Speed of Light in Vacuum
m/s 299792458c
1-Way Propagation
Linear Motion of Photon
Fast Motion + Non-Relativistic
000 ttt VXX
c0V
Source
Observer
t = t0
t = t1
photon
Passive Observables
Arrival Time
Incoming Direction
Received Wavelength
1t
1d
1
Equation of Light Time within Solar System Departure Time Arrival Time Light Time = Travel Time
Obtain Light Time
R
V
S
O01 tt
0t1t
Derivation of Eq. of Light Time Beginning/End of Photon Motion
2 1 2 1t t x x V
Taking the norm
Assumption: Body Motions are known
21R V
tt OS xx ,
Derivation (contd.)
V c
1 1 2 2
1 1
, ,
,
O S
S
t t
R t
x x x x
R R x x
Velocity Expression (Newtonian)
Velocity Expression (Special Relativity)
1S tV c
R
v R
Solving Eq. of Light Time
Newton Method
0 RVf
'*
f
ff
''
''*
VRV
VRRf
Approximate Solution Initial Guess: Infinite c = Zero Solution First Newton Corrector
Further Correction: General Relativity
111111
1111
*1
,
, ,
0'0
00
tRV
tR
Vc
R
RV
Rf
SSSOSSSO
SSSSO
SO
SO
vvxxvv
xxxx
Light Direction
Aberration: Observer’s Velocity Parallax: Offset of Observer’s Position Periodic: Annual, Diurnal, Monthly, … Correction for Light Time: within Solar
System
R
R
V
Vd
1
1
Aberration Finiteness of Speed of Light Bradley (1727) Track of Raindrops on Car’s Side Window
c
V
V
dvdvd
vd
vd
vV
vVd
11
1
1
11
11
1
1'
Annual Aberration Order of Magnitude = Aberration Constant
Angle Expression
"2010km/s 103
km/s 30 45
c
vE
sin'c
vE
S
E0
’
E1
vE
Annual Aberration (contd.) Adopting Ecliptic Coordinates Approximate Formula
Mean Longitude of Sun: L Aberration Ellipse
L
L
A
A
coscos
sinsin
1
sin
cos22
AA
Diurnal Aberration Adopting Equatorial Coordinates Approximate Formula
Sidereal Rotation Angle: Geocentric Latitude:
coscos''cos
sinsincos''
A
A
"3.0106.1m/s103
m/s480' 6
8
c
R EE
Parallax Offset of Observer’s Position Bessel (1838): 81 Cyg Direction Difference between L&R Eyes
0
01010
010
010
10
10
r
r
r
R
dxdxd
xd
xd
xx
xxRd
Annual Parallax
Order of Magnitude = Parallax
Angle Expression
0
AU 1
r
00 sin Sun E
S
0
Annual Parallax (contd.) Ecliptic Coordinates Approximate Formula
90°Phase Shift from Aberration Parallactic Ellipse
00
00
sincos
cossin
L
L
1
sin
cos2
0
2
0
Diurnal (Geocentric) Parallax Very close objects only: Moon Adopting Equatorial Coordinates Approximate Formula
Geocentric Parallax
sincos''cos
cossincos''
51 104AU1
sin'
EE R
r
R
Doppler Shift Newtonian Approximation
Outgoing = Red shift Incoming = Blue shift
c
zdvv
10
0
01
Approximate Doppler Shift Order of Magnitude = Aberration Constant Annual Doppler
Diurnal Doppler
Lz sincos
Θz sincoscos''
Propagation Delay/Diffractions Vacuum (= Gravitational)
– Wavelength independent
Non-Vacuum – Eminent in Radio wavelength– Intrergalactic, Interstellar, Solar corona– Ionospheric, Tropospheric– Atmospheric
Wavelength-Dependent Delay
Cancellation by 2 waves measurements– Geodetic VLBI: S-, X-bands– GPS: L1-, L2-bands– Artificial Satellites: Up- and Down-links
Empirical Model– Solar corona, Ionospheric, Tropospheric
2f
C
f
BAf
Delay Models Solar Corona (Muhleman and Anderson 19
81)
Tropospheric (Chao 1970)
dsNcf e2CORONA
3.40 6r
ANe
045.0cot0014.0
cos
ns7TROP
zz
Atmospheric Refraction Variation of Zenith Distance
Saastamoinen (1972)
P: Pressure in hP,
PW: Water Vapor Press. T: Temperature in K
zbzaz 3tantan
z
T
PPa W156.0
271".16
Multi-Way Propagation Variation of 1-Way Propagation Series of Light-Time Eq. Ex.: t3, t2, t1, t0
Transponder Delay– Optical: 0– Radio: Constant
Source
Observer
Transponder 1
Transponder 2
t0
t1
t2
t3
Round Trip Propagation Typical Active Observation Emission/Arrival Times No Need of Target Motion Info Sum of 1-Way Propagations Cancellation of 1-st Order
Effects
Observer
Target
t2
t0
t1
Round Trip Light Time Approximate Mid-Time
Approximate Distance at Mid-Time
2 ,
202
2
120
1
tt
c
VOt
ttt
11
2
02 ,2
ttR
c
V
R
RttcR
OSSO
SO
SOSO
xx
Simultaneous Propagation
t2
Almost Simultaneous Arrivals Summed Light Time Eq. Light Time of Mid-Point
Baseline Vector b Mid-Direction k
t1
t0
Observer 1
Observer 2
Source
b
k
212 tt
Summed Light Time Eq. Approximate Equation
2
210 2
,
c
VO
RRc
xxxR
R
Simult. Propagation (contd.)
t2
Differenced Light Time Eq. Arrival Time Delay
Baseline Vector b Mid-Direction k t1
t0
Observer 1
Observer 2
Source
b
k
12 tt
Eq. of Interferometric Obs.
1 2
c b k
b x x
Approximate Equation
= Equation of VLBI Observation