signal propagation
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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
RV
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
'*
fff
''''*
VRVVRRf
Approximate Solution Initial Guess: Infinite c = Zero Solution First Newton Corrector
Further Correction: General Relativity
111111
1111
*1
,
, ,
0'000
tRV
tR
VcR
RVRf
SSSOSSSO
SSSSO
SO
SO
vvxxvvxxxx
Light Direction
Aberration: Observer’s Velocity Parallax: Offset of Observer’s Position Periodic: Annual, Diurnal, Monthly, … Correction for Light Time: within Solar
System
RR
VVd
1
1
Aberration Finiteness of Speed of Light Bradley (1727) Track of Raindrops on Car’s Side Window
c
VV
dvdvd
vdvd
vVvVd
11
1
1
11
11
1
1'
Annual Aberration Order of Magnitude = Aberration Constant
Angle Expression
"2010km/s 103
km/s 30 45
cvE
sin'cvE
S
E0
’
E1
vE
Annual Aberration (contd.) Adopting Ecliptic Coordinates Approximate Formula
Mean Longitude of Sun: L Aberration Ellipse
LL
A
A
coscossinsin
1sin
cos22
AA
Diurnal Aberration Adopting Equatorial Coordinates Approximate Formula
Sidereal Rotation Angle: Geocentric Latitude:
coscos''cossinsincos''
A
A
"3.0106.1m/s103
m/s480' 68
cR EE
Parallax Offset of Observer’s Position Bessel (1838): 81 Cyg Direction Difference between L&R Eyes
0
01010
010
010
10
10
r
rr
R
dxdxd
xdxd
xxxxRd
Annual Parallax
Order of Magnitude = Parallax
Angle Expression
0
AU 1r
00 sin Sun E
S
0
Annual Parallax (contd.) Ecliptic Coordinates Approximate Formula
90°Phase Shift from Aberration Parallactic Ellipse
00
00
sincoscossin
LL
1sin
cos2
0
20
Diurnal (Geocentric) Parallax Very close objects only: Moon Adopting Equatorial Coordinates Approximate Formula
Geocentric Parallax
sincos''coscossincos''
51 104AU1
sin'
EE R
rR
Doppler Shift Newtonian Approximation
Outgoing = Red shift Incoming = Blue shift
c
z dvv
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
2fC
fBAf
Delay Models Solar Corona (Muhleman and Anderson 19
81)
Tropospheric (Chao 1970)
dsNcf e2CORONA
3.40 6rANe
045.0cot0014.0cos
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 W156.0271".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
1tt
cVOtttt
11
202 ,
2ttR
cV
RRttcR
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
,
cVO
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
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