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