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