1999-2000 Lecture Notes on Astrometry

Spheroidal Coordinates

(Geographic) Latitude: Longitude : Height from reference ellipsoid: h

cos

sinsin

cossin

Z

N

N

z

y

x

1999-2000 Lecture Notes on Astrometry

Geographic Latitude Geocentric Latitude: Geographic (=Geodetic) Latitude:

Equator

Pole

P

r

Geocenter H

Zenith

Nadir

Horizon

1999-2000 Lecture Notes on Astrometry

Spheroidal Coordinates (contd.)

Curvature Radius along/across Meridian: Normal of Meridian: N

22

2

sin1 ,

1 ,

edd

aN

hNehN ZN

1999-2000 Lecture Notes on Astrometry

Ellipse

Semi-major axis: a Semi-minor axis: b 1

2

2

2

2

2

2

b

z

a

y

a

x

a

b

1999-2000 Lecture Notes on Astrometry

Flattening Factor

Flattening Factor: f Eccentricity: e, Co-Eccentricity: e’

22

222

2

2

11' ,

ffa

bae

fea

be

a

baf

1999-2000 Lecture Notes on Astrometry

Spheroidal to Rectangular

cos

sinsin

cossin

Z

N

N

z

y

x

22

2

sin1 ,

1 ,

edd

aN

hNehN ZN

1999-2000 Lecture Notes on Astrometry

Rectangular to Spheroidal

Not-so-easy Longitude Latitude Equation

Eliminating h

),atan2 x(y

zheN

yxphN

cos1

sin2

22

1999-2000 Lecture Notes on Astrometry

Latitude Equation

2

22

where

sin1

cossinsincos

aeC

e

Czp

1999-2000 Lecture Notes on Astrometry

Modified Latitude Eq. New Variable

New Equation

tant

2

2

1 where

01

)(

eg

pgt

Ctzttf

1999-2000 Lecture Notes on Astrometry

Derivation

22

22

2

22

22

22

11

1

1

11

111

1

1cos,

1sin

te

Ctztp

tt

t

tt

e

C

t

zt

t

ptt

t

1999-2000 Lecture Notes on Astrometry

Solving Modified Latitude Eq.

t

g

Cztfpf

0

0)( ,0)0(

Reduction of Argument Domain Variable Domain

Newton Method Initial Guess

Cz

pt

0

z0

1999-2000 Lecture Notes on Astrometry

Newton Method

Most Effective to Solve Nonlinear Eq.

Linear Approx. Newton Iteration

0)( xf

x

y=f(x)

x0 x1x

)('

)()(*

xf

xfxxf )(* xfx

y

1999-2000 Lecture Notes on Astrometry

Newton Method (contd.)

Second Order– Doubling Number of Correct Digits

Fast but Unstable Slow when Multiple Roots Key Points

– Bracketing … Simple Root– Stable Starter

1999-2000 Lecture Notes on Astrometry

Good Starter for Newton Method

Bracketed Solution

Assumption 1

Assumption 2

Stable Starter = Upper Bound

RL xxx

RL xfxf 0

0)('',0)(' xfxfxxx RL

1999-2000 Lecture Notes on Astrometry

Application (Mod. Lat. Eq.) Preparation

01

3)(''

01

)('

0)0(

1)(

52

32

2

gt

Cgttf

gt

Cztf

fpf

pgt

Ctzttf

1999-2000 Lecture Notes on Astrometry

Application (contd.) Newton Correction

Stable Starter = Lower Bound

Cz

pft

)0(0 *

0

Cgtz

Cgtgtp

tf

tfttf

3

2

33

2*

1

1

)('

)()(