天文觀測 I

Optical Telescope

Telescope

• The main purposes of astronomical telescope:– To collect the weak light (photons) from sky.

– To map the sky to image

– To enhance the angular separations among the astrophysical objects.

• The developments of telescopes do not only depend on the telescope developments themselves but also on the improvements of techniques of the analyzers, detectors or even space and computer science.

Telescope

• Basic equipments of a telescope:– Telescope

• Mirrors (reflector), lenses (refractor).

– Analyzer• Filter, spectrograph, polarimeter

– Detector• Photographic plate, photoelectric device (CCD,

PMT, photodiode)

Light Collection and Limiting Magnitude

• One of the major purposes is to collect the light from astrophysical objects.

• The light collection ability is proportional to the area of primary mirror (lens).

• Limiting magnitude – the faintest star can be seen

• The limiting magnitude for the wide-open, dark-adopted human eyes is +6.

• However, if the eyes are well collimated (FOV ~5 arcmin), the limit magnitude can be improved to +8.5

• mlim (human eyes) = +8.5(V) ~200 photons/s

Limiting Magnitude

• The telescope with eyepiece

2 2

2 2

lim lim lim lim

limlim lim

lim

lim

Telescope diameter: D; Pupil's diameter: d~8mm

Flux:

6 2.5log ; 2.5log

2.5log

6 5log 16 5log1

For a 1/3 meter

tt e

e

e e t t

tt e

e

t

F D DF F

F d d

m F const m F const

Fm m

F

D Dm

d m

limtelescope, 13.6tm

Limiting Magnitude

• For the professional telescope (with detector)

human eyes

lim 2

lim lim

Photons current (cannot be integrated)

Detector on telescope can do the integration

1

2.5log

5log 2.5log1 1

The improvement of limiting magnitu

FD T

m F const

D tconst

m s

de is very inefficient with

increasing of exposure time

Imaging• Basically, the profession astronomers prefer reflecting

telescope telescopes rather than refracting ones mainly owing to the chromatic aberration and also the mechanical consideration.

• However, for easily drawing, I will use lens instead of mirror to describe the optical properties of telescope.

chromatic aberrationspherical aberration

parabolic mirrorspherical mirror

Imaging – Geometric Optics

• Ray tracing/ matrix method:

Define a column matrix

; <<1x

α

x

Reference line(Optical axis)

α:+

-x:

about the reference line:+

below the reference line: -

Imaging – Geometric Optics

x

x

M

11 12

21 22

11 12

21 22

Matrix M

m mM

m mx x x x

m m x

x m m x

Ray Tracing – Translation

α

x

α’

x’

tan

1 0 1 0

1 1

x x D x D

Tx D x D

D

Ray Tracing – Lens

• For a lens with focal length f and thickness t 0

• (2) Light from focus parallel light

• (1) Parallel light concentrated on focus

f

f

Ray Tracing – Refraction (1)

11 12

21 22

12 12

22 22

11

21

No translation (D=0)

=0; tan =- 1 =-

0 0

1

1

1

1

x x

f f

x x

xm m

f Rm mx x x

x

xm x m

f f

x m x m

mfR

m

Ray Tracing – Refraction (2)

11

21

11 11

21 21

No translation (D=0)

; =0

10

1

0 1

0

11

0 1

x

f

x x

x xm

ff fRx x

mx x

x xm m

f f

xx m x m

f

fR

Ray Tracing – Single Lens

f

First refraction then translation (D=f)

111 0

10 1

11

0

(not important)

fTRx x f x

fx

f

f

x f

x

Focal plane

Star light

Telescope

• The most important matrix elements in the combined matrix are– m21= f (focal length)

– m22=0

– So the x’=fαand independent of x (where the light incident on th elens)

• No matter how complex the optical system is, the combined matrix m21=f (effective focal length) and m22=0.

Convert the incident angle

to displacement on focal plane

x f

x

Telescope with Eye Piece

Telescope with Eye Piece

• Two lenses with focal lengths of f1 (primary lens) and f2 (eye piece).

• The distance between two lenses is D=f1+f2.

Star lightf1 f2

D

Eye pieceLens 1 Lens 2

Telescope with Eye Piece

2 1

1

22 1

1 2 21

1

1

2 1

2 21

1

1

2

Combined matrix

01 11 11 0

10 1 0 1

0

Angular magnification= 1

Minus sign me

M R TR

f

ff f

f f ff f

f

f

f f

x f x ff f

f

f

f

1 2

1 1

2 2

ans the image is reversed

If the eye piece is concave lens and

The image is not reversed

D f f

f f

f f

Off-axis Aberration

• The derivations above are only valid for the small incident angle (angle between star light and optical axis)

• For the large field of view, higher order terms make off- axis aberration.

• Off-axis aberrations:– Coma

– Astigmatism

– distortion

2 3

1 2 3

coma astigmatism distortion

For point source, : size of image(dueto aberation)z

z c c c

PSR B0540-69

LMC X-1Extended source?

Off-axisaberration

Coma

Astigmatism

Coma

Focal Ratio (F-number)

• All the images of astrophysical source on the focal plane have finite size even for the point source because– Diffraction

– Seeing

– Off-axis aberration

f

Focal planeDiameter: D

Angular size

Radius r f

Focal Ratio (F-number)

22

22

2 22

Incident intensity:

Flux on focal plane:

2

1 1 1 1 1

4 4 4

and are the properties of astrophysical object

: property of telescope

d

d

d

I

F

DF r I

D DF I I I

r f fD

I

f

D

Focal Ratio (F-number)

• f/D = focal ratio, written as f/#, called f-number.– f/3.5 f/D=3.5

• Smaller f/# gives larger image flux

• For a faint extended source (e.g. distant galaxy)– Large f/# (small Fd) D small or f large or both

– Small D small number incident photons

– Large f image spreads out over large area

– Need a small f/#

• For bright source (e.g. planet)– The flux is not a problem. To resolve the fine structure of the

source, large f/# is better.

Point-Spread Function (PSF)

• Even for a point source, the image on the focal plane would spread out to finite area due to diffraction and seeing.

• The extended source would be “smeared”.

0 0

0 0

, : sky image; , : image on focal plane

, : Point-spread function

, , ,

For point source, , ,

: function

, ,

o

O I

P

I P O d d

O O

I P

Point-Spread Function (PSF)

• Diffraction: size of Airy disk: δθ=1.22λ/D. For D=1m, λ=500nm, δθ=0.1 arcsec

• Seeing: due to the disturbance of the atmosphere– Size less than 1 arc second to

several arc second, highly dependent on weather and site.

Airy disk

Short exposure

Longexposure

Seeing disk

Point-Spread Function (PSF)

• IRAF gives 6 functions to model PSF

22 2 2

2

2

Gaussian exp ;2

1Lorentz (with wings)

1

Penny1 Gaussian Core & Lorentz wings

Penny2 Gaussian Core & Lorentz wings

rr x y

r

2 2

2 22

which can free tilted

1Moffat15 ; ; 1.5

1

(oblique PSF)

Moffat25 Same as Moffat15 but =2.5

xyx y

x yz xy

z

Point-Spread Function (PSF)

Gaussian

Lorentzα=1β=1

Lorentzα=1β=2

Lorentzα=2β=1

Detector – CCD

• CCD -- Charge Coupling Device.

• Use photoelectric effect

• Unlike the X-ray to measure the energy of photoelectron, the CCD for optical is just “count” the number of photons. In principle 1 photon 1 photoelectron.

• Most of photons hit the CCD can be converted into photoelectrons but only a part of them can be collected. However, for the CCD equipped with astronomical telescope, the efficiency (called quantum efficiency (QE)) is very high (>90%), and thus, high sensitivity.

• The pixel size can be made very small (~20 μm) so the spatial resolution can be very high.

Detector – CCD

• The CCDs have been used in LOT

CCD type Pixel Number Pixel Size

(μm x μm )

Plate Scale

(arcsec/pixel)

ADC

(bits)

FLI IMG1024S 1024 x 1024 24 x 24 0.62 16

Apogee AP-8 1024 x 1024 24 x 24 0.62 16

PI 1300 B 1340 x 1300 20 x 20 0.52 16

-6 -6

LOT 8000 mm

18020 m 8000mm=2.5 10 rad 2.5 10 60 60 0.52 arcsec

1300 0.52=670 arcsec=11.17 arcmin

1340 0.52=697 arcsec=11.61 arcmin

f

CCDSemi conductor: band structure

Conducing band(empty)

Valence band(full)

Band gapEg< 1 eV

Visible light photon energy : 1.7 eV to 3 eV

Photon

Photoelectron

hole Voltageapplied

Detector – CCD

ExposureIncident Photon

Potentialwell

Photoelectrons

Detector – CCD Reading

CCD – Dark Current

Thermal excitation

If the thermal electrons were stroed in CCD equilibrium

However, the thermal electrons keeping extracting

from potential well equilibrium broken

More and more

gE

B kT

A

Ne

N

thermal electrons would be generated

Dark Current

Lower temperature lower ( ) value

low dark current

50

iEBkT

A

CCD

Ne

N

T C

Thermal electrons

CCD – Dark Current

16

1 1

For room temperature (T 300 K)

1 10

>> ;

For room temperature (T 300 K)

For CCD operating temperature (T=-50 C=223 K)

g

g

r

g

c

g

c r

E

B kT

A

A B A A B

E

kTB

A

E

kTB B

A A

E

k T TB

B

Ne

N

N N N N N

Ne

N

N Ne

N N

Ne

N

63 10

The dark current is reduced by more than a factor of ten thousand !!

CCD – Flat Filed & Bias• The quantum efficiencies (QEs) may different from pixel

to pixel and also depend on the wavelength.

ADC

n

CCD analog signal digitized signal

ADC: Analog-Digital Converter

For a signal being digitized to n bits, there are

2 different values.

For example, n=16 0 - 65535

However, the baseline of analog

signal may change

with time and ADC can not measure the negative signal.

The baseline is usually intentionally adjusted to a positive

value so the ADC can convert it to meaningful value bias

CCD – Digital Output

ADC

Analog signal Audio-to-DigitalConverter

Digitized signal

exposure

From photoelectrons

T

exposure

From dark current

T

From bias

constant with time

Flat field

Data Size of Image

For PI1300B CCD,

1340 1300 =1742000 pixels

16 bits/pixel = 2 byte/pixel

1742000 2=348400 bytes=3.3226 Mbytes

(not including the header)

1 min exposure, 10 hours observation

3.3226 Mbytes 60 10=1993.56 MByte

s

=1.968 Gbyte/night

Photo Flux Estimation

-2 -12

-2 -1

2 -1

Human eyes : +8.5(V) ~200 photons/s

Pupil's diameter: d~8mm

200Photo flux= 400 ph cm s

0.4

4008.5 2.5log

For a star V=18.5 =0.04 ph cm s

For 1m telescope

50 314 ph s

For 10 min expo

m

m

m

mF

F

F

sure

314 600 188496 photons

Photo Flux Estimation

22

2 5 2

5 2

2

Area on focal plane=2

0.04 cm 4.8 10 cm

4.8 10 cmwhich cover 12 pixels

0.002 cm

18849615708 ph/pixel

12If the photons spread to 1024 pixel

188496184 ph/pixel

1024

r f

Optical Spectrograph

• For point source:– Photometry : collecting photons and try to concentrate

them to focal plane as much as possible

– Spectrography: the collected photons have to be reassigned according to the photon wavelength (i.e. spread them out)

• Thus to make the optical spectrum of star: – Large telescope is required

– Small telescope only for bright sources

Optical Spectrograph – Prism

• Prism : use the index of refraction as a function of wavelength to separate the light

Optical Spectrograph– Prism

sin sin

sin sin

In the equation

Only is the function of , and thus

D i r A

i n r

r n i

i A r

D i r A

r n

dndD dr dr

d d dn d

Optical Spectrograph—Prism

2 2

2 2

2 222 2

From the 4 equations, we have

sin sin sin cos sin

cos sinsin

1sin

sin 1 sin sin cos sin

Clearly, is not a linear function of

r A n i A i

n dndrr Ad dn i

n dndrA

d dn i A n i A i

r

Optical Spectrograph—Prism

• In addition to the non-linearity, there are other drawbacks for the prism.– Absorption

– Reflection: when the light pass through the media with different index of refraction, there must be reflection happening on the boundary

Optical Spectrograph—Grating

• The telescope for profession astronomers usually adopt grating spectrometer rather than the prism to observe the spectra from astrophysical objects.

• The grating spectrometer uses the interference of the light to separate the light with different wavelength.

• The reflecting grating spectrometers are more often seen than the transmission one.

Optical Spectrograph—Grating

Focal plane

Detector(e.g. CCD)

Huygen’s Principle

• Wavefront: the subspce of the wave with same phase.

Huygen’s Principle

• For a wavefront at t and t+Δt– Each point on the wavefront at t can be

considered as a point source.– The wavefront at t+Δt can be considered as the

“envelop” of outgoing wave from point sources

Huygen’s Principle – Reflection and Refraction

Wave Equation

22 2

A light (EM) wave in vacuum satisfied the

following equation

1

where is the wave function. In the EM wave,

it can be

(1)Electric field E

(2)Magnetic field B

(3)Vector potential A

(4)Scala

c t

��������������

��������������

��������������

r potential

EM Wave

2 2

2 22

2 2 2 2

2 2

2 22

2 2 2 2

Source free (vacuum) Maxwell equation

0 0

1 1

1 1 1

1 1 1

E B

B EE B

c t c t

E E E E

B E EE

c t c t c t

B B B B

E B BB

c t c t c t

Wave Equation – Linear Superposition

1 2

1 2

22 1

1 2 2

22 2

2 2 2

If both and are satisfied wave equation

The linear combination, = +

where and are constant, are also

satisfied wave equation

1

1

Add the

c t

c t

21 22

1 2 2 2

two equations

+1+

c t

Wave Equation – 1D

2 2

2 2 2

2

2

In 1-d, the wave equation is

, ,1

A function satisfied

,

can be the solution of the equation

Let

RHS:

x t x t

x c t

x t x ct

x ct u

x u uu

x u x u

x x u

x x x x u u

2

2

2 2 2

2

2 2 2 2

2 22

2 2 2 2

1 1 1LHS:

1 1 1

1 1

x u

x u

x u uuc

c t c u t c u

x x uc

c t c t t c t u

x u uc cc u t c u u

Wave – 1D

1

1

1

At t t

x ct

x

2 2 1

2

2

At t t t t

x ct

x

1 1 2 2

1 1 2 2

2 1

2 1

x ct x ct

x ct x ct

x xc

t t

Wave – 1D

0 0

Monochromatic wave can be generally expressed as

cos cos

where is a constant phase factor

For an observer at fixed point ( const),

the "wave" acting as a simple harmonic oscill

A k x ct A kx kct

x

0

ator, like

cos where 2

2 2

2, called wave number or wave vector

A t f

ckc f

k

Wave – 1D

0

0 0

0 0

2 20

cos

We like to express the sinusoidal function

in complex form: cos sin

cos Re

Now is a complex number.

Power amplitude

i

i kx t

i kx t i kx t i kx ti

A kx t

e i

A kx t A e

A e A e e Ae

A

A

2 2

20 0 0

i kx t i kx t

i i

Ae A e AA A

A e A e A

Re z

Im zz

Wave – 3D : Plane Wave

exp

ˆ ˆ ˆ

ˆ ˆ ˆ

At a fixed time is a constant

constant phase = wave front

constant

which is a plane

2,

Direction of propagation direction of t

x y z

x y z

A i k r t

r xx yy zz

k k x k y k z

t

k r k x k y k z

k

k

he wave

ˆIf expz zk k z A i k z t

Interference

1 2

1 2

1 2

2

2 2

1 2 1 2 1 2

1 1 2 2 1 2 1 2

2 2

1 2 1 2

Power Power Interference of of

The wave function obeys

the linear superposion principle.

: combination of and

The power

2Re

term

(cross term)

The power does not obey

the linear superposion principle.

The power may be boosted up

or dimished by the interference t e .r m

Interference

1 1 0 1

2 2 0 2

0

2 2

1 2 1 2

2 20 0

20 1 2

2 2 20 0 0 1 2

20

exp exp

exp exp

where is real

Power 2Re

2 Re exp exp

2 Re exp

2 1 cos

A i kx t A i kx t

A i kx t A i kx t

A

A A

A i kx t i kx t

A A A i

A

1 2

Interference

1 2

20

20

The power of the combined wave depends on the phase

difference =

0 power 4 (in phase)

power=0 (out of phase)

3, power 2

2 2

The phase difference may be caused by the different path

A

A

kx t

length

2

2k x x x

Optical Spectrograph—Grating

• Although the reflecting grating is more often seen, I will use transmission grating to show how it works because it is easier to make the plots.

The light at allslots are in samephase

LightNormal incident

• Slit width a 0

• d: distance between slits

• N: # of slits

Considered asa point source

Interference

L

Screen(detector)

S

1

2

3

4

to N

Interference

β

β

d

d sinβ

Phase difference

2 sin

between any two adjacent slits

d

In phase

Interference

120

0

2

0

2

0

2

0

Wave at S may be written as

1

1

1

Power at S:

1 1

1 1

1 cos

1 cos

sin2

sin2

N ii i is

iNi

i

s s s s

iN iNi i

oi i

A A e e e e

eA e

e

P A A A

e eA e A e

e e

NA

N

A

2

Interference

0

0

0

At 2 integer

has maximum values (principle maxima)

For 2 0

2sinsin

22lim

2sin sin

2 2

sin cos cos sin2 2lim

sin cos cos sin2 2

sin 12lim lim

sin 12

s

m m

P

m

N mN

m

N NNm Nm

m m

N

0

0

0 0

sin2

sin2

sin2 2lim

sin2 2

sinsin 2 2lim 1 limsin

2 2

2sin

22

sin2

x

N

N

NN

Nx

N Nx N

N m

Nm

Interference22

0

(linear relation)

at 2

2 22 sin

for small ,

The minima right beside the mth maximum

22 ,

m

s

m m

m m

m

P N A m

d dm

m

dd

m d dN

m

d Nd

m

d d

Optical Spectrograph—Resolution

2 1 2 1

2 1

2 1 1

1

Resolution: , 0

Two wavelength can be resolved if

maximum of minimum of

Large N good resolution

Quality of spectrometer:

# of slits 100 to1000

m m

m m

d d Ndm

d Nd

mN

R

3 4

per mm

: 1000 to 50000 tatal

: 10 to 5 10

N

R

Grating Spectrometer – General Principle

Grating equation

Principle maximum from

interference

sin sin

: distance between successive,

equal spacing groove or slit

: incident angle

: diffraction angle

: +: reflection grating

-: transm

m d

d

ission grating

Grating Spectrometer: finiteL

S

Screen(detector)

f

Δx

β=Δx / f

Diffraction

• a 0. The most of incident flux is absorbed (reflection) or blocked (transmission) by the grating.

• Too few flux less sensitive

• a large : more flux but diffraction

Diffraction

x=0

x=a

Amplitude of each pointB0dx

2 sin

relative to 0

x

x

Diffraction

00

0

0

Wave function on S from the slot

2 sinexp

2 sinexp 1

2 sin

2 sinLet

exp 1

a is

i

i

s

A B e i x dx

B ei a

i

a

iB aeA i

Diffraction + Interference

12

0

2

0 0

Wave at S for N slots may be written as

1

1 11

1 1

Power at S:

1 11 1

1 1

N ii is s

iiN iNi

s i i

s s s s

i iiN iNi i

i

A A e e e

iB aee eA e

e e

P A A A

iB ae iB aee ee e

e e

2 20 2

22

2 20

2 2cos 1 cos

1 cos

sinsin22

sin2 2

i

NB a

N

B a

Diffraction + Interference

2

2

2

sin 2f , 1:

2

Maximun at =0 =0, f=1

Local maxima:

3 2=0 =0, f=1 , = 3 = ,f= ,

2 3

5 2= 5 = ,f= ,

2 5

minima: f=0

= 2 = , 4 = 2 ,

a

a

a a

Diffraction2

sin 2

2

Diffraction2

sin 2

2

Diffraction + Interference

22

2 20

Diffraction Interference

int

dif

sinsin22

sin2 2

Interference part, interval between two zeros

2

Diffraction part, interval between two zeros

2

,

N

B a

Nd

ad a

int difNd a

Diffraction + Interference• Smaller slit width (a)

Diffraction + Interference• Larger slit width (a)

Diffraction + Interference

dif

1

2

1 dif

Diffrection part: maximun at =0

but m=0, no resolution

Diffrection part, two zeros beside =0

Interference part, m= 1

Large , more flux (flux power )

but

The flux at m=

m

m

a

d

a a

a d

1 0

Unblazed Reflection Grating

• The calculation above is for the unblazed reflection Grating

Blazed Reflection Grating

• The purpose for the blazed reflection grating is that for a specified angle Δ=0 but m≠0.

• However, such condition is only exactly fit a certain wavelength, which is called “blazed wavelength” and usually the wavelength the observer most interested (e.g. Hα)