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 m lim (human eyes) = +8.5(V) ~200 photons/s

Limiting Magnitude The telescope with eyepiece

Limiting Magnitude For the professional telescope (with detector)

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 aberration spherical aberration parabolic mirror spherical mirror

Imaging – Geometric Optics Ray tracing/ matrix method: α x Reference line (Optical axis) α:α: + - x: about the reference line:+ below the reference line: -

Imaging – Geometric Optics M

Ray Tracing – Translation α x α’α’ x’ 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)

Ray Tracing – Refraction (2)

Ray Tracing – Single Lens f Focal plane Star light

Telescope The most important matrix elements in the combined matrix are –m 21 = f (focal length) –m 22 =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 m 21 =f (effective focal length) and m 22 =0.

Telescope with Eye Piece

Two lenses with focal lengths of f 1 (primary lens) and f 2 (eye piece). The distance between two lenses is D=f 1 +f 2. Star light f1f1 f2f2 D Eye piece Lens 1Lens 2

Telescope with Eye Piece

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

PSR B LMC X-1 Extended source?

Off-axis aberration

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 plane Diameter: 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 F d )  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 ”.

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 Long exposure Seeing disk

Point-Spread Function (PSF) IRAF gives 6 functions to model PSF

Point-Spread Function (PSF) Gaussian Lorentz α=1β=1 Lorentz α=1β=2 Lorentz α=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 typePixel NumberPixel Size (μm x μm ) Plate Scale (arcsec/pixel) ADC (bits) FLI IMG1024S1024 x x Apogee AP x x PI 1300 B1340 x x

CCD Semi conductor: band structure Conducing band (empty) Valence band (full) Band gap E g < 1 eV Visible light photon energy : 1.7 eV to 3 eV Photon Photoelectron hole Voltage applied

Detector – CCD Exposure Incident Photon Potential well Photoelectrons

Detector – CCD Reading

CCD – Dark Current Thermal electrons

CCD – Dark Current

CCD – Flat Filed & Bias The quantum efficiencies (QEs) may different from pixel to pixel and also depend on the wavelength.

CCD – Digital Output ADC Analog signal Audio-to-Digital Converter Digitized signal Flat field

Data Size of Image

Photo Flux Estimation

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

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

EM Wave

Wave Equation – Linear Superposition

Wave Equation – 1D

Wave – 1D

Wave – 3D : Plane Wave

Interference

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 all slots are in same phase Light Normal incident Slit width a  0 d: distance between slits N: # of slits Considered as a point source

Interference Screen (detector) S to N

Interference β β d d sinβ In phase

Interference

Optical Spectrograph — Resolution

Grating Spectrometer – General Principle

Grating Spectrometer S Screen (detector) f ΔxΔx 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 point B 0 dx

Diffraction

Diffraction + Interference

Diffraction

Diffraction + Interference

Smaller slit width (a)

Diffraction + Interference Larger slit width (a)

Diffraction + Interference

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. H α )