Download Diffraction limited resolution - X-ray and Observational Astronomy

DEPARTMENT OF PHYSICS AND ASTRONOMY
3671 Multi-Wavelength Astronomy
Lecture 3: Telescopes
Dr Matt Burleigh
In this lecture we will cover:
• Design of modern, large optical (& IR)
telescopes
• Diffraction limited resolution
 Qmin=1.22l/D
• Influence of atmosphere – “seeing”
• Adaptive optics
– Overcoming “seeing”
• Magnification and plate scale
Dr. Matt Burleigh
8m Gemini North on Hawaii
• Opened early 2001
Dr. Matt Burleigh
8m Gemini North on Hawaii
• Opened early 2001
Dr. Matt Burleigh
Diffraction limited resolution
• A fundamental limit exists in our ability
to resolve objects
• This limit arises by diffraction
• Consider a single slit width D
• Any ray passing through this aperture
and arriving at a specific point in the
focal plane is associated with another
ray passing through the aperture one
half slit width away and arriving at the
same point.
Dr. Matt Burleigh
Diffraction limited resolution
• If the two rays are one-half wavelength (l/2) out of phase,
destructive interference occurs:
– (D/2) sin q = l/2
– Or sin q = l/D
• Now consider dividing the aperture into four equal segments
• A ray from the edge of the opening pairs up with one passing
through a point one-quarter of a slit width away
• For destructive interference to occur:
– (D/4) sin q = l/2
– Or sin q = 2l/D
• This analysis may be continued by considering dividing the
aperture into 6 segments, then 8, 10 etc
• In general, for minima to occur as a result of destructive
interference from light passing through a single slit
– Sin q = m l/D
– Where m = 1, 2, 3 … for dark fringes
Dr. Matt Burleigh
Diffraction limited resolution
• The analysis for light passing through a circular
aperture like a telescope is more complex
• Due to the symmetry of the problem, the diffraction
pattern appears as concentric rings:
Dr. Matt Burleigh
Diffraction limited resolution
• The solution to this problem was first obtained in
1835 by Sir George Airy
• The central bright spot is known as an Airy Disk, the
rings as Airy Rings
• A similar equation to our ideal slit describes the
location of diffraction minima, but m is no longer an
integer
Ring
m
Imax / I0
Central max
0.00
1.00
First min
1.22
2nd max
1.635
2nd min
2.233
3rd max
2.679
Dr. Matt Burleigh
0.0175
0.0042
Diffraction limited resolution
• When the diffraction patterns of two sources are sufficiently
close together, the diffraction rings are no longer distinguished
• The two images are said to be unresolved when the central
max of one image falls inside the first minimum of the other
• This arbitrary resolution condition is called the Rayleigh
Criterion
• Assuming qmin is quite small, and invoking the small angle
approximation:
qmin = 1.22 l / D
• q in radians
• note to convert radians to arcsecs x by 206265
• Resolution improves with increasing telescope size and at
shorter wavelengths
Dr. Matt Burleigh
Seeing
• Unfortunately, the resolution of ground-based telescopes
does not improve without limit as the mirror size
increases
• This is due to the turbulent nature of the Earth’s
atmosphere
• Local changes in atmospheric T and density over small
distances create regions where the light is refracted in
random directions
• This causes a point source to become blurred
• Since stars are effectively point sources, they twinkle
• The quality of the image of a star at a given observing
location and time is called “seeing”
– A measure of resolution allowed by atmosphere
• The best seeing at major observatories like Hawaii & La
Palma can be below 0.5 arcseconds
Dr. Matt Burleigh
Seeing, image size & point spread function
• The light from a point source like a star spreads itself across the
detector (photographic plate, CCD, etc) in a roughly circular
pattern
– Obviously in poor conditions or if the object is moving the image
can become elongated
• This is called the point spread function
• A cross-section of the PSF is approximately gaussian
• We measure the size of the image (PSF) in arcseconds
–
–
–
–
–
1 arcsecond = 1/60 arcminute = 1/3600 degree
Human eye resolution ~1arcminute
Pair of car headlights 100km away ~1arcsecond apart
A man on the moon ~1/1000 arcsecond
Size of Betelgeuse (red giant star) ~ 1/20 arcsecond
Dr. Matt Burleigh
Seeing
• Betelgeuse seen with the
4m William Herschel
Telescope on La Palma
in about 1” seeing
Dr. Matt Burleigh
Choosing an observing site
• A steady atmosphere to minimise times of poor seeing
• Away from sources of scattered light (ie street lights) and
sources of dust (industrial areas, sandy deserts)
• For infra-red astronomy, want atmospheric water vapour
content as low as possible
• Good weather!
– Remote site
– High altitude
– Little rainfall
– Atacama, Chile
– Oceanic islands – La Palma, Hawaii
– Poorer sites: Anglo-Australian Telescope
– Best site – Space!! (HST)
Dr. Matt Burleigh
HST + WFPC 2
• HST / WFPC2 image of a multiple star system including a white dwarf
•Aa (solar-like star) – Ab (white dwarf) = 0.4”
• Image made in ultraviolet at 170nm
• Diffraction limited resolution of HST (2.4m) = 0.02”
• In reality, resolution is more like 0.08”. Why?
Dr. Matt Burleigh
Adaptive optics
• Advanced computers and new technologies now allow
astronomers to compensate for the effects of seeing in real time
• It’s called Adaptive Optics
• First AO systems designed by the military for use with spy
satellites (and their own ground-based telescopes!)
• 1990s declassified material plus efforts within astronomical
community has led to AO systems being installed at world’s
major observatories
• Only now becoming really effective
• AO system consists of 3 principle components
– A deformable mirror (wavefront corrector)
– A wavefront sensor
– Control system (real-time computer)
Dr. Matt Burleigh
Adaptive optics
Dr. Matt Burleigh
The wave front sensor
• Estimates the distortion of the
atmosphere along the line-of-sight to
the target
• Requires a bright source near the target
– either a star (Gemini telescopes need stars
brighter than 13th mag)
– Or a fake star created by a laser
– Laser stimulates sodium atoms in a layer
at an altitude of about 90km
Dr. Matt Burleigh
The control system
• Calculates the corrections required
based on inputs from the wave front
sensor
• Commands the actuators which deform
the mirror
• Calculations are performed in submillisecond range to keep up with
changing atmosphere
Dr. Matt Burleigh
The deformable mirror
• Piezoelectric actuators deform the
mirror
• The number of actuators required for
near-perfect corrections in the optical is
several thousand – unrealistic
• AO works best in the IR, where fewer
actuators are required
Dr. Matt Burleigh
Quantifying AO performance
• The Freid parameter, r0
–
–
–
–
A measure of atmospheric turbulence
r0 decreases as turbulence increases
Also, r0 is proportional to l**6/5
r0 can be viewed as the size of a telescope that would
give the same resolution as the atmospheric seeing
• So if seeing = 1”, r0= 10cm in the optical (using l/D)
• i.e. r0 = l / seeing in radians
– Since r0 l-dependent, then if r0 = 10cm at 550nm it would
be 70cm at 3.4 microns in the IR
– So it is easier to perform AO in the IR than in the optical
Dr. Matt Burleigh
Quantifying AO performance
• The Strehl ratio R is the ratio of the quality of the
image obtained to that of a theoretically perfect
point source image (Airy disk)
• R is proportional to r0 and R improves (ie the image
quality improves) with larger r0
• 0 < R < 1; R=1 is a perfect image
– For detection of extra-solar planets, need R>0.9
• As R increases, most of the light is concentrated in
the central core and little in the Airy rings
• Of course, R is l-dependent
– e.g. an AO system that gives an R of say 0.9 in the IR will
only give 0.1-0.2 in the visual
Dr. Matt Burleigh
AO at the USAF
Starfire facility
Binary Kappa Peg:
Uncompensated
on 1.5m telescope
Dr. Matt Burleigh
756-actuators:
0.3” resolution
3D plot: uncompensated
and with full AO (right)
AO on Gemini North
Globular cluster NGC6934 observed with the 8m Gemini
North telescope on Hawaii, in the visible (left) and in the IR
(right) with the Hokopu’a AO system
Dr. Matt Burleigh
Magnification, resolution and plate scale
• If F= focal length (mm) and D = diameter of telescope,
then focal ratio f=F/D
• If q=angular resolution in arcsecs then
– Image size s = F tan q or Fq assuming small angles
• An increase in focal length increases the scale of a
pattern, but not the resolution
• Magnification – changes the size of an image
• Resolution – necessary to increase D to see more
detail
• Plate scale – no. of arcsec per mm in image plane
– Plate scale dq/ds = 1/F = 206265 / fD
– Rem. 206265 arcsecs/radian
Dr. Matt Burleigh
Telescopes: summary
• After this lecture you should know and
understand
–

–
–
The definition of diffraction limited resolution
q = 1.22l/D
The limitations imposed by atmospheric seeing
Overcoming seeing with Adaptive Optics
• Definition of Fried parameter and Strehl ratio
– Definitions of image size magnification and plate
scale
Dr. Matt Burleigh