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AST111 Lecture 4a
Telescopes
4m Mayall telescope of NOAO on Kitt Peak
What a Telescope does
• Light gathering power, so we can see fainter
objects. Telescopes can also be made to
gather light at wavelengths that we can’t see
with our eyes.
• Provides angular resolution. Greater detail.
We will begin by considering optical ground
based telescopes.
The Light Bucket
• The larger the number of photons or energy gathered, the
easier it is to detect a source.
• Units of flux: erg cm-2 s-1 or photons cm-2 s-1.
• Total number of photons detected is flux × area × time.
• To improve your ability to detect a faint source you can
integrate longer, or you can have a larger light bucket.
• The total number of photons depends on the area of the
light bucket, so is proportional to its diameter squared.
Scaling from your Eye
Your eye has a pupil of a few mm and an integration
time of about 1/30 of a second.
You can see stars that have mv~6 mag.
1m
= 200 times larger.
A 1m telescope has a diameter
5mm
th
In a 1/30 of a second integration on a 1m telescope you
can detect an object 5 log10200=11.5 mag fainter that
your eye or 17.5 mag.
Why did I use 5 log instead of 2.5 log? Because the
light collection ability depends on the AREA of the
telescope which goes as the square of the diameter.
Remember magnitudes are -2.5log f.
Scaling from your eye overestimates
your ability to detect objects.
• On a 1m if you integrated for an hour, you
would gain 30x60x60~105 more photons
corresponding to a change in magnitude of
2.5 x log 105=12.5 mag.
• 17.5+12.5 = 30. However, it is not possible
to detect objects at 30th magnitude on a 1m
diameter telescope.
• Why not?
Sky Background
• The sky emits light (and is brighter during solar maxim than solar
minimum)
• On a moonless night at midnight and at solar minimum, the sky is about
22 mag/(”)2 bright in V band.
• Note the units: mag/square arcsecond is a surface brightness.
• The sky is very bright past 1 micron because of thermal radiation.
• This makes it difficult to observe at near and mid-infrared wavelengths.
• Because the sky is so bright at infrared wavelengths, infrared
astronomers are often given what is called ``bright time’’ (when the
moon is up). ``Dark time’’ is when there is little or no moon.
• You can often figure out what kind of astronomer a person is by when
they are given observing time.
• Light pollution limits the ability of a telescope to detect faint objects.
• Optical telescopes are best located in dark dry sites away from cities.
Sky background as a source of noise
• Background radiation is a source of noise.
• If the noise is bigger than the flux from the object you are
trying to detect, then you cannot detect your object.
• We quantify this in terms of S/N or the Signal to Noise
ratio. If S/N<=1: you can’t detect your object.
• Very faint, possibly believable detections can be done with
S/N=3. If you are fitting a very complicated model to your
data you might need higher signal to noise or S/N=100 or
more.
• When we write observing proposals we justify the
integration time required by discussing the S/N and what
we plan to do with the data.
Poisson Statistics
• There are many sources of noise.
• The Poisson distribution describes events that are random in
space and times.
• Photon detection often obeys Poisson statistics.
Nphotons detected
uncertainty or standard deviation
=
p
Nphotons
• When few photons are detected (such as in X-ray
measurements) the uncertainty in measuring your
source is given by Poisson stats.
• Often background radiation can also be characterized by
Poisson stats.
Poisson Statistics
N photons detected
The measurement: N ±
p
N
the error
Poisson Statistics (continued)
Suppose you integrate for 20 seconds and detect
100 photons.
What is your measurement for the count rate?
100/20 = 5 photons/second
error is √100 = 10
detected 100 ± 10 photons
Count rate measurement:
5 ± 0.5 photons per second
Integration and Signal to Noise
Suppose the number of photons is a constant
times the integration time
N photons = Ct where C is the count rate
The Noise
Signal/Noise
=
p
Nphotons =
S
Ct
= p
N
Ct
p
Ct
p
S
/ t
N
If you multiply the integration time by 4 you
improve signal to noise only by a factor of 2
Example using Poisson statistics
X-ray astronomy: few photons, often not much
background
Suppose you integrate for 100 seconds on a 1m
telescope in space and detect 100 photons from a
particular source
Q: How long would you need to integrate on a 2m
telescope (using same bandwidth and efficiency) to
measure the flux with a signal to noise of 10?
Example using Poisson statistics
Integrate for 100 seconds on a 1m telescope in space and
detect 100 photons
Noise = √100 = 10
Signal = 100
S/N = 100/10 = 10
Q: How long would you need to integrate on a 2m
telescope to measure the flux with a signal to noise of 10?
We want the same S/N (that means the same number of
photons) but we have 4 times the light collecting area. So
we need to integrate 1/4 the time.
A: 100/4 = 25 seconds
Diffraction
• One phenomenon that
affects the angular
resolution of a
telescope is diffraction.
• Diffraction is the
spreading of light when
it passes through an
aperture.
Diffraction Limit
Light is spread out over a
particular angle which depends on
the size of the aperture and the
wavelength of light.
Because light is spread out there is
a limit on the angular resolution
which can be resolved through a
particular size telescope.
Stars which are closer than this
resolution cannot be separated,
they appear smeared together.
Diffraction limit
Point sources (stars) are
smeared over an angle
✓ ⇠ /D
D aperture diameter
𝝀 wavelength
D, 𝝀 are in same units
Δ𝜃 in radians
Diffraction limited
• Ideally we would like any given telescope to be ``diffraction
limited.” This means that nothing else in the system is smearing
out images to a worse angular resolution that physically possible
with the telescope.
• The Hubble Space Telescope is “diffraction limited”. It has a 2m
diameter mirror which means at optical wavelengths,
λ / D ~ 550nm / 2m~550 × 10−9 m/2m
~2 × 10−7 radians
Remember 1" ≈ 5 × 10−6 radians,
λ / D ~ 0.04"
Atmospheric Seeing
Ground based telescopes are limited to 1” by blurring caused by the
atmosphere. The Hubble space telescope has angular resolution that is
better by a factor of 10 better than typical ground based seeing.
Image formation by a camera
• A camera focuses a plane
wave (light from a very
distant star, for example) to
a point.
• A direction is turned into a
location on a film or camera.
This is like taking a Fourier
transform.
• Each direction focuses to a
different point on the focal
x = f✓
plane.
𝜃 angular difference between
• The distance from a lens to
the focal plane is the focal
two sources in radians
length, f.
x distance on focal plane
f, x same units
Plate and Pixel Scale
x = f✓
Camera has pixels that have a particular size in
microns
For optical/near IR ground based observing you want
to match your pixel size to the seeing
Arcseconds per micron on focal plane = plate scale
Arcseconds per pixel = pixel scale
1” on the sky corresponds to a few to 20 microns
typically
Angular
magnification
Incoming light
at angle θ
x = f objθ
This is an optical
layout of a refracting
telescope.
There are objective
and eyepiece lenses.
x = f eyeϕ
The angle of the
light has changed, is
now φ
Angular magnification
x = f objθ
ϕ=
f obj
f eye
x = f eyeϕ
θ
M = f obj / f eye
is the angular magnification
magnification
Angular magnification
ϕ=
f obj
f eye
θ
M = f obj / f eye
is the angular magnification
What is the new plate scale if the camera has focal
length f?
Answer: x = M f𝜃
you multiply the plate scale by the magnification
Chromatic aberration
Disadvantages of refracting telescopes:
1. They suffer from chromatic aberration.
2. Lenses also can absorb certain wavelengths of light (UV in
particular) reducing the efficiency of the telescope.
3. The objective lens must be supported from its edges. This is
hard to do if the lens is large.
Reflecting telescopes do not have these disadvantages.
Telescope
dimensions
• Aperture. Diameter of primary mirror or lens. Determines
light collecting capability
• Focal length. Length it takes the incoming light to
converge to a point. Short focal lengths give smaller
telescopes and larger magnification but require better
optics.
• Magnification: Ratio of focal length of eyepiece to that of
telescope
• F ratio. Focal length divided by aperture diameter
(sometimes called F-number).
Astroscan
Reflecting
Telescopes
Mees
Prime focus
Heavy
spectrographs
It’s difficult to make large
Newtonian telescopes
because the instrument
must be supported high up.
Cass focus allow you to
support the heavy mirror in
the same place as the
instrument.
For really heavy
instruments, the Coude
focus is often used.
For small light instruments
sometimes prime focus is
used.
Mounts
• Fixed at zenith:
Aricebo
• fixed altitude: HobbyEberle
Mounts
• Alt-Az: Altitude and azimuth are each
separate directions of motions. Tracking
done with two motors.
• Equatorial: Telescope aligned with north so
that tracking can be done with a single
motor.
Equatorial Mounts
German
English, fork, or
horseshoe mounts
Examples of
Telescope
What kind of
telescope is the 4m
on Kitt Peak?
Interferometers
From very far away, light
from a point source
appears to be a plane wave
with wave fronts slanted at
a particular angle.
Antennas detect the
location of the wave peaks
or the phase of the waves.
By comparing the phase
lag between wave peaks at
different antennas, the
angle on the sky of the
source can be determined.
Interferometers
The diffraction limit for an
interferometer depends on
the distance between the
antennas and the
wavelength of light.
Δθ ≈ λ / d
As the earth rotates, the
orientation of the
telescopes change with
respect to the source,
allowing the source to be
mapped.
Question: which focus is typically used for
radio receivers?
Interferometers
•
•
•
There are interferometers arrays
(SMA Submillimeter Array in
Hawaii and VLA, Very Large
Array, in New Mexico). ALMA
(Atacama Large Millimeter Array)
is a large array which will be
constructed in the desert in Chile.
Interferometers are built with
antennas covering the entire earth
in the radio (VLBA and VLBI,
very long base-line array and
interferometer).
Interferometers are also built at
shorter wavelengths such as the
optical (e.g. SIM, Space
Interferometer Mission).
However it is a lot harder to get
them to work.
Review
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Diffraction limit
Background radiation as a source of noise
Poisson statistics
Focal length, angular magnification.
Refracting and reflecting telescopes
Cass, Newtonian, Coude, Prime focuses.
Interferometers