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 EyeYour 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.

A 1m telescope has a diameter times larger. In a 1/30th 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.

1m 2005mm

=

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

• 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.

Nphotons detected uncertainty or standard deviation

• There are many sources of noise. • The Poisson distribution describes events that are random in

space and times. • Photon detection often obeys Poisson statistics.

� =p

Nphotons

Poisson Statistics

N photons detected

The measurement: N ±pN

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

S

N=

CtpCt

S

N/

pt

The Noise

Signal/Noise

If you multiply the integration time by 4 you improve signal to noise only by a factor of 2

� =p

Nphotons

=pCt

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 aperture diameter 𝝀 wavelength D, 𝝀 are in same units Δ𝜃 in radians

�✓ ⇠ �/D

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,

9

7

6

/ ~ 550nm / 2m~550 10 m/2m ~2 10 radiansRemember 1" 5 10 radians,

/ ~ 0.04"

D

D

λ

λ

−

−

−

×

×

≈ ×

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 plane.

• The distance from a lens to the focal plane is the focal length, f.

𝜃 angular difference between two sources in radians x distance on focal plane f, x same units

x = f✓

Plate and Pixel Scale

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

x = f✓

Angular magnification

Incoming light at angle θ

objx f θ=

The angle of the light has changed, is

now φ

eyex f ϕ=

This is an optical layout of a refracting telescope. There are objective and eyepiece lenses.

Angular magnification

objx f θ= eyex f ϕ=

obj

eye

ff

ϕ θ= / is the angular magnification

obj eyeM f f=

is the angular magnification

Angular magnification

obj

eye

ff

ϕ θ=

/ is the angular magnification

obj eyeM f f=

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).

Reflecting Telescopes

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.

Mees

Astroscan

Heavy spectrographs

Prime focus

Mounts• Fixed at zenith:

Aricebo • fixed altitude: Hobby-

Eberle

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?

InterferometersFrom 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.

InterferometersThe diffraction limit for an interferometer depends on the distance between the antennas and the wavelength of light.

As the earth rotates, the orientation of the telescopes change with respect to the source, allowing the source to be mapped.

/ dθ λΔ ≈

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• 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