Astronomy 302

Lecture 1 & 2

Telescopes, Optics, & Imaging

(Read Karttunen Fundamental Astronomy, 4th ed. Chap 3)

1.0 The Atmosphere

Clearly the first issue that our light faces (after a long journey) is the Earth's atmosphere.
The atmosphere has several effects:

1) It absorbs some frequencies of the the light
Hence, the transmission of the atmosphere is a strong function of lambda

Figure 1: the atmosphere (Karttunen)

As you can see there are several windows in the atmosphere -- these bands define the astronomical bands

Fig 2: close up of the UV-radio sections

Fig 3: The J (1.05-1.4 um), H (1.5-1.8), and K (2-2.5 um) Near IR bands (gemini)

2) The atmosphere also introduces small phase errors in the wavefront of the incoming light

Fig 4: (from Michael Richmond ) shows the effects of seeing on the position of a star

3) The background light (light pollution) can make deep optical images difficult to obtain

Fig 5: effects of light pollution (Karttunen)

2.0 Characteristics of a good site

1) It should be high (minimizes light loss to Atmosphere, and leads to best seeing)
2) It should be dry (maximizes sky transmission in IR)
3) It should be dark (minimizes Optical background)
4) It should be clear (maximizes sky throughput)
5) It must have low winds, and low turbulence, and a large r_o (good seeing)
6) It must have minimal snow load

In the end there are *very* few places on Earth that are truly exceptional for telescopes.

The best site is arguably Mauna Kea which has the best seeing of any site (why is that?)

there are also outstanding sites in Chile, like ESO's Paranal, and UA's twin 6.5m Magellan Telescopes

3.0 TELESCOPE OPTICS

Fig 6: The basic types of telescopes for the eye (Karttunen)

Fig 7: The principal of angular magnification of a Refractor (Karttunen)

The diameter of the objective lens (D) is the "aperture of the telescope"

the focal length the is lens is f
the ratio of f/D = focal ratio = F
example: a 1 m lens with a focal length of 10 m would be an f/10 lens

the smaller the focal ratio the "faster" the optic, the stronger the
curvature of the lens, the more expensive it is to build

3.1 PLATESCALE

One of the most important concepts of a telescope is the platescale of the focal plane
of the telescope. By this we mean how many mm does an object of angle u make at the
focal plane.

When the Object is seen at angle u on the sky it forms an image of height s (see fig 7).
Therefore:

s = f tan(u)
since u is a small angle (in radians) we can use the small angle approximation and have
s=fu

for a professional telescope (like the 61inch) we may not know f. But we do know the f/ratio (F) of the secondary mirror
and the size of the primary.

A Cassegrain telescope like the 61 inch. (Karttunen)

and so a more universal formula is (where the diameter of M1 is D):
s=FDu

example: How big is a 10 arcsec image on the sky in mm for the 61inch (D=1.54 m) with the F=f/13.5 cassegrain secondary?
s = 13.5 * 1.54e3 * (10./ (60*60)*3.1415/180)
= 1.007 mm

But what about the magnification of the telescope?

3.2 ANGULAR MAGNIFICATION

It is clear that a refractor (see Fig 7) produces an angular magnification of:

Mag = u'/u ~ f/f'

so the angular magnification is simply the ratio of the Objective focal length (f) to
the focal length of the eyepiece (f').

example: What is the angular magnification of the 61" if we placed a 20 mm eyepiece on it?
Mag = 13.5 * 1.54e3 / 20 = 1039x
which is pretty good.

Note that Mag only depends on f and not D. In other words one could build a 1 mm telescope with a focal length

of 1 m that would give the same angular magnification as the 61 inch.
So why build big telescopes (Especially if cost ~ D³) ?

3.3 DIFFRACTION

Light is both a wave and a particle
Therefore it must obey diffraction
Here is a good site about the nature of diffraction
All diffraction limited images are based on the fourier transform of the input aperture
To be exact the electric vector of the EM wave is the transform. Since images are intensity
we actually detect the square of the amplitude of the EM wave.

The image detected is actually the forward Fourier transform of the autocorrelation of the
pupil (uniformly illuminated for a flat/perfect wavefront).

In the special case of a circular aperture (which is the optimal shape for making lenses and mirrors)
yields an Airy pattern as the PSF (point spread function).

EXAMPLE (from Dr. Close's and Jin Park's coronograph test bed, Park & Close et al. 2006 PASP 118, p1591)

Optical layout:

Input pupil

Output image

Fig 10: The cross-section of an Airy pattern in the focal plane of telescope (from here)

Now from Fig 10 we see that the first minima is at a radius of s = 1.22 lambda f/D
and we know that on the sky this angular size u =s/f so the angular size of the Airy Pattern on the sky is:

radius to first minima = 1.22 lambda/ D radians

Real Airy PSFs in the Mid-IR from the MMT AO system (Close et al. 2003 ApJ)

How does this effect resolution?

3.4 DIFFRACTION-LIMITED RESOLUTION

Resolution is the critical issue for many applications of imaging.

fig 11: Resolution with Airy PSFs (Karttunen)

There is the classic (but too conservative) Rayleigh limit of 1.22lambda/D (see Fig 11 (c))
but we can do better (we are astronomers!)

Fig12: Adaptive optics (AO) images from the MMT (Close et al. 2003, ApJ)

The Dawes limit of the FWHM (Full Width at Half Maxima) of a single PSF is really closer
to the true limit of resolution (see fig 11 (e)). For a typical diffraction-limited Cassegrain telescope
with a central obscuration of ~0.2, the FWHM of the Airy pattern is 0.98lamda/D or ~lambda/D, therefore:

Resolution of a diffraction limited telescope is ~lambda/D radians

example: At the MMT with the AO system we can achieve a resolution of
2.2/6.5e6*206264 = 0.069" or 69 mas (milliarcseconds) in the K-band (2.2 microns)

What happens in the optical (or in the IR without AO) ?

3.5 SEEING LIMITED RESOLUTION

Now if we don't use AO then the atmosphere has messed up the images.
Although the starlight is still coherent it has strong departures from a flat
wavefront. Therefore, only little parts of the telescope pupil interfere and so
there are many diffraction-limited "speckles" that move around on millisecond
timescales. So any images that have exposure times longer than ~50 milliseconds
will be seeing limited!

Look here for more about the above movie
(and how speckle interferometry can help!)

Now most objects are too faint for speckle interferometry and so almost all
ground-based observations in the the optical are seeing-limited.

This means:

FWHMseeing ~ lambda/r_o radians

where the freid parameter (r_o ) is typically 10cm in the optical (V band)
hence FWHMseeing ~ 1" at V
But if there is a lot of turbulence in the atmosphere in the direction
the telescope is looking then r_o can be as small as 2 cm (5" seeing!). On the other
hand if night is really good at a good site r_o can be as big as 20 cm (0.5" seeing!).
Most good sites have seeing from 0.4-1.5", with medians of 0.6-0.7".
The 61" has 1-2" seeing typically, while the MMT is closer to 0.5-1.5".

NOTES: since ro ~ lambda^6/5 the seeing is slightly better at longer lambda
FWHMseeing ~ lambda^-1/5

Will talk more about these issues in the AO lectures.

3.6 WIDE FIELD SEEING LIMITED IMAGING

Since the time averaged turbulence is similar over the field of view (FOV) of
most cameras, it is safe to assume that the PSFs of all the stars in the FOV will
have a similar shape (to within noise limits) and so a long image at the 61inch
might look like:

Fig 14: 61inch f/13.5 Cassegrain B,V, and I filter images of NGC 6791 from Dr. Betsy Green
(this is an open cluster and the Blue stragglers and O/B subdwarfs are marked)

3.7 Color Magnitude Diagram

By measuring the flux (or magnitude) of stars in a cluster we can construct a color
magnitude diagram (like in Project #1).

Below is an example of a color magnitude diagram for the globular cluster M55.

from here.

Note since all the stars are roughly the same age they trace an isochrone.
You should be able to trace stars of increasing mass up the main sequence (MS)
to the MS turn-off then to red giant branch (RGB), the horizontal branch (HB) then the Asymptotic Giant Branch (AGB). See here for more info

We will learn how to take and reduce similar data in this class!