**By David Whysong **

**After reading all the discussion about how a central obstruction
changes the Airy disk, I decided to work it out for myself. So I opened
up my Jackson (the standard physics book on electromagnetic fields), turned
to the treatment of diffraction, and spent a couple hours trying to figure
it all out. In case you're wondering, yes there IS black magic involved.
Really. :-)**

**Some of you may have seen these graphs before if you have a telescope
optics book; I don't, so I did it myself. If you haven't seen these graphs,
they are very instructive. This is all theory (that's all I'm good for,
right now).**

**I'll spare you the math (if you want the details, email me). I plotted
the intensity of an Airy disk versus angle theta from the optical axis
for five different "ideal" telescopes: an 8" refractor, a 4" refractor,
a 5" refractor, an 8" SCT, and a 9.25" SCT. I know the 9.25" isn't a common
aperture, but I was interested because I'm planning on ordering one soon.**

**My graphs have one important drawback: I haven't bothered to integrate
the total power in the Airy disks, and so the y-axis scales (the intensities)
for different 'scopes are arbitrary, and they are all different. I can
try to do the more detailed calculations to get the intensities all scaled
correctly, but it will take time.**

**The equations and graphs are available as a Mathematica notebook
or in html/gif format. I can send them in a zip file by email. If you want
a plot like this for a different instrument, I can probably do the calculation
pretty easily.**

**The results:**

**An ideal 8" SCT with a 34% central obstruction should always outperform
an ideal 4" refractor. The airy disk of the SCT is more compact by a factor
of 2 (obvious from Rayleigh's criterion) and the first diffraction ring
is still within the outer edge of the central maximum of the refractor.
However, a 5" refractor has a very slightly more compact outer edge to
its Airy disk than the 8" SCT. The 8" SCT does appear to have a tighter
Airy disk overall though.**

**By the way, about that 99% obstruction -- it would narrow the central
peak of the Airy disk, but the first central maximum has about 20% the
intensity of the peak. After that, the intensity of the diffraction patterns
falls off as roughtly 1/theta. Not a useful situation for most purposes,
unless maybe you can do a lot of complicated math with the resultant image
and filter out the "bad" spatial frequencies. This might be nice for splitting
binaries. An 8" SCT with a 99.9% central obstruction has a central maximum
of about 1.8 microradians, whereas with a 34% obstruction the central maximum
is around 2.6-2.7 microradians. You gain some resolution with the huge
obstruction, but not a tremendous amount.**

**Notes: I had to adjust the power for each graph by hand. Mathematica
had trouble integrating the Bessel functions to get the total intensity.
As a result of this, each different instrument's graph has a different,
arbitrary scaling factor in intensity. I have roughly normalized each graph
so that the intensity at the center of the Airy disk is one. Think of it
this way: the following graphs assume that the peak intensity of the Airy
disk is always the same, regardless of the aperture of the instrument.
Each graph describes the intensity of the diffracted light as a function
of angle, from zero (the optic axis) to 2 arc seconds.**

**For all graphs, the incoming light is assumed to have wavelength
of 500 nm. The y-axis is intensity (with dimensions of power per unit solid
angle), and has arbitrary scale as mentioned above. The x-axis is angle
in radians from the optic axis.**

**For an ideal 102 mm refractor with incoming plane waves, this is
the transmitted intensity I as a function of theta:**