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Reaching Rayleigh Limit, Dawes Limit



 
 
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Old December 29th 03, 05:55 PM
edz
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Default Reaching Rayleigh Limit, Dawes Limit

I've been researching past posts on Resolution Limits to compile
information for an upcoming article. I keep seeing some of the same
misinformation coming up, (mis-stated values for Dawes as one
example). The question arises, "What success have user's had reaching
Dawes Limit with their scopes?" Generally the user is questioning why
a black space cannot be seen between two components of a double with a
separation measured at Dawes Limit. The important note here is that
neither Rayleigh nor Dawes is intended to represent a separation
showing two stars with a black space separating them, although there
may be conditions where a black space can be seen at these limits. An
understanding of Rayleigh and Dawes Limits is needed to use this
information for determining what is the closest stars that can be
split.

There are some 2-3 year old posts with excellent information on the
topic, but the threads can no longer be posted to bring them up front.
David Knisely has helped me a great deal to understand resolution.
Look up some of the past threads searching on Dawes Limit or Detail in
Saturn's Rings.

User's should understand the terms and the various limits and clearly
understand what these limits impose on their equipment and
observations. So I excerpted this brief from what I've written to
post here. I believe you will find it useful.

http://www.cloudynights.com/ubbthrea...5&o=&fpart= 1

This link provides about 50 targets to illustrate doubles at Rayleigh
or Dawes Limits for various scopes and the difficulty associated with
various doubles, bright/bright, bright/faint, faint/faint. A full
explanation of the affects of these various conditions will be found
in the article.

edz

Rayleigh Limit = 5.45 / D inches (or 138 / Dmm) is a measure of the
ability of the scope aperture to split a double star.


Likewise, Dawes Limit 4.56 / D inches (or 116 / Dmm) is another
measure.

Rayleigh Limit states you should be able to tell that a double is two
stars if the centers of the diffraction disks of the two stars
(commonly referred to as the Airy Disks, but see below) are separated
by a dimension equal to the radius of the first diffraction
interspace. That's the radius from the center of the Airy disk to the
minimum of the space between the disk and the first diffraction ring.
(This is important, I will refer to this a little further down). This
calculation is directly tied to optics theory and the ability of a
lens to resolve detail based on the wave nature of light. The limit of
a lens to resolve is determined by the diameter of the lens and the
wavelength of light. Take note that this limit, which has the centers
of two disks separated by the radius of a disk would not provide for
any black space between the two components.

Dawes limit was determined by actual field-testing of many and varied
double stars. It states the lens should be capable of seeing the
double as two components when the centers of the two components are
separated by a dimension defined by 116/Dmm. Similar to Rayleigh, that
allows you to see a notch, not a complete black space, between the two
components. So Dawes limit says you can see doubles closer than
Rayleigh limit.

Generally, it is held that Dawes should only be applied to equal 6th
magnitude doubles. My understanding is that although Dawes performed
his testing on many and varied doubles, the stated limit is simply an
average of his various results. Although all else here is commonly
accepted, this averaging explanation warrants further reading.

It is true that you can tell there are two components to a double
before you have reached a point where they are completely split with a
black space between them. I keep my notes when I use various
eyepieces, something like elongated, elongated pointed, notched,
barely touching, thin black line, clear black space.

Some very good telescopes are capable of exceeding both of these
limits. Conversely, some lesser quality scopes will not be able to
even reach these limits. But these are good indications of what a good
telescope should be able to see. Rayleigh or Dawes limits usually
cannot be reached when viewing doubles that are very bright, have
widely varying magnitudes or are very faint. These are all more
difficult conditions.

Pi Aquilae is magnitude 6.1 and 6.9 at a separation of 1.4 arc
seconds. This is a good double to test the ability to achieve Dawes
limit. It's not exactly equal in magnitude, but it doesn't vary too
widely and it is neither too bright nor too faint. I've observed this
double recently on two different occasions using my TV85 scope. The
TV85 calculates to a Rayleigh and a Dawes limit of: Ray Lim
5.45/3.35=1.63" and Daw Lim 4.56/3.35=1.36". I was not able to
completely split this double to a black space, but I was able to
identify it as a double at several magnifications. So I did reach
Rayleigh and Dawes limits. But is this scope capable of exceeding
these limits? I'll explain that in a bit.

I was not able to completely split lambda Ophiuchi at 1.5", but I was
able to completely split 69 (Tau) Ophiuchi at 1.7" to a thin black
line. These are stars that seem to be right at the limit of this 85mm
aperture and by all indications they clearly seem to agree with
expectations based on the above formulae. Knowing the quality and the
aperture of your scope and the limits implied by these formulae will
help you solidify your expectations of your scope's performance.

Now back to the passage I referred to as important. Let me further
explain. First I will repeat what I said earlier. Rayleigh Limit
states you should be able to tell that a double is two stars if the
centers of the diffraction disks of the two stars is separated by a
dimension equal to the radius of the first diffraction interspace.
That's the radius from the center of the Airy disk to the minimum of
the space between the disk and the first diffraction ring.

What if I want to see a double star with at least a thin black space
between the components? What are my limits? What should I expect of my
scope?

Based on the definition above, Rayleigh limit is a measure of a
radius. It is the measure from the center of the bright central dot,
or the central disk, out to the minimum of the "black space" between
the central dot and the first bright diffraction ring that surrounds
the Airy disk. If you want to see two stars as completely separated
with a thin black space between them it is necessary for the centers
of the Airy disks of the two components to be separated so this "black
space" overlaps and becomes visible between them. That separation
dimension is approximately equal to two radii or the diameter of the
Airy disk. Rayleigh Limit for my 85mm scope is 138/85 or 1.6
arcseconds radius. Therefore the Airy disk diameter is 2 x 1.6" or 3.2
arcseconds.

The central disk itself is slightly smaller than the Airy disk
dimension since the Airy disk is measured out to the minima of the
first dark space. It varies with magnitude, but say it's 85% for a
bright star and for a faint star it may be less than 50% of the
diameter of the Airy disk is occupied by the central bright spot. So
the equivalent separation of something less than twice the Rayleigh
Limit, we'll use 60% for this example, is needed to have a black space
between two stars. If Rayleigh Limit for my 85mm scope is 1.6
arcseconds radius, then any time I can cleanly split doubles of less
than 2 x 1.6" or 3.2" x 60% or 1.9 arcseconds, to at least a thin
black space between them, then I have exceeded Rayleigh Limit. For
brighter stars the central bright spot is larger, therefore I might
multiply by 85% rather than 60%.

Dawes is not a measure dependant on the wave nature of light. It was
empirically determined to represent a point of minimum separation
where a double can be noticed as two components. To say that I have
exceeded Dawes, I should be able to show that I can notice stars as
double when they have a separation of less than 116/Dmm.

Do not confuse the definition of the Airy disk as the bright central
dot in the diffraction pattern. This is really not correct and this
term is very often confused in much of the literature in print. The
Airy disk is measured out to the minimum of the first diffraction
interspace. The central dot is correctly referred to as the spurious
disk. There is no true measurement for the spurious disk itself. The
measurement 5.45/D based on the wavelength of light (and specific only
to yellow light at 550 nanometers) is out to the first minima. The
edges of the central disk usually cannot be seen as the light falls
off to zero towards the first minima as we move from the center of the
central disk out into the first diffraction interspace where the
minima occurs. The dimension of the Airy disk varies with the
wavelength of light, being larger for red light and smaller for blue
light. Therefore it may be slightly easier to split two blue stars
than two yellow stars and both are easier than two red stars would be.

edz
 




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