Moons as Disks, Shadow Transits and Saturn's Divisions

In the past few weeks, with the optimum placement for viewing of both
Jupiter and Saturn, these questions have come up a few times; Which
moons can be seen and with what scopes? and What does it take to
resolve the moons to a disk? and I think I can see the Encke
division, is it possible?

This is a collection of notes from a series of replies I posted to
answer those questions on the Cloudy Nights planetary forum and the
Amart Equip forum. There are further discussions on the CN planetary
forum related to

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

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

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


In addition to the excellent explanations on The Rings of Saturn and
Their Divisions by David Knisely
http://www.weatherman.com/ see Astro Product Reviews and articles

and The Encke Minima and Encke Division in Saturn's A-Ring by Eric
Jamison
http://home.fiam.net/ericj/encke.html

many observers should find this information useful in the coming
weeks. Enjoy!

edz


DISK VS. AIRY DISK

How can I tell if I'm seeing moons resolved as a disk?
A perfect way to observe the difference would be when the extended
object is in the field of view of a star. Focus precisely and then
compare the difference between the disk and the star.

Assuming for the moment that this star would be moderately bright,
let's say for example 5th mag, then the star will provide you with the
near perfect size Airy disk. The central bright spot in the middle of
the Airy disk will be nearly equal to the Rayleigh limit calculation
for your scope. The size of that spot does vary slightly with the
magnitude of stars, so although the Airy disk is always the same size
in the scope, the central bright spot varies with magnitude.

All stars produce the same size Airy disk in your scope. Only the
central bright visible spot within the Airy disk varies slightly.
However, extended objects have an infinite number of points that give
off light. So all the edges around a moon disk produce an Airy disk in
your scope. The image formed from a moon disk is the result of a
circle of an infinite number of Airy disks.

Continue with the assumption that you can see a moderately bright star
near the moon disk. For a moon disk, the image in the scope will be
larger than the Airy disk of a star. The scope will show the moon disk
fattened up by producing the Airy disks all around the edges. The
image size is slightly smaller than the sum of the Rayleigh Limit plus
the object diameter. A little further on we'll discuss specifically
how big the image is.


SEEING JUPITER'S MOONS AS DISKS

Jupiter is 88,700miles in diameter. At 5AU it's disk would appear
39.3 arcsec.
The sizes (at 5AU) and the magnitudes of Jupiter's moons a
Ganymede 3,270 = 1.45 arcsec, mag 4.6
Callisto 2,980 = 1.32 arcsec, mag 5.6
Io 2,260 = 1.00 arcsec, mag 5.0
Europa 1,940 = 0.86 arcsec, mag 5.3

Ganymede, at 3,270 miles in diameter, at a distance of 5 A.U., would
appear 1.45 arcseconds across. This will vary slightly as Jupiter gets
closer or further away from Earth. Jupiter varies from about 4.25AU
to about 6AU.

Dawes Limit is an inappropriate criterion to measure whether an object
will appear larger or smaller than the Airy disk produced by the
scope. Dawes Limit is simply an empirical measure at which two
components of a double star can be noticed as double because a notch
identifies them. Dawes is not equivalent to Airy disk size. The
correct measure for the radius of the Airy disk for your scope is
Rayleigh Criterion, 5.45/Dinches or 138/Dmm.

Rayleigh Criteria gives the radius of the Airy disk. The central
bright spot, or the visible disk portion of the Airy disk, for a
moderately bright star (assumed 5th-6th mag) is approximately one half
the Airy disk diameter. The Airy disk radius for a 80mm scope is
138/80 = 1.72 arcseconds. The Airy disk for ALL 80mm scopes is 1.72
radius, therefore diameter = 1.72 x 2 = 3.44 arcseconds.

If the light is only moderately bright, such as from a 5th - 6th
magnitude star, then the central bright spot, or the visible disk
within the Airy disk, is about one half of the full diameter of the
Airy disk. Therefore, in a 80mm scope, the diameter of the central
bright disk for a moderately bright star would be 1.72 arcseconds,
equal to the Rayleigh Limit.

If the object is brighter, say 4th or 3rd magnitude, there is more
light in the visible central disk, maybe on the order of 60% to 75% of
all the light, up to a maximum of 84% for the brightest stars.
Therefore the central bright disk may be on the order of 60% to 75% of
the diameter of the Airy disk for fairly bright objects. It may be
less than 50% of the diameter of the Airy disk for a faint star. How
much of the light falls into the central disk and how much is thrown
into the diffraction rings is dependant on the magnitude.

For an object to be resolved, the angular dimension of the object must
be larger than the angular dimension of the Airy disk. Otherwise the
scope will simply fatten up the image and make it appear larger than
it truly is. The special condition of a disk as an extended object
slightly changes the size of the "unresolved" image.

Ganymede's moon disk is 1.45 arcsec across. It is smaller than the
1.72 arcsec Airy disk, the resolution limit of the 80mm scope, so it
will not be resolved. But it will form an image in the scope larger
than the Airy disk. Only a point source will produce an image the size
of the Airy disk. A moon disk is an extended object. All points on
the 1.45 arcsec moon disk may be considered point sources. Each point
source gives off light that forms an Airy disk.

The image in the scope of a true Airy disk, from a star too far away
to have any perceptible dimension, is the Airy disk. The airy disk has
dimension. Ganymede, a moon disk, has an infinite number of Airy
disks that can be considered to emanate from everywhere on the 1.45
arcsec moon disk, including centered on all the edges. If the light
from each point is equal and near 5th magnitude, then each point
produces an Airy disk with a bright central visible disk 1.72 arcsec
diameter. With the center of a visible diffraction disk on the very
edge of the moon's disk, half of each visible diffraction disk extends
beyond the moon disk. Therefore, at first pass, Ganymede will produce
an image in the scope equal to the width of Ganymede's disk plus
Rayleigh Limit (the Airy disk).

Rather than an Airy disk of 1.72 arcsec, Ganymede will produce an
apparent image disk of 1.45 + 1.72 or 3.17 arcseconds. This object
itself, the disk of Ganymede, is too small for the resolution of the
80mm scope and still is not resolved. But the image due to the special
condition of the extended object is wider than an Airy disk. But the
size of this image will be further qualified by integrated magnitudes.

Two stars very close together will have an integrated magnitude
brighter than each of the individual stars. It is reasonable to
assume that the integrated light of the moon disk is made up of an
infinite number of points, each having less light than the full
integrated magnitude of the moon disk itself. The visible spot
portion of the Airy disks, including those formed at the edges of the
moon disk, if truly formed by fainter light, may be somewhat smaller
than predicted above. Since it is difficult to know exactly what the
brightness of components of the integrated magnitude really are, how
small is difficult to determine, but it is reasonable to assume the
overall dimension of the image disk is smaller than 1.45 + 1.72
arcsec, maybe smaller by only 10% to 20% of the Airy disk radius.

A reasonable assumption is for a faint component, 40% of the energy
resides in the central bright spot. For a scope with a Rayleigh Limit
of 1.72 arcsec, 40% of the energy in the central disk would result in
a central bright spot with a diameter of 1.38 arcsec. It is
reasonable to assume the (central bright spot in) Airy disks formed by
an infinite number of points are all somewhat smaller and fainter than
would be the Airy disk for the integrated magnitude of all the points.
In fact the light overhanging the edges may be even smaller and
fainter than the 40% energy value I claculate here. This would result
in an image 1.45 + 1.38 = 2.83 arcsec wide in an 80mm scope.

Said a different way, for any scope to be able to resolve an extended
object, the scope must have a resolution smaller than the object.
Otherwise, the scope will simply show the object fattened up by
producing the Airy disks all around the edges. The image size is
slightly smaller than the sum of the Rayleigh Limit plus the object
diameter.

None of Jupiter's moons can be resolved with a 80mm scope. Ganymede,
and possibly Callisto, but no others, may be resolved with a 100mm
scope.


JUPITER'S MOONS IN A 6" SCOPE

An f8 150mm refractor has a Rayleigh Limit of 5.45/6 or 138/150 = 0.92
arcseconds. It is capable of resolving moderately bright doubles as
close as 0.9 arcsec. I have confirmed that it is capable of doing so.
With my CR150 I have "cleanly split" 4 different doubles, all with
components between mag5 and 6, three at 0.9 and one at 0.8 arcsec. One
was seen at 300x split, one at 370x and two required 480x to see a
clean split between the two components. I have detected a 0.7 arcsec
double, but not seen a split in anything below 0.8.

The sizes (at 5AU) and the magnitudes of Jupiter's moons a
Ganymede 3,270 = 1.45 arcsec, mag 4.6
Callisto 2,980 = 1.32 arcsec, mag 5.6
Io 2,260 = 1.00 arcsec, mag 5.0
Europa 1,940 = 0.86 arcsec, mag 5.3

The magnitudes are very well placed for assuming none are too bright
or too faint to fit the normal (Rayleigh Limit) amount of light in the
central visible bright spot of the Airy disk. Ganymede may be just a
bit bright, and this might just enlarger the central spot a little.

Ganymede at 3,270 miles in diameter, at a distance of 5 A.U. would
appear 1.45" arcseconds across. 3270/5AU = y/x = tangent theta =
0.0004029 degrees = 1.45"
Jupiter can range from less than 4.5AU to just over 6AU from Earth.
These calculations are based on a distance of 5AU from Earth.

Even though I have acuity of 150 arcsec, I find I need a much larger
apparent size to see objects near the resolution limit. It has been
well documented that as doubles approach the Rayleigh Limit, it
becomes more difficult to see them. Take note of the magnifications it
took to see doubles of 0.9 and 0.8. In all but one, It took 370x to
480x. It took 480 x 0.8 = 384 apparent arcsec size to see a 0.8 arcsec
double. It took 370 x 0.9 = 333 apparent arcsec to see a 0.9 arcsec
double. As a comparison, it takes only about 130x to 150x to see
doubles of about 2 arcsec (260-300 arcsec) and only 75x to 100x to see
doubles near 2.5 arcsec (187-250 arcsec).

The images of the moons in the scope are all wider than the moon
disks. Since the edges of the moon disk give off light and create
Airy disks in the scope image, the dimension of the image is nearly
the width of the moon disk plus the Airy disk (half airy disk
overhanging the edges). Because the images are larger than an Airy
disk, they will be easier to see.

Based on that, I estimate magnifications to see these as disks in the
6" f8 refractor.
200x to see Europa 0.86 arcsec, image disk about 1.6
190x to see Io 1.0 arcsec, image disk about 1.7
150x to see Callisto 1.32 arcsec, image disk about 2.1 and
140x to see Ganymede 1.45 arcsec image disk about 2.2.


SEEING SATURN'S MOONS

At close approach Saturn is just over 8 AU from Earth. Currently it is
9+ AU from Earth. These calcs are based on 9AU

Titan is 5150km (3193 miles) mag 8.4, 0.78 arcsec diameter
Rhea is 1528km (947 miles) mag 9.7, 0.23 arcsec
Iapetus is 1436km (890 miles) mag 8.6 to 11.5, 0.22 arcsec
Dione is 1120km (694 miles) mag 10.4, 0.17 arcsec
Tethys is 1046km (650 miles) mag 10.3, 0.16 arcsec
Enceladus is 512km (317 miles) mag 11.8
Mimas is 421km (255 miles) mag 12.9
Hyperion is 360km (223 miles) mag 14.2

You can see here most of the angular dimensions are very tiny. For any
scope up to 20" aperture every moon except Titan shows up almost as a
point source.

Titan is just barely resolvable to a disk with an 8" under the best
possible conditions. The image disk will be larger than an Airy disk,
making the need for magnification a little less than if it were a
point source, but the faint magnitude will require additional
magnification to see Titan resolved. I believe magnification on the
order of 275x would be required to see Titan as a disk in an 8". It is
not resolved with anything smaller. Under good conditions it could
probably be resolved at 250x with a 10".

I have seen four moons with my G5 125mm SCT. I have seen 5 with my
CR150. The 5th was Enceladus. I have never seen Iapetus or Mimas.
Hyperion is beyond my scope capabilities.


SHADOW TRANSITS OF JUPITER'S MOONS

The sizes (at a distance of 5AU) of Jupiter's moons a
Ganymede 3,270 = 1.45 arcsec
Callisto 2,980 = 1.32 arcsec
Io 2,260 = 1.00 arcsec
Europa 1,940 = 0.86 arcsec

Jupiter can range from less than 4.5AU to just over 6AU from Earth.
These calculations are based on a distance of 5AU from Earth.

A black spot on a white background, a shadow of a moon, is a perfect
example of a special extended object resolution. If the stated
resolution of your telescope is R, this special condition can
sometimes be seen at a size of R/2 to R/3. That means if you have a 8"
scope with a Rayleigh Limit of resolution calculated at 5.45/8 or 0.68
arcseconds, your R/2 and R/3 limits would range from 0.34 to 0.23
arcseconds.

For Jupiter, let's base the calculations on a 80mm scope. That means
if you have a 80mm scope with resolution calculated at 138/80 = 1.72
arcseconds, your R/2 and R/3 limits would range from 0.86 to 0.57
arcseconds. An 80mm scope is quite enough to see even Europa transit.
The only difficulty you might need to overcome would be the contrast
of the shadow drifting across any dark equatorial bands on the disk.

Shadow disks act differently than bright moon disks. The light from a
moon disk is explained to emanate from an infinite number of points,
hence making the image disk in the scope larger than the actual disk.
Shadows do not give off light, they hide light.

The image of a shadow disk may appear smaller than the actual
dimension of the true shadow disk. The light surrounding the shadow
disk has the properties that we previously used to describe the moon
disks. The points along the edges of the light around the shadow all
produce Airy disks that infringe upon the image of the shadow disk
image. Therefore, a shadow disk image may appear smaller than it's
true dimension by a small amount. The light gradient across the
border between the shadow and the light has a gray area and the
contrast between the bright and dark areas is much less than fully
dark to fully light.

Ganymede's and Callisto's shadows might be seen with magnifications as
low as 125x, even with the 80mm scope.
You might expect magnifications to be
160x-180x to see Io's shadow and
175x-200x to see Europa's shadow


SHADOW TRANSITS OF SATURIAN MOONS

The distance to Saturn when we are both on the same side of the Sun is
about 8 A.U. Currently Saturn is about 9AU from Earth.

For a distance of 9AU from Earth,
Titan is 5150km (3193 miles) 0.78 arcsec diameter
Rhea is 1528km (947 miles) 0.23 arcsec
Iapetus is 1436km (890 miles) 0.22 arcsec
Dione is 1120km (694 miles) 0.17 arcsec
Tethys is 1046km (650 miles) 0.16 arcsec

Titan is 3200 miles in diameter. Titan would appear to be 0.78
arcseconds in diameter when Saturn is 9AU from Earth, 0.86 arcsec when
it is 8AU. Titan is close in size to Cassini. The Cassini division is
2,800 miles wide and at 8.A.U. it appears 0.75 arcseconds wide at its
widest point, at the ansae.
The second largest of these moons, Rhea, will appear in our scopes as
only 0.23 arcseconds in angular diameter. We see the moons as point
sources, but their shadows are seen as a special form of extended
object.

A black spot on a white background, a shadow of a moon, is a perfect
example of a special extended object resolution. If the stated
resolution of your telescope is R, this special condition can
sometimes be seen at a size of R/2 to R/3. That means if you have a 8"
scope with a Rayleigh Limit of resolution calculated at 5.45/8 or 0.68
arcseconds, your R/2 and R/3 limits would range from 0.34 to 0.23
arcseconds.

With the second largest moon to Titan casting a shadow only 0.23
arcseconds in diameter, you would need everything to absolutely be the
best possible condition in the best sample of equipment with the best
possible contrast to see any moon shadow other than Titan on the disk
of Saturn with an 8" scope. (This ignores the small reduction in the
size of the shadow due to the light cone from the very distant Sun).

You would have a slightly better chance with a 10" scope. At 5.45/10 =
0.545 arcseconds, R/2 and R/3 would be 0.22 to 0.18 arcseconds. At
least a 10" scope could possible see a dark spot caused by the two
largest of these other moons. Dione's and Tethys' shadows would be too
small to see in a 10" under the best possible conditions.

Titan, if it were passing close enough could be casting a shadow 0.78
to 0.86 arcseconds. That would appear just a bit wider than the
Cassini division.

Rarely would we ever see Saturn's moons transit. Because of the
plane of tilt that so nicely shows us the rings, the moons appear to
follow an elliptical orbit around the planet. However, when the rings
near edge on, we may have opportunity to see a Saturnian moon transit.
We'll have to wait a few years (2009?) for that.


WHAT DOES IT TAKE TO SEE THE ENCKE DIVISION?

Encke is assumed to have a width somewhat less than 1000km and
possible as narrow as 500km. Some sources quote 350Km. For
comparison the Cassini division is 4200km. Cassini, at the current
distance to Saturn has an angular measure of 0.75 arcseconds.
Therefore, Encke has an angular measure of a maximum 0.18 arcseconds
and may possibly be as small as 0.1 arcseconds.

In order to see small objects or features you must take into
consideration several things; the size of the object or feature, the
contrast with adjacent features or dark sky, the limitations of
resolution of the instrument and the limitations of the eyes.

We'll leave atmospheric conditions out of this and make all
assumptions that observations are being made under the best possible
sky conditions.

We've identified the size of the feature. We'll use the most
optimistic value and assume Encke measures 0.18 arcseconds. How about
contrast? For Encke it's actually not very high, like Cassini. Unlike
Cassini, which is bordered by the brightest features in the entire
ring system and has one of the highest contrasts of any extended
object, the Encke division has far less contrast being much more
subdued in the less bright regions of the A ring. Contrast is not a
big help for Encke.

How about resolution? It all depends on your optics. We can work
various sizes down to see if they would be able to resolve Encke. Or
we could work our way up using resolution criteria to determine what
size scope might be able to see Encke. We'll do a little of both.

Resolution is calculated using Rayleigh criteria. Some will say we can
use Dawes Limit, but that is an incorrect assumption. Dawes, based on
being able to identify that a double has two components, does not
provide for seeing a clear separation between to objects. Rayleigh
criterion IS the formula for the Airy disk and is the basis for
calculation of resolution in all optics. Rayleigh Limit = 138/Dmm or
5.45/Dinches.

Let's try an 8" scope. Resolution limit for an 8" scope is 5.45/8 =
0.68 arcseconds. That's not going to be good enough to see Encke, is
it? Well there's a bit more to it.

Encke fits the classification of an extended linear feature, black on
white (although without high contrast), and this gives the observer a
benefit. It's much easier to see this particular kind of feature than
any other type of extended feature. In fact, the benefit can be 3.5x
to 5x greater than the resolution limit of the scope. An 8" scope can
detect linear features as small as 0.68/3.5 = 0.19 arcseconds to
0.68/5 = 0.14 arcseconds wide.

Is the 8" enough to see Encke? We assume Encke measures between 1000km
and 500km and has an angular measure of a maximum 0.18 arcseconds and
may possibly be as small as 0.1 arcseconds. The 8" scope using the
criteria R/3.5 or R/5, can see somewhere around 0.19 to 0.14
arcseconds at best.

If Encke truly is at the widest dimension suspected AND the 8" scope
is seeing extended linear features to the maximum benefit, then the 8"
IS just barely enough to see this feature. However every one of those
assumptions are in the favor of making the claim it can be seen with
8".

Suppose any single one of those assumptions needs to go with B instead
of A. For example, if Encke is only 500km wide (0.1 arcseconds) and
the 8" scope is seeing extended linear features only 3.5x better
(0.68/3.5 = 0.19 arcseconds), then an 8" wouldn't be anywhere near
enough scope to see Encke. Even if the scope were seeing features at
1/5 the normal dimension of resolution, 8" still would not be enough.

If the scope is seeing features at 1/5th the normal dimension (0.68/5
= 0.14 arcseconds), it's still not enough if Encke is only 500km wide.

So, there are four possible combinations due to uncertainty, and in
three of those conditions an 8" scope cannot see the Encke division.
Only if you took every possible advantage and then had the best
conditions (and a precise collimation) could an 8" scope see the Encke
division.

Even a 10" scope will meet the conditions in only two out of the four
possible combinations. If Encke is only 500km wide, neither of these
scopes are enough scope under any conditions.

OK. Let's say, by whatever possibility, you met the slim conditions
and your scope can see Encke. What will it take for your eyes to see
it? This is acuity. Normal acuity for most people dictates that a
feature must be enlarged to a dimension of about 2.5 to 3 arcmin
before your eyes can see it. An exceptionally few people can see
features when the apparent size of an object is magnified to only 2
arcmin. But, again that extended linear feature is going to help you.
In fact, it would seem under excellent conditions, you may be able to
see linear features if the feature crosses at least just two receptors
in the eye. This may mean you might need to magnify it only to an
apparent size of 1 arcmin or 60 arcseconds. So again, we will assume
the most optimistic condition and allow that only enough magnification
is needed to reach an apparent size of 60 arcseconds.

If Encke is 1000km wide (0.18 arcsec) then magnification required to
see it is 60/0.18 = 333x. If Encke is only 500km wide then you need
60/0.1 = 600x magnification to see it.

The likely-hood that you could get every one of these conditions in
your favor is extremely slim. In fact it is very unlikely.

So here is a set of very reasonable assumptions from all these facts
and conditions:
It is not likely 8" is enough scope to see Encke. It's even a stretch,
but it's possible, for a 10" scope to see Encke. It's not likely it
would be seen by anyone, even someone with exceptional acuity, at
anything less than about 350x. It's more likely that a magnification
of around 450x might be required.

So the most likely combination to allow one to see the Encke division
is a 10" scope at over 400x magnification. But don't let that stop
you from trying!

edz