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Seeing Jupiter's moons as disks

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#1 EdZ

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Posted 03 March 2004 - 10:00 PM

Disk vs. Airy Disk
SEEING JUPITER’S MOONS AS DISKS

I answered this question, which was posted on the Astromart Equipment talk forum under the title “Disk vs. Airy Disk” on 2/12/04

“Does anyone here actually know what the laws of physics say with regard to observation of Jovian and Saturnian satellites, and specifically, what the laws of physics say the minimum aperture necessary to resolve the disks thereof might be? It seems to me of late that the four Galilean satellites of Jupiter are each resolvable (under conditions of good seeing) into their own, discrete disk, using only an humble 80mm f/11.4 refractor. I am curious to know, is this merely wishful and imaginative viewing on my part, or are my subjective observations likely to be accurate (in at least this specific instance)?”

My response,
I wrote a complete response to a very similar question, Saturn's moons, just a few weeks back.

See the thread
Seeing Moons and Shadow Transits

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 are:
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, Ganymede will produce an image in the scope equal to the width of Ganymede's disk plus the diffraction disk from the Airy disks at the edges.

Rather than an Airy disk of 1.72 arcsec, Ganymede may produce an apparent image disk as large as 1.45 + 1.72 or 3.17 arcseconds. The 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 diffraction in the extended object, is wider than an Airy disk.

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 of a fainter magnitude. The visible spot portion of the Airy disks, including those formed at the edges of the moon disk, if formed by fainter light, may be somewhat smaller than predicted above. How small is difficult to determine, but it may be 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, therefore approximately 2.8 arcsec.

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.

However, the casting of a shadow across the planetary disk is another matter entirely. For that you must read the answer at the link above.

"Is there a term for the type of 'partial resolution' wherein an object, though not resolved into a discrete, visible disk, nonetheless gives a non-Airy disk, distinctly non-stellar appearance?"

The only possible term I can think of that would fit this description is "Extended Object". Solar System moons that are large enough to be resolved to a disk, where a scope is large enough to produce an image of a disk, would be considered extended objects. Solar system moons far enough away, with a small enough diameter so as to elude resolution, are still considered extended objects. For both, the disk in the image will follow the form stated above. I know of no term to differentiate between the two.

You may consider that there are two types of objects, point sources and extended objects. All point sources are truly small enough that the image formed is an Airy disk. Anything larger than a true dimensionless point source is an extended object.

Of course, you can now get into the classifications of type of extended objects, moons-bright disks, dark spots-shadow transits, dark lines- Cassini, circles-craters, faint diffuse light-nebula. Each has it's own criteria. That's another lengthy study.

edz


#2 Scott Beith

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Posted 03 March 2004 - 10:51 PM

Ed,
Being too lazy to do the math, I will just ask you. Could my 150mm f/8 refractor resolve all four as disks?

Any idea of the magnification needed to accomplish this?

Jupiter is a serious point of intrest for me, and I would like to know a little more about the results I can expect on a night of good seeing.

Thank you Sir.

Scott

#3 EdZ

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Posted 03 March 2004 - 11:47 PM

Scott,

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 are:
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.

edz


#4 Scott Beith

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Posted 03 March 2004 - 11:54 PM

Thanks Ed. I can push to 400x right now, but will soon have the capability to go a little higher. Great info Sir. Thanks for the research.

Scott

#5 Jure

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Posted 04 March 2004 - 07:31 AM

Under excellent to almost perfect seeing on one night I saw with my CR150HD at 240x all the moons as distinct disks of different diameters. And I wasn't really paying attention to them, I just happened to notice it.
CS!Jure

#6 EdZ

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Posted 04 March 2004 - 09:11 AM

At most times Jupiter is not close enough to Earth for Europa to be resolved by a 6" telescope. Jupiter would need to be 4.7 AU or closer for Europa to be large enough to exceed the diameter of the Airy disk created by a 6" scope. Check a reference for the date you observed. Jupiter is currently about 4.5AU and will reach closeest approach next month at 4.4AU. It can be as far away as 6.2AU from Earth.

Unless that condition is met, you did not resolve Europa. The scope would simply fatten up the image until it appears a little larger than an Airy disk. It can be helpful to have a star in the same field of view as the moons. The highly magnified image can be compared to that of the star and you may be able to compare the disk image to the Airy disk produced by the star. it would need to be about twice as large as the Airy disk to indicate resolution.

edz



#7 sixela

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Posted 20 September 2005 - 08:52 AM

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, Ganymede will produce an image in the scope equal to the width of Ganymede's disk plus the diffraction disk from the Airy disks at the edges.

Just to pick a nit (and it doesn't invalidate the rest of your analysis)...

But illumination drop-off is more severe than in the original Airy disc pattern:

-on the very edge of that "extended" image only one infinitesimal point of the moon disk will contribute to light from its "Airy disc". As a point has zero area, the light intensity after integration will be zero

-in the middle, the contribution comes from a very large area of points.

In other words, while theoretically the size of the image is as large as you indicate, you're not likely to *see* something as large, at least if you don't count the parts that, in practice, are indistinguishable from the background.

#8 LivingNDixie

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Posted 20 September 2005 - 12:54 PM

I have read where people have seen the Moons as disks in scopes as small as four inch APOs.


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