For decades, and – if we omit the Dark Age – for centuries, astronomers and scientifically literates had a strong presentiment that stars that harbor planets are not only common but probably ubiquitous. However, until the last few decades, the biggest challenge for Astronomers was still to find ways to observe, resolve and render objects of our own solar system.
Fortunately, in this last decade, the Europeans with their Very Large Telescope in Chili, the American observatories in Hawaii, the many collaborations on large telescopes around the world, the Keck Interferometer and the Hubble Telescope above it all have finally given to humanity a tremendously richer and deeper understanding of our universe. If technics in light gathering have made huge progress, it is in the domain of resolution where Astrophysicists teaming up with instrumentalists have made the greatest leaps in the beginning of this millennium. Instrumentalists have given us giant leaps in instrumentation optics; technologically, like adaptive optics, but with human ingenuity that has given us interferometry, coronography and maskings.
Collecting vast amount of photons is crucial in discovering faint objects orbiting about a star, but the capacity to resolve these objects is even more important if one seeks to analyze these celestial objects and seeks to turn these analysis into solid science. By peering in the infrared as to penetrate inside thick clouds of dust, by using maskings to block the light of the parent star (coronography), by taking advantage of the diffractive property of light (non-redundant masking), and with the help of mathematic tools created by Jean-Baptiste Fourier, astronomers do not need to wait anymore until a planet passes between its parent star and shall be big enough in order to be detected.
Indeed, the search for exoplanets has matured quite a bit in the last five years. Astronomers have done a great job these last decade in focusing on instrumentation rather than theorizing on a blackboard about the type of objects that has yet to be captured on film or on a CCD chip. Indeed, astronomy has the wonderful privilege to invariably deliver to the public, literate or not, a phone book size list of discoveries over the course of every single year. Every years the universe unveils objects far more amazing than we can predict. And that's good: today's academics and scientists who chose to study exoplanets, for instance, are now armed with solid instrumentation, and can present very exiting and useful science.
Some Theory...
All stars form by gravitational collapse of gas and dust in cold interstellar clouds fed by supernovas and dying stars. These clouds are not spread homogeneously. Therefore lumps of matter and gases start to accumulate in some regions of these clouds, lumps which core is heated dramatically by the tug of war between pressure and gravity. As more mass gathered at a tinier region of space gravitational forces dramatically increases. The process accelerates until hydrogen atoms collapse so hard onto each other that they fuse. When four nucleus of hydrogen collapse, they form an atom of helium. However, the resultant helium nucleus is only 0.7 percent the mass of 4 hydrogen atoms! Where all this mass went? Well, thanks to Albert Einstein E = mc2 we do know now: it was converted into a lot of energy. Multiply this process by a few billions of moles of hydrogen, we will have here all the ingredients for a thermonuclear process to be unleashed and a star is born.
A star however do not need to swallow all the matter about itself in order to start burning fuel and start sending a flurry of photons. It often remains around a star enough matter to mold a few planets. This material is dragged along in the gravitational dance around the star, and the whole thing orbit in concert. As the rotation increases its pace after a vast amount of matter has been swallowed by the star (think about the ice-skater spinning while folding her arm), a disk of matter is formed around the star, using the same beautiful process involving the tag of war between kinetics and gravity that forms a galaxy like our own. This rotation leads to a flattened disk orbiting the protostar.
A protostar is a young star that often shows itself still cradled in a bed of rock, gases and dust. In this disk, further gravitational attraction occurs. Sticking and collisions among the solid particles often leads to the growth of planetisimals and genuine planets, forming a protoplanetary disk. Some area in the disk however can remain crowded with dust and rocks as the region populated with asteroids, between Mars and Jupiter, can demonstrate.
These protoplanetary disks were inferred to be present around protostars and pre-main sequence stars by modeling their infrared excesses and their bipolar outflows. These disks have now been directly imaged using millimeter interferometers and the Hubble Space Telescope. It has been therefore verified and made evident that most stars are born surrounded by a protoplanetary disk, although the ubiquity of these disks does not offer the guarantee that planets may form.
So, How many?
In october 2009, About 374 planetary systems have been discovered around nearby solar-type stars. Most are single Jovian systems with orbit sizes from 0.02 to 4 A.U. But this month, the astronomers of the European Southern Observatory have harvested their first terrestrial planet3, a five Earth-mass planets fast rotating at less than 2 A.U. The exoplanets discovered up until now are located in the Sun's neighborhood, with distances of less than 100 parsecs – one shall compare with our galaxy that has a radius of 10 000 parsecs! It is believe that 10 percent of the stars have at least one giant planet with a period of less than 10 years. If there are about 100 billion stars in the galaxy, there should be of the order of ten billion planets in our galaxy!
So far, we have discovered planets around only ten percent of a nearly complete sample of une thousand nearby solar-type stars. But the true fraction of stars with planets is undoubtedly larger, as the Doppler method is technologically limited to finding relatively massive stars with relatively close orbits. Planetary masses range from ~0.3 to 15 jupiter-mass but the frequency tend to decrease with mass. The lower mass limit is only due to limitations in the detection method, but as the last discovery of a terrestrial planet has shown, this lower limit will certainly lower even more. However, no one can explain still the absence of more massive planets, and the decrease in number as mass increases.
Among the jovian type planets that has been found so far, two types of planetary orbits found: `hot Jupiters' in circular orbits, and colder Jupiters with eccentric orbits. The possible explanations:
The gravitational interactions - or scattering - among several Jovian-type planets can produce eccentric orbits. A phenomenon that explains Pluto's unusual orbit in our solar system. An explanation for close orbits are the inward migration of Jovian planet due to gas drag and gravitational resonances with a young gaseous disk in which the young planets travels through. In such planetary systems, any inner terrestrial planets would probably be destroyed by collision with a Jovian planet spiraling down toward the star.
We also need, of course, to pay attention to gravitational scattering by nearby stars which can readily produce eccentric Jovian orbits, but such encounters are rare except in very dense stellar clusters or toward the center of the galaxy. Stars hosting planets also tend to systematically have higher metallicities than stars without planets. The cause of this is well explain yet: a stars born with more metals have disks with more interstellar dust which are more effective in forming planets. Or, perhaps, stars with inner Jovian planets have already accreted other planets (via inward migration) which have contaminated their surfaces with additional heavy elements.
Far Field / Near Field
Tuesday, September 22, 2009
Last week, I was introduced to the concept of Far field. I heard the term before but never thought to investigate the term. So, this is what I have learned so far. And please correct me with a nice comment if I am wrong. Near and far field are regions around the source where different parts of the field are relatively more or less important. The far field is basically a region around the source which radius is larger, far larger than the wavelength of an electromagnetic wave emitted from that source.
When the light that has left a object from very far away arrives to a telescope in a near flat field manner, the lens used to "bring the object closer" to us must also be far larger than the wavelength of this light, in order to preserve this far field property. The concept of far field is only for convenience as it allow the person who desires to analyze this system to employ far simpler mathematical jargons and formula than that of a near field system.
When the light that has left a object from very far away arrives to a telescope in a near flat field manner, the lens used to "bring the object closer" to us must also be far larger than the wavelength of this light, in order to preserve this far field property. The concept of far field is only for convenience as it allow the person who desires to analyze this system to employ far simpler mathematical jargons and formula than that of a near field system.
The Airy Disk
Friday, August 28, 2009
Written by Oldham Optical UK.
An Explanation Of What It Is, - And Why You Can’t Avoid It
It can be extremely surprising and annoying to find out that the big parabolic mirror you have just paid a lot of money for cannot focus all the incoming light down to an infinitesimally small point, - but instead puts it into a finite sized disk with some faint rings around it.
You have been told the Airy Disk cannot be avoided and that it’s due to diffraction and the wave nature of light, – but every explanation you look up soon disappears off into the deeper realms of mathematics. It all seems a bit of a fiddle!
Well! – I am afraid a lot of the mathematics cannot be avoided if you need to go into the subject deeply, but it is possible to get a working knowledge of what’s happening without too much of the maths and with just a bit of thinking in the bath and a bit of work on the kitchen table with a ruler and compass.
Perhaps the easiest way to start to visualise what’s happening with your parabolic mirror is by considering the wavefronts of light coming in from a distant star. The star is so far away that the light rays approaching can be considered parallel, and the approaching wavefronts can be considered flat plane lines, across the incoming parallel rays.
If you live at the seaside, - stop and look at a series of waves marching in towards the beach. If the tide is well up and they hit a concrete sea wall, - you will often see them reflect back. If the sea wall happens to be curved, you may even see some of the reflected waves converge, (focus), towards a point.
Unfortunately, - Not many sea walls are designed with the perfect parabolic shape to be an example of how a parabolic mirror works! - but any concave curve should show some focusing effect enough for you to get an idea of what's happening.
So getting back to light and a parabolic mirror, - The flat wavefronts march in to the mirror like waves in the sea and are reflected. The parabolic shape of the mirror converts the incoming flat wavefronts into reflected spherical wavefronts. These spherical wavefronts converge at the focal point of the mirror. See the diagram on the right. If the mirror was perfect, light was not comprised of waves and diffraction did not exist, the focal point where all the light converged would be infinitesimally small.
Sorry! - It's just not like that!
Diffraction is something that happens at edges. When something like light meets the edge of a solid object, some of the energy tries to bend round it. The usual diagram seen in Physics textbooks to illustrate Diffraction is of flat wavefronts approaching a screen with holes in it. The holes are small and close to the wavelength of the light used. The light passes through the holes and then part of it tries to bend around the edges of the holes as it emerges.
If the holes are small and not many wavelengths of light wide, then the resulting wavefront from each is spherical. It appears as if the light is being generated from point sources situated at the holes themselves, rather than the actual source which is somewhere behind the screen.
The really interesting thing is where you have two or more holes all acting as point sources, the light from each hole will interfere with each other. That is what happens with a Parabolic mirror to form the Airy Disk and the rings.
There is an experiment you can do in the bathroom at home that illustrates the same wave effect, but note this health Warning first, - It might not be a good idea to let your wife find out you are carrying out experiments of this nature!
So sneak into the bathroom armed with blocks of wood and float them in the bath. They should be spaced with a small gap between them. Make waves on one side and watch how they propagate through the hole. If you don’t think you dare try this in the bath, you can do the same thing just as well in the garden pond with a couple of rocks, - but do not let the neighbours see you! – They might tell the wife!
Diffraction does not need a hole, it only needs an edge. In the case of our parabolic mirror it is the edge of the mirror, (The rim), that that will give diffraction.
Imagine the individual ray of light that comes all the way in from a star that strikes the sharp edge (rim), of the mirror; This will reflect, but because it is right on the sharp edge of the mirror it will be scattered in a similar manner to the rays of light emerging from a hole in a screen. The reflected light is scattered off the rim in an expanding spherical wavefront just as if coming from a point source situated on the rim of the mirror.
In real life, the reflected light from the edge of the mirror merges with the rest of the wavefront reflecting from the main surface of the mirror. Most of the combined wavefront is spherical from the main body of the mirror but bends back in the opposite direction towards the edges. It is the departures at the edges that form the rings around the central Airy Disk.
Although it sounds very complicated, its actually very simple to demonstrate what is happening just by sitting down at the kitchen table and drawing out the wavefronts on a piece of paper with a ruler and a compass.
Start by drawing a mirror say 2 “ in Diameter. Then with a compass you are going to draw out concentric arcs from each rim of the mirror to represent the wavefronts as in the diagram adjacent. Each arc must be regularly spaced as it represents a wavelength.
So draw say 10 arcs from each rim of the mirror starting at say 2" radius, and at every ½" out to say 7½". The arcs from each rim will intersect.
If these pencil arcs were waves of light they would interfere with each other. Where two arcs cross there would be a bright patch of light and where one arc is half way between two other arcs from the second rim, they would cancel out and give a dark patch.
If you then look at what you have drawn you will see you can draw a series of straight lines through the intersections. All lines begin from roughly the centre of the mirror. These lines represent the bright patches of light.
They are the black dotted lines in the diagram, with the central bright spot shown in a heavy black line.
Now steadily go through the drawing and put a mark where one line from one rim is half way between two lines from the other rim. Once finished the marks should lie on a second series of lines that again start at roughly the centre of the mirror and run out through the marks. This series of lines is the dark patches that exist between the Airy Disk and the Rings.
The first two defining the Airy Disk are the blue dotted lines in the diagram.
Although on flat plain paper these are drawn as lines, - on a three dimensional parabolic mirror they represent cones of light. Where these cones intersect the focal plane of your telescope they will appear as a bright central disk with rings around it. The central disk is of course the Airy Disk. In practice, when looking through a telescope the rings rapidly get faint and it is usually only the first ring and perhaps the second that can be seen in practice.
The angle of the Disk and of the first Ring is illustrated on the diagram above. The first Airy ring is easy to work out. It is the first set of bright intersections away from the central spot. The Airy Disk itself is not quite as easy to see at first. The accepted convention for the size of the Airy Disk is to measure to the centre of the first dark patche either side of the central spot. The diagram should give you the lead.
Instead of using the term "angle to the disk", - the term more commonly used in Astronomical circles is "Angular Diameter". So please note that when the rest of this document uses "Angular Diameter", it means the angle to the Airy disk (or rings), as described above.
If you repeat the drawing with different diameter mirrors, say 1” & 3”, while still using ½” rings to represent the same wavelength, you will find that as you increase the diameter of the mirror, the Angular diameters of the Disk and the Ring reduce. Since the angular diameter reduces, the physical diameter of the Disk and rings also reduce.
Although this illustration is just something done on the kitchen table, It is exactly the same mathematics that apply with light to form the real Airy Disk and Rings. The angular diameter of the Disk and rings is calculated with exactly the same rules. It follows that the Airy Disk gets smaller as the Diameter of the mirror increases.
Before the people who really understand Mathematics jump in and complain that the intersections are not really on straight lines, - they are on hyperbolic curves, - it is agreed that they really are on slight curves rather than straight lines! The lines come in from infinity and do bend very slightly as they approach the mirror. So evidence of a slight curve is likely to be noticed on the kitchen table using wavelengths of ½" with a 2" mirror.
However light has a wavelength of about 500nM, so the line for the first Airy Ring will have come in from infinity and will bend out of line near the mirror a maximum of ½ a wavelength, which is about 250nM. It is submitted that coming all the way from infinity and only bending 250nM out of line is still pretty damn straight!
In any case, - the accepted formulas used for working out Airy Disk size and angular diameter assume the lines are straight, - so no reason this description cannot be allowed to do the same!
The description above only covers light from the rim of the mirror - so what about the light from the rest of the mirror? - Don’t forget the rest of the mirror surface is pouring light into the central spot. The very simple explanation is that this light merges with the light scattered from the rim of the mirror to give the bright Airy Disk.
The only practical thing a telescope constructor can do to make the Airy Disk smaller is to make the mirror diameter larger. This makes the angular diameter of the Airy Disk smaller. The angle is fixed for any one diameter of mirror. Once the angle is fixed, the physical size of the Airy Disk is determined solely by the focal length of the mirror.
A common formula used for working out the size of the Airy Disk uses the Focal ratio, which includes both the diameter of the mirror and the focal length in one term.
One formula for the Airy disk Diameter is:
D = 2.43932 x λ x Focal Ratio
D = Diameter of Airy Disk in mm; λ = Wave Length in mm (e.g. 546nM = 0.000546mm), (If Focal ratio = F/4 & 546nM used then D = 0.00533mm for example).
One formula for the angular diameter of the Airy Disk is:
A = 7200(Arc Tan(1.21966 x λ /d))
A = Angular Diameter of the Airy Disk in Arc Seconds; d = Diameter of Mirror in mm.
Ok! - so you now know what the Airy Disk is!, but from the description above you should be thinking there’s a easy way to get rid of the damn thing! – Surely if it’s just the light at the edge of the mirror that causes diffraction, then why not just mask off the edge of the mirror with a cardboard ring? – This will stop light hitting the edge of the mirror. Wont this get rid of the Airy disk?
Sorry – this does not work! You now get diffraction from the edge of the mask instead! – What you have done by masking the mirror edge is add what is called an “Aperture Stop” to your optical system.
By doing so you slightly raise the focal ratio of the “system” and the Airy Disk actually gets bigger. This also explains why in your telescope you need to keep the spider holding the elliptical flat or secondary mirror as small and unobtrusive as practical. Any obstruction causes additional diffraction.
Finally, - here is the killer - let’s suppose you could magically mask off the edge of the mirror without causing extra diffraction. You have managed to limit the beam of light falling on the mirror to less than the mirror diameter so the edge of the mirror is not illuminated. You must really think you have the problem cracked now?
Sorry – This still does not work! The edge of the beam of light falling on the mirror surface is itself a discontinuity. When it strikes the mirror and reflects it behaves exactly as if it was a real physical edge. There is diffraction from the edge of the beam exactly the same as if it was the physical rim of the mirror. For this situation there is no easy and simple "kitchen sink" explanation. This is where I “cop out” and say this is due to the wave nature of light and you do need to start digging into the physics text books if you want to understand it better.
Exactly the same thing happens if you paint the outer edge of your mirror with black paint. All this does is cause diffraction to occur where the black paint finishes and the reflective surface starts. All you have done with the paint is reduce the diameter of the mirror. The result is that the Airy Disk gets bigger.
The message is that you cannot avoid the Airy Disk.
However you can make it smaller: You can either increase the diameter of the mirror, which reduces the angular diameter of the Airy Disk; - or reduce the focal length of the mirror, - which reduces the physical size of the Airy Disk.
The Airy disk size is suddenly becoming more important to Amateur Astronomers due to the recent improvements in Digital cameras where the CCD sensor pixel size has now reduced to the point where it is comparable or smaller than the Airy Disk size. In future, Telescope mirror diameter and focal length may have to be chosen to better match the CCD pixel size?
To close, - You could completely avoid the Airy disk by using an infinitely large mirror? – but I would respectfully suggest a long talk with your wife or perhaps your doctor before trying to order one of these.
An Explanation Of What It Is, - And Why You Can’t Avoid It
It can be extremely surprising and annoying to find out that the big parabolic mirror you have just paid a lot of money for cannot focus all the incoming light down to an infinitesimally small point, - but instead puts it into a finite sized disk with some faint rings around it.
You have been told the Airy Disk cannot be avoided and that it’s due to diffraction and the wave nature of light, – but every explanation you look up soon disappears off into the deeper realms of mathematics. It all seems a bit of a fiddle!
Well! – I am afraid a lot of the mathematics cannot be avoided if you need to go into the subject deeply, but it is possible to get a working knowledge of what’s happening without too much of the maths and with just a bit of thinking in the bath and a bit of work on the kitchen table with a ruler and compass.
Perhaps the easiest way to start to visualise what’s happening with your parabolic mirror is by considering the wavefronts of light coming in from a distant star. The star is so far away that the light rays approaching can be considered parallel, and the approaching wavefronts can be considered flat plane lines, across the incoming parallel rays.
If you live at the seaside, - stop and look at a series of waves marching in towards the beach. If the tide is well up and they hit a concrete sea wall, - you will often see them reflect back. If the sea wall happens to be curved, you may even see some of the reflected waves converge, (focus), towards a point.
Unfortunately, - Not many sea walls are designed with the perfect parabolic shape to be an example of how a parabolic mirror works! - but any concave curve should show some focusing effect enough for you to get an idea of what's happening.
So getting back to light and a parabolic mirror, - The flat wavefronts march in to the mirror like waves in the sea and are reflected. The parabolic shape of the mirror converts the incoming flat wavefronts into reflected spherical wavefronts. These spherical wavefronts converge at the focal point of the mirror. See the diagram on the right. If the mirror was perfect, light was not comprised of waves and diffraction did not exist, the focal point where all the light converged would be infinitesimally small.Sorry! - It's just not like that!
Diffraction is something that happens at edges. When something like light meets the edge of a solid object, some of the energy tries to bend round it. The usual diagram seen in Physics textbooks to illustrate Diffraction is of flat wavefronts approaching a screen with holes in it. The holes are small and close to the wavelength of the light used. The light passes through the holes and then part of it tries to bend around the edges of the holes as it emerges.
If the holes are small and not many wavelengths of light wide, then the resulting wavefront from each is spherical. It appears as if the light is being generated from point sources situated at the holes themselves, rather than the actual source which is somewhere behind the screen.
The really interesting thing is where you have two or more holes all acting as point sources, the light from each hole will interfere with each other. That is what happens with a Parabolic mirror to form the Airy Disk and the rings.
There is an experiment you can do in the bathroom at home that illustrates the same wave effect, but note this health Warning first, - It might not be a good idea to let your wife find out you are carrying out experiments of this nature!
So sneak into the bathroom armed with blocks of wood and float them in the bath. They should be spaced with a small gap between them. Make waves on one side and watch how they propagate through the hole. If you don’t think you dare try this in the bath, you can do the same thing just as well in the garden pond with a couple of rocks, - but do not let the neighbours see you! – They might tell the wife!
Diffraction does not need a hole, it only needs an edge. In the case of our parabolic mirror it is the edge of the mirror, (The rim), that that will give diffraction.
Imagine the individual ray of light that comes all the way in from a star that strikes the sharp edge (rim), of the mirror; This will reflect, but because it is right on the sharp edge of the mirror it will be scattered in a similar manner to the rays of light emerging from a hole in a screen. The reflected light is scattered off the rim in an expanding spherical wavefront just as if coming from a point source situated on the rim of the mirror.
In real life, the reflected light from the edge of the mirror merges with the rest of the wavefront reflecting from the main surface of the mirror. Most of the combined wavefront is spherical from the main body of the mirror but bends back in the opposite direction towards the edges. It is the departures at the edges that form the rings around the central Airy Disk.
Although it sounds very complicated, its actually very simple to demonstrate what is happening just by sitting down at the kitchen table and drawing out the wavefronts on a piece of paper with a ruler and a compass.
Start by drawing a mirror say 2 “ in Diameter. Then with a compass you are going to draw out concentric arcs from each rim of the mirror to represent the wavefronts as in the diagram adjacent. Each arc must be regularly spaced as it represents a wavelength.
So draw say 10 arcs from each rim of the mirror starting at say 2" radius, and at every ½" out to say 7½". The arcs from each rim will intersect.
If these pencil arcs were waves of light they would interfere with each other. Where two arcs cross there would be a bright patch of light and where one arc is half way between two other arcs from the second rim, they would cancel out and give a dark patch.
If you then look at what you have drawn you will see you can draw a series of straight lines through the intersections. All lines begin from roughly the centre of the mirror. These lines represent the bright patches of light.
They are the black dotted lines in the diagram, with the central bright spot shown in a heavy black line.
Now steadily go through the drawing and put a mark where one line from one rim is half way between two lines from the other rim. Once finished the marks should lie on a second series of lines that again start at roughly the centre of the mirror and run out through the marks. This series of lines is the dark patches that exist between the Airy Disk and the Rings.
The first two defining the Airy Disk are the blue dotted lines in the diagram.
Although on flat plain paper these are drawn as lines, - on a three dimensional parabolic mirror they represent cones of light. Where these cones intersect the focal plane of your telescope they will appear as a bright central disk with rings around it. The central disk is of course the Airy Disk. In practice, when looking through a telescope the rings rapidly get faint and it is usually only the first ring and perhaps the second that can be seen in practice.
The angle of the Disk and of the first Ring is illustrated on the diagram above. The first Airy ring is easy to work out. It is the first set of bright intersections away from the central spot. The Airy Disk itself is not quite as easy to see at first. The accepted convention for the size of the Airy Disk is to measure to the centre of the first dark patche either side of the central spot. The diagram should give you the lead.
Instead of using the term "angle to the disk", - the term more commonly used in Astronomical circles is "Angular Diameter". So please note that when the rest of this document uses "Angular Diameter", it means the angle to the Airy disk (or rings), as described above.
If you repeat the drawing with different diameter mirrors, say 1” & 3”, while still using ½” rings to represent the same wavelength, you will find that as you increase the diameter of the mirror, the Angular diameters of the Disk and the Ring reduce. Since the angular diameter reduces, the physical diameter of the Disk and rings also reduce.
Although this illustration is just something done on the kitchen table, It is exactly the same mathematics that apply with light to form the real Airy Disk and Rings. The angular diameter of the Disk and rings is calculated with exactly the same rules. It follows that the Airy Disk gets smaller as the Diameter of the mirror increases.
Before the people who really understand Mathematics jump in and complain that the intersections are not really on straight lines, - they are on hyperbolic curves, - it is agreed that they really are on slight curves rather than straight lines! The lines come in from infinity and do bend very slightly as they approach the mirror. So evidence of a slight curve is likely to be noticed on the kitchen table using wavelengths of ½" with a 2" mirror.
However light has a wavelength of about 500nM, so the line for the first Airy Ring will have come in from infinity and will bend out of line near the mirror a maximum of ½ a wavelength, which is about 250nM. It is submitted that coming all the way from infinity and only bending 250nM out of line is still pretty damn straight!
In any case, - the accepted formulas used for working out Airy Disk size and angular diameter assume the lines are straight, - so no reason this description cannot be allowed to do the same!
The description above only covers light from the rim of the mirror - so what about the light from the rest of the mirror? - Don’t forget the rest of the mirror surface is pouring light into the central spot. The very simple explanation is that this light merges with the light scattered from the rim of the mirror to give the bright Airy Disk.
The only practical thing a telescope constructor can do to make the Airy Disk smaller is to make the mirror diameter larger. This makes the angular diameter of the Airy Disk smaller. The angle is fixed for any one diameter of mirror. Once the angle is fixed, the physical size of the Airy Disk is determined solely by the focal length of the mirror.
A common formula used for working out the size of the Airy Disk uses the Focal ratio, which includes both the diameter of the mirror and the focal length in one term.
One formula for the Airy disk Diameter is:
D = Diameter of Airy Disk in mm; λ = Wave Length in mm (e.g. 546nM = 0.000546mm), (If Focal ratio = F/4 & 546nM used then D = 0.00533mm for example).
One formula for the angular diameter of the Airy Disk is:
A = Angular Diameter of the Airy Disk in Arc Seconds; d = Diameter of Mirror in mm.
Ok! - so you now know what the Airy Disk is!, but from the description above you should be thinking there’s a easy way to get rid of the damn thing! – Surely if it’s just the light at the edge of the mirror that causes diffraction, then why not just mask off the edge of the mirror with a cardboard ring? – This will stop light hitting the edge of the mirror. Wont this get rid of the Airy disk?
Sorry – this does not work! You now get diffraction from the edge of the mask instead! – What you have done by masking the mirror edge is add what is called an “Aperture Stop” to your optical system.
By doing so you slightly raise the focal ratio of the “system” and the Airy Disk actually gets bigger. This also explains why in your telescope you need to keep the spider holding the elliptical flat or secondary mirror as small and unobtrusive as practical. Any obstruction causes additional diffraction.
Finally, - here is the killer - let’s suppose you could magically mask off the edge of the mirror without causing extra diffraction. You have managed to limit the beam of light falling on the mirror to less than the mirror diameter so the edge of the mirror is not illuminated. You must really think you have the problem cracked now?
Sorry – This still does not work! The edge of the beam of light falling on the mirror surface is itself a discontinuity. When it strikes the mirror and reflects it behaves exactly as if it was a real physical edge. There is diffraction from the edge of the beam exactly the same as if it was the physical rim of the mirror. For this situation there is no easy and simple "kitchen sink" explanation. This is where I “cop out” and say this is due to the wave nature of light and you do need to start digging into the physics text books if you want to understand it better.
Exactly the same thing happens if you paint the outer edge of your mirror with black paint. All this does is cause diffraction to occur where the black paint finishes and the reflective surface starts. All you have done with the paint is reduce the diameter of the mirror. The result is that the Airy Disk gets bigger.
The message is that you cannot avoid the Airy Disk.
However you can make it smaller: You can either increase the diameter of the mirror, which reduces the angular diameter of the Airy Disk; - or reduce the focal length of the mirror, - which reduces the physical size of the Airy Disk.
The Airy disk size is suddenly becoming more important to Amateur Astronomers due to the recent improvements in Digital cameras where the CCD sensor pixel size has now reduced to the point where it is comparable or smaller than the Airy Disk size. In future, Telescope mirror diameter and focal length may have to be chosen to better match the CCD pixel size?
To close, - You could completely avoid the Airy disk by using an infinitely large mirror? – but I would respectfully suggest a long talk with your wife or perhaps your doctor before trying to order one of these.
What is Aperture masking interferometry?
Thursday, August 27, 2009
Aperture Masking Interferometry is a form of speckle interferometry, allowing diffraction limited imaging from ground-based telescopes. This technique allows ground based telescopes to reach the maximum possible resolution, allowing ground-based telescopes with large diameters to produce far sharper images than the Hubble Space Telescope. The principal limitation of the technique is that it is limited to relatively bright astronomical objects. A mask is placed over the telescope which only allows light through a small number of holes. This array of holes acts as a miniature astronomical interferometer. The method was developed by John E. Baldwin and collaborators in the Cavendish Astrophysics Group.
In the aperture masking technique, the bispectral analysis (speckle masking) method is typically applied to data taken through masked apertures, where most of the aperture is blocked off and light can only pass through a series of small holes (subapertures). The aperture mask removes atmospheric noise from these measurements, allowing the bispectrum to be measured more quickly than for an un-masked aperture.
For simplicity the aperture masks are usually either placed in front of the secondary mirror or placed in a re-imaged aperture plane. The masks are usually categorized either as non-redundant or partially redundant. Non-redundant masks consist of arrays of small holes where no two pairs of holes have the same separation vector (the same baseline - see aperture synthesis). Each pair of holes provides a set of fringes at a unique spatial frequency in the image plane.
Partially redundant masks are usually designed to provide a compromise between minimizing the redundancy of spacings and maximizing both the throughput and the range of spatial frequencies investigated. Although the signal-to-noise of speckle masking observations at high light level can be improved with aperture masks, the faintest limiting magnitude cannot be significantly improved for photon-noise limited detectors.
a) shows a simple experiment using an aperture mask in a re-imaged aperture plane. b) and c) show diagrams of aperture masks which were placed in front of the secondary mirror of the Keck telescope by Peter Tuthill and collaborators. The solid black shapes represent the subapertures (holes in the mask). A projection of the layout of the Keck primary mirror segments is overlaid.
References:
* Peter Tuthill's PhD thesis on aperture masking
* Baldwin et al. (1986)
* Buscher & Haniff (1993)
* Haniff et al. (1987)
* Haniff et al., 1989
* Buscher et al. 1990
* Haniff & Buscher, 1992
* Tuthill et al. (2000)
* Young et al. (2000)
In the aperture masking technique, the bispectral analysis (speckle masking) method is typically applied to data taken through masked apertures, where most of the aperture is blocked off and light can only pass through a series of small holes (subapertures). The aperture mask removes atmospheric noise from these measurements, allowing the bispectrum to be measured more quickly than for an un-masked aperture.
For simplicity the aperture masks are usually either placed in front of the secondary mirror or placed in a re-imaged aperture plane. The masks are usually categorized either as non-redundant or partially redundant. Non-redundant masks consist of arrays of small holes where no two pairs of holes have the same separation vector (the same baseline - see aperture synthesis). Each pair of holes provides a set of fringes at a unique spatial frequency in the image plane.
Partially redundant masks are usually designed to provide a compromise between minimizing the redundancy of spacings and maximizing both the throughput and the range of spatial frequencies investigated. Although the signal-to-noise of speckle masking observations at high light level can be improved with aperture masks, the faintest limiting magnitude cannot be significantly improved for photon-noise limited detectors.
a) shows a simple experiment using an aperture mask in a re-imaged aperture plane. b) and c) show diagrams of aperture masks which were placed in front of the secondary mirror of the Keck telescope by Peter Tuthill and collaborators. The solid black shapes represent the subapertures (holes in the mask). A projection of the layout of the Keck primary mirror segments is overlaid.
References:
* Peter Tuthill's PhD thesis on aperture masking
* Baldwin et al. (1986)
* Buscher & Haniff (1993)
* Haniff et al. (1987)
* Haniff et al., 1989
* Buscher et al. 1990
* Haniff & Buscher, 1992
* Tuthill et al. (2000)
* Young et al. (2000)
Scales and Distances
Tuesday, August 25, 2009
by Sebastien Parmentier
Space is insanely gigantic.
Its proportions are too unimaginable for the common folks. However, before talking about anything else, it is utterly important to situate our place in this universe. And for the student who read this lines, and wants, I hope, to learn a bit by following me along in this adventure, getting a good sense of scales is sine qua non to get a good understanding of astronomy and interferometry.
Interferometry is fascinating in a way that it merges two branches of optics: one that is employed to look at infinitely large distances and one that peers into phenomena in the nano-scale.
Let's start with the large scales first. If the Sun was scaled down into a basket ball, Earth scaled down by the same ratio would become the size of a marble of about a centimeter in diameter. If we scale down the distance separating those two bodies by the same ratio, we would still have them separated by about 125 feet! Impressive, but still not that bad. Now... where would Mars be situated from that basketball if it were scaled down the same way? About 204 feet. Jupiter? 638 ft. Saturn, which would be the size of a prune next to that basketball? About... 1171 ft! What about Pluto, the size of a speck of dust compared to the big ball? Well... Pluto has a very elongated (or “eccentric”) orbit. At its Aphelion, what we call when the planet is at the farthest from the sun, this speck of dust would be at 6074 feet!
Now let's scale down our solar system in the order of a thousand more and let this whole solar system fit right on a quarter of a dollar. What would be the size of the Milky Way scaled down by the same ratio? The actual real life size of the whole United States! Now imagine... the Grand Canyon, the Mississippi, the Appalachian, all this millions of acres of land, and one tiny quarter dollar laying on the floor, somewhere, in the middle of all this...
Some stars are so massive in matters and size, our sun is the one looking like a speck of dust next to a basketball. Some of those basketball do rotate in about one our, while our sun, now that speckle of dust, takes 27 days to cover 360°....
Distance in space is really a big problem in astronomy, as much as Nuclear physicists have their respective problems to peers into atoms. Gathering light is relatively easy. However, resolving objects is the big challenge of astronomers. Take a binary star. Not two stars born in the same vicinity from the same dust cloud but a real binary system of one star traveling around a parent star. If these two stars were separated by 50 AU (50 times the distance between Earth and the Sun), a naked eye would only have the resolving power of having them separated into two distinct points of light if this binary star was no farther than 5.6 light years.
One group of star are in fact closer: Alpha Centauri (α Cen / α Centauri) is indeed the brightest star system in the southern constellation of Centaurus. And although it appears as a single point to the naked eye, Alpha Centauri is actually a system of three stars, one of which is the fourth brightest star in the night sky. The two brightest components of the system are too close to be resolved as separate stars by the naked eye and so are perceived as a single source of light with a total visual magnitude of about −0.27 (brighter than the third brightest star in the night sky, Arcturus).
The closest star system to the Sun is the Alpha Centauri system at 4.39 light-years away (about 277, 600 AU). Of the three stars in the system, the dimmest -- called Proxima Centauri -- is actually the nearest star, 4.26 light-years distant. But the bright stars Alpha Centauri A and B form a close binary as they are separated by only 23 times the Earth- Sun distance - merely greater than the distance between Uranus and the Sun. Therefore, it is impossible to see two distinct points with an unaided eye.
The Alpha Centauri system is not visible in much of the northern hemisphere. Alpha Centauri A, also known as Rigil Kentaurus, is the brightest star in the constellation of Centaurus and is the fourth brightest star in the night sky. Sirius is the brightest and a binary system, but it is more than twice as far away. Alpha Centauri A is the same type of star as our Sun, probably born from the same gas cloud, and perhaps as suitable for life as our Sun has showed to be.
Now, if binary systems as close as these ones, formed by very shiny objects, get to be so difficult still to resolve, how the heck can we detect objects so incredibly less bright, such as planets in the vicinity of stars? We will talk about it very soon, but not before a little talk about resolving power and what to do when an object is too small to be resolved into a nice Airy disk.... in my next post.
Its proportions are too unimaginable for the common folks. However, before talking about anything else, it is utterly important to situate our place in this universe. And for the student who read this lines, and wants, I hope, to learn a bit by following me along in this adventure, getting a good sense of scales is sine qua non to get a good understanding of astronomy and interferometry.
Interferometry is fascinating in a way that it merges two branches of optics: one that is employed to look at infinitely large distances and one that peers into phenomena in the nano-scale.
Let's start with the large scales first. If the Sun was scaled down into a basket ball, Earth scaled down by the same ratio would become the size of a marble of about a centimeter in diameter. If we scale down the distance separating those two bodies by the same ratio, we would still have them separated by about 125 feet! Impressive, but still not that bad. Now... where would Mars be situated from that basketball if it were scaled down the same way? About 204 feet. Jupiter? 638 ft. Saturn, which would be the size of a prune next to that basketball? About... 1171 ft! What about Pluto, the size of a speck of dust compared to the big ball? Well... Pluto has a very elongated (or “eccentric”) orbit. At its Aphelion, what we call when the planet is at the farthest from the sun, this speck of dust would be at 6074 feet!
Now let's scale down our solar system in the order of a thousand more and let this whole solar system fit right on a quarter of a dollar. What would be the size of the Milky Way scaled down by the same ratio? The actual real life size of the whole United States! Now imagine... the Grand Canyon, the Mississippi, the Appalachian, all this millions of acres of land, and one tiny quarter dollar laying on the floor, somewhere, in the middle of all this...
Some stars are so massive in matters and size, our sun is the one looking like a speck of dust next to a basketball. Some of those basketball do rotate in about one our, while our sun, now that speckle of dust, takes 27 days to cover 360°....
Distance in space is really a big problem in astronomy, as much as Nuclear physicists have their respective problems to peers into atoms. Gathering light is relatively easy. However, resolving objects is the big challenge of astronomers. Take a binary star. Not two stars born in the same vicinity from the same dust cloud but a real binary system of one star traveling around a parent star. If these two stars were separated by 50 AU (50 times the distance between Earth and the Sun), a naked eye would only have the resolving power of having them separated into two distinct points of light if this binary star was no farther than 5.6 light years.
One group of star are in fact closer: Alpha Centauri (α Cen / α Centauri) is indeed the brightest star system in the southern constellation of Centaurus. And although it appears as a single point to the naked eye, Alpha Centauri is actually a system of three stars, one of which is the fourth brightest star in the night sky. The two brightest components of the system are too close to be resolved as separate stars by the naked eye and so are perceived as a single source of light with a total visual magnitude of about −0.27 (brighter than the third brightest star in the night sky, Arcturus).
The closest star system to the Sun is the Alpha Centauri system at 4.39 light-years away (about 277, 600 AU). Of the three stars in the system, the dimmest -- called Proxima Centauri -- is actually the nearest star, 4.26 light-years distant. But the bright stars Alpha Centauri A and B form a close binary as they are separated by only 23 times the Earth- Sun distance - merely greater than the distance between Uranus and the Sun. Therefore, it is impossible to see two distinct points with an unaided eye.
The Alpha Centauri system is not visible in much of the northern hemisphere. Alpha Centauri A, also known as Rigil Kentaurus, is the brightest star in the constellation of Centaurus and is the fourth brightest star in the night sky. Sirius is the brightest and a binary system, but it is more than twice as far away. Alpha Centauri A is the same type of star as our Sun, probably born from the same gas cloud, and perhaps as suitable for life as our Sun has showed to be.
Now, if binary systems as close as these ones, formed by very shiny objects, get to be so difficult still to resolve, how the heck can we detect objects so incredibly less bright, such as planets in the vicinity of stars? We will talk about it very soon, but not before a little talk about resolving power and what to do when an object is too small to be resolved into a nice Airy disk.... in my next post.
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