Saturday, 11 July 2020

#127: Science Fiction Recommendation


One of the most inspiring sights available to amateur astronomers is the planet Saturn, especially when the rings are opened to their widest angle. Along with most views of this strange world comes the smaller pinprick of light known as Titan, Saturn’s largest moon, and second largest moon in the solar system. It is half as big again as Earth’s moon. Titan is often referred to as a planetary moon, and has proven itself to have one of the most interesting surfaces in the solar system, mainly thanks to the Cassini mission, which began mapping Titan in 2004.

Five years before this, in 1999, Hal Clement’s novel Half-Life was published. Hal Clement (real name Harry Clement Stubbs, 1922-2003) was a hard science fiction writer who really focused on the science angle, and the methods used to gain answers to puzzling questions. The novel details an expedition from Earth to study the atmosphere and surface of Titan, searching for biochemical clues as to why humans are rapidly going extinct. The expedition consists of 23 scientists, men and women, and all of them quite ill, and we follow their means and methods used to search for answers. While an amateur telescope won’t reveal too much about the surface of Titan, Clement’s novel will, and the book will likely increase your interest greatly in this alien world. Of course in 1999 we knew far less about Titan than we do now, but this only makes the novel even more fascinating, as Clement uses every little known scrap of contemporary information about the moon in his book. Reading Clement is almost like reading transcripts from an actual space expedition to Titan, and I’m certain many astronomy enthusiasts, especially those with sturdy science backgrounds, would find his work fascinating. Clement earned a degree in astronomy from Harvard, and went on to earn an M. Ed from Boston, and later an M. S. in chemistry from Simmons.

Not only is he a well educated man, but he is also a master writer, earning the Grand Master title from the Science Fiction Writers of America in 1999, the same year Half-Life was published (the author would have been 77). And if this was the only great book he had written, he would still deserve the title. However, though Half-Life contains no aliens, many of his books do. The way humans and aliens interact in his stories, searching for answers to scientific puzzles, is equally fascinating. Some of his best early writing concerns the planet Mesklin, with its eccentric orbit and even more eccentric inhabitants. Even Clement’s fictional planets have strong basis in science fact, and the way he explores them and attempts to solve their mysteries seems to invite readers along for the ride. Here are a few more titles by Clement that I highly recommend, though it is by no means his entire list. Many of these titles are available on Kindle, and will likely be read by the purchaser more than once.

Mission of Gravity (serialized beginning in 1950)
Iceworld (1951)
Cycle of Fire (1957)
Close to Critical (1958)
Still River (1987)
Half-Life (1999)


Messier of the Moment: M 5, in Serpens Caput

NGC 5904 is one of the finest globular clusters in the sky, and it is quite suitable for smaller apertures. My first view was July 19th, 1985, when I saw it from Lake Penage, west of Sudbury. I was using my 8” Edmund scope, for which I had made stops of 4” and 6”. Even at 4” of aperture I was able to resolve some stars, though not in the central region. At 72x about 10 stars were winking in and out. Moving up to 6” of aperture, the center now becomes intensely bright. 56X gave good resolution, but again only in the outer haze. At 112x the center was hinting at breaking up, and star patterns began emerging around it. At full aperture, resolution was good even at 36x. At 112x stars begin to finally flicker in the central area. The core is literally blazing now, and streams and spirals of stars surround it. The core showed irregularity at 169x, and at 254x fainter stars to the core can be glimpsed. A very small “inner” inner core can now be seen.

Jump to May 20th, 2020, from a dark site in Kent County, using a 12” scope. While Messier 5 is a rewarding object for a 6” and 8” mirror, in a 12” scope it can be a bit overwhelming. At 60x the cluster is very large, and there is already good resolution from the core outwards to the limiting edges. At 100x the brilliant core is breaking up, with even more stars seen closer to it. Brighter stars seem to circle the core. At 136x and 187x (the best viewing range, in my opinion) uncountable resolving stars seem to resemble a face-on spiral galaxy that is resolving along its winding arms. Other viewers have also commented on this effect.

At 272x the core appears to thrust upward towards the viewer, giving the impression of viewing a snow-covered volcanic cone from directly above. This is a truly wondrous object for amateur telescopes! Optimum viewing time is 10 pm July 3rd, though it is still high enough several weeks before and after to make a journey there worthwhile.

Messier 5: Size 23’; Visual mag. 5.7; Brightest star mag. 12.2

Mapman Mike


Sunday, 22 March 2020

#126: March, Life Lists

The following post updates an earlier one in this series; it was also published in Aurora, the newsletter of the Windsor Branch of the Royal Astronomical Society of Canada.

In earlier posts I talked about astronomers who work their way methodically through object lists, often to the detriment of the object. A quick glance, a note or two, maybe a quick sketch and then they are off to the next object. All well and fine if you are looking at 14th magnitude galaxies, but if your list is packed with the brightest deep sky gems, what is the hurry? 400 top objects could well last your entire life.

To make certain that I never finish my life list, or run out of options on any given clear night, I chose the NGC list a long time ago as my main deep sky emphasis. I will not pass a Collinder or Berkeley cluster without a look, but my longest viewing time is spent on the NGC list. In a lifetime of observing I have seen and logged many of them. Ones seen previously with the 8” scope are being reviewed with the 12”. And some of the best ones are being glimpsed with my 2” refractor. It’s fun to compare the notes of past viewed objects when using a different scope.
The current edition of Uranometria 2000 claims that more than 30,000 non-stellar objects are on its charts. That's over 22,000 more than are in the NGC list. Good grief. Makes me kind of thankful that I am not overly obsessed. Of those 30,000 objects, nearly 26,000 of them are galaxies. Which means that nearly 5 in 6 non-stellar objects viewable in the night sky with amateur scopes are galaxies. Most of those are pretty faint, too. Here is the breakdown, as reported on the atlas' back cover, remembering that this encompasses the north and south skies:

     25,883 galaxies
     671 galaxy clusters (Abell)
     14 star clouds
     1,613 open clusters, including those in the Magellanic Clouds
     170 globular clusters
     355 bright nebulae
     367 dark nebulae
     1,145 planetary nebulae
     260 radio sources
     35 x-ray sources

I remember being very surprised when I first learned how many galaxies were available to amateur astronomers (all the ones listed in Uranometria are mag. 15 or brighter). I mean that I was very surprised. Growing up loving the Milky Way area and all of its clusters and nebulae, it was easy to conclude that clusters of stars far outnumber galaxies. That, to say the least, was an erroneous conclusion. If one decides to choose the NGC for a life list, one must be prepared to see some galaxies. I don't just mean the ones in the Messier catalog, either.
Finding and doing justice to galaxies requires the very darkest of dark skies. I used to be able to pick off 12th mag. galaxies from my back deck in Anderdon with my 8" scope. Light pollution has increased so much in my area that this is now almost impossible, even with a 12”. If I want to do a decent job of observing in Sextans this spring, I have to be prepared to travel with the scope. This brings on an interesting conundrum. Do I observe the brighter galaxies from my home, saving the fainter ones for country skies? Though this is a feasible plan, I try to imagine what brighter objects might look like from a dark sky. If I can even glimpse it from home, then it should be that much more impressive from a dark sky. Should I save all my faint objects for darker skies? What about the Messier list? What would these objects look like from a really dark sky site, if only seen from light polluted ones?

One way to help tackle this problem, which I'm certain many of us face all too often, is to keep notes and report on the brighter ones as seen from home suburban skies, but also return to them in dark skies and enjoy another, more lingering look. Open clusters and globulars that resolve, for example, will show many more stars from a dark sky, and galaxies can be expected to show a larger area and appear brighter, especially towards the center. Light pollution filters work extremely well in light polluted skies. However, their future under LED lights remains in flux.

Double star work can carry on quite well, too, from poorer skies, often even during full moon nights, though the rich background seen in a dark sky might reveal a hidden gem or two to the very observant viewer that would be otherwise unnoticed. However, I am determined to do as much deep sky work in the darkest skies I can reach, though I realize that time is running out for the Windsor area. I've been spoiled by my northern Ontario early years of observing, and even by Hallam when it was at its best.

Messier Of The Month: M 43

M43 is also commonly known as De Mairan’s Nebula, as well as NGC 1982. De Mairan was a French scientist who lived from 1678-1771. He observed the nebula in 1731, which Messier later added to his now famous list. The sword area of Orion, which also includes the more famous and much larger M 42, contains no less than 10 separate NGC numbers, and makes for a fascinating hour of observing to tell one from the other. This area shall be the subject of a future At The Eyepiece article.

M43 is not only overshadowed by its bigger deep sky mate M 42, it is actually a part of it. M 43 lies just barely north of M 42, separated from it by a narrow dark lane, and is almost touching the main nebula. Look for an ivory coloured 7th magnitude star north of the big nebula. M 43 encircles this star, with more of it showing south of the star than north. A fainter elongated segment lies north again, towards another star. The nebula is easy to see without a filter, and should be visible in scopes of 4”. Although noted in years past with the 4.5” Tasco reflector, the above description was made using the 12” Orion Dobsonian, from Hallam, on January 5th, 2019. Next time you are observing in this area, make sure to take a look at Messier 43. An interesting article could also be written about Messier objects that can be viewed with other Messier objects in the same field of view. This is one of them.

Messier 43 (NGC 1982): 20’ x 15’; Emission and Reflection Nebula.

Clear skies, and happy observing.
Mapman Mike

Thursday, 30 January 2020

#125: Crab Nebula

The following is an article I wrote for our astronomy club newsletter, for January 2020.  I belong to the Windsor branch of the RASC, and write articles five times per year.  To see previous newsletters, link to this address on our homepage.  There are also other fine articles by other members, as well as some outstanding amateur photos.

This time of year very little time is spent at the telescope eyepiece. I did enjoy a fine night on December 21st, but as of early January that has been it, other than a few naked eye peeks at Betelgeuse. Instead, I will talk briefly about the 2019 highlights and lowlights, and say a bit about 2020.

For me, 2019 will be remembered as the year I had to abandon the club's Hallam site for my observing program. The greenhouse lights to the south have reached a level of insanity I could not have imagined even a few years ago. While in the long term I am optimistic that the stray light problem from Leamington will be addressed, in the short term Hallam is now reserved (for me) for lunar and planetary work. I have had to move my deep sky observing to Chatham Kent County, where I can at least enjoy good skies to the north, east, and south. The success of Detroit’s streetlight program has made the west and northwest sky much more light polluted than several years ago, when nearly half the streetlights in that city did not function.

And yet another big problem lies in the near future, as over six thousand new satellites will be orbiting the Earth within two or three years. These satellites will be used to bring fast internet to the entire planet. They are supposed to be highly reflective, too. In 2020, 1500 of them will be launched. The first few batches of 60 are already up there. Even using remote telescopes for photography will be greatly affected. Needless to say, it isn’t only amateurs who are worried; major observatories will have some serious obstacles to overcome as well. And what happens to these satellites if Earth is struck by a major solar flare? If you think we have a lot of space junk up there now, just wait.

My NGC observing project continued in 2019, as I managed to get out under clear skies with my telescope 34 times, totaling over 100 hours of observing. That is considerably better than 2018, one of the cloudiest years I have ever seen. One of those observations was the total lunar eclipse of January 20th, possibly the coldest observation I have ever recorded from Essex County. It was a very fine and clear night, however.

And in Spring 2019 I finally managed to complete my viewing of the entire NGC list in the constellation boundaries of Leo! That was a seven year project that took me to some fabulous sights, a few of which I have shared in these pages. My newest Spring observing adventure lies within the border of Coma Berenices, and might take me just as long. Will I ever get to tackle Virgo someday? If I do, you will read about it here.

Winter months are an excellent time to plan a viewing program for the warmer months. I wish you all many happy observing planning sessions, and excellent skies with which to undertake those plans.

Messier of the Month: M 1, “The Crab Nebula”

Approximately 7500 years ago, a star exploded in the direction of the constellationTaurus. Seen and recorded over a thousand years ago by the Chinese in 1054 A.D., the remnant from this supernova was the first astronomical object to be identified as such. The Earl of Rosse drew the object in 1840, observing it with his 36” telescope. His drawing was said to resemble a crab, and thus the nickname. This is one of the most studied objects in the sky, and provides for some fascinating on-line reading. Stunning photos reveal the object in breathtaking detail. What is highly unusual about M 1 is that on very detailed professional photos, significant changes can be noted in less than a month. Even so, I still like to view it in a small scope. In a really dark sky it can be seen with binoculars, though my views have been only through telescopes. And it continues to expand, being significantly larger now in photos than when I first began the hobby.

My first look came in March, 1973, using my 4.5” Tasco reflector. With fresh snow on the ground and an unshielded streetlight only a few meters away, it wasn’t the ideal first look at such an object. A much better view came from my back deck in Anderdon Township in March, 1994. I was using my trusty Edmund 8” reflector. At lower powers it reminded me somewhat of a very large and bright galaxy. The edges appeared smooth, even at 169x. Skip ahead to late November, 2016. I viewed it with the 12” Dob from Hallam, and found it impressively large, very bright, and quite elongated. The large central area is much brighter than the outer areas. Several faint stars appear to me very close, or even touching the outside edges. The south-following end is very patchy, whereas the north end tapers and fades gradually.

My Space Eye view came in October, 2015, from Hallam. I did not have high expectations of seeing M 1 with the 2” refractor, but it was located at 30x! The nebula was really faint, but quite unmistakable. It was oval, and though fairly small, averted vision really helped bring it out.
Messier 1 (NGC gn 1952): 6’ x 4’ emission nebula in Taurus.

Mapman Mike

Tuesday, 3 September 2019

#124: Messier 34


Perseus is home to some of the finest clusters in the sky. Messier 34 is only one of the showpieces in this constellation sitting atop the Milky Way, and in January the cluster is ideally suited for observation, being almost overhead in the early evening. I have had the good fortune to observe this bright open cluster with three different scopes, including Space Eye, my 2” refractor. M 34 is a fine object for very small scopes, and is seen well at 30x. It is a large group filled with bright stars, and the eye is kept busy. The central area resolves well, and there are also several bright stars in the outskirts. That observation was done in October of 2015.

Back in October of 2002 I viewed it from Comber with my Edmund 8” reflector. At 36x it appeared as a very loose, medium-rich cluster. There are numerous bright stars, some in pairs. The cluster is best viewed at low power. 56X resolves the entire group, though views were also lovely at 72x. 

In January of this year I finally got to look at it with the 12” scope. It fits in the field of view at 60x, its many bright stars becoming almost blindingly white. The cluster includes some fine doubles, and the overall large and loose structure invites the eye to search all around for interesting pairings and other details. One aspect that struck me was a fascinating string of faint stars on the south end, intersecting with another line coming through the center of the cluster. There are about 12 stars in the first line, some with very faint companions. 100X shows the central area in good detail, though at this range the full cluster size is lost. Since M 34 is right in the Milky Way, it is fun to sweep the area surrounding the cluster. A rich knot of faint stars can be seen just south following, and I also came across a wide U of stars not far away.
It’s often too cold in January to do much more than a brief observation. This time is usually reserved for a look at the Orion Nebula. However, M 34 can be appreciated on a quick visit, and will also reward those with time for a closer view.

M 34 (NGC 1039): Size 25’; overall mag. 5.2; about 60 stars; brightest star mag. 9.

 https://skyview.gsfc.nasa.gov/current/cgi/runquery.pl?Interface=bform&VCOORD=ngc+1039&SURVEY=Digitized+Sky+Survey&SCOORD=Equatorial&EQUINX=2000&MAPROJ=Gnomonic&SFACTR=0.35&ISCALN=Linear&GRIDDD=No&COLTAB=B-W+Linear&PIXELX=600&PIXELY=600  

Clear skies!

Mapman Mike

#123: A Second Method For Tackling Unknown Skies


In blog #120 (M 103) I talked about a way of getting to know unfamiliar parts of the sky by using a bright deep sky object as a jumping off point for telescope users.  Exploring slowly and carefully in different directions has always been a worthwhile way to spend time with the night sky.

 This time I would like to talk about a second method of preparing for a night's observing.  However, a star atlas is now required, and perhaps some books describing deep sky objects, variable stars, and double stars.  I like the books put out by the Webb Society, but I still often use the 3-volume Burnham's Celestial Handbook.  As to sky atlases, the only one to even consider for advanced amateurs is the Uranometria All-Sky Edition, 2nd Ed, revised.  For the price of a cheap eyepiece ($60), this will be the best investment you ever made.  Cloudy nights suddenly become fun with this atlas.  And it comes with free shipping.
If you are among those who purchased the first edition way back when, as I did, then realized the maps were backwards and difficult to use in the field, have no fear.  Things are corrected and much improved now!  And an all-in-one edition means only one volume is now needed to study Orion in depth!  Before the 2nd edition of Uranometria came about, I used to do the following for my 2nd Method of preparing for a session.  Afterwards I will tell you am alternate, improved way, thanks to the revised atlas format.

1)      Pick an area of sky of interest to you.  I will pick oc 2232 in Monoceros, for reasons that should become apparent if you glance at the atlas (it's a simple enough area to start with).  However, any area will do, as long as you begin somewhere you will recognize.

2)   Without looking at any photos of objects to be observed (wait until after you have actually searched for them at the eyepiece), write down some info about the main object from one of your resources.  (oc 2232: 29'; stars mag 6--8, about a dozen).  I use the companion volume to the Uranometria atlas (see the link, above).

    3)   Search on the map (chart #116 in Uranometria) for neighbouring points of interest.  Write down any info about these objects you may have.  You should notice oc 2219 preceding, oc 2250 following, and oc Cz 26 north following.  These clusters are within easy star-hop range of oc 2232.  Their relative obscurity may or may not be deserved.  You can decide for yourself.  The size of your mirror and the darkness of your sky will help you determine this, along with your experience as an observer.  A few unnamed doubles are also indicated nearby, as well as a handful of variables, including GL, V723, and IM. Don't ignore the brightest star within oc 2232, which is labeled as a double star.
 https://upload.wikimedia.org/wikipedia/commons/f/f4/NGC_2232.png

What we have just done is prepare our session more in depth than with the previous method, where we only needed the location of one bright object (and no atlas) to get started.  This time we have gone target hunting, looking for specific objects near our bright starting point. One of the reasons I chose 2232 is that despite a good star-hop search using Method 1, those three other open clusters still may not have been found.  They are faint, and at low power may be missed (if you found them using Method 1, then congratulations!).  Using Method 2, we are now searching in an exact location for specific objects, hopefully with some success.  Use neighbouring stars leading from 2232 and marked on the atlas to guide you directly.  This is called guided star hopping, and with practice can be a very quick and efficient method of getting around. Thus Method 2 works similarly to Method 1, but it is now required to choose an area of sky with a few other objects of interest near the main object.  If you have the right atlas, this will pose no difficulty.  Choose your area based on your sky conditions, horizon, and season.  This is how I prepare for a session using Method 2.  As mentioned at the beginning of this article, however, I will sometimes do things differently now. 

Since the publication of the 2nd edition of Uranometria it includes, in addition to 220 star maps that show every NGC and IC object down to Mag. 15, 24 close-up maps at the back of the atlas.  These highlight and enlarge x2 or x3 things like galaxy clusters (Virgo, Coma, Hercules and others), extra rich areas of the Milky Way (Cygnus, Sagittarius, Scorpius, etc) and the Magellenic Clouds.  There is a lifetime of observing just within these supplemental maps!  I can't wait to get at Cygnus this summer with Charts A1 and A2, and in the spring with the Leo galaxy charts (A10, 11).  And once you see the Large Magellenic Cloud Chart and how many NGC objects reside there (a 2-page spread), you will want to book an astronomy B & B in Australia or Chile for a month!

     Soon I will detail Planning Method 3.  May you have clear skies and successful star hunting until then!
 
Mapman Mike

Monday, 2 September 2019

#122: Messier 39


I first made the acquaintance of this wide open star cluster in late July of 1971.  I was observing with my Tasco Lunagrosso 4.5” Reflector.  It was also the summer of a very favourable opposition of Mars.  Life was good!  I wrote at the time: “A very large open cluster of stars, mostly white.  The object filled about half of my field of view at 22x, and all of my 45x eyepiece.  At the lower power I counted about 25 stars, several of them 6th and 7th magnitude, although some were much fainter.  M 39 is a very pleasant cluster to view, and easy to locate.”

Cut to late August, 1978.  I am now using my Edmund 8” Reflector, the big red beast!  My notes from then:  "M 39 is a cluster so bright and widely scattered that it is seen very well in the 9 x 30 finderscope [virtually a 1” scope].  This also makes it a very worthwhile object for binoculars.  36x and 56x suited this very loose group.  I stopped the scope down to 4”, and the group was splendid to view!  Indeed, full aperture added nothing but more brightness, and a few much fainter stars.  At full aperture, an interesting way to view this cluster is to put it well out of focus.  My favourite view was at 56x and 4” of aperture, however.  A lovely wide pair of stars lies near the very center (ARN 78: 7.6-8.8/52”).”  Two other doubles with bright primary stars are just south.

In late September of 2013 I observed it with my Orion 12” Dob.  “The cluster looks best at low power.  It is suitable for a 2” refractor and anything larger.  I used 43x and 60x, where the cluster appears like a slightly more condensed version of the Pleiades.  The stars are nearly blinding in a 12” mirror.  Not a rich cluster, but very bright and well scattered.”  And I did indeed view it in Space Eye more recently, my wonderful 2” refractor, declaring it a minor showpiece for that instrument.

 https://skyview.gsfc.nasa.gov/current/cgi/runquery.pl?Interface=bform&VCOORD=ngc+7092&SURVEY=Digitized+Sky+Survey&SCOORD=Equatorial&EQUINX=2000&MAPROJ=Gnomonic&SFACTR=0.35&ISCALN=Linear&GRIDDD=No&COLTAB=B-W+Linear&PIXELX=600&PIXELY=600

I consider M 39 a great summer object to show people who are new to telescopes.  It makes an immediate impact, and rewards longer views with the bright double stars, as well as the many fainter members.  For experienced observers an interesting side trip can be made to oc Platis 1, marked as Anon Platis on Unronometria Chart 32.  Lying ½ degree north and 2' preceding (west) the Messier object, this tiny cluster will reward viewers with scopes of 8" or higher.  The region around Messier 39 is littered with small clusters and interesting star fields, and makes a good starting point for some low power meandering.
M 39 (oc 7092):  Size 31’; Mag. 4.6; Br. star mag. 7.
Oc Platis 1:  Size 10’; Br. star mag. 8.9.

Clear skies.

Mapman Mike

#121: Planning Observing Sessions: A Few Ideas*


For the next few articles, I would like to discuss observing plans.  The dark sky is a very large neighbourhood, filled with all manner of stars, clusters, nebula, and galaxies.  It can be a bit overwhelming.  Where to start?  Most of us began with the moon, the planets, and the brightest Messier objects.  I still don't think there is a better place to start.  However, this series of articles is not aimed at beginners.  I am aiming at the seasoned amateur, perhaps someone who has already logged the Messier catalogue and bagged a number of objects from other lists.  This kind of activity is certainly to be encouraged, as there are many different types of objects to see, and any number of variations among them.  And lists are prevalent.  Even though checking off astronomy object life lists can be a good thing, there can be downsides.

     One downside to using popular lists is that often only the biggest and brightest objects make it there.  What's wrong with that?  Nothing is wrong with it.  However, by including a number of interesting fainter and more difficult objects, those brighter ones, if saved for just the right moment, become even more impressive.  Training the eye to see fainter objects will certainly make a brighter one pop out of the eyepiece even more.  Many of my greatest observing accomplishments have been spotting galaxies at the very limit of my telescope's, and my eye's, ability.  Afterwards, it is always amazing to check out M 31 or M 33 again!

     'Ah,' you say, 'I not one of those observers that like to look for faint fuzzies--no thanks; not for me.'  ‘Don’t be hasty,’ I say.  Viewing fainter objects helps one appreciate the brighter things even more, not to mention the degree of challenge and the navigation skills you will acquire.  But they really do improve your observing skills.

     A second problem with lists is that we want to finish them.  If you are ticking off the 400 brightest Herschel objects, you are going to want to get to #400 someday.  Again, this is all well and good.  Unless you are hoping to tick off 30 or 40 of the objects during the next clear night.  Sorry, but that is not observing.  That is merely using a checklist.  If you are going to only observe the best and the brightest, then the urge to race through the list should be at the very back of your mind.  Often it isn't.  Locate.  See object. Check it off.  Multiply by 400.

     A third problem is that areas of unfamiliar sky around the listed objects are often avoided, in a haste to get on to the next thing on the list.  What else might be nearby?  A lovely double star, perhaps?  A nebula?  Maybe a rich star field?  Or it could be a stunning red carbon star.  All missed because the observer has moved on to the next object without bothering to check the surrounding field.

     So, are there other fun ways of approaching the night sky besides using lists of popular objects?   In my previous article (M 103) I would talked briefly about one of three methods I like to use.  Let me know what you think.  

* This article is an update of one I wrote for this blog, near its very beginning.