MOPFAQ (My Own Personal Frequently Asked Questions) for SAA
Brian Tung <>


Latest update (2008-07-23)
    * most amateur telescopes can only reveal four satellites of Jupiter

The following is a list of questions that I see asked reasonably often
on sci.astro.amateur.  It's certainly not an exhaustive list.  When I
see a question I think I can answer pretty well, I make a note of it
and add it to this list.  If I can't, I discard it and let someone else
answer it.

There may be some overlap with the other SAA FAQs (although I bet it's
pretty small, actually), but that's OK, since only in this FAQ list do
these questions get answered in my own inimitable (or is it inimical?)

    | Note: Many of the answers include a so-called ASCII or text |
    | diagram.  You should use a constant-width font when you     |
    | read this FAQ so these diagrams come out right.  If the box |
    | around this text looks right, your font choice is fine.     |

The List of Questions

    Q.  What causes the seasons?
    Q.  What is the purpose of an equatorial mount?
    Q.  If I have a correctly aligned equatorial mount, why do I need
        a hand controller/auto-guider/other device to keep my telescope
        pointed in the right direction?
    Q.  What causes field rotation?
    Q.  What are some good telescope choices?
    Q.  Are there any good introductory astronomy books?
    Q.  What are some good inexpensive planetarium programs?
    Q.  How should I select a star atlas?
    Q.  How do you measure distances in the sky?
    Q.  What is the distinction between true field of view and apparent
        field of view?
    Q.  How can I measure the apparent field of view of my eyepiece?
    Q.  What is an orthoscopic eyepiece?
    Q.  What does "parfocal" mean?
    Q.  What is a Barlow lens, and how do I use it?  And how does it
        work, anyhow?
    Q.  Is a fast telescope (small f/ratio) brighter than a slow
        telescope (large f/ratio)?
    Q.  What is the limiting magnitude for my scope?
    Q.  How is seeing related to twinkling?
    Q.  What does "diffraction-limited" mean?
    Q.  I often seen optical quality measured as "1/8-wave or better."
        What does this mean?  Is, for example, 1/8-wave better than
    Q.  What is the star test, and how does it work?
    Q.  What do "over-corrected" and "under-corrected" mean?
    Q.  What is astigmatism, and what causes it?
    Q.  What do the terms achromat, apochromat, and semi-apochromat mean?
    Q.  Does a central obstruction damage image quality or not?
    Q.  Does a Newtonian's spider damage image quality or not?
    Q.  What is prime focus?
    Q.  What does [insert nutty acronym] mean?
    Q.  What is the deal with [insert nut]?
    Q.  How do mirrors keep their shape?  I thought glass was a liquid.
    Q.  My mirror/lens is scratched--is this going to be a problem?
    Q.  What's the difference between a mirror diagonal and a prism
    Q.  What does it mean to offset a secondary, and why would I want
        to do that?
    Q.  How the heck is my image oriented?
    Q.  How much magnification can I get out of my telescope?
    Q.  How do I figure out the magnification when I take an astrophoto?
    Q.  What can I expect to see when I use my telescope for the first
    Q.  Did I just see Jupiter's satellites in my binoculars?
    Q.  Did I see a fifth (sixth, seventh) satellite of Jupiter in my
    Q.  What are these words "following" and "preceding" (or "leading"
        and "trailing") and what do they have to do with directions?
    Q.  Is it true that looking at the Moon through a telescope will
        harm your eyes?
    Q.  What interesting astronomy-related science projects can my
        child do?
    Q.  Is Pluto still a planet?
    Q.  What happened before the Big Bang?

Q.  What causes the seasons?

A.  The seasonal variation of weather on the Earth is affected by many
things, but the principal factor is the tilt of the Earth's axis with
respect to its orbital plane.

You often hear (correctly) that the Earth's axis is tilted, at an angle
of about 23.4 degrees.  But--23.4 degrees from what?  Up and down?  Up
and down from what?  Up and down from the ecliptic plane.

The Earth travels around the Sun in an almost circular ellipse, with a
major axis (long diameter) of about 300 million kilometers (about 186
million miles).  The average distance of the Earth from the Sun is thus
half of that, or 150 million kilometers (about 93 million miles).  The
orbit is slightly elliptical, though, with the Sun not quite at the
center of the ellipse, but offset by about 2.5 million kilometers (1.5
million miles) along the major axis.  As a result, the actual distance
at any time varies by about 5 million kilometers (3 million miles)
throughout the year.  It so happens that in our epoch, the distance is
at a minimum in early January, and at a maximum in early July.

      Earth in                        Earth in
     o January      O Sun            o  July

     |<- 147.5 Gm ->|<-- 152.5 Gm -->|

     Earth and Sun throughout the year (not to scale)

In this diagram, the Earth's orbital plane is coming out of the screen,
so that in January, the Earth is moving directly "toward you," and then
sweeps around, and is moving directly "away from you" in July.  The Gm
stands for gigameter--another way to say million kilometers.  From this
diagram, you'd expect the Earth to receive more heat from the Sun in
January than in July, which it does, and for the weather to be warmer in
January than in July, which is only true for the southern hemisphere.

Why is that?  Why should the weather be different in the two
hemispheres?  Because there is another factor--the tilt of the Earth's
axis.  In the diagram above, we have the Earth revolving around the Sun
in a horizontal plane, which we call the ecliptic plane, or simply the
ecliptic.  The north and south ecliptic poles are up and down from the
ecliptic.  If the Earth's axis were also straight up and down, the
weather would indeed be warmer in January than in July, all over the
Earth, since the only thing that changes for any point on the Earth
throughout the year is its distance from the Sun.

     |                               |
     o              O                o
     |                               |

But the Earth's axis is not aligned with the ecliptic poles--is not
straight up and down from the ecliptic.  Instead, it's canted at an
angle, 23.4 degrees.  By coincidence, the north end of the axis points
most directly away from the Sun in late December, at the northern winter
solstice, and most directly toward the Sun in late June, at the northern
summer solstice.  Very approximately, then, the northern end of the axis
is pointed away from the Sun at a time when the Earth is closest to the
Sun, and toward it when the Earth is furthest:

    \                               \ 
     o              O                o
      \                               \

It's important to note that although either end of the axis points
sometimes toward the Sun and sometimes away, that's not because the axis
actually "wobbles."  The effect is really caused by the relationship
between the axis and the Earth's motion around the Sun.

Axial tilt has two major effects.  One, when a hemisphere's end of the
axis is pointed away from the Sun, it receives light from the Sun, when
it gets it at all, only obliquely, only at an angle.  This oblique
heating is less effective than when it strikes a point on the Earth
squarely.  Two, at such times, the hemisphere pointed away from the Sun
is also illuminated for a smaller portion of the day--indeed, the north
pole receives no sunlight for several months centered around the winter
solstice.  Fewer hours of illumination, and less effective illumination
when you do get it, means colder weather on the whole.

These effects are naturally reversed when the hemisphere points back
toward the Sun.  We see now why the seasons are reversed in the two
hemispheres: it's because when one hemisphere's pole is pointed toward
the Sun, the other must be pointed away.  There's more to it than
that--in particular, the Earth's atmospheric, oceanic, and land heat
capacity causes the temperature extremes to lag behind the solstices by
a few weeks, depending on location.  But the principal driver appears to
be the tilt of the Earth's axis.

Does this all mean that the variation in the Earth's distance from the
Sun has no effect?  Not at all!  It's just that any variation in weather
induced by this variation is mostly swamped by the axial tilt.

Q.  What is the purpose of an equatorial mount?

A.  An equatorial mount makes it easier for a telescope to follow the
stars as they appear to revolve around us throughout the night.

What it is, actually, is the Earth's rotation.  If the Earth didn't
rotate, then all the stars would stay in place, and observing the night
sky at high power would be easy.  Of course, day and night would last
about 4,380 hours each, leading to baking temperatures during the day
and way below freezing at night.

But the Earth does rotate ("E pur si muove," as Galileo was alleged to
have said), making our days and nights tolerable, for the most part.  It
also makes the stars appear to revolve around us.  That revolution is
only an illusion, though.  The Earth, you see, is like a carnival
carousel.  A typical carousel rotates counter-clockwise as seen from
above, just like the Earth as seen from high above the north pole.  But
when you're *on* the carousel, you see the outside world revolving the
other way--that is, clockwise.  That's why the stars appear to revolve
around us.

Now, if you try to focus on any outside object--say, a friend taking
your picture--you can't do so if you keep still on the carousel.  No,
you have to swivel your head in order to follow your friend.  Moreover,
you can't just swivel in any old direction; you have to swivel around
the same axis and as fast as the carousel does, only in the opposite
direction.  If the carousel is rotating counter-clockwise, you have to
swivel your head clockwise, to compensate. Notice that it doesn't matter
if you're at the center of the carousel or at the edge, nor does it
matter where the object is that you're looking at, to first order--all
that matters is that you swivel appropriately to counter the carousel's

In the same way, as the Earth rotates once a day, it's necessary to
swivel your telescope around to follow any individual object, whether
it's a planet, a star, a cluster, or a galaxy.  And again, you can't
just swivel your telescope any which way--you have to swivel it on the
same axis as the Earth, at the rate of one rotation every day, only in
the opposite direction.

But where *is* the axis of the Earth?  If you were at the north pole,
the axis of the Earth would point straight up and down; if you were to
take a walking stick and spike it straight into the ground, it would be
parallel with (in fact, the same as, really) the Earth's axis.  If you
were to then walk some 110 kilometers (70 miles) away, to a latitude of
89 degrees north, and again plant your walking stick straight into the
ground, it would no longer be parallel to the Earth's axis.  Instead,
because of the Earth's curvature, it would be off by 1 degree.  In order
to get it parallel to the Earth's axis, you would have to tilt it by
that 1 degree, back toward the direction you came from--that is, to the

A similar situation applies at any other latitude.  If you're, say, 50
degrees away from the north pole--that is, at latitude 40 degrees
north--you would have to tilt your walking stick down by 50 degrees, and
pointing to the north.  Since there are 90 degrees between the northern
horizon (or any part of the horizon) and the zenith, another way of
putting it is that your stick would have to be pointed up from the
horizon by 90 minus 50, or 40 degrees.  And if you wanted your telescope
to remain pointed at a star, you would have to slowly swivel it around
an axis that is tilted up from the horizon by 40 degrees.  The axis
would then be parallel to the Earth's axis.

Well, that is all an equatorial mount does--it changes the axis around
which a telescope swivels, so that the axis is parallel to the Earth's
axis.  An ordinary camera tripod swivels left to right, around a
vertical axis.  Except at the two poles, that would never do.  The
easiest way to fix that would be to simply tilt the tripod over.
However, it ought to be obvious that that situation is completely
unstable and will send your telescope crashing to the ground.  Generally
speaking, then, the tripod stays where it is, it is only the joint where
the telescope attaches to the tripod that is tilted over.  Depending on
how the telescope is attached to this joint, its own weight may cause
instability, so a counterweight is often needed, opposite from the
telescope, in order to keep everything balanced and stable.

Despite all this machinery, the mount doesn't move the telescope on its
own.  It only makes it easier for you to do it by hand.  Alternatively,
you can attach a motor drive to the mount, and that *does* move the
telescope automatically, to follow or "track" whatever object you choose.

Incidentally, the term "equatorial" comes from the fact that the motion
of the telescope is parallel to the Earth's equator.

Q.  If I have a correctly aligned equatorial mount, why do I need a
hand controller/auto-guider/other device to keep my telescope pointed in
the right direction?

A.  Because your mount may not be perfectly aligned, because the motor
doesn't run perfectly uniformly, and because the Earth has an atmosphere.

Suppose the axis of your equatorial mount is not perfectly parallel to
the Earth's axis, but is instead off by (say) 1 degree.  Then, even if
the star you're following is initially centered in the eyepiece or
camera field of view, it will gradually drift away from the center,
despite your best efforts to follow it across the sky.  A hand
controller or auto-guider can help adjust the motor drive so that the
object stays in the center of the eyepiece or camera field of view.

Then, too, motor drives and gears are imperfect.  They may be designed
to rotate the telescope at a rate of 1 rotation each day, but that is
only the average rate.  In actuality, the rotation rate of the gears
alternately speeds up and slows down.  The amount of variation may be
tiny, but it is usually enough to make the target object appear to sway
back and forth.  Again, a hand controller or auto-guider is used to
compensate.  In some cases, the swaying is periodic, and a feature
called periodic error correction can be used instead.

Finally, the Earth's atmosphere refracts light as it comes from the
distant stars and galaxies.  An overhead star is, generally speaking,
just about overhead in reality, but a star that ought to be below the
horizon by all rights can have its light bent so much by the thick
layers of the Earth's atmosphere that it appears above the horizon.
The amount by which a star's position appears to be shifted by the
atmosphere varies in a complex way from zenith to horizon, so a single
speed motor drive cannot accurately track.  There are motor drives that
have a special rate, called the King rate, designed to partially address
this problem, but even the King rate is optimized for a given angle
above the horizon and cannot adequately compensate in all cases.  For
truly accurate tracking, some active correction is still needed.

Q.  What causes field rotation?

A.  Field rotation is the effect that even when you accurately keep an
object at the center of the field of view, objects at the periphery
appear to rotate around it.  It is caused by misalignment of the
telescope, and can be defined essentially as the difference between
north and up.

Suppose you're looking at two stars with the same right ascension--say,
the westernmost pair of stars in the square of Pegasus.  (That's in the
northern hemisphere.  Find a similar pair of stars in the southern
hemisphere.)  When they rise in the east, depending on your latitude,
those stars are side-by-side, more or less--something like the



       +-----------------------------------+ horizon

When they cross the meridian, though, beta will be nearly directly above
(that is, toward the zenith from) alpha, as follows: 



And when they set in the west, beta will now be to the upper right of
alpha, like so:


       +-----------------------------------+ horizon

If you were to track alpha Pegasi using a (very wide-FOV) alt-az
mounted scope (or a pair of binoculars), you would see beta Pegasi make
a broad, circular arc above alpha.  That is the field rotation at work.
There is nothing magical about being at the same right ascension, by
the way; the entire field around alpha will appear, from the point of
view of an alt-az scope, to rotate clockwise throughout the night.

A GOTO scope essentially figures out the instantaneous alt-az
coordinates of a given object.  By using the same algorithm to figure
out the instantaneous alt-az coordinates of a position just north of
that object, it can differentiate and determine the amount of field
rotation.  (Yes, I know that a GOTO scope can work in equatorial
mode--I believe the Celestron Compustar only worked that way--but
most of them nowadays are sold to users who will put them in alt-az

Incidentally, you may occasionally see the angle between "up" and
"north" referred to as the "parallactic angle."

Q.  What are some good telescope choices?

A.  Before I begin, let me lay down some telescope law.  Maximum power
has little to do with how good a scope is.  It's like rating a stereo
amplifier by how far the volume knob goes.  It's easy to make that make
the volume arbitrarily loud, but it does you no good for it to "go to
11" if the sound coming out of the speakers stinks.

From a practical perspective, telescope quality is largely determined by
its aperture (how wide its main lens or mirror is), its mount stability,
and its optical quality--roughly in that order.  A run-of-the-mill
6-inch scope will do better than a high-quality 2-inch on just about
anything you care to test them on.  (One exception is well-corrected
wide fields.)

The disadvantage with increasing aperture is bulk.  A 10-inch telescope
may not be the easiest thing in the world to handle for a budding
astronomer.  With all this in mind, here are some reasonable choices for
a first telescope purchase:

    1.  A 6-inch or 8-inch "dobsonian," which is a reflecting telescope
        (it uses a mirror as its main light-gathering element), mounted
        in a rocker box.  Its main attributes are that it is reasonably
        easy to point; the mount is stable and inexpensive, allowing
        most of the cost to go to the optics; and it is large enough to
        allow you to see enough objects to keep yourself in astronomical
        clover for some time.

    2.  An 80 mm "short tube" refractor, which is a refracting telescope
        (it uses a lens as its main light-gathering element), typically
        but not always placed on a simple alt-azimuth mount.  Most but
        not all of these are sold on stable tripods.  Its main points
        are that it is portable; it has a wide field of view, making it
        easy to find astronomical targets; and it just plain looks like
        a telescope (which a dobsonian does not, to many people), which
        is useful for keeping children interested.

    3.  A 3-1/2 to 5-inch compound scope (which uses both lenses and
        mirrors).  Not exactly my first choice, but these typically do
        well at high powers, and if you live in the city and cannot get
        outside it very often, and don't mind focusing (ahem) on the
        Moon and the planets, it's not a bad selection.

Most of these can come with some sort of computerized aid, which will
either move the telescope to point at selected targets at the push of a
button, or will guide you in pointing at them yourself.  These can help,
but they naturally add to the cost of the system.  In some cases, the
computer can be added retroactively, but this is not the usual case
(yet), so be sure to ask and make certain.

In the U.S., decent telescopes can be had from Celestron, Meade, Orion
Telescopes, and Hardin Optical.  They can be purchased direct from the
web sites, or from a reputable dealer.  In many cases, a dealer can be
identified as reputable from the manufacturer web sites.

Q.  Are there any good introductory astronomy books?

A.  As it so happens, I like astronomy books, to the extent that I'll
buy one even if I know most of what's in it already, just because it's
well written.  This means that I come in contact with most of the good
introductory books, as long as they're sold in North America.

The current favorite (not just for me, but for many folks) is Terence
Dickinson's Nightwatch.  It covers both the science and the practice of
amateur astronomy, which I find is a good way to maintain the wonder of
astronomy.  It is fairly specific when it comes to naming brands and
models, so it does need to be updated every few years.

Dickinson also wrote a somewhat more advanced book, along with Alan
Dyer, called The Backyard Astronomer's Guide.  BAG covers much of the
same ground as Nightwatch, but in greater detail.  It has also been more
recently revised than Nightwatch (as of this writing in November 2005),
so it includes merchandise that was not available when Nightwatch was

In the pocket book category, I like Ian Ridpath and Wil Tirion's Stars
and Planets, which gives a constellation-by-constellation guide to the
sky, complete with star maps and an observing guide to the celestial

Finally, for more help in purchasing telescope equipment, I recommend
Phil Harrington's Star Ware.  It gives a bit of the history and design
of telescopes, and also names names on telescopes and accessories.

Q.  What are some good inexpensive planetarium programs?

A.  For the Windows platform, two well-regarded free programs are Cartes
du Ciel, by Patrick Chevalley, and Hallo Northern Sky, by Han Kleijn.
Here are their web sites:

Despite its name, HNS covers the entire celestial sphere.  A similar
program for the Macintosh platform is Night Sky.

A program for Windows, Macintosh, and Linux whose emphasis is on
photo-realism rather than as a star-mapping utility is Stellarium, with
web site at

Another program that is not really planetarium software in the usual
sense is Celestia.  Celestia's strength is not its object coverage,
which is quite modest by today's standards, but in its photo-realism
and its novel ability to navigate its point of view to any point in
the solar neighborhood; you can, for instance, observe the Sun as it
would be seen from Sirius.  It runs under Windows, Mac, and Linux, and
its web site is

For PalmOS users, there are a number of inexpensive choices:
Planetarium ($24), PleiadAtlas ($10), 2sky ($30), and Astromist ($16),
with all prices in U.S. dollars.  Their web sites:

For PocketPC, the choices are Pocket Deepsky 2000 ($20), Pocket Stars
($15), Pocket Universe ($30), and The Sky Pocket Edition ($49).  Their
web sites:

Q.  How should I select a star atlas?

A.  That's a tough and personal question to answer.  The right choice
depends a lot on how you intend to use it, and what you're interested in

One common use of a star atlas is as a guide for star-hopping to various
deep-sky objects (DSOs).  Now, DSOs are extended objects; unlike stars,
their light is spread out over an area, rather than concentrated in a
tiny point.  This tends to make them harder to see than stars of the same

For that reason, you might expect that an atlas might contain, say, all
stars down to magnitude 8, and all DSOs down to magnitude 6.  But in
fact, exactly the opposite is true.  An atlas that contains all stars
down to magnitude 6.5, say, might contain all or most DSOs down to
magnitude 10, and some even dimmer than that.  Partly, this is because
there are far fewer relatively bright DSOs than there are relatively
bright stars, but it is also because the atlas publishers don't expect
you to match the eyepiece field, star by star, when you're hunting down
DSOs.  Rather, they give you only what they think is needed to find the
target, and using a variety of means.  This may frustrate those
prospective star-hoppers who don't happen to be particularly good at
pattern matching.

Because of this, I think it's best to have a star atlas that matches, at
least approximately, whatever view of the sky you have when you navigate.
For example, if you navigate using the unaided eye (or with a unit-power
finder like the Telrad), then your signposts are unaided-eye stars--that
is, stars down to magnitude 6.5.  You should therefore use an atlas that
goes down to about magnitude 6.5, because any atlas that goes deeper is
wasted for your purposes.  (You can't navigate using 8th-magnitude stars
if you can't see any!)  Atlases in this category include the Edmund Mag
6 Atlas, Cambridge Star Atlas, Bright Star Atlas, and the Norton Star
Atlas.  All cost in the neighborhood of $10 to $20 U.S.

If, on the other hand, you navigate using a small finderscope (or maybe
a larger one under light-polluted skies), then an atlas that goes down
to magnitude 8.5 or so will best match your finderscope view.  The
principal one in this range is Sky Atlas 2000.0.  The unaided-eye stars
are plotted more prominently than the dimmer ones, so it's still
possible to get your bearings with the unaided eye, and then fine-tune
your navigation with the 7th and 8th-magnitude stars you see through
the finder.  Another atlas in this range is the Herald-Bobroff Atlas,
which has not one but *six* sets of charts.  The main set of charts
covers stars down to about magnitude 9.0.  A little deeper still is
Uranometria 2000.0, which has stars down to magnitude 9.5.  These
atlases run you in the range of $40 to $120 U.S.

Those with larger finders, or possibly navigating through the eyepiece,
will want something even deeper.  The deepest *paper* atlas is the
Millennium Star Atlas, which contains stars down to magnitude 11.0.  MSA
is a big beast--it comes in three sizable volumes--and costs a similarly
hefty $250 U.S.!  You can tell that this is about the upper limit when
it comes to paper atlases.  (One can, of course, use any of the computer
planetarium programs to get stars down to about magnitude 15.  These can
usually print out custom charts.  It isn't as convenient in many ways,
of course.)

However, DSO hunting is only one possible application.  Another one is
variable star observing.  In this case, your targets may not be terribly
hard to see, but they might be hard to identify.  A 6th-magnitude atlas
might put you in the right spot of the sky, but you can't figure out
which 9th-magnitude dot is the cataclysmic variable you're looking for.
In this case, a dedicated atlas, such as the AAVSO atlas, is probably
your best bet.  It generally contains stars down to magnitude 9.0 or so,
but more importantly, it has the variable stars clearly indicated, with
comparison magnitudes of stable stars.

Or perhaps you use a GOTO telescope.  In that case, you are guided
entirely by the selection of objects of interest, not by how you
navigate.  A typical magnitude-6.5 sky atlas contains objects that can
be seen under decent conditions by about a 4-inch scope; Sky Atlas
2000.0 contains objects visible in a 6-inch scope, and MSA those visible
in an 8 to 10-inch scope.  Under these circumstances, the object guides
that are packaged separately in the case of Sky Atlas and Uranometria
become all the more important, since it is these books that tell you
which objects are of particular interest, among all the thousands of
objects visible in such a telescope.

Q.  How do you measure distances in the sky?

A.  When people are first asked to describe the separation, say, between
two stars, they often say something like, "Oh, they're about a foot
apart," because that's the sort of measurement we're used to down on
Earth.  It makes perfect sense to say that two nails are a foot apart on
a piece of wood.

However, it doesn't make any sense at all in the night sky, because the
sky has no depth of field--all the stars look like they're at about the
same distance.  If all the stars were a mile away, a foot would appear
very small--it would be very difficult for your eye to separate two
stars separated by only a foot.  And if they were 160 kilometers (100
miles) away, it would be plainly impossible.

As it happens, the stars are all more than 40 trillion kilometers (25
trillion miles) away and at all different distances, so the measurement
of a foot is useless.  What we need is some kind of measurement that
doesn't depend on how far away the stars are.  One such measurement is
the angle of separation.  If we draw two lines, one from each star down
to us, so that they intersect at our eyes, those lines meet at an angle.
The distances to the stars doesn't enter it at all--it doesn't matter
where each star is along its line, the angle stays constant.  Therefore,
the angle of separation is a perfectly suitable measure of distances in
the sky.

The unit of angular measure is the degree.  There are 360 degrees in a
circle, and the sky is a hemisphere, so there are 180 degrees from one
point on the horizon all the way to the opposite point on the horizon,
or 90 degrees from the zenith overhead down to any point on the horizon.
If one star is on the horizon, and another is, say, one-fifth of the way
up from the horizon to the zenith, then their angular separation is
one-fifth of 90 degrees, or 18 degrees.  We can also measure sizes this
way.  About 180 full Moons would fit, stacked end to end, from the
horizon to the zenith, so the angular size of the Moon is 90 degrees,
divided by 180, or 1/2 degree.

Some angles are too small to measure conveniently using degrees, so we
split up the degree into smaller units.  A minute is 1/60 of a degree,
so the Moon can also be said to be 30 minutes across.  A second, in
turn, is 1/60 of a minute.  (It's called a second, because it's the
second division of a degree.  There is also such a thing as a third,
which is--you guessed it--1/60 of a second, but that unit now only exists
as a curiosity and isn't used by anyone.)  Stars in binary systems can
often be distinguished only through a telescope, and their separations
are often measured in seconds.  For example, the Double Double in the
constellation of Lyra the Lyre is composed of two binary systems.  In
each binary, the two stars are separated by about 2.5 seconds.

Because minutes and seconds are also units of time (and of right
ascension, too), the angular variety are often called arcminutes (or
minutes of arc) and arcseconds (or seconds of arc).

Q.  What is the distinction between true field of view and apparent
field of view?

A.  In short, the true field of view tells you how much of the sky you
see, and the apparent field of view tells you how big that much sky
looks when magnified by the telescope (or binoculars).

If you look through a telescope at the Moon, say, at 100x, you'll find
that the Moon more or less fills the field of view.  In wide-angle
eyepieces, there may be quite a bit of dark space around the Moon, and
in other eyepieces, you may not be able to squeeze the whole Moon in at
100x, but we can ignore these differences for the moment.

If the Moon exactly fills the eyepiece field of view, the true field of
view is 30 arcminutes, since that is the angular size of the Moon, and
the Moon is filling the field of view.  That's how much of the sky you
can see through the eyepiece.  It doesn't make any difference that the
patch of sky you happen to be looking at is all Moon--if you rotate the
telescope over to a featureless expanse of sky, the true field of view
is still the same old 30 arcminutes.

However, the Moon certainly doesn't look only 30 arcminutes across.  It
looks that big to the unaided eye, but the point of using a telescope
on the Moon is to make it look bigger.  How much bigger?  Well, 100x
bigger, and 100 times 30 arcminutes, or half a degree, is 50 degrees.
That's how big the Moon, or whatever patch of sky you're looking at,
seems to you, and that therefore is the *apparent* field of view.

That's also the relationship in general:

    apparent FOV = true FOV * magnification

This is only an approximate formula, and to be more precise is affected
by a number of optical vagaries, but we don't need to worry about that

Incidentally, the apparent field of view is inherent in the eyepiece.
In fact, if you just look through the eyepiece, without inserting it
first into any telescope, you'll see a sharply defined circle.  That
circle's angular size is the apparent field of view.  The true field of
view is related to the angular field of view by the magnification, and
so it is *not* inherent in the eyepiece--it also depends on the
telescope you're using the eyepiece with.

Q.  How can I measure the apparent field of view of my eyepiece?

A.  Here's a method that's worked for me.  You need a tape measure and
the ability to use your two eyes for slightly different purposes.

As mentioned in the answer to the preceding question, if you look
through an eyepiece just on its own, without a telescope, you'll usually
see a sharp edge to the field of view.  That's because you're seeing the
field stop, which lies at the focal plane of the eyepiece.  If the
barrel itself is being used as the field stop, the edge might not be as
clean, but you can still use this method.

Set up in front of an object of known height (or width) h.  I use a
doorway that's 80 inches tall.  Holding both eyes open, look through
the eyepiece with one eye.  Be sure to keep the eyepiece level with
the midpoint of the object.  You should see an indistinct field with
the eyepiece eye, and the object with your other eye.

Now, move back and forth until the object is just as tall (or wide)
as the field of view.  Measure the distance d between your *eye* (not
eyepiece) and the object.  The apparent field of view is then measured
directly as

    aFOV = 2*atan ---

Here's a rough ASCII diagram:

              / ^
             /  |
            /   |
           /    |
          /     |
         /      |
    eye<EP      h
         \      |
          \     |
           \    |
            \   |
             \  |
              \ v

      |<-- d -->|

For example, my 6 mm Radian yielded a distance d of 69 inches, so the
aFOV was 2*atan(80/138) = about 60 degrees.  I estimate the error on
this method to be on the order of 1.5 degrees, plus or minus.

Q.  What is an orthoscopic eyepiece?

A.  An orthoscopic eyepiece is simply one which exhibits no distortion.
If an object 2 degrees across looks exactly twice as wide as one that's
only 1 degree across, then you're looking through an orthoscopic

Otherwise, the eyepiece exhibits distortion.  If the 2-degree object
looks *less* than twice as wide as the 1-degree object, then the
eyepiece has barrel distortion.  This is because through such an
eyepiece, a square object looks as though its sides were bowed out,
like a barrel.  On the other hand, if the 2-degree object looks *more*
than twice as wide as the 1-degree object, then the eyepiece has
pincushion distortion.  This is because through that eyepiece, a square
object looks as though its sides were bowed in, and this supposedly
reminds people of pincushions.  (Does that make any sense to you?  It
doesn't to me.)

Anyway, the term "orthoscopic," used generically, doesn't refer to a
specific design, but rather a characteristic which any design might
or might not have.  Plossls, for example, are orthoscopic eyepieces.
Tele Vue Radians are nearly orthoscopic, but not quite; they exhibit a
small amount of pincushion distortion.  Naglers and other wide-angle
eyepieces have a relatively large amount of pincushion distortion.  This
may make navigating while using them a little like walking while wearing
coke bottle glasses.  The image is clear, so long as you keep the scope
pointed at one particular object, but once you start moving it, some
people get a little "motion sick."  Others aren't bothered by it at all.

There is a specific design, called an Abbe orthoscopic (after optical
designer Ernst Abbe, who designed it in or around 1800).  It consists
of a positive triplet field lens (the part nearer the "bottom" of the
eyepiece) and a positive singlet eye lens (the part nearer the "top").
It is, of course, orthoscopic in the general sense, and furthermore, it
is high in contrast (allegedly because it only has 4 elements and 4
air-glass interfaces).  It is a popular choice among planet observers.

Q.  What does "parfocal" mean?

Parfocal essentially means that you don't have to change focus when
you change eyepieces.  Eyepieces are said to be parfocal to others, or
alternatively, two or more eyepieces can be said to be parfocal.  One
eyepiece on its own cannot be simply parfocal; such a statement makes
no sense.

For example, suppose you have three eyepieces: a 40 mm, a 25 mm, and a
10 mm.  Suppose that whenever you have something in focus with the 25
mm eyepiece, you can swap it out for the 10 mm eyepiece, and the object
will still be in focus, without having to adjust anything.  Suppose
further that that isn't the case with the 40 mm eyepiece; whenever you
switch from either the 25 mm or the 10 mm to the 40 mm eyepiece, you
have to adjust focus.  Then, the 25 mm and 10 mm eyepieces are said to
be parfocal (to each other); the 40 mm eyepiece is not parfocal to
either of them.

This means that parfocality, if I can mangle the word in such a way,
is transitive: if A is parfocal to B, and B is parfocal to C, then A is
also parfocal to C.

There is one special circumstance in which a single eyepiece can be
said to be parfocal.  A zoom eyepiece may or may not require refocusing
as you change the focal length (and hence the magnification); if it
doesn't require adjustment, it can reasonably be said to be parfocal
(to itself).

Q.  What is a Barlow lens, and how do I use it?  And how does it work,

A.  A Barlow lens is an accessory that boosts the magnification of any
eyepiece you use with it.  Many Barlows are nominally 2x, meaning that
if you normally get 100x with a specific eyepiece in a specific scope,
adding the Barlow will get you 200x.  However, Barlows (and other power
amplifiers) come in a variety of different "magnifications."

The normal way of using a Barlow lens is to insert it into the focuser,
in place of the usual eyepiece.  The eyepiece is then inserted into the
Barlow lens.  If you compare using an eyepiece in isolation with using
it in conjunction with the Barlow, you will ordinarily find that you
have to refocus the telescope, often considerably.  Occasionally, you
may have to refocus too much--your telescope won't reach focus with that
particular combination.  If you *don't* have to refocus, or at least
don't have to refocus very much, the Barlow is often called "parfocal"
(see the preceding question).

In principle, a Barlow lens is only a negative lens in a long tube.  The
lens is usually a doublet, although it may be a triplet, as in the
Celestron Ultima or the Meade "apochromatic" Barlow, or it may be a
combination of a negative doublet and a positive doublet, as in the case
of the Tele Vue Powermates.

Light approaches the objective of a telescope as a parallel beam, and
the objective focuses this beam into a light "cone."  This cone has a
characteristic shape--some telescopes have fat light cones, and others
have thin ones.  The fatness or thinness of the light cone is measured
by dividing its length by the diameter of the base.  The length of the
cone is equal to the focal length of the objective, and the diameter of
the cone's base is equal to the aperture of the objective.  The ratio
of these two numbers is called the focal ratio, or f/ratio for short.

A telescope objective makes light converge, so we say that it is a
positive lens or mirror; it has a positive focal length.  A negative
lens, such as the one in a Barlow, has a negative focal length; it makes
light *diverge*.  Therefore light that enters the Barlow in a parallel
beam will spread out when it emerges.

If the light beam is not quite parallel when it enters the Barlow, but
is already converging slightly, then the beam will still spread out upon
emerging, but it will spread out more "slowly."  If we make the light
converge even more before entering the Barlow, we will get to a point
where the light is not spreading out at all when it leaves the Barlow,
but is instead a parallel beam, or even converging slightly.

In such a situation, you might have a light cone going in that is quite
fat--that is, with a small focal ratio--but comes out relatively thin,
with a large focal ratio.  And if you have a larger focal ratio with the
same aperture (the Barlow doesn't affect the effective aperture of the
telescope), the effective focal length must be increased in proportion.
Finally, if the effective focal length is increased, then the power or
magnification of any given eyepiece must be increased as well, since the
power of an eyepiece is equal to

            focal length of telescope
    power = -------------------------
            focal length of eyepiece

Incidentally, it's a commonly held conception that a Barlow lens, by
increasing the effective focal ratio of the telescope, inherently makes
most of the (longitudinal) color error go away.  That generally isn't
the case.  In an achromatic telescope, the colors are almost all the way
spread out by the time the light gets to the Barlow lens--it just can't
get all the colors to focus back into one place at that point.  What
does appear to be the case is that inserting a Barlow makes it possible
to make better use of eyepiece designs that work well only at long focal
ratios.  These tend to be less expensive than designs that work well at
short (and long) focal ratios.

Q.  Is a fast telescope (small f/ratio) brighter than a slow telescope
(large f/ratio)?

A.  Roughly speaking, the answer is yes if you're talking about taking
long-exposure astrophotos, and no if you're talking about actually
looking through the telescope at a given magnification.

What really determines the amount of light gathered by a telescope is
the aperture.  There are also certain variations based on the design,
involving the central obstruction (which clearly blocks some light
from entering the telescope in the first place) and the reflectivity or
transmission of the optical elements (which cause light to be lost once
it has entered the telescope).  Let's ignore those for now, and assume
that we are comparing a fast telescope with a slow telescope of roughly
the same design.

In that case, the amount of light gathered by the telescope varies as
the square of the aperture (diameter).  For example, a telescope with an
aperture twice as wide gathers 2 squared or 4 times as much light.

Now, for visual use--looking through the scope with an eyepiece--that's
all that matters.  If you have an 8-inch f/4 (that's fast) and an 8-inch
f/10 (that's slow), and you use both at, say, 100x, then images will
look equally bright through both telescopes, because the same amount of
light is being gathered by both, and is spread out over the same area in

Of course, in order to get 100x in the fast telescope, you need about an
8 mm eyepiece, and in the slow one, only a 20 mm eyepiece.  If you were
to use a 20 mm eyepiece on the fast telescope, that would yield a power
of only 40x.  Things would look 2/5 as large as they do through the same
eyepiece on the slow scope, and since the same amount of light is being
squeezed into only 2/5 squared, or 4/25, of the area, things look much
brighter (provided you can fit the 5 mm exit pupil of this set-up into
your eye).

In the case of (prime-focus) astrophotography, there is no eyepiece.
What determines the size of the image is the focal *length* of the scope
and not the focal ratio.  The image scale is directly proportional to
the focal length.  Since the 8-inch f/4 has a focal length of 32 inches,
and the 8-inch f/10 one of 80 inches, images will be only 2/5 times as
large in the former as in the latter--precisely as when using a single
given eyepiece in visual use.  And for the same reason, things will look
25/4 (that is, 6-1/4) times as bright, although they will also cover
6-1/4 times smaller an area.

In short, in astrophotography, scopes of the same focal *length* yield
images with the same *scale*; scopes of the same focal *ratio* yield
images with the same *surface brightness* (see the following question).

Astrophotographers can take advantage of this enhanced brightness to
reduce the length of exposures.  To get the same surface brightness, one
can take images in 4/25 the time in any f/4 as in any f/10.  (Strictly
speaking, because of those other design factors mentioned at the start
of this answer, we should be using so-called "t/ratios," which take them
into account.)

Q.  What is the limiting magnitude for my scope?

A.  That depends a lot on how acute your eyes are, what kind of skies
you have, and what kind of object you're looking at.

One way to address the first two factors is to figure out what the
unaided-eye limiting magnitude is for you on any given night.  If you
can see to magnitude 6 with the unaided eye on any given night, and we
can determine that you should be able to see 7 magnitudes deeper with
your telescope than you can without it, then the limiting magnitude is
13 on that night.  If, on some other night, the unaided eye limiting
magnitude is 5.5, then the limiting magnitude on this second night
would be 12.5.

The formula for determining how much deeper you should be able to see
with your telescope than without it depends a little on your pupil size
(the pupil in your eye, that is).  However, we can assume a 5 mm pupil
without too much error, and give the adjustment as

    5 log (D/5)

where D is the aperture of your telescope, in millimeters.  Thus, a 50
millimeter scope gives an adjustment of 5 log 10, or 5 magnitudes

However, there is another factor to consider, and that is the type of
object you're looking at.  If you're under magnitude 6 skies--that is,
you can see stars down to magnitude 6--then the above formula tells you
that you should be able to see down to magnitude 11 with a 50 mm
telescope.  But those are just magnitude 11 *stars*.  You will not be
able to see many galaxies, for example, with magnitudes as high as 11,
because galaxies are extended objects, not dots.  Their light is
therefore spread out over an area, and although they would be visible
if only that light were squeezed into a dot, spreading it that thin
makes them invisible.

Under these circumstances, a hard rule simply doesn't exist.  You will
have to learn from experience what your telescope will see when it
comes to extended objects like galaxies and nebulae.  There is a rule
of thumb, though, that might help.  It involves a certain property of
the object, called surface brightness, which you can think of as the
brightness per unit area, and the unit area in this case is square
arcminute.  What we ordinarily call the magnitude of the galaxy is more
precisely called the *integrated* magnitude--that is, the brightness of
the whole galaxy, summed up (or "integrated") over its entire area.  If
a galaxy is magnitude 10, but has an area of 100 square arcminutes,
then each square arcminute contains 100 times less light than the whole
galaxy.  Since a factor of 100 is equal to 5 magnitudes, the surface
brightness of the galaxy is 10+5, or 15.

The rule of thumb, which I'll call the Rule of 13 (even though I didn't
come up with this rule), states:

    Add the integrated magnitude to the surface brightness, and
    subtract 13.  The object should then be about as visible as
    stars whose magnitude is this number.

In the case of the galaxy above, its integrated magnitude is 10, and
its surface brightness is 15.  Add those together, and subtract 13,
and you get 12.  If you can see stars through your telescope down to
magnitude 12, then this rule suggests that you have a decent chance of
spotting the galaxy.

Again, though, the Rule of 13 is just a guideline, and not really a hard
and fast rule.  You might use it to prioritize an observing list, but I
wouldn't omit a target solely because the Rule of 13 suggests that you
won't be able to see it.  One particular issue with the Rule of 13 is
that most catalogues don't list peak surface brightness numbers.  The
upshot is that if an object has a strong central condensation, using
the average surface brightness will lead to an underestimation of its

Q.  How is seeing related to twinkling?

A.  Seeing and twinkling are related, but they are not the same thing.
The term "seeing" refers to the effect of turbulence on the view
through the eyepiece, how it distorts the image.  Twinkling is the
blinking on and off of stars; another name for it is "scintillation."
It too is caused by atmospheric turbulence.

An oft-quoted rule of thumb states that when the stars twinkle, the
seeing is poor, and conversely, when the stars do not twinkle, the
seeing is good.  That, however, is not always the case, and I'll try to
explain why.

We can think of a star emitting light rays in all directions, only some
of which land here on the Earth, or in our eyes.  If we were to trace
those rays back where they came from, we would find that they converge,
quite naturally, back at the star.  However, the stars are so far away
from us, and our eyes so small in comparison, that those light rays are
as good as parallel by the time we see them.

If the Earth had no atmosphere, the light rays would continue to be
parallel until the moment they struck the ground, or our eyes.  If you
could see the places where the light rays struck the ground, you could
imagine that the ground would be littered with countless little points
of light, and with no atmosphere, the points would be perfectly evenly

However, the Earth *does* have an atmosphere.  In any transparent
medium that is not a vacuum (such as the atmosphere), light travels
slower than it does in a vacuum.

This is not a problem so long as the atmosphere is uniform all the way
down to the ground.  All light rays are equally slowed and their motion
is unaffected except for speed.  You can see that if the light rays
were merely slowed down, they would travel to the ground in exactly the
same paths as before, and their points of impact would still be
perfectly evenly spaced.

Unfortunately, the Earth's atmosphere is not uniform.  Some pockets of
air are hotter or colder, denser or less dense.  This not only slows
the light rays down by different amounts as they traverse the atmosphere,
but it also bends them, and the more the atmosphere varies in density
or temperature, the greater the bending.

If the atmosphere changed willy-nilly as the light descended to the
Earth, who knows what would happen to the image of the star.  But the
very fact that we can see the stars at all indicates that that doesn't
happen.  Luckily for astronomers, the turbulence layers are generally
rather limited in extent.

Let's consider, for a moment, if there's just one layer of turbulence,
high up in the air.  When the light rays hit that layer, they are still 
a long way from hitting the ground (in human terms).  The light rays
may only be bent a little by the turbulence layer, but since they are
so high up, their points of impact may be changed significantly, perhaps
as much as a few centimeters.  Moreover, the light rays are bent
randomly and are not all bent in the same direction.  Instead of a
pattern of perfectly evenly spaced points of impact, we would see
places where more light rays were hitting the ground, and the light
there would be brighter.  In other places, fewer light rays would strike
the ground, and there the light would be dimmer, or perhaps even absent.

What does this have to do with twinkling?  Stars twinkle because as
the atmosphere swirls around, occasionally the light rays get bent so
far as to miss your eye entirely, or mostly.  Your eye's pupil, or
opening, is about 5 millimeters across at nighttime.  If your eye
happens to be in one of those holes in the pattern, where fewer light
rays are meeting, the star will suddenly appear to dim.  Then, as the
atmosphere moves around again, the light rays will return to your eye
and the star will re-brighten.  These brightness changes are not
inherent in the star; they are a product solely of how the atmosphere
brings the light rays to your eye.  The changes take place on the
order of perhaps a second or so, and this is what we see as twinkling,
or scintillation.

On the other hand, suppose that the layer of turbulence is close to the
ground.  In that case, by the time the light rays hit the layer, they
are already so close to striking the Earth that their points of impact
are changed only slightly, perhaps just a millimeter or so.  In that
case, you can see that there can't be so much as a single hole 5 mm
wide where no light strikes.  The holes can only be twice as wide as
the bending is--and twice a millimeter is only 2 millimeters.

In other words, given the same degree of turbulence, but different
heights, high-altitude turbulence causes the stars to twinkle
noticeably, while low-altitude turbulence causes them to twinkle only
modestly, or not at all.  But this is only to the unaided eye.  What
about telescopes?

What about telescopes indeed!  A telescope is generally at least 50
millimeters across at the objective, and usually greater than that.
Even with high turbulence, the chances that there will be a circle more
than 50 millimeters across where no light rays hit is minuscule.  I
suspect that it is so close to zero that there is essentially no way
for stars to entirely twinkle out (though they might appear to dim and

However, just because stars don't twinkle at the eyepiece doesn't mean
that the image is unaffected.  Suppose you're observing Mars, and you
have it centered in the eypiece.  Light rays from the center of Mars get
to the objective along a line perfectly parallel to the axis of the
telescope; these light rays are called on-axis rays.  Light rays from
the edge of Mars, say, get to the objective along a line at a very
slight angle to the axis of the telescope; these light rays we call
off-axis rays.

If light from Mars hits a turbulence layer, though, all bets are off.
The actual shifting of the light may not matter very much, since as we
noted, it's very unlikely that there will be a hole as large as the
telescope's objective.  However, it also gets bent--that is, its angle
changes.  On-axis light rays become off-axis light rays, off-axis light
rays become on-axis light rays, and the image of Mars appears therefore
to shift a little in the eyepiece.  Worse yet, the bending may happen
unequally to the different parts of Mars, meaning that the image appears
to wrinkle and sway, almost as though we were observing it from the
bottom of a swimming pool, and it is *this* effect that we call seeing.

All this leads to the interesting question: How much does the altitude
of the turbulence affect seeing?

Not very much at all!  Since what determines the seeing is the angle
through which the light rays are bent, and not their displacement, it
doesn't matter whether the turbulence layers are way up there, or all
the way down here, the angle of deflection is the same.  Seeing is to
first order independent of the altitude of the turbulence.

The upshot of all this is that if the turbulence is high up but weak,
the twinkling may be quite noticeable, but the seeing still reasonably
good, whereas if the turbulence is low down but strong, the twinkling
may be absent, but the seeing will be poor.  Of course, if the
turbulence is high and strong, or low and weak, then the two phenomena
agree, but that's not always the case.  Twinkling just isn't a reliable
indicator of seeing.

By the way, the fact that bad seeing resembles the view from the bottom
of a swimming pool has led some people to the idea that the two
phenomena are similar.  Other than the visual effect, however, that
simply isn't true.  Images from the bottom of a swimming pool waver not
because of turbulence within the water, but because the surface of the
water has waves, and light from outside objects therefore hits the
water at constantly changing angles.  This affects how much the light
is bent, and that's what causes the boiling effect in this case.  The
same effect on the image of Mars, on the other hand, has little to do
with the top "surface" of the atmosphere, and is primarily dependent on
turbulence layers within the atmosphere.

Q.  What does "diffraction-limited" mean?

A.  Literally, it means that the performance of a set of optics, such
as a telescope, is limited by the diffractive nature of light, and not
by other factors such as the quality of the optics.  However, as we'll
see later, this term is so often abused that it is nearly meaningless.

Let's consider a Newtonian reflector.  The same basic ideas apply in
other kinds of telescopes, but then there are other factors that obscure
the basic principle at work.  Ideally, a Newtonian's primary mirror has
a paraboloidal shape--that is, it's the result of rotating a parabola
around its axis of symmetry.  Why "ideally"?  Because when you shine a
beam of light rays down on a mirror, all parallel to that axis of
symmetry, that is the one shape that will focus all of those rays down
to a point, appropriately enough called the focal point.

That at least was the hope of its designer, the English scientist Isaac
Newton (1646-1727).  It was later discovered that because light has a
wave nature to it, the light rays interfere with each other noticeably
when they get close to the focal point.  The result is that instead of
getting a neat, infinitesimal point of light at the focus, what you get
instead is a ball.  This is the effect of light diffraction.  It must
be emphasized that this has nothing at all to do with not making the
mirror good enough.  Even if the mirror is absolutely perfect, you can't
get a point of light.  In other words, you're not fighting imperfect
engineering techniques, you're fighting physics--usually a losing

When viewed through an eyepiece fitted to the telescope, this ball has
the appearance of a disc, surrounded by a series of successively dimmer
rings.  This disc is named the Airy disc, after George Airy, the English
astronomer and physicist who first gave the proper explanation for it,
and the entire pattern is sometimes called the Airy pattern.  The view
of any celestial object (or terrestrial, for that matter) is affected by
the Airy pattern.  Instead of being a sharp point of light, stars at
high power are Airy discs.  These discs are *not* the discs of the stars
themselves--again, they are only artifacts of the physics of light.

This same effect is visited on extended objects such as galaxies and
planets, too.  Every single point of detail is smeared around according
to the Airy pattern, just as the light of a star is.  The upshot is
that patches of detail that are finer than the size of the Airy disc
are washed out--all you get in these cases are uniform patches.  This
is especially a problem with the planets, which are full of patches of
fine detail.

The diffraction of light, however, is only one of many ways for detail
to be lost.  Another is the atmosphere; if the air above the telescope
is turbulent, that can cause light rays to bend even before they enter
the telescope, and that destroys some detail.  (See the preceding
question for more information.)  The optics themselves may radiate some
heat as the night progresses, and that is destructive, too.  Finally,
even if the optics are properly acclimated, and the atmosphere is
steady, the optical quality may also degrade the image.

Out of these factors, the one that is under the control of the telescope
(or rather, the telescope manufacturer) is optical quality.  It is also
a major selling point; naturally, higher-quality telescopes will sell
better than lower-quality ones, all other things remaining equal.  On
the other hand, other things do not remain equal: it costs more to make
a higher-quality telescope.  As the quality increases, at some point, it
just doesn't make any sense to put any more money into the optics; the
improvement in the image isn't worth it.

The question is, at what point does that happen?  How do you judge when
the optics are "good enough"?  The common criterion is the diffractive
effect of light.  At any given moment, the atmosphere may suddenly calm,
the optics may be all cooled, but light diffraction never rests.  It is
*always* in effect.  On the best of nights, the detail you can see is
limited by either the optical quality, or the diffraction of light.  If
it's the latter, then the optics *are* "good enough"; that is, they are
"diffraction-limited."  This is the first definition of the term.

This is, however, just subjective hand-waving.  After all, at no point
does higher quality lose *all* effect--it's just that the effect may be
tiny compared to that of diffraction.  For that reason, astronomers and
opticians have tried to quantify "diffraction-limited."  One way is to
specify the necessary accuracy of the curve, or "figure," of the optics.
This is the source of the Rayleigh criterion, which states that the
telescope is diffraction-limited if the light waves miss focus by no
more than 1/4 the wavelength of light.  However, this criterion totally
ignores the distribution of the error.  It considers identical two sets
of optics, one which has only two small patches of error, and one that
is completely covered with error patches.

Another criterion, due to Marechal, considers a telescope diffraction-
limited if the RMS--the "root mean square"--of the error is less than
1/14 the wavelength of light.  This takes into account the distribution
of the error, so that the telescope with two small error patches rates
better than the one completely covered with error patches.

The Rayleigh and Marechal criteria are a result of observations; that
is, experienced astronomers tested a number of telescopes with known
quality, and tried to establish a point at which the image was "good."
Another way to quantify "diffraction-limited" is to examine how light
is distributed in the Airy pattern.  In an ideal set of optics, about
84 percent of the light goes into the Airy disc itself.  The remaining
16 percent goes into the rings, most of that being in the first ring.
The light in the rings means that the intensity at the very center of
the Airy disc is limited to a certain theoretical maximum for that
particular aperture and design.

However, imperfect optics can throw even more light into the rings and
away from the center, so that the central intensity drops even further.
The *Strehl ratio* of a telescope is the actual intensity at the center
as compared to the ideal central intensity.  Using this metric, a scope
is said to be diffraction-limited if the Strehl ratio is over 80 percent.

These various ways to quantify "diffraction-limited" represent,
collectively, the second definition of the term.  It turns out that they
are in remarkable agreement with each other.  In other words, under
normal circumstances, optics that satisfy one criterion will come at
least close to satisfying the others (although there are exceptions).
Unfortunately, there is a third definition.  This last definition is how
the term is used in advertising copy, and here it is so overused as to
be nearly meaningless.  The nominal definition used is the old peak-to-
valley definition (1/4-wave), but this can be interpreted loosely (see
the following question).  What's more, the old phrase "guaranteed to be
diffraction-limited" means just that: we'll exchange it if it isn't
diffraction-limited, but it's up to you to initiate the process.  In
that way, many customers who either don't know or don't care enough end
up with subpar telescopes.

Many authors who should know better recommend that beginners look for
telescopes that are rated as "diffraction-limited."  It's not wrong to
look for diffraction-limited optics, of course; the problem is that you
can't always trust what you read, and you may have to do some hunting
around before you find the real deal.

Q.  I often seen optical quality measured as "1/8-wave or better."  What
does this mean?  Is, for example, 1/8-wave better than 1/4-wave?

It depends.  All other things being equal, yes, 1/8-wave is better than
1/4-wave.  But all other things are not always equal.

Light from a distant star arrives at the objective (lens, mirror, or
both) in parallel rays.  A perfect objective would focus all of these
light rays down to one point, but perfection is impossible: invariably,
some of the light rays overshoot the mark, some undershoot it.  One
measure of quality of an objective is how far the light rays miss the

The precision of even a so-so set of telescope optics is astonishing:
the error is on the order of 0.00001 inches. Now obviously, measuring
something that small in inches or even millimeters would be a pain, so
instead, the wavelength of light is used. Then that error of 0.00001
inches can be said instead to be about 1/2 wave, since the wavelength
of visible light is on the order of 0.00002 inches.

Now I'm going to make that same error be about 1/6-wave.

The trick is that we forgot to specify the color, or wavelength, of
light that we used as our yardstick.  Light of a blue-green color is
about 500 nanometers, or about 0.00002 inches, and using that
wavelength as our yardstick, the error of our objective is indeed
1/2 wave.  But let's suppose we use red light of about 700 nanometers.
Then the error drops to 1/3-wave, simply because the wave we're using
as our yardstick is longer. With me so far?

(Actually, it's worse than that in some cases.  The above analysis works
for reflectors, but in the case of refractors, they're often very
well-shaped for some colors, but not for others.  For visual use,
refractors should be corrected in the middle of the visible light range,
about 550 nanometers.  Testing them in red light may set you off by much
more than the 50 percent or so in the case of mirrors.)

Then there's concern over where we're measuring the error.  I've
measured it at the wavefront--what happens after the light has already
passed through the objective.  But let's suppose we measure the error
at the surface of, say, an objective mirror.  If we have a pit of depth
x in the mirror, then light hitting that pit is delayed by 2x--x on the
way in, and x on the way out.  In other words, the surface error is half
of the wavefront error.  Now our 1/3 wave error drops to just 1/6 wave.

(This "half" rule only works for mirrors--it doesn't apply to objective
lenses, so refractors aren't subject to quite such a drop.)

Wait, it gets better.  I've talked about measuring the extent to which
light rays overshoot or undershoot the mark, called P-V (peak-to-valley)
error.  What if what we want to know is how smooth the wavefront is, not
in the worst cases, but just on average?  Then we measure the error in
terms of RMS (root mean square).  Because of the way the RMS error is
derived, it's impossible to make a hard rule about the relation between
it and P-V error, but we might reasonably see that a P-V of 1/6-wave
might become a 1/20-wave RMS.

So, if you want to be hard on an objective, you measure it P-V, at the
wavefront, in a short wavelength like blue-green.  If you want to be
kind to the objective, you measure it RMS, at the surface, in a long
wavelength like red.  The difference can be something like an entire
order of magnitude, and you need to be sure how your particular error is
being measured.  Measuring errors RMS without mentioning it is pretty
slimy and I don't think anyone big does that, but measuring at the
surface in a long wavelength is probably at least somewhat common.

Q.  What is the star test, and how does it work?

A.  The star test is a test of optical quality and other factors
affecting image quality, and it can be run by anyone.  The definitive
reference regarding star testing in the amateur community is Harold
Suiter's Star Testing Astronomical Telescopes.  What follows is a very
brief description of how and why it works.

In the star test, you start with a star in focus at the center of the
field of view, at very high power--ideally something exceeding 40x per
inch of aperture.  (For example, use 160x or more on a 4-inch
telescope.)  Now, ever so slightly rack the star in and out of focus,
on both sides of focus.  When out of focus, where you might expect just
a fuzzy ball of light, what you actually get is a small bullseye pattern
of a central dot and some concentric rings.  This is due to the
diffractive nature of light, which for reasons of space we won't get
into here.

If everything is working properly, the bullseye pattern should look
exactly identical on both sides of focus.  However, if there is an error
somewhere, this will be reflected in a difference between the two
out-of-focus images.  For example, one possible error is called
spherical aberration, and it is caused by a misshapen mirror or lens.
In order to focus light to a point, a mirror (say) must be shaped like
a paraboloid--that is, a parabola rotated around its axis.  If the shape
is more like a section of a sphere than a paraboloid, then the rays of
light entering near the periphery of the mirror will be focused too
close to the mirror, and those that enter near the center will be
focused too far from the mirror.  The result is that instead of a sharp
point of focus, what you get is a kind of oozing funnel, with the broad
cup of the funnel pointing toward the mirror and the drain tip pointing
away.  See the following question for more on this.

(There's a name for the funnel, by the way.  It's called the "caustic
horn."  Horn for the shape, clearly, and caustic because it's at the
focus, and therefore "hot."  I prefer "oozing funnel," myself.)

Spherical aberration creates fuzziness in images.  When just observing,
its effect may be difficult to distinguish from bad focus or
atmospheric instability, but it reveals itself in a characteristic way
in the star test.  Since the light at the center focuses too far away
from the mirror, when you go out of focus in one direction, you see the
drain tip of the focus "funnel," and the center gets brighter and the
outer diffraction rings get dimmer.  In the other direction, you see
the cup of the "funnel," and the center gets dimmer and the outer rings
get brighter.  The two out-of-focus images are no longer identical, and
this particular effect on the star test can be identified, in
isolation, as spherical aberration.

Unfortunately, there are many different errors possible, and these
combine in some non-intuitive ways, making diagnosis non-trivial.
What's worse, the images can also be affected by factors that are not
the fault of the optics at all, such as turbulence.  The end result is
that reading the star test is not easy.  It is easy to identify a good
telescope, but not so easy to identify a bad one--what you interpret as
errors may be factors out of the telescope's control.  If you want to
be a conscientious user of the star test, you should read Suiter's

Q.  What do "over-corrected" and "under-corrected" mean?

A.  The terms refer to correction of spherical aberration.

The "ideal" reflector mirror is paraboloidal, at least in the case of a
Newtonian reflector.  Only a paraboloid--that is, the 3-D shape you get
by revolving a parabola around its axis--will focus a parallel, on-axis
beam of light to a point.  (Well, more or less.  See the question on
diffraction-limited telescopes, above.)

However, it is not simple to grind a paraboloidal mirror.  It doesn't
have equal curvature everywhere; it's less curved on the edges than it
is in the center.  It's rather easier to grind a spherical mirror, which
has the same amount of curvature everwhere on its surface.

A spherical mirror, unlike a paraboloidal one, does not focus the
on-axis light to a point, however.  Instead, the light rays striking the
edge of the mirror focus closer in to the mirror than do the rays
striking the center.  As a result, even point sources do not appear
exactly pointlike; instead, what you get is a fuzzy ball, due to the
imperfect focusing of the spherical mirror.

Because a spherical mirror is easier to make, however, one often begins
by grinding a spherical mirror, and then "correcting" its shape to that
of a paraboloidal one.  If one doesn't correct enough, then the light
rays at the edge continue to focus closer in than the ones at the center
(although perhaps not as much as with a perfectly spherical mirror).  In
this case, the image is still a little fuzzy, and the mirror is said to
be "under-corrected."

On the other hand, if you go too far, and the mirror is too "sharp," 
then the light rays at the edge now focus further out than the ones in
the center.  The mirror is then said to be "over-corrected," although
the resulting image quality is just about the same as it is with an 
equivalently under-corrected mirror.

Only in between, when the mirror is precisely corrected to a
paraboloidal shape, do you get sharp, on-axis images.  It should be
remarked that even this mirror has an off-axis aberration, called coma,
in which stars out from the center of the field of view appear
comet-like, with their tails trailing out to the edge of the field of
view.  Coma is only objectionable in the faster (i.e., smaller focal
ratio) scopes--perhaps f/5 or faster--and even then, it can be largely
reduced by devices such as Tele Vue's Paracorr.

Q.  What is astigmatism, and what causes it?

A.  Astigmatism is a directional flaw in the optics, and therefore in
the image presented by those optics.

One way to think of the way a telescope works is to imagine parallel
rays entering the telescope, which are then focused to a point on the
image plane by the objective (whether that's a lens in the case of a
refractor, or a mirror in the case of a reflector).  That image is
then magnified by the eyepiece so that you can see it.

That's what happens ideally, at least.  What occasionally happens is
that the light in one direction (say, up and down) focuses too short,
and the light in the other direction (left and right) focuses too far.
That's astigmatism.  Because the deviation of the optics from the 
ideal is positive in one direction and negative in the other, this is
sometimes called potato-chip error.

With astigmatism, at no place is all the light in focus.  In the star
test (see the preceding two questions), if you focus in, then the
up-down light is in focus, but the left-right light is out of focus, and
instead of a point, you get a left-right bar.  If you focus out, then
the left-right light is in focus, but the up-down light is out of focus,
so you get an up-down bar.

In between, when both are slightly out of focus, you get a plus or cross
shape.  That's often the point of best focus for that telescope, but it
may not be very good, depending on how astigmatic the telescope is. 

Astigmatism can come from anywhere in the optical system, from your
eyes to the objective.  You can identify where it's coming from by
process of elimination.  For example, if you try looking into the scope
from two different directions, but the astigmatism still points the same
way (with respect to the telescope), then you know the problem is not
in your eyes.

Next, rotate your eyepiece.  If the astigmatism doesn't move, it's not
in the eyepiece.  If you're using a reflector, then try rotating your
primary.  If the astigmatism *still* doesn't move, then it's not in your
primary.  And so on.

Astigmatism is sometimes ground into the optics, or come about as a
result of the way the glass formed.  If so, you're out of luck, I'm
afraid--you'll have to either live with the flaw, or get the optics
replaced.  However, more often than not, it's a result of the way the
optics are supported.  Mirror sag or pinching support clips on the
primary are principal suspects, as are similar clips on the secondary.

Q.  What do the terms achromat, apochromat, and semi-apochromat mean?

A.  These terms usually refer to the degree of color correction that a
refracting telescope possesses.  An achromat is a refractor that has
only a little bit of chromatic aberration, in which the colors that make
up white light are split up by the objective lens.  An apochromat has
almost no chromatic aberration at all.  A semi-apochromat, to the extent
that the term means anything at all, lies somewhere in between.

When refracting telescopes were first made, 400 years ago, their
objectives were made from a single piece of glass.  This lens refracted
the incoming light to a focus, so that it could be magnified by the
eyepiece.  The problem was that different colors or wavelengths of light
were refracted by differing amounts, and therefore did not come to focus
at the same distance.  This problem is called longitudinal chromatic
aberration (often shortened to just chromatic aberration).

As a result, if you pointed the telescope at a white star, only one of
the wavelengths of light could be in focus at any given time.  If you
focused on the red light, the other wavelengths would all be out of
focus, and could be seen as a fuzz of light that gradually grew bluer
toward the edges.  This robbed the image of considerable detail.  The
effect could be reduced somewhat by making the telescope longer, and so
immensely long telescopes were designed and built, with the high-water
mark being represented by Johannes Hevelius's refractor, which measured
over 60 meters (about 200 feet) long.

Then, in 1733, the English barrister Chester Hall discovered that by
making the objective from two pieces of glass with different properties,
much of the variation in focus of the different colors could be
eliminated.  The basic design is still used today, and reduces the
chromatic aberration by about 98 percent.  What is left, the last 2
percent, is called the secondary spectrum of the objective.

Hall's invention (and its subsequent popularization by John Dollond)
made it possible to produce well-corrected telescopes that were only a
couple of meters long, rather than nearly a hundred.  They were *the*
telescope of choice for about a century, culminating in Joseph von
Fraunhofer's refractors in the early 19th century.

Eventually, the cost of making exquisitely fashioned lenses larger than
Fraunhofer's caught up to the achromat design, and the heyday of the
reflecting telescope began.  It wasn't until the late years of the 19th
century that the German physicist Ernst Abbe designed the first truly
new refractor design in a century and a half, with the apochromat.  This
design used (typically) three glasses to produce an objective with even
better correction for chromatic aberration.  It reduced the secondary
spectrum of the achromat by a further 80 to 90 percent, so that only
a few tenths of a percent of the original aberration in the single-lens
design remains--too small to be seen on virtually any target in the
night sky.

Abbe also gave strict definitions to what qualified as an achromat or an
apochromat.  These definitions felt the impact of the photographic era,
however, so that an Abbe apochromat could actually seem worse than an
achromat when used visually, even though this difference couldn't be
seen in the film image.  Today, most amateur astronomers aren't familiar
with Abbe's definitions, and use the terms to refer only to the degree
of color correction.  (In fact, he first designed apochromatic lenses
for microscopes, where chromatic aberration is also a problem, and only
later transferred it to telescopes, along with his partner, Carl Zeiss.)

Both achromatic and apochromatic telescopes are commercially available,
with apochromats commanding a significant premium--a 4-inch apochromat
(or "apo," as it is often called) typically costs well over $2,000 U.S.,
whereas a similarly sized well-executed achromat may cost only $500.  It
should be pointed out that apochromats convey several benefits besides
mere color correction--they produce wide, flat fields, and are also
well-corrected for spherical aberration (which see)--which are more
important for astrophotography than they are for visual use.

The Abbe definitions did not allow for an intermediate class of color
correction.  An achromat, according to Abbe, brings two wavelengths to
focus at the same point; an apochromat, three.  Since the terms today
refer mostly to the reduction of the secondary spectrum, however, one
could define a semi-apochromat to be a telescope that reduces the
secondary spectrum of an achromat by about half (or about 1 percent of
the original one-lens design).  No single-objective design exists with
this level of correction, although the compound, Petzval configuration
does; this design has been used by Tele Vue in some of its refractors.

Q.  Does a central obstruction damage image quality or not?

A.  Yes, it does, but nearly everyone overestimates the damage done,
especially those using telescopes without central obstructions.

There is a rule of thumb running around (so to speak) to the effect that
a reflector is only as good (in terms of contrast and resolution) as a
refractor whose aperture is equal to the reflector's aperture minus the
diameter of the reflector's central obstruction.

I happen to think some of that is what happens when people stop thinking
and instead speak only in aphorisms such as "A stitch in time saves
nine" and "A reflector is only as good as a refractor of reduced
aperture."  But I'll leave that alone.

It really depends on what you're looking at and what the specific
instruments are.  It also could depend on what kind of skies you observe
under.  To reduce it to, "A 6-inch refractor is as good as an 8-inch
reflector," say, is to simplify matters beyond sense.

The *origin* of that phrase is in the modulation transfer function, or
MTF, which just measures how well the optical system preserves feature
contrast, as a function of feature size.  It's equal to 100 percent at a
given feature size (measured in arcseconds) if the brightness difference
between feature and background is completely preserved; 50 percent if
the brightness difference is reduced to 50 percent; and so forth.  The
MTF of a telescope with a central obstruction (such as a reflector) is
*like* (but not identical to) that of a smaller, unobstructed telescope;
the smaller aperture is said to be the aperture of the reflector minus
the diameter of the obstruction.

Well, consider three telescopes:

    A.  An 8-inch refractor.
    B.  An 8-inch reflector, with a 2-inch obstruction.
    C.  A 6-inch refractor.

Let us assume that all three are smooth and well-corrected.  Telescopes
A and C also have to be well-corrected for color error.  The MTF of
telescope B will be lower than that of telescope A, and about equal to
that of telescope C.  That is the origin of that aphorism.

This has to be taken with a grain of salt, however.  First of all, it
is not true for the finest of features.  The MTF of an obstructed scope
at the highest frequencies is essentially indistinguishable from that
of an unobstructed scope.  Telescopes A and B will perform just about
equally well on small detail--say, smaller than about 1 arcsecond across.

Secondly, *all* telescopes do well with larger detail, and this "larger"
doesn't have to be very large at all.  The MTF of all three telescopes
on details of a few arcseconds and larger are high, and while telescope
B will perform worse than A (and about the same as C) on these larger
details, the difference doesn't hurt that much.

The greatest difference is in the mid-frequency (that is, mid-size)
range, between perhaps 1 and 3 arcseconds in size, where the MTFs of all
three telescopes are falling, and the difference between A and B is most
apparent.  In this range, telescopes B and C might preserve only 3/4 as
much contrast as telescope A.  A great many features on Jupiter, for
example, fall here--festoons, barges, ovals--but not all.

Thirdly, this MTF hurts more on bright objects such as the planets,
where the eye doesn't have to work hard on mere detection.  With most
DSOs, you are working hardest on just seeing the objects, and even on
detail, the increased light-gathering power of the 8-inch telescope
helps you more than its somewhat decreased contrast hurts you.

Fourthly, this rule of thumb is swamped by differences in optical
quality.  If either telescope has rough optics, that will matter as much
as, if not more than, a central obstruction.  Reflectors are more prone
to turned down edges, but refractors are more prone to chromatic
aberration.  "All other things being equal" doesn't happen very much in
the real world.

So, to be a bit more accurate, one should say, "A 6-inch refractor is as
good as an 8-inch reflector, but not on fine detail, and really just on
the planets, and only if both are made to the same optical standards,
and on larger details, it doesn't matter very much."  However, that is
starting to get a bit wordy, so many people just mention the first part.
To be generous, they may *know* the whole thing, but forget to say it.

Q.  Does a Newtonian's spider damage image quality or not?

A.  This question gets asked almost as much as the previous one.  The
truth is that it does, but the overall magnitude of the effect is much
smaller than even that of the central obstruction.

The reason that it nonetheless gets asked very often is that the effect,
although relatively small, is quite noticeable.  Bright stars sprout
spikes of light which correspond to the spider arms.  Each arm creates
two diametrically opposed spikes, so that a three-arm spider creates six
spikes, while a four-arm spider creates four spikes, not eight, because
each of the four spikes is doubly reinforced by a pair of diametrically
opposed arms.  The explanation for this is beyond the scope of this FAQ;
I have written a more extensive explanation of diffraction and its
effects at

Some people find this irritating; others find it inconsequential or even
kind of charming.  (I find it charming, myself.)  Most people concede
that it is largely a matter of personal preference--most everyone except
double star observers, that is, for whom the chance that a dim companion
star is hiding somewhere behind a bright star's spike is frustrating.

What worries more people is the effect on planets.  For example, Jupiter
may appear to be at the center of a hazy cross of light, and it is clear
that a noticeable amount of light is being smeared around as a result of
the spider support.  Since any smearing results in *some* loss of image
contrast and detail, it's quite reasonable to wonder how large that loss

Because the effects of diffraction are most easily noticed at the edges
of obstructions and stops, people get the idea that diffraction is an
"edge phenomenon," that diffraction is somehow caused by light striking
an edge and "forgetting" where to go.  Although that image is not quite
entirely wrong, it can easily lead to some mistaken conclusions.  One
might deduce, for example, that since diffraction is associated with the
edge or perimeter of an obstruction, the magnitude of the effect must be
proportional to the length of that perimeter.  And since a spider's
perimeter is long way out of proportion to its area, the perception
arises that the spider's contribution to diffraction is much greater
than you'd guess based on its area.

As a matter of optical fact, however, the fraction of light that goes
somewhere where it doesn't belong (due solely to diffraction at the
spider) is proportional to the fraction of the aperture's area that the
spider blocks.  If a spider blocks 0.1 percent of the aperture, by area,
then 0.1 percent of the light arriving from a distant planet or star
never enters the telescope in the first place.  Of the light that does
get in, a further 0.1 percent goes where it shouldn't, due to
diffraction.  It is this latter 0.1 percent that leads to the smearing
and resulting loss of contrast; the former 0.1 percent only causes the
image to be (unnoticeably) dimmer.  It's only because the 0.1 percent
smearing is concentrated along very tight lines that it's so easy to

Note that the same reasoning applies to the central obstruction as well,
except that the area covered by the obstruction is typically in the 4 to
15 percent range.  As a result, 4 to 15 percent of the light is blocked
from entering the system altogether, resulting in a dimmer image; then,
4 to 15 percent of the remaining light is smeared around where it
shouldn't go, resulting in a less contrasty image.  This effect is an
order of magnitude worse than the spider, and since the obstruction's
effect is noticeable but hardly fatal, the contrast loss due to the
spider can be safely neglected.

The aesthetic damage caused by the spider can be resolved using either
curved spiders, which typically cause greater diffraction but spread it
around more evenly, rather than concentrating it along narrow lines, or
by using an optical window to support the secondary.  However, optically
flat windows (that is, flat to within 1/4 wave, which see) aren't easy
to make.

Q.  What is prime focus?

A.  Speaking loosely, it is the point where your telescope creates an
image, so that you can either view it with your eyepiece, or record it
on film.

In order to get further into this, it's necessary to explain a little
about how a telescope works.  When you observe a distant star, the
objective of your telescope (whether it's a mirror or a lens) causes
light from that star to converge to a point.  "After" that point, of
course, light begins to diverge, very much as though there were a little
carbon copy of that star in your telescope.  Your eyepiece magnifies
that copy, and the magnified image is what you see when you look through
the telescope.  Because the image emits light as a star would if it were
actually there, the image is called a "real image."

Now, light from the star focuses to a point because, for all intents and
purposes, the star is a point source.  If you observe the Moon, on the
other hand, light from that focuses only to a disc; this is the real
image of the Moon formed by the objective.  Again, your eyepiece
magnifies this real image, and when you look through the telescope, you
see a magnified image of the Moon.

If, rather than magnifying the real image with an eyepiece, you instead
capture it with camera film, you can then develop the film to get a big
picture of the Moon.  Because the image is recorded at the principal
focal point of the objective, this is called "prime focus" photography.
This is to contrast it with other kinds of astrophotography, such as the

    * Piggyback: The camera is mounted on top of a tracking telescope.
      This permits it to take a long-exposure, wide-angle photo without
      recording star trails (which is what you get if you simply take a
      long-exposure photo with a stationary camera).

    * Afocal: The camera peers directly into the telescope, just as the
      human eye would.  It is called afocal, because light exiting the
      eyepiece does not reach focus (in principle).

To be austere, prime focus is really the point toward which the
objective converges light from an object at infinity; if something, such
as another mirror, gets in the way, then the focus is not prime focus.
For example, in a Newtonian telescope, the light path is reflected by
the secondary mirror toward the eyepiece, which magnifies the image at
Newtonian focus; similarly, the focal point of a Schmidt-Cassegrain
telescope is called the Cassegrain focus; and there are other focus
points called the coude focus, or the Naismyth focus, or whatever.  All
of these distinctions are vital to a pedant (so it's useful to know of
them in case you find yourself in an argument), but generally speaking,
most reasonable people will know what you mean by "prime focus," if you
reference it to a specific telescope.

Q.  What does [insert nutty acronym] mean?

A.  There are a few acronyms and abbreviations that crop up persistently
on SAA.  I'll try to collect them here as I encounter them.

FL = Focal Length.  The distance between a mirror or lens and the point
at which it focuses parallel light rays to a point.  If the mirror or
lens diverges rather than converges light, the focal length is negative
and represents the distance between the mirror or lens and the point
from which the diverging rays appear to emanate.

FOV = Field Of View.  How much sky can be seen in the eyepiece (true
FOV), or alternatively, how big that view looks when magnified by the
eyepiece (apparent FOV); measured in degrees or fractions thereof.  See
also the question elsewhere in this FAQ.

FS = Field Stop.  The washer-shaped ring in the underside of an eyepiece
that determines the field of view.  Also the inside diameter (usually in
mm) of the field stop.

O-III = Doubly-ionized oxygen, with two extra electrons.  (O-I refers to
un-ionized oxygen.)  A common emission line for nebulae, it resides at
500.7 nm, in the blue-green section of the spectrum.

OTA = Optical Tube Assembly.  The optics, plus the tube that holds them
together.  Includes the objective and any auxiliary lenses, but not the

SAA = Sci.Astro.Amateur.  The Usenet newsgroup on amateur astronomy.

SCT = Schmidt-Cassegrain Telescope.  A compound telescope with a concave
primary mirror and a convex secondary mirror that puts the image near
the rear of the telescope (that's the Cassegrain part), and a complex
correcting lens at the front that allows a spherical primary to be used
rather than the more difficult-to-make paraboloid (that's the Schmidt

TINSFA = There Is No Substitute For Aperture.  Aperture determines how
much light is collected by the telescope, and somewhat less obviously,
it also is the primary determinant of the best resolution and contrast
the telescope can achieve.

Q.  What is the deal with [insert nut]?

A.  SAA (sci.astro.amateur) has its quota of nuts.  Those who have been
around the Sun a few times in the newsgroup know who they are (though
they might have trouble restraining themselves from responding), but for
those who don't, here's a quick recap.

Daniel Min is an astrology-obsessed would-be neo-con.  I say would-be,
because the neo-conservatives would be embarrassed to count him amongst
their number.  He is a rabid pro-Bush, fundamentalist xenophobe (don't
bother pointing out the inconsistency with astrology) who cheerfully
supports the extermination of people who don't think the way he does.
Daniel posts using anonymous Usenet servers, so that it is difficult to
killfile him; however, his posts are always pretty obvious to the eye.
He has an odd fondness for posting in (and translating) ecclesiastical
Latin, which is always good for a laugh for those of us who actually
read Latin.

Gerald Kelleher (aka oriel36) is a celestial mechanics nut who thinks
that astronomy should have stopped with Kepler, and never gone on to the
analytical universe of forces and accelerations put forth by Newton.  He
seems quite taken with the elegance and beauty of the music of the
ellipses that Kepler recorded.  Whenever a thread on celestial mechanics
(or indeed anything having to do with celestial motions) comes up, he
can be counted on to throw in his two cents worth, usually denigrating
any competent contributor who's moved beyond the 17th century.  He
wastes no time reading any rebuttals to his posts, so there's no point
wasting any time writing them.  He also starts a few threads of his own,
whenever it seems that interest in 16th-century celestial mechanics
might be flagging.

Ed Conrad bursts forth from time to time, ringing the changes on his
idea that MAN IS AS OLD AS COAL.  (In case you're wondering, most coal
deposits are an order of magnitude or two older than the oldest known
hominids.)  His posts are pretty obvious, inasmuch as he signs them with
his own name and his subject lines are generally in all caps.  He is
known for gratuitously slapping himself on the back for being right
(although that never happens to be the actual case).

Nancy Lieder, sometimes known as just Nancy, is an alien enthusiast who
insists that she channels advanced beings who hail from zeta Reticuli.
(Reticulum is a real constellation, by the way; it lies in the far south
of the celestial sphere, so that in point of fact, it can't be seen from
most of the United States, including where Nancy apparently lives.)  She
became extraordinarily active when the great comets Hyakutake and Hale
Bopp came around, and every now and then, she still promotes her notion
of Planet X, the twelfth planet.  (Whatever happened to ten and eleven
is anyone's guess.)  As of this writing (April 2006), it's been a while
since she's spoken up, but her name still strikes fear in the hearts of
many an old-timer.

Brad Guth belongs to the "NASA never landed on the Moon" camp, but with
the added wrinkle that he believes that Venus harbors life.  It's hard
to imagine a more inhospitable place for life, but never mind that.  It
turns out that Venus's cloudtops may well harbor life, and when that bit
of news came out, you can bet that Brad was there to trumpet his earlier
"prediction," despite the fact that much of the evidence came from the
same organization--NASA--that steadfastly maintains that it *did* land
men on the Moon.  Like many, he claims to have seen the light of
skepticism, but actually, it's the train's lamp of conspiracy.

Mick (aka Mike, aka MTA, aka TMA, aka ATM, aka a number of other
aliases) is a Canadian gadfly whose usual modus operandi is to take
others to task for being sloppy in their posts, despite the fact that he
consistently is sloppy in his.  Quite frequently he quotes his quarry's
entire post, following up with a single line to the effect of "What do
you know about this any way?" [sic]  Mostly harmless, although he does
manage to offend those who haven't been exposed to his peevish ways.
He does start some reasonable astronomy-related threads, usually by
posting a link with a provocative subject line.  Naturally, he has found
time to lambaste others for starting reasonable astronomy-related
threads by posting a link with a provocative subject line.

Q.  How do mirrors keep their shape?  I thought glass was a liquid.

A.  Glass isn't a liquid.  One fact often cited as evidence that glass
flows (and is hence a kind of super-viscous liquid) is that old church
windows are thicker at the bottom than they are the top.

However, the method used to form these windows was not perfect (I think
it was spin casting), and often resulted in uneven sheets of glass.
Now, if you were laying glass windows, and you had such an uneven sheet,
which end would *you* put on the bottom?  Surely not the narrow end.
No--in order to improve stability, you would put the thick end at the
bottom.  What's more, careful examination of these windows fails to
reveal telltale signs of downward flow, such as glass piling up around
and out of the window casing.

That explanation alone doesn't prove that no glass flows measurably in
hundreds of years, but the fact is that sensitive experiments *have*
been conducted on glass, and it doesn't flow fast enough to be
considered a liquid.  It can deform elastically (like stretching a
rubber band), and it can deform plastically (like bending a metal bar),
but only under continuously maintained stress.  Also, telescope glass
is stronger than ordinary window glass--particularly from the medieval
period.  Any telescope you buy will be safe over your lifetime as well
as those of your descendants for centuries to come.

The reason that glass can flow at all is because it is amorphous--it
doesn't have a regular crystalline lattice.  The various phases inside
the glass can therefore (*very* slowly) shift amongst one another and
produce creep.

But just because glass flows doesn't mean it's not a solid.  The alkali
metals--lithium, sodium, potassium, and so on--are very malleable and
can be said to flow, probably much more so than glass does.  Yet they
are all considered by chemists to be solid at room temperature, and
rightly so.  Glass has too many more similarities to solids than to
liquids: it can break just like a crystal; it can be melted, although
the melting point is really a range of temperatures; left to its own, it
doesn't collapse into a puddle; and so forth.  It could be considered
another state of matter, but *not* a liquid.

Q.  My mirror/lens is scratched--is this going to be a problem?

A.  As long as the glass isn't actually cracked, a single scratch is
not likely to be a problem.  A lot of scratches will eventually render
the optics unusable.

Your mirror or lens is designed to focus the light that falls on it
down to a point.  A scratch throws a monkey wrench into that process;
it throws the light all over the place, where it doesn't belong.  In
principle, then, a scratch can ruin the image and its contrast.
However, a scratch is essentially a one-dimensional aberration on a
two-dimensional surface.  Only a tiny percentage of the light falling
on the lens is affected--most of it still goes in the right place.

All bets are off when a bunch of scratches combine to cover a large
fraction of the optical surface, so don't smooth out your lens with
60-grade coarse sandpaper!  If you have any doubts about the glass, try
it out on the planets.  If you can't see at least some detail on
Jupiter, for example, you probably have a problem.

Q.  What's the difference between a mirror diagonal and a prism

A.  Both diagonals reflect light through an angle of 90 degrees to make
viewing more comfortable.  A mirror diagonal uses a single mirror
canted at 45 degrees to do the job, whereas a prism diagonal does it by
using the long side of a 45-90-45 prism.

                 ^                     ^
                 |                     |
                 | /                 +-|-/
                 |/                  | |/
    -------------/       --------------/
                /                    |/
               /                     /

     mirror diagonal       prism diagonal

One difference between them is that a mirror usually has a reflectivity
in the range of 90 percent, meaning that the remaining 10 percent is
lost.  A prism diagonal relies on total internal reflection.  As long
as the prism is made of a glass whose refraction index (a number,
greater than 1, that indicates how much light slows down in the glass)
is more than the square root of 2 (about 1.414), *all* the light that
strikes the long side of the prism will be reflected upward.  There is
some loss of light from having to go through all that glass in the
prism, but it's quite a bit smaller--perhaps 1 or 2 percent--so the
light transmission is still larger for the prism, by about 8 or 9
percent.  However, a difference of 8 or 9 percent only amounts to about
0.1 magnitudes, so in general, that difference isn't terribly

Another difference has to do with how the light enters the diagonal.
In the diagrams above, I've shown the light coming in straight from the
left, and reflecting straight upward.  However, not all light comes in
that way.  Only a bit of the light coming from the center of the field
of view comes in that way.  The rest of the light comes in mostly from
the left, but at a slight angle.  With a mirror diagonal, light coming
in from the left, plus a slight angle, is simply reflected upward, plus
the same slight angle.  The slight angles have no effect on a mirror

With a prism diagonal, though, it's slightly different.  The glass,
with its refraction index of around 1.5, slows down the light to about
2/3 of its original speed.  This also has the effect of bending the
light slightly, so that light coming in from the left side, plus a
slight angle, is instantly changed, at the first glass surface, to
light coming in from the left, plus a somewhat smaller angle.  The long
side of the prism then reflects this light upward, plus the same
somewhat smaller angle.

Fortunately, when this light reaches the upper surface of the prism,
it speeds up again and bends back outward, so that it becomes light
going upward, plus the original slight angle.  In other words, the two
errors induced by entering and exiting the glass compensate for one
another.  In passing through the glass, it is shifted somewhat closer
to the center than it should be, but that only means that you need to
refocus a little.  Again, the amount of refocusing is so small as to be
unnoticeable by just the eye.

There is one more effect of going through glass, though.  It turns out
that the refraction index for the glass is not constant for all colors.
Instead, there is a small dependence on color--that is, frequency.  All
other things being equal, violet light is bent more than red light.
Along the center line, this makes no difference since the light goes
straight in and is not bent, but anywhere the light is entering the
glass at an angle, the component colors of the light are separated by
small angles.  These angles persist throughout the glass, are maintained
by the total internal reflection, and are *not* compensated for when the
light exits the upper surface of the prism.  In other words, the prism
induces its own color error, aside from any color error that might be
present in the rest of the telescope.

Is this significant?  If the angles of the incident light are small, as
they are in slow telescopes (i.e., those with long focal ratios, in
excess of f/8 or so), then the separation of colors is small and
probably unnoticeable.  On the other hand, in very fast telescopes
(those with short focal ratios, less than f/5 or so), there is a much
greater separation of colors, and the effect may be noticed.  In many
cases, though, these telescopes are short refractors, which may exhibit
color errors of their own, and the one error may be mistaken for the

In any event, with slow telescopes, you can generally ignore the color
effect and concentrate on price and optical quality (that is, whether
the glass is smooth--a factor that affects both types).  On telescopes
of moderate focal ratios, between f/5 and f/8, the effect is small, but
perhaps noticeable to you; however, it shouldn't be an overriding

Q.  What does it mean to offset a secondary mirror, and why would I want
to do that?

A.  Here's a quick explanation of offset.

What goes into the telescope, toward the primary mirror, is a cylinder
of light.  Actually, it's not quite a cylinder of light--that would be
just the rays that enter parallel to the axis of the telescope.  These
are called on-axis rays.  If you include all of the off-axis rays, you
you get a kind of wobbly cylinder, and bouncing off the primary, you
get a wobbly cone.  But it doesn't affect the geometric explanation of

The primary bounces this light back toward the front of the telescope,
and also focuses the cylinder toward a point, so that the light coming
back from the mirror looks like a cone--in fact, it's often called the
"cone of light."  The secondary should be placed in order to intercept
the whole cone.  However, this secondary also blocks light going to the
primary, so you don't want to make it any bigger than you need to.  So,
the secondary should be exactly the same size and shape as a
cross-section of the cone of light, but not just any cross-section--it
has to be set at a 45-degree angle, so it can reflect the light up to
your eyepiece.

If you cut a cylinder at a 90-degree angle, you get a circular
cross-section, with the center of the circle along the axis of the
cylinder.  If you instead cut the cylinder at a 45-degree angle, you
get an ellipse--a stretched-out circle, whose center is also along the
axis of the cylinder.  Now, if you cut a *cone* at a 45-degree angle,
you also get an ellipse, but now its center is *not* along the axis of
the cone.  From the point of view of an observer at the eyepiece, the
center is *offset* a small distance away from the cone's axis.  In
order to maximize your coverage of the light cone, you need to offset
the secondary by this same amount.  It will look approximately centered
if you look straight through the focuser, but it won't look centered if
you look down the OTA toward the primary.

Q.  How the heck is my image oriented?

A.  If you're using a pair of binoculars, it's right-side-up and not
mirror-reversed.  If you're using a refractor (or Schmidt-Cassegrain,
or some other telescope used straight through), without a star diagonal,
the image is upside-down and not mirror-reversed.

After that, it gets a little complicated; it's not as simple as many
introductory astronomy books make it out.  The way they talk, you'd
guess that images can be either right-side-up and non-reversed (like
binoculars), upside-down and non-reversed (like refractors without star
diagonals), or right-side-up and reversed (like refractors with star
diagonals).  In truth, there are other possibilities.

It's impossible to give an ironclad rule, because in most cases, there
are too many variables.  First of all, what's right side up in the sky?
Generally speaking, up is toward the zenith, and down is toward the
horizon.  But what's up in the eyepiece?  That depends on how you're
looking at the eyepiece.  On a Newtonian, for example, how do you orient
your head when you look through the telescope?

Then, too, there are many other factors outside the observer.  Is the
telescope equatorial or alt-azimuth?  If it has a diagonal, how is it
inserted into the focuser--is it sideways, up and down, or somewhere in
between?  If the telescope is a Newtonian, is the focuser on top, on the
side, on the bottom, where?  All these can affect the image orientation.

About the only further thing that can be said is that if the telescope
is a Newtonian, however mounted, the image is non-reversed, and possibly
rotated by an arbitrary amount.  If it's a refractor or other
straight-through telescope, plus a star diagonal, the image is reversed,
and possibly rotated by an arbitrary amount.  That and the first
paragraph should cover most of the cases.

Actually, one of the most disorienting things about the image presented
in telescopes is not just the way they're oriented, it's the way the
image "moves" as you move the telescope.  You may find stars moving the
opposite direction to the way you moved the telescope (as happens in a
pair of binoculars), or they might move the *same* way, or they might
move perpendicular to the way you moved the telescope.  It can be quite
startling the first time you encounter it, and it takes a while to get
used to it.  Some people, in fact, never get used to it, and it's one
of the reasons they have a hard time star-hopping.

Q.  How much magnification can I get out of my telescope?

A.  The short answer is that small telescopes are limited by their
size--they can get perhaps 50x per inch of aperture, or 2x per mm of
aperture.  Large telescopes are limited by atmospheric turbulence, which
typically (but not always) limits useful magnification to around 200x to
400x, something in that vicinity.

It comes as a surprise to many beginners that high magnification is not
the most important property of a telescope.  The most important is that
it has a large aperture, or opening, which allows it to obtain both more
light and more detail.  However, high magnification is needed for you to
be able to see that detail, so let's examine the magnification issue a
bit more closely.

As I described in the question on diffraction-limited telescopes, each
telescope produces an image whose detail is limited by the diffractive
nature of light (hence, "diffraction-limited").  That effect can be more
effectively countered the more area you gather light across, so larger
telescopes are less affected by diffraction than smaller ones.

To be more precise, a telescope twice as wide as another will create an
image that's smeared around by diffraction just half as much.  It will
therefore stand to twice as much magnification before the diffractive
smearing becomes objectionable.  Through trial and error, observers have
found that the maximum comes into play at around 50x per inch.  You can
see that this rule does give a maximum magnification that is twice as
high for a telescope that's twice as large.

There's nothing theoretical about this rule, by the way (except for the
fact that we know it comes about because of diffraction); it's a purely
empirical rule.  So the fact that you might be able to push it beyond
50x per inch doesn't mean that you or your telescope has somehow defied
the laws of nature; it just means that your eye doesn't see the effects
of diffraction until the higher magnifications.  In fact, the more acute
your eye, the *lower* the maximum magnification--not higher.

There's a limit to this progression, however, caused by atmospheric
turbulence, which causes the effect we call "seeing" (which see).  The
negative effect of seeing does not depend very strongly on seeing
(although its qualitative description does), so it effectively caps the
maximum magnification somewhere in the few hundreds.  Turbulence is
dynamic, so the longer you're willing to wait, the more likely you'll
catch a moment of still air, and the higher your maximum magnification.

Q.  How do I figure out the magnification when I take an astrophoto?

A.  Astrophotos don't have magnifications, because you don't use an
eyepiece.  They do have what is called the image scale, and you can
figure that out from just the focal length of the objective.

To explain that, let's look closely (ahem) at what magnification really
means.  The object in the sky has a certain angular size (which see).
When you look at it through the eyepiece, it has another, bigger angular
size.  The ratio between those angular sizes is the magnification, and
it has no units; it's just a number.

The Moon, for instance, has an angular size of about half a degree.  If
you see an image that's 50 degrees across when you look in the eyepiece,
that means the magnification is 50 degrees, divided by half a degree, or
100x.  The x is not a unit; it just means "times," as in "100 times

What you're looking at, when you observe through the eyepiece, is the
real image formed by the objective.  That real image has a specific
linear size; it's just floating in mid-air at the focal plane of the
objective.  You look at it through the eyepiece--basically a very well-
corrected magnifying glass--just as though it were a real object inside
the telescope.

When you take an astrophoto, on the other hand, you don't use an
eyepiece.  You record that real image directly onto film, or using a
CCD, or whatever.  Your image or photo retains that linear size, but no
angular size; if you measure the image size, you get an answer in mm,
not degrees.  Therefore, the division you used for magnification when
you observed through the eyepiece no longer returns a unitless number;
now you get mm divided by degrees, or mm/degrees.  That's called the
image scale.

If you image the Moon, for instance, with an objective whose focal
length is 1000 mm, you'll get an image about 9 mm across.  That means
the image scale is 9 mm, divided by half a degree, or 18 mm/degree.  You
can use the image scale to figure out how big any object will be in the
image, by multiplying by the image scale.  If you image the Andromeda
Galaxy, which has a length of, say, 4 degrees, it'll cover 18 mm/degree,
times 4 degrees, or 72 mm--too big for a 35 mm camera, but it'll fit in
a medium-format frame.

The formula for magnification is, as you may already know, focal length
of objective divided by focal length of eyepiece.  For image scale, it's
focal length of objective divided by 57.3 degrees.  That weird number
is the number of degrees in a radian, and it's equal to 180 divided by
pi.  So another way to figure image scale is to multiply the focal
length by pi (about 3.14), and divide by 180.  I'm sure some of you know
pi to more digits, but you won't be able to use that kind of precision
(and you won't need it anyway).

Q.  What can I expect to see when I use my telescope for the first

A.  That depends a lot on what you're looking at, what you're looking
through, and where you're looking from.  I'll take care of the last
factor by assuming that you're observing from a typical suburban
setting.  That's not true for everybody, of course, but it should give
you an idea of what's possible.  The light pollution doesn't greatly
affect the views of the planets (aside from Pluto)--only galaxies,
nebulae, and the like.  I'll also assume that you're using a small or
medium-sized telescope, 8 inches or smaller.

First of all, the Moon will reveal more detail than you can possibly
keep track of, in any telescope.  By the time you have exhausted the
detail visible on the Moon through your telescope, you will no longer
need to read this Q&A (or perhaps the rest of this FAQ, either).

Secondly, you will not be able to see all the detail all at once.  The
atmosphere and your personal perception conspire to require patience on
your part to make the most out of your observing session.  Count on
five to ten minutes *minimum* (and probably longer) to see even a tenth
of what your telescope will eventually be able to reveal on any specific

Thirdly, different targets require different magnifications.  Details on
the planets typically require higher power--start at perhaps 25x per
inch of aperture, and use what the atmosphere will give you.  Deep sky
objects *generally* require lower power, and the larger the object, the
lower the power, but there are exceptions.  Try starting out at 10x per
inch of aperture, and try to frame the object well in the field of view.

If you're using a small telescope (4 inches or smaller), you need only
a negligible amount of time for the telescope to cool down.  You should
be able to see at least two belts on Jupiter, its four big satellites
(unless these are obscured by Jupiter or its shadow) and the rings
around Saturn.  Toward the high end, you should also be able to see at
least a third belt on Jupiter (the NTB, or north temperate belt), the 
Cassini division in Saturn's rings (especially in the early years of the
21st century), and some detail on Mars.

You should be able to see some of the brighter galaxies: some targets to
try are M31 in Andromeda and M81/M82 in Ursa Major.  Some nebula are
also within your range: M42, the Great Orion Nebula, is of course the
first to try, but also M8, a large bright nebula in Sagittarius.  There
are also the planetary nebulae, the dying breaths of stars: M27, the
Dumbbell Nebula in Vulpecula, and M57, the Ring Nebula in Lyra, are the
two easiest, especially if you have a nebula filter.

Better bets are the open and globular clusters.  The Messier catalogue
contains almost all of the brighter globulars, and plenty are within
your reach: M2, M3, M5, M13, M22, and M53 are good starters for the
globulars.  For the open clusters, M7, M41, M44, and M45 (the Pleiades)
are easily seen in the smallest of telescopes, although M44 in the heart
of dim Cancer may be hard to find.

If you're using a medium-sized telescope (4 to 8 inches), you may need
15 minutes to an hour for the telescope to cool down properly, depending
on the temperature differential between storage and use.  You should be
able to see at least three belts on Jupiter, the Great Red Spot (or at
least the hollow in the south equatorial belt where it resides), details
inside the belts, and some incipient granularity in the polar regions.
Saturn should show color differences in its subtle banding, the Cassini
division, and in the higher end, the Encke minimum, the slight dimming
in the outer or A ring.  Mars should show a moderate to substantial
amount of detail: dark areas such as Syrtis Major, the polar caps, the
twilight clouds, and perhaps the larger storms.

Under suburban skies, a medium-sized telescope should be able to show
you most of the Messier objects, although they might be difficult,
especially at the small end of the range.  At the high end, you may be
able to make out the beginnings of detail in the galaxies, especially
with M82 and M51.  Many of the nebulae are now visible: try M20, if you
have a larger telescope.  Most planetary nebulae can be seen with a
filter, although most of them are nearly stellar; some good exceptions
are the four Messier planetaries, as well as NGC 7009, the Saturn

Somewhere between half and all of the globulars in the Messier catalogue
can be seen, and also most if not all of the open clusters should be
visible.  With a larger telescope, you may be able to make out some
nice combinations, such as M35/NGC 2158, a pairing of two open clusters,
similar in size, but with NGC 2158 several times further (and therefore
considerably dimmer, too).

If you can't see everything you're "supposed" to, don't despair.  It may
simply be that conditions aren't right, or that your skies aren't dark
enough.  If you can, have someone experienced test out your telescope.
And if you can see *more* than you're supposed to, wonderful!

Q.  Did I just see Jupiter's satellites in my binoculars?

A.  Very likely, you did.  People are often surprised that a lowly pair
of 7x35 or 10x50 binoculars can make out Jupiter's satellites.  They
usually expect the satellites to be either too dim, or too close to

One thing they are not, for certain, is too dim.  The four big
satellites, from innermost to outermost, are Io, Europa, Ganymede, and
Callisto.  Each is no dimmer than magnitude 5.5, and therefore quite
possible to see, even by the unaided eye, from reasonably dark skies.
There are even reports of some people who *have* seen them with the
unaided eye.

Unfortunately, they are usually too close to Jupiter to see that way,
most of the time.  Callisto, the outermost, is still never more than
about 26.5 arcminutes away from Jupiter--less than the width of the
full Moon.  And usually, the separation is somewhat less.  With Jupiter
ordinarily glowing at a respectable magnitude -2 or brighter, the
planet's light typically drowns out the satellites.

But since you can *just about* see the satellites with the unaided eye,
it stands to reason that it should be almost easy with a pair of
binoculars.  Unless a Jovian satellite is either hidden or in front of
Jupiter, or very nearly so, you should be able to see it.  The only
trick is to hold the binoculars steady, but that can be managed by
bracing them against a car or fence or other sturdy object.

Q.  Did I see a fifth (sixth, seventh) satellite of Jupiter in my

A.  Chances are, if you have to ask, you didn't.  The four Galilean
satellites are easy to see (see preceding question), but after that,
there's a long gap to the next brightest satellite, Amalthea.  It was
discovered visually--the last satellite of any planet to be so
discovered--but it is very dim.  At magnitude 14.1, it is dimmer than
Pluto is currently (in 2008), and since Amalthea is always found close
to Jupiter, it is exceedingly difficult to see.  It took E.E. Barnard,
possibly the most talented visual observer who ever lived, a 36-inch
telescope to discover it in 1892.  You won't see it by accident.

It is common to see what appears to be a fifth satellite of Jupiter,
even one that appears to be in line with the other, big four satellites.
These are invariably background field stars that are aligned with the
satellites by chance.

Q.  What are these words "preceding" and "following" (or "leading" and
    "trailing") and what do they have to do with directions?

A.  They indicate celestial west and east, respectively.

The next question is, why not use west and east?  But there's a problem.
The first time you were introduced to celestial east and west--on a star
atlas, say--you may have noticed something unusual about the order of
the directions.  Specifically, moving clockwise from north, you read
north, *west*, south, *east*.  This is mirror-reversed from terrestrial
maps, where the order is north, east, south, west.

The reason for this is that when you're looking at a terrestrial map,
you're looking at the Earth from above it--that is, outside it.  But
when you look at a star atlas, you're looking at the imaginary celestial
sphere from within it.  (In fact, since the celestial sphere *is*
imaginary, you can't possibly look at it from outside, anyway.)  So
naturally the directions are reversed: if you could look at the Earth
from within, the directions would be reversed, too.

What has this all got to do with those weird terms?  When you look at
the planet Jupiter, for example, if north is up, then celestial west is
to your right, and celestial east to your left.  But we also concern
ourselves with features on Jupiter, meaning that we use a system of
Jupiter longitudes.  (Actually, there are three systems, but never mind
that for now.)  Following a convention that a planet rotates west to
east, Jupiter west is to the *left* and Jupiter east is to the *right*.
That makes "east" and "west" ambiguous.

How about using "left" and "right"?  Alas, those too are ambiguous, for
they depend on what instrument you're using.  Depending on whether your
telescope or binoculars invert, reverse, or rotate the image, "left"
could be one way or the other.

For these reasons, the unambiguous terms "preceding" and "following"
are used.  They are interpreted as follows: Observe the planet through
the telescope (or binoculars), with any motor drive turned off.  The
planet will appear to drift because of the rotation of the Earth.  The
edge of the planet that disappears first is the preceding edge.  The
edge that disappears last is the following edge.

This interpretation also has the advantage that features on Jupiter and
Mars, for example, move from following edge to preceding edge, meaning
that the preceding features, too, disappear first, and the following
features disappear last.

Personally, I dislike the terms "preceding" and "following," and use
"leading" and "trailing" instead, in my own reports.  But following a
convention is a good thing in certain circumstances, and if I were to
file a report with ALPO, I would be best off using "preceding" and
"following," like everyone else.

Q.  Is it true that looking at the Moon through a telescope will harm
your eyes?

A.  You cannot harm your eyes by looking at the Moon through a
telescope.  It may be uncomfortably bright, and you may can improve the
visibility of detail by either adding a neutral density filter (a gray
screw-on filter) to the eyepiece, or by increasing the magnification.
But there is no safety risk.

You may wonder how this can be, since the telescope gathers so much
more light than your eye.  However, it also magnifies the Moon, so that
the extra light is spread out over a greater area.  Each part of the
Moon's image is seen by just one portion of your eye, and as far as
damage is concerned, the critical factor is the intensity of light
falling, per individual portion of your eye.  If your eye's pupil is
5 mm across, and your telescope is 100 mm across, then the telescope
gathers 20 squared, or 400 times more light than your eye alone.  But
if you're using a magnification of 20x or greater, then that light is
spread out over an image at least 400 times larger, so that the actual
brightness seen by any portion of your eye is no greater, and usually
less, than when you observe the Moon with the unaided eye.

What if you observe the Moon at less than 20x--say, 10x?  Shouldn't the
light be spread out over a smaller area, and thus more concentrated?  At
10x, the 400 times more light is spread out over an image that is only
100 times larger, so it seems as though each part of the image should be
4 times as bright as when seen by the unaided eye.

However, consider that each portion of the Moon can be thought of as
pouring down light, out of which only a shaft 100 mm across--as wide as
your telescope--actually enters the optics.  In the process of
magnification, that shaft is reduced to fit into your eye's pupil, and
the factor of reduction is equal to the magnification.  In other words,
if you magnify by only 10x, the 100 mm shaft of light is shrunk down to
10 mm.  The result is that only part of the light--a smaller shaft that
is 5 mm across--as big as your eye's pupil--actually gets in.  The rest
of it falls uselessly (at least as far as image brightness is concerned)
on the surface of your eyeball.  Since a circle 5 mm across has 1/4 the
area of a circle 10 mm across, only 1/4 of the light gets into your eye,
and this precisely compensates for the extra intensity from lowering the

Of course, it *feels* as though the Moon is about to blind us, for two
reasons.  One is that we typically observe the Moon by night.  The same
phase by day is just as bright, but it doesn't feel blindingly bright
through the telescope because our eyes are then accustomed to daytime
light levels.  Another reason is that the Moon *is* magnified by the
telescope, and at the same intensity throws more total light onto your
retina.  By way of an analogy, if I shine a flashlight into your eye at
a distance of 10 centimeters (4 inches), it's uncomfortably bright,
whereas if I put a mask on the flashlight that only lets through a tiny
spot of light, it's merely annoying.  The total light output is much
smaller, but the intensity of that tiny spot is just as great as before.

Incidentally, some people may ask, why then is observing the Sun through
a telescope so dangerous?  After all, although we don't stare at the Sun
(at least, we shouldn't), its light still comes through our eye.  If
looking at the Moon through a telescope is no more dangerous than
looking at it without the telescope, why isn't the same true for the

The answer is that the Sun is so bright that each portion of its image
is enough to create some heating in the eye.  (So does the Moon, but its
light is about 400,000 times less intense and the heating is completely
negligible.)  If any given part of your eye is subjected to that heating
for long enough, permanent damage will result.  Your eyes avoid this by
moving around, so that the image of the Sun doesn't stay in place, and
the part of your eye that is getting heated by the Sun one moment has a
chance to cool down the next.

However, if you were to be so foolish as to observe the Sun through a
telescope, each portion of your eye gets heated the same amount, but now
moving the eye doesn't help, since it is still likely to be heated by
the Sun.  Moreover, with a small image of the Sun (as when seeing it
with the unaided eye), the fraction of your eye being heated is small,
and it can dissipate heat rather easily to slow down the damage.  With a
magnified image, the fraction of your eye being heated is much larger,
and there is now nowhere for the heat to go.  You can as a result burn
out your retina with startling and tragic speed.

without proper safety precautions, such as an appropriate filter.  Do
not use solar filters that screw onto the eyepiece.  The focused heat at
the eyepiece is too intense and will crack the filter, sending all that
concentrated light and heat into your eye.  The light must be filtered
before entering the telescope.  (Exception: A Herschel wedge can be
safely used.  If you don't know what a Herschel wedge is, though, don't
guess--just use a proper solar filter.)

Q.  What interesting astronomy-related science projects can my child do?

A.  It depends on how old your child is (astronomy requires a decent
attention span), and how much equipment you have.  I'll try to suggest
a few that require only modest equipment (or none at all).  Note that
the background given below is only a summary.  The student is still
responsible for gathering the primary sources (such as articles in an
encyclopedia set or astronomy magazines).

1.  Topic: The eclipsing binary, Algol (beta Persei).

    Background: Eclipsing binaries are pairs of stars that orbit each
    other (like a basketball official's fists calling travelling), in
    such a way that from time to time, one of the stars blocks the
    other.  Ordinarily, we see the light from both stars, blended
    together, and the brightness of the pair is the brightnesses of
    the two stars added together.

    Occasionally, though, the dimmer star blocks the brighter star.
    (This seems weird, but what's happening is that the dim star is
    dim and *large*, whereas the brighter star is bright but small.)
    We then see only the dimmer star, or perhaps the dimmer star plus
    a fraction of the brighter star, and the combined brightness
    diminishes.  You may also, more rarely, see the brighter star
    block the dimmer star.  However, this is harder to detect, because
    the reduction in brightness is smaller.

    Eclipsing binaries seem like variables.  However, they are
    distinguished from true variables (single stars that actually get
    brighter and dimmer) by two things.  First, eclipsing binaries are
    perfectly regular.  Only *some* true variables, such as Cepheids,
    are regular--others, such as Mira-type variables, are only
    semi-regular; that is, they go up and down, but sometimes there
    may be 300 days between peaks, and then other times there are 350

    Secondly, an eclipsing binary usually stays at a constant
    brightness, which is the combined brightnesses of the two stars.
    Only when one star is blocked does the pair dim.  In contrast,
    true variables (other than recurrent novae, which are stars that
    burp hydrogen fusion every now and then) typically ramp up and
    down in brightness.

    Task: Your job here is to observe the brightness of Algol, by
    comparing it with other stars in the night sky, to see if it's an
    eclipsing binary or not.  (I'll give the parent a hint: It is,
    and it has a period of about 2.87 days.)  Only a pair of binoculars
    is required--I find it difficult personally to compare brightnesses
    without them, although you may find differently.

2.  Topic: The height of the Moon as the phases change.

    Background: You may have noticed that the Sun is high in the sky at
    noon in summer, and low in the sky at noon in winter.  That is
    because the *declination* of the Sun changes throughout the year.
    If you imagine a plane that cuts through the Earth at the equator,
    and extends in all directions to infinity, that plane is called the
    *celestial equator*.  The declination of an object such as the Sun
    is the angle of the object above or below the celestial equator, as
    viewed from the Earth.

    The declination of the Sun is around +23.4 degrees on or about June
    21 of each year, and around -23.4 degrees on or about December 21.
    At the equinoxes, on or about March 21 and September 21, the Sun's
    declination is 0 degrees.  This might give you the impression that
    the Sun is bobbing up and down like a buoy as the year progresses,
    but that's not the case.  If you were to imagine a second plane that
    cuts the Earth at an angle of 23.4 degrees to the equator, that cut
    would reach a maximum latitude of 23.4 degrees north, a minimum
    latitude of 23.4 degrees south, and it would intersect the equator
    at two points in between, but it would be as straight a cut as you
    could make on the spherical Earth.  It certainly would not bob up
    and down.

    This second plane, which the Sun appears to follow throughout the
    year is called the ecliptic.  Along the ecliptic in the sky are the
    constellations of the zodiac.  As you might expect, however, it's
    hard to tell that the Sun goes through the zodiac constellations,
    since when the Sun is up, you can't see the stars, and vice versa.
    All the same, ancient astronomers managed, by keeping careful track
    of where the various constellations were just before dawn and just
    after dusk.

    The Moon, as it turns out, also follows an orbit that, although not
    *quite* on the ecliptic, is close enough for our purposes.  (The
    difference is about 5 degrees.)  The advantage of observing the
    Moon is that unlike the Sun, it is often up at night and one has an
    easier time of it tracking exactly where in the sky it is, relative
    to the stars.  (One Persian story has it that a wise old vizier was
    asked which was more important, the Sun or the Moon.  "The Moon,"
    he said, "because the Sun shines during the day, when it's light out

    For that reason, each month, around the time of the new Moon, when
    the Moon is between the Earth and the Sun, the declination of the
    Moon is very close to that of the Sun.  If the Sun has a positive
    declination--that is, if it's directly overhead somewhere in the
    northern hemisphere--then the Moon, very likely, has a positive
    declination, too, and within 5 degrees of the Sun's declination.
    And if the Sun's declination is negative, then the Moon's is, too,
    probably.  However, it may be difficult to tell, if the Moon is so
    close to the Sun.  (See the preceding question, about observing the
    Moon through the telescope.)

    It's different at full Moon.  Then the Moon is on the opposite side
    of the Earth from the Sun, and it's very easy to see.  This also
    means that if the Sun's declination is positive--again, remember that
    this means that it's directly overhead somewhere in the northern
    hemisphere--then the Moon is probably directly overhead on the
    opposite side of the Earth, in the *southern* hemisphere.  That
    means that it has a *negative* declination.  And vice versa: If the
    Sun's declination is negative, the Moon's is probably positive.

    In short, the path that the Sun follows throughout the *year*, the
    Moon follows throughout the *month*.  (Therefore, for example, at
    first quarter, a quarter of a month after the new Moon, the Moon is
    roughly where the Sun will be in a quarter of a year.)

    Task: Follow the Moon for a month, whenever you can.  This should
    be easy during the first part of the cycle after the new Moon, when
    the Moon is up during the early evening.  As the month progresses,
    the Moon rises later and later, until just before the next new Moon,
    it rises only just before dawn.  At that point, you'll have to get
    up early to see the Moon, in order to see it against the stars.
    (You can see it during the day, too, but then you can't see the
    stars.)  Plot the position of the Moon on a star atlas (or on a
    copy), and see how the declination changes throughout the month.
    What months do you expect the first quarter Moon to be high in the
    sky?  What about the last quarter Moon?

I'll add more projects here as I think of them.

Q.  Is Pluto still a planet?

A.  Not anymore.  To be more precise, as of August 24, 2006, Pluto is no
longer considered a major planet by the International Astronomical Union
(IAU).  However, this classification is still in flux, so the answer to
this question may change in the near future.

Pluto was discovered by Clyde Tombaugh in 1930, but people first began
searching for it in the 19th century after it was noted that Uranus and
Neptune weren't behaving quite as they ought to.  Instead, they seemed
to be under the gravitational influence of some unseen body further out
than Neptune.  Based on the perturbations seen in the orbits of these
two planets, the ninth planet was estimated to be smaller than either
of them, but still significantly more massive than the Earth.  (Both
Uranus and Neptune are about 16 times as massive as the Earth.)

Soon after Pluto was discovered, it was found to be smaller than
expected.  In fact, every new measurement of Pluto seemed smaller than
the last!  At first, it was thought to be about the size of the Earth,
about 13,000 kilometers (8,000 miles) across.  By 1960, based on visual
appearance and assumptions of actual brightness, its estimated size had
dropped to about 6,000 kilometers (3,700 miles).

The final straw came in 1978, when Pluto's satellite, Charon, was
discovered.  By observing the motion of a planet's satellite, scientists
can determine the mass of the planet.  Pluto turned out to be a whopping
500 times *less* massive than the Earth.  It is only 1/6 as massive as
our own Moon, in fact.  The current best estimate of Pluto's size is
about 2,300 kilometers (1,400 miles) in diameter.  More recently, the
observations of Uranus and Neptune have been re-examined in light of
better mass figures for those two planets, and with those corrections,
the discrepancies have been completely explained.  No further planet is

Back to Pluto.  As a result of its diminutive size--Mercury, the next
smallest planet, is still about 4,900 kilometers (3,000 miles) in
diameter--and its unlikely location out beyond the gas giants, many
astronomers proposed changing Pluto's status.  The group of planets
includes not only the familiar nine from Mercury to Pluto, but also
asteroids and comets.  These are divided into major planets and minor
planets, ostensibly based on size, but also based on tradition.

What some people were proposing to do was to declare Pluto a minor
planet.  This proposal became stronger as it became evident that there
were a bunch of similarly composed objects beyond Neptune, at around
Pluto's distance from the Sun.  None of them was as big as Pluto, but
it might just be a matter of time until another body that large was
discovered.  One version of the proposal suggested that Pluto be
labelled minor planet number 10,000 (it would retain its proper name,
too, of course, just like the asteroids).

The matter never got to a vote.  Once word of the proposal got out,
there was such a tremendous backlash that the IAU promptly sent out a
statement that Pluto's status was unchanged.  Apparently, there is still
a sense of pride that someone in the 20th century should have discovered
a planet.  The minor planet number 10,000 was assigned to a different
body altogether.

However, the matter was re-engaged in 2005 after a larger body *was*
discovered, with the unpreposessing name of 2003 UB313.  Was it to be
the tenth planet?  Or would Pluto be removed from the list?  What was a
planet, anyway--was it something of sufficient size, or did it depend on
how it interacted with its environment?  After all, the other planets do
not revolve around the Sun along with many companions, the way Pluto
does.  Or, perhaps, was "planet" really a historical term, only to be
applied in an informal setting?

On August 24, 2006, the IAU formally defined a planet to be an object in
orbit around a star, of sufficient size that it was spherical, more or
less, and so that it dominated its orbital environment.  Pluto qualified
on the first two counts, but because of all the other objects that
revolve around the Sun in much the same environment as Pluto, some of
which were almost as large as Pluto, it was "demoted" to the status of
dwarf planet, which are objects that, like Pluto, are spherical and
orbit a star, but that do not dominate their environment.  The retinue
of major planets therefore returns to its condition before 1930.

However, there is some dissatisfaction with the hasty manner in which
the classification was arrived at.  There seems to be some call for
revoking the re-classification, so the saga of Pluto the planet is not
yet done.  Stay tuned!

Q.  What happened before the Big Bang?

A.  Nobody knows what happened before the Big Bang.  It is even likely
that the question doesn't make sense (at least, not as it is usually

The usual answer is that the Big Bang is not an explosion in space, at
some epoch in time, but an explosion *of* space *and* time.  That is to
say, the Big Bang created both space and time, with the galaxies and
other matter in the universe being largely by-products.  For whatever
reason, most people find it easier to accept that space was created in
the Big Bang than that time was created in the Big Bang, too.

Part of the problem is that it's simply not intuitive that there could
have been a time without time, as it were.  It's revealing that we don't
ask what was in front of the Big Bang, or to the left of it, but we do
ask what came before it.  Part of that must be our curiosity about the
ultimate cause of things (something that science really can't answer),
but part of it is also that we don't view time as being interchangeable
with the other three, spatial dimensions.

Einstein's general theory of relativity, however, does treat time as
being very closely related with the three spatial dimensions.  In a way
that only makes sense mathematically, the time dimension is sort of
like a spatial dimension times i (the square root of -1).  As counter-
intuitive as this seems, it gives physicists some confidence that as
the Big Bang created space, it also created time.

One way to approach this might be to think of the Big Bang as sort of
the opposite of a black hole--the ultimate "white hole," so to speak.
In everyday terms, once something goes inside an imaginary boundary of
a black hole called the event horizon, it can never leave.  Physicists
would say that within the event horizon, all worldlines end at the
singularity at the center of the black hole; everything's future, both
in time and in space, ends there.  In short, the singularity lies in the
infinite future.

The curious thing that general relativity tells us about a black hole is
that from the perspective of someone outside the black hole, space and
time are in some sense inverted; stretches of time become expanses of
space, and vice versa.  As we approach the event horizon from the
outside, a faraway observer will see us less and less falling forward in
space, and more and more falling forward in *time*, until they see us
frozen in place right at the event horizon.

From our own perspective, though, we see space and time behaving as they
usually do, as we go on falling until we hit the singularity at some
definite point in time.  (It's a little like travelling from the north
pole to the equator.  From the point of view of an observer at the north
pole, you go from walking forward to walking "downward," but from your
point of view, you're always walking forward.)

The Big Bang would be like that, but with everything reversed in time;
instead of everything's worldline ending at the singularity that is the
Big Bang, they all *start* there.  From our own perspective, perhaps, we
see space and time behaving as they usually do, so that (for instance)
we perceive the Big Bang as having taken place at some definite point in
time--about 14 billion years ago--but for an imaginary observer outside
our universe, perhaps they see space and time reversed, so that for
them, our Big Bang singularity lies in the infinite past.

In any case, as successful as it may be, we know that general relativity
has striking limitations, conditions under which it *must* break down,
and those conditions come into play, unfortunately, at the moment of the
Big Bang.  At the start of the universe, so much energy was compressed
into so small a volume of space that classical laws like general
relativity can no longer be valid, and quantum mechanics must be brought
to bear.  (In physics, the term "classical" simply means non-quantum.
Thus, even a difficult theory such as general relativity qualifies as

However, physicists implicitly assume that there was no time at which
quantum mechanics stopped governing and general relativity took over.
This isn't as big a leap of faith as it might seem.  It's akin to
assuming that there was no point in time at which you suddenly became an
entirely different person; rather, you are and have always been a single
person, with one set of behaviors when you were an infant, and another
set of behaviors when you became an adult, and a more or less smooth
transition between the two (however bumpy it might have seemed at the

In the same way, there must be a single theory of gravity, which looks
like general relativity in the normal conditions that apply almost
everywhere in the universe today, and yet in the high-energy,
small-scale limit that applied for the Big Bang, is a wholly quantum-
mechanical theory.  Similar theories have been created for the other
forces in nature (electromagnetism, and the two nuclear forces), but
gravity has thus far defied "unification."  Ultimately, the synthesis of
a quantum theory of gravity will be a watershed moment in physics, and
may well permit us to plumb the universe's history, right to the moment
of its creation at the Big Bang.


Copyright (c) 2001-2008 Brian Tung