John Pazmino
 NYSkies Astronomy Inc
 2001 August 1
    To follow a binary star's motions over the years you need a means
of actually assessing the orientation and distance of the pair. It is 
out of the realm of home astronomy to build or buy the traditional 
filar micrometer. This is an expensive device and requires a permanent 
scope with stable mount, accurate clockdrive, and excellent optics. 
This micrometer also demands quite clear and keen eyesight, patience, 
and endurance. And then some maths to make sense of the measures.
    After asking around among double stars observers, Dr Martin 
Gaskell, University of Nebraska, recalled a technique I actually 
played with some thirty years ago. He pointed me back to the 
diffraction micrometer, made from a hunk of cardboard! 
Constructing the micrometer 
    Yes, cardboard. You can make it in an afternoon and observe binary 
stars with it on that very night. 
    The method does involve some maths, but that's what your 
calculette is for. As long as it's a scientific or engineering model, 
with the trigonometric functions on it, you're set.
    Cut a round disc of stiff smooth thin opaque cardboard to 
completely cover the front end of your telescope, like a cap. Paper is 
too flimsy and board is too heavy. Cardboard from gift boxes is about 
perfect. If the front end has a well or seat, like a very short dew 
shield or a bezel for the corrector plate, make the disc fit smoothly 
within in. If not, you can tape a low collar of cardboard around the 
tube for the disc to sit in. Be sure the disc can be turned smoothly 
and easily without binding or sticking, yet tight enough so it doesn't 
blow or shake loose. 
    For noncircular tubes you'll have to build a carriage or bed for
the disc, which itself is seated on the front end of the tube. This 
may be of any sturdy material, even of cardboard. A bit of ingenuity 
is handy in this situation.
    In this disc cut carefully, using a hobby knife, a grid of 
parallel slits. Leave enough land all around to keep the disc rigid. 
The solid and open strip of each slit are equal in width so the 'fill' 
ratio is quite 50%. The width of each slit (either the open or solid 
zone) is from 1 to 2 centimeters. Use which ever width gives you the 
most slits across the aperture of your scope while maintaining 
structural integrity of the disc. (If this shtick of really getting 
data out of binaries grabs you, you'll later make several discs for 
different width of slits.) 
    Your scope may have a central obstruction. Make the solid zone of 
a slit sit over this obstruction to minimize its intrusion into the 
open area of the slit. If this obstruction has a spindle, you may cut 
a hole for it so the disc can rotate on this spindle. 
Testing the micrometer
    Let's test this thing. 
    Aim the scope a known and resolvable double star. Then plop this 
slotted disc -- the diffraction micrometer -- on the scope. First off, 
you'll see the stars are much dimmer, by about a full magnitude. 
That's partly due to the blockage of the solid strips in the disc. 
    Closer inspection reveals that the stars, all of them in the field 
except for the dimmer ones, have 'satellites'! These, one on each side 
of the main image, are tiny spectra of the stars. Very crude and 
likely not colored, but they are true spectra. You're seeing the two 
'1st order' spectra. There are 2nd, 3rd, and higher order spectra 
farther away from the central image but they are for sure far too 
faint to notice. 
    The double star will, then, consist of six dots, the primary 
star's central image, its two attendant spectra, the secondary star's 
central image, and its spectra. Note that the three images of all of the 
stars are parallel. 
    To see these images more plainly do pump up the magnification a 
bit. Use the highest power the scope, eye, and sky will permit. This 
is more than the working maximum power of 1D (D is the scope aperture 
in mm). Get the stars well apart even if they are a little swollen. 
You may go to 2D or even 2.5D. 
    Now try rotating the disc. Gently turn it in its well by nudging 
it around. You may want to glue cardboard ears on the disc to 
facilitate this chore. The images in the eyepiece rotate, too. That's 
the secret. You can align all six binary star images in a straight line
or move them to sit exactly side by side. 
Some maths
    The little spectra, seen as dots next to the central image stand 
angularly away from that image by delta arcseconds, where delta is 
    (delta) = (lambda/width)*(206265 arcsec/radian) 
    width is the pitch between the slits, in millimeters, spanning one 
open and one solid zone. You know this from the construction of the disc. 
Ideally you should have a slit width that gives a delta comparable to 
the actual angular separation of the binary star, but for now go with 
what you have. Later you can make extra discs. 
    lambda is the wavelength of the star's light. This is NOT a single 
value due to the continuum spectrum given out by the stars. However, 
the eye-brain perception and the fact that stars have their maximum 
emission within the visual sensitivity range, conspire to let us take 
this as 570 nanometers.
    If you are concerned about such an assumption, you may tape over 
the slits green gelatin filters, like Wratten 58, to barricade out the 
far blue and red rays and let in just the green ones around the peak 
of the eye's spectrosensitivity. This is a wide bandpass filter, not a 
real monochromatic one. Can you use a monochromatic, hydrogen-alpha, 
filter? Yes but the stars will so dim, from turning away so much of 
their light, that it's just unworkable. 
    delta is in arcseconds due to that extra factor to shift radians 
into arcseconds. This delta is a property of the disc and can be 
written on it permanently. Make sure you properly juggle the metric 
prefixes! milli- means 1/thousand and nano- means 1/billion. 
    For a specific example suppose we made the slits 18 mm apart. Then 
delta, the angular separation of the satellites spectra from the main 
image, is 
    (delta) = (570e-9 m)/(18e-3 m)
            = 3.167e-5 rad 
            = 6.525 arcsec
You can label this particular disc 'delta = 6.525 arcsec' for good. 
    Now you need an index mark on the diffraction micrometer disc. 
Any place on the circumference is fine, but it's easier to find at 
night if you put it at one end of the slit that passes thru the center 
of the disc. Make it good and bold for easy reading at night. 
    After this first test you have to add a compass scale to the 
scope. This is easiest done on a paper collar marked off in degrees 
all around the circumference of the tube. You can lay this out by 
geometry on a large sheet of paper. Or plot out the markings with a 
computer in BASIC or PostScript and a LaserJet printer. If lazy, get a 
giant protractor, the open circle kind, from an art supply store and 
snip out the inner struts so it fits snugly over the tube. Please 
accent the degree marks on the collar so you can read them at night. 
    On the collar the degrees must run clockwise as you look down into 
the tube from the eye of a star. If the degrees run anticlockwise, 
flip the collar over. Or make a correct new one. 
    Examine the star in the scope with the clockdrive shut off. Rotate 
the disc so the three images of each star line up exactly with the 
diurnal drift. After several runs you'll get surprisingly good 
alignment. Note the angle of the disc; write this down. It should be 
270 degrees, the direction to west. 
    It sure will not be 270 due to the arbitrary way you attached the 
collar. Leave the collar alone and just record that this actual degree 
reading stands for 270 degrees. In general you'll have to go thru this 
process for each session. The telescope tube may be rotatable in its 
rings, you may have an altazimuth instrument, you may not precisely 
polar align the mount. All these (plus a few more) will cause field 
rotation during the session and require you to establish the east-west 
offset for each double star. 
    You could make the collar a slipring; slide it around so 270 deg 
lines up with the index. I find that for a degree ring made of paper, 
the extra handling will soon loosen and tear it. Leave it fixed in place 
and just note the offset.  
    Note that life is far easier if you seat the disc so the index is 
closer to the top of the tube, nearer to its north point, so you read 
off of the upper edge of the collar. This immensely reduces the chance 
of reading an angle from the under side of the scale. 
First measurements 
    Now we're ready for our first measurement of of a binary star. 
Examine the star under the highest tolerable power. Let's call the 
primary star and its satellite spectra a', a, and a". Those for the 
secondary are b', b, and b" a and b are the central images, Please 
remember that as you rotate the disc the central images of the stars a 
and b remain fixed while the satellites a', a", b', and b" rotate in 
step with the disc. They align perpendicular to the slits as seen from 
the focal plane. 
    Pick, it doesn't matter which, one of the satellite images, say, 
a". Rotate the disc so the angle formed by line a-b and line b-a" is 
as exact a right angle as you can tell. The right angle is at point b. 
    You will find two orientations of the disc that give this aspect. 
Ve sure you line up with the SAME satellite! Note the angle of each 
orientation on the degree collar, say they are alpha1 and alpha2. 
    Double check this setup. The leg between the central image a and 
its satellite a" must be the hypotenuse. If it's one of the legs of 
the right angle your setting is all wrong. It may be more comfortable 
to use a crosshair eyepiece to line the images against. Or you may 
find the hairs too distracting and prefer a clear eyepiece. 
    The sketches here show how the images must line up in the two 
orientations. I show them nice and big and spread out here. In the 
eyepiece they'll be packed into the center of the field. You do need 
that high magnification.
                    +          |       + 
                    a'         |       a' 
          a  *       +         |       +        * a 
           / |       b'        |       b'       | \ 
         /   |                 |                |   \ 
       /     |                 |                |     \ 
     +-------*                 |                *-------+ 
     a"      b                 |                b       a" 
     + b"                      |                     b" +   
   orientation #1     * = central image       orientation #2 
   --------------     + - satellite image     --------------
    You may find that delta, by the construction of your disc, just 
will not make that right triangle. It is too small or too large. Now 
you see why it is well to have several discs with different delta. 
Separation and position angle
    The position angle of the double, theta, is 
    (theta) = (alpha1 + alpha2)/2 
which is the average of the two angles. 
    There will be the ambiguity of 180 deg; this is resolved by a 
quick look at an orbit chart of the star. Because we did not zero the 
collar, be sure to add or subtract the offset, as found bt the drift 
method above. 
    The separation of the two stars, rho, is 
    (rho) = (delta)*cos((alpha1 - alpha2)/2)
First the alphas are subtracted, then divided by two, then the cosine 
is taken. Leave the algebraic signs alone! Don't try to 'fix' them. 
The calculette will sort things out for you. 
    Of course, errors will creep in. Do the same exercise for each of 
the other three satellites in turn, yielding four sets of data. they 
should agree in position angle and separation within, believe it or 
not, a fraction of one percent. 
Why this works
    It's a lot easier to see what's happening by joining the two 
diagrams into one along the line a-b: 
                                 / | \
                               /   |   \
                             /     |     \
                         a"(1)     b     a"(2)
    We constructed, in two steps, the isoceles triangle of a, 
a"(1), a"(2). The altitude of this triangle is the separation 
between a and b. The two equal sides are each delta. The purpose of 
building this triangle in two steps is to get at the apex angle, the 
one at a. angle a is that included between alpha1 and alpha2, or is
(alpha1 - alpha2). Then, for either of the two right triangles within 
this big triangle the angle a/2 is then (alpha1 - alpha2)/2.  
Simplified method 
    Do you really have to got thru all this? Well, if you just want to 
casually see a double star go thru its rounds you can bypass the 
maths. Rotate the micrometer disc until all six images are exactly in 
a straight line. Read the degree collar. This, with the offset, is the 
very position angle. Proportion the separation between a and b against 
the delta along this line of images. That's the separation. 
    This is really a cunning and simple method of assessing binary 
star aspects! This method of diffraction micrometry was developed 
apparently first by Schwarzschild (the blackhole guy) in 1896. He used 
a variation of this method to measure many binaries. In his device, 
the width of the slits was adjustable, and thus delta, to make the 
images stand exactly as far apart as the components of the binary. 
Thus the delta for this aspect was the very separation of the 
    He varied the slit size by oblique perspective. The disc was a 
'tent' of two slitted plates set on a carriage atop the telescope. By 
raising or lowering the peak of the tent, the telescope 'saw' narrow 
or wide slits, there by adjusting the delta. Guess what? he built his 
contraption out of, erm, cardboard. 
    The diffraction micrometer has limitations. First off, as you saw 
when you first tried it out, the images are far dimmer than with no 
disc. This comes from the obstruction by the solid parts of the slits 
and the dispersion of light into several images for each star. You'll 
probably be limited to studying double stars only as faint as two or
three magnitudes above your normal threshold for the scope.
    If the double star components are too different in brightness the 
images of the dimmer will be overwhelmed by those of the brighter. It 
seems that the two should be within two magnitudes of each other for 
easiest work. This limitation does depend on keenness of your vision.
    You do need at least a good size aperture. I tried the micrometer 
on small scopes, 75mm to 100mm aperture, the ones you see with me at 
our starviewing meets. Apart from the frail disc that remains after 
cutting the slits, the images are just too weak. When I played with 
the diffraction micrometer in the 1960s and 1970s I used a 150mm 
Newton and then a 200mm Schmidt-Cassegrain. Both gave very pleasing 
    A small scope also lacks the raw resolution to split the closer
binaries. Based on the rule of (res) = (120 in 
mm), an 80mm scope resolves 1.5 arcsec at best. The stars's images 
touch and are barely distinquishable under the best of seeing. On the 
whole the larger the scope, the better for increased number of slits, 
tighter resolution, and brighter images. 
    Atmospheric seeing, blessedly, is not that much of an obstacle. 
The images may be soft or swollen but as long as they are actually 
round you can get good fixes on them. Never the less, the steadier is 
the air, the more certain is the measurement. 
    Cardboard is not all that durable. It works superbly for gentle 
use and care of handling. If after a while you find yourself hooked by 
binary star observing, consider making a micrometer of sheet metal 
with the aid of a mechanic or metal shop. 
    Each disc has one geometry of slit and one delta value. Being that 
binary stars have a diversity of separations, you'll find that for 
a given star your disc has too small or too large a delta. You then 
have to make a set of discs for a stepped range of delta. Don't even 
think of building a Schwarzschild micrometer! Or any other kind with 
slides, rails, hinges, gears, &c to vary the delta. 
    The micrometer does NOT, as some authors may tell you, bring out 
more details in planets or nebulae. It does NOT resolve stars into 
discs. It does NOT vastly increase the resolving power of your 
telescope. It does enable you to consciously measure very small 
separations of point sources, the double stars. 
Homebased binary star work 
    Yet, after all is said and done, why isn't the diffraction 
micrometer in every observer's toolkit? It is, to be fair, very seldom 
described, even in litterature for double stars. The material I read 
dates from the 1980s at the latest to all the way back to Schwarzschild's 
own paper in 1896. There seems to be no recent papers on it for the 
home astronomer, while there is lots and lots on diffraction (and 
other) micrometers for observatories. 
    There's also the matter of disincentivity for double stars. Look 
in any observing book, What's emphasized? What gets the awards and 
attention? Not double stars. The data in such books is often 
incompetent, copied from ancient sources heedless of the real motion 
of the stars. And let's not think about the color descriptions applied 
to the stars! Lipstick manufacturers could take lessons from them. 
    On the other hand, binary star observing is one of the regions 
where home astronomers can, like for variable stars, make genuine 
worthy contributions. The sheer simplicity of the equipment and 
technique should appeal to just about anyone who is otherwise disposed 
to do some useful work in astronomy. The combination of many 
observer's reports, like for variable stars, will smooth out the 
inevitable random dispersion of measurements. 
    Are binary star observations really needed? Here's the rub. Yes 
and no. Yes in that with so many stars as binaries -- about 1/3 of all 
the stars you see in the sky are binaries! -- there's plenty of work 
for everyone. So much so that most binaries are observed infrequently, 
some were observed only once at discovery. 
    No, in that there is no credible bona-fide solicitation for double 
star measures, setting observing standards, collecting and compiling 
data, encouraging observers, perfecting techniques, distributing data 
to other astronomers, and so on. 
    So, this is a new century (get used to it, OK?). Can our profession 
kick off a collaborative effort for home and campus astronomers to 
mainstream double star observing?