Therefore, this web page was created to merge the important sources of Cassegrain formulas that I have come across. I chose some common names for variables to make things easier to read, since every author seems to have different variables for the same quantities! I also give a couple formulas I have derived that show how sensitive the Cassegrain design is to certain quantities. These are fairly simple to derive, but I don't see them in the amateur publications that I have. So, I put them here.

The differences between the various types of Cassegrains are the figures of the primary and secondary mirrors, and they will be covered in the next section, followed by other useful facts and procedures that I have seen fit to include.

Figure 1: A diagram of a Cassegrain telescope is shown above. The distances between the mirrors and other quantities are illustrated.

The important quantities here are:

the diameters of the primary and secondary mirrors,

the focal length of the primary

the primary-secondary separation,

the system focal point to secondary distance,

The purpose of a Cassegrain secondary mirror is to slow the convergence of the light cone from the primary (thereby increasing the system focal length), direct it to the focal plane, and possibly compensate for some of the image aberrations caused by the primary. The secondary has a "magnification factor" that is referred to as

Here is a handy equation - how to calculate the radius of curvature of the secondary if you know p and q. This is required if you already have a primary that you wish to use in a Cassegrain. After you choose a magnification, you know p and q, and all you need is the secondary's radius of curvature, R2. Below is a manipulated form of the equation for R2 given by Richard Buchroeder from page ~135 of "Advanced Telescope Making Techniques, Vol. 1". The first equation is solved for R2. The second equation is the same relationship solved for q.*F = f1(System focal length)M

/M = q("Magnification" of system)p

p = (f1+e)/ (M+(secondary mirror to focal plane distance)1)

2*R2 =*p-M / (M(Radius of curvature of the secondary mirror)1)

*D2 = D1(Diameter of secondary required to fully illuminate ONLY the center point of the field.)p / f1

(Defined for convenience for later formulas)s = q / d

The second equation is important because2R2 =/ (*(1/p) - (1/q))

q = R22*p / (R2 -p)

dFor example, for my 12.5" F/12.5 classical Cassegrain,dq /p = R2^{2}/(4*p^{2}4*-*pR2+R2^{2})d

dq //R2 = p(2*R2 -p)- R2*p/(-R22*p)^{2}

The second number means that an error of plus (or minus) one millimeter in

The first number means that for every unit of length (say a millimeter) that the secondary is moved towards the primary mirror, the focal plane moves 8.86 millimeters farther behind the primary. This derivative (rate of change) is only valid for the position of the secondary, and it will change if the secondary is moved. (The

Now, you might wonder why the focal plane moves so much more than we expect. Common sense seems to indicate that, for the 12.5" F/12.5 Cass, with an F/4.2 primary and a secondary magnification of 12.5/4.2 = 3, moving the secondary 0.01" toward the primary should move the focal plane back 0.01" * 3 = 0.03". But this is NOT true. Instead, it moves 0.0886" back. So where does the extra sensitivity come from? Well, when you decrease the spacing between the secondary and primary, the focal length of the system actually gets longer (!), and this adds dramatically to the sensitivity to spacing changes. This is why adjusting the spacing between an aspheric secondary and the primary changes the system correction (see tip a below).

The sensitivity to changes in these quantities increases as the magnification rises. For a 14.25" F/21 Cassegrain, d

Now, you don't really need this next formula, but some of the more math-oriented ATMs might enjoy seeing it. It was also obtained from "ATMT Vol 1.", and I will add the author's name and the page number soon. A general formula for the shape of a cross section of a mirror's surface is given by:

The optical axis lies along thez =(y^{2}/R)/(1 +[1-(1+b)(y/R)^{2}]^{1/2})(Don't worry - you don't need this to design your Cassegrain!)

For a sphere,

For an ellipse, 0 >

For a parabola,

For a hyperbola,

Basically, the parameter

"Flavor" of Cassegrain |
for primary b mirror |
b for
secondarymirror |

Dall-Kirkham |
( s-1)(M+1)M^{2
} ------------ - 1 M^{3}(+M)s |
0 (sphere) |

Classical Cassegrain |
-1 (parabola) |
-4M ------ - 1 (-1)M^{2} |

Ritchey-Chrétien |
-2s -----
- 1 M^{3} |
-4(M-1)
- 2(M+M)s ----------------- - 1
(-1)M^{3} |

So how to figure the mirrors? Calculate

Let's say for an R-C primary,

a) Correction issues for Cassegrains with hyperboloidal secondary mirrors (classical, R-C):

The correction of a Cassegrain optical system with an aspheric secondary mirror (classical, R-C) is dependent on the primary-secondary distance. This is because the secondary's conic constant, b is a function of how far it is from the primary mirror and the distance to the focal plane. (This is why it is difficult to match a secondary mirror that is already finished to a primary mirror - the chances of the secondary having the right value of b are small.) If the secondary has the right conic constant and it is placed at the proper design location, the system will be perfectly corrected for spherical aberration. However, if you see spherical aberration, digest the following:

-If the secondary mirror is moved AWAY
from the primary mirror, the correction of the system will INCREASE.

-If the secondary is moved TOWARD the primary mirror, the correction of the system will DECREASE. (This assumes it is large enough to be moved and still intercept all of the light from the primary)

-If the secondary is moved TOWARD the primary mirror, the correction of the system will DECREASE. (This assumes it is large enough to be moved and still intercept all of the light from the primary)

-If you are figuring a Cassegrain
secondary using star testing, increasing the correction of the
secondary will decrease the correction of the system.

So, if you see a bit of overcorrection in your classical or R-C,
move the secondary mirror closer to the primary and deal with the
focus shift, or figure the secondary to have a bit more
correction. Why is this true? Think about it this way - if
we move the secondary farther from the primary, we use only a smaller,
central portion of the secondary. In effect, the hyperbola in the
center of the secondary has less correction than that of the full
mirror, and this causes the light in the outer zones of the mirror to
converge slightly more slowly, which is the same as increasing the
correction of the whole system. So, the effect on a Cassegrain
system of changing the correction of the secondary mirror is the
OPPOSITE of changing the correction of the primary mirror.b) Correction issues for Dall-Kirkhams:

For Dall-Kirkhams, the position of the (spherical) secondary makes a difference, too, as evidenced by the presence of the variables s and M in the equation for the value of b for the primary mirror. Those two variables are dependent on q, d and p, which vary with the location of the secondary mirror. A quick numerical calculation using the mirror spacings for my 12.5" classical Cassegrain indicate that if it were a Dall-Kirkham, then moving the secondary toward the primary would require a slightly less corrected primary. So, moving the secondary toward the primary should slightly increase the correction of a Dall-Kirkham system in the system. This is the opposite of what happens in the other Cassegrains with hyperboloidal secondaries. I believe this system is less sensitive to movement of the secondary than a classical or R-C. I still need to double-check these results, though.

c) I collimate my Nasmyth-focus classical Cassegrain by inserting a laser collimator and using the following procedure:

- First, I adjust tilt of the focuser to aim the laser dot at the center of the tertiary mirror, adjusting the mechanical positions of the tertiary and focuser if necessary, and making sure that the focuser is square to the tube. This adjustment should not need to be done very often.
- Second, I adjust the tertiary's tilt to send the laser dot to the
exact center of the secondary mirror. Before I install the
secondary, I use a permanent marker to draw a ring exactly
centered on the secondary's surface. This makes the tertiary tilt
adjustment (or focuser alignment, for those without a tertiary) much
easier and more accurate.

- Third, I adjust the secondary's tilt to return the laser beam to the center of the tertiary and the center of the laser collimator.
- Fourth, and lastly, I remove the laser collimator and put in an
eyepiece. I find a bright
star or planet in the eyepiece. While looking at the star in the
telescope, I adjust the primary's tilt until I remove all of the coma
from the image at the center of the field of view. This takes a
little practice, but once you get the hang of it it is simple, quick,
and very accurate.

d) Size of illuminated field:

In order for an area of the focal plane to be "fully illuminated", it must get light from all of the primary and secondary mirrors. For a Cassegrain this means that the secondary mirror must be large enough to allow the entire primary mirror to be "seen" from the parts of the focal plane that we wish to fully illuminate. For example, if we know the location of the focal plane and we put our eye there at its center, we should see the reflection of the primary mirror centered in the secondary mirror. If only the very center of the field is fully illuminated (say the central 0.05"), then we should see the reflection of the edge of the primary mirror right at the edge of the secondary mirror. If the fully illuminated field is larger, say 0.5" in diameter, then we should be able to move our eye off the optical axis by 0.25" and still see the entire primary mirror in the secondary mirror, though it will no longer be centered in the secondary. (This will soon be illustrated below.)

More sections will probably be added in the future as I figure out what information is useful to people.