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**
8.1.2. Newtonian collimation**

The difficulty in aligning
optical elements of the Newtonian reflector arises from its diagonally
oriented flat mirror, causing apparent shift of the projection of its
surface away from the primary, in the view of the focuser opening. Due
to this effect, in a perfectly aligned instrument (**FIG. 116**, top),
with the center of the flat coinciding with the point of intersection of
primary's and focuser axes - positioned at right angle - observing with
the eye at the focal point, the flat appears off the focuser axis,
shifted away from the primary. Also, image of the primary mirror in the
reflected view of the flat appears decentered with respect to the flat,
while in fact their respective centers do lie on the optical axis, and
are perfectly aligned with the focuser axis.

There are two drawbacks of
this type of alignment: (1) uneven illumination of the outer field, and
(2) difficulty of determining the correct visual alignment of the three
components. The former is pretty much insignificant in visual use, since
the uneven illumination falloff at the two opposite sides of the field
are unlikely to be noticed. The latter does present practical
difficulty, although it is possible to determine and produce an image of
the proper visual alignment for a particular telescope, and use it as a
guide.

Much more common
alternatives are those that are based on near-concentric visual
alignment of the focuser, diagonal, primary mirror in the diagonal's
view and diagonal's reflection in the primary. One of them achieves this
by offsetting the diagonal by sliding it
down along its surface plane, until it becomes visually centered in the
focuser's view. It also evens up field illumination, without introducing
any form of mechanical.

The other is a conventional
3-step collimation procedure, in which the diagonal is centered in the
focuser by shifting it toward primary. It also achieves near-concentric visual
alignment and even field illumination, but at a price of introducing
some forms of misalignment, some generally negligible, some others
possibly not (**FIG. 116**, bottom). This collimation mode is
sometimes referred to as *partial offset*, as opposed to *offset*,
or *full offset* when the flat shifts diagonally along its surface.
A better terminology, suggested by Don Pensack, is *unidirectional*
and *bidirectional diagonal offset*, respectively.

**FIGURE 116**: TOP:
Visual alignment of the focuser, diagonal and primary mirror in a
perfectly aligned Newtonian reflector with unoffset diagonal flat. Since
the diagonal is not centered in the axial cone, so is not reflection of
the primary in it. Diagonal itself, being on primary's axis, is
projected to the focus centered around it, as indicated by a ray from
the top diagonal point.
BOTTOM: Conventional
3-step collimation procedure, with the forms of misalignment induced at
each step, as well as those remaining after it is completed.

The starting assumption is that all three elements are in near-perfect
mechanical alignment, with the diagonal center nearly coinciding with
the focuser and tube axis point of intersection, and the primary nearly
centered around tube axis as well. Focuser axis is perpendicular to the
tube axis. Mirrors are likely to be deviating somewhat from their
perfect orientation but, for the sake of illustration, we'll assume they
are close to it (diagonal deviations from 45-degree angle - or in the plane
orthogonal to it - will affect the position of primary's reflection in
it, while primary's deviation from orthogonal to the axis position will
affect position of diagonal's reflection in it, which is centered around
primary's optical axis).

Following calculation, assuming surface rotation around their center
point (except when acknowledged otherwise), and omitting negligible
factors (such as depth of the primary) is not strict in mathematical
sense, but should give close approximations practically as good as exact
values for the range of amateur Newtonian telescopes.

The first step of this procedure (**FIG. 116** bottom, 1) is centering
the diagonal in the focuser's view by shifting it axially toward the
primary. The needed axial shift **∆**a
(to distinguish it from the offsetting shift **∆**) to equalize
apparent visual angles** α**
and **β** for the top and bottom half of the diagonal, respectively,
resulting in the apparent and actual center of the diagonal coincide,
requires the equality (a-∆a)/(H-a+∆a)=(a+∆a)/(H+a+∆a),
which solves to:

where **A=2a** is the diagonal's minor axis, and **H** the
height of focus above the diagonal's center. By neglecting **∆a**
in brackets with three parameters (i.e. neglecting that the
focus-to-diagonal separation will increase by **∆a
**after primary's tilt), it simplifies to a close approximation ∆**a**~a2/H**
**= A2/4H.

The shift has, however, separated reflected (imaginary) axis of the
primary from the focuser axis, inducing decenter error. At this point,
primary's axis is shifted toward the primary by **∆**a
linearly, which brings to the field center off-axis field point of
identical height, with the P-V wavefront error of coma, according to **
Eq. 73.1**, given by ω=6.7∆a/F3.
For 300mm *f*/5 Newtonian with the
focus height H=250mm, diagonal's minor semi-axis a=35mm and
corresponding axial shift ∆a=4.9mm,
this gives ω=0.26 wave RMS.

The next step, with the diagonal centered in the focuser's view, is to
bring reflection of the primary in the diagonal's view (i.e. on the
surface of the diagonal) to the diagonal's center, making the two
concentric. Since the axial shift resulted in nearly equalized apparent
angles **α**
and **β**, the primary reflection is also nearly centered in the
secondary, and only slightly shifted toward tube opening due to focuser
axis being displaced upward from primary's axis. Centering of the
primary reflection is accomplished by tilting the diagonal clockwise, by
an angle given as:

γ = ∆a/2(f-H)
(74.1)

in radians (for this small angles, practically identical to tan**γ**), **
f** being the mirror focal length.
Obviously, an imaginary ray coinciding with the focuser axis will be
now, after reflection from the diagonal, directed toward the center of
the primary.

However, the diagonal tilt directs the "reflected" primary's axis - with the best image
point on it - further away from the field center (**FIG. 116**, bottom
2). At this point, with both, diagonal and primary in its view centered
in the focuser's view, the total RMS wavefront error of coma in the
field center is that of decenter and tilt
combined, or ω=(6.7∆a/F3)+13.4γH/F3
(the latter from
Eq. 73,
substituting ** γ** for **τ**
and multiplying with 57.3x60 for the conversion from arc minutes to
radians, with the result reduced by a factor H/*f*
for the focuser). For the same 300mm *f*/5
Newtonian, that comes to γ=0.00196 and
the combined RMS wavefront error of coma ω=0.31
wave.

The final step is bringing the deflected optical axis of the primary -
i.e. the focal point - back to the field center, in the middle of the
focuser. This is accomplished by tilting the primary clockwise,
so that its optical axis tilts upwards until it nearly coincides with
the focuser axis "reflected" from the diagonal. The required angle of
tilt equals 2γ. Since diagonal's reflection in the primary is centered around
primary's axis (since the diagonal is axially centered itself), bringing
primary's axis to the center of the focuser is, in terms of the view in
the focuser, bringing diagonal's reflection in the primary to the center
of the view, that is, making it concentric with both, diagonal and
primary's reflection in it.

As a result of primary's axis now coinciding with the focuser axis
"reflected" from the diagonal, the two axes hit nearly identical point
at the diagonal, and nearly coincide in the focuser. In other words, the
point of best image is brought to nearly coincide with the focuser axis,
that is, with the field center (note that the focal point is now
slightly higher, effectively by the amount of diagonal's axial
shift).

However, the consequence of partial offset is that the
primary's axis and tube axis are at an angle. This angle equals 2**γ**,
hence it is given by:

**δ**
= a2/(f-H)H
= A2/4(f-H)H
(74.2)

in radians, with **a** being, as before, the diagonal's semi-axis, ** ***f*
the mirror focal length, **H** the focus-to-diagonal axial separation,
and **A** the diagonal's minor axis, A=2a (if
we designate the diagonal-to-primary separation as **S**, it simplifies to
δ=a2/HS=A2/4HS).
For the above 300mm *f*/5 Newtonian, it would come to 0.0039, or 0.22
degrees.

Axial disparity between primary's and tube axis -
assuming tube axis aligned with mechanical axes of the mount - causes pointing error.
Pointing error induces tracking error, and negatively affects efficiency of go-to
guiding.

Note that this calculation is nearly accurate for the
focuser perpendicular to the tube/structure axis. If the focuser is at
an angle, it will either add to, or deduct from the axial disparity
caused by collimating with only partially offset diagonal. The change in
pointing error due to the focuser tilt angle **τ**
in radians (i.e. as tan**τ**) is given by Δδ=Lτ/2S,
with **L** being the separation from geometric center of diagonal's
surface to its effective center of rotation (approximately the point of
intersection of diagonal's center axis and the plane of position
adjustment), and **τ** being positive
for the tilt away, and negative for focuser tilt toward primary (hence,
increasing pointing error for the former, and reducing it for the
latter). Change in the pointing error is a consequence of the effective
decenter of the diagonal vs. tube/primary, given by Lτ/2
(since the diagonal tilts by about τ/2 in order to compensate for the
focuser tilt and bring primary's axis to nearly coincide with the
focuser axis).

For the above 300mm *f*/5
Newtonian, assuming L=1.5A=105mm, a 2° focuser tilt away from the
primary (τ=2) would add 0.08° (Δδ=0.0015)
to its pointing error, raising it to ~0.3°. A 2° focuser tilt toward
primary would result in Δδ=-0.0015,
thus ~0.14° pointing error.

Focuser tilt in the plane perpendicular to the vertical
plane containing tube axis also adds decenter error to the focuser's axis
"reflected" toward primary. As a result, the sideways tilt of the
diagonal is needed to bring it to the center of the primary (it also
requires to decenter diagonal sideways by τH
with respect to the tube axis, in order to have the diagonal centered in
the focuser's view; this means that the diagonal's reflection in the
primary is no more centered). Now the diagonal tilt has two components:
(1) the upward angle resulting from the axial shift toward primary
alone, and (2) the sideways angle resulting from the diagonal decenter
due to focuser tilt. Hence, the effective disparity between the axes of
the tube and the primary - pointing error - is given by:

**δ'=(δ2+τ'2)0.5**
(74.3)

with **τ'** - the angle of
diagonal's sideways tilt needed to bring "reflected" axis of the focuser
to primary's center - given by τ'=τH/2S.
For the above 300mm *f*/5 Newtonian
and sideways focuser tilt of 2 degrees (τ=0.035),
τ'=0.0035 and the effective pointing error
is 0.0052 in radians, or 0.3°.

For a given focuser tilt in the plane in between the one
containing tube axis and that perpendicular to it, the value of
resulting pointing error is somewhere between **
δ** and **
δ'**.

All considered, offsetting diagonal bidirectionally is preferable to
the partial offset, i.e. with it being axially centered without offsetting vertically. While the apparent alignment
of the reflective surfaces in the focuser is not entirely concentric
with the diagonal being offset as well -
reflection of the diagonal in the primary is shifted slightly toward
primary, due to it being lowered with respect to primary's axis (its
relative shift is still considerably less than in optimally aligned
system without offsetting) - no other tradeoffs are involved. That
said, pointing error inherent to the conventional (axially offset)
collimation is small enough to be generally insignificant for visual use
without computerized guiding.

These considerations are valid for collimating from the
proximity of the focal plane. It is the most logical vantage point for
visual use, since it is where the eyepiece is positioned. For other
vantage points, the above relation for the needed shift centering the
diagonal in the focuser's view, given as A2/4H
is still valid, assuming that the actual point height is substituted for **H**.
Using higher vantage points reduces the needed shift, and by the same
amount the extent of slightly decentered field of full illumination. The
field is exactly centered when the vantage point coincides with the tip
of the cone formed by extending the lines connecting edges of the
primary and diagonal (FIG. 74.1);
from this point, the apparent circles of the diagonal and primary
coincide. Since this point is roughly ~10cm above the focal plane in
Newtonians - specific value is closely approximated by {S/[(D/A)-1]}-H - it
does make
collimation more complicated, and potentially less accurate. With
knowledge and experience in collimating Newtonian, however, it should not be a significant
concern. Publication that details this collimation mode - including more general
information on Newtonian collimation and use - is Vic Menard's "*New
Perspectives on Newtonian Collimation*".

Also, collimating without centering the diagonal in the
focuser is not as difficult as it may seem. The only non-concentric
element is the diagonal, and the proportion of primary reflection's
distance from the diagonal's inner (toward the primary) and outer edge is given by
{[a*f*/(H-a)]-d}/{[a*f*/(H+a)]-d}, **d** being the aperture
radius.