In modern telescopes the secondary mirror (often called M2) is attached to electro-mechanic actuators which allow fine adjustment of the optical alignment and focusing. The latest telescopes projects also provide for fast motion of the secondary mirror, either to switch rapidly between the observed object and the sky background, or to correct guiding errors due to wind loading. All these systems produce some heat which, if it is not carried away by a cooling circuit, will dissipate right into the telescope light-beam and may then cause a seeing degradation.
In order to evaluate this issue,
Jacques Beckers of ESO
performed a series of measurements at the ESO 2.2-m telescope
using a CCD array to measure FWHM image size ([Beckers 92]).
For the tests three
different types of heating gadgets were used to produce heat: a
metal square box 22
22
22 cm and a metal plate
220
27.5
1.1 cm. They were mounted on top of the 2.2-m
telescope, the plate across the light-beam, simulating two opposite
spiders (cf. fig. 
),
and the box mounted on it on thermal insulating legs.
The measurements were performed on stars about 30
from zenith
while either the box or the plate were heated. The box was heated
with 135 Watts and the plate with 560 Watts. Both wattages resulted
in a surface-air temperature difference of about 60 K.
The mean heat transfer coefficient was respectively 7.5 W/m
for the
box and 7.8 W/m
for the plate. The wind velocity was low during
the experiment (3 m/s or less), which lets assume that the air flow
motion near the top of the telescope was around 1 to 2 m/s (although
no measurements were done inside the dome).
A measurement cycle
(heating to maximum temperature starting from ambient and cooling off
to near ambient) lasted 2 hours.
The CCD images were fit with a Gaussian profile and the FWHM
were recorded. The natural seeing was recorded by the DIMM
seeing
monitor operating at La Silla
and was near 0.8 arcsec during the three nights of the experiments.
The experimenters had expected to record a significant increase of
seeing with the heating elements in the telescope light-beam, and were
surprised when the effects were very small, barely noticeable through
the variability of natural seeing.
The measurements were analyzed by least square fitting of
where 
was the natural seeing measured by the DIMM
and 
a general
fixed error contribution from the telescope.
The factor a was the
quantity of interest, as the FWHM contribution was assumed to be
linear with 
, and was evaluated as:
arcsec/K, for the box
arcsec/K, for the plate/spiders
= 60 K
the seeing contributions were respectively
0.22 arcsec for the box dissipating 135 Watts
and 0.36 arcsec for the plate (560 Watts).
Figure: Seeing caused by a free convection plume from the M2 unit.
We can transpose these experimental data also to telescopes of
different sizes provided that a geometry similarity
is approximately maintained in the proportions of the test.
Consider the telescope configuration sketched in
fig. .
Heat is dissipated by the secondary mirror unit causing a vertical
plume across the light-beam.
Let us assume that the plume spreads in proportion to its distance
from its origin. A circumferential cross-section at a distance
r from the optical axis has then dimensions 
along the line of
sight and 
across it. We assume further
that the thermal turbulence in the plume is described by a
relationship of type ().
The seeing spread through a line of sight at a distance
r from the optical axis (see fig. )
will be proportional to:
where 
and 
are respectively the
average temperature structure coefficient and heat flux along a
line of sight across the plume.
The total seeing spread of the light beam will be obtained by
integrating over the plume cross-section seen by the telescope,
normalized by the mirror area:
where D is the pupil (mirror) diameter.
Since both 
and 
are proportional to r, it is:
Noting that the total heat flow Q from
the secondary unit is proportional to 
, we
obtain:
Therefore the scaling relationship based on the experimental results is:
This relationship may be used to specify the maximum allowable
heat dissipation for a given seeing requirement.
For instance, if we consider a 8-m
telescope and a requirement
that the seeing contribution shall be 
0.025 arcsec,
the allowable heat dissipation from the secondary mirror unit
will be about 108 Watts.
