Equation (
) may also be used for
a simple order-of-magnitude estimate of the effect on seeing caused by
free convection from a warmer dome floor.
Consider the geometry of fig. .
If the dome diameter is 
, the mean height
of the air volume crossed by the optical beam with respect to the
heat exchange surface will be about 
.
Assuming for the constant b the value of 2.68 determined for the
atmospheric surface layer (cf. equation ()) and
taking 
C, we find:
Considering that the light beam travels three times between the
primary and the secondary mirror
,
the total light path inside the protected volume of the
dome will be 
and from equation
(),
a rough estimate of the dome seeing 
will be given by
Figure: Order-of-magnitude value of seeing FWHM caused by a
warmer dome floor.
This is plotted in fig. for two typical
values of enclosure size:
free convection flow in the enclosure volume will
begin to cause significant seeing effects (
0.4 arcsec) in
the light beam for heat fluxes of the order of 20 W/m
.
Considering that a
typical free convection heat transfer rate would be 3 W/m
K at
the floor, one should then expect a seeing contribution of about 0.06
to 0.08 arcsec per deg K of floor-air temperature difference.
Relationship of the same quantitative order
between heat flux, distance from the exchange
surface and seeing are likely applicable also to
other potential sources of free convection located inside the
enclosure (e.g. items G and I in fig. ).
However, in view of the smaller exchange surface areas of walls and
other heat dissipating objects with respect to the inner air volume,
the seeing rate per deg K of surface-air temperature difference
will be quite lower than for the floor.
The rate of dome seeing per floor-air 
of
0.06 to 0.08 arcsec/K
which is estimated here indeed explains the high seeing values
experienced during the first years of operation by many
telescopes of 4-m class built in the 70s, like the ESO 3.6-m
and the CFHT.
These telescopes are enclosed in large concrete/steel
buildings with no natural ventilation and,
as the inner dome thermal environment was hardly or poorly controlled,
the different heat capacities of the concrete base, the primary mirror, the
telescope structure and the dome inevitably caused large differences of
temperatures. Measurements at the ESO 3.6-m telescope
done by the author [Zago 84] and
[Schmider] before the installation of an effective thermal control
system in the dome,
show that positive differences of 5 K between the dome floor,
and dome ambient air are typical and raise at times up to 10 K.
As we have already mentioned in section above,
the thermal conditioning of these domes has been improved in the
meantime, so that surface-air 
now hardly exceed 1 or 2 K
in the worst cases, while the newer telescopes have been built inside
lighter enclosures which also allow natural ventilation. As a consequence,
dome seeing, in the proper meaning of seeing created by
large convection flow patterns in
the dome enclosure,
is no more a critical factor of telescope performance nowadays.
Therefore there is not much opportunity nor perhaps
interest to investigate in detail a
matter which appears technically solved.
