The LMT Didactical experiment
J. Surdej and A. PospieszalskaSurdej
What is a liquid
mirror?
As we all know, a perfect reflecting parabola represents the
ideal contour for a mirror to focus parallel rays of light into a
single point.
It is therefore of the greatest importance to have realized that
a rotating liquid surface takes the shape of a parabola under the
constant pull of gravity and a centrifugal acceleration, which grows
stronger as the distance from the central axis increases (E.
Cappoci, 1850). The parabolic shape occurs because a liquid surface
always sets it local surface perpendicular to the net acceleration
it experiences, which in this case becomes stronger and less
inclined with distance from the central axis (see Figure
1).


Figure 1: Acting forces on a liquid set in a
rotating container 
Figure 2: Parabolic shape of the spinning
liquid with the distribution of forces acting on a single
liquid particle 
Indeed, inspection of the above figure clearly shows that the
tangent dy/dx to the liquid surface may be expressed as (see
Figure 2)
dy/dx = tg() = ^{2}x / g,
(1)
where ^{2}x and g represent the centrifugal force and
gravity acting on a mass unit (see the red dot in Figure 1and
Figure 2). Integration of the above differential equation
leads to the solution
y = ^{2}x^{2}/(2g),
(2)
which, by analogy with the equation of a parabola caracterized by
the focal length f, i.e.
y = x^{2}/(4f),
(3)
implies that the focal length of the liquid mirror is merely
given by the relation
f = g/(2^{2}).
(4)
A focal length of approximately 40 cm is obtained for a value of
the angular velocity of 33 turns per minute.
"Homemade" liquid mirror
telescope
A gramophone with a turntable of 20 cm in diameter may thus
provide us with a perfect liquid mirror characterized by a focal
ratio f/D ~ 2, a very acceptable value for an astronomical mirror.
For large mirrors of practical interest, the periods of rotation are
typically of the order of several seconds and the linear velocities
at the rims of the mirrors range between 2 and 10 km/h, very near
the velocity of a walking man.
It is very instructive to construct a small size liquid mirror by
pouring in a small cylindrical container, set on the turntable of a
gramophone, some polyurethane. After polimerisation of this resine,
in the form of a preshaped paraboloid, one may conveniently fill
the latter with some used car oil (or mercury in the case of a real
astronomical mirror). By means of a small eyepiece set at the focus
of the spinning paraboloid, it then becomes possible to observe
distant objects located just above the gramophone (see Figure
3). More professional liquid mirrors have been constructed at
various locations.
Figure 3: "Homemade" liquid mirror telescope: Cylindrical
container, set on the turntable of a gramophone filled with oil
Figure 4: 1.4m diameter mercury mirror
constructed by N. Ninane and her collaborators at AMOS and CSL
(Liège, Belgium)
