User:Cleonis/Sandbox/Mercury mirror

A mercury mirror is a vessel filled with mercury that is rotating so that the mercury assumes a parabolic shape. A parabolic shape is precicely the shape that the primary mirror of a telescope must have. Compared to a solid glass mirror that must be cast and ground and polished, a rotating mercury mirror is much cheaper to manufacture.

A telescope with a mercury mirror can only look straight up, so it is not suitable for investigations where the telescope must remain pointing at the same location of space.

To reduce weight, a rotating mercury mirror is a relatively thin layer that is distributed over the surface of a dish that is as close to the necessary parabolic shape as possible.

Currently, the mercury mirror of the Large Zenit Telescope is the largest mercury mirror in operation. It has a diameter of 6 meter, and it rotates at a rate of about 6 revolutions per minute.

Explanation of the equilibrium
Every rotation rate has a shape that is specific to that rotation rate. If the dish of a mercury mirror is spinning even slightly too fast, the mercury will part in the center. If the dish of the mercury mirror is rotating too slow, the mercury will slump down. When the rotation rate of the dish exactly matches the shape, then the mercury distributes into an even layer.

In fluid mechanics, the state when no part of the fluid has motion relative to any other part of the fluid is called 'solid body rotation'. When the mercury mirror has reached a state of solid body rotation, then the dynamic equilibrium can be understood as a balance of two energies: gravitational potential energy, and rotational kinetic energy. When a fluid is in solid body rotation it is the lowest state of energy that is available, because in a state of solid body rotation there is no friction to dissipate any of the energy



The dynamic equilibrium cannot be understood in terms of an equilibrium of forces, for when the mercury mirror is rotating, there is an unbalanced force acting on the mercury. The force of gravity is acting in vertical direction, the surface of the parabolic dish exerts a normal force on the mercury resting on it. The resultant force of those two provides the required centripetal force.

The following discussion is for the case of the mercury mirror as it is rotating in solid body rotation.

The kinetic energy of a parcel of mercury given by the formula:
 * $$ E_{kin.} = \frac{1}{2} m v^2 $$

In the case of circling motion the relation $$ v = \omega r $$ holds, hence


 * $$ E_{kin.} = \frac{1}{2} m \omega^2 r^2 $$

The formula for the potential energy can be calculated by integrating the force over distance to the center of rotation. The shape of the parabolic dish is such that at every distance to the center of rotation the required centripetal force is provided.

The required centripetal force is given by the following formula:
 * $$F_c = m \omega^2 r$$

Integrating that over r, the following expression for the gravitational potential energy for a parcel of mercury resting on the surface of the dish is obtained:


 * $$E_{pot.} = \frac{1}{2} m \omega^2 r^2$$

This shows that in solid body rotation the rotational kinetic energy and the potential energy are equal in magnitude. The cross section of the slope of the parabolic dish is such that mercury at a distance 2R from the center of rotation has 4 times more potential energy than mercury at distance R

Dissipation of energy
To understand the dynamics of energy it is also helpful to consider what happens when the operators of the mercury mirror stop driving the dish, in order to replace the mercury.

Let the rotating dish not be driven anymore, and let a gentle braking force be applied to the rotating dish. Friction between the dish and the mercury will tend to reduce the rotation rate of the mercury. As the mercury sags to the center, gravitational potential energy is converted to rotational kinetic energy. The conversion of potential energy tends to sustain the angular velocity. More precisely: when the mercury is giving in to the centripetal force, the centripetal force is doing work. When the amount of braking force that is applied is measured precisely, then it can be seen how much energy must be dissipated in order for the mercury mirror to lose its angular velocity. The amount of energy that must be dissipated is the total amount of energy: the rotational kinetic energy plus the gravitational potential energy.

Another interesting situation is a rotating dish filled with Helium, cooled to a temperature where the Helium becomes superfluidic. If the Helium is not only rotating, but also sloshing about somewhat, then the sloshing component will remain, for when the Helium is superfluidic there is no friction to dissipate any energy. The sloshing is an oscillation of some of the energy between kinetic energy and potential energy. If the dish with Helium is allowed to rise to a temperature where there is some friction then this friction will dissipate energy until the Helium is in solid body rotation. The friction makes Helium reach an energetically more favorable state. When the Helium is in solid body rotation there is still some potential energy and some rotational kinetic energy left, but there is no friction to dissipate that energy.