Appendix A: Derivation of the Secondary Sag as a Function of Telescope Altitude

One-half of the secondary support structure consists of the two rods shown in Figure 9 plus the opposing rods on the other side of the secondary cage. The total tension force on a single rod can be decomposed into a thermal component and a base tension that is set at the time when the rods are originally tensioned with the turnbuckles. The base tension of a single rod is not a fixed quantity, but is rather a variable which responds to the force balance between all of the rods attached to the secondary cage. I assume here that the cage is aligned symmetrically so that only half of the rods need to be considered at a time in this force balance.

The force on a single rod is given by

where

F_T = the base tension on the rod,

= the thermal tension given by changes in the rod temperature from the original rod installation, and

= the tension changes in the rod owing to a variable gravity load as the telescope changes orientation.

Note that the thermal changes are defined such that positive values of correspond to increases of rod temperature. I will define "axial" forces as those in the direction of the telescope primary. Radial forces are those perpendicular to the telescope boresight. The axial force, F_a, on a single rod is given by

where a = the angle between the rod and the radial direction. Figure 9 shows that a=13.6º for the bottom rods and a=18.2º for the top rods. I will call these angles a₁ and a₂, respectively.

To keep track of each rod I will use the subscripts lb., lt, rb, and rt which refer to left-bottom, left-top, right-bottom, and right-top, respectively. For example, F_lb denotes the total force on the left-bottom rod and F_lbT denotes the base tension on this same rod.

For convenience, I will also assume that the rods considered are aligned with the horizon when the telescope is at zero degree altitude. The end results derived here are independent of this assumption. The alignment of the rods with respect to the gravity vector only affects how the radial loads are distributed on the rods at the horizon and the actual angular dependence of the sag with altitude. It does not affect the total magnitude of the sag.

At the horizon all of the forces will assumed to be zero. This can be done because the starting focus point is arbitrary for this calculation. The axial force balance equation is then given by

(1)     

At the zenith the gravity forces act in opposition to the axial tension in the bottom rods and in parallel to the axial force in the top rods. I account for this by writing the force for the top rods as

and by writing the force for the bottom rods as

With this convention the force of gravity is always considered positive and the axial force balance at zenith becomes

(2)     

where W is the weight of the secondary cage including the mirror and electronics. I have assumed in the equation above that the cage is support symmetrically by both halves of the support system of rods. Combining equations (1) and (2) above gives

(3)     

Note that all of the thermal terms have canceled. In other words, as long as there are no temperature changes between the time the telescope is at the zenith and when it is at the horizon, thermal changes will not alter the differential sag that takes place as the telescope moves in altitude. This is true even if all of the rods are not at the same temperature. This, however, is not the whole picture because this calculation does not tell you where the starting point of the secondary lies, it will only give you the differential position between horizon and zenith orientations. But, it is instructive to finish the calculation that we have started before moving on to address this issue. I will address the issue of the absolute position of the secondary in Appendix B.

If the cage is stiff, then we can assume that all of the rods feel the same gravitational force. In other words, we have

We make this simplification in equation (3) and solve for ђF_g to give

(4)     

The definition of stress gives us the equation

(5)     

where F is the force on a rod, A_rod is the cross-sectional area of a rod, E is the elastic modulus of the rod steel, is the length change of the rod, and L is the original length of the rod. We can substitute for F in equation (5) if we interpret to be the change in length from the original tension balanced position. Doing this, combining equations (4) and (5), and solving for gives

(6)     

Equation (6) gives the change in length of a rod due to gravity as the telescope moves from the horizon to zenith. We must now convert this to the equivalent change in the axial position of the secondary cage. Figure A1 shows the geometry required to make this conversion and the derivation of the equation

(7)     

which gives the relationship between the axial motion of the secondary cage, , as a function of the change in length of a support rod, . We now combine equations (6) and (7) to solve for the total secondary sag expected as the telescope moves from the horizon to the zenith. This gives

(8)     

were L is the length of any rod of your choice and a is the angle of this rod to the direction of the telescope radius. Note that if the cage were symmetric with all of the rod lengths equal and a₁ = a₂, then we would have

(9)     

For the case of the 3.5-m, equation (9) can be used in place of equation (8) with only minor errors (10%) if L is the average length of both top and bottom rods and a is the average angle of both top and bottom bars.

As a recap, the variables that go into equations (8) and (9) are

L = the length of a secondary support rod

W = the weight of the secondary cage, mirror, and electronics

A_rod = the cross sectional area of a single rod

E = the elastic modulus (Young's modulus) of the rod steel

a₁ = the angle that the lower rods make with respect to the telescope radial direction (i.e. with respect to horizontal when the telescope is at the zenith)

a₂ = the angle that the upper rods make with respect to the telescope radial direction