Appendix B: The Effects of Thermal Changes on the "Absolute" Position of the Secondary

In Appendix A I showed how to calculate the differential secondary motion that comes from changes in the telescope altitude. Here I will show how I calculate the effects of thermal changes on the absolute position of the secondary. The term "absolute position" is admittedly a little misleading. I use it in this discussion primarily to distinguish these motions from those caused by changes in the telescope altitude. The motions discussed here are superimposed on the altitude sag discussed in Appendix A. Given static conditions, they can be thought of as changes in the starting point of the secondary as the telescope is at the horizon.

I will use the same nomenclature for the rod forces here as I used in Appendix A.

In Appendix A, the axial force balance equation at the horizon was shown to be

(1)     

Likewise, the radial force balance equation at the horizon is given by

(2)     

Multiplying equation (1) by cos( a₁), equation (2) by sin( a₁) and then subtracting the two resultant equations gives the fairly messy equation

(3)     

We are dealing with small angles for a₁ and a₂ . If we convert these angles to radians, we can approximate cos(a) and sin(a) as the first term of their Maclaurin series expansions

For the angles used on the APO 3.5-m secondary cage, these substitutions can be made with errors of only 5%. Doing so in equation (3) results in

(4)     

Rearranging equation (4) gives

(5)    

We can use the defining relationship between stress and strain to write equation (5) in terms of the elongation of each rod from its unstressed equilibrium length. This equation is

where F is the force on a rod, A is the cross-sectional area of the rod, E is the elastic modulus of the rod's steel, L is the equilibrium length of the rod, and is the elongation of the rod caused by the applied stress. The change in force on a rod caused by thermal changes is given by combining the stress-strain equation with the equation that defines the coefficient of expansion, C_te,

Using these two relationships in equation (5) gives

(6)    

If you have been following the derivation up to this point, you know that the s in the equation above refer to the total elongation that the rods are under. The quantity that we really wish to derive is , where is the rod elongation at some initial epoch (i.e. when the rods were first tensioned) and where is the rod elongation at some later time after the rod temperatures had changed. We are free to assume that at the initial epoch we have isothermal conditions and that we will consider to represent the changes in the rod elongation from this isothermal state. Therefore, at the initial epoch we have . At the initial epoch, equation (6) becomes

We can then subtract this equation from equation (6) without changing the form of the right hand side of equation (6). The only change to the left-hand side of equation (6) is a substitution of everywhere you see . In other words, we can simply reinterpret what we mean by ђL. From now on, we will interpret to mean the change in rod elongation from some initial isothermal state and is the change in rod temperature from this initial state.

We now make the assumption that at the second epoch, the secondary rods are again isothermal. This means that we now assume that. We also make the simplifying approximation that L_lb = L_lt = L_rt = L. This isn't necessary to find a solution, but it makes the calculations much easier at the expense of only a 2% error in the final results. Adding this assumption and approximation to equation (6) gives

(7)    

From the symmetry of the secondary cage supports we can make one further simplification to equation (7). We know that for isothermal conditions we have the same change in elongation for both top rods. This means that we have . With this simplification we can now drop the dual subscript notation and simply refer to the change in elongation of the bottom rods, , and the change in elongation of the top rods, . Clearly, we have and . Making these substitutions in equation (7) and rearranging terms gives us

(8)    

We now need to convert the rod elongations in equation (8) into the equivalent axial sag of the secondary cage. In Figure A1 I show how to a very good approximation, this conversion is given by the equations

and

I also assume now that the secondary cage is stiff. Under this assumption we can say that . Substitution of these equations into equation (8) and solving for gives us the relationship that we have been looking for

(9)    

It is necessary to make a few comments about equation (9). First, it is applicable only when we have a₁ unequal to a₂. This is very important. If the cage is completely symmetric and if the cage rods are all isothermal, then you can see from equation (5) that there will be no temperature dependence to the secondary cage position! Equation (9) becomes invalid because you will have divided by zero in the derivation. It is important to realize that if the cage is completely symmetric, then the only temperature dependence that you will see will come from non-isothermal conditions. In particular, if you have temperature differences between the top and bottom rods, you can still get a temperature dependence to the cage position.