6 Estudio De Mercado
6.1 Población o Mercado Objetivo
We will end this section with a brief discussion of the considerations that go into the design of a telescope so that the error analysis procedures described above will produce accurate results. This requires that the telescope be designed in such a way that the device itself does not introduce error in the raw data that goes beyond a specified tolerance for measurement error suited to the purposes for which the telescope is to be used. Thus, a complex error analysis task is involved in the design of a telescope for adequate performance, viz., to have the imaging system and camera effectively independent of the state of the device and any internally or externally produced physical perturbations. And, much like the case of error analysis just considered, a great deal of it also relies on modeling rather than statistics.
In order to ensure that a telescope system is capable of measurements that are accurate to a given precision ν0, it is necessary to ensure that forces exerted inter-
nally or externally to the telescope do not lead to displacements in the image plane of size larger than ν0. This is a constraint on possible designs of the telescope in much
the same way that the hamilton equations are a constraint on possible mathematical models. The basic components of an optical telescope are its optical imaging system, the mounting of the optical system that allows for its orientation with respect to some celestial coordinate system, the electrical drive mechanism that rotates the mounting to match the rate of rotation of the Earth (called a sidereal drive mechanism), and a mounting for an imaging instrument like a CCD camera. Given this simple specifi- cation of components and following the discussion in Roth et al. (1994), this general
design constraint gives rise to three more specific but simple design constraints:
I. The static constraint: static forces (depending, e.g., on the orientation of the mounting system) shall not cause displacements in the focal plane larger than
ν0;
II. The kinetic constraint: vibrations resulting from any source shall not excite amplitudes of over ν0/2 in the focal plane and should be attenuated as soon as
possible;
III. The kinematic constraint: image drift in the focal plane caused by kinematic in- accuracies (from,e.g., the sidereal drive mechanism) shall not exceed the amount
ν0 for a given observing time tb.
These three constraints act much like the basic equations of a theory, providing general constraints on the design of an optical telescope that can ensure accuracy to the precision ν0. This is so because this system of constraints is highly undercon-
strained. There is a wide range of possibilities of kinds of optical systems (refracting, reflecting, both), kinds of mountings (e.g., horizontal mountings, equatorial mount- ings), kinds of drives (different drive mechanisms, motors, control circuitry, etc.), supporting electrical systems, guiding systems for locating stars, signal processing,
etc. Any number of combinations of such designs can meet the above three design cri- teria. Thus, much like the process of model construction detailed in the last chapter, additional constraints must be introduced to construct a design model.
Of course one would select a number of ideal criteria, such as what kind of control of the orientation of the optical system is possible, but these kinds of constraints are flexible. There are other constraints that are less flexible or inflexible, including available finances, available sources of raw materials and components, and accessible facilities for manufacturing and machining along with the technical expertise in these areas. These constraints help to significantly narrow down the possibilities for what design models will be feasible. Aside from these more obvious constraints, the three design constraints above can be used to compute a variety of more detailed constraints on a design model that is able to meet the ν0-precision design constraint.
To ensure that it is even possible to meet the first two basic design criteria (I and II), it is necessary to model how a given mounting design responds to internally and
externally generated forces. The response of the system to forces must be analyzed in terms of vibrations and displacements of the focal plane. Accordingly, the model of the mounting system is a kind of error model, used to model the measurement error that will result from the measurement device itself. This is another case where an error model does not rely significantly on statistics, but rather on models of elasticity from continuum mechanics. Simple elasticity models allow more detailed design constraints to be computed that guide the solution of the design problem.
It can be shown that thestiffness of materials is the dominant concern for meeting design constraints I and II. For a simple linear model of elasticity, the stiffness is simply the constant c of proportionality in the hooke equation
F=cx,
where c has units of newtons per metre.10 There are two reasons for this: stiffness
determines the organization of all of the parts so that constraint I is satisfied; and stiffness also plays an important role in determining the response of the telescope to vibration and can be used to control vibration (constraint II) (Rothet al.,1994, 148). In terms of the static constraint, it can be shown using results from the theory of elas- ticity thatn parts connected in series must haven times the required system stiffness and that the most important stiffness constraint is high flexural stiffness, i.e., resis- tance to bending. In terms of constraint II it can be similarly shown that the design of the mounting should simultaneously meet two conditions: 1) aim for high natural frequencies (which improve vibrational damping properties) and high mechanical im- pedence (to insulate the system from external vibrations), and 2) design every part with maximal stiffness and minimal mass and avoid unnecessary attachments coupled to the image plane.
In meeting constraint III, statistics becomes a more dominant concern, since meet- ing this condition has much to do with parts of the system being manufactured, machined and connected to spec, which also requires accurate measurements of the properties of the materials and their arrangement. Consequently, these concerns fall within the domain of concern of the standard theory of measurement and error.
The interest in considering how error is handled in the cases of image reduction and telescope design is that a great deal of the analysis of error in these cases is handled using error models that are more similar to the mathematical models of the target phenomenon, i.e., the principled models of the “top half” of the model hi- erarchy, than the models of error used in statistical analysis on the “bottom half”. Statistical methods of error analysis are based on mathematical models of random- ness and uncertainty, but they are applied as mathematical algorithms for processing raw data into a data model and as algorithms for analyzing the error introduced in direct measurements. Error models for image reduction and for meeting the static and kinetic design constraints for a telescope, on the other hand, are presented as constraint problems for which a solution is to be computed.
In the design case, the combination of principled models of the components of the measurement system and their couplings together with a formulation of the design problem as a constraint problem guides the design process in a way that makes solving the problem of finding a design that meets all of the strict constraints, including the ν0 level of precision, feasible. Moreover, it provides assurance that this level of
precision isstable, since it allows the designer, or operator after the fact, to know what assumptions this level of precision relies upon. This then can guide a maintenance schedule to ensureν0precision performance and the troubleshooting process of finding
the source of a detected failure to meet the precision requirement. Having a reliable assurance that the precision constraint is met under given operating conditions, the operator can then be assured that the six-parameter error model for image reduction is sufficient for ensuring that the measured value of the celestial position of an observed object is indeed accurate to within around the calculated measurement error. Thus, having a way of knowing whether the measurement system is operating to spec, by knowing how it responds to internal and external physical perturbations, it is possible to ensure that the operator of the device can interface with it using only a very simple error model.