After the Palomar 5 -m telescope was completed, astronomers...

After the Palomar 5 -m telescope was completed, astronomers estimated that the mass of larger telescopes would increase as the (mirror diameter) $^{3}$ or (mirror diameter) $^{4}$ because of the greater mass of glass and support needed to keep the entire telescope stiff. If the 10 -m Keck telescopes followed this rule, how many times more massive would they have been than the Palomar telescope?

Key Concepts

Scaling Laws

Scaling laws describe how physical properties such as mass, volume, or strength change in relation to the size of an object. In the context of telescopes, if the mass of a telescope is proportional to the mirror diameter raised to a certain power, doubling the diameter can lead to an exponential increase in mass, such as eight times (for the cube) or sixteen times (for the fourth power). This concept helps in estimating the material and engineering challenges that arise with larger telescopic mirrors.

Structural Engineering in Telescope Design

As telescope size increases, the structural demands also increase because larger mirrors need additional support to maintain their shape and precision. This extra support structure adds to the telescope's overall mass. The design must account for the stiffness and stability required to counteract gravitational forces and potential deformations, which is why the mass might scale with a higher power than the mirror diameter alone.

Polynomial Scaling in Engineering

Polynomial scaling refers to relationships where one quantity changes as a power of another; in this case, the telescope mass scales as a polynomial function of the mirror diameter. An exponent of 3 or 4 indicates a steep increase in mass with diameter, signifying not only the increased mirror material but also the added structural components required to support the larger size. This is a fundamental concept in engineering design, where scaling considerations directly impact feasibility, cost, and materials selection.

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