Problem 56 - Algebra and Trigonometry

Question

Reflecting Telescope The Hale telescope at the Mount Palomar Observatory has a \(200-\) in. mirror, as shown in the figure. The mirror is constructed in a parabolic shape that collects light from the stars and focuses it at the prime focus, that is, the focus of the parabola. The mirror is 3.79 in. deep at its center. Find the focal length of this parabolic mirror, that is, the distance from the vertex to the focus.

Step-by-Step Solution

Answer

The focal length of the parabolic mirror is approximately 659.65 inches.

Step 1: Identify The Function Of The Parabola

The mirror is shaped like a parabola and can be represented by the function \( y = ax^2 \), where \( y \) is the depth of the mirror and \( x \) is the half-width from the center of the mirror. The vertex is at the origin \((0,0)\) since the center of the mirror is \(3.79\) inches below the edge.

Step 2: Determine The Limits

The diameter of the mirror is \(200\) inches, so the radius \( x \) is \(100\) inches. Thus, when \( x = 100 \), \( y = 3.79 \).

Step 3: Plug Values Into Parabola Equation

Substitute \( x = 100 \) and \( y = 3.79 \) into the parabola equation: \( 3.79 = a(100)^2 \).

Step 4: Solve For 'a'

Rearrange the equation: \( a = \frac{3.79}{100^2} = \frac{3.79}{10000} = 0.000379 \). This tells us the value of \( a \) which defines the parabola.

Step 5: Calculate The Focal Length

The formula for the focal length \( f \) of a parabola \( y = ax^2 \) is given by \( f = \frac{1}{4a} \). Substitute \( a = 0.000379 \) into this formula: \( f = \frac{1}{4 \times 0.000379} \).

Step 6: Compute and Simplify the Focal Length

Calculate \( f = \frac{1}{0.001516} \approx 659.65 \). Thus, the focal length of the mirror is approximately \( 659.65 \) inches.

Key Concepts

Focal Length

In the study of parabolic mirrors, the focal length is a vital measurement. It represents the distance from the vertex of the parabola to its focus. For a mirror, this is where the light converges after being reflected off the surface. To find the focal length of a parabolic mirror, you can use the formula:

• Focal Length, \( f = \frac{1}{4a} \)

Here, \( a \) is a key parameter from the equation of the parabola. A parabolic mirror's efficiency in focusing light depends largely on accurately determining the focal length. For practical applications in telescopes, such as the huge Hale telescope, understanding and precisely calculating the focal length ensures that the collected light is sharply focused for optimal viewing.

Parabola Equation

The equation of a parabola is the mathematical expression that describes its shape. In the case of the reflecting telescope mirror, the parabola is described by the equation:

• \( y = ax^2 \)

In this formula:

• \( y \) represents the depth of the mirror from the vertex.
• \( x \) is the horizontal distance from the vertex (or half of the diameter).
• \( a \) is a constant that determines the curvature of the parabola.

This equation is pivotal when designing any parabolic mirror because it defines how the mirror will reflect light. A careful calculation of \( a \) is paramount as it influences the mirror's ability to properly focus the light at the focal point.

Vertex of a Parabola

The vertex of a parabola is its peak or the point where it changes direction. For a parabolic mirror, this is typically at its deepest point. The vertex is often considered the starting point for defining a parabola's attributes. Most commonly placed at the origin of a coordinate system,

• For a parabolic mirror in telescopes, the vertex is crucial because it provides the foundation for calculations.
• The vertex at the origin simplifies the equation by centering it around (0,0), making calculations involving the radius and height more manageable.
• In practical designs like telescope mirrors, placing the vertex accurately ensures the parabolic shape will function as intended, efficiently directing all incoming light to the focal point.

Reflecting Telescope

Reflecting telescopes use parabolic mirrors to gather and focus light. They differ from refracting telescopes, which use lenses. The design of reflecting telescopes has significant advantages due to its optical properties:

• The parabolic shape of the primary mirror allows reflecting telescopes to collect more light and focus it precisely on a focal point.
• Reflecting telescopes, like the Hale telescope, are less prone to chromatic aberration, a common issue in lenses where colors can be improperly focused.
• The light-gathering power of a reflecting telescope is crucial for viewing distant celestial objects with clarity and detail.
• Such telescopes are valuable for astronomical studies where clarity can reveal more detailed structures in outer space, enabling important discoveries.