Question

Part A Listed following are the names and mirror diameters for six of the world’s greatest reflecting telescopes used to gather visible light. Rank the telescopes from left to right based on their light-collecting area from largest to smallest. For telescopes with more than one mirror, rank-based on the combined light-collecting area of the mirrors.


Part B Shown following are the primary mirror arrangements and total light-collecting area of five different telescopes. Each mirror uses a different segmented arrangement, but assume that they are all equivalent in quality and in their ability to focus light. Also assume that the telescopes use identical detectors and have the same observing conditions. Rank the telescopes from left to right based on their ability to detect very dim objects, from greatest to least. To rank two (or more)telescopes as equal, drag one on top of the other(s).

Part C Shown following are the primary mirror arrangements and total light-collecting area of five different telescopes. Notice that although the arrangements look similar to those in Part B, the light-collecting areas are not the same. Also listed is an amount of time (exposure time) that each telescope will be pointed at the same distant galaxy. Again assume that the quality of these mirrors, the detectors, and the observing conditions are identical. Rank the telescopes from left to right based on the brightness of the image each telescope will take of the galaxy in the time indicated, from brightest to dimmest. To rank two (or more) telescopes as equal, drag one on top of the other(s).

Answer 1

Concept and reason

The concepts used to solve this problem are light collecting area and the relation between the brightness and the surface area.

The expression for the light collecting area of the telescope is used to rank the telescopes on the basis of their light collecting area.

The expression for the surface brightness is used to rank the telescopes on the basis of their brightness.

Fundamentals

The expression for the light collecting area of the telescope is given by,

$A = n\pi \left( {\frac{{{D^2}}}{4}} \right)$

Here, D is the diameter of the mirror and n is the number of mirrors in the telescope.

The expression for the surface brightness is given by,

$SB = \left( t \right)\left( A \right)$

Here, $SB$ is the surface brightness, t is the exposure time, and A the light collecting area.

(A)

The light collecting area of the telescope is given by,

$A = n\pi \left( {\frac{{{D^2}}}{4}} \right)$

The Large Binocular telescope consists of two mirrors each having the diameter of 8.4 m.

Substitute 8.4 m for D and 2 for n in above equation as follows:

$\begin{array}{c}\\A = \left( 2 \right)\pi \left( {\frac{{{{\left( {8.4{\rm{ m}}} \right)}^2}}}{4}} \right)\\\\ = 110.8{\rm{ }}{{\rm{m}}^2}\\\end{array}$

The Hobby- Eberly telescope consists of one mirror having the diameter of 9.2 m.

Substitute 9.2 m for D and 1 for n in above equation as follows:

$\begin{array}{c}\\A = \left( 1 \right)\pi \left( {\frac{{{{\left( {9.2{\rm{ m}}} \right)}^2}}}{4}} \right)\\\\ = 66.5{\rm{ }}{{\rm{m}}^2}\\\end{array}$

The Magellan II telescope consists of one mirror having the diameter of 6.5 m.

Substitute 6.5 m for D and 1 for n in above equation as follows:

$\begin{array}{c}\\A = \left( 1 \right)\pi \left( {\frac{{{{\left( {6.5{\rm{ m}}} \right)}^2}}}{4}} \right)\\\\ = 33.2{\rm{ }}{{\rm{m}}^2}\\\end{array}$

The Keck I telescope consists of one mirror having the diameter of 10 m.

Substitute 10 m for D and 1 for n in above equation as follows:

$\begin{array}{c}\\A = \left( 1 \right)\pi \left( {\frac{{{{\left( {10{\rm{ m}}} \right)}^2}}}{4}} \right)\\\\ = 78.5{\rm{ }}{{\rm{m}}^2}\\\end{array}$

The Subaru telescope consists of one mirror having the diameter of 8.3 m.

Substitute 8.3 m for D and 1 for n in above equation as follows:

$\begin{array}{c}\\A = \left( 1 \right)\pi \left( {\frac{{{{\left( {8.3{\rm{ m}}} \right)}^2}}}{4}} \right)\\\\ = 54.1{\rm{ }}{{\rm{m}}^2}\\\end{array}$

The Gemini North telescope consists of one mirror each having the diameter of 8 m.

Substitute 8 m for D and 1 for n in above equation as follows:

$\begin{array}{c}\\A = \left( 1 \right)\pi \left( {\frac{{{{\left( {8{\rm{ m}}} \right)}^2}}}{4}} \right)\\\\ = 50.3{\rm{ }}{{\rm{m}}^2}\\\end{array}$

(B)

The ability to detect the object depends on the surface area of the mirrors.

The surface area for all the telescopes is equal. All the telescopes have equal ability to detect very dim objects.

(C)

The relation between the brightness and the surface area of the telescope is given by,

$SB = \left( t \right)\left( A \right)$

Substitute $5{\rm{ }}{{\rm{m}}^2}$ for A and 4 min for t in above equation to calculate the brightness of the telescope 1 as follows:

$\begin{array}{c}\\SB = \left( {4{\rm{ min}}} \right)\left( {5{\rm{ }}{{\rm{m}}^2}} \right)\\\\ = 20{\rm{ W/}}{{\rm{m}}^2}\\\end{array}$

Substitute $1{\rm{ }}{{\rm{m}}^2}$ for A and 20 min for t in above equation to calculate the brightness of the telescope 2 as follows:

$\begin{array}{c}\\SB = \left( {20{\rm{ min}}} \right)\left( {1{\rm{ }}{{\rm{m}}^2}} \right)\\\\ = 20{\rm{ W/}}{{\rm{m}}^2}\\\end{array}$

Substitute $2{\rm{ }}{{\rm{m}}^2}$ for A and 10 min for t in above equation to calculate the brightness of the telescope 3 as follows:

$\begin{array}{c}\\SB = \left( {10{\rm{ min}}} \right)\left( {2{\rm{ }}{{\rm{m}}^2}} \right)\\\\ = 20{\rm{ W/}}{{\rm{m}}^2}\\\end{array}$

Substitute $20{\rm{ }}{{\rm{m}}^2}$ for A and 1 min for t in above equation to calculate the brightness of the telescope 4 as follows:

$\begin{array}{c}\\SB = \left( {1{\rm{ min}}} \right)\left( {20{\rm{ }}{{\rm{m}}^2}} \right)\\\\ = 20{\rm{ W/}}{{\rm{m}}^2}\\\end{array}$

Substitute $10{\rm{ }}{{\rm{m}}^2}$ for A and 2 min for t in above equation to calculate the brightness of the telescope 5 as follows:

$\begin{array}{c}\\SB = \left( {2{\rm{ min}}} \right)\left( {10{\rm{ }}{{\rm{m}}^2}} \right)\\\\ = 20{\rm{ W/}}{{\rm{m}}^2}\\\end{array}$

Ans: Part A

The order of the telescopes from the largest collecting area to the smallest collecting area is $1 > 4 > 2 > 5 > 6 > 3$.

Part B

The order of the telescopes based on the ability to detect the very dim objects is $1 = 2 = 3 = 4 = 5$.

Part C

The order of the telescopes based on the brightness is $1 = 2 = 3 = 4 = 5$.