# Investigation into Resolution and Sensitivity

## Syllabus Requirement

As part of bold point one, students must:

*identify data sources, plan, choose equipment or resources for, and perform an investigation to demonstrate why it is desirable for telescopes to have a large diameter objective lens or mirror in terms of both sensitivity and resolution*(skill)

## Introduction

Modern telescopes have large primary mirrors so as to be more sensitive, that is gather more photons from distant celestial objects. Theoretically, increasing the size of the primary also increases the resolution of a telescope although in practice atmospheric effects degrade the actual performance of ground-based telescopes. What is the actual relationship between size of a primary and a) sensitivity? and b) resolution?

## Part I: Sensitivity

There are three ways you can simulate the effect of primary mirror size on sensitivity.

### Method 1: Filling up the "Light Bucket"

- Draw circles representing primary mirrors. Start with a radius of 1cm and draw several up to 15cm radius. Cut them out then paste them on thick card or plywood.
- Cut out a 1cm wide length of paper equal in length to the circumference of the mirror plus 1cm. Glue it around the circumference of the mirror so that it makes a raised edge. Repeat for the other "mirrors".
- Using dried peas, round lentils or something similar to represent "photons", fill up each mirror so that it is full.
- Either count up up the total number of "photons" in each mirror or weigh their total mass or volume as an analogue of total number.
- Tabulate you data and then plot your results. Can you determine a relationship between mirror size and sensitivity? You may need to plot more than one graph to verify the actual mathematical relationship.

If you want to use some experimental data, the table below provides some actual values from a simulation using dried lentils. Use this data to try and determine a relationship between mirror size and sensitivity.

Mirror Diameter (cm) | Vol (mL) | Mass (g) |
---|---|---|

2 | 4 | 1.97 |

4 | 10 | 6.19 |

6 | 28 | 15.94 |

8 | 32 | 23.25 |

10 | 54 | 38.02 |

12 | 80 | 60.7 |

14 | 100 | 77.53 |

16 | 140 | 99.91 |

20 | 230 | 160.77 |

30 | 450 | 367.68 |

### Method 2: Tiddly Winks

- Draw circles representing primary mirrors. Start with a radius of 1cm and draw several up to 15cm radius. Label them and cut them out.
- Use tiddly winks or some are similar small, circular marker such as paper from a hole puncher to represent "photons". Place the "photons side by side on each mirror until they are all filled up.
- Count the number of photons each mirror has collected and tabulate your data.
- Plot the number of photons against mirror size and determine what relationship, if any, there is between the two.

### Method 3: Mathematical Model

- Construct a table similar to the one below for a variety of telescopes from the last four centuries. Allocate a small number of "photons" as the relative number that a human eye can receive. In the example below, three have been selected. You can either perform your calculations manually or set up a spreadsheet to help with your calculations and plotting.

Telescope Diameter of

Primary (m)Radius

(m)Number of "photons" Sensitivity compared to eye human eye 0.008 0.004 3 1 Galilean

refractor0.03 0.015 4-inch refractor 0.1 0.05 8-inch Dobsonian 0.2 0.1 60-cm reflector 0.6 0.3 Yerkes Refractor 1 0.5 Herschel's reflector 1.2 0.6 Lord Rosse's 1.83 0.915 ANU 2.3m 2.3 1.15 AAT 3.9 1.95 Hale 5 2.5 Bolshoi Teleskop 6 3 Gemini 8.1 4.05 Keck 10 5 CELT 30 15 OWL 100 50 - What other columns may you need to add to the table? (Hint: What effect does increasing the diameter of the primary have on its surface area?)
- If you want to check your results with a sample solution, you can download an
*Excel*spreadsheet file here (30KB). - What is the mathematical relationship between the size of a primary mirror and its sensitivity?

## Part II: Resolution

### Method 1: Mathematical Model

A simple investigation for resolution involves a mathematical model using the resolution equation:

- Using the information supplied for the same telescopes as in the sensitivity model above, calculate the theoretical resolution each telescope can achieve. Assume observations are at a wavelength of 550nm, ie peaking in yellow light. Remember, θ = resolution and D = diameter of primary. D and λ need to be in the same units.
- Calculate the relative resolution of each compared to the human eye.
- Plot your results and verify the mathematical relationship graphically.

Telescope | Diameter of Primary (m) |
Radius (m) |
Theoretical resolution (arcsec) |
milli- arcsec | Resolution compared to eye |
---|---|---|---|---|---|

human eye | 0.008 | 0.004 | 1 | ||

Galilean refractor |
0.03 | 0.015 | |||

4-inch refractor | 0.1 | 0.05 | |||

8-inch Dobsonian | 0.2 | 0.1 | |||

60-cm reflector | 0.6 | 0.3 | |||

Yerkes Refractor | 1 | 0.5 | |||

Herschel's reflector | 1.2 | 0.6 | |||

Lord Rosse's | 1.83 | 0.915 | |||

ANU 2.3m | 2.3 | 1.15 | |||

AAT | 3.9 | 1.95 | |||

Hale | 5 | 2.5 | |||

Bolshoi Teleskop | 6 | 3 | |||

Gemini | 8.1 | 4.05 | |||

Keck | 10 | 5 | |||

CELT | 30 | 15 | |||

OWL | 100 | 50 |

To check your calculations you can download an *Excel* spreadsheet file with values calculated here (30KB)