C) More on Magnitudes

By the 1900’s, astronomers wanted to record and discuss with each other the brightness of objects in the sky. To do this, they adapted the magnitude scale so it became a measurement that is not dependent on the judgment of the observer, the instrument being used, or how clear the sky is. Let’s listen to Chris discuss the magnitude scale with his friend Rachel in this 6:05 video. The information is reviewed in the text below the video.

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The first advancement came when astronomers agreed to define a first magnitude star as 100 times brighter than a sixth magnitude star. In other words, a difference of 5 magnitudes equals a difference in brightness of 100.

Imagine a normal staircase. Each step increases your height above the first step by the same amount. You just keep adding (stepping up) by the same amount to determine the total height increase. Now, imagine a staircase where your total height after each step is multiplied by the same amount. What would it feel like walking up those steps? (Comment in journal below).

The magnitude scale is like the second staircase. It is a multiplicative scale, whereas the first scale is an additive scale. Each step is 2.512 TIMES higher than the previous step from the floor. The 2.512 comes from defining the difference between 1st magnitude and 6th magnitude as 100, as mentioned above.

JOURNAL QUESTION:

a. Can you figure out the mathematical relationship between 100, 2.512 and the magnitude difference between one and six? Comment in the JOURNAL below.

QUORUM ACTIVITY

A PDF of a textfile of the commands of this Part is available at this link:

Let’s explore further, using Quorum to construct our “steps”. Either use the Quorum Box at the bottom of the page, or use the link below to open a Quorum Box in a new tab, which you can easily navigate to using the keyboard shortcut “ctrl tab#”. Or arrange both windows on your screen side by side. Use what works best for your vision.

The highlighted lines below are the Quorum commands.

First, define a number variable equal to 2.512. “x” is used in our example as this number variable:

number x = 2.512

Now define a new number variable called stepheight1 that is equal to x:

number stepheight1 = x

Do you see why the first step height is simply x? Say or Output the variable x in a friendly format:

output “Step Height 1 is ” + x

The next step height is x times x, so:

number stepheight2 = x*x

output “Step Height 2 is ” + stepheight2

Continue for steps 3-10. (Steps 3-6 are done below for you:)

number stepheight3 = x*x*x

output “Step Height 3 is ” + stepheight3

number stepheight4 = x*x*x*x

output “Step Height 4 is ” + stepheight4

number stepheight5 = x*x*x*x*x

output “Step Height 5 is ” + stepheight5

number stepheight6 = x*x*x*x*x*x

Finish the code for steps 7-10.

JOURNAL QUESTIONS

1. Everyone should use the screen reader mode (or “voiceover” if using a Mac) to listen to the command line to calculate the height of step 10. If you have never used this before, ask for help in locating it on your computer. Record, in the JOURNAL Box below, three adjectives that describe the experience.

2. Imagine now that you rely on screen readers to review code. Think of another way to code this that would be easier to listen to. Discuss with your group.

3. Is the height of that final step above the floor what you expected?

Copy and paste the “Console Output” window into your Journal. Then look below the Journal for several more activities to choose from to help understand this multiplicative scale.

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JOURNAL BOX

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OPTIONAL HANDS ON ACTIVITIES

1. Spaghetti Noodle Break

– a) A package of dry, long spaghetti noodles are divided into a bundles of 1, 2.5, 6.3, 16, 40, and 100 noodles, representing the multiplicative scale (multiplying by 2.5) for the apparent magnitudes of 1 through 6.

– b) The energy needed to break each bundle in half represents the energy the CCD camera receives from magnitude one (100) to magnitude six (1) stars.

– c) IF the magnitude scale were an additive scale, one noodle would instead be ADDED for each step down to a first magnitude star: 1 noodle = magnitude 6, 2 noodles = magnitude 5, 3 noodles = magnitude 4, etc. Try this. Do you now have a feel for the energy difference between a additive scale and a multiplicative scale?

2. Glass gems drop:

– a) 100 glass gems (purchased from the dollar store, or ones used in histogram, etc.) are dropped, all at once, into a metal container to represent the energy from a 1st magnitude star. Then 40, 16, 6.3, 2.5, and 1. Break one gem with a tap from a hammer to get the 0.3 and 0.5 fractions! **Note: you can also use pennies, just round the numbers to whole numbers.

– b) Try this as an additive scale, dropping, 6, 5, 4, 3, 2, then 1 gem. It should be easy to hear the difference between these two types of scales, the multiplicative scale (in this case, 2.512), and the additive scale (just plus one for each step).

3. Spring Push (a 3D model has been developed, please order from GLAS Education if interested!)

– a) Five (or more) different springs are mounted in a 3D printed box. Each has a spring constant (resistive force) representing a 2.5X increase.

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