Fomalhaut, or Alpha Pisces Austrinis, as seen from Stellarium software.

One of the fundamental requirements for astronomy to work as a science is the need to accurately measure the distances to objects. Astronomers have developed a series of methods for measuring stellar and cosmic distances, which fit together and inform each other. Taken together, they’re referred to as the Distance Ladder, as each one provides the basis for the next step out in distance.

Website for our MIT BLOSSOMS video lesson plan on using the parallax method to find the distances to nearby stars.

For the first rung on the Distance Ladder, for objects closer than about 100 light years, we use the parallax method. I developed a lesson plan to teach this and my students and I made a video of this lesson for the MIT BLOSSOMS project. You can go to their website to see this video and download the lesson plan at:

https://blossoms.mit.edu/videos/lessons/parallax_activity_measuring_distances_nearby_stars 

I’ve also written a previous post about the parallax method here:

https://spacedoutclassroom.com/2012/12/07/the-parallax-method/.

As I developed lesson plans to use on my poster for the American Astronomical Society conference, I decided to revise my lesson that uses absolute and apparent magnitudes to determine the distances to stars. It would be a good way to introduce collecting and using astronomical data. My students were already familiar with the constellations, stellar classifications, and the Hertzsprung-Russell Diagram, they had been gathering data about the stars using Stellarium software, and had completed the parallax lesson, so they were ready to go. But this lesson requires some explanation:

Stellar Magnitudes:

Beyond 100 or so light years, we have to use a method called the Distance Modulus formula. To use it, one has to measure how bright a star appears (apparent magnitude, or m) with high accuracy. How bright a star appears is based on two things: how close it is and how much light the star actually emits. Astronomers eliminate differences in distance as a factor by pretending to move all stars to same distance: 32.6 light years or ten parsecs. Their brightness from this distance is called their absolute magnitude (M).

Separating out a spectrum of the star’s wavelengths provides a fingerprint that identifies the star’s classification, using the pattern of absorption lines and flux densities based on Wien’s Law. We’ve learned enough about each type and sub-type of star to know how much total light it gives off. This is called its luminosity, and is measured compared to our sun.

Hipparchus, who created the first accurate star catalog about 130 B.C. He provided magnitude numbers from 1 for the brightest stars to 6 for the dimmest that could be seen with the unaided eye.

Hipparchus, Herschel, and Newton:

To figure out the distance to a star is therefore to compare how bright a star really is (absolute magnitude) with how bright a star appears (apparent magnitude). Now this isn’t quite as straightforward as it seems. Sir Isaac Newton discovered that light follows an inverse square law – that the brightness of a light falls with the square of its distance. In other words, a light that is twice as far away will be one fourth as bright as before. It is an exponential curve.

Greek astronomers, as painted by Raphael in The School of Athens. Hipparchus is holding the celestial sphere.

Another problem is that the original magnitude scale was developed by the Greek astronomer Hipparchus in about 150 B.C. He created the first star catalog and assigned the stars numbers based on their perceived brightness, with the brightest star (Sirius) given the number 1 and the dimmest star visible the number 6. This inverted scale has stuck with us and can be a bit tricky to understand. The important thing is that the higher the magnitude number, the dimmer the star is. Lower numbers mean brighter stars.

Sir William Herschel, who discovered that five differences in magnitude are about 100 times difference in brightness.

Once Newton put light on a mathematical basis, astronomers wanted to standardize the magnitude system so that mathematical formulas could be used. William Herschel, along with his sister Caroline, cataloging thousands of stars (and discovered Uranus along the way). They discovered that a magnitude 1 star was roughly 100 times brighter than a magnitude 6 star, or that five orders of magnitude produce a 100 fold change in brightness. Using this, the magnitude scale was adjusted to make it come out exactly 100 times, so some stars such as Sirius now have negative apparent magnitudes.

The Modulus Formula:

With the magnitude scale adjusted to fit a logarithmic curve, you can now say that one star is exactly so many times brighter than another. You can represent this relationship with the formula: (M – m – 5)/-5 = logD , where M is the absolute magnitude of the star, m is the apparent magnitude, and D is the distance in parsecs.

Let’s work this through for the star Fomalhaut. It has an apparent magnitude of 1.15, and an absolute magnitude of 1.72. So plugging in the numbers gives us: (1.72 – 1.15 – 5)/-5 = 0.886 = logD. Taking the antilog of 0.886 gives us 7.69 parsecs. Since there are 3.26 light years in a parsec, the distance to Fomalhaut is therefore 25.1 light years. Now, since this is also a nearby star, we can use the parallax method to double check the distance. The methods in the Distance Ladder back each other up.

Modulus Method Lesson Page 1

It gets harder when stars are so far away that you can’t accurately measure their apparent magnitude, or if they are in distant galaxies, etc. There are other methods in the Distance Ladder, such as Hubble’s Law, that can give a distance in an expanding universe based on the degree of a galaxy’s red shift. In between, there is a method first pioneered by Henrietta Leavitt based on the precise period-luminosity function of Cepheid variable stars. Hubble used her work to determine the distance to the Andromeda galaxy.

The Magnitude – Luminosity Function. Since luminosity varies with the inverse of the distance squared, it is an exponential curve. In this case, if you know the ratio of luminosity between two stars, you can use the curve to determine the differences in magnitudes. The modulus formula uses logarithms to do the same calculations.

The Lesson Plan:

Now that I’ve explained all that, its time to try out the lesson plan. All three pages are attached here and can be downloaded. I had an older version of it in the form of often duplicated printouts, but no remaining digital copy, so I scanned the pages in and revised the explanation and data tables to make the process work better, then put student samples on my poster for AAS.

In addition to using the Modulus Formula directly, the lesson shows how to do the same using the inverse square light curve. With the curve, one can figure out the magnitude differences (between apparent and absolute or between two different stars, such as Fomalhaut and Alpha Centauri) and determine the difference in brightness. Or go the other direction – knowing the differences in brightness, one can determine the differences in magnitudes.

Modulus lesson plan Page 3

One trick is remembering whether a star has to be moved forward or back to get it to 32.6 light years. If forward, its absolute magnitude will be a lower number than its apparent magnitude. If the star is closer than 32.6 light years, then the apparent magnitude will be less that the absolute magnitude (it appears brighter because it is close to us – remember that it is a reversed scale).

After we did the parallax simulation activity together as a class, I had my students learn the Modulus method. I then taught them how to make the representative color images using WISE data presented in my last post. Most of them were able to grasp the concepts well, based on my feedback quizzes and assessment forms.

I’ve made a few more modifications for the purpose of posting the pages here, such as adding extra columns on the first page to facilitate doing more calculations. Eventually I need to make brand new digital versions that are easier to edit. Although effective, I’m still not quite satisfied with the flow or layout of the lesson.