A few years ago I received this photo from my cousin, Doug, who at the time was living at Casey Station, Antarctica. Doug is a communications technician with the Australian Antarctic Division. Over the past decade he has mostly lived and worked at one or another of the four A.A.D. research bases. This year he is at Macquarie Island. He sends a weekly newsletter, adorned with magnificent photos of pristine icy landscapes, extraordinary wildlife and colourful and mind-blowing captures of the night sky that include Aroura Australis and time-lapse star-trails like the photograph here.
I find all his stories and photography amazing, but the mathematician in me drew special interest in this South Pole time-lapse specimen. There is no set point in the picture that identifies exactly where the south celestial pole (SCP) lies, but it’s not too difficult to pose a guess when considering the circle centre from all these arcs. The arcs themselves are quite interesting, because they are the actual stars, with their movement recorded over a slice of time. From our perspective, it would seem that any given star would complete a full circle around the SCP in 24 hours. However, we know that what’s really rotating is our Earth against the fixed background of stars. Here then comes the obvious maths problem: For how long was the camera shutter open in order to take this photo?
Here’s how I did it:
- I started by importing the photo to a Graphs page of my TI-Nspire CAS.
- I then moved the axes to place the origin at where I estimated the SCP to be.
- Next I selected one of the star-trails. You can see that my selection is towards the lower-left of the photo, but really any of them would do.
- I graphed f1(x)=1 and then used the line-rotation feature until the line ran across one end of the star-trail arc (blue line).
- I graphed f2(x)=1 and used the line-rotation feature again until the line ran across the other end of the star-trail arc (red line).
- The CAS indicates the equations of the two lines and I extracted the gradient of each into a Notes page.
- Continuing on the Notes page I used the two gradients and a known formula to compute the angle of the arc. (Note that my document is set to degrees)
- This angle as a fraction of 360 degrees equates to the exposure time as a fraction of a full day. I have made my calculation of the exposure time in seconds further down the Notes page. (1593 s in this example – about 26.5 mins)
And now the best part of it all: I can check my result another way! Remember that my cousin Doug had sent me this digital photo, which as a JPG file, I can not only view it, import it into calculator software, but I also have the original metadata. Here is the details page for the file. Note that among bountiful information, the exposure time is indicated: 1652 s. My calculation was but 59 s out.
Want to see more classroom resources from the deep south? Classroom Antarctica is a comprehensive online teaching resource produced by the Australian Antarctic Division, with lesson plans aimed at grades 3 to 8. Ideas contained in Classroom Antarctica will stimulate your students’ interest in real-world applications for science, mathematics and studies of society and environment, inspiring and engaging your students in learning.