As part of a course for work I’ve been tasked with completing a small project using one of a range of pieces of physics software. I picked one called Tracker, which allows you to track various physical properties in a video – including movement, acceleration, and light intensity.

 

Being able to track light intensity means we can simulate the “transit photometry” technique for detecting exoplanets – observe a distant star and look for periodic dips in the brightness as an exoplanet travels between us and the star, blocking some of the light.

 

To simulate this I used a spherical table lamp and a ball on a string hanging from the ceiling.  Note that the exposure setting on the camera is changed half-way through – the 50Hz flicker in the lamp due to the mains AC cycle is more obvious and produces more noise at lower exposure settings. Also note the upward spikes in luminosity just after the “planet” transits – this is due to the auto-exposure on the camera overcompensating, fixed exposure should ideally be used for this demonstration.

 

 

Because a ball on a string is effectively a pendulum, and because all “orbits” (including a pendulum) have the same period regardless of eccentricity (see tweet below), we can use a simple pendulum equation to get a good estimate of the period: T = 2Pi sqrt(L/g) , where L is the length of the string and g is local gravity.

 

Another neat fact is that the falling object and a satellite orbiting just above the Earth’s surface will have the exact same oscillation period! pic.twitter.com/g7fgYSlmM6

— 〈 Berger | Dillon 〉 (@InertialObservr) August 1, 2019

 

With a string length of 1.63m we predict a period of 2.56s

 

Taking measurements of the minimal luminosity points on the right hand half of the graph we get the following periods:

 

2.49 , 2.53 , 2.53 , 2.53 , 2.53 , 2.53

 

Remarkably consistent, and taking an average of 2.52s it varies from our pendulum estimate by 0.04s, or 1.6% out.

 

Our main sources of experimental error are the 50Hz flicker on the lamp, the frame rate on the camera not necessarily catching the moments of maximal luminosity, and the autoexposure on the camera providing biased data. These could all be minimised by using a smoothing circuit or DC lamp for the flicker, and by using a high frame-rate camera with fixed exposure.  The latter solution could also reveal the characteristic details of the light curve that allow calculation of semi-major axis, star mass, star radius, planet radius, eccentricity, and inclination of orbit.

 

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Note: The official installer for Tracker fails on an Ubuntu 20.04 system (previous versions on 18.04 worked), but it can be installed using the .deb file located here. I can’t guarantee unofficial sources, but opensuse.org are usually reliable.

In 2010 I was working in a bookshop and teaching myself the web scripting language PHP. A good way to learn a language is to find problems you can solve with it, and as I had recently been working with ISBNs, the identifying numbers given to printed books, I decided to write a program to search Pi for them.

I found the first three books in Pi. Then, a few years later, it got picked up by a few popular Twitter accounts and went viral, including a write-up in the science page of Der Spiegel.

What’s the point? There isn’t, other than a programming exercise and the pleasure of finding things out. But it does raise a few interesting things:

Is Pi “normal”?

Normal in this sense means “does it have the same numbers of each digit, 0-9?”
If Pi is normal then you’d expect every possible number sequence to appear in it. If it’s not (say, if you get to a certain point and then there’s never a 9 again) then it won’t. It’s generally thought that Pi is normal, but it’s still an open question in mathematics.

How does it fare with the old 10 digit ISBN?

As you’d expect, there’s about a thousand times more ISBNs if you search for the 10 digit version, as it’s a shorter string and more likely to appear in any random set of digits. Some ISBN-10s end in an X, which obviously doesn’t appear in Pi, but that can be ignored, we’re just messing about here. David J Fianders went to the effort of tracking down ISBN-10s.

It’s now out of date!

ISBNs are constantly being added for new publications, so it may well be that “my” first three books have been superseded. Feel free to try searching for new ones!