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.
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.