How do astronomers measure distances to the stars? Using a digital camera to record parallax shift is an accurate and authentic method that can be used in a classroom.
Astronomers are remote observers, separated by great distances from the objects they study. Knowing the distance of celestial objects is crucial: it provides a key factor in distinguishing a very luminous, distant object from one that is much closer but inherently less bright – and thus in working out what the object is. Even single stars and whole galaxies can look similar – until we know that one is a billion times further away than the other, and thus in reality trillions of times brighter.
The most important astronomical method for determining distances from Earth to the stars is parallax – the apparent shift in position of a viewed object when the observer changes position. We notice the parallax effect when we look out a train window: nearby objects seem to move past much faster than distant ones. Using this effect, we can work out the distance of an object by measuring the apparent change in its position when our position as observers changes.
The challenge, of course, is accuracy. How can we make distance measurements based on parallax as accurate as possible? In this second article on measuring distances using parallax, we use an authentic method very similar to that used by astronomers, which has been adapted for use in a classroom (for the previous article, see Pössel, 2017). Instead of using angle-measuring devices (as in the previous article, based on theodolite measurements) to work out the distance to a ‘star’, here we take photographs from different positions and use these to provide the measurements we need to calculate our star’s distance.
A variation of this photographic method, described later in this article, is even more authentic and accurate, although slightly more complicated. It involves using a fixed reference point outside the classroom – just as astronomers use very distant background objects (e.g. active galaxies called quasars) as fixed reference points when measuring stellar parallax, rather than relying on their telescopes pointing in the same direction for each observation.
For these activities, you will need a digital camera, mounted (if possible) on an optical bench (see figure 1). The activities can be carried out by groups of students. Allow around 30 minutes for setting up and taking measurements, and an additional 30 minutes for the analysis and calculations.
Digital camera with lens of known focal length (at least 50 mm)
Optical bench (or similar setup for moving the camera sideways and measuring the shift, such as a flat table top and a long ruler)
Model star (a small sphere no more than 1 cm in diameter or an LED, mounted on a stick)
Object of known length with clear markings (e.g. a metre rule), as a calibration object
Image processing programme, e.g. Adobe Photoshop, GIMP
The experimental situation is shown in figure 2. Here, b represents the shift distance along the camera baseline (the optical bench or table) between the first and second positions (A and B), while C is the position of the model star, and d is the distance from the camera baseline to the star – that is, the distance we are trying to calculate.
The steps for calculating the distance d are set out below.
Unlike theodolites, cameras do not allow us to measure angles directly. We thus need to relate the location of the images of the star on the camera’s image detector screen to the angle of the light rays from the star when the camera is in different positions.
The viewpoint of the camera is shown in figure 3. The star’s apparent location will be at location CA when the camera is at the first position (A), and its image will be located at DA on the detector screen. Similarly, after moving the camera to B, the star’s apparent location will be at CB and its image at DB. (Here, the line segment OQ represents the distance between the camera’s image plane and the parallel plane that contains the star.) The length CBCA is the distance b in figure 2, while the angles a and ß remain the same.
The formula is:
d = distance to star
L = actual length of the calibration object
b = actual distance the camera was moved (which corresponds to the distance from CA to CB)
dL = actual distance of the calibration object from the camera baseline (along line OQ)
p = distance as the number of pixels between the star images (at DA and DB)
pL= length as the number of pixels of the image of the calibration object
When you have worked out a value for d, you have successfully used parallax measurement to determine the distance to the star.
Now use the tape measure to find the distance d directly, and compare this to the value calculated from the parallax measurement. How accurate was the calculated measurement?
You can repeat this activity with the ‘star’ placed at different distances, to find out whether the accuracy of parallax measurements changes with distance (see section below, ‘What accuracy can we expect?’).
For even more astronomical realism, we can adapt the parallax photography method to use a reference object outside the classroom, which should be considerably further away than the ‘star’. With this procedure, instead of relying on the camera pointing in the same direction after having been moved from A to B, we choose a distant reference object that is visible in each of the two images. We then measure the star image’s pixel distance from the reference object in each image. This alternative approach, which we describe here, should yield more accurate results.
So, how accurate are the results obtained using this improved method? Our data suggest that they can be remarkably accurate (compared with directly measured distances), as shown in figure 5. The largest relative error amounts to just 3.2%.
Note that, at larger distances, there is an increase in the relative error as well as the actual error. This is because the geometry changes: the distance to the model star becomes larger compared with the distance of the reference object, so the error introduced by the parallax of the reference object becomes ever greater.
With the simple angle-measuring method described in the previous article (Pössel, 2017), the accuracy of distance measurements was significantly lower – generally within about 10%, as shown in figure 6. So the method described in the current article provides a significant improvement in accuracy from the previous method, where the dominant sources of error are the angular measurements.
Pössel M (2017) Parallax: reaching the stars with geometry. Science in School 39: 40-44.
For information on how to carry out real astronomical parallax measurements with small instruments, see:
Cenadelli D et al. (2016) Geometry can take you to the Moon. Science in School 35: 41-46.
Cenadelli D et al. (2009) An international parallax campaign to measure distance to the Moon and Mars. European Journal of Physics 30: 35-46. doi: 10.1088/0143-0807/30/1/004
Hirshfeld AW (2013) Parallax: The Race to Measure the Cosmos. Mineola, NY, USA: Dover Publications. ISBN: 9780486490939
Variations of these experiments have been in use in astronomy laboratory courses for a long time. For example:
De Jong ML (1972) A stellar parallax exercise for the introductory astronomy course. American Journal of Physics 40(5): 762-763. doi: 10.1119/1.1986635
Deutschman WA (1977) Parallax without pain. American Journal of Physics 45(5): 490. doi: 10.1119/1.11009