Image: The constellation Orion. Each of the seven principal stars of this constellation, forming Orion's shoulders, belt, and feet, is between a few thousand and a few hundred thousand times more luminous than the Sun. Credit: Mouser, deep sky image of the constellation Orion, CC BY-SA 3.0 https://creativecommons.org/licenses/by/3.0/legalcode
Astronomers are distant observers. With very few exceptions inside our own solar system, we cannot travel to our objects of study. Instead, we need to infer the properties of stars, nebulae and planets from our observations. Knowing an object's distance is key to such cosmic detective work. If all we knew were an object's apparent brightness in the sky, we couldn't distinguish between objects that are fairly near but not very bright and those that are far away but emit lots of light!
This is evident when it comes to some of the most basic celestial objects: stars. Our direct experience on a clear night is that stars are tiny pinpricks of light. A simple flashlight will give us much more illumination than all the thousands of stars we see in a starry night taken together. But the Sun is a star, too, and it is the brightest object most of us will ever experience – so bright that it poses a danger to our eyes, and we shouldn’t look at it directly! Yet some of the stars we see at night emit much more light than our Sun. The key factor that makes them appear much less bright, for an observer here on Earth, is distance. The only difference is distance. Even the closest star, Alpha Centauri, is more than 265 000 times far away from us compared to the Sun.
Estimating cosmic distances is a difficult task. Even distances within our own Solar System are sizeable by everyday standards. The astronomical unit, corresponding to the average distance of the Earth from the Sun, is about 150 million kilometres.
The nearest star, Alpha Centauri, is 4·10 13 km away. Such giant numbers are unwieldy, and astronomers have introduced an alternative way of stating distances. Nothing moves faster than light, and astronomers have taken to using light travel times as their measure of the distances to the nearest stars (parsecs, and its derivatives, as well as redshifts are used for greater distances). The Sun, for instance, is about 8 light-minutes away from us: light takes 8 minutes to travel from the Sun to us. Alpha Centauri is 4.2 light-years away.
(Do not be confused by the occurrence of words like ‘minute’ or ‘year’ in these units. Light-minutes and light-years are measures of distance, not of time.)
The most distant objects we know are much farther away. Light takes billions of years to cover the distance between those objects and us, so these objects are billions of light-years away (corresponding to tens of sextillions, or 10 22 , of kilometres). No single method can cover this immense range of distances. Instead, astronomers rely on what they call the cosmic distance ladder: a set of complementary methods of determining distances, where methods applied to more distant objects are calibrated using methods applicable to less distant objects.
Apparent brightness: inverse square formula
A number of key methods of measuring astronomical distances involve the following basic principle: Assume that we know how much light an object emits (this is called 'luminosity'). We can measure the object's apparent brightness in the sky. Comparison of luminosity and apparent brightness is directly related to the object's distance from us. Quantitatively, assume that the objects emits energy per second L, and that this emission does not favour any particular direction (isotropic emission). L is called the object's luminosity. At a distance d from the object, this total energy will have spread out over a spherical surface of area 4 pi d 2 . Imagine that our detector – for instance, our telescope mirror – covers an area A, as shown in the following figure, which is at a distance d from the radiating object O.
Since the object's light emission is spread evenly over the whole sphere, our detector will only receive a fraction A/(4 pi d 2 ) of that emission; in other words, our detector will receive energy per second
F = L·A/(4 pi d 2 )
from the object. We can rewrite this formula by dividing out the detector area to yield the intensity of the radiation
I = F/A = L/(4 pi d 2 ),
that is, the amount of energy per second per unit detector area reaching us from a particular object. This is the famous ‘inverse square law’ for radiation. Typical classroom demonstrations of the inverse square law involve a photodetector, such as a small solar cell, placed at varying distances from a light source (e.g. Stanger 2008).
Image: Supernovae of type Ia, such as this one in 1994 in the galaxy NGC 4526, are standard candles that allow astronomers to deduce the distance of far-away galaxies. Credit: NASA, ESA, The Hubble Key Project Team, and The High-Z Supernova Search Team. CC BY-SA 3.0 https://creativecommons.org/licenses/by/3.0/legalcode
The inverse square law links the quantities I, L, and d. Imagine a class of objects whose members all have the same luminosity L. Such objects are called standard candles. Whenever we observe a standard candle, we know its luminosity L, can measure the intensity I, and use the inverse square law to calculate the object's distance.
So-called supernovae of type Ia are the most important standard candles for very distant galaxies, in the context of cosmology, the science of the universe as a whole. These supernovae are violent, thermonuclear explosions of white dwarf stars. They are extremely bright, and visible over great distances. A supernova's type can be identified from the properties of the light we receive from the explosion (more concretely, from the spectrum of the supernova). Once it is clear that one is indeed observing a supernova of this type, the (maximum) luminosity of the explosion can be deduced.
For most astronomical standard candles, L is not constant for all objects within the class, but instead is correlated with a measurable property of objects of this kind. The most famous example are Cepheid variable stars, which show periodic changes in brightness. The period of these changes is correlated with the luminosity of the stars; measure the period, and you can deduce the luminosity L. This correlation was first noticed and exploited by Henrietta Swan Leavitt in 1908-1912.
Standard candles like this have played a key role in the history of astronomy. In the early 20th century, Cepheids were used to show that our galaxy is just one among many. And in the final years of the 20th century, type Ia supernovae were used to show that cosmic expansion is accelerating – the discovery of ‘Dark Energy’, which was rewarded with the 2011 Nobel Prize in Physics.
The ideal standard candle would be exceedingly bright, and thus visible over large distances, it would be easily identifiable (e.g. via the determination of a spectrum), and it would be reasonably common to allow for a wide range of distance determinations. Ideally, we would have some very close examples of this type of standard candle in our cosmic neighbourhood, which allows for calibrating the standard candle (that is, measuring its luminosity L) and many very distant examples that allow for distance determinations of galaxies as well as for cosmological measurements.
In reality, no single standard candle meets all the criteria at once. Instead, astronomers have built a distance ladder of standard candles. For instance, for nearly 300 Cepheid variables, their distance can be measured directly using basic geometry (stellar parallax). These known distances can be used to calibrate the Cepheid period-luminosity relation. Once this relation is known, we can examine nearby supernovae of type Ia in galaxies that harbour Cepheids. Using the Cepheid distances, we can determine the peak luminosities of supernovae of type Ia. Once we know these luminosities, we can use supernovae of type Ia as standard candles that are so bright they can be seen to substantial extragalactic and cosmological distances.
The activity presented here allows your students to discover and explore the key principles of standard candles for themselves, using a simple example in an everyday setting.
Standard candles and the inverse square law
In practice, quantitatively measuring intensities is a challenging task, which requires careful calibration of one's instruments markedly beyond the scope of this activity. Instead, we will make use of the fact that we are making our measurements of various sources with one and the same piece of equipment, our digital camera.
Assuming that we receive light of intensity I from an object, our camera will gather a total amount of light per second corresponding to P = I·A·η, where A is the collecting area of the camera and η < 1 is a dimensionless constant that allows us to encode (a) that some light will be absorbed within the camera lens and (b) some light might not reach the camera chip but be scattered elsewhere. The total energy deposited on the chip is E = P·t, where t is the exposure time. Assume that in our image the object in question spans a certain pixel region and assume a linear response of the chip and linear processing, then E will be proportional to the sum S of the pixel values for that region. (An optional part of the activity involves testing this linearity.)
What greatly simplifies our task is that the pixel value sum S depends linearly on the intensity I. As long as we take care to take all our images under the same conditions (same exposure time, same lens, same settings), this linearity means that we can compare the intensities I 1,2 of different sources by comparing the sums S 1,2 of the pixel values for the images of these sources,
S 1 /S 2 = I 1 /I 2 .
It does not get simpler than this, and this simple formula, together with the inverse square formula linking luminosity, intensity and distance, will be the foundation of the following activity.
How to undertake this activity?
This activity can be undertaken at different levels, depending both on the degree of independence of the activity (i.e. how much is prepared beforehand by the teacher) and on the level of analysis.
As far as preparations are concerned, at the most basic level, the teacher scouts one or more likely locations, takes care to set up the necessary software and prepares simplified recipes for using the software. Students can then concentrate on the science, namely on the measurements and their evaluation. This level of preparation allows for the quickest completion of the exercise. If, on the other hand, the exercise is set up as a completely free inquiry, students need to find their own location, do research on what software they need (e.g. to convert raw images either to FITS or another suitable format for their analysis) and install what they need. This makes for a much more realistic experience, as such preparations are a standard part of astronomical research. Naturally, it also makes the exercise that much more time-intensive.
The most basic level of analysis directly uses the inverse square formula to relate measured brightness to distance, using a reference object for which the distance has been measured by conventional means (either directly or using a map, e.g. Google Maps). This version of the exercise concentrates on the fundamental concept to be learned and allows for quickest completion.
As an additional activity, the role of the digital camera can be explored. As shown above, use of the camera to measure apparent brightness via simple ratios of pixel value sums relies on a linear relation between the amount of light received from a certain region of the scenery and brightness values for the corresponding image pixels. This linearity can be checked in optional supplemental activities that, at the same time, can be the first steps towards more advanced astrophotography activities. At a more advanced level, students should be encouraged to think about causes of the deviation of their derived distances from the directly measured distances. Two fundamental causes they are likely to encounter are intrinsic brightness variations (that is, deviations from the standard candle assumption) and obscuration (an object’s light being dimmed by intervening matter). Both have their analogues in astronomy, where the simple standard candle assumption (same intrinsic brightness) often needs to be refined (e.g. for Supernovae of type Ia, by making use of a correlation between the supernova light curve’s evolution over time and its peak brightness), and where dust and gas clouds can dim the light of a distant source. In this exercise, we are in the fortunate situation of being able to get close to (‘travel to’) our light sources, and measure their intrinsic brightness directly. Students can make these measurements and apply a corresponding correction to their distance derivation; this should reduce the deviation considerably. We can also identify obstructions in the worked-out example; for instance, those street lights that appear dimmer than expected indeed turn out to be obscured by tree branches. While the more advanced level takes considerably more time, it teaches valuable skills in analysing data and error sources.
A worked-out example with brightness measurements and corrections, including sample images and a sample spreadsheet, can be found at http://www.haus-der-astronomie.de/materials/distances/street-lights