from the Book of Genesis: ‘We are now approaching lunar sunrise, and for all the people back on Earth, the crew of Apollo 8 has a message that we would like to send to you,’ began Anders. ‘In the beginning God created the heaven and the Earth. And the Earth was without form, and void; and darkness was upon the face of the deep.’ Borman concluded with a sentence clearly spoken by a lonely man 400,000 kilometres from home. ‘And from the crew of Apollo 8, we close with goodnight, good luck, a Merry Christmas – and God bless all of you, all of you on the good Earth.’
The mission’s most potent legacy, however, is NASA image AS8-14-2383, snapped by Bill Anders on a Hasselblad 500 EL at f/11 and a shutter speed of 1/250th of a second on Kodak Ektachrome film. It was, in other words, a very bright photograph. The image is better known as Earthrise. When viewed with the lunar surface at the bottom, Earth is tilted on its side with the South Pole to the left, and the equator running top to bottom. Little landmass can be seen through the swirling clouds, but the bright sands of the Namib and Saharan deserts stand out salmon pink against the blackness beyond. Just 368 years and 10 months after a man was burned at the stake for dreaming of worlds without end, here is Earth, a fragile crescent suspended over an alien landscape, the negative of a waxing Moon in the friendly skies of Earth. This is an unfamiliar, planetary Earth, no longer central; just another world. When Kennedy spoke of Apollo as a journey to an unknown celestial body, he meant the Moon. But we discovered Earth and, in the words of T. S. Eliot, came to know the place for the first time.
Newton’s laws are the keys to understanding our place in our local neighbourhood. Coupled with precision observations of the motion of the planets and moons, they allow the scale and geometry of the solar system to be deduced, and their positions to be calculated at any point in the future. The nature and location of the stars, however, requires an entirely different approach because at first sight they appear to be point-like and fixed. The observation that the stars don’t appear to move is important if you know something about parallax, as the ancients did. Parallax is the name given to a familiar effect. Hold your finger up in front of your face and alternately close each of your eyes, keeping your finger still. Your finger appears to move relative to the more distant background, and the closer your finger is to your face, the more it appears to move. This is not an optical illusion; it’s a consequence of viewing a nearby object from two different spatial positions; in this case the two slightly different positions of your eyes. We don’t normally perceive this parallax effect because the brain combines the inputs from the eyes to create a single image, although the information is exploited to create our sense of depth. Aristotle used the lack of stellar parallax to argue that the Earth must be stationary at the centre of the universe, because if the Earth moved then the nearby stars would be observed to move against the background of the more distant ones. Thousands of years later, Tycho Brahe used a similar argument to refute the conclusions reached by Copernicus. Their logic was completely sound, but the conclusion is wrong because the nearby stars do move relative to the more distant background stars as the Earth orbits the Sun, and indeed as the Sun orbits the galaxy itself. You just have to look extremely carefully to see the effect.
Amongst the thousands of stars visible to the naked eye, 61 Cygni is one of the faintest. It’s not without interest, being a binary star system of two orange K-type dwarf stars, slightly smaller and cooler than the Sun, orbiting each other at the lethargic rate of around 700 years. Despite the pair’s relative visual anonymity, however, 61 Cygni has great historical significance. The reason for this quiet fame is that this faint star system was the first to have its distance from Earth measured by parallax.
Friedrich Bessel is best known to a physicist or mathematician for his work on the mathematical functions that bear his name. Pretty much any engineering or physical problem that involves a cylindrical or spherical geometry ends up with the use of Bessel functions, and, in blissful ignorance, you will probably encounter some piece of technology that has relied on them in the design process at some point today. But Bessel was first and foremost an astronomer, being appointed director of the Königsberg Observatory at the age of only 25. In 1838, Bessel observed that 61 Cygni shifted its position in the sky by approximately two-thirds of an arcsecond over a period of a year as viewed from Earth. That’s not very much – an arcsecond is one 3600th of a degree. It is enough, however, to do a bit of trigonometry and calculate that 61 Cygni is 10.3 light years away from our solar system. This compares very favourably with the modern measurement of the distance, 11.41 ± 0.02 light years. Parallax is so important in astronomy that there is a measurement system completely based on it, which allows you to do these sums in your head. Astronomers use a distance measurement known as a parsec – which stands for ‘per arcsecond’. This is the distance of a star from the Sun that has a parallax of 1 arcsecond. One parsec is 3.26 light years. Bessel’s measurement of the parallax of 61 Cygni was 0.314 arcseconds, and this immediately implies that it’s around 10 light years away.
Even today, stellar parallax remains the most accurate way of determining the distance to nearby stars, because it is a direct measurement which uses only trigonometry and requires no assumptions or physical models. On 19 December 2013 the Gaia space telescope was launched on a Soyuz rocket from French Guiana. The mission will measure, by parallax, the positions and motions of a billion stars in our galaxy over five years. This data will provide an accurate and dynamic 3D map of the galaxy, which in turn will allow for an exploration of the history of the Milky Way, because Newton’s laws, which govern the motions of all these stars under the gravitational pull of each other, can be run backwards as well as forwards in time. Given precise measurements of the positions and velocities of 1 per cent of the stars in the Milky Way, it is possible to ask what the configuration of the stars looked like millions or even billions of years ago. This enables astronomers to build simulations of the evolution of our galaxy, revealing its history of collisions and mergers with other galaxies over 13 billion years, stretching back to the beginning of the universe. Newton and Bessel would have loved it.
Stellar parallax, when deployed using a twenty-first-century orbiting observatory, is a powerful technique for mapping our galaxy out to distances of many thousands of light years. Beyond our galaxy, however, the distances are far too great to employ this direct method of distance measurement. In the mid-nineteenth century, this might have appeared an insurmountable problem, but science doesn’t proceed by measurement alone. As Newton so powerfully demonstrated, scientific progress often proceeds through the interaction between theory and observation. Newton’s Law of Gravitation is a theory; in physics this usually means a mathematical model that can be applied to explain or predict the behaviour of some part of the natural world. How might we measure the mass of a planet? We can’t ‘weigh’ it directly, but given Newton’s laws we can determine the planet’s mass very accurately if it has a moon. The logic is quite simple – the moon’s orbit clearly has something to do with the planet’s gravity, which in turn has something to do with its mass. These relationships are encoded in Newton’s law, and careful observation of the moon’s orbit around the planet therefore allows for the planet’s mass to be determined. For the more mathematical reader, the equation is:
where a is the (time-averaged) distance between the planet and the moon, G is Newton’s gravitational constant and P is the period of the orbit. (This equation is in fact Kepler’s third law, discovered empirically by Kepler in 1619. Kepler’s laws can be derived from Newton’s law of gravitation.) Under the assumption that the mass of the planet is far larger than the mass of the Moon, this equation allows for the mass of the planet to be measured. This is how theoretical physics can be used to extract measurements from observation, given a mathematical model of the system. To measure the distance to objects that are too far away to use parallax, therefore, we need to find a theory or mathematical relationship
