Astronomy Down Under

Astronomy 101 - Lesson 1 - Astronomical Numbers

18 Jul at 21:53 by Wilson

Much of astronomy has to deal with large numbers; astronomical numbers, in more senses than one. For example, the average distance from the Earth to the Sun is 149,597,870.691 kilometres — or, more commonly, 150 million kilometres. The mass of the Sun is approximately 1,989,100,000,000,000,000,000,000,000,000 kg, and I’m not even going to try to write that in words. The approximate age of the Earth is 4.6 billion years, while the age of the universe is closer to 15 billion years.

It is probably very clear from the paragraph above that these large numbers are very hard to work with — or even to write. It’s even hard to get an instinctive grasp of how large they really are; at some point, really, all you have are way too many zeroes. That’s the reason why scientists use a more compact notation for very large — or very small — numbers. This notation (not surprisingly known as “scientific notation”) divides the number in two parts and uses the magic of the powers of ten to get rid of all those zeros. Those two parts are the mantissa and the exponent.

In short, the mantissa is the part of the number that we are reasonably sure of. Look again at that number describing the mass of the Sun; it should be pretty obvious that those zeroes are just “filler”; we don’t know what all those digits are supposed to be, but we are fairly confident about the first five non-zero digits. Those five digits will form the mantissa.

The exponent indicates how large the number is. It tells us how many zeroes go after the mantissa or, more generally, how much we need to move the decimal point of the mantissa to get to the real number. In other words, it tells us which power of ten must be multiplied by the mantissa to get the number we’re trying to express.

Let’s get back to the examples above. As a general rule, the mantissa will always be kept between 1 and 10; so, the average distance from the Earth to the Sun would be written as 1.49597870691 x 10⁸ km — or, if want to be a bit more compact, 1.5 x 10⁸ km — meaning that, starting from 1.5, we move the decimal point 8 positions to the right to arrive at the intended number. The mass of the Sun, in the same notation, is merely 1.9891 x 10³⁰ kg, which is certainly much more readable than the other version.

What the scientific notation does is to allow us to write and, more importantly, read numbers much more efficiently. It keeps all those zeroes out of the way and allows us to concentrate on what we actually know about the quantity we’re talking about. It also makes it much easier to compare large numbers; the exponent will tell you at a glance the magnitude of the numbers in question.

And, as I mentioned, it also works the other way: a negative exponent will tell you how many positions to the left you need to move the decimal point, allowing you to easily represent very small numbers. The diameter of a hydrogen atom, for example, is 0.0000000000106 metres — or 1.06 x 10^-11 metres.

The next lesson will be about units, and it will focus on what we can measure with these numbers, with special attention to the astronomically relevant units of measurement.