### Video Transcript

In this video, we will be learning about a unit of distance, rather surprisingly, known as the light-year. As well as this, we’ll be looking at when and why we’d need to use the light-year in the first place.

So let’s start by thinking about the SI unit of distance. The SI base unit of distance is the metre. It’s a very useful unit for measuring distances that we deal with in day-to-day life. In fact, the metre and its prefixes are very convenient for measuring anything from the thickness of a coin to the distance between your house and the nearest supermarket. However, as soon as we start thinking about astronomical distances, the metre and its most well-known prefixes become inconvenient and too small. For example, the average distance between the Sun and the Earth is about 150 billion metres, or about 150 gigametres. Now, giga is already one of the largest prefixes that we use in day-to-day physics. And we’re already having to use this prefix just for the distance between the Sun and the Earth. This, by the way, is by no means even close to the largest distance that we’d have to consider when studying astronomy or astrophysics.

For this reason, we need to find a unit that is defined so that one unit of that distance is a very, very, very large distance. There are a few such units that are commonly in use by astronomers and astrophysicists, such as the astronomical unit or a parsec, which is actually much, much bigger than this. But the unit that we’ll be studying in this video is the light-year. So why have we confusingly called a unit of distance a light-year? To answer this, let’s first look at the definition of a light-year. A light-year is defined as the distance travelled by light in a vacuum in the period of one year.

Now we can recall that the speed of light in a vacuum is a constant value. This constant speed of light, which is often called 𝑐, is approximately three times 10 to the power of eight metres per second. In other words, light can travel about 300 million metres every second. And, of course, this speed of light is defined as the speed of light in a vacuum. That speed decreases very slightly when we’re talking about light moving through the atmosphere. But this minute decrease in the speed of light within the atmosphere is not large enough for us to worry about. However, the point still stands. The original definition of a light-year talks about light travelling through a vacuum. So we should use a constant speed of light in a vacuum.

But, anyway, so we’ve seen that light can travel a very large distance, specifically this many metres, every second. And so the distance that light travels in a vacuum in one year is absolutely immense. We can actually work out what this distance is in metres. And to do this, we need to recall that the speed of light is equal to the distance travelled by light divided by the time taken for light to travel that distance. And, of course, this equation is just a special case of the equation for speed, where the speed of any object is defined as the distance travelled by the object divided by the time taken for the object to travel that distance.

But, anyway, so we see that the speed of light is defined as such. And we can rearrange the equation by multiplying both sides by the time taken, 𝑡. This way, 𝑡 cancels on the right-hand side. And so, what we’re left with is that the speed of light multiplied by a certain period of time is equal to the distance travelled by light within that period of time. So now we can sub in some values. We know that the speed of light in a vacuum is a constant, three times 10 to the power of eight metres per second. And we’re trying to work out the distance that is a light-year, which means the time that we need to plug in is one year. Because, remember, we’re trying to find the distance travelled by light in a vacuum, of course, in a year.

However, when we look at the units, we’ve got seconds in the denominator in this fraction but a year in the numerator here. We want these to cancel, so we can convert the year into seconds. To do this, we can recall that one year is equal to 365 days because there are 365 days in a year. But then in every single one of these days, there are 24 hours because there are 24 hours in a day. And so, we can say that one year is equal to 365 days multiplied by 24 hours per day. Because as we said earlier, there are 24 hours in a day. And, therefore, the days in the numerator cancels with the days in the denominator. This way, what we’re left with is that one year is equivalent to 365 times 24 hours. And that is equal to 8760 hours.

Then we can recall that every single hour has 60 minutes in it. And so, we can multiply this fraction by 60 minutes per hour. And, of course, the reason we can do this is because 60 minutes is equivalent to one hour. And so, multiplying by this fraction simply means that we’re multiplying by one. We’re not changing the fraction at all. All we’re doing is we’re converting units because, once again, we see a unit cancelling in the numerator and denominator. And so, one year is equivalent to 8760 multiplied by 60 minutes. That leaves us with 525600 minutes in a year.

Then we can repeat this trick again by recalling that there are 60 seconds in every minute. This way, we get a cancellation of the minutes unit. And we finally found that within a year, there are 31536000 seconds. So we found the conversion factor between a year and seconds. Therefore, we can sub this in instead of one year. And then we see that this unit of seconds in the denominator will now cancel with this unit of seconds in the numerator. This way, what we’re left with is simply metres in a numerator. And this is great because we’re trying to find the distance 𝑑, which is a light-year, in metres. And what we find when we calculate this is that the distance, which is one light-year, is equal to 9.46 times 10 to the power of 15 metres, to three significant figures.

So that is a massive distance. And at this point, we’re going to substitute the letter 𝑑 for one light-year. And, by the way, the shorthand unit of light-years is lowercase l lowercase y, just like how the metre is denoted with a lowercase m. And so, now we see that we’ve got what we wished for. We wanted a unit of distance where one of those units was equal to a very large distance. And here, one light-year is equal to 9.46 times 10 to the power of 15 metres. So we can comfortably express astronomical distances in light-years.

Now, the way to remember that a light-year is not a unit of time but is instead a unit of distance is to not think of it as a light-year but, instead, to think of it as a light-year. In other words, place the emphasis on the light not on the year. Because from there, we can take this to mean light speed–year which will allow us to remember that we’re talking about the speed of light and a time interval of a year. And even better, we can remember this as the light speed 𝑐 multiplied by the amount of time which is a year. And that brings us back to this equation here. The speed of an object — in this case, the speed of light actually — multiplied by the amount of time that we’re thinking about, which in this case is a year, must be equal to a distance. And so, a light-year is basically a speed, that’s the speed of light, multiplied by a time, that’s a year. And, therefore, this whole thing must be a unit of distance not time.

So that’s a neat little trick to remember that a light-year is a unit of distance. And, in fact, this unit of distance is often used to talk about distances between stars or distances between galaxies. For example, if we wanted to talk about the distance between Earth and the nearest star to Earth apart from the Sun, then we can say that this distance, the distance between the Earth and Alpha Centauri, is just over four light-years. Which, if we wanted to describe in metres, it would be about 38 times 10 to the power of 15 metres. Which becomes very clunky compared to the concise description that is four light-years. And so, now that we’ve seen how we can use light-years to label astronomical distances, let’s take a look at an example question.

The Milky Way is about 100000 light-years in diameter. One light-year is equal to 9.46 times 10 to the power of 15 metres. What is the diameter of the Milky Way in metres?

Okay, so this question is talking about the Milky Way which is the galaxy that we are in. Now, this galaxy is a spiral galaxy. But if we approximate it to be a circle, then we can say that the diameter, which is the distance from one end to the other end whilst going through the centre, is about 100000 light-years. So we can call the diameter 𝑑 and say that 𝑑 is equal to 100000 light-years. Now, we’ve also been told that one light-year is equal to 9.46 times 10 to the power of 15 metres. And we’ve been asked to find the diameter of the Milky Way in metres.

So one light-year is 9.46 times 10 to the power of 15 metres. And the diameter of the Milky Way is 100000 lots of this distance. So we can say that the diameter 𝑑 is equal to 100000 multiplied by one light-year which is the same thing as 9.46 times 10 to the power of 15 metres. Therefore, we’ve taken the unit of light-year and, instead, we’ve substituted in this value. Then when we evaluate the right-hand side of this equation, we can, first of all, see that we can write 100000 as 10 to the power of five. And then we can multiply 10 to the power of five by this number here.

All we’ll be doing then is adding the exponent five over here to this 15 over here. Because when we’ve got the same base, that’s 10, in both cases with different exponents and we multiply these two numbers together, then the exponents add together. And so, what we’re left with is 9.46 multiplied by 10 to the power of 15 plus five. And so, our final answer is that the diameter of the Milky Way in metres is 9.46 times 10 to the power of 20 metres. Let’s take a look at another example.

A laser beam was fired off into space from an observatory on Earth’s surface. The beam was fired into an empty region of space, so it would not interact with anything. How far, in light-years, has the beam travelled three years later?

Okay, so in this question, we’ve got an observatory on the surface of the Earth which fires a laser beam off into space. Now we’ve been told that this beam of light is fired into an empty region of space so it doesn’t interact with anything. In other words, the beam just keeps going at the same speed for a very long time. And we’ve been asked to find the distance, in light-years, travelled by the beam in a period of three years.

So let’s say, first of all, that this is the distance that we’re trying to find. In other words, a beam fired from here, three years later is this far from the Earth. Now, naturally, this diagram is absolutely not to scale. But that doesn’t really matter. It shouldn’t hinder us from calculating the correct value. So let’s call the distance that we’re trying to calculate 𝑑. And let’s also recall the definition of a light-year. We can recall that one light-year is the distance travelled by light in a vacuum in one year. Now, in this scenario, we’ve basically got the light travelling in a vacuum for almost its entire journey.

We say almost because, of course, when the laser was fired from the surface of the Earth, for a very short period of time, it was travelling through the atmosphere of the Earth. However, the atmosphere does not massively slow down any light travelling through it. And even if it did, this distance is so small that compared to the total distance travelled in a vacuum, we can ignore it completely. We don’t need to worry about it. In other words, we can safely say that the distance that we’re trying to calculate, 𝑑, is basically equivalent to if the light had travelled through a vacuum for its entire journey. We can ignore any atmospheric effects.

So with that in mind, we know that one light-year is the distance travelled by light in a vacuum in one year. And so, three years after the laser was fired, because the light is travelling in a vacuum, the light will have travelled a distance of three light-years. In other words, in the first year of the light’s journey, the light will have travelled a distance of one light-year. In the second year of the light’s journey, by definition, the light would have travelled another light-year. And then, finally, in its third year of travel, it would have travelled a third light-year. And so, essentially, this question is asking us how many light-years does a light travel in three years. And the answer to that is three light-years. Okay, so now that we’ve had a look at a couple of example questions, let’s summarise what we’ve talked about in this lesson.

We firstly saw that for astronomical distances, the metre and its common prefixes, such as the kilometre or megametre or gigametre or terametre, are all too small and fairly inconvenient. In other words, we needed to find a unit which was defined such that one of those units was equivalent to a very large distance. We saw that one such unit of distance was the light-year, abbreviated ly, which is defined as the distance travelled by light in a vacuum in one year. And, finally, we saw that a good way to remember that a light-year is a unit of distance is to think of it as a light speed–year. Which, in other words, means that when multiplying the speed of light by a unit of time, which is the year, and so these multiplied together must be a unit of distance. And that is why the confusingly named light-year is actually a unit of distance and not time.