### 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.