### Video Transcript

In this video, we’re going to be
looking at Hubble’s Law. Now, Hubble’s law is named after
the American astronomer Edwin Hubble. Coincidentally, that’s the same
astronomer after whom the Hubble Space Telescope is named. Now as an astronomer, he spent a
lot of time studying galaxies and noticed something really intriguing.

He noticed that on average galaxies
seem to be moving away from Earth. And not only that, but that the
further away the galaxy was away from Earth, the faster it seems to be moving
away. In other words, on average, all
galaxies seem to be moving away from earth. But the ones that were nearer to
Earth weren’t moving away quite as quickly, whereas further galaxies from Earth were
moving away really, really fast.

Now, at this point, we can ask the
question “How did Edwin Hubble know how quickly a galaxy was moving away from
Earth?” Well, to answer this question, we
need to think about the Doppler effect.

Let’s start by imagining that we’re
looking at an ambulance from above. Now, this ambulance is
stationary. But it has a siren on and it’s
making a lot of noise. We can think of the sound emitted
by the ambulance as little spheres spreading outwards from the ambulance over
time. These spheres can be used to
represent the peaks of the sound waves emitted by the ambulance.

Now, the reason that we say they’re
spheres is because the sound is emitted in all directions. And each one of these peaks is
emitted at a regular interval. Let’s say that this peak occurs
whenever we hear the “nee” in the “nee-nah” of the ambulance. So a person is standing over here
say, would hear a repeated nee-nah, nee-nah from the ambulance as the waves spread
outwards and pass the point at which the person was standing.

Now, naturally, this sound will
have a specific frequency or a specific set of frequencies. For simplicity, we can say that the
frequency is inversely proportional to the period of these waves because remember
that’s the definition of frequency. It’s the inverse of the time period
of a wave. And in this case, the time period
is the amount of time taken for an entire cycle of the wave to pass a particular
point.

For example, coming back to our
observation point, we can say that the period of the wave is the amount of time
taken between, for example, this peak of the wave to pass the point of observation
and this peak of the wave. But because the ambulance is
emitting sound waves at regular intervals, this frequency remains constant. However, this is not the case once
the ambulance starts moving.

Let’s say now that our ambulance is
moving towards the right as we’ve drawn it with a speed 𝑣. And let’s now consider two
different observation points. Let’s think about firstly an
observation point ahead of the ambulance. Now remember the ambulance is still
going to emit spherical sound waves at regular intervals.

However, the fact that it’s moving
is going to be really important because if we think about an earlier point in time
when the previous wave is emitted that wave will spread further out but it’s going
to look something like this. Now, the reason for this is because
when this wavefront was emitted, the ambulance was a little bit further back. And so, this point is the center of
this circle here.

However, by the time, this wave is
emitted, the ambulance has moved forward to this point over here. And if we think even further back
in time the ambulance would have been over here. And so, the wavefront emitted by
the ambulance at that point look something like this. Remember that circle has had enough
time to spread out quite a lot. But it still going to spread out
evenly about the point at which the ambulance was when that particular wavefront was
emitted.

And so, the end result of this is
that the wavefronts in front of the ambulance are very squashed together. Hence, at our observation point
over here, we’re going to see wavefronts arriving pretty quickly. It doesn’t take long after the
first wavefront has passed to the second one to pass as well.

But then, remember we said that the
frequency of the sound waves was directly dependent on how far these wavefronts are
far from each other. That’s because they’re directly
dependent on the time period or how long it takes between one wavefront passing our
observation point and the next one passing as well.

And so, because we’ve got
wavefronts passing really quickly, the time period is small and hence the frequency
of the sound waves is large. Compare that with an observation
point we place over here for example behind the ambulance. We know that the waves travelling
in this direction as the circles expand are going to result in a long time between
wavefronts passing the observation point. And hence, in this situation, the
time period is large and hence the frequency of sound that’s heard is small.

So this is the Doppler effect. If the ambulance, that’s the thing
emitting the sound waves, is moving towards the observer, then the frequency of the
sound waves is high. However, if the ambulance is moving
away from the observer, then the frequency of the sound waves is low. That’s also why we hear cars going
niaaaaaaaw. We hear a distinct drop in
frequency as they go past us. And this doesn’t just apply to
sound; it applies to light as well.

If we’ve got a galaxy that’s moving
away from Earth, then the light that we see coming from this galaxy is going to have
a larger period or time between peaks as compared to the galaxy wasn’t moving away
from Earth, in which case we’d see something like this where the peaks are a lot
closer together.

Now, essentially, what this means
is that the light coming from galaxies that are moving away from Earth is shifted to
the red end of the spectrum. Because remember the period of red
light is longer than the period of say for example green light or blue light. Or another way to put it is that
the wavelength of red light is larger than the wavelength of other colors such as
blue or green. And this is exactly what Edwin
Hubble noticed.

He noticed that the light from most
galaxies was shifted to the red end of the spectrum. This is called redshift. And furthermore, he realized that
the further the galaxy was from Earth, the more redshifted the light was. In other words, the further the
galaxy was from us, the faster it was moving away from Earth.

Now, at this point, it’s worth
mentioning by the way that the Earth is not in a special place in the universe. Everything is moving away from
everything else. But because we are on Earth, to us
it seems like everything is moving away from Earth. In reality, space is expanding and
everything is moving away from everything else.

Now, Hubble also quantified what
was going on with these galaxies and their velocities. He said that the velocity with
which a galaxy was moving away from us was directly proportional to the distance
between that galaxy and Earth. We can say that in this drawing,
that’s the distance 𝐷, the distance between the galaxy and Earth.

Now, at this point, we can take the
expression and add a constant of proportionality. We’ll call this 𝐻 subscript zero
and we multiply that by 𝐷 to give us what’s known as Hubble’s law. Now 𝐻 subscript zero is known as
Hubble’s constant. And many scientists nowadays are
trying to accurately find the value of Hubble’s constant.

So just to clarify, Hubble’s law
tells us that on average galaxies are moving away from Earth. And the velocity with which they
move away from Earth is directly proportional to the distance between the galaxy and
Earth with the constant of proportionality being Hubble’s constant. With this information, Hubble
showed that the universe was expanding and that to an ever-accelerating rate because
further galaxies are moving away from earth faster and faster.

This not only provided evidence for
the expansion of the universe, but for the Big Bang theory as well. If the universe is expanding, then
in the past it makes sense for it to have been a lot smaller. Everything must have been a lot
closer together. And the final point we should make
is that Hubble’s law is empirical. This means that it’s based on
experiment rather than derived from some theory for example.

So now that we’ve learned about
Hubble’s law, let’s apply this to a few examples.

The galaxy NGC 87 has been observed
to be moving away from Earth at a speed of 3420 kilometers per second. Using a value of 20.8 kilometers
per second per mega light years for the value of the Hubble constant, find the
distance between NGC 87 and Earth. Give your answer to two significant
figures and in units of megalight-years.

Okay, so in this question, we’ve
got the galaxy NGC 87. So here is that galaxy. And we’ve been told that is moving
away from Earth at a speed, which we’ll call 𝑣 of 3420 kilometers per second. That’s really quite fast, isn’t
it? Now as well as this, we’ve been
told to use a value of 20.8 kilometers per second per megalight-years for the value
of the Hubble constant.

Hence, we can say that’s 𝐻
subscript zero, the Hubble constant, is equal to 20.8 kilometers per second per
megalight-year. Now kilometers per second per
megalight-year is actually a really weird way to write a unit because both
kilometers and megalight-years are units of distance. So if we wanted to, we could
convert either megalight-years into kilometers or kilometers into megalight-years
and then those would just cancel each other out.

However, the reasons that we write
the Hubble constant in terms of kilometers per second per mega light year becomes
apparent when we consider what Hubble’s law actually is. Hubble’s law tells us that the
velocity with which a galaxy moves away from Earth is equal to the Hubble constant
multiplied by the distance between the galaxy and Earth. That’s this distance here, which
we’ll call 𝐷. And it can accidentally happens to
be the distance between NGC 87, the galaxy, and Earth.

Now, if we rearrange Hubble’s law
so that we have the Hubble constant is equal to the velocity of the galaxy relative
to Earth divided by the distance between the galaxy and Earth, then we can see that
the most convenient unit of measurement of the velocity of the galaxy from Earth is
kilometers per second and the most convenient unit of measuring the distance between
the Earth and the galaxy is megalight-years because remember these distances are
huge.

And so, because a light year is a
large distance, that’s the distance travelled by light in one year, a megalight-year
is absolutely massive. But anyway, so this happened to be
most convenient units: kilometers per second for the velocity and megalight-years
for the distance.

Now we’ve been asked to find this
distance. And we’ve been given the value of
𝑣, the velocity of the galaxy relative to Earth, and the value of the Hubble
constant. So we need to rearrange Hubble’s
law once again. We can do this by dividing both
sides of the equation by the Hubble constant which means that it cancels on the
right-hand side. This means that we’re left with the
velocity of the galaxy divided by the Hubble constant is equal to the distance
between the galaxy and Earth.

At this point then, all we need to
do is to substitute in the values for the velocity and the Hubble constant. We get that the distance between
the galaxy and Earth is equal to 3420 kilometers per second — that’s the velocity —
divided by 20.8 kilometers per second per megalight-year.

And if we keep the units of the
calculation, we can see that kilometers per second cancel in the numerator and the
denominator. And so, what we have left is one
divided by megalight-years in the denominator of our fraction, which luckily is the
same thing as megalight-years in the numerator of our fraction. And remember we need to give our
answer in units of megalight-years. So already, we’re making good
progress. All we need to do now is to find
the value of this fraction.

Now, the value actually ends up
being 164.42 dot dot dot megalight-years. But remember we’ve been asked to
give our answer to two significant figures. So here’s the first significant
figure and here’s the second. Now, the next value is a four. Four is less than five. And so, our second significant
figure is going to stay the same. And hence to two significant
figures, our answer is that the distance between NGC 87 and Earth is 160
megalight-years.

So now that we’ve considered this,
let’s look in a little bit more detail at the units in which we give the Hubble
constant.

The Hubble constant is an important
constant in astronomy. It is often measured in units of
kilometers per second per megalight-year. Which of the following units could
Hubble constant also be expressed in? A) Meters per second, B) meter
squared per second, C) per meter, D) per second, E) meters.

Now in this question, we’ve been
told that the Hubble constant — which we’ll call 𝐻 subscript zero — is given in
units of kilometers per second per megalight-year. By the way, this convention is
often used to represent units. In other words, the square
parentheses around the 𝐻 subscript zero represent the units of 𝐻 subscript
zero.

So anyway, we’ve been given the
units of the Hubble constant and we need to find which of the following units could
the Hubble constant also be expressed in. In other words, this set of units
has to be equivalent to one of these. Now to work out which one it is,
let’s first think of these units as kilometers per second being a fraction and then
being divided by megalight-years.

In other words, what we have is a
fraction kilometers per second being divided by megalight-years. But then, this is the same as
multiplying by one over that thing. In other words, what we’ve got is
kilometers per second multiplied by one divided by megalight-years. At which point, we can multiply the
numerators and multiply the denominators. What this leaves us with is just
kilometers in the numerator because we had kilometers multiplied by one and in the
denominator, we’ve got seconds multiplied by megalight-years.

Now at this point, we can recall
that a kilometer is a unit of distance and a second is a unit of time. So what does the unit
megalight-years represent? Well, a light-year is the distance
travelled by light in one year. And hence, a light-year is a unit
of distance.

So a megalight-year is simply 10 to
the power of six light-years because remember the prefix mega just means 10 to the
power of six. And hence, what we have is another
unit of distance in the denominator. So at this point, we’ve got the
units of the Hubble constant, which are equal to the units of distance divided by
the unit of time and unit of distance.

So we could feasibly convert
kilometers into megalight-years or megalight-years into kilometers; they both
measure distance. At which one, we’d have some
numerical value to multiply or divide by depending on which we converted to
which. But that’s not relevant.

The point is that we could convert
megalight-years, for example, to kilometers, at which point they would both cancel
out of course leaving the numerical factor. But the unit that would remain is
one divided by seconds because at this point when the kilometers cancel, we’ve got
nothing in the numerator. So we stick a one there. And in the denominator, we still
have a unit of seconds.

Now, once again, we’re ignoring the
numerical factor that we had earlier because it’s not relevant. It’s only going to change the
number in front of the unit, which we could conventionally write of course as the
numerical factor multiplied by whatever 𝐻 nought was in kilometers per second per
megalight-years. But then the remaining unit is one
divided by seconds.

And of course, another way to write
one divided by seconds is seconds to the power of negative one because anything
raised to a negative power is the same as one divided by that object to the positive
power. A way to say this mathematically is
that one divided by 𝑎 to the power of 𝑛 is the same thing as 𝑎 to the power of
negative 𝑛.

And so at this point, we’ve arrived
at the final answer. We can see that out of the options
we’ve been given, the correct one is option D: the Hubble constant is often given in
units of kilometers per second per megalight-year. However, another way to express the
Hubble constant is in units of per second.

So, let’s now summarize what we’ve
talked about in this video. We can start off by saying firstly
that galaxies have been observed to be moving away from us, that is the Earth, and
Hubble realized this because he saw that the light coming from these galaxies is
redshifted. Now of course, this is not true for
all galaxies, but it does seem to be the case on average.

Most galaxies are moving away from
us. And what’s more important is that
the amount of redshift and therefore the speed with which the galaxies retreat from
Earth is directly proportional to that distance from Earth. In other words, the further away
our galaxy is from Earth, the faster it moves away from Earth. Another way to state this is that
𝑣, the velocity with which the Galaxy moves away from Earth, is directly
proportional to 𝐷, the distance between the galaxy and Earth.

And finally, we can say that the
constant of proportionality is 𝐻 subscript zero; that’s the Hubble constant. This gives 𝑣 is equal to 𝐻
subscript zero multiplied by 𝐷. This equation is known as Hubble’s
law.