Video: Hubble’s Law

In this video, we will learn about the formula 𝑣 = 𝐻₀𝑑 (Hubble’s law) and how it can be used to calculate the distance to faraway galaxies using the recession velocity.

14:24

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.

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