The Whirlpool Galaxy is 23 million light years away from Earth. Using a value of 20.8 kilometres per second per mega light year for the value of the Hubble constant, find the value of the radial velocity of the Whirlpool Galaxy. Give your answer to two significant figures and in units of kilometres per second.
Okay, so what we’ve been told in this question here is that the Whirlpool Galaxy is 23 million light years away from Earth. So we can say that the distance, which we’ll call 𝐷, between the Earth and the galaxy is 23 million light years. Now, as well as this, we’ve been told to use a value of 20.8 kilometres per second per mega light year for the value of the Hubble constant. So let’s say that the Hubble constant, which we’ll call 𝐻 subscript zero, is 20.8 kilometres per second per mega light year.
In this question, we’ve been asked to find the value of the radial velocity of the Whirlpool Galaxy. So what do we even mean by the radial velocity. Well, let’s zoom out of this picture a little bit and assume that we’ve got the Earth as a tiny speck at the centre of a sphere. Let’s assume that a Whirlpool Galaxy is on the surface of the sphere. In a situation like this, if we wanted to measure the distance between these two, then we’d have to essentially measure the radius of the sphere.
But then, if we were to start measuring from Earth and measure towards the galaxy, then we’re moving in a certain very specific direction. And the radial velocity of the galaxy is simply the velocity of this galaxy in that same direction. In other words, a radial velocity simply means that the galaxy is moving directly away from Earth. We’re not talking about any velocity of the galaxy in this direction, for example, or this direction or anything like that. We’re talking about the direction, as we said earlier, directly away from Earth. As luck would have it, however, Hubble’s law specifically talks about the radial velocity of a galaxy.
Hubble’s law tells us that the velocity of a galaxy away from Earth, or in other words the radial velocity, is equal to the Hubble constant multiplied by the distance between the Earth and the galaxy. So we can use this equation to work out the velocity that we’ve been asked to find in the question. All what we need to do is to plug in the value for 𝐷 and the value for 𝐻 nought.
However, before we do this, let’s have a quick look at the units. We know that 𝐻 nought has been given to us in kilometres per second per mega light year. However, the distance 𝐷 has been given to us in light years. So if we want to be consistent with our set of units, we either need to convert 𝐷 into mega light years. Or we need to convert 𝐻 nought into kilometres per second per light year. It doesn’t really matter which we do as long as we’re consistent with the values we use.
So let’s choose to convert our distance into mega light years. To do this, we need to recall that a mega light year is defined as 10 to the power of sixth or one million light years, because the prefix mega represents 10 to the power of six or one million. And the distance we’ve been given between the Earth and the Whirlpool Galaxy is 23 million light years. Or, in other words, it’s equal to 23 mega light years because there are 23 lots of one million light years in 23 million light years. And so we can say now that the distance between the galaxy and Earth is 23 mega light years, at which point we’ve now got a consistent set of units. So we can plug these values into our equation for Hubble’s law, which looks something like this. We’ve got the velocity, which is equal to 𝐻 nought, multiplied by 𝐷.
And if we keep the units in our calculation, we can see that we’ve got a mega light years in the numerator and a mega light years in the denominator. So they cancel. This means that we’re left with the unit of kilometres per second as our velocity. And this is a good thing because we need to give our answer in units of kilometres per second. So to find the answer, we’d need to multiply 20.8 by 23. When we evaluate the right-hand side of our equation, we find that it’s equal to 478.4 kilometres per second.
However, that’s not our final answer. We’ve been asked to give our answer to two significant figures. So we need to round this. So here’s significant figure number one. And here’s significant figure number two. We need to look at the next one, which happens to be an eight, to tell us what will happen to the second one. Now, eight is larger than five. So this means that our second significant figure is going to round up. And at this point, we found our final answer.
The radial velocity of the Whirlpool Galaxy is 480 kilometres per second to two significant figures. We’ve also fulfilled all the required conditions in that the answer needs to be to two significant figures and in kilometres per second.