The galaxy NGC 87 has been observed to be moving away from Earth at a speed of 3420 kilometres per second. Using a value of 20.8 kilometres per second per mega light year 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 this is one of those long questions, which means we need to start by underlining all of the important information. This way we can digest it a bit easier. So we know that we’ve got a galaxy. It’s called NGC 87 and it’s been observed to be moving away from Earth at a speed of 3420 kilometres per second.
What we need to do is to use a value of 20.8 kilometres per second per megalight year for the value of the Hubble constant. We need to find the distance between NGC 87 and Earth. We also need to give our answer to two significant figures — two sig figs — and in units of megalight years.
So how do we go about linking together all of the stuff that we’ve been given? Now, the next thing to do is to write down all the numerical values we’ve been given on the side and to assign symbols to them. For example, the speed or velocity in this case 𝑉 of NGC 87 away from Earth is 3420 kilometres per second.
We could have also used 𝑆 for speed by the way. But the question tells us the direction that the galaxy is moving. It’s moving away from Earth. So we know the speed and the direction, which means we know the velocity of the galaxy.
Also as we’ll see in a second, conventionally, we use 𝑉 in this kind of calculation. We also know that Hubble’s constant, written capital 𝐻 sub nought, is equal to 20.8 kilometres per second per megalight year. And we’re asked to find the distance between Earth and NGC 87. That’s 𝑑. So we need a relationship that links together the velocity of the galaxy, the Hubble constant, and the distance between the galaxy and Earth.
Hubble’s law is the relationship we’re looking for, what it says that the velocity 𝑉 of any galaxy travelling away from Earth is directly proportional to its distance 𝑑 from Earth, with the proportionality constant being the Hubble constant. Take a second to think about that by the way. This is quite profound.
What it’s saying is that as galaxies get further away from Earth, they’re travelling away faster and faster. This is not true just from the perspective of Earth by the way. If we were safe for example on another galaxy, we would see Earth travelling away from us faster and faster and also all of the galaxies. This is related to the expansion of the universe and the fact that it’s not only expanding, but the expansion is getting faster and faster. That’s quite a mind-boggling fact that it takes some getting used to. But for now, let’s get back to the question.
We’ve been asked to find the distance between Earth and NGC 87. So we need to rearrange the equation a bit. If we divide both sides of the equation by the Hubble constant, it gives us just the distance on the right-hand side. This is exactly what we want. We can add, just substitute in our values for the velocity and for the Hubble constant. And we should get the distance is being 164.42 dot dot dot and we need to worry about the units as well.
Since in this calculation, we did the velocity divided by the Hubble constant, we need to put in the unit of the velocity divided by that of the Hubble constant. Luckily, the kilometres per second cancel out on the numerator and the denominator, which just leaves us with one over megalight years in the denominator, which also simplifies to megalight years in the numerator.
We also need to give our answer to two significant figures by the way. So here’s the first significant figure and here’s the second. We’ll look at the figure after the second significant figure to see if the second one rounds up or down. In this case, that one is a four, which means the second significant figure has to round down or in other words, it stays exactly the same. Therefore, to two significant figures and in units of megalight years, our answer is that galaxy NGC 87 is 160 megalight years from Earth.