Video: Understanding 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] m/s B] m²s [C] m⁻¹ [D] s⁻¹ [E] m

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Video Transcript

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.

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